The Standard Model Effective Field Theory and

The Standard Model Effective Field Theory
and
Dimension 6 Operators
Aneesh Manohar
University of California, San Diego
17 Mar 2015 / Vienna
A Manohar (UCSD)
17 Mar 2015 / Vienna
1 / 46
Outline
The SMEFT
Equations of Motion
Operators
Naive Dimensional Analysis
µ → eγ and MFV
Holomorphy
A Manohar (UCSD)
17 Mar 2015 / Vienna
2 / 46
Talk based on:
Rodrigo Alonso, Elizabeth Jenkins, Michael Trott, AM
C. Grojean, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Scaling of Higgs Operators
and h → γγ Decay, JHEP 1304:016 (2013).
E.E. Jenkins, A.V. Manohar and M. Trott, On Gauge Invariance and Minimal Coupling, JHEP 1309:063
(2013).
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model
Dimension Six Operators I: Formalism and λ Dependence, JHEP 1310:087 (2013).
E.E. Jenkins, A.V. Manohar and M. Trott, Naive Dimensional Analysis Counting of Gauge Theory Amplitudes
and Anomalous Dimensions, Phys. Lett. B726 (2013) 697-702.
E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model
Dimension Six Operators II: Yukawa Dependence, JHEP 1401:035 (2014).
R. Alonso, E.E. Jenkins, A.V. Manohar and M. Trott, Renormalization Group Evolution of the Standard Model
Dimension Six Operators III: Gauge Coupling Dependence and Phenomenology, JHEP 1404:159 (2014).
R. Alonso, E.E. Jenkins and A.V. Manohar, Holomorphy without Supersymmetry in the Standard Model
Effective Field Theory, arXiV:1409.0868 [hep-ph].
SMEFT
A model independent way to include the effects of new physics. The
assumption is that there are no new particles at the electroweak scale.
This is what the data indicates.
Fields are the SM fields. hHi breaks SU(2) × U(1).
L = LSM +
L5
L6
+ 2 + ...
Λ
Λ
Need to include the + . . .. Cannot just stop at a given dimension.
A Manohar (UCSD)
17 Mar 2015 / Vienna
4 / 46
H=
√1
2
ϕ+
v + h + iϕ0
!
ϕ+ , ϕ0 give mass to W , Z , and h is the Higgs scalar.
†
2
H HW →
1 2
1 2
v + hv + h W 2
2
2
Couplings of h fixed because it is part of H.
An alternate approach is to assume SU(2) × U(1) → U(1) broken
symmetry, i.e. a chiral field U, and a scalar field h, with arbitrary
couplings.
A Manohar (UCSD)
17 Mar 2015 / Vienna
5 / 46
Notation
Fields are three generations of fermions
L : qr , lr ,
R : ur , dr , er
r = 1, . . . , ng = 3
the scalar doublet H, and SU(3) × SU(2) × U(1) gauge fields.
L = LSM +
1
1 (5)
L + 2 L(6) + . . . note the dots
Λ
Λ
Λ is the scale of new physics, and assume Λ > v
Power Counting
L(6) ∼
2
mH
Λ2
L(8) ∼
4
mH
Λ4
L(6) × L(6) ∼
4
mH
Λ4
mH a typical electroweak scale (v , mt , mZ )
A Manohar (UCSD)
17 Mar 2015 / Vienna
6 / 46
SMEFT: dim 5
L5 = crs H
†i
†j
liαr H ljβs αβ
i : SU(2) index
α : Lorentz index
r : flavor index
Violates lepton number ∆L = 2. This is the operator that gives
Majorana mass to the neutrino.
The scale is the seesaw scale and is high. Not relevant for physics at
LHC.
A Manohar (UCSD)
17 Mar 2015 / Vienna
7 / 46
SMEFT: dim 6
Lots of operators at dimension six. 59 operators that preserve baryon
number, and 4/5 that violate baryon number.
Buchmuller and Wyler, Nucl.Phys. B268 (1986) 621
Grzadkowski et al. JHEP 1010 (2010) 085
Flavor indices run over ng = 3 values.
2499 baryon number conserving operators, including flavor indices.
