#### A functional RG equation for the c-function

Alessandro Codello
3
Giulio D'Odorico
1
Carlo Pagani
0
2
0
INFN sez. di Trieste, via Bonomea 265, I-34136, Trieste,
Italy
1
Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics
, Heyendaalseweg 135, 6525 AJ Nijmegen,
the Netherlands
2
SISSA - International School for Advanced Studies
, via Bonomea 265, I-34136, Trieste,
Italy
3
CP3-Origins & Danish IAS University of Southern Denmark
, Campusvej 55, DK-5230 Odense M,
Denmark
After showing how to prove the integrated c-theorem within the functional RG framework based on the effective average action, we derive an exact RG flow equation for Zamolodchikov's c-function in two dimensions by relating it to the flow of the effective average action. In order to obtain a non-trivial flow for the c-function, we will need to understand the general form of the effective average action away from criticality, where nonlocal invariants, with beta functions as coefficients, must be included in the ansatz to be consistent. Then we apply our construction to several examples: exact results, local potential approximation and loop expansion. In each case we construct the relative approximate c-function and find it to be consistent with Zamolodchikov's c-theorem. Finally, we present a relation between the c-function and the (matter induced) beta function of Newton's constant, allowing us to use heat kernel techniques to compute the RG running of the c-function. ArXiv ePrint: 1312.7097
Contents
1 Introduction 2 3 4
The integrated c-theorem
2.1 Weyl-invariant quantization and functional measures
2.2 CFT action on curved background
2.3 Anomaly matching from the path-integral
2.4 Proof of the integrated c-theorem
Flow equation for the c-function
3.1 The fRG flow equation for the c-function
3.2 fRG derivation of the integrated c-theorem
General form of the effective average action
4.1 The local ansatz and its limitations
4.2 Nonlocal ansatz and the scale anomaly Applications
5.1 Checking exact results
5.1.1 Massive deformation of the Gaussian fixed point
5.1.2 Massive deformation of the Ising fixed point
5.2 The c-function in the local potential approximation
5.2.1 Flow between the Gaussian and Ising fixed points
5.2.2 Sine-Gordon model
5.3 The c-function in the loop expansion
5.3.1 Zamolodchikovs metric: diagrammatics
5.3.2 Diagonal contributions
5.3.3 Non-unitary theories
The c-function and Newtons constant
6.1 Relation between ck and Gk
6.2 Minimally coupled scalar
6.3 Self-interacting scalar 7 Conclusions A Loop expansion from the fRG
The renormalization group (RG) underlies most of our modern understanding of quantum
and statistical field theories [1, 2]. There are different ways to implement the RG procedure.
Whereas standard sliding scale arguments (Gell-mann-Low) are particularly suitable for
weakly coupled computations, it is only with Wilsons ideas that non-perturbative insights
have been possible.
The arena where the RG acts is theory space. This space is parametrized by all
couplings corresponding to terms which are consistent with symmetries of the systems.
The beta functions for the couplings define a vector field in theory space, and the RG flow
can be seen in geometrical terms as a certain trajectory in this space.
From this point of view the infrared physics depends upon the differential equation
governing the flow as well as on the boundary conditions. If the initial point sits at a finite
scale (for example, it is a bare action depending on some UV cutoff ), one is considering
an effective field theory, whose range of validity is limited by the cutoff scale. However, if
we want a theory to be called fundamental, we would like to be able to push the initial
scale to arbitrarily high values, eventually to infinity. The only known way to perform this
limit is to hit a UV fixed point.
Fixed point theories do not depend on any intrinsic scale since they are scale invariant.
As a consequence they can be used to model systems at criticality. These theories are
characterized by dimensionless couplings and physical quantities exhibit scaling relations which
can be observed in experiments. These relations arise in very different systems sharing the
same dimensionality, symmetry and field content. This is what is usually referred to as
the concept of universality, the independence of the critical properties of a system from
its microscopic details. The RG offers a simple and intuitive explanation of universality:
the critical properties of a system are determined by the fixed point, microscopic actions
defined at different scales that flow to the same fixed point, or equivalently that belong to
the same basin of attraction of a fixed point, will describe the same criticality.
We see that in this light the problem of understanding the critical properties realized in
nature boils down to the classification of all the different fixed points. In two dimensions we
know that every unitary scale invariant theory is also conformal invariant, so the problem
further reduces to the classification of all possible conformal field theories (CFT). This
can be done via algebraic methods, exploiting the properties of the associated Virasoro
algebra [3, 4].
A fixed point theory can then be deformed by adding weakly coupled operators that
trigger a nontrivial flow out of the fixed point. By considering the linearization of this
flow we can obtain all the remaining CFT data (like scaling dimensions and other critical
exponents) that characterize the physical system and the way in which it responds to
deformations. This is also the main idea of conformal perturbation theory.
So far our discussion has been limited to the neighborhood of a fixed point. The next
natural step is to try to gain more information on the global properties of theory space.
Such information is provided by Zamolodchikovs c-theorem [5], which states that in every
unitary Poincare invariant theory there exists a function of the coupling constants, the
c-function, that decreases from CFTUV to CFTIR, and that is stationary at the endpoints
of the flow, where its value equals the central charge of the corresponding CFT. Note
that the difference between the two central charges is an intrinsic quantity (intrinsic
meaning independent of spurious contributions like scheme dependence of the renormalization
procedure), so the content of the theorem is highly nontrivial.
In this case a complete RG analysis requires the ability to follow the flow arbitrarily
far away from a fixed point. Unless the two fixed points are sufficiently close to each other
we cannot rely on perturbative schemes. The non-perturbative framework we will use to
address these issues is the functional renormalization group (fRG) based on the effective
average action (EAA) [6]. The EAA is a functional whose scale dependence is given by an
exact flow equation [7] which, being exact, allows to explore non-perturbative aspects. A
first application of exact RG equations to the c-function has been explored in [8, 9].
