Vanishing Regulators

Alessio Baldazzi111e-mail address: abaldazz@sissa.it,∗ Roberto
Percacci222e-mail address: percacci@sissa.it,∗ and Luca Zambelli333e-mail
address: luca.zambelli@bo.infn.it,†

∗International School for Advanced Studies, via Bonomea 265, I-34136 Trieste,
Italy and INFN, Sezione di Trieste, Italy

†INFN-Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy

Abstract

The non-perturbative functional renormalization group equation depends on the
choice of a regulator function, whose main properties are a “coarse-graining
scale” $k$ and an overall dimensionless amplitude $a$. In this paper we shall
discuss the limit $a\to 0$ with $k$ fixed. This limit is closely related to
the pseudo-regulator that reproduces the beta functions of the
$\overline{\text{MS}}\,$scheme, that we studied in a previous paper. It is not
suitable for precision calculations but it appears to be useful to eliminate
the spurious breaking of symmetries by the regulator, both for nonlinear
models and within the background field method.

##  1 Introduction

The Functional Renormalization Group (FRG) [1, 2, 3, 4] is a powerful tool to
study Quantum Field Theories (QFT) and their applications in statistical
mechanics, condensed matter theory, high energy physics [5]. It describes a
continuous interpolation between an UV action describing some microscopic
physics and the Effective Action (EA) where all the quantum fluctuations have
been integrated out. The functional that provides this interpolation is called
the Effective Average Action (EAA) and is denoted $\Gamma_{k}[\phi]$, where
$\phi$ are the fields, $k$ is a coarse-graining scale and $\Gamma_{0}=\Gamma$
is the EA. The EAA can be defined by a functional integral with a cutoff
suppressing the contribution of low-momentum modes, thus realizing Wilson’s
idea of integrating out high momentum modes first. The cutoff itself is
implemented by adding to the action the term

$\Delta S_{k}[\phi]=\frac{1}{2}\int d^{d}x\,\phi R_{k}(-\partial^{2})\phi\ ,$
(1.1)

leading to the functional differential equation

$k\frac{d\Gamma_{k}}{dk}=\frac{1}{2}{\rm
Tr}\left(\frac{\delta^{2}\Gamma_{k}}{\delta\phi\delta\phi}+R_{k}\right)^{-1}k\frac{dR_{k}}{dk}\
.$ (1.2)

This provides a non-perturbative definition of RG that reduces to the
perturbative one in the appropriate domain [6, 7, 8, 9, 10, 11, 12, 13]. In
this context, comparison with the results of dimensional regularization become
meaningful. In Ref. [14] we have discussed a two-parameter family of
regulators $R_{k}(a,\epsilon)$ that includes (for $\epsilon=0$) a popular
class of regulators used in the FRG literature. On the other hand, taking the
limit $a\to 0$ and $\epsilon\to 0$ (in this order), it reproduces the beta
functions of $\overline{\text{MS}}\,$. In this paper we shall discuss what
happens when the limits are taken in the opposite order (see Fig. 1). We next
discuss the motivation for this study.

Figure 1: Blue, continuous curve: a path that reproduces the beta functions of
dimensional regularization. Red, dashed curve: the limit of vanishing
regulator. For a more detailed discussion see Sec. 3.

The notation $\Gamma_{k}[\phi]$ emphasizes the important dependence of this
functional on the scale $k$, but $\Gamma_{k}$ also depends on the shape of the
cutoff function $R_{k}$. The notation $\Gamma[\phi,R_{k}]$ would thus be more
appropriate, and (1.2) could be replaced by a functional equation where the
derivatives with respect to $k$ are replaced by functional derivatives with
respect to $R_{k}$. As mentioned above, all the relevant physical information
is contained in the EA and therefore a priori, all the dependence on $R_{k}$
is unphysical, including the dependence on $k$. However, there are situations
where $k$ can be identified with a physical parameter that acts in the theory
as an IR cutoff. In these cases, the dependence on $k$ can assume a physical
meaning.

Even though in such cases the dependence on $k$ reproduces the dependence on
physical parameters, the dependence on the shape of $R_{k}$ still remains
unphysical. Thus any observable must be independent of this shape. On the
other hand, when one makes approximations, even physical observables will
exhibit some spurious dependence on the shape of the cutoff. We will refer to
this as “cutoff dependence”. 444It is distinct from, but closely related to
the “scheme dependence” of renormalized perturbation theory. For example, in
statistical physics, the position of a fixed point is not universal, but the
critical exponents are. Still, when one calculates the critical exponents, one
must use some approximation and the results always depend on the shape of the
regulator. In a specific calculation, one can then try to exploit this cutoff
dependence to optimize the cutoff, i.e. to find the cutoff that yields the
best possible value for the observables. This is, in practice, an important
aspect of FRG studies [15, 17, 16, 18, 19, 20].

The main motivation for this study comes from another issue that arises in
certain applications of the FRG. The central idea is simple and can be stated
in great generality. Suppose that the action at the microscopic level is
invariant under certain transformations. Since the symmetry reflects physical
properties of the system, one would like to maintain it in the course of the
RG flow. 555Since the RG flow interpolates between a classical bare action and
a renormalized EA, this is like saying that the symmetry should be preserved
under quantization. However, for technical reasons, it may be difficult to
construct a regulator that has the symmetry, and in this case the EAA will not
have it either. This will give rise to unpleasant complications. Intuitively,
we may try to minimize the breaking of the symmetry by making the regulator as
“small” as possible. Let us make this notion a bit more precise. For
dimensional reasons, we can write the regulator as $R_{k}(z)=k^{2}r_{a}(y)$,
where $r_{a}=ar_{1}$ and $r_{1}$ is a dimensionless function of the
dimensionless variable $y=z/k^{2}$, that is assumed to satisfy the
normalization condition $r_{1}(0)=1$ and $a$ is a positive real number. 666
Consider a fixed shape function $r$, such that $r(y)=0$ for $y>1$. The limit
$a\to\infty$ is expected to completely remove from the path integral all the
fluctuations with momenta $q^{2}<k^{2}$. This is often referred to as the
sharp cutoff limit. Numerically optimal results are usually obtained for
$a\approx 1$. In this paper we shall mainly confine ourselves to the following
regulator choice

$R_{k}(z)=a(k^{2}-z)\theta(k^{2}-z)\,,$ (1.3)

and be concerned with the limit $a\to 0$, which we call the limit of vanishing
cutoff. 777Thus, vanishing cutoff should not be misinterpreted as $k\to 0$.
One expects that in this limit the spurious effects due to the breaking of the
symmetry by the regulator can be removed, or at least minimized. It may seem
that this limit is trivial, because for $a=0$ there is no cutoff, and the
right-hand-side (rhs) of the exact FRG equation vanishes, but we shall see
that some important physical information remains available even in the limit.

In order to better explain the problems arising from the use of vanishing
regulators, and ways to circumvent them, it is best to focus on simple and
well-understood systems. In Sec. 2 we consider the harmonic and anharmonic
oscillator. Some of the features of vanishing regulators appear already in
these cases. In Sec. 3 we deal with the paradigm of statistical mechanical
systems, the $\mathbb{Z}_{2}$–invariant scalar field (representing the Ising
universality class). We find that the main features of the Wilson-Fisher (WF)
fixed point remain accessible in the limit of vanishing regulator, but the
best approximation is obtained with the simplest truncation, that only
involves relevant couplings (the mass and the quartic coupling). There we also
discuss the relation between the vanishing-$a$ limit of (1.3) and the constant
(momentum-independent) regulator, as well as the subtleties concerning the
application of vanishing regulators in an even number of dimensions.

In Sec. 4 we address the $O(N+1)$ nonlinear sigma model, using a particular
coordinate system on the sphere $S^{N}$. This is an example of a system where
the regulator breaks the symmetry of the theory (respecting only the subgroup
$O(N)$) but in the limit of vanishing regulators the symmetry is seen to be
restored. In Sec. 5 we discuss a similar problem that arises in applications
of the background field method. It is generally the case that the regulator
breaks the symmetry of the classical action consisting of equal and opposite
shifts of the background and quantum fields. Also this symmetry is seen to be
restored in the limit of vanishing regulators. We conclude in Sec. 6 with a
brief discussion of our results and some outlooks. Some auxiliary formulas and
analyses are provided in two appendices.

##  2 Quantum oscillators

In this section we shall consider a very simple application of the FRG
equation as a tool to compute the EA at $k=0$. This will allow us to
investigate the effect of the vanishing regulators on the calculation of some
physical observable. We shall consider first the simple harmonic oscillator
and then the anharmonic one.

The general bare action we are interested in reads

$S=\int
dt\left(\frac{1}{2}\dot{x}^{2}+\frac{1}{2}\omega^{2}x^{2}+\frac{\lambda}{4!}x^{4}\right)\
.$ (2.1)

In the Local Potential Approximation (LPA), which is the first term in a
derivative expansion, the EAA is approximated by

$\Gamma_{k}=\int dt\left(\frac{1}{2}\dot{x}^{2}+V_{k}(x)\right)\ .$ (2.2)

Using the regulator in Eq. (1.3) we get the following flow equation for the
potential

$\partial_{k}V_{k}=\frac{1}{\pi}\left(\frac{a\,k\arctan\left(k\sqrt{\frac{1-a}{ak^{2}+V^{\prime\prime}_{k}}}\right)}{\sqrt{(1-a)\left(ak^{2}+V^{\prime\prime}_{k}\right)}}-\sqrt{\frac{a}{1-a}}\arctan\left(\sqrt{\frac{1-a}{a}}\right)\right)\,.$
(2.3)

The second term on the rhs is equal to the first term evaluated at $V_{k}=0$.
This subtraction is not ad hoc, and is actually always meant to be present in
the FRG equation [21], although in most applications it is dropped since it
only affects the ground-state energy. 888The correct counterpart of this term
in applications to gravity, especially within the background-field
formulation, is still uncertain. This term is due to the regularization of the
functional measure of the path integral, which ensures complete suppression of
the functional integral in the $k\to\infty$ limit. As it is uniquely defined
to reproduce the Weyl ordering prescription, it gives rise to the conventional
ground-state energies of the non-gravitating quantum mechanical models [22,
23].

Expanding the potential into a Taylor series

$V_{k}(x)=E_{k}+\frac{1}{2}\omega_{k}^{2}x^{2}+\frac{\lambda_{k}}{4!}x^{4}+\ldots\
,$ (2.4)

the beta functions are

$\displaystyle
k\partial_{k}E_{k}=\frac{k}{\pi}\left[\frac{a\,\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}_{k}^{2}}}\right)}{\sqrt{(1-a)\left(a+\tilde{\omega}_{k}^{2}\right)}}-\sqrt{\frac{a}{1-a}}\arctan\left(\sqrt{\frac{1-a}{a}}\right)\right]\
,$ (2.5a) $\displaystyle k\partial_{k}\omega_{k}^{2}=-\frac{a\lambda_{k}}{2\pi
k}\frac{\frac{\sqrt{1-a}\sqrt{a+\tilde{\omega}_{k}^{2}}}{1+\tilde{\omega}_{k}^{2}}+\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}_{k}^{2}}}\right)}{(a+\tilde{\omega}_{k}^{2})^{3/2}\sqrt{1-a}}\
,$ (2.5b) $\displaystyle
k\partial_{k}\lambda_{k}=\frac{3a\lambda_{k}^{2}}{4\pi
k^{3}}\frac{\frac{\sqrt{1-a}\sqrt{a+\tilde{\omega}_{k}^{2}}(3+2a+5\tilde{\omega}_{k}^{2})}{(1+\tilde{\omega}_{k}^{2})^{2}}+3\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}_{k}^{2}}}\right)}{(a+\tilde{\omega}_{k}^{2})^{5/2}\sqrt{1-a}}\
,$ (2.5c)

where $\tilde{\omega}_{k}=\omega_{k}/k$.

###  2.1 Harmonic oscillator

We start by addressing the computation of the vacuum energy of the harmonic
oscillator. When $\lambda_{k}=0$, Eq. (2.5b) shows that $\omega_{k}$ is
independent of $k$, thus we shall simply write $\omega_{k}=\omega$. The
solution of (2.5a) is

$E_{k}=\frac{\omega}{2}-\frac{k\sqrt{a+\tilde{\omega}^{2}}\left(\tilde{\omega}\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\arctan\left(\frac{1}{\tilde{\omega}}\right)-\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\right)\right)}{\pi\sqrt{1-a}}-\frac{k\sqrt{a}\arctan\left(\sqrt{\frac{1-a}{a}}\right)}{\pi\sqrt{1-a}}\,.$
(2.6)

This function is plotted in Fig. 2 for various values of $a$. First of all we
see that $E_{0}=\omega/2$ for any $a$. The $a$-independence of the result is
just an example of a more general phenomenon: while the $k$ dependence of any
quantity along an RG trajectory is sensitive to the functional form of
$R_{k}$, the boundary values at $k\to+\infty$ and $k\to 0$ are not.