1350 CP-even and 1149 CP-odd operators.
Alonso, Jenkins, AM, and Trott, JHEP 1404 (2014) 159.
Notation:
L : qr , lr ,
R : ur , dr , er
r = 1, . . . , ng = 3
e j = ij H † j
Hj , H
A
I
Xµν : Gµν
, Wµν
, Bµν
A Manohar (UCSD)
17 Mar 2015 / Vienna
8 / 46
Fierz Identities
Write down all possible gauge invariant operators of dimension 6.
Use Fierz identities:
ψ 1 γ µ PL ψ2 ψ 3 γ µ PL ψ4 = ψ 3 γ µ PL ψ2 ψ 1 γ µ PL ψ4
So in the four-quark operators:
λk
λk µ
αi
µ
q αi
p γ qr βj q s γµ qtσl → q s γ qr βj q p γµ qtσl
q γ µ Γ q q γµ Γ q,
Γ ⊗ Γ ∝ 1 ⊗ 1, T A ⊗ T A , τ I ⊗ τ I , T A τ I ⊗ T A τ I
Two independent contractions:
(1)
1⊗1
(3)
τI ⊗ τI
Qqq = q p γ µ qr q s γµ qt
Qqq = q p γ µ τ I qr q s γµ τ I qt
A Manohar (UCSD)
17 Mar 2015 / Vienna
9 / 46
not needed:
(8)
Qqq = q p γ µ T A qr q s γµ T A qt
(3,8)
Qqq
= q p γ µ τ I T A qr q s γµ τ I T A qt
h iα h iλ
1
1 α λ
TA
TA
= δσα δβλ −
δ δ
2
2Nc β σ
β
σ
h ii h ik
τI
= 2δli δjk − δji δlk
τI
j
l
1 ↔ swap both
A Manohar (UCSD)
TA ⊗ TA
τIT A ⊗ τIT A
∝ 1 + swap color
∝ 1 + swap weak
swap weak ↔ swap color
17 Mar 2015 / Vienna
10 / 46
SM Equations of Motion
D 2 Hk − λv 2 Hk + 2λ(H † H)Hk + q j Yu† u jk + d Yd qk + e Ye lk = 0 ,
for the Higgs field (mix H and fermion operators)
e j + Y † d Hj ,
/ qj = Yu† u H
iD
d
/ lj = Ye† eHj ,
iD
/ d = Yd qj H † j ,
iD
e†j ,
/ u = Yu qj H
iD
/ e = Ye lj H † j ,
iD
for the fermion fields, and
[D α , Gαβ ]A = g3 jβA ,
[D α , Wαβ ]I = g2 jβI ,
D α Bαβ = g1 jβ ,
for the gauge fields, where [D α , Fαβ ] is the covariant derivative in the
adjoint representation.
A Manohar (UCSD)
17 Mar 2015 / Vienna
11 / 46
Equations of Motion
Can use classical equations of motion in a quantum field theory.
No quantum corrections needed.
Green’s functions can change, but the S-matrix does not.
The two versions of the operator do not agree diagram by diagram.
Only the total contribution to the S-matrix is unchanged.
A Manohar (UCSD)
17 Mar 2015 / Vienna
12 / 46
SMEFT Operators
Leading higher dimension operators are d = 6.
Assuming B and L conservation, there are 59 independent
dimension-six operators which form complete basis of d = 6
operators.
59 operators divided into eight operator classes.
1 : X3
2 : H6
3 : H 4D2
4 : X 2H 2
5 : ψ2H 3
6 : ψ 2 XH
7 : ψ2H 2D
8 : ψ4
A
I
X = Gµν
, Wµν
, Bµν
ψ = q, l, u, d, e
Buchmuller & Wyler 1986
Grzadkowski, Iskrzynski, Misiak and Rosiek 2010
A Manohar (UCSD)
17 Mar 2015 / Vienna
13 / 46
Dimension Six Operators
1 : X3
QG
QGe
f
ABC
GµAν GνBρ GρCµ
f
ABC
e Aν GBρ GCµ
G
ρ
ν
µ
IJK
2 : H6
QH
†
3 : H 4D2
3
(H H)
QH
QHD
†
5 : ψ 2 H 3 + h.c.
