Computing the effective action with the functional renormalization group
Eur. Phys. J. C
Computing the effective action with the functional renormalization group
Alessandro Codello 2
Roberto Percacci 0 1
Lesław Rachwał 4
Alberto Tonero 3
0 Sezione di Trieste, INFN , Trieste , Italy
1 SISSA , via Bonomea 265, Trieste 34136 , Italy
2 CP3Origins and the Danish IAS University of Southern Denmark , Campusvej 55, 5230 Odense , Denmark
3 ICTPSAIFR and IFT , Rua Dr. Bento Teobaldo Ferraz 271, 01140070 São Paulo , Brazil
4 Department of Physics, Center for Field Theory and Particle Physics, Fudan University , 200433 Shanghai , China
The “exact” or “functional” renormalization group equation describes the renormalization group flow of the effective average action k . The ordinary effective action 0 can be obtained by integrating the flow equation from an ultraviolet scale k = down to k = 0. We give several examples of such calculations at oneloop, both in renormalizable and in effective field theories. We reproduce the fourpoint scattering amplitude in the case of a real scalar field theory with quartic potential and in the case of the pion chiral Lagrangian. In the case of gauge theories, we reproduce the vacuum polarization of QED and of YangMills theory. We also compute the twopoint functions for scalars and gravitons in the effective field theory of scalar fields minimally coupled to gravity.

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . 1
2 The nonlocal heat kernel expansion . . . . . . . . . 3
3 Real scalar . . . . . . . . . . . . . . . . . . . . . . 6
4 Quantum electrodynamics . . . . . . . . . . . . . . 8
5 Yang–Mills . . . . . . . . . . . . . . . . . . . . . . 11
6 The chiral model . . . . . . . . . . . . . . . . . . . 13
7 Gravity with scalar field . . . . . . . . . . . . . . . 16
8 Conclusions . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . 23
1 Introduction
The functional renormalization group (FRG) is a way of
studying the flow of infinitely many couplings as functions of
an externally imposed cutoff. The idea originates from
Wilson’s understanding of the renormalization group (RG) as
the change in the action that is necessary to obtain the same
partition function when the ultraviolet (UV) cutoff is lowered
[1,2]. Early implementations of this idea were based on
discrete RG transformations, but soon there appeared equations
describing the change of the action under continuous shifts of
the cutoff. The first such equation was the Wegner–Houghton
equation [3], which has been widely used to study statistical
models and, in a particle physics context, to put bounds on the
Higgs mass [4]. Another related equation that has been used
originally to gain new insights in the renormalizability of φ4
theory is the Polchinski equation [5]. In particle physics one
is usually more interested in the effective action (EA) than in
the partition function, so one may anticipate that an equation
describing the flow of the generator of 1PI Green functions
may be of even greater use. For this purpose, the convenient
functional to use is the effective average action (EAA) k . It
is defined in the same way as the ordinary effective action,
with the following modifications. First, one adds to the bare
action S[φ] a cutoff term characterized by a cutoff scale k,
of the form
1
Sk [φ] = 2
dd q φ (−q) Rk (q2) φ (q).
Thus, the partition function becomes
eWk [ j] =
(dφ)e−S[φ]− Sk [φ]− dx j φ .
Second, after performing the Legendre transform one
subtracts the same term: k [ϕ] = −Wk [ j ] + dx j ϕ − Sk [ϕ],
where ϕ = φ . For general reviews see e.g. [6–9]. The effect
of this term is to suppress the propagation of low
momentum modes leaving the vertices unchanged. The cutoff
ker(
1
)
(
2
)
nel Rk (z) is required to go to zero fast when its argument
z (which in flat spacetime applications can be thought of as
momentum squared) is greater than the cutoff scale k2. In
typical application this decay could be a polynomial of
sufficiently high degree or an exponential. The cutoff kernel is
also required to tend to zero (for all z) when k → 0. This
implies that when k → 0 the EAA reduces to the ordinary
effective action.
The kdependence of the EAA is described by the
Wetterich equation [10–12]
(
3
)
1
∂t k = 2 S Tr
δ2 k
δϕδϕ + Rk
−1
∂t Rk ,
where t = log(k/ k0), k0 is an arbitrary reference scale and
the supertrace in the r.h.s. stands (in flat spacetime) for an
integration over coordinate and momentum space and a trace
over any representation of internal and spacetime symmetries
that the fields may carry. Due to the fast falloff of the cutoff
kernel, also the function ∂t Rk , which appears inside the trace
in the r.h.s. of (
3
), decays fast for large z. This makes the
trace in the r.h.s. of (
3
) convergent.
The functional renormalization group equation (FRGE)
has been widely used in studies of the infrared (IR)
properties of statistical and particle physics models, in particular
of phase transitions and critical phenomena. It has also been
used to study the ultraviolet behaviour of gravity, in
particular to establish the existence of a nontrivial fixed point which
may be used to define a continuum limit [13–15]. Here we
would like to discuss some examples taken mostly from
particle physics where the Wetterich equation is used instead as
a tool to compute the effective action.
The basic idea is as follows. Assume that k is the most
general functional of the given fields which is invariant under
the assumed symmetries of the system. In many applications
it is justified to assume that it is a semilocal functional [19,
20], meaning that it admits an expansion into infinitely many
local terms constructed with the fields and their derivatives
of arbitrary degree. We call “theory space” the space of these
functionals. Equation (
3
) defines a vector field on this space
whose integral lines are the RG trajectories. We can now fix
an arbitrary initial point in theory space and identify it as the
“bare” action of the theory at some UV scale . Typically one
will choose this bare action to be local and simple, but this is
not essential. One can integrate the RG flow in the direction
of decreasing t and the IR endpoint of the flow for t → −∞
represents the effective action. The couplings in the effective
action can be interpreted as renormalized couplings, and the
integral of their beta functions from k = 0 to k = is the
relation between bare and renormalized couplings.
One can also ask what would happen if we tried to take
the limit → ∞. This is equivalent to solving the FRGE in
the direction of increasing t with the same initial condition.
(Since the initial condition at the original value of is kept
fixed, also the effective action will remain fixed, so this is very
similar to the Wilsonian procedure of changing the action at
the UV cutoff “keeping the physics fixed”.) There is a variety
of possible behaviours. If some coupling blows up at finite
t (a Landau pole), the RG flow stops there and one has to
interpret the theory as an effective field theory with an upper
limit to its validity. On the other hand if the trajectory ends
at a fixed point, one may expect all physical observables to
be well behaved. In this case the theory is UV complete. The
main point is that by integrating the flow towards the UV one
can study the UV divergences of the theory and argue about
its UV completeness.
Below we will calculate the r.h.s. of the Wetterich
equation for several theories, and then we integrate the flow
down to k = 0 to obtain the effective action. Of course,
given that the effective action of any nontrivial theory is
infinitely complicated, we can only obtain partial
information as regards such a theory, and then only in certain
approximations. Here we will exploit the great flexibility of the
FRGE with regards to approximation schemes. In typical
previous applications of the FRGE, for example in the study
of the Wilson–Fisher fixed point, it is often enough to retain
only the zeromomentum part of the effective action, but it
is important to retain the full field dependence. In particle
physics one usually considers the scattering of a few
particles at a time and the full field dependence is not needed.
On the other hand, there one is interested in the full
momentum dependence. Clearly, a different type of approximation
is needed.
In the following, unless otherwise stated, we will calculate
the r.h.s. of the flow equation keeping k fixed at its
ultraviolet form .1 In perturbation theory this is equivalent to
working at oneloop. The oneloop EAA is given by
1
k(
1
)[ϕ] = S[ϕ] + 2 Tr log
δ2 S
δϕδϕ + Rk
and satisfies the equation
(
1
)
∂t k
1
= 2 Tr
δ2 S
δϕδϕ + Rk
−1
∂t Rk .
(
4
)
(
5
)
We will use known results on the nonlocal heat kernel to
compute the trace on the r.h.s. and in this way obtain the flow
of the nonlocal part of the EAA. A similar calculation in the
full flow equation is beyond currently available techniques.
Integrating the flow we will derive the nonlocal, finite parts
of the effective action containing up to two powers of the
1 Notice that the action S that is used in the functional integral to define
the theory actually differs from . We refer to [16–18] for a discussion
of the “reconstruction problem” which gives the exact relation between
and S.
the same form as in the original action. The effective action
is organized as an expansion in powers of p/Fπ , where p is
momentum and Fπ is the pion decay constant. We compute
in the oneloop approximation the fourpoint function and
we show that it reproduces the wellknown result of Gasser
and Leutwyler [45].
Finally in Sect. 7 we consider the theory of a scalar field
coupled to dynamical gravity. We compute the FRGE
keeping terms with two field strengths but all powers of
momentum. This calculation is justified by an expansion in powers
of p/MPlanck. In this case we obtain for the first time
unambiguous covariant formulae for the logarithmic terms in the
EA.
