Higher Order Perturbative Calculations in Strong

Transcription

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UNIVERSITY OF CYPRUS
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PHYSICS DEPARTMENT
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Higher Order
on
Perturbative Calculations in
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Strong Interaction Physics
with Improved Discretized Actions
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for Quarks and Gluons
PhD DISSERTATION
MARTHA CONSTANTINOU
APRIL 2008
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UNIVERSITY OF CYPRUS
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PHYSICS DEPARTMENT
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PhD Dissertation
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of
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Martha Constantinou
Advisor
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Haralambos Panagopoulos
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Submitted in partial fulfillment of the requirements for the
degree of Doctor of Philosophy
in the Physics Department at the University of Cyprus
c 2008 by Martha Constantinou
Copyright All rights reserved
April 2008
PhD Candidate: Martha Constantinou
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Title of Dissertation: Higher order perturbative calculations in Strong Interaction
Physics with improved discretized actions for quarks and gluons
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Dissertation Committee:
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Prof. Haralambos Panagopoulos, Research Supervisor
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Prof. Constantia Alexandrou
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Prof. Giorgios Archontis, Committee Chairman
Prof. Athanassios Nicolaides
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Prof. Ettore Vicari
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Defended on April 14, 2008
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Dedicated to Ventsi who has been an inspiration to me
Acknowledgments
First of all, I would like to express my sincerest gratitude to my advisor, Professor Haris
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Panagopoulos, whose guidance made this Thesis possible. I am grateful for all interesting
computation that he proposed to me and trusted me for their completion. I thank him
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for his infinite patience and his continuous encouragement throughout my PhD studies.
While working with him, I surely benefited from his large experience and knowledge. His
perfectionism was a motivation for improving myself as a researcher and developing my
autonomy. It was certainly a privilege to be his student.
Secondly, I want to thank Apostolos Skouroupathis and Fotos Stylianou for our collab-
an
oration in many of the computations presented here. Special thanks go to Apostolos for
being not only a good collaborator but a true friend. He was always available for interesting conversations and to hear my thoughts and concerns about our projects. I should
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also thank my office mates Andreas Christou, Giannis Koutsou, Savvas Polydorides and
Phanourios Tamamis for providing a pleasant and enjoyable working environment during
my graduate studies.
I do not have enough words to thank my parents for their unconditional love and
everlasting support. I am grateful to them for being close to me, for emphasizing on the
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importance of education, and insisting at the same time that there are more in life than
work. Most of all, I owe them a huge thank you for creating the perfect family environment
that kept me balanced at many difficult times. So mom and dad, I take this chance to let
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you know how much I love you and that you have been row models for me.
A very special acknowledgment goes to Ventsi Ivanoff, for bringing joy to my life and
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for his spiritual support. His unique way of dealing with difficulties, helped me be stronger
when facing one. His productive criticism, even though I often complained about it, has
gradually improved me. I should thank him for his patience and forbearance whilst spending hundreds of hours working on this Thesis. Ventsi, this work is dedicated to you, because
I consider it as a success of the both of us.
I also feel the need to thank those who contributed in different ways to the realization
of this work:
⋆ Sofia Papanastasiou who encouraged me to join the Lattice QCD community.
⋆ Maria Christoforou for being a good friend with genuine interest on the progress of my
work.
⋆ Yiota Andreou with her positive way of thinking that was transferring to me as well.
Our brief lunch breaks were extremely refreshing for me.
⋆ Chrysanthi Demetriou and Zoe Demetriou for the fun time we had during their graduate
studies the Physics.
⋆ My younger sisters Sotia and Maria-Mikaella for their endless love and care, that really
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contributed to who I am.
⋆ Stavroulla Andreou who I accidentally met a few years ago, and since then she has
been next to me, supporting me through rough times. Most of all, she is a good listener
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whenever I need it.
⋆ All those of you, who have been by my side during my graduate studies. Your support
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is definitely appreciated.
Dear friends, you will always have a special place in my heart.
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Last but not least, I thank the Research Promotion Foundation of Cyprus for financial
support over the last two and a half years.
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Abstract
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In this Thesis we address a number of perturbative calculations in Quantum Chromodynamics, formulated on the lattice. We employ a variety of improved fermion and gauge
field actions, which are currently used in numerical simulations. The calculations that we
present are the following:
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• The evaluation of the relation between the bare coupling constant g0 and the renormalized one in the MS scheme, gMS . This computation is performed to 2 loops in perturbation theory, employing the standard Wilson action for gluons and the overlap action for
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fermions. We also derive the 3-loop coefficient of the bare β-function (βL (g0 )) and provide
the recipe for extracting the ratio of energy scales, ΛL /ΛMS . Moreover, we generalize our
results for fermions in an arbitrary representation of the gauge group SU(N).
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• We extend a systematic improvement method of perturbation theory for gauge fields
on the lattice, to encompass all possible gluon actions made of closed Wilson loops. The
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improvement procedure entails resummation of an infinite, gauge invariant class of Feynman diagrams. Two different applications are presented: The additive renormalization of
fermion masses, and the multiplicative renormalization ZV (ZA ) of the vector (axial) cur-
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rent. In many cases where nonperturbative estimates of renormalization functions are also
available for comparison, the agreement with improved perturbative results is significantly
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better as compared to results from bare perturbation theory.
• We study the critical value of the hopping parameter, κc , up to 2 loops in perturbation
theory. This quantity is a typical case of a vacuum expectation value resulting in an additive
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renormalization. The clover improved action is employed for fermions and the Symanzik
improved action for gluons. In order to compare our results to nonperturbative evaluations
of κc coming from Monte Carlo simulations, we employ our improved perturbation theory
method for improved actions.
• An ongoing project regards the improvement of the fermion propagator and quark
operators, to second order in the lattice spacing a, in 1-loop perturbation theory. The
1
Abstract
2
computations are performed using clover fermions and Symanzik improved gluons. Our
calculation has been carried out in a general covariant gauge. The higher order terms
allow us to specify the corrections which must be applied to the quark operators, in order
for them to be O(a2 ) improved. Our results are applicable also to the case of twisted
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mass fermions, which are currently being studied intensely by a number of collaborations
worldwide.
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Perlhyh
Sta plasia ti paroÔsa Diatrib pragmatopoioÔntai diataraktiko upologismo sthn
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Kbantik Qrwmodunamik , sto formalismì tou Plègmato. Efarmìzoume beltiwmène fermionikè kai gklouonikè drsei, oi opoe qrhsimopoioÔntai eurèw sti arijmhtikè prosomoi¸sei. Oi upologismo pou parousizontai enai oi akìloujoi:
Exagwg th sqèsh metaxÔ th gumn (bare) stajer sÔzeuxh
nakanonikopoihmènh (renormalized) sto sq ma
an
•
MS, gMS .
kai th epa-
H diexagwg tou upologismoÔ
gnetai sthn txh diìrjwsh dÔo brìgqwn, me th qr sh fermionwn
overlap
kai gklouonwn
Ta apotelèsmat ma odhgoÔn ston kajorismì tou suntelest tri¸n brìgqwn
th gumn β
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Wilson.
g0
βL (g0 ), kai parèqoume th suntag gia thn eÔresh tou lìgou th
ΛL /ΛMS. Epiplèon, genikeÔoume ta apotelèsmat ma gia fermiìnia
sunrthsh,
energeiak klmaka,
SU(N).
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se opoiad pote anaparstash th omda bajmdo
• Epèktash mia susthmatik mèjodou beltwsh th jewra diataraq¸n gia peda baj-
mdo sto plègma, oÔtw ¸ste na enai efarmìsimh gia opoiad pote drsh pou apoteletai
Wilson. H diadikasa beltwsh sunepgetai thn jroish mia olìdiagramtwn Feynman, ta opoa enai anallowta upì metasqhmatismoÔ
klhrh kathgora
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apì kleistoÔ brìgqou
bajmdo. DÔo diaforetikè efarmogè parousizontai: H prosjetik epanakanonikopohsh
mcr
a
th krsimh fermionik mza
kai h pollaplasiastik epanakanonikopohsh
ZV (ZA )
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tou dianusmatikoÔ (axonikoÔ) reÔmato. Se arketè peript¸sei ìpou up rqan ektim sei
twn stajer¸n epanakanonikopohsh apì arijmhtikè prosomoi¸sei, h sumfwna tou me
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ta dik ma beltiwmèna apotelèsmata enai emfan¸ kalÔterh se sqèsh me ta apotelèsmata
th gumn jewra.
• Melèth th krsimh tim th paramètrou hopping, κc , mèqri dÔo brìgqou sth jewra
diataraq¸n. H en lìgw posìthta enai tupik perptwsh anamenìmenh tim kenoÔ pou odh-
ge se prosjetik epanakanonikopohsh. Gia ta fermiìnia epilèqjhke h beltiwmènh drsh
clover,
en¸ gia ta gklouìnia h beltiwmènh drsh
telesmtwn ma gia to
Symanzik.
Gia lìgou sÔgkrish twn apo-
κc me ta antstoiqa twn prosomoi¸sewn Monte Carlo, qrhsimopoioÔme
3
Perlhyh
4
th mèjodo beltwsh th jewra diataraq¸n pou anaptÔxame gia beltiwmène drsei.
•
H teleutaa ergasa ma anafèretai se èna upologismì o opoo brsketai se exèli-
xh kai afor th beltwsh tou fermionikoÔ diadìth, kaj¸ kai fermionik¸n telest¸n mèqri
2η txh th stajer plègmato
a
kai se jewra diataraq¸n enì brìgqou. Oi upologi-
smo diexgontai me qr sh twn drsewn
clover
kai
Symanzik
kai pragmatopoioÔntai se mia
genik sunallowth bajmda. Oi uyhlìterh txh ìroi ma epitrèpoun na kajorsoume ti
ou
diorj¸sei pou prèpei na efarmostoÔn stou fermionikoÔ telestè, oÔtw ¸ste na beltiw-
O(a2 ). Ta apotelèsmat ma enai efarmìsima kai sthn perptwsh twn fermionwn
twisted mass, pou qrhsimopoioÔntai entatik apì arketè ereunhtikè omde pagkosmw.
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Contents
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List of figures
List of tables
12
12
16
2 The Wilson formulation and O(a) improvements
2.1 Standard Wilson quarks and gluons . . . . . . . . . . . . . . . . . . . . . .
21
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O(a) improved actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The clover fermion action . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 The Symanzik improved gluon action . . . . . . . . . . . . . . . . .
26
26
29
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2.2
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Introduction to the Standard Model . . . . . . . . . . . . . . . . . . . . . .
Perturbative calculations using improved actions . . . . . . . . . . . . . . .
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1.1
1.2
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1 Introduction
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3 O(a) improved overlap action
32
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Nielsen-Ninomiya No-Go theorem . . . . . . . . . . . . . . . . . . . .
3.3
3.4
3.5
Locality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Ginsparg-Wilson relation . . . . . . . . . . . . . . . . . . . . . . . . .
Overlap action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.1
3.2
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35
37
44
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4 Twisted mass action
32
33
4.1
4.2
4.3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The lattice twisted mass action for degenerate quarks . . . . . . . . . . . .
Calculations with twisted mass QCD . . . . . . . . . . . . . . . . . . . . .
44
45
48
4.4
4.5
Twisted mass QCD for nondegenerate quarks . . . . . . . . . . . . . . . .
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
51
5
CONTENTS
6
5 The background field formalism
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
55
Background fields in the continuum theory . . . . . . . . . . . . . . . . . .
The lattice background field method . . . . . . . . . . . . . . . . . . . . .
56
58
5.4
Vertices of the overlap action in the background field method . . . . . . . .
61
6 The running coupling and the β-function
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Renormalization group equation and β-function . . . . . . . . . . . . . . .
65
65
67
The step scaling function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
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6.3
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5.2
5.3
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7 QCD with overlap fermions: Running coupling and the 3-loop β-function 75
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.2 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Description of the calculation . . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
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7.5
Generalization to an arbitrary representation . . . . . . . . . . . . . . . . .
7.5.1 The strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5.2 Adjoint representation . . . . . . . . . . . . . . . . . . . . . . . . .
93
93
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7.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.3
7.4
98
The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2.1 Dressing the propagator . . . . . . . . . . . . . . . . . . . . . . . . 107
8.2.2 Dressing vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
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8.2
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8 Improved perturbation theory for improved lattice actions
105
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
8.4
8.5
Dressing QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
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8.3
8.2.3 Numerical values of improved coefficients . . . . . . . . . . . . . . . 112
8.2.4 The improvement procedure in a nutshell . . . . . . . . . . . . . . . 119
Application: 1-loop renormalization of fermionic currents . . . . . . . . . . 120
9 Two-loop additive mass renormalization with clover fermions and Symanzik
improved gluons
125
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 127
CONTENTS
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9.3
9.4
Computation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Improved perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . 135
9.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
10 O(a2 ) improvements
145
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
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10.2 Improvement to the fermion propagator . . . . . . . . . . . . . . . . . . . . 146
10.2.1 Basic divergent integrals . . . . . . . . . . . . . . . . . . . . . . . . 148
10.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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10.3 Improved operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11 Conclusions
162
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Appendix B: Numerical integration
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Appendix A: Notation
166
A.1 Continuum QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
A.2 Lattice QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
171
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B.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B.2 The integrator routine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B.3 A particular example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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Bibliography
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Appendix C: The algorithm for improving the Symanzik coefficients
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188
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List of Figures
The dependence of αs on the energy scale µ. . . . . . . . . . . . . . . . . .
2.1
A plaquette on a 2-dimensional slice of the hypercubic lattice. . . . . . . .
22
2.2
2.3
Graphical representation of Qµν (Eq. (2.22)) appearing in the clover action.
The 4- and 6-link loops contributing to the gauge action of Eq. (2.26). . . .
28
30
3.1
The poles of Eq. (3.29) and the integration region C. . . . . . . . . . . . .
40
4.1
A plot of the pion propagator against time separation for the quenched
approximation on a 323 × 64 lattice with coupling β = 6.2 ([24]). . . . . . .
45
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1.1
14
r0 fPS as a function of (r0 mPS ) for β = 3.9 and β = 4.05 . . . . . . . . . .
Nucleon mass as a function of m2π for β = 3.9, 4.05 . . . . . . . . . . . . .
52
53
5.1
The multiple character of a vertex in the background field method. . . . .
61
6.1
The value of αs (MZ 0 ) as derived from various processes and the average of
these measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
7.1
7.2
7.3
Fermion contributions to the 1-loop function ν (1) . . . . . . . . . . . . . . .
Fermion contributions to the 2-loop function ν (2) . . . . . . . . . . . . . . .
Plot of the total 1-loop coefficient k (1) versus the overlap parameter ρ. . . .
82
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7.4
7.5
Plot of the total 2-loop coefficient c(1,−1) versus ρ. . . . . . . . . . . . . . .
Plot of the total 2-loop coefficient c(1,1) versus ρ. . . . . . . . . . . . . . . .
91
91
7.6
7.7
The 3-loop coefficient bL2 (Eq. (7.40)), plotted against ρ, for N = 3 and
Nf = 0, 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The non-pointlike nature of an overlap vertex. . . . . . . . . . . . . . . . .
92
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7.8
A particular example of a 2-loop fermionic diagram. . . . . . . . . . . . . .
94
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4.2
4.3
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LIST OF FIGURES
7.9
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The ρ dependence of the ratio ΛL /Λ MS in the adjoint representation for
N = 3 and Nf = 0, 1/2, = 1. . . . . . . . . . . . . . . . . . . . . . . . . . .
97
A cactus diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Improved coefficient c̃0 for N=2 and N=3 (plaquette action). . . . . . . . . 117
8.3
8.4
8.5
Improved coefficients c̃0 and c̃1 (tree-level Symanzik improved action). . . . 117
Improved coefficients c̃0 and c̃1 (Iwasaki action). . . . . . . . . . . . . . . . 118
Coefficients c0 , c1 , c3 and their dressed counterparts c̃0 , c̃1 , c̃3 for different
8.6
values of β c0 = 6 c0 /g02 (TILW actions). . . . . . . . . . . . . . . . . . . . . 118
1-loop contribution to the amputated Green’s function (bV,A ). . . . . . . . 121
8.7
8.8
1-loop contribution to the quark self-energy (bΣ ). . . . . . . . . . . . . . . 121
dr
Plots of ZV,A and ZV,A
for the plaquette, Iwasaki and TILW actions. . . . . 123
9.1
1-loop diagrams contributing to dm(1−loop) . . . . . . . . . . . . . . . . . . . 128
9.2
9.3
2-loop diagrams contributing to dm(2−loop) . . . . . . . . . . . . . . . . . . . 129
Total value of dm to 2 loops, for N = 3, Nf = 0 and c2 = 0. . . . . . . . . 134
9.4
9.5
9.6
Improved and unimproved values of dm up to 2 loops, as a function of cSW ,
9.7
for the plaquette action (β = 5.29, N = 3, Nf = 2). . . . . . . . . . . . . . 137
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for the Iwasaki action (β = 1.95, N = 3, Nf = 2). . . . . . . . . . . . . . . 137
for the DBW2 action (N = 3, Nf = 2). We set β = 0.87 and β = 1.04. . . . 138
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9.8
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8.1
8.2
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10.1 1-loop diagrams contributing to the improvement of the fermion propagator. 146
10.2 1-loop diagrams contributing to the improvement of the bilinear operators. 160
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10.3 1-loop diagrams contributing to the extended operators. . . . . . . . . . . . 161
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A.1 The interaction vertices of quarks and gluons. . . . . . . . . . . . . . . . . 167
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List of Tables
Numerical results for k (1) , c(1,−1) , c(1,1) . . . . . . . . . . . . . . . . . . . . . 88
(0)
(1)
Per diagram breakdown of the 1-loop coefficients ki and ki for various ρ
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3
7.4
Contribution to ci
of diagrams 1, 4, 5, 3+14+15, 6+12, 8+18. . . . . . 100
(0,1)
Contribution to ci
of diagrams 1, 2+13+16, 3+14+15, 4, 5. . . . . . . . 101
7.5
7.6
7.7
Contribution to ci
of diagrams 6+12, 7+11, 8+18, 9+17, 10. . . . . . . 101
(1,−1)
Contribution to ci
of diagrams 1, 4, 5. . . . . . . . . . . . . . . . . . . 102
(1,−1)
Contribution to ci
of diagrams 3+14+15, 6+12, 7+11, 8+18. . . . . . . 102
7.8
7.9
Contribution to ci
of diagrams 1, 2+13+16, 3+14+15, 4, 5. . . . . . . . 103
(1,1)
Contribution to ci
of diagrams 6+12, 7+11, 8+18, 9+17, 10. . . . . . . 103
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7.1
7.2
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(1,1)
(2,−1)
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7.10 The overlap parameter dependence of the 2-loop coefficients ci
. . . . . 104
(2,1)
7.11 Numerical results for the 2-loop coefficients ci
for different values of the
overlap parameter ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Input parameters β, c0 , c1 , c3 , c̃0 , c̃1 , c̃3 (c2 = 0). . . . . . . . . . . . . . . 119
dr
Results for ZV,A , ZV,A
(Eq. (8.39), (8.40)) using ρ=1.0, ρ=1.4 . . . . . . . . 122
9.1
9.2
9.3
Input parameters c0 , c1 , c3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Total 1-loop coefficients ε(1) , ε(2) , and ε(3) . . . . . . . . . . . . . . . . . . . 140
Total 2-loop contribution to dm of order O(N 2 , c02 ). . . . . . . . . . . . . . 140
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a
8.1
8.2
Total 2-loop contribution to dm of order O(N 0 , c02 ). . . . . . . . . . . . . . 141
2-loop coefficients of dm containing closed fermion loops. . . . . . . . . . . 141
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9.4
9.5
9.6
9.7
9.8
Total dm coefficient containing the parameter c2 (part 1). . . . . . . . . . . 141
Total dm coefficient containing the parameter c2 (part 2). . . . . . . . . . . 142
1-loop contribution to dm for the Iwasaki action. . . . . . . . . . . . . . . 142
9.9
2-loop results for dm coming from diagrams 3, 4, 6, for the Iwasaki action.
10
142
LIST OF TABLES
11
9.10 2-loop results for dm coming from diagrams 7-11, 14-18, 24, 26, for the
Iwasaki action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.11 2-loop results for dm coming from diagrams 12, 13, 19, 20, for the Iwasaki
action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.14 1- and 2-loop results, and nonperturbative estimates for κc .
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9.12 2-loop results for dm coming from diagrams 21-23, 25, 27, 28, for the Iwasaki
action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.13 Results for dmdr
(1−loop) (Eq. (9.23)), with N = 3. . . . . . . . . . . . . . . . 144
. . . . . . . . 144
10.1 The ε(0,i) coefficients of Eq. (10.28) for different actions. . . . . . . . . . . . 158
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10.2 The ε(1,i) coefficients of Eq. (10.28) for different actions. . . . . . . . . . . . 158
10.3 The ε(2,i) coefficients of Eq. (10.28) for different actions (part1). . . . . . . 158
10.4 The ε(2,i) coefficients of Eq. (10.28) for different actions (part2). . . . . . . 159
(i,1)
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10.5 The dependence of ε̃1 (Eq. (10.29)) on the Symanzik parameters. . . . . 159
(i,1)
10.6 The dependence of ε̃2 (Eq. (10.30)) on the Symanzik parameters. . . . . 159
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Chapter 1
Introduction to the Standard Model
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1.1
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Introduction
The Standard Model (SM) is a Theory which successfully describes, in terms of Quantum
st
Fields, three of the four fundamental interactions: The electromagnetic, the weak and the
strong; it leaves out the gravitational force which is negligible in the atomic and subatomic
level. It was developed through 1970-1973 by the contributions of different scientists and
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it is consistent with both quantum mechanics and special relativity.
A lot earlier, in 1950’s and 1960’s, a huge number of new particles had been detected
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in accelerators and there was an urgent need for a theory to explain their appearance.
In 1964 M. Gell-Mann and G. Zweig proposed an underlying structure of fundamental
elements, the so-called quarks, that could group in different combinations and form all
a
particles, known as hadrons. Today we know that quarks come in six flavors (up, down,
charm, strange, top and bottom) and for each quark there is a corresponding antiquark.
They also carry an additional unobserved quantum number, the color, in order to avoid
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conflict with Pauli’s exclusion principle. For each flavor the quark can have any of the three
colors (red, blue, green) and quarks with different flavors have different masses. The quark
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model separates hadrons in two categories: a. The mesons, with integer spin, consisting of
a quark-antiquark pair and b. the baryons, with half-integer spin, which are bound states
of three quarks or three antiquarks. This model not only could explain the already found
particles, but could also predict the existence of new ones.
More than a decade later, S. Weinberg in collaboration with S. Glashow, and A.
Salam independently created the electroweak theory, unifying the electromagnetic and
weak forces. The theory involves six leptons (electron, muon, tau, with their massless and
12
1.1. Introduction to the Standard Model
13
chargeless neutrinos) and the force carriers: Photon for electromagnetism and the W ± , Z
bosons for the weak interaction. It is important to say that the above models are far from
a mere hypothesis; the existence of both quarks and neutrinos has been experimentally
verified, and the models have already passed very stringent experimental tests.
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M. Gell-Mann, H. Fritzsch and H. Leutwyler introduced the color force as the interaction
between the fundamental constituents of the strong force, which are the quarks and gluons
(the interaction mediators). Both quarks and gluons have color as an additional degree of
freedom; indeed gluons are self-interacting. With the formulation of the theory of the color
force, Quantum Chromodynamics (QCD for short), the Standard Model was theoretically
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completed. Another fundamental feature of the SM is the prediction of the existence of a
Higgs particle responsible for the nonvanishing masses of W ± and Z. The existence of the
Higgs mechanism has not been confirmed experimentally yet, but it is hoped that this will
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be achieved in the LHC facilities, currently starting their operation at CERN.
We dedicate the rest of the section to the strong force and discuss different aspects
of its theoretical formulation. When the quark model of QCD was first proposed, a lot
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of doubts arised concerning its validity since no quark was seen in experiments until that
time. In 1973 G. ’t Hooft, H. Politzer, D. Gross and F. Wilczek studied two of the main
features of the strong interactions: Asymptotic freedom and confinement. That is, when
nucleons are given a lot of energy, the strong interaction actually weakens and the quarks
can hardly interact with each other (asymptotic freedom). On the contrary, when the
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quarks are pulled apart, they appear to be tightly bound to each other within the nucleons
(confinement). Thus, it would take infinite energy to separate them, explaining why they
have never been seen as isolated entities.
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a
Quantum Chromodynamics has all the necessary features to describe the strong interactions. To start with, it is a nonabelian gauge theory based on the group SU(3) (the
number 3 indicates the number of colors carried by quarks). This group has eight gener-
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ators corresponding to the number of massless gluons. Moreover, the nonabelian nature
of QCD explains asymptotic freedom and the gluon self-interaction; it also leads to quark
confinement, as has been numerically shown. The beauty of QCD lies in the fact that only
a few parameters are needed to explain the whole spectrum of strong interactions; these are
the coupling constant of the force and the quark masses. Despite its simple mathematical
form, it is extremely complicated to solve it, even to perform fundamental computations
like the hadron masses.
Among the features of QCD is the ‘running’ of the fundamental coupling constant αs
14
with the energy scale. The term ‘constant’ is actually a misnomer since αs depends on the
energy exchange µ, in each process under consideration. At high energies, or equivalently
at small separations of the quarks, the coupling constant is small and the theory can be
studied perturbatively. On the other hand, at low energies (<
∼ 1GeV) the coupling becomes
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large and perturbation theory breaks down completely. This behavior almost resulted in
abandoning QCD: It was considered totally incapable of addressing the strong interactions
because the coupling was too strong for perturbation theory to be of any use.
In the Lagrangian of QCD (at least in the absence of fermions, or in presence of massless
fermions) there is no particular dimensionful quantity in terms of which one could express
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dimensionful observables, such as the bound state spectrum. A solution to that, is to
specify the value of the energy scale µ at which αs ∼ 1. This is the energy (known as
ΛQCD ) at which the perturbative method cannot be applicable as we come down from high
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energies. The behavior of the coupling with respect to the energy scale is shown in Fig. 1.1.
Note that ΛQCD merely sets the scale against which other quantities are to be measured.
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1
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Αs
ßQCD
Μ
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a
Figure 1.1: The dependence of αs on the energy scale µ.
The failure of the perturbative expansion in the low energy sector is a major problem,
since at that region quarks and gluons bind together into composite states, the hadrons. To
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study the world of hadrons and prove confinement, it is necessary to search for alternative,
nonperturbative tools. This is where Lattice QCD (LQCD) comes into play.
Lattice QCD was first suggested in 1974 by K. Wilson [1] as the mathematical tool to
calculate observables in the nonperturbative region of QCD. The numerical computations
are performed by means of Monte Carlo simulations, with input the bare coupling and the
bare quark masses without having to tune additional parameters. The theory is formulated
15
on a finite-sized hypercubic space-time lattice, characterized by the spacing a (the distance
between neighboring lattice points). By definition, the lattice introduces a momentum
cutoff proportional to a−1 , excluding in this way the high frequencies and making the
theory finite. Space and time are further allocated a finite extent, so that the number
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of degrees of freedom is finite. However, this number is still large, making simulations
computationally very demanding; it is necessary to use some approximations in order to
deal with it. Simulations are performed for a variety of lattice spacings and the continuum
limit is reached by extrapolating the results to a → 0. An essential constraint in numerical
calculations is the range of the lattice spacing that can be used. Typical algorithms slow
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down proportionally to a5 and thus simulations use values for a limited within the interval
0.05fm ≤ a ≤ 0.01fm. Performing numerical simulations is crucial both for predicting the
dependence of the hadronic quantities on the bare parameters and for checking that QCD
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is the correct theory of the strong interactions, by comparing with experiments. For an
appropriate comparison, the bare quantities must be renormalized.
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Although the idea of the lattice formulation was introduced to cope with the low energies, it can also serve the region of energies well above the energy scale relevant to hadronic
properties. At that region we employ perturbation theory for the computations, which is
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useful for investigating the behavior of lattice theories near the continuum limit. It is also
the key in connecting simulation results to physical quantities. For this to be achieved,
the Lagrangian parameters and the matrix elements must be renormalized. Traditionally,
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the renormalization factors are defined perturbatively, even in many studies of the low
energy sector. Ideally, for a truly nonperturbative result, the renormalization factors need
a
to be calculated using simulations. Unfortunately, the range of physical distances covered
by a single lattice is limited and it is very difficult to calculate the large scale differences
of nonperturbative renormalizations. Furthermore, the mixing between operators at the
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quantum level is often too small to be distinguishable with numerical techniques. The
above problems make perturbation theory necessary for the correct connection between
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experiments and theory and to verify that a single Lagrangian describes the high energy
sector and explains at the same time the spectrum of light hadrons.
In this Thesis, we use Lattice QCD in perturbation theory to investigate main issues
regarding the running coupling, the quark masses, improvement methods of perturbation
theory, and improved measurable quantities (quark propagator, quark operators). All
computations have been performed with improved actions, the importance of which will
be explained next. Each of our computations is motivated in the following section.
1.2. Perturbative calculations using improved actions
1.2
16
Perturbative calculations using improved actions
We have previously discussed the discretization of the continuum action in the framework
of Lattice QCD. Starting from the continuum action for a single fermion, its naive discretization gives rise to 2d = 16 fermions rather than one; the extra unphysical fermions
are called fermion doublers. This doubling phenomenon is demonstrated by the fermion
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propagator, which has sixteen poles for zero fermion mass, within the integration interval (first Brillouin zone). The main problem is that the doublers destroy the chirality
properties (left-right symmetry) of the continuum theory. This failure is explained in the
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Nielsen-Ninomiya theorem [2] which leaves us with the choice of either having the unwanted doubler fermions on the lattice or eliminating the doubler modes at the expense
an
of breaking chiral symmetry. The first attempt to resolve the doubling problem was given
by Wilson [1], who added an extra term in the action. Besides all the advantages of the
Wilson action, there are a number of drawbacks. One of them is the fact that lattice
artifacts in physical quantities are proportional to a, rather that a2 . This will prove to be
a great disadvantage.
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In principle, the only restriction while discretizing the QCD Lagrangian, is the recovery
of the continuum limit. Yet, it is worthwhile to construct improved actions for a better
behavior at all lattice spacings. One motivation is the discretization errors appearing in
the lattice formalism, which affect simulation results and have to be removed; these can be
dramatically reduced if one uses improved actions. While these errors are to be removed
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in the continuum limit, it is advantageous to have them under control at nonzero lattice
spacing for two reasons: a. For large errors, it is not clear how to perform a safe and
reliable extrapolation to a → 0 and b. unimproved actions force us to work with very
a
small lattice spacings which is computationally very demanding (particularly in full QCD
where simulations are very expensive in terms of computer time). So, it would be beneficial
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to work with improved actions at larger values of a and still have results of the same quality.
Moreover, improved actions are expected to preserve more symmetries of the continuum
theory.
Over the years, many improved actions have been constructed, both for fermion and
gauge fields. The most frequently used discretizations of the continuum fermion action are
the clover [3], the staggered [4], and the Ginsparg-Wilson fermions (overlap fermions [5,
6, 7] and domain wall [8, 9]). All of the actions mentioned above have discretization
errors of order O(a2 ) and reach the continuum limit faster than Wilson fermions. In the
following chapters we will concentrate on the actions used for our calculations: Clover and
17
overlap fermions. Regarding the gluon part of the action, there is the Symanzik improved
action [10], which includes all possible closed loops made of 4 and 6 links (see Fig. 2.3).
This action has 4 parameters that can have different values. In Table 8.1 we tabulate the
most commonly used sets of these coefficients.
We are particularily interested in the relationship between the bare coupling and the
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cutoff, an essential ingredient in lattice calculations. Usual Monte Carlo simulations require the knowledge of this relationship far from the critical point, where corrections to
asymptotic scaling could become relevant. By definition, the β-function is the quantity
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that dictates this relationship between running coupling and the lattice spacing. Due to
asymptotic freedom of QCD, the β-function can be expanded in terms of the coupling con-
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stant, with perturbative evaluations for the coefficients. The 1- and 2-loop coefficients are
universal (regularization and renormalization scheme independent), while the rest of them
depend on the regulator. We calculate the 3-loop coefficient using the overlap fermions,
one of the most frequently used actions in simulations at the present time. Knowledge of
the coupling is of high importance because its precise determination would fix the value
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of a fundamental parameter in the Standard Model. As a by-product of the β-function
calculation, one can derive the Λ parameter of QCD, a quantity needed to convert dimensionless quantities coming from numerical simulations into measurable predictions for
physical observables. It is mathematically defined as a particular solution of the renormalization group equation. This parameter is dimensionful and cannot be directly calculated
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on the lattice. Instead, one can obtain the ratio between the lattice Λ parameter, ΛL, and
the scale parameter in some continuum renormalization scheme such as MS, ΛL /ΛMS.
a
Another question we address is the critical value of the hopping parameter, κc , for clover
fermions and Symanzik gluons. The demand for strict locality and absence of fermion dou-
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blers, results in the explicit breaking of chiral symmetry. Thus, setting the bare quark mass
equal to zero is not sufficient to ensure chiral symmetry in the quantum continuum limit,
because quantum corrections introduce an additive renormalization to the fermionic mass.
For the renormalized mass to be zero, the bare mass must be renormalized multiplicatively
and additively. The hopping parameter is an adjustable quantity used in simulations, that
is directly related to the fermion mass. When it is assigned its critical value, chiral symmetry is restored. The computation of the hopping parameter is a typical case of a vacuum
expectation value resulting in an additive renormalization; as such, it is characterized by
a power (linear) divergence in the lattice spacing, and its calculation lies at the limits of
18
applicability of perturbation theory.
The nature of perturbation theory indicates that when performing higher order calculations, the quantities under study are deduced with increased accuracy. At the same time,
the calculations become notoriously complicated and demanding (i.e. there is a notable
increase, both in the number of Feynman diagrams involved, and in the number of terms
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in each diagram), as compared to lower order studies. It is important to develop a method
for improving the perturbative expansion, so that one can get improved results from lower
order calculations. The method we propose, is called cactus improvement and sums up
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a whole subclass of tadpole diagrams, to all orders in perturbation theory. The effect of
resummation is to replace various parameters in the action (coupling constant, Symanzik
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coefficients, clover coefficient) by ‘dressed’ values; the latter are solutions to certain coupled integral equations, which are easy to solve numerically. Some positive features of this
method are: a. It is gauge invariant, b. it can be systematically applied to improve (to
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all orders) results obtained at any given order in perturbation theory, c. it does indeed
absorb in the dressed parameters the bulk of tadpole contributions.
This Thesis contains work carried out over the past three years and it is laid out as
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follows. Chapters 2-6 provide some background information for the work covered in the
rest of the Thesis. Particularly, in Chapter 2 we explore the Wilson action (both for quarks
and gluons) as well as two of the most widely used improved actions: The clover fermion
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action and the Symanzik gluon action. Chapter 3 is devoted to another class of improved
actions of great complexity, the overlap fermions. Besides some standard material, we also
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derive the required vertices that are needed for a subsequent Chapter. A brief overview
of the twisted mass fermion action is given in Chapter 4, while Chapters 5-6 regard the
background field method and the running coupling (with emphasis on the β-function),
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respectively.
The main calculations of the Thesis start in Chapter 7 with the evaluation of the relation
between the bare coupling constant g0 and the renormalized one in the modified minimal
subtraction scheme MS, gMS . For convenience, we have worked with the background field
technique, which only requires evaluation of 2-point Green’s functions for the problem at
hand. The computation was performed to 2 loops in perturbation theory, employing the
standard Wilson action for gluons and the overlap action for fermions. Our results depend
explicitly on the number of fermion flavors (Nf ) and colors (N) and are tabulated for
different values of the overlap parameter ρ in its allowed range (0 < ρ < 2). We also derive
19
the 3-loop coefficient of the bare β-function (βL (g0 )) and provide the recipe for extraction
of the ratio of energy scales, ΛL /ΛMS. Moreover, we generalize our results to fermions in
an arbitrary representation of the gauge group SU(N).
In Chapter 8 we extend a previous systematic improvement of perturbation theory
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for gauge fields on the lattice, to encompass all possible gluon actions made of closed
Wilson loops. Two different applications are presented: The additive renormalization of
fermion masses (in Chapter 9), and the multiplicative renormalization ZV (ZA ) of the
vector (axial) current. In many cases where nonperturbative estimates of renormalization
functions are also available for comparison, the agreement with improved perturbative
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results is significantly better as compared to results from bare perturbation theory.
In Chapter 9 we study the critical value of the hopping parameter, κc , up to 2 loops in
perturbation theory. We employ the clover improved action for fermions and the Symanzik
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improved gluon action with 4- and 6-link loops. Our results are polynomial in cSW (clover
parameter) and cover a wide range of values for the Symanzik coefficients ci . The depen-
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dence on the number of colors N and the number of fermion flavors Nf is shown explicitly.
In order to compare our results to nonperturbative evaluations of κc coming from Monte
Carlo simulations, we employ our improved perturbation theory method for improved ac-
on
tions.
The work covered in Chapter 10 is an ongoing project on the improvement of the
fermion propagator and several quark operators, to order a2 in 1-loop perturbation theory.
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We employ improved actions for both fermions (clover) and gauge fields (Symanzik). The
dependence on the gauge parameter is explicitly shown. The terms of order a2 can be
a
used to specify the required corrections to the quark operators, in order to achieve O(a2 )
improvement. Our results are applicable also to the case of twisted mass fermions, which
are currently being studied intensely by a number of collaborations worldwide.
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Finally in Chapter 11 we summarize and conclude. The Appendices contain supplemental material that has been left out of the main body of the Thesis in order to improve
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readability.
The work of Chapters 8-9 was carried out in collaboration with Apostolos Skouroupathis
and the project of Chapter 10 is in progress with Fotos Stylianou. Most of the results
presented here have already been published in the following papers:
• M. Constantinou, H. Panagopoulos, A. Skouroupathis, Improved Perturbation Theory
for Improved Lattice Actions, Phys.Rev. D74 (2006) 074503, [hep-lat/0606001]
• M. Constantinou, H. Panagopoulos, QCD with overlap fermions: Running coupling
20
and the 3-loop beta-function, Phys. Rev. D76 (2007) 114504, [arXiv:0709.4368]
• M. Constantinou, H. Panagopoulos, Gauge theories with overlap fermions in an arbi-
trary representation: Evaluation of the 3-loop beta-function, Phys. Rev. D77 (2008)
57503, [arXiv:0711.4665]
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• A. Skouroupathis, M. Constantinou, H. Panagopoulos, Two-loop additive mass renormalization with clover fermions and Symanzik improved gluons, Phys. Rev. D77
(2008) 014513, [arXiv:0801.3146]
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and conference proceedings:
• M. Constantinou, H. Panagopoulos, A. Skouroupathis, Improving perturbation theory
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with cactus diagrams, PoS LAT2006 (2006) 155, [hep-lat/0612003]
• A. Skouroupathis, M. Constantinou, H. Panagopoulos, 2-loop additive mass renormalization with clover fermions and Symanzik improved gluons, PoS LAT2006 (2006)
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162, [hep-lat/0611005]
• M. Constantinou, H. Panagopoulos, The three-loop beta-function of SU(N) lattice
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a
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on
gauge theories with overlap fermions, PoS LAT2007 (2007) 247, [arXiv:0711.1826]
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Chapter 2
2.1
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The Wilson formulation and O(a)
improvements
Standard Wilson quarks and gluons
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In this chapter we provide the standard Wilson action for both fermions and gluons, as well
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as O(a) improvements on this action. For a brief introduction on the lattice formulation
of QCD and our notation, the reader should see Appendix A. For completeness allow us
to repeat here some of this notation that is necessary for this section. In the formulation
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of Lattice QCD, the fermion fields Ψ(x), Ψ(x) live on the lattice sites x and carry color
(i, j, ... = 1, ..., N), flavor (f = 1, ..., Nf ) and Dirac indices (α, β, ... = 1, ..., 4). We recall
a
that N, Nf are the number of fermion colors and fermion flavors, respectively. The variables
U(x, µ) are defined on the links connecting two neighboring lattice sites. The index µ =
0, ..., 3 labels the direction of the link and µ̂ is the unit vector in the µth direction.
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The link variables are not linear in the gauge fields Gaµ (x), but they are defined in a
way that the continuum action is recovered when setting a → 0
Uµ (x, x + aµ̂) ≡ Uµ (x) = eiag0 T
a Ga (x+ aµ̂ )
µ
2
(2.1)
({T a } (a = 1, ..., N 2 − 1) are the SU(N) generator matrices). By convention, the argument
of Gaµ is defined in the midpoint of the link (without affecting the continuum limit or the
simulations) and U is an N × N unitary matrix satisfying
U(x, x − aµ̂) ≡ U−µ (x) = e−iag0 T
21
a Ga (x− aµ̂ )
µ
2
= U † (x − aµ̂, x)
(2.2)
2.1. Standard Wilson quarks and gluons
22
The local gauge transformation Λ(x) (being in the same representation as U) is employed
on the fermion and gauge fields through the relations
Ψ(x) → Λ(x)Ψ(x)
Ψ(x) → Ψ(xΛ† (x))
Uµ (x) → Λ(x)Uµ (x)Λ† (x + aµ̂)
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(2.3)
One requires that the lattice action be invariant under the gauge transformations and
thus, its gluon part must be constructed by gauge invariant objects. The simplest choice
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is the trace of the 1 × 1 loop, called plaquette
Pµν ≡ Uµ (x)Uν (x + aµ̂)Uµ† (x + aν̂)Uν† (x)
(2.4)
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This is the usual product of link variables along the perimeter of a plaquette originating at
x in the positive µ − ν directions. As can be realized from Fig. 2.1, there are two different
on
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orientations for each plaquette, which are Hermitian conjugate to each other. Thus, taking
the trace over color indices ensures gauge invariance, whilst the real part is taken to average
over the loop and its complex conjugate.
U~n†+ν̂, ~n+µ̂+ν̂
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~n + ν̂
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a
U~n†, ~n+ν̂
~n + µ̂ + ν̂
U~n+µ̂, ~n+µ̂+ν̂
~n + µ̂
~n
U~n, ~n+µ̂
ν
µ
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Figure 2.1: A plaquette on a 2-dimensional slice of the hypercubic lattice.
The formulation of the action in terms of the link variables, rather than the gauge fields
directly, serves to uphold gauge invariance. This is a symmetry we are not willing to give
up, because otherwise we would have more parameters to tune (the different couplings for
the quark-gluon, 3-gluon and 4-gluon interactions, the gluon mass that would not be zero).
There would also arise many more operators at any given order in a, leading to increase of
23
the discretization errors. Finally, the existing proofs of perturbative renormalizability of
QCD, defined on the lattice, rely on strict gauge invariance [11]. In fact, it is recommended
to preserve as many symmetries of the theory as possible at all values of a. By construction,
the discretization breaks continuous rotational, Lorentz and translational symmetry.
XX
f
f
Ψ (x)(DW + mf0 )Ψf (x) +
x
1
2N X X
(1 − ReTr[Pµν ])
2
g0 µ<ν x
N
(2.5)
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SW [U, Ψ, Ψ] = a4
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Wilson on 1974 proposed in his famous paper [1] one of the most popular lattice actions
to overcome the problem of the fermion doubling. Below, we present Wilson’s lattice action
characterizing the quarks and gluons
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where the discretization of the continuous Dirac fermion action was performed by replacing
the derivative with the symmetrized difference. To avoid heavy notation, the Dirac indices
are not written, but a sum over them is implied. The mass mf0 is to be understood as
an element of a diagonal matrix in flavor space.1 The insertion of U between the terms
Ψ(1 ± γ5 )Ψ is needed to maintain the gauge symmetry. The Wilson-Dirac operator is
3
st
−
→
←
−
←
− −
→
1X
{γµ ( ∇ µ + ∇ µ ) − a r ∇ µ ∇ µ }
2 µ=0
(2.6)
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DW =
The term proportional to r is the Wilson term, with r the so called Wilson parameter. It
assigns a heavy mass proportional to r/a to the extra fifteen species of quarks, that do
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not survive in the continuum limit. Regarding the masses of the real fermions, these are
affected by corrections vanishing proportionally to a. The parameter’s standard value is 1,
a
which is the one adopted in all of our perturbative calculations. The forward and backward
covariant derivatives acting on the fermion-antifermion fields are
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−
→
1
[U(x, µ)Ψ(x + aµ̂) − Ψ(x)]
∇ µ Ψ(x) =
a
←
−
1
∇ µ Ψ(x) =
[Ψ(x) − U −1 (x − aµ̂, µ)Ψ(x − aµ̂)]
a
−
→
1
Ψ(x) ∇ µ =
[Ψ(x) − Ψ(x − aµ̂)U(x − aµ̂, µ)]
a
←
−
1
Ψ(x) ∇ µ =
[Ψ(x + aµ̂)U −1 (x, µ) − Ψ(x)]
a
1
(2.7)
(2.8)
(2.9)
(2.10)
In practice, the mass can be√replaced by the hopping parameter κ = 1/(2a m0 + 8r) and the fermion
fields are rescaled by a factor of 2κ/a3/2 . A comprehensive description of the hopping parameter appears
in Chapter 9.
24
The choice of the plaquette in the action instead of any alternative closed loop (involving
more U’s) is entirely for practical reasons: Computations with small loops are faster to
perform having the discretization errors under control.
Besides the absence of fermion doublers and the existence of gauge symmetry, the
ou
Wilson action has a number of properties:
1. It is invariant under translations by a.
2. The transformations of charge conjugation C, parity P and time reversal T , leave the
action invariant.
3. The Wilson-Dirac operator DW has γ5 -Hermicity, that is, γ5 DW γ5 = D † .
tin
4. Eq. (2.5) includes only nearest-neighbor interactions, leading to vertices with compact
form and easy to work with.
an
The above properties of Eq. (2.5) go along with the following disadvantages:
1. The massive Wilson-Dirac operator DW + m0 is not protected against zero modes. As a
result, the quark mass gets an additive and multiplicative renormalization and the critical
mass has a linear divergence (mc ∼ 1/a).
on
st
2. Chiral symmetry is explicitly broken at order a by the Wilson term, and it is restored
only in the continuum limit.
3. The leading cutoff effects with Wilson type fermions are of order a, rather that a2 . In
the next sections we will discuss O(a) improvements of the standard Wilson action.
4. The axial current transformations are not an exact symmetry and the nonsinglet axial
C
current requires a nontrivial multiplicative renormalization to restore current algebra up
to O(a) effects.
a
This analysis is consistent with the famous No-Go theorem of Nielsen and Ninomiya stating
that it is not possible to define a local, translationally invariant, hermitian lattice action
M
ar
th
that preserves chiral symmetry and does not have doublers. The theorem will be explained
in more detail in the next chapter.
At this point it would be useful to derive the fermion and gluon propagator from
Eq. (2.5), an essential quantity for both perturbative calculations and simulations. Their
extraction is carried out as follows:
a. We perform a Taylor expansion of the link variables up to O(g02 ) and isolate the terms
leading to the propagator, that is
X
x,y
Ψ(x)M f (x, y)Ψ(y) ,
X
µν
g
Gµ (x)Mµν
(x, y)Gν (y)
25
b. According
to Wick’s theorem, the fermion (gluon) propagator is the inverse matrix of
f
g
M (x, y) Mµν . Since the M matrices are not diagonal but satisfy the relation
a4
X
M f (x, y)M f (y, z) = (2π)4
y
δx,z
,
a4
a4
X
g
g
Mµν
(x, y)Mνρ
(y, z) =
ψ,ν
δµρ δx,z
a4
the fields is
Gµ (x) =
Z
1
(2π)4
π/a
d4 p eiap·x Ψ̃(p)
−π/a
π/a
˜
d4 p e−iap·x Ψ(p)
−π/a
Z
tin
1
(2π)4
Ψ(x) =
Z
π/a
d4 p eiap·x G̃µ (p)
an
1
Ψ(x) =
(2π)4
ou
it is convenient to find the propagator in momentum space. The Fourier transformation of
−π/a
P
2 apµ
sin(apµ ) + m0 + µ 2r
sin
(
)
a
2
2
P
2 apµ
sin2 (apµ ) + m0 + µ 2r
sin
(
)
a
2
γµ
a
on
P
−i µ
S f (p, k) = (2π)4 δ 4 (p − k) P
1
st
Thus, one arrives at the expressions below for the fermion and gluon propagator respectively
µ a2
4δµν X
apµ
4
apµ
apν
sin(
) − 2 sin(
) sin(
)
2
a
2
a
2
2
µ
C
g
Sµν
(p, k)−1 = (2π)4 δ 4 (p + k)
(2.11)
(2.12)
The existence of r in the denominator of Eq. (2.11), secures that in the massless case
th
a
(m0 = 0) there are no poles at the edges of the Brillouin zone. Thus, the fermion doublers
are absent, which was the initial purpose of the construction of this action.
M
ar
c. The transition back to position space is based on the inverse Fourier transformation
Z Z
d4 p d4 k −ia(p.x+k.y) f
e
S (p, k)
(2π)4 (2π)4
Z
d4 p iap.(x−y) g
g
Mµν (x, y) =
e
Sµν (p)
(2π)4
f
M (x, y) =
As always in perturbation theory, we must introduce an appropriate gauge-fixing term
2.2. O(a) improved actions
26
to the action; in terms of the gauge field Gµ (x) it reads
SF P =
1 X −
Tr ∆µ Qµ (x)∆−
ν Qν (x) ,
1−ξ x,µ,ν
∆−
µ Qν (x) ≡ Qν (x − µ̂) − Qν (x)
(2.13)
fixing produces the following action for the ghost fields c and c
ou
Having to compute a gauge invariant quantity, we can, for convenience, choose to work
either in the Feynman gauge (ξ = 0) or in the Landau gauge (ξ = 1). Covariant gauge
n
1 +
†
+
ig
Q
(x),
∆
c(x)
Tr (∆+
c(x))
∆
c(x)
+
ig
[Q
(x),
c(x)]
+
µ
µ
µ
µ
µ
2
x
µ
o
1 − g 2 Qµ (x), Qµ (x), ∆+
+
·
·
·
,
∆+
c(x)
µ c(x) ≡ c(x + µ̂) − c(x). (2.14)
µ
12
XX
tin
Sgh = 2
XX
1
Tr {Qµ (x)Qµ (x)} + · · ·
Ng 2
12
x
µ
(2.15)
st
Sm =
an
Finally, the change of integration variables from links to vector fields yields a Jacobian
that can be rewritten as the usual measure term Sm in the action
on
The terms Sm and SF P must be added to the total action.
