JHEP08(2017)054
In the functional method for integrating out heavy fields, keeping only the light fields
in an effective action, one evaluates the path integral for the quadratic part of the action
in Φ expanded around its minimum (or b ackground field value),
5
e
iS
eff
[φ]
=
Z
[DΦ]e
iS[φ,Φ]
=
Z
[Dη]e
i
S[φ,Φ
c
]+
1
2
δ
2
S
δΦ
2
Φ=Φ
c
η
2
+O(η
3
)
≈ e
iS[φ,Φ
c
]
"
det
−
δ
2
S
δΦ
2
Φ=Φ
c
!#
−
1
2
= e
iS[φ,Φ
c
]−
1
2
Tr ln
−
δ
2
S
δΦ
2
Φ=Φ
c
,
where Φ
c
is defined as
δS
δΦ
Φ=Φ
c
= 0. This standard procedure relies on the Gaussian
form of the functional integral for the quadratic term, and an identity for the determinant
written in terms of a logarithmic operator in the action. The one-loop effective Lagrangian
correspondin g to S
eff
[φ] is then
L
1-loop
EFT
[φ] = ic
s
Z
d
d
q
(2π)
d
tr ln
−P
2
+ M
2
+ U
H
, (2.2)
which applies equally for heavy-only contributions from bosons and f er mions, if U
H
is
block-d iagonal with respect to bosons vs. fermions.
6
The lower-case trace is over all in-
ternal indices and the spacetime trace gives the momentum integral, expressed here in d
dimensions. The numerical pre-factor c
s
depend s on the type of boson or fermion being
integrated out [
23].
The expansion of the logarithm in terms of a ser ies of local operators suppressed by
a heavy mass scale can be performed by a variety of techniques. As mentioned in the
intr oduction, recent developments have led to a simple and systematic way of dir ectly
writing down the terms in this expansion using covariant diagrams. Regardless of the
method used to evaluate the logarithm expansion, it can be done once-and-for-all, and the
result is the same and universal in th e sense that the final expression is in dependent of the
details of the UV Lagrangian, which remain encapsulated in the U
H
matrix of light fields,
covariant derivative P
µ
, and mass matrix M. This heavy-only universal on e-loop effective
action (UOLEA) can then be written as [
24]
L
UOLEA
=−ic
s
tr
n
f
i
2
U
H
ii
+f
i
3
G
′µν
i
G
′
µν,i
+f
ij
4
U
H
ij
U
H
ji
+f
i
5
[P
µ
,G
′
µν,i
][P
ρ
,G
′ρν
i
]+f
i
6
G
′µ
ν,i
G
′ν
ρ,i
G
′ρ
µ,i
+f
ij
7
[P
µ
,U
H
ij
][P
µ
,U
H
ji
]+f
ijk
8
U
H
ij
U
H
jk
U
H
ki
+f
i
9
U
H
ii
G
′µν
i
G
′
µν,i
+f
ijkl
10
U
H
ij
U
H
jk
U
H
kl
U
H
li
+f
ijk
11
U
H
ij
[P
µ
,U
H
jk
][P
µ
,U
H
ki
]
5
We assume here a real scalar but this applies equally to complex or ghost scalar fields, as well as vectors
and fermions [
23, 24, 29].
6
Additional care is required to take into account more complex structures that can potentially arise, as
we discuss in section
2.3.
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