2 The nonlocal heat kernel expansion
The r.h.s. of Eq. (
3
) is the trace of a function of an operator .
In the simplest cases this operator is a secondorder
Laplacetype operator. In the presence of general background fields
(gravity, Yang–Mills fields) this operator will be a covariant
Laplacian, related to the inverse propagator of the theory in
question. In general it will have the form
= −D21 + U,
field strength. (By “field strength” we mean here in general
the curvature of the metric or of the Yang–Mills fields or the
values of the scalar condensates.) Such terms can be used to
describe several physical phenomena, such as selfenergies
and, in some cases, scattering amplitudes. In each of the cases
that we shall consider one can find some justification for the
approximations made, as we shall discuss below.
We now give a brief overview of the main results and of
the content of the subsequent sections. In Sect. 2 we review
the mathematical results for the nonlocal heat kernel
expansion of a function of a Laplacetype operator of Eq. (
6
). The
operator will generally depend on background fields such as
metric, Yang–Mills field (if it acts on fields carrying nonzero
charges of the gauge group) or scalar condensates. The trace
of the heat kernel of the operator admits a wellknown
asymptotic expansion whose coefficients are integrals of powers of
the field strengths. The nonlocal heat kernel expansion is a
sum of infinitely many such terms, containing a fixed
number of field strengths but arbitrary powers of derivatives. It
can thus be viewed as a vertex expansion of the heat kernel.
One can expand the trace of any operator, and hence also
the trace on the r.h.s. of Eq. (
3
), as a sum of these nonlocal
expressions, with coefficients that depend on the particular
function that is under the trace. There are certain ambiguities
in these calculations: one may choose to regard the r.h.s. as
a function of different operators, and one has the freedom of
choosing different cutoff functions Rk . We shall see,
however, that physical results are independent of these choices.
As a warmup in Sect. 3 we will begin by using this
technique to calculate the EA of a scalar field. We will see that the
integration of the FRGE yields the familiar relations between
the bare and renormalized couplings and that in the limit
→ ∞ there are only three divergences. Integration of the
flow equation down to k = 0 yields an EA that encodes the
familiar formula for the oneloop scattering amplitude.
In Sect. 4 we compute the EA for photons in QED which is
obtained by integrating out the fermion fields. We reproduce
the known vacuum polarization effects, and, within an
expansion in inverse powers of the electron mass, the fourphoton
interactions described by the Euler–Heisenberg Lagrangian.
In Sect. 5 we calculate the vacuum polarization effects in
Yang–Mills theory. In this case, unlike all other cases
considered in this paper, due to IR divergences it is not possible
to integrate the flow equation down to k = 0. We thus have to
restrict our attention to a finite range of momenta for which
the theory remains in a neighbourhood of its asymptotically
free UV fixed point.
The remaining two sections are devoted to examples of
effective field theories (EFTs). In Sect. 6 we consider the
chiral nonlinear sigma model, which describes the low energy
interactions of pions (and also, in a different interpretation,
the low energy scattering of longitudinal W bosons). As
expected, in this case we find divergences that are not of
(
10
)
where D is a covariant derivative with respect to all the
background fields and U is a nonderivative part that is a matrix
in the appropriate representations of all the symmetry groups
that are carried by the fields (it thus carries both internal and
spacetime indices).
Before discussing any physical application, we outline
here the heat kernel method we employ in the calculation
of the trace. The typical expression that we need to trace is
hk ( , ω) =
If we insert Eq. (
9
) in the r.h.s. of Eq. (
8
), by linearity the
trace goes through the integral and we remain with
(
6
)
(
7
)
(
8
)
(
9
)
One can now use the asymptotic expansion for the trace of
the heat kernel
(
11
)
(
12
)
(
14
)
(
15
)
(
16
)
Tr e−s
1
= (4π s)d/2
dd x √gtr
where the spacetime curvatures are constructed using the
LeviCivita connection and μν = [Dμ, Dν ] is the field
strength tensor. The first d/2 terms in Eq. (
11
) come with an
overall negative power of s, while all subsequent terms have
positive powers. When we insert this expansion in Eq. (
10
)
we can write
∂t k = 21 (4π1)d/2 dd x √g tr
×{b0( )Q d [hk ] + b2( )Q d2 −1[hk ]
2
+b4( )Q d2 −2[hk ] + · · · + bd ( )Q0[hk ] + · · · }
(
13
)
where the “Qfunctionals” are defined by
0
For n a positive integer one can use the definition of the
Gamma function to rewrite (
14
) as a Mellin transform
∞
dss−n f˜(s).
while for m a positive integer or m = 0
Q−m [ f ] = (−1)m f (m)(0).
This expansion is useful to study the UV divergences,
which are always given by local expressions. In particular,
one finds that the first d terms in the expansion (
13
) give rise
to divergences in the effective action.
In order to calculate nonlocal, finite parts of the effective
action we need a more sophisticated version of the heat kernel
expansion which includes an infinite number of heat kernel
coefficients. This expansion has been developed in [21–24]
and retains the infinite number of heat kernel coefficients in
the form of nonlocal “structure functions” or “form factors”.
For an alternative derivation see [25]. Keeping terms up to
second order in the fields strengths, the nonlocal heat kernel
expansion reads as follows:
Tr e−s
where the basic heat kernel structure function f (x ) is defined
in terms of the parametric integral
Using in Eq. (
18
) the Taylor expansion of the basic
strucx x2
ture function f (x ) = 1 − 6 + 60 + O(x 4), we obtain the
following “short time” expansion for the structure functions:
(
18
)
(
19
)
(
20
)
1 x x 2
fRic(x ) = 60 − 840 + 15120 + O(x 4)
1 x 11x 2
f R (x ) = 120 − 336 + 30240 + O(x 4)
1 x x 2
f RU (x ) = − 6 + 30 − 280 + O(x 4)
1 x x 2
fU (x ) = 2 − 12 + 120 + O(x 4)
1 x x 2
f (x ) = 12 − 120 + 1680 + O(x 4).
If we insert Eq. (
20
) in Eq. (
17
), the first term
reproduces the coefficients of the local heat kernel expansion
discussed previously. If we compare with Eq. (
11
) we see
that not all coefficients match exactly. This is because the
local heat kernel expansion is derived by calculating the
unintegrated coefficients while the nonlocal heat kernel
expansion is derived by calculating the integrated ones. So the
coefficients derived by expanding the structure functions Eq. (
18
)
may differ from the local ones Eq. (
12
) by a total derivative
or a boundary term. For example, only two of the three
possible curvature square invariants present in Eq. (
11
) appear in
Eq. (
17
), the third one has been eliminated using Bianchi’s
identities and discarding a boundary term. For this reason
also the total derivative terms in the coefficient B4( ) are
not present in the nonlocal expansion. Thus, in general, a
straightforward series expansion of the nonlocal heat kernel
structure functions will not reproduce exactly the same heat
kernel coefficients of the local expansion. See [22] for more
details on this point.
Inserting Eq. (
17
) in Eq. (
10
) we obtain
1 1
∂t k = 2 (4π )d/2
+O(R3)
0
∞
0
∞
∞
0
∞
ds h˜k (s, ω)s−d/2
ds h˜k (s, ω)s−d/2+1
ds h˜k (s, ω)s−d/2+1
ds h˜k (s, ω)s−d/2+2 fRic(sz) Rμν
ds h˜k (s, ω)s−d/2+2 f R (sz) R
ds h˜k (s, ω)s−d/2+2 f RU (sz) U
ds h˜k (s, ω)s−d/2+2 fU (sz) U
ds h˜k (s, ω)s−d/2+2 f (sz)
μν
1 1
= 2 (4π )d/2
dd x √g
Q d [hk ] tr1
2
+tr
μν g
R
+Q d2 −1[hk ] tr 6 1 − U
+ R gR R tr1 + R gRU trU + tr (UgU U)
μν
+ Rμν gRic Rμν tr1
+ · · · .
(
21
)
Here and in the following z = −D2. The first three terms
are local and have been rewritten in terms of Qfunctionals
as in the first terms of Eq. (
13
). In the remaining ones we
have defined
gA = gA(z, ω, k) =
ds h˜k (s, ω)s−d/2+2 f A(sz), (
22
)
0
∞
for A = {Ric, R, RU, U, }. From the definition of the
Laplace transform we see that shifting the argument of hk
by a is the same as multiplying the Laplace transform by
e−sa . Then:
∞
0
where
ds s−ne−sa h˜k (s, ω) = Qn[hk,ω],
a
hak,ω(z) ≡ hk (z + a, ω); hk,ω(z) ≡ hk (z, ω).