The main purpose of this paper is to move the first steps necessary in order to give a
bridge between these two general results: the c-theorem and the computation of universal
quantities related to the integrated flow between fixed points (that is, to global
properties of theory space), and the fRG formalism based on the exact flow for the EAA. Our
approach will be mainly a constructive one. We will give a general recipe to construct
a c-function compatible with Zamolodchikovs theorem within the fRG framework. After
identifying a natural candidate for a scale dependent c-function, ck, we will be able to
write an exact non-perturbative flow equation for it. Of course, there are only few cases
in which the exactness of the flow equation can be used and one usually needs to resort
to approximations. However, we will see that already for a simple truncation as the local
potential approximation the flow equation gives results compatible with the c-theorem.
Our viewpoint will be based upon a curved space construction. The reason for this is
twofold. First, this avoids having to resort to algebraic techniques or OPE analysis: the
central charge, for instance, becomes the coefficient of the conformal anomaly, which in
curved space becomes an operator anomaly in the one point function. Second, this is more
suitable for functional techniques, as the derivation of the trace anomaly matching condition
will show, and more useful to write a general effective action. Indeed, this construction will
require an investigation of what is the general form of the EAA away from fixed points,
since the usual expansion in local operators is incapable of giving a nonzero running for
ck. Working in curved space the natural candidate for ck is the coefficient of the Polyakov
action. We will take this as our definition for the c-function and leave for further study the
mapping between our approach and the one based on local RG with spacetime dependent
couplings [10, 11].
The paper will be organized as follows. In section 2 we will construct a Weyl-invariant
functional measure and discuss the form of a CFT on curved background. This will lead us
to a re-derivation of the trace anomaly matching condition, from which the integrated
c-theorem follows from known results [12]. We will then move on to discuss the scale
dependent c-function, and obtain our flow equation for it, in section 3. This construction
uses the EAA as the main tool, so in section 4 we will investigate its general form. In
section 5 we discuss various applications of our formalism while in section 6 we put forward a
simple relation between the beta function of Newtons constant and the running c-function.
Section 7 is devoted to the conclusions.
The integrated c-theorem
We start by reviewing the integrated c-theorem expressing the change of the central charge
c = cUV cIR along a RG trajectory connecting two fixed point theories, or equivalently
two CFTs. We will work in curved space where the central charge, or equivalently the
conformal anomaly, can be seen as the coefficient of the Polyakov term in the effective
action. When we specify the background metric to be of the specific form g = e2 ,
with the dilaton, c becomes the coefficient of the operator R and can be easily
extracted. But before we need to briefly discuss functional measures in curved space,
Weyl-invariant quantization and the form of the effective action for a CFT on a curved
background.
Weyl-invariant quantization and functional measures
The standard diffeomorphism invariant path integral measure in curved space [13], denoted
here DgI , is Weyl-anomalous: under a Weyl transformation of the background metric g
e2 g and of the fields ew , where w is the conformal weight of the field,1 one
encounters the conformal anomaly:
where c is the central charge of the CFT, which we want to use as UV action in the path
integral, and W Z [, g] is the Wess-Zumino action:
1 Z
which, upon Weyl variation, gives back (2.1):
SP [g] = 961 Z d2xgR 1 R ,
SP [e2 g] SP [g] = W Z [, g] .
The Polyakov action generates to the following quantum energy-momentum tensor,
which is anomalous:
This is the conformal anomaly, in the two dimensional case. In curved space, where it can
be written in terms of curvature invariants, the conformal anomaly manifests itself already
in the one-point function (2.5), while in flat space it is seen only starting from the two-point
function. For example, in flat space the two point function of the energy-momentum tensor
obtained from the Polyakov action, when written in complex coordinates, reproduces the
standard CFT result [3]:
1 c/2
hTzzTwwi = (2)2 (z w)4 . (2.6)
This relation shows the equivalence between the central charge and anomaly coefficient.
We can use the Polyakov action to define, formally, a new measure in the following way:
Now using (2.1) and (2.4) one can show that indeed (2.7) is Weyl-invariant:
DeI2I g (ew ) = De2 g (ew ) ecSP [e2 g]
I
= DgI ecW Z[,g]ecSP [g]+cW Z[,g]
With these definitions, we now look at the effective action. First we define the standard
Weyl non-invariant effective action:2
If the bare or UV action is conformally invariant S[ew , e2 g] = S[, g], this is not so for
the standard effective action, which instead satisfies the Wess-Zumino relation:
I [ew , e2 g] I [, g] = cW Z [, g] .
1P I
gives rise to a Weyl-invariant effective action:
Equation (2.12) is valid only when II [, g] = S[, g], but still is important from the RG
point of view: it is possible to obtain a Weyl-invariant effective action only if there are
no perturbations to the UV action and thus no induced RG flow. Thus the (bare) UV
action and the (effective) IR action are the same in this case. Said in other words, the
path integration amounts to the substitution of the quantum field with the average field.
A purely Gaussian theory provides an example where one can check explicitly the validity
of equation (2.12).
Similar reasoning has been made in [14] with the exception that in that work a
Stuckelberg trick was used to maintain Weyl-invariance for any UV action.
2We define R1P I R eR g (1,0)[,g] where (1,0)[, g] [,g] .
CFT action on curved background
We have seen how to define, at least formally, a Weyl-invariant effective action starting from
a Weyl invariant UV action via the functional measure (2.7), which is to be understood as
the measure we will use from now on. Nevertheless on a curved background the effective
action of a CFT is not Weyl-invariant since every CFT with c 6= 0 is anomalous, and
thus its action must contain a Polyakov term. Still, in absence of relevant perturbations,
quantization will just give the IR effective action equal to the UV action.
These considerations lead to the following split form for the effective action of a
general CFT in presence of a background metric:
Here SCF T [, g] is the curved space generalization of the flat space CFT action SCF T []
SCF T [, ], defined by its Taylor series expansion in terms of correlation functions of ,
these being, in principle, exactly known. Very few CFT actions can be written in local
form, these are the Gaussian, the Ising model (in the fermion representation) and the
WessZumino-Witten AKM actions [4]. SP [g] is the Polyakov action and c its central charge.