The second point to notice is that the convergence of the flow towards the IR
becomes faster for decreasing $a$. This can be understood as follows. The
regulator term is effective in suppressing the propagation when it becomes
comparable or larger than the kinetic term, i.e. for $R_{k}(q^{2})>q^{2}$. For
the regulator (1.3), this happens when

$q^{2}<k_{\text{eff}}^{2}\equiv\frac{a}{1+a}k^{2}\ .$ (2.7)

Thus decreasing $a$ has the same effect as decreasing $k_{\text{eff}}$.

Figure 2: The continuous curves are the RG trajectories (2.6) for the harmonic
oscillator ground-state energy $E_{k}$ for the regulator (1.3) and various
values of $a$. From bottom to top: $a\to\infty$ (sharp cutoff, green), $a=1$
(black), $a=1/10$ (blue), $a=1/100$ (red), $a=1/10000$ (pink). The dashed
curve is the flow (2.12). The horizontal axis has been rescaled by the
function $k=\tan(\pi x/2)$.

From this discussion, there seems to be no issue with the limit $a\to 0$. Some
subtlety appears however when we try to construct the RG trajectory from the
$a\to 0$ limit of Eq. (2.5a), which reads

$k\partial_{k}E_{k}\sim-\frac{k}{2}\sqrt{a}+O(a)\ .$ (2.8)

This way of taking the limits is of course nonsensical, as the resulting beta
functions would be identically vanishing. The $\omega$ independence is a
consequence of the fact that the numerator of the rhs of the FRG equation is
already proportional to $a$. However, we obtain a nontrivial equation if we
rescale

$\displaystyle\omega^{2}=a\,\hat{\omega}^{2},\quad\quad E_{k}=\sqrt{a}\
\hat{E}_{k}\ ,$ (2.9)

and then take the $a\to 0$ limit of Eq. (2.5a). This leads to the finite
result

$k\partial_{k}\hat{E}_{k}=\frac{1}{2}\left(\frac{k}{\sqrt{k^{2}+\hat{\omega}^{2}}}-1\right)+O(\sqrt{a})\
,$ (2.10)

which is the same flow equation one would find with a constant (often called a
“Callan-Symanzik”) regulator

$R_{k}=k^{2}.$ (2.11)

The latter leads to the flow

$E_{k}=\frac{1}{2}\left(\sqrt{k^{2}+\omega^{2}}-k\right)\ ,$ (2.12)

which is plotted as the dashed curve in Fig. 2.

The rescaling (2.9) is formulated as an “active” transformation of the
couplings $E_{k}$ and $\omega_{k}$ throught a factor $\sqrt{a}$, while the RG
coordinate $k$ stays independent of $a$. Within the flow equations of the
dimensionless couplings $\tilde{E}_{k}\equiv E_{k}/k$ and
$\tilde{\omega}^{2}$, it is also possible to reinterpret it as a “passive”
transformation, in which the dimensionful couplings $E_{k}$ and $\omega_{k}$
are independent of $a$, while the RG coordinate changes by
$k=\hat{k}/\sqrt{a}$. This second interpretation is consistent with the
observation that $k_{\mathrm{eff}}\to\hat{k}$ for $a\to 0$ according to Eq.
(2.7).

###  2.2 Anharmonic oscillator

Next, we turn to the anharmonic oscillator with $\lambda\neq 0$. We will
consider only the first order of the expansion in $\lambda$. We see from
(2.5c) that at this order the beta function of $\lambda_{k}$ is zero.
Therefore $\lambda_{k}=\lambda$ at all scales. Expanding the vacuum energy
parameter to first order in $\lambda$

$E_{k}=E_{k}\Big{|}_{\lambda=0}+\frac{dE_{k}}{d\lambda}\Big{|}_{\lambda=0}\lambda+\ldots$
(2.13)

and solving the flow equation with the initial condition that
$\lim_{k\to\infty}\Gamma_{k}=S$ is the bare action (2.1), we find that the
first order correction to the energy is

$\displaystyle\frac{dE_{k}}{d\lambda}\Big{|}_{\lambda=0}$ $\displaystyle=$
$\displaystyle\frac{1}{32\omega^{2}}+\frac{\arctan\left(\frac{1}{\tilde{\omega}}\right)\left[2\tilde{\omega}\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\right)+(a-1)\left(\pi+\arctan\left(\frac{1}{\tilde{\omega}}\right)^{2}\right)\right]}{8\pi^{2}(a-1)\omega^{2}}$
(2.14)
$\displaystyle-\frac{\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\right)\left(\pi\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}+\frac{\tilde{\omega}}{a+\tilde{\omega}^{2}}\arctan\left(\sqrt{\frac{1-a}{a+\omega^{2}}}\right)\right)}{8\pi^{2}(a-1)k\omega}\,.$

On the other hand for the frequency we find to first order in $\lambda$

$\displaystyle\omega_{k}^{2}$ $\displaystyle=$
$\displaystyle\omega^{2}+\frac{\lambda}{4\omega}\left(1+\frac{2\tilde{\omega}\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\arctan\left(\sqrt{\frac{1-a}{a+\tilde{\omega}^{2}}}\right)}{\pi(1-a)}-\frac{2}{\pi}\arctan\left(\frac{1}{\tilde{\omega}}\right)\right)\,.$
(2.15)

The quantities $\frac{dE_{k}}{d\lambda}\Big{|}_{\lambda=0}$ and $\omega_{k}$
given in (2.14-2.15) are the solutions of the flow equations at arbitrary $k$.
They interpolate between the initial conditions
$\lim_{k\to\infty}\frac{dE_{k}}{d\lambda}\Big{|}_{\lambda=0}=0$ and
$\lim_{k\to\infty}\omega_{k}=\omega$ and the corresponding parameters in the
EA at $k=0$.

Also in this case, it is not possible to directly take the limit $a\to 0$ in
the flow equations, because then the beta functions simply vanish. However,
having solved the flow equations one can take the limits $a\to 0$ and $k\to 0$
in any order obtaining

$\displaystyle\omega_{0}^{2}=\omega^{2}+\frac{\lambda}{4\omega}\,,$ (2.16a)
$\displaystyle E_{0}=\frac{\omega}{2}+\frac{\lambda}{32\omega^{2}}\,.$ (2.16b)

One can take the limit $a\to 0$ in the flow equations provided the potential
is rescaled to:

$\displaystyle V_{k}(x)$ $\displaystyle=\sqrt{a}\ \hat{V}_{k}(\hat{x})\ ,$
(2.17a) $\displaystyle x$ $\displaystyle=a^{-1/4}\ \hat{x}\ .$ (2.17b)

Expanding around $a=0$ the flow equation becomes

$\partial_{k}\hat{V}_{k}=\frac{1}{2}\left(\frac{k}{\sqrt{k^{2}+\hat{V}_{k}^{\prime\prime}}}-1\right)\,.$
(2.18)

This is the flow equation of the potential for a constant regulator.
Projecting the latter on a polynomial truncation of the potential as in Eq.
(2.4), we deduce the beta functions

$\displaystyle\partial_{k}E_{k}=\frac{1}{2}\left(\frac{k}{\sqrt{k^{2}+\omega_{k}^{2}}}-1\right)\
,$ (2.19a)
$\displaystyle\partial_{k}\omega_{k}^{2}=-\frac{1}{4}\frac{k}{\left(k^{2}+\omega_{k}^{2}\right)^{3/2}}\lambda_{k}\
,$ (2.19b)
$\displaystyle\partial_{k}\lambda_{k}=\frac{9}{8}\frac{k}{\left(k^{2}+\omega_{k}^{2}\right)^{5/2}}\lambda_{k}^{2}\
.$ (2.19c)

Solving these equations one reobtaines (2.16).

##  3 The Ising universality class

In this section we shall consider the theory of a single,
$\mathbb{Z}_{2}$-invariant scalar field $\phi$ in the LPA

$\Gamma_{k}=\int d^{d}x\left[\frac{1}{2}(\partial\phi)^{2}+V_{k}(\phi)\right]\
.$ (3.1)

While the FRG equation allows us to treat the potential as a whole, it will be
instructive to further expand

$V_{k}(\phi)=\sum_{n=0}^{\infty}\frac{\lambda_{2n}}{(2n)!}\phi^{2n}\ .$ (3.2)

The term $n=0$ is the vacuum energy and can usually be ignored, but we shall
need it later in our discussion. Then, from the FRG equation we can derive
infinitely many beta functions
$\beta_{2n}=k\frac{\partial\lambda_{2n}}{\partial k}$. For arbitrary
regulator, and in any dimension, for the first few couplings this leads to

$\displaystyle\beta_{0}$
$\displaystyle=\frac{1}{2(4\pi)^{d/2}}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{P_{k}+\lambda_{2}}\right]\,,$
(3.3a) $\displaystyle\beta_{2}$
$\displaystyle=-\frac{1}{2(4\pi)^{d/2}}\lambda_{4}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{2}}\right],$
(3.3b) $\displaystyle\beta_{4}$
$\displaystyle=\frac{1}{2(4\pi)^{d/2}}\left(6\lambda_{4}^{2}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{3}}\right]-\lambda_{6}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{2}}\right]\right),$
(3.3c) $\displaystyle\beta_{6}$
$\displaystyle=\frac{1}{2(4\pi)^{d/2}}\left(-90\lambda_{4}^{3}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{4}}\right]+\right.$
$\displaystyle\left.+30\lambda_{4}\lambda_{6}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{3}}\right]-\lambda_{8}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{(P_{k}+\lambda_{2})^{2}}\right]\right)\
,$ (3.3d)

where

$Q_{n}\left[W\right]=\frac{1}{\Gamma(n)}\int_{0}^{\infty}dz\,z^{n-1}W(z)$
(3.4)

are momentum integrals ($z={q}^{2}$). These integrals can be evaluated in
closed forms by using the optimized regulator (1.3). The $Q$ functionals are
then given by hypergeometric functions

$Q_{n}\left(\frac{\partial_{t}R}{(P+m^{2})^{\ell}}\right)=k^{2(n+1-\ell)}\frac{2a}{\Gamma(n+1)(a+\tilde{m}^{2})^{\ell}}\phantom{A}_{2}F_{1}\left(\ell,n,1+n,\frac{a-1}{a+\tilde{m}^{2}}\right)\
.$ (3.5)

These are plotted in Fig. 3 for $d=3$ and $d=4$. In particular, in the
massless case and in the limit of vanishing regulator we obtain

$\lim_{a\to
0}Q_{n}\left(\frac{\partial_{t}R}{P^{\ell}}\right)=\begin{cases}0\quad\mathrm{for}\quad\ell<n+1\
,\\\ 1\quad\mathrm{for}\quad\ell=n+1\ ,\\\
\infty\quad\mathrm{for}\quad\ell>n+1\ .\end{cases}$ (3.6)

Clearly, the beta functions will not be finite. 999Note that these are
infrared divergences: in the massive case all $Q$ functionals go to zero for
$a\to 0$. For this reason an additional regularizing device is needed to make
sense of vanishing regulators. In Ref. [14] we have discussed a family of
regulators depending on an additional parameter $\epsilon$ that, in the limit
$a\to 0$ and $\epsilon\to 0$ (in this order) reproduces the results of
dimensional regularization. In the $a$-$\epsilon$ plane the limit had to be
taken along a curve of the general form shown in Fig. 1. In this paper we
shall instead try to take the limits in the inverse order. In fact, we shall
not even talk about the parameter $\epsilon$ and try to take the limit $a\to
0$ along the path $\epsilon=0$ (red, dashed) in Fig. 1.

Figure 3: The functionals (3.5) with $n=3/2$ ($d=3$, left) and $n=2$ ($d=4$,
right), with $\tilde{m}=0$ (top line) and $\tilde{m}=0.25$ (bottom line). In
each figure $\ell=1,2,3,4$, from bottom to top.

Returning to the beta functions (3.3), we note that if we set $\lambda_{2}=0$,
the beta functions of the relevant couplings go to zero, those of the marginal
couplings are independent of $a$ and those of the irrelevant couplings diverge
in the limit of vanishing regulator. Given this rather singular behavior, one
may fear that all physical information gets lost in this limit. Actually, this
is not so, as we intend to show in $d=3$, where the system is know to have a
nontrivial fixed point.