†
(H H)(H H)
∗ †
H † Dµ H
H Dµ H
WµIν WνJρ WρK µ
QW
QW
e
f Iν W Jρ W K µ
IJK W
µ
ν
ρ
QeH
(H † H)(¯lp er H)
QuH
e
¯p ur H)
(H † H)(q
QdH
¯p dr H)
(H † H)(q
QeB
(¯lp σ µν er )HBµν
QHW
H H
I
Wµν
W Iµν
QuG
e GA
¯p σ µν T A ur )H
(q
µν
QHe
QH W
e
f I W Iµν
H †H W
µν
QuW
e WI
¯p σ µν ur )τ I H
(q
µν
QHq
QHB
H † H Bµν B µν
QuB
e Bµν
¯p σ µν ur )H
(q
QHq
QH Be
e µν B µν
H †H B
QdG
A
¯p σ µν T A dr )H Gµν
(q
QHu
QHWB
I
H † τ I H Wµν
B µν
QdW
I
¯p σ µν dr )τ I H Wµν
(q
QHd
7 : ψ2 H 2 D
←
→
(H † i D µ H)(¯lp γ µ lr )
→I
† ←
(H i D µ H)(¯lp τ I γ µ lr )
←
→
¯p γ µ er )
(H † i D µ H)(e
←
→
¯ p γ µ qr )
(H † i D µ H)(q
←
→
¯p τ I γ µ qr )
(H † i D Iµ H)(q
←
→
¯ p γ µ ur )
(H † i D µ H)(u
←
→
(H † i D µ H)(d¯p γ µ dr )
QH W
eB
f I B µν
H †τ I H W
µν
QdB
¯p σ µν dr )H Bµν
(q
QHud + h.c.
e † Dµ H)(u
¯p γ µ dr )
i(H
4 : X 2H 2
†
6 : ψ 2 XH + h.c.
QHG
H H
A
Gµν
GAµν
QeW
QH Ge
e A GAµν
H †H G
µν
†
A Manohar (UCSD)
I
(¯lp σ µν er )τ I HWµν
(1)
QHl
(3)
QHl
(1)
(3)
17 Mar 2015 / Vienna
14 / 46
Dimension Six Operators
¯ LL)
¯
8 : (LL)(
Qll
(¯lp γµ lr )(¯ls γ µ lt )
(1)
Qqq
¯p γµ qr )(q
¯s γ µ qt )
(q
(3)
Qqq
¯p γµ τ I qr )(q
¯s γ µ τ I qt )
(q
¯ s γ µ qt )
(¯lp γµ lr )(q
(1)
Qlq
(3)
Qlq
¯ RR)
¯
8 : (LL)(
¯
¯
8 : (RR)(
RR)
Qee
¯p γµ er )(e
¯ s γ µ et )
(e
Qle
¯ s γ µ et )
(¯lp γµ lr )(e
Quu
¯p γµ ur )(u
¯ s γ µ ut )
(u
Qlu
¯ s γ µ ut )
(¯lp γµ lr )(u
Qdd
(d¯p γµ dr )(d¯s γ µ dt )
Qld
(¯lp γµ lr )(d¯s γ µ dt )
Qeu
¯p γµ er )(u
¯ s γ µ ut )
(e
Qqe
¯p γµ qr )(e
¯s γ µ et )
(q
¯s γ µ τ I qt ) Qed
(¯lp γµ τ I lr )(q
¯p γµ er )(d¯s γ µ dt )
(e
(1)
Qqu
¯p γµ qr )(u
¯s γ µ ut )
(q
(1)
Qud
(8)
Qud
¯p γµ ur )(d¯s γ µ dt )
(u
(8)
Qqu
¯p γµ T A qr )(u
¯s γ µ T A ut )
(q
(1)
Qqd
(8)
Qqd
¯p γµ qr )(d¯s γ µ dt )
(q
¯ p γµ
(u
T Au
¯
¯ + h.c.