We can use this to write the functions gA in terms of
Qfunctionals of shifted arguments:
1 1
gRic(z, ω, k) = 6z Q d2 −1[hk,ω] − z2 Q d2 [hk,ω]
1 1
+ z2 0 dξ Q d2 [hkzξ, ω(1−ξ)]
7 1
gR (z, ω, k) = − 48z Q d2 −1[hk,ω] + 8z2 Q d2 [hk,ω]
where z˜ = z/ k2. It has the virtue that the Qfunctionals
can be evaluated in closed form. In d = 4 we will need the
functionals Q2, Q1 and Q0 for both unshifted and shifted
argument. The unshifted Qfunctionals are
Qn [hk ] =
2k2n 1
(n + 1) 1 + ω˜
where ω˜ = ω/ k2. The parametric integrals of the shifted
functionals can be calculated using
for n = 0, 1, 2 . . .
(
27
)
0
1
dξ Q0[hkzξ(1−ξ)
2k2
] = k2 + ω
⎡
dξ Q1[hkzξ(1−ξ)]
2k4 ⎡
Plugging these expressions into Eq. (
25
) we see that (in
d = 4) the functions gA depend only on z˜ and ω˜ . In this
case it is convenient to define gA(z˜, ω˜ ) ≡ gA(z, ω, k). These
functions are explicitly given by
θ (z˜ − 4)
(
31
)
θ (z˜ − 4)
1 1
gRic(z˜, ω˜ ) = 30 1 + ω˜
1
gR (z˜, ω˜ ) = 1 + ω˜
1
+ 24
1
+ 240
1 −
1 − z
˜
With these relations we can now compute the functional
traces on the rhs of the FRGE.
3 Real scalar
We begin considering Euclidean scalar theories defined by
the following bare action:
S[ϕ] =
=
dd x
E
The fieldindependent term has been put in for later
convenience but it is unimportant as long as gravity can be
neglected. The restriction to a quartic potential is not
dictated by arguments of renormalizability and to deal with
arbitrary potential is not problematic in the context of the FRGE.
In general, all the higher dimensional operators—which are
generated by the quartic interaction—have an effect on the
running of the quartic coupling itself. We ignore these terms
because we are interested in reproducing the standard result
for the oneloop fourpoint amplitude.
Using Eq. (
36
), the first step is to compute the Hessian
entering in the oneloop RG flow of Eq. (
5
)
δ S
δϕδϕ = −
λ
+ m2 + 2
ϕ2.
In order to properly account for threshold effects, it is
convenient to choose the argument of the cutoff Rk ( ) to be
the operator = − + λ2 ϕ2. Thus we have U = λ2 ϕ2. In
this way the function to be traced in the flow equation assumes
the standard form of Eq. (
7
) discussed in the previous section.
Equation (
10
) reads
1
∂t k [ϕ] = 2 Tr hk
− + λ2 ϕ2, m2
where z = − . Using Eqs. (
14
) and (
25
) we obtain the
following form for the beta functional:
At this point we make the following ansatz for the EAA
entering in the l.h.s. of Eq. (
38
), bearing in mind that we are
interested in terms up to fourth order in the scalar fields:
with Fk (0) = λk . Plugging the ansatz for k [ϕ] in Eq. (
40
)
we read off the beta functions
1 1
∂t Ek = 2 (4π )d/2 Q d2 [hk ] ,
∂t Zk = 0,
1 λ
∂t mk2 = − 2 (4π )d/2 Q d2 −1 [hk ]
and the flow of the structure function
3 λ2
∂t Fk (z) = 2 (4π )d/2
0
1
dξ Q d2 −2 hzξ(1−ξ) .
k
Taking z → 0 in the last equation we obtain the beta
function for the selfinteraction coupling constant
3 λ2
∂t λk = 2 (4π )d/2 Q d2 −2 [hk ] .
From now on we restrict ourselves to d = 4, using Eq. (
27
)
to evaluate the Qfunctionals, we see that the beta functions
for mk2 and λk are
λ k2
∂t mk2 = − (4π )2 k2 + m2 ,
3λ2 k2
∂t λk = (4π )2 k2 + m2 .
One can now perform the RG improvement and make the
substitutions m → mk , λ → λk in the r.h.s. of the beta
functions and show, for example, that the theory (
36
) in
trivial. We will not repeat this discussion here, since we are
interested to show how to compute the fourpoint amplitude
in the context of the FRGE. Introducing the dimensionless
mass m˜ k2 = k−2mk2 and expanding for small m˜ k2 we get the
standard perturbative and scheme independent result:
∂t m˜ k2 = −2m˜ 2
1 λ
− (4π )2 1 + m˜ 2
=
λ
−2 + (4π )2
m2
˜
λ
− (4π )2 + · · · ,
3 λ2 3λ2
∂t λk = (4π )2 1 + m˜ 2 = (4π )2 + · · ·
We compute now the finite part of the EAA by integrating
the flow of the structure function Eq. (
43
). Using Eq. (
27
)
and (
34
) to compute the beta functional of Eq. (
40
) we get
⎭
⎭
k [ϕ] =
dd x
Ek + Z2k ∂μϕ∂μϕ
(
48
)
(
49
)
The oneloop effective action 0[ϕ] is recovered by
integrating Eq. (
47
) from k = down to k = 0
0[ϕ] = [ϕ] − 41 (4π1 )2 d4x
×
⎩
−λ
⎧ # 1
⎨
2
#
4
−
2m2 + m4 log
2 − m2 log
2 + m2 $
m2
ϕ2
If we try to take the limit → ∞ this expression contains
quartic, quadratic and logarithmic divergences. The
renormalized action is of the form (
41
), with finite
“renormalized” couplings E0, m0, λ0. The relation between these and
the “bare” couplings E , m , λ is contained in
renormalization conditions, which in the present context amount to a
choice of initial condition . The finite part of the
renormalized couplings is arbitrary and has to be determined by
fitting the theory to experimental observations. Here we choose
renormalization conditions that simply remove all the local
terms contained in the integral in (
48
):
E
m2
λ
1 1 #1
= E0 + 4 (4π )2 2
1 λ
= m20 − 4 (4π )2
1 λ2
= λ0 + 8 (4π )2
#
#
4
− 2m2 +m4 log
2 − m2 log
2 + m2 $
2 + m2 $
,
1 + log
2 + m2 $
m2
m2
.
At this point the EA 0 contains a local part of the same
form of (
36
) except for the replacement of the subscripts
by subscripts 0, plus a nonlocal part that is given by the last
line of (
48
). In this part, using the perturbative logic, the bare
couplings can also be replaced by renormalized ones, up to
terms of higher order. It is clear that this step only makes
mathematical sense if is bounded (the bound depending
on the smallness of the coupling). In any case the EA then
4m20
1 + −w
The scattering amplitude for the process ϕϕ → ϕϕ is
obtained by taking four functional derivatives of the
effective action with respect to ϕ after performing the analytic
continuation to Minkowski space. Evaluating the expression
in Fourier space, we get
A(s, t, u; m0) = λ0 +
*
w=s,t,u (4π )2
λ02
1 + 4m20 ArcTanh
−w
where s = ( p1 + p2)2, t = ( p1 + p3)2 and u = ( p1 + p4)2
and all momenta are taken to be incoming.
Notice that the expression entering in the r.h.s. of Eq. (
51
)
can be written in terms of the following Feynman integral,
which results from the computation of oneloop bubble
diagrams:
takes the form
For an FRGE computation beyond the oneloop
approximation see for example [27].
4 Quantum electrodynamics
We now consider Euclidean Quantum Electrodynamics
(QED) in d = 4, which is a perturbatively renormalizable
theory characterized by the following bare action:
S[ A, ψ¯ , ψ ] =
[ A, ψ¯ , ψ ]
=
d4x
1
4e2 Fμν F μν + ψ¯ (D/ + me)ψ ,
where e is the bare electric charge, me ≡ me, is the bare
electron mass, D/ = γ μ Dμ is the Dirac operator, Dμ =
∂μ + i Aμ is the covariant derivative and Fμν is the photon
field strength tensor
μν = +Dμ, Dν , = i Fμν = i (∂μ Aν − ∂ν Aμ).
(
53
)
(
54
)
+ 1 .
(
52
)
1
k [ A, ψ¯ , ψ ] = S[ A, ψ¯ , ψ ] + 2 Tr log
Notice that in Eq. (
59
) we choose the argument of the
gauge cutoff function to be the flatspace Laplacian −∂2,
while for the fermion cutoff we take the covariant operator
.
The oneloop flow equation is obtained by
differentiating Eq. (
59
) with respect to the RG parameter t . Here we
are interested only in the fermion contribution to the photon
effective action k [ A] ≡ k [ A, 0, 0], we have
and σ μν = 2i +γ μ, γ ν ,. We will work in the gauge α = 1
where the oneloop EAA for QED can be obtained
introducing the cutoff kernels directly into Eq. (
56
):
To quantize the theory we have to introduce a gaugefixing
term which can be taken to be
where α is the gaugefixing parameter. Notice that the
Faddeev–Popov determinant can be safely discarded because
on a flat spacetime the ghost term decouples. The oneloop
effective action is given by
The flow equation for [ A] is now of the form of Eq. (
8
).
