JHEP02(2023)063
Published for SISSA by
Springer
Received
:
December 21, 2022
Accepted
:
January 10, 2023
Published
:
February 7, 2023
Renormalization of the Standard Model Effective
Field Theory from geometry
Andreas Helset,
a
Elizabeth E. Jenkins
b
and Aneesh V. Manohar
b
a
Walter Burke Institute for Theoretical Physics, California Institute of Technology,
1200 East California Boulevard, Pasadena, CA 91125, U.S.A.
b
Physics Department 0319, University of California San Diego,
9500 Gilman Drive, La Jolla, CA 92093-0319, U.S.A.
E-mail:
ahelset@caltech.edu
,
ejenkins@ucsd.edu
,
amanohar@ucsd.edu
Abstract:
S
-matrix elements are invariant under field redefinitions of the Lagrangian.
They are determined by geometric quantities such as the curvature of the field-space man-
ifold of scalar and gauge fields. We present a formalism where scalar and gauge fields are
treated together, with a metric on the combined space of both types of fields. Scalar and
gauge scattering amplitudes are given by the Riemann curvature
R
ijkl
of this combined
space, with indices
i,j,k,l
chosen to be scalar or gauge indices depending on the type of
external particle. One-loop divergences can also be computed in terms of geometric invari-
ants of the combined space, which greatly simplifies the computation of renormalization
group equations. We apply our formalism to the Standard Model Effective Field The-
ory (SMEFT), and compute the renormalization group equations for even-parity bosonic
operators to mass dimension eight.
Keywords:
Effective Field Theories, Renormalization Group, SMEFT
ArXiv ePrint:
2212.03253
Open Access
,
c
©
The Authors.
Article funded by SCOAP
3
.
https://doi.org/10.1007/JHEP02(2023)063
JHEP02(2023)063
Contents
1 Introduction
1
2 Field-space manifold
2
3 Renormalization
5
3.1 First order variation
7
3.2 Second order variation
7
3.3 One-loop counterterms
10
4 Standard Model Effective Field Theory
11
5 Renormalization group equations
15
5.1 NDA
16
6 Geometric zeros
17
7 Conclusion
18
A
SU(2)
L
×
SU(2)
R
generators
19
B Notation and operator relations
20
B.1 Dimension 0
21
B.2 Dimension 2
21
B.3 Dimension 4
21
B.4 Dimension 6
21
B.4.1
H
4
D
2
22
B.4.2
XH
2
D
2
22
B.5 Dimension 8
23
B.5.1
H
6
D
2
23
B.5.2
XH
4
D
2
23
B.5.3
X
2
H
2
D
2
24
B.5.4
H
4
D
4
26
C Renormalization group evolution in the SMEFT to dimension eight
27
C.1 Field anomalous dimensions
27
C.2 Dimension 0
29
C.3 Dimension 2
30
C.4 Dimension 4
31
C.5 Dimension 6
31
C.5.1
H
6
31
C.5.2
H
4
D
2
32
C.5.3
X
2
H
2
32
– i –
JHEP02(2023)063
C.6 Dimension 8
32
C.6.1
X
4
32
C.6.2
H
8
33
C.6.3
H
6
D
2
34
C.6.4
H
4
D
4
34
C.6.5
X
2
H
4
35
C.6.6
X
3
H
2
36
C.6.7
X
2
H
2
D
2
36
C.6.8
XH
4
D
2
36
1 Introduction
An important property of the
S
-matrix is its invariance under field redefinitions [
1–4]. The
Lagrangian and correlation functions (Green’s functions) change under field redefinitions;
however, the scattering amplitudes and physical observables remain unchanged.
1
Field
redefinitions which do not include derivatives, such as
φ
(
x
)
→
F
(
φ
(
x
))
for a scalar field, can
be viewed as a change of coordinates on the manifold where the scalar fields live, which does
not change the dynamics of the theory. An example is chiral perturbation theory, where
for two light flavors, the Goldstone boson manifold is the group
SU(2)
which is isomorphic
to the three-sphere
S
3
. Two common field choices are to use Cartesian coordinates
̃
π
=
( ̃
π
1
,
̃
π
2
,
̃
π
3
,
̃
π
4
)
with the constraint
̃
π
·
̃
π
= 1
for
S
3
, or the exponential parameterization
exp(
i
π
·
τ
/f
)
with
π
= (
π
1
,π
2
,π
3
)
for the corresponding
SU(2)
group element. The two
forms lead to different off-shell correlation functions, but the same
S
-matrix elements.
The geometric approach was used to compute scattering amplitudes, and to charac-
terize deviations from the Standard Model (SM) in terms of the curvature of the scalar
manifold of the Higgs field [
6, 7 ]. It was shown that deviations from the SM model for
Higgs Effective Field Theory (HEFT) or Standard Model Effective Field Theory (SMEFT)
have a simple universal form in terms of the curvature [
6, 7 ]. Further work can be found
in refs. [
8, 9 ]. Recently, the geometric structure of scattering amplitudes under field redef-
initions has been extended to include field redefinitions with derivatives and higher-spin
fields through several approaches [
10, 11 ].