Other possible Weyl-invariant terms depending on the metric alone are not present in d = 2,
but appear in higher dimensions.
We now give an explicit example of this construction. The Gaussian theory has c = 1
and is the simplest example of a CFT:
where the second term is due to the integration of the fluctuations, while the Polyakov
term with the minus sign comes from the Weyl-invariant measure (2.7). The two cancel
since the 12 Tr log = SP [g]. In order to have UV 6= IR one needs to add a relevant
perturbation triggering the RG flow.
Anomaly matching from the path-integral
Starting from UV [, g] = SUV [, g] + cUV SP [g] plus relevant operators, we can consider
the IR effective action obtained by integrating out fluctuations:
Dg eSUV [+,g]cUV SP [g]+relevant
Since the metric is non-dynamical we passed the Polyakov term through the path integral.
Here by relevant we mean, depending on the case, massive deformations or marginally
eSIR[,e2 ]e(cUV cIR)W Z[,] = Z
De2 eSUV [+,e2 ]+relevant ,
1P I
where we used (2.4) on flat space W Z [, ] = SP [e2 ]. In order to recover the flat space
measure we first shift ew and ew and then use the invariance (2.7):
eSIR[ew ,e2 ]e(cUV cIR)W Z[,] = Z (2.18)
D eSUV [ew (+),e2 ]+relevant .
1P I
Then we use the conformal invariance properties of the actions, i.e. we substitute SUV [ew ,
e2 ] = SUV [] and SIR[ew , e2 ] = SIR[] since both actions are Weyl-invariant:
eSIR[]e(cUV cIR)W Z[,] = Z (2.19)
1P I
Note that D D is the flat space measure. The only remaining dependence on is
due to the relevant terms, which make the path integral non-trivial. The last equality tells
us that the dilaton effective action (generated by matter loops) compensates exactly the
difference between the anomalies in the UV and IR. This is precisely the anomaly matching
condition considered in [12, 15].
Proof of the integrated c-theorem
1P I
we can read off c from the terms of the dilaton two-point function quadratic in momenta.
The relevant terms can be expanded in powers of :
Z
relevant =
which is the integrated version of the c-theorem. From here one simply notices that the
integral is positive due to reflection positivity and concludes that c 0 [4, 5].
Flow equation for the c-function
The c-theorem states [5] that for a two-dimensional unitary quantum field theory, invariant
under rotations and whose energy-momentum tensor is conserved, there exists a function
c of the coupling constants which is monotonic along the RG flow and, at a fixed point,
is stationary and equal to the central charge of the corresponding CFT. Therefore this
function c is such that tc < 0 (where the RG time is given by the logarithm of the
radius t = log r, so the flow is towards the infrared for r , hence the minus sign). The
differential equation for c can be integrated from r = 0 to r = and gives back (2.24).
A natural trial definition for an interpolating c-function is given by taking (2.24) with the
integral which has been cut off at some scale (see for instance [16]):
We will follow a different approach. Instead of cutting off directly in real space we will
cutoff in momentum space. This will allow us to naturally connect with the framework of
the functional Renormalization Group (fRG) and to derive an exact RG flow equation the
c-function.
The fRG flow equation for the c-function
To construct the c-function is to consider a Wilsonian RG prescription. A clever way to
do the momentum shell integration in a smooth way, is to introduce a suppressing factor
in the path integral via Dg Dg eSk[,g]. The role of the cutoff action Sk[, g] is
to restrict the integration to modes above the IR scale k. In this way we obtain a scale
dependent effective action k[, g], which, using (2.13), can be decomposed as:
where Sk[, g] is defined by Sk[0, g] = 0 and ck is the scale dependent c-function. By
gravitational terms we mean the purely geometrical terms depending on the metric alone,
like R g or R gR, generated by fluctuations. The collection of the k[, g] for all k
constitute the RG trajectory connecting UV [, g] to IR[, g]; a cartoon of this shown in
figure 1. If we now repeat the steps leading to equation (2.20), but with the cutoff term
added, we arrive at:
eSk[,e2 ]e cU2V4ck R = Z
D eSUV [+]+relevanteSk[ew ,e2 ] .
Now a derivative of (3.3) with respect to the RG time t = log k gives the RG flow of the
central charge:
tck = 24 tSk[ew , e2 ]
in which the expectation value is calculated within the regularized path integral. We see
that we obtain the flow of the c-function if we are able to evaluate the r.h.s. of (3.4),
after specifying the form of the cutoff action. The running of ck is related to the
coarsegrained dilaton two-point function. To understand how to handle this equation, we need
to introduce the effective average action (EAA).
In the functional RG framework, one considers an IR regulator quadratic in the fields:
1 Z
chosen to suppress field modes in a covariant way: if n is an eigenfunction of the covariant
Laplacian n = nn, Rk will act as a mass insertion for modes with n k2, while
leaving unchanged the ones with n k2. In this way we obtain a scale-dependent
partition function:
Zk[J, g] = eWk[J,g] =
Z D eS[,g]Sk[,g]+R gJ .
The effective average action is then defined as the (shifted) Legendre transform:
d2xgJ Wk [J, g] Sk [, g] ,
D eSUV [+,g]cUV SP [g]Sk[,g] .
The main virtue of these definitions is that the EAA satisfies an exact RG flow equation [7].
A scale derivative of (3.8) gives:
in which the expectation values are calculated with the fRG-regularized path integral.
Using the fact that the Legendre transform of the generator of connected correlation functions
is k + Sk, we have:
1
Substituting back in the previous expression we find the functional RG equation satisfied
by the EAA:
1
tRk[g] .
This equation is well defined, exact and offers a way to define QFTs non-perturbatively [6].
From the exact flow equation for the EAA we obtain a corresponding equation for the
c-function. In particular, we can express the r.h.s. of (3.4) using (3.9):
tck = 24 tk[ew , e2 ] R .