The first trace in Eq. (
60
) does not depend on the photon field
and thus will not generate any Aμ contribution to ∂t k [ A].
This reflects the fact that QED is an abelian gauge theory
with no photon selfinteractions. Thus to oneloop order, all
the contributions to the running of the gauge part of the EAA
stem from the fermionic trace. From now on we will discard
the gauge trace.
We calculate the fermion trace in r.h.s. of Eq. (
60
) using
the nonlocal heat kernel expansion in Eq. (
17
). From Eq.
(
58
) we see that is the generalized Laplacian operator of
Eq. (
6
) with U = −σ μν Fμν /2. The function to be traced is
We can now specialize Eq. (
21
) to the QED case:
hk ( , me2) =
∂t Rk ( )
+ me2 + Rk ( )
1 1
∂t k [ A] = − 2 (4π )2
+ Fμν
0
∞
0
∞
d4x tr1
ds s−2h˜k (s, me2)
ds h˜k (s, me2) f F2 (sz) F μν ,
where z = −D2. The structure function f F2 (x ) is given by 2
2
f F2 (x ) = 2 fU (x ) − 4 f (x ) = f (x ) + x [ f (x ) − 1]
Plugging Eq. (
63
) into Eq. (
62
) and using Eq. (
23
) we get
1 1
∂t k [ A] = − 2 (4π )2
d4x
= 4
dξ ξ(1 − ξ ) e−xξ(1−ξ).
0
1
×
0
1
× 4 Q2[hk ] + 4 Fμν
(
61
)
(
62
)
(
63
)
(
66
)
1
Z A,k = e2 .
k
The quantity k (−D2) is the running photon polarization
which is a function of the gaugecovariant Laplacian. The t
derivative of Eq. (
65
) gives
1
∂t k [ A] = ∂t Z A,k 4
= − 2π1 2 0 dξ ξ(1 − ξ ) Q0[hkzξ(1−ξ)]. (
68
)
Since k (0) = 0, the beta function for the wavefunction
renormalization of the photon field is obtained by evaluating
Eq. (
68
) at z = 0:
Notice that in the limit me k the fraction in Eq. (
70
)
becomes equal to one and we recover the standard beta
function found in perturbation theory with a mass independent
regularization scheme [28]. On the other hand, for k me
the denominator becomes large and the beta function goes
to zero. This threshold behaviour is the manifestation of the
decoupling of the electron at low energy.
If we integrate the beta function for the electric charge in
Eq. (
70
) from an UV scale down to an IR scale k, we find
(
65
)
2
ek =
e2
e2
1 + 12π2 log 1+ 2/me2
1+k2/me2
The first constant piece is the renormalization of the
vacuum energy and we will drop it here. To proceed we need to
specify the form of the ansatz for k [ A], to be inserted in the
l.h.s. of Eq. (
62
). We choose
where Z A,k is the photon wavefunction renormalization
which is related to the electric charge via the following
identification:
2 We used tr U = 0, tr U2 = 2Fμν Fμν and tr μν μν = −4Fμν Fμν .
Equation (72) is interesting for several reasons. First, it
shows the screening effect of the vacuum: electron–positron
pairs polarize the vacuum around an electric charge so that the
effective electric charge ek , at the scale k, is smaller than the
electric charge e at the higher scale . Second, for k → 0,
it gives the relation between the bare electric charge e and
the renormalized electric charge e0:
2
e0 =
e2
.
Third, it shows that QED, as defined by the bare action in
Eq. (
53
), is a trivial quantum field theory: if we take the limit
→ ∞ in Eq. (73), at fixed finite e , we get a zero
renormalized electric charge e0. Conversely, if we solve Eq. (73)
for the bare charge e2 and we set the renormalized charge
e02 to some fixed value, then the bare coupling will diverge at
the finite “Landau pole” scale
These are the two faces of QED’s triviality. So, even if
the theory is perturbatively renormalizable, it cannot be a
fundamental theory valid at arbitrarily high energy scales.
To find an explanation for the success of QED, we have to
take the effective field theory point of view.
We now come back to consider the full momentum
structure of the r.h.s. of Eq. (
68
). Using Eqs. (
66
) and (
69
) we can
read off the running of the photon polarization function
(73)
(74)
ek2 k2
∂t k (z) = 6π 2 k2 + me2
ek2
× [1+ k (z)]− 2π 2
0
1
dξ ξ(1−ξ ) Q0 hz ξ(1−ξ) . (
75
)
k
We can find the oneloop renormalized polarization
function 0(x ) integrating Eq. (
75
) from the UV scale down
to k = 0 after having set the coupling ek to its bare value e .
Notice that the term proportional to k (z) in the r.h.s. of Eq.
(
75
) is at least of order e4 and we will discard it in
performing the integration since we are interested in reproducing the
oneloop result. As we did in the section for the real scalar,
we use the optimized cutoff of Eq. (
26
) to evaluate the
Qfunctional entering in Eq. (
75
). Performing the integral we
obtain
(z) −
e2
0(z) = 2π 2
0
1
dξ ξ(1 − ξ ) log
z
× 1 + ξ(1 − ξ ) me2 .
Notice that Eq. (
76
) does not contain divergent pieces,
which are local.
Setting the initial condition (z) = 0, we see that the
renormalized photon vacuum polarization function is given
by
e2
0(z) = − 2π 2
0
1
z
dξ ξ(1 − ξ ) log 1 + ξ(1 − ξ ) me2 .
(
76
)
(
77
)
Inserting Eq. (
77
) in Eq. (
65
) and redefining Aμ →
Aμ/Z 1/2 we obtain the following oneloop photon effective
A
action:
Equation (78) is the full QED oneloop effective action for
the photon which is quadratic in the field strength. Although
the polarization function in Eq. (78) is a function of −D2,
in an abelian theory like QED it boils down to a function of
just the flat Laplacian −∂2 and thus does not give a nonzero
contribution to higher vertices of the effective action.
With similar methods one can calculate the local terms in
the EAA which are of quartic order in the field strength (and
in the derivatives):
k [ A]F4 = ak
d4x Fμν F μν 2
+bk
d4x Fμν F να Fαβ F βμ,
(79)
where ak and bk are the Euler–Heisenberg coefficients
with negative quartic mass dimension. We can compute the
fermionic trace in Eq. (
60
) using the local heat kernel
expansion of Eq. (
11
). Contributions of order F 4 are given by the
coefficient b8( ) of the expansion, which, for constant field
strength, has the following form [29]:
b8(−D2) = 214 U4 − 61 U2
μν
μν
1
+ 288 ( μν
(78)
where we imposed the initial condition a∞ = b∞ = 0.
These values coincide with the wellknown result for the
Euler–Heisenberg coefficients.
By plugging these values back into Eq. (79) and combining
them with the O(∂4) term of Eq. (78) we obtain
which is the p4 part of the photon effective action [28]. The
first term in Eq. (84) is responsible of the Uehling effect
[30], while the other two describe the low energy scattering
of photons mediated by virtual electrons [31]. For a
nonperturbative use of the FRGE in QED see for example [32].
5 Yang–Mills
The situation for the nonabelian case is quite similar to the
abelian one, except for the fact that the gauge bosons are now
interacting. We begin by considering the Euclidean Yang–
Mills action for the gauge fields Aiμ in dimension d:
1
SY M [ A] = 4
dd x F μiν F μνi .
In Eq. (85) the quantity F μiν is the gauge field strength
tensor defined by
F μiν = ∂μ Aiν − ∂ν Aiμ + i g f i jk A μj Akν ,
where f i jk are the structure constants of the gauge symmetry
group and g is the coupling constant. The EAA is constructed
using the background field method [33,34]. The gauge field
is split as follows:
Aμ = A¯μ + aμ,
where aμ parametrizes the gauge fluctuations around the
background field A¯μ. In the following we will remove the
bar and we will denote the background field simply by Aμ.
In order to properly quantize the theory we choose as a
gaugefixing condition χ i = Dμaμi , where D is the covariant
,
(83)
(84)
(85)
(86)
(87)
function for evaluating the Qfunctionals, we obtain
1 e4
a0 = 36 (4π )2 0
7 e4
b0 = − 90 (4π )2 0
∞ dk
4k2
∞ dk
4k2
1
derivative constructed with the background connection
acting on fields in the adjoint representation. The gaugefixing
action then reads
dd x Dμaμi Dν aνi ,
where α is the gaugefixing parameter. The background ghost
action reads
Sgh [a, c¯, c; A] =
dd x Dˆ μc¯i Dμci ,
(88)
(89)
where c¯ and c are the ghost fields and Dˆ is the covariant
derivative constructed with the full field. The total action is
then obtained by summing the three contributions
S[a, c¯, c; A] = SY M [ A + a]+Sg f [a; A] + Sgh [a, c¯, c; A].
