Perturbative versus Non–Perturbative Renormalization

Approximated functional renormalization group (FRG) equations lead to regulator-dependent β - functions, in analogy to the scheme-dependence of the perturbative renormalization group (pRG) approach. A scheme transformation redefines the couplings to relate the β -functions of the FRG method with an arbitrary regulator function to the pRG ones obtained in a given scheme. Here, we consider a periodic sine-Gordon scalar field theory in d = 2 dimensions and show that the relation of the FRG and pRG approaches is intricate. Although both the FRG and the pRG methods are known to be sufficient to obtain the critical frequency β 2 c = 8 π of the model independently of the choice of the regulator and the renormalization scheme, we show that one has to go beyond the standard pRG method (e.g., using an auxiliary mass term) or the Coulomb-gas representation in order to obtain the β -function of the wave function renormalization. This aspect makes the scheme transformation non-trivial. Comparing flow equations of the two-dimensional sine-Gordon theory without any scheme-transformation, i.e., redefinition of couplings, we find that the auxiliary mass pRG β -functions of the minimal subtraction scheme can be recovered within the FRG approach with the choice of the power-law regulator with b = 2, which therefore constitutes a preferred choice for the comparison of FRG and pRG flows.

The renormalization group (RG) concept has its roots in theoretical physics, and it plays a pivotal role in understanding the behavior of physical systems across different scales.The idea of scale transformations and scale invariance has been present in physics for decades.The exact, functional renormalization group (FRG) method has been constructed to perform renormalization non-perturbatively [1][2][3][4][5][6].The FRG is an implementation of the RG concept, particularly useful when dealing with strongly interacting systems in quantum and statistical field theory.It bridges the gap between the known microscopic laws (describing fundamental particles and interactions) and the complex macroscopic phenomena (such as phase transitions and collective behavior) in physical systems.The FRG combines functional methods from quantum field theory with the intuitive RG idea proposed by Wilson [1,2].Wilson envisioned a way to interpolate smoothly between the microscopic and macroscopic descriptions of physical systems.Figuratively, the FRG acts as a microscope with adjustable resolution.Unlike perturbative methods that rely on small coupling constants, the FRG is non-perturbative.It doesn't require an expansion in a small parameter.Instead, it directly explores the system's behavior across scales [7][8][9][10][11][12].
The history of the FRG approach runs back over decades starting from the Wegner-Houghton RG equation [3] which is based on the Wilson-Kadanoff blocking [1,2], through the Polchinski RG equation [4] to the modern form of FRG [5,6].The central object in FRG is the scale-dependent effective action (also called average action or flowing action).It depends on a sliding scale (the RG scale) that continuously decreases as we move from microscopic to macroscopic scales.The FRG is based on an exact functional flow equation derived by Wetterich and Morris in 1993 which describes the dependence of the effective action on the RG scale and reads as [5,6], , where is the effective action depending on the running scale k, Γ k [φ] denotes its second functional derivative and the trace Tr stands for an integration over all the degrees of freedom of the scalar field φ.A regulator (infrared cutoff) R k , the choice of which is arbitrary within certain limits [see Eqs. ( 13)- (15) of Ref. [12]], is introduced to decouple slow modes with low momenta while leaving high momentum modes unaffected.The regulator ensures that the effective action captures all relevant fluctuations.Although the explicit form of the R k regulator function is not fixed, the solution of the exact FRG equation ( 1) is independent of the particular choice of R k in the low-energy limit k → 0 [13].However, once approximations have been used, the solution does depend on the regulator function (in the IR limit, too) which weakens the predictive power of the method.Because the FRG equation cannot be solved without approximations, the regulator-dependence is a serious drawback of the method which requires special attention and the search for optimization methods is still an active research field, see e.g., Ref. [14].
One could argue that the standard perturbative renormalization group (pRG) procedure is completely free of any problem similar to the regulator-dependence of the non-perturbative approach.However, this is not true, since perturbative RG equations are derived with a particular choice of the renormalization scheme, and thus, β-functions (at least higher-loop coefficients) are scheme-dependent.As an example, for the β function of quantum electrodynamics, only the first two leading coefficients (one-loop and two-loop) are scheme-independent.An explicit derivation of this fact is given around Eq. (19.92) in Chap.19 of Ref. [15].It is a natural expectation that the scheme-dependence of pRG is connected with the regulator-dependence of FRG method [16][17][18].Indeed, the universality of the FRG method was discussed in Ref. [17].It was proven in Ref. [18] that the FRG flow equation admits a perturbative solution; a scheme transformation was given which was used to obtain the β function of the FRG method with a special choice of the regulator function from the perturbative β function obtained in the modified minimal subtraction (MS) scheme.The β functions of the FRG approach are not universal because the FRG method leads to a mass-dependent scheme which manifests itself through the nontrivial coupling of mass.In other words, the explicit mass makes the relation between the FRG and pRG β functions nontrivial.If the quantum field theory has no explicit mass, this scheme transformation is usually but always simple.For a single real scalar field it is proven to have a trivial form, however, it is not necessary the case for multi-component fields such as the sigma model which is the continuum limit of the n-vector model which consists of N real scalar fields coupled by a ϕ 4 interaction that is symmetric under rotations of the N fields.The sigma model has important physical realisations and it has been investigated by the FRG method, too, as a recent example see e.g., Ref. [19] where the linear sigma model is coupled to quarks.The connection between the FRG method and the perturbative RG with MS scheme has also been studied in [20] where a regulator function is proposed which reproduces the results of dimensional regularization at one and two loops.Thus, the resulting flow equations can be seen as non-perturbative extensions of the MS scheme.In addition, the connection between dimensional regularization and Wilsonian RG in regard of the Naturalness/Hierarchy problem is investigated in [21,22].
Let us demonstrate the nontrivial mass dependence of the FRG method for a 4-dimensional φ 4 scalar field theory; for this simple choice, we have the Euclidean action (The Einstein summation convention is used.)First, we consider its perturbative renormalization around the Gaussian fixed point (m 2 = g 4 = 0) where the pRG flow equation for the quartic coupling in the minimal subtraction (MS) scheme, at 1-loop order has been givne in Eq. (10.52) of [29] or Eq. ( 45) of [30] for ϵ ≡ 0) with k ≡ µ.Its reads as follows, (1-loop, MS-scheme:) Its solution signals the appearance of the Landau pole at very high energies while in the low energy (k → 0) limit the quartic coupling tends to zero.The pRG equation ( 3) can be compared to its non-perturbative counterpart obtained from Eq. (1).To determine this, the first step is to use the derivative expansion of the effective action at the lowest order which is the local potential approximation (LPA), where the Wetterich equation ( 1) reduces to a differential equation for the scale dependent potential V k (φ) The momentum integral in the RG equation ( 5) can be performed analytically with the Litim [31] and with the sharp-cutoff [3] regulators which results in the following forms in d = 4 dimensions, As a next step, by substituting the potential , one can derive flow equations for the couplings m 2 k and g k using the Taylor expansion of both sides of the above RG equations.Thus, the non-perturbative flow equations for the quartic coupling with the Litim and the sharp-cutoff regulators read as [6/32 = 3/16] (Litim cutoff:) The difference is in the power of the expression in the denominator.Let us eliminate the nontrivial mass-dependence of ( 10) and ( 11) by expanding them around the Gaussian fixed point (m 2 k = g 4 ≡ 0).In the leading order, both the Litim and the sharp cutoff regulator can reproduce the perturbative flow equation (3).In addition, it is also evident that the leading-order term of the β-function is found to be identical for both the Litim and the sharp-cutoff regulators.One can show that this holds for any choice of the regulator function.Thus, the leading-order term is regulatorindependent.In general, the massless FRG flow equation for the (dimensionful) coupling g 2n of the self-iteraction φ 2n is regulator-independent in d = 2n dimensions.In this case, there is no need for any scheme transformations because the β functions of the FRG and pRG methods are identical [18].
If one considers the φ 4 theory for d < 4 dimensions, the prefactor of the g 2 4,k term in the RG equations for the dimensionful β functions becomes regulator-dependent in the FRG.However, the coupling g 4,k of φ 4 carries a dimension for d < 4 but the RG flow equation should stand for dimensionless couplings, so a trivial tree-level term appears in the flow equation when one switches from the dimensionful g 4,k to the dimensionless g4,k coupling.This trivial term becomes the leading-order term in the pRG flow equation, and it is naturally regulator-independent and is responsible for the critical behaviour around the Gaussian fixed point.An alternative choice is to consider the φ 6 self-interaction in d = 3 where the coupling g 6 is dimensionless and the RG evolution is regulator-dependent, but still the scheme transformation is expected to be simple if there is no explicit mass in the model.A very suitable choice for the study of the scheme-dependence would be a theory where the coefficients of the β function are regulator-dependent for a dimensionless coupling with no explicit mass.Here, we suggest the periodic scalar theory (i.e., the sine-Gordon model) which fulfils these requirements, thus, it seems to be a good test model for scheme-dependence.
So, let us try to modify the scalar model in d = 2 dimensions by extending its essential symmetry to a periodic self-interaction.Indeed, replacing the mass term and the quartic interactions by a cosine of the field variable one arrives at the 2-dimensional sine-Gordon (SG) scalar model defined by the Euclidean action [32] In addition to the reflection (Z 2 ) symmetry, the action of the SG model given in Eq. ( 12) remains unchanged under the transformation and thus, it has another discrete symmetry; namely, it is periodic in the field variable.Due to this additional symmetry, one expects changes in the phase structure compared to the φ 4 model.Indeed, the SG model has two phases in d = 2 dimensions; it is known to undergo an infinity-order (topological) Kosterlitz-Thouless-Berezinski (KTB) [33,34] type phase transition which is controlled by the critical value of the frequency β 2 c = 8π, which separates the two phases.Let us note that both the periodicity and the reflection symmetry have been broken spontaneously in the broken phase of the model.The frequency β 2 can be seen as the inverse of the wave function renormalization which is obtained by the rescaling of the field φ → βφ.Thus, one can derive flow equations for the Fourier amplitude and the wave function renormalization (z = 1/β 2 ).The Fourier amplitude is dimensionful but the wave function renormalization is dimensionless and its flow equation is regulator-dependent.This makes the SG model suitable for a detailed study of the regulator-dependence.
The non-perturbative RG generates higher harmonics cos(nβφ) like the φ 2n terms generated in case of the φ 4 theory.However, it is known that higher harmonics play no role in the phase structure of the SG model because the SG model can be mapped onto a two-dimensional Coulomb or vortex-gas, and there, the vortices (charges) with higher vorticity (multiple charges) are found to be irrelevant, just like the higher harmonics in the SG theory.In addition, the perturbative RG treatment is possible for the SG model, but not in a traditional way, when the potential is expanded in Taylor series and only a finite number of terms are kept.In the case of the SG model, either one follows the scenario of Refs.[32,44] with all terms of the Taylor expansion summed up, or one uses the idea of an auxiliary mass term presented in Refs.[30,46,47].The two-dimensional SG model is suitable for a detailed study and comparison of the scheme-and regulator-dependence.
Before the discussion of the periodic sine-Gordon model, let us compare the perturbative and non-perturbative RG study of the φ 4 and φ 6 polynomial theories in 2 ≤ d ≤ 4. In Ref. [18], an explicit scheme transformation (see Eq. ( 17) of [18]) is given which relates the φ 4 couplings obtained by the FRG method (Litim regulator) and the pRG method (MS scheme) in d = 4 dimensions, where F is a function of the explicit mass and, again, the k ≡ µ identification is made.This is a perturbative relation obtained by solving the Wetterich equation (1) using the loop expansion.Differentiating with respect to k and neglecting higher-order terms, one obtains in d = 4 dimensions, where the β function for the Litim regulator in the FRG approach (10) is indeed recovered from the perturbative MS scheme.Thus, one expects identical results for the pRG and FRG β functions in d = 4 dimensions for vanishing mass.In Ref. [18], it is argued that a similar relation exists for all regulator types.Let us discuss the case of d < 4 dimensions.
It is illustrative to compare the perturbative and non-perturbative RG studies of the φ 4 model in 2 ≤ d ≤ 4 dimensions.Here we focus on the flow equation for the quartic coupling g which is dimensionless in d = 4 dimensions but carries a dimension if d < 4. The β function, i.e., the pRG flow equation at 1-loop order in the MS scheme in d = 4 − ϵ dimensions is well known [see, e.g., Eq. ( 10.52) of Ref. [29] or Eq. ( 45) of Ref. [30]] and reads as, (1-loop, MS-scheme:) For the details of the ϵ-expansion we refer to Refs.[27] and [28].The important difference between Eq. ( 16) and Eq. ( 3) is the linear term −ϵg k .The quartic coupling g 4,k is dimensionful in d < 4 but the flow equation must be given for dimensionless couplings, so, a trivial tree-level term (d − 4) g4,k appears in the pRG equation when one switches from the dimensionful g 4,k to the dimensionless g4,k .One notes that a priori, Eq. ( 16) valid in the limit of vanishing ϵ.However, one can use it to extrapolate to the cases ϵ = 1 and ϵ = 2.At least, the tree-level term (d − 4)g 4,k is correct in any dimension.The prefactor of g2 4,k may depend on the dimension which is not taken into account in Eq. ( 16) but as we argued the prefactor of the leading-order, linear term is correct which means that Eq. ( 16) correctly reproduces the critical behaviour around the Gaussian fixed point but non-linear terms become important near the Wilson-Fisher fixed point.
Let us now consider the FRG flow equation for g4,k in 2 ≤ d ≤ 4 dimensions with various choices for the regulator function.We use the sharp cutoff which gives results equivalent to the Wegner-Houghton FRG equation, the Litim regulator where the obtained flow equation is related to the Polchinski FRG equation via a Legendre transformation, and the power-law regulator function which could be referred to as the Morris-type regulator [6].The FRG flow equations read as, (Sharp cutoff:) Here, R k (p) = p 2 (k 2 /p 2 ) b is the power-law regulator.All of the above equations give the same result for d = 4 in both orders of g4,k , and they also agree with Eq. ( 16).Furthermore, the linear term is the same across all equations for arbitrary d, which was expected, since it comes from its dimensions.The lack of d dependence in the second order term of the perturbative solution Eq. ( 16) prevents it to be extended and compared to the corresponding term of the FRG results in other dimensions in any meaningful way.This is, of course, an expected outcome, since the perturbative expression was derived around d = 4 in the first place.Note that in Eq. ( 19) the factor (d − 4)/ sin(π d/4) has to taken in the appropriate limit for d → 4.
For the φ 6 model, the coupling g 6 is dimensionless in d = 3 dimensions and the leading order term of its β-function is expected to be regulator-dependent.Thus, we extended our potential as Ṽk ( φ) = 1  4! g4,k φ4 + 1 6! g6,k φ6 .The flow equations for g4,k are the same as in the previous section (of course, the g 6 coupling appears in these flow equations but not at the leading order).So, the focus is on the β-functions of the g 6 coupling.We use the same regulator functions as for the φ 4 model.The result for the sharp cutoff is .
The result for the Litim cutoff is .
We evaluate the special forms of the general expression for d = 3 and d = 4 for convenience.One can imediately see, that unlike the φ 4 case, we do not have a regulator independence in either dimension.However, if we assume that the two couplings are equally small, then we can say that the g6,k term is the leading, the g4,k g6,k term is subleading and the g3 4,k term is even smaller (sub-subleading).For d = 4, we find regulator independence for the first two orders.For d = 3, there is no such general behavior, and one can look for scheme-transformations which relate the FRG β-functions (with various choices for the regulator) to the pRG flow equations with various schemes.However, the mass term is absent, and it was shown in Ref. [18] that the scheme-transformation is trivial for vanishing mass in d = 4 dimensions, thus one expects trivial scheme transformations in d = 3 dimensions, too.In addition, the polynomial model without the explicit mass has no direct physical realization.So, let us look for a model where one finds no explicit mass but important physical applications, and consider whether the scheme transformation between the FRG and pRG schemes are trivial or not.An ideal choice is the SG scalar model in d = 2 dimensions, where one of its coupling (the frequency which is related to the inverse of the wave function renormalization) is dimensionless and its β function is regulator dependent [48].

III. PERTURBATIVE AND NON-PERTURBATIVE RG STUDY OF O(N ) MODELS IN 2 ≤ d ≤ 4 A. Orientation
Before we go into the details of the renormalization of the (periodic) SG scalar theory it is illustrative to compare the perturbative and non-perturbative RG study of (polynomial) O(N ) models in 2 ≤ d ≤ 4 dimensions.Polynomial and periodic models belong to different universality classes and their phase transitions are different, too.Let us study the critical behavior around the Wilson-Fisher fixed point of O(N ) models in the framework of the perturbative and non-perturbative RG approaches.Before doing this, we would like to draw the attention of the reader to a special case where one considers O(N ) scalar field theories with nonpolynomial potentials.For example, it was pointed out in Refs.[23] and [24] that the linearized RG equations of Wegner and Houghton [3] supported nonpolynomial normal modes of the perturbative RG flow in dimensions d > 2. The resulting theories reduce to the sine-Gordon theory at the d = 2 boundary.This was further connected to the exact RG methodology in [25].However, the full picture of why these modes do not lead to new universality classes for physical RG behavior was not laid out for quite some time, until [26], and the explanation is closely tied to the nontrivial connection between the perturbative and non-perturbative theory regimes.

B. Perturbative RG approach for O(N ) models
The O(N ), i.e., the N -vector model [28] is given by the action where ϕ = (φ 1 , φ 2 , ..., φ N ) is an O(N ) multiplet.The perturbative β function (for dimensionless couplings) is wellknown, see e.g., [28,38] where higher-loop corrections were taken into account.We give the first two orders, which was shown in a general overview Ref. [30] where g is the dimensionless quartic coupling.For d < 4 there is a non-trivial fixed point, i.e., the Wilson-Fisher fixed point given by gc = (4 − d) Let us note that a more accurate value for this fixed point is known in the literature where higher-loop corrections were taken into account, see e.g., [28,38].For d = 4 one finds a trivial fixed point g = 0 only.In general, the n-th order term in β(g) is given by n!g n , for example where the factor 1/(8π 4 ) is included in g.The critical exponents around the Wilson-Fisher fixed point can be calculated based on the β function: η is given by e.g.Eq. (70) of Ref. [30], and ν is given by e.g.Eq. (92) of Ref. [30], In the mean-field approximation one finds ν = 1/2.It is interesting to note that, in d = 2 dimensions, the perturbative RG approach does not exclude the existence of a nontrivial zero of the β function close to, but not equal to, zero coupling.This zero would otherwise correspond to the Wilson-Fisher fixed point in d = 2 dimension (see, e.g., [38]).While the Mermin-Wagner theorem excludes long-range order, in the sense of a long-range alignment of spins in the O(N ) model, the existence of other type of long-range ordering could be possible due to hidden order parameters [42].For example in 2d melting, the translational order parameter vanishes at all non-zero temperatures, however, the system sustains long-range orientational order at finite temperatures, thus, the order parameter associated to orientational order is not vanishing [42].
C. Non-perturbative RG approach for O(N ) models The Wetterich FRG equation provides us the possibility to perform the renormalization non-perturbatively, although approximations are required to obtain its solution.For example, one can use the gradient expansion (4) where the lowest order is the LPA.The Wetterich FRG equation (1) at LPA reduces to (5) which is a partial differential equation for the potential function and its great advantage is that various functional form of the potential can be easily investigated.For example, it is possible to consider a Taylor expansion of the effective potential around its minimum.Keeping only the quadratic and quartic terms one finds the following dimensionless potential (with dimensionless couplings and dimensionless field variable) where ρ = ϕ 2 /2.It is possible to relate the values of the couplings m 2 and g with the values of the coupling λ and the running minimum ρ 0 .These relations give the correct result for the Wilson-Fisher fixed point, but they are not working for the Gaussian fixed point, which is m 2 = g = 0 because no solution of the fixed point equations for ρ 0 and λ has a vanishing ρ 0 .By inserting (29) into the RG equation ( 5) one can obtain flow equations for the couplings for general d and N , see e.g., [39][40][41] where we used dimensionless quantities ( λk and ρ0,k ) and A d is a constant which depends on the dimension.From these RG flow equations the Wilson-Fisher fixed point can be determined.It was shown in Ref. [43] that the value of the minimum ρ0 is well defined (positive) for every value of d as long as d > 2 for every N .For d > 4 the solution for λ is negative, and, again, this is true for every N .Figure 6 of Ref. [43] shows that the minimum value ρ0 diverges for d = 2 for every N .The Mermin-Wagner-Coleman theorem [35][36][37] states that a continuous, i.e., the O(N ) symmetry for N ≥ 2 cannot be broken spontaneously in two dimensions.Of course, the N = 1 case is different because there the symmetry is discrete.Thus, the use of the FRG equation with the functional form ( 29) is suitable to produce us results in agreement with the Mermin-Wagner-Coleman theorem for N ≥ 2, i.e., no spurious Wilson-Fisher fixed points appear in d = 2 but fails for the case N = 1 since it is well-known that the Ising model has two phases in d = 2 dimensions, so one must find spontaneous symmetry breaking (SSB) for the case N = 1, but the FRG method with the functional form (29) signals the absence of SSB also for N = 1 which is not correct [43].
As a summary, one can conclude that scalar models with O(N ) symmetry do not serve as an ideal choice to compare perturbative and non-perturbative renormalization: (i) the scheme transformation between β-functions is non-trivial due to the presence of the mass term, (ii) the functional forms of the potential used in the perturbative (expansion around the Gaussian fixed point) and the non-perturbative (expansion around the running minimum) approaches are different.

IV. PERTURBATIVE AND NON-PERTURBATIVE RG OF THE SG AND CG MODELS A. Perturbative RG study of the SG model without auxiliary mass terms
Let us first present an overview of the standard perturbative study of the SG model by using various regularization methods [45] without the inclusion of any auxiliary mass terms.Let us note that the perturbative renormalization of the SG model usually be done in the massive case and then one has to consider the massless limit to obtain RG flow equations.We refer to this procedure as "renormalisation with an auxiliary mass term".In this subsection, we overview perturbative results where findings were given without the use of an auxiliary mass term.The perturbative RG treatment is based on the Taylor expansion of the interaction potential which generates the vertices of the theory.In the framework of the conventional perturbation theory, when all interaction vertices of the Taylor expansion are treated individually, one must truncate the expansion and keep only a finite number of terms but then periodicity is violated which is the essential symmetry of the SG model.If the interaction vertices of the Taylor expansion are not treated individually, and one can relate them to each other and be able to sum up all terms to get back the cosine function, periodicity is restored.This procedure can be done if the renormalisation results in an overall multiplicative factor of the couplings of each vertices which is the case in d = 2 dimensions when normal-ordering is sufficient to remove UV divergences [29,32,44].This leads to the determination of critical frequency which separates the phases of the model.(A different strategy is when one uses an auxiliary mass term [30,46,47] where RG flow equations can be derived for the wave function renormalization but this is not discussed in this section.) The most important conclusion is that one can obtain the critical frequency β 2 c = 8π of the SG model in the framework of the standard perturbative approach at one-loop order.This result is independent of the choice of the renormalization scheme.Similarly, one finds no regulator-dependence in the determination of the critical frequency in the framework of the non-perturbative FRG approach [48].It is also important to note that no RG flow equation is derived for the frequency, i.e., the wave function renormalization.One could assume that this can be done at higher loop order.However, it can be shown that it is not possible in the standard perturbative approach [49].In order to be able to derive RG flow equation for the frequency, one has to go beyond the standard perturbative treatment.One possible way is to consider the so-called Coulomb gas representation of the SG model.The other scenario is to extend the original SG model on the basis of an auxiliary mass term, perform its perturbative RG study and then take the massless limit of the derived RG flow equation for the frequency.Before we present an overview of these non-standard perturbative results, let us discuss the non-perturbative RG study of the SG model.

B. Non-perturbative RG flow equations of the SG model taken at leading order
Let us discuss the non-perturbative RG study of the two-dimensional SG model where the exact RG flow equations are expanded in terms of the Fourier amplitude and only the leading order (LO) terms are kept.In order to look for the solution of the FRG equation ( 1) beyond LPA, we consider the following ansatz for the SG model where the local potential contains a single Fourier mode and the wave function renormalization z k is assumed to be field-independent.This approximation is denoted by LPA' (watch the prime!).On the one hand the FRG method retains the symmetries of the model, so the length of period and the frequency remains unchanged over the RG flow.
On the other hand, the field can be rescaled as θ = βφ and the inverse of the squared frequency appears in front of the kinetic term.So, it can be seen as an RG scale-dependent wave function renormalization.The conclusion is that either one considers the Fourier amplitude and the wave function renormalization (u k , z k ) or the Fourier amplitude and the frequency (u k , β 2 k ) as RG scale-dependent parameters.Thus, the critical frequency β c and the critical value for the wave function renormalization z c are related to each other inversely, z c = 1/β 2 c = 1/(8π).The FRG study of the SG model beyond LPA has been discussed in [48,50,51].In [50] the wave function renormalization has both field-dependent and independent parts but the field-dependent part plays no important role in the phase structure (similarly to the higher harmonics of the potential), so, here we focus on the field-independent wave function renormalization.So let us summarize briefly the main results of [51] since our new results presented in the next section are based on that work.FRG flow equations for the Fourier amplitude and wave function renormalization are derived at LPA' level in Ref. [51] and read as follows, where P = z k p 2 + R k .Since the dimensionful frequency is scale-independent, it is convenient to merge it with the scale-dependent wave function renormalization z k , so, we introduced rescaled quantities ẑk = z k /β 2 , Rk = R k /β 2 and P = P/β 2 = ẑk p 2 + Rk .A very general regulator function is the so called CSS (Compactly Supported Smooth) one [52] which is defined as Its advantage is to reproduce all major types of regulators in various limits of its parameters, (Optimised Litim-type:) lim c→0,h→1 (Power-law Morris-type:) lim c→0,h→0 (Exponential Wetterich-type:) lim c→1,h→0 Here, b is a free parameter.The sharp cutoff-regulator can also be reached by the CSS (e.g. if one takes the limit b → ∞ for the power-law case) but this type of regulator cannot be applied beyond LPA because it requires derivatives of the regulator which is problematic for the sharp cutoff since it is non-differentiable and not continuous.The Litim-type regulator has similar problem, although it can be applied at LPA' but not beyond that.The exponential and the power-law regulators have no such problems, they can be applied at any order of the gradient expansion.Momentum integrals of the FRG equation have to be performed numerically, except the expanded form of Eqs. ( 33), (34) around the Gaussian fixed point where analytical results available.This requires a special choice for the regulator function R k such as the power-law [6] regulator, so, in this work we perform calculations by this type of regulator.In general, the regulator function beyond LPA should be given by the inclusion (multiplicative approach) or the exclusion (additive approach) of the field independent wavefunction renormalization z k , In the multiplicative approach, the rescaled regulator Rk contains the rescaled wavefunction renormalization ẑk .In the additive approach, the frequency can be absorbed by the overall multiplicative constant of the rescaled regulator or can be chosen arbitrarily since it is a scale-independent free parameter of the model.It is important to note, that the additive approach requires the use of the power-law regulator function.The phase structure should be independent whether we use the multiplicative or additive approaches and of its parameters such as b.
By using the mass cutoff in the additive approach, i.e. power-law type regulator with b = 1, the momentum integrals of ( 33) and ( 34) can be performed and the RG equations reads as [48], with the dimensionless coupling ũ = k −2 u.Analytic solutions are always available for the power-law type regulator if one considers the approximated flow equation where the exact RG equation ( 33) and (34) are expanded in Taylor series with respect to u k around zero.In the additive approach, the leading order flow equations have the following forms (additive, power-law, arbitrary b:) where c 2 (b) > 0 for b > 1.These flow equations result in a KTB type (i.e.infinite order) phase transition with ẑc = 1/(8π).In the multiplicative approach one finds very similar leading order RG flow equations, (multiplicative, power-law, arbitrary b:) It is important to note that the leading-order flow equation for the Fourier amplitude ũk are the same in the multiplicative and additive approaches, which is responsible for the determination of the critical value 1/ẑ c = 8π which separates the phases of the model.This is true for any choices of the regulator function, so the critical value z c is found to be scheme-independent.Thus, the scaling of dimensionless Fourier amplitude ũk has been determined by trivial tree-level (∼ k −2 ) and the non-trivial (∼ k 1/(4π ẑ) ) scalings in the above two schemes, which result in a regulator-independent expression, ũk ∼ k −2+1/(4π ẑ) , from which one can read off the critical value 1/ẑ c = 8π.Similarly, prefactors of the leading order flow equations obtained for the wave function renormalization are the same in the multiplicative and additive approaches.The difference is due to the power of the wave function renormalization thus it is regulator-dependent.
A similar approach has been done in [53] where the RG flow equations above ( 45) of [53] were obtained by using the Wilson-Kadanoff blocking relation up to leading order terms and reads as where the identifications g ≡ ũ and ∂ l ≡ −k∂ k are used.In order to compare them to the leading order RG equations ( 41),( 42) and ( 43), (44) of the SG model one has to use the following identification β 2 = 1/ẑ, where the additive case reads as (additive, power-law, arbitrary b:) and for the multiplicative case one finds (multiplicative, power-law, arbitrary b:) For b = 2, one finds agreement between the additive power-law flow equations Eqs. ( 46) and ( 47) and Eq. ( 45).However, in [53] the same non-perturbative (Wilson-Kadanoff) RG method was used, so, all what one can conclude is that a preferred choice for the regulator functions has been identified.Thus, in the next section let us discuss the Coulomb gas representation of the SG model and its perturbative RG study.

C. Perturbative RG study of the CG model in the dilute gas representation
The mapping between the CG and SG models holds in arbitrary dimension [54] (and it is exact in case of pointlike charges).Thus, the RG study of the SG model can be directly used to map out the phase structure of the CG model and vice versa.In the framework of the real (or coordinate) space RG approach one can use the dilute gas approximation for the CG model which is equivalent to the low fugacity, i.e., small Fourier amplitude limit of the SG model.In this RG approach the charges (vortices) are considered as rigid discs with finite diameter, so, this can be seen as sharp cutoff version performed in the coordinate space which corresponds to a smooth cutoff version in the momentum space.The real space RG equations of the CG model in arbitrary dimension is given in [55] which have the following form in d = 2 dimensions By using the following identifications, ∂ l = −k∂ k , y ∼ ũk and x ∼ 1/ T ∼ 1/ẑ k where T is the temperature and y is the fugacity, Eq. ( 50) can be rewritten as where c u , c z are constants that depend on the actual choice for the exact relation between the parameters of the CG and SG models.It is clear that the SG flow equations of the additive case ( 41) and ( 42) in the limit b → 1 have identical functional form to (51).Since the critical value 1/ẑ c = 8π is independent of the actual choice of the regulator, one has to define c u = 1/(4π).By the rescaling of the Fourier amplitude, the value of the other constant c z can be chosen to be identical to c(b = 1)/(8π) = 1/(24π) which results in exactly identical flow equations to the additive power-law case with b = 1.Thus, it suggests the choice b = 1.Indeed, the power-law regulator is a smooth cutoff in the momentum-space RG and this is found to be identical to the sharp-cutoff of the real space RG approach as expected.
Another real space RG study of the CG model can be found in [56] using again the dilute-gas approximation.The RG equations (3.2.8) and (3.2.9) of [56] can be rewritten in d = 2 dimensions where we introduced the following identifications 2z ≡ ũ, α ≡ β and ∂ l ≡ −k∂ k where K 2 , I 1 and B are constants.
Once the RG flow equations of the multiplicative case are rewritten by using β 2 k = 1/ẑ k , see Eqs. ( 48) and ( 49), one finds flow equations identical to (52) if either the coefficient c(b) and/or the rescaling of ũ are chosen properly.The additive case, see Eqs. (46) and (47) gives identical functional form only in the so called sharp cutoff limit, i.e., b → ∞ but then the coefficient c(b = ∞) cannot be defined unambigously, since the sharp cutoff confronts to the derivative expansion (it is problematic beyond LPA).So, the multiplicative power-law gives better result then the additive one and again one can conclude that the sharp-cutoff real space RG corresponds to a smooth cutoff regulator of the momentum RG.
However, the perturbative RG study of the equivalent CG representation of the SG model is not the ideal choice to find connections between renormalization schemes and regulators because one compares different models.A better choice is the direct RG study of the SG model.Thus, let us discuss in the next section the perturbative RG approach for the SG model by using an auxiliary mass term.

A. Orientation
The perturbative renormalization of the SG model has been investigated by using the inclusion of an auxiliary mass term, see e.g., [30,46,47,57].Thus, the perturbative renormalization of the SG model is done in the massive case when one adds an exilic mass term m 0 to the original periodic SG theory and RG flow equations of the massless SG model are given by considering the limit m 0 → 0.
The basic idea is to extend the original periodic SG model by an explicit mass term.Following [46] one can consider the Euclidean Lagrangian where m 0 is an explicit mass serves as an IR regulator and a is the UV cutoff (of dimension length) and this is related to the momentum RG scale a ∼ 1/k.One can choose two different strategies.In the first case the correlation functions can be calculated by using an UV regularised form where K 0 is the modified Bessel function.In the second case, one can consider the renormalization of the massive SG model and treat m 0 as an IR regulator take the limit m 0 → 0 at the UV regularized level before UV renormalization, see e.g., Ref. [47].In any case, β-functions of the original SG model is given in the massless limit, i.e., the renormalisation is performed with an auxiliary mass term and in this section we briefly review these results.

B. Leading order perturbative results
In Ref. [30] one finds a standard pRG approach for the SG model but using the method of an auxiliary mass term.The original SG model (without the auxiliary mass) has the following parametrisation in Ref. [30], with the following pRG flow equations, see Eqs. ( 159) and ( 160) in [30] where αk = ũk β 2 k and t k = β 2 k , which are identical (keeping only the leading order terms) to the additive power-law flow equations Eqs. ( 46), ( 47) with b = 2.Of course, one can use again a scheme transformation to relate the FRG flow equations with arbitrary choice of the regulator to the pRG flow equations.However, in this case the transformation is not trivial since one has to rescale the Fourier amplitude by the frequency, so, practically one should consider different models to be able to relate the FRG and pRG β functions.
Here, we rather propose to fix the model and look for a particular choice of the regulator by which the FRG flow equations reproduce the functional form of the pRG β functions.Based on this logic, we have found the additive power-law flow equations Eqs. ( 46) and ( 47) with b = 2 a suitable choice.In the next subsection, we discuss results at next-to-leading order.

C. Next-to-leading order perturbative results
Let us consider the renormalization of the SG model at next-to-leading order [46,47] where the method of the auxiliary mass term has been used, so, this is again not the standard perturbative approach.The action used for this pRG calculation (without the auxiliary mass) reads as, where a is a length scale introduced solely to make α dimensionless.Since a is in coordinate space, it is related to momentum as a ∼ 1/k.By using the following parametrizations, the perturbative RG flow equations of [46,47] reads as where A 1 = 5/4 and B 1 = −1 and it was argued that B 1 + 2A 1 = 3/2 is a universal number.The above flow equations which are at next-to-leading order can be rewritten as, and by using the identification α k = ũk β 2 k the pRG flow equations are the followings, It is evident that the leading-order FRG flow equations, ( 46),( 47) and ( 48), (49) are not suitable to reproduce the nextto-leading term propotional to ũ3 k in (63).Thus, one has to go beyond the leading-order RG flow equations obtained by the FRG approach in order to compare them to (63) which is one of our goals in this work.At this point it is important to clarify our motivation for the study and comparison of higher order terms of the RG flow equations.One can argue that these terms have no importance since they do not influence the critical scaling behaviour around the Coleman fixed point ũ⋆ = 0 and β 2 ⋆ = 8π 2 .However, in Ref. [47] it has been shown that one can construct a quantity from certain combination of the coefficients of these terms of the RG flow which could be universal.Thus, it is relevant to study the flow equations with NLO terms which is the goal of the next section.
The main result of this section is that one can use the RG study of the SG model to compare regulator (i.e., scheme) dependence of non-perturbative and perturbative approaches and our conclusion is that the power-law type regulator (with b = 2) of the FRG method gives identical functional form to the perturbative RG flow equations obtained by the application of the minimal subtraction scheme.

VI. NON-PERTURBATIVE RG FLOW EQUATIONS OF THE SG MODEL TAKEN AT NEXT-TO-LEADING ORDER
Let us now calculate next-to-leading order terms for the flow equations ( 41),( 42) and ( 43), (44).To do this one has to expand Eqs. ( 33), (34) around the Gaussian fixed point taking into account higher order terms in ũk .Then momentum integrals can be performed analytically by the use of the power-law regulator [6] similarly to the leading-order case.As we have already discussed, the regulator function beyond LPA can be given by the inclusion (multiplicative approach) or the exclusion (additive approach) of the field independent wavefunction renormalization z k .
In the additive approach, the next to leading order (NLO) FRG flow equations have the following forms (additive, power-law, arbitrary b:) The coefficients are as follows, c 1 . It is important to note that c i (b) > 0 for b > 1.In the multiplicative approach one finds very similar NLO RG flow equations, (multiplicative, power-law, arbitrary b:) .
It is useful to rewrite these equations in terms of the frequency β2 k = 1/ẑ k .In the additive case one finds (additive, power-law, arbitrary b:) while in the multiplicative case (multiplicative, power-law, arbitrary b:) The most important observation is that ũ3 k and ũ4 k terms are generated by the FRG approach at NLO but these terms do not violate the KTB type (i.e.infinite order) phase transition at ẑc = 1/(8π).The ũ3 k terms of (68) and (70) comes with certain power of β 2 k which differs from (63).However, the additive case (68) with b = 2 gives identical functional form for the flow equation of the Fourier amplitude found in (57).To demonstrate this, on Fig. 1 we compare the flow diagram based on the pRG equation (57) to that of obtained by the FRG equations ( 68 the case b = 2 are the closest to the black lines which are the pRG trajectories. There are ũ4 k terms in (69) and (71) which are absent in (63).Thus, one can conclude that the FRG flow equations at NLO cannot reproduce the functional form of the perturbative RG flow equations of (63) obtained at NLO but one finds agreement with (57) where a ũ3 k term is present in the flow equation of the Fourier amplitude (although ( 57) is considered as a leading order result).

VII. SUMMARY
In this article, we have attempted to find connections between perturbative and non-perturbative renormalization group equations, with a particular emphasis on the sine-Gordon model in 2 dimensions.In [18] it was shown that one can always find a suitable scheme transformation that maps an FRG β-function (using a given regulator) to a perturbative result obtained in a given scheme and vice-versa.It was argued in [18] that this transformation is simplified if there is no explicit mass term in the model (at least for a single field).It was claimed that this is due to the fact that the FRG handles the mass in a non-conventional way, i.e., the FRG method leads to a mass-dependent scheme which manifests itself through the nontrivial coupling of mass.[18].Based on these facts, one could argue that the massless ϕ 2n model is not a suitable test model for the comparison of perturbative and non-perturbative renormalization because, of the absence of the expilict mass Ref. [18].
An interesting question is whether one finds this "trivial" scheme transformation for all massless models.With reference to this, here we proposed to investigate the sine-Gordon scalar model where there is no explicit mass term, it is a physically relevant model, and it is known that the β-function obtained for the frequency (the inverse of the wave function renormalization) is regulator and scheme-dependent.For the sine-Gordon model, concentrating on the running of the coupling parameter u alone, we showed that in the framework of the standard perturbative approach, by using dimensional regularization, one can already obtain the exact value β 2 c = 8π for the critical frequency which separates the phases of the SG model in d = 2 dimensions.Furthermore, this result was found to be independent of the renormalization scheme.This is in perfect agreement with results of the non-perturbative analysis.However, we also argued that it is not possible to derive RG flow equation for the wave function renormalization by the standard perturbative renormalization.Thus, in order to consider the connection between renormalization schemes and FRG regulators one has to go beyond the usual, standard perturbative treatment by for example introducing an auxiliary mass term to the original SG model.In this way, one can derive RG flow equations for the wave function renormalization, i.e., for the frequency.For the latter, the β function turns out to be scheme dependent compare for example Eqs. ( 57) and (63).
We have found that if one relates the FRG and the auxiliary mass term pRG β-functions of the SG model, one obtains a non-trivial scheme transformation which requires the rescaling the Fourier amplitude by the frequency.Although, it seems to be in contradiction to the general picture based on the results of [18] where any massless model is assumed to require a "trivial" scheme transformation it is actually a consequence of the fact that both the FRG and the auxiliary mass term pRG approaches are mass-dependent schemes which manifests through the nontrivial coupling of mass.Thus, even if the scheme transformation turns out to be somewhat less "trivial" than initially assumed, strictly speaking, the conclusions of Ref. [18] are not violated.
However, if one goes a bit further and compares flow equations obtained by the FRG and by the auxiliary mass term pRG approaches for the same SG theory, we showed that one can choose a particular regulator (power-law with parameter b = 2) which gives identical results.In this context, the power-law (Morris) regulator with b = 2 turned out to be a choice which leads to the highest degree of agreement between perturbative and nonperturbative RG flows.In other words, the power-law regulator with b = 2 constituted a preferred choice for the comparison of FRG and pRG flows, as one compares, e.g., Eqs. ( 57) and (68).It is important to note, that the choice for the regulator suggested by our analysis in this work is not a result of an optimisation procedure.However, it is known (see e.g., Ref. [58]) that the optimal choice for the parameter b of the power-law regulator is b = 2. Indeed, in Ref. [58] the principle of minimal sensitivity was used for the optimization of the CSS regulator and confirms that the functional form of a regulator first proposed by Litim is optimal within the LPA.It was known that the Litim regulator provides us critical exponents (for the polynomial scalar field theory) identical to that of obtained by Polchinski's RG method at LPA.It was also known, however, that the Polchinski RG equation has drawbacks in case of the sine-Gordon [59] and the multi-layer sine-Gordon models [60] at LPA'.In addition, Litim's exact form leads to a kink in the regulator function, so it confronts with higher order terms of the gradient expansion.Thus, the choice of the regulator function is an open question beyond LPA while the results of this work support the use of the power-law regulator with b = 2 at LPA'.

5 FIG. 1 .
FIG. 1. RG flow diagram of the SG model indicates the fixed point β 2 c = 8π.Black lines are trajectories given by the numerical solution of the pRG equation (57).Colored lines are trajectories given by the numerical solution of the FRG equations (68) and (69) derived by the (additive) power-law regulator with various values for the parameter b.