Equation (3.12) is the exact flow equation for the c-function in the fRG framework.
Using (3.11) in the r.h.s. leads to the following explicit form:
where we defined [, ] [ew , e2 ] and Rk[ ] Rk[e2 ]. The exact RG flow equation
for the c-function is the main result of this section.
To write more explicitly the flow equation for the c-function we define the regularized
propagator Gk[ ] ((2,0)[, ] + Rk[ ])1, perform two functional derivatives of (3.12)
with respect to the dilaton, set = 0 and extract the term proportional to :
GktRk
1 Tr Gk (2,2) + R(2)
GktRk
Tr Gk (2,1) + R(1)
k k
GktRk(1) +
Tr GktRk(2)o
where all quantities are evaluated at = = 0. Note that in (3.14) we had to derive
the cutoff kernel Rk, since this depends explicitly on the dilaton. As shown in [18], these
additional terms in the flow equation for the proper-vertices are crucial in maintaining
background symmetry when employing the background field method.
The flow equation in the form (3.14) is a bit cumbersome so we introduce a compact
notation to rewrite it in a simpler way. If we introduce the formal operator t = tRk Rk ,
we can rewrite the flow equation (3.12) for the c-function as:
tck = 12 Tr t log Gk[ ] R ,
Figure 2. Diagrammatic representation of the two terms in the r.h.s. of the flow equation (3.16)
for the c-function.
where we used the following simple relations:
tGk[ ] = Gk[ ]tRk[ ]Gk[ ]
t log Gk[ ] = Gk1[ ]tGk[ ] = Gk[ ]tRk[ ] .
Now we can rewrite the flow equation (3.14) in the following compact form:
where again all quantities are evaluated at = = 0. This is the form that we will use
in applications in section 5. Finally, we can represent diagrammatically the two terms on
the r.h.s. of (3.16) as in figure 2 and switch to momentum space to evaluate the diagrams
by employing the techniques presented in [18]. In particular, continuous lines represent
matter regularized propagators Gk[0], while vertices with m-external wavy lines are the
matter-dilaton vertices (2,m)[, ] + Rk(m)[ ]. Finally, each loop represents a R d2x t or a
k
fRG derivation of the integrated c-theorem
We now rederive both the integrated c-theorem and the exact flow equation for the
cfunction using a fRG theory space perspective.
Away from a fixed point, apart for a Wess-Zumino term with running coefficient,
that for the moment we call Ck, there must be many additional terms spoiling the fixed
point Wess-Zumino relation (2.10). Since these terms vanish at a fixed point they must be
proportional to the (dimensionless) beta functions. We can thus make the following ansatz:
k[ew , e2 g] k[, g] = CkW Z [, g] + terms .
This relation can be read as a generalized running Wess-Zumino action. The terms
indicate terms proportional to (at least one) dimensionless beta function which vanish at
the CFTs and are generated along the flow by the fact that we are moving away from
criticality.
If we now use the Weyl-invariant measure to construct the EAA, then at the UV fixed
point, that is for k , we must have CUV = 0. On the other hand, if we are not
quantizing in a Weyl-invariant manner, we should reproduce the Wess-Zumino relation
both at k = and k = 0. This tells us that in fact Ck = ck cUV if the UV theory
is quantized in a Weyl-invariant manner and Ck = ck if not. Weyl-invariant quantization
corresponds, in the EAA formalism, to a constant shift of Ck.
IR[ew , e2 g] IR[, g] = (cIR cUV ) W Z [, g] .
If we now set g = and expand IR[, ] IR[ew , e2 ] in powers of the dilaton
we find:
1 Z
The functional derivatives of the effective action are related to the traces of the
energymomentum tensor:
The first relation is identically zero at a CFT, i.e. hxiIR = 0 . Inserting the second
relation in (3.19) and expanding, as before, y around x using (2.23) gives immediately the
integrated c-theorem (2.24). This derivation represents a consistency of the ansatz (3.17).
It is now clear that from the Wess-Zumino relation at finite k (3.17) we can easily read
off the flow of the central charge. In this way, since tck = tCk, from the coefficient of
R in tk[ew , e2 ] we recover the exact RG flow equation for the c-function (3.12).
Another way to see that the flow of the c-function is given by (3.12) is to recognize that
Ck is nothing more than the coupling constant of the Polyakov action. As we said, when
working on curved backgrounds one should always add the Polyakov term to a truncation.
Thus the Wess-Zumino action on the r.h.s. of (3.17) derives from the presence of the
Polyakov action, with coefficient Ck, in the EAAs on the l.h.s. of the same equation. Then,
as just seen in the previous paragraph, a t-derivative relates tck to the two-point function
of the dilaton. In principle one can obtain the flow of Ck directly as the coefficient of
R gR 1 R but this is more laborious. Finally, note that the inclusion of the Polyakov
action with running central charge makes the truncation consistent with the conformal
anomaly both in the UV and in the IR. To understand the -terms we will consider, in
the next section, the scale anomaly.
General form of the effective average action
In this section we put forward some requirements which an ansatz for the EAA should
satisfy. These requirements are motivated from the fact that the EAA should reproduce
some generic features of QFTs, namely the scale and the conformal anomaly. In particular
we will try to shed light on the nature of the -terms introduced in equation (3.17).
The local ansatz and its limitations
When studying truncations of the EAA, one generally starts by expanding the functional
in terms of local operators compatible with the symmetries of the system:
Z
d2x g Oi[, g] .
This equation defines the running coupling constants gi,k, which become the coordinates
that parametrize theory space in the given operator basis.
A class of operators, which is not complete, but allows many computations to be
performed analytically, is the one composed of powers of the field, i.e. Oi[, g] = 2i and
gi,k = (22ii,)k! . In this approximation, one usually re-sums the field powers into a running
effective potential Vk() and equivalently considers the following ansatz for the EAA:
d2xg 21 + Vk() ,
known as local potential approximation (LPA). Within this this truncation the exact flow
equation (3.11) becomes a partial differential equation:
with cd1 = (4)d/2(d/2 + 1). Even such a simple truncation is able to manifest
qualitatively all the critical information relative to the theory space of scalar theories and in
particular the fixed point structure [19, 20].
