Multi-meson Yukawa interactions at criticality
MMulti-meson Yukawa interactions at criticality
Gian Paolo Vacca ∗ INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna
Luca Zambelli † Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, D-07743 Jena, Germany
The critical behavior of a relativistic Z -symmetric Yukawa model at zero temperatureand density is discussed for a continuous number of fermion degrees of freedom and ofspacetime dimensions, with emphasis on the role played by multi-meson exchange in theYukawa sector. We argue that this should be generically taken into account in studies basedon the functional renormalization group, either in four-dimensional high-energy models or inlower-dimensional condensed-matter systems. By means of the latter method, we describethe generation of multi-critical models in less then three dimensions, both at infinite andfinite number of flavors. We also provide different estimates of the critical exponents of thechiral Ising universality class in three dimensions for various field contents, from a couple ofmassless Dirac fermions down to the supersymmetric theory with a single Majorana spinor. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - t h ] M a r . INTRODUCTION In this paper we will study the renormalization group (RG) flow of a simple Yukawa modeldescribing relativistic fermions interacting through the exchange of scalar fluctuations. We willdiscuss some of its critical properties in a continuum of spacetime dimensions 2 < d ≤ d = 3 case. The class of models we want to consider isdescribed by the following generic bare Lagrangian L = 12 ∂ µ φ∂ µ φ + V ( φ ) + ¯ ψγ µ i∂ µ ψ + i H ( φ ) ¯ ψ ψ . (I.1)where we have N f copies of fermions, whose representation will be kept general in the following,and one real scalar field. The requirement of power-counting renormalizability would furtherrestrict the interactions inside the potentials V and H (and would generically require the in-clusion of derivative interactions too) but we are not going to impose such conditions, since weare interested in describing the possible conformal models in this family, even if strongly cou-pled. In case the potentials V and H are even and odd respectively, the system is characterizedby a chiral Z symmetry, besides the U( N f ) symmetry. For this reason, the model with barepotentials V ( φ ) = ¯ m φ + ¯ λ φ , H ( φ ) = ¯ yφ (I.2)is often called Gross-Neveu-Yukawa model, since it shares these symmetries with the purelyfermionic Gross-Neveu model [1] and can be obtained from it by means of a Hubbard-Stratonovich transformation.Even for more general bare Lagrangians that are not related by any bosonization technique,the Yukawa models and chiral fermionic models remain deeply connected. The three dimensionalGross-Neveu model shows a second order quantum phase transition that separates the phasewith preserved chiral symmetry from the one where this is spontaneously broken and a chiralcondensate of fermions appears. The latter can be effectively described as a scalar degreeof freedom, therefore this transition can be unveiled also as a dynamical effect in interactingscalar-spinor systems. Indeed, it is found that the critical properties of the Gross-Neveu modelin 2 < d < (cid:15) -expansions [2–6], large N f expansions [4, 7, 8], lattice simulations [9–13]and functional RG equations [14–19].These critical properties have great physical relevance for the description of several systems.In condensed matter, three dimensional relativistic fermionic systems, such as QED and theThirring model, play the role of building blocks for theories of high- T C superconductivity [20],and for the description of electrons in graphene [21]. Understanding the phase diagram andcritical properties of these models at variable N f represents pretty much the same challengeas the one posed by the Gross-Neveu and Yukawa model, and one can even address them in aunified picture [22]. Even the simple Yukawa model discussed in this work can find applicationsto extremely nontrivial phenomena in condensed matter. For the case of two massless Diracfermions, its quantum critical phase transition in d = 3 might be a close relative of the putativetransition between the semi-metallic and the Mott-insulating phases of electrons in graphene [18].For a single Dirac field instead, it is considered to be in the same universality class of spinlessfermions on the honeycomb lattice with repulsive nearest neighbors interactions [13].For a single Majorana spinor, it is a precious example of a three dimensional model showingemergent supersymmetry. Indeed, it is known that in this case the critical theory not onlyenjoys N = 1 supersymmetry, but also possesses only one relevant component, which meansthat by tuning a single macroscopic parameter one can discriminate between two distinct phaseswith preserved or spontaneously broken supersymmetry [17, 23]. On these grounds, a potentialexperimental realization of supersymmetry was proposed in [23], at the boundary of topologicalsuperconductors. A similar phenomenon occurs for Yukawa systems with complex scalars andspinors, which have been argued to give rise to an emergent N = 2 supersymmetry [24].The phase diagram of Gross-Neveu and Yukawa models has been analysed in d < d → H ( φ ), that essentiallydescribes vertices with two fermions and an arbitrary number of scalars. This kind of interactionshave been neglected in the FRG studies of fermionic models for a long time. Only recently theyhave been discussed in other works considering more complicated models and different butrelated questions. For example, in [30] the flow equations for this Yukawa system coupled toquantum gravity were derived, but only the linear coupling H ( φ ) = ¯ yφ was considered in explicitstudies of these equations. Most prominently, in [31] the effect of higher Yukawa couplings onthe chiral phase structure of QCD at finite temperature and chemical potential was analyzed bymeans of an effective quark-meson model. It was observed, within polynomial truncations of aYukawa potential H ( φ ), that higher order quark-meson interactions are quantitatively importantin the description of the chiral transition.A similar but different study will be performed here, for the present Z -symmetric Yukawamodel, in lower dimensionality and for a generic number of flavors. We will confine ourselves tothe study of the zero-temperature system at criticality, looking for scaling solutions for various d and N f , and comparing the results obtained with different methods. In Sect. III we start withthe leading order of the 1 /N f -expansion, reproducing known results in three dimensions, andgeneralizing them to multi-critical theories below three dimensions. Technical details regardingthis analysis are sketched in App. B. In Sect. IV we turn to a finite number of fermions and,by neglecting the wave function renormalization of the fields, we observe how critical Yukawatheories arise while continuously lowering the dimensionality towards two. To this end, we con-sider the FP equations for the two generic functions V ( φ ) and H ( φ ), and solve them numericallywithout resorting to any truncation. In Sect. V, still neglecting the wave function renormal-izations, we adopt a different strategy for the numerical integration of the FP equations, andcompute the global FP potentials in three dimensions, for various flavor numbers. For the caseof a single Majorana spinor, we also apply these numerical methods to the computation of thecritical exponents and perturbations. In Sect. VI we discuss polynomial truncations, showinghow these can give results in satisfactory agreement with the global numerical analysis. As aconsequence, we use them for a self-consistent inclusion of the wave function renormalizations,and produce estimates of the critical exponents in three dimensions and for various numberof fermions, which we compare with some of the existing literature. Finally, in Sect. VII weaddress the d → II. THE RG FLOW OF A SIMPLE YUKAWA MODEL WITHMULTI-MESON-EXCHANGE.
The functional renormalization group (FRG) is a representation of quantum dynamics basedon Wilson’s idea of floating cutoff k . In this work we will adopt its formulation in terms of ascale-dependent 1PI effective action, called average effective action [32]. For a given system, theform of this action is determined by the field content Φ and by the symmetry properties, as wellas by an initial condition (bare action) and boundary conditions for the integration of the flowequation ˙Γ k [Φ] = 12 STr (cid:20)(cid:16) Γ (2) k [Φ] + R k (cid:17) − ˙ R k (cid:21) . (II.1)Here (Γ (2) k [Φ]+ R k ) − represents the matrix of regularized propagators, while R k is a momentum-dependent mass-like regulator. Since the dot stands for differentiation with respect to the RGtime t = log k , this flow equation comprehends the infinite set of beta functions for the infinitelymany allowed interactions inside Γ k . Extracting them amounts to projecting both sides ofthe equation on each separate interaction functional. In practical computations, one dropsinfinitely many operators, thus performing a nonperturbative approximation called truncationof the theory space. To this end, several systematic strategies are available and appropriate indifferent circumstances, such as the vertex expansion or the derivate expansion. For reviewssee [33].In this work we will consider the following truncation:Γ k (cid:2) φ, ψ, ¯ ψ (cid:3) = (cid:90) d d x (cid:18) Z φ,k ∂ µ φ∂ µ φ + V k ( φ ) + Z ψ,k ¯ ψγ µ i∂ µ ψ + i H k ( φ ) ¯ ψ ψ (cid:19) . (II.2)Here φ is a real scalar field, while ψ denotes N f copies of a spinor field with d γ real components.The latter parameter is related to the symmetries of the system and plays therefore a crucialrole in pure fermionic as well as in fermion-boson models. Yet, as long as we truncate thetheory space to the ansatz of Eq. (II.2), focusing on the mechanism of Z -symmetry breaking,we can simply deal with the total number of real Grassmannian degrees of freedom X f = d γ N f ,considering it as an arbitrary real number. As soon as X f ≥ Z φ and Z ψ , whichwould appear in the next-to-leading (first) order of the derivative expansion. In the following wewill call the ansatz of Eq. (II.2) a local potential approximation (LPA) for this simple Yukawamodel, whenever the wave functions renormalizations are neglected ( Z φ,k = Z ψ,k = 1), andtherefore the fields have no anomalous dimensions η φ,ψ = − ∂ t log Z φ,ψ . The inclusion of thelatter will be named LPA (cid:48) . Our justification for the choice of this truncation is in the exhaustiveevidence that similar ans¨atze give a good description of the existence and properties of conformalmodels in 2 < d ≤ φ −→ k ( d − / Z / φ φ , ψ −→ k ( d − / Z / ψ ψ since the new dimensionless renormalized field would then be constant at criticality. As aconsequence we will focus on the potentials for these fields v k ( φ ) = k − d V k (cid:18) Z / φ φk ( d − / (cid:19) , h k ( φ ) = k − Z ψ H k (cid:18) Z / φ φk ( d − / (cid:19) . In this new set of variables the flow equations read˙ v = − dv + d − η φ φ v (cid:48) + 2 v d (cid:110) l (B) d ( v (cid:48)(cid:48) ) − X f l (F) d ( h ) (cid:111) (II.3)˙ h = h ( η ψ −
1) + d − η φ φ h (cid:48) + 2 v d (cid:110) h ( h (cid:48) ) l (FB) d , ( h , v (cid:48)(cid:48) ) − h (cid:48)(cid:48) l (B) d ( v (cid:48)(cid:48) ) (cid:111) (II.4) η φ = 4 v d d (cid:110) ( v (3) ) m (B) d ( v (cid:48)(cid:48) ) + 2 X f ( h (cid:48) ) (cid:104) m (F) d ( h ) − h m (F) d ( h ) (cid:105)(cid:111) φ (II.5) η ψ = 8 v d d (cid:110) ( h (cid:48) ) m (FB) d , ( h , v (cid:48)(cid:48) ) (cid:111) φ (II.6)where v d = (2 d +1 π d/ Γ( d/ − , the threshold functions l ( F/B ) d and m ( F/B ) d on the right handside denote regulator-dependent contributions from loops containing fermionic or bosonic prop-agators, and the equations for the anomalous dimensions are to be evaluated at the minimum φ of the scalar potential. Their definition can be found in App. A, together with the explicit form6hey take for the linear regulator, which is our choice in this work since it allows for a simpleanalytic computation of such integrals. For this linear regulator the flow equations of the twopotentials, read˙ v = − dv + d − η φ φ v (cid:48) + C d (cid:32) − η φ d +2 v (cid:48)(cid:48) − X f − η ψ d +1 h (cid:33) (II.7)˙ h = h ( η ψ −
1) + d − η φ φ h (cid:48) + C d h (cid:0) h (cid:48) (cid:1) (cid:32) − η ψ d +1 (1 + h ) (1 + v (cid:48)(cid:48) ) + 1 − η φ d +2 (1 + h ) (1 + v (cid:48)(cid:48) ) (cid:33) − h (cid:48)(cid:48) (cid:16) − η φ d +2 (cid:17) (1 + v (cid:48)(cid:48) ) (II.8)where we have denoted for convenience C d = 4 v d /d .A simple way of facilitating the stability of the vacuum is the requirement of Z symmetry, i.e.invariance over φ → − φ . For standard Yukawa system, with a linear bare Yukawa interaction H ( φ ) = yφ , this requires a discrete chiral symmetry ψ → iψ and ¯ ψ → i ¯ ψ . A generalization oflocal interactions with such a symmetry then requires an odd H ( φ ). There is also the possibilityto let the spinors unchanged under the transformation, which would require an even function H ( φ ).The goal of this work is to construct global FP solutions of the flow equations compatiblewith the symmetry conditions, and to study the properties of the RG flow in their neighborhood.The FPs, which describe scaling solutions, are computed by solving the coupled system of twoordinary differential equations ˙ v = 0 and ˙ h = 0 or, in some cases, from the equivalent systemfor the quantities ( v , y = h ). The dependence of such scaling solutions on the two parameters d and X f is one of the main themes discussed in the literature as well as in the present work.Regarding the former, we will assume 2 < d ≤ d , but we will especially concentrate on the properties of the d = 3system. For the latter, we restrict ourselves to non-negative number of degrees of freedom, andwe start from the two simple limiting cases one can address. The simplest is X f →
0. In thiscase, the fermion sector remains nontrivial, see Eqs. (II.4,II.6), but is not allowed to influencethe scalar dynamics, which is therefore identical to the fermion-free model, see Eqs. (II.3,II.5).Hence, as far as criticality is concerned, we expect to observe the same pattern of FPs that canbe observed without fermions, with the same critical exponents in the scalar sector, even if atgenerically nonvanishing values of the Yukawa couplings. The second limit which brings radicalsimplifications is X f → ∞ , and it is discussed in the next Section.7 II. LEADING ORDER LARGE − X f EXPANSION.
Large- N f methods are a traditional and successful way to analyze the strongly coupled do-main of the three dimensional Gross-Neveu model, which is renormalizable at any order in a1 /N f -expansion [4, 7, 8]. As a consequence, any other nonperturbative method is challenged toreproduce known results in this limit. For this reason, before moving to the finite- X f resultsprovided by the FRG, let us start with discussing the behavior of this simple Yukawa model withmany fermionic degrees of freedom, within the basic parameterization of its dynamics providedby Eq. (II.2), in a continuous set of dimensions 2 < d <
4. This FRG analysis, for the case ofa linear Yukawa function, has already been performed in [16]. Our results can be considered asan extension of it, to include a generic function h ( φ ). As we will see, the main advantage thatthis brings at large- X f is the possibility to describe also multi-critical models in d < v with X f v , as well as η φ with X f η φ , and look at the leadingorder in 1 /X f . The first simplification is the fact that only canonical scaling terms and purefermion loops survive. Therefore the flow equations at this order reduce to˙ v = − dv + d − η φ φ v (cid:48) − v d l (F) d ( h ) (III.1)˙ h = h ( η ψ −
1) + d − η φ φ h (cid:48) (III.2) η φ = 4 v d d ( h (cid:48) ) (cid:104) m (F) d ( h ) − h m (F) d ( h ) (cid:105) (III.3) η ψ = 0 . (III.4)Let us draw some general considerations about the FP solutions, by postponing the task ofconsistently solving the flow equation for η φ . The equation for h is almost regulator-independent(apart for the value of η φ ) and the solution is a simple power h ( φ ) = c h φ / ( d − η φ ) . (III.5)This is real only if the exponent is rational and with an odd denominator. Furthermore it issmooth only if the exponent is a positive integer. The FP solution for v is instead regulatordependent. Adopting the linear regulator, in 2 < d < v ( φ ) = c v φ d/ ( d − η φ ) − v d d F (cid:18) , − d − d − h ( φ ) (cid:19) . (III.6)The function F (cid:0) , − d ; 1 − d , − x (cid:1) , which actually can be reduced to a Hurwitz-Lerch function − d Φ (cid:0) − x, , − d (cid:1) , has a logarithmic singularity at x = −
1, therefore the condition that h ( φ ) be8eal entails that this singularity is always avoided, and that the potential is globally defined. Onthe other hand, the smoothness of v is not for granted. Since F (cid:18) , − d − d − x (cid:19) = 1 − dd − x − d − d x + O ( x ) (III.7)and since this function is always convex, the leading φ -dependence of v at its minimum, i.e. atthe origin, is provided by h ( φ ) itself. Hence, the latter must be a smooth function, becausewe want the couplings associated to the derivatives of the potential at the minimum to bewell defined at the FP. The same reasoning, if applied to the Yukawa couplings, leads to therequirement that h ( φ ) be smooth at the origin. This translates into a quantization condition onthe dimensionality of the scalar field d − η φ n , n ∈ N (III.8)which is a consequence of the large- X f limit.We find it helpful, for the interpretation of this relation, to consider a similar condition atthe purely scalar FPs, with trivial Yukawa interaction. With this we mean the limit X f → ∞ followed by c h →
0, which is clearly not the same as the fermion-free model; yet, by consistency,this limit should describe the classical properties of the latter model. Indeed, if c h = 0 the onlycondition left is that the homogeneous part of the FP scalar potential be smooth and stable,that is d − η φ d n , n ∈ N . (III.9)The meaning of this constraint is well known. By neglecting the quantum corrections, hencesetting η φ = 0, one would deduce that the smooth bounded solutions v ( φ ) = c v φ n are allowedonly in d n = 2 nn − n − , n ∈ N . (III.10)This is the usual tree-level counting according to which the interaction φ n is marginal in d n and becomes relevant for d < d n . From the quantum point of view, these dimensions are thecorresponding upper critical dimensions for multi-critical universality classes. For any n , below d n a new FP with nontrivial η φ branches from the Gaußian FP and survives for 2 ≤ d
1, highertruncations are needed. Thus, the ¯ n -th FP can provide UV completion for theories approachingthe n -th FP in the IR, only if n < ¯ n . The detailed study of the global flows among these FPs isin principle a straightforward task in the large- X f approximation, but it is out of the purposesof the present work. We confine ourselves to sketching some properties of the FP potentials andof the linearized perturbations in vicinity of the FPs, which can be found in App. B, togetherwith some comments on how these nontrivial critical theories disappear in d = 4. IV. LPA AT FINITE X f AND GENERIC d . SOME FEATURES FROM NUMERICALINVESTIGATIONS. In the previous Section we described how the large- X f expansion supports the expectationthat, as the number of dimensions is lowered from d = 4 towards d = 2, across the upper criticaldimensions of Eq. (III.11), new universality classes become accessible in the theory space ofYukawa models. In this Section we are going to present evidence that this happens also at finite X f . Here and in the rest of this work, we restrict our analysis to the subset of theory spacewhich enjoys a conventional Z -symmetry, such that v is even and h is odd. Furthermore weadopt the LPA and neglect the flow equations for the wave function renormalization of the fields.As it was argued in the previous Section, as well as in App. B with more details, one cannotexpect this approximation to perform well for any n and X f . Therefore the following studies11hould be understood as a first step towards a proper description of these universality classes.Only the d = 3 chiral Ising universality class will be later analyzed also in the LPA (cid:48) , by resortingto polynomial truncations of the potentials, see Sect. VI.Since we look for odd Yukawa potentials, we can restrict the list of the operators that becomerelevant at the corresponding critical dimensions: φ n : d vc ( n ≥
2) = 2 nn − , , , , · · · φ n +1 ¯ ψψ : d hc ( n ≥
0) = 4( n + 1)2 n + 1 = 4 , , · · · (IV.1)In order to reveal the new universality classes appearing below these dimensions, we follow thestrategy developed in [36], that was already successfully applied to the purely scalar modelin continuous dimensions [35]. This consists in solving the FP condition, which is a Cauchyproblem involving a system of two coupled second order ODEs, by a numerical shooting method,i.e. varying the initial conditions in a space of parameters which is two dimensional, since two ofthe four boundary conditions are fixed by the symmetry requirements ( v (cid:48) (0) = 0 and h (0) = 0).For the potential v we choose as parameter σ = v (cid:48)(cid:48) (0), relating it to v (0) using the differentialequation. For h we use h = h (cid:48) (0). Trying to numerically solve the non linear differentialequations with generic initial conditions, one typically encounters a singularity at some valueof φ c ( σ, h ) where the algorithm stops. Such value increases in a steep way close to the initialconditions which correspond to a global solution, even if the numerical errors mask partially thisbehavior. As a consequence, in our case a three-dimensional plot for φ c ( σ, h ) is very useful togain a first understanding of the positions of the possible FPs.In Fig. 1 we show the results of this analysis, for X f = 1 and for several dimensions: d =5 , , . , . , , , − , , . For d = 5 and d = 4, as it is expected, we see a single spike in( σ, h ) = (0 ,
0) which corresponds to the Gaußian solution. More details on this are given, for X f <
1, in Sect VII. In 3 < d < φ and φ ¯ ψψ become relevant, as is shown in Eq. (IV.1). In this interval, it is evident from thefigure that we find three new spikes. One is characterized by h = 0 and σ < h , and correspond to the chiral Ising universality class. Theyhave σ <
0, which suggests that also these scaling solutions are in a broken regime for X f = 1,at least in the LPA approximation. Moving to < d < φ , but no other operators involving fermions have to be added to the set of therelevant ones. This corresponds to the appearance of the tricritical theory in the pure scalarsector, as we see from the new spike which develops with σ > h = 0. Once d < alsothe new operators φ and φ ¯ ψψ become relevant and new critical solutions may appear. Indeed,in the left and the central plot of the third line of Fig. 1 we see two new spikes, which againoccur at opposite values of h and are therefore equivalent, this time with σ >
0. Finally inthe lower-right plot, where we present the case d = , which is lower than enough to clearlysee the effects of the new relevant scalar operator φ , one can appreciate the third new spike at σ < h = 0. The latter FP corresponds to the quadricritical scalar model as describedfor example in [35, 37]. The former solutions, already assuming that they globally exist, definewhat one could call the chiral quadricritical Ising universality class, since they originate fromthe Gaußian FP together with the purely scalar quadricritical model.We don’t show more plots with lower values of d , since the pattern is pretty clear. Pushingfurther this analysis towards dimensions close to d = 2, though conceptually straightforward,would probably anyway require more than the LPA. To provide the reader with some moredetails, in Fig. 2 we zoom in the panel of Fig. 1 that refers to d = − . The three non trivialspikes which appeared at higher values of d > h (cid:54) = 0, can also be visualized by a plot at constant value of σ , approximatelycorresponding to the position of the peaks, see Fig. 3. Here the range of h is wider than inFig. 2, so that one can see also a trace of the FPs generated at d <
4, which are neverthelesslocated at a different value of σ .The analysis we discussed in this Section can be repeated for other values of X f , thus gettinga qualitative understanding of the position of the FPs as a function of both d and X f . However,because of the uncertainties in the location of these peaks, it is hard to get a good qualitativeknowledge of this function. Nevertheless, the latter is needed to prove that the argumentspresented in this Section are rigorous, that each of the peaks corresponds to one FP, and tocompute the corresponding critical exponents. For this reason, in the next Section we are goingto adopt a different numerical method that will allow us to precisely answer these questions,focusing on d = 3 for definiteness, but allowing for a generic X f .13 IG. 1: Spike plots for X f = 1 on varying the dimension: d = 5 , , . , . , , , − , , , from left toright and from top to bottom. V. d = 3 LPA AT FINITE X f . NUMERICAL SOLUTION OF THE FP EQUATIONS In this Section we construct, for some specific cases, the numerical solutions for v and h of theFP differential equations, obtained by setting Eqs. (II.7) and (II.8) equal to zero, in a domainfor the dimensionless field φ that covers the asymptotic region. This is what might be calleda global scaling solution. For convenience, we have actually considered the equivalent systemfor the quantities v ( φ ) and y ( φ ) = h ( φ ). We focus here on d = 3 for which, from the analysisat X f = 1 performed in the previous Section, we expect a FP with non-trivial scalar potentialand Yukawa function. In the following we are going to take several values of X f into account.After having found the corresponding nontrivial FP potentials, we determine the associatedcritical exponents and eigenperturbations. The knowledge of the global scaling solutions will14 IG. 2: Spike plot for d = − and X f = 1, zoomed area around the origin.FIG. 3: Spike plot for d = − and X f = 1, zoomed area around the origin. be important for a study of the quality of polynomial expansions, presented in Sect. VI . Thelatter approach is very useful especially in the case of the LPA (cid:48) , which gives us access to aself-consistent computation of the anomalous dimensions without enlarging the truncation to afull next-to-leading order of the derivative expansion. Clearly this programmatic analysis canbe repeated for other values of d .We choose to construct a global numerical solution by starting from the knowledge of theasymptotic behavior allowed by the FP equations. Once the asymptotic expansions are deter-mined