Nucleon to Pion Transition Distribution Amplitudes in a Light

Transcription

Nucleon to Pion
Transition Distribution Amplitudes
in a Light-Cone Quark Model
Manuel Pincetti
Department of Nuclear and Theoretical Physics, University of Pavia
and INFN, Section of Pavia
work done in collaboration with S. Boffi and B. Pasquini
Transversity ’08 — Ferrara, 28-31 may 2008
Manuel Pincetti (DFNT and INFN Pavia)
N → π TDAS in a LCCQM
Transversity ’08 - May 31, 2008
1 / 29
Outline
1
Motivation
2
Nucleon DAs
3
Meson Cloud
4
From GPDs to TDAs
“Historical path”
Interpretation of the TDAs
5
Mesonic TDAs
6
Baryonic TDAs
Nucleon to Pion TDAs
2 / 29
Motivation
Motivation
Nucleon to Pion TDAs encoded the non-perturbative transition between two
hadronic states that takes place in the processes...
3 / 29
Motivation
Motivation
γ ∗ N → N 0 π: pion electroproduction...EXPs: JLab, Hermess & Compass;
3 / 29
Motivation
Motivation
P̄ P → γ ∗ π: proton-antiproton annihilation...EXP: GSI-FAIR.
3 / 29
Motivation
Motivation
P̄ P → γ ∗ π: proton-antiproton annihilation...EXP: GSI-FAIR.
⇒ TDAs represent the most direct source of information
about the Pion-Cloud of the Nucleon!!!
3 / 29
Nucleon DAs
A step backwards...Nucleon DAs
Nucleon DAs
⇓
Standard Tool to investigate exclusive processes in
QCD...
4 / 29
Nucleon DAs
Within the “Convolution Approach” (Brodsky & Lepage; Efremov & Radyuskin) they
encapsulate the bound-state dynamics (process INdep.) factorized from the
hard-scattering amplitude (process dep.)
2
Z
1
Z
[dx]
GM (Q ) =
0
1
∗
2
[dy] DA(yi , Q ) h(xi , yi , Q2 )DA(xi , Q2 )
0
4 / 29
Nucleon DAs
At leading twist starting from the FT of the matrix element of the trilocal
operator 3 DAs appear
Braun et al. NPB589 (’00)
4F
=
h
h0|ijk uiα (z1 n)[z1 ; z0 ]i0 i ujβ (z2 n)[z2 ; z0 ]j 0 j dkγ (z3 n)[z3 ; z0 ]k0 k |P (p)i
5 +
N
5
+
N
µ 5 +
fN V N (p
/C)αβ (γ N )γ + A (p
/γ C)αβ (N )γ + T (σpµ C)αβ (γ γ N )γ
i
4 / 29
Nucleon DAs
4F h0|ijk uiα (z1 n)[z1 ; z0 ]i0 i ujβ (z2 n)[z2 ; z0 ]j 0 j dkγ (z3 n)[z3 ; z0 ]k0 k |P (p)i
=
h
5 +
N
5
+
N
µ 5 +
fN V N (p
i
In the A+ = 0 gauge can be interpreted as probability amplitude for the nucleon
to
P
consist of valence quarks with longitudinal momentum fraction xi ∈ [0, 1], i xi = 1
4 / 29
Nucleon DAs
=
h
5 +
N
5
+
N
µ 5 +
fN V N (p
i
to
P
⇓
Best way to test the lower Fock state component of the Nucleon state
|N i = |qqqi + |qqqq q̄i + · · ·
4 / 29
Nucleon DAs
=
h
5 +
N
5
+
N
µ 5 +
fN V N (p
i
to
P
⇓
|N i = |qqqi + |qqqq q̄i + · · ·
A lot of theoretical work has been done...
Chernyak & Zhitnitsky(’84), King & Sachrajda (’87), CZ+Oglobin (’88), Dziembowski (’88), Dziembowski &
Franklin (’90), Stefanis & Bergmann (’93), Bolz & Kroll (’96), Braun et et al. (’99)
4 / 29
Nucleon DAs
=
h
5 +
N
5
+
N
µ 5 +
fN V N (p
i
to
P
⇓
|N i = |qqqi + |qqqq q̄i + · · ·
Chernyak & Zhitnitsky(’84), King & Sachrajda (’87), CZ+Oglobin (’88), Dziembowski (’88), Dziembowski &
Franklin (’90), Stefanis & Bergmann (’93), Bolz & Kroll (’96), Braun et et al. (’99)...