###  3.1 The Wilson-Fisher fixed point: relevant couplings

In order to make our point it will be enough, as a first step, to consider a
truncation that contains only the relevant couplings (we are now in $d=3$):

$V_{k}=\frac{\lambda_{2}}{2}\phi^{2}+\frac{\lambda_{4}}{24}\phi^{4}\ .$ (3.7)

Defining the dimensionless variables

$\tilde{\lambda}_{2n}=k^{-d+n(d-2)}\lambda_{2n}\,,$ (3.8)

the beta functions are

$\displaystyle\tilde{\beta}_{2}=-2\tilde{\lambda}_{2}-\frac{a}{6\pi^{2}\left(a+\tilde{\lambda}_{2}\right)^{2}}\,_{2}F_{1}\\!\left(2,\frac{3}{2},\frac{5}{2};-\frac{1-a}{a+\tilde{\lambda}_{2}}\right)\tilde{\lambda}_{4}\,,$
(3.9a)
$\displaystyle\tilde{\beta}_{4}=-\tilde{\lambda}_{4}+\frac{a}{\pi^{2}\left(a+\tilde{\lambda}_{2}\right)^{3}}\,_{2}F_{1}\\!\left(3,\frac{3}{2},\frac{5}{2};-\frac{1-a}{a+\tilde{\lambda}_{2}}\right)\tilde{\lambda}_{4}^{2}\,.$
(3.9b)

Expanding in $\tilde{\lambda}_{2}$

$\displaystyle\tilde{\beta}_{2}=-2\tilde{\lambda}_{2}-\frac{\tilde{\lambda}_{4}}{4\pi^{2}(1-a)}\left[a\left(\frac{\arctan\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{(1-a)a}}-1\right)+\frac{1}{2}\left(2a-1-\frac{\arctan\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{a(1-a)}}\right)\tilde{\lambda}_{2}\right],$
(3.10a)
$\displaystyle\tilde{\beta}_{4}=-\tilde{\lambda}_{4}+\frac{\tilde{\lambda}_{4}^{2}}{\pi^{2}a^{2}}\,_{2}F_{1}\\!\left(\frac{3}{2},3;\frac{5}{2};\frac{a-1}{a}\right)\,.$
(3.10b)

The WF fixed point is now located at

$\displaystyle\tilde{\lambda}_{2}^{*}=\frac{2a^{3}\left(1-\frac{\arctan\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{(1-a)a}}\right)}{a^{2}\left(2a-1-\frac{\arctan\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{(1-a)a}}\right)+16(1-a)\,_{2}F_{1}\left(\frac{3}{2},3;\frac{5}{2};\frac{a-1}{a}\right)}\
,$ (3.11a)
$\displaystyle\tilde{\lambda}_{4}^{*}=\frac{\pi^{2}a^{2}}{\,{}_{2}F_{1}\\!\left(\frac{3}{2},3;\frac{5}{2};\frac{a-1}{a}\right)}\
.$ (3.11b)

If we expand the critical couplings around $a=0$

$\displaystyle\tilde{\lambda}_{2}^{*}\mathrel{\underset{a\to
0}{\scalebox{2.0}[1.0]{$\sim$}}}-\frac{2a}{5}+o\left(a^{3/2}\right)\,,$
(3.12a) $\displaystyle\tilde{\lambda}_{4}^{*}\mathrel{\underset{a\to
0}{\scalebox{2.0}[1.0]{$\sim$}}}\frac{16\pi\sqrt{a}}{3}+o(a^{3/2})\,.$ (3.12b)

Thus the WF fixed point merges with the Gaussian fixed point. Note that since
$\tilde{\lambda}_{2}=\tilde{m}^{2}$ is linear in $a$ for $a\to 0$ at the WF
fixed point, the Q functional (3.5) does not go to zero and this entails that
the quantum contribution to the critical exponents will be nontrivial for
$a\to 0$.

Indeed, the position of the fixed point is not physically significant. If we
consider the stability matrix at the nontrivial fixed point

$M=\left(\frac{\partial\tilde{\beta}_{i}}{\partial\tilde{\lambda}_{j}}\right)_{*}=\left(\begin{array}[]{cc}-\frac{5}{3}&\frac{4a\,_{2}F_{1}\\!\left(\frac{3}{2},3;\frac{5}{2};\frac{a-1}{a}\right)\left(1-\frac{\tan^{-1}\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{(1-a)a}}\right)}{\pi^{2}a^{2}\left(2a-\frac{\tan^{-1}\left(\frac{\sqrt{1-a}}{\sqrt{a}}\right)}{\sqrt{(1-a)a}}-1\right)+16(1-a)\,_{2}F_{1}\\!\left(\frac{3}{2},3;\frac{5}{2};\frac{a-1}{a}\right)}\\\
0&1\\\ \end{array}\right)$

we see that the component $(1,2)$ of $M$ goes to zero for $a\to 0$ and so the
stability matrix becomes diagonal. The eigenvalues of $M$, that is minus the
critical exponents $\theta_{i}$, are actually independent of $a$, in
particular $\nu=(\theta_{1})^{-1}=0.6$. We see that even though the WF fixed
point collapses towards the Gaussian one, it keeps its distinct character in
the limit $a\to 0$ and a different critical exponent $\nu$. In fact, the
numerical value is not very bad, considering the drastic approximation.

###  3.2 The Wilson-Fisher fixed point in the LPA

Let us now treat the potential as a whole [24]. Inserting (3.1) in the FRGE we
obtain the “beta functional”

$\partial_{t}V_{k}=\frac{1}{2(4\pi)^{d/2}}Q_{d/2}\left[\frac{\partial_{t}R_{k}}{P_{k}+V_{k}^{\prime\prime}}\right]\
.$ (3.13)

Using the regulator (1.3), setting $d=3$ and rescaling

$\displaystyle\phi$ $\displaystyle=\frac{k^{1/2}}{\pi\sqrt{6}}\tilde{\phi}\ ,$
(3.14a) $\displaystyle V_{k}(\phi)$
$\displaystyle=\frac{k^{3}}{6\pi^{2}}v(\tilde{\phi})\ ,$ (3.14b)

the beta function of the dimensionless potential $v$ becomes

$\partial_{t}v=-3v+\frac{1}{2}\tilde{\phi}v^{\prime}+\frac{a}{a+v^{\prime\prime}}\
{}_{2}F_{1}\\!\left(1,\frac{3}{2},\frac{5}{2};\frac{a-1}{a+v^{\prime\prime}}\right)\,.$
(3.15)

We look for even scaling solutions shooting from the origin with initial
condition $v^{\prime\prime}(0)$ and $v^{\prime}(0)=0$. There are only two
values of $v^{\prime\prime}(0)$ which can be identified as fixed-point
solutions: $v^{\prime\prime}(0)=0$, that corresponds to the Gaussian fixed
point, and some negative value that corresponds to the WF fixed point. As in
the preceding section, for decreasing values of $a$, the WF fixed point moves
towards the Gaussian one. We see that also in the functional treatment, the WF
fixed point collapses into the Gaussian one.

This is confirmed by shooting from infinity. The potential for the WF solution
has the following asymptotic behavior for large field

$v=A\tilde{\phi}^{6}+a\left(\frac{1}{150A\tilde{\phi}^{4}}-\frac{2a+3}{31500A^{2}\tilde{\phi}^{8}}+\frac{8a^{2}+12a+15}{8505000A^{3}\tilde{\phi}^{12}}-\frac{a}{67500A^{3}\tilde{\phi}^{14}}+O\left(A^{-4}\tilde{\phi}^{-16}\right)\right)\,.$

The free parameter $A$ can be fixed as function of $a$ by requiring
$\mathbb{Z}_{2}$ symmetry for vanishing field [25]. We find that in the limit
$a\to 0$, $A$ tends to $A\approx 0.0015$. 101010 The asymptotic parameter is
$A=0.001$ for $a=1$ and it increases monotonically for $a\to 0$.

The scaling exponents $\theta_{i}$ are obtained by linearizing the flow
equation around the fixed point and calculating the spectrum of
eigenperturbations. The analysis has to be done numerically. For the Gaussian
fixed point the spectrum is independent of $a$. Figure 4 gives $\nu$ of the WF
fixed point as a function of $a$ for $10^{-5}<a<10^{5}$. As expected, the best
value is obtained for $a\approx 1$, while in the limit of vanishing regulator
$\nu$ appears to approach $\nu=1$. Besides the correlation-length exponent
$\nu=(\theta_{1})^{-1}$, we also find positive eigenvalues, as reported in
Tab. 1.

| $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$ | $\theta_{5}$
---|---|---|---|---|---
$a=1$ | $1.539$ | $-0.656$ | $-3.180$ | $-5.912$ | $-8.796$
$a=0$ | $1$ | $-1$ | $-3$ | $-5$ | $-7$
Table 1: The first few critical exponents at the Wilson-Fisher fixed, point
computed in the local potential approximation for the regulator (1.3). We
report the most common choice $a=1$ and the limiting case of the vanishing
regulator. Figure 4: The dots represent the values of the critical exponent
$\nu$ as a function of $a$. For comparison we have drawn the values of $\nu$
for the sharp regulator and the constant (mass) regulator, as well as the
conformal-bootstrap value [26]. This figure extends Fig. 12 in Ref. [16] to
low values of $a$.

For vanishing $a$ all the scaling exponents are odd integers. This coincides
with the spectrum of the $O(N)$ model in the limit of large $N$, which is
known, and we have indeed checked, to be independent of $a$ [27, 28].

###  3.3 Vanishing regulators and constant regulators

At this point it is relevant to recall that the critical exponent $\nu=1$ is
known to result also from the LPA equations for a constant regulator (2.11)
[16]. Together with the findings of Sec. 2, this observation points at a more
general result which we detail in this section.

So far we have first solved the fixed point equations for generic $a$ and then
sent $a\to 0$. On the other hand, we are now going to argue that when the
vanishing-regulator limit is taken on the LPA beta functions, i.e. before
integrating the flow, it results, in general non-even $d$, in the flow
equations of the constant regulator.

The first way of reaching this conclusion is based on a redefinition of the RG
scale $k$, which we have already introduced in Sec. 2. Suppose that in
addition to the parameter $a$ we also introduce a parameter $b$ rescaling the
cutoff $k$

$R_{k}(z)=a\left(bk^{2}-z\right)\theta\left(bk^{2}-z\right)\,.$ (3.16)

This rescaling can be motivated as follows. First of all, it should not change
the scaling solutions. Furthermore, as discussed in Sec. 2.1, we can define an
“effective” cutoff scale $k_{\text{eff}}$ as the momentum scale where the
cutoff term $R_{k}$ becomes comparable to the kinetic term. If we decrease
$a$, the effective cutoff scale also decreases. It was suggested in Ref. [16]
that the decrease of $a$ should be compensated by choosing $b$ so that at some
conventional scale $z_{0}<k^{2}$, the regulator is normalized:
$R_{k}(z_{0})=k^{2}$. This fixes $b=\frac{1}{a}+\frac{z_{0}}{k^{2}}$, leading
to the regulator

$R_{k}(z)=\left(k^{2}-a\left(z-z_{0}\right)\right)\theta\left(k^{2}-a\left(z-z_{0}\right)\right)\,.$
(3.17)

Now we see that in the limit $a\to 0$, the regulator becomes a constant as in
Eq. (2.11). The latter leads to the dimensionless flow equation

$\partial_{t}v=-dv+\left(\frac{d}{2}-1\right)\tilde{\phi}v^{\prime}+\frac{\pi}{(4\pi)^{d/2}\Gamma\left(\frac{d}{2}\right)\sin\left(\frac{d\pi}{2}\right)}\left(1+v^{\prime\prime}\right)^{\frac{d}{2}-1}\
.$ (3.18)

In $d=3$ and after the rescaling $v\to v/(4\pi)$ and
$\tilde{\phi}\to\tilde{\phi}/\sqrt{4\pi}$ this takes the simple form

$\partial_{t}v=-3v+\frac{1}{2}\tilde{\phi}v^{\prime}-\sqrt{1+v^{\prime\prime}}\
.$ (3.19)

An alternative way of arguing that the $a\to 0$ limit reduces the LPA flow
equation for the regulator (1.3) to the constant regulator case (3.18) is by
performing an $a$-dependent rescaling as in Sec. 2. Namely, by redefining

$\displaystyle\tilde{\phi}$ $\displaystyle=a^{(d-2)/4}\hat{\phi}\ ,$ (3.20a)
$\displaystyle v(\tilde{\phi})$
$\displaystyle=a^{d/2}\hat{v}(\hat{\phi})+a\frac{1}{(d-2)(4\pi)^{d/2}\Gamma(1+d/2)}\
,$ (3.20b)

in the flow equations for the regulator (1.3) and then taking the $a\to 0$
limit at fixed $\hat{\phi}$ and $\hat{v}$, we again find Eq. (3.18). For
instance in $d=3$ this rescaling entails that the prefactor $a$ in (3.15) goes
away.