8 : (LR)(
RL)
Qledq
(¯lpj er )(d¯s qtj )
¯
r )(ds
γµT Ad
t)
¯p γµ T A qr )(d¯s γ µ T A dt )
(q
¯
¯ + h.c.
8 : (LR)(
LR)
Qquqd
(1)
¯pj ur )jk (q
¯sk dt )
(q
(8)
Qquqd
(1)
Qlequ
(3)
Qlequ
¯pj T A ur )jk (q
¯sk T A dt )
(q
¯sk ut )
(¯lpj er )jk (q
¯sk σ µν ut )
(¯lpj σµν er )jk (q
Buchmuller & Wyler 1986
Grzadkowski, Iskrzynski, Misiak and Rosiek 2010
ψ 4 → JJ, (LR)(LR), (LR)(RL)
In the broken phase:
"
H=
√1
2
X 2H 2 :
ψ 2 XH :
#
h → γγ, h → gg
l r σµν es F µν
ψ2H 3 :
A Manohar (UCSD)
ϕ+
v + h + ϕ0
µ → eγ
higgs coupling
3
=
mass
1
17 Mar 2015 / Vienna
16 / 46
Power Counting for the RGE
Amplitudes and anomalous dimensions obey power counting:
d (6)
C ∝ C (6)
dµ
h
i2
∝ C (8) + C (6)
µ
µ
d (8)
C
dµ
In the SM, because of the dimension two operator H † H, have
µ
d (4)
2 (6)
C + ...
C ∝ C (4) + mH
dµ
2 → v2
equivalently mH
? SM parameter RG evolution affected.
A Manohar (UCSD)
17 Mar 2015 / Vienna
17 / 46
Anomalous Dimension Matrix
g3X 3
g3X 3
H6
H 4D2
g2X 2H 2
y ψ2H 3
gy ψ 2 XH
ψ2H 2D
ψ4
1
2
3
4
5
6
7
8
1
g2
0
0
1
0
0
0
0
H6
2
g6λ
λ, g 2 , y 2
g 4 , g 2 λ, λ2
g6, g4λ
λy 2 , y 4
0
λy 2 , y 4
0
H 4D2
3
g6
0
g 2 , λ, y 2
g4
y2
g2y 2
g2, y 2
0
g2X 2H 2 4
g4
0
1
g 2 , λ, y 2
0
y2
1
0
y ψ2H 3
5
g6
0
g 2 , λ, y 2
g4
g 2 , λ, y 2
g 2 λ, g 4 , g 2 y 2
g 2 , λ, y 2
λ, y 2
gy ψ 2 XH 6
g4
0
0
g2
1
g2, y 2
1
1
0
g2, y 2
g4
y2
g2y 2
g 2 , λ, y 2
y2
0
0
0
0
g2y 2
g2, y 2
g2, y 2
ψ2H 2D
7
g6
ψ4
8
g6
Structure of anomalous dimension matrix.
Jenkins, Trott, AM: 1309.0819
Many entries exist because of EOM
A Manohar (UCSD)
17 Mar 2015 / Vienna
18 / 46
Naive Dimensional Analysis
Georgi, AM: NPB234 (1984) 189
Jenkins, Trott, AM: 1309.0819
Related recent work by Buchalla et al. arXiv:1312.5624
2 2
L=f Λ
ψ
√
a f Λ
H
f
b yH
Λ
c d D
gX e
,
Λ
Λ2
NDA weight w ≡ powers of f 2 in denominator.
6
(H † H)3
2 2 H
L=f Λ
= Λ2
,
f
f4
γij ∝
λ
16π 2
nλ y2
16π 2
ny g2
16π 2
Λ = 4πf
w =2
ng
,
N = nλ + ny + ng
N = 1 + wi − wj
A Manohar (UCSD)
17 Mar 2015 / Vienna
19 / 46
Familiar Example: b → sγ
Use
Oq = bγ µ PL u uγµ PL s
g
Og =
mb b σ µν Gµν PL s
16π 2
w =1
w =0
Then
d
µ
dµ
Cq
Cg
=
L
L+1
L−1
L
Cq
Cg
where L is the number of loops of the diagram.