...nevertheless (so far) moderate success in phenomenology.
4 / 29
Nucleon DAs
=
h
5 +
N
5
+
N
µ 5 +
fN V N (p
i
to
P
⇓
|N i = |qqqi + |qqqq q̄i + · · ·
...nevertheless (so far) moderate success in phenomenology.
Last month → QCDSF/UKQCD coll., 0804.1877 [hep-lat]
“Nucleon distribution amplitudes from lattice QCD”...
Lattice predictions for the moments of the Nucleon DAs.
4 / 29
Nucleon DAs
Nucleon DAs are based on the assumption that the Fock states with more than a
minimum number of costituents play no significant role...
5 / 29
Nucleon DAs
⇒ NO need of higher Fock states component in a model calculation!
5 / 29
Nucleon DAs
In our Light-Cone approach we truncate the expansion of the Nucleon state at
first order...
|N i = Ψ(3q) |N (qqq)i + · · ·
5 / 29
Nucleon DAs
In our Light-Cone approach we truncate the expansion of the Nucleon state at
first order...
|N i = Ψ(3q) |N (qqq)i + · · ·
We describe the nucleon by means of the LCWFs Ψf3q expressed in terms of the
canonical WFs Ψc3q , solutions of the instant-form Hamiltonian, through the
relation
p
P
⇒ Ψf3q = ω1 ω2 ω3 /x1 x2 x3 M0 {λ0 } h{λi }|R† |{λ0i }iΨc3q
i
where R is a Generalized Melosh Rotation and
M0 =
N q
X
m2i + ~ki2
i
is the free-mass operator
5 / 29
Nucleon DAs
♠ Instant-Form Wave Function: Ψ = ΦI ⊗ ΦS ⊗ Ψ̃({~ki })
Spin and Isospin component: SU(6) symmetric
x0 , time;
x1 , x2 , x3
space
Momentum Space component:
p 2
P s-wave
N
,
with
M
=
Ψ̃ = (M 2 +β
ki + m2q
0
2 )γ
i
0
3 parameters: mq , β and γ fitted to reproduce the
magnetic moment of the nucleon and gA :
mq = 263 MeV, β= 607 MeV and γ = 3.5
Schlumpf, PhD Thesis, hep-ph/9211255
6 / 29
Nucleon DAs
p 2
P s-wave
N
Ψ̃ = (M 2 +β
ki + m2q
2 )γ , with M0 =
i
x0 , time;
x1 , x2 , x3
space
0
mq = 263 MeV, β= 607 MeV and γ = 3.5
♠ Ligh-Front wave function
+
x time;
x− , x1 , x2
space
Breaking of SU(6) symmetry
Non-zero quark orbital momentum
6 / 29
Nucleon DAs
p 2
P s-wave
N
ki + m2q
Ψ̃ = (M 2 +β
2 )γ , with M0 =
i
x0 , time;
x1 , x2 , x3
space
0
mq = 263 MeV, β= 607 MeV and γ = 3.5
x+ time;
x− , x1 , x2
space
Melosh Rotation
=⇒
↑
qLC
=
w[(k+ + m)qI↑ + (k1 + ik2 )qI↓ ]
↓
qLC
=
w[(k1 + ik2 )qI↑ + (k+ + m)qI↓ ]
6 / 29
Nucleon DAs
p 2
P s-wave
N
ki + m2q
Ψ̃ = (M 2 +β
2 )γ , with M0 =
i
x0 , time;
x1 , x2 , x3
space
0
mq = 263 MeV, β= 607 MeV and γ = 3.5
+
x time;
x− , x1 , x2
space
Melosh Rotation
=⇒
↑
qLC
=
w[(k+ + m)qI↑ + (k1 + ik2 )qI↓ ]
↓
qLC
=
w[(k1 + ik2 )qI↑ + (k+ + m)qI↓ ]
⇒ NON trivial SPIN STRUCTURE and a correlation between quark SPIN and quark
ORBITAL ANGULAR MOMENTUM.
6 / 29
Nucleon DAs
Results for the Nucleon DAs
Boffi, Pasquini and M.P., drafting
Defining
.