O(a) improved actions
2.2
C
There is a variety of improved fermion and gluon actions, but we focus on those we have
used in our perturbative calculations. Next, we briefly discuss the clover improved fermion
th
a
action and the Symanzik improved gluon action. For the overlap action we dedicate the
whole next chapter since it includes the derivation of the overlap vertices with more than
two gluons.
The clover fermion action
M
ar
2.2.1
The widely used clover action was originally studied by Sheikholeslami and Wohlert [3] to
remove the O(a) contributions of the Wilson fermion action. In order to understand the
origin of the clover action, we introduce the Effective Field Theory, according to which
high energy effects can be described as perturbations in the Standard Model’s Lagrangian.
27
For a → 0 the lattice theory can be seen as an effective theory described by the action [10]
2
Sef f = S0 + a S1 + a S2 + ... =
Z
d4 x(L0 + a L1 + a2 L2 + ...)
(2.16)
O1 = Ψ iσµν Gµν Ψ
−
→ −
→
←
− ←
−
O2 = Ψ ( ∇ µ ∇ µ − ∇ µ ∇ µ ) Ψ
tin
O3 = m Tr[Gµν Gµν ]
−
→
←
−
O4 = m Ψ (γµ ∇ µ − ∇ µ γµ ) Ψ
ou
The only operators that can appear in L1 are
O5 = m2 Ψ Ψ
(2.17)
an
−
→ →
−
where Gµν = i/g [ ∇ µ , ∇ ν ] and σµν = 1/2 [γµ, γν ]. The forward and backward covariant
−
→ ←
−
derivatives ∇ µ , ∇ µ have been previously defined in Eqs. (2.7) - (2.10). One may use the
equations of motion to eliminate some of the operators, for instance O2 , O4 , provided that
st
proper renormalization conditions have been employed
O4 + 2O5 = 0
(2.18)
on
O1 − O2 + 2O5 = 0 ,
The remaining operators of Eqs. (2.17) can be used to construct an improved action by
adding a suitable counterterm to the Wilson action. We assume that L1 is a linear combi-
C
nation of the operators and can be cancelled out by adding the following terms
δS = a5
X
(2.19)
a
x
(c1 Ô1 (x) + c3 Ô3 (x) + c5 Ô5 (x))
th
bn (x) is some lattice representation of the field On . To make things even simpler,
where O
M
ar
we choose to represent Tr[Gµν Gµν ] and Ψ Ψ by the plaquette term and the local scalar
density that already appear in the Wilson action. This way, O3 , O5 correspond to a reparametrization of the bare coupling and bare mass by a factor of the form 1 + O(am).
Thus, the only independent operator is O1 and the improved action can be written as
f
SSW = SW
+ a5
icsw X X f
Ψ (x)σµν Fbµν (x)Ψf (x),
4 f x,µ,ν
(2.20)
28
The advantages of the clover action is that it is local and leaves perturbation theory
tractable. The addition of the clover term is only about a 15% overhead on Wilson fermion
simulations [12, 13]. The first term of Eq. (2.20) is the fermion part of the Wilson action
(Eq. (2.5)) and cSW is the clover parameter. The tensor Fbµν is a lattice representation of
the gluon field tensor, defined through
Qµν is the sum of the plaquette loops
tin
Qµν = Ux, x+µ Ux+µ, x+µ+ν Ux+µ+ν, x+ν Ux+ν, x
(2.21)
ou
1
Fbµν (x) = 2 (Qµν (x) − Qνµ (x))
8a
+ Ux, x+ν Ux+ν, x+ν−µ Ux+ν−µ, x−µ Ux−µ, x
an
+ Ux, x−µ Ux−µ, x−µ−ν Ux−µ−ν, x−ν Ux−ν, x
+ Ux, x−ν Ux−ν, x−ν+µ Ux−ν+µ, x+µ Ux+µ, x
on
st
as shown in Fig. 2.2.
(2.22)
µ
C
ν
a
Figure 2.2: Graphical representation of Qµν (Eq. (2.22)) appearing in the clover action.
th
Let us recall that the clover parameter CSW , multiplying the additional term in Eq. (2.20)
must be chosen properly to achieve the improvement; this is not an easy task to accomplish. A correct choice for cSW removes O(a) errors in on-shell quantities such as hadron
M
ar
masses. Its tree-level value is cSW = 1 and Heatlie et al. [14] showed that all terms that
are effectively of order a are removed in the 1-loop matrix elements of the quark currents.
Wohlert estimated this coefficient up to O(g02 ), finding [15]
csw = 1 + 0.26590(7) × g02 + O(g04)
(2.23)
where the Wilson gauge action is assumed. Since then, many studies of the clover parameter appeared in the literature, either using improved gluons [16, 17], or nonperturbative
29
calculations through the requirement that chiral symmetry is preserved up to terms of order
a2 . This kind of calculations has been performed by ALPHA Collaboration. Their data in
the quenched approximation for 0 ≤ g0 ≤ 1 are represented by the rational expression [18]
while in the Nf = 2 case by [19]
1 − 0.454 g02 − 0.175 g04 + 0.012 g06 + 0.045 g08
1 − 0.720 g02
(2.25)
tin
cSW =
(2.24)
ou
cSW
1 − 0.656 g02 − 0.152 g04 − 0.054 g06
=
1 − 0.922 g02
It is important to mention that in order to achieve improvement on off-shell quantities,
an
one must add irrelevant terms of higher dimensionality, to the operator under study. These
terms must have appropriate improved coefficients and the operators have to be normalized
as well. Such a study is described in Chapter 10.
The Symanzik improved gluon action
st
2.2.2
Having in mind the improvement of the fermion action, we can proceed in the same manner
on
to the improvement of the gauge action. The gauge part of the Wilson action is constructed
by 1 × 1 plaquettes which are the smallest possible closed loops. We can generalize the
Wilson action by including all loops with 4 and 6 links, as shown in Fig. 2.3. In standard
X
X
2h
= 2 c0
Re Tr {1 − Uplaquette } + c1
Re Tr {1 − Urectangle }
g0
plaquette
rectangle
i
X
X
+c2
Re Tr {1 − Uchair } + c3
Re Tr {1 − Uparallelogram }
a
SG
2
C
notation (see, e.g., Ref. [20]), the action reads
th
chair
(2.26)
parallelogram
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ar
The coefficients ci can in principle be chosen arbitrarily, subject to the following normalization condition which ensures the correct classical continuum limit of the action
2
c0 + 8c1 + 16c2 + 8c3 = 1
(2.27)
1 × 1 plaquette, 1 × 2 rectangle, 1 × 2 chair (bent rectangle), and 1 × 1 × 1 parallelogram wrapped
around an elementary 3-d cube.
30
Some popular choices of values for ci used in numerical simulations will be considered
in Subsection 8.2.3 (Table 8.1); they are normally tuned in a way as to ensure O(a)
st
an
tin
ou
improvement.
on
Figure 2.3: The 4- and 6-link loops contributing to the gauge action of Eq. (2.26).
The lowest order expansion of this action, leading to the gluon propagator, is
1
=
2
Z
π/a
d4 k X a
ξ
Aµ (k) Gµν (k) −
k̂µ k̂ν Aaν (−k)
4
(2π) µν
ξ−1
C
(0)
SG
−π/a
(2.28)
a
where ξ is the gauge fixing parameter (see Eq. (2.13) ) and
th
X
Gµν (k) = k̂µ k̂ν +
ρ
− k̂µ k̂ρ δρν
dµρ ,
k̂µ =
2
akµ
sin
,
a
2
k̂ 2 =
h
i
= (1 − δµν ) C0 − C1 a2 k̂ 2 − C2 a2 (k̂µ2 + k̂ν2 )
M
ar
dµν
k̂ρ2 δµν
X
k̂µ2
µ
(2.29)
The coefficients Ci are related to ci by
C0 = c0 + 8c1 + 16c2 + 8c3 , C1 = c2 + c3 , C2 = c1 − c2 − c3
(2.30)
In momentum space, the gluon propagator Dµν is given by a set of linear equations of the
form
X
Gµρ (k) −
ρ
31
ξ
k̂µ k̂ρ Dρν (k) = δµν
ξ−1
(2.31)
In perturbative calculations we need to know Dµν for arbitrary dimensions, but Dµν can
ou
be given in closed form only for integer dimensions. Only for the special case C2 = 0 one
can derive explicit forms for arbitrary dimensions. To overcome this, it is convenient to
split the gluon propagator into two parts: A singular part, which can easily be extended
to arbitrary dimensions, and a finite part, which does not need to be regularized. Thus,
plaquette
Dµν = C0−1 Dµν
+ ∆Dµν
k̂ 2
δµν − ξ
k̂µ k̂ν
k̂ 2
!
(2.33)
an
plaquette
Dµν
(k) =
1
tin
where
(2.32)
is the Wilson propagator (c0 = 1, c1 = c2 = c3 = 0). One can observe that Dµν and
st
plaquette
C0−1 Dµν
in Eq. (2.32) have the same infrared singularity. The finite part ∆Dµν is
plaquette
, in four dimensions.
obtained by solving Eq. (2.31) for the difference Dµν − C0−1 Dµν
In computations involving the Symanzik action, the final results cannot be expressed
M
ar
th
a
C
on
explicitly in terms of ci , but must be tabulated for different sets of the Symanzik coefficients.
There are about 10 popular sets used in simulations (Table 8.1) which we have employed
in our perturbative calculations (Chapters 8, 9, 10).
Introduction
an
3.1
tin
O(a) improved overlap action
ou
Chapter 3
In this chapter we introduce another kind of O(a) improved fermions, described by the
st
overlap action. Its distinction from the actions mentioned in Chapter 2 is the fact that
it conserves chiral symmetry. In QCD, this is a global symmetry and forbids an additive
fermion mass renormalization, while Goldstone bosons (with their interactions) appear
on
from its spontaneous breakdown. Concerning electroweak interactions, the role of chirality
is more essential, as it is a local gauge symmetry, necessary for a renormalizable theory.
C
Whilst trying to regularize these theories on the lattice, one finds 2 difficulties:
(a) According to the No-Go theorem of Nielsen and Ninomiya [2, 21] (see next section),
chiral symmetry does not appear on the lattice without losing locality, translational invari-
a
ance, or introducing fermion doublers.
(b) The loss of chirality causes the appearance of O(a) lattice artifacts, the mixing between
matrix elements of operators with different chirality, as well as other finite lattice spacing
th
effects.
Various attempts have been made to maintain chiral invariance on the lattice at the
M
ar
cost of losing translation invariance, locality, or the correct free fermion spectrum. The
search for appropriate solutions to the problem of global chirality, leads to the necessity
of defining a new type of fermion action: The overlap fermions [5, 6, 7] and domain-wall
fermions [8, 9] (fermions in 4+1 dimensions). Here we will only study the overlap action,
since it was used for our β-function calculation (Chapter 7). Next, we briefly present the
No-Go theorem, the meaning of locality and the Ginsparg-Wilson relation. Then, we focus
on the overlap action and explain in detail the extraction of its fermion-gluon vertices with
32
3.2. The Nielsen-Ninomiya No-Go theorem
33
up to 4 gluons; this part of the chapter is original work.
3.2
The Nielsen-Ninomiya No-Go theorem
As any regularization, the lattice breaks some properties of the underlying continuous
theory. The Nielsen-Ninomiya theorem imposes nontrivial limitations on those properties of
ou
the fermion action that can be maintained on the lattice. In particular, this theorem states
that if a lattice Dirac operator provides correct pole structure, it cannot simultaneously
be chirally invariant, translation invariant and local. As an example, consider the naively
an
tin
discretized Dirac operator on the lattice
γµ −
→
←
− D=
∇µ + ∇µ
(3.1)
2
−
→ ←
−
where ∇ µ ( ∇ µ ) is the nearest-neighbor forward (backward) difference operator (Eqs. (2.7)
- (2.10)). Indeed, this choice for D anticommutes with γ5 , leading to an invariant action
under chiral transformations. In momentum space the operator can be written in the form
|pµ | ≤ π/a
st
D̃(p) = iγµ sin(a pµ ) ,
(3.2)
which obviously has 16 zeros in four dimensions, leading to a proliferation of fermion
on
species. Thus, besides the physical fermion, D̃(p) is propagating the so called doubler
modes. In the free theory this may not be a problem, but in the interacting theory, the
C
latter would contribute in the loop corrections.
A massless fermion in the lattice regularization is described by the action
a
S = a8
X
x,y
Ψ(x) D(x − y) Ψ(y)
(3.3)
th
where D(x − y) is a Dirac operator which ideally has the following properties:
M
ar
1. It is local in the sense that it is bounded by C e−γ|x−y|/a , where C and γ are constants
that do not depend on the lattice spacing a. A Dirac operator with such exponential behavior, is as good as an ultralocal operator involving only nearest neighbor interactions. The
locality of D implies that its Fourier transformation D̃(p) is an analytic periodic function
of the momenta pµ with period 2π/a.
2. D̃(p) satisfies D̃(p) = iγµ pµ + O(a2 p2 ) for |p| << π/a, so that the correct continuum
limit is obtained.
3. D̃(p) is invertible for p 6= 0 (absence of fermion doublers).
3.3. Locality
34
4. It anticommutes with γ5 , ensuring that the fermion action is invariant under continuous
chiral transformations.
According to the Nielsen-Ninomiya theorem, the above points cannot be fulfilled simultaneously. The simpler example is the Wilson-Dirac operator
→
←
− ar ←
− −
→
γµ −
∇µ + ∇µ − ∇µ ∇µ
2
2
(3.4)
ou
D=
with the second term introduced to avoid the presence of massless fermion doublers. As a
tin
consequence, this irrelevant operator breaks chiral symmetry explicitly, resulting in extra
problems:
• At finite lattice spacing, we lose chiral symmetry due to O(a) lattice artifacts; only in
the continuum limit is re-established.
• For the symmetry to be restored in the quantum theory, one must include an additive
3.3
st
an
mass renormalization.
• Operators in different chiral representations get mixed.
Locality
on
Before proceeding to Ginsparg-Wilson fermions, it is helpful to state the precise definition
of the term locality, as it is used here. First, let us consider a Lagrangian of the general
form
(3.5)
C
L = αΦ(x)∂µ2 Φ(x) + βΦ(x)∂µ4 Φ(x) + · · ·
The Lagrangian is called ultralocal if there is a finite number of derivatives in Eq. (3.5) and
a
the corresponding Dirac operator D(x−y) according to the causality relation, must be zero
for (x − y)2 < 0. When using the lattice as a regulator, ultralocality entails interactions
involving a finite number of lattice points; their couplings drop to zero beyond a finite
M
ar
th
number of lattice spacings.
A strictly local (ultralocal) Dirac operator D, has nonzero contributions to the sum
X
D Ψ(x) = a4
D(x − y) Ψ(y)
(3.6)
y
coming from the points y in a finite neighborhood of x. Moreover, the kernel D(x − y)
should only depend on the gauge field variables existing near x. The definition of locality,
as it is adopted here, is that the Dirac operator in an arbitrary gauge field background,
obeys the more relaxed bound
3.4. The Ginsparg-Wilson relation
35
||D(x, y; U)|| ≤ C e−γ||x−y||
(3.7)
where C, γ are positive constants independent of the gauge field U. Eq. (3.7) shows that
D is a local operator with localization range 1/γ. As long as γ is proportional to the
cutoff 1/a, Eq. (3.7) is as good as strict locality and the sum in Eq. (3.6) is dominated
by the contributions from the bounded region around x. Furthermore, it is assumed that
D(x, y; U) depends negligibly on U’s far apart from x and y, that is
ou
δ
D(x, y; U)
δ Uµ (z)
(3.8)
tin
also falls exponentially in |x−y|, |x−z| and |z −y|. Eqs. (3.6) - (3.7) satisfy the principle of
universality, according to which, different regulators lead to the same quantum limit of the
an
continuum. The above definition of locality provides a criterion to accept or reject Dirac
operators. Indeed, in real systems, we cannot consider only nearest neighbor interactions,
or those with strictly finite support, since the interaction coefficients fall exponentially. On
The Ginsparg-Wilson relation
on
3.4
st
the contrary, interactions decreasing slower than exponentially are considered as nonlocal
and should not be acceptable; universality is not expected to hold in these cases.
In the early eighties, Ginsparg and Wilson [22] proposed a way to preserve chiral symmetry
on the lattice without giving up any of the four properties stated in Section 2. This is
C
feasible because chiral symmetry can be realized in a different way than the one assumed
when proving the Nielsen-Ninomiya theorem. Their proposal was that the Dirac operator
a
no longer anticommutes with γ5 , but instead satisfies
th
γ5 D −1 + D −1 γ5 = 2aRγ5
⇒ Dγ5 + γ5 D = 2aDRγ5 D ,
D † = γ5 Dγ5
(3.9)
M
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In the above relation, known as the Ginsparg-Wilson (GW) equation, a is the lattice
spacing, and R is a local operator, which commutes with γ5 . The restriction [R, γ5 ] = 0
arises from the invariance of the action to O(ǫ) for a lattice modified chiral transformation
(see Eqs. (3.10) - (3.11)). Although the Dirac operators of Eq. (3.9) cannot be ultralocal,
they may be local (the interaction range of an ultralocal operator is bounded by some finite
lattice distance, while for a local operator is exponentially bounded). The GW equation
ensures that the effects of the chiral symmetry breaking terms in the Dirac operator appear
3.4. The Ginsparg-Wilson relation
36
only in local terms and are physically irrelevant. Eq. (3.9) ensures that the fermion action is
invariant under certain infinitesimal transformations (Eq. (3.10)) which can be considered
as the lattice form of the usual chiral rotation. The propagating states must be effectively
chiral, so that the presence of R is not felt in distances larger than its range. A common
ou
choice for R is the constant value 1/2.
Apparently, the anticommutation property {γ5 , D} = 0 is recovered in the continuum
limit, and the r.h.s. of Eq. (3.9) is zero for D Ψ = 0. A desired consequence of exact
chirality on the lattice is that there are no O(a) lattice artifacts, to any order of g. This
can be explained in the following way: The O(a) lattice artifacts appear when adding a
tin
lattice version of Ψ σµν Fµν Ψ. Any lattice regularization of such a term would violate chiral
symmetry, hence the O(a) artifacts should be absent from the action. However, O(a2 )
lattice artifacts could be large at particular lattice spacings and different for alternative
an
solutions of the GW relation.
Having an exact chiral symmetry of the action, makes it easy to introduce left- and
right-handed fields. Starting from the infinitesimal transformation
Ψ → Ψ + ǫ γ5
st
Ψ → Ψ + ǫ γ5 (1 − 2a R D) ,
(3.10)
(where D satisfies the GW equation) one can prove that the action of Eq. (3.3) is invariant
on
to first order in ǫ,
X
Eq. (3.9)
Ψ(x)[γ5 D(x − y) + D(x − y)γ5(1 − 2a R D)]Ψ(y) = 0
δS = a8 ǫ
x,y
(3.11)
C
On the lattice it is useful to define the operator γ̂5 ≡ γ5 (1 − 2a R D) with the properties
(γˆ5 )† = γˆ5 , (γˆ5 )2 = 1, γ5 D = −D γˆ5
(3.12)
th
a
The chiral projectors for fermion and antifermion fields are
1
1
P̂± = (1 ± γˆ5 ), P± = (1 ± γ5 )
2
2
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ar
respectively; they satisfy
D P̂+ = P− D ,
D P̂− = P+ D
(3.13)
(3.14)
and allow the decomposition
D = P+ D P̂− + P− D P̂+
The left- and right-handed chiral components of the fields are
(3.15)
3.5. Overlap action
37
ΨL = P̂− Ψ , ΨL = Ψ P+
ΨR = P̂+ Ψ , ΨR = Ψ P−
(3.16)
and they transform under lattice chiral rotations in the same way as the corresponding
fields in the continuum theory. Actually, the left- and right-handed fields decouple in the
action Eq. (3.3)
Ψ D Ψ = ΨL D ΨL + ΨR D ΨR
(3.17)
ou
analogously to the continuum case. It is obvious that we can eliminate the right-handed
components by employing the constraints P̂− Ψ = Ψ , ΨP+ = Ψ. Both conditions along
Overlap action
an
3.5
tin
with the action of Eq. (3.3) are local and gauge invariant; this guarantees a complete
definition of the theory at the classical level.
In 1998 Neuberger starting from the Wilson discretization of the Dirac action, constructed
a fermion action [6] preserving chiral symmetry on the lattice. In the same publication, he
st
showed that the Dirac operator of the overlap formalism does satisfy the GW relation (for
R = 1/(2ρ)), and the overlap action can be written in the following compact way
X
on
Soverlap = a8
Ψ̄(n) DN (n, m) Ψ(m)
(3.18)
n,m
C
where DN (n, m) is a local operator (it remains local in a gauge field background as well),
the so called overlap-Dirac operator
#
1
δn,m
,
DN (n, m) = ρ
− X√
a4
X † X nm
X=
1
(DW − ρ)
a4
(3.19)
th
a
"
and DW is the massless Wilson-Dirac operator of Eq. (2.6) (the Wilson parameter r is set
M
ar
to 1).
The operator X is hermitian, X † X = 1, and the overlap parameter ρ is restricted by
the condition 0 < ρ < 2 to guarantee the correct pole structure of DN (for ρ ≤ 0 there
are no massless fermions, while for ρ ≥ 2 there appear more than one). The coupling
constant is nontrivially included in the link variables, present in the definition of X, and in
the framework of perturbation theory, one takes the expansion of X in powers of g0 . This
3.5. Overlap action
38
expansion in momentum space has the form
X(p′ , p) = χ0 (p)(2π)4 δP (p′ − p) +X1 (p′ , p) + X2 (p′ , p) +X3 (p′ , p) + X4 (p′ , p) +O(g05)
{z
}
|
tree−level
|
{z
}
1−loop
|
{z
}
2−loop
ou
(3.20)
where χ0 is the inverse fermion propagator and Xi are the vertices of the Wilson fermion
action (p (p′ ): Fermion (antifermion) momentum). The construction of all overlap vertices
relevant to the β-function computation (see Eqs. (3.34) - (3.38) below) makes use of χ0
tin
and X1 − X4 ; these quantities can be written as
C
on
st
an
ρ
1 X
iX
χ0 (p) =
γµ sin(apµ ) +
1 − cos(apµ ) −
a µ
a µ
a
Z
p′ + p X
X1 (p′ , p) = g0 d4 kδ(p′ − p − k)
Aµ (k)V1,µ
2
µ
Z
p′ + p X
g2
d4 k1 d4 k2 ′
X2 (p′ , p) = 0
δ(p
−
p
−
k
−
k
)
A
(k
)A
(k
)V
1
2
µ 1
µ 2 2,µ
2
(2π)4
2
µ
Z 4 4 4
3
3
h
p′ + p i
X
XY
d k1 d k2 d k3 ′
g03
′
2
δ(p − p −
ki )
Aµ (ki ) − a V1,µ
X3 (p , p) =
3!
(2π)8
2
µ i=1
i=1
Z 4 4 4 4
4
4
h
p′ + p i
X
XY
g04
d k1 d k2 d k3 d k4 ′
′
2
X4 (p , p) =
δ(p
−
p
−
k
)
A
(k
)
−
a
V
(3.21)
i
µ i
2,µ
4!
(2π)12
2
µ i=1
i=1
a
where
V2, µ (p) = −i γ µ a sin(ap µ ) + a cos(ap µ )
(3.22)
th
V1, µ (p) = i γ µ cos(ap µ ) + sin(ap µ ),
A µ represents a gluon field; later on we will have to generalize Eqs. (3.21) to the case
M
ar
where both a background and a quantum gluon field are present, see Eqs. (5.29).
At this point we can proceed with the perturbative expansion of DN in powers of
g0 . This leads to the propagator of zero mass fermions and to gluon-fermion-antifermion
vertices (with up to 4 gluons for the needs of our work). The much simpler case of vertices
with up to 2 gluons (and no background) can be found in Ref. [23].
Now, we will explain the procedure for constructing the vertices, by expanding the
√
terms 1/ X † X, appearing in the overlap-Dirac operator, in powers of g0 , using a procedure
3.5. Overlap action
39
introduced by Y. Kikukawa and A. Yamada [23].
In an integral representation, the expression √
√
1
X †X
=
Z
∞
−∞
1
X †X
can be written as
dt
1
π t2 + X † X
(3.23)
ou
In fact, Eq. (3.23) is valid for any operator X provided that X † X has no vanishing eigenvalues. We begin the desired expansion of the overlap-Dirac operator in powers of g0 , by
setting
O(g00 )
(3.24)
tin
X † X = X0† X0 +Z
| {z }
The first term of Eq. (3.24) corresponds to the inverse fermionic propagator, while Z leads
an
to the vertices; for our 2-loop calculation of the β-function we need to write Z up to O(g04 )
Z = (X0† X1 + X1† X0 ) + (X0†X2 + X1† X1 + X2† X0 ) + (X0† X3 + X1† X2 + X2† X1 + X3† X0 )
{z
} |
{z
} |
{z
}
|
O(g01 )
O(g02 )
O(g03 )
st
+ (X0† X4 + X1† X3 + X2† X2 + X3† X1 + X4† X0 ) +O(g05 )
|
{z
}
on
O(g04 )
(3.25)
Using the above equations, we write the denominator on the r.h.s. of Eq. (3.23) as
(3.26)
C
1
1
1
1
1
=
1
−
Z
+
Z
Z
+
...
t2 + X † X
t2 + X0† X0
t2 + X0† X0
t2 + X0† X0 t2 + X0† X0
From this point forward it is easier to work in momentum space since, taking the Fourier
th
a
transform, the denominator becomes diagonal
F.T.[
1
t2 +
X0† X0
]=
t2
1
+ ω 2(p)
(3.27)
M
ar
(ω 2 (p) defined in Eq. (3.33)). Combining Eqs. (3.23) and (3.26) we derive the Taylor
1
expansion of √
in momentum space
X †X
3.5. Overlap action
Z
40
Z ∞
dt 2πδ(p′ − p)
dt
1
1
√
(p , p) =
−
Z(p′ , p) 2
2
2
′
2
2
′
†
t + ω 2 (p)
X X F.T.
−∞ π t + ω (p )
−∞ π t + ω (p )
Z ∞
Z
1
1
1
dt ∞ dk
+
Z(p′ , k) 2
Z(k, p) 2
+ ... (3.28)
4
2
2
′
2
t + ω (k)
t + ω 2 (p)
−∞ π −∞ (2π) t + ω (p )
1
′
∞
ou
The first term in the r.h.s. leads to the inverse propagator, while the rest represent the
overlap vertices with a fermion-antifermion pair and a certain number of gluons (second
term: 1 gluon, third term: 2 gluons, and so on). The integral over t can now be performed
tin
by applying the residue theorem in complex analysis, a powerful tool to evaluate line
integrals over closed curves. Let us consider as an example the third term on the r.h.s. of
an
Eq. (3.28), which can be evaluated in the following way.
Z’s are not involved in the integral over t, for the reason that they depend only on the
momenta. Further, the ω’s commutes with Z’s, so we can change the ordering between
∞
−∞
dk
Z(p′ , k)Z(k, p)
(2π)4
Z
∞
dt
1
1
1
2
2
′
2
2
2
π t + ω (p ) t + ω (k) t + ω 2(p)
on
Z
st
1/(t2 + ω 2(k)) and Z(k, p) as long as the Z’s are kept in the order appearing in Eq. (3.28).
Thereby, we can write the selected term as
−∞
!
(3.29)
C
The integral under consideration has six poles in the complex t-plane: ±iω(p), ±iω(k),
±iω(p′ ). We can easily calculate it by closing the contour around the upper complex
M
ar
th
a
t-plane, as shown in Fig. 3.1.
Im[t]
C
i w(p)
i w(k)
i w(p’)
−i w(p’)
−i w(k)
Re[t]
−i w(p)
Figure 3.1: The poles of Eq. (3.29) and the integration region C.
3.5. Overlap action
41
According to the residue theorem, one must only take into account the poles encompassed
in the region bounded by this contour. The prescription reads
Z
∞
−∞
X
dt
1
1
1
2πi
=
Res(f, tn )
π t2 + ω 2 (p′ ) t2 + ω 2 (k) t2 + ω 2 (p)
π
′)
n=ω(p),ω(k),ω(p
{z
}
|
(3.30)
f (t)
∞
−∞
1
1
1
dt
ω(p′ ) + ω(k) + ω(p)
=
π t2 +ω 2 (p′ ) t2 +ω 2 (k) t2 +ω 2(p)
ω(p′ )ω(k)ω(p) [ω(p′)+ω(k)] [ω(k)+ω(p)] [ω(p)+ω(p′ )]
tin
Z
ou
Here, Res(f, tn ) denotes the residue of f (t) at the poles tn and one directly gets
Similarly we integrate all terms of Eq. (3.28) over t and this leads to Eqs. (3.32) - (3.38).
After laborious analytical manipulations, the overlap-Dirac operator is expanded into terms
an
with up to 4 gluons as
(3.31)
χ0 (k1 )
ω(k1)
(3.32)
DN (k1 , k2 ) = D0 (k1 ) (2π)4 δ 4 (k1 − k2 ) + Σ(k1 , k2 )
st
D0 (k1 ) is the inverse propagator,
where
s
X
2
µ
sin (pµ ) + ρ − 2r
C
ω(p) =
on
D0 (k1 ) = 1 +
X
2
sin (pµ /2)
µ
2
(3.33)
a
and Σ(k1 , k2 ) are the overlap vertices
V11 (k1 , k2 ) + V12 (k1 , k2 ) + V22 (k1 , k2 )
| {z } |
{z
}
th
Σ(k1 , k2 )
=
ρ
1−gluon vertex
2−gluon vertex
M
ar
+ V13 (k1 , k2) + V23 (k1 , k2 ) + V33 (k1 , k2 )
|
{z
}
3−gluon vertex
+ V14 (k1 , k2) + V24 (k1 , k2 ) + V34 (k1 , k2 ) + V44 (k1 , k2 ) +O(g05 )
|
{z
}
where we have set a = 1.
4−gluon vertex
(3.34)
3.5. Overlap action
42
V1i -V4i are given below in the most compact way
#
"
1
1
V1i (k1 , k2 ) =
Xi (k1 , k2 ) −
χ0 (k1 ) Xi†(k1 , k2 ) χ0 (k2 )
ω(k1 ) + ω(k2)
ω(k1 )ω(k2)
X
{j>0,k>0}
{j+k=i}
d4 k3
1
1
1
×
4
(2π) ω(k1 ) + ω(k3) ω(k1) + ω(k2 ) ω(k2) + ω(k3 )
"
ou
=
Z
− Xj (k1 , k3 ) χ†0 (k3 ) Xk (k3 , k2 )
− Xj (k1 , k3 ) Xk† (k3 , k2 ) χ0 (k2 )
− χ0 (k1 ) Xj† (k1 , k3 ) Xk (k3 , k2 )
tin
V2i (k1 , k2 )
(3.35)
an
#
ω(k1) + ω(k2) + ω(k3)
+
χ0 (k1 )Xj† (k1 , k3 )χ0 (k3 )Xk† (k3 , k2 )χ0 (k2 ) (3.36)
ω(k1 )ω(k2)ω(k3)
!
4
4
Y
d
k
d
k
1
1
3
4
V3i (k1 , k2 ) =
×
(2π)4 (2π)4 4 pǫS ω(kp1 ) + ω(kp2 )
4
!
"
X
1 X
ω(kp1 )ω(kp2 )ω(kp3 ) Xj (k1 , k3 )Xk† (k3 , k4 )Xl (k4 , k2 )
−
6 pǫS
{j>0,k>0,l>0}
on
st
Z Z
4
{j+k+l=i}
C
+ ω(k1 ) + ω(k3) + ω(k4 ) + ω(k2) ×
h
χ0 (k1 )Xj† (k1 , k3)Xk (k3 , k4 )Xl† (k4 , k2 )χ0 (k2 )
a
+ χ0 (k1 )Xj† (k1 , k3 )Xk (k3 , k4 )χ†0 (k4 )Xl (k4 , k2 )
th
+ χ0 (k1 )Xj† (k1 , k3 )χ0 (k3 )Xk† (k3 , k4)Xl (k4 , k2 )
+ Xj (k1 , k3 )χ†0 (k3 )Xk (k3 , k4 )Xl† (k4 , k2 )χ0 (k2 )
+ Xj (k1 , k3 )χ†0 (k3 )Xk (k3 , k4 )χ†0 (k4 )Xl (k4 , k2 )
M
ar
i
+ Xj (k1 , k3 )Xk† (k3 , k4 )χ0 (k4 )Xl† (k4 , k2 )χ0 (k2 )
!
X ω(kp1 )ω(kp2 ) ω(kp1 )/2 + ω(kp3 )/3
−
×
ω(k
1 )ω(k3 )ω(k4 )ω(k2 )
pǫS
4
χ0 (k1 )Xj† (k1 , k3)χ0 (k3 )Xk† (k3 , k4 )χ0 (k4 )Xl† (k4 , k2 )χ0 (k2 )
#
(3.37)
3.5. Overlap action
43
!
4
4
4
Y
d
k
d
k
1
d
k
1
4
5
3
V44 (k1 , k2 ) =
×
(2π)4 (2π)4 (2π)4 12 pǫS ω(kp1 ) + ω(kp2 )
5
"
!
X
ω(kp1 )ω(kp2 )ω(kp3 )ω(kp4 ) ω(kp1 )/6 + ω(kp5 )/30
×
Z Z Z
pǫS5
+ X1 (k1 , k3 )X1† (k3 , k4 )X1 (k4 , k5)χ†0 (k5 )X1 (k5 , k2 )
+ X1 (k1 , k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )X1 (k5 , k2 )
+ X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )X1† (k4 , k5 )X1 (k5 , k2 )
an
tin
i
+ χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )X1† (k4 , k5 )X1 (k5 , k2 )
!
1 X
ω(kp1 )ω(kp2 ) ω(kp1 ) + ω(kp3 )
×
−
6 pǫS
5
h
X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
ou
h
(X1 (k1 , k3 )X1† (k3 , k4 )X1 (k4 , k5 )X1† (k5 , k2)χ0 (k2 )
+ X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2 )
st
+ X1 (k1 , k3 )χ†0 (k3 )X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 )
+ X1 (k1 , k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
on
+ χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )X1 (k5 , k2 )
+ χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2)
C
+ χ0 (k1 )X1† (k1 , k3)χ0 (k3 )X1† (k3 , k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 )
+ χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )χ†0 (k5 )X1 (k5 , k2)
+ χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )χ†0 (k4 )X1 (k4 , k5 )X1† (k5 , k2 )χ0 (k2 )
a
X
ω 2 (kp1 )ω(kp2 )ω(kp3 )
×
ω(k1)ω(k2 )ω(k3)ω(k4)ω(k5 )
i
th
+
χ0 (k1 )X1† (k1 , k3)X1 (k3 , k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
+
pǫS5
!
ω(kp2 )[ω(kp1 )/2 + ω(kp3 )/6] + ω(kp4 )[ω(kp1 )/3 + ω(kp2 ) + ω(kp5 )/3]
×
M
ar
χ0 (k1 )X1† (k1 , k3 )χ0 (k3 )X1† (k3 , k4 )χ0 (k4 )X1† (k4 , k5 )χ0 (k5 )X1† (k5 , k2 )χ0 (k2 )
(Sn : permutation group of the numbers 1 through n)
#
(3.38)
ou
Chapter 4
Introduction
an
4.1
tin
Twisted mass action
In this chapter we present a new approach to lattice fermions, that is known under the
st
name of twisted mass QCD (tmQCD). As previously mentioned, the Wilson action breaks
chiral symmetry, which can be restored with the introduction of appropriate counterterms
(for instance, additive fermion mass renormalization). The result of the absence of chiral
on
symmetry for nonzero lattice spacing, is that the Wilson-Dirac operator is not protected
against zero modes, unless the bare quark mass is positive. However, due to additive mass
C
renormalization, the masses of the light quarks correspond to negative bare masses. One
of the consequences of the zero modes is the following: After integration over the fermion
and anti-fermion fields in the functional integral, there is a small eigenvalue of the Wilson-
a
Dirac operator in the fermionic determinant and the fermion propagators appearing in the
correlation functions. Thus, in the quenched approximation, where the fermionic determinant is ignored, this eigenvalue in the quark correlation is not cancelled out upon division.
th
The results are large fluctuations in particular measurables that compromise the ensemble
average. The gauge field configurations at which this happens, are called exceptional. For
M
ar
this to become more obvious, allow us to present a plot from Ref. [24] showing the ensemble
average of the pion propagator over all gauge configurations (dots) besides an exceptional
one (dashed line). One can see that the dashed line deviates dramatically from the average,
thus, the inclusion of the exceptional configuration in the ensemble average would lead to
large errors. On the other hand, its omission makes the Monte Carlo simulations invalid.
44
4.2. The lattice twisted mass action for degenerate quarks
45
2
10
1
10
0
ou
10
−1
−2
10
10
20
30
40
t
50
60
an
0
tin
10
st
Figure 4.1: A plot of the pion propagator against time separation for the quenched approximation on a 323 × 64 lattice with coupling β = 6.2 ([24]).
on
A solution to this problem is the addition of a mass term [25] to the standard Wilson
action, that is given the role of protecting the Dirac operator against these configurations.
The resulting action has the benefit that certain observables are automatically free of O(a)
C
lattice artifacts. Some additional advantages of this action are the efficient simulations
and that the operator mixing appears to be the same as in the continuum. The twisted
4.2
th
a
mass action can be used to study quarks at small masses, where the Wilson action would
fail. Also, the properties and the interactions of hadrons can be probed nonperturbatively
from first principles.
The lattice twisted mass action for degenerate
M
ar
quarks
The twisted mass lattice action for a doublet of Nf = 2 mass degenerate quarks, written
in the so called twisted basis {χ, χ̄}, is
{χ}
Stm = a4
X
x
h
i
χ̄(x) DW + m0 + iµq γ 5 τ 3 χ(x)
(4.1)
46
with m0 real and positive. DW is the Wilson-Dirac operator as given in Eq. (2.6). The last
term with the twisted mass parameter µq protects the Dirac operator against zero modes
for any finite µq , since the twisted Dirac operator has positive determinant
det(DW + m0 + iµq γ 5 τ 3 ) = det(Q2 + µ2q )
(4.2)
ou
where Q = γ 5 (DW + m0 ) is the hermitian Wilson operator; hence, the twisted Dirac operator does not have any zero eigenvalues. The isospin generator τ 3 acts in flavor space and its
tin
appearance means that isospin is no longer conserved (i.e. the up and down quark have opposite signs of the twisted mass leading to flavor symmetry breaking). Moreover, the twist
term breaks parity symmetry (due to γ 5 ). These symmetries are restored in the continuum
limit. The action remains invariant under the flavor-dependent axial transformations
5 τ3
2
χ
an
Ψ = eiωγ
3
iωγ 5 τ2
Ψ̄ = χ̄ e
(4.3)
st
with the mass parameters mixed to each other as
on
m′ = µq sin(ω) + m cos(ω)
µ′q = µq cos(ω) − m sin(ω)
(4.4)
C
In the full twist case, ω = π/2, the flavor symmetry is restored at a rate O(a2 ). This case
is useful, since there is automatic cancellation of O(a) effects in quantities like energies and
operator matrix elements. The action can be written in the physical basis {Ψ, Ψ̄}, where
a
the µq term has been eliminated
th
{Ψ}
Stm = a4
X
x
h
i
Ψ̄(x) DW tm + M Ψ
(4.5)
M
ar
DW tm is the twisted Wilson operator
DW tm
3
−
→
←
−
− −
→
1X
5 3 ←
{γµ ( ∇ µ + ∇ µ ) − a r e−iωγ τ ∇ µ ∇ µ }
=
2 µ=0
and M is the polar mass
M=
q
m20 + µ2q
(4.6)
(4.7)
47
In the continuum limit, where the last term of Eq. (4.6) vanishes, tmQCD can be seen
as a change of variables which leaves the physical content of the theory unchanged if the
rotation angle ω satisfies
tan(ω) =
µq
m0
(4.8)
Thus, in the continuum limit the axial rotation of the fermion fields (Eq. (4.3)) relates
ou
tmQCD to the standard QCD.
In terms of the fields {χ, χ̄} the axial (A) and vector (V ) currents along with the
pseudoscalar (P ) and scalar (S) densities are given by
3
3
Vµa = χ̄γµ τ2 χ
S 0 = χ̄χ
tin
Aaµ = χ̄γµ γ 5 τ2 χ
3
P a = χ̄γ 5 τ2 χ
(4.9)
an
Using the transformations of Eq. (4.3) we derive the expressions of the above composite
fields in the physical basis

cos(ω)Aa + ε3ab sin(ω)V b
τ
µ
µ
a
A′ µ ≡ Ψ̄γµ γ 5 Ψ=
3

2
Aµ

cos(ω)V a + ε3ab sin(ω)Ab
τa
µ
µ
a
V ′ µ ≡ Ψ̄γµ Ψ =
V 3
2
(a = 1, 2),
on
st
a
µ
(4.10)
(a = 3),
(a = 1, 2),
(4.11)
(a = 3),
a
C
The same can be applied to the densities P, S leading to the following equalities
τ
Ψ=
cos(ω)P 3 + i sin(ω) 1 S 0
2
2
a
P ′ ≡ Ψ̄γ 5

P a
a
(a = 1, 2)
0
S ′ ≡ Ψ̄Ψ = cos(ω)S 0 + 2i sin(ω)P 3
th
(4.12)
(a = 3)
(4.13)
M
ar
The rotated physical fermion fields {Ψ, Ψ̄} satisfy the standard Ward Identities
a
a
∂µ A′ µ = 2MP ′ ,
a
∂µ V ′ µ = 0
(4.14)
while the twisted fields {χ, χ̄} satisfy the partially conserved axial current (PCAC) and
4.3. Calculations with twisted mass QCD
48
partially conserved vector current (PCVC) relations
∂µ Aaµ = 2mq P a + iµq δ 3a S 0
(4.15)
∂µ Vµa = −2µq ε3ab P b
(4.16)
We are particularily interested in the action written in the twisted basis, because it
ou
is the one used in simulations. This is due to the fact that the renormalization of gauge
invariant correlation functions is simpler for the twisted fields {χ, χ̄}. The expression for
the twisted mass propagator is
◦
p◦ 2µ + M(p)2 + µ2q
◦
◦
pµ =
1
sin(apµ ),
a
an
with p and M defined through
tin
G(p) =
−iγµ p µ + M(p) − iµq γ 5 τ 3
r
M(p) = m0 + ap̂2µ ,
2
p̂µ =
2
apµ
sin(
)
a
2
(4.17)
(4.18)
st
The tree-level expression can be extracted by taking the Taylor expansion for small values
on
of the lattice spacing a and keeping terms up to O(a), obtaining
G0 (p) = p2 + m20 + µ2q + am0 rp2
(4.19)
C
The first observation is that for zero bare mass (or even for m0 = am̃0 ), the theory is
free of O(a) effects, but this picture changes once we take into account the interactions
a
between quarks. Moreover, the inclusion of the twisted mass parameter does not affect the
O(a) improvement of the m0 = 0, am̃0 cases. The absence of µq in the O(a) term can be
explained from the origin of the O(a) correction terms. The am0 rp2 term in Eq. (4.19)
M
ar
th
comes from M(p)2 and no µq is involved because it points in a different direction in flavor
space than the other terms.
4.3
Calculations with twisted mass QCD
In the previous section we provided the axial transformations Eq. (4.3) that relate the
continuum tmQCD to the standard QCD. However, these transformations are not an exact
symmetry of the theory, thus the equivalence holds up to cutoff effects for renormalized
correlation functions. The counterterms that enter the renormalization of the bare quark
4.3. Calculations with twisted mass QCD
49
masses are
χ̄χ,
m0 χ̄χ,
iµq χ̄γ 5 τ 3 χ
(4.20)
The lattice symmetries forbid the additive renormalization of the twisted mass µq , so
it is renormalized only multiplicatively, while the bare mass m0 undergoes additive and
mR = Zm (g02 , aµ)(m0 − mcr )
µq,R = Zµ (g02 , aµ)µq
ou
multiplicative renormalization
(4.21)
tin
µ denotes the renormalization scale dependence of the renormalization constants Z. The
critical mass mcr is the value that m0 must be given, so that the untwisted renormalized
an
mass mR vanishes (recovery of the chiral limit).
The renormalization factors Z can be related to those of the composite fields, using the
PCAC and PCVC relations (Eqs. (4.15), (4.16)) as normalization conditions. This connec-
st
tion is established by choosing renormalization schemes such that the PCAC and PCVC
relations hold, with renormalized currents, densities and renormalized masses. In partic-
on
ular, the vector Ward Identity implies that the renormalization factor of the pseudoscalar
density ZP and Zµ obey
ZP Zµ = 1
(4.22)
C
Moreover, the renormalization constants of the untwisted mass and the scalar density may
be shown to satisfy
ZS 0 Zm = 1
(4.23)
th
a
Using the above equations, the twist angle ω can be defined from the renormalized mass
parameters
µq,R
ZS 0
µq
(4.24)
=
tan(ω) =
µR
ZP m0 − mcr
M
ar
Let us now include gauge fields in our theory, so that the total action is denoted by
Stotal = SG + SW tm , where SG is the purely gluon part and SW tm is the fermion Wilson
action (Eq. (4.1)) with the twisted mass parameter µq . The partition function is given by
Z=
Z
D[χ̄, χ]D[U] e−S
(4.25)
4.4. Twisted mass QCD for nondegenerate quarks
50
and the n-point correlation function is
1
< O >=
Z
Z
D[χ, χ̄]D[U] e−S O
(4.26)
where O is a product of local gauge invariant composite fields. By renormalizing the theory
as indicated above, and performing the continuum limit, the correlation functions in the
< O′ [ψ, ψ̄] >QCD =< O′ [χ, χ̄] >tmQCD
ou
physical and in the twisted basis are equivalent
(4.27)
an
tin
It is, of course, implied that < O′ >QCD has been computed in standard QCD with
renormalized quark mass MR
q
MR = m2R + µ2q,R
(4.28)
while < O >tmQCD regards the computation in tmQCD with renormalized masses mR , µq,R .
Thus, a correlation function in QCD can be expressed as a linear combination of correlation
st
functions in tmQCD at a particular twist angle ω.
A computation of an expectation value in the twisted basis starts by performing the
inverse axial transformations (Eq. (4.3)) to the physical fields {Ψ, Ψ̄} that take us to the
on
{χ, χ̄} basis. One then computes the resulting correlation function using the twisted mass
action (Eq. (4.1)) with a choice of quark masses. Finally, the continuum limit is taken and
C
the result is the desired continuum QCD correlation function with quark mass equal to
MR of Eq. (4.28).
Twisted mass QCD for nondegenerate quarks
a
4.4
th
So far we have discussed the Nf = 2 case of degenerate light quarks, but the action
can be generalized to include a further doublet of non-degenerate quarks [26, 27]. Such
M
ar
a generalization arises from the need to describe the heavier quarks, charm and strange.
Since we want to use this action in simulations of full tmQCD, we must maintain the reality
and positivity of the quark determinant. Thus, in the action we add a flavor off-diagonal
splitting
{χ}
Stm = a4
X
x
h
i
χ̄(x) DW + m0 + iµq γ 5 τ 3 + ǫq τ 1 χ(x)
(4.29)
4.5. Discussion
51
where ǫq is the mass splitting parameter and we demand µq , ǫq > 0. The additional term
retains theq
properties of tmQCD at full twist and it keeps the quark determinant real and
positive if m20 + µ2q > ǫq .
The transition to the physical basis is achieved with the following field transformations
τ 1 1
) √ (1 + iτ 2 ) χ
2
2
1
τ1 Ψ̄ = χ̄ √ (1 − iτ 2 ) exp(−iωγ 5 )
2
2
exp(−iωγ 5
{Ψ}
Stm = a4
X
x
tin
The action in this basis is now
(4.30)
ou
Ψ =
h
i
Ψ̄(x) DW tm + M Ψ
(4.31)
(4.32)
an
p
where M = m2q + µ2q is again the polar mass. The partial conservation equations can be
obtained in the same manner as in the degenerate case
(4.33)
∂µ Vµa = −2µq ε3ab P b + iǫq ε1ab S b
(4.34)
on
st
∂µ Aaµ = 2m0 P a + iµq δ 3a S 0 + ǫq δ a1 P 0
where
τa
χ
(4.35)
2
For the description of the heavy doublet charm and strange (c,s) we associate the physical
S a = χ̄
C
P 0 = χ̄γ 5 χ,
quark mass with the mass parameter M, that is
mstrange = M − ǫq
(4.36)
th
a
mcharm = M + ǫq
M
ar
and the fermion determinant is positive if M > ǫq .
4.5
Discussion
Twisted mass QCD is frequently used in simulations due to the nice properties of the
theory. The twisted mass term introduces an infrared bound on the spectrum of the WilsonDirac operator and as a result the quenched and partially quenched approximations are
well-defined. In the continuum case, an axial rotation of the fermion fields can eliminate
4.5. Discussion
52
the twisted mass term, while on the lattice the twisted mass action and the standard
Wilson action cannot be related by a change of variables; consequently they have different
discretization errors. Moreover, tmQCD at maximal twist is automatically O(a) improved.
The European Twisted Mass Colaboration (ETMC) is currently performing large scale
ou
simulations for two flavors of light quarks with degenarate mass, using maximal twist
LQCD. A review can be found in Ref. [28]. Among their work, some recent studies are the
following:
tin
1. Charged pseudoscalar mass and decay constant [29, 30]: The charged pseudoscalar
mass meson mass, denoted by amPS , is extracted from the time exponential decay of the
pseudoscalar correlation function, covering a range of values 300MeV <
∼ mPS <
∼ 550MeV.
Moreover, the charged pseudoscalar decay constant fPS is determined from
2µq
|h0|P a|πi| ,
m2PS
a = 1, 2
(4.37)
an
fPS =
due to the exact lattice PCVC relation with no need to calculate any renormalization
st
constant. In order to compare results at different values of the lattice spacing, one measures
the hadronic scale r0 /a (r0 : Sommer parameter), which is defined via the force between
on
static quarks at intermediate distance. The results for r0 fPS are plotted in Fig. 4.2 as a
function of r0 mPS , for β = 3.9, 4.05.