A Manohar (UCSD)
17 Mar 2015 / Vienna
20 / 46
Tree-Loop Mixing (incorrect) Claim
Operators can be classified into “tree” and “loop”
Based on some notion of “minimal coupling”
The mixing of “tree” and “loop” operators vanishes at one loop
Bauer et al. for HQET provides a counterexample
A Manohar (UCSD)
17 Mar 2015 / Vienna
21 / 46
γ68 as a counterexample
"
#
d
1
(3)
CeB =
4g
N
(y
+
y
)C
[Y
]
u
q
u ts + . . .
1 c
lequ
dµ pr
16π 2
prst
"
#
1
d
(3)
µ CeW =
−2g
N
C
[Y
]
u ts + . . .
2 c lequ
dµ pr
16π 2
prst
"
#
1
d
(3)
4g1 (ye + yl ) Clequ [Ye ]ts + . . .
µ CuB =
dµ pr
16π 2
stpr
"
#
1
d
(3)
−2g2 Clequ [Ye ]ts + . . . ,
µ CuW =
dµ pr
16π 2
stpr
µ
Class 6 are the magnetic dipole operators:
QeB = (¯lp,a σ µν er )H a Bµν
pr
Class 8 are four-fermion operators
(3)
¯sk α uαt ) − 8(¯lpj uαt )jk (q
¯sk α er )
¯sk σ µν ut ) = −4(¯lpj er )jk (q
Qlequ = (¯lpj σµν er )jk (q
(1)
¯sk α er )
= −4Qlequ − 8(¯lpj uαt )jk (q
A Manohar (UCSD)
17 Mar 2015 / Vienna
22 / 46
Full 2499 × 2499 anomalous dimension matrix computed, including all
Yukawa couplings, not just yt .
Alonso, Jenkins, Trott, AM:
JHEP 1310 (2013) 087, JHEP 1401 (2014) 035, arXiv:1312.2014
Group theory can be quite complicated — the penguin graph gives an
anomalous dimension 7 pages long.
Concentrate on a small part of the result.
A Manohar (UCSD)
17 Mar 2015 / Vienna
23 / 46
There are some big numbers:
The evolution of the H 6 coefficient is
i
d
1 h
2
2
µ CH =
108
λ
C
−
160
λ
C
+
48
λ
C
H
H
HD + . . .
dµ
16π 2
Independent of normalization of CH .
For mH ∼ 126 GeV, 108 λ/(16π 2 ) ≈ 0.1.
A Manohar (UCSD)
17 Mar 2015 / Vienna
24 / 46
X 2 H 2 (Gauge-Higgs) Operators
Grojean, Jenkins, Trott, AM: JHEP 1304 (2013) 016
Look at only 4 operators (since there are 59 operators)
CHW → cW , etc. rescaled by gi gj /(2Λ2 )
g32 †
H H GµA ν GA µ ν ,
2 Λ2
g2
= 22 H † H Wµa ν W a µ ν ,
2Λ
OG =
OW
g12 †
H H Bµ ν B µ ν ,
2 Λ2
g1 g2 † a
=
H τ H Wµa ν B µ ν ,
2 Λ2
OB =
OWB
Also CP-odd versions, which do not interfere with the SM amplitude.
When you expand them out in the broken phase, H → h + v ,
get h → γγ, h → γZ and gg → h.
A Manohar (UCSD)
17 Mar 2015 / Vienna
25 / 46
Considered these operators in an earlier work, and an explicit model
that produced these operators.
AM, M.B. Wise, PLB636 (206) 107, PRD74 (2006) 035009
An exactly solvable model that produces only the OW , OWB and OB
operators and the W 3 operator: AM: PLB 726 (2013) 347
L = LSM + Dµ S †α D µ Sα − V ,
the usual Standard Model Lagrangian LSM , the Sα , α = 1, . . . , N
kinetic energy term, and the potential
λ1 †
λ2
H H S †α Sα + H † τ a H S †α τ a Sα
N
N
λ3 †α
λ
4
+ S Sα S †β Sβ + S †α τ a Sα S †β τ a Sβ
N
N
V = mS2 S †α Sα +
A Manohar (UCSD)
17 Mar 2015 / Vienna
26 / 46
cW =
(λ1 /λ3 )
48 log
Λ23
hΦi
,
cB =
(λ1 /λ3 )YS2
12 log
cW 3 =
Λ23
hΦi
,
cWB =
(λ2 /λ4 )YS
Λ2
4
24 log hΦi
N g23
.