M λαβ,γ = 4F h0|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p)i
λ Proton Helicity
7 / 29
Nucleon DAs
Defining
.
λ Proton Helicity
p
V =
Ap =
1 √
1
+ − 32
fN 4 2 (p )
3
1 √
1
+ −2
fN 4 2 (p )
T p = − f1N
↑
M12,1
+
↑
M21,1
↑
↑
M21,1
− M12,1
↑
1
+ − 32
√
M11,2
4 (p )
2
7 / 29
Nucleon DAs
Defining
M λαβ,γ
.
j
ijk i
k
= 4F h0| uα (z1 n)uβ (z2 n)dγ (z3 n)|P (p)i
λ Proton Helicity
3
↑
↑
M12,1
+ M21,1
↑
↑
1
+ − 32
√
M
−
M
4 (p )
21,1
12,1
2
Vp =
1 √
1
+ −2
fN 4 2 (p )
Ap =
1
fN
T p = − f1N
↑
1
+ − 23
√
M11,2
4 (p )
2
The symmetry of the WFs and the
identity of the 2 u − quarks
⇓
Only 1 of the 3 DAs is indipendent...
.
Historically Φ = V − A
7 / 29
Nucleon DAs
Defining
.
λ Proton Helicity
↑
↑
+ M21,1
M12,1
↑
↑
1
+ − 32
√
−
M
M
4 (p )
12,1
21,1
2
Vp =
1 √
1
+ − 23
fN 4 2 (p )
Ap =
1
fN
T p = − f1N
↑
1
+ − 23
√
M11,2
4 (p )
2
λ
M αβ,γ
= −√
N,[f ]
×Ψ̃λ
24
x1 x2 x3
The symmetry of the WFs and the
identity of the 2 u − quarks
⇓
Only 1 of the 3 DAs is indipendent...
.
Historically Φ = V − A
Z Y
3
dκi⊥ X λ1
λ2
+ λ3
+
u+α (x1 p+
1 )u+β (x2 p1 )u+γ (x3 p1 )
3 ]2
[2(2π)
i=1
λ1,2,3
X
3
{x1 , κ1⊥ ; λ1 , u}{x2 , κ2⊥ ; λ2 , u}{x3 , κ3⊥ ; λ3 , d} δ (2)
κi⊥
i=1
7 / 29
Nucleon DAs
DA Φ from Chernyak, Ogloblin
& Zhitnisky, Yad. Fiz. 48, 841 (’88)
QCD Sum Rule
(Different POV to emphasize the structure)
DA Φ from Bolz & Kroll,
Z. Phys A 356, 327 (’96)
Data Fit
Our model calculation for the DA Φ
8 / 29
Nucleon DAs
Comparison of the moments of the DA Φ: l + m + n ≤ 2
.
Φ(l,m,n) =
Z
Z
1
n
2
[dx]xl1 xm
2 x3 Φ(x1 , x2 , x3 , Q )
0
(l, m, n)
0 0 0
1 0 0
0 1 0
0 0 1
2 0 0
0 2 0
0 0 2
1 1 0
1 0 1
0 1 1
COZ
1
0.54 − 0.62
0.18 − 0.20
0.20 − 0.25
0.32 − 0.42
0.065 − 0.088
0.09 − 0.12
0.08 − 0.10
0.09 − 0.11
−0.03 − 0.03
1
[dx]Φ(x1 , x2 , x3 , Q2 )
0
SB
1
0.572
0.184
0.244
0.338
0.066
0.170
0.139
0.096
0.018
DF
1
0.582
0.213
0.207
0.367
0.085
0.083
0.108
0.106
−0.021
BKfit
1
0.381
0.309
0.309
0.179
0.125
0.125
0.101
0.101
0.083
Our
1
0.346
0.331
0.323
0.152
0.142
0.137
0.099
0.096
0.091
LAT
1
0.394
0.302
0.304
0.18
0.132
0.138
0.113
0.112
0.05
COZ: Chernyak, Ogloblin & Zhitnitsky, Yad.Fiz.48 (’88); SB: Stefanis & Bergman, Phys.Rev.D47
(’93); DF: Dziembowski & Franklin, Phys.Rev.D42 (’90); BK: Bolz & Kroll, Z.PhysA356 (’96);
LAT: QCDSF/UKQCD coll., arXiv:0804.1877 [hep-lat].