C
r0 fPS
a
0.42
th
0.38
β = 4.05
β = 3.9
M
ar
0.34
0.30
(r0 mPS )2
0.26
0.0
0.5
1.0
1.5
2.0
Figure 4.2: r0 fPS as a function of (r0 mPS )2 for β = 3.9 and β = 4.05
4.5. Discussion
53
2. Nucleon Mass and other baryon masses:
Ref. [31] presents simulation results on the nucleon mass and the ∆ baryon masses. Their
evaluation was performed at four quark masses (corresponding to a pion mass 300MeV
<
∼ mπ <
∼ 690MeV) and different lattice sizes. The masses of the nucleon and the ∆’s are
C
on
st
an
tin
ou
extracted from 2-point correlators using standard interpolating fields. Fig. 4.3 shows the
nucleon mass as a function of the squared pion mass. The simulation’s parameters are
given in Ref. [31].
th
a
Figure 4.3: Nucleon mass as a function of m2π for β = 3.9 on a lattice of size 243 × 48
(filled triangles) and on a lattice of size 323 × 64 (open triangles). Results at β = 4.05 are
denoted by stars. More details can be found in Ref. [31].
There are many other applications of tmQCD, among them the pion mass mπ and
the pion decay constant Fπ , both obtained from the long distance behavior of the 2-point
M
ar
function [32, 33]
(AR )10 (x)(PR )1 (y)a (M
R ,0)
= cos(ω) (AR )10 (x)(PR )1 (y) (M ,ω)
D
E R
2
1
+ sin(ω) Ṽ0 (x)(PR ) (y)
(MR ,ω)
(4.38)
Twisted mass QCD has been also employed for the determination of the chiral condensate from the local scalar density. An analogous computation with Wilson fermions has
4.5. Discussion
54
never been performed, due to the cubic divergence that appears in the chiral limit. This
evaluation is based on the relation
(PR )3 (x) (M
R ,ω)
= cos(ω) (PR )3 (x) (M
R ,0)
−
i
sin(ω) (SR )0 (x) (M ,0)
R
2
(4.39)
It is worth mentioning that using tmQCD there is no general recipe for by-passing the
M
ar
th
a
C
on
st
an
tin
ou
lattice renormalization problems of the Wilson fermions. One must study each computation
individually and decide whether it is advantageous to use some variants of tmQCD.
ou
Chapter 5
Introduction
an
5.1
tin
The background field formalism
The background field method was first introduced by B. DeWit [34], and some years later,
st
J. Honerkamp [35] and G. ’t Hooft [36] independently discussed several issues of the same
technique. The aim of this method is the simplification of quantum computations related to
gauge and gravitational theories; the study of renormalization constants is a good example
on
demonstrating the effectiveness of the method. In the last decade, the background field
technique found application in the Standard Model as well [37, 38].
C
A characteristic of the classical limit of gauge field theories is the gauge symmetry,
which is broken by quantum corrections. A particular example is the effective action,
which for a given classical action sums up all quantum corrections, and can only be de-
a
termined in perturbation theory. A way to overcome this problem is the introduction of a
background field; the resulting effective action is then gauge invariant with respect to gauge
transformations of the background field. Therefore, numerical studies become technically
th
easier. Considering the lattice regularization, a gauge theory is renormalizable to all orders
in perturbation theory [39, 40]. In addition, the implementation of the background field
M
ar
does not require any further counterterms besides those already needed in its absence, as
it happens in the continuum case.
Although the background fields lead to a gauge invariant theory, it is common to give
up this symmetry at intermediate levels of a computation (by performing calculations in
momentum space). This originates from the perturbative treatment of the interaction with
the background field.
One of the first applications of the particular method in lattice gauge theories is the
55
5.2. Background fields in the continuum theory
56
computation of the matching between different couplings; it has been proven to be the
most efficient and economical way. Indeed, the method was utilized for the evaluation of
the coefficients d1 (µ̄a) [42, 43, 44] and d2 (µ̄a) [45] of the equation relating the running
coupling in the MS renormalization scheme (α MS (µ̄)) to the bare lattice coupling in the
pure SU(N) gauge theory (α0 ), given by
(5.1)
ou
α MS (µ̄) = α0 + d1 (µ̄a)α02 + d2 (µ̄a)α03 + ...
tin
(α0 = g02/4π, α MS = g 2MS /4π, a : lattice spacing, µ̄ : scale parameter).
Next, we discuss the continuum version of the background field technique, the sequential additions to the total action and the possible choices for the gauge transformations.
Furthermore, we provide expressions for the functional integral and the effective action.
We then analyze the technique on the lattice and show how the action is altered. We also
Background fields in the continuum theory
on
5.2
st
an
describe two different ways of expressing the gauge transformations and explain why only
one is preferable. Finally, we quote examples of the X’s (Eqs. (3.21)) appearing in the
overlap action in the background field formalism, as well as the 3-gluon vertex.
Having introduced the general principles of the background field technique, let us now
a
C
describe its mathematical formalism in the continuum theory. An SU(N) gauge potential
Gµ = Gaµ T a ({T a } being the generators of the algebra) is described by the Yang-Mills
action
Z
1
S[G] = − 2 dD xTr[Fµν (x) Fµν (x)]
(5.2)
2g0
M
ar
th
where g0 is the bare gauge coupling constant and Fµν (x) the field strength tensor (Fµν =
∂µ Gν − ∂ν Gµ + [Gµ , Gν ]). The main idea of the background field method, is that the gauge
field is decomposed into two parts, the quantum (Q) and the background (A) field
Gµ = Aµ + g0 Qµ
(5.3)
Aµ is a smooth external source field, which is not required to satisfy the Yang-Mills equa-
tions, while the quantum field is the integration variable of the functional integral. An
infinitesimal gauge transformation of Gµ with parameter Λ can be distributed in many
ways over Aµ and Qµ , but the most convenient choices are
5.2. Background fields in the continuum theory
57
a. the ‘background transformation’
δAµ = Dµ Λ,
δQµ = [Qµ , λ]
(5.4)
b. and the ‘quantum transformation’
δQµ = Dµ Λ + [Qµ , λ]
(5.5)
ou
δAµ = 0,
One has to reexpress the gauge action S[G] in terms of Aµ , Qµ (Gµ is replaced by
Z
dD x Tr[Dµ Qµ (x)Dν Qν (x)]
(5.6)
an
1
Sgf [A, Q] = −
ξ0
tin
Eq. (5.3)) and add the gauge fixing term which breaks the quantum gauge invariance. The
latter is chosen in such a way that it preserves the background gauge invariance
SF P [A, Q, c̄, c] = −2
Z
st
In the above equation, ξ0 is the bare gauge parameter and Dµ = ∂µ + i Aµ is the covariant
derivative. The ghost action is now
dD x Tr[Dµ c̄(x) (Dµ + ig0 Qµ (x)) c(x)]
(5.7)
on
External quantities are coupled only to Qµ , so that Aµ is always invariant under gauge
transformations. Of course, for vanishing background field the total action S[A + g0 Q] +
C
Sgf [A, Q]+SF P [A, Q, c̄, c], coincides with its standard form (with the usual covariant derivative). The partition function Z[J, η̄, η] (Jµ (x), η̄, η: classical source fields) changes in the
presence of Aµ and is given by the formula
Z
D[Q]D[c̄]D[c]e−Stotal[A,Q,c̄,c]+(J,Q)+(η̄,c)+(c̄,η)
th
a
1
Z[A, J, η̄, η] =
N
(5.8)
with the normalization factor N ensuring that Z[0, 0, 0, 0] = 1. Note that Aµ is not coupled
M
ar
to the source J. The scalar product(A, B) between two fields A, B of the same type is
(A, B) =
Z
dD xAaµ (x) Bµa (x)
(5.9)
The partition function is invariant under the gauge symmetries of Aµ . Further, Z can be
expanded in powers of A, J, η̄, η, with the coefficients being expectation values of products
constructed by local operators, at vanishing sources. In what follows, Z[A, J, η̄, η] will be
considered as a well defined formal power series of A, J, η̄, and η. The expectation value
5.3. The lattice background field method
58
of an operator in the presence of a background field is defined through
1
< O >B =
NB
Z
D[Q]D[c̄]D[c] O[A, Q, c̄, c] e−Stotal[A,Q,c̄,c]
(5.10)
(NB is chosen so that < 1 >B = 1). In order to write an expression for the effective action
W [A, J, η̄, η] = ln(Z[A, J, η̄, η])
ou
Γ[A, J, η̄, η], one must find the expansion of the generating functional for the connected
diagrams W
(5.11)
tin
Following the procedure described in Ref. [40] we derive the effective action
Γ[A, Q⋆ , C̄, C] = W [A, J, η̄, η] − (J, Q⋆ ) − (η̄, C) − (C̄, η)
Q⋆ =
δW
,
δJ
c⋆ =
δW
,
δ η̄
δW
δη
an
where
c̄⋆ = −
(5.12)
(5.13)
st
Γ corresponds to the background field effective action considered as a functional of A and
evaluated at Q = 0. It can be obtained from the calculation of the 1-particle-irreducible,
Green’s functions of the background field. Z and W are invariant under background
on
gauge transformations whereas all sources and ghost fields transform in the same way as
Q does (see Eq. (5.4)). The vertex functions Γ(j,k,l) (j, k, l correspond to the number
of background, quantum and ghost fields respectively, appearing in the vertex) can be
The lattice background field method
a
5.3
C
obtained by the series expansion of Eq. (5.12) in powers of its arguments.
th
On the lattice, the background field method can be approached in more than one ways.
Many choices can be made for the action, and the difference between them is irrelevant
M
ar
in the continuum limit (for lattice theories with all symmetries necessary for renormalizability). Consider a 4 − D lattice, whose sides are labelled by a four-vector x and each
dimension is characterized by its unit vector µ̂ (|µ̂|: one lattice spacing in direction µ). The
variable Uµ (x) relates the link connecting the points x and x + aµ̂. The gauge fields are
introduced via the link variables appearing in the action and a convenient decomposition
of the links in the presence of a background field is
Uµ (x) = eiag0 Qµ (x) · eiaAµ (x)
(5.14)
59
where Qµ (x) and Aµ (x) are noncommuting N × N hermitian matrices satisfying
Qµ (x) = Qaµ (x)T a ,
Aµ (x) = Aaµ (x)T a ,
Tr[T a T b ] =
δ ab
2
(5.15)
ou
Since the dependence of the link variable on Q and A appears in different exponentials,
the gauge transformations can be viewed in two alternative ways:
a. The quantum field Qµ can be seen as a matter field with covariant transformation,
while the background field transforms as a pure gauge field. This is equivalent to the
continuum background transformations of the previous section
tin
an
⇒
QΛµ (x) = Λ(x)Qµ (x)Λ−1 (x)
1
AΛµ (x) =
ln(Λ(x)eiaAµ Λ−1 (x + aµ̂))
ih a
ih
i
Λ
iag0 Qµ (x) −1
iaAµ (x) −1
Uµ (x) = Λ(x)e
Λ (x) Λ(x)e
Λ (x + aµ̂)
(5.16)
st
When the forward and backward lattice derivatives act on a matrix valued function f
defined on lattice sites, one gets the relations
on
−
→A
D µ f (n) = eiaAµ (n) f (n + µ̂)e−iaAµ (n) − f (n)
←
−
−iaAµ (n−µ̂)
DA
f (n − µ̂)eiaAµ (n−µ) − f (n)
µ f (n) = e
(5.17)
(5.18)
C
These are covariant upon substituting Eqs. (5.16), that is
⇋
DA
µ
Qν (n)
Λ
= Λ(n)
⇋
DA
µ
Qν (n) Λ−1 (n)
(5.19)
th
a
b. The second interpretation of the gauge transformation is analogous to the continuum
quantum transformation. This is derived by keeping the background field invariant with
the transformation attributed to the gauge field
M
ar
Λ Λ h
i
UµΛ (n) ≡ eiag0 Qµ (n)
eiaAµ (n) = Λ(n)Uµ (n)Λ−1 (n + µ̂)e−iaAµ (n) eiaAµ (n)
(5.20)
The introduction of the background field is accompanied by a gauge fixing term in the
action to absorb the gauge transformation of Eq. (5.20) and thus ensures a finite functional
60
integral. A proper choice is
Sgf =
a4 X X
Tr[Dµ− Qµ Dν− Qν ]
ξ0 µ,ν x
(5.21)
The lattice derivatives appearing in Sgf are covariant, so that the gauge fixing term is
SF P = 2a
XX
µ
x
† −1
Tr[Dµ ĉ(x) (M ) Qµ (x) · Dµ + ig0 Qµ (x) c(x)]
with M the matrix (more details can be found in Ref [41])
eiag0 Qµ − 1
iag0 Qµ
an
M(Qµ ) =
tin
4
ou
invariant under the background field transformations of Eqs. (5.16). The ghost action
must also be modified and take the form
(5.22)
(5.23)
Note that the measure term is not affected by the presence of a background field and
Ng02 X X
Tr[Qµ (x)Qµ (x)] + O(g04 )
12 µ x
(5.24)
on
Sm =
st
can be written as
The background field and coupling constant renormalization is determined by the 2point function of the background field; no renormalization for the quantum and ghost
C
field is needed. The reason for this is that these fields appear only within the loops of a
diagram (external lines correspond to background fields) and their renormalization factor
M
ar
th
a
would be cancelled with those of the propagators. The gauge fixing parameter also needs
to be redefined, since the longitutinal part of the propagator must be renormalized. The
renormalized quantities can be written with respect to the bare ones
B0µ = ZB BRµ
1/2
(5.25)
g02 = Zg gR2
(5.26)
ξ 0 = Zξ ξ R
(5.27)
with BR , gR , ξR being the renormalized quantities. The fact that exact gauge invariance
is preserved in the background field formalism, leads to the following relation between the
renormalization constants ZB , Zg
1/2
Zg ZB = 1
(5.28)
5.4. Vertices of the overlap action in the background field method
5.4
61
Vertices of the overlap action in the background
field method
The main consequence of the introduction of the background field is the appearance of
different vertices in perturbative calculations. For each vertex with gauge fields, we must
ou
take into account all variants of quantum and background fields. Depending on the calculation performed, some of the variations of the vertices can be excluded. For instance,
each vertex in a diagram cannot have more background fields than the external lines of the
tin
diagram. Thus, if the diagrams involved in a computation have only two external gluons
and a certain number of loop gluons, the 3-gluon vertex cannot have all gluons as background fields. Let us demonstrate the variations of a fermion-antifermion-2gluons vertex
st
an
which has 3 contributions, as shown in Fig. 5.1.
on
Figure 5.1: The multiple character of a vertex in the background field method. Dashed
lines represent gluon fields; those ending on a cross stand for background gluons. Solid
lines represent fermions.
The use of the background field technique implies that instead of the generic gluon fields
C
(in the case of the overlap action these appear in Xi ’s of Eqs. (3.21)), one must consider
all different combinations of background (A) and quantum (Q) fields which originate in
th
a
the links of Eqs. (2.6), (5.14). As already illustrated, in the definition of the link variable,
the quantum field appears on the left of the background field and the algebra generators
of A, Q do not commute. Thus, while taking all possible permutations of {A, Q} we
M
ar
eliminate the variants where the background fields are placed on the left of the quantum
fields. Hence, for the β-function calculation Eqs. (3.21) are written as
X1 (p′ , p) = X1Q (p′ , p) + X1A (p′ , p)
X2 (p′ , p) = X2QQ (p′ , p) + X2QA (p′ , p) + X2AA (p′ , p)
X3 (p′ , p) = X3QQQ(p′ , p) + X3QQA (p′ , p) + X3QAA (p′ , p) + X3AAA (p′ , p)
X4 (p′ , p) = X4QQQQ(p′ , p) + X4QQQA(p′ , p) + X4QQAA (p′ , p)+X4QAAA (p′ , p)+X4AAAA (p′ , p) (5.29)
Eqs. (5.29) must be reinserted into Eqs. (3.35) - (3.38) to obtain the components of the
62
desired fermion-antifermion-gluon vertices (Eq. (3.34)). For the calculation of the third
coefficient of the β-function, we must consider all vertices with a fermion-antifermion pair
and up to 4 gluons. The involving 2-loop diagrams have 2 external gluons (A) and 2 internal
(loop) gluons (Q). This allows us to to also exclude from Eqs. (5.29) the parts including
ou
more than two Q’s, or two A’s, that is X3QQQ , X3AAA , X4QQQQ, X4QQQA and X4AAAA . The
expressions for Xi ’s including only one kind of gluons, can be directly read from Eqs. (3.21)
(for background gluons we set g0 = 1). The next equations correspond to the remaining
M
ar
th
a
C
on
st
an
tin
nontrivial vertices.
Z 4 4
d k1 d k2 ′
g0
=
δ(p − p − k1 − k2 ) ×
2
(2π)4
"
p′ +p X
V2,µ
Qµ (k1 )Aµ (k2 ) + Aµ (k2 )Qµ (k1 )
2
µ
#
p′ +p Qµ (k1 )Aµ (k2 ) − Aµ (k2 )Qµ (k1 )
+iaV1,µ
2
63
X2QA (p′ , p)
Z 4 4 4
g02
d k1 d k2 d k3 ′
=
δ(p − p − k1 − k2 − k3 ) ×
4
(2π)8
"
p′ +p X
−a2 V1,µ
Qµ (k1 )Qµ (k2 )Aµ (k3 ) + Aµ (k3 )Qµ (k2 )Qµ (k1 )
2
µ
#
p′ +p Qµ (k1 )Qµ (k2 )Aµ (k3 ) − Aµ (k3 )Qµ (k2 )Qµ (k1 )
+iaV2,µ
2
an
tin
X3QQA (p′ , p)
ou
(5.30)
st
Z 4 4 4
g0
d k1 d k2 d k3 ′
δ(p − p − k1 − k2 − k3 ) ×
=
4
(2π)8
"
p′ +p X
−a2 V1,µ
Qµ (k1 )Aµ (k2 )Aµ (k3 ) + Aµ (k3 )Aµ (k2 )Qµ (k1 )
2
µ
#
p′ +p +iaV2,µ
Qµ (k1 )Aµ (k2 )Aµ (k3 ) − Aµ (k3 )Aµ (k2 )Qµ (k1 )
2
(5.31)
(5.32)
C
on
X3QAA (p′ , p)
M
ar
th
a
Z 4 4 4
g02
d k1 d k2 d k3 2d4 k4 ′
δ(p − p − k1 − k2 − k3 − k4 ) ×
=
8
(2π)12
"
p′ +p X
−a2 V2,µ
Qµ (k1 )Qµ (k2 )Aµ (k3 )Aµ (k4 ) + Aµ (k4 )Aµ (k3 )Qµ (k2 )Qµ (k1 )
2
µ
#
p′ +p −ia3 V1,µ
Qµ (k1 )Qµ (k2 )Aµ (k3 )Aµ (k4 ) − Aµ (k4 )Aµ (k3 )Qµ (k2 )Qµ (k1 )
(5.33)
2
X4QQAA(p′ , p)
Upon substituting the expression for Xi ’s in the overlap vertices, the latter become
extremely lengthy and complicated. For instance, the vertex with Q-Q-A-Ψ-Ψ consists of
9,784 terms, while the vertex with Q-Q-A-A-Ψ-Ψ has 724,120 terms.
As an example, we present below the vertex with Q-Q-A-Ψ-Ψ, written in the language
of Xi ’s.
V13,QQA
64
"
#
1
1
QQA
QQA†
X3
χ0 (k1 ) X3
+
+
=
(k1 , k2 ) −
(k1 , k2 ) χ0 (k2 )
ω(k1 ) + ω(k2 )
ω(k1 )ω(k2 )
Z 4
d k3
1
1
1
×
+
4
(2π) ω(k1 ) + ω(k3 ) ω(k1 ) + ω(k2 ) ω(k2 ) + ω(k3 )
"
h
−X1Q (k1 , k3 ) χ†0 (k3 ) X2QA (k3 , k2 ) − X2QQ (k1 , k3 ) χ†0 (k3 ) X1A (k3 , k2 )
V23,QQA
V33,QQA
†
†
†
†
− X1Q (k1 , k3 ) X2QA (k3 , k2 ) χ0 (k2 ) − X2QQ (k1 , k3 ) X1A (k3 , k2 ) χ0 (k2 )
− X1A (k1 , k3 ) X2QQ (k3 , k2 ) χ0 (k2 ) − X2QA (k1 , k3 ) X1Q (k3 , k2 ) χ0 (k2 )
†
†
†
†
− χ0 (k1 ) X1A (k1 , k3 ) X2QQ (k3 , k2 ) − χ0 (k1 ) X2QA (k1 , k3 ) X1Q (k3 , k2 )
†
†
†
†
χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X2QA (k3 , k2 )χ0 (k2 ) + χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X2QQ (k3 , k2 )χ0 (k2 )
+
†
†
χ0 (k1 )X2QQ (k1 , k3 )χ0 (k3 )X1A (k3 , k2 )χ0 (k2 )
Z Z
d4 k3 d4 k4 1
(2π)4 (2π)4 4
Y
1
ω(kp1 ) + ω(kp2 )
!"
+
†
†
χ0 (k1 )X2QA (k1 , k3 )χ0 (k3 )X1Q (k3 , k2 )χ0 (k2 )
!
1 X
−
ω(kp1 )ω(kp2 )ω(kp3 ) ×
6
#
i
st
h
ω(k1 ) + ω(k2 ) + ω(k3 )
×
ω(k1 )ω(k2 )ω(k3 )
an
+
i
tin
− χ0 (k1 ) X1Q (k1 , k3 ) X2QA (k3 , k2 ) − χ0 (k1 ) X2QQ (k1 , k3 ) X1A (k3 , k2 )
ou
− X1A (k1 , k3 ) χ†0 (k3 ) X2QQ (k3 , k2 ) − X2QA (k1 , k3 ) χ†0 (k3 ) X1Q (k3 , k2 )
on
+
pǫS4
pǫS4
h
i
†
†
†
X1Q (k1 , k3 )X1Q (k3 , k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )X1A (k3 , k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )X1Q (k3 , k4 )X1Q (k4 , k2 )
+ ω(k1 ) + ω(k2 ) + ω(k3 ) + ω(k4 ) ×
h
†
†
†
χ0 (k1 )X1Q (k1 , k3 )X1Q (k3 , k4 )X1A (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1Q (k1 , k3 )X1Q (k3 , k4 )χ†0 (k4 )X1A (k4 , k2 )
†
†
†
C
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )X1A (k4 , k2 )χ0 (k2 )
†
†
+ X1Q (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )χ†0 (k4 )X1A (k4 , k2 ) + X1Q (k1 , k3 )X1Q (k3 , k4 )χ0 (k4 )X1A (k4 , k2 )χ0 (k2 )
†
†
†
a
+ χ0 (k1 )X1Q (k1 , k3 )X1A (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1Q (k1 , k3 )X1A (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 )
†
†
†
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1A (k3 , k4 )X1Q (k4 , k2 ) + X1Q (k1 , k3 )χ†0 (k3 )X1A (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 )
†
†
th
+ X1Q (k1 , k3 )χ†0 (k3 )X1A (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) + X1Q (k1 , k3 )X1A (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 )
†
†
†
+ χ0 (k1 )X1A (k1 , k3 )X1Q (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 ) + χ0 (k1 )X1A (k1 , k3 )X1Q (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 )
†
†
†
M
ar
†
†
+ χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )X1Q (k4 , k2 )χ0 (k2 )
i
†
†
+ X1A (k1 , k3 )χ†0 (k3 )X1Q (k3 , k4 )χ†0 (k4 )X1Q (k4 , k2 ) + X1A (k1 , k3 )X1Q (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 )
!
X ω(kp1 )ω(kp2 ) ω(kp1 )/2 + ω(kp3 )/3
×
−
ω(k1 )ω(k3 )ω(k4 )ω(k2 )
pǫS4
h
†
†
†
χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )χ0 (k4 )X1A (k4 , k2 )χ0 (k2 )
†
+ χ0 (k1 )X1Q (k1 , k3 )χ0 (k3 )X1A (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 )
+
†
†
†
χ0 (k1 )X1A (k1 , k3 )χ0 (k3 )X1Q (k3 , k4 )χ0 (k4 )X1Q (k4 , k2 )χ0 (k2 )
i
#
(5.34)
ou
Chapter 6
6.1
an
tin
The running coupling and the
β-function
Introduction
st
In the Standard Model’s Lagrangian, the interactions of the fundamental constituents
on
of matter are described by different coupling constants, characterizing the strength of
the electromagnetic, weak or strong forces. QCD is the gauge field theory of the strong
interaction between quarks and gluons which is fully defined by a few parameters, namely
C
the strong coupling constant αs and the quark masses; here we will focus on αs . A common
attribute of all couplings is that they are not really constants, but they depend on the
a
energy transfer µ in the interaction process. This explains why we often refer to them as
running couplings (they ‘run’ with the energy scale).
QCD has specific features, asymptotic freedom and confinement, which determine the
th
behavior of quarks and gluons in particle reactions at high and at low energy scales. It
predicts that αs decrease with increasing energy or momentum transfer, and vanishes at
M
ar
asymptotically high energies. For procedures with energy transfer within the interval 10100GeV, the predictions of the theory can be worked out as an expansion in powers of
the corresponding coupling constant; the so-called perturbative expansion. For the strong
interaction, at energy scale µ ∼ 100GeV, the coupling constant has been found to be
αs = 0.12. At this scale, the perturbation theory works very well. However, if the energy
scale decreases from 100GeV, the αs increases and at µ ∼ 1GeV it is so large that the
perturbative expansion is not reliable. Nevertheless, low energy experiments are important
since we can observe the world of hadrons, which perturbation theory has completely failed
65
6.1. Introduction
66
to describe. A nonperturbative solution of the theory is required to deal with the large
αs regime. Over the years, numerical techniques have been developed, the numerical
simulations of QCD formulated on a discrete lattice of space-time points. This method
allows the nonperturbative study of the theory for low-energy scales.
ou
At the low-energy regime, the lattice formulation in cooperation with numerical simulations are used to systematically extract predictions of QCD. An enormous number of
publications appear for the nonperturbative estimation of the strong coupling constant.
Many of these studies have been performed by the ALPHA Collaboration [46]. One of the
main goal of the group is the calculation of the strong coupling constant, using numerical
tin
simulations combined with perturbative renormalization. The emphasis of its long-term
projects is given on precision and control of the systematic errors. The aim is not only
to compute the running coupling over a wide range of low energies, but to also reach a
an
range of high energies in order to make comparisons with perturbation theory. Some of the
nonperturbative studies on the strong coupling constant in the quenched approximation
st
of QCD, but also in the unquenched, can be found in Refs. [47, 48, 49, 51, 52, 53]. The
coefficients of the perturbative expansion of αs and of β-function are known for various
fermion and gluon actions [54, 55, 56, 57, 58, 59, 60, 45].
on
A number of experimental tests of the strong coupling behavior are summarized in
Ref. [61]. A world summary of measurements of αs , leads to an unambiguous verification
of the running of αs and of asymptotic freedom, in excellent agreement with the predictions
C
of QCD. It has become standard to evaluate αs choosing the energy scale to be the mass
of the Z-boson, µ = MZ 0 . Averaging the set of measurements shown in Fig. 6.1, balanced
a
between different particle processes and the available energy range, one finds αs (MZ 0 ) =
0.1189 ± 0.0010. The error estimates of Fig. 6.1 include the theoretical uncertainties. It is
interesting to see that the lattice contribution to αs (MZ 0 ) has relatively small error bars
M
ar
th
and it is very close to the average value of all processes. Onwards we omit the index s of
αs for simplicity.
67
st
an
tin
ou
6.2. Renormalization group equation and β-function
Renormalization group equation and β-function
C
6.2
on
Figure 6.1: The value of αs (MZ 0 ) as derived from various processes and the average of these
measurements [62]. This graph has been taken from Review of Particle Physics (2006).
The introduction of the lattice as a regulator, causes the appearance of a new parameter, the
lattice spacing a, expressing the distance between neighboring lattice points. Naturally, for
a
lattice theories, the measurable quantities depend on a, but must recover their continuum
value when setting a → 0. One of the most important quantities to study is the coupling
M
ar
th
constant and its connection with a and other observables. Unavoidably, one has to define
the bare and renormalized value for each quantity. When referring to the bare value,
we have in mind the one involved in the Lagrangian, while the renormalized is the one
compared to experimental measurements. In general, bare values depend on a. Let us
study a system living on a 4-dimensional lattice. A fluctuation of the lattice spacing,
changes the number of lattice points and links. The real system is not affected by the
mathematical formalism of the lattice, so the bare parameters must be tuned with a, so
that the observables are not affected.
We now define a measurable quantity O with dimensions of mass (or inverse length)
68
dO. On the lattice, it takes a dimensionless form Ô depending on the bare parameters of
the theory. The existence of the continuum limit implies that
O(g0 , a) =
Ô
adO
(6.1)
requiring at the same time
lim O(g0 (a), a) = Ophysical
(6.2)
ou
a→0
For small a’s, Eqs. (6.1) - (6.2) become equal, therefore Ô = adO Ophysical . Knowledge of
tin
the function Ô allows the determination of g0 with respect to a and Ophysical ; hence, the
coupling constant must be accompanied by a measurable quantity. For a small enough,
there is a global g0 (a) applicable to all quantities. This is convenient, since we can choose
an
a particular quantity to define g0 (a), for instance the quark-antiquark static potential. We
assume that the quark-antiquark pair is separated by a distance R measured in physical
units, and its lattice version is given by
st
1 R
V (R, g0 (a), a) = V̂ ( , g0(a))
a a
(6.3)
on
where R/a is their separation distance in lattice units. This is our initial point: Despite
the variation of the lattice spacing the observable V (R, g0 (a), a) must be invariant. This
is ensured by
d
V (R, g0 , a) = 0
da
"
!
#
∂
∂g0
∂
⇒ a
− −a
V (R, g0 , a) = 0
∂a
∂a ∂g0
th
a
C
a
(6.4)
M
ar
The above equation is known as the Renormalization group equation, from which the
Callan-Symanzik β-function is defined in its lattice form
∂g0 βL (g0 ) = −a
∂a (6.5)
g, µ̄
where µ̄ is the renormalization scale and g (g0 ) the renormalized (bare) coupling constant.
The lattice β-function expresses how the g0 changes under a variation of a. The renormalized coupling constant g depends on the renormalization scheme (parameterized by a scale
69
µ), and this dependence is given in the same manner, by the renormalized β-function
β(g) ≡ µ
dg
dµ
(6.6)
We will adopt the modified minimal subtraction MS scheme, the most commonly used
scheme for the analysis of experimental data. MS involves momentum integrals in D =
ou
4 − 2ǫ dimensions and subtracts off the resulting 1/ǫ poles and also ln(4π) − γE (γE is the
Euler-Mascheroni constant).
tin
We relate α0 to the renormalized coupling constant α MS as defined in the MS scheme
at a scale µ̄; at large momenta, these quantities are associated as follows
α MS (µ̄) = α0 + d1 (µ̄a)α02 + d2 (µ̄a)α03 + ...
(6.7)
an
(α0 = g02 /4π, α MS = g 2MS /4π). The running coupling is a QCD quantity with high impor-
st
tance and its precise determination would fix the value of a fundamental parameter in the
Standard Model. Moreover, the observables of an experimental process depending on an
overall energy scale µ and some kinematical variables y, can be computed in a perturbative
on
series which is usually written in terms of the MS coupling constant
2
O(µ, y) = αMS (µ) + c(y)αMS
(µ) + ...
(6.8)
C
From now on, we will denote the renormalized coupling constant g MS simply by g. It is
well known that in the asymptotic limit for QCD (g0 → 0), one can write the expansion
of the β-function in powers of the coupling constants, that is
th
a
βL (g0 ) = −b0 g03 − b1 g05 − bL2 g07 − ...
β(g) = −b0 g 3 − b1 g 5 − b2 g07 − ...
(6.9)
(6.10)
M
ar
The coefficients b0 , b1 are universal constants (regularization independent) provided in
Eqs. (6.11), and are computed from 1- and 2-loop diagrams. On the contrary, bLi (i ≥ 2)
depends on the regulator; it must be determined perturbatively. There are alternative
methods to determine b0 , b1 , among them, the static quark potential approach [63] (in the
quenched approximation of QCD), involving a phenomenological parametrization of the
70
interquark potential. Their expressions for any color (N) and flavor (Nf ) number is
11
2
N − Nf
3
3
34 2
13
1
1
b1 =
N − Nf
N−
(4π)4 3
3
N
1
b0 =
(4π)2
ou
The coefficient b2 in the MS scheme is
(6.11)
(6.12)
tin
1
2857 3
1709N 2 187
56N
1
11
2
b2 =
+ Nf
N + Nf −
+
+
−
(4π)6 54
54
36
4N 2
27
18N
Coefficients d1 , d2 of Eq. (6.7) can be extracted from b0 , b1 , b2 , bL2 using Eqs. (6.9), (6.10),
(7.5) and (7.6).
an
For Nf < 11N/3 the expansion of the β-function begins with a negative term, in other
words, in the asymptotic high energy regime, the coupling decreases logarithmically with
increasing energy. It reflect the observation that at high energy, quarks behave like free
st
particles. The universality of b0 , b1 means that they are independent of the definition of the
renormalized coupling constant, as long as the bare coupling is small (gR = g0 + O(g02 )).
on
This statement can be proven in the following way by considering two renormalization
couplings gA and gB , which can be expanded in powers of each other. Since the only
dimensionless parameters are gA , gB , then gA is a function only of gB and vice versa
C
gA (gB ) = gB + c gB2 + O(gB3 )
(6.14)
a
gB (gA ) = gA − c gA2 + O(gA3 )
(6.13)
M
ar
th
The leading coefficient is set to unity because of the condition that both gA , gB are equal
to g0 at leading order. The two β-functions can be related using the definition of Eq. (6.6)
and Eq. (6.13)
∂gA
dgA (6.13) ∂gB ∂gA (6.14)
βA (gA ) = µ
= µ
= βB (gB )
(6.15)
dµ
∂µ ∂gB
∂gB
where
∂gA (6.13)
(6.14)
= 1 + 2c gB + O(gB2 ) = 1 + 2c gA + O(gA2 )
∂gB
(6.16)
On the other hand, an analogous equation can be written for βB (gB ) in terms of gA
(6.14)
2
B 3
4
B 2
B
B
3
4
βB (gA ) = −bB
0 gB − b1 gB + O(gB ) = −b0 gA − (b1 − 2c b0 ) gA + O(gA )
(6.17)
71
Substituting the necessary ingredients back to Eq. (6.15), one finds
2
B
B
3
4
2
βA (gA ) = [−bB
0 gA − (b1 − 2c b0 ) gA + O(gA )] · [1 + 2c gA + O(gA )]
2
B 3
4
= −bB
0 gA − b1 gA + O(gA )
(6.18)
B
A
B
proving that bA
0 = b0 and b1 = b1 , that is they do not depend on the regularization nor
∂
∂
Λ=0
−µ
+ β(g)
∂µ
∂g
This is expressed by the exact relation
e
Z
exp −
0
g
dg
′
1
1
b1
+
− 2 ′
′
′
3
β(g ) b0 (g )
b0 g
an
Λ = µ b0 g
2
2 −b1/(2b0 ) −1/(2b0 g 2 )
(6.19)
tin
invariant parameter Λ
ou
the renormalization scheme.
The solution to Eqs. (6.6) contains an integration constant, the renormalization group
(6.20)
and can be expanded in powers of g ′ or equivalently g0 . This implies that the energy
relation
on
st
dependence of α can be specified completely in terms of the Λ parameter measured in
energy units. The basic lattice mass was introduce as a lattice parameter ΛL . By definition,
it is connected with the lattice spacing and the bare coupling via the following asymptotic
1
aΛL = exp −
2b0 g02
2 (b0 g02 )−b1 /2b0 1 + qg02 + O g04
C
where is q called the correction factor
a
q=
b21 − b0 bL2
2b30
(6.21)
(6.22)
th
Knowledge of the first correction to the 2-loop approximation of ΛL is important in order
M
ar
to verify the asymptotic prediction, Eq. (6.21).
For large µ, the asymptotic solution of the renormalization group equation reads
h
4π
2b1 ln[ln(µ2 /Λ2 )]
1
−
b0 ln(µ2 /Λ2 )
b20 b20 ln(µ2 /Λ2 )
4b2
1
b2 b0 5 i
+ 4 2 12 2 (ln[ln(µ2 /Λ2 )] − )2 + 2 −
2
8b1
4
b0 ln (µ /Λ )
µ→∞
α2 (µ) ∼
(6.23)
(6.24)
and the integration constant Λ can be regarded as a fundamental parameter of QCD; once
6.3. The step scaling function
72
it is known, the running coupling of the strong interactions is fixes at all scales. Eq. (6.24)
illustrates the asymptotic freedom property (αs → 0 for µ → ∞) and shows that QCD
become strongly coupled at µ ∼ Λ. Although Λ and b2 depend on the renormalization
scheme, they can be easily transformed between different schemes through the 1- and
2-loop coefficient relating the strong couplings in those schemes
⇒
−dA
1
ΛA
= e 8πb0
ΛB
⇒
A
A 2
B
A
bA
2 = b2 − b1 d1 + b0 (d2 − (d1 ) )
tin
ou
2
A
3
αA (µ) = αB (µ) + dA
1 (µ)αB (µ) + d2 (µ)αB (µ) + ...
(6.25)
an
Note that both couplings are taken at the same energy scale and the coefficients in their
perturbative relation are pure numbers.
st
The significance of the Λ parameter is that it can be given as an input parameter for
perturbative predictions of jet cross sections and compare to high energy experiments, in
order to test the agreement between theory and experiment. QCD on the lattice is renor-
The step scaling function
C
6.3
on
malized through the hadron spectrum and such a calculation would reveal the connection
between low and high energies.
a
As mentioned in the introduction, numerical simulations are necessary for calculations in
the high energy region, in order to make comparisons with estimates for αs (or Λ param-
th
eter) coming from perturbation theory. It is difficult to study this region by numerical
simulations, due to the large scale separation involved. Actually, it requires simulations
M
ar
with a cutoff a−1 much larger than the largest energy scale and a lattice with large size
L4 (L larger than a pion’s Compton wavelength). That is, to avoid discretization and
finite-size errors a, µ, L must obey
a−1 >> µperturbative >> µhadronic >> L−1
meaning that numerical simulations have to be performed on a lattice with L/a >> 70. The
method that has been developed involves an intermediate finite-volume renormalization
73
scheme, in which the scale evolution of the coupling is computed recursively from low to
very high energies.
The connection between the perturbative region of QCD and the nonperturbative
hadronic regime requires a nonperturbative definition of the coupling constant. Using
ou
this definition, one should be able to calculate the coupling on the lattice with a small
error. It is also important to have small cutoff effects and a perturbative expansion which
is relatively easy to calculate up to 2 loops. A popular definition is that given by the
Schrödinger functional (SF), which is the propagation amplitude for going from some field
configuration at the time x0 = 0 to another configuration at x0 = T ; an introduction to
tin
SF can be found in Ref. [64].
In this scheme, the coupling’s dependence on the energy scale µ is associated with the
running of a coupling ḡ 2 (L) where L = µ−1 . Starting from the low energy scale (L → Lmax ),
an
at which ḡ 2 (Lmax ) ≡ u0 is fixed at some value, we double the size of L.3 The coupling at
the scale 2L is related to ḡ 2(L) through a unique function, the step scaling function
(6.26)
st
ḡ(2L) ≡ u1 = σ(ḡ(L))
σ(ḡ 2 (L)) is the discrete version of the β-function, describing how the coupling constant
on
alters under scale variations. Using this function, one can compute recursively the coupling
at scales 2−n Lmax
un = ḡ 2 (2−n L)
(6.27)
C
over several orders of magnitude in µ, while keeping the lattice size at a manageable level.
The step scaling function is computed with Monte Carlo simulations. The starting
th
a
point is a simulation on a lattice with a certain number of lattice points in each direction,
L/a, and the bare coupling constant must tuned to such that the renormalized coupling
ĝ has the desired value u. The next step is a simulation at 2L/a using the same bare
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ar
parameters. This provides an estimate for the coupling constant called Σ(un , a/L) which is
an approximation of the step scaling function σ(un ). In fact, there appear lattice artifacts,
that is
σ(un ) = Σ(un , a/L) + O(a2 )
(6.28)
The above procedure is repeated for different resolutions L/a and σ(un ) can be obtained
by extrapolating to the continuum limit. Note that the whole procedure of doubling the
3
In general, we can change the value of L by a factor s, that is L → sL. Here, we use the most common
choice, s = 2.
74
lattice size is equivalent to keeping the L fixed and changing the lattice spacing.
The couplings at L and 2L are related through the integral
ln(2) =
Z
2L
L
dL′
=−
L′
Z
ḡ(2L)
ḡ(L)
dg
β(g)
(6.29)
(6.30)
tin
σ(un ) = un + s0 u2n + s1 u3n + s2 u4n + ...
ou
such that the perturbative expansion of σ(u) can be derived from the asymptotic expansion
of the β-function (Eq. (6.9))
where
s0 = 2b0 ln(2)
(6.31)
st
an
s1 = s20 + 2b1 ln(2))
5
s2 = s30 + s0 s1 + 2b2 ln(2)
2
on
From the determination of the scaling function, the evolution of the coupling can be
computed straightforwardly by solving the recursive equations
u0 = ḡ 2 (Lmax ), b un = σ(un+1)forn = 0, 1, 2, ...
(6.32)
C
As an example, let us take u0 = 3.480 and demand that σ(u) should be used only in the
range of couplings covered by the data, the recursion after six steps gives
a
ḡ = 1.053(12),
L = 2−6 Lmax
th
The error estimates comes from propagating the statistical errors to the fit polynomial and
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solving the recursion using this function.
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Chapter 7
7.1
an
tin
QCD with overlap fermions: Running
coupling and the 3-loop β-function
Introduction
st
In later years, use of nonultralocal actions which preserve chiral symmetry on the lattice
on
has become more viable for numerical simulations. The two actions which are being used
most frequently are overlap fermions [5, 6, 7] based on the Wilson fermion action and
domain-wall fermions [8, 9]. Overlap fermions are notoriously difficult to study, both
C
numerically and analytically. Many recent promising investigations involving simulations
with overlap fermions have appeared; see, e.g., Refs. [65, 66, 67, 68, 69, 70, 71]. Regarding
a
analytical computations, the only ones performed thus far have been either up to 1 loop,
such as Refs. [58, 72, 73, 74, 75, 76, 77], or vacuum diagrams at higher loops [78, 79]. The
calculation presented in this chapter is the first one involving nonvacuum diagrams beyond
th
the 1-loop level.
We compute the 2-loop renormalization Zg of the bare lattice coupling constant g0 in
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the presence of overlap fermions and Wilson gluons. For convenience, we have worked with
the background field technique, which only requires evaluation of 2-point Green’s function
for the problem at hand. We relate g0 to the renormalized coupling constant g MS as defined
in the MS scheme at a scale µ̄; at large momenta, these quantities are related as follows
α MS (µ̄) = α0 + d1 (µ̄a)α02 + d2 (µ̄a)α03 + ...
(7.1)
(α0 = g02/4π, α MS = g 2MS /4π, a : lattice spacing). The 1-loop coefficient d1 (µ̄a) has been
75
7.2. Theoretical background
76
known for a long time; several evaluations of d2 (µ̄a) have also appeared in the past ∼10
years, either in the absence of fermions [45, 80], or using the Wilson [59] or clover [81, 82]
fermionic actions. Knowledge of d2 (µ̄a), together with the 3-loop MS-renormalized βfunction [83], allows us to derive the 3-loop bare lattice β-function, which dictates the
ou
dependence of lattice spacing on g0 . In particular, it provides a correction to the standard
2-loop asymptotic scaling formula defining ΛL . Ongoing efforts to estimate the running
coupling from the lattice [64, 48, 84, 85] have relied on a mixture of perturbative and nonperturbative investigations. As a particular example, relating α MS to αSF (SF: Schrödinger
Functional scheme, as advocated by the ALPHA Collaboration), entails an intermediate
tin
passage through the bare coupling α0 ; the conversion from α MS to α0 is then carried out
perturbatively.
While the main application of SU(N) gauge theories on the lattice regards QCD, where
an
fermions are in the fundamental representation of SU(3), there has recently been some interest in gauge theories with fermions in other representations and with N 6= 3. Such
st
theories are being studied in various contexts [86, 87, 88, 89, 90, 91, 92, 93], e.g., supersymmetry, phase transitions, and the ’AdS/QCD’ correspondence.
Our results depend explicitly on the number of fermion flavors (Nf ) and colors (N).
on
Since the dependence of Zg on the overlap parameter ρ cannot be extracted analytically,
we tabulate our results for different values in the allowed range of ρ (0 < ρ < 2), focusing
on values which are being used most frequently in simulations. We also provide expressions
Theoretical background
a
7.2
C
for our results for fermions being in an arbitrary representation, which has led to a separate
publication [94].
th
The renormalized β-function describes the dependence of the renormalized coupling con-
M
ar
stant g on the scale inherent in the renormalization scheme (chosen to be the MS scheme).
A more extended introduction on the running coupling and the β-function is given in
Chapter 6. A bare β-function is also defined for the lattice regularization (βL (g0 ))
dgMS β(gMS ) = µ̄
,
dµ̄ a,g0
dg0 βL (g0 ) = −a
da gMS , µ̄
(7.2)
where a is the lattice spacing, µ̄ the renormalization scale and gMS (g0 ) the renormalized
(bare) coupling constant. In the asymptotic limit, one can write the expansion of Eq. (7.2)
77
in powers of g0
βL (g0 ) = −b0 g03 − b1 g05 − bL2 g07 − ...
(7.3)
3
5
7
β(gMS ) = −b0 gMS
− b1 gMS
− b2 gMS
+ ...
(7.4)
with the coefficients b0 , b1 being universal constants and regularization independent (Eq. (6.11)).
ou
On the contrary, bLi (i ≥ 2) depends on the regulator; it must be determined perturbatively.
Here we present the calculation of bL2 using the overlap fermionic action and Wilson gluons.
tin
Bare (βL (g0 )) and renormalized (β(g)) β-functions can be related using the renormalization function Zg , defined through g0 = Zg (g0 , aµ̄)g , that is 4
−1
∂ ln Zg2
2
β (g0 ) = 1 − g0
Zg β(g0 Zg−1 )
∂g02
an
L
Computing Zg2 to 2 loops
(7.6)
st
Zg2 (g0 , aµ̄) = 1 + g02 (2b0 ln(aµ̄) + l0 ) + g04 (2b1 ln(aµ̄) + l1 ) + O(g06)
(7.5)
and inserting it in Eq. (7.5), allows us to extract the 3-loop coefficient bL2 . The quantities
on
b0 , b1 , b2 and l0 have been known in the literature for quite some time [83, 58]; b0 and b1
are the same as those of the bare β-function, Eq. (6.11), and b2 in the MS scheme is
C
2857 3
56N
1
1709N 2 187
1
11
2
+ Nf
b2 =
N + Nf −
+
+
−
(4π)6 54
54
36
4N 2
27
18N
(7.7)
th
a
The constant l0 is related to the ratio of the Λ parameters associated with the particular
lattice regularization and the MS renormalization scheme
l0 = 2b0 ln (ΛL /Λ MS)
(7.8)
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ar
For overlap fermions the exact form of l0 appears in Ref. [58]
1
5
(1)
l0 =
− 0.16995599N + Nf −
− k (ρ)
8N
72π 2
(7.9)
where k (1) (ρ) is the convergent part of the 1-loop fermionic contribution (denoted by kf (ρ)
Zg could be denoted: ZgL, MS
renormalization scheme (MS).
4
to indicate its dependence on the regulator (L: lattice) and on the
78
in Ref. [58]), presented in Table 7.1.
Eq. (7.5) is valid order by order in perturbation theory and expanding it in powers of
g02 the first nontrivial relation is
bL2 = b2 − b1 l0 + b0 l1
(7.10)
ou
Thus, the evaluation of bL2 requires only the determination of the 2-loop quantity l1 . A
direct outcome of our calculation is the 2-loop corrected asymptotic scaling relation between
valid (see Eq. (5.28))
an
ZA (g0 , aµ̄)Zg2 (g0 , aµ̄) = 1
tin
a and g0 (Eq. (6.21)) where all quantities in the correction term q, except bL2 , are known.
The most convenient and economical way to proceed with calculating Zg (g0 , aµ̄) is to use
the background field technique described in Chapter 5, in which the following relation is
(7.11)
where ZA is the background field renormalization function
(7.12)
st
Aµ (x) = ZA (g0 , aµ̄)1/2 AµR (x)
on
(Aµ (AµR ): bare (renormalized) background field). In the notation of Chapter 5, the quantum field is represented by Qµ (x). In this framework, instead of Zg (g0 , aµ̄), it suffices to
compute ZA (g0 , aµ̄) with no need to evaluate any 3-point functions. For this purpose,
C
we consider the background field one-particle irreducible (1PI) 2-point function, both in
ab
the continuum (dimensional regularization, MS subtraction): ΓAA
R (p)µν and on the lattice:
ab
ΓAA
L (p)µν . In the notation of Ref. [45], these 2-point functions can be expressed in terms
of scalar functions νR (p), ν(p)
th
a
ab
ab
ΓAA
δµν p2 − pµ pν (1 − νR (p)) /g 2
R (p)µν = −δ
(1)
(7.13)
(2)
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νR (p) = g 2 νR (p) + g 4νR (p) + ...
X
µ
ab
ab
ΓAA
p2 [1 − ν(p)] /g02
L (p)µµ = −δ 3b
(p̂µ = (2/a) sin(aqµ /2)).
ν(p) = g02 ν (1) (p) + g04ν (2) (p) + ...
(7.14)
79
There follows
ZA =
1 − νR (p, µ̄, g)
1 − ν(p, a, g0 )
(7.15)
The gauge parameter λ must also be renormalized (up to 1 loop), in order to compare
lattice and continuum results
(1)
ZQ = 1 + g02 zQ + ...