2880π 2 hΦi
disproves claims that these are “loop-supressed” operators, and
smaller than other contributions such as four-fermion operators.
All terms depend on RG invariant quantities.
A Manohar (UCSD)
17 Mar 2015 / Vienna
27 / 46
Constraint on cWB from the S parameter.
cWB = −
1 Λ2
S.
8π v 2
The gg → h amplitude gets contribution from cG
For h → γγ,
cγγ = cW + cB − cWB
For h → γZ ,
cγZ = cW cot θW − cB tan θW − cWB cot 2θW ,
A Manohar (UCSD)
17 Mar 2015 / Vienna
28 / 46
Contribution to the Higgs Decay Rate
2
4π 2 v 2 cγγ Γ(h → γγ)
' 1−
Λ2 I γ ΓSM (h → γγ) and I γ ≈ −1.64. Note the 4π 2 .
Similar expressions for gg → h and h → γZ .
A Manohar (UCSD)
17 Mar 2015 / Vienna
29 / 46
Brief Experimental Summary
ATLAS: µγγ = 1.17 ± 0.27
CMS: µγγ = 1.14 ± 0.25
Naive combination of these results (not recommended) gives
µγγ ' 1.155 ± 0.18
If due to cγγ :
v2
cγγ (Mh ) ' −0.086, 0.003 ± 0.003
Λ2
The second solution is preferred. The first solution is when cγγ
switches the sign of the standard model h → γγ amplitude.
The experiments are sensitive to these effects if the new physics scale
Λ is near a few TeV. (4.5 TeV)
A Manohar (UCSD)
17 Mar 2015 / Vienna
30 / 46
Magnetic Dipole Operator
Ceγ =
rs
1
1
CeB − CeW
g1 rs
g2 rs
ev
L = √ Ceγ er σ µν PR es Fµν + h.c.
2 rs
where r and s are flavor indices ({ee , eµ , eτ } ≡ {e , µ , τ }) and
9
1
Y (s) + e2 12 − csc2 θW + sec2 θW
Ceγ
rs
rs
4
4
1
+ 2Ceγ [Ye Ye† ]vs +
+2 cos2 θW [Ye† Ye ]rw Ceγ + e2 (12 cot 2θW ) CeZ
rv
ws
2
rs
†
†
− (2 sin θW cos θW ) [Ye Ye ]rw CeZ − cot θW [Ye ]rs CHWB + iCH W
fB
ws
8
5
+ e2 [Ye† ]rs Cγγ + i Ceγγ + e2 cot θW − tan θW [Ye† ]rs CγZ + i CeγZ
3
3
C˙eγ =
(3)
+ 16[Yu ]wv C lequ
rsvw
as corrected by Signer and Pruna, arXiv:1408:3565
A Manohar (UCSD)
17 Mar 2015 / Vienna
31 / 46
The current experimental limit BR(µ → eγ) < 5.7 × 10−13 from the
MEG experiment implies
√
v
2 me
Ceγ . 2.7 × 10−4 TeV−2
µe
at the low energy scale µ ∼ mµ .
This bound implies
mt (3)
C
. 1.4 × 10−3 TeV−2
me lequ
µett
using the estimate ln(Λ/mH )/(16π 2 ) ∼ 0.01 for the renormalization
group evolution, and assuming that this term is the only contribution to
Ceγ at low energies.
µe
A Manohar (UCSD)
17 Mar 2015 / Vienna
32 / 46
The anomalous magnetic moment of the muon is
4mµ v
δaµ = − √ Re C eγ
µµ
2
which yields the limits
−2
|CHWB | . 0.6 TeV
,
−2
|Cγγ | . 4 TeV
,
m
t
(3) Re Clequ . 7 TeV−2 ,
mµ
µµtt
assuming that each of these is the only contribution to C eγ .