Both kind of arguments however are applicable only for non-exceptional $d$. In
particular, the momentum integral leading to Eq. (3.18) is convergent for
$d<2$, and using analytic continuation the result has a meromorphic structure
with poles for even values of $d$. On the other hand, if we try to directly
take the limit $a\to 0$ with the regulator (1.3), and expand the $Q$
functionals (3.5), with $n=d/2$, $d$ even and $\tilde{m}=0$, in $a$ around
$a=0$, there appear terms with $\log a$. As a consequence, we expect that the
vanishing regulator limit of the LPA flow equation will enjoy special
properties in even dimensions.

###  3.4 Beta functions in even dimensions

As we argued at the end of the previous section, in the case of even
dimensions the limit of vanishing regulators has a more intricate structure.
Therefore, in this section we analyze these special cases in more detail.

We start with $d=2$, where the flow equation of the LPA reads

$\partial_{t}v=-2v+\frac{a}{4\pi(1-a)}\log\\!\left(\frac{1+v^{\prime\prime}}{a+v^{\prime\prime}}\right).$

Defining

$v(\tilde{\phi})=a\hat{v}(\tilde{\phi})$ (3.21)

and simplifying a factor $a$ from the flow equation, in the $a\to 0$ limit we
are left with

$\displaystyle\partial_{t}\hat{v}=-2\hat{v}-\frac{1}{4\pi}\log
a-\frac{1}{4\pi}\log\\!\left(1+\hat{v}^{\prime\prime}\right).$ (3.22)

The potential must be shifted by a factor that contains $\log a$, i.e.
$\hat{v}\to\hat{v}-\frac{1}{8\pi}\log a$ , in order to eliminate this
divergent term for the limit $a\to 0$ . We observe that the coefficient of the
$\log a$ term matches exactly the coefficient of the $1/\epsilon$ pole of the
expansion of (3.18) for $d=2+\epsilon$. 111111This correspondence between
$\log a$ singularities of the flow equations for the regulator (1.3) and
$1/\epsilon$ poles of (3.18) holds also in higher even dimensions. The finite
logarithmic contribution coincides with the one in Eq. (3.18). Therefore, up
to a field-independent shift of the potential, in $d=2$ the vanishing-
regulator limit agrees with the constant regulator.

We then turn to the LPA in $d=4$. We first truncate the potential to a
polynomial expansion around vanishing fields as in Eq. (3.2). For continuity
with the previous sections, we also turn to the dimensionless couplings
defined in Eq. (3.8). By considering the leading contributions to the beta
functions $\tilde{\beta}_{2n}$ for vanishing $a$, we construct an ansatz based
on the following scaling

$\displaystyle\hat{\lambda}_{2}$ $\displaystyle=a^{-1}\tilde{\lambda}_{2}\ ,$
(3.23a) $\displaystyle\hat{\lambda}_{4}$
$\displaystyle=\log(a)\tilde{\lambda}_{4}\ ,$ (3.23b)
$\displaystyle\hat{\lambda}_{2n}$ $\displaystyle=a^{n-2}(\log
a)^{n}\tilde{\lambda}_{2n},\quad n>2\ .$ (3.23c)

Assuming the $\hat{\lambda}_{2n}$ couplings can be kept fixed in the $a\to 0$
limit, results in the following set of beta functions

$\displaystyle\partial_{t}\hat{\lambda}_{2}$
$\displaystyle=-2\hat{\lambda}_{2}+\frac{\hat{\lambda}_{4}}{16\pi^{2}}\left[1+\frac{1+\log\\!\left(1+\hat{\lambda}_{2}\right)}{\log
a}\right],$ (3.24a) $\displaystyle\partial_{t}\hat{\lambda}_{4}$
$\displaystyle=\frac{1}{\log
a}\left[\frac{3}{16\pi^{2}}\frac{\hat{\lambda}_{4}^{2}}{1+\hat{\lambda}_{2}}+\frac{1}{16\pi^{2}}\hat{\lambda}_{6}\right],$
(3.24b) $\displaystyle\partial_{t}\hat{\lambda}_{6}$
$\displaystyle=2\hat{\lambda}_{6}-\frac{15}{16\pi^{2}}\frac{\hat{\lambda}_{4}^{3}}{(1+\hat{\lambda}_{2})^{2}}+\frac{1}{16\pi^{2}}\hat{\lambda}_{8}$
$\displaystyle+\frac{\hat{\lambda}_{8}}{16\pi^{2}}\frac{1+\log\\!\left(1+\hat{\lambda}_{2}\right)}{\log
a}+\frac{15}{16\pi^{2}}\frac{\hat{\lambda}_{4}\hat{\lambda}_{6}}{(1+\hat{\lambda}_{2})\log
a}\ ,$ (3.24c)

and similar results for higher couplings. Notice that terms of order $(\log
a)^{-1}$ could be neglected as sub-leading in all beta functions apart for the
second one, where such term is in fact the leading one.

In order to include the beta functions of all couplings in a functional
treatment, we turn to the task of including the definitions (3.23) in a
rescaling of the effective potential. It is impossible to achieve this goal by
a two-parameters rescaling of the kind studied in the previous sections.
However, Eq. (3.23c) trivially lends itself to a functional rescaling. Hence,
we can treat the first two couplings on a special footing, and embed the
remaining ones in a functional which is related to higher derivatives of
$v(\phi)$.

First, to simplify notations, it is convenient to define

$\displaystyle\tilde{\rho}$ $\displaystyle=\tilde{\phi}^{2}/2\ ,$ (3.25)
$\displaystyle u(\tilde{\rho})$ $\displaystyle=v(\tilde{\phi})\ .$ (3.26)

Next, we define

$f(\tilde{\rho})=u^{\prime}(\tilde{\rho})-\tilde{\lambda}_{2}-\frac{\tilde{\lambda}_{4}}{3}\tilde{\rho}\
.$ (3.27)

So by construction $f(0)=f^{\prime}(0)=0$, while
$f^{(n)}(0)\propto\lambda_{2(n+1)}$. The functional flow equation for $f$ can
be obtained from the functional equation for $u^{\prime}$ by

$\partial_{t}f(\tilde{\rho})=\partial_{t}u^{\prime}(\tilde{\rho})-\tilde{\beta}_{2}-\frac{\tilde{\beta}_{4}}{3}\tilde{\rho}\
.$ (3.28)

and then replacing $u_{k}^{\prime}$ through the definition (3.27). The
identities $\partial_{t}f_{k}(0)=\partial_{t}f_{k}^{\prime}(0)=0$ also follow
from this definition. By the rescaling

$\displaystyle f(\tilde{\rho})$ $\displaystyle=\frac{a}{\log
a}\hat{f}(\hat{\rho})\ ,$ (3.29) $\displaystyle\tilde{\rho}$
$\displaystyle=a\log a\,\hat{\rho}\ ,$ (3.30)

together with the previous definitions of $\hat{\lambda}_{2}$ and
$\hat{\lambda}_{4}$, we recover the full tower of relations (3.23). By
inserting the previous definitions in the flow equation for $u^{\prime}(\rho)$
one can deduce the following functional flow equation

$\displaystyle\partial_{t}\hat{f}(\hat{\rho})$
$\displaystyle=-2\hat{f}(\hat{\rho})+2\hat{\rho}\hat{f}^{\prime}(\hat{\rho})+\frac{3}{16\pi^{2}}\hat{f}^{\prime}(\hat{\rho})+\frac{1}{8\pi^{2}}\hat{\rho}\hat{f}^{\prime\prime}(\hat{\rho})+\frac{5}{16\pi^{2}}\hat{\rho}\hat{f}^{\prime\prime}(0)-\frac{1}{16\pi^{2}}\frac{\hat{\rho}\hat{\lambda}_{4}^{2}}{1+\hat{\lambda}_{2}}$
$\displaystyle-\frac{1}{16\pi^{2}}\hat{\lambda}_{4}\log\\!\left(1+\hat{\lambda}_{2}\right)+\frac{1}{16\pi^{2}}\hat{\lambda}_{4}\log\\!\left(1+\hat{\lambda}_{2}+\hat{\lambda}_{4}\hat{\rho}\right).$
(3.31)

This functional flow generates the leading terms in Eq. (3.24c), and similar
beta functions for the higher-order couplings, upon truncating it to a
polynomial ansatz regular at the origin. However, we stress again that Eq.
(3.31) does not include Eqs. (3.24a,3.24b), which therefore have to be
supplemented to exhaust the LPA flow equations.

These flow equations are different from those of a constant regulator. Indeed,
the latter are formally UV divergent. More specifically, in
$\tilde{\beta}_{2n}$ the contribution linear in $\tilde{\lambda}_{2n+2}$
corresponds to a momentum integral with dimension $2$ which is not regularized
by the constant regulator (2.11). Similar discrepancies arise in
$d=6,8,\dots$. The flow equation for the constant regulator in $d=4-\epsilon$
reads

$\displaystyle\partial_{t}{v}=$ $\displaystyle\
-4{v}+2\tilde{\rho}{v}^{\prime}+\frac{\left(2\tilde{\rho}{v}^{\prime\prime}+{v}^{\prime}+1\right)\left[\log\\!\left(2\tilde{\rho}{v}^{\prime\prime}+{v}^{\prime}+1\right)-1\right]}{16\pi^{2}}$
$\displaystyle+\frac{\left(2\tilde{\rho}{v}^{\prime\prime}+{v}^{\prime}+1\right)}{16\pi^{2}}\left[\gamma-\log(4\pi)-\frac{2}{\epsilon}\right].$
(3.32)

The second line in this equation arises from the expansion of the sine in the
denominator of Eq. (3.18). It provides contributions to the
${\tilde{\lambda}}_{2n+2}$ term inside ${\tilde{\beta}}_{2n}$. Such terms
would be absent in the $\overline{\text{MS}}\,$scheme. If we perform an ad hoc
subtraction of the second line, the flow equation (3.32) leads to the
following beta functions

$\displaystyle{\tilde{\beta}}_{2}$
$\displaystyle=-2{\tilde{\lambda}}_{2}+\frac{{\tilde{\lambda}}_{4}\log\\!\left({\tilde{\lambda}}_{2}+1\right)}{16\pi^{2}}\
,$ (3.33a) $\displaystyle{\tilde{\beta}}_{4}$
$\displaystyle=\frac{3{\tilde{\lambda}}_{4}^{2}}{16\pi^{2}\left({\tilde{\lambda}}_{2}+1\right)}+\frac{{\tilde{\lambda}}_{6}\log\\!\left({\tilde{\lambda}}_{2}+1\right)}{16\pi^{2}}\
,$ (3.33b) $\displaystyle{\tilde{\beta}}_{6}$
$\displaystyle=2{\tilde{\lambda}}_{6}-\frac{15{\tilde{\lambda}}_{4}^{3}}{16\pi^{2}\left({\tilde{\lambda}}_{2}+1\right)^{2}}+\frac{{\tilde{\lambda}}_{8}\log\\!\left({\tilde{\lambda}}_{2}+1\right)}{16\pi^{2}}+\frac{15{\tilde{\lambda}}_{6}{\tilde{\lambda}}_{4}}{16\pi^{2}\left({\tilde{\lambda}}_{2}+1\right)}\
.$ (3.33c)

A comparison with Eq. (3.24) immediately reveals several differences. Apart
for the scaling (classical) terms, the first two quantum terms are equal, up
to the fact that the $\lambda_{2}$ dependence of the $\lambda_{2n+2}$ term has
been washed away in Eq. (3.24) by the $a\to 0$ limit, and up to the crucial
$\log a$ dependence of Eq. (3.24b). However, all the additional quantum terms
in Eq. (3.33) are absent in Eq. (3.24).

The peculiar simplicity which the Eqs. (3.24) attain in the $a\to 0$ limit,
together with the $1/\log a$ dependence of Eq. (3.24b), raises the question as
to whether these beta functions retain enough physical information for being
practically useful. As a first step towards addressing this question, we limit
ourselves to a simple observation. Namely, as long as the subleading
logarithmic $a$-dependence is retained in Eq. (3.24), the $\phi^{4}$-theory
beta function and other universal physics is still present. For instance, we
can study the WF fixed point in $d=4-\epsilon$. In order to employ Eq. (3.24)
in this study, we need to prescribe that the $\epsilon\to 0$ limit be taken
before the $a\to 0$ one. This means in practice that the vanishing-regulator
limit is taken on the $d=4$ FRG equations. Had we sent $a\to 0$ in $d<4$, we
would have found different equations for $\hat{\lambda}_{2n}$ and precisely
the constant-regulator ones, as already mentioned in Sec. 3.3.