(7.16)
ou
λ = Z Q λ0 ,
(ZQ : renormalization function of the quantum field). Using the quantum field 1PI 2-point
QQ
ab
ab
function in the continuum (ΓQQ
R (p)µν ) and on the lattice (ΓL (p)µν ) through
δµν p2 − pµ pν (1 − ωR (p)) + λpµ pν
tin
ab
ab
ΓQQ
R (p)µν = −δ
(1)
µ
ab
ab 2
ΓQQ
b [3 (1 − ω(p)) + λ0 ]
L (p)µµ = −δ p
(7.18)
st
X
an
ωR (p) = g 2 ωR (p) + O(g 4 )
(7.17)
on
ω(p) = g02 ω (1) (p) + O(g04 )
(1)
one can obtain the coefficient zQ
(1)
(1)
zQ = ω (1) (p, a, g0 ) − ωR (p, µ̄, g)
C
(7.19)
form
a
In terms of the perturbative expansions Eqs. (7.13), (7.14), (7.17), (7.18), Zg2 takes the
th
(1) i
h
(1)
(2)
(1) ∂ν
Zg2 = 1 + g02 (νR − ν (1) ) + g04 (νR − ν (2) ) + λ0 g04 (ω (1) − ωR ) R
∂λ λ=λ0
(7.20)
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The fermion part of ω (1) coincides with that of ν (1) , and similarly for the fermion part of
(1)
(1)
ωR and νR . Consequently, one may write
(1)
(1)
(1)
(1)
ω (1) − ωR = [ω (1) − ωR ]Nf =0 + [ν (1) − νR ] − [ν (1) − νR ]Nf =0
(7.21)
Since the quantities of interest are gauge invariant, we choose to work in the bare Feynman
gauge, λ0 = 1, for convenience. In order to compute ZA we need the expressions for
(1)
(2)
80
(1)
νR , νR , zQ , ν (1) , ν (2) . The MS renormalized functions necessary for this calculation to
2 loops are [45, 59]
(1)
νR (p, λ)
N
11 p2 205
Nf 2 p2
3
1
10
=
− ln 2 +
+
+
+
ln
−
16π 2
3
µ̄
36
2λ 4λ2
16π 2 3 µ̄2
9
(1)
ωR (p, λ)
N
=
16π 2
p2 577
N2
−8 ln 2 +
= 1) =
− 6ζ(3) +
µ̄
18
(16π 2 )2
ou
13
p2 97
Nf 2 p2
1
1
1
10
ln 2 +
+
(7.23)
− +
+
+
ln
−
6
2λ
µ̄
36 2λ 4λ2
16π 2 3 µ̄2
9
tin
(2)
νR (p, λ
(7.22)
an
401
Nf
p2
1
p2 55
N 3 ln 2 −
+
− ln 2 +
− 4ζ(3)
µ̄
36
N
µ̄
12
(16π 2 )2
(7.24)
st
For the lattice quantities, the gluonic contributions (Nf = 0) have been presented in
on
previous works [45, 80] (for the Wilson gauge action)
5N
1
ln (a2 p2 ) −
+ 0.137286278291N
2
48π
8N
(7.25)
ν (1) (p, λ0 = 1) = −
1
11N
ln (a2 p2 ) −
+ 0.217098494367N
2
48π
8N
(7.26)
ν (2) (p, λ0 = 1) = −
N
3
ln (a2 p2 ) +
− 0.01654461954 + 0.0074438722N 2 (7.27)
4
32π
128N 2
a
C
ω (1) (p, λ0 = 1) = −
th
The 1- and 2-loop coefficients d1 (µa), d2 (µa) relating the bare and renormalized coupling constants (Eq. (7.1)) can be directly evaluated from the above quantities
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h
i
(1)
(1)
d1 (µa) = −4π νR (p) − ν (p)
d2 (µa) =
λ=λ0
d21 (µa)
(1) i
h
(2)
(1) ∂νR
(2)
(1)
− 4π νR (p) − ν (p) + λ0 (ω − ωR )
∂λ λ=λ0
2
(7.28)
(7.29)
The fermionic contributions are associated with the diagrams of Fig. 7.1 and Fig. 7.2. In
the present work, ν (2) is perturbatively calculated for the first time using overlap fermions
81
and Wilson gluons. For completeness, we also compute the coefficient ν (1) and compare it
with previous results. The 1-loop diagrams (Fig. 7.1) correspond to ν (1) , and the 2-loop diagrams (Fig. 7.2) lead to ν (2) . Dashed lines ending in a cross represent the background field,
while those inside loops denote the quantum field. Solid lines correspond to fermion fields
ou
and a dot stands for the mass counterterm. Note that, for overlap fermions, the mass
counterterm equals zero, by virtue of the exact chiral symmetry of the overlap action;
consequently, diagrams 19 and 20 both vanish. Certain 2-loop diagrams have infrared di-
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th
a
C
on
st
an
tin
vergences and become convergent only when grouped together (6+12, 7+11, 8+18, 9+17).
82
1
2
2
5
6
3
an
1
4
8
11
12
14
15
16
18
19
20
on
st
7
10
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th
13
a
C
9
tin
ou
Figure 7.1: Fermion contributions to the 1-loop function ν (1) . Dashed lines ending on a
cross represent background gluons. Solid lines represent fermions.
17
Figure 7.2: Fermion contributions to the 2-loop function ν (2) . Dashed lines represent
gluonic fields; those ending on a cross stand for background gluons. Solid lines represent
fermions. The filled circle is a 1-loop fermion mass counterterm.
7.3. Description of the calculation
7.3
83
Description of the calculation
In recent years, overlap fermions are being used ever more extensively in numerical simulations, both in the quenched approximation and beyond. This fact, along with the desirable
properties of the overlap action, was our motivation to calculate the β-function with this
type of fermions. The important advantage of the overlap action is that it preserves chiral
ou
symmetry while avoiding fermion doubling. It is also O(a) improved. The main drawback
of this action is that it is necessarily nonultralocal; as a consequence, both numerical simulations and perturbative studies are extremely difficult and demanding (in terms of human,
tin
as well as computer time). Let us remind that the overlap action is given by Eq. (3.18), its
overlap parameter ρ is restricted by the condition 0 < ρ < 2 and the coupling constant is
an
included in the link variables, present in the definition of X. The perturbative expansion
of X in powers of g0 leads to the vertices needed for our calculation, a procedure described
in Chapter 3. The resulting vertices are extremely lengthy and the implementation of the
background field technique enlarges the expressions even more.
st
• Algebraic manipulations:
on
For the algebra involving lattice quantities, we make use of our symbolic manipulation
package in Mathematica, with the inclusion of the additional overlap vertices, written in
C
the background field language. These are enumerated to 4 different vertices, giving a total
of 743,968 terms when expanded. The first step to evaluate the diagrams is the contraction
among vertices, a step performed automatically once the vertices and the ‘incidence matrix’
a
of the diagram are specified. The outcome of the contraction is a preliminary expression
for the diagram under study; there follow simplifications of the color dependence, Dirac
th
matrices and tensor structures. We use symmetries of the theory (permutation symmetry
and lattice rotational invariance), or any other additional symmetry that may appear in
individual diagrams, to keep the size of the expression down to a minimum. A significant
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feature of the overlap vertices is their non-pointlike nature. This becomes obvious if we
go back to Eq. (3.34) and realize that each vertex comprises different Vji ’s (the index i
symbolizes the number of gluon in the vertex and j represents the number of integrals over
dummy momenta). The most convenient and computer RAM saving way to manage each
diagram is to divide it into subdiagrams, coming from different parts (Vji ’s) of the vertices.
For example, let us illustrate how diagram 5 of Fig. 7.2 is drawn apart into subdiagrams.
84
The particular diagram is constituted of two identical vertices: Q-A-Ψ-Ψ. This vertex
consists of the parts V12 , V22 and the resultant subdiagrams are four, coming from the
contraction of {V12 , V12 }, {V12 , V22 }, {V22 , V12 }, or {V22 , V22 }. Of course, the second and third
subdiagrams, have been worked out together, because they are effectively equivalent. Some
ou
subdiagrams, due to oversized expressions, need to be divided into smaller contributions
according to the appearing order of the background and gluons fields (for example, the
2-loop diagram 1 the two background fields arise from the same vertex, and these could
tin
belong to the same Xi or not. This gives us a criterion to organize the diagram into smaller
contributions that are manipulated separately).
• Logarithmic Contributions:
diagram may in principle depend on q as follows
2
2
2
2
2
2
2 2
an
The external momentum q appears in arguments of trigonometric functions and each
4
µ qµ
q2
+ O(q 4 , q 4 ln a2 q 2 )
st
α0 + α1 q + α2 q ln a q + α3 q (ln a q ) + α4
P
(7.30)
on
where the coefficients αi are typically 2-loop integrals with no external momenta, which
must be evaluated numerically.
(a.) The most demanding calculation is that of α2 and α3 , due to the required procedure;
C
these terms are isolated from the convergent ones. Certain diagrams (15-18) have superfi-
a
cially divergent terms, which are responsible for the double logarithms. Diagrams 2, 3, 8, 9,
10 and 13-18 have subdivergences (a few thousand terms) leading to single logarithms. All
divergent terms are obtain by applying 2 different types of subtractions: First we replace
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ar
th
the fermion propagator with the gluon propagator, via
1
1
1
1 =
+
−
D0 (k)
D0 (k) k̂ 2
k̂ 2
(7.31)
where k can be the loop momentum p or p + aq. D0 (k) is the inverse overlap propagator
P
(Eq. (3.32)) and k̂ 2 ≡ 4 µ sin2 (kµ /2) is the inverse gluon propagator (in the Feynman
gauge). Eq. (7.31) is necessary, since the divergent integrals that appear in the literature
are given in terms of k̂ 2 . The terms in parenthesis have 2 more powers of momentum
than the l.h.s and in most cases are directly added to the rest of the convergent part.
For some terms the repetition of this subtraction is needed until the overlap propagator
85
is totally eliminated from all divergent terms. Then follows another kind of subtraction
in order to extract the external momentum from the gluon propagators (which cannot be
accomplished with naive Taylor expansion)
1
1
1
1 =
+
−
(p̂)2
(p\
+ aq)2
(p\
+ aq)2 (p̂)2
(7.32)
ou
Eq. (7.32) might need to be applied in the parenthesis term (double subtraction), in order
to gain enough powers of aq, while the first term on the r.h.s yields factorized 1-loop
expressions, whose q dependence is easily extracted. For our calculation, some diagrams
tin
require up to triple subtraction. After performing both types of subtractions, all divergent
contribution is expressible in terms of known integrals.
(b.) For the convergent terms we employ naive Taylor expansion in aq up to O(q 2 ) leading
an
to the evaluation of α0 , α1 . This extraction makes explicit the functional dependence of
each diagram on q; the coefficients of terms proportional to q 2 are integrals over the two
internal momentum 4-vectors. Although this procedure seems to be straight forward, it
• Numerical integration:
on
st
involves many millions of terms in intermediate stages. This results a huge amount of
human and CPU time, and it is too costly on computer RAM.
C
The required numerical integrations are performed by optimized Fortran programs
which are generated by our Mathematica ‘integrator’ routine. The algorithm along with
a
details on the improvements that have been made for the purposes of the present work, are
discussed in Appendix B. Each integral is expressed as a sum over the discrete Brillouin
zone of finite lattices, with varying size L, and evaluated for different values of the overlap
th
parameter ρ. The average length of the expression for each diagram, after simplifications,
is about 2-3 hundred thousand terms, so that diagrams must be split into parts (usually
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of 2000 terms) to be integrated. The numerical values of these parts must then be added
together to avoid running into systematic errors or spurious divergences.
• Extrapolation:
Finally, we extrapolate the results to L → ∞; this procedure introduces an inherent
86
systematic error, which we can estimate quite accurately. This error is estimated in the
following way: 51 different extrapolations are automatically performed by our extrapolation
routine, using a broad spectrum of functional forms of the type
X
ei,j L−i lnLj
(7.33)
i,j
tin
ou
For the nth form, the deviation dn is calculated using alternative criteria for quality of fit.
P −2
These deviations are used to assign weights d−2
n
n dn to each extrapolation, producing
a final value together with the error estimate. Regarding the extrapolation of the 1-loop
results, these errors are of order of magnitude 10−10 − 10−12 (the expressions are relatively
simple and we integrated them for lattices with up to 128 lattice points per direction). On
the other side, 2-loop results are usually integrated for L ≤ 28, giving errors of order of
an
magnitude 10−6 − 10−8 . Apparently, the difference between the two cases is substantial
and there has been a major effort to minimize the errors by implementing alternative ways
of extrapolating; they will be listed below.
st
The majority of the diagrams, are composed of several subdiagrams, which in their
turn are split into many Fortran files to be integrated. There is no fundamental reason to
on
claim the same color dependence for all subdiagrams; on the contrary, the color structure
arises from the combination of the algebra generators {T a }. They could for instance have
a prefactor of the type N, 1/N, (N 2 −1)/N or (N 2 −2)/N. Infrared divergent diagrams
C
must be summed up before carrying out the extrapolation. On the other hand, for infrared
convergent diagrams there is no obligation to proceed in the same way. Having as a goal
a
the error minimization, we experiment with different strategies of extrapolating:
1. The numerical values of a certain diagram can be expressed in terms of N and 1/N.
Right after, we may extrapolate them to L → ∞ and get the desired result of the diagram
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ar
th
for different ρ’s.
2. Alternatively, one can add results with the same color prefactor. This way, a diagram
can has the form ai N + a2 /N + a3 (N 2 −1)/N + ... . We then extrapolate each one of the
ai ’s and bring the results into the form a N + b/N.
3. Another choice, is to extrapolate the contribution of each file separately and then write
them as a N + b/N.
One observes that methods 2 and 3 are equivalent for diagrams with a single color
prefactor. For each diagram, we performed all three variations of extrapolating and for each
choice of ρ we selected the signal with the lowest error estimate. Furthermore, particular
7.4. Results
87
sets of diagrams, {2, 13, 16} and {3, 4, 15}, are grouped together although each one of them
is infrared convergent. Their extrapolation is performed for each set (as if they were one
diagram), as well as separately, using the 3 methods explained above. Once again we
compare the final results coming from the different method and choose the one with the
smallest error.
Results
ou
7.4
(1)
We denote the contribution of the ith 1-loop Feynman diagram to ν (1) (q) as νi (q); sim(2)
(2)
"
(0)
aq
b 2 νi (q)=Nf ki + a2 q 2
2 2o
(1)
(2) ln a q
+ O((aq)4 )
ki + ki
(4π)2
#
(7.34)
st
X 

qµ4 

#


2


2 2
2 2
ln
a
q
ln
a
q
µ
(4)
(3)
(0)
(1)
(2)
+ci
+ci
ci +a2 q 2 ci +ci
+O((aq)4 ) (7.35)
2
2
2 )2 

(4π)
(4π)
(q




on
aq
b 2 νi (q)=Nf
n
an
(1)
"
tin
ilarly, contributions of 2-loop diagrams to ν (2) (q) are indicated by νi (q). The quantities
(1)
(2)
νi (q), νi (q) depend on N, Nf , ρ and aq according to the following formulae (λ0 = 1)
C
P
(j)
(j)
where qb2 = 4 µ sin2 (qµ /2). The index i runs over diagrams, and the coefficients ki , ci
(j)
depend on the overlap
parameter ρ. Moreover,
ci contains the color structure and can be
h
i
(j)
(j,−1)
(j,1)
written as ci = ci
/N + ci N . Comparison with continuum results and usage of
Ward Identities requires
i
X
(0)
ki
X
= 0,
(2)
ki
2
= ,
3
(4)
ci
=0
M
ar
•
i
X
(0)
ci = 0
(gauge invariance)
i
th
•
X
a
•
X
(2)
ci =
i
1
1
(3N − )
2
16π
N
(Lorentz invariance)
i
(3)
• c15 =
1
,
3N
4
(3)
c16 = N,
3
5
(3)
c17 = − N,
3
(3)
c18 =
N2 − 1
3N
(7.36)
We have checked that all the above conditions are verified by our results. Inserting these
conditions in Eqs. (7.34), (7.35), the expressions for the total fermionic contribution to
7.4. Results
88
ν (1) (q) and ν (2) (q), after addition of all diagrams, take the form
(1)
k
0.020377(7)
0.01581702(2)
0.0133504717(2)
0.0116910952(1)
0.0104621922(2)
0.0095058191(2)
0.00874441051(7)
0.00813753230(4)
0.00766516396(3)
0.00732057894(3)
0.00710750173(2)
0.00703970232(7)
0.0071425543(2)
0.0074569183(2)
0.0080467046(1)
0.0090134204(1)
0.010526080(2)
0.0128914(2)
#
1
1 ln a2 q 2
+
(3N− )
+O((aq)2 ) (7.38)
16π 2
N (4π)2
c(1,−1)
-0.0096(6)
-0.0044(1)
-0.00321(6)
-0.00244(4)
-0.00191(1)
-0.001606(6)
-0.001397(3)
-0.001241(1)
-0.001107(1)
-0.000979(1)
-0.000849(2)
-0.000706(3)
-0.000543(4)
-0.000335(7)
-0.00005(1)
0.00034(1)
0.00093(6)
0.0020(1)
ou
N
i
a
ρ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
i
(1,1)
+Nci
c(1,1)
0.124(3)
0.0118(5)
0.0045(1)
0.0030(1)
0.0022(6)
0.00176(2)
0.00145(1)
0.00124(1)
0.001051(9)
0.000872(3)
0.000710(8)
0.00052(1)
0.00033(3)
0.00007(1)
-0.0002(1)
-0.0004(1)
-0.0021(5)
-0.02(3)
tin
+Nf
X c(1,−1)
(7.37)
an
Nf =0
i
"
#
st
(2)
(2)
ν (q) = ν (q)
+ Nf
2 ln a2 q 2
(1)
ki +
+ O((aq)2 )
2
3 (4π)
on
Nf =0
"
X
C
ν (1) (q) = ν (1) (q)
th
Table 7.1: Numerical results for k (1) ≡
P
i
(1)
ki , c(1,−1) ≡
P
(1,−1)
,
i ci
c(1,1) ≡
P
(1,1)
.
i ci
M
ar
P (1)
In Table 7.1 we tabulate the total 1-loop contribution k (1) ≡ i ki for 18 values of the
overlap parameter (0 < ρ < 2). Each diagram was integrated for lattice size L4 , L ≤ 128
and then extrapolated to L → ∞. In all Tables and Figures, the errors accompanying
(1)
our results are entirely due to this extrapolation. The coefficients ki (ρ) do not depend
on the number of colors N, nor on the choice of regularization for the pure gluonic part
of the action (Symanzik, Iwasaki, etc.). 1-loop diagram 1 is entirely responsible for the
(2)
logarithmic contribution and it is independent of ρ, k1 =2/3. The numbers of k (1) in
Table 7.1 are in agreement with corresponding numbers from Ref. [58].
7.4. Results
89
The 2-loop calculation of ν (2) (q) was accomplished for the same values of ρ and for
P (1,−1) (1,1)
P (1,1)
L ≤ 28. Table 7.1 also presents the coefficients c(1,−1) ≡
, c
≡
,
i ci
i ci
versus ρ. Due to the extremely large size of the vertices involved, it is almost impossible
to extend the results to larger L. Typically, the integration of 2000 terms is completed in
(0)
ou
∼ 7 days on 1 CPU; the present calculation comprises approximately 3500×2000 terms.
Thus, if only a single CPU were available, our work would have required ∼50 years. In
certain cases with large systematic errors we extended the integration up to L = 46. The
(1)
per diagram results for ki and ki are collected in Table 7.2. The per diagram 2-loop
(0,1)
(0,−1)
contributions for ci , ci
are given in Tables 7.3 - 7.5 for 0.2 ≤ ρ ≤ 1.8. The rest of
(1,1)
(1,−1)
tin
the convergent coefficients, ci , ci
, can be found in Tables 7.6 - 7.9. Although the
total single logarithmic contributions are ρ independent, the corresponding per diagram
(2,1)
(2,−1)
coefficients, ci , ci
, depend onρ; their values are provided in Tables 7.10 - 7.11. In
an
general, the overlap action leads to coefficients which are very small for most values of
ρ. As a consequence, systematic errors, which are by and large rather small, tend to be
st
significant fractions of the signal for ρ > 1.4.
From Eqs. (7.6), (7.20) - (7.25), (7.27) we find the following expression for l1 in terms
of N, Nf , k (1) , c(1,−1) , c(1,1)
3
+ 0.018127763034 − 0.007910118514 N 2
128"N 2
#
55
N (1) c(1,−1)
1
N 481
+ Nf
− 4ζ(3) −
− 2k −
+ Nc(1,1)
(7.39)
(16π 2 )2 N 12
(16π 2)2 36
8π
N
C
on
l1 = −
a
We can write the final form of the 3-loop coefficient bL2 for the β-function (Eq. (7.10)),
including gluonic and fermionic contributions, using Eqs. (6.11), (7.7), (7.9) and (7.39)
11
+ 0.000364106020 N − 0.000092990690 N 3
2
2048π N
"
(4π 2 − 1)2
+ Nf
− 0.000046883436 − 000013419574 N 2
4(16π 2)3 N 2
!
Nf
23
8ζ(3) 37 N
+
−
+
+
(16π 2 )3
9N
3N
6
M
ar
th
bL2 = −
#
(4 N 3 + N − 3 N 2 N )
(11 N − 2 Nf ) c(1,−1)
f
f
−
+ c(1,1) N +
k (1) (7.40)
48π 2
N
(16π 2 )2 N
7.4. Results
90
A large variety of possible numerical checks has been performed, as mentioned above:
a. The total contribution to the gluon mass adds to zero, as expected. b. The coefficients
of the non-Lorentz invariant terms cancel. c. The terms with double logarithms correspond
to the continuum counterparts. This has been checked diagram by diagram. d. Terms
ou
with single logarithms add up to their expected value, which is independent of ρ (although
the expressions per diagram are ρ-dependent).
In Fig. 7.3 we plot the 1-loop coefficient k (1) with respect to ρ. Note that the errors are
too small to be visible at this scale. The 2-loop coefficients c(1,−1) and c(1,1) are plotted in
Figs. 7.4 - 7.5, respectively, for different values of the overlap parameter. The extrapolation
an
tin
errors are visible for ρ ≤ 0.4 and ρ ≥ 1.7. Substituting k (1) , c(1,−1) and c(1,1) into Eq. (7.40),
we find the numerical results for the 3-loop contribution, bL2 , of the β-function. These are
plotted in Fig. 7.6, choosing N = 3 and Nf = 0, 2, 3.
0.026
st
0.024
0.022
on
0.020
0.016
0.014
a
0.012
C
k(1)
0.018
0.010
th
0.008
M
ar
0.006
0.0
0.2
0.4
0.6
0.8
1.0
ρ
Figure 7.3: Plot of the total 1-loop coefficient k (1) ≡
ρ.
1.2
P
i
1.4
(1)
ki
1.6
1.8
2.0
versus the overlap parameter
7.4. Results
91
0.004
0.002
0.000
ou
c(1,-1)
-0.002
-0.004
tin
-0.006
-0.008
0.2
0.4
0.6
0.8
1.0
ρ
1.2
1.4
1.6
st
-0.012
0.0
an
-0.010
on
Figure 7.4: Plot of the total 2-loop coefficient c(1,−1) ≡
0.005
0.003
(1,−1)
i ci
2.0
versus ρ.
C
0.004
P
1.8
a
0.001
th
c(1,1)
0.002
M
ar
0.000
-0.001
-0.002
-0.003
-0.004
0.0
0.2
0.4
0.6
0.8
1.0
ρ
1.2
1.4
Figure 7.5: Plot of the total 2-loop coefficient c(1,1) ≡
1.6
P
(1,1)
i ci
1.8
2.0
versus ρ.
92
ou
7.4. Results
tin
0.002
0.001
an
0
-0.001
st
b2L
Nf=0
-0.002
-0.003
Nf=3
-0.005
0.4
0.6
a
0.2
C
-0.004
on
Nf=2
0.8
1
1.2
1.4
1.6
1.8
ρ
M
ar
th
Figure 7.6: The 3-loop coefficient bL2 (Eq. (7.40)), plotted against ρ, for N = 3 and Nf = 0
(horizontal red line), Nf = 2 (green line) and Nf = 3 (blue line).
7.5. Generalization to an arbitrary representation
7.5
93
Generalization to an arbitrary representation
7.5.1
The strategy
Here we provide the prescription that generalizes our results for ν (1) and ν (2) (Eqs. (7.37),
(7.38)) to an arbitrary representation r, of dimensionality dr . For the calculation under
ou
study, only the fermion part of the action is affected, with the link variables assuming the
form
Ux, x+µ = exp(i g0 Aaµ (x) Tra )
(7.41)
X
a
Tra Tra ≡ 1̂ cr ,
tr(Tra Trb ) ≡ δ ab tr = δ ab
In the fundamental representation F , one has
a
≡T ,
N2 − 1
cF =
,
2N
dF = N,
tF =
1
2
(7.42)
(7.43)
st
TFa
d r cr
N2 − 1
an
[Tra , Trb ] = i f abc Trc ,
tin
where Tra denote the generators in the representation r, and satisfy the relations
on
Studying the color structures for each diagram with a fermionic loop reveals the appropriate
substitutions one should make, in order to recast the results in an arbitrary representation.
The generalization prescription can be summarized in what follows.
th
a
C
For the 1-loop contribution in the fundamental representation (Eq. (7.37)), the color
structure is
1
tr(T a T b ) = δ ab tF = δ ab
(7.44)
2
Since diagrams with a closed fermion loop are always accompanied by a factor of Nf , the
straightforward substitution
Nf −→ Nf · (2 tr )
(7.45)
M
ar
gives the desired results in an arbitrary representation.
For the 2-loop fermion contribution to ν (2) (Eq. (7.38)), things get a bit more complicated, because there are different types of color structures. Fortunately, they all obey a
general pattern, which will be given below. In all our diagrams, vertices with a fermionantifermion pair and with two or more gluons contain also contributions involving integrations over internal momenta, and they are best depicted diagrammatically as non-pointlike
vertices, as shown in Fig. 7.7. There is no propagator (and thus no poles) associated to the
bold fermion lines. It is important to note that the color structures corresponding to such
94
Figure 7.7: The non-pointlike nature of an overlap vertex. Dashed lines represent gluon
fields; those ending on a cross stand for background gluons. Solid lines represent fermions.
ou
vertices are identical to those in ultralocal theories (with bold lines replaced by ordinary
propagators). The diagram of Fig. 7.8 actually contains two subdiagrams, arising from the
an
tin
non-pointlike contributions of the vertex of Fig. 7.7.
As an example, the diagram of Fig. 7.8 actually contains two subdiagrams, arising from
the non-pointlike contributions of the vertex appearing in Fig. 7.7. Subdiagram A has a
AB
B
st
A
C
color dependence of the type
on
Figure 7.8: A particular example of a 2-loop fermionic diagram. Dashed (solid) lines
represent gluon (fermion) fields.
tr(T a T c T c T b ) = cr tr δ ab
(7.46)
th
a
(a, b color indices of the external lines), while subdiagram B has
tr(T a T c T b T c ) = tr δ ab (cr −
N
)
2
(7.47)
M
ar
Therefore, the color structure of the diagram in Fig. 7.8 has the form
δ
ab
N α cr tr + β tr (cr − )
2
(7.48)
In the fundamental representation, this expression becomes
N2 − 1
1 δ α
+ β (−
)
4N
4N
ab
(7.49)
95
Thus, starting from our result for this diagram, which has the following color dependence:
(α′ N + β ′/N) δ ab , the prescription for converting it to another representation is
′
′
1
−1
′
′
− 4(α + β )(−
) δ ab
=
4α
4N
4N
N
′
′
′
→
4α cr tr − 4(α + β )(cr − ) tr δ ab
2
′N
2
(7.50)
ou
(α N + β /N) δ
ab
tin
One may check that all diagrams follow the formula above.
For the computation of bL2 , we need also the expressions for b0 , b1 , b2 in an arbitrary
representation
st
an
11
1
4
(7.51)
b0 =
N − tr Nf
(4π)2 3
3
1
34 2
20
b1 =
N − tr Nf
N + 4cr
(7.52)
(4π)4 3
3
2857 3
1415 2
79
1
11
2 205
2
2
N + 2 tr Nf cr −
cr N −
N +4 tr Nf
cr + N
(7.53)
b2 =
(4π)6 54
18
54
9
54
In order to calculate the ratio ΛL /Λ MS, the quantity l0 is necessary. For overlap fermions,
it equals
on
1
5
(1)
l0 =
− 0.16995599N + 2 tr Nf −
−k
8N
72π 2
(7.54)
(2)
(q)
N =0
" f
th
(2)
a
C
Moreover, according to the prescriptions given in Eqs. (7.45), (7.50), the results for ν (1)
and ν (2) become
"
#
2 2
2
ln
a
q
+ O((aq)2 )
ν (1) (q) = ν (1) (q)
+ 2 tr Nf k (1) +
(7.55)
Nf =0
3 (4π)2
ν
(q) = ν
M
ar
+4 tr Nf
#
2 2
N
N
ln
a
q
c(1,1) − c(1,−1) cr −
+ (cr +N)
+O((aq)2 )
(7.56)
2
2
(4π)4
Finally, Eqs. (7.55), (7.56) lead to the 3-loop coefficient of the bare β-function, which
in an arbitrary representation has the form
96
1
11
+ 0.000364106020 N − 0.000092990690 N 3
2
128 (4π) N
"
1
1
1
cr
2 c2r
+ tr Nf
+
+
− 0.00011964262 − 0.00003220865 cr N
32 (4π)2 N 2 2 (4π)4 N
(4π)6
bL2 = −
184
130
− 64 ζ(3) cr +
+ 32 ζ(3) N
3
3
ou
tr Nf
− 0.00001086180 N +
3 (4π)6
2
c(1,−1)
k (1)
2
−
(c
+
N)
t
N
+
N
(2 cr − N) (−4 tr Nf + 11 N)
+
r
r
f
32 π 4
24 π 2
#
(1,1)
c
N (4 tr Nf − 11 N)
(7.57)
+
24 π 2
an
7.5.2
tin
+
Adjoint representation
st
As a particular application, let us focus on the adjoint representation, A. The latter is
on
encountered, e.g., in the standard supersymmetric extension of gauge theories in terms of
vector superfields, where the gluinos are Majorana fermions in the adjoint representation,
C
thus similar in many respects to Nf = 1/2 species of Dirac fermions.
The generators now take the form
(TAa )bc ≡ i fbac
(7.58)
th
a
and the dimensionality of A is dA = N 2 − 1. Moreover,
cA = N,
tA =
d A cA
=N
N2 − 1
(7.59)
M
ar
Eqs. (7.55) and (7.56) for ν (1) and ν (2) now read
#
2 2
ln
a
q
4
+ Nf N 2 k (1) +
+ O((aq)2 )
(7.60)
3 (4π)2
Nf =0
"
#
2 2
ln
a
q
(2)
νadj (q) = ν (2) (q)
+ Nf N 2 2(c(1,1) − c(1,−1) ) +
+ O((aq)2 ) (7.61)
Nf =0
32π 4
(1)
νadj (q) = ν (1) (q)
"
97
It is interesting to find the numerical values of ΛL /Λ MS in the adjoint representation
(the corresponding results in the fundamental representation appear in Ref. [58]), which
can be done using our 1-loop results for k (1) . This ratio is defined through Eq. (7.8), where
l0 in this case is
1
5
(1)
=
− 0.16995599N + 2 N Nf −
−k
8N
72π 2
(7.62)
ou
l0adj
Our results for ΛL/Λ MS are plotted in Fig. 7.9 for N = 3 and Nf = 0, 1/2, 1. One
may compare Fig. 7.9 with an analogous figure pertaining to fermions in the fundamental
Nf = 1/2
st

0.008
on
(ΛL/ΛMS)adjoint
0.010
an
0.012
tin
representation (Ref. [58]); in that case, one obtains 0.02 ≤ ΛL /Λ MS ≤ 0.025, for Nf = 1.
0.006
Nf = 0 (scaled down by a factor of 10)
C
0.004
Nf = 1
a
0.002
th
0.000
0.0
M
ar
0.2
0.4
0.6
0.8
1.0
ρ
1.2
1.4
1.6
1.8
2.0
Figure 7.9: The ρ dependence of the ratio ΛL /Λ MS in the adjoint representation for N = 3
and Nf = 0 (horizontal line scaled down by a factor of 10 from its value 0.034711), Nf = 1/2
and Nf = 1.
7.6. Discussion
7.6
98
Discussion
We have calculated the 2-loop coefficient of the coupling renormalization function Zg , for
the Yang-Mills theory with gauge group SU(N) and Nf species of overlap fermions. We
used the background field method to simplify the computation; in this method there is no
need of evaluating any 3-point functions. This is the first 2-loop calculation using overlap
ou
fermions with external momenta, and it proved to be extremely demanding in human and
CPU time; this is due to the fact that we had to manipulate very large expressions (millions
of terms) in intermediate stages.
tin
We used our numerical results of Zg to determine the 3-loop coefficient bL2 of the bare
lattice β-function; the latter dictates the asymptotic dependence between the bare coupling
an
constant g0 and the lattice spacing a, required to maintain the renormalized coupling at a
given scale fixed. Knowledge of bL2 provides the correction term to the standard asymptotic
scaling relation between a and g0 , via Eq. (6.21).
The dependence of Zg and bL2 on N and Nf is shown explicitly in our expressions. On
the other hand, dependence on the overlap parameter ρ cannot certainly be given in closed
on
st
form; instead, we present our results for a large set of values of ρ in its allowed range.
The 3-loop correction is seen to be rather small: This indicates that the perturbative
series is very well behaved in this case, despite the fact that it is only asymptotic in nature.
Furthermore, around the values of ρ which are most often used in simulations (1 ≤ ρ ≤ 1.6),
fermions bring about only slight corrections to the 3-loop β-function, even compared to
C
pure gluonic contributions, as can be seen from Fig. 7.6.
The only source of numerical error in our results has its origin in an extrapolation
to infinite lattice size. Compatibly with the severe CPU constraints, numerical 2-loop
a
integration had to be performed on lattices typically as large as 284 , or even up to 464
in cases where an improved extrapolation was called for. An intermediate range for ρ
M
ar
th
(0.6 ≤ ρ ≤ 1.3) showed the most stable extrapolation error, and this may be a sign of their
suitability for numerical simulations.
The present study, being the first of its kind in calculating 2-loop diagrams with overlap
vertices and external momentum dependence, had a number of obstacles to overcome. One
first complication is the size of the algebraic form of the Feynman vertices; as an example,
the vertex with 4 gluons and a fermion-antifermion pair contains ∼724,000 terms when
expanded. Upon contraction these vertices lead to huge expressions (many millions of
terms); this places severe requirements both on the necessary computer RAM and on the
efficiency of the computer algorithms which we must design to manipulate such expressions
7.6. Discussion
99
automatically. Numerical integration of Feynman diagrams over loop momentum variables
is performed on a range of lattices, with finite size L, and subsequent extrapolation to
L → ∞. As it turns out, larger L are required for an accurate extrapolation in the present
case, compared to ultralocal actions. In addition, since the results depend nontrivially
ou
on the parameter ρ of the overlap action, numerical evaluation must be performed for a
sufficiently wide set of values of ρ, with an almost proportionate increase in CPU time.
Extreme values of ρ (ρ >
∼ 2) show unstable numerical behavior, which is attributable
∼ 0, ρ <
to the spurious poles of the fermion propagator at these choices; this forces us to even larger
L. A consequence of all complications noted above is an extended use of CPU time: Our
M
ar
th
a
C
on
st
an
tin
numerical integration codes, which ran on a 32-node cluster of dual CPU Pentium IV
processors, required a total of ∼50 years of CPU time.
Tables
(0)
(0,−1)
c{3+14+15}
(0,−1)
c4
0.0030297(6)
0.003287(3)
0.0035578(5)
0.0038238(4)
0.0040929(3)
0.00436473(6)
0.00464517(2)
0.00493494(1)
0.00523630(2)
0.00555174(5)
0.00588422(6)
0.00623695(4)
0.0066142(2)
0.0070210(5)
0.007464(2)
0.007950(1)
0.008493(1)
0.009111(1)
C
0.00260(3)
0.00222(2)
0.001645(6)
0.001253(2)
0.000930(1)
0.000669(1)
0.0004522(5)
0.0002740(3)
0.0001310(2)
0.0000194(4)
-0.0000625(7)
-0.000115(1)
-0.000142(1)
-0.000147(1)
-0.0001289(9)
-0.000099(2)
-0.000070(1)
-0.000060(8)
th
a
-0.00173(3)
-0.00186(4)
-0.00160(1)
-0.001497(4)
-0.001477(4)
-0.0014686(5)
-0.0014870(3)
-0.0015297(3)
-0.0015952(3)
-0.0016846(5)
-0.0017974(6)
-0.001937(2)
-0.002100(2)
-0.002289(1)
-0.002515(3)
-0.002762(6)
-0.00309(2)
-0.00344(3)
M
ar
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
(0,−1)
c1
on
Table 7.2: Per diagram breakdown of the 1-loop coefficients ki
ρ
(1)
k2
0.019344(6)
0.018038350(6)
0.01753314922(4)
0.017322163470(3)
0.017278696275(4)
0.0173605843730(1)
0.017556782951(6)
0.017872114285(1)
0.018322547098(2)
0.0189345964405(7)
0.019747192643(9)
0.020815997561(1)
0.02222124823(1)
0.024081810674(6)
0.02658161963(1)
0.03002396495(1)
0.0349588593(1)
0.0425532(1)
ou
(1)
k1
0.001034(3)
-0.00222133(1)
-0.0041826775(2)
-0.00563106832(2)
-0.0068165041(2)
-0.0078547653(2)
-0.00881237245(7)
-0.00973458199(4)
-0.01065738314(3)
-0.01161401750(3)
-0.01263969091(2)
-0.01377629524(7)
-0.0150786939(2)
-0.0166248923(2)
-0.01853491504(4)
-0.0210105446(1)
-0.024432779(2)
-0.0296618(1)
tin
(0)
k2
-0.04028528(3)
-0.04397848498(6)
-0.047732030565(5)
-0.05156042074(3)
-0.055479104515(8)
-0.05950493780(1)
-0.06365663388(1)
-0.06795525969(2)
-0.07242482258(3)
-0.07709299443(1)
-0.081992032002(4)
-0.08715997701(2)
-0.09264225671(1)
-0.098493871766(7)
-0.104782467614(10)
-0.11159278151(1)
-0.11903332401(1)
-0.1272468911(5)
an
(0)
k1
0.04028529(3)
0.043978484984(9)
0.04773203057(2)
0.05156042075(1)
0.05547910452(1)
0.059504937801(3)
0.063656633886(10)
0.06795525970(2)
0.07242482265(4)
0.07709299444(2)
0.08199203200(1)
0.08715997701(2)
0.09264225672(2)
0.09849387180(3)
0.10478246764(3)
0.111592781506(3)
0.11903332403(2)
0.127246891(2)
st
ρ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
100
(0,−1)
Table 7.3: Contribution to ci
(1)
and ki
for various ρ values.
(0,−1)
c{6+12}
(0,−1)
c{8+18}
-0.0025115(4)
-0.0026708(4)
-0.0028296(3)
-0.00298808(5)
-0.00315083(6)
-0.0.003319(1)
-0.0034919(1)
-0.00367242(8)
-0.0038611(1)
-0.0040591(1)
-0.00426755(5)
-0.0044881(2)
-0.0047220(2)
-0.00497085(9)
-0.0052374(4)
-0.0055220(5)
-0.0058287(5)
-0.0061601(5)
-0.004250(9)
-0.00434(2)
-0.004557(2)
-0.004696(3)
-0.004814(1)
-0.0049262(1)
-0.0050348(1)
-0.0051445(1)
-0.00525858(4)
-0.00538102(4)
-0.00551473(6)
-0.0056641(1)
-0.0058342(1)
-0.0060321(7)
-0.0062660(7)
-0.006555(3)
-0.006902(2)
-0.007360(4)
0.00289(1)
0.00331(2)
0.003778(2)
0.004128(2)
0.0044170(7)
0.0046792(3)
0.0049171(3)
0.0051380(3)
0.0053482(3)
0.0055528(4)
0.0057565(4)
0.0059643(2)
0.0061828(2)
0.0064197(4)
0.0066865(5)
0.006999(2)
0.007386(3)
0.007896(3)
c5
(0,−1)
of diagrams 1, 4, 5, 3+14+15, 6+12, 8+18.
Tables
c{3+14+15}
(0,1)
0.00240(3)
0.00262(5)
0.00245(1)
0.002454(4)
0.002535(4)
0.0026300(6)
0.0027545(4)
0.0029060(3)
0.0030829(3)
0.0032871(5)
0.0035185(6)
0.003782(2)
0.004074(2)
0.004402(3)
0.004768(4)
0.005178(6)
0.00568(2)
0.00623(2)
-0.000714(2)
-0.0008117(9)
-0.000916(2)
-0.0010195(6)
-0.0011269(3)
-0.0012371(3)
-0.0013485(2)
-0.0014616(2)
-0.0015763(1)
-0.0016922(1)
-0.0018092(1)
-0.0019274(1)
-0.0020469(2)
-0.0021684(4)
-0.0022931(7)
-0.0024212(2)
-0.0025571(7)
-0.002703(2)
-0.00270(3)
-0.00233(3)
-0.001780(6)
-0.001404(3)
-0.0010990(9)
-0.0008551(7)
-0.0006539(6)
-0.0004929(3)
-0.0003671(2)
-0.0002733(4)
-0.0002106(8)
-0.000178(1)
-0.000172(1)
-0.000195(2)
-0.000240(2)
-0.000304(2)
-0.000382(4)
-0.000439(7)
(0,1)
(0,1)
(0,1)
-0.0030895(6)
-0.003362(3)
-0.0036544(6)
-0.0039430(4)
-0.0042365(3)
-0.00453901(6)
-0.00485238(2)
-0.00517919(1)
-0.00552207(3)
-0.00588395(5)
-0.00626832(6)
-0.00667913(4)
-0.0071215(2)
-0.0076015(5)
-0.008127(2)
-0.008707(1)
-0.009357(1)
-0.010099(2)
c5
(0,1)
(0,1)
(0,1)
c{6+12}
c{7+11}
c{8+18}
c{9+17}
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0.004250(9)
0.00434(2)
0.004557(2)
0.004696(3)
0.004814(1)
0.0049262(1)
0.0050348(1)
0.0051445(1)
0.00525858(4)
0.00538102(4)
0.00551473(6)
0.0056641(1)
0.0058342(1)
0.0060321(7)
0.0062660(7)
0.006555(3)
0.006902(2)
0.007360(4)
-0.0006320(2)
-0.000682(1)
-0.0007205(2)
-0.0007459(3)
-0.00076162(2)
-0.00076893(9)
-0.00076874(6)
-0.0007622(1)
-0.00075053(9)
-0.0007350(1)
-0.00071691(4)
-0.00069776(9)
-0.00067891(5)
-0.0006628(2)
-0.0006509(3)
-0.0006449(5)
-0.0006450(2)
-0.000649(1)
-0.00289(1)
-0.00331(2)
-0.003778(2)
-0.004128(2)
-0.0044170(7)
-0.046792(3)
-0.0049171(3)
-0.0051380(3)
-0.0053482(3)
-0.0055528(4)
-0.0057565(4)
-0.0059643(2)
-0.0061828(2)
-0.0064197(4)
-0.0066865(5)
-0.006999(2)
-0.007386(3)
-0.007896(3)
0.0009153(5)
0.001002(1)
0.0010774(2)
0.0011410(3)
0.0011955(1)
0.0012415(2)
0.0012800(2)
0.0013115(2)
0.0013371(2)
0.0013576(3)
0.0013738(1)
0.00138700(6)
0.00139837(8)
0.0014091(1)
0.0014208(2)
0.0014346(3)
0.0014488(2)
0.001462(1)
M
ar
th
a
C
ρ
(0,1)
0.0022883(3)
0.0024238(4)
0.0025593(2)
0.00269701(6)
0.00283979(4)
0.00298856(3)
0.00314442(4)
0.00330843(7)
0.0034817(1)
0.00366520(8)
0.00386023(4)
0.00406807(4)
0.00429008(4)
0.0045278(1)
0.0047832(2)
0.0050574(2)
0.0053534(2)
0.0056730(6)
of diagrams 1, 2+13+16, 3+14+15, 4, 5.
on
(0,1)
c4
ou
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
(0,1)
tin
c{2+13+16}
an
(0,1)
c1
st
ρ
101
(0,1)
c10
0.0001472(3)
0.0001734(1)
0.0002008(2)
0.00022928(3)
0.00025940(4)
0.00029154(4)
0.00032602(3)
0.00036319(3)
0.00040346(3)
0.00044735(2)
0.00049550(3)
0.00054880(3)
0.00060838(2)
0.00067577(2)
0.00075316(8)
0.0008431(2)
0.0009496(2)
0.0010789(4)
of diagrams 6+12, 7+11, 8+18, 9+17, 10.
Tables
102
(1,−1)
c{3+14+15}
-0.0109(2)
-0.00436(2)
-0.00248(1)
-0.001493(8)
-0.000913(1)
-0.0004935(9)
-0.0001571(3)
0.0001413(4)
0.0004281(4)
0.000723(1)
0.001041(2)
0.001407(2)
0.001846(2)
0.002398(4)
0.003130(9)
0.004180(4)
0.00578(4)
0.0086(1)
-0.0494(3)
-0.01523(4)
-0.00672(2)
-0.00322(1)
-0.001529(7)
-0.000637(5)
-0.000141(1)
0.0001316(7)
0.000264(1)
0.0003028(7)
0.0002729(5)
0.000177(1)
0.000015(3)
-0.000214(1)
-0.000548(4)
-0.001052(1)
-0.00189(2)
-0.00338(3)
(1,−1)
ou
-0.00184(6)
-0.00128(4)
-0.00112(2)
-0.001082(7)
-0.0010521(8)
-0.0010420(6)
-0.0010455(3)
-0.0010611(1)
-0.0010885(2)
-0.0011286(1)
-0.0011836(4)
-0.0012561(6)
-0.001355(3)
-0.0014852(5)
-0.001669(3)
-0.001940(6)
-0.00233(3)
-0.00307(6)
st
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
(1,−1)
c4
tin
c1
an
ρ
(1,−1)
(1,−1)
ρ
th
M
ar
c{6+12}
(1,−1)
c{7+11}
c{8+18}
0.00870(6)
0.00436(1)
0.002966(3)
0.0022765(2)
0.0018592(1)
0.001581059(7)
0.00138402(2)
0.00123931(5)
0.00113125(4)
0.00105148(4)
0.00099516(4)
0.00096112(8)
0.0009511(3)
0.0009714(4)
0.001036(1)
0.0011762(4)
0.001474(2)
0.002209(2)
-0.00034964(1)
-0.00039464(6)
-0.0004396(1)
-0.00048364(6)
-0.00052645(6)
-0.00056740(4)
-0.00060633(4)
-0.00064287(4)
-0.00067671(4)
-0.00070758(4)
-0.00073529(4)
-0.00075955(3)
-0.00078014(4)
-0.00079695(4)
-0.00080976(2)
-0.00081839(2)
-0.00082252(6)
-0.0008218(1)
0.0437(4)
0.01214(9)
0.00426(5)
0.00126(4)
-0.00004(1)
-0.000738(4)
-0.001127(3)
-0.0013538(7)
-0.0014841(4)
-0.0015584(2)
-0.0016033(4)
-0.0016329(4)
-0.0016608(8)
-0.001703(6)
-0.001756(4)
-0.001876(6)
-0.00210(3)
-0.00266(2)
C
0.00049(3)
0.000363(5)
0.0003226(6)
0.0003011(4)
0.0002924(6)
0.0002911(4)
0.0002954(3)
0.00030475(8)
0.00031883(9)
0.0003385(1)
0.00036414(9)
0.0003974(1)
0.0004399(2)
0.0004947(9)
0.0005673(5)
0.000667(1)
0.000812(4)
0.00107(2)
a
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
c5
of diagrams 1, 4, 5.
on
(1,−1)
(1,−1)
(1,−1)
of diagrams 3+14+15, 6+12, 7+11, 8+18.
Tables
c{3+14+15}
(1,1)
0.0103(2)
0.00393(2)
0.00208(1)
0.001105(8)
0.000518(1)
0.0000815(9)
-0.0002805(3)
-0.0006127(4)
-0.0009418(4)
-0.001289(1)
-0.001672(2)
-0.002118(2)
-0.002656(2)
-0.003335(4)
-0.004230(9)
-0.005497(3)
-0.00740(4)
-0.0106(1)
0.115(3)
0.0072(3)
0.00144(5)
0.00038(2)
-0.000026(9)
-0.000179(4)
-0.000255(3)
-0.000302(2)
-0.000329(1)
-0.0003415(9)
-0.000345(2)
-0.000336(4)
-0.000310(8)
-0.000280(7)
-0.00018(1)
-0.00007(3)
-0.0005(5)
-0.02(3)
-0.0494(3)
-0.01523(4)
-0.00672(2)
-0.00322(1)
-0.001529(7)
-0.000637(5)
-0.000141(1)
0.0001316(7)
0.000264(1)
0.0003028(7)
0.0002729(5)
0.000177(1)
0.000015(3)
-0.000214(1)
-0.000548(4)
-0.001052(1)
-0.00189(2)
-0.00338(3)
(1,1)
(1,1)
c{6+12}
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
-0.00870(6)
-0.00436(1)
-0.002966(3)
-0.0022765(2)
-0.0018592(1)
-0.001581059(7)
-0.00138402(2)
-0.00123931(5)
-0.00113125(4)
-0.00105148(4)
-0.00099516(4)
-0.00096112(8)
-0.0009511(3)
-0.0009714(4)
-0.001036(1)
-0.0011762(4)
-0.001474(2)
-0.002209(2)
(1,1)
c{7+11}
C
ρ
M
ar
th
a
0.0003652(1)
0.00040934(7)
0.00045220(8)
0.000493101(9)
0.00053206(4)
0.00056860(5)
0.00060266(5)
0.00063401(5)
0.00066253(4)
0.00068809(3)
0.00071059(3)
0.00072999(3)
0.00074631(4)
0.00075949(3)
0.00076959(2)
0.00077654(2)
0.0007803(1)
0.0007800(1)
(1,1)
(1,1)
c5
0.00187(6)
0.00129(4)
0.00112(2)
0.001080(7)
0.001053(1)
0.0010465(6)
0.0010560(2)
0.0010791(1)
0.0011158(2)
0.00116676(8)
0.0012348(3)
0.0013231(6)
0.001442(3)
0.0015955(6)
0.001810(3)
0.002110(8)
0.00256(3)
0.00342(6)
-0.00021(3)
-0.000213(3)
-0.000155(4)
-0.000127(4)
-0.0001182(5)
-0.0001121(4)
-0.00010878(6)
-0.00010816(9)
-0.0001091(1)
-0.0001125(1)
-0.0001182(1)
-0.0001269(1)
-0.0001389(2)
-0.0001569(2)
-0.0001823(3)
-0.0002212(3)
-0.000286(3)
-0.000403(7)
of diagrams 1, 2+13+16, 3+14+15, 4, 5.
on
(1,1)
c4
ou
(1,1)
tin
c{2+13+16}
an
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
(1,1)
c1
st
ρ
103
(1,1)
c{8+18}
(1,1)
c{9+17}
-0.0437(4)
0.00049(7)
-0.01214(9)
0.0003(4)
-0.00426(5)
-0.0000(1)
-0.00126(4)
0.00033(6)
0.00004(1)
0.00047(6)
0.000738(4)
0.00047(2)
0.001127(3)
0.000462(6)
0.001354(7)
0.000467(1)
0.0014841(4) 0.000467(9)
0.0015584(2) 0.000459(3)
0.0016033(4) 0.000468(8)
0.0016329(4) 0.00046(1)
0.0016608(8) 0.00046(3)
0.001703(6)
0.000444(7)
0.001756(4)
0.00047(1)
0.001876(6)
0.00066(1)
0.00210(3)
0.00018(6)
0.00266(1)
-0.0000(2)
(1,1)
c10
0.0002(1)
0.00013(4)
0.000123(3)
0.000065(2)
0.000042(2)
0.0000261(8)
0.00001216(4)
-0.0000010(1)
-0.00001360(1)
-0.00002609(2)
-0.0000389(1)
-0.00005355(3)
-0.0000712(2)
-0.0000933(6)
-0.0001250(1)
-0.0001744(1)
-0.000266(4)
-0.00038(5)
of diagrams 6+12, 7+11, 8+18, 9+17, 10.
