Limit of vanishing regulator in the functional renormalization group

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 $a\to0$ 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.

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 → 0 with k fixed. This limit is closely related to the pseudo-regulator that reproduces the beta functions of the 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.

I. INTRODUCTION
The Functional Renormalization Group (FRG) [1][2][3][4] is a powerful tool to study quantum and statistical field theories 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/statistical fluctuations have been integrated out. The functional that provides this interpolation is called the Effective Average Action (EAA) and is denoted Γ k [φ], where φ are the fields, k is a coarse-graining scale and Γ 0 = Γ 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 leading to the functional differential equation 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, ) that includes (for = 0) a popular class of regulators used in the FRG literature.
On the other hand, taking the limit a → 0 and → 0 (in this order), it reproduces the beta functions of MS .
In this paper we shall discuss what happens when the limits are taken in the opposite order (see Fig. 1). * abaldazz@sissa.it † percacci@sissa.it ‡ luca.zambelli@bo.infn.it We next discuss the motivation for this study. The notation Γ k [φ] emphasizes the important dependence of this functional on the scale k, but Γ k also depends on the shape of the cutoff function R k . The notation Γ[φ, R k ] would thus be more appropriate, and (I.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 approxi-mations, even physical observables will exhibit some spurious dependence on the shape of the cutoff. We will refer to this as "cutoff dependence". 1 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][16][17][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. 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.
To be more precise, the classical symmetry of the bare action is translated into a "quantum" symmetry of the EAA, which is deformed by the presence of the regulator. The latter symmetry is only implicitly determined, as the corresponding regulator-dependent Ward identity cannot in general be analytically and exactly solved [21] . 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) = k 2 ar 1 (y) (I. 3) 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. 2 In many applications it is convenient to choose a shape function r 1 depending on some of the parameters appearing in the ansatz adopted for the EAA. The most common example is the insertion of an overall wave-function-renormalization factor Z k . In this paper we shall mainly neglect these subtleties, as in most of our studies we will truncate the effective action to a scale-dependent local effective potential, and will be concerned with the limit a → 0, which we call the limit of vanishing cutoff. 3 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-handside (rhs) of the exact FRG equation vanishes, but we shall see that some important physical information remains available even in this limit. Even though many of the challenges and properties of the vanishing-regulator limit can be expected to characterize large families of shape functions r 1 , in this paper we mainly focus on the following regulator choice as in several interesting cases it is hardly feasible to study the vanishing regulator limit without having first specified a shape function. The reasons for this are explained in Sec. III C and further discussed in Sec. VI. 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. II we consider the harmonic and anharmonic oscillator. Some of the features of vanishing regulators appear already in these cases. In Sec. III we deal with the Z 2 -invariant scalar field theory in d ≥ 2 Euclidean dimensions, and its RG fixed point (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 (after this limit is taken and among all possible polynomial truncations of the potential) for the correlation-length critical exponent ν 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 (I.4) 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. IV 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. V 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 fluctuation fields. Also this symmetry is seen to be restored in the limit of vanishing regulators. We conclude in Sec. VI with a brief discussion of our results and some outlooks. Some auxiliary formulas and analyses are provided in two appendices.

II. 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 In the Local Potential Approximation (LPA), which is the first term in a derivative expansion, the EAA is approximated by Using the regulator in Eq. (I.4) we get the following flow equation for the potential 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 [22], although in most applications it is dropped since it only affects the ground-state energy. 4 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 → ∞ 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/statistical mechanical models [23,24]. Expanding the potential into a Taylor series The correct counterpart of this term in applications to gravity, especially within the background-field formulation, is still uncertain.
the beta functions are