9 / 29
Meson Cloud
Does Meson Cloud matter?
The role of a nonperturbative Pion Cloud surronding the nucleon is well explained
in QCD as a consequence of the spontaneously-broken chiral symmetry. The
Meson Cloud picture accounts for some unexpected experimental results...
¯
Gottfried sum rule, dū;
Thomas, PLB 126 (’83)
Proton (neutron) charge density, long range positively (negatively) charged
component;
Kelly, PRC 66 (’02)
Neutron form factor GnE (Q2 ), pronunced bump structure;
Friedrich & Walcher,
EPJ A 17 (’03)
Proton Spin, replacement of quark spin by quark and anti-quark OAM.
Myhrer & Thomas,
[hep-ph] 0709.4067 (’07)
10 / 29
Meson Cloud
Meson Cloud Model
Pasquini & Boffi, PRD 73 (’06)
In such a model the physical nucleon Ñ is made of a bare nucleon N dressed by a
surrounding meson cloud...
X (BM )
Ψ(3q)(qq̄) |B(qqq)M (q q̄)i + · · ·
|Ñ i = Ψ(3q) |N (qqq)i +
B,M
11 / 29
Meson Cloud
Meson Cloud Model
X (BM )
|Ñ i = Ψ(3q) |N (qqq)i +
Ψ(3q)(qq̄) |B(qqq)M (q q̄)i + · · ·
B,M
Light-cone Hamiltonian: HLC =
HLC |p̃N , λ; Ñ i =
P
B,M
B
H0 (q) + H0M (q) + HI (N, BM )
2
p2N ⊥ + MN
|p̃N , λ; Ñ i.
+
pN
11 / 29
Meson Cloud
Meson Cloud Model
X (BM )
|Ñ i = Ψ(3q) |N (qqq)i +
Ψ(3q)(qq̄) |B(qqq)M (q q̄)i + · · ·
B,M
Light-cone Hamiltonian: HLC =
HLC |p̃N , λ; Ñ i =
P
B,M
B
H0 (q) + H0M (q) + HI (N, BM )
2
p2N ⊥ + MN
|p̃N , λ; Ñ i.
+
pN
Expanding the nucleon WF in terms of the eigenstates of H0 ≡ H0B (q) + H0M (q),
|p̃N , λ; Ñ i =
√
Z
|p̃N , λ; N i +
X0 |n1 ihn1 |HI |p̃N , λ; N i
EN − En1 + i
n1
X 0 |n2 ihn2 |HI |n1 ihn1 |HI |p̃N , λ; N i
+ ···
+
(EN − En2 + i)(EN − En1 + i)
n ,n
1
√
Z renormalization factor
!
2
Speth & Thomas, Adv.Nucl.Phys. 24, (’98)
11 / 29
Meson Cloud
In the one-meson approximation, truncating the series expansion to the first order
in HI ...
|p̃N , λ; Ñ i =
√
Z|p̃N , λ; N i +
X Z dyd2 k⊥
B,M
2(2π)3
1
y(1 − y)
p
X
λ (N,BM )
φλ0 λ00
(y, k⊥ )
λ0 ,λ00
+
0
00
× |yp+
N , k⊥ + ypN ⊥ , λ ; Bi |(1 − y)pN , −k⊥ + (1 − y)pN ⊥ , λ ; M i
where:
λ (N,BM )
φλ0 λ00
Vλλ0 ,λ00 (N,BM )
1
2 −M 2
M
(y,k⊥ )
y(1−y)
N
BM
(y, k⊥ ) = √
is the probability amplitude for a nucleon with helicity λ to fluctuate into a
virtual BM system with the baryon having helicity λ0 , longitudinal
momentum fraction y and transverse momentum k⊥ , and the meson having
helicity λ00 , longitudinal momentum fraction 1 − y and transverse momentum
−k⊥ ;
+
0
00
|yp+
N , k⊥ + ypN ⊥ , λ ; Bi |(1 − y)pN , −k⊥ + (1 − y)pN ⊥ , λ ; M i
Baryon
↔
Meson
Fluctuation
12 / 29
From GPDs to TDAs
Transition Distribution
Amplitudes
13 / 29
From GPDs to TDAs
From GPDs...to TDAs
For u DVCS, the non-perturbative part does not describe anymore a H → H
transition, but rather a Meson to Photon transition
14 / 29
From GPDs to TDAs
From GPDs...to TDAs
For u DVCS, the non-perturbative part does not describe anymore a H → H
transition, but rather a Meson to Photon transition
or
a Baryon to Photon
&
Baryon to Meson Transition
14 / 29
“Historical path”
History and Authorship
First idea given by Frankfurt, Polyakov & Strikman in hep-ph/9808449 and F.P.S.