Tables
104
(2,−1)
c{8+18}
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
-0.013366(6)
-0.006513(9)
-0.004235(7)
-0.0031009(8)
-0.0024235(1)
-0.00197417(6)
-0.00165489(8)
-0.00141675(4)
-0.00123263(4)
-0.001086232(8)
-0.00096723(3)
-0.00086874(3)
-0.00078598(2)
-0.000715570(9)
-0.00065502(4)
-0.00060247(1)
-0.00055648(5)
-0.0005169(5)
0.013333(2)
0.006473(4)
0.004195(2)
0.0030608(3)
0.0023834(1)
0.00193408(3)
0.00161479(2)
0.00137665(2)
0.00119252(1)
0.00104613(1)
0.00092713(2)
0.000828632(5)
0.000745877(6)
0.000675469(7)
0.000614920(6)
0.00056237(1)
0.00051638(2)
0.00047587(1)
st
an
tin
c{3+14++15}
ou
(2,−1)
ρ
(2,−1)
(2,1)
(2,1)
(2,1)
ρ
c{2+13++16}
c{3+14+15}
c{8+18}
c{9+17}
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0.0007574(4)
0.00062213(4)
0.0005500758(2)
0.0005026441(2)
0.0004686248(3)
0.0004433894(4)
0.00042474477(5)
0.0004116462(2)
0.00040370895(4)
0.0004010140(2)
0.0004040517(2)
0.0004137557(3)
0.0004316242(3)
0.0004599683(2)
0.0005023841(1)
0.0005646959(2)
0.000657053(8)
0.00080(1)
C
(2,1)
on
Table 7.10: The overlap parameter dependence of the 2-loop coefficients ci
-0.013333(2)
-0.006473(4)
-0.004195(2)
-0.0030608(3)
-0.0023834(1)
-0.00193408(3)
-0.00161479(2)
-0.00137665(2)
-0.00119252(1)
-0.00104613(1)
-0.00092713(2)
-0.000828632(5)
-0.000745877(6)
-0.000675469(7)
-0.000614920(6)
-0.00056237(1)
-0.00051638(2)
-0.00047587(1)
-0.0005641(2)
-0.00042061(1)
-0.000342511(4)
-0.000289971(1)
-0.0002510600(5)
-0.0002207785(4)
-0.0001966701(8)
-0.000177455(1)
-0.0001624980(7)
-0.000151587(1)
-0.000144841(1)
-0.0001426940(4)
-0.0001459505(5)
-0.0001559042(3)
-0.0001745785(4)
-0.0002051875(3)
-0.0002530826(2)
-0.0003280(2)
M
ar
th
a
0.0132578(3)
0.006399(1)
0.0041184(8)
0.0029817(3)
0.00230196(1)
0.00185038(3)
0.00152810(2)
0.00128886(2)
0.00110281(1)
0.000954557(1)
0.00083376(1)
0.00073352(2)
0.000649062(9)
0.000576995(2)
0.000514822(6)
0.000460677(3)
0.000413133(3)
0.0003710854(7)
(2,1)
Table 7.11: Numerical results for the 2-loop coefficients ci
overlap parameter ρ.
.
(2,1)
c10
-0.000049(2)
-0.0000(2)
-0.0000(3)
-0.00001(2)
-0.000016(4)
-0.000019(2)
-0.0000220(4)
-0.0000260(4)
-0.0000312(2)
-0.0000375(4)
-0.0000455(2)
-0.0000557(3)
-0.000069(2)
-0.000086(3)
-0.000107(3)
-0.000137(2)
-0.00018(2)
-0.00023(5)
for different values of the
ou
Chapter 8
8.1
an
tin
Improved perturbation theory for
improved lattice actions
Introduction
st
Since the earliest studies of quantum field theories on a lattice, it was recognized that quan-
on
tities measured through numerical simulation are characterized by significant renormalization effects, which must be properly taken into account before meaningful comparisons to
corresponding physical observables can be made.
C
As has been rigorously demonstrated [11], the renormalization procedure can be formally carried out in a systematic way to any given order in perturbation theory. However,
a
calculations are notoriously difficult, as compared to continuum regularization schemes;
furthermore, the convergence rate of the resulting asymptotic series is often unsatisfactory.
A number of approaches have been pursued in order to improve the behavior of per-
th
turbation theory, among them Refs. [98, 99]. These approaches share in common the aim
to reorganize perturbative series in terms of an expansion coefficient which would be more
M
ar
suitable than the bare coupling constant g0 ; the definition of such a “renormalized” coupling constant is not unique, but can depend on the observables under study and on an
energy scale. It is expected that such a definition will reabsorb a large part of the tadpole
contributions which are known to dominate lattice perturbation theory.
Some years ago, a method was proposed to sum up a whole subclass of tadpole diagrams,
dubbed “cactus” diagrams, to all orders in perturbation theory [97, 100]; this procedure
has a number of desirable features:
• it is gauge invariant
105
8.1. Introduction
106
• it can be systematically applied to improve (to all orders) results obtained at any
given order in perturbation theory
• it does indeed absorb the bulk of tadpole contributions into an intricate redefinition
of the coupling constant
ou
In cases where nonperturbative estimates of renormalization coefficients are also available
for comparison, the agreement with cactus improved perturbative results is significantly
better as compared to results from bare perturbation theory.
tin
Here we extend the improved perturbation theory method of Refs. [97, 100], to encompass the large class of actions which are used nowadays in simulations of QCD, a work
an
published in Physical Review D [101]. This class includes Symanzik improved gluon actions with any arbitrary combination of closed Wilson loops, combined with any fermionic
action. The effect of resummation is to replace various parameters in the action (coupling constant, Symanzik coefficients, clover coefficient) by ”dressed” values; the latter are
solutions to certain coupled integral equations, which are easy to solve numerically.
on
st
In this chapter we explain the procedure of dressing the gluon propagators, the gluon
and fermion vertices, as a result of the summation of cactus diagrams to all orders. We
show how these dressed constituents are employed to improve 1-loop and 2-loop Feynman
diagrams coming from bare perturbation theory. In the end of the chapter we apply our
resummation procedure to improve the multiplicative renormalization ZV and ZA of the
C
vector and axial current respectively, using overlap fermions and Symanzik gluons. Another
application is presented in the next chapter, the additive renormalization of the fermion
masses up to 2 loops, employing clover fermions and Symanzik gluons. In cases where
th
a
nonperturbative estimates of renormalization functions are also available for comparison,
the agreement with improved perturbative results is significantly better as compared to
results from bare perturbation theory. Finally, the improvement of QED is briefly discussed
M
ar
in Section 8.4.
Clearly, all resummation procedures, whether in the continuum or on the lattice, bear
a caveat: A one-sided resummation could ruin desirable partial cancellations which might
exist among those diagrams which are resummed and others which are not; what is worse,
the end result might depend on the gauge. As we shall see, no partial cancellations will be
ruined in our procedure, due to the distinct N-dependence of the resummed diagrams (N
is the number of colors); furthermore, our results will be gauge independent.
8.2. The method
8.2
107
The method
In this section, following the outline of Ref. [97], we start illustrating our method by
showing how the gluon propagator is dressed by the inclusion of cactus diagrams. We will
then dress gluon and fermion vertices as well. Finally, we will explain how this procedure is
applied to Feynman diagrams at a given order in bare perturbation theory, concentrating
8.2.1
ou
on the 1- and 2-loop case.
Dressing the propagator
tin
We consider, for the sake of definiteness, the Symanzik improved gluon action involving
an
Wilson loops with up to 6 links, as presented in Chapter 2 (Eq. (2.26)); we recall that
there is a set of four coefficients ci satisfying c0 + 8c1 + 16c2 + 8c3 = 1.
We apply the usual parameterization of links in terms of the continuum gauge fields
Aµ (x)
Aµ (x) = Aaµ (x) T a ,
Tr (T a T b ) = 21 δ ab
(8.1)
st
Ux,µ = exp i g0 a Aµ (x + aµ̂/2) ,
on
where a is the lattice spacing (set to 1 from now on), µ̂ is the unit vector in direction µ and
T a is the generator of the SU(N) algebra. Use of the Baker-Campbell-Hausdorff (BCH)
formula, leads to the following form for Ui
C
(1)
2 (2)
3 (3)
4
Ui = exp i g0 Fi + i g0 Fi + i g0 Fi + O(g0 )
(1)
In the above equation Fi
(8.2)
is simply the sum of the gauge fields on the links of loop i. For
a
this to become more clear, let us consider the plaquette as an example, where one has
(1)
(j)
= Aµ (x+µ̂/2) + Aν (x+µ̂+ν̂/2) − Aµ (x+ν̂+µ̂/2) − Aν (x+ν̂/2)
th
F0
(8.3)
M
ar
Fi (j > 1) are j-th degree polynomials in the gauge fields, constructed from nested
commutators. The resummation procedure involves diagrams made of vertices containing
(1)
only Fi .
We may now define the cactus diagrams which dress the gluon propagator: These are
gauge invariant tadpole diagrams which become disconnected if any one of their vertices
is removed (see Fig. 8.1). The original motivation of this procedure is the well known observation of ‘tadpole dominance’ in lattice perturbation theory. Each vertex is constructed
8.2. The method
108
(1)
pure gluon parts of the action. This fact implies that, the longitudinal
solely from the Fi
tin
ou
parts of all propagators will always cancel. Therefore, the effect of dressing is the same in
all covariant gauges; this will be mathematically proven in Eq. (8.11).
an
Figure 8.1: A cactus diagram.
A diagrammatic equation for the dressed gluon propagator (thick line) in terms of the bare
propagator (thin line) and 1-particle irreducible (1PI) vertices (solid circle) reads
+
+···
+
st
=
(8.4)
on
Our goal is to solve Eq. (8.4), so that we write the dressed propagator in terms of the bare
parameters od the action. The 1PI vertex obeys the following recursive equation
+
M
ar
th
a
C
=
+ ...
+
+
+
+
+
+.
..
+
+.
.
.
+.
..
(8.5)
Let us mention that each part of the above diagrammatic equations is made of 4 terms;
each one coming from the plaquette, rectangle, chair or parallelogram contribution of the
action. The tadpole diagrams of Eq. (8.5) are a contraction of vertices that exclude all
8.2. The method
109
anticommutators; one needs the Taylor expansion of Ui only up to O(g0 ). In order to put
Eq. (8.5) into a mathematical form and solve it, let us first write down the bare inverse
propagator D −1 resulting from the Symanzik gluon action (Eq. (2.26)), and from the gauge
fixing term (Eq. (2.13))
=
X
k̂ρ2 δµν
− k̂µ k̂ρ δρν
ρ
dµρ +
1
k̂µ k̂ν
1−ξ
(8.6)
ou
−1
Dµν
(k)
For unexplained notation, see Subsection 2.2.2. The inverse propagator can thus be put in
1
k̂µ k̂ν
1−ξ
tin
the form
−1
(1)
(2)
(3)
Dµν
(k) ≡ c0 G(0)
µν (k) + c1 Gµν (k) + c2 Gµν (k) + c3 Gµν (k) +
(8.7)
X
an
The matrices G(i) (k) (Eq. (2.29) ) are symmetric and transverse, i.e. they satisfy
G(i)
µν (k) k̂ν = 0
(1) st
ν
(1)
Each of them originates from a corresponding term Tr Fi
Fi
(8.8)
of the gluon action. Con-
on
sequently, each of the diagrams on the r.h.s. of Eq. (8.5), being the result of a contraction
(1)
with only two powers of Fi left uncontracted, will necessarily be equal to a linear combi-
G1PI (k) ≡
C
nation of G(i) (k); this implies that the 1PI vertex G1PI (k) (the l.h.s. of Eq. (8.5)) can be
written as
= α0 G(0) (k) + α1 G(1) (k) + α2 G(2) (k) + α3 G(3) (k)
(8.9)
a
Each of the quantities αi will in general depend on N, g0 , c0 , c1 , c2 , c3 , but not on the
M
ar
th
momentum. We must now turn Eq. (8.5) into a set of 4 recursive equations for αi .
Eq. (8.4) leads to the following expression for the dressed propagator D dr (k)
D dr ≡
= D + D G1PI D + D G1PI D G1PI D + · · · = D
⇒ (D dr )−1 = (1̂ − G1PI D) D −1 = D −1 − G1PI
= c̃0 G(0) + c̃1 G(1) + c̃2 G(2) + c̃3 G(3) +
1
k̂µ k̂ν ,
1−ξ
!
1̂
(8.10)
1̂ − G1PI D
c̃i ≡ ci − αi
(8.11)
We observe that dressing affects entirely the transverse part of the inverse propagator,
8.2. The method
110
replacing the bare coefficients ci with improved ones c̃i , and leaves the longitudinal part
intact. The same property carries over directly to the propagator itself; the consequence
of this will be that our method leads to the same results in all covariant gauges.
In terms of the dressed propagator, Eq. (8.5) can be drawn as
+
+ ...
+
(8.12)
ou
=
To proceed, we must evaluate the generic tadpole diagram of Eq. (8.12) coming from an
n-point vertex of the action, in which n − 2 lines have been pairwise contracted. This
8.2.2
an
tin
calculation is explained in Subsection 8.2.3, after deriving the expressions for the improved
vertices (Subsection 8.2.2). This leads to the numerical values of the dressed coefficients
c̃i .
Dressing vertices
st
• We will begin by considering the 3-gluon vertex, coming from the action, Eq. (2.26).
This vertex results from a Taylor expansion of Ui to 3rd order in g0 . Expressing Ui as in
(1)
(2)
C
on
Eq. (8.2), we see that only terms of the form Tr(Fi Fi ) will appear in this vertex, since
(3)
(1) 3 Tr(Fi ) and Tr (Fi ) will vanish.
By analogy with Eq. (8.12), the dressed 3-gluon vertex equals
+
+
+ ...
(8.13)
a
=
Consistently with the dressing of propagators, each (2l + 1)-point vertex in Eq. (8.13) is a
M
ar
th
sum of 4 parts (one from each type of Wilson loop in the action), made up of
(1) 2l−1
Tr (Fi )
Denoting the bare 3-gluon vertex by
(0)
V 3 = c0 V 3
(1)
+ c1 V 3
(2) Fi
(2)
+ c2 V 3
(8.14)
(3)
+ c3 V 3
it is relatively straightforward to see from Eq. (8.13) that the dressed vertex, V3dr , is given
8.2. The method
111
by
V3dr
=
3
X
∞
X
(i g0 )2l+1
ci
i=0
l=0
2
F (2l+2; N) βil
2
(2l+1)! (N −1)
!
(i)
(i g0 )−1 V3
(8.15)
The summations inside parentheses are a mere multiple of those in Eq. (8.29); consequently,
the result for V3dr turns out very simple
(1)
+ c̃1 V3
(2)
+ c̃2 V3
(3)
+ c̃3 V3
(8.16)
ou
(0)
V3dr = c̃0 V3
tin
• We turn now to the 3-point fermion-antifermion-gluon vertex [100]. In the cases of
Wilson and overlap fermions, these vertices remain unaffected, since the fermion actions
=
+
an
do not contain any closed Wilson loops on which the BCH formula might be applied. The
vertex from the clover action (Eq. (2.20)), on the other hand, is amenable to improvement;
we write
+ ...
+
(8.17)
st
where fermions are denoted by a dotted line. Just as in Eq. (8.15), we find
=
l=0
on
∞
X
(i g0 )2l
2
F (2l+2; N) β0l
·
2
(2l+1)! (N −1)
!
=
c̃0
·
c0
(8.18)
C
• Vertices with more fields would seem a priori more difficult to handle. To illustrate
th
a
the complications that may arise, let us consider the 4-gluon vertex. The BCH expansion
(1) 4 (2) 2 (1) (3)
of Tr(Ui ) contributes to this vertex in the form Tr (Fi ) , Tr (Fi ) and Tr(Fi Fi ).
Such terms may in principle dress differently from each other. In addition, the dressed
(1) 4 vertex produced from Tr (Fi ) will not be a multiple of its bare counterpart; rather, it
will be a linear combination of two color tensors (which are independent for N > 3)
M
ar
Tr{T a T b T c T d + permutations} and (δ ab δ cd + δ ac δ bd + δ ad δ bc )
(8.19)
This issue has been resolved in Ref. [97], and it generalizes directly to the present case.
Actually, such complications will not appear while dressing 1- and 2-loop diagrams in
(1) 4 typical cases: Terms of the type Tr (Fi ) must simply be omitted in order to avoid
double counting, since their contribution is already included in dressing diagrams with one
(2) 2 (1) (3)
less loop. Thus, one is left only with: Tr (Fi ) and Tr(Fi Fi ); for both of these terms
8.2. The method
112
it is straightforward to show, just as in Eqs. (8.15, 8.16), that their dressing amounts to
replacing ci by c̃i .
• The same considerations as above apply to all higher vertices from both the gluon and
fermion actions as well.
Numerical values of improved coefficients
ou
8.2.3
Let us now evaluate a typical diagram on the r.h.s. of Eq. (8.12) in order to derive numerical
values for c̃i . Before contraction the vertex reads
an
tin
4
(ig0 )n X X n (1) n o
−S →
ci
tr Fi,x,µν
n! g02 i=1 x,µν
Z
dq1
dqn
(ig0 )n
...
· ei(q1 +···+qn )x tr{T a1 T a2 . . . T an }
=
2
4
n! g0
(2π)
(2π)4
n
XY
c0
q̂ iµ Aaνi (qi ) − q̂ iν Aaµi (qi )
ai
\ ai
−q̊iµ Aaνi (qi ) + (q\
iµ +qiν )Aν (qi ) + (qiµ −qiν )Aν (qi )
+2c1
x,µν i=1
n X Y
ai
−iqiν /2 ai
iqiµ /2 ai
2
(q\
+q
)A
(q
)
−
q̂
e
A
(q
)
−
q̂
e
A
(q
)
iµ
iν
i
iρ
i
iρ
i
ρ
µ
ν
x,µνρ
µ6=ν6=ρ6=µ
on
+c2
st
x,µν i=1
n XY
i=1
n Y
i=1
n Y
ai
−iqiµ /2 ai
−(q\
Aν (qi ) + q̂iρ e−iqiν /2 Aaµi (qi )
iµ −qiν )Aρ (qi ) − q̂iρ e
a
+
ai
iqiµ /2 ai
(q\
Aν (qi ) − q̂iρ eiqiν /2 Aaµi (qi )
iµ −qiν )Aρ (qi ) + q̂iρ e
C
+
i=1
n X Y
ai
ai
ai
\
\
−(q\
+q
)A
(q
)
+
(q
−q
)A
(q
)
+
(q
+q
)A
(q
)
iν
iρ
i
iµ
iρ
i
iµ
iν
i
µ
ν
ρ
th
+c3
x,µνρ
µ6=ν6=ρ6=µ
M
ar
i=1
+
n
1 Y
3
i=1
ai
\ ai
\ ai
(q\
iν −qiρ )Aµ (qi )−(qiµ −qiρ )Aν (qi )+(qiµ −qiν )Aρ (qi )
!
(8.20)
(q̂ µ = 2i sin(qµ /2), q̊µ = 2i sin(qµ )).
Each of the diagrams under consideration is a sum of 4 terms, one term for each of
the Wilson loops Ui (i = 0, 1, 2, 3) in the action, from which its n-point vertex may have
originated. There are (n − 2)/2 1-loop integrals in the diagram; each of them corresponds
8.2. The method
113
(1)
to the contraction of two powers of Fi
via a dressed propagator, and will contribute one
power of βi (c̃0 , c̃1 , c̃2 , c̃3 ), where
β2 =
β3 =
Z
Z
−π
π
−π
π
−π
π
−π
d4 q
(2π)4
d4 q
(2π)4
d4 q
(2π)4
d4 q
(2π)4
dr
dr
2 q̂µ2 Dνν
(q) − 2 q̂µ q̂ν Dµν
(q)
dr
dr
dr
(4q̂ν2 − q̂ν4 ) Dµµ
(q) + q̂µ2 (4−q̂ν2 ) Dνν
(q) − 2 q̂µ q̂ν (4−q̂ν2 ) Dµν
(q)
2
dr
2
dr
2
q̂µ (8−q̂ν ) Dρρ (q)/2 − q̂µ q̂ρ (8−q̂ν ) Dµρ (q)/2
ou
β1 =
Z
π
dr
dr
3 q̂µ2 (4−q̂ν2 ) Dρρ
(q)/2 − 3 q̂µ q̂ν (4−q̂ρ2 ) Dµν
(q)/2
(8.21)
tin
β0 =
Z
We remind the reader that the index i corresponds to plaquette (i=0), rectangle (i=1),
an
chair (i=2) and parallelogram (i=3). Distinct values are assumed for µ, ν, ρ; no summation
implied. Once again, we note that βi are gauge independent, since the longitudinal part
on
F (2; N) = δa1 a2 Tr {T a1 T a2 }
st
cancels in the loop contraction.
For the contraction of the SU(N) generators we first evaluate F (n; N), which is the
sum over all complete pairwise contractions of Tr{T a1 T a2 . . . T an }
F (4; N) = (δa1 a2 δa3 a4 + δa1 a3 δa2 a4 + δa1 a4 δa2 a3 ) Tr {T a1 T a2 T a3 T a4 }
X
1
δa1 a2 δa3 a4 . . . δan−1 an Tr T P (a1 ) T P (a2 ) . . . T P (an ) (8.22)
F (n; N) = n/2
2 (n/2)! P ∈S
C
n
(F (2n + 1; N) ≡ 0; Sn is the permutation group of n objects). The generating function
th
a
G(z; N) for this quantity, defined via
G(z; N) ≡
∞
X
zn
n=0
n!
F (n; N)
(8.23)
M
ar
has been computed explicitly in Ref. [97], using Gaussian integration over the space of
traceless Hermitian matrices [102], with the result
G(z; N) = ez
2 (N −1)/(4N )
L1N −1 (−z 2 /2)
(8.24)
(Lαβ (x) : Laguerre polynomials). By differentiating Eq. (8.23), one gets the desired expres-
8.2. The method
114
sion for F (n; N)
dn
G(z;
N)
(8.25)
z=0
dz n
Since 2 out of n generators remain uncontracted in our case, color contraction does not
F (n; N) =
lead to F (n; N), but rather to
n F (n; N)
2(N 2 −1)
(8.26)
ou
Thus, upon contraction, an n-leg diagram in Eq. (8.12), with its vertex coming from the
term Ui of the Lagrangian (i = 0, 1, 2, 3), will merely result in the following multiple of G(i)
tin
ci (i g0 )n n F (n; N)
(n−2)/2 (i)
4 βi
G
2
2
g0 n! 2(N −1)
(8.27)
3
X
αi G
(i)
=
i=0
∞
X
ci (i g0 )n n F (n; N)
(n−2)/2 (i)
4 βi
G
2
2
g
n! 2(N −1)
n=4,6,8,... 0
(8.28)
st
i=0
3
X
an
We are finally in a position to set Eq. (8.12) in a mathematical form
on
Unknown in Eq. (8.28) are the coefficients αi ; they appear on the l.h.s., as well as inside the
integrals βi of the r.h.s, by virtue of Eqs. (8.21), (8.11). We recall that G(i) are functions
of the external momentum k and if these are independent of each other, then Eq. (8.28)
amounts to 4 equations for the 4 coefficients αi . Actually, G(2) is not independent of the
rest 3 functions G(i) , so that we have 3 equations for 3 coefficients, but this causes no
C
complication. In any case, c2 is typically set to zero in simulations.
The generalization of our procedure for improved gluon actions with arbitrary numbers
and types of Wilson loops is now evident. It is crucial to check at this stage that all
th
a
combinatorial weights are correctly incorporated in Eq. (8.28); this is indeed the case.
Splitting Eq. (8.28) into 4 separate equations, and making use of Eq. (8.25), we can
recast the infinite summations in closed form
∞
X
1 (i g0 )n n F (n; N)
(n−2)/2
4 βi
2
2
g
n! 2(N −1)
n=4,6,8,... 0
!
∞
X
(i g0 )n
2(i g0 )
n/2
−1/2
= 1+
F (n+1; N) βi
βi
2
2
n!
g0 (N −1)
n=0
2
= 1−
G′ (z; N)
1/2
2
z (N −1)
z=(i g0 βi )
M
ar
αi
=
ci
(8.29)
8.2. The method
115
2 ′
ci −αi
2
(N −1) =
⇒
G (z; N)
1/2
ci
z
z=(i g0 βi )
N−1 1
2
2
−βi g02 (N −1)/(4N )
2
LN −1 (g0 βi /2) + 2 LN −2 (g0 βi /2)
(8.30)
= e
N
ou
In solving Eqs. (8.30), each choice of values for (ci , g0 , N) leads to a unique set of
values for c̃i ≡ ci − αi . The latter are no longer normalized in the sense of Eq. (2.27);
one may equivalently choose, however, to express the results of our procedure in terms
of a normalized set of improved coefficients, c̃i /C̃0 and an improved coupling constant
g̃02 = g02/C̃0 , where C̃0 = c̃0 + 8c̃1 + 16c̃2 + 8c̃3 . In fact, it is convenient to treat bare and
ci
,
g02
γ̃i ≡
c̃i
,
g02
β̃i (c̃0 , c̃1 , c̃2 , c̃3 ) ≡ g02 βi (c̃0 , c̃1 , c̃2 , c̃3 ) = βi (γ̃0 , γ̃1, γ̃2 , γ̃3 ) (8.31)
an
γi ≡
tin
improved coefficients on an equal footing, by defining rescaled quantities as follows
st
The dressed propagators in β̃i will now contain a rescaled gauge parameter (1−ξ) →
g02 (1−ξ), which is irrelevant since the longitudinal part does not contribute. Insertion
of Eqs. (8.31) into Eq. (8.30), leads to the coupled equations satisfied by the rescaled
on
quantities γ̃i
1
γ̃i = 2
γi e−β̃i (N −1)/(4N )
N −1
N−1 1
LN −1 (β̃i /2) + 2 L2N −2 (β̃i /2)
N
(8.32)
C
For the gauge groups SU(2) and SU(3), the Laguerre polynomials have a simple form,
making Eqs. (8.32) more explicit
th
a
(N = 2) : γ̃i = γi e−β̃i /8
M
ar
(N = 3) :
γ̃i = γi e−β̃i /6
β̃i
1−
12
!
β̃i β̃ 2
1− + i
4
96
!
(8.33)
Given the highly nonlinear nature of Eqs. (8.32), it is not a priori clear that a solution
for γ̃i always exists. The converse, of course, is trivial: Finding the bare values γi which
lead to a given set of dressed values γ̃i is immediate, since the integrals β̃i only depend on
γ̃i , not γi. As it turns out, this is always the case, for all physically interesting values of
ci , and for all values of g0 ranging from g0 = 0 up to a certain limit value, well inside the
8.2. The method
116
strong coupling region.
Fortunately, numerical solutions of Eqs. (8.32), (8.33) can be found very easily. We use
a fixed point procedure, applicable to equations of the type x = f (x)
γ̃i = fi (γ̃i )
⇒
(m)
γ̃i = lim γ̃i
m→∞
(0)
,
where : γ̃i
= γi ,
(m+1)
γ̃i
(m)
= fi (γ̃i
) (8.34)
ou
In order for the procedure to converge (attractive fixed point), it must be that: |∂fi /∂γ̃i | <
1 in a neighborhood of γ̃i . This has been verified in a number of extreme cases. The algo-
tin
rithm developed for the evaluation of γ̃i appears in Appendix C. In what follows, we present
the values of the improved coefficients for several gluon actions of interest, as derived from
the algorithm of Appendix C.
• Plaquette action:
an
Let us start with the plaquette action for which the undressed coefficients are c0 =1,
c1 =c2 =c3 =0 [97]. In this case, Eqs. (8.32) reduce to only one equation, for γ̃0 , while
γ̃i =γi=0 (i=1, 2, 3). This equation is further simplified greatly since the integral β̃0 can now
st
be evaluated in closed form, which is β̃0 = 1/(2γ̃0 ). Employing the above into Eq. (8.33)
we obtain c̃0 for N = 2 and N = 3
2
c̃0 = e−g0 /(12 c̃0)
C
(N = 3) :
g2
1− 0
24 c̃0
on
(N = 2) :
2
c̃0 = e−g0 /(16 c̃0)
g2
g04
1− 0 +
8 c̃0 384 c̃20
(8.35)
(8.36)
a
In Fig. 8.2 we plot c̃0 as a function of g02 , for N = 2 and N = 3. The range of g0 values, for
√
which solutions exist, extends from g02 = 0 (where c̃0 = 1) up to 16 e/3 ≃ 3.23 (N = 2)
M
ar
th
and 1.558 (N = 3); this covers the whole region of physical interest.
8.2. The method
117
1.00
N =2
0.90
c̃0
0.80
0.60
0.2
0.4
0.6
0.8
g02
1.0
1.2
1.4
1.6
tin
0.50
0.0
ou
N =3
0.70
an
Figure 8.2: Improved coefficient c̃0 for N=2 and N=3 (plaquette action).
Additionally, in Table 8.1 there are listed the unimproved and improved coefficients for
several actions and different choices of β.
on
st
• Tree-level Symanzik action:
The tree-level Symanzik improved action [10] corresponds to c0 =5/3, c1 = − 1/12 and
c2 =c3 =0. The dressed coefficients c̃0 , c̃1 are now found using the algorithm of Appendix
C (no closed form can be provided) and are shown in Fig. 8.3 for N = 3. In order to plot
both c̃0 and c̃1 together, c̃1 is multiplied be a factor of -20. A similar scaling appears in
1.8
a
1.6
C
the next figures, whenever is necessary.
c̃0
1.4
c0 = 1.667
c1 = −0.833
th
1.2
1.0
M
ar
0.8
−20 · c̃1
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
2.5
g02
Figure 8.3: Improved coefficients c̃0 and c̃1 (tree-level Symanzik improved action).
8.2. The method
118
• Iwasaki action:
The Iwasaki set of parameter values [103] is c0 =3.648, c1 = − 0.331 and c2 =c3 =0. In
principle, c0 and c1 depend on g0 , but they are typically kept constant in simulations. The
corresponding dressed values are plotted in Fig. 8.4 (N = 3).
4.0
ou
c0 = 3.648
c1 = −0.331
3.5
c̃0
3.0
2.5
−10 · c̃1
tin
2.0
1.0
0.0
0.5
1.0
an
1.5
1.5
2.0
g02
2.5
3.0
3.5
st
Figure 8.4: Improved coefficients c̃0 and c̃1 (Iwasaki action).
on
• Tadpole improved Lüscher-Weisz (TILW) actions:
Another class of gluon actions based on Symanzik improvement are the TILW actions [104,
105]. In this case, the coefficients c0 , c1 , c3 are optimized for each value of β = 2N/g02
2.6
2.4
C
separately, while c2 = 0 for all cases. In Fig. 8.5 we show the values of ci and of their
dressed counterparts c̃i in a typical range for β : 8.0 ≤ β c0 ≤ 8.6 and N = 3.
c0
c̃0
a
2.2
th
2.0
1.8
1.6
M
ar
1.4
1.2
−10 · c1
−100 · c3
−10 · c̃1
−100 · c̃3
1.0
7.8
7.9
8.0
8.1
8.2
8.3
β · c0
8.4
8.5
8.6
8.7
Figure 8.5: Coefficients c0 , c1 , c3 (red/blue/green dots, respectively) and their dressed
counterparts (dots joined by a line), for different values of β c0 = 6 c0 /g02 (TILW actions).
8.2. The method
119
• DBW2 action:
Finally, the DBW2 gluon action [106] corresponds to c2 =c3 =0, and β-dependent values for
c0 , c1 . Some standard values for c0 and c1 (obtained starting from β c0 = 6.0 and 6.3), as
well as c̃0 and c̃1 are shown in Table 8.1.
0
0
0
0
0
0
-0.0128098
-0.0134070
-0.0142442
-0.0147710
-0.0154645
-0.0163414
0
0
0
0
0
c̃1
c̃3
ou
0
0
0
-0.083333
-0.083333
-0.083333
-0.151791
-0.154846
-0.159128
-0.161827
-0.165353
-0.169805
-0.331
-0.331
-0.331
-1.4086
-0.53635
c̃0
0.6797
0.70507
0.74977
1.32778
1.33311
1.39048
1.81690
1.82811
1.84700
1.85801
1.87425
1.89626
2.59313
2.72733
2.88070
8.80696
3.39826
0
0
0
-0.05489
-0.05530
-0.05982
-0.09631
-0.09703
-0.09835
-0.09908
-0.10021
-0.10177
-0.17151
-0.18930
-0.21048
-0.7313
-0.22528
tin
1.0
1.0
1.0
1.6666667
1.6666667
1.6666667
2.3168064
2.3460240
2.3869776
2.4127840
2.4465400
2.4891712
3.648
3.648
3.648
12.2688
5.29078
c3
an
5.0
5.3
6.0
5.0
5.07
6.0
8.60/c0
8.45/c0
8.30/c0
8.20/c0
8.10/c0
8.00/c0
1.95
2.2
2.6
0.6508
1.1636
c1
st
c0
on
β
0
0
0
0
0
0
-0.00831
-0.00860
-0.00902
-0.00927
-0.00961
-0.01005
0
0
0
0
0
C
Action
Plaquette
Plaquette
Plaquette
Symanzik
Symanzik
Symanzik
TILW
TILW
TILW
TILW
TILW
TILW
Iwasaki
Iwasaki
Iwasaki
DBW2
DBW2
The improvement procedure in a nutshell
th
8.2.4
a
Table 8.1: Input parameters β, c0 , c1 , c3 , c̃0 , c̃1 , c̃3 (c2 = 0).
M
ar
The steps involved in the resummation of cactus diagrams can now be described quite
succinctly:
• Substitute gluon propagators in Feynman diagrams by their dressed counterparts.
The latter are obtained by the replacement ci → c̃i = g02 γ̃i , where γ̃i are the solutions
of Eqs. (8.32).
• Perform the same replacement, ci → c̃i , on the 3-gluon vertex.
8.3. Application: 1-loop renormalization of fermionic currents
120
• Account for dressing of the 3-point vertex from the clover action by adjusting the
clover coefficient cSW : cSW → cSW · (c̃0 /c0 ) .
• In dressing subleading-order diagrams, avoid double counting, i.e., subtract terms
which were included in dressing leading-order diagrams. These are very easy to
ou
identify and subtract: Writing a general subleading-order result (aside from an overall
prefactor) as: a/N 2 + b + c Nf /N (Nf : number of fermion flavors), the quantity to
subtract will include all of a/N 2 (because terms with BCH commutators are higher
tin
order in N), and it will be a multiple of (2N 2 − 3); thus subtraction boils down to
the substitution
2
2
a/N + b + c Nf /N → b + a + c Nf /N
(8.37)
3
Application: 1-loop renormalization of fermionic
st
8.3
an
(see Refs. [97, 107, 108, 109] for different applications of this). The remaining subleading vertices dress exactly as the propagators and 3-point vertices above.
on
currents
In the last section of this chapter, we turn to an application of cactus improvement:
C
The 1-loop renormalization of the vector (Vµ (x) ≡ Ψ̄(x)γµ Ψ(x)) and axial (Aµ (x) ≡
Ψ̄(x)γ5 γµ Ψ(x)) currents using the overlap action. We employ Symanzik improved gluons;
a
hence, our results (depending also on the overlap parameter ρ) are presented for various
sets of Symanzik coefficients and specific values of ρ. As mentioned in the introduction, a
different application of our improvement method is that of the additive mass renormaliza-
th
tion for clover fermions. Both the unimproved and improved calculations are presented in
the next chapter.
M
ar
The renormalization constants ZV , ZA are defined through
V S (p) ≡ ZVS V (a) ,
AS (p) ≡ ZAS A(a)
(8.38)
where S is the renormalization scheme, p is the external momentum for zero lattice spacing,
V S (p), AS (p) are the continuum operators and V (a), A(a) the lattice operators. Bare 1loop results for ZV , ZA have been computed in the literature [72, 20, 77] using overlap
action. Using this action, one can show that the renormalization constants ZV and ZA are
121
equal [72], due to chiral symmetry (preserved by overlap fermions). In the MS scheme the
constants read
ZV,A (a, p) = 1 − g02 z1V,A ≡ 1 − g02
CF
(bV,A + bΣ ) ,
16π 2
CF =
N2 − 1
2N
(8.39)
following the notation of Refs. [20, 77]. bV,A and bΣ are 1-loop results pertaining to the
an
tin
ou
amputated 2-point function of the current and to the fermion self-energy, respectively;
since ZV = ZA , we can write bV = bA .
st
Figure 8.6: 1-loop contribution to the amputated Green’s function (bV,A ).
a
C
on
There is only one diagram contributing to bV,A (Fig. 8.6), and 2 diagrams needed for the
calculation of bΣ (Fig. 8.7).
th
Figure 8.7: 1-loop contribution to the quark self-energy (bΣ ).
M
ar
Using cactus improvement, Eq. (8.39) becomes
dr
dr
ZV,A
(a, p) = 1 − g02 z1V,A
(8.40)
dr
To compute z1V,A
we dress the Symanzik coefficients and the propagators as described in
dr
the previous section. In Table 8.2 the values of ZV,A and ZV,A
are presented for different
sets of the Symanzik coefficients, choosing ρ = 1.0, ρ = 1.4. Systematic errors are too
dr
small to affect any of the digits appearing in the table. The dependence of ZV,A and ZV,A
on the overlap parameter ρ is more clearly shown in Fig. 8.8, where we plot our results
122
for three actions: Plaquette, Iwasaki and TILW. Note that improvement is more apparent
for the case of the plaquette action. Indeed, from Table 8.2 one can clearly see that the
effect of dressing is smaller for improved gluon actions. This, of course, could have been
expected, since these actions were constructed in a way as to reduce lattice artifacts, in
the first place.
tin
ou
dr
dr
ZV,A
(ρ=1.0) ZV,A (ρ=1.4) ZV,A
(ρ=1.4)
1.35247
1.14707
1.19615
1.29231
1.13574
1.16207
1.28735
1.13386
1.15932
1.23484
1.11311
1.13019
1.31941
1.15259
1.17690
1.32764
1.15613
1.18146
1.33677
1.16012
1.18651
1.34298
1.16282
1.18995
1.34972
1.16579
1.19369
1.35705
1.16908
1.19774
1.44921
1.21724
1.24847
1.38773
1.19256
1.21440
1.31940
1.16293
1.17656
1.45362
1.27543
1.25057
an
ZV,A (ρ=1.0)
1.26427
1.24502
1.24164
1.20418
1.27581
1.28223
1.28946
1.29434
1.29973
1.30569
1.39343
1.34872
1.29507
1.49631
st
β=6/g02
6.00
5.00
5.07
6.00
3.7120
3.6018
3.4772
3.3985
3.3107
3.2139
1.95
2.20
2.60
0.6508
on
Action
Plaquette
Symanzik
Symanzik
Symanzik
TILW
TILW
TILW
TILW
TILW
TILW
Iwasaki
Iwasaki
Iwasaki
DBW2
C
dr
Table 8.2: Results for ZV,A , ZV,A
(Eq. (8.39), (8.40)) using ρ=1.0, ρ=1.4 .
a
A comparison between our improved ZV,A values and some nonperturbative estimates
[114], shows that improvement moves in the right direction. Cactus dressing had already
M
ar
th
been tested using clover fermions [100], and it turns out to be as good as standard tadpole
improvement [99], but still not very close to nonperturbative results.
8.4. Dressing QED
123
2.0
dr
ZV,A (Iwasaki, β=1.95)
1.9
ZV,A (Iwasaki, β=1.95)
1.8
ZV,A (Plaquette, β=6.0)
dr
dr
ZV,A (TILW, β⋅c0=8.45)
1.7
ZV,A (TILW, β⋅c0=8.45)
ZV,A (Plaquette, β=6.0)
ou
1.6
1.5
tin
1.4
1.3
1.1
1.0
0.6
0.8
1.0
1.2
an
1.2
1.4
1.6
1.8
2.0
st
ρ
8.4
on
dr
Figure 8.8: Plots of ZV,A and ZV,A
for the plaquette, Iwasaki and TILW actions. Labels
have been placed in the same top-to-bottom order as their corresponding curves.
Dressing QED
C
Cactus improvement can be easily carried over to Lattice Quantum Electrodynamics. In
a
this case, link variables commute; hence the first order BCH formula is exact and dressing
includes the full contribution of diagrams with cactus topology.
Dressing the propagator now proceeds precisely as in Eq. (8.29). The only difference is
th
that the result of contracting
n−2 out of n generators of SU(N): n F (n; N)/(N 2 −1), must
n
now be replaced by:
(n − 3)!! (the number of ways to pair n−2 out of n objects); this
2
results in:
M
ar
αi
=
ci
∞
X
n=4,6,8,...
1 (i g0 )n n (n−2)/2
−βi g02 /2
(n
−
3)!!
2
β
=
1
−
e
i
g02 n!
2
2
⇒ c̃i ≡ ci − αi = ci e−βi g0 /2
(8.41)
(βi are the c̃-dependent integrals defined in Eqs. (8.21)).
As before, the 4 coupled equations (8.41) for c̃i assume their simplest form in the case
8.5. Discussion
124
of the plaquette action (c0 = 1, c1 = c2 = c3 = 0); in this case, β0 = 1/(2c̃0 ) and we obtain
2
c̃0 = e−g0 /(4c̃0 ) ,
c̃1 = c̃2 = c̃3 = 0
(8.42)
Dressing vertices is simpler than in the nonabelian case. For a bare (2m)-point vertex,
(0)
(1)
(2)
(3)
denoted as: V2m = c0 V2m + c1 V2m + c2 V2m + c3 V2m , instead of contracting group genera-
=
i=0
3
X
(i)
V2m ci
∞ X
(i g0 )2l+2m
(2m)!
2l + 2m
l
βi
(2l − 1)!!
(2l+2m)!
(i g0 )2m
2m
l=0
(i)
V2m ci
2
e−βi g0 /2 =
i=0
3
X
i=0
(i)
V2m c̃i
tin
=
3
X
(8.43)
an
dr
V2m
ou
tors,
one must
simply count the number of distinct pairings of 2l objects out of (2l+2m):
2l + 2m
(2l − 1)!!. The dressed vertex becomes
2m
8.5
Discussion
on
st
The above relations allow us to summarize the dressing procedure for QED very briefly:
• Replace ci by c̃i (as given in Eq. (8.41)) throughout
• Omit all diagrams which contain any cactus subdiagram, to avoid double counting
In closing, we remark that resummation of cactus diagrams is readily applicable to any ob-
a
C
servable in lattice gauge theories. This procedure for improving bare perturbation theory
is gauge invariant, and can be applied in a systematic fashion to improve (to all orders)
results obtained at any given order in perturbation theory. Another positive feature which
th
reveals the simplicity of our method is the fact that the resummation procedure is applied
by replacing the action parameters (coupling constant, Symanzik coefficients, clover co-
M
ar
efficients) by dressed values. In cases where the results are given as a polynomial of the
parameters mentioned above, the improvement can directly be applied in the bare results,
with no need of additional calculations. This leads to both human and computer time
saving.
The computation of the dressed renormalization factors ZV,A and the comparison with
available nonperturbative results, shows that the improvement moves in the right direction
and it is as good as the Lepage-Mackenzie tadpole improvement.
ou
Chapter 9
tin
Two-loop additive mass
renormalization with clover fermions
Introduction
st
9.1
an
and Symanzik improved gluons
on
Here, we calculate the additive renormalization of the fermion mass in Lattice QCD, using
clover fermions and Symanzik improved gluons. The calculation is carried out up to 2
loops in perturbation theory and it is directly related to the determination of the critical
C
value of the hopping parameter, κc .
The clover fermion action [3] (SW) successfully reduces lattice discretization effects and
approaches the continuum limit faster. This justifies the extensive usage of this action in
th
a
Monte Carlo simulations in recent years. The coefficient cSW appearing in this action is a
free parameter for the current work and our results will be given as a polynomial in cSW .
Regarding gluon fields, we employ the Symanzik improved action [10], which also aims
M
ar
at minimizing finite lattice spacing effects. For the coefficients parameterizing the Symanzik
action, we consider several choices of values which are frequently used in the literature.
The lattice discretization of fermions introduces some well known difficulties; demanding strict locality and absence of doublers leads to breaking of chiral symmetry. In order to
recover this symmetry in the continuum limit one must set the renormalized fermion mass
(mR ) equal to zero. To achieve this, the mass parameter m◦ appearing in the Lagrangian
must approach a critical value mc , which is nonzero due to additive renormalization.
The mass parameter m◦ is directly related to the hopping parameter κ used in simula125
9.1. Introduction
126
tions. Its critical value, κc , corresponds to chiral symmetry restoration
κc =
1
2 mc a + 8 r
(9.1)
1
mB ≡ mo − mc =
2a
1
1
−
κ κc
ou
where a is the lattice spacing and r is the Wilson parameter. Using Eq. (9.1), the nonrenormalized fermion mass is given by
(9.2)
tin
Thus, in order to restore chiral symmetry one must consider in simulations the limit mo →
mc . This fact points to the necessity of an evaluation of mc .
an
The perturbative value of mc is also a necessary ingredient in higher-loop calculations of
the multiplicative renormalization of operators (see, e.g., Ref. [115]). In mass independent
schemes, such renormalizations are typically defined and calculated at zero renormalized
mass, and this entails setting the value of the Lagrangian mass equal to mc .
The quantity which we study is a typical case of a vacuum expectation value resulting
on
st
in an additive renormalization; as such, it is characterized by a power (linear) divergence
in the lattice spacing, and its calculation lies at the limits of applicability of perturbation
theory. Previous studies of the hopping parameter and its critical value have appeared in
the literature for Wilson fermions - Wilson gluons [107] and for clover fermions - Wilson
gluons [108, 116]. The procedure and notation in our work is the same as in the above
C
references.
Our results for κc (and consequently for the critical fermion mass) depend on the
number of colors (N) and on the number of fermion flavors (Nf ). Besides that, there is
th
a
an explicit dependence on the clover parameter cSW which, as mentioned at the beginning,
is kept as a free parameter. On the other hand, the dependence of the results on the
choice of Symanzik coefficients cannot be given in closed form; instead, we present it in a
M
ar
list of Tables and Figures. In order to compare our results to nonperturbative evaluations
of κc coming from Monte Carlo simulations, we employ an improved perturbation theory
method for improved actions.
In Section 9.2 we formulate the problem and describe our calculation of the necessary
Feynman diagrams. Section 9.3 is a presentation of our results. Finally, in Section 9.4
we apply to our 1- and 2-loop results an improvement method, proposed by our group
[117, 118, 101]. This method resums a certain infinite class of subdiagrams, to all orders in perturbation theory, leading to an improved perturbative expansion. We end this
9.2. Formulation of the problem
127
section with a comparison of perturbative and nonperturbative results. Our findings are
summarized in Section 9.5.
9.2
Formulation of the problem
We employ the Wilson formulation of the QCD action on the lattice, with Nf flavors of
ou
degenerate clover (SW) [3] fermions (Eq. (2.20)). Regarding gluons, we use the Symanzik
improved gauge field action (Eq. (2.26)), involving Wilson loops with 4 and 6 links and
four parameters, c0 , c1 , c2 , c3 . In the full action we have to include the ghost (Eq. 2.14)
tin
and the measure terms (Eq. 2.15).
The bare fermion mass mB must be set to zero for chiral invariance in the classical
an
continuum limit. Terms proportional to r in the action, as well as the clover terms, break
chiral invariance. They vanish in the classical continuum limit; at the quantum level, they
induce nonvanishing, flavor-independent fermion mass corrections. Numerical simulation
algorithms usually employ the hopping parameter,
1
2 mo a + 8 r
st
κ≡
(9.3)
on
as an adjustable input. Its critical value, at which chiral symmetry is restored, is thus 1/8r
classically, but gets shifted by quantum effects.
The renormalized mass can be calculated in textbook fashion from the fermion self–
C
energy. Denoting by ΣL (p, mo , g) the truncated, one particle irreducible fermion 2-point
function, we have for the fermion propagator
◦
−1
i p/ + m(p) − ΣL (p, mo , g)
1 X
2r X 2 µ
◦
p/ =
γµ sin(apµ ), m(p) = m◦ +
sin (ap /2)
a µ
a µ
a
S(p) =
th
where :
(9.4)
M
ar
To restore the explicit breaking of chiral invariance, we require that the renormalized mass
vanish
S
−1
(0)
=0
m◦ → mc
=⇒
mc = ΣL (0, mc , g)
(9.5)
The above is a recursive equation for mc , which can be solved order by order in perturbation
theory.
We denote by dm the additive mass renormalization of m◦ : mB = m◦ − dm. In
128
order to obtain a zero renormalized mass, we must require mB → 0, and thus m◦ → dm.
Consequently,
mc = dm = dm(1−loop) + dm(2−loop)
(9.6)
The tree level value of the critical mass is zero, mc = 0.
2
tin
1
ou
Two diagrams contribute to dm(1−loop) , shown in Fig. 9.1. In these diagrams, the
fermion mass must be set to its tree level value, mo → 0.
an
Figure 9.1: 1-loop diagrams contributing to dm(1−loop) . Wavy (solid) lines represent gluons
(fermions).
st
The quantity dm(2−loop) receives contributions from a total of 26 diagrams, shown in
Fig. 9.2. Genuine 2-loop diagrams must again be evaluated at mo → 0; in addition,
on
one must include to this order the 1-loop diagram containing an O(g 2 ) mass counterterm
(diagram 23). Certain sets of diagrams, corresponding to 1-loop renormalization of propagators, must be evaluated together in order to obtain an infrared convergent result: These
M
ar
th
a
C
are diagrams 7+8+9+10+11, 12+13, 14+15+16+17+18, 19+20, 21+22+23.