Within the simplest truncation corresponding to retaining only
$\tilde{\lambda}_{2}$ and $\tilde{\lambda}_{4}$, where we add the classical
scaling term $-\epsilon\lambda_{4}$ to $\hat{\beta}_{4}$ to account for the
shift of dimensionality, the WF fixed point to first order in $\epsilon$ is
located at

$\displaystyle\hat{\lambda}_{2}=$ $\displaystyle\
\frac{1}{6}\epsilon\left(1+\log a\right)\ ,$ (3.34)
$\displaystyle\hat{\lambda}_{4}=$ $\displaystyle\
\frac{16}{3}\pi^{2}\epsilon\log a\ .$ (3.35)

These fixed-point couplings have to be interpreted as small, even if they
seemingly blow up for $a\to 0$, because the limit $\epsilon\to 0$ has to be
taken first. Notice that keeping the sub-leading order-$(\log a)^{-1}$
contribution to $\hat{\beta}_{4}$ is essential for revealing the fixed point.
By computing the corresponding critical exponents, we find the universal one-
loop result

$\theta_{1}=2-\frac{\epsilon}{3}\ ,\qquad\theta_{2}=-\epsilon\ .$ (3.36)

##  4 $\boldsymbol{O(N+1)}$ symmetry in nonlinear models

As a first example of a symmetry that is broken by the regulator, we shall
consider here the two dimensional $O(N+1)$ nonlinear sigma model in a
particular coordinate system. We start from the order-$\partial^{2}$ expansion
of $\Gamma_{k}$ for a $O(N)$-invariant multiplet of scalars

$\Gamma_{k}[\phi]=\int
d^{2}x\left[U_{k}(\rho)+\frac{1}{2}Z_{k}(\rho)\partial_{\mu}\phi_{a}\partial^{\mu}\phi^{a}+\frac{1}{4}Y_{k}(\rho)\partial_{\mu}\rho\partial^{\mu}\rho\right]\
,$ (4.1)

where the $N$ fields $\phi^{a}$ are in the fundamental representation of
$O(N)$, and $\rho=\phi^{a}\phi^{a}/2$ is the corresponding local invariant. We
further define the radial wave function renormalization

$\tilde{Z}_{k}\left(\rho\right)=Z_{k}\left(\rho\right)+\rho
Y_{k}\left(\rho\right),$ (4.2)

The beta functions for $Z_{k}$, $\tilde{Z}_{k}$ and $U_{k}$ are given in App.
A.

If we make the assumptions

$Z_{k}(\rho)=\frac{Z_{k}}{g_{k}^{2}}\
,\quad\tilde{Z}_{k}(\rho)=\frac{1}{g_{k}^{2}}\left(\frac{1}{Z_{k}}-2\rho\right)^{-1}\
,\quad U_{k}=0\ ,$ (4.3)

the EAA becomes

$\Gamma_{k}[\phi]=\int
d^{d}x\frac{Z_{k}}{2g_{k}^{2}}\left(\delta_{ab}+\frac{\phi_{a}\phi_{b}}{\frac{1}{Z_{k}}-2\rho}\right)\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{b}\
.$ (4.4)

The tensor in parentheses is the metric of the $N$-dimensional sphere of
radius $Z_{k}^{-1/2}$, written in a coordinate system that consists in
projecting a point of the sphere orthogonally on the equatorial plane. In this
way the northern hemisphere is mapped to the domain
$\phi^{a}\phi^{a}<1/Z_{k}$. The symmetry group is extended to $O(N+1)$.

A standard cutoff

$\Delta S_{k}(\phi)=\frac{Z_{k}}{2g_{k}^{2}}\int
d^{2}x\,\phi_{a}R_{k}(-\partial^{2})\phi^{a}$

breaks $O(N+1)$ invariance, while preserving $O(N)$. Therefore, if we start at
some scale $k$ with an EAA of the form (4.4), the flow will immediately
generate $O(N+1)$-violating terms, and thus will take place in the larger
theory space parameterized by (4.1).

This can be seen already by projecting the flow generated by the ansatz (4.4)
on the local potential, i.e. by considering Eq. (A.2). For nonvanishing $a$
and for field-dependent wave function renormalizations, the choice $U_{k}=0$
is not preserved by the RG flow. However, in the $a\to 0$ limit it indeed
becomes a consistent ansatz, as in $\partial_{t}U_{k}$ the rhs behaves like
$a\log a$ when $a\to 0$.

Let us then inspect the flow of the wave function renormalizations. Inserting
the previous ansatz in the flow equation (A.3) for $\tilde{Z}_{k}(\rho)$,
121212Note that now $Z_{k}(\rho=0)=Z_{k}/g_{k}^{2}$, so inside the formulae
for the $Q$ functionals we must send $Z_{k}\to Z_{k}/g_{k}^{2}$ and
$\eta_{k}\to\eta_{k}+2\partial_{t}g_{k}/g_{k}$. in the limit $a\to 0$ we
obtain

$-\frac{2Z_{k}\partial_{t}g_{k}}{g_{k}^{3}\left(1-2Z_{k}\rho\right)}-\frac{Z_{k}\eta_{k}}{g_{k}^{2}\left(1-2Z_{k}\rho\right)^{2}}=\frac{1}{4\pi}\frac{Z_{k}(2\partial_{t}g_{k}+(\eta_{k}-2)g_{k})(2(N-1)Z_{k}\rho+1)}{g_{k}(1-2Z_{k}\rho)^{2}}.$

As the functional $\rho$ dependence on both sides of the equation is
comparable, this equation can be algebraically solved for $\partial_{t}g_{k}$
and $\eta_{k}$, resulting in

$\displaystyle\partial_{t}g_{k}$
$\displaystyle=-\frac{(N-1)g_{k}^{3}}{4\pi+g_{k}^{2}}\ ,$ (4.5a)
$\displaystyle\eta_{k}$ $\displaystyle=-\partial_{t}\log
Z_{k}=\frac{2Ng_{k}^{2}}{4\pi+g_{k}^{2}}\ .$ (4.5b)

These are the correct one-loop beta functions, augmented by RG resummations
due to the dependence of the regulator on $Z_{k}$ and $g_{k}$. The same result
can be derived by considering the flow equation for $Z_{k}(\rho)$. Thus,
within the present truncation the nonlinearly realized $O(N+1)/O(N)$ symmetry
is preserved by taking the limit $a\to 0$. Essentially the same flow equations
have been obtained in Ref. [14] with a pseudo-regulator reproducing the
$\overline{\text{MS}}\,$scheme. 131313The present result is obtained by
setting $\sigma=1$ in the beta functions of Ref. [14].

The assumption $U_{k}=0$, although justified by the observation that only a
trivial potential is compatible with the nonlinearly realized symmetry, can be
easily relaxed as long as this explicit symmetry-breaking term is treated as
an external source. The simplest of such terms is a linear coupling to the
$O(N+1)/O(N)$ variation of $\phi^{a}$, i.e. $\phi^{N+1}$

$U_{k}=-H\sqrt{\frac{1}{Z_{k}}-2\rho}\ .$ (4.6)

This ansatz, comprehending an arbitrary source $H$, was observed to be
compatible with the flow equation in the case of an
$\overline{\text{MS}}\,$pseudo regulator [14]. This linear term can also be
used to construct an exact FRG equation which manifestly preserves the full
$O(N+1)$ symmetry for every regulator function $R_{k}$, see App. B. For the
present standard FRG implementation and regularization scheme, the ansatz
(4.6) is not compatible with the flow equation of the potential, neither for
$a\neq 0$ nor in the $a\to 0$ limit. Only by assuming that $H$ be a function
of $a$ vanshing faster than $a$ itself, closure of the $O(\partial^{2})$ RG
flow on the ansatz (4.4) is recovered. To understand this phenomenon it is
necessary to study how the modified master equation for the $O(N+1)/O(N)$
symmetry behaves in the $a\to 0$ limit, which is the topic of Appendix B.
There we show how the construction of a nonvanishing potential term for the
nonlinear sigma model is indeed a complicated problem which requires the
simultaneous solution of both the flow equation and the modified master
equation. As explained in Appendix B, in solving this problem the $a\to 0$
limit is of limited use.

##  5 Background field issues

When one splits the field into a classical background and a quantum
fluctuation

$\phi=\phi_{B}+\varphi\ ,$ (5.1)

the action, being a function of $\phi$, is invariant under the shift symmetry

$\displaystyle\phi_{B}\mapsto\phi_{B}+\epsilon\ ,$ (5.2a)
$\displaystyle\varphi\mapsto\varphi-\epsilon\ .$ (5.2b)

This can be expressed by the identity

$\frac{\delta S}{\delta\phi_{B}}-\frac{\delta S}{\delta\varphi}=0\ .$ (5.3)

On the other hand, the regulator only depends on the quantum field and is
therefore not invariant under the split symmetry. In particular in gauge
theories, in order to preserve background gauge invariance, the cutoff is
usually written as a function of the background covariant derivative:
$R_{k}(-\bar{D}^{2})$. This effect can be mimicked in the scalar case by
artificially introducing a dependence of $R_{k}$ on $\phi_{B}$. For example,
Morris and collaborators considered regulators of the general form [25]

$r(y)=(1-h(\tilde{\phi}_{B})-y)\theta(1-h(\tilde{\phi}_{B})-y)\ .$ (5.4)

The EA then becomes a functional $\Gamma_{k}[\varphi,\phi_{B}]$, i.e. it has a
separate dependence on these two arguments. The breaking of the shift symmetry
results in a modified Ward identity

$\frac{\delta\Gamma_{k}}{\delta\phi_{B}}-\frac{\delta\Gamma_{k}}{\delta\varphi}=\frac{1}{2}{\rm
Tr}\left[\left(\frac{\delta^{2}\Gamma_{k}}{\delta\varphi\delta\varphi}+R_{k}\right)^{-1}\frac{\delta
R_{k}}{\delta\phi_{B}}\right]\ .$ (5.5)

It has been shown that such background dependence in the regulator can either
destroy physical fixed points or create artificial ones [25]. On the other
hand, when the FRG equation (1.2) is solved together with the Ward identity
(5.5), the correct physical picture can be reconstructed. While this can be
achieved in the scalar case [25], it is much harder in the case of gauge
theories, and in particular for gravity [29]. It is therefore desirable to
find other ways around this obstacle. The form of the equation (5.5) suggests
that in the limit of vanishing regulator the shift symmetry is restored. One
would therefore expect that in this limit the aforementioned pathologies
should also disappear. In this section we will see how this actually happens
in the scalar theory.

We begin by briefly reviewing some results of Ref. [25]. We consider the same
system as in Sec. 3.2, in $d=3$, but we use the regulator (5.4). In a single-
field approximation one identifies $\phi_{B}=\varphi$. The corresponding flow
equation for the potential reads

$\partial_{t}v=-3v+\frac{1}{2}\tilde{\varphi}v^{\prime}+\frac{\left(1-h\right)^{3/2}}{1-h+v^{\prime\prime}}\left(\\!1\\!-\\!h-\\!\frac{1}{2}\partial_{t}h\\!+\\!\frac{1}{4}\tilde{\varphi}h^{\prime}\\!\right)\theta(1-h)\,.$
(5.6)

Two special cases for $h$ have been considered. The first case is
$h=\alpha\tilde{\varphi}^{2}$. In this case, for $\alpha<0$ the Heaviside
theta on the rhs of Eq. (5.6) is equal to one. Solving the fixed point
equation, one finds that the Gaussian fixed point becomes interacting and an
increasing number of fake fixed points appear, as $\alpha$ becomes more
negative. For example, Tab. 2 presents the nontrivial fixed points and the
associated relevant critical exponents for two negative values of $\alpha$.

FP | $\theta_{1}$ | $\theta_{2}$
---|---|---
$1$ | $1.17$ | -
$2$ | $2.11$ | $0.82$

$\boldsymbol{\alpha=-1/2}$

FP | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$ | $\theta_{4}$
---|---|---|---|---
$1$ | $0.89$ | - | - | -
$2$ | $2.35$ | $0.76$ | - | -
$3$ | $2.02$ | $1.43$ | $0.60$ | -
$4$ | $2.10$ | $1.69$ | $1.08$ | $0.39$

$\boldsymbol{\alpha=-2}$

Table 2: The nontrivial fixed-point solutions of Eq. (5.6) with
$h=\alpha\tilde{\phi}^{2}$, and the corresponding relevant critical exponents,
for ${\alpha=-1/2}$ (left panel) and ${\alpha=-2}$ (right panel). The entries
which are left blank correspond to irrelevant deformations. FP1 is the Wilson-
Fisher fixed point, while FP2 is a "deformed Gaussian" fixed point as it
possesses two relevant directions.

In both cases FP2 is the deformed Gaussian fixed point. For $\alpha>0$ because
of the Heaviside theta function on the rhs of Eq. (5.6),
$v=A\tilde{\varphi}^{6}$ for $\tilde{\varphi}>1/\sqrt{\alpha}$. The Gaussian
fixed point is always absent, and for $\alpha>0.08$ also the WF fixed point
disappears.