A. Harmonic oscillator
We start by addressing the computation of the vacuum energy of the harmonic oscillator. When λ k = 0, Eq. (II.5b) shows that ω k is independent of k, thus we shall simply write ω k = ω. The solution of (II.5a) is This function is plotted in Fig. 2 for various values of a. First of all we see that E 0 = ω/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 → +∞ and k → 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 (I.4), this happens when Thus decreasing a has the same effect as decreasing k eff . From this discussion, there seems to be no issue with the limit a → 0. Some subtlety appears however when we try to construct the RG trajectory from the a → 0 limit of Eq. (II.5a), which reads This way of taking the limits is of course nonsensical, as the resulting beta functions would be identically vanishing. The ω 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 and then take the a → 0 limit of Eq. (II.5a). This leads to the finite result which is the same flow equation one would find with a constant (often called a "Callan-Symanzik") regulator The latter leads to the flow which is plotted as the dashed curve in Fig. 2.
The rescaling (II.9) is formulated as an "active" transformation of the couplings E k and ω k throught a factor √ a, while the RG coordinate k stays independent of a. Within the flow equations of the dimensionless couplingsẼ k ≡ E k /k andω 2 , it is also possible to reinterpret it as a "passive" transformation, in which the dimensionful couplings E k and ω k are independent of a, while the RG coordinate changes by k =k/ √ a. This second interpretation is consistent with the observation that k eff →k for a → 0 according to Eq. (II.7).

B. Anharmonic oscillator
Next, we turn to the anharmonic oscillator with λ = 0. As our interest is in comparing different possible prescriptions for taking the a → 0 limit with the available solutions for some interesting quantities, we do not address the numerical analyses needed to compute the energy levels, finding analytical expressions more instructive. We therefore consider only the first order of the expansion in λ. We see from (II.5c) that at this order the beta function of λ k is zero. Therefore λ k = λ at all scales. Expanding the vacuum energy parameter to first order in λ and solving the flow equation with the initial condition that lim k→∞ Γ k = S is the bare action (II.1), we find that the first order correction to the energy is (II.14) On the other hand for the frequency we find to first order in λ The quantities dE k dλ λ=0 and ω k given in (II.14-II. 15) are the solutions of the flow equations at arbitrary k. They interpolate between the initial conditions lim k→∞ dE k dλ λ=0 = 0 and lim k→∞ ω k = ω and the corresponding parameters in the EA at k = 0.
Also in this case, it is not possible to directly take the limit a → 0 in the flow equations, because then the beta functions simply vanish. However, having solved the flow equations one can take the limits a → 0 and k → 0 in any order obtaining (II.16b) One can take the limit a → 0 in the flow equations provided the potential is rescaled to: Expanding around a = 0 the flow equation becomes 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. (II.4), we deduce the beta functions Solving these equations one reobtaines (II.16).

III. THE ISING UNIVERSALITY CLASS
In this section we shall consider the theory of a single, Z 2 -invariant scalar field φ in the LPA While the FRG equation allows us to treat the potential as a whole, it will be instructive to further expand 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 β 2n = k ∂λ2n ∂k . For arbitrary regulator, and in any dimension, for the first few couplings this leads to are momentum integrals (z = q 2 ). These integrals can be evaluated in closed forms by using the optimized regulator (I.4). The Q functionals are then given by hypergeometric functions Q n ∂ t R (P + m 2 ) = 2ak 2(n+1− ) Γ(n + 1)(a +m 2 ) × 2 F 1 , n, 1 + n, a − 1 a +m 2 . In particular, in the massless case and in the limit of vanishing regulator we obtain (III.6) Clearly, the beta functions will not be finite. 5 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 that, in the limit a → 0 and → 0 (in this order) reproduces the results of dimensional regularization. In the aplane 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 and try to take the limit a → 0 along the path = 0 (red, dashed) in Fig. 1.
Returning to the beta functions (III.3), we note that if we set λ 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.