129
5
6
7
9
10
11
12
13
15
16
17
21
22
8
ou
4
14
an
tin
3
19
20
24
25
26
C
23
28
a
27
on
st
18
M
ar
th
Figure 9.2: 2-loop diagrams contributing to dm(2−loop) . Wavy (solid, dotted) lines represent
gluons (fermions, ghosts). Crosses denote vertices stemming from the measure part of the
action; a solid circle is a fermion mass counterterm.
The calculation of a diagram is similar to the procedure described in Chapter 7. Here
are summarized the required steps:
• The appropriate vertices and the ‘incidence’ matrix are specified for the diagram
under study, so that the contraction is performed.
• The color dependence, Dirac matrices and tensor structures are simplified and sym-
9.3. Computation and results
130
metries of the theory (permutation symmetry, lattice rotational invariance) are used,
to minimize the length of the expression.
• For the particular study, the external momentum is set to zero by definition of the
critical mass calculation (Eq. (9.5)) and there are no logarithmic contributions.
ou
• Each diagram is a sum of trigonometric products (sines, cosines) of the loop momenta
and it is convenient to bring it into a compact and canonical form.
• We then numerically integrate over the internal momenta using our ‘integrator’ rou-
tin
tine, which generates a highly optimized Fortran code. The final expression for most
of the diagrams consists of many thousand terms and they cannot be integrated in
an
a single Fortran program. In such cases, we split the expression into parts of approximately 2000 terms and integrate separately each one of them. Of course, these
are added together right after the integration. The numerical results are given for
lattices of varying finite size L; for 1-loop diagrams: 4 ≤ L ≤ 128, while for 2-loop
diagrams: 4 <= L <= 40. They are also given for different values of the Symanzik
st
coefficients.
on
• The remaining step, is the extrapolation of our results to lattice with infinite size,
L → ∞. As it is well known, the extrapolation introduces systematic errors that
are calculated quite accurately. The infrared divergent diagrams mentioned above
(7-11, 12-13, 14-18, 19-20, 21-23) must be considered as a group, and then proceed
Computation and results
th
9.3
a
C
with the extrapolation. The particularity of diagram 23 is the mass counterterm that
contains; we multiply it by the 1-loop result of the critical mass and then add it to
diagrams 21 and 22.
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Given that the dependence of mc on the Symanzik coefficients ci cannot be expressed
in closed form, we chose certain sets of values for ci , presented in Table I, which are
in common use [119, 103, 10, 104, 105, 106]: Plaquette, Symanzik (tree level improved),
Tadpole Improved Lüscher-Weisz (TILW), Iwasaki and DBW2. Actually, since the gluon
propagator contains only the combinations C1 and C2 (Eq. (2.30)), all results for mc
can be recast in terms of C1 , C2 and one additional parameter, say, c2 ; in this case the
dependence on c2 (at fixed C1 , C2 ) is polynomial of second degree.
131
The contribution dml of the lth 1-loop diagram to dm, can be expressed as
2
dml =
(N 2 − 1) 2 X i
(i)
g ·
cSW εl
N
i=0
(9.7)
(i)
can be written in the form
(N 2 − 1) 4 X i
(i,j,k)
g ·
cSW N j ck2 el
2
N
i,j,k
(9.8)
tin
dml =
ou
where εl are numerical 1-loop integrals whose values depend on C1 , C2 . The dependence
on cSW is seen to be polynomial of degree 2 (i = 0, 1, 2).
The contribution to dm from 2-loop diagrams that do not contain closed fermion loops,
an
where the index l runs over all contributing diagrams, j = 0, 2 and k = 0, 1, 2 (since up to
two vertices from the gluon action may be present in a Feynman diagram). The dependence
(i,j,k)
st
on cSW is now polynomial of degree 4 (i = 0, · · · , 4). The coefficients el
(as well as
(i)
ẽl of Eq. (9.9) below) are 2-loop numerical integrals; once again, they depend on C1 , C2 .
Finally, the contribution to dm from 2-loop diagrams containing a closed fermion loop, can
be expressed as
4
on
X
(N 2 − 1)
(i)
dml =
Nf g 4 ·
ciSW ẽl
N
i=0
(9.9)
where the index l runs over diagrams 12-13, 19-20. Summing up the contributions of all
X
l
dml =
(N 2 − 1) 2 X i
(N 2 − 1) 4 X i
g ·
cSW ε(i) +
g ·
cSW N j ck2 e(i,j,k)
2
N
N
i
i,j,k
a
dm =
C
diagrams, dm assumes the form
X
(N 2 − 1)
Nf g 4 ·
ciSW ẽ(i)
N
i
th
+
(9.10)
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ar
In the above, ε(i) , e(i,j,k), ẽ(i) are the sums over all contributing diagrams of the quantities:
(i)
(i,j,k)
(i)
εl , el
, ẽl , respectively (cf. Eqs. (9.7), (9.8), (9.9) ).
The coefficients ε(i) lead to the total contribution of 1-loop diagrams. Their values are
listed in Table 9.2, for the ten sets of ci values shown in Table 9.1. Similarly, results for
the coefficients e(i,j,k) and ẽ(i) corresponding to the total contribution of 2-loop diagrams,
are presented in Tables 9.3-9.7.
In order to enable cross-checks and comparisons, numerical per-diagram values of the
(i)
(i,j,k)
constants εl , el
132
(i)
and ẽl are presented in Tables 9.8 - 9.12, for the case of the Iwasaki
action. For economy of space, several vanishing contributions to these constants have
simply been omitted. A similar breakdown for other actions can be obtained from the
authors upon request.
dmIwasaki
(1−loop)
dmDBW2
(1−loop)
2
= g −0.434285489(1)+0.1159547570(3) cSW+0.0482553833(1) cSW
(9.11)
= g 2 −0.2201449497(1)+0.0761203698(3) cSW+0.0262264231(1) c2SW (9.12)
= g 2 −0.0972070995(5)+0.0421775310(1) cSW+0.01141359801(1) c2SW (9.13)
2
tin
dmPlaquette
(1−loop)
ou
The total contribution of 1-loop diagrams, for N = 3 can be written as a function of
the clover parameter cSW . In the case of the Plaquette, Iwasaki, and DBW2 actions, we
find, respectively
Nf = 3 :
4
dmPlaquette
(2−loop) = g
− 0.1255626(2) + 0.0203001(2) cSW + 0.00108420(7) c2SW
− 0.00116538(2) c3SW − 0.0000996725(1) c4SW
(9.14)
− 0.1192361(2) + 0.0173870(2) cSW + 0.00836498(8) c2SW
− 0.00485727(3) c3SW − 0.0011561947(4) c4SW
(9.15)
C
=g
4
st
Nf = 2 :
dmPlaquette
(2−loop)
=g
4
on
Nf = 0 :
dmPlaquette
(2−loop)
an
A similar process can be followed for 2-loop diagrams. In this case, we set N = 3, c2 = 0
and we use three different values for the flavor number: Nf = 0, 2, 3. Thus, for the
Plaquette, Iwasaki and DBW2 actions, the total contribution is, respectively
− 0.1160729(2) + 0.0159305(2) cSW + 0.0120054(1) c2SW
3
4
− 0.00670321(3) cSW − 0.0016844558(6) cSW
(9.16)
th
a
M
ar
Nf = 0 :
dmIwasaki
(2−loop)
=g
4
Nf = 2 :
4
dmIwasaki
(2−loop) = g
Nf = 3 :
4
dmIwasaki
(2−loop) = g
− 0.0099523(2) − 0.0024304(5) cSW − 0.00232855(4) c2SW
− 0.00032100(2) c3SW − 0.0000419365(1) c4SW
(9.17)
− 0.0076299(2) − 0.0040731(5) cSW + 0.00102758(6) c2SW
− 0.00242924(3) c3SW − 0.000457690(2) c4SW
(9.18)
− 0.0064687(2) − 0.0048944(5) cSW + 0.00270565(7) c2SW
3
4
− 0.00348335(3) cSW − 0.000665567(2) cSW
(9.19)
4
dmDBW2
(2−loop) = g
Nf = 2 :
dmDBW2
(2−loop)
Nf = 3 :
dmDBW2
(2−loop)
=g
4
=g
4
+ 0.005099(2) − 0.0053903(7) cSW − 0.0011157(1) c2SW
− 0.00004482(2) c3SW − 0.0000111470(2) c4SW
(9.20)
+ 0.005944(2) − 0.0061840(7) cSW + 0.0002046(2) c2SW
− 0.0010177(3) c3SW − 0.000125065(3) c4SW
(9.21)
+ 0.006366(2) − 0.0065809(7) cSW + 0.0008648(2) c2SW
− 0.0015042(4) c3SW − 0.000182023(5) c4SW
(9.22)
ou
Nf = 0 :
133
tin
In Figs. 9.3, 9.4 and 9.5 we present the values of dm(2−loop) for Nf = 0, 2, 3, respectively;
the results are shown for all choices of Symanzik actions which we have considered, as a
function of cSW (N = 3, c2 = 0). In all cases, the dependence on cSW is rather mild. One
an
observes that dm(2−loop) is significantly smaller for all improved actions, as compared to
the plaquette action; in particular, in the case of DBW2, dm(2−loop) is closest to zero and
st
it vanishes exactly around cSW = 1.
Another feature of these results is that they change only slightly with Nf , especially
in the range cSW < 1.5 . This is due to the small contributions of diagrams with closed
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th
a
C
on
fermion loops (diagrams 12, 13, 19, 20). By the same token, in the case of nondegenerate
flavors, dm(2−loop) is expected to depend only weakly on the mass of the virtual fermion.
134
0.02
0.00
-0.06
ou
-0.04
-0.08
-0.10
tin
DBW2
Iwasaki
TILW,β=8.00
TILW,β=8.10
TILW,β=8.20
TILW,β=8.30
TILW,β=8.45
TILW,β=8.60
Symanzik
Plaquette
-0.12
-0.14
0.0
0.5
1.0
an
dm2-loop / g4
-0.02
1.5
2.0
st
cSW
on
Figure 9.3: Total value of dm to 2 loops, for N = 3, Nf = 0 and c2 = 0. Legends appear
in the same top-to-bottom order as the corresponding lines.
C
0.02
-0.04
-0.06
DBW2
Iwasaki
TILW,β=8.00
TILW,β=8.10
TILW,β=8.20
TILW,β=8.30
TILW,β=8.45
TILW,β=8.60
Symanzik
Plaquette
M
ar
dm2-loop / g4
th
-0.02
a
0.00
-0.08
-0.10
-0.12
0.0
0.5
1.0
1.5
2.0
cSW
9.4. Improved perturbation theory
135
0.02
0.00
ou
-0.04
-0.08
-0.10
-0.12
0.0
tin
DBW2
Iwasaki
TILW,β=8.00
TILW,β=8.10
TILW,β=8.20
TILW,β=8.30
TILW,β=8.45
TILW,β=8.60
Symanzik
Plaquette
-0.06
0.5
1.0
an
dm2-loop / g4
-0.02
1.5
2.0
st
cSW
9.4
on
Improved perturbation theory
C
We now apply our method of improving perturbation theory [117, 118, 101], based on
a
resummation of an infinite subset of tadpole diagrams, termed ‘cactus’ diagrams. In Ref.
[101] we show how this procedure can be applied to any action of the type we are considering
here, and it provides a simple, gauge invariant way of dressing, to all orders, perturbative
th
results at any given order (such as the 1- and 2-loop results of the present calculation).
Some alternative ways of improving perturbation theory have been proposed in Refs. [98,
M
ar
99]. In a nutshell, our procedure involves replacing the original values of the Symanzik and
clover coefficients by improved values, which are explicitly computed in [101]. Applying at
first this method to 1-loop diagrams, the improved (“dressed”) value dmdr of the critical
mass (N = 3, c2 = 0) can be written as
dmdr
(1−loop)
=
2
X
i=0
(i)
εdr ciSW
(9.23)
136
(i)
In comparing with ε(i) of Eq. (9.10), the quantity εdr is the result of 1-loop Feynman diagrams with dressed values for the Symanzik parameters, and it has already been multiplied
(i)
by g 2 (N 2 − 1) /N. The dependence of εdr on g is quite complicated now, and cannot be
(i)
given in closed form; instead εdr must be computed numerically for particular choices of g.
(i)
ou
Listed in Table 9.13 are the results for εdr along with the value of β = 2N/g 2 corresponding
to each one of the 16 actions used in this calculation.
An attractive feature of this improvement procedure is that it can be applied also
to higher loop perturbative results, with due care to avoid double counting of the cactus
diagrams which were already included at one loop. Ideally, of course, one loop improvement
tin
should already be adequate enough, so as to obviate the need to consider higher loops;
indeed, we find this to be the case and, consequently, we limit our discussion of 2-loop
improvement to only the plaquette action (β = 5.29, N = 3, Nf = 2), the Iwasaki action
an
(β = 1.95, N = 3, Nf = 2) and the DBW2 action (β = 0.87 and β = 1.04, N = 3, Nf = 2).
Using these values, the contribution to dmdr
(2−loop) is a polynomial in c SW
st
2
dmdr
(2−loop), plaquette = −0.77398(8) + 0.16330(4) cSW + 0.06224534(1) cSW
(9.24)
−0.00767090(8) c3SW − 0.001160923(1) c4SW
(9.25)
−0.0044006(9) c3SW − 0.00073780(6) c4SW
on
2
dmdr
(2−loop), Iwasaki = −0.0813302(9) + 0.043030(3) cSW + 0.0308196(2) cSW
C
2
dmdr
(2−loop), DBW2(β=0.87) = −0.044906(1) + 0.029449(4) cSW + 0.0239522(2) cSW
−0.0082231(1) c3SW − 0.001218955(4) c4SW
(9.26)
th
a
2
dmdr
(2−loop), DBW2(β=1.04) = −0.031260(1) + 0.021793(2) cSW + 0.0188027(2) cSW
−0.00705284(9) c3SW − 0.001055657(1) c4SW
(9.27)
M
ar
dr
The comparison between the total dressed contribution dmdr = dmdr
(1−loop) + dm(2−loop)
and the unimproved contribution, dm, for the plaquette action is exhibited in Fig. 9.6,
as a function of cSW . Similarly, dmdr for the Iwasaki and the DBW2 actions is shown in
Fig. 9.7 and Fig. 9.8, respectively.
137
-0.1
-0.2
-0.3
dm
ou
-0.4
-0.5
dmdr
-0.7
-0.8
0.0
1.0
cSW
1.5
an
0.5
tin
-0.6
2.0
on
st
Figure 9.6: Improved and unimproved values of dm up to 2 loops, as a function of cSW , for
the plaquette action (β = 5.29, N = 3, Nf = 2).
-0.1
-0.3
dm
a
-0.4
C
-0.2
-0.5
th
-0.6
dmdr
M
ar
-0.7
-0.8
-0.9
0.0
0.5
1.0
cSW
1.5
2.0
the Iwasaki action (β = 1.95, N = 3, Nf = 2).
9.5. Discussion
138
0.0
-0.1
-0.2
dmdr
-0.3
dm
-0.4
dm
-0.6
-0.7
0.0
1.0
cSW
1.5
an
0.5
tin
-0.5
ou
dmdr
2.0
on
st
the DBW2 action (N = 3, Nf = 2). We set β = 0.87 (solid lines) and β = 1.04 (dotted
lines).
Finally, in Table 9.14, we present a comparison of dressed and undressed results, for
C
some commonly used values of β, Nf , cSW , and we also compare with available nonperturbative estimates for κc [111, 110, 112, 120, 113]. We observe that improved perturbation
theory, applied to 1-loop results, already leads to a much better agreement with the non-
Discussion
th
9.5
a
perturbative estimates.
M
ar
To recapitulate, we have calculated the critical mass mc , and the associated critical hopping
parameter κc , up to 2 loops in perturbation theory, using the clover action for fermions
and the Symanzik improved gluon action with 4- and 6-link loops. The perturbative
value of mc is a necessary ingredient in the higher-loop renormalization of operators, in
mass independent schemes: Such renormalizations are typically defined and calculated at
vanishing renormalized mass, which amounts to setting the Lagrangian mass equal to mc .
In our calculations, we have chosen for the Symanzik coefficients ci a wide range of
Tables
139
values, which are most commonly used in numerical simulations. The dependence of our
results on the number of colors N and the number of fermion flavors Nf is shown explicitly.
The dependence on the clover parameter cSW is in the form of a fourth degree polynomial
whose coefficients we compute explicitly; it is expected, of course, that the most relevant
ou
values for cSW are those optimized for O(a) improvement, either at tree level (cSW = 1),
or at one loop [3], or nonperturbatively [111].
Since mc is gauge invariant, we chose to calculate it in the Feynman gauge. The
propagator appearing in Feynman diagrams is the inverse of a nondiagonal matrix; while
this inverse can be written down explicitly, it is more convenient, and more efficient in
tin
terms of CPU time, to perform the inversion numerically. Integrations over loop momenta
were performed as momentum sums on lattices of finite size L, where typically L <
∼ 40;
extrapolation to L → ∞ introduces a systematic error, which we estimate quite accurately.
an
Our results for mc are significantly closer to zero in the case of Symanzik improved
actions, as compared to the plaquette action. In particular, the DBW2 action stands
st
out among the rest, in that mc vanishes exactly for a value of cSW around 1. Thus,
improved actions seem to bring us quite near the point of chiral symmetry restoration.
The dependence of mc on the number of flavors is seen to be very mild. This fact would
on
also suggest that, in the case of nondegenerate flavors, mc should depend only weakly on
the mass of the virtual fermion.
Finally, we have made some comparisons among perturbative and nonperturbative
C
results for κc . While these are expected to differ for a power divergent additive renormalization, such as the quantity under study, we nevertheless find a reasonable agreement.
a
This agreement is further enhanced upon using an improved perturbative scheme, which
entails resumming, to all orders in the coupling constant, a dominant subclass of tadpole
diagrams. The method, originally proposed for the Plaquette action (see Ref. [117]), was
th
extended in Ref. [101] to encompass all possible gluon actions made of closed Wilson loops,
and can be applied at any given order in perturbation theory. As would be desirable, 1-loop
M
ar
improvement is seen to be already adequate to give a reasonable agreement among perturbative and nonperturbative values. Indeed, our results for κdr
1−loop are significally closer
to the nonperturbative evaluations, as shown in Table 9.14; in fact, the 2-loop dressing
procedure introduces no further improvement to the comparison.
Tables
140
= 8.60
= 8.45
= 8.30
= 8.20
= 8.10
= 8.00
1.0
1.6666667
2.3168064
2.3460240
2.3869776
2.4127840
2.4465400
2.4891712
3.648
12.2688
c1
c3
0
-0.083333
-0.151791
-0.154846
-0.159128
-0.161827
-0.165353
-0.169805
-0.331
-1.4086
0
0
-0.0128098
-0.0134070
-0.0142442
-0.0147710
-0.0154645
-0.0163414
0
0
tin
Plaquette
Symanzik
TILW, βc0
TILW, βc0
TILW, βc0
TILW, βc0
TILW, βc0
TILW, βc0
Iwasaki
DBW2
c0
ou
Action
Action
ε
ε(1)
ε(2)
0.0434830339(1)
0.0378314931(2)
0.03408560232(6)
0.0339409375(1)
0.0337409869(2)
0.0336166372(1)
0.03345591621(5)
0.03325593631(8)
0.0285451387(1)
0.01581657412(5)
0.01809576875(4)
0.01476335801(5)
0.01265991972(4)
0.01258108895(1)
0.012472434543(4)
0.0124050416(1)
0.012318127134(5)
0.012210297749(7)
0.00983490867(5)
0.004280099253(2)
on
st
-0.1628570582(5)
-0.12805490528(8)
-0.10821568768(4)
-0.10749185625(3)
-0.1064962872(3)
-0.1058799831(2)
-0.1050866191(1)
-0.10410447893(3)
-0.08255435613(4)
-0.0364526623(2)
C
Plaquette
Symanzik
TILW (8.60)
TILW (8.45)
TILW (8.30)
TILW (8.20)
TILW (8.10)
TILW (8.00)
Iwasaki
DBW2
(0)
an
Table 9.1: Input parameters c0 , c1 , c3 .
(0,2,0)
e
M
ar
Action
th
a
Table 9.2: Total contribution of 1-loop diagrams.
Plaquette
Symanzik
TILW (8.60)
TILW (8.45)
TILW (8.30)
TILW (8.20)
TILW (8.10)
TILW (8.00)
Iwasaki
DBW2
e(1,2,0)
e(2,2,0)
-0.01753602(2) 0.00259963(2) -0.000155894(8)
-0.00810366(1) 0.00095046(2) -0.000404510(9)
-0.00437013(7) 0.00019403(5) -0.00045894(1)
-0.00425575(7) 0.00016978(6) -0.00045962(1)
-0.00410086(7) 0.00013682(7) -0.00046040(1)
-0.00400636(6) 0.00011666(8) -0.00046080(1)
-0.00388630(6) 0.00009097(9) -0.00046123(1)
-0.00374009(6) 0.00005958(9) -0.000461601(9)
-0.00112957(2) -0.00052964(6) -0.000436966(5)
0.0008481(2) -0.00085301(8) -0.00018540(1)
e(3,2,0)
e(4,2,0)
-0.000163242(2)
-0.000107348(2)
-0.000078117(3)
-0.000077102(3)
-0.000075713(3)
-0.000074857(3)
-0.000073760(3)
-0.000072410(3)
-0.000045009(3)
-0.000006164(3)
-0.00001721759(2)
-0.00001275904(1)
-0.00001020820(1)
-0.00001011451(1)
-0.00000998564(1)
-0.00000990584(1)
-0.00000980314(1)
-0.00000967600(1)
-0.00000682353(1)
-0.00000173502(3)
Table 9.3: Total 2-loop contribution to dm of order O(N 2 , c02 ).
Tables
141
Plaquette
0.01656633(2) -0.00055904(1)
Symanzik
0.00605656(1) 0.000935801(6)
TILW (8.60) 0.00202637(3) 0.00157890(3)
TILW (8.45) 0.00190729(3) 0.00159800(3)
TILW (8.30) 0.00174666(3) 0.00162375(2)
TILW (8.20) 0.00164901(3) 0.00163939(2)
TILW (8.10) 0.00152532(3) 0.00165917(2)
TILW (8.00) 0.00137535(4) 0.00168310(3)
Iwasaki
-0.00103022(1) 0.00203254(1)
DBW2
-0.0018961(2)
0.0016130(3)
e(2,0,0)
e(3,0,0)
e(4,0,0)
0.002622771(7)
0.002120980(9)
0.001790242(9)
0.001777415(9)
0.001759689(9)
0.001748661(9)
0.001734421(9)
0.00171671(1)
0.001313076(3)
0.000413397(9)
0.000158125(2)
0.000104973(2)
0.000076167(2)
0.000075164(3)
0.000073791(3)
0.000072944(3)
0.000071859(3)
0.000070522(3)
0.000043949(3)
0.000005057(3)
0.00004282674(2)
0.00002971553(1)
0.00002260669(1)
0.00002235603(1)
0.00002201243(1)
0.00002180041(1)
0.00002152826(1)
0.00002119259(1)
0.00001423324(1)
0.00000307480(3)
ou
e(1,0,0)
e
tin
(0,0,0)
Action
ẽ(1)
0.00118621(2)
0.00081496(1)
0.00063643(1)
0.00063033(1)
0.00062198(1)
0.00061684(1)
0.00061025(1)
0.00060214(1)
0.00043546(1)
0.00015833(3)
-0.000546197(8)
-0.000448276(6)
-0.000389464(5)
-0.000387269(5)
-0.000384243(5)
-0.000382366(5)
-0.000379946(5)
-0.000376945(5)
-0.00030800(1)
-0.00014883(4)
ẽ(2)
ẽ(3)
ẽ(4)
0.001365146(9)
0.001041379(8)
0.000857737(3)
0.000851127(3)
0.000842047(3)
0.000836433(3)
0.000829214(4)
0.000820289(4)
0.000629274(8)
0.00024756(2)
-0.000692228(3)
-0.000574521(3)
-0.000500011(5)
-0.000497194(5)
-0.000493307(5)
-0.000490894(5)
-0.000487781(4)
-0.000483915(4)
-0.000395294(3)
-0.00018242(5)
-0.00019809791(7)
-0.0001453370(2)
-0.0001148491(1)
-0.0001137544(1)
-0.0001122515(1)
-0.0001113227(1)
-0.0001101288(1)
-0.0001086536(1)
-0.0000779538(3)
-0.0000213595(6)
on
a
Plaquette
Symanzik
TILW (8.60)
TILW (8.45)
TILW (8.30)
TILW (8.20)
TILW (8.10)
TILW (8.00)
Iwasaki
DBW2
C
ẽ
st
(0)
Action
an
Table 9.4: Total 2-loop contribution to dm of order O(N 0 , c02 ).
th
Table 9.5: Total contribution of 2-loop diagrams containing closed fermion loops.
(0,0,1)
e(1,0,1)
e(2,0,1)
0.077167(3)
0.034929(2)
0.020247(1)
0.019816(1)
0.019235(1)
0.018881(1)
0.018433(1)
0.017888(1)
0.0087615(7)
0.0007907(2)
-0.019808(3)
-0.010895(2)
-0.007117(2)
-0.006998(2)
-0.006835(2)
-0.006736(2)
-0.006609(2)
-0.006454(2)
-0.003656(1)
-0.0004889(3)
-0.0085415(2)
-0.0041454(2)
-0.0024559(1)
-0.0024050(1)
-0.0023362(1)
-0.0022942(1)
-0.0022410(1)
-0.0021762(1)
-0.00107856(8)
-0.00008343(2)
M
ar
Action
Plaquette
Symanzik
TILW (8.60)
TILW (8.45)
TILW (8.30)
TILW (8.20)
TILW (8.10)
TILW (8.00)
Iwasaki
DBW2
e
e(0,2,1)
e(1,2,1)
-0.047102(4)
0.010439(3)
-0.017940(2)
0.004491(2)
-0.008702(1)
0.002251(1)
-0.008448(1)
0.002185(1)
-0.0081078(6) 0.0020973(9)
-0.0079023(7) 0.002044(1)
-0.0076431(9) 0.0019761(8)
-0.0073300(6) 0.0018940(6)
-0.0027484(4) 0.0006646(5)
0.0001308(2) -0.0001587(3)
Table 9.6: Total contribution of 2-loop diagrams containing the parameter c2 (part 1).
a
th
M
ar
tin
an
st
on
C
ou
Tables
141
Tables
142
(2,2,1)
e(0,2,2)
e(1,2,2)
e(2,2,2)
-0.0000842143(1)
-0.0000454986(1)
-0.00002872341(6)
-0.00002818123(6)
-0.00002744385(5)
-0.00002699223(5)
-0.00002641646(5)
-0.00002571231(5)
-0.00001249281(2)
-0.00000050404(9)
-0.09448252(9)
-0.03417549(2)
-0.017374635(6)
-0.016917713(6)
-0.016304614(5)
-0.015933835(5)
-0.015466270(5)
-0.014902324(4)
-0.00596123(2)
-0.00028731(2)
0.02755993(3)
0.01248953(1)
0.007205477(3)
0.007049188(2)
0.006838088(3)
0.006709626(4)
0.006546741(4)
0.006348924(5)
0.00295502(1)
0.00020317(4)
0.010521016(1)
0.0041047891(2)
0.0021218443(2)
0.0020666192(2)
0.0019924047(3)
0.0019474604(2)
0.0018907121(3)
0.0018221643(3)
0.0007286816(4)
0.0000278810(8)
an
Plaquette
0.0039245(3)
Symanzik
0.0014622(1)
TILW (8.60) 0.0006472(1)
TILW (8.45) 0.0006251(1)
TILW (8.30) 0.0005954(1)
TILW (8.20) 0.0005775(1)
TILW (8.10) 0.0005550(1)
TILW (8.00) 0.0005279(1)
Iwasaki
0.00015719(6)
DBW2
-0.00002436(1)
ou
e(3,2,1)
e
tin
Action
(i)
i
(i)
ε2
-0.05602636832(2) -0.02652798781(3)
0
0.0285451387(1)
0
0.00983490867(5)
on
0
1
2
ε1
st
Table 9.7: Total contribution of 2-loop diagrams containing the parameter c2 (part 2).
j
k
(i,j,k)
e3
(i,j,k)
e4
(i,j,k)
e6
th
i
a
C
Table 9.8: Contribution of 1-loop diagrams, for the Iwasaki action.
0
2
0
2
0
2
0
0
0
0
0
0
M
ar
0
0
1
1
2
2
-0.0003923686(9) -0.000743134(3) -0.0000714882(8)
0.0002615791(6) 0.000495422(2)
0.0000357441(4)
0
0.001900337(2)
0
0
0.0017774410(9) 0
0
-0.0010339720(2) 0
0
-0.001041123(1)
0.0002799238(4)
Table 9.9: Contribution of diagrams 3, 4, 6, for the Iwasaki action.
Tables
143
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
3
3
0
0
2
2
2
0
0
2
2
2
0
0
2
2
2
2
2
0
1
0
1
2
0
1
0
1
2
0
1
0
1
2
0
1
(i,j,k)
(i,j,k)
e7−11
(i,j,k)
e14−18
0.00042802(1)
0.0057103(7)
-0.00111995(2)
-0.0022472(3)
-0.00371263(2)
0
0
0
0
0
0
0
0
0
0
0
0
(i,j,k)
e24
-0.000195263(2)
0
0.0030512(2)
0
-0.00029748(1)
0
-0.0008718(2)
0
-0.00224859(1)
0
0.00064534(1)
0
-0.003656(1)
0
0.00011079(6)
-0.000144897(2)
0.0006450(5)
0.000248682(4)
0.00295502(1)
0
-0.000000974(1)
0
-0.00107856(8)
0
0.000141960(3)
0.000042314(2)
0.00039546(6)
0.00002909398(7)
0.0007286816(4) 0
0
0
0
0
e26
0
0
-0.000298742(2)
0.0003705893(7)
0
0
0
0.000429899(1)
-0.00022905(1)
0
0
0
0.0003303085(7)
-0.000267364(2)
0
-0.000019835(1)
-0.00001249281(2)
ou
k
tin
j
an
i
(i)
i
(i)
ẽ12−13
ẽ19−20
0.000261920(6)
0.000173538(9)
-0.0000308339(1) -0.00027717(1)
0.000370942(2)
0.000258332(8)
0
-0.000395294(3)
0
-0.0000779538(3)
a
C
0
1
2
3
4
on
st
Table 9.10: Contribution of diagrams 7-11, 14-18, 24, 26, for the Iwasaki action.
M
ar
th
Table 9.11: Contribution of diagrams 12, 13, 19, 20, for the Iwasaki action.
i
j
k
0
0
1
1
2
2
3
3
4
4
0
2
0
2
0
2
0
2
0
2
0
0
0
0
0
0
0
0
0
0
(i,j,k)
e21−23)
(i,j,k)
e25
(i,j,k)
e27
(i,j,k)
e28
0.000373419(3)
-0.000158621(4) -0.000094848(3)
-0.0001759336(5)
-0.000373419(3)
0.000079311(2)
0
0.0000879668(3)
-0.000887295(1)
0.0001396819(4) 0.000045158(4)
0.0001893113(5)
0.000887295(1)
0.000085189(2)
0
-0.000120480(1)
0.000194437(1)
-0.0000319392(3) 0.000168506(2)
-0.0000509266(2)
-0.000194437(1)
-0.000005787(2)
0
0.0000098758(1)
0.000059183(3)
0
-0.000015234(1)
0
-0.000059183(3)
0.0000172022(5) 0
0.0000168072(6)
0.00000682353(1) 0
0.000007409712(6) 0
-0.00000682353(1) 0
0
0
Table 9.12: Contribution of diagrams 21-23, 25, 27, 28, for the Iwasaki action.
Tables
144
(1)
εdr
(2)
-0.579221119(2)
-0.4869797578(8)
-0.478756110(2)
-0.3915226522(2)
-0.5358770348(7)
-0.5497415338(3)
-0.5651407386(9)
-0.5756111531(9)
-0.5870122772(4)
-0.599415804(1)
-0.757856451(1)
-0.6555102085(5)
-0.541348980(1)
-0.7749943512(7)
-0.574781578(1)
-0.4822863343(9)
0.1159547570(3)
0.1121369999(4)
0.11072412996(5)
0.0947962001(5)
0.1265917638(3)
0.1291104644(3)
0.1319263769(1)
0.1337937558(7)
0.1358437825(6)
0.138085996(2)
0.1671007819(8)
0.1537748193(6)
0.1359882440(3)
0.1847244889(1)
0.1575688409(9)
0.1412499230(5)
0.03618067788(9)
0.03538605357(4)
0.03507238306(5)
0.03124138429(9)
0.03813963851(4)
0.0386337113(1)
0.0391695069(1)
0.03951713046(7)
0.0398899143(3)
0.0402877133(4)
0.044746728234(1)
0.04293183656(3)
0.03967626495(6)
0.04731717866(3)
0.04281261980(1)
0.039186543574(5)
ou
6.00
5.00
5.07
6.00
3.7120
3.6018
3.4772
3.3985
3.3107
3.2139
1.95
2.20
2.60
0.6508
0.8700
1.0400
εdr
tin
Plaquette
Symanzik
Symanzik
Symanzik
TILW (8.60)
TILW (8.45)
TILW (8.30)
TILW (8.20)
TILW (8.10)
TILW (8.00)
Iwasaki
Iwasaki
Iwasaki
DBW2
DBW2
DBW2
(0)
εdr
an
β
st
Action
Plaquette
Plaquette
0
0
Plaquette
β
6.00
6.00
2
5.29
2
0
0
2
2
2
2
1.95
3.7120
3.2139
0.87
0.87
1.04
1.04
M
ar
th
Iwasaki
TILW (8.60)
TILW (8.00)
DBW2
DBW2
DBW2
DBW2
cSW
κ1−loop
κ2−loop
κdr
1−loop
κdr
2−loop
κnon−pert
[Ref.]
c
1.479
1.769
0.1301
0.1275
0.1335
0.1306
0.1362
0.1337
0.1362
0.1332
0.1392
0.1352
1.9192 0.1262
0.1307
0.1353
0.1341
1.53
1.0
1.0
0.0
1.0
0.0
1.0
0.1368
0.1370
0.1387
0.1384
0.1372
0.1375
0.1348
0.1388
0.1378
0.1397
0.1460
0.1379
0.1421
0.1352
0.1379
0.1384
0.1406
0.1479
0.1379
0.1434
0.1352
C
Nf
a
Action
on
Table 9.13: Results for dmdr
(1−loop) (Eq. (9.23)), with N = 3.
0.1292
0.1339
0.1348
0.1502
0.1352
0.1454
0.1334
Table 9.14: 1- and 2-loop results, and nonperturbative estimates for κc .
[110]
[111]
0.1373 [112]
0.1363 [120]
0.1421 [113]
ou
Chapter 10
Introduction
an
10.1
tin
O(a2) improvements
It is well known that the lattice formulation of QCD leads to systematic errors in the
st
results obtained from numerical simulations, due to the finiteness of the lattice spacing
a. A solution to this problem is to use O(a) improved actions, so that the lattice spacing
affects the results to O(a2 ). Although this is the most popular way out, it only improves
on
on-shell quantities, for instance the hadron masses; action improvement is not sufficient
to remove O(a) terms from operator matrix elements, due to their short-distance contact
C
terms. The off-shell improvement is accomplished by adding to each operator higher dimensional ‘irrelevant’ terms with appropriately chosen coefficients and then normalizing
these operators.
a
For many years, computations were performed using only on-shell correlation functions,
avoiding the off-shell quantities. However, it is desirable to employ off-shell improvement
for various reasons. For instance, in the non-perturbative renormalization proposed in
th
Ref. [121], continuum perturbative results are compared with lattice off-shell Green’s functions. In this way, one can relate lattice measurements to renormalization schemes like
M
ar
MS. It is preferable to keep the discretization errors down to a minimum before comparing
with the continuum results, by improving the lattice quantities. The effect of such an
improvement, that we study here, is the removal of all corrections that in the continuum
limit are effectively of order a or a2 . No O(a2 ) 1-loop computations exist to date.
In this chapter we present a 1-loop perturbative calculation of the quantum corrections
to the fermion propagator and to a complete basis of local fermion bilinear currents, using
massless fermions described by the clover action (Eq. (2.20)). We also use a 3-parameter
145
10.2. Improvement to the fermion propagator
146
family of Symanzik improved gluon actions (Eq. (2.26)), which comprises all common gluon
actions (Wilson, Iwasaki, DBW2, Lüscher-Weisz, etc.). We provide results up to O(a2 )
with the dependence on the gauge parameter shown explicitly. The terms of order a2 can
be used to specify the required modifications of the quark operators in order to achieve
ou
O(a2 ) improvement. This is a project we have been working on for the last few months and
will continue beyond the completion of this Thesis. In the following section we provide our
results for the fermion propagator and in the last section we discuss the O(a2 ) corrections
to matrix elements of the quark operators that have the form Ψ̄ΓΨ, where Γ denotes all
possible distinct products of Dirac matrices (Eq. (10.31)). We consider both flavor singlet
Improvement to the fermion propagator
an
10.2
tin
and nonsinglet operators.
The fermion propagator is the most common example of an off-shell quantity suffering from
st
O(a) effects. Capitani et al. [122] have calculated the first order terms in the lattice spacing
for massive fermions, and as a first attempt we repeated this computation; our results are
on
in perfect agreement with those of Ref. [122]. We carried out this calculation beyond the
first order correction, taking into account all terms up to O(a2 ). The clover coefficient cSW
has been considered to be a free parameter and our results are given as a polynomial of
cSW . Moreover, the dependence on the number of colors N, the coupling constant g and
the gauge fixing parameter λ, is shown explicitly. The Symanzik coefficients, ci , appear in
th
a
C
a nontrivial way and thus, the dependence of the propagator on ci cannot be given in a
closed form; we tabulate these results for different choices of ci .
1
2
M
ar
Figure 10.1: 1-loop diagrams contributing to the improvement of the fermion propagator.
Wavy (solid) lines represent gluons (fermions).
The 1-loop Feynman diagrams that enter our 2-point Green’s function calculation, are
shown in Fig. 10.1, where solid lines denote fermion fields and wavy lines represent gluon
fields. The procedure of calculating each diagram is, in many ways, similar to the one
described in Section 3 of Chapter 7:
147
• Wick contraction of the vertices, followed by several simplification of the color dependence, the Dirac matrices and the tensor structures. We fully exploit the symmetries of the
theory (periodicity, reflection, conjugation, hypercubic, etc.) to contain the proliferation
on the algebraic expressions.
ou
• Extraction of all functional dependencies on the external momentum p ((sub)-divergent,
convergent terms) and the lattice spacing (terms of order a0 , a, a2 ). The isolation of the
logarithmic and non-Lorentz invariant terms requires careful manipulation, with subtrac-
the exact equalities
∆D (q) =
4
4
µ
sin (qµ /2) − 4
P
µ
2
sin (qµ /2)
2 q̃ 2 q̂ 2
)
o
2 sin(qµ /2) 2 sin(qν /2) n
δµν
µν
−
(1
−
λ)
+
∆D
(q)
q̂ 2
(q̂ 2 )2
st
µν
P
n
o
σν
δ σν
µσ 2
µσ ∆D (q)
δ q̂ − G
+
δ q̂ − G
(q̂ 2 )2
q̂ 2
µσ 2
µσ
on
D µν (q) =
an
1
1
= 2+
2
q̃
q̂
(
tin
tions among the propagators so that the divergent part (initially depending on the fermion
and the Symanzik propagator) is written in terms of the Wilson gluon propagator, using
(10.1)
(10.2)
(10.3)
where q̃ 2 is the denominator of the fermion propagator
◦2
qµ +
rX 2
q̂
M+
2 µ µ
,
◦
q µ = sin(qµ ),
q̂ 2 = 4
X
µ
sin2 (
qµ
)
2
(10.4)
a
µ
!2
C
q̃ 2 =
X
th
D µν is the Symanzik propagator, and Gµν (∆D µν (q)) is given by Eq. (2.28) (Eq. (2.32)).
Terms in curly brackets are less IR divergent by two powers in the momentum. The
arguments of the propagators in the above relations can be either k or k + a p (k (p): loop
M
ar
(external) momentum).
The convergent terms can be treated by Taylor expansion in a p to the desired order.
Instead, the extraction of the a p dependence is performed using iteratively the subtraction
trick
f (k + a p) = f (k) + (f (k + a p) − f (k))
(10.5)
that leads to the following exact relations
k^
+ ap
2
1
k\
+ ap
2
P
1
µ sin(2kµ + a pµ ) sin(a pµ )
=
−
2
k̃ 2
k^
+ a p k̃ 2
P
P
P
a pµ
a pµ
2 kν
2 kν +a pµ
4 µ sin(kµ + 2 ) sin( 2 )
)
ν sin ( 2 ) +
ν sin (
2
−
2
k^
+ a p k̃ 2
=
1
k̂ 2
−
4
P
µ
a pµ
) sin( a 2pµ )
2
2
k\
+ a p k̂ 2
sin(kµ +
(10.6)
ou
1
148
(10.7)
tin
Here we should point out that for dimensional reasons, there is a global prefactor 1/a
multiplying our expressions, and thus, the O(a2 ) correction is achieved by considering all
an
terms up to O(a3 ).
• Conversion of the algebraic expressions for the loop integrands (a total of ∼ 40,000 terms)
st
into highly optimized Fortran code for numerical integration. The integration over the internal momentum was performed on different finite lattices (44 ≤ lattice size ≤ 1284 ) and
for 10 popular sets of the Symanzik coefficients corresponding to the Plaquette, Symanzik,
on
Iwasaki, TILW and DBW2 action. Their values are listed in Table 9.1.
• Extrapolation of the numerical results to infinite lattice size. This procedure entails a
Basic divergent integrals
a
10.2.1
C
systematic error, which is reliably estimated, using a sophisticated inference technique; for
1-loop quantities we expect a fractional error smaller that 10−8 .
th
This evaluation differs from the rest of our calculations, because we must augment our
programs for automatic manipulations of Feynman diagrams, with procedures for handling higher order corrections in a. Even though 1-loop computations are considered to be
M
ar
relatively easy, the present one had a lot of complications. In particular, for the manipulation of the logarithmic and the non-Lorentz invariant terms, we analytically evaluated an
extensive basis of superficially divergent Feynman integrals (Eqs. (10.8) - (10.17)). This
computation was performed in a noninteger number of dimensions d ≥ 4, where ultraviolet
(sub-)divergences are explicitly isolated à la Zimmermann and evaluated; the remainders
are d−dimensional parameter-free lattice integrals which can be recast in terms of Bessel
functions, and finally expressed as sums of a pole part plus known numerical constants. A
149
few of these integrals were calculated in Ref. [123]. Integrals in Eqs. (10.8), (10.9), (10.12)
are the most demanding in the list, because they must be evaluated to two further orders
in a, beyond the order at which an IR divergence initially sets in; as a consequence, their
evaluation requires going to d > 6 dimensions. These particular integrals are a sufficient
ou
basis for all integrals which can appear in any O(a2 ) 1-loop calculation; that is, any such
calculation can be recast in terms of the integrals (10.8), (10.9), (10.12), plus other integrals which are more readily handled. The correct way to evaluate (10.8), (10.9), (10.12)
−π
tin
π
P 4
1
d4 k
µ pµ
2
=
0.036678329075
+
0.0000752406(3)
p
+
2
(2π)4 k̂ 2 · k[
384π 2p2
+p
•
Z
π
h
d4 k sin(kµ )
2
2
2 = pµ −0.008655827648 − 0.0002215402(2) p + 0.000014360819 pµ
4
(2π) k̂ 2 · k[
+p
P 4
p2µ i
ln(p2 )
p2
µ pµ
−
+
1+
−
+ O(p5 )
(10.9)
2
2
2
768π p
32π
24 12
C
−π
(10.8)
on
•
Z
ln(p2 )
+ O(p4 )
16π 2
st
−
an
•
Z
was previously unknown in the literature, despite their central role in O(a2 ) calculation,
and this had prevented 1-loop calculations to O(a2 ) thus far. The set of these integrals are
P
listed below (k̂ 2 = 4 µ sin2 (kµ /2))
th
−π
h
d4 k sin(kµ ) sin(kν )
2
2
=
δ
µν 0.014966695116 − 0.001256484446 p − 0.001027789631 pµ
2
4
(2π)
k̂ 2 · k[
+p
a
π
M
ar
+
•
Z
π
−π
h
p2 ln(p2 ) i
ln(p2 ) i
−
+
p
p
0.003970508789
+ O(p4 )
µ
ν
192π 2
48π 2
(10.10)
h
d4 k sin3 (kµ )
2
2
=
p
µ −0.006184131744 + 0.001102333439 p − 0.00174224479 pµ
2
(2π)4 k̂ 2 · k[
+p
2
i
ln(p2 )
p
2
+
− + pµ + O(p5 )
(10.11)
2
64π
2
•
Z
π
−π
150
h
d4 k sin(kµ ) sin(kν )
2
2
2 = δµν 0.004327913824 + 0.00011077012(8) p + 0.000442830335 pµ
(2π)4 (k̂ 2 )2 · k[
+p
P
p4µ
p2µ
p2
ln(p2 )
1+
+
−
−
1536π 2 p2
64π 2
24 12
P 4
h
1
µ pµ
+pµ pν −0.0003788538(2) +
+
2
2
32π p
768π 2 (p2 )2
π
−π
ou
pµ pν pρ
+ O(p3 )
2
2
48π p
π
−π
C
Z
(10.14)
d4 k sin(kµ ) sin(kν ) sin(kρ ) sin(kσ )
ln(p2 ) =
(δ
p
p
)
0.000227848225
−
µν
ρ
σ
S
2
(2π)4
384π 2
(k̂ 2 )2 · k[
+p
+δµνρσ −0.001675948042 + 0.000186391491 p2 + 0.000410290033 p2µ
th
•
−π
P
P 4
4 pµ
sin
(
)
d4 k
µ
µ pµ
2
2
+ O(p4 )
2 = 0.004050096698 − 0.000107954163 p +
4
2 p2
(2π) (k̂ 2 )2 · k[
1024π
+p
(10.13)
a
•
π
on
st
+0.001027789631δµνρ pµ −
Z
(10.12)
h
ln(p2 ) i
d4 k sin(kµ ) sin(kν ) sin(kρ )
=
(δ
p
)
−0.000728769948
+
νρ µ S
2
(2π)4
192π 2
(k̂ 2 )2 · k[
+p
an
•
Z
p2µ
p2ν
ln(p2 ) i
−
+
+ O(p4 )
2
2
2
2
2
384π p
384π p
768π
tin
−
µ
M
ar
p2 ln(p2 )
2
+(δµν δρσ )S 0.001589337971 − 0.000245852737p +
768π 2
−0.000372782983(δµνρ pµ pσ )S − 0.000062130497(δµν δρσ p2µ )S
+
pµ pν pσ pτ
+ O(p4 )
64π 2 p2
(10.15)
•
Z
π
−π
d4 k
(2π)4
P
µ
sin4 ( k2µ ) sin(kν )
(k̂ 2 )2 · k[
+p
2
h
= pν −0.000800034900 − 0.000258089450 p2 − 0.000056815263 p2ν
P
sin4 ( p2ρ )
ln(p2 )
−
−
1280π 2p2
2560π 2
P
µ
µ
) sin(kν ) sin(kρ )
sin4 ( kµ +p
2
2
(k̂ 2 )2 · (k[
+ p )2
(10.16)
h
= δνρ 0.000400017450 + 0.000129044725p2
tin
−π
d4 k
(2π)4
i
p2
2
− pν
+ O(p5 )
2
P 4 i
P 4
2
2
p
p
p
ln(p
)
p
ν
ρ
µ
µ
µ pµ
( − 3p2ν ) +
+
+0.000351195932p2ν +
2
2
2
2
5120π 2
2560π p
5120π (p2 )2
an
•
π
ou
ρ
Z
151
p3ν pρ + p3ρ pν pν pρ ln(p2 )
+0.000334438049pν pρ −
+
+ O(p4 )
2560π 2 p2
5120π 2
(10.17)
st
In the above integrals, ( )S means sum over inequivalent permutations and p must be read
as the product a p. No summation over the indices µ, ν, ρ, σ, τ is implied, unless otherwise
on
stated.
C
As an example, let us describe the evaluation of Eq. (10.16)
π
−π
d4 k
(2π)4
P
a
A1 (a) ≡
Z
µ
sin4 ( k2µ ) sin(kν )
(10.18)
2
(k̂ 2 )2 · k\
+ ap
th
from which we must extract all terms up to O(a3 ), and thus the superficial degree of
divergence for A1 becomes -4. So, it is sufficient to evaluate this integral in D = 4 − 2ε
dimensions (ε < 0, |ε| < 1/2). The extraction of the desired powers of a is performed using
M
ar
the following subtractions
A1 (a) =
"
A1 (a) − A1 (a)
dA1 (a) −a
da a=0
a=0
dA1 (a) +a
a=0
da + A1 (a)
a=0
a d A1 (a) −
2 da2 2
2
a=0
a d A1 (a) +
2 da2 2
2
a=0
a d A1 (a) −
6 da3 3
3
a=0
a d A1 (a) +
6 da3 3
#
cont
3
a=0
(10.19)
152
The expression in the square brackets is UV finite and can be computed in the continuum
limit, by setting a → 0. The rest of the terms in the r.h.s. are lattice contributions, responsible for the appearance of logarithms and non-Lorentz invariant terms. The derivatives
appearing above are provided here
P
sin4 ( k2µ ) k ν
◦
µ
(10.20)
(k̂ 2 )3
Z π D P
4 kµ ◦
dA1 d k
µ sin ( 2 ) k ν
2ε
= (a κ)
D
da (k̂ 2 )4
−π (2π)
a=0
Z π D P
4 kµ ◦
d k
d2 A1 µ sin ( 2 ) k ν
2ε
= (a κ)
D
da2 (k̂ 2 )4
−π (2π)
a=0
−2
X
−2
X
◦
pρ k ρ
ρ
ou
−π
dD k
(2π)D
!
(10.21)
tin
a=0
π
p2ρ cos(kρ ) +
ρ
8 X
k̂ 2
ρ
◦
pρ k ρ
an
Z
2ε
A1 = (a κ)
2
!