The second case is $h=\alpha v^{\prime\prime}$. The Gaussian 141414Note that
the Gaussian fixed point corresponds to the point
$(v(0),v^{\prime}(0))=(1/3,0)$ . and the WF fixed points always exist, but
when $\alpha$ is increased, new fixed points appear near the Gaussian one
151515In particular for $\alpha\geq 0.85$ a first additional fixed point
appears.  and move away from it as $\alpha$ becomes bigger: for example, for
$\alpha=1$ there is a spurious fixed point and for $\alpha=2$ there are three
of them.

In Ref. [25] the authors solve the anomalous Ward identity for shift symmetry
and show how to recover the physical results. Instead, we shall discuss here
the effect of taking the limit of vanishing regulator. To this end, we first
introduce the parameter $a$ in (5.4)

$r(y)=a(1-h(\tilde{\phi}_{B})-y)\theta(1-h(\tilde{\phi}_{B})-y)\ .$ (5.7)

Within a single-field LPA truncation this leads to the flow equation

$\displaystyle\partial_{t}v=-3v+\frac{1}{2}\tilde{\varphi}v^{\prime}+$
$\displaystyle\theta(1-h)a\frac{\left(1-h\right)^{3/2}\left(1-h-\frac{1}{2}\partial_{t}h+\frac{1}{4}\tilde{\varphi}h^{\prime}\right)}{a(1-h)+v^{\prime\prime}}$
$\displaystyle\times{}_{2}F_{1}\\!\left(1,\frac{3}{2},\frac{5}{2};\frac{(a-1)(1-h)}{a(1-h)+v^{\prime\prime}}\right)\,.$
(5.8)

Again, we discuss separately the two choices for the function $h$.

Figure 5: Spike plots with $h=-2\tilde{\varphi}^{2}$ and with different values
of $a$: $a=0.5$ (blue), $a=0.4$ (purple), $a=0.16$ (red) and $a=0.07$
(orange). For each curve, the rightmost spike is the deformed Gaussian fixed
point, the leftmost one is the Wilson-Fisher fixed point. Decreasing $a$
further, both these fixed points move towards the origin. In the red and in
the orange curves one and two of the fake fixed points have disappeared
correspondingly.

#### First case: $\boldsymbol{h=\alpha\tilde{\varphi}^{2}}$.

Following Ref. [25] we start with a quadratically-field-dependent regulator.
However we slightly depart from that reference in that we find it more
convenient to portrait the landscape of fixed points by a different numerical
method, a shooting from the origin. This consists in constructing numerical
solutions for each possible value of the boundary condition
$v^{\prime\prime}(0)$. The generic solutions however terminate at a finite
value $\tilde{\varphi}_{S}$ of the field which corresponds to a movable
singularity of the fixed-point equation. In this process one obtains a plot of
$\tilde{\varphi}_{S}$ as a function of $v^{\prime\prime}(0)$ (also known as
spike plot). Sharp maxima in the latter variable are in one-to-one
correspondence with global fixed points. The result is presented in Fig. 5. We
can see that decreasing $a$ the spurious fixed points disappear and the
physical fixed points converge to the origin. This is the same phenomenon that
we observed in Secs. 3.1 and 3.2.

At these fixed points, we compute the spectrum of critical exponents with the
same method used in Ref. [25], namely by shooting from infinity, as we did in
Sec. 3.2. This means that we first construct an asymptotic expansion of the
fixed-point potential as well as of the eigenfunction of the linearized flow
around the fixed point. For $\alpha<0$ the Heaviside theta on the rhs of
equation (5.8) is equal to one, and the potential has the following behavior
at infinity

$\displaystyle v$ $\displaystyle=$ $\displaystyle
A\tilde{\varphi}^{6}+\frac{a|\alpha|^{5/2}}{150A}|\tilde{\varphi}|+\frac{a|\alpha|^{3/2}(525A-(3+2a)\alpha^{2})}{31500A^{2}|\tilde{\varphi}|}$
$\displaystyle+\frac{a\sqrt{-\alpha}\left(212625A^{2}-\alpha^{2}(3780aA+5670A)+\left(16a^{2}+24a+30\right)\alpha^{4}\right)}{17010000A^{3}|\tilde{\varphi}|^{3}}+O\left(|\tilde{\varphi}|^{-5}\right)\,.$

Shooting on $A$ and on a corresponding asymptotic parameter for the
perturbation, and by demanding $\mathbb{Z}_{2}$ parity at the origin, we
determine the location of the fixed point as well as the quantized values of
the critical exponents. In the $a\to 0$ limit the latter become independent of
$\alpha$ and agree with the spectrum discussed in Sec. 3.2.

For $\alpha>0$, because of the Heaviside theta one the rhs of Eq. (5.8)
$v=A\tilde{\varphi}^{6}$ for $\tilde{\varphi}>1/\sqrt{\alpha}$. Therefore for
$\tilde{\varphi}<1/\sqrt{\alpha}$ the potential as a function of
$\delta=\left(\frac{1}{\sqrt{\alpha}}-\tilde{\varphi}\right)^{1/2}$ has the
following asymptotic behavior

$v=\frac{A}{\alpha^{3}}-\frac{6A}{\alpha^{5/2}}\delta^{2}+\frac{15A}{\alpha^{2}}\delta^{4}+\frac{2\sqrt{2}\,a\,\alpha^{13/4}}{75A}\delta^{5}-\frac{135000A^{4}+a^{2}\alpha^{10}}{6750\alpha^{3/2}A^{3}}\delta^{6}+o\left(\delta^{7}\right)\
.$

Shooting from infinity and decreasing $a$ we recover the Gaussian and the WF
fixed points. In particular, for $\alpha=1/25$ the Gaussian fixed point
reappears for $a\lesssim 10^{-2}$, while for $\alpha=1/9$ the WF fixed point
reappears for $a\lesssim 0.35$ and the Gaussian one for $a\lesssim 4\cdot
10^{-3}$. Also in this case the critical exponents of the Gaussian and WF
fixed points approach the values found for vanishing $a$ in Sec. 3.2.

#### Second case: $\boldsymbol{h=\alpha v^{\prime\prime}}$.

We then move on to consider a regulator which depends on the second derivative
of the effective potential, through a constant $\alpha>0$. In this particular
case shooting from the origin is not convenient for technical reasons,
therefore we shoot from large field values.

This time $v=A\tilde{\varphi}^{6}$ for
$\tilde{\varphi}>\tilde{\varphi}_{c}\equiv(30A\alpha)^{-1/4}$ provided
$v^{\prime\prime}>1/\sqrt{\alpha}$. Below $\tilde{\varphi}_{c}$ the potential
can be expanded in $\delta=\tilde{\varphi}_{c}-\tilde{\varphi}$ as follows

$\displaystyle v=$ $\displaystyle\
A\left(30A\alpha\right)^{-3/2}-6A\left(30A\alpha\right)^{-5/4}\delta+\frac{1}{2\alpha^{2}}\delta^{2}+F(\delta)\,,$
$\displaystyle F=$ $\displaystyle\
\delta^{16/5}\left(-\frac{25\sqrt{5}A^{1/10}\alpha^{-17/10}}{88\sqrt{2}\,3^{3/10}a^{2/5}}+\frac{125\
5^{3/4}A^{-1/20}\alpha^{-53/20}}{5984\sqrt[4]{2}\,3^{17/20}a^{4/5}}\delta^{1/5}\right.$
$\displaystyle\left.-\frac{71875A^{-1/5}\alpha^{-18/5}}{246445056\
3^{2/5}a^{6/5}}\delta^{2/5}+o(\delta^{3/5})\right)\,.$

Shooting on $A$ and searching for values which correspond to a vanishing
$v^{\prime}(0)$ one can reveal several spurious fixed points at nonvanishing
$\alpha$ and $a$. More and more of them are generated from the Gaussian fixed
point for bigger and bigger values of $\alpha$. We find that decreasing $a$ at
fixed $\alpha>0$ reduces the number of spurious fixed points, and in the $a\to
0$ limit all of them disappear while the Gaussian and the WF fixed points
merge. We verify that also in this case the critical exponents tend to the
values obtained in Sec. 3.2 for $a\to 0$.

###  5.1 Ward Identity for the shift symmetry

Going beyond a single-field approximation, i.e. keeping both $\varphi$ and
$\phi_{B}$ as distinct, the LPA truncation becomes 161616The mixing term
$\partial_{\mu}\phi_{B}\partial^{\mu}\phi$ is ruled out by the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry on the arguments of the EAA.

$\Gamma_{k}[\varphi,\phi_{B}]=\int
d^{d}x\left(\frac{1}{2}\left(\partial_{\mu}\varphi\right)^{2}+\frac{1}{2}\left(\partial_{\mu}\phi_{B}\right)^{2}+V_{k}(\varphi,\phi_{B})\right)\
.$ (5.9)

Using the regulator
$R_{k}=ak^{2}\left(1-y-h(\tilde{\phi}_{B})\right)\theta\\!\left(1-y-h(\tilde{\phi}_{B})\right)$
the modified Ward identity (5.5) and the flow equation become

$\displaystyle\partial_{\tilde{\varphi}}v-\partial_{{\tilde{\phi}}_{B}}v=c_{d}\frac{h^{\prime}}{2}\,\frac{a(1-h)^{d/2}}{a(1-h)+\partial_{\tilde{\varphi}}^{2}v}\
{}_{2}F_{1}\\!\left(1,\frac{d}{2},\frac{d}{2}+1;-\frac{(1-a)(1-h))}{a(1-h)+\partial_{\tilde{\varphi}}^{2}v}\right),$
(5.10)
$\displaystyle\partial_{t}v+d\,v-\frac{(d-2)}{2}\left({\tilde{\varphi}}\partial_{\tilde{\varphi}}v+{\tilde{\phi}}_{B}\partial_{{\tilde{\phi}}_{B}}v\right)=$
(5.11) $\displaystyle
c_{d}\,\frac{a(1-h)^{d/2}}{a(1-h)+\partial_{\tilde{\varphi}}^{2}v}\left(1-h-\frac{1}{2}\partial_{t}h+\frac{1}{4}\left(d-2\right){\tilde{\varphi}}h^{\prime}\right)\
{}_{2}F_{1}\\!\left(1,\frac{d}{2},\frac{d}{2}+1;-\frac{(1-a)(1-h))}{a(1-h)+\partial_{\tilde{\varphi}}^{2}v}\right)\\!,$

where $c_{d}=\left((4\pi)^{d/2}\Gamma\left(\frac{d}{2}+1\right)\right)^{-1}$ .
We rescale all the quantities in the following way

$\displaystyle{\tilde{\varphi}}$
$\displaystyle=a^{(d-2)/4}\,\hat{\varphi}\,,\hskip 56.9055pt$
$\displaystyle{\tilde{\phi}}_{B}=a^{(d-2)/4}\,\hat{\phi}_{B}\,,$ (5.12)
$\displaystyle v(\tilde{\varphi})$
$\displaystyle=a^{d/2}\,\hat{v}(\hat{\varphi})+a\frac{1}{(4\pi)^{d/2}(d-2)\,\Gamma\left(\frac{d}{2}+1\right)}\,,$
$\displaystyle h=a^{\gamma}\,\hat{h}\ .$ (5.13)

This set of definitions agrees with the one in Eq. (3.20). Here $\gamma$
depends on the choice of $h$: for example $\gamma=1$ for both
$h=\alpha{\tilde{\phi}}_{B}^{2}$ and $h=\alpha v^{\prime\prime}$. For the sake
of generality we shall keep $\gamma$ free for the time being. Expanding for
small $a$ and assuming $2<d<4$, the Ward identity and the flow equation become

$\displaystyle\partial_{\hat{\varphi}}{\hat{v}}-\partial_{{\hat{\phi}}_{B}}{\hat{v}}=\frac{a^{\gamma+1-d/2}}{d(4\pi)^{d/2}\Gamma\left(\frac{d}{2}\right)}\,\hat{h}^{\prime}+\cdots\,,$
$\displaystyle\partial_{t}{\hat{v}}+d\,{\hat{v}}-\frac{(d-2)}{2}\left({\hat{\varphi}}\partial_{\hat{\varphi}}{\hat{v}}+{\hat{\phi}}_{B}\partial_{{\hat{\phi}}_{B}}{\hat{v}}\right)=$
$\displaystyle\hskip
28.45274pt\frac{\Gamma\left(\frac{d}{2}-1\right)}{(4\pi)^{d/2}}\,(1+\partial_{\hat{\varphi}}^{2}{\hat{v}})^{d/2-1}-\frac{a^{\gamma+1-d/2}}{d(4\pi)^{d/2}\Gamma\left(\frac{d}{2}\right)}\,\left(\partial_{t}\hat{h}+d\,\hat{h}-\frac{(d-2)}{2}{\hat{\phi}}_{B}\,\hat{h}^{\prime}\right)+\cdots\,,$

where the dots denote quantities that goes to zero for $a\to 0$ .
From the modified Ward identity we see that to have a well defined vanishing-
regulator limit we must demand $\gamma\geq\frac{d}{2}-1$. If
$\gamma>\frac{d}{2}-1$ ,
$\partial_{\hat{\varphi}}{\hat{v}}=\partial_{{\hat{\phi}}_{B}}{\hat{v}}$ :
this implies that
${\hat{v}}({\hat{\varphi}},{\hat{\phi}}_{B})={\hat{v}}({\hat{\varphi}}+{\hat{\phi}}_{B})$
and so we recover the shift symmetry and the flow equation without background.
If $\gamma=\frac{d}{2}-1$, the modified Ward identity gives

${\hat{v}}({\hat{\varphi}},{\hat{\phi}}_{B})={\hat{v}}_{s}({\hat{\varphi}}+{\hat{\phi}}_{B})-\frac{1}{d(4\pi)^{d/2}\Gamma\left(\frac{d}{2}\right)}\hat{h}({\hat{\phi}}_{B})\,.$
(5.14)

Inserting this result into the flow equation, we recover again the equation
without background.