However, the effective action usually contains also nonlocal terms. Some of these
nonlocal terms are directly related to the finite part of the effective action [21, 22], which
generally has a complicated form encoding all the information contained in the correlation
functions or amplitudes. These terms are not present in the LPA which can be seen as the
limit where we discard all the momentum structure of the vertices.
Nevertheless there are other nonlocal terms that are non-zero only away from a fixed
point: these are the -terms introduced in equation (3.17). As we will explain in this
section these terms are needed to recover known results and will play a central role in our
computations. If we limit ourselves to the local truncation ansatz (4.1), then one finds
that the flow equation for the c-function is driven only by the classical non Weyl-invariant
terms, which is not correct. This is not due to the fact that the flow equation (3.12) is
wrong, rather, it is the truncation ansatz (4.1) that is insufficient. Fluctuations induce the
-terms of equation (3.17) and we will see that these are crucial in driving the flow of the
c-function.
We will argue that these nonlocal terms have a precise form. We will do this requiring
the EAA to reproduce the scale anomaly.
Nonlocal ansatz and the scale anomaly
It is easy to understand the origin of the terms on the r.h.s. of (3.17) which are linear in
: they are related to the scale anomaly. To see this let us rescale the fields and expand
the EAA in powers of the dilaton:
hik = k[, ] 0 , (4.5)
defines the scale dependent energy-momentum tensor trace. In the IR the EAA reduces to
the standard effective action, which generally is scale anomalous. If we start with some UV
action deformed by terms of the form Pj gj R d2x g Oi, the corresponding scale anomaly
in flat space reads:
where di are the dimensions of the coupling constants. The expression in brackets is nothing
but the beta function of the dimensionless coupling:
This is a standard result known from both ordinary and conformal perturbation theories [4].
Now we consider again the -terms on the r.h.s. of (3.17). They come from the
conformal variation of the EAA which should include also the terms due to the scale anomaly.
Therefore it is natural to generalize the above equation for a generic k:
to insert in the r.h.s. of the exact flow equation (3.16).
We now propose a covariant form for (4.9) using the following properties:
Xi gi,k Z g Oi[, g] 12 Xi i Z g Oi[, g] 1 R + ,
If we insert this into (4.4) we find:
This expression gives a non trivial flow of the c-function since we now have the vertex
=
reproduces (4.9) to linear order in . In order to get an ansatz consistent also with the
conformal anomaly we need to add to (4.12) the Polyakov term with as coefficient the
running central charge Ck:
The form (4.13) represents a parametrization of the EAA consistent with (3.17) to linear
order in the beta functions and hints to what could be the general for of the EAA away
from criticality. For the time being we will not improve further our ansatz, since we will
see in the next section, that the understanding of the linear terms in the beta functions is
already sufficient to build the c-function in some non-trivial cases. We hope to come back
to the issue of higher order terms in , which may play a role in making a bridge between
the fRG perspective adopted here and the ideas related to the local RG [10, 11].
Applications
Checking exact results
Here we provide two examples where the c-function and the difference cUV cIR are
computed and can be compared to known exact results. We will consider a free scalar field
and a free (Majorana) fermionic field whose fixed point actions are perturbed by a mass
term, so they flow to cIR = 0.
Massive deformation of the Gaussian fixed point
We consider a scalar field with Gaussian action and cUV = 1 perturbed by a mass
term. Since the beta function of the mass is zero (there are no interactions), our
general ansatz (4.13) for the EAA reads:
or when we rescale the fields:
1 Z
d2xg ( + m2) 9c6k Z gR 1 R ,
1 Z
Its clear that the only interaction between and is the one induced by the dimension of
the mass. In order to avoid possible vertices coming from the cutoff action we use the mass
cutoff Rk(z) = ak2 which has the advantage of having no dependence with respect to the
background metric. We have introduced the parameter a to check the cutoff independence
of the result. After a short computation3 we find the following flow:
where m is the dimensionful mass. This RG flow occurs along trajectoryI of figure 3.
3We need to evaluate the first diagram of figure 2, for more details see section 5.2.
tck =
Integrating the above differential equation, with the initial condition c = 1 (the
central charge of the Gaussian fixed point) we find:
In the k 0 limit this gives c0 = 0 which implies c = 1 independently of the cutoff
parameter a. As expected a massive deformation of the Gaussian fixed point leads in the
IR to a theory with zero central charge.
5.1.2 Massive deformation of the Ising fixed point
In this example we make a massive deformation of the Ising fixed point. The critical
Ising model is described by a free Majorana fermion and a massive deformation of this
correspond to consider T > Tc [4]. According to our general ansatz (4.13) and considering
that, as before, the mass beta function is zero, the EAA reads:
Z
d2xg / + m 9c6k Z gR 1 R ,
tck =
which gives c0 = 0 and c = 12 as expected.
The c-function in the local potential approximation
If we now pass to dimensionless variables, = kw and Vk() = k2Vk(), then the second
and third terms in the above equation, to linear order in , become Vk() k2tVk() ,
so that the scalar-dilaton interaction is proportional to the dimensionless scale derivative
of the potential.
To obtain the flow equation for the c-function we use (3.16) and the mass cutoff
Rk(z) = k2 so that all cutoff vertices drop out. Only the first diagram of figure 2 contributes
terms of order p2 in the external momenta, more specifically we need to evaluate the
integral:
or after the rescaling:
d2x / + e m 2c4k Z .
The computation proceeds along the lines of the scalar case. Once again we use the mass
cutoff Rk = ak and we find:
with the following regularized propagator:
Figure 4. tck in the ( m2k, k) plane. We marked with a red dot the position of the Gaussian and
Ising fixed points.
provided that Vk00(0) > 1, since otherwise the momentum integral does not converge.