+ Pobylitsa in PRD60 (’99). To study γ ∗ + p → B + M one has to introduce what
they called Skewed Distribution Amplitudes (SDAs).
’04 Pire & Szymanowski gave the definitions for, what they called, the Meson to
Photon Transition Distribution Amplitudes (TDAs) [appeared in PRD71, (’05)],
M M → γ ∗ γ.
’05 Pire & Szymanowski → definitions of the 8 indipendent Baryon to Meson
TDAs, PLB622 (’05), p̄N → γ ∗ π.
’05 First model estimations for the mesonic TDAs, π → γ, by Tiburzi PRD72.
’06 Lansberg, Pire & Szymanowski → definitions of the 16 indipendent Baryon to
Photon TDAs, γ ∗ N → N 0 γ, NPA 782 (’06).
’07 First estimations for 3 (of 8) Baryon to Meson TDAs, γ ∗ N → N 0 π, Lansberg,
Pire & Szymanowski PRD75 (’07).
’07 First paper entitled “Transition Distribution Amplitudes”, making systematic
classification, Lansberg, Pire & Szymanowski, 0709.2567 [hep-ph].
’08 Model estimations for all the 8 Nucleon to Pion TDAs,
Boffi, Pasquini & M.P., drafting.
15 / 29
Mesonic Sector
The mesonic TDAs possess an interpretation at the amplitude level and provide
with information on how a meson and a photon “look alike”...
16 / 29
Mesonic Sector
Baryonic Sector
The baryonic TDAs rather provide information on how one can find a meson or a
photon in the baryon...
16 / 29
Mesonic Sector
Baryonic Sector
The baryonic TDAs rather provide information on how one can find a meson or a
photon in the baryon...
Crucial to test the Pion Cloud contribution!!!
16 / 29
Mesonic TDAs
Few words on Mesonic TDAs (mTDAs)...not our business
Z
dz − ixP + z−
−z
z
e
hπ(pπ )|ψ̄(
)Γψ( )|γ(pγ , ε)i|z+ =z =0
⊥
2π
2
2
γµ →
Γ: γ µ γ5 →
σ µν
1 µεP ∆⊥ π
V (x, ξ, t)
fπ ε
Vectorial
1
fπ (ε
· ∆)P µ Aπ (x, ξ, t) Axial
h
→ εµνρσ Pσ ερ T1 (x, ξ, t)
i
− f1π (ε · ∆)∆⊥ρ T2 (x, ξ, t) Tensorial
4 indipendent mTDAs in spite of 2 indipendent mGPDs;
Polinomiality condition OK! ⇒ Double Distributions;
Initial state 6= Final ⇒ No conditions from Time Rev. ⇒ odd-powers of ξ;
Models: DD, Tiburzi ’05; Lansberg et al. ’06; Spectral Model, Broniowski et
al. ’07; NJL, Courtoy & Noguera ’07; χQM, Kotko & Praszalowicz ’08.
EXP. SIDE:
mTDAs complete the Kinematical domain of γ ∗ γ → M1 M2
Data exist from LEP & CLEO for ρρ states
17 / 29
Baryonic TDAs
Baryonic TDAs (bTDAs)
For both meson and photon case 3 quarks are exchanged in the t−channel,
satysfing the relation x1 + x2 + x3 = 2ξ (ξ ≥ 0)
18 / 29
Baryonic TDAs
=⇒ bTDAs are FT of matrix elements of the same operator that appears in the
usual baryonic Distribution Amplitudes (bDAs),
ijk qαi (z1 n)qβj (z2 n)qγk (z3 n)
18 / 29
Baryonic TDAs
usual baryonic Distribution Amplitudes (bDAs), ijk qαi (z1 n)qβj (z2 n)qγk (z3 n)
Same operator → same ren. group eqs., but different kin. → different evol. eqs!