##  6 Discussion

We have discussed the effect of an overall suppression of the regulator with a
constant factor $a$, and in particular the limit $a\to 0$, that we called the
limit of vanishing regulator. Let us summarize the main results.

As is clearly seen already in the case of the quantum mechanical oscillator,
decreasing $a$ has the effect of accelerating the flow, in the sense that
already a small decrease of $k$ leads very fast to the effective action.
Thereafter things remain nearly constant with $k$. However, the quantum-
mechanical study shows that in general different results are obtained
depending on whether the $a\to 0$ limit is performed on the solutions of the
flow equations, or on the beta functions themselves, see Fig. 2. While the
former way of taking the limit is rather straightforward, obtaining meaningful
results from the latter process requires a suitable $a$-dependent rescaling of
the couplings.

In the case of the Wilson-Fisher fixed point, we have first studied the first
form of the vanishing-regulator limit, by analysing the $a$ dependence of the
fixed-point solution. Decreasing $a$ has the effect of shifting the fixed
points towards the Gaussian one, but the scaling exponents remain distinct
even in the limit $a\to 0$. Here we have limited our analysis to the leading
order of the derivative expansion.

In a polynomial approximation of the potential, the values of the scaling
exponents become progressively worse as one increases the order of the
polynomial. This is in agreement with the statement in Ref. [16] that the
radius of convergence of the Taylor expansion of $V$ is proportional to $a$.
We have avoided this problem by also considering the functional treatment
(LPA), but in this case one gets the exponent $\nu=1$, which is worse than for
any polynomial and coincides with the upper boundary conjectured in [16].

We have then analyzed the second form of the vanishing-regulator limit, taking
it on the LPA beta functional of scalar field theory, finding agreement with
the first kind of limit as far as the critical exponents are concerned,
although the locations of the fixed point differ. Despite some pre-existing
arguments were suggesting that the limit of vanishing regulator might
generally reproduce the results of a constant (momentum independent) mass-like
regulator, we have observed that in the LPA this is the case only away from
even dimensions. In the special case of even integer $d$, singularities in the
momentum integrals lead to a more intricate structure. As a consequence, the
vanishing-regulator limit remains different from the constant regulator in
$d=4$. We expect this conclusion to hold also in higher even dimensions. It
remains to be seen whether these conclusions are robust against enlargements
of the truncation. For instance, at the second order of the derivative
expansion, there might be a nontrivial interplay between the momentum-
derivatives of the regulator and the $a\to 0$ limit, resulting in further
differences between the constant and the vanishing regulators.

For all these reasons, it will be quite interesting to systematically study
the next order of the derivative expansion, including a field-dependent wave
function renormalization $Z(\phi)$. In this paper, this level of approximation
has been analyzed only for the two-dimensional nonlinear sigma model, as in
this case it is the first nontrivial order of the derivative expansion. It is
also known that in the case of quantum critical points the convergence of this
expansion requires an increasingly accurate tuning of $a$. Therefore it might
be expected that the limit of vanishing regulators cannot be of much use when
one moves in that direction.

We should mention however, that the amplitude $a$ is only one of an infinite
series of free parameters within the regulator $R_{k}$. In this work we have
not allowed for such residual freedom, having fixed the regulator to a
piecewise linear form. Optimization criteria over the remaining parameters
might be essential to obtain accurate results in the vanishing-regulator
limit. It might also be possible to take advantage of these additional
parameters, with their associated free limiting behavior, to construct
alternative flow equations resulting from the vanishing-regulator limit. For
instance, in the so-called LPA′ truncation, this kind of additional freedom
allowed to construct a one-parameter family of $\overline{\text{MS}}\,$-like
schemes within the FRG [14].

In spite of the poor results of the $a\to 0$ limit of the LPA for the
benchmark case of the Wilson-Fisher fixed point, we think that this limit may
be useful in simple approximations, in problems where a symmetry is broken by
the regulator. As a first example we have discussed the $O(N+1)$-nonlinear
sigma model, in a formulation where the regulator breaks the global symmetry
to $O(N)$. In this case we have shown that in the limit of vanishing regulator
the beta functions converge to those of the $O(N+1)$-symmetric theory.

We have then considered the shift symmetry arising in the background field
treatment of a scalar theory. When this symmetry is broken by the regulator,
this can either generate unphysical fixed points or, what is worse, destroy a
physical fixed point. We have verified that the Ward identities of the shift
symmetry are restored in the limit of vanishing regulator, and that all the
unphysical features of the flow disappear when $a$ becomes sufficiently small.

The main motivation of this work was the hope that vanishing regulators, or
perhaps just “sufficiently small regulators”, may be useful in the application
of the FRG to gauge theories and gravity, where the background field method is
almost always used. Our results suggest that this may be possible, but that
the usefulness of this idea may be restricted to the simplest truncations.

Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 754496.

##  Appendix A Flow equations in the $\boldsymbol{O(\partial^{2})}$ derivative
expansion

We introduce the following notations

$\displaystyle G_{0}$ $\displaystyle=$
$\displaystyle\left(Z_{k}(\rho)q^{2}+R_{k}(q^{2})+U_{k}^{\prime}(\rho)\right)^{-1},$
(A.1a) $\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle\left(\tilde{Z}_{k}(\rho)q^{2}+R_{k}(q^{2})+U_{k}^{\prime}(\rho)+2\rho
U_{k}^{\prime\prime}(\rho)\right)^{-1},$ (A.1b) for the Goldstone-bosons and
radial-mode propagators.

The flow equations for $U_{k}$ and $\tilde{Z}_{k}$, which is defined in Eq.
(4.2), are

$\displaystyle\partial_{t}U_{k}$
$\displaystyle=\frac{1}{2(4\pi)^{d/2}}\left(Q_{\frac{d}{2}}\left[G_{1}\partial_{t}R_{k}\right]+(N-1)Q_{\frac{d}{2}}\left[G_{0}\partial_{t}R_{k}\right]\right),$
(A.2) $\displaystyle\partial_{t}\tilde{Z}_{k}$
$\displaystyle=-\frac{\left(\tilde{Z}_{k}^{\prime}+2\rho\tilde{Z}_{k}^{\prime\prime}\right)}{2(4\pi)^{d/2}}Q_{\frac{d}{2}}\left[G_{1}^{2}\partial_{t}R_{k}\right]-(N-1)\frac{\left(Z_{k}^{\prime}+\rho
Y_{k}^{\prime}\right)}{2(4\pi)^{d/2}}Q_{\frac{d}{2}}\left[G_{0}^{2}\partial_{t}R_{k}\right]$
$\displaystyle+\frac{2\rho\left(\tilde{Z}_{k}^{\prime}\right)^{2}}{(4\pi)^{d/2}}\left[\frac{2d+1}{2}Q_{\frac{d}{2}+1}\left[G_{1}^{3}\partial_{t}R_{k}\right]+\frac{(d+2)(d+4)}{4}\left(Q_{\frac{d}{2}+2}\left[G_{1}^{2}G_{1}^{\prime}\partial_{t}R_{k}\right]\right.\right.$
$\displaystyle\left.\left.+Q_{\frac{d}{2}+3}\left[G_{1}^{2}G_{1}^{\prime\prime}\partial_{t}R_{k}\right]\right)\right]+\frac{2\rho\left(3U_{k}^{\prime\prime}+2\rho
U_{k}^{\prime\prime\prime}\right)^{2}}{(4\pi)^{d/2}}\left(Q_{\frac{d}{2}}\left[G_{1}^{2}G_{1}^{\prime}\partial_{t}R_{k}\right]+Q_{\frac{d}{2}+1}\left[G_{1}^{2}G_{1}^{\prime\prime}\partial_{t}R_{k}\right]\right)$
$\displaystyle+\frac{2\rho\tilde{Z}_{k}^{\prime}\left(3U_{k}^{\prime\prime}+2\rho
U_{k}^{\prime\prime\prime}\right)}{(4\pi)^{d/2}}\left[\left(d+2\right)\left(Q_{\frac{d}{2}+1}\left[G_{1}^{2}G_{1}^{\prime}\partial_{t}R_{k}\right]+Q_{\frac{d}{2}+2}\left[G_{1}^{2}G_{1}^{\prime\prime}\partial_{t}R_{k}\right]\right)\right.$
$\displaystyle\left.+2Q_{\frac{d}{2}}\left[G_{1}^{3}\partial_{t}R_{k}\right]\right]+(N-1)\frac{\rho
Y_{k}}{(4\pi)^{d/2}}\left(2U^{\prime\prime}_{k}Q_{\frac{d}{2}}\left[G_{0}^{3}\partial_{t}R_{k}\right]+dZ^{\prime}_{k}Q_{\frac{d}{2}+1}\left[G_{0}^{3}\partial_{t}R_{k}\right]\right)$
$\displaystyle+(N-1)\frac{2\rho\left(Z_{k}^{\prime}\right)^{2}}{(4\pi)^{d/2}}\left[\frac{(d+2)(d+4)}{4}\left(Q_{\frac{d}{2}+2}\left[G_{0}^{2}G_{0}^{\prime}\partial_{t}R_{k}\right]\\!+\\!Q_{\frac{d}{2}+3}\left[G_{0}^{2}G_{0}^{\prime\prime}\partial_{t}R_{k}\right]\right)\right.$
$\displaystyle\left.+\frac{1}{2}Q_{\frac{d}{2}+1}\left[G_{0}^{3}\partial_{t}R_{k}\right]\right]+(N-1)\frac{2\rho\left(U_{k}^{\prime\prime}\right)^{2}}{(4\pi)^{d/2}}\left(Q_{\frac{d}{2}}\left[G_{0}^{2}G_{0}^{\prime}\partial_{t}R_{k}\right]+Q_{\frac{d}{2}+1}\left[G_{0}^{2}G_{0}^{\prime\prime}\partial_{t}R_{k}\right]\right)$
$\displaystyle+(N-1)\frac{2\rho
Z_{k}^{\prime}U_{k}^{\prime\prime}}{(4\pi)^{d/2}}\left(d+2\right)\left(Q_{\frac{d}{2}+1}\left[G_{0}^{2}G_{0}^{\prime}\partial_{t}R_{k}\right]+Q_{\frac{d}{2}+2}\left[G_{0}^{2}G_{0}^{\prime\prime}\partial_{t}R_{k}\right]\right).$
(A.3)

Using the regulator $R_{k}=aZ_{k}(k^{2}-z)\theta(k^{2}-z)$ 171717We defined
$Z_{k}=Z_{k}(\rho=0)$ . we have

$\displaystyle Q_{n}\left[G^{\ell}\partial_{t}R_{k}\right]=$ $\displaystyle\
\frac{1}{\Gamma(n)}\left((2-\eta_{k})k^{2}\,q_{n,\ell}\left(a,\omega,\zeta\right)+\eta_{k}\,q_{n+1,\ell}\left(a,\omega,\zeta\right)\right)\
,$ $\displaystyle Q_{n}\left[G^{\ell}G^{\prime}\partial_{t}R_{k}\right]=$
$\displaystyle\ -\frac{\left(\zeta-
aZ_{k}\right)}{\Gamma(n)}\left((2-\eta_{k})k^{2}\,q_{n,\ell+2}\left(a,\omega,\zeta\right)+\eta_{k}\,q_{n+1,\ell+2}\left(a,\omega,\zeta\right)\right)\
,$ $\displaystyle
Q_{n}\left[G^{\ell}G^{\prime\prime}\partial_{t}R_{k}\right]=$ $\displaystyle\
\frac{2\left(\zeta-
aZ_{k}\right)^{2}}{\Gamma(n)}\left((2-\eta_{k})k^{2}\,q_{n,\ell+3}\left(a,\omega,\zeta\right)+\eta_{k}\,q_{n+1,\ell+3}\left(a,\omega,\zeta\right)\right)$
$\displaystyle\
-\frac{a^{2}\,Z_{k}^{2}\,k^{2(n-\ell-2)}}{\Gamma(n)\left(\zeta+\omega/k^{2}\right)^{\ell+2}}\
,$

where

$\displaystyle G\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\left(\zeta
z+R_{k}+\omega\right)^{-1}\ ,$ $\displaystyle
q_{n,\ell}\left(a,\omega,\zeta\right)\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{1}{n}\frac{a\,Z_{k}\,k^{2(n-\ell)}}{\left(aZ_{k}+\omega/k^{2}\right)^{\ell}}\,\,_{2}F_{1}\\!\left(\ell,n,n+1;\frac{aZ_{k}-\zeta}{aZ_{k}+\omega/k^{2}}\right)\
,$

and $G$ can be $G_{0}$ or $G_{1}$ depending on the choice of $\zeta$ and
$\omega$, in particular