Inserting this back in (5.10) finally gives:
tck =
which is the flow equation for the c-function in the LPA with a mass cutoff. This the main
result of this section. Note that since (5.13) is valid under the condition Vk00(0) > 1, the
c-theorem tck 0 is indeed satisfied within the LPA.
Flow between the Gaussian and Ising fixed points
We now consider the simple case where there are just two running couplings parametrizing
theory space, i.e. we expand the running effective potential in a Taylor series:
1 1
Vk() = 2! m2k2 + 4! k4 + . . .
where m2k is the mass and k the quartic self-interaction. Inserting (5.14) in the flow
equation for the effective potential (4.3) and projecting out the flow of the two couplings
gives, after passing to dimensionless variables m2k = k2m 2k and 2k = k2k, the following
system of beta functions:
This system has two fixed points: the Gaussian ( m2k, k) = (0, 0) and the Ising ( m2k, k) =
( 14 , 32 ). The Gaussian fixed point has two IR repulsive directions, while the Ising fixed
point has one IR repulsive and one IR attractive direction. The trajectories starting along
these directions are shown in figure 3, in particular trajectoryIII connects the two fixed
points. One can see that this last trajectory is the set of points where the dimensionless
mass beta function vanishes k = 8 m2k (1 + m2k).
We can now use (5.13) to evaluate the c-function in this truncation. This turns out to
be simply related to the square of the dimensionless mass beta function:
1 1
tck = (1 + m2k)3 tm 2k 2 = (1 + m2k)3
As for (5.13), the result is only valid for m2k > 1, so in this range we do have tck 0,
which is consistent with the c-theorem. The flow (5.16) is similar to the one given in [9],
which was there found by trial and error. Equation (5.16) is the first non-trivial example
of explicit flow equation for the c-function obtained using the procedure presented in this
work. In figure 4 we plot tck in the plane ( m2k, k): one can see that the magnitude of
tck is smaller along a valley containing the two fixed points. Along this valley lies the
trajectory connecting them, trajectoryIII of figure 3.
We would like to compute c by integrating the flow of the central charge along
the path connecting the Gaussian and Ising fixed points, but in this simple truncation
trajectoryIII is defined by the vanishing of the dimensionless mass beta function, and
thus tck is zero along it. To find a non-trivial result, we need to consider a more refined
truncation ansatz for the running effective potential. We leave these studies to future work.
Sine-Gordon model
We now consider the Sine-Gordon model which, in the continuum limit, is described by
the following action [4]:
tm 2k =
8 1 + m2 2
k
where m2k = m2k/k2 is the dimensionless mass. Inserting the Sine-Gordon running
potential (5.18) into the flow equation (5.13) now gives:
tck = me4k k2 8 1 + m2k 2 . (5.19)
162 1 + m2 5
k
We solved the system of equations numerically imposing cUV = 1 finding c ' 0.998, in
satisfactory agreement with the exact result c = 1. In figure 5 we plot the running of ck
as well as its beta function.
tck
Figure 5. Flow of the Sine-Gordon model: the continuos line shows the running of the c-function
and the dotted line has a bell shape meaning that the beta function of ck is zero at the endpoints
of the flow.
The c-function in the loop expansion
The last approximation we will consider is the loop expansion. The exact flow equation
for the EAA (3.11) can be solved perturbatively [23, 24] loop by loop. We review this in
the appendix. In the first part of this section we will look at the various contributions
diagrammatically, while in the second part we will explicitly evaluate one subclass of these.
Zamolodchikovs metric: diagrammatics
Using relation (A.4) we can compute the running of the EAA at each order in the loop
expansion. The running of the L-th term tL,k, say, will contain a contribution to the
running of ck that we will call tcL,k. The term cL,k arises only from diagrams with L
matter loops and two dilaton external lines. In this way we can build a loop expansion for
the c-function.
We can start by applying this construction step by step so to make clear how everything
works. We will work with a Z2-symmetric scalar theory, so that the part linear in the dilaton
of our general ansatz (4.13) takes the form:
where 2 is the mass beta function, 4 is the 4 coupling beta function, and so on.
At one loop, we have only the following diagram, obtained from (A.5) of the appendix
by functional derivation with respect to the dilaton,
Here we adopt the same diagrammatic rules of section 3.2 where the continuous line
represents the regularized propagator (in this case given in equation (A.6) of the appendix),
while the wavy line represents the dilaton. On every diagram the operator t acts, but in
this case it is just t. In this diagram the vertices, are derived from (5.20), is the mass
beta function, so this contribution goes like 22 and we recover the LPA result (5.13) as one
would expect.
From the flow of the two-loop contribution, (A.11) of the appendix, we obtain different
terms. We get the non-diagonal contribution (we will make this jargon clear in a second):
proportional to 2 4. Together with this, we also have the following 2-loop diagonal
contributions:
which are proportional to 2 22. These represent a diagonal but coupling-dependent
contribution, in the sense that couplings do not only appear through the beta functions. When
going to 3-loops, 4-loops and so on, corresponding diagrams must be considered for all the
diagonal contributions.
At three loops (remember we are considering a Z2-symmetric theory, so there are no
scalar odd power interactions) we get again the diagonal contributions:
proportional to 2 6. From these first diagrams we clearly see that from the structure of
the loop expansion we only get terms quadratic in the beta functions.
We can indeed follow Zamolodchikov and define the metric gij through:
Our construction gives a diagrammatic representation of it within the loop expansion. It
is also clear now what we meant by diagonal or nondiagonal contributions: they refer to
the entries of this metric. In principle one can evaluate all these diagrams for a generic
cutoff Rk(z) but this turns out to be a difficult analytical task. In the next section we will
be able to evaluate analytically one particular class of diagonal entries.4
Diagonal contributions
At L-loop order, the simplest coupling-independent diagonal contribution comes from the
following diagram:
corresponding to the expression:
4 Z
d2x Z d2y xy t [Gk (x y)]L+1
(which generalizes equation (A.15) of the appendix). In the above equation the 2(L + 1)!
comes from the symmetry factor of the diagram, and the minus sign from the fact that
we are acting with an overall t. To recover the contribution to tck is simple: expand y
around x as in equation (2.23), and isolate the proper term according to equation (3.12).