The geometric view of scattering amplitudes also has practical advantages. It reor-
ganizes the calculation of amplitudes in terms of geometric invariants. Many terms in a
Feynman-diagram expansion are organized into geometric quantities, leading to a more
efficient calculation of the amplitude. It provides a universal description of some scatter-
ing amplitudes — Higgs and longitudinal
W
scattering in BSM models, soft scattering
amplitudes for spontaneously broken theories [
7, 12 ], and the renormalization group equa-
tions [
6, 7 , 13 ] in terms of the curvature.
1
Scattering amplitudes refers to
S
-matrix elements, i.e., the on-shell amplitudes including external leg
(wavefunction) corrections. A simple derivation of
S
-matrix invariance is given in ref. [
5], implementing
field redefinitions as a change of variables in the functional integral.
– 1 –
JHEP02(2023)063
The previous results were extended to include gauge fields and combine the scalar and
gauge sectors in a unified framework [
14]. Kinetic terms for the scalar and gauge fields
are unified into a combined metric tensor with both scalar and gauge indices. The results
unify scalar and gauge amplitudes, so that
φφ
→
φφ
,
φφ
→
AA
and
AA
→
AA
are different
components of a single curvature tensor. Even though the starting metric is block-diagonal,
with scalar and gauge components, the curvature tensor is not. Terms in the curvature
tensor such as
Γ
i
jr
Γ
r
kl
have internal index sums which run over both scalar and gauge
indices. They give terms in the scattering amplitude from diagrams with internal scalar
and gauge exchange. The geometric analysis can be used to compute one-loop anomalous
dimensions. We apply the methods in this paper to reproduce the renormalization group
equations (RGEs) for the dimension-six even-parity bosonic operators in the SMEFT [
15–
17] as a check on the formalism. We then obtain the RGEs for dimension-eight even-parity
bosonic operators in the SMEFT. Parts of the dimension-eight RGEs have been computed
previously [
18–20], but a lot of terms are new. We agree with the previous results for the
terms common to both calculations.
We will use the standard EFT power counting in
1
/M
, where
M
is a mass scale.
Dimension-six contributions to the Lagrangian or RGE are proportional to
1
/M
2
, dimen-
sion-eight contributions to
1
/M
4
, etc. Section
2 discusses the geometric formulation we use,
including the combined scalar-gauge metric, covariant derivatives, and curvature. Section
3
computes the second variation of the action using geodesic coordinates for the fluctuations,
and the one-loop renormalization counterterms in terms of curvatures and field-strength
tensors. The SMEFT Lagrangian to dimension eight, and the expressions for the metric
and Killing vectors in the SMEFT are given in section
4 . The formalism of sections
2 and 3
is applied to compute the RGEs in section
5 . Operator counterterms have to be reduced
to the canonical dimension-eight basis. These reduction expressions are lengthy, and given
in appendix
B , and the RGEs in the canonical basis are given in appendix
C . Section
6
discusses the implications of our results for geometric zeros in the anomalous dimensions.
We conclude in section
7 .
2 Field-space manifold
Consider a theory of scalar and gauge bosons with interactions with at most two deriva-
tives,
2
and ignore CP-violating interactions for simplicity. The general gauge-invariant
Lagrangian takes the form
L
=
1
2
h
IJ
(
φ
)(
D
μ
φ
)
I
(
D
μ
φ
)
J
−
V
(
φ
)
−
1
4
g
AB
(
φ
)
F
A
μν
F
Bμν
,
(2.1)
where
h
IJ
(
φ
)
,
V
(
φ
)
, and
g
AB
(
φ
)
depend on the scalar fields, and
(
D
μ
φ
)
I
=
∂
μ
φ
I
+
A
B
μ
t
I
B
(
φ
)
, F
B
μν
=
∂
μ
A
B
ν
−
∂
ν
A
B
μ
−
f
B
CD
A
C
μ
A
D
ν
,
(2.2)
2
Higher-derivative interactions are linked to higher-derivative field redefinitions, which is outside the
scope of this work. They have been considered in refs. [
10, 11 ].
– 2 –
JHEP02(2023)063
where
t
I
A
(
φ
)
are Killing vectors of the scalar manifold, so they generate a symmetry.
3
The
Lie derivative of the scalar metric
h
IJ
vanishes,
(
L
t
A
h
)
IJ
=
t
K
A
h
IJ,K
+
h
KJ
t
K
A,I
+
h
IK
t
K
A,J
= 0
,
(2.3)
where
h
IJ,K
=
∂
K
h
IJ
and
t
I
A,J
=
∂
J
t
I
A
. The Killing vectors satisfy the Lie bracket relations
[
t
A
,t
B
]
I
=
f
C
AB
t
I
C
,
(2.4)
and the relation
∇
J
t
IA
=
−∇
I
t
JA
,
(2.5)
where
t
IA
=
h
IJ
t
J
A
. The gauge coupling constant is included in
t
I
A
, and hence also in the
structure constants
f
C
AB
.
The kinetic term coefficient for the scalars,
h
IJ
(
φ
)
, can be interpreted as a metric in
scalar field space [
21], and transforms as a metric under field redefinitions. The kinetic
term coefficient for the gauge fields,
g
AB
(
φ
)
, which depends on the scalars, is symmetric
under
A
↔
B
, and transforms as an invariant tensor with two adjoint indices under action
by the Killing vector
t
I
A
,
g
AB,I
t
I
C
−
f
D
CA
g
DB
−
f
D
CB
g
AD
= 0
.