A. 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): Defining the dimensionless variables the beta functions arẽ (III.9b) Expanding inλ 2 The WF fixed point is now located at If we expand the critical couplings around a = 0 Thus the WF fixed point merges with the Gaussian fixed point. Note that sinceλ 2 =m 2 is linear in a for a → 0 at the WF fixed point, the Q functional (III.5) does not go to zero and this entails that the quantum/statistical contribution to the critical exponents will be nontrivial for a → 0.
Indeed, the position of the fixed point is not physically significant. If we consider the stability matrix at the nontrivial fixed point we see that the component (1, 2) of M goes to zero for a → 0 and so the stability matrix becomes diagonal. The eigenvalues of M , that is minus the critical exponents θ i , are actually independent of a, in particular ν = (θ 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 → 0 and a different critical exponent ν. In fact, the numerical value is not very bad, considering the drastic approximation.

B. The Wilson-Fisher fixed point in the LPA
Let us now treat the potential as a whole [25]. Inserting (III.1) in the FRGE we obtain the "beta functional" (III.14) Using the regulator (I.4), setting d = 3 and rescaling the beta function of the dimensionless potential v becomes (III.16) We look for even scaling solutions shooting from the origin with initial condition v (0) and v (0) = 0. There are only two values of v (0) which can be identified as fixed-point solutions: v (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 (III.17) The free parameter A can be fixed as function of a by requiring Z 2 symmetry for vanishing field [26]. We find that in the limit a → 0, A tends to A ≈ 0.0015. 6 The scaling exponents θ 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 ν 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 ≈ 1, while in the limit of vanishing regulator ν appears to approach ν = 1. Besides the correlation-length exponent ν = (θ 1 ) −1 , we also find positive eigenvalues, as reported in Tab. I.
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 [28,29]. 6 The asymptotic parameter is A = 0.001 for a = 1 and it increases monotonically for a → 0. regulator, as well as the conformal-bootstrap value [27]. This figure extends Fig. 12 in Ref. [16] to low values of a. Table I. The first few critical exponents at the Wilson-Fisher fixed, point computed in the local potential approximation for the regulator (I.4). We report the most common choice a = 1 and the limiting case of the vanishing regulator.

C. Vanishing regulators and constant regulators
At this point it is relevant to recall that the critical exponent ν = 1 is known to result also from the LPA equations for a constant regulator (II.11) [16]. Together with the findings of Sec. II, 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 → 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. II. Suppose that in addition to the parameter a we also introduce a parameter b rescaling the cutoff k R k (z) = a bk 2 − z θ bk 2 − z .
(III. 18) This rescaling can be motivated as follows. First of all, it should not change the scaling solutions. Furthermore, as discussed in Sec. II A, we can define an "ef-fective" cutoff scale k 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: (III.19) Now we see that in the limit a → 0, the regulator becomes a constant as in Eq. (II.11). The latter leads to the dimensionless flow equation This argument can actually be easily generalized to arbitrary shape functions r 1 , as defined in Eq. (I.3). We first include the parameter b in the regulator, to account for the possibility to rescale k R k (z) = bk 2 a r 1 (y/b). (III.22) Then we choose b = 1/a such that the regulator becomes Then the a → 0 limit of Eq. (III. 22), results in the constant regulator. 7 An alternative way of arguing that the a → 0 limit reduces the LPA flow equation for the regulator (I.4) to the constant regulator case (III.20) is by performing an a-dependent rescaling as in Sec. II. Namely, by redefining in the flow equations for the regulator (I.4) and then taking the a → 0 limit at fixedφ andv, we again find 7 By comparing this with the original regulator in Eq. (I. 3) we see that we have effectively cast the regulator as a function of k 2 eff = ak 2 , rather then of k itself, and then considered k eff as a independent.
Eq. (III. 20). For instance in d = 3 this rescaling entails that the prefactor a in (III.16) goes away. Both kind of arguments however are applicable only for non-exceptional d. In particular, in some cases removing the momentum dependence of the regulator by sending a → 0, as in Eq. (III.19) and Eq. (III.23), is not possible, because the a → 0 limit and the momentum integral cannot be exchanged. This happens whenever the integral corresponding to the constant regulator is divergent. In fact, the momentum integral leading to Eq. (III.20) is convergent only for d < 2. 8 If in the scalar LPA we adopt the constant regulator in d ≥ 2, using analytic continuation as a tool for the definition of the momentum integral 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 → 0 with the regulator (I.4), and expand the Q functionals (III.5), with n = d/2, d even andm = 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. As a matter of fact, if analytic continuation is not adopted in the definition of the loop integrals, the arguments we just outlined point to the conclusion that the vanishing-regulator limit does not need to reproduce the constant-regulator case in the whole range d ≥ 4.