18 / 29
Baryonic TDAs
As done for GPDs one can define different regions:
ERBL: xi ≥ 0; DGLAP1: x1 ≥ 0; x2 ≥ 0; x3 ≤ 0; DGLAP2: x1 ≥ 0; x2 ≤ 0; x3 ≤ 0;
18 / 29
Baryonic TDAs
As done for GPDs one can define different regions:
ERBL: xi ≥ 0; DGLAP1: x1 ≥ 0; x2 ≥ 0; x3 ≤ 0; DGLAP2: x1 ≥ 0; x2 ≤ 0; x3 ≤ 0;
So far only ERBL region solved → asymptotic form, NOT phenomenologically
usefull as for DAs (unfair description of Form Factors)
Pire & Szymanowski, PLB622 (’05)
18 / 29
Baryonic TDAs
Our Focus: Nucleon to Pion TDAs
Starting from the leading twist decomposition of the FT of the matrix element
hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i
19 / 29
Baryonic TDAs
one obtains 8 indipendent functions:
2 Vectorials
0
V1pπ
and
0
V2pπ ;
0
0
0
2 Axials Apπ
and Apπ
; and 4 Tensorials Tipπ (i=1,...,4)
1
2
19 / 29
Baryonic TDAs
one obtains 8 indipendent functions:
0
0
0
0
0
2 Vectorials V1pπ and V2pπ ; 2 Axials Apπ
and Apπ
; and 4 Tensorials Tipπ (i=1,...,4)
1
2
4F hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i
i
=
fN h pπ0
+
pπ 0
5
5 +
V1 (p
/C)αβ (N )γ + A1 (p
/γ C)αβ (γ N )γ
fπ
0
0
+
+T1pπ (σpµ C)αβ (γ µ N + )γ + M −1 V2pπ (p
/C)αβ (∆/T N )γ
0
0
0
0
5
5
+
−1 pπ
+
+M −1 Apπ
(p
/γ C)αβ (γ ∆/T N )γ + M T2 (σp∆T C)αβ (N )γ
2
+M −1 T3pπ (σpµ C)αβ (σ µ∆T N + )γ + M −2 T4pπ (σp∆T C)αβ (∆/T N + )γ
i
19 / 29
Baryonic TDAs
Nucleon to Pion TDAs ↔ Nucleon DAs
4F hπ
i
0
(pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i
=
fN h pπ0
pπ 0
+
5
5 +
V
(p
/C)αβ (N )γ + A1 (p
fπ 1
0
0
+
0
0
+
−1 pπ
5
5
+
+M −1 Apπ
2 (p
0
0
i
20 / 29
Baryonic TDAs
4F hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i =
i
fN h pπ0
pπ 0
+
5
5 +
V1 (p
/C)αβ (N )γ + A1 (p
fπ
0
0
+
0
0
5
5
+
−1 pπ
+
+M −1 Apπ
2 (p
0
0
i
Restricting to the case with ∆T = 0, where ∆ = pπ − pN , only 3 TDAs survive...
20 / 29
Baryonic TDAs
4F hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i =
i
fN h pπ0
pπ 0
+
5
5 +
V1 (p
/C)αβ (N )γ + A1 (p
fπ
0
0
+
0
0
5
5
+
−1 pπ
+
+M −1 Apπ
2 (p
0
0
i
Restricting to the case with ∆T = 0, where ∆ = pπ − pN , only 3 TDAs survive...
4F hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i
=
|∆ =0
T
i
i
fN h pπ0
+
pπ 0
5
5 +
pπ 0
µ +
V1 (p
/C)αβ (N )γ + A1 (p
/γ C)αβ (γ N )γ + T1 (σpµ C)αβ (γ N )γ
fπ
in complete analogy to the usual Nucleon DAs!!!