(10.22)
Z π D P
4 kµ ◦
X
d A1 d k
24 X 2
◦
µ sin ( 2 ) k ν
2ε
3 ◦
=
(a
κ)
+2
p
k
+
pρ cos(kρ ) pσ k σ
ρ
ρ
3
D
2
4
2
da (k̂ )
k̂ ρ,σ
−π (2π)
ρ
a=0
!
48 X ◦ 3
pρ k ρ
(10.23)
−
(k̂ 2 )2
ρ
on
st
3
◦
where k µ ≡ sin(kµ ). The integrands of Eqs. (10.20) and (10.22) are odd under the trans-
C
formation k → −k and give zero upon integration. So, the calculation of A1 reduced to
the computation of the continuum part and Eqs. (10.21) - (10.23).
a
A. In the continuum part of Eq. (10.19) (square brackets) we change the integration
th
variable, k → a k and then take the limit a → 0. This is allowed only for integrals with
superficial degree of divergence equal to -4 (D = 4 − 2ε). We finally employ the formula
of Chetyrkin and Tkachov (Eq. (A.1) of Ref. [124])
Z
∞
!
P 4
dD k
pν p2 1 1 61
µ kµ kν
=
+
− γE + ln(4πκ2 /p2 )
(2π)D (k 2 )2 (k + p)2
16π 2 320 ε 15
!
p3ν 1 1 26
−
+
− γE + ln(4πκ2 /p2 )
16π 2 160 ε 15
P 4
pν
µ pµ
−
(10.24)
16π 2 80p2
M
ar
κ2ε
=
16
Acont
1
−∞
153
B. The computation of Eq. (10.21) involves Bessel functions and has no logarithmic contributions. This was to be expected since the superficial degree of divergence for this integral
is -2. The symmetry of the theory implies that the indices ν and ρ must be equal for a non
vanishing result. Moreover, one must take into account all possible values of the index µ,
as shown in the following relation
tin
ou
Z π D P
4 kµ ◦ 2
dA1 (a) d k
µ sin ( 2 ) k ν
2ε
= −2 pν (a κ)
D
da (k̂ 2 )4
−π (2π)
a=0
Z π D
d k sin4 ( k21 ) + (D − 1) sin4 ( k22 ) sin2 (k1 )
2ε
= −2 pν (a κ)
D
(k̂ 2 )4
−π (2π)
◦2
◦2
In the last equation, the product pν k ν was equivalently written as pν k 1 , since the result
Z
∞
dγe−γ x γ n =
Γ(n + 1)
xn+1
st
0
an
is the same for all values of ν. The solution of Eq. (10.21) can be found by introducing an
additional integration variable, due to the equality
In the formula above we substitute x by k̂ 2 and directly use it to rewrite Eq. (10.21) as
−π
dk1 −4 γ sin2 ( k1 ) 2
2
e
sin (k1 )
2π
a
π
th
+(D − 1)
Z
C
on
Z
dA1 (a) −2 pν (a κ)2ε ∞
=
dγ γ 3 ×
da
Γ(4)
0
a=0
"Z
D−1
Z π
π
dk1 −4 γ sin2 ( k1 ) 4 k1
dk2 −4 γ sin2 ( k2 )
2
2
2
e
sin ( ) sin (k1 )
e
2
−π 2π
−π 2π
Z
π
−π
dk2 −4 γ sin2 ( k2 ) 4 k2
2
e
sin ( )
2π
2
Z
π
−π
dk3 −4 γ sin2 ( k3 )
2
e
2π
D−2 #
The integrals over ki are expressed in terms of the modified Bessel I0 (2 γ) and I1 (2 γ)
M
ar
Z
dA1 (a) −2 pν (a κ)2ε ∞
dγ γ 3 ×
=
da Γ(4)
0
a=0
"
D−1 e−2 γ
1
3
3
−2 γ
e
I0 (2 γ)
I1 (2 γ) 1 + +
− I0 (2 γ) 1 +
4γ
γ 4 γ2
4γ
−2 γ
−2 γ #
e
e
1
D−2
+(D − 1) e−2 γ I0 (2 γ)
I1 (2 γ)
I0 (2 γ) − I1 (2 γ) 1 +
2γ
2
4γ
154
The integral over γ, being convergent, can be solved numerically in 4 dimensions, leading
to a final expression for Eq. (10.21)
dA1 (a) da a=0
= −0.000800034899585846 pν
(10.25)
ou
C. We now continue with the evaluation of the term with the third derivative (Eq. (10.23)).
The basic idea is the same as explained in part B, but with an essential difference: Some
of the terms appearing in this integral have degree of divergence equal to -4. We cannot
tin
perform numerical integration over γ in D = 4 because there is a pole in γ → ∞, and we
must find an alternative way to extract the logarithms. This is achieved with a subtraction
an
of the pole. This computation compared to the procedure described for the evaluation of
Eq. (10.21) is more complicated: (a) Many different convergent and divergent integrals
appear when taking all possible values of the indices that are summed, (b) a lot of trigono-
st
metrical simplifications are necessary, and (c) the indices of the momenta k and p must be
independent, so that we proceed with the numerical evaluation.
Here we only present in detail one of the superficially divergent integral, and then give
on
the final expression for Eq. (10.23). After manipulating the initial form for this integral as
explained above, we arrive in the following expression
Z π D
d A1 (a) 24(p2 pν − p3ν )
d k sin4 ( k21 )
2
2ε
(a κ)
= conv + 4 p pν −
D
da3 D−1
(k̂ 2 )4
−π (2π)
a=0
(D + 2)p3ν − 3 p2 pν
D−1
a
− 8
C
3
th
− 8 (D +
2)p3ν
2
(a κ)
− 3 p pν (a κ)
Z
2ε
π
−π
2ε
Z
π
−π
dD k sin4 ( k21 ) sin4 (k1 )
(2π)D
(k̂ 2 )6
dD k sin4 (k1 ) sin4 ( k22 )
(2π)D
(k̂ 2 )6
(10.26)
M
ar
where conv is a sum of various convergent integrals that are evaluated exactly like Eq. (10.21)
conv = 0.0003484123625292182 p2 pν − 0.001480904688657616 p3ν
while the rest three integrals are divergent. Next we present only the computation of the
last term of Eq. (10.26); the other terms are evaluated in the same manner
155
We define the integral
P1 = (a κ)
2ε
Z
π
−π
dD k sin4 (k1 ) sin4 ( k22 )
(2π)D
(k̂ 2 )6
and as previously, P1 can be expressed in terms of the modified Bessel functions I0 , I1 , and
I2
Z
∞
D−2
5
dγ γ (I0 (2 γ))
0
3 e−2 D γ
I2 (2 γ)
8 γ2
1
I0 (2 γ) − I1 (2 γ) 1 +
4γ
ou
(a κ)2ε
P1 =
Γ(6)
integration region into two parts
∞
dγ =
0
Z
1
dγ +
0
Z
∞
dγ
an
Z
tin
The integral over γ has a pole in γ → ∞, thus we proceed with the separation on the
1
where for γ ǫ [0, 1] the integral is evaluated numerically in 4 dimensions
0
1
3 e−4 γ
dγ γ (I0 (2 γ))
I2 (2 γ)
8 γ2
2
5
1
I0 (2 γ) − I1 (2 γ) 1 +
= 5.8496317 10−7
4γ
st
Z
on
1
Γ(6)
th
a
C
On the contrary, for the region [1, ∞) we must subtract the pole of the integrand, as shown
below
"
#
Z
−2 D γ
(a κ)2ε ∞
3
e
1
9
D−2
dγ γ 5 (I0 (2 γ))
I2 (2 γ) I0 (2 γ) − I1 (2 γ) 1 +
−
Γ(6) 1
8 γ2
4γ
4096 π 2 γ 6−ε
|
{z
}
D=4
Z
(a κ)2ε ∞
9
dγ γ 5
+
Γ(6) 1
4096 π 2 γ 6−ε
where the terms in square brackets are evaluated in D=4, giving −1.9884876 10−7. The last
term includes the logarithms of P1 , and its solution is analytical. The result is a function
M
ar
of ε and we perform a Taylor expansion in ε keeping terms up to O(ε0 ), that is
(a κ)2ε
Γ(6)
Z
∞
9
1
3
dγ γ
=−
2
6−ε
2
4096 π γ
16 π 640
5
1
1
+ ln(4 π κ a2 )
ε
156
Employing the same procedure to all divergent terms of Eq. (10.26), we finally find
d3 A1 (a) 1
1
2
2
+ ln(4 π κ a )
= p pν −0.0003271432211431204 +
da3 5120 π 2 ε
a=0
−0.0000110577592320742 −
1
5120 π 2
1
+ ln(4 π κ a2 )
ε
(10.27)
ou
+p3ν
10.2.2
tin
Finally, we substitute Eqs. (10.24), (10.25), (10.27) in Eq. (10.19) to derive the expression
of Eq. (10.16).
Results
an
Next, we provide the total expression for the inverse propagator S −1 as a function of
g, N, cSW , λ. The quantities ε(i,j) are numerical coefficients depending on the Symanzik
parameters; the index i denotes the power of the lattice spacing a that they multiply
th
a
C
on
st
a2
a
S −1 (p) = i 6 p + p2 − i 6 p3
2
6
i
g 2 CF h (0,1)
(0,2)
(0,3) 2
2 2
− i 6p
ε
−
4.792009568(6)
λ
+
ε
c
+
ε
c
+
λ
ln(a
p
)
SW
SW
16 π 2
h
i
2
1
(1,1)
2 g CF
(1,2)
(1,3) 2
2 2
ε
− 3.86388443(2) λ + ε
cSW + ε
cSW −
3 − 2 λ − 3 cSW ln(a p )
− ap
16 π 2
2
i
g 2 CF h (2,1)
1 (2,2)
(2,3) 2
(2,4)
2 2
− i a2 6 p3
ε
+
1.024635179(9)
λ
+
ε
c
+
ε
c
+
ε
−
λ
ln(a
p
)
SW
SW
16 π 2
6
h
2
g CF
ε(2,5) + 2.55131292(9) λ + ε(2,6) cSW + ε(2,7) c2SW
− i a2 p2 6 p
2
16 π
i
13
λ + cSW + c2SW
+ ε(2,8) −
ln(a2 p2 )
4 2
P 4
h
2
p
5 i
µ µ g CF
(2,9)
ε
−
λ
(10.28)
− i a2 6 p
p2 16 π 2
48
M
ar
where CF = (N 2 − 1)/(2N) and 6 p3 =
P
µ
γµ p3µ . The value λ = 1 (0) corresponds to the
Feynman (Landau) gauge. The parameters ε(i,j) are tabulated in Tables 10.1 - 10.4. We
observe that the O(a) logarithms as well as all terms multiplied by λ, are independent
of the Symanzik coefficients; on the contrary O(a2 ) logarithms have a mild dependence
P
on the Symanzik parameters. Several non-Lorentz invariant tensors ( µ p4µ , 6 p3 ) appear in
O(a2 ) correction terms, compatibly with hypercubic invariance. Finally, our O(a) results
for the Plaquette action, are in agreement with Eq. (37) of Ref. [122]).
157
To enable cross-checks and comparisons, the per-diagram contributions d1 (p), d2 (p) are
(i,1)
given below, with the numerical values for the coefficients ε̃j presented in Tables 10.5 10.6. The tadpole diagram 1 of Fig. 10.1 is free of logarithmic terms and independent of
cSW ; its final expression is
g 2 CF h (0,1)
(1,1)
2
i
6
p
ε̃
+
3.05026254(1)
λ
+
a
p
ε̃
+
1.52913127(1)
λ
1
1
16 π 2
i
(2,1)
+ i a2 6 p3 ε̃1 − 0.50971042(1) λ
(10.29)
ou
d1 (p) =
(i,1)
(i,1)
tin
where the numerical values for the Symanzik dependent coefficients ε̃1 are given in Table 10.5. The main contribution to the propagator correction comes from diagram 2, as
g 2 CF h (0,1)
(0,2)
(0,3) 2
2 2
i
6
p
ε̃
−
7.850272109(6)
λ
+
ε
c
+
ε
c
+
λ
ln(a
p
)
SW
2
SW
16 π 2 1
(1,1)
2
(1,2)
(1,3) 2
2 2
3 − 2 λ − 3 cSW ln(a p )
+ a p ε̃2 − 5.39301570(2) λ + ε
cSW + ε
cSW −
2
1 (2,1)
2 3
(2,2)
(2,3) 2
(2,4)
2 2
+ i a 6 p ε̃2 + 1.534345602(9) λ + ε
cSW + ε
cSW + ε
− λ ln(a p )
6
+ i a2 p2 6 p ε(2,5) + 2.55131292(9) λ + ε(2,6) cSW + ε(2,7) c2SW
13
(2,8)
2
2 2
+ ε
−
ln(a p )
λ + cSW + cSW
4 2
P 4
i
5
µ pµ
(2,9)
ε
−
λ
(10.30)
+ i a2 6 p
p2
48
a
C
on
st
d2 (p) =
an
can be seen from the following expression, with ε̃2 listed in Table 10.6. The remaining
terms with coefficients ε(i,j) are the same as in Eq. (10.28).
th
Using our results for the fermion propagator, we can compute the multiplicative renormalization function of the quark field (ZΨ ) which is required in order to relate the matrix
elements, as extracted numerically from lattice simulations, to the physical finite matrix
M
ar
elements. We are interested in the computation of these quantities in the two most widely
used renormalization schemes, MS and RI′ , and for a general covariant gauge. For this
future work, we will follow the procedure of Refs. [125, 115].
Our O(a2 ) corrected results can be applied to the twisted mass QCD by setting the
clover parameter equal to zero. This is useful since the twisted mass action (Chapter 4) is
intensively being studied by international scientific groups.
(0,2)
ε
-2.24886853(7)
-2.01542504(4)
-1.85472029(6)
-1.84838009(3)
-1.83959982(6)
-1.83412923(5)
-1.82704771(6)
-1.81821854(5)
-1.60101088(7)
-0.96082198(5)
(0,3)
ε
-1.39726711(7)
-1.24220271(2)
-1.13919759(2)
-1.13513794(1)
-1.12951598(5)
-1.12601312(2)
-1.12147952(3)
-1.11582732(3)
-0.97320689(3)
-0.56869876(4)
ou
(0,1)
ε
16.6444139(2)
13.02327272(7)
10.90082304(6)
10.82273528(9)
10.71525766(9)
10.6486809(1)
10.56292631(3)
10.45668970(6)
8.1165665(2)
2.9154231(2)
tin
Action
Plaquette
Symanzik
TILW, βc0 = 8.60
TILW, βc0 = 8.45
TILW, βc0 = 8.30
TILW, βc0 = 8.20
TILW, βc0 = 8.10
TILW, βc0 = 8.00
Iwasaki
DBW2
158
(1,1)
(1,2)
ε
-5.20234231(6)
-4.7529781(1)
-4.4316083(2)
-4.4186677(2)
-4.40071157(1)
-4.38950279(4)
-4.37497018(8)
-4.35681290(3)
-3.88883584(9)
-2.2646221(1)
(1,3)
ε
-0.08172763(4)
-0.075931174(1)
-0.07178771(1)
-0.07160078(1)
-0.071339052(3)
-0.07117418(3)
-0.070959405(2)
-0.070688697(3)
-0.061025650(8)
-0.03366740(1)
C
on
st
Action
ε
Plaquette
12.8269254(2)
Symanzik
10.69642966(8)
TILW, βc0 = 8.60 9.3381342(2)
TILW, βc0 = 8.45 9.2865455(1)
TILW, βc0 = 8.30 9.2153414(1)
TILW, βc0 = 8.20 9.17111769(1)
TILW, βc0 = 8.10 9.1140228(1)
TILW, βc0 = 8.00 9.0430829(2)
Iwasaki
7.40724287(1)
DBW2
3.0835163(2)
an
Table 10.1: The ε(0,i) coefficients of Eq. (10.28) for different actions.
th
a
Table 10.2: The ε(1,i) coefficients of Eq. (10.28) for different actions.
(2,1)
ε
-5.10464931(2)
-4.6761807(2)
-4.30504709(4)
-4.2903739(2)
-4.2700655(2)
-4.2574226(2)
-4.24104736(2)
-4.2206409(1)
-3.8352281(1)
-2.3937137(2)
M
ar
Action
Plaquette
Symanzik
TILW, βc0 = 8.60
TILW, βc0 = 8.45
TILW, βc0 = 8.30
TILW, βc0 = 8.20
TILW, βc0 = 8.10
TILW, βc0 = 8.00
Iwasaki
DBW2
(2,2)
(2,3)
ε
ε
0.02028705(5) 0.10348577(3)
0.05136635(6) 0.07865292(7)
0.05733870(8) 0.06695681(3)
0.05751390(9) 0.06651692(3)
0.05775197(7) 0.06590949(1)
0.05789811(5) 0.06553180(2)
0.05808114(5) 0.06504530(6)
0.05830392(9) 0.06444077(4)
0.08249970(7) 0.04192446(4)
0.1024452(2) -0.00343999(2)
(2,4)
ε
101/120
0.3447625905(1)
0.8468299712(1)
0.8469212812(1)
0.8470492593(1)
0.8471299589(1)
0.8472351894(1)
0.8473680083(1)
0.8539636810(1)
0.8939977071(1)
Table 10.3: The ε(2,i) coefficients of Eq. (10.28) for different actions (part1).
(2,5)
ε
-3.0455303(1)
-2.82592090(5)
-2.70488383(7)
-2.70084109(6)
-2.69534558(8)
-2.6919846(2)
-2.6876939(1)
-2.6824541(1)
-2.7092247(1)
-4.1942127(5)
(2,7)
ε
0.534320852(7)
0.49783419(2)
0.46915700(3)
0.467966296(9)
0.46630972(2)
0.46527307(3)
0.463925850(6)
0.462237852(9)
0.41846440(4)
0.23968038(4)
(2,8)
ε
59/240 0.24147089523(1)
0.23790881578(1)
0.23774989722(1)
0.23752715138(1)
0.23738675027(1)
0.23720333782(1)
0.23697175965(1)
0.22850572224(1)
0.17209414004(1)
(2,9)
-
ε
31/80
0.02916667(1)
0.02360188(1)
0.02335610(1)
0.02301162(1)
0.0227944(1)
0.02251115(1)
0.022153640(1)
0.004400000(1)
0.103360000(1)
ou
(2,6)
ε
0.70358496(5)
0.65343092(3)
0.62190916(4)
0.62061757(5)
0.61882111(4)
0.61769697(3)
0.61623801(3)
0.61441084(7)
0.55587473(6)
0.34886590(2)
tin
Action
Plaquette
Symanzik
TILW, βc0 = 8.60
TILW, βc0 = 8.45
TILW, βc0 = 8.30
TILW, βc0 = 8.20
TILW, βc0 = 8.10
TILW, βc0 = 8.00
Iwasaki
DBW2
159
(0,1)
on
(i,1)
th
a
Table 10.5: The dependence of ε̃1
M
ar
Action
Plaquette
Symanzik
TILW, βc0 = 8.60
TILW, βc0 = 8.45
TILW, βc0 = 8.30
TILW, βc0 = 8.20
TILW, βc0 = 8.10
TILW, βc0 = 8.00
Iwasaki
DBW2
(1,1)
ε̃1
4.5873938103(5)
3.535587351(2)
2.951597245(2)
2.930489282(8)
2.901480060(5)
2.883535318(1)
2.860450351(1)
2.831895997(2)
2.211832365(2)
0.93454384(2)
st
ε̃1
9.174787621(1)
7.071174701(5)
5.903194489(4)
5.86097856(2)
5.80296012(1)
5.767070636(3)
5.720900703(3)
5.663791993(4)
4.423664730(5)
1.86908767(4)
C
Action
Plaquette
Symanzik
TILW, βc0 = 8.60
TILW, βc0 = 8.45
TILW, βc0 = 8.30
TILW, βc0 = 8.20
TILW, βc0 = 8.10
TILW, βc0 = 8.00
Iwasaki
DBW2
an
Table 10.4: The ε(2,i) coefficients of Eq. (10.28) for different actions (part2).
(0,1)
(Eq. (10.29)) on the Symanzik parameters.
ε̃2
7.4696262(2)
5.95209802(7)
4.99762855(6)
4.96175672(9)
4.91229754(9)
4.8816102(1)
4.84202561(3)
4.79289770(6)
3.6929018(2)
1.0463355(2)
(i,1)
Table 10.6: The dependence of ε̃2
(2,1)
ε̃1
-1.5291312701(2)
-1.1785291169(8)
-0.9838657482(6)
-0.976829761(3)
-0.967160020(2)
-0.9611784393(4)
-0.9534834504(5)
-0.9439653322(7)
-0.7372774550(8)
-0.311514612(6)
(1,1)
ε̃2
8.2395316(2)
7.16084231(8)
6.3865370(2)
6.3560562(1)
6.3138614(1)
6.287582370(9)
6.2535725(1)
6.2111869(2)
5.19541051(1)
2.1489724(2)
(2,1)
ε̃2
-3.57551804(2)
-3.4976516(2)
-3.32118134(4)
-3.3135441(2)
-3.3029055(2)
-3.2962441(2)
-3.28756391(2)
-3.2766755(1)
-3.0979507(1)
-2.0821991(2)
(Eq. (10.30)) on the Symanzik parameters.
10.3. Improved operators
10.3
160
Improved operators
In the context of this work we also study the O(a2 ) improvement of the fermion operators
that have the form Ψ̄ΓΨ. Γ corresponds to the following set of products of the Dirac
matrices
1
σµν = [γµ , γν ]
2
Γ = 1̂, γ 5 , γµ , γµ γ 5 , σµν γ 5 ,
(10.31)
ou
for the scalar (OS ), pseudoscalar (OP ), vector (OV ), axial (OA ) and tensor (OT ) operator,
respectively. We restrict ourselves only to local operators, and forward matrix elements
(2-point Green’s functions) due to the requirement of momentum conservation.
an
tin
The Feynman diagrams that correspond to the calculation of the above operators are
shown in Fig. 10.2. A cross stands for the insertion of the appropriate Dirac matrices
(Eq. (10.31)) that correspond to the operator under study.
2
st
1
C
on
3
4
5
Figure 10.2: 1-loop diagrams contributing to the improvement of the bilinear operators.
Wavy (solid) lines represent gluons (fermions). A cross denotes the Dirac matrices 1̂
(scalar), γ 5 (pseudoscalar), γµ (vector), γµ γ 5 (axial) and σµν γ 5 (tensor).
a
Alternatively, one can write the general form for the improved operators based, for
instance, on symmetry constraints. In particular, for O(a) improvement, Capitani et al.
th
provide the expressions for the bilinear operators using Wilson fermions, which for massless
quarks and considering terms relevant only to forward matrix elements are
imp
=
M
ar
OS
O
P imp
V imp
Oµ
OµA
T
Oµν
imp
imp
=
=
=
=
imp
↔
1
= Ψ̄Ψ − a k1 Ψ̄ D
6 Ψ
2
imp
Ψ̄γ5 Ψ
= Ψ̄γ5 Ψ
imp
↔
1
Ψ̄γµ Ψ
= Ψ̄γµ Ψ − a k1 Ψ̄D µ Ψ
2
imp
↔
1
Ψ̄γµ γ5 Ψ
= Ψ̄γµ γ5 Ψ − a i k1Ψ̄σµλ γ5 D λ Ψ
2
↔
imp
↔
1
Ψ̄σµν γ5 Ψ
= Ψ̄σµν γ5 Ψ + a i k1 Ψ̄ γµ D ν −γν D µ γ5 Ψ
2
Ψ̄Ψ
(10.32)
(10.33)
(10.34)
(10.35)
(10.36)
10.3. Improved operators
↔
→
161
←
→
←
where D ≡ D − D is the symmetric covariant derivative, with D , D defined in Eqs. (2.7) (2.10). The terms of the form Ψ̄ΓΨ represent local operators, while the terms proportional
to k1 are extended operators. The latter involve also diagrams 6 - 8 of Fig. 10.3. In the
same manner, we can add terms of order a2 and the improvement of all the above operators
can be expressed in a single equation, that is
= Ψ̄ΓΨ + a
n
X
′
k1i
Ψ̄ Qi1
2
Ψ+a
i=1
n
X
k2i Ψ̄ Qi2 Ψ
i=1
(10.37)
ou
O
imp
tin
where the first term is the unimproved operator and ΨQk1 Ψ (ΨQk2 Ψ) are operators with the
same symmetries as the original, but dimension one (two) higher. To eliminate all O(a2 )
an
terms, we will evaluate Ψ̄ΓΨ to O(a2 ), Ψ̄ Qi1 Ψ to O(a) and Ψ̄ Qi2 Ψ to O(a0 ). Then one
must choose k1i and k2i appropriately, in order to cancel out the O(a) and O(a2 ) terms.
on
st
6
7
8
Figure 10.3: 1-loop diagrams contributing to the extended operators. Wavy (solid) lines
represent gluons (fermions) and a cross denotes the Dirac matrices of Eq. (10.31).
⋆⋆⋆⋆⋆
We are currently using the strategy we employed for the propagator, to compute O(a2 )
a
C
corrections for matrix elements of the aforementioned quark operators. Our final results
will be given as a polynomial of the clover parameter, in a general covariant gauge. Since
their dependence on the Symanzik coefficients, ci , cannot be written in a closed form, we
M
ar
th
will tabulate our results for a variety of choices for ci , in order to cover the whole range of
values that are used in both perturbative calculations and numerical simulations.
ou
Chapter 11
tin
Conclusions
an
In this last chapter we discuss the conclusions of the Thesis, where we have used Lattice Quantum Chromodynamics as a tool for perturbative calculations regarding the high
st
energy regime of the strong interactions. Each computation was performed employing improved actions for both fermions (overlap, clover action) and/or gluons (Symanzik action).
In the main part of this Thesis we have presented the following computations:
on
• In Chapter 7 we describe the evaluation of the 2-loop coefficient of the coupling
renormalization function Zg , for the gauge group SU(N) and Nf species of overlap
C
fermions. For an economical and convenient computation we employed the background field technique. This is the first 2-loop calculation using overlap fermions
with external momenta, and it proved to be extremely demanding in human and
CPU time; due to the fact that overlap action involves complicated expressions.
th
a
Our numerical results of Zg have been used for the determination of the 3-loop coefficient bL2 of the bare lattice β-function. Thus, we provided the asymptotic dependence
of the bare coupling constant g0 on the lattice spacing a up to O(g07). Knowledge of
M
ar
bL2 provides the correction term to the standard asymptotic scaling relation between
a and g0 . The dependence of Zg and bL2 on N and Nf is shown explicitly in our
expressions, which are presented for a large set of values of the overlap parameter ρ
in its allowed range.
The 3-loop correction term to β-function is seen to be rather small: This indicates
that the perturbative series is very well behaved in our calculation, despite the fact
that it is only asymptotic in nature. Furthermore, around the values of ρ which are
most often used in simulations, fermions bring about only slight corrections to the
162
11. Conclusions
163
3-loop β-function, even compared to pure gluonic contributions.
The only source of numerical error in our results has its origin in an extrapolation
to infinite lattice size. An intermediate range for ρ (0.6 ≤ ρ ≤ 1.3) showed the most
stable extrapolation error, and this may be a sign of their suitability for numerical
simulations.
ou
As a by product of the present work, we have produced the lengthy expressions
corresponding to all overlap vertices which can arise in a 2-loop computation, and
presented them in a rather compact form. Further computations of similar complexity, for example the 2-loop renormalization of operators in the overlap action (such
tin
as fermion currents), only require the vertices which we have presented here.
an
• Our cactus method developed to improve the perturbative series has been discussed
in Chapter 8. This is the resummation of a dominant subclass of Feynman diagrams
to all orders of the bare coupling constant. The method, originally proposed for the
Plaquette action, was extended here to encompass all possible gluon actions made
perturbation theory.
on
st
of closed Wilson loops. We remark that this method is gauge invariant and can
be applied to any observable in lattice gauge theories. Moreover, it can be used in
a systematic way to improve (to all orders) results obtained at any given order in
C
Another positive feature which reveals the simplicity of our method is the fact that
the resummation procedure is applied by replacing the action parameters (coupling
constant, Symanzik coefficients, clover coefficients) by dressed values. In cases where
a
the results are given as a polynomial of the parameters mentioned above, the improvement can directly be applied in the bare results, with no need of additional
calculations. This leads to both human and computer time saving.
M
ar
th
Two different applications have been employed to test our improvement procedure:
The 2-loop additive renormalization of the fermion critical mass mcr and the 1-loop
renormalization of fermion currents ZV,A . The comparison with available nonperturbative results, shows that the improvement moves in the right direction and it is as
good as the Lepage-Mackenzie tadpole improvement.
• In Chapter 9 we calculated the critical mass mc , and the associated critical hopping parameter κc , up to 2 loops in perturbation theory, using the clover action for
fermions and the Symanzik improved gluon action. The perturbative value of mc
11. Conclusions
164
is a necessary ingredient in the higher-loop renormalization of operators, in mass
independent schemes.
In our calculations, we have chosen for the Symanzik coefficients ci a wide range of
degree polynomial whose coefficients we compute explicitly.
ou
values, which are most commonly used in numerical simulations. The dependence of
our results on the number of colors N and the number of fermion flavors Nf is shown
explicitly. The dependence on the clover parameter cSW is in the form of a fourth
Our results for mc are significantly closer to zero in the case of Symanzik improved
tin
actions, as compared to the plaquette action. In particular, the DBW2 action stands
out among the rest, in that mc vanishes exactly for a value of cSW around 1. Thus,
weakly on the mass of the virtual fermion.
an
improved actions seem to bring us quite near the point of chiral symmetry restoration.
The dependence of mc on the number of flavors is seen to be very mild. This fact
would also suggest that, in the case of nondegenerate flavors, mc should depend only
st
Finally, we have made some comparisons among perturbative and nonperturbative
results for κc . While these are expected to differ for a power divergent additive
renormalization, such as the quantity under study, we nevertheless find a reasonable
on
agreement. This agreement is further enhanced upon using our improved perturbative scheme, previously discussed. As would be desirable, 1-loop improvement is
C
seen to be already adequate to give a reasonable agreement among perturbative and
nonperturbative values. Indeed, our results for κdr
1−loop are significally closer to the
nonperturbative evaluations; in fact, the 2-loop dressing procedure introduces no
a
further improvement to the comparison.
th
• In Chapter 10 we discuss the improvement to second order in the lattice spacing a,
in 1-loop perturbation theory. In particular, we focused on the fermion propagator
M
ar
and quark operators of the form Ψ̄ΓΨ, employing the improved clover fermion action
and Symanzik improved gluons.
We provided the general expression for the inverse fermion propagator S −1 as a
function of the coupling constant, the number of colors, the gauge fixing parameter
and the clover parameter. The dependence on the Symanzik coefficient cannot be
shown explicitly, thus we presented our results for 10 sets of their values.
The effect of improvement is the removal of all corrections of order a and a2 that
have appeared in our results. This is achieved through the addition of irrelevant
11. Conclusions
165
interaction term to the fermion part of the action by improving the lattice expression
of fermion operators so as to cancel unwanted contributions in their matrix elements.
Our O(a2 ) corrected results are applicable to the widely used twisted mass QCD by
M
ar
th
a
C
on
st
an
tin
ou
setting the clover parameter equal to zero. These results will be useful for many
collaborations worldwide.
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Appendix A: Notation
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In this Appendix we present a brief introduction to the basic equations that govern QCD
and are necessary for the completeness of the Thesis. We begin with the continuous
Lagrangian, explaining all mathematical symbols. The expression for the action and the
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partition function are also provided, as well as the interaction vertices. Then we switch to
Lattice QCD and show how the discretization is performed and its consequences on the
A.1
Continuum QCD
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action of QCD.
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In the continuous Lagrangian, quarks (antiquarks) are Dirac 4-spinors denoted by Ψf (x)
f
(Ψ (x) = (Ψf (x))† γ0 ) and can have one of the 6 flavors, denoted by Nf ; they are SU(3)
triplets in color space. Gluons are gauge bosons represented by 8 (more general N 2 − 1)
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gauge fields Aµ (x) ǫ SU(3). In particular, they can be written as Aµ = Aaµ T a where T a
are traceless hermitian 3 × 3 matrices and a = 1, .., N 2 − 1. The Lagrangian density is the
sum of a fermionic and a purely gluonic part
a
Nf
X
1
f
Ψ (x)(iγ µ Dµ (x) − mf0 )Ψf (x) − Tr[Gµν (x)Gµν (x)]
2
f =1
(A.1)
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LQCD (x) =
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The trace of the second term is taken over color indices. The gluon field strength tensor
appearing in the Lagrangian is defined by
Gµν (x) =
i
[Dµ (x), Dν (x)] ,
g0
Dµ (x) = ∂µ − i g0 Aµ (x)
(A.2)
where Dµ is the covariant derivative and g0 is the bare coupling of the strong interaction.
The indices µ, ν label the space time coordinates, and a summation over repeated indices
is implied. Moreover, the bare quark mass m0 differs for quarks with different flavor.
166
A.1. Continuum QCD
167
Both terms of Eq. (A.1) are invariant under a local gauge transformation Λ(x) ǫ SU(3)
Ψ(x) → Λ(x)Ψ(x)
Ψ(x) → Ψ(x)Λ† (x)
Aµ (x) → Λ(x)Aµ (x)Λ† (x) −
1
∂µ Λ(x)Λ† (x)
g0
(A.3)
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There are 3 interaction vertices arising from Eq. (A.1): The quark-antiquark-gluon vertex,
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the 3- and 4-gluon vertex, as demonstrated in Fig. A.1.
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Figure A.1: The interaction vertices of quarks and gluons. Solid (wavy) lines represent
fermions (gluons).
For the introduction of the lattice formulation it is necessary to switch to Euclidean
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space, by performing a Wick’s rotation: t → it, so that time is purely imaginary. The
product of two 4-vectors is now given by
a
xµ y µ = x0 y0 + x1 y1 + x2 y2 + x3 y3
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with x0 , y0 the time coordinates. In what follows and in the main body of the Thesis
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the metric of the Euclidean space is applied. One must define the partition function Z,
necessary for the calculation of Green’s functions and the normalization of expectation
values, defined by the path integral
Z=
Z
DΨ] DΨ DA e−S[Ψ,Ψ,A]
(A.4)
where S is the QCD action in Euclidean space
S[Ψ, Ψ, A] =
Z
d4 x LQCD (x)
(A.5)
A.2. Lattice QCD
168
The expectation values for physical quantities, O, can be represented by operators built
from quark and gluon fields
1
< O >=
Z
Z
DΨ DΨ DA O e−S[Ψ,Ψ,A]
(A.6)
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Clearly, the action is dimensionless and thus the Lagrangian has dimensions 5 [length]−4 ,
or equivalently [mass]4 . From the mass term we note that the fermion fields have dimen-
A.2
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sion [length]−3/2 and using the pure gluon term one can see that the coupling constant is
dimensionless; thus the gauge fields have dimension [length]−1 .
Lattice QCD
nµ ǫ Z ,
a : lattice spacing
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xµ → nµ a ,
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In a hypercubic space-time lattice (ultraviolet regulator), the continuum Euclidean coordinate xµ is replaced by a variable having discrete values
This discretization introduces a momentum cutoff which is inverse to the lattice spacing,
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since the momenta are restricted in the finite interval −π/a ≤ p ≤ π/a (first Brillouin
zone). Thus, the integrals transform to finite sums
d4 x → a4
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Z
X
n
and all quantities calculated in the lattice are finite. The first thing that needs to be done
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a
is to convert the fermion and gauge fields into the lattice language. The discretized quarks
are now described by Grassmann variables Ψ(n) and are placed on the lattice sites n, while
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the gluons live in the links between two neighboring lattice points. This way, gluons carry
the interaction among quarks and at the same time they interact with each other. The
lattice gauge fields are represented by the variable Uµ (n) defined as
Uµ (n) = ei a g0 Aµ (n)
(A.7)
In many cases, it is convenient to work with dimensionless quantities, and this can be
achieved by absorbing the dimension through appropriate powers of the lattice spacing.
5
We work in units where ~ = c = 1
A.2. Lattice QCD
169
For instance, the quark field has to be multiplied by a−3/2
Ψ(n) → a−3/2 Ψ(n)
In order to write a lattice version of the QCD action6 , we discretize the derivative using
the naive differences
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(A.8)
(A.9)
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−
→
1
∇ µ Ψ(n) =
[Uµ (n)Ψ(n + aµ̂) − Ψ(n)]
a
←
−
1
∇ µ Ψ(n) =
[Ψ(n) − Uµ (n − aµ̂)−1 Ψ(n − aµ̂)]
a
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where µ̂ is the unit vector in direction µ. One of the desired properties of the lattice
action is the gauge invariance for the reasons mentioned in Chapter 2. The lattice gauge
transformations take the form
Ψ(n) → Λ(n)Ψ(n)
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Ψ(n) → Ψ(n)Λ† (n)
Uµ (n) → Λ(n)Uµ (n)Λ† (n)
(A.10)
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For this purpose, the pure gluonic part of the action must be constructed by gauge invariant
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elements. The simplest one is a product of link variables along the perimeter of a plaquette
originating at n in the positive µ − ν directions (see Fig. 2.1). The ‘naive’ lattice action
can thus be written as
S[Ψ, Ψ, U] = a4
XX
n
a
f
f
Ψ (n)(D + mf0 )Ψf (n)
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where
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2N X X
1
+ 2
(1 − ReTr[Uµ (n)Uν (n + aµ̂)Uµ† (n + aν̂)Uν† (n)])
g0 µ<ν n
N
(A.11)
3
1X
−
→
←
−
D=
{γµ ( ∇ µ + ∇ µ )}
2 µ=0
(A.12)
The interaction vertices can be extracted by taking the Taylor expansions in g0 that
appear in the exponential of the link variable. The expansion can be taken up to the order
6
In principle there are alternative ways to discretize the continuous action, but all must give the correct
limit when taking a → 0. More details are provided in Chapter 2.
A.2. Lattice QCD
170
of g0 relevant to the calculations performed. For high order computations, the number of
vertices is much larger than the one appearing in the continuum. In the limit a → 0, the
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only ones surviving are the three vertices of the continuum (Fig. A.1).
Theory
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B.1
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Appendix B: Numerical integration
The integrator is a routine converting our 1- and 2-loop expressions into a Fortran code for
numerical integration over loop momenta, on lattices with finite size L. When evaluating
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the Feynman diagrams in Mathematica, we arrive to lengthy expressions constituted of
trigonometric functions, propagators and action parameters, manipulated in a symbolic
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way. This is given as an input to the integrator, which converts all symbolic functions
into their trigonometric analogues, given in Fortran form. It is also capable of managing
multiple values for the action parameters.
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Two-loop diagrams lead to long-winded expressions and thus they can rarely be integrated into a single Fortran file (the compilation of the Fortran code has a limitation of
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maximum 2500 terms, or less for very complicated expressions). In such a case, we split
the integrand into sets of approximately 2000 terms; each one is separately integrated, and
then added to the rest of the contributions of a certain diagram.
a
The integration region is the first Brillouin zone (−π ≤ p ≤ π), but we use symmetries
of the theory and the momenta, to reduce it to the one forth of it, 0 ≤ p ≤ π/2, for
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execution time saving. To make this possible, we make sure that the integrand is indeed
even with respect to the loop momenta, when performing trigonometric simplifications.
The algorithm is composed of nested loops, with the innermost being the most expensive
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in execution time, so the integrand is organized as an inverse tree: For the integrator to be
more optimized, parts with the same functional dependence on the innermost integration
variable are grouped together, so that the integral is performed only once for each group.
As mentioned above, the integrator recognizes the action parameters (i.e. the clover
parameter cSW ) and groups together terms multiplied by the same parameter; the final
result is presented as a polynomial of these parameters (in the routine, these appear as
var[i]’s). Additionally, for parameters whose values are read as an input (masses, overlap
171
B.1. Theory
172
parameter, Symanzik coefficients), the code runs in parallel for different sets of their values
to avoid computing the same quantity several times.
Starting from an already highly optimized version for the integrator, for the needs of
the calculations presented in this Thesis, we consolidated a few improvements to make it
even more efficient and less CPU time consuming.
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• As shown in the main part of the Thesis, 2-loop computations with overlap fermions
lead to very complicated and oversized expressions for the diagrams, resulting a huge
amount of Fortran files for numerical integration. For the purposes of the β-function
calculation (Chapter 7) we defined some new symbolic functions while calculating the
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diagrams to reduce the length of their expressions. In addition to that, we modified
the intagrator routine, in order to recognize the newly introduced functions. This
inclusion makes the Fortran code a bit more time consuming, but it is more economic
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than substituting these functions in the integrand before integrating.
• One of the factors responsible for significant increase of the execution time is the
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total iterations of the innermost integral. For instance, the integration of 2000 terms
coming from a calculation using overlap fermions with 10 iterations in the inner loop
(and for 4 ≤ L ≤ 28) requires approximately 4 days on a single CPU; this increases
to 15 days when changing the iterations to 150 for the inner loop. That was our
motivation to modify the algorithm in the following way: The loop momenta are
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denoted by p1 and p2 and the integrator is constructed in such a way that p2 is the
momentum corresponding to the inner integration, while p1 is integrated last. The
integrand as provided by Mathematica, is organized by default in an alphabetic form
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a
with respect to p1 , p2 . In cases where the dependence on p2 is more complicated than
that on p1 , too many function appear in the innermost loop making the integration
more time demanding. For that reason, we altered the integrator to perform a check
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before placing the p2 dependence inside the inmost loop. First, it counts the number
of terms appearing in the inner loop, leaving p1 and p2 as it is. Then, it changes
the variables, p1 ↔ p2 , and counts again. By comparing the two numbers, it decides
which one is favorable (less iterations in the innermost loop) and builds the Fortran
code based on that form.
• In extremely complicated integrands for overlap fermions, an alternative procedure
was used to minimize the execution time, since the above step failed to save CPU
time. This required a preliminary work on the integrand, in which we factored
B.2. The integrator routine
173
particular combinations of functions that appear many times (typically more than
200). These combinations create the so called xlist and are numerically calculated
just once. The xlist does not involve any summation over dummy indices. Although
this procedure increases the number of iterations in the inner loop, we have checked
The integrator routine
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B.2
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that there is a decrease of the total integration time. Of course, the integrator is
intelligent enough to exclude from the inner loop, combinations that depend only on
the variable integrated last.
Next appears the latest version of our 2-loop integrator, which contains definitions relevant
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to overlap fermions, and the Iwasaki gluonic propagator. The program assumes distinct
values for the Lorentz indices (no summation is implied), and each summand in the integrand is multiplied by a linear combination of var[i]; accordingly, the result is also given
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as a linear combination. To save CPU time, the program evaluates the integral for up to
300 sextuplets of values for (r, m, c0, c1, c2, c3) (Wilson parameter, Overlap parameter
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and the Symanzik parameters) simultaneously. When calling the integrator, an optional
argument (‘gauge’) may be used for the gauge parameter ξ (gauge=0: Feynman gauge
(default), gauge=1: Landau gauge). Before providing the integrator code, let us define
some symbolic functions appearing in the routine.
s2(p, µ) ≡ sin(pµ /2) ,
•
s2sq(p) ≡
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•
•
•
•
X
a
c2sq(p) ≡
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•
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•
s2qu(p) ≡
c2(p, µ) ≡ cos(pµ /2)
sin(pµ /2)2 ,
µ
X
cos(pµ /2)2 ,
µ
X
sin(pµ /2)
sisq(p) ≡
cisq(p) ≡
4
X
sin(pµ )2
µ
X
cos(pµ )2
µ
µ
X
hat2(p) ≡ 1/ 4
sin2 (kµ /2)
µ
fhat(p) ≡ 1/(hat2−1 (p) + m2 )
mO(p) ≡ m − 2r s2sq(p) ,
bO(p) ≡ ρ − 2r
X
µ
sin2 (pµ /2)
omegaO(p) ≡ ω(p) =
X
•
dprop(p, a, b) ≡ ∆Da b ,
1/2
omegabinvO(p) ≡ 1/(bO(p) + ω(p))
Eq. (13)) of Ref. [20]
Eq. (14) of Ref. [20]
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prop(p, a, b) ≡ Dab ,
2
omegaplusinvO(p1 , p2 ) ≡ 1/(ω(p1) + ω(p2 ))
omegabO(p) ≡ bO(p) + ω(p) ,
•
sin (pµ ) + bO (p)
µ
• omegainvO(p) ≡ 1/ω(p) ,
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•
174
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where m is the fermion mass and ρ is the overlap parameter. Moreover, cprop0, cprop1,
cprop2, cprop3 correspond to the Symanzik coefficients c0 , c1 , c2 , c3 .
integrator2[integrand_, propagators_, file_, options___]:=
Block[
{integrandlocal = integrand, propagatorslocal = propagators,
expr,variables,variablelist,varlist,varlength,rulelist,factorlist,proplist,
dummy1,dummy2,expr1,expr2,expr3,expr4,index1,index2,index3,index,optionlist,
gauge,gaugelocal,tlist,tlistlocal,tlistTrueFalse,xlist,xlistlocal,xlistTrueFalse,
xlistlocalorig,temp,defaultlist,substlist,exprS,expr1S,expr2S,expr3S,expr4S,
index1S,index2S,index3S,indexS,
proplistS,tlistlocalS,xlistlocalS,xlistlocalorigS,factorlistS,flagp1p2=0},
defaultlist = {gauge -> 0, tlist -> {}, xlist -> {}};
optionlist = List[options];
gaugelocal = gauge /. optionlist /. defaultlist;
tlistlocal = tlist /. optionlist /. defaultlist;
xlistlocalorig = xlist /. optionlist /. defaultlist;
xlistlocal = xlistlocalorig;
tlistTrueFalse = Table[!FreeQ[integrandlocal,t[i]],
{i,Length[tlistlocal]}];
xlistTrueFalse = Table[!FreeQ[integrandlocal,x[i]],
{i,Length[xlistlocal]}];
w[a__] := WriteString[ToString[file], "
", a, "\n"];
wc[a__]:= WriteString[ToString[file], "
&", a, "\n"];
Label[p1p2];
expr = List /@ (If[Head[integrandlocal]===Plus, List@@integrandlocal, List[integra
ndlocal]]);
expr = Drop[#,1]& /@ expr;
expr = expr /. (a_[b_,rho[n_]] :>
a[b /. {p[1] :> ToExpression[StringJoin["i",ToString[n]]],
p[2] :> ToExpression[StringJoin["j",ToString[n]]]}]);
substlist = {s2sq[p[1]]->s2sq1, s2sq[p[2]]->s2sq2, s2sq[p[1]+p[2]]->s2sq12,
sisq[p[1]]->sisq1, sisq[p[2]]->sisq2, sisq[p[1]+p[2]]->sisq12,
s2qu[p[1]]->s2qu1, s2qu[p[2]]->s2qu2, s2qu[p[1]+p[2]]->s2qu12,
c2sq[p[1]]->c2sq1, c2sq[p[2]]->c2sq2, c2sq[p[1]+p[2]]->c2sq12,
cisq[p[1]]->cisq1, cisq[p[2]]->cisq2, cisq[p[1]+p[2]]->cisq12,
omegaO[p[1]]->omegao1[jov[ir]],
omegainvO[p[1]]->omegainvo1[jov[ir]],
omegabO[p[1]]->omegabo1[jov[ir]],
omegabinvO[p[1]]->omegabinvo1[jov[ir]],
mO[p[1]]->mo1[jov[ir]],
omegaO[p[2]]->omegao2[jov[ir]],
omegainvO[p[2]]->omegainvo2[jov[ir]],
omegabO[p[2]]->omegabo2[jov[ir]],
omegabinvO[p[2]]->omegabinvo2[jov[ir]],
mO[p[2]]->mo2[jov[ir]],
omegaO[p[1]+p[2]]->omegao12[jov[ir]],
omegainvO[p[1]+p[2]]->omegainvo12[jov[ir]],
omegabO[p[1]+p[2]]->omegabo12[jov[ir]],
omegabinvO[p[1]+p[2]]->omegabinvo12[jov[ir]],
mO[p[1]+p[2]]->mo12[jov[ir]],
omegaplusinvO[0,p[1]]->omegaplusinvo1[jov[ir]],
omegaplusinvO[0,p[2]]->omegaplusinvo2[jov[ir]],
omegaplusinvO[0,p[1]+p[2]]->omegaplusinvo12[jov[ir]],
omegaplusinvO[p[1],p[2]]->omegaplusinvo1p2[jov[ir]],
omegaplusinvO[p[1],p[1]+p[2]]->omegaplusinvo1p12[jov[ir]],
omegaplusinvO[p[2],p[1]+p[2]]->omegaplusinvo2p12[jov[ir]],
m[p[1]]->m1[jov[ir]], m[p[2]]->m2[jov[ir]],
m[p[1]+p[2]]->m12[jov[ir]],
s2[2 a__]^2 :> si2[a], c2[2 a__]^2 :> ci2[a],
s2[a__]^2 :> s22[a], c2[a__]^2 :> c22[a],
s2[2 a__] :> si[a], c2[2 a__] :> ci[a],
s2[a__]^4 :> s24[a], c2[a__]^4 :> c24[a],
fhat[p[1]]-> fhat1[jov[ir]], hat2[p[1]]->hat1,
fhat[p[2]]-> fhat2[jov[ir]], hat2[p[2]]->hat2,
fhat[p[1]+p[2]]-> fhat12[jov[ir]], hat2[p[1]+p[2]]->hat12};
expr = expr /. substlist;
xlistlocal = xlistlocalorig /. substlist;
xlistlocal = xlistlocal /. mO -> m[jov[ir]];
expr = expr /. {dprop[p[1],a__] :> dprop["i",a],
dprop[p[2],a__] :> dprop["j",a],
dprop[p[1]+p[2],a__] :> dprop["ij",a]};
expr = expr /. dprop[a_,rho[i_],rho[j_]] :>
dpropn[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"n"]]& /@ {i,j}
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variables = Select[Variables[integrandlocal],(FreeQ[#,p[2]] && FreeQ[#,t] && FreeQ
[#,x])&];
variablelist = Table[Cases[variables,_[_,rho[j]]],{j,4}];
variablelist =
Append[Drop[variablelist,-1],
Join[variablelist[[-1]],
Select[variables,(!FreeQ[#,p[1]] && (FreeQ[#,rho] || (!FreeQ[#,dprop
]) || (!FreeQ[#,prop])))&]]];
variables = Complement[Variables[integrandlocal],variables];
variablelist = Join[variablelist,Table[Cases[variables,_[_,rho[j]]],{j,4}]];
variablelist =
Append[Drop[variablelist,-1],
Join[variablelist[[-1]],
Select[variables,(FreeQ[#,rho] || (!FreeQ[#,dprop]) || (!FreeQ[#,pro
p]))&]]];
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proplist = proplist /.