$\displaystyle G=G_{0}\hskip 28.45274pt\mbox{if}\hskip
28.45274pt\begin{cases}\omega=U_{k}^{\prime}(\rho)\ ,\\\ \zeta=Z_{k}(\rho)\
,\end{cases}$ (A.4) $\displaystyle G=G_{1}\hskip 28.45274pt\mbox{if}\hskip
28.45274pt\begin{cases}\omega=U_{k}^{\prime}(\rho)+2\rho
U_{k}^{\prime\prime}(\rho)\ ,\\\ \zeta=\tilde{Z}_{k}(\rho)\ .\end{cases}$
(A.5)

For the constant regulator (2.11) one finds

$\displaystyle Q_{n}\left[G^{\ell}\partial_{t}R_{k}\right]\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!k^{2(n-\ell+1)}Z_{k}(2-\eta_{k})\frac{\Gamma(\ell-n)}{\Gamma(n)\Gamma(\ell)}\zeta^{-n}\left(Z_{k}+\omega/k^{2}\right)^{n-\ell}\
,$ $\displaystyle
Q_{n}\left[G^{\ell}G^{\prime}\partial_{t}R_{k}\right]\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!-\zeta\,Q_{n}\left[G^{\ell+2}\partial_{t}R_{k}\right]\
,$ $\displaystyle
Q_{n}\left[G^{\ell}G^{\prime\prime}\partial_{t}R_{k}\right]\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!2\zeta^{2}\,Q_{n}\left[G^{\ell+3}\partial_{t}R_{k}\right]\
.$

##  Appendix B Master equation for the nonlinear $\boldsymbol{O(N\\!+\\!1)}$
model

In Sec. 4 the use of a vanishing regulator for the two dimensional nonlinear
$O(N+1)$ model has been discussed. We have observed that a nonvanishing
potential term of the form (4.6) is not preserved by the flow equation in the
$a\to 0$ limit. In this section we provide more details about the contraints
on a general local potential $U_{k}(\rho,H)$. Here we show how the non-
compatibility of the ansatz (4.6) with the flow equation is encoded in the
modified master equation for the $O(N+1)/O(N)$ symmetry.

Our starting point is indeed the following modified master equation

$\frac{\delta\Gamma_{k}}{\delta\phi^{a}}\frac{\delta\Gamma_{k}}{\delta
H}+H\,\phi_{a}=\mathrm{Tr}\left\\{\left[R_{k}\left(\Gamma_{k}^{(2)}+R_{k}\right)^{-1}\right]_{ab}\frac{\delta^{2}\Gamma_{k}}{\delta
H\delta\phi^{b}}\right\\}.$ (B.1)

This identity, which differs from the standard master equation for a
nonvanishing rhs, can be derived for instance from a functional integral
representation of $\Gamma_{k}$, in presence of a linear source term of the
form (4.6) in the bare action, by performing a change of integration variable
corresponding to a $O(N+1)/O(N)$ infinitesimal transformation. It is
straightforward to prove that this functional identity is compatible with the
exact RG flow equation [30], meaning that it defines an RG-invariant
hypersurface in theory space. However, truncations of the theory space often
spoil this property, such that the truncated master equation becomes an
additional requirement on the RG flow, to be enforced at every $k$.

Whenever the regularization preserves the (unmodified) nonlinear $O(N+1)/O(N)$
symmetry, the one-loop regulator-dependent term on the rhs of Eq. (B.1)
vanishes identically. The modified master equation then reduces to the
standard master equation, which is a tree-level identity. In this case the
equation is straightforward to solve, independently from and prior to the
analysis of the RG flow equation. For an introduction to the role played by
this identity in the construction of a renormalized perturbation theory in two
dimensions see for instance Ref. [31]. Before analysing in details the shape
that this constraint takes for vanishing $a$, one can already apply its form
of Eq. (B.1) to the truncation we assumed in Sec. 4. There we took $U=0$ and
$H=0$. This combination trivially fulfills the modified master equation. It
should however be noted that Eq. (B.1) represents the constraint of nonlinear
$O(N+1)/O(N)$ symmetry only in the theory space of generic functionals of
$\phi^{a}$ and $H$. If a nonvanishing $H$ is never introduced in the effective
action, i.e. on the subspace where $H=0$, there nevertheless is a functional
constraint encoding the nonlinear $O(N+1)/O(N)$ symmetry, and it can be
obtained from Eq. (B.1) by replacing derivatives involving $H$ with the
expectation values of the corresponding composite operators. The analysis of
this kind of modified master equation is therefore highly nontrivial and will
not be addressed in this work.

We then address the constraints that Eq. (B.1) imposes on a truncation similar
to the one in Eq. (4.4), but with an arbitrary nonvanishing $U_{k}(\rho,H)$.
181818This general ansatz can be made compatible with the assumed linear $H$
dependence of the bare action, by requiring the linearity of the potential at
the UV cutoff scale $k=\Lambda$. For this truncation Eq. (B.1) becomes

$\partial_{\rho}U_{k}\,\partial_{H}U_{k}+H=\partial_{\rho}\partial_{H}U_{k}\frac{1}{4\pi}\int_{0}^{\infty}dz\frac{R_{k}(z)}{\tilde{Z}_{k}\,z+R_{k}(z)+\partial_{\rho}U_{k}+2\rho\partial_{\rho}^{2}U_{k}}\
.$ (B.2)

For the Litim regulator the loop integral is readily evaluated leading to

$\displaystyle
4\pi\frac{\partial_{\rho}U_{k}\,\partial_{H}U_{k}+H}{\partial_{\rho}\partial_{H}U_{k}}=$
$\displaystyle-\frac{ak^{2}}{a-\frac{g_{k}^{2}}{Z_{k}}\tilde{Z}_{k}}-\frac{a}{\left(a-\frac{g_{k}^{2}}{Z_{k}}\tilde{Z}_{k}\right)^{2}}\frac{g_{k}^{2}}{Z_{k}}\left(\tilde{Z}_{k}k^{2}+\partial_{\rho}U_{k}+2\rho\partial_{\rho}^{2}U_{k}\right)$
$\displaystyle\times\log\\!\left(\frac{\tilde{Z}_{k}k^{2}+\partial_{\rho}U_{k}+2\rho\partial_{\rho}^{2}U_{k}}{a\frac{Z_{k}}{g_{k}^{2}}k^{2}+\partial_{\rho}U_{k}+2\rho\partial_{\rho}^{2}U_{k}}\right).$
(B.3)

The loop contribution to the modified master equation is a nonlinear function
of derivatives of $U_{k}$ up to second order. Therefore solving this equation
for $U_{k}$ is a difficult task. Even more so, as this solution must be
required to also obey the RG flow equation. As the LPA projection breaks
compatibility of the modified master equation with the RG flow equation, the
latter is an independent nonlinear second order partial differential equation
for $U_{k}$. This illustrates the difficulty of dealing with modified Ward
identities in the FRG framework. For a discussion of these issues in the
context of gauge theories, see for instance [32, 33]

Can the limit $a\to 0$ be of any help in solving this complex problem? In
addressing this question we need to specify the behavior of the functions
$U_{k}$ and $\tilde{Z}_{k}$ for $a\to 0$. For definiteness, we assume the
scaling

$\partial_{\rho}U\sim a\ ,\quad\quad\tilde{Z}_{k}\sim a^{0}\
,\quad\quad\phi\sim a^{0}\ .\\\ $ (B.4)

Considering then Eq. (B.3), it is natural to assume

$H\sim a\ ,$ (B.5)

which allows the linear source term to be interpreted as being part of the
potential. However, inspection of the rhs of Eq. (B.3) reveals that the
leading behavior of the one-loop contribution is in fact $a\log a$. As a
consequence we provide an ansatz encoded in the following definitions

$\displaystyle H$ $\displaystyle=a\hat{H}\ ,$ (B.6) $\displaystyle
U_{k}(\rho,H)$ $\displaystyle=a\,\hat{U}_{k}(\rho,\hat{H})-a\log
a\,F_{0}(\rho)-a\log(-\log a)\,F_{1}(\rho)\ .$ (B.7)

Notice that we choose an ansatz with $F_{0}$ and $F_{1}$ independent of $H$.
This might lead us to a particular solution of the modified master equation.
The modified master equation then can be projected on three distinct
equations, each showing a different small $a$ asymptotic behavior. The
$O(a\log(a))$, $O(a\log(-\log a))$ and $O(a)$ terms in this equation
respectively lead to

$\displaystyle F_{0}^{\prime}(\rho)\hat{U}^{(0,1)}(\rho,H)$
$\displaystyle=-\frac{Z_{k}k^{2}}{4\pi
g^{2}_{k}}\frac{\hat{U}^{(1,1)}(\rho,H)}{\tilde{Z}_{k}(\rho)}\ ,$ (B.8)
$\displaystyle F_{1}^{\prime}(\rho)\hat{U}^{(0,1)}(\rho,H)$
$\displaystyle=-\frac{Z_{k}k^{2}}{4\pi
g^{2}_{k}}\frac{\hat{U}^{(1,1)}(\rho,H)}{\tilde{Z}_{k}(\rho)}\ ,$ (B.9)
$\displaystyle H+\hat{U}^{(0,1)}(\rho,H)\hat{U}^{(1,0)}(\rho,H)$
$\displaystyle=-\frac{Z_{k}k^{2}}{4\pi
g^{2}_{k}}\frac{\hat{U}^{(1,1)}(\rho,H)}{\tilde{Z}_{k}(\rho)}\left(\log\bigg{(}\frac{\tilde{Z}_{k}k^{2}}{F_{0}^{\prime}(\rho)+2\rho
F_{0}^{\prime\prime}(\rho)}\bigg{)}-1\right)\ .$ (B.10)

It is evident how the $a\to 0$ limit does not relief the nonlinearity of the
modified master equation. While the first two equations can be
straightforwardly solved for $F_{0}$ and $F_{1}$, once $U(\rho,H)$ is known,
the third equation is highly nontrivial. In fact, Eq. (B.8) can be replaced
inside Eq. (B.10) to obtain a second order nonlinear partial differential
equation for $\hat{U}$. While the construction of the most general solution is
a very complex task, which we expect in general to be possible only by
numerical methods, a particular solution can be found by assuming the ansatz

$\hat{U}(\rho,H)=\pm\hat{H}\sqrt{\frac{1}{Z_{k}}-2\rho}+F_{2}(\rho)\ .$ (B.11)

This leads to a first order ordinary differential equation for $F_{2}$ which
can be easily solved. The determination of the corresponding $F_{0}$ and
$F_{1}$ results in the following particular solution

$\displaystyle F_{0,1}(\rho)$
$\displaystyle=c_{0,1}+\frac{Z_{k}k^{2}\rho}{4\pi}\ ,$ (B.12) $\displaystyle
F_{2}(\rho)$
$\displaystyle=c_{2}-\frac{Z_{k}k^{2}\rho}{4\pi}\log\\!\left(\frac{g^{2}}{4\pi}\right)+\frac{k^{2}}{8\pi}(1-2Z_{k}\rho)\log(1-2Z_{k}\rho)\
,$ (B.13)

where $c_{0,1,2}$ are integration constants which can depend on $k$. Having an
analytic formula for a particular solution of the master equation, is of
course a nice result, which is possible only thanks to the simplifications
brought by the $a\to 0$ limit. However, in itself this result is of limited
use, for two main reasons. First, in general there is no reason to expect that
this anstaz be closed under the RG flow. Given the compatibility of the flow
equation with the master equation, any particular solution is free to flow
into the most general solution during an infinitesimal RG step. Second, in the
LPA case even this compatibility is lost. As a consequance, the solution of
Eqs. (B.7), (B.12), and (B.13) will flow into a potential which does not
fulfill the modified master equation. Therefore, this solution would be useful
only if accompanied by a prescription for projecting the latter potential back
onto a functional of the same form as the particular solution itself. We do
not explore possible prescriptions for this projection in this work.

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