(2.6)
We extend the notion of a field-space manifold to include gauge fields, where
g
AB
(
φ
)
will take center stage, and unify the scalar and gauge sectors, so eqs. (
2.3) and (
2.6) are
components of a single equation. This also provides a unified description of scalar and
gauge amplitudes.
We group the scalars and gauge bosons into real multiplets
φ
I
and
A
Bμ
B
, where
I,J,K,...
are scalar indices and
(
Aμ
A
)
,
(
Bμ
B
)
,...
are gauge and Lorentz indices, treated
as a combined index. We will use
i,j,k,...
to run over both scalar and gauge indices. We
define a combined metric
̃
g
ij
=
(
h
IJ
0
0
−
η
μ
A
μ
B
g
AB
)
(2.7)
from the scalar and gauge kinetic terms. The quadratic part of the gauge kinetic term is
L
=
−
1
2
g
AB
(
φ
)
[
(
∂
μ
A
A
ν
)(
∂
μ
A
B
ν
)
−
(
∂
μ
A
A
μ
)(
∂
ν
A
B
ν
)
]
.
(2.8)
The first term in the square brackets motivates the choice in eq. (
2.7). The second term is
cancelled by the gauge-fixing term.
In earlier works [
6, 7 ], the metric used was the scalar metric
h
IJ
. This metric gives
the Christoffel symbol
Γ
I
JK
=
1
2
h
IL
(
h
JL,K
+
h
LK,J
−
h
JK,L
)
,
(2.9)
and Riemann curvature
R
IJKL
=
h
IM
(
∂
K
Γ
M
LJ
−
∂
L
Γ
M
KJ
+Γ
M
KN
Γ
N
LJ
−
Γ
M
LN
Γ
N
KJ
)
.
(2.10)
3
More details can be found in ref. [
7].
– 3 –
JHEP02(2023)063
Covariant derivatives using the connection in eq. (
2.9) are denoted by
∇
I
, where only
the scalar indices are treated as active indices. We can compare these with quantities
derived from the metric in eq. (
2.7), which we denote with a tilde superscript. The various
components of the Christoffel symbol
̃
Γ
i
jk
are
̃
Γ
I
JK
= Γ
I
JK
,
(2.11a)
̃
Γ
(
Aμ
A
)
JK
=
̃
Γ
I
(
Aμ
A
)
K
=
̃
Γ
(
Cμ
C
)
(
Aμ
A
)(
Bμ
B
)
= 0
,
(2.11b)
̃
Γ
I
(
Aμ
A
)(
Bμ
B
)
=
1
2
h
IJ
∇
J
g
AB
η
μ
A
μ
B
,
(2.11c)
̃
Γ
(
Aμ
A
)
I
(
Bμ
B
)
=
1
2
g
AC
∇
I
g
CB
δ
μ
A
μ
B
,
(2.11d)
where
∇
I
g
AB
=
g
AB,I
is the covariant derivative using the connection
∇
I
, since
A,B
are
not active indices for
∇
I
. Christoffel symbols with an odd number of gauge indices vanish.
We will also use the notation
̃
Γ
I
(
Aμ
A
)(
Bμ
B
)
≡
Γ
I
AB
(
−
η
μ
A
μ
B
)
,
Γ
I
AB
=
−
1
2
h
IJ
∇
J
g
AB
,
(2.12a)
̃
Γ
(
Aμ
A
)
I
(
Bμ
B
)
≡
Γ
A
IB
δ
μ
A
μ
B
,
Γ
A
IB
=
1
2
g
AC
∇
I
g
CB
,
(2.12b)
where
η
μ
A
μ
B
and
δ
μ
A
μ
B
have been factored out. Even though the metric in eq. (
2.7) is block
diagonal, we get non-zero mixed Christoffel symbols with both scalar and gauge indices.
The Riemann curvature tensor components
̃
R
ijkl
are computed from the Christoffel
symbols
̃
Γ
i
jk
, and the summation over indices runs over both scalar and gauge indices. The
components of
̃
R
ijkl
are
̃
R
IJKL
=
R
IJKL
,
(2.13a)
̃
R
(
Aμ
A
)
JKL
=
R
(
Aμ
A
)(
Bμ
B
)(
Cμ
C
)
L
= 0
,
(2.13b)
̃
R
IJ
(
Aμ
A
)(
Bμ
B
)
=
(
1
4
(
∇
I
g
AC
)
g
CD
(
∇
J
g
BD
)
−
1
4
(
∇
J
g
AC
)
g
CD
(
∇
I
g
BD
)
)
η
μ
A
μ
B
,
(2.13c)
̃
R
I
(
Aμ
A
)
J
(
Bμ
B
)
=
(
1
2
∇
I
∇
J
g
AB
−
1
4
(
∇
J
g
AC
)
g
CD
(
∇
I
g
BD
)
)
η
μ
A
μ
B
,
(2.13d)
̃
R
(
Aμ
A
)(
Bμ
B
)(
Cμ
C
)(
Dμ
D
)
=
−
1
4
(
∇
I
g
AC
)
h
IJ
(
∇
J
g
BD
)
η
μ
A
μ
C
η
μ
B
μ
D
+
1
4
(
∇
I
g
AD
)
h
IJ
(
∇
J
g
BC
)
η
μ
A
μ
D
η
μ
B
μ
C
.