{fhat[p[1]]->fhat1[jov[ir]], fhat[p[2]]->fhat2[jov[ir]],
hat2[p[1]]-> hat1, hat2[p[2]]-> hat2,
fhat[p[1]+p[2]]->fhat12[jov[ir]],
hat2[p[1]+p[2]]-> hat12};
proplist = FortranForm /@ proplist;
integrator2::wrongArgument = "integrator2 called with inappropriate argument"
integrator2[integrand_, ___]:=
Message[integrator2::wrongArgument] /;
!(Complement[Union[If[Head[#]===Symbol,#,Head[#]]& /@ Variables[integrand]],
{s2,c2,s2sq,sisq,s2qu,c2sq,cisq,r,m,var,hat2,fhat,prop,dprop,cprop0,cprop1,
cprop2,cprop3,mO,omegaO,omegainvO,omegaplusinvO,omegabO,omegabinvO,x,t}]==={})
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proplist = {Select[propagatorslocal*dummy1*dummy2, FreeQ[#,p[2]]&]
/. dummy1 -> 1 /. dummy2 -> 1};
proplist = {proplist[[1]], propagatorslocal/proplist[[1]]};
),
Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@
Complement[Range[4],{i,j}]
),
jiw[ir]]
/; (!(i===j));
expr = expr /. dprop[a_,rho[i_],rho[i_]] :>
dpropd[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ {i}),
Complement[Range[4],{i}]),
jiw[ir]];
expr = expr /. {prop[p[1],a__] :> prop["i",a],
prop[p[2],a__] :> prop["j",a],
175
Do[rulelist = Rule[#,1]& /@ (variablelist[[-j]]);
expr = (Join[{(#[[1]]) /. rulelist},
{(#[[1]]) / ((#[[1]]) /. rulelist)},
Drop[#,1]])& /@ expr,
{j,Length[variablelist]}];
varlist=Union[integrandlocal[[Sequence @@ #]]&/@ Position[integrandlocal,var[_]]];
varlength = Max[(#[[1]])& /@ varlist, 1];
factorlist = ((#[[1]])& /@ expr);
factorlist = Table[# /. var[j]->1 /. var[_]->0,{j,varlength}]& /@ factorlist;
factorlist = factorlist /.
r -> r[jov[ir]] /. m -> m[jov[ir]] /. mO -> m[jov[ir]] /.
cprop0 -> cprop0[jiw[ir]] /. cprop1 -> cprop1[jiw[ir]] /.
cprop2 -> cprop2[jiw[ir]] /. cprop3 -> cprop3[jiw[ir]];
factorlist = Map[FortranForm,factorlist,{2}];
jiw[ir]]
/; (!(i===j));
expr = expr /. prop[a_,rho[i_],rho[i_]] :>
propd[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"h"]]& /@ {i}),
Complement[Range[4],{i}]),
jiw[ir]];
expr = Map[FortranForm,expr,{2}];
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expr1 = Union[Take[#,{5,8}]& /@ expr];
index3 = Table[Position[expr4,Drop[expr3[[j]],1]][[1,1]],{j,Length[expr3]}];
index = Table[Position[expr1,Take[expr[[j]],{5, 8}]][[1,1]],
{j,Length[expr]}];
If[flagp1p2 == 0, flagp1p2 = 1;
integrandlocal = integrandlocal /. {p[1]->p[2], p[2]->p[1]};
propagatorslocal = propagatorslocal /. {p[1]->p[2], p[2]->p[1]};
tlistlocalS = tlistlocal;
tlistlocal = tlistlocal /. {p[1]->p[2], p[2]->p[1]};
xlistlocalS = xlistlocal;
xlistlocalorig = xlistlocalorig /. {p[1]->p[2], p[2]->p[1]};
exprS = expr;
expr1S = expr1; expr2S = expr2; expr3S = expr3; expr4S = expr4;
index1S = index1; index2S = index2; index3S = index3;
indexS = index;
proplistS = proplist; factorlistS = factorlist;
Goto[p1p2]];
If[Length[expr4S] < Length[expr4],
tlistlocal = tlistlocalS;
xlistlocalorig = xlistlocalorig /. {p[1]->p[2], p[2]->p[1]};
xlistlocal = xlistlocalS;
expr = exprS;
expr1 = expr1S; expr2 = expr2S; expr3 = expr3S; expr4 = expr4S;
index1 = index1S; index2 = index2S; index3 = index3S; index = indexS;
proplist = proplistS; factorlist = factorlistS];
Print["Inner loop: ",Length[expr4]," iterations"];
If[Length[expr4]>200,Print["
*****"]];
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),
Complement[Range[4],{i,j}]
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,
w["dimension dpropn(-20:20,-20:20,0:20,0:20,50)"];
w["dimension propn(-20:20,-20:20,0:20,0:20,50)"];
w["dimension modn(-39:80), modh(-39:80)"];
w["dimension g(4,4), g2(4,4), ginv(4,4)"];
w["integer indx(4)"];
w["dimension omegainvo1(50), omegabinvo1(50), omegaplusinvo1(50)"];
w["dimension omegao1(50), mo1(50), omegabo1(50)"];
w["dimension omegainvo2(50), omegabinvo2(50), omegaplusinvo2(50)"];
w["dimension omegainvo12(50),omegabinvo12(50),omegaplusinvo12(50)"];
w["dimension omegaplusinvo1p2(50),omegaplusinvo1p12(50)"];
w["dimension omegaplusinvo2p12(50)"];
w["dimension result(",varlength,")"];
If[Length[tlistlocal]>0, w["dimension t(",Length[tlistlocal],")"]];
If[Length[xlistlocal]>0, w["dimension x(",Length[xlistlocal],")"]];
w[""];
w["pi = 4.0*atan(1.d0)"];
WriteString[ToString[file], "CCCC ",
"Enter minimum and maximum length of lattice (even, < 41)", "\n"];
w["read(*,*) nmin, nmax"];
"Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one se
t per line", "\n"];
"NB: No more than 50 different values for overlap parameters; idem for
Iwasaki", "\n"];
w["ir
= 1"];
w["iov
= 1"];
w["iiw
= 1"];
w["iovflag = 1"];
w["iiwflag = 1"];
WriteString[ToString[file], " 1
read(*,*,end=2) rtemp, mtemp, \n"];
wc["
c0temp, c1temp, c2temp, c3temp"];
w["do iov2 = 1, iov - 1"];
w[" if((rtemp-r(iov2)).lt.1.d-9 .and. (mtemp-m(iov2)).lt.1.d-9) then"];
w[" jov(ir) = iov2"];
w[" iovflag = 0"];
w[" endif"];
w["enddo"];
w["if(iovflag.eq.1) then"];
w[" r(iov) = rtemp"];
w[" m(iov) = mtemp"];
w[" jov(ir) = iov"];
w[" iov = iov + 1"];
w["endif"];
w["iovflag = 1"];
w["do iiw2 = 1, iiw - 1"];
w[" if((c0temp-cprop0(iiw2)).lt.1.d-9 .and. "];
wc[" (c1temp-cprop1(iiw2)).lt.1.d-9 .and. "];
wc[" (c2temp-cprop2(iiw2)).lt.1.d-9 .and. "];
wc[" (c3temp-cprop3(iiw2)).lt.1.d-9) then"];
w[" jiw(ir) = iiw2"];
w[" iiwflag = 0"];
w[" endif"];
w["enddo"];
w["if(iiwflag.eq.1) then"];
w[" cprop0(iiw) = c0temp"];
w[" cc1(iiw) = cprop2(iiw) + cprop3(iiw)"];
w[" cc2(iiw) = cprop1(iiw) - cprop2(iiw) - cprop3(iiw)"];
w[" jiw(ir) = iiw"];
w[" iiw = iiw + 1"];
w["endif"];
st
prop[p[1]+p[2],a__] :> prop["ij",a]};
expr = expr /. prop[a_,rho[i_],rho[j_]] :>
propn[Sequence @@ (ToExpression[StringJoin[a,ToString[#],"n"]]& /@ {i,j})
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w["program main"];
w["implicit real*8 (a-h,o-z)"];
w["real*8 k2p1, k4p1, k2p2, k4p2, k2p12, k4p12"];
w["real*8 m, m1, m2, m12, mo1, mo2, mo12"];
w["real*8 rtemp, mtemp, c0temp, c1temp, c2temp, c3temp"];
w["parameter (l = ",Length[expr],")"];
w["dimension a(l,4,300), b(l,4,300)"];
w["dimension s2(-500:500), c2(-500:500), s22(-500:500), c22(-500:500)"];
w["dimension s24(-500:500), c24(-500:500)"];
w["dimension si(-500:500), ci(-500:500), si2(-500:500), ci2(-500:500)"];
w["dimension r(50), r24(50), r12(50), m(50)"];
w["dimension cprop0(50), cprop1(50), cprop2(50), cprop3(50)"];
w["dimension cc1(50), cc2(50)"];
w["dimension jov(300), jiw(300)"];
w["dimension fhat1(50), fhat2(50), fhat12(50)"];
w["dimension m1(50), m2(50), m12(50)"];
w["dimension dpropd(0:20,0:20,0:20,0:20,50)"];
w["dimension propd(0:20,0:20,0:20,0:20,50)"];
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w[" enddo"];
w[" enddo"];
w[" do mu = 1,4"];
w[" do nu = 1,4"];
w["
ginv(mu,nu) = 0.d0"];
w[" enddo"];
w[" ginv(mu,mu) = 1.d0"];
w[" enddo"];
w[" call ludcmp(g,4,4,indx,d)"];
w[" do nu = 1, 4"];
w[" call lubksb(g,4,4,indx,ginv(1,nu))"];
w[" enddo"];
w[" propn(i1,i2,i3,i4,ir) = ginv(1,2)- ", gaugelocal, " * g2(1,2)"];
w[" dpropn(i1,i2,i3,i4,ir) = ginv(1,2)"];
w[" if (i1.ge.0.and.i2.ge.0) then"];
w[" propd(i1,i2,i3,i4,ir) = ginv(1,1)- ", gaugelocal, " * g2(1,1)"];
w[" dpropd(i1,i2,i3,i4,ir) = ginv(1,1) - hat1"];
w[" endif"];
w["enddo"];
WriteString[ToString[file], " 1001 ", "continue", "\n"];
w["enddo"];
w["enddo"];
w["enddo"];
w["enddo"]];
w[""];
w["do i = 1,l"];
w["do ir = 1,nr"];
w["a(i,1,ir) = 0.d0"];
w["enddo"];
w["enddo"];
w["do i1 = nhalf,n"];
w[" i1n = modn(i1)"];
w[" i1h = modh(i1)"];
w[" do i = 1,l"];
w[" do ir = 1,nr"];
w[" a(i,2,ir) = 0.d0"];
w[" enddo"];
w[" enddo"];
w[" do i2 = nhalf,n"];
w[" i2n = modn(i2)"];
w[" i2h = modh(i2)"];
w[" do i = 1,l"];
w[" do ir = 1,nr"];
w[" a(i,3,ir) = 0.d0"];
w[" enddo"];
w[" enddo"];
w[" do i3 = nhalf,n"];
w["
i3n = modn(i3)"];
w["
i3h = modh(i3)"];
w["
do i = 1,l"];
w["
do ir = 1,nr"];
w["
a(i,4,ir) = 0.d0"];
w["
enddo"];
w["
enddo"];
w["
do i4 = nhalf,n"];
w["
i4n = modn(i4)"];
w["
i4h = modh(i4)"];
w["
s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))"];
w["
sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4))"];
w["
c2sq1 = (c22(i1)+c22(i2)+c22(i3)+c22(i4))"];
w["
cisq1 = (ci2(i1)+ci2(i2)+ci2(i3)+ci2(i4))"];
If[!FreeQ[{expr,xlistlocal},s2qu1],
w["
s2qu1 = (s24(i1)+s24(i2)+s24(i3)+s24(i4))"]];
w["
if (s2sq1.gt.1e-9) then"];
If[!FreeQ[{expr,proplist,xlistlocal},hat1],
w["
hat1 = 0.25d0/s2sq1"]];
w["iiwflag = 1"];
w["ir = ir + 1"];
w["goto 1"];
WriteString[ToString[file], " 2
nr = ir - 1 \n"];
w["iov = iov - 1"];
w["iiw = iiw - 1"];
w["do ir = 1, iov"];
w[" r24(ir) = 4.d0*r(ir)**2"];
w[" r12(ir) = 2.d0*r(ir)"];
w["enddo"];
w[""];
w["do 1000 n = nmin, nmax, 2"];
w["nmod2 = mod(n,2)"];
w["nhalf = n/2"];
w["n4 = n*4"];
w["do i = -12*n,12*n"];
w[" s2(i) = sin(i*pi/n)"];
w[" c2(i) = cos(i*pi/n)"];
w[" s22(i) = s2(i)**2"];
w[" c22(i) = c2(i)**2"];
w[" s24(i) = s2(i)**4"];
w[" c24(i) = c2(i)**4"];
w[" si(i) = sin(2*i*pi/n)"];
w[" ci(i) = cos(2*i*pi/n)"];
w[" si2(i) = si(i)**2"];
w[" ci2(i) = ci(i)**2"];
w["enddo"];
w[""];
w["do i = 1-n, 2*n"];
w[" itemp = mod(i+n,2*n)-n"];
w[" if(itemp.gt.0) itemp = min(itemp, n-itemp)"];
w[" if(itemp.lt.0) itemp = max(itemp,-n-itemp)"];
w[" modn(i) = itemp"];
w[" modh(i) = min(mod(i+n,n), n - mod(i+n,n))"];
w["enddo"];
w[""];
If[!FreeQ[integrand,prop] || !FreeQ[integrand,dprop],
w["do i1 = -nhalf, nhalf"];
w["do i2 = -nhalf, nhalf"];
w["do i3 = 0, nhalf"];
w["do i4 = 0, nhalf"];
w["if (i1.eq.0.and.i2.eq.0.and.i3+i4.eq.0) goto 1001"];
w["do ir = 1, iiw"];
w[" s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))"];
w[" hat1 = 0.25d0/s2sq1"];
w[" k2p1 =
4* (s22(i1)+s22(i2)+s22(i3)+s22(i4))"];
w[" k4p1 =
16* (s24(i1)+s24(i2)+s24(i3)+s24(i4))"];
w[" dummy = (1-cc1(ir)*k2p1)"];
w[" do mu = 1, 4"];
w[" if (mu.eq.1) imu = i1"];
w[" if (mu.eq.2) imu = i2"];
w[" if (mu.eq.3) imu = i3"];
w[" if (mu.eq.4) imu = i4"];
w[" do nu = 1, mu"];
w["
if (nu.eq.1) inu = i1"];
w["
w["
w["
w["
g(mu,nu) = (1-dummy) * (2*s2(imu))*(2*s2(inu))"];
wc["
+ cc2(ir) *(2*s2(imu))**3*(2*s2(inu))"];
wc["
+ cc2(ir) *(2*s2(imu))*(2*s2(inu))**3"];
w["
g2(mu,nu) = (2*s2(imu))*(2*s2(inu))/k2p1**2"];
w["
if (mu.eq.nu) g(mu,nu) = g(mu,nu) + dummy * k2p1"];
wc["
- cc2(ir) * k4p1 - cc2(ir) * k2p1 * (2*s2(imu))**2"];
w["
if (nu.lt.mu) g(nu,mu) = g(mu,nu)"];
w["
if (nu.lt.mu) g2(nu,mu) = g2(mu,nu)"];
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w["
do j4 = 1,n"];
w["
j4n = modn(j4)"];
w["
j4h = modh(j4)"];
w["
ij4n = modn(i4+j4)"];
w["
ij4h = modh(i4+j4)"];
w["
s2sq12 = (s22(j1+i1)+s22(j2+i2)+s22(j3+i3)+s22(j4+i4))"];
w["
s2sq2 = (s22(j1)+s22(j2)+s22(j3)+s22(j4))"];
w["
sisq12 = (si2(j1+i1)+si2(j2+i2)+si2(j3+i3)+si2(j4+i4))"];
w["
sisq2 = (si2(j1)+si2(j2)+si2(j3)+si2(j4))"];
w["
c2sq12 = (c22(j1+i1)+c22(j2+i2)+c22(j3+i3)+c22(j4+i4))"];
w["
c2sq2 = (c22(j1)+c22(j2)+c22(j3)+c22(j4))"];
w["
cisq12 = (ci2(j1+i1)+ci2(j2+i2)+ci2(j3+i3)+ci2(j4+i4))"];
w["
cisq2 = (ci2(j1)+ci2(j2)+ci2(j3)+ci2(j4))"];
w["
s2qu12 = (s24(j1+i1)+s24(j2+i2)+s24(j3+i3)+s24(j4+i4))"];
w["
s2qu2 = (s24(j1)+s24(j2)+s24(j3)+s24(j4))"];
w["
if (s2sq12.gt.1e-9 .and. s2sq2.gt.1e-9) then"];
w["
hat2 = 0.25d0/s2sq2 "];
w["
hat12 = 0.25d0/s2sq12"];
Do[If[tlistTrueFalse[[i]] && (!FreeQ[tlistlocal[[i]],p[2]]),
w["
t(",i,") = 0.d0"];
Do[wc["
+ ",
FortranForm[tlistlocal[[i]]
/. s2[b_] :> s2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[
k]]],
p[2] :> ToExpression[StringJoin["j",ToString[
k]]]}]
/. c2[b_] :> c2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[
k]]],
k]]]}]]],
{k,4}]],
{i, Length[tlistlocal]}];
w["
do ir = 1,iov"];
If[!FreeQ[proplist,fhat2] || !FreeQ[expr,fhat2],
w["
m2(ir) = m(ir) + r12(ir)*s2sq2"];
w["
fhat2(ir) =1.d0/(m2(ir)**2 + sisq2 )"]];
If[!FreeQ[proplist,fhat12] || !FreeQ[expr,fhat12],
w["
m12(ir) = m(ir) + r12(ir)*s2sq12"];
w["
fhat12(ir)=1.d0/(m12(ir)**2 + sisq12)"]];
If[!FreeQ[{integrand,xlistlocalorig},mO] || !FreeQ[{integrand,xlistlocalorig},omeg
aO] ||
!FreeQ[{integrand,xlistlocalorig},omegainvO] || !FreeQ[{integrand,xlistlocalori
g},omegaplusinvO] ||
!FreeQ[{integrand,xlistlocalorig},omegabO] || !FreeQ[{integrand,xlistlocalorig}
,omegabinvO],
w["
mo2(ir) = m(ir) - r12(ir)*s2sq2"];
w["
omegao2(ir) = sqrt(sisq2 + mo2(ir)**2)"];
w["
omegainvo2(ir) = 1.d0/omegao2(ir)"];
w["
omegaplusinvo2(ir) = 1.d0/(omegao2(ir)+ m(ir))"];
w["
omegabo2(ir) = (omegao2(ir)-mo2(ir))"];
w["
omegabinvo2(ir) = 1.d0/(omegao2(ir)-mo2(ir))"];
w["
mo12(ir) = m(ir) - r12(ir)*s2sq12"];
w["
w["
w["
w["
w["
omegabinvo12(ir) = 1.d0/(omegao12(ir)-mo12(ir))"];
w["
omegaplusinvo1p2(ir) = 1.d0/(omegao1(ir)+ omegao2(ir))"];
w["
omegaplusinvo1p12(ir) = 1.d0/(omegao1(ir)+ omegao12(ir))"];
w["
omegaplusinvo2p12(ir) = 1.d0/(omegao2(ir)+ omegao12(ir))"]];
w["
enddo"];
w["
do ir = 1, nr"];
Do[If[xlistTrueFalse[[i]],
w["
x(",i,") = 0.d0"];
If[Head[xlistlocal[[i]]]===Plus,
Do[
w["
do ir = 1,iov"];
If[!FreeQ[{expr,proplist,xlistlocal},fhat1],
w["
m1(ir) = m(ir) + r12(ir)*s2sq1"];
w["
fhat1(ir) = 1.d0/(m1(ir)**2 + sisq1)"]];
If[!FreeQ[{integrand,xlistlocalorig},mO] || !FreeQ[{integrand,xlistlocalorig},omeg
aO] ||
!FreeQ[{integrand,xlistlocalorig},omegainvO] || !FreeQ[{integrand,xlistlocalori
g},omegaplusinvO] ||
!FreeQ[{integrand,xlistlocalorig},omegabO] || !FreeQ[{integrand,xlistlocalorig}
,omegabinvO],
w["
mo1(ir) = m(ir) - r12(ir)*s2sq1"];
w["
w["
w["
w["
w["
omegabinvo1(ir) = 1.d0/(omegao1(ir)-mo1(ir))"]];
w["
enddo"];
Do[If[tlistTrueFalse[[i]] && (FreeQ[tlistlocal[[i]],p[2]]),
w["
t(",i,") = 0.d0"];
Do[wc["
+ ",
FortranForm[tlistlocal[[i]]
/. s2[b_] :> s2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[
k]]],
k]]]}]
/. c2[b_] :> c2[b /. {p[1] :> ToExpression[StringJoin["i",ToString[
k]]],
k]]]}]]],
{k,4}]],
{i, Length[tlistlocal]}];
w["
do i = 1,",Length[expr1]];
w["
do ir = 1,nr"];
w["
b(i,1,ir) = 0.d0"];
w["
enddo"];
w["
enddo"];
w["
do j1 = 1,n"];
w["
j1n = modn(j1)"];
w["
j1h = modh(j1)"];
w["
w["
w["
w["
do ir = 1,nr"];
w["
b(i,2,ir) = 0.d0"];
w["
enddo"];
w["
enddo"];
w["
do j2 = 1,n"];
w["
j2n = modn(j2)"];
w["
j2h = modh(j2)"];
w["
w["
w["
w["
do ir = 1,nr"];
w["
b(i,3,ir) = 0.d0"];
w["
enddo"];
w["
enddo"];
w["
do j3 = 1,n"];
w["
j3n = modn(j3)"];
w["
j3h = modh(j3)"];
w["
w["
w["
w["
do ir = 1,nr"];
w["
b(i,4,ir) = 0.d0"];
w["
enddo"];
w["
enddo"];
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w["
do ir = 1,nr"];
w["
do i = 1,l"];
w["
a(i,4,ir) = a(i,4,ir) * 0.5d0"];
w["
enddo"];
w["
enddo"];
w["
endif"];
w["
do ir = 1,nr"];
Do[If[expr[[j,3]]===FortranForm[1],
w["a(",j,",3,ir) = a(",j,",3,ir) + a(",j,",4,ir)"],
w["a(",j,",3,ir) = a(",j,",3,ir) + a(",j,",4,ir)*"];
wc[" ",expr[[j,3]]]],
{j,Length[expr]}];
w["
enddo"];
w[" enddo"];
w[" if (i2.eq.n .or. (i2.eq.nhalf .and. nmod2.eq.0)) then"];
w["
do ir = 1,nr"];
w["
do i = 1,l"];
w["
a(i,3,ir) = a(i,3,ir) * 0.5d0"];
w["
enddo"];
w["
enddo"];
w[" endif"];
w[" do ir = 1,nr"];
w["a(",j,",2,ir) = a(",j,",2,ir) + a(",j,",3,ir)"],
w["a(",j,",2,ir) = a(",j,",2,ir) + a(",j,",3,ir)*"];
wc[" ",expr[[j,2]]]],
{j,Length[expr]}];
w[" enddo"];
w[" enddo"];
w[" if (i1.eq.n .or. (i1.eq.nhalf .and. nmod2.eq.0)) then"];
w[" do ir = 1,nr"];
w[" do i = 1,l"];
w[" a(i,2,ir) = a(i,2,ir) * 0.5d0"];
w[" enddo"];
w[" enddo"];
w[" endif"];
w[" do ir = 1,nr"];
w["a(",j,",1,ir) = a(",j,",1,ir) + a(",j,",2,ir)"],
w["a(",j,",1,ir) = a(",j,",1,ir) + a(",j,",2,ir)*"];
wc[" ",expr[[j,1]]]],
{j,Length[expr]}];
w[" enddo"];
w["enddo"];
w["do ir = 1,nr"];
Do[w["result(",k,") = 0.d0"];
Do[If[!(factorlist[[j,k]] === FortranForm[0]),
w["result(",k,") = result(",k,") + a(",j,",1,ir) * "];
wc["(",N[factorlist[[j,k]],16],")"]],
{j,Length[expr]}];
w["result(",k,") = result(",k,") * 2**4 / float(n)**8"],
{k,varlength}];
w[" write(*,*) n,r(jov(ir)), m(jov(ir)),"];
wc["
cprop0(jiw(ir)), cprop1(jiw(ir)),"];
wc["
cprop2(jiw(ir)), cprop3(jiw(ir)), result"];
w["enddo"];
w["call flush(6)"];
WriteString[ToString[file], " 1000 ", "continue", "\n"];
w["stop"];
w["end"]; ]
If[Head[xlistlocal[[i,j]]]===Times && Length[xlistlocal[[i,j]]]>5,
wc[" +", FortranForm[Take[xlistlocal[[i,j]],4]]];
If[Length[Drop[xlistlocal[[i,j]],4]]>4,
wc[" *", FortranForm[Take[Drop[xlistlocal[[i,j]],4],3]]];
wc[" *", FortranForm[Drop[Drop[xlistlocal[[i,j]],4],3]]],
wc[" *", FortranForm[Drop[xlistlocal[[i,j]],4]]]],
wc[" +", FortranForm[xlistlocal[[i,j]]]]],
{j,Length[xlistlocal[[i]]]}],
wc[" +", FortranForm[xlistlocal[[i]]]]]],
{i, Length[xlistlocal]}];
w["
prop = ", proplist[[2]]];
Do[If[proplist[[2]]===FortranForm[1],
w["b(",j,",4,ir) = b(",j,",4,ir) + "];
If[Head[expr4[[j,1,1]]]===Times && Length[expr4[[j,1,1]]] > 4,
wc[" ",Take[#,4]& /@ expr4[[j,1]],"*"];
temp = Drop[#,4]& /@ expr4[[j,1]];
If[Head[temp[[1]]]===Times && Length[temp[[1]]] > 4,
wc[" ",Take[#,4]& /@ temp,"*"];
wc[" ",Drop[#,4]& /@ temp],
wc[" ",temp]],
wc[" ",expr4[[j,1]]]],
If[expr4[[j,1]]===FortranForm[1],
w["b(",j,",4,ir) = b(",j,",4,ir) + prop"],
w["b(",j,",4,ir) = b(",j,",4,ir) + prop*"];
wc[" ",expr4[[j,1]]]]],
{j,Length[expr4]}];
w["
enddo"];
w["
endif"];
w["
enddo"];
w["
do ir = 1,nr"];
Do[If[expr3[[j,1]]===FortranForm[1],
w["b(",j,",3,ir) = b(",j,",3,ir) + b(",index3[[j]],",4,ir)"],
w["b(",j,",3,ir) = b(",j,",3,ir) + b(",index3[[j]],",4,ir)*"];
wc[" ",expr3[[j,1]]]],
{j,Length[expr3]}];
w["
enddo"];
w["
enddo"];
w["
do ir = 1,nr"];
wc[" ",expr2[[j,1]]]],
{j,Length[expr2]}];
w["
enddo"];
w["
enddo"];
w["
do ir = 1,nr"];
wc[" ",expr1[[j,1]]]],
{j,Length[expr1]}];
w["
enddo"];
w["
enddo"];
w["
do ir = 1,nr"];
w["
prop = ", proplist[[1]]];
w["
if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0))"];
wc["
prop = 0.5d0 * prop"];
w["a(",j,",4,ir) = a(",j,",4,ir) + prop*b(",index[[j]],",1,ir)"],
w["a(",j,",4,ir) = a(",j,",4,ir) + prop*b(",index[[j]],",1,ir)*"];
wc[" ",expr[[j,4]]]],
{j,Length[expr]}];
w["
enddo"];
w["
endif"];
w["
enddo"];
w["
if (i3.eq.n .or. (i3.eq.nhalf .and. nmod2.eq.0)) then"];
B.3. A particular example
B.3
180
A particular example
It is useful to take a typical term that appears in a certain diagram and present the Fortran
Code, generated by the integrator. The term is
−6 c2(p1 + 2p2 , rho(1))2 hat2(p1 ) omegabinvO(p2 ) omegabinvO(p1 + p2 )
ou
omegainvO(p2 )2 omegainvO(p1 + p2 )2 omegaplusinvO(p2 , p1 + p2 )2
s2(2p2 , rho(2)) s2(2p2 , rho(3))2 s2(2p1 + 2p2 , rho(2)) s2(2p1 + 2p2 , rho(4))2
M
ar
th
a
C
on
st
an
tin
All the above symbolic functions are defined in the previous section.
do 1000 n = nmin, nmax, 2
nmod2 = mod(n,2)
nhalf = n/2
n4 = n*4
do i = -12*n,12*n
s2(i) = sin(i*pi/n)
c2(i) = cos(i*pi/n)
s22(i) = s2(i)**2
c22(i) = c2(i)**2
s24(i) = s2(i)**4
c24(i) = c2(i)**4
si(i) = sin(2*i*pi/n)
ci(i) = cos(2*i*pi/n)
si2(i) = si(i)**2
ci2(i) = ci(i)**2
enddo
on
st
2
an
tin
ou
enddo
if(iiwflag.eq.1) then
cprop0(iiw) = c0temp
cc1(iiw) = cprop2(iiw) + cprop3(iiw)
cc2(iiw) = cprop1(iiw) - cprop2(iiw) - cprop3(iiw)
jiw(ir) = iiw
iiw = iiw + 1
endif
iiwflag = 1
ir = ir + 1
goto 1
nr = ir - 1
iov = iov - 1
iiw = iiw - 1
do ir = 1, iov
r24(ir) = 4.d0*r(ir)**2
r12(ir) = 2.d0*r(ir)
enddo
program main
implicit real*8 (a-h,o-z)
real*8 k2p1, k4p1, k2p2, k4p2, k2p12, k4p12
real*8 m, m1, m2, m12, mo1, mo2, mo12
real*8 rtemp, mtemp, c0temp, c1temp, c2temp, c3temp
parameter (l = 1)
dimension a(l,4,300), b(l,4,300)
dimension s2(-500:500), c2(-500:500), s22(-500:500), c22(-500:500)
dimension s24(-500:500), c24(-500:500)
dimension si(-500:500), ci(-500:500), si2(-500:500), ci2(-500:500)
dimension r(50), r24(50), r12(50), m(50)
dimension cprop0(50), cprop1(50), cprop2(50), cprop3(50)
dimension cc1(50), cc2(50)
dimension jov(300), jiw(300)
dimension fhat1(50), fhat2(50), fhat12(50)
dimension m1(50), m2(50), m12(50)
dimension dpropd(0:20,0:20,0:20,0:20,50)
dimension propd(0:20,0:20,0:20,0:20,50)
dimension dpropn(-20:20,-20:20,0:20,0:20,50)
dimension propn(-20:20,-20:20,0:20,0:20,50)
dimension modn(-39:80), modh(-39:80)
dimension g(4,4), g2(4,4), ginv(4,4)
integer indx(4)
dimension omegainvo1(50), omegabinvo1(50), omegaplusinvo1(50)
dimension omegao1(50), mo1(50), omegabo1(50)
dimension omegainvo12(50),omegabinvo12(50),omegaplusinvo12(50)
dimension omegaplusinvo1p2(50),omegaplusinvo1p12(50)
dimension omegaplusinvo2p12(50)
dimension result(1)
C
a
M
ar
th
do i = 1-n, 2*n
itemp = mod(i+n,2*n)-n
if(itemp.gt.0) itemp = min(itemp, n-itemp)
if(itemp.lt.0) itemp = max(itemp,-n-itemp)
modn(i) = itemp
modh(i) = min(mod(i+n,n), n - mod(i+n,n))
enddo
do i = 1,l
do ir = 1,nr
a(i,1,ir) = 0.d0
enddo
enddo
do i1 = nhalf,n
i1n = modn(i1)
i1h = modh(i1)
do i = 1,l
do ir = 1,nr
a(i,2,ir) = 0.d0
enddo
enddo
do i2 = nhalf,n
i2n = modn(i2)
i2h = modh(i2)
do i = 1,l
do ir = 1,nr
181
pi = 4.0*atan(1.d0)
Enter minimum and maximum length of lattice (even, < 41)
read(*,*) nmin, nmax
CCCC Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one set per line
CCCC NB: No more than 50 different values for overlap parameters; idem for Iwasaki
ir
= 1
iov
= 1
iiw
= 1
iovflag = 1
iiwflag = 1
1
read(*,*,end=2) rtemp, mtemp,
&
c0temp, c1temp, c2temp, c3temp
do iov2 = 1, iov - 1
if((rtemp-r(iov2)).lt.1.d-9 .and. (mtemp-m(iov2)).lt.1.d-9) then
jov(ir) = iov2
iovflag = 0
endif
enddo
if(iovflag.eq.1) then
r(iov) = rtemp
m(iov) = mtemp
jov(ir) = iov
iov = iov + 1
endif
iovflag = 1
do iiw2 = 1, iiw - 1
if((c0temp-cprop0(iiw2)).lt.1.d-9 .and.
& (c1temp-cprop1(iiw2)).lt.1.d-9 .and.
& (c2temp-cprop2(iiw2)).lt.1.d-9 .and.
& (c3temp-cprop3(iiw2)).lt.1.d-9) then
jiw(ir) = iiw2
iiwflag = 0
endif
CCCC
on
st
an
tin
ou
ij4h = modh(i4+j4)
s2sq12 = (s22(j1+i1)+s22(j2+i2)+s22(j3+i3)+s22(j4+i4))
s2sq2 = (s22(j1)+s22(j2)+s22(j3)+s22(j4))
sisq12 = (si2(j1+i1)+si2(j2+i2)+si2(j3+i3)+si2(j4+i4))
sisq2 = (si2(j1)+si2(j2)+si2(j3)+si2(j4))
c2sq12 = (c22(j1+i1)+c22(j2+i2)+c22(j3+i3)+c22(j4+i4))
c2sq2 = (c22(j1)+c22(j2)+c22(j3)+c22(j4))
cisq12 = (ci2(j1+i1)+ci2(j2+i2)+ci2(j3+i3)+ci2(j4+i4))
cisq2 = (ci2(j1)+ci2(j2)+ci2(j3)+ci2(j4))
s2qu12 = (s24(j1+i1)+s24(j2+i2)+s24(j3+i3)+s24(j4+i4))
s2qu2 = (s24(j1)+s24(j2)+s24(j3)+s24(j4))
if (s2sq12.gt.1e-9 .and. s2sq2.gt.1e-9) then
hat2 = 0.25d0/s2sq2
hat12 = 0.25d0/s2sq12
do ir = 1,iov
mo2(ir) = m(ir) - r12(ir)*s2sq2
omegao2(ir) = sqrt(sisq2 + mo2(ir)**2)
omegainvo2(ir) = 1.d0/omegao2(ir)
omegaplusinvo2(ir) = 1.d0/(omegao2(ir)+ m(ir))
omegabo2(ir) = (omegao2(ir)-mo2(ir))
omegabinvo2(ir) = 1.d0/(omegao2(ir)-mo2(ir))
mo12(ir) = m(ir) - r12(ir)*s2sq12
omegaplusinvo1p2(ir) = 1.d0/(omegao1(ir)+ omegao2(ir))
enddo
do ir = 1, nr
prop = 1
b(1,4,ir) = b(1,4,ir) +
& hat2*omegabinvo12(jov(ir))*omegainvo12(jov(ir))**2*omegaplusinvo1p12(jov(ir))*
*2*
M
ar
th
a
C
&
s22(2*i4 + 2*j4)
enddo
endif
enddo
do ir = 1,nr
b(1,3,ir) = b(1,3,ir) + b(1,4,ir)
enddo
enddo
do ir = 1,nr
b(1,2,ir) = b(1,2,ir) + b(1,3,ir)*
& s2(2*i2 + 2*j2)
enddo
enddo
do ir = 1,nr
b(1,1,ir) = b(1,1,ir) + b(1,2,ir)*
& c22(2*i1 + j1)
enddo
enddo
do ir = 1,nr
prop = 1
if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0))
&
prop = 0.5d0 * prop
a(1,4,ir) = a(1,4,ir) + prop*b(1,1,ir)*
& omegabinvo1(jov(ir))*omegainvo1(jov(ir))**2
enddo
endif
enddo
if (i3.eq.n .or. (i3.eq.nhalf .and. nmod2.eq.0)) then
do ir = 1,nr
do i = 1,l
182
a(i,3,ir) = 0.d0
enddo
enddo
do i3 = nhalf,n
i3n = modn(i3)
i3h = modh(i3)
do i = 1,l
do ir = 1,nr
a(i,4,ir) = 0.d0
enddo
enddo
do i4 = nhalf,n
i4n = modn(i4)
i4h = modh(i4)
s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))
sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4))
c2sq1 = (c22(i1)+c22(i2)+c22(i3)+c22(i4))
cisq1 = (ci2(i1)+ci2(i2)+ci2(i3)+ci2(i4))
if (s2sq1.gt.1e-9) then
do ir = 1,iov
mo1(ir) = m(ir) - r12(ir)*s2sq1
enddo
do i = 1,1
do ir = 1,nr
b(i,1,ir) = 0.d0
enddo
enddo
do j1 = 1,n
j1n = modn(j1)
j1h = modh(j1)
ij1n = modn(i1+j1)
ij1h = modh(i1+j1)
do i = 1,1
do ir = 1,nr
b(i,2,ir) = 0.d0
enddo
enddo
do j2 = 1,n
j2n = modn(j2)
j2h = modh(j2)
ij2n = modn(i2+j2)
ij2h = modh(i2+j2)
do i = 1,1
do ir = 1,nr
b(i,3,ir) = 0.d0
enddo
enddo
do j3 = 1,n
j3n = modn(j3)
j3h = modh(j3)
ij3n = modn(i3+j3)
ij3h = modh(i3+j3)
do i = 1,1
do ir = 1,nr
b(i,4,ir) = 0.d0
enddo
enddo
do j4 = 1,n
j4n = modn(j4)
j4h = modh(j4)
ij4n = modn(i4+j4)
183
ou
an
tin
M
ar
th
a
C
on
st
a(i,4,ir) = a(i,4,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,3,ir) = a(1,3,ir) + a(1,4,ir)*
& si2(i3)
enddo
enddo
do ir = 1,nr
do i = 1,l
a(i,3,ir) = a(i,3,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,2,ir) = a(1,2,ir) + a(1,3,ir)*
& si(i2)
enddo
enddo
do ir = 1,nr
do i = 1,l
a(i,2,ir) = a(i,2,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,1,ir) = a(1,1,ir) + a(1,2,ir)
enddo
enddo
do ir = 1,nr
result(1) = 0.d0
result(1) = result(1) + a(1,1,ir) *
&(-6.)
result(1) = result(1) * 2**4 / float(n)**8
write(*,*) n,r(jov(ir)), m(jov(ir)),
&
cprop0(jiw(ir)), cprop1(jiw(ir)),
&
cprop2(jiw(ir)), cprop3(jiw(ir)), result
enddo
call flush(6)
1000 continue
stop
end
tin
ou
Appendix C: The algorithm for
improving the Symanzik coefficients
In Chapter 8 we discussed an improvement method of perturbation theory for the Symanzik
gluon action, that has the effect of replacing the bare parameters of the action with im-
an
proved ones. This dressing is based on defining improved Symanzik coefficients c̃i that
depend on the bare ones, ci , on the coupling constant g0 and on the number of colors N.
The dressed coefficients obey the recursive Eqs. (8.32) that can be solved using a fixed
st
point procedure. In this Appendix we present the algorithm that we have developed and
takes as an input a set of values for the parameters g0 , N, ci and evaluates c̃i .
on
Before providing our algorithm, let us set up the necessary notation:
• nc is the number of colors N
C
• betacoupling is related to the coupling constant g0 through the equation
2 nc
g02
a
betacoupling =
th
• c(0), c(1), c(2), c(3) denote the bare Symanzik coefficients
• gtilde(i) are the γ̃i ’s of Eq. (8.32)
M
ar
• cimpr(i) are the dressed Symanzik parameters given by (Eq. (8.31))
cimpr(i) =
184
2 nc gtilde(i)
betacoupling
CCCC
do i = 0, 3
gtildeold(i) = gtilde(i)
enddo
gtildenorm = gtilde(0) + 8*gtilde(1) + 16*gtilde(2) + 8*gtilde(3)
call beta(gtilde(0)/gtildenorm,gtilde(1)/gtildenorm,
&
gtilde(2)/gtildenorm,gtilde(3)/gtildenorm,btilde,n)
do i = 0, 3
btilde(i) = btilde(i)/gtildenorm
enddo
do i = 0, 3
gtilde(i) = gamma(i) * exp(-btilde(i)*(nc-1)/(4.d0*nc)) *
&
(1.d0 - btilde(i)/12.d0 - (nc-2)*btilde(i)/6.d0
&
+ (nc-2)*btilde(i)**2/96.d0)
enddo
call flush(6)
if((abs(gtilde(0)-gtildeold(0)).gt.1.d-7).or.
(abs(gtilde(1)-gtildeold(1)).gt.1.d-7).or.
(abs(gtilde(2)-gtildeold(2)).gt.1.d-7).or.
(abs(gtilde(3)-gtildeold(3)).gt.1.d-7) ) goto 3
2
goto 1
continue
stop
end
c %%%%%%%%%%%%%%%%%%%% beta subroutine %%%%%%%%%%%%%%%%%%%%
subroutine beta(z0,z1,z2,z3,b,n)
implicit integer*4 (i-n)
real*8 k2p1, k4p1
real*8 m, m1, mo1
parameter (l = 11)
dimension a(l,4,300)
dimension b(0:3)
dimension s2(-1600:1600), c2(-1600:1600), s22(-1600:1600)
nmod2 = mod(n,2)
nhalf = (n+nmod2)/2
n4 = n*4
do i = -12*n,12*n
s2(i) = sin(i*pi/n)
c2(i) = cos(i*pi/n)
s22(i) = s2(i)**2
c22(i) = c2(i)**2
s24(i) = s2(i)**4
c24(i) = c2(i)**4
si(i) = sin(2*i*pi/n)
ci(i) = cos(2*i*pi/n)
si2(i) = si(i)**2
ci2(i) = ci(i)**2
enddo
do i = 1,l
do ir = 1,nr
a(i,1,ir) = 0.d0
enddo
enddo
do i1 = nhalf,n
do i = 1,l
do ir = 1,nr
a(i,2,ir) = 0.d0
enddo
enddo
do i2 = nhalf,n
do i = 1,l
do ir = 1,nr
a(i,3,ir) = 0.d0
enddo
enddo
do i3 = nhalf,n
do i = 1,l
do ir = 1,nr
a(i,4,ir) = 0.d0
enddo
enddo
do i4 = nhalf,n
s2sq1 = (s22(i1)+s22(i2)+s22(i3)+s22(i4))
185
M
ar
th
write(*,*)
write(*,*) c, betacoupling
write(*,*) cimp
call flush(6)
a
do i = 0, 3
cimp(i) = gtilde(i) * 2 * nc / betacoupling
enddo
pi = 4.0*atan(1.d0)
Write up to 300 sextuplets of values for r, m, c0, c1, c2, c3, one set per line
r(1) = 0.
!! Dummy value, will never need it
m(1) = 0.
!! Dummy value, will never need it
cc1(1) = z2 + z3
cc2(1) = z1 - z2 - z3
nr = 1
do ir = 1, nr
r24(ir) = 4.d0*r(ir)**2
r12(ir) = 2.d0*r(ir)
enddo
C
&
&
&
ou
read(*,*,end=2) c(0), c(1), c(2), c(3), betacoupling
do i = 0, 3
gamma(i) = c(i) * betacoupling/2/nc
gtilde(i) = gamma(i)
enddo
st
3
colors, we present the unimproved’
c0, c1, c2, c3, ’
constant beta, and’
coefficients cimp0, cimp1, cimp2, cimp3’
on
1
’For ’, nc, ’
’coefficients
’the coupling
’the improved
an
tin
write(*,*)
write(*,*)
write(*,*)
write(*,*)
dimension c22(-1600:1600), s24(-1600:1600), c24(-1600:1600)
dimension si(-1600:1600), ci(-1600:1600), si2(-1600:1600)
dimension ci2(-1600:1600)
dimension r(300), r24(300), r12(300), m(300)
dimension cc1(300), cc2(300)
dimension fhat1(300), m1(300)
dimension dprop1(4,4), prop1(4,4), g(4,4), ginv(4,4)
integer indx(4)
dimension omegao1(300), omegabo1(300), mo1(300)
dimension result(4)
C. The algorithm for improving the Symanzik coefficients
program main
implicit integer*4 (i-n)
dimension btilde(0:3), c(0:3), gamma(0:3),
&
gtilde(0:3), gtildeold(0:3), cimp(0:3)
parameter(nc=3)
n = 64
ou
an
tin
st
on
C
a
M
ar
th
186
a(2,3,ir) = a(2,3,ir) + a(2,4,ir)
a(3,3,ir) = a(3,3,ir) + a(3,4,ir)
a(4,3,ir) = a(4,3,ir) + a(1,4,ir)*
& c22(i3)
a(5,3,ir) = a(5,3,ir) + a(1,4,ir)*
& s22(i3)
enddo
enddo
do ir = 1,nr
do i = 1,l
a(i,3,ir) = a(i,3,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,2,ir) = a(1,2,ir) + a(2,3,ir)
a(2,2,ir) = a(2,2,ir) + a(2,3,ir)*
& c22(i2)
a(3,2,ir) = a(3,2,ir) + a(3,3,ir)*
& c22(i2)
a(4,2,ir) = a(4,2,ir) + a(1,3,ir)*
& s2(i2)
a(5,2,ir) = a(5,2,ir) + a(4,3,ir)*
& s2(i2)
a(6,2,ir) = a(6,2,ir) + a(5,3,ir)*
& s2(i2)
a(7,2,ir) = a(7,2,ir) + a(1,3,ir)*
& c22(i2)*s2(i2)
enddo
enddo
do ir = 1,nr
do i = 1,l
a(i,2,ir) = a(i,2,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,1,ir) = a(1,1,ir) + a(1,2,ir)*
& s22(i1)
a(2,1,ir) = a(2,1,ir) + a(4,2,ir)*
& s2(i1)
a(3,1,ir) = a(3,1,ir) + a(1,2,ir)*
& c22(i1)*s22(i1)
a(4,1,ir) = a(4,1,ir) + a(2,2,ir)*
& s22(i1)
a(5,1,ir) = a(5,1,ir) + a(7,2,ir)*
& s2(i1)
a(6,1,ir) = a(6,1,ir) + a(1,2,ir)*
& s22(i1)
a(7,1,ir) = a(7,1,ir) + a(3,2,ir)*
& s22(i1)
a(8,1,ir) = a(8,1,ir) + a(5,2,ir)*
& s2(i1)
a(9,1,ir) = a(9,1,ir) + a(6,2,ir)*
& s2(i1)
a(10,1,ir) = a(10,1,ir) + a(3,2,ir)*
& s22(i1)
a(11,1,ir) = a(11,1,ir) + a(5,2,ir)*
& s2(i1)
enddo
enddo
do ir = 1,nr
result(1) = 0.d0
result(1) = result(1) + a(1,1,ir) *
sisq1 = (si2(i1)+si2(i2)+si2(i3)+si2(i4))
s2qu1 = (s24(i1)+s24(i2)+s24(i3)+s24(i4))
k2p1 =
4* (s22(i1)+s22(i2)+s22(i3)+s22(i4))
k4p1 =
16* (s24(i1)+s24(i2)+s24(i3)+s24(i4))
if (s2sq1.gt.1e-9) then
hat1 = 0.25d0/s2sq1
do ir = 1,nr
dummy = (1-cc1(ir)*k2p1)
do mu = 1, 4
if (mu.eq.1) imu = i1
do nu = 1, mu
if (nu.eq.1) inu = i1
g(mu,nu) = (1-dummy) * (2*s2(imu))*(2*s2(inu))
&
+ cc2(ir) *(2*s2(imu))**3*(2*s2(inu))
&
+ cc2(ir) *(2*s2(imu))*(2*s2(inu))**3
if (mu.eq.nu) g(mu,nu) = g(mu,nu) + dummy * k2p1
&
- cc2(ir) * k4p1 - cc2(ir) * k2p1 * (2*s2(imu))**2
if (nu.lt.mu) g(nu,mu) = g(mu,nu)
enddo
enddo
do mu = 1,4
do nu = 1,4
ginv(mu,nu) = 0.d0
enddo
ginv(mu,mu) = 1.d0
enddo
call ludcmp(g,4,4,indx,d)
do nu = 1, 4
call lubksb(g,4,4,indx,ginv(1,nu))
enddo
do mu = 1, 4
do nu = 1, mu
prop1(mu,nu) = ginv(mu,nu)
dprop1(mu,nu) = ginv(mu,nu)
if (mu.eq.nu) dprop1(mu,nu) = dprop1(mu,nu) - hat1
if (nu.lt.mu) prop1(nu,mu) = prop1(mu,nu)
if (nu.lt.mu) dprop1(nu,mu) = dprop1(mu,nu)
enddo
enddo
prop = 1
if (i4.eq.n .or. (i4.eq.nhalf .and. nmod2.eq.0))
&
prop = 0.5d0 * prop
a(1,4,ir) = a(1,4,ir) + prop*
& prop1(1,2)
a(2,4,ir) = a(2,4,ir) + prop*
& prop1(2,2)
a(3,4,ir) = a(3,4,ir) + prop*
& prop1(3,3)
enddo
endif
enddo
do ir = 1,nr
do i = 1,l
a(i,4,ir) = a(i,4,ir) * 0.5d0
enddo
enddo
endif
do ir = 1,nr
a(1,3,ir) = a(1,3,ir) + a(1,4,ir)
+ a(2,1,ir) *
* 2**4 / float(n)**4
ou
+ a(3,1,ir) *
+ a(4,1,ir) *
+ a(5,1,ir) *
* 2**4 / float(n)**4
an
tin
+ a(6,1,ir) *
+ a(7,1,ir) *
+ a(8,1,ir) *
+ a(9,1,ir) *
* 2**4 / float(n)**4
+ a(10,1,ir) *
+ a(11,1,ir) *
187
M
ar
th
a
C
on
st
* 2**4 / float(n)**4
&(8.)
result(1) = result(1)
&(-8.)
result(2) = 0.d0
&(16.)
&(16.)
&(-32.)
result(3) = 0.d0
&(8.)
&(8.)
&(-16.)
&(-8.)
result(4) = 0.d0
&(24.)
&(-24.)
b(0) = result(1)
b(1) = result(2)
b(2) = result(3)
b(3) = result(4)
enddo
return
end
ou
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