µµ
The bound on the electric dipole moment of the electron translates to the limits
C f . 2 × 10−4 TeV−2 , Ceγγ . 1 × 10−3 TeV−2 , mt Im Clequ . 2 × 10−5 TeV−2
me
HW B
eett
using the recently measured upper bound
de < 8.7 × 10−29 e cm
from the ACME collaboration, again assuming each of these terms is the only
contribution.
A Manohar (UCSD)
17 Mar 2015 / Vienna
33 / 46
Minimal Flavor Violation
MFV based on the structure of flavor violation in the SM.
⊗i SU(3)i
i = Q, U, D, L, E
YD → UQ YD UD†
The RGE preserves MFV, since it is a symmetry
But if MFV is violated by dimension-six operators, then MFV feeds into
all the sectors via the RGE
Allows one to test MFV in a model-independent way
MFV not compatible with GUTS. Accidental symmetry of the SM.
Recent polarization B modes indicate a GUT scale.
A Manohar (UCSD)
17 Mar 2015 / Vienna
34 / 46
Empirical Observation
C˙eγ ∝ Cγγ + i Ceγγ , Ceγ
no
∗
Cγγ − i Ceγγ , Ceγ
f (z) depends only on z but not z ∗
Find this after adding all graphs and using the EOM. Individual
contributions are not holomorphic, and only the total respects
holomorphy.
Observation: 1-loop anomalous dimension matrix respects
holomorphy to a large extent.
Using EOM, so equivalent to computing (on-shell) S-matrix
elements.
A Manohar (UCSD)
17 Mar 2015 / Vienna
35 / 46
Holomorphy
Recall d = 6 operator classes
1 : X3
2 : H6
3 : H 4D2
4 : X 2H 2
5 : ψ2H 3
6 : ψ 2 XH
7 : ψ2H 2D
8 : ψ4
Divide d = 6 Operators into Holomorphic, Antiholomorphic and
Non-Holomorphic Operators
±
Xµν
=
1
eµν ,
Xµν ∓ i X
2
eµν ≡ µναβ X αβ /2
X
e ± = ±iX ± ,
X
µν
µν
e
e µν = −Xµν
X
X → X±
ψ → L, R
Complex self-duality condition in Minkowski space.
A Manohar (UCSD)
17 Mar 2015 / Vienna
36 / 46
Holomorphy
Definition
The holomorphic part of the Lagrangian, Lh , is the Lagrangian
constructed from the fields X + , R, L, but none of their hermitian
conjugates.
These transform as (0, 21 ) or (0, 1) under the Lorentz group, i.e. only
under the SU(2)R part of SU(2)L × SU(2)R .
Holomorphy
Ld=6 = Lh + Lh + Ln = Ch Qh + Ch Qh + Cn Qn
o
n
3
2
Qh ⊂ X + , X + H 2 , Lσ µν R X + H, (LR)(LR)
n
o
3
2
Qh ⊂ X − , X − H 2 , Rσ µν L X − H, (RL)(RL)
n
o
Qn ⊂ H 6 , H 4 D 2 , ψ 2 H 3 , ψ 2 H 2 D, (LR)(RL), JJ
Some of the n operators are complex.
A Manohar (UCSD)
17 Mar 2015 / Vienna
38 / 46
Holomorphy
X
d
C˙ i ≡ 16π 2 µ Ci =
γij Cj ,
dµ
i = h, h, n
j=h,h,n


γhh γhh γhn


 γhh γhh γhn 
γnh γnh γnn
γhh = γhh
∗
γhh
= γhh
∗
γhn
= γhn
only need to look at:
γhh γhh γhn
γnh γnh γnn
!
0: Vanishes by NDA, i.e. NDA gives a negative loop order
@: There is no one-loop diagram (including from EOM)
hF : Holomorphic. Nonholomorphic terms forbidden by NDA and flavor
symmetry
→ 0: Vanishes by explicit computation, after adding all contributions.