D. Beta functions in two and four 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 Defining v(φ) = av(φ) (III.26) 8 However the UV divergence in 2 ≤ d < 4 affects only the fieldindependent part of the effective potential and in these cases it could be removed by implementing the standard subtraction as in Sec. II, see e.g. Eq. (II.18). Notice that this subtraction would introduce an IR divergence in d = 2. For some values of d the limit a → 0 cannot be taken at the level of the integrands. In these cases we have first of all to compute the integrals, and this requires to specify the shape function r 1 . This is the main reason why in the present paper we focus on the special regulator choice of Eq. (I.4).
More general results holding for arbitrary shape functions can be deduced once the field-theory model, the number of Euclidean dimensions d and the truncation of the EAA is specified. and simplifying a factor a from the flow equation, in the a → 0 limit we are left with The potential must be shifted by a factor that contains log a, i.e.v →v − 1 8π log a , in order to eliminate this divergent term for the limit a → 0 . We observe that the coefficient of the log a term matches exactly the coefficient of the 1/ pole of the expansion of (III.20) for d = 2 + . 9 The finite logarithmic contribution coincides with the one in Eq. (III.20). 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. (III.2). For continuity with the previous sections, we also turn to the dimensionless couplings defined in Eq. (III.8). By considering the leading contributions to the beta functionsβ 2n for vanishing a, we construct an ansatz based on the following scalinĝ λ 2n = a n−2 (log a) nλ 2n , n > 2 . (III.28c) Assuming theλ 2n couplings can be kept fixed in the a → 0 limit, results in the following set of beta functions (III.29c) and similar results for higher couplings. Notice that terms of order (log a) −1 could be neglected as subleading 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 (III.28) in a rescaling of the effective potential. It is impossible to achieve this goal by a twoparameters rescaling of the kind studied in the previous sections. However, Eq. (III.28c) 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(φ).
First, to simplify notations, it is convenient to definẽ Next, we define .  The peculiar simplicity which the Eqs. (III.29) attain in the a → 0 limit, together with the 1/ log a dependence of Eq. (III.29b), 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. (III.29), the φ 4 -theory beta function and other universal physics is still present. For instance, we can study the WF fixed point in d = 4 − . In order to employ Eq. (III.29) in this study, we need to prescribe that the → 0 limit be taken before the a → 0 one. This means in practice that the vanishing-regulator limit is taken on the d = 4 FRG equations. Had we sent a → 0 in d < 4, we would have found different equations forλ 2n and precisely the constant-regulator ones, as already mentioned in Sec. III C.
Within the simplest truncation corresponding to retaining onlyλ 2 andλ 4 , where we add the classical scaling term − λ 4 toβ 4 to account for the shift of dimensionality, the WF fixed point to first order in is located These fixed-point couplings have to be interpreted as small, even if they seemingly blow up for a → 0, because the limit → 0 has to be taken first. Notice that keeping the sub-leading order-(log a) −1 contribution toβ 4 is essential for revealing the fixed point. By computing the corresponding critical exponents, we find the universal one-loop result 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-∂ 2 expansion of Γ k for a O(N )-invariant multiplet of scalars where the N fields φ a are in the fundamental representation of O(N ), and ρ = φ a φ a /2 is the corresponding local invariant. We further define the radial wave function renormalizatioñ The beta functions for Z k ,Z k and U k are given in App. A. If we make the assumptions (IV.4) The tensor in parentheses is the metric of the Ndimensional sphere of radius Z −1/2 k , written in a co-ordinate 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 φ a φ a < 1/Z k . The symmetry group is extended to O(N + 1).
A standard cutoff Therefore, if we start at some scale k with an EAA of the form (IV.4), the flow will immediately generate O(N + 1)-violating terms, and thus will take place in the larger theory space parameterized by (IV.1). This can be seen already by projecting the flow generated by the ansatz (IV.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 → 0 limit it indeed becomes a consistent ansatz, as in ∂ t U k the rhs behaves like a log a when a → 0.
Let us then inspect the flow of the wave function renormalizations. Inserting the previous ansatz in the flow equation (A.3) forZ k (ρ), 10 in the limit a → 0 we obtain (IV.5) As the functional ρ dependence on both sides of the equation is comparable, this equation can be algebraically solved for ∂ t g k and η k , resulting in (IV.6b) 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 (ρ). Thus, within the present truncation the nonlinearly realized O(N + 1)/O(N ) symmetry is preserved by taking the limit a → 0. Essentially the same flow equations have been obtained in Ref. [14] 10 Note that now Z k (ρ = 0) = Z k /g 2 k , so inside the formulae for the Q functionals we must send Z k → Z k /g 2 k and η k → η k + 2∂tg k /g k . with a pseudo-regulator reproducing the MS scheme. 11 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 φ a , i.e. φ N +1 This ansatz, comprehending an arbitrary source H, was observed to be compatible with the flow equation in the case of an 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 (IV.7) is not compatible with the flow equation of the potential, neither for a = 0 nor in the a → 0 limit. Only by assuming that H be a function of a vanshing faster than a itself, closure of the O(∂ 2 ) RG flow on the ansatz (IV.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 → 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 → 0 limit is of limited use.