To see more explicitly the form that the metric of Zamolodchikov takes, we need some
preliminary results. Using a mass cutoff Rk = k2, the zero mass running renormalized
propagator (A.6) will be the same as the standard massive one, only with k2 in place of
the mass m2, and the cutoff vertices play no role. In real space the propagator reads:
tf [Rk] = 2a f ak2
The different contributions are then calculated after expanding y around x using (2.23).
We find:
d2y 2(2y)2L+1 a hK0 |y| ak2
These diagonal terms can be written to all orders, they give a contribution to the flow
equation for ck of the form:
in which we defined the quantity
3 Z
dx x4 [K0 (x)]L K1 (x) .
Note the interesting thing that contributions at loop order L are proportional to the square
of the beta function of the coupling L+1,k. Thus the flow of ck receives contributions from
all loops (as it is inherently non-perturbative) but a given interaction starts to contribute
only at a given loop order. All the AL can be evaluated numerically and they turn out
to be positive. The numerical values of the first AL are shown in table 1. Note the fast
decrease relative to the one-loop value.
We can now write down the contribution of this class of diagrams to the running of
the c-function at all loops in the Z2-symmetric case:
tc(kdiagonal) = X A2i1 22i ,
i=1
which also gives the explicit form for the diagonal entries of the Zamolodchikov metric.
Since this sum is manifestly positive, we can say that the c-theorem is satisfied to all loops
by the diagonal terms considered.
As we have seen previously, the entries of Zamolodchikovs metric contain a
couplingindependent piece, plus further pieces proportional to increasing powers of the coupling
constants, as we increase the loop order. The positivity properties of the metric are far
from trivial when all these terms are involved. However, when the couplings are sufficiently
small, the positivity will be determined solely by the coupling independent terms.
Non-unitary theories
Finally we make a comment on when the c-theorem is not satisfied, i.e. the case when
tck < 0. We know that the c-theorem does not hold without the unitarity assumption [5].
This can indeed be checked explicitly. Its easy to see that when one considers interactions
with complex couplings then the coefficients in the loop expansion turn negative. For
instance, one notable example is the Lee-Yang model [4], in which one introduces the
non-unitary complex interaction:
which turns out to have the wrong sign to be consistent with the c-theorem:
since A2 > 0, as reported in table 1.
The c-function and Newtons constant
In this section we derive an interesting relation between the c-function and the matter
induced beta function of Newtons constant.5 This can then be used to obtain another
form of the flow of the central charge tck.
1 1 1
d2xg 16Gk R 4 t 16Gk
We recognize that the Polyakov term above is the same that we included in our general
anstaz for the EAA (4.13). Thus we infer that there is a relation between the beta function
of Newtons constant and the running c-function:
This is a nontrivial statement by itself. It tells us that the running c-function for a certain
matter field type can also be computed from the contributions of that kind of matter to
the beta function of Newtons constant. In fact a derivative of (6.2) with respect to the
RG scale gives (remember that tCk = tck):
tck =
where Gk tGk is the Newtons constant beta function. We will check the consistency
of relation (6.2) in the case of a minimally coupled and a self-interacting scalar.
Minimally coupled scalar
Consider a minimally coupled scalar describing a massive deformation of the Gaussian
fixed point as discussed in section 5.1.1. The action is given in (5.1) and the exact flow
equation (3.11) for this case reads:
Note that the dilaton plays no role now, since we are free to set = 0. Instead, to find
ck using (6.2) we need to extract the terms in the trace on the r.h.s. of (6.4) that are
5In what follows we identify the Newtons constant as the coupling in front of the Ricci scalar. In a non
linear sigma model on curved target space this coupling is equivalently identified as the dilaton constant
mode.
6We need here 1/4 instead of 1/2 because of the the further symmetry we have in exchanging the two Rs.
tck
7We are not using Weyl quantization so Ck = ck
Figure 6. ck and tck as a function of k for a massive deformation of a minimally coupled scalar.
Mass (a = 1), optimized (a = 1) and exponential (a = 1, b = 1) cutoffs (upper curves), exponential
(a = 1, b = 12 ) cutoff (middle curves), exponential (a = 1, b = 32 ) cutoff (lower curves). In all cases
we set m2 = 1.
proportional to the invariant R gR. As usual, this can be done using the heat kernel
tRk(z)
expansion [25]. Defining hk(z) = z+m2+Rk(z) , one finds:
R gR
d2xgR ,
which, when compared with the scale derivative of 161Gk R gR on the l.h.s. of (6.4),
gives:
Thus our formula (6.2) leads to:7
Note that this relation is valid for arbitrary cutoff function Rk(z), as opposed to the result
of section 5.1.1 valid only for the mass cutoff. For both the mass cutoff Rk(z) = ak2 and
the optimized cutoff Rk(z) = a(k2 z)(k2 z) we find the following form:
In all cases and for all values of the parameters a and b we find that cUV = 1 and cIR = 0
as expected. A derivative of (6.9) gives the flow of the c-function:
tck =
The interpolating ck of equation (6.9) and the flow of the last equation are shown in figure 6.
We clearly see that the flow is scheme dependent, but the integral of it along a trajectory,
giving c, is universal.
Self-interacting scalar
We consider now an interacting scalar, i.e. the LPA action (5.9) of section 5.2. We can
obtain ck directly from equation (6.9) by just making the replacement m2 Vk00(0):
A scale derivative now gives:
tck =
abk2 (tVk00(0) 2Vk00(0))
ak2 + bVk00(0) 2
We need to decide the value of 0 where to evaluate this expression. In this case it
is important to distinguish the ordered from the broken phase. If the running effective
potential has the polynomial form (5.14), then we have 0 = 0 in the ordered phase and
0 = q6m2k/k in the broken phase, the two phases being separated by trajectoryIII
and its continuation. Inserting these expressions in (6.12) gives the following form for the
Conclusions
In this work we have explored a new way to study the flow of the c-function within the
framework of the functional RG based on the effective average action (EAA). This function
interpolates between the UV and IR central charges of the corresponding CFTs and is thus
a global feature of the flow, related to the integration of it along a trajectory connecting
two fixed points, independent of scheme ambiguities.