(90)
The background effective action k [a, c¯, c; A] which is
constructed using the background field method is a
functional of the background field ( Aμ) and of the classical fields
conjugated to the currents coupled to the quantum
fluctuations, which we denote again by (aμ, c¯, c). The background
EA is invariant under the simultaneous gauge transformation
of both. One can define the gauge invariant EAA by setting
the classical fields to zero k [ A] ≡ k [0, 0, 0; A]. In the
following we will study the RG flow of k [ A]. (Note that in
this case Dˆ can be replaced by D in the ghost action.)
The exact RG equation for k [ A] can be found in [35].
Using Eq. (90) we obtain
We need now to make an ansatz for the l.h.s. of the flow
equation. Retaining terms up to second order in the field
strength, but with arbitrary momentum dependence, the EAA
has the form
where Z A,k = 1/gk2 and k (z) is the running vacuum
polarization function.
Notice that the background wavefunction
renormalization constant, and so the gauge coupling, enters the flow of
k (x ) only as an overall factor.
Comparing the expression of Eq. (94) with Eq. (92) and
using Eq. (
25
), we get the flow equation for the running
vacuum polarization function:
where z stands for the covariant Laplacian −D2. Evaluating
the Qfunctionals for the optimized cutoff (
26
) in dimension
d = 4 gives
N
∂t Z A,k + ∂t +Z A,k k (z), = (4π )2
⎡ 22
× ⎣ 3 −
22
3 + 3z
8k2
1 − z
4k2
k (0) = 0 we can rewrite the above equation as
where we defined α
in perturbation theory.
We now go back to the running of the vacuum polarization
in Eq. (98). The term ηA,k k (z) is at least of order gk4 and
we will discard it here since we are interested in reproducing
the oneloop result. Moreover, we set the running coupling
to its bare value g . We can now integrate the flow of k (z)
in Eq. (98) from an UV scale down to an IR scale k. We
get
(z) −
g2 N
k (z) = 2(4π )2
The integral in Eq. (104) is finite in the limit → ∞
and no renormalization is needed. The function gF2 has no
constant term so for every z and k big enough the flow of
k (z) is zero and no divergences can develop. In this limit,
the vacuum polarization function goes to its boundary value,
i.e. (z) = 0. Using the general integrals of Eqs. (
28
)–
(
30
), the vacuum polarization function at the scale k is finally
found to be
g2 N ⎨⎧
k (z) = − (4π )2
22 ⎡ 1 z
− 3 ⎣ 2 log k2 + log
1 + %
4k2 ⎤
1 − z
2
+
⎩
64
9 + 9z
8k2
(100)
for z/ k2 ≥ 4 and k (z) = 0 for z/ k2 < 4. From Eq.
(105) we see that we cannot send k → 0, since the first
logarithm diverges in this limit. For k2 z, Eq. (105) gives
the following contribution to the gauge invariant EAA:
g2 N
64π 2
d4x F μiν
11 log (−D2)ab
3 k2
We can interpret the obstruction to taking the limit k → 0
in Eq. (105) as a signal of the breakdown of the approximation
used in its derivation, where we considered the flow of k (z)
as driven only by the operator 41 F 2. In order to be able to
continue the flow of the EAA in the deep IR, we need the
full nonperturbative power of the exact RG flow equation
that becomes available if we insert the complete ansatz (94)
in the r.h.s. side of it [36–44].
6 The chiral model
In the previous sections we have considered perturbatively
renormalizable theories. In the remaining two we shall
consider nonrenormalizable ones. The standard way of treating
nonrenormalizable theories is the method of effective field
theories [28]. We shall see here how to recover some
wellknown results of the EFT approach using the FRGE.
Previous application of the FRGE to the nonlinear sigma models
have been discussed in [46–51]. The dynamics of Goldstone
bosons is described by the nonlinear sigma model, a
theory of scalar fields with values in a homogeneous space. In
particular in QCD with N massless quark flavours the
Goldstone bosons of spontaneously broken SUL (N ) × SUR (N )
symmetry correspond to the meson multiplet. These theories
are known as the chiral models. They have derivative
couplings and their perturbative expansion is illdefined in the
UV. A related and phenomenologically even more pressing
issue is the high energy behaviour of the tree level
scattering amplitude, which grows like s/Fπ2, where s is the c.m.
energy squared and Fπ (the “pion decay constant”) is the
inverse of the perturbative coupling. This leads to
violation of unitarity at energies of the order ∼4π Fπ , which is
usually taken as the first sign of the breakdown of the
theory.
The chiral NLSM that we consider here is a theory of
three scalar fields π α(x ), called the “pions”, parametrizing
(in a neighbourhood of the identity) the group SU (
2
).
Geometrically, they can be regarded as normal coordinates on
the group. We call U the fundamental representation of the
group element corresponding to the field π α: U = exp( f π ),
π = π α Tα, Tα = 2i σα, Tα† = −Tα, tr (Tα Tβ ) = − 21 δαβ ,
α = 1, 2, 3. The coupling f is related to the pion decay
constant as Fπ = 2/ f . The standard SU (
2
)L ×SU (
2
)R invariant
action for the chiral model is3
1
S[π ] = − f 2
d4x tr U −1∂μU U −1∂μU.
(107)
3 Since U −1 = U † we have tr U −1∂μU U −1∂μU = −tr ∂μU ∂μU †.
This is the term with the lowest number of derivatives. Terms
with more derivatives will be discussed later. Introducing the
above formulae and keeping terms up to six powers of π we
get
1
S[π ] = 2
Note that the pion fields π α are canonically normalized
and the metric is dimensionless, whereas the fields ϕα are
dimensionless.
Following [52–55] we use the background field method
and expand a field ϕ around a background ϕ¯ using the
exponential map: ϕ(x ) = expϕ¯(x) ξ(x ) where the quantum field
ξ is geometrically a vector field along ϕ¯. The EAA will be,
in general, a function of the background field and the
Legendre transform of sources coupled linearly to ξ , which we will
denote by the same symbol hoping that this will cause no
confusion. We also omit the bar over the background field so that
we can write the EAA as k [ξ ; ϕ]. For our purposes it will be
sufficient to compute this EAA at ξ = 0: k [ϕ] ≡ (
2
)[ϕ]. In
k [0; ϕ].
The RG flow for k [ϕ] is driven by the Hessian k
the oneloop approximation that we shall use, this is equal
to
where ≡ ∇μ∇μ, ∇μ is the covariant derivative with
respect to the Riemannian connection of hαβ , Uαβ =
−R αηβ ∂μϕ ∂μϕη. We have expressed the second
variation in terms of the dimensionless background fields ϕα,
which produces the overall factor 1/ f 2. In the oneloop
approximation the running of couplings in the r.h.s. of the
FRGE is neglected and f has to be kept fixed along the
flow.
Since geometrically SU (
2
) is a threesphere with radius
two [47], the Riemann tensor can be written in the form
(108)
(109)
(110)
Rαβγ δ = 21 (hαγ hβδ−hαδ hβγ ). The appearance of the
covariant Laplacian suggests the choice of = − + U as
argument of the cutoff kernel function. In this way the cutoff
combines with the quadratic action to produce the function
hk ( , ω) given in Eq. (
7
), with ω = 0.
Evaluation of the trace follows the steps outlined in Sect. 2
and one arrives at the beta functional:
∂t k [π ] = 21 (4π1 )2 d4x k2 f 2 ∂μπα∂μπ α − 112 k2 f 4
+ 81 f 4 ∂μπ β ∂ν πα g (− / k2)∂μπ α∂ν πβ
− ∂μπα∂ν π β g (− / k2)∂μπ α∂ν πβ
+O(π 6) ,
(112)
where we have used μν αβ = ∂μϕ ∂ν ϕη Rαβ η and Eqs.
(
34
)–(
35
). The form factors gU (− / k2) and g (− / k2)
correspond to gU (− , 0, k) and g (− , 0, k) in the notation
of Sect. 2. It is important to stress that in the derivation of this
result no regularization was needed: the integrals we had to
perform were IR and UV finite. This is a general property of
the beta functional ∂t k .
Another important fact to note is that the first two terms
appear in the same ratio as in the original action of Eq. (108).
This is just a consequence of the fact that the cutoff preserves
the SUL (
2
) × SUR (
2
) invariance of the theory. As a result,
in the RG flow the metric hαβ (ϕ) gets only rescaled by an
overall factor when expressed in terms of the dimensionless
field ϕ.
We see that in addition to terms of the same type of the
original action, quantum fluctuations generate new terms
with four derivatives of the fields. To this order one can write
an ansatz for the EAA that contains generic fourderivative
terms:
k [π ] =
d4x
21 ∂μπα∂μπ α − 214 fk2
+ ∂μπα∂μπ α fk4γU,k ( )∂ν πβ ∂ν π β
+ ∂μπ α∂μπ β fk4γU,k ( )∂ν πα∂ν πβ
− ∂μπ α∂ν π β fk4γ ,k ( )∂μπα∂ν πβ
+ ∂μπ β ∂ν π α fk4γ ,k ( )∂μπα∂ν πβ
+O(π 6) .