V. BACKGROUND FIELD ISSUES
When one splits the field into a classical background and a quantum/statistical fluctuation the action, being a function of φ, is invariant under the shift symmetry (V.2b) 11 The present result is obtained by setting σ = 1 in the beta functions of Ref. [14].
This can be expressed by the identity On the other hand, the regulator only depends on the background 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 (−D 2 ). This effect can be mimicked in the scalar case by artificially introducing a dependence of R k on φ B . For example, Morris and collaborators considered regulators of the general form [26] it has a separate dependence on these two arguments. The breaking of the shift symmetry results in a modified Ward identity (V.5) It has been shown that such background dependence in the regulator can either destroy physical fixed points or create artificial ones [26]. On the other hand, when the FRG equation (I.2) is solved together with the Ward identity (V.5), the correct physical picture can be reconstructed. While this can be achieved in the scalar case [26], it is much harder in the case of gauge theories, and in particular for gravity [30]. It is therefore desirable to find other ways around this obstacle. The form of the equation (V.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. [26]. We consider the same system as in Sec. III B, in d = 3, but we use the regulator (V.4). In a singlefield approximation one identifies φ B = ϕ. The corresponding flow equation for the potential reads , v = Aφ 6 for ϕ > 1/ √ α. The Gaussian fixed point is always absent, and for α > 0.08 also the WF fixed point disappears.
The second case is h = αv . The Gaussian 12 and the WF fixed points always exist, but when α is increased, new fixed points appear near the Gaussian one 13 and move away from it as α becomes bigger: for example, for α = 1 there is a spurious fixed point and for α = 2 there are three of them.
In Ref. [26] 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 (V.4) (V.7) Within a single-field LPA truncation this leads to the flow equation Again, we discuss separately the two choices for the function h. a. First case: h = αφ 2 . Following Ref. [26] 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 (0). The generic solutions however terminate at a finite valueφ S of the field which corresponds to a movable singularity of the fixed-point equation. In this process one obtains a plot ofφ S as a function of v (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. III A and III B. 12 Note that the Gaussian fixed point corresponds to the point (v(0), v (0)) = (1/3, 0) . 13 In particular for α ≥ 0.85 a first additional fixed point appears.
At these fixed points, we compute the spectrum of critical exponents with the same method used in Ref. [26], namely by shooting from infinity, as we did in Sec. III B. 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 α < 0 the Heaviside theta on the rhs of equation (V.8) is equal to one, and the potential has the following behavior at infinity Shooting on A and on a corresponding asymptotic parameter for the perturbation, and by demanding 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 → 0 limit the latter become independent of α and agree with the spectrum discussed in Sec. III B.
For α > 0, because of the Heaviside theta one the rhs of Eq. (V.8) v = Aφ 6 forφ > 1/ √ α. Therefore forφ < 1/ √ α the potential as a function of has the following asymptotic behavior Shooting from infinity and decreasing a we recover the Gaussian and the WF fixed points. In particular, for α = 1/25 the Gaussian fixed point reappears for a 10 −2 , while for α = 1/9 the WF fixed point reappears for a 0.35 and the Gaussian one for a 4 · 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. III B.
b. Second case: h = αv . We then move on to consider a regulator which depends on the second derivative of the effective potential, through a constant α > 0. In this particular case shooting from the origin is not convenient for technical reasons, therefore we shoot from large field values. Shooting on A and searching for values which correspond to a vanishing v (0) one can reveal several spurious fixed points at nonvanishing α and a. More and more of them are generated from the Gaussian fixed point for bigger and bigger values of α. We find that decreasing a at fixed α > 0 reduces the number of spurious fixed points, and in the a → 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. III B for a → 0.