Our main result is an RG exact equation for the running c-function based on the
identification of it with the coefficient of the running Polyakov action. This equation
relates the flow of the central charge to the exact flow of the EAA. To solve the equation
for non-trivial cases we built a suitable ansatz requiring the EAA to reproduce generic
features of QFTs, namely the scale and the conformal anomalies. In its own right this is
an interesting result since it teaches us that a consistent ansatz for the EAA off criticality
should include some nonlocal terms proportional to beta functions. Of course we do not
claim full generality for this ansatz, but we found that it is sufficiently accurate to trigger
the flow of the c-function in non-trivial cases. Explicit computations, within the local
potential approximation and the loop expansion, have been presented in section 5 showing
the compatibility of our framework with the c-theorem.
Moreover we have put forward a relation between the beta function of Newtons
constant and the running conformal anomaly. This relation comes from internal consistency of
the generic ansatz for the EAA we proposed and allows us to use heat kernel techniques to
compute the RG running of the c-function. We also checked this other relation in explicit
cases, finding it consistent. Nevertheless we point out that our analysis is not complete.
The works [10, 11, 16] highlight that there are some subtleties related to the definition of
the c-function. A complete mapping between the local RG approach and the fRG is still
lacking and further study is needed in this direction. Another issue, which has not been
touched at all, is the generalization of these ideas to the higher dimensional case, in
particular d = 4 where one can consider similar constructions for the a-function [10, 11, 26, 27],
which we leave to future work.
Loop expansion from the fRG
The exact flow equation (3.11) satisfied by the EAA can be solved iteratively. If we
choose as seed for the iteration the bare action, then the iteration procedure reproduces
the renormalized loop expansion [23, 24].
One starts with 0,k S, where S is the UV or bare action, and sets up an iterative
solution (the subscript 0 indicates the order of the iteration, is the UV cutoff and k is the
RG scale) by plugging 0,k into the r.h.s. of the flow equation and integrates the resulting
differential equation with the boundary condition 1, = S. The solution k,1 is then
plugged back into the r.h.s. of the flow equation and the procedure is be repeated.
To see this let us introduce } as a loop counting parameter and expand the EAA:
The bare action is k-independent tS = 0. The exact flow equation (3.11) now takes the
form:
} t1,k [] + }2t2,k [] + . . . = 2 tRk
} Tr S(2) [] + Rk + } (12,k) [] + }2(22,k) [] + . . .
The original flow equation (3.11) is finite both in the UV and IR: to maintain these
properties the bare action S has to contain counterterms to cancel the divergencies that may
appear in the L,k. Thus we define:
where each counterterm SL, is chosen to cancel the divergent part of L,0. Since this
divergent part is the same as the divergent part of L,k (we refer to [24] for more details on
this point), this choice renders the denominator of (A.2) finite. Here S0 is the renormalized
action, i.e. the bare action with renormalized fields, masses and couplings. From (A.2) we
can read off the flow of the L-th loop contribution:
1 L1 tk []
tL,k [] = (L 1)! }L1 }
The one-loop equation is straightforward:
depends on k only trough the cutoff Rk. Thus, within the loop expansion, the operator t,
introduced in section 3.2, is equivalent to t.
Gk[] =
Z dk0 1 Z dk0
k0 t0 1,k0 = 2 k k0 Tr Gk0 t0 Rk0
k
1 Z
Note that in the second line we have exchanged the order of the trace and the derivative.
This has been possible since we inserted an additional UV regulator (one can also use
dimensional regularization [24]). In the following all manipulations are intended with an
implicit UV cutoff .
We now choose SL, = [L,0]div and define the renormalized one-loop contribution:
Obviously, this limit is finite only if the theory is perturbatively renormalizable.
Now let us consider the two-loop contribution:
(12,k) = 21 GkS0(3)GkS0(3) + 12 S0(4)Gk ,
where we suppressed all indices. Using the above equation we get:
t2,k = 12 12 GkS0(3)GkS0(3) + 12 S0(4)Gk ren
= 21 t 3 1 2 GckdS0(3)adeGekf S0(3)bfcGkab +
[tGk]ba
1 S(4)abcdGckdGkab
where we used relations (3.16) to extract the overall scale derivative. Integrating and
renormalizing (A.5) as before gives:
In the limit k 0 we recovered the usual two-loop result with the correct coefficients and
in (nested) renormalized form. We can represent diagrammatically these contributions by
adopting the same rules of section 3.2 with the difference that a continuous line represents
a renormalized regularized propagator and vertices are constructed from the renormalized
action S0(m). To each loop we associate an integration R d2x in coordinate space or R (2d2q)2 in
momentum space and we act overall with t. Proceeding along these lines all the standard
loop expansion can be recovered at any loop order. From now on, for notational simplicity
So we find:
that we will use and generalize in section 5.3. We start from the following three-loop term
flow:
GckdS0(4)ademGekf S0(4)bfcnGkab + GckdS0(4)adenGekf S0(4)bfcmGkab
Gkaa1 S0(4)a1a2mnGk,a2b S(4)abcdGckd+GkabS0(4)abcd
0
1
t3,k = 2
recalling tGkG(k1) = GktRk we pick up the contribution of the diagram we are
interested in:
1 Gqm
GqkmGckdS0(4)ademGekf S0(4)bfcnGkabGnq
k
= t
where we used the cyclicity of the trace. Note that the symmetry factor of the three-loop
contribution to the effective action is automatically recovered. Similarly one can easily
obtain all the higher loop diagrams of this form.
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