(113)
Here the coupling fk and the dimensionless form factors
γU,k and γ ,k are functions of the scale k. The local parts of
the form factors γU,k (0) and γ ,k (0) can be combined as
12,k = γU,k (0) + γ ,k (0) and
22,k = γU,k (0) − γ ,k (0). (115)
Up to this point we have interpreted an infinitesimal RG
transformation as the result of an integration over an infinitesimal
shell of momenta ak < p < k, with a = 1 − and > 0.
In the literature on the Wilsonian RG, however, this is
complemented by two additional transformations: a rescaling of
all momenta by a factor a and a rescaling of all fields so
that they remain canonically normalized. The rescaling of
the momenta is effectively taken into account by the
rescaling of the couplings f˜k = k fk , etc. We will now explicitly
implement the rescaling of the fields:
(114)
(116)
,
d4x
Comparing the t derivative of Eq. (113) with Eq. (117) we
see that, by construction, the kinetic term does not change
along the RG flow while for the second term we find instead
∂ˆt fk2 = − 1k62πf 42 .
For the nonlocal form factors, one obtains
1 1
∂ˆt γU,k ( ) = 16π 2 32 gU (− / k2),
1 1
∂ˆt γ ,k ( ) = 16π 2 16 g (− / k2).
In order to recover the standard perturbative result for the
effective action we integrate Eq. (117) from some initial UV
scale k = , which we can view as the “UV cutoff”, down to
k = 0, keeping the Goldstone coupling f fixed at the value
it has in the bare action ( f ) and neglecting corrections of
order O( f 4 f˜2) to the flow of the form factors. In fact, from
the integration of Eq. (118) we will see that the coupling f 2
changes by a factor 1 + 32f˜π2 2 , which is a number close to one
in the domain of validity of the effective field theory (even if
≈ f −1, f changes only by a few percent). Thus we have
− 0 =
=
dk
k ∂ˆt k [π ]
dk d4x
k
× ∂μπα∂μπ απβ π β − ∂μπ α∂μπ β παπβ
+ f 4 ∂μπ α∂μπα∂ˆt γU,k ( )∂ν π β ∂ν πβ
+∂μπ α∂μπβ ∂ˆt γU,k ( )∂ν π β ∂ν πα
+∂μπ β ∂ν πα∂ˆt γ ,k ( )∂μπ α∂ν πβ
−∂μπα∂ν π β ∂ˆt γ ,k ( )∂μπ α∂ν πβ +O(π 6) .
0
0
0
The effective action is then obtained by integrating Eqs. (118)
and (119). We get
dk 2 1 f 4
k ∂ˆt fk = − 2 16π 2
2
2
− f0 =
and
γU, ( ) − γU,0( )
dk
= 0 k
1 1
= 32 16π 2
∂ˆt γU,k ( )
1 − 21 log −
2
,
(123)
(124)
(125)
(118)
(119)
(120)
(121)
(122)
.
In this effective field theory approach the need to
renormalize does not arise so much from having to eliminate
divergences, since corresponds to some finite physical scale and
all integrals are finite anyway. Instead, it is dictated by the
desire to eliminate all dependence on high energy couplings,
which are unknown or only poorly known, and to express
everything in terms of quantities that are measurable in the
low energy theory.4 So, in order to eliminate the quadratic
dependence on , we define the renormalized coupling to be
f02 = f
2
Here the couplings with subscripts can be interpreted
as the “bare” couplings and the ones with subscripts 0 can
be interpreted as the “renormalized” ones. Notice that, once
subtracted, the quadratic dependence completely disappears,
but due to the need to compensate for the dimension of the
Laplacian in the logarithms, one has to define the
renormalized 1 and 2 at some (infrared) scale μ and therefore there
unavoidably remains a residual scale dependence.
After this renormalization procedure, the effective action
can be written, in perturbation theory, as a function of the
renormalized coupling:
d4x
21 ∂μπα∂μπ α − 214 f 2
× ∂μπα∂μπ απβ π β − ∂μπ α∂μπ β παπβ
1 4
+ 2 f ∂μπα∂ν π α∂μπβ ∂ν π β
2 4
+ 2 f ∂μπα∂μπ α∂ν πβ ∂ν π β
1 f 4
− 32 16π 2 ∂μπα∂μπ α
× 1 − 21 log −μ2
∂ν πβ ∂ν π β
4 Ultimately, this produces the same effect, for when the high energy
parameters have been eliminated one may as well send to infinity.
potential [60–63]. As we shall see, some of these terms are
related to the low energy ones by simple properties, so that
the two calculations partly overlap. From a different point
of view, the flow of scalar couplings due to gravity has also
been discussed in [64], and with the aim of establishing the
existence of an UV fixed point, in [65,66]. Previous
application of the FRGE to the calculation of some terms in the
gravitational EA have been given in [67–71].
Action All calculations will be done in the Euclidean field
theory. We will study the flow of the EAA driven by an action
of the form
+Sgh [h, C¯ , C ; g],
k [h, C¯ , C ; g] = SH [g + h] + Sm [g + h, φ] + Sg f [h; g]
(128)
where hμν is the metric fluctuation, C¯ μ and Cμ are
anticommuting vector ghosts for diffeomorphisms, gμν is the
background metric. The term
dd x .det(g + h)R(g + h),
(129)
− 312 16fπ4 2 ∂μπ α∂μπβ 1− 21 log −μ2
where we have eliminated the subscripts from the couplings.
The renormalized couplings f , 1 and 2 have to be measured
experimentally, but the nonlocal terms are then completely
determined and constitute therefore low energy predictions
of the theory.
These nonlocal terms enter in the computation of the
Goldstone boson scattering amplitude A(παπβ → πσ πρ ) =
A(s, t, u)δαβ δσρ + A(t, s, u)δασ δβρ + A(u, t, s)δαρ δβσ , which
is obtained by taking four functional derivatives of the
effective action in Eq. (126) with respect to π α after performing
the analytic continuation to Minkowski space and evaluating
the expression at the particles external momenta. We get
A(s, t, u; μ) =
2s2 log −s
μ2 + t (t − u) log
−t
μ2 + u(u − t ) log −u
μ2
,
(127)
where s = ( p1 + p2)2, t = ( p1 + p3)2 and u = ( p1 + p4)2
and all momenta are taken to be incoming. This result is well
known in the literature on chiral models [45].
7 Gravity with scalar field
In this section we will consider another example of
effective action for an effective field theory, namely a scalar
coupled minimally to gravity. As first argued in [56,57] and
confirmed by explicit calculations [58,59] low energy
gravitational scattering amplitudes can be calculated
unambiguously in this effective field theory, in spite of its perturbative
nonrenormalizability. The reason is that low energy effects
correspond to nonlocal terms in the effective action that are
nonanalytic in momentum. Such terms are not affected by
the divergences, which manifest themselves as terms
analytic in momentum. Here we shall follow the logic of
previous chapters and derive the terms in the effective action
containing up to two powers of curvature, by integrating the
flow equation. There have been several calculations of
divergences for a scalar coupled to gravity, including also a generic
1
× 2
(g + h)μν ∂μφ∂ν φ + V (φ)
(130)
(131)
is the Feynman–de Dondertype gaugefixing term (the gauge
parameter α is set to one). We will only need the ghost action
for h = 0, in which case it has the form
Sgh [0, C¯ , C ; g] =
dd x √g C¯ μ − δνμ − Rνμ C ν . (132)
The covariant derivative ∇ and the curvature R are
constructed with the background metric and we denote =
∇μ∇μ.
Using Eq. (
5
), this action generates all possible
diffeomorphisminvariant terms. We retain only those that
are quadratic in ”curvatures”, where we include among
curvatures the Riemann tensor and its contractions, terms with
two derivatives acting on one or two scalar fields the
potential and its derivatives.5 Within this class of EAA’s we can
calculate the RG flow and integrate it analytically to obtain
an EA, which consists of both local and nonlocal terms. In
5 To some extent this parallels the treatment of mass terms in chiral
perturbation theory.
a more accurate treatment these new terms would all
contribute to the r.h.s. of the flow equation, but such calculations
would be far more involved.
Hessian Arranging the fluctuation fields hμν , δφ in a d(d +
1)/2 + 1dimensional column vector δ , the total quadratic
part can be written as
1
2κ
where the dot refers to the components in the space spanned
by δφ. The Hessian H has the form
H = K(− ) + 2Vδ∇δ + U.