Individual graphs need not vanish.
h: Holomorphic, by explicit computation
∗: Non-zero
I
QeW = (¯lp σ µν er )τ I HWµν
(1)
¯sk ut )
Qlequ = (¯lpj er )jk (q
(1)∗
C˙ eW ∝ Ye Ye Yu† Clequ
but NDA says only order y 2 .
A Manohar (UCSD)
17 Mar 2015 / Vienna
40 / 46
Holomorphy
(X + )3 (X + )2 H 2 ψ 2 X + H
(LR)(LR)
(LR)(RL)
JJ
ψ2H 3 H 6 H 4D2 ψ2H 2D
(X + )3
h
→0
0
0
0
0
0
0
0
0
(X + )2 H 2
h
h
h
0
0
@
0
0
→0
→0
ψ2X +H
h
h
h
hF
→0
→0
→0
0
@
→0
(LR)(LR)
→0
@
hF
hF
†
Yu† Ye,d
†
Yu† Ye,d
@
@
@
→0
(LR)(RL)
→0
@
→0
Yu Yd , Yu† Ye†
hF
∗
@
@
@
→0
JJ
→0
@
→0
Yu Ye,d
∗
∗
@
@
@
∗
ψ2H 3
→0
†
Yu,d,e
h
h
∗
∗
∗
@
∗
∗
H6
→0
∗
@
@
@
@
∗
∗
∗
∗
H 4D2
→0
→0
→0
@
@
@
→0
@
∗
∗
ψ2H 2D
→0
→0
→0
→0
→0
∗
→0
@
∗
∗
The 1 1 block is holomorphic
γhh = 0
The 1 2 block vanishes except for the red terms proportional to
Yu Ye or Yu Yd .
e j − l j Ye† e Hj + h.c.
LY = −q j Yd† d Hj − q j Yu† u H
e j = ij H †j
H
ψ 2 H 3 behaves to some extent like a holomorphic operator.
one entry ∗ present even if Yukawa couplings set to zero.
A Manohar (UCSD)
17 Mar 2015 / Vienna
42 / 46
RGE of SM parameters
Recall that
µ
d (4)
2 (6)
C ∝ mH
C
dµ
d
d
µ τ =µ
dµ
dµ
θX
4π
−i
2
2π
gX
!
=
2
2mH
CHX ,+
πgX2
e and
where θ-terms are normalized as L ⊃ (θX gX2 /32π 2 )X X
X ∈ {SU(3), SU(2), U(1)}.
τ is the SUSY holmorphic gauge coupling
A Manohar (UCSD)
17 Mar 2015 / Vienna
43 / 46
∗ Entry: Some Numerology
C˙ H = −3g22 g12 + 3g22 − 12λ Re(CHW ,+ )
− 3g12 g12 + g22 − 4λ Re(CHB,+ )
− 3g1 g2 g12 + g22 − 4λ Re(CHWB,+ ) + . . .
The CHB,+ and CHWB,+ terms vanish if g12 + g22 = 4λ:
2
mH
= 2mZ2 = (129 GeV)2 ,
and the CHW ,+ term vanishes if g12 + 3g22 = 12λ:
2
mH
=
2 2 4 2
m + m = (119 GeV)2 ,
3 Z 3 W
If g12 + g22 = 4λ:
C˙ H = 6g12 g22 Re(CHW ,+ ) + . . .
A Manohar (UCSD)
17 Mar 2015 / Vienna
44 / 46
No explanation for this at present.
A Manohar (UCSD)
17 Mar 2015 / Vienna
45 / 46
Summary
Complete RGE of dimension-six operators of SM EFT computed
for first time.
Contribution of dimension-six operators to running of SM
parameters calculated for first time.
RG evolution of dimension-six operators important for Higgs
boson production gg → h and decay h → γγ and h → Z γ, which
occur at one loop in SM.
Significant constraints on flavor structure of SM EFT. Test of MFV
hypothesis.
Approximate holomorphy of 1-loop anomalous dimension matrix
of dimension-six operators.
I
I
Does it hold in a more general gauge theory?
Does any of it extend beyond one loop?
A Manohar (UCSD)
17 Mar 2015 / Vienna
46 / 46