(2.13e)
Curvature components with an odd number of gauge indices vanish. Here
∇
I
∇
J
g
AB
=
g
AB,IJ
−
Γ
K
IJ
g
AB,K
,
(2.14)
since only the scalar indices are active indices for
∇
I
. The gauge curvature obeys the
Bianchi identities
̃
R
(
Aμ
A
)(
Bμ
B
)
IJ
+
̃
R
(
Aμ
A
)
IJ
(
Bμ
B
)
+
̃
R
(
Aμ
A
)
J
(
Bμ
B
)
I
= 0
,
(2.15)
– 4 –
JHEP02(2023)063
Figure 1
. Generic one-loop graph. The internal (dashed) lines are fluctuation fields,
η
I
and
ζ
A
μ
,
and the external (solid) lines are the background fields
Φ
I
and
A
Bμ
B
. All interaction vertices are
quadratic in the fluctuations.
and similarly for
̃
R
IJKL
and
̃
R
(
Aμ
A
)(
Bμ
B
)(
Cμ
C
)(
Dμ
D
)
.
We will also use the covariant derivative
̃
∇
I
using the Christoffel connection
̃
Γ
i
jk
in
eq. (
2.11), where scalar and gauge indices are both active indices. One quantity which
enters in helicity amplitudes is [
14]
̃
∇
I
∇
J
(
g
AB
η
μ
A
μ
B
) =
(
∇
I
∇
J
g
AB
−
1
2
∇
I
g
AC
g
CD
∇
J
g
BD
−
1
2
∇
J
g
AC
g
CD
∇
I
g
BD
)
η
μ
A
μ
B
,
(2.16)
where
A,B
are active indices for the combined covariant derivative
̃
∇
, but not for the
scalar covariant derivative
∇
. As in eq. (
2.12), it is convenient to factor out
η
μ
A
μ
B
from
both sides,
̃
∇
I
∇
J
g
AB
≡
(
∇
I
∇
J
g
AB
−
1
2
∇
I
g
AC
g
CD
∇
J
g
BD
−
1
2
∇
J
g
AC
g
CD
∇
I
g
BD
)
=
̃
∇
J
∇
I
g
AB
.
(2.17)
These geometric quantities arise in the calculation of the renormalization group equations.
3 Renormalization
The one-loop renormalization of the Lagrangian in eq. (
2.1) can be computed using the
background field method. The scalar fields are written as the sum of a background field
Φ
plus fluctuation
η
,
φ
I
→
Φ
I
+
η
I
. The one-loop renormalization is computed by expanding
the Lagrangian to second order in the fluctuations, and then integrating over the fluctua-
tions. A generic one-loop graph that contributes to the RGEs is shown in figure
1 . This
method was used for the dimension-six SMEFT operators in ref. [
22].
The expansion
φ
I
→
Φ
I
+
η
I
is not a covariant expansion to second order in the fluctu-
ation, and it is better to use instead an expansion in geodesic coordinates [
6, 7 , 23 , 24 ]
φ
I
= Φ
I
+
η
I
−
1
2
Γ
I
JK
η
J
η
K
+
...
(3.1)
This results in a covariant second variation of the action. In our case, we use geodesic
coordinates for both scalar and gauge fluctuations,
η
i
=
(
η
I
ζ
Aμ
A
)
,
(3.2)
– 5 –
JHEP02(2023)063
with the connection derived from the combined metric in eq. (
2.7). The expansions of the
fields are
φ
I
= Φ
I
+
η
I
−
1
2
̃
Γ
I
jk
η
j
η
k
+
...
= Φ
I
+
η
I
−
1
2
̃
Γ
I
JK
η
J
η
K
−
1
2
̃
Γ
I
(
Aμ
A
)(
Bμ
B
)
ζ
Aμ
A
ζ
Bμ
B
+
...
= Φ
I
+
η
I
−
1
2
Γ
I
JK
η
J
η
K
+
1
2
Γ
I
AB
ζ
Aμ
ζ
B
μ
+
... ,
(3.3a)
A
Bμ
B
=
A
Bμ
B
+
ζ
Bμ
B
−
1
2
̃
Γ
(
Bμ
B
)
jk
η
j
η
k
+
...
=
A
Bμ
B
+
ζ
Bμ
B
−
1
2
̃
Γ
(
Bμ
B
)
(
Cμ
C
)
K
ζ
Cμ
C
η
K
−
1
2
̃
Γ
(
Bμ
B
)
J
(
Cμ
C
)
η
J
ζ
Cμ
C
+
...
=
A
Bμ
B
+
ζ
Bμ
B
−
Γ
B
CK
ζ
Cμ
B
η
K
+
... ,
(3.3b)
where
Φ
I
and
A
Bμ
B
are the background fields. After expanding the action, we will simply
use
φ
I
and
A
Bμ
B
for the background fields when there is no ambiguity. Note that with
the choice in eq. (
3.3), there is mixing between the scalar and gauge fluctuations at second
order.
The computation of the variation of the action to second order is a lengthy calculation.