The coefficient of the secondderivative term is a quadratic
form in field space,
K αβμν 0
0
κ
is the DeWitt metric. (We denote by δαβ μν the identity in the
space of symmetric tensors.) Furthermore,
0
κ K μν γ δ∇γ φ
−κ K αβ γ δ∇γ φ
0
,
U αβμν
2κ K μνγ δ∇γ ∇δφ + 21 κgμν V (φ)
21 κgαβ V (φ)
κ V (φ)
U αβμν = K αβμν R + 21 (gμν Rαβ + Rμν gαβ )
K =
where
K αβμν
Vδ =
U =
with
− 41 gαμ Rβν + gαν Rβμ + gβμ Rαν + gβν Rαμ
− 21 (Rαμβν + Rανβμ) + κ − 21 K αβμν (∇φ)2
1
− 4 (gαβ μ ∇ φ∇β φ)
∇ φ∇ν φ + gμν α
1
+ 4
gαμ∇β φ∇ν φ + gαν ∇β φ∇μφ
+gβμ∇αφ∇ν φ + gβν ∇αφ∇μφ − K αβμν V , .
= I(− ) + 2Yδ∇δ + W
=
αβ μν
μν
·
αβ ·
··
It is convenient to extract an overall factor of K and write the
Hessian as H = K where
is a linear operator in field space and therefore has the index
structure
The coefficients in the operator are related to those of the
Hessian by Yδ = K−1Vδ, W = K−1U, where
Kμ−ν1αβ = 2δμν αβ
Note W need not to be symmetric, in fact the question is
not even well posed because of the different position of the
indices. Explicit calculation leads to the following
expressions:
,
Wαβ μν
2K μν γ δ∇γ ∇δφ + 21 gμν V (φ)
− d −22 κgαβ V (φ) ,
V (φ)
Yδ =
W =
where
Wαβ μν = 2 Uαβ μν
Rμν
− 21 R gμν
κ
− 2 gαβ gμν V .
d − 4
− d − 2 gαβ
κ
− 2
∇μφ∇ν φ − 21 gμν (∇φ)2
(140)
(141)
(142)
(143)
(144)
(145)
(146)
(147)
(148)
(149)
Completing the square In order to use the standard heat
kernel formulae for minimal Laplacetype operators, we have to
eliminate the firstorder terms Y · ∇. This can be achieved by
absorbing them in a redefinition of the covariant derivative:
∇˜ μ = ∇μI − Yμ. Then in (141) can be rewritten as
= −∇˜ μ∇˜ μ + W˜
W˜ = W − ∇μYμ + YμYμ.
To compute W˜ we need the following intermediate results:
∇μYμ =
YμYμ =
and
0
K μν γ δ∇δ∇γ φ
−κ∇α∇β φ
0
−κδαβ γ δ K μν δ∇γ φ∇ φ
0
0
−κ(∇φ)2
Kγ δ δ = d4 δγ δαβ γ δ K μν δ∇γ φ∇ φ
= − 21 δ((αμ∇β)φ∇ν)φ + 41 gμν ∇αφ∇β φ.
Collecting, we find
Aαβ μν Bαβ
Cμν D
W˜ =
with
Aαβ μν = Wαβ μν − 21 κδ((αμ∇β)φ∇ν)φ + 4 κgμν∇αφ∇β φ
1
Cμν = K μν αβ ∇α∇β φ + 21 gμν V (φ)
D = −κ(∇φ)2 + V (φ).
The curvature of covariant derivatives, ˜ μν = [∇˜ μ, ∇˜ ν ], is
related to the curvature of the original covariant derivative,
= [∇μ, ∇ν ] by
˜ μν =
μν − ∇μYν + ∇ν Yμ + YμYν − Yν Yμ.
(151)
To compute ˜ κλ we need
0
2∇[κ Yλ] = 2K μν γ δ g[λδ∇κ]∇γ φ
−2κδαβ γ δ g[λδ∇κ]∇γ φ
0
and
2Y[κ Yλ] =
Then we find
˜ κλ =
where
−2κδαβ γ [κ Zμνλ]∇γ φ∇ φ
0
( ˜ κλ)αβ μν = ( κλ)αβ μν − 2κδαβ γ [κ K λ]μν ∇γ φ∇ φ,
Heat kernel coefficients We compute first the local heat
kernel coefficients of the operator , using (
12
). The first two
are
tr b0( ) = 11 ,
25
tr b2( ) = − 6 R + 2κ((∇φ)2)2 − V + 10κ V .
For the calculation of b4 we need a few preliminary results.
Using tr I = 11 and defining P = W˜ − R6 I, the heat kernel
coefficient (
12
) can be rewritten, in four dimensions, in the
more compact form
tr b4( ) = 11810 (Riem2 − Ric2) + 21 tr P2 + 112 tr ˜ 2. (156)
(150)
(152)
(153)
(154)
(155)
In the evaluation of the last two terms we use the following
traces:
tr P2 = tr W˜ 2 − 31 R tr W˜ + 3161 R2 ,
tr W˜ = Aμν μν + D ,
Aμν μν = 6R − κ(∇φ)2 − 10κ V ,
tr W˜2 = Aμν αβ Aαβ μν + 2Bαβ Cαβ + D2,
Aμν αβ Aαβ μν = 3Rμνρσ Rμνρσ − 6Rμν Rμν + 5R2
.
tr ˜ μν ˜ μν = ( ˜ μν )αβ γ δ( ˜ μν )γ δαβ
+2( ˜ μν )αβ ( ˜ μν )αβ ,
where
( ˜ μν )αβ γ δ( ˜ μν )γ δαβ = −6Rμνρσ Rμνρσ
(157)
+κ R(∇φ)2 + 2κ Rμν ∇μφ∇ν φ
3
− 2 κ((∇φ)2)2
×2( ˜ μν )αβ ( ˜ μν )αβ
= κ((∇2φ)2 − 4∇μ∇ν φ∇μ∇ν φ).
We thus arrive at
tr b4( ) = 118901 Riem2 − 158501 Ric2 + 17129 R2
+ 45 κ2((∇φ)2)2 − 6 κ∇2φ∇2φ
1
+ 61 κ∇β ∇αφ∇β ∇αφ + 61 κ Rαβ ∇αφ∇β φ
1
+κ κ V (φ) − 3 R − V (φ) (∇φ)2
+κ V (φ)∇2φ
13
+5κ2V 2(φ) − 2κ V (φ)2 − 3 κ R V (φ)
R
− 6 V (φ) +
which hold under an integral modulo surface terms, one can
rewrite
where we used again (160). The coefficient of s2 in (
17
) is
Rμν (4 fRic + fU − 4 f )Rμν + R(4 f R − f RU + f )R.
(164)
We are now ready to write the flow equation.
Flow equations We write the oneloop flow equation for the
“single metric” bEAA k [g] ≡ k [0, 0, 0; g]. It consists of
two terms:
∂t k [g] = 21 Tr ∂+tRkR(k ( ) ) − Tr
∂t Rk ( gh)
gh + Rk ( gh)
.
(165)
The first comes from the graviton and scalar fluctuations,
the second from the ghosts. Using Eq. (
21
) and the heat kernel
discussed in the previous section, the flow equation for the
bEAA is
where the functions ga, a = 1, . . . , 10, are linear
combinations of the functions gRic, gR, gU , gRU , g given in the
second column of the following table:
a
ga
1 3gRic + 4gU − 16g
2 3gR + 2gU + 8gRU + 4g
3 −2gRU − 23 gU + g
4 2g
5 gRU
6 −10gRU − 12gU
7 gU − 4g
8 − 21 gU + g
9 141 gU − 23 g
10 2gU
Aa
43
310
202
− 3
1
3 1
− 326
−1 3
3 1
− 3
5
2
2
Ba
−3711458
5 2
− 3
4
3
2
3 20
− 83
− 3
2
3
−1
0
Ca
8
− 5
1
5
0
0
0
0
0
0
0
0
Note that only the first two receive contributions from the
ghosts. Next we recall that using the optimized cutoff, the
functions gRic, …, g are given by Eqs. (
31
)–(
35
). In the
present case there is no dependence on ω, so dropping this
argument the functions ga(z˜) = ga(z, k) can all be written
in the form
Ba Ca
ga (z˜) = Aa + − Aa + z˜ + z˜2
where the coefficients Aa , Ba , and Ca are given in the
remaining columns of the table above.
Integration of the flow We have found that the classical
action (128) generates the flow (166). The flow is finite but
integrating over k up to some scale and taking the limit
→ ∞ it generates divergences in the effective action. The
first line is proportional to terms that are already present in
the action and, upon integration over k, correspond to quartic
and quadratic divergences. The remaining terms were not
originally present in the action. Equation (167) allows us to
split these terms into local parts (the first term) and nonlocal
parts (the rest). The local parts correspond to logarithmic
UV divergences. The nonlocal parts vanish for k2 > z/4,
signalling its infrared character.
The new terms in the flow equation force us to consider
a more general class of EAA’s that, in addition to the terms
that were originally present in (128) also contains for each
ga (z˜) ≡ ga(z, k), a = 1, . . . , 10, in (166), a corresponding
form factor fa,k (z). In principle all the terms in the last two
lines of (166) could appear with different couplings, but in
the present truncation they all renormalize in the same way.