A. Ward Identity for the shift symmetry
Going beyond a single-field approximation, i.e. keeping both ϕ and φ B as distinct, the LPA truncation becomes 14 Using the regulator (V.7) the modified Ward identity (V.5) and the flow equation become We rescale all the quantities in the following waỹ This set of definitions agrees with the one in Eq. (III.24). Here γ depends on the choice of h: for example γ = 1 for both h = αφ 2 B and h = αv . For the sake of generality we shall keep γ free for the time being. Expanding for small a and assuming 2 < d < 4, the Ward identity and the flow equation become where the dots denote quantities that go to zero for a → 0 . From the modified Ward identity we see that to have a well defined vanishing-regulator limit we must demand γ ≥ d 2 − 1. If γ > d 2 − 1 , ∂φv = ∂φ Bv : this implies thatv(φ,φ B ) =v(φ +φ B ) and so we recover the shift symmetry and the flow equation without background.
Inserting this result into the flow equation, we recover again the equation without background.

VI. DISCUSSION
We have discussed the effect of an overall suppression of the regulator with a constant factor a, and in particular the limit a → 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 → 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 → 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 ν = 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. Even though some naive arguments suggest 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 when the constant-regulator momentum integrals are convergent. As we adopted analytic continuation in the definition of the integrals, this excludes even integer values of d ≥ 4 (the d = 2 case can indeed be reduced to the constant-regulator case by a field-independent shift in the potential). 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, if analytic continuation is used, or in the whole range d ≥ 4 without analytic continuation. 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 → 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(φ). In this paper, this level of approximation has been analyzed only for the twodimensional 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. For the three-dimensional Wilson-Fisher fixed point, this tuning process is expected to converge to optimal values within the range 0.5 < a < 1 [19]. Hence, it appears very unlikely that at the special point a → 0 the derivative expansion might be convergent.
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. This choice has been justified as follows. In some circumstances, depending on the theory (or approximation) under study, as well as on the number of Euclidean dimensions d, the argument of the momentum integral might be non-integrable in the a → 0 limit. Nonetheless the integral might allow for a finite a → 0 limit, i.e. the limit and the integral cannot be exchanged. Whenever this happens, one must first clearly define the momentum integrals by choosing a specific shape function and when applicable a unique analytic continuation, and then investigate the possible behavior of these integrals in the parametric a → 0 limit. In all other cases, namely when the a → 0 limit can be brought inside the momentum integrals, one can easily generalize the discussion to arbitrary shape functions r 1 , as done in Sec. III C. Still, 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 MS -like schemes within the FRG [14].
Indeed, as we explained in Sec. I the limit of vanishing regulator shares several features with the more specific case of the MS -like pseudo-regulators discussed in Ref. [14]. In that reference, and in particular in Sec. VI, we observed that the best way of capturing the effect of quantum/statistical fluctuations beyond one loop is not adopting the derivative expansion, but rather accounting for the momentum dependence of vertices as in a vertex expansion. Because of their similarities, it is reasonable to expect that this behavior of MSlike pseudo-regulators against the choice of truncation scheme might be shared by the larger class of vanishing regulators.
In spite of the poor results of the a → 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.