The substitution in eq. (
3.3) is used for the fields, and then the action is expanded to second
order in
η
and
ζ
. There is considerable simplification when using the expansion in eq. (
3.3)
and the symmetries in eqs. (
2.3) and (
2.6). Using geodesic fluctuations gives the resultant
fluctuations in terms of geometric quantities. In computing variations of the action, it is
useful to define various covariant derivatives. The gauge covariant derivative of
φ
is
(
D
μ
φ
)
I
=
∂
μ
φ
I
+
A
B
μ
t
I
B
(
φ
)
,
(3.4)
and the gauge covariant derivative of a gauge adjoint such as
F
A
αβ
or
ζ
Aα
is
(
D
μ
F
αβ
)
A
=
∂
μ
F
A
αβ
−
f
A
BC
A
B
μ
F
C
αβ
,
(
D
μ
ζ
α
)
A
=
∂
μ
ζ
Aα
−
f
A
BC
A
B
μ
ζ
Cα
.
(3.5)
The derivative of the scalar fluctuation that is covariant w.r.t. gauge transformations and
scalar manifold coordinate transformations is [
7]
(
D
μ
η
)
I
=
∂
μ
η
I
+
t
I
B,K
A
B
μ
η
K
+Γ
I
JK
(
D
μ
φ
)
J
η
K
.
(3.6)
Since
(
D
μ
φ
)
I
transforms like the fluctuation
η
I
, the covariant second derivative of
φ
is
(
D
ν
D
μ
φ
)
I
=
∂
ν
(
D
μ
φ
)
I
+
t
I
B,J
A
B
ν
(
D
μ
φ
)
J
+Γ
I
KL
(
D
ν
φ
)
K
(
D
μ
φ
)
L
.
(3.7)
Generalizing to the combined metric in eq. (
2.7), we define a covariant derivative
̃
D
w.r.t.
both the gauge field and the background metric
̃
g
ij
analogous to the definition of
D
in
ref. [
7]. Let
Z
i
μ
=
[
(
D
μ
φ
)
I
F
Aμ
A
μ
]
(3.8)
– 6 –
JHEP02(2023)063
be the analog of
D
μ
φ
in the combined scalar-gauge space, and define
(
̃
D
μ
η
)
I
=
∂
μ
η
I
+
t
I
B,K
A
B
μ
η
K
+
̃
Γ
I
jk
Z
j
μ
η
k
=
∂
μ
η
I
+
t
I
B,K
A
B
μ
η
K
+Γ
I
JK
(
D
μ
φ
)
J
η
K
+
̃
Γ
I
(
Aμ
A
)(
Bμ
B
)
F
Aμ
A
μ
ζ
Bμ
B
=
∂
μ
η
I
+
t
I
B,K
A
B
μ
η
K
+Γ
I
JK
(
D
μ
φ
)
J
η
K
−
Γ
I
AB
F
A
μν
ζ
Bν
= (
D
μ
η
)
I
−
Γ
I
AB
F
A
μν
ζ
Bν
,
(3.9)
and similarly
(
̃
D
μ
ζ
)
(
Aμ
A
)
=
∂
μ
ζ
Aμ
A
−
f
A
CD
A
C
μ
ζ
Dμ
A
+
̃
Γ
(
Aμ
A
)
ij
Z
i
μ
η
j
=
∂
μ
ζ
Aμ
A
−
f
A
CD
A
C
μ
ζ
Dμ
A
+
̃
Γ
(
Aμ
A
)
I
(
Bμ
B
)
(
D
μ
φ
)
I
ζ
Bμ
B
+
̃
Γ
(
Aμ
A
)
(
Bμ
B
)
I
F
B μ
B
μ
η
I
= (
D
μ
ζ
μ
A
)
A
+Γ
A
IB
(
D
μ
φ
)
I
ζ
Bμ
A
+Γ
A
BI
F
B μ
A
μ
η
I
.
(3.10)
The final expressions for the one-loop counterterms simplify greatly when written in terms
of
̃
D
μ
η
and
̃
D
μ
ζ
.
3.1 First order variation
The first variation of the action in eq. (
2.1) is
δ
η
S
=
∫
d
4
x
{
−
h
IJ
(
D
μ
D
μ
φ
)
J
−
1
4
g
AB,I
F
A
μν
F
Bμν
−∇
I
V
}
η
I
(3.11)
under scalar fluctuations, and
δ
ζ
S
=
∫
d
4
x
{
h
IJ
t
I
B
(
D
ν
φ
)
J
+
g
AB,I
(
D
μ
φ
)
I
F
A
μν
+
g
AB
(
D
μ
F
μν
)
A
}
ζ
Bν
(3.12)
under gauge fluctuations, and
δ
η
S
= 0
,
δ
ζ
S
= 0
are the classical equations of motion.
3.2 Second order variation
Obtaining the second order variation of the action is a tedious computation, with many
terms, which collapse into a covariant expression when using the symmetry conditions in
eqs. (
2.3) and (
2.6). The second order terms can be divided into the scalar variation
δ
ηη
, the
gauge variation
δ
ζζ
, and the mixed variation
δ
ηζ
. We have used the geodesic fluctuations
in eq. (
3.3) to compute the second order variation, which results in a covariant expression
and simplifies the final result. To the second variation, we have added a gauge-fixing term,
eq. ( 3.20), to eliminate terms linear in
(
D
μ
ζ
μ
)
A
. The gauge-fixing term is included in the
expressions below.