The functions ga are the beta functions of the form factors
fa :
1
∂t fa,k (z) = 32π 2 ga(z˜).
Even though our main interest is in infrared physics, it is
useful to consider first the divergent part of the action. For a
fixed z and for k2 > z/4, the step function in (167) vanishes.
In this case the integration of the flow only gives power and
logarithmic terms. We find for large
(168)
1
[g] = 32π 2
d4x √g 3 4
+
29
− 3 R + 4κ(∇φ)2 − 2V
+ 3403 Rμν Rμν + 210 R2 − 23 κ R(∇φ)2
26 1
− 3 κ R V (φ) − 3 RV (φ)
+ 25 κ2((∇φ)2)2 + 2κ2V (φ)(∇φ)2
−4κ V (φ)(∇φ)2
+ 20κ V
2
+10κ2V 2(φ) − 4κ V (φ)2 + V (φ)2 log
+finite terms ,
2
μ2
(169)
where we have introduced an arbitrary renormalization scale
μ. The last three lines agree with the results of [63],
specialized to a single component field. The finite terms have
the same structure as the ones that are written but with
independent, finite, coefficients. These correspond to the
arbitrariness in the choice of the initial conditions for the
flow.
We now integrate the flow equations from the UV scale
down to zero. The first line in (166), which depends on
powers of the cutoff scale k, produces simply the terms in
the first line of (169).
Next we discuss the logarithmic terms. For each form
factor the integration is to be performed keeping z fixed. For a
generic IR scale k we have
where in the second step we changed variables of integration
to u = z/ k2. Now we insert the explicit form (167). We
assume that z/ 2 4 and z/ k2 4. The integration can
be done explicitly and in the limit k → 0 one finds
fa,0(z) = fa, (z) − 64Aπa 2 log
2
μ2
+ 64Aπa 2 log
z
μ2
1
− 32π 2
Note the following fact: if we had just integrated the local
part of the form factor, namely the first term in (167), from
k to , we would have obtained 32Aπ2 log( 2/ k2), which is
both UV and IR divergent. When we integrate the full form
factor, for k → 0 in (167) z˜ becomes large, the theta function
is one and the square root tends to one, so that the Aterms
cancel. Thus, by integrating the full nonlocal form factor,
we obtain a result that is UV divergent but IR finite.
While the nonlocal terms are finite and entirely
unambiguous, the local terms in the effective action 0 are not
fixed and have to be determined by matching the form of the
EA with experimental data. This is achieved by means of
renormalization conditions. In the present formalism, these
correspond to the choice of .
The initial conditions for the form factors can be chosen
to eliminate the dependence in the EA. This requires
(170)
.
(171)
(172)
fa, (z) = 64Aπa 2 log
2
μ2
+ γa ,
E0 = E
μ2
V + V log
−
μ2
V
(174)
The local terms are operators of the form appearing in
(169), with arbitrary finite coefficients. These coefficients
are related to the “bare” couplings by renormalization
conditions. To discuss them it is more transparent to specify the
form of the potential, e.g.
V (φ) = E + 21 m2φ2 + 214 φ4.
Recalling that all the couplings in the EAA are kdependent,
we denote by subscripts and 0 the “bare” and
“renormalized” couplings, respectively. Then we find that the terms that
are already present in the action are renormalized as follows:
where γa are arbitrary constants, corresponding to the finite
terms in (169). Then the form factors in the EA are
fa,0(z) = 64Aπa 2 log
z
μ2
,
and the EA has the form
ξ2φ2 R + ξ4φ4 R + τ (∇φ)2 R + w(∇φ)4
+ α R2 + β Rμν Rμν ,
with effective couplings
,
μ2
2
μ2
,
.
,
,
The last term in (181) is due entirely to scalar loops and would
also be present if gravity was treated as an external field. In
this context this term has been discussed several times in the
literature; for example, see [72,73].
We note that all the renormalized couplings depend on
the reference scale μ. This dependence encodes a different
μ2
2
μ2
1 1
κ0 = κ
+
notion of renormalization group. For the cosmological
constant and Newton’s constant this has been discussed e.g. in
[74–77].
Finally let us comment on the finite local terms in (174).
The choice γa = 321π2 Aa + B12a + 1C2a0 has the effect that
all the local terms vanish. The choice γa = 0 leaves a
residual finite term that can easily be calculated from (173) and
the table. For example the first term in the last line of (166)
would leave in this case 8π1 2 κm4φ2. We see that the
gravitational correction to the scalar mass is suppressed by the
ratio (m/MPlanck)2, as one would naturally guess from the
weakness of gravity at low energy. In any case the constants
γa cannot be calculated but have to be fixed by comparison
to experiment.
From the point of view of the effective field theory
approach, the action (174) contains part of the terms needed
to reconstruct the scalar–graviton vertex in a perturbative
expansion about flat space. More precisely, it contains the
terms with two generalized curvatures, which correspond
to Feynman diagrams with a threegraviton and twoscalar–
twograviton vertices. Other contributions corresponding to
triangle diagrams are encoded in terms with three generalized
curvatures, which we have not evaluated.
8 Conclusions
We have presented several calculations of oneloop effective
actions in quantum field theories of increasing complexity:
from a simple linear scalar field to gauge theories (QED and
nonabelian Yang–Mills theory), to chiral sigma models and
finally to gravity coupled to a scalar. In each case we have
derived the first few terms in an expansion of the action in
powers of generalized “field strengths”, meaning either the
potential or curvatures.
Instead of calculating directly a functional integral, we
have obtained the effective action following Wilson’s idea of
integrating out field modes one momentum shell at a time.
The FRGE gives us a formula for the “beta functional” of
the theory, i.e. the derivative of the EAA k with respect
to the cutoff k. The ordinary effective action is recovered
by integrating the flow from some initial condition at a
UV scale k = down to k = 0. The choice of
corresponds to the choice of renormalization conditions. The
main conceptual advantage of the method used here is that
one never encounters divergent quantities. The r.h.s. of the
flow equation (
3
) is both ultraviolet and infraredfinite, due
to the falloff properties of the cutoff term (
1
). To be sure, the
divergences of the QFT are still present: they appear if one
tries to send → ∞, which corresponds to integrating the
flow equation towards the UV. However, they appear only in
this final stage of the integration and they do not arise as an
obstacle in the calculation of the effective action. By
postponing the integration over the momentum cutoff to the
last step, the structure of the QFT is conceptually clearer, not
being marred by the issue of the divergences.
Ideally, in this program, one would like to use the exact
FRGE (
3
). In practice, one has to make some approximation.
Here we have restricted ourselves to the oneloop
approximation, which amounts to keeping k = in the r.h.s. of
the FRGE. This approximation is dictated by the current
status of the techniques for calculating functional traces, which
allow us to compute the trace of a function of a Laplacetype
operator. This is the case when k is a local functional of
the fields containing up to two derivatives. Integration of the
FRGE immediately generates all sorts of operators,
including nonlocal terms, which would lead outside the domain of
applicability of current heat kernel techniques.
Another limitation of the work presented here is that we
studied mainly terms up to quadratic in field strengths in
the effective action; however, we got their exact oneloop
momentum dependence. This was motivated by our main
interest in models of particle physics, where only few
particles scatter at a time. Of course, we admit that general
oneloop effective action (in any interacting QFT) possesses the
structure with operators up to infinitely many derivatives as
well as infinitely many curvatures. Again due to the
computational techniques used we restricted ourselves to the
simplest case of terms up to quadratic in curvatures.
However, within this approximation, we were able to find some
scattering amplitudes, like a fourpoint one in scalar field
theory.
Still, within the oneloop approximation, we have been
able to derive the flow equations for the nonlocal structure
functions, which upon integration give nontrivial finite parts
of the effective action. The origin of these nonlocal terms
from the IR part of the momentum integration, and their
independence of UV physics, is particularly transparent. Another
application of such an alternative method of computation
leading to EA is that the oneloop approximation can be taken
as a first step towards a reformulation of the QFT, where
nonperturbative effects are taken into account. As an example of
such computations, in [27] the running of the couplings in
the r.h.s. of the flow equation for threedimensional scalar
theory has been taken into account, improving the
calculation of Sect. 3. This leads to a different expression for the
fourpoint amplitude that is automatically finite also in the
limit → ∞, curing both the UV and the IR divergences. It
would be very interesting to perform this type of calculation
also for the other models considered here.
Acknowledgments This work is based in part on the Ph.D. theses
presented by the first author at the Gutenberg University in Mainz and
by the last two authors at SISSA. A.C. and A.T. would like to thank
their supervisors, Martin Reuter and Marco Fabbrichesi, for their
continuous support. The work of A.C. was partially supported by the
Danish National Research Foundation DNRF:90 grant. The work of A.T.
was supported by the São Paulo Research Foundation (FAPESP) under
grants 2011/119734 and 2013/024041.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
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Funded by SCOAP3.
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