It is important to stress the difference between this logic and the following one that is sometimes found in the FRG literature: the RG flow equations are solved first (and independently of the Ward identities) for a parametric family of regulators; then the latter param-eters are tuned such that the violation of some finitedimensional subset of the Ward identities is minimized. This procedure, when applied to the parameter a of Eq. (I.3), typically results in some nonvanishing value which is close to the value maximizing the rate of convergence of the chosen truncation scheme (a ∼ 1). This approach has been studied for instance in the case of conformal symmetry [31]. In this reference the Ward identities for special conformal transformation, either in their quantum or classical form (i.e. regulator dependent or independent respectively), are not solved as functional constraints. 15 By contrast, in the studies we presented in Secs. IV and V, the ansätze for the EAA included exact solutions of the classical Ward identites for O(N + 1) and shift symmetry respectively, which are easy to solve independently from the RG equations. It is thus not surprising that the symmetry breaking induced by the RG flow is minimized for a → 0. In fact, one might expect that the quantum Ward identities reduce to their classical counterparts when a → 0. Thus, because of the different strategy followed in the choice of the initial ansatz for the EAA, the authors of Ref. [31] could only minimize the unavoidable symmetry breaking, whereas in this work we could tune it to zero by taking the limit of vanishing regulators.
It is interesting that a study similar to the one of Ref. [31] was performed in Ref. [32], where the symmetry expected to emerge at the RG fixed point is supersymmetry rather than conformal symmetry. In this latter work the ansatz for the EAA does indeed include an exact solution of the classical supersymmetric Ward identity. The minimization of the breaking of supersymmetry at the fixed point by means of the optimiza-tion of the regulator was also studied, but unfortunately the limit of vanishing regulator was not within the parametric space considered in this reference. In fact, we expect the application of the vanishing-regulator limit to supersymmetric models to be interesting and useful.
The main motivation of this work was the hope that vanishing regulators, or perhaps just "sufficiently small regulators", may be useful also in the application of the FRG to gauge theories and gravity, where the background field method is almost always adopted. Our results suggest that this may be possible, but that the usefulness of this idea may be restricted to the simplest truncations.

ACKNOWLEDGMENTS
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.
For the constant regulator (II.11) one finds In Sec. IV 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 (IV.7) is not preserved by the flow equation in the a → 0 limit. In this section we provide more details about the contraints on a general local potential U k (ρ, H). Here we show how the noncompatibility of the ansatz (IV.7) 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 (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 Γ k , in presence of a linear source term of the form (IV.7) 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 [33], meaning that it defines an RGinvariant 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 oneloop 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. [34]. 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. IV. 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 φ 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. (IV.4), but with an arbitrary nonvanishing U k (ρ, H). 17 For this truncation Eq. (B.1) becomes For the Litim regulator the loop integral is readily evaluated leading to 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 [35,36] Can the limit a → 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 andZ k for a → 0. For definiteness, we assume the scaling ∂ ρ U ∼ a ,Z k ∼ a 0 , φ ∼ a 0 . Z k (ρ) log Z k k 2 F 0 (ρ) + 2ρF 0 (ρ) − 1 .

(B.10)
It is evident how the a → 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 (ρ, 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 forÛ . 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 ansatẑ U (ρ, H) = ±Ĥ 1 Z k − 2ρ + F 2 (ρ) .

(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 F 0,1 (ρ) = c 0,1 + Z k k 2 ρ 4π , (B.12) 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 → 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.