20 / 29
Baryonic TDAs
Nucleon DAs
4F hπ 0 (pπ )|O|P (p1 , s1 )i
i
=
|∆ =0
T
fN h pπ0
+
V
(p
/C)αβ (N )γ
fπ
0
+T pπ (σpµ C)αβ (γ µ N + )γ
=
h
5 +
fN V p (p
/C)αβ (γ N )γ
5
+
+Ap (p
/γ C)αβ (N )γ
0
5
5 +
+Apπ (p
4F h0|O|P (p1 , s1 )i
i
+T p (σpµ C)αβ (γ µ γ 5 N + )γ
i
∆T = 0 ⇒ 1:1 correspondence among Nucleon to Pion TDAs and Nucleon DAs in the
SOFT-PION LIMIT: ξ → 1 Lansberg et al., PRD75 (’07)
Soft Pion Theorem
0
i
hπ|O|P i = − h0|[Qa5 , O]|P i
fπ
+ nucleon pole term
V pπ (x1 , x2 , x3 , ξ, ∆2 )
0
Apπ (x1 , x2 , x3 , ξ, ∆2 )
0
T pπ (x1 , x2 , x3 , ξ, ∆2 )
=
=
=
1 p x1 x2 x3
V ( ,
,
)
4ξ
2ξ 2ξ 2ξ
1 p x1 x2 x3
A ( ,
,
)
4ξ
2ξ 2ξ 2ξ
3 p x1 x2 x3
T ( ,
,
)
4ξ
2ξ 2ξ 2ξ
A similar relation T DAs(xi ) ↔ DAs(xi /2ξ) has been found by Braun et al. in the study
of the Pion-Nucleon GDAs h0|O|N πi, Phys. Rev. D 75 (’07)
21 / 29
Baryonic TDAs
Our Results...
α, β, γ = 1, ..., 4
λ = ±1/2
4 × 4 × 4 × 2 = 128
but 8 IND.
.
Dλαβ,γ = 4F hπ 0 (pπ )|ijk uiα (z1 n)ujβ (z2 n)dkγ (z3 n)|P (p1 , s1 )i
1
V1pπ0 = −i 21/4 √1+ξ(P
+ )3/2
fπ
fN
1
0
Apπ
= i 21/4 √1+ξ(P
+ )3/2
1
fπ
fN
1
T1pπ0 = i 21/4 √1+ξ(P
+ )3/2
fπ
fN
↑
↑
D12,1
+ D21,1
↑
↑
D12,1
− D21,1
↑
D11,2
+
fπ
fN
M
1
0
√
Apπ
= −i (∆T 1 +i∆
2
T 2 ) 21/4 1+ξ(P + )3/2
fπ
fN
2
T3pπ0 = i ∆M2
21/4
T
T
T4pπ0
= i (∆
√
1/4
√
fπ
1
1+ξ(P + )3/2 fN
fπ
1
1+ξ(P + )3/2 fN
2M 2
T 1 +i∆T 2 )
2
21/4
(∆T 1 −i∆T 2 )
D↑
(∆T 1 +i∆T 2 ) 22,2
M
1
√
V2pπ0 = −i (∆T 1 +i∆
T 2 ) 21/4 1+ξ(P + )3/2
T2pπ0 = i ∆M2
↑
↑
D12,2
+ D21,2
↑
↑
D12,2
− D21,2
↑
↑
− (∆T 1 + i∆T 2 )D11,1
(∆T 1 − i∆T 2 )D22,1
↑
↑
(∆T 1 − i∆T 2 )D22,1
+ (∆T 1 + i∆T 2 )D11,1
fπ
1
√
1+ξ(P + )3/2 fN
↑
D22,2
22 / 29
Baryonic TDAs
Dλαβ,γ = − √
24
x1 x2 x3
1
2ξ
32 X
u+α (x1 P + )u+β (x2 P + )u+γ (x3 P + )
λ1,2,3
Z Y
3
p
dyd2 k⊥
P+
λ(N,N π)
2
y(1 − y)δ y − 2ξ +
d κi⊥ φλ0 0
y, k⊥
×
[2(2π)3 ]2
p1
i=1
λ0
3
X
p+
×δ 1 − y − π+ δ (2) pπ⊥ + k⊥ δ (2)
κi⊥
p1
i=1
N,[f ]
×Ψ̃λ0
{x1 /2ξ, κ1⊥ ; λ1 , u}{x2 /2ξ, κ2⊥ ; λ2 , u}{x3 /2ξ, κ3⊥ ; λ3 , d}
XZ
⇓
DA :
M λαβ,γ
∝
N,[f ]
Ψ̃λ0
N,[f ]
T DA : Dλαβ,γ ∝ Ψ̃λ0
{x1 , κ1⊥ ; λ1 , u}{x2 , κ2⊥ ; λ2 , u}{x3 , κ3⊥ ; λ3 , d}
{
x1
x2
x3
, κ1⊥ ; λ1 , u}{ , κ2⊥ ; λ2 , u}{ , κ3⊥ ; λ3 , d}
2ξ
2ξ
2ξ
Same functional dependence of Pire et al., Phys.Rev.D75, 074004 (’07) and Braun et
al.,Phys.Rev.D75, 014021 (’07)
23 / 29
Baryonic TDAs
The choice of the Dirac indices α, β, γ automatically fixes the quark helicities...