ηη
:
the scalar variation is
δ
ηη
S
=
1
2
∫
d
4
x
{
h
IJ
(
̃
D
μ
η
)
I
(
̃
D
μ
η
)
J
+
[
−
̃
R
IKJL
(
D
μ
φ
)
K
(
D
μ
φ
)
L
−
(
∇
I
∇
J
V
)
−
1
4
(
∇
I
∇
J
g
AB
−
Γ
C
IA
g
CB,J
−
Γ
C
IB
g
AC,J
)
F
Aμν
F
B
μν
−
h
IK
h
JL
g
AB
t
K
A
t
L
B
]
η
I
η
J
}
,
(3.13)
– 7 –
JHEP02(2023)063
which can be written in the more compact form
δ
ηη
S
=
1
2
∫
d
4
x
{
h
IJ
(
̃
D
μ
η
)
I
(
̃
D
μ
η
)
J
+
[
−
̃
R
IKJL
(
D
μ
φ
)
K
(
D
μ
φ
)
L
−
(
∇
I
∇
J
V
)
−
1
4
(
̃
∇
I
∇
J
g
AB
)
F
Aμν
F
B
μν
−
t
IA
t
A
J
]
η
I
η
J
}
.
(3.14)
The covariant derivative
̃
∇
I
∇
J
g
AB
is given in eq. (
2.17). Scalar and gauge indices on the
Killing vector
t
I
A
are lowered and raised by the metrics
h
IJ
and
g
AB
and their inverses,
t
IA
=
h
IJ
t
J
A
, t
A
I
=
g
AB
h
IJ
t
J
B
.
(3.15)
ηζ
:
the mixed variation is
δ
ηζ
S
=
∫
d
4
x
[
(
h
KJ
∇
I
t
J
A
−
h
IJ
∇
K
t
J
A
)
(
D
μ
φ
)
K
−
1
2
(
∇
J
∇
I
g
AB
)(
D
ν
φ
)
J
F
B
μν
−
1
2
g
BD
g
AD,I
h
LJ
t
L
B
(
D
μ
φ
)
J
+
1
2
g
BD
g
AD,I
g
CB,L
(
D
ν
φ
)
L
F
C
μν
+
g
BA,K
(
D
μ
φ
)
K
h
IL
g
BG
t
L
G
−
1
4
(
D
ν
φ
)
J
g
AB,J
g
BD
g
DC,I
F
C
μν
]
η
I
ζ
Aμ
.
(3.16)
The first term can be rewritten using the identity
h
KJ
∇
I
t
J
A
−
h
IJ
∇
K
t
J
A
= 2
h
KJ
∇
I
t
J
A
= 2
∇
I
t
KA
,
(3.17)
which follows from eq. (
2.5) since
t
A
is a Killing vector. The entire expression reduces to
δ
ηζ
S
=
∫
d
4
x
{[
2(
∇
I
t
JA
)+
t
B
I
(
∇
J
g
AB
)
−
1
2
t
B
J
(
∇
I
g
AB
)
]
(
D
μ
A
φ
)
J
+
(
−
̃
R
(
Aμ
A
)
I
(
Bμ
B
)
J
+2
̃
R
IJ
(
Aμ
A
)(
Bμ
B
)
)
F
Bμ
B
ρ
(
D
ρ
φ
)
J
}
η
I
ζ
Aμ
A
(3.18)
using the combined curvature
̃
R
defined in eq. (
2.13).
ζζ
:
the gauge variation is
δ
ζζ
S
=
1
2
∫
d
4
x
{
−
g
AB
η
μ
A
μ
B
(
̃
D
μ
ζ
)
Aμ
A
(
̃
D
μ
ζ
)
Bμ
B
+
[
t
IA
t
I
B
η
μν
−
̃
R
I
(
Aμ
)
J
(
Bν
)
(
D
α
φ
)
I
(
D
α
φ
)
J
+
1
2
g
AB,I
(
(
̃
D
μ
D
ν
φ
)
I
+(
̃
D
ν
D
μ
φ
)
I
)
+
(
∇
I
∇
J
g
AB
−
g
AD,I
g
DG
g
GB,J
)
(
D
μ
φ
)
I
(
D
ν
φ
)
J
+
1
2
(
g
DB
f
D
CA
−
g
DA
f
D
CB
+2
g
CD
f
D
AB
)
F
C
μν
−
1
4
g
DB,K
h
KL
g
CA,L
F
C
αμ
F
D
αν
+
1
8
h
IM
g
AB,M
g
CD,I
F
C
αβ
F
Dαβ
η
μν
+
1
2
h
IM
g
AB,M
V
,I
η
μν
]
ζ
Aμ
ζ
Bν
}
.
(3.19)
– 8 –
JHEP02(2023)063
Gauge-fixing term:
the gauge-fixing term, which has been included in the above second
variation of the action, is
S
g.f.
=
−
1
2
∫
d
4
x g
AB
G
A
G
B
,
G
A
= (
̃
D
μ
ζ
)
Aμ
+
1
2
g
AC
g
CB,I
(
D
μ
φ
)
I
ζ
Bμ
−
h
IJ
g
AB
t
J
B
η
I
= (
̃
D
μ
ζ
)
Aμ
+Γ
A
IB
(
D
μ
φ
)
I
ζ
Bμ
−
t
A
I
η
I
.
(3.20)
This is an extension of the gauge-fixing term in ref. [
25]. The gauge-fixing term in eq. (
3.20)
has been chosen to eliminate terms linear in
(
̃
D
μ
ζ
)
Aμ
in the second variation of the action,
and to make the
ζ
kinetic term invertible. Physical results do not depend on the choice of
gauge-fixing term.