s
u+ (xi P + , ↑) =
0 1
1
xi P + B
0 C
C
√ B
@
1 A
2
s
and
u+ (xi P + , ↓) =
0
1
0
xi P + B
1 C
C
√ B
@
0 A
2
−1
0
Combinations of helicities, λi , and OAM, Lz , carried by the active partons
↑
D12,1
↑
D21,1
↑
D11,2
↑
D11,1
↑
D12,2
↑
D21,2
↑
D22,1
↑
D22,2
λ
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
λ1
1/2
−1/2
1/2
1/2
1/2
−1/2
−1/2
−1/2
λ2
−1/2
1/2
1/2
1/2
−1/2
1/2
−1/2
−1/2
λ3
1/2
1/2
−1/2
1/2
−1/2
−1/2
1/2
−1/2
OAM
No
No
No
1
1
1
1
2
D↑ unsuppressed
D↑ suppressed
D↑ doubly-suppressed
24 / 29
Baryonic TDAs
↑
D12,1
↑
D21,1
↑
D11,2
↑
D11,1
↑
D12,2
↑
D21,2
↑
D22,1
↑
D22,2
λ
1/2
1/2
1/2
1/2
1/2
1/2
1/2
1/2
λ1
1/2
−1/2
1/2
1/2
1/2
−1/2
−1/2
−1/2
λ2
−1/2
1/2
1/2
1/2
−1/2
1/2
−1/2
−1/2
λ3
1/2
1/2
−1/2
1/2
−1/2
−1/2
1/2
−1/2
OAM
No
No
No
1
1
1
1
2
D↑ unsuppressed –D↑ suppressed – D↑ doubly-suppressed
TDA V1 calculated at ξ = 0.5 and
for ∆2 = −2GeV2
V1pπ0 = −i
1
fπ ↑
↑
√
D
+D
12,1
21,1
21/4 1 + ξ(P + )3/2 fN
24 / 29
Baryonic TDAs
Summary...
DAs sector:
Model Calculation of the 3 Leading twist DAs pretty close to the Bolz & Kroll
Fit to data...
Moment predictions for the DA Φ close to the brand new Lattice calculations
and Bolz & Kroll Fit.
TDAs sector:
General Overlap decomposition in the ERBL region...
First Time calculation of the 8 TDAs Nucleon to Pion
25 / 29
Baryonic TDAs
Summary...
DAs sector:
Model Calculation of the 3 Leading twist DAs pretty close to the Bolz & Kroll
Fit to data...
Moment predictions for the DA Φ close to the brand new Lattice calculations
and Bolz & Kroll Fit.
TDAs sector:
General Overlap decomposition in the ERBL region...
First Time calculation of the 8 TDAs Nucleon to Pion
...and Outlook
Impact Parameter representation (as for GPDs) → Transverse localization of the
virtual meson in the nucleon
Proton Decay: p → π 0 e+ & p → π + ν̄... GUT → the matrix elements
hπ 0 |ijk (ui Cdj )ukγ |P i = A1 Nγ
hπ + |ijk (ui Cγdj )ukγ |P i = A2 Nγ
Martinelli et al., Nucl.PhysB312 (’89)
25 / 29
Baryonic TDAs
OK...Now YOU deserve a coffee break!!!!
26 / 29
Baryonic TDAs
BACK-UP SLIDES
27 / 29
Baryonic TDAs
Hamiltonian for the N − −π interaction
Z
HI (N, πN ) = −igπN N
dx− d2 x⊥
ψ̄N (x)γ5 φ(x)ψN (x),
2
φ(x) and ψN (x) are the pion and nucleon fields respectively,
φ(x) ≡
3
X
τi φi (x) =
i=1
φ3
φ1 + iφ2
φ1 − iφ2
−φ3
≡
0
√π −
2π
√
2π +
−π 0
;
28 / 29
Baryonic TDAs
Nucleon Wave Function
29 / 29

Nucleon to Pion Transition Distribution Amplitudes in a Light

Transcription

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