Ghosts:
there is also a ghost Lagrangian which depends on the gauge variation of the
gauge-fixing term in eq. (
3.20). Under a gauge transformation with parameter
θ
A
,
δη
I
=
t
I
A,J
η
J
θ
A
, δζ
A
μ
=
−
∂
μ
θ
A
−
f
A
BC
θ
B
(
A
C
μ
+
ζ
C
μ
)
,
(3.21)
the ghost Lagrangian takes the form
S
ghost
=
∫
d
4
x
c
A
δ
G
A
δθ
B
c
B
=
∫
d
4
x
{
(
D
μ
c
)
A
(
D
μ
c
)
B
+(
D
μ
c
)
A
f
A
BC
ζ
C
μ
c
B
−
2
c
A
Γ
A
IB
(
D
μ
φ
)
I
(
D
μ
c
)
B
−
2
c
A
Γ
A
IB
(
D
μ
φ
)
I
f
A
BC
ζ
C
μ
c
B
−
c
A
h
IJ
g
AC
t
J
C
t
I
B
c
B
−
c
A
h
IJ
g
AC
t
J
C
t
I
B,K
η
K
c
B
}
,
(3.22)
where
c
and
c
are the anticommuting ghost and anti-ghost fields. The covariant derivative
of the ghost and anti-ghost analogous to eq. (
3.10) is
(
̃
D
μ
c
)
A
=
∂
μ
c
A
−
f
A
BC
A
B
μ
c
C
+Γ
A
IB
(
D
μ
φ
)
I
c
B
= (
D
μ
c
)
A
+Γ
A
IB
(
D
μ
φ
)
I
c
B
,
(
̃
D
μ
c
)
A
=
∂
μ
c
A
−
c
C
f
C
BA
A
B
μ
+
c
B
Γ
B
IA
(
D
μ
φ
)
I
= (
D
μ
c
)
A
+
c
B
Γ
B
IA
(
D
μ
φ
)
I
,
(3.23)
in terms of which the ghost action is
S
ghost
=
∫
d
4
x
{
(
̃
D
μ
c
)
A
(
̃
D
μ
c
)
A
+
c
A
[
(
1
2
g
AE
∇
I
∇
J
g
EB
−
1
4
g
AE
g
EC,I
g
CD
g
DB,J
)
(
D
μ
φ
)
I
(
D
μ
φ
)
J
+Γ
A
IB
(
D
μ
D
μ
φ
)
I
−
t
IA
t
IB
]
c
B
+(
D
μ
c
)
A
f
A
BC
ζ
C
μ
c
B
−
2
c
A
Γ
A
IB
(
D
μ
φ
)
I
f
A
BC
ζ
C
μ
c
B
−
c
A
h
IJ
g
AC
t
J
C
t
I
B,K
η
K
c
B
}
,
(3.24)
which can be written in the simpler form
S
ghost
=
∫
d
4
x
{
(
̃
D
μ
c
)
A
(
̃
D
μ
c
)
A
+
c
A
[
g
AC
1
2
̃
∇
I
∇
J
g
CB
(
D
μ
φ
)
I
(
D
μ
φ
)
J
+Γ
A
IB
(
D
μ
D
μ
φ
)
I
−
t
IA
t
IB
]
c
B
+
[
(
D
μ
c
)
A
f
A
BC
ζ
C
μ
−
2
c
A
Γ
A
IB
(
D
μ
φ
)
I
f
A
BC
ζ
C
μ
−
c
A
t
A
I
t
I
B,K
η
K
]
c
B
}
.
(3.25)
The last line of the ghost action is cubic in the fluctuation fields, and not needed for the
one-loop functional integral over fluctuations.
– 9 –
JHEP02(2023)063
3.3 One-loop counterterms
The divergent one-loop contributions are calculated from the second variation of the action.
The general form was first computed in ref. [
26] and extended to a kinetic term with non-
trivial metric in ref. [
7]. In the purely scalar case, if the second variation has the form
δ
ηη
S
=
1
2
∫
d
4
x
{
h
IJ
(
D
μ
η
)
I
(
D
μ
η
)
J
+
X
IJ
η
I
η
J
}
,
(3.26)
then the infinite part of the one-loop functional integral in
4
−
2
dimensions is
∆
S
=
1
32
π
2
∫
d
4
x
{
1
12
Tr [
Y
μν
Y
μν
]+
1
2
Tr
[
X
2
]
}
,
(3.27)
where
[
Y
μν
]
I
J
= [
D
μ
,
D
ν
]
I
J
,
X
I
J
=
h
IK
X
KJ
.
(3.28)
In our case, we can use the above results treating
X
and
Y
as matrices in the com-
bined scalar-gauge space, and subtract the corresponding expression for the ghosts. The
components of
X
,
X
=
[
[
X
ηη
]
I
J
[
X
ηζ
]
I
(
Bμ
B
)
[
X
ηζ
]
(
Aμ
A
)
J
[
X
ζζ
]
(
Aμ
A
)
(
Bμ
B
)
]
,
(3.29)
can be read off from eqs. (
3.14), ( 3.18), and (
3.19), and
X
A
B
for the ghosts from eq. (
3.24).
The covariant derivative in the combined scalar-gauge space is
̃
D
μ
[
η
I
ζ
A
λ
]
=
∂
μ
[
η
I
ζ
A
λ
]
+
[
t
I
C,J
A
C
μ
+Γ
I
LJ
(
D
μ
φ
)
L
−
Γ
I
CB
F
C
μσ
Γ
A
CJ
F
C
μλ
−
f
A
CB
A
C
μ
η
λσ
+Γ
A
LB
(
D
μ
φ
)
L
η
λσ
][
η
J
ζ
B
σ
]
,
(3.30)
and the commutator of covariant derivatives
̃
D
takes a very simple form,
[
̃
D
μ
,
̃
D
ν
]
i
j
=
[
̃
Y
μν
]
i
j
=
̃
R
i
jkl
Z
k
μ
Z
l
ν
+
̃
∇
j
̃
t
i
C
F
C
μν
,
(3.31)
extending ref. [
7, (3.45)], where the combined Killing vector is
̃
t
i
B
=
[
t
I
B
−
δ
A
B
∂
μ
A
+
f
A
CB
A
C
μ
A
]
.
(3.32)
This grouping of the Killing vectors was introduced in ref. [
14]. The commutator of covari-
ant derivatives for the ghosts is
[
̃
D
μ
,
̃
D
ν
]
A
B
= [
Y
μν
]
A
B
=
̃
R
A
BKL
(
D
μ
φ
)
K
(
D
ν
φ
)
L
+
̃
∇
B
̃
t
A
C
F
C
μν
=
̃
R
A
BKL
(
D
μ
φ
)
K
(
D
ν
φ
)
L
−
f
A
CB
F
C
μν
+Γ
A
LB
t
L
C
F
C
μν
.
(3.33)
The divergent contribution in eq. (
3.27) allows us to compute the anomalous dimension
of the effective Lagrangian. The remaining computation is purely algebraic. Evaluate
Y
μν
and
X
in terms of the metrics and potential in the Lagrangian, and then take the traces in
eq. ( 3.27). Note that matrix multiplication and traces are over the combined scalar-gauge
space. The ghost contribution is subtracted, since ghosts are anticommuting. We discuss
the application of our results to the SMEFT in the next section.
– 10 –
JHEP02(2023)063
4 Standard Model Effective Field Theory
Although the construction and main results of this paper apply to a general effective field
theory for scalars and gauge fields, it is of particular interest to apply it to the SMEFT. In
the SMEFT, the only scalar field is the Higgs doublet, which we write as four real scalars,
as in eq. (
A.1),
H
=
1
√
2
(
φ
2
+
iφ
1
φ
4
−
iφ
3
)
,
(4.1)
and the scalar indices
I,J,...
take values from 1 to 4. We group all gauge fields of the full
gauge group
SU(3)
c
⊗
SU(2)
L
⊗
U(1)
Y
into the multiplet
A
B
μ
=
G
A
μ
W
a
μ
B
μ
.
(4.2)
The corresponding field-strength tensors are
G
A
μν
,
W
a
μν
, and
B
μν
. Unless otherwise speci-
fied,
a
runs from 1 to 3. At times we will combine the electroweak
SU(2)
L
⊗
U(1)
Y
gauge
groups, and let
a
run from 1 to 4, where
W
4
μν
=
B
μν
, and denote this explicitly.
The operators in the starting SMEFT Lagrangian are those that can be included as
terms in the metrics or potential. All fermions are dropped. The terms in the SMEFT
Lagrangian to dimension four are the SM terms
L
=
−
1
4
G
A
μν
G
A
μν
−
1
4
W
a
μν
W
aμν
−
1
4
B
μν
B
μν
+
D
μ
H
†
D
μ
H
−
λ
(
H
†
H
−
1
2
v
2
)
2
=
−
1
4
G
A
μν
G
A
μν
−
1
4
W
a
μν
W
aμν
−
1
4
B
μν
B
μν
+
1
2
(
D
μ
φ
)
I
(
D
μ
φ
)
I
−
1
4
λ
(
φ
I
φ
I
−
v
2
)
2
.
(4.3)
From eq. (
4.3), we can read off the potential
V
(
φ
) =
1
4
λ
(
φ
I
φ
I
−
v
2
)
2
,
(4.4)
and gauge covariant derivative [
6]
(
D
μ
φ
)
I
=
∂
μ
φ
1
φ
2
φ
3
φ
4
+
1
2
0
g
2
W
3
μ
+
g
1
B
μ
−
g
2
W
2
μ
g
2
W
1
μ
−
g
2
W
3
μ
−
g
1
B
μ
0
g
2
W
1
μ
g
2
W
2
μ
g
2
W
2
μ
−
g
2
W
1
μ
0
g
2
W
3
μ
−
g
1
B
μ
−
g
2
W
1
μ
−
g
2
W
2
μ
−
g
2
W
3
μ
+
g
1
B
μ
0
φ
1
φ
2
φ
3
φ
4
.
(4.5)
The Killing vectors
t
a
can be read off from eq. (
4.5)
t
1
=
1
2
g
2
φ
4
φ
3
−
φ
2
−
φ
1
, t
2
=
1
2
g
2
−
φ
3
φ
4
φ
1
−
φ
2
, t
3
=
1
2
g
2
φ
2
−
φ
1
φ
4
−
φ
3
, t
4
=
1
2
g
1
φ
2
−
φ
1
−
φ
4
φ
3
,
(4.6)
using
t
1
,
2
,
3
for the
SU(2)
L
generators and
t
4
for the
U(1)
Y
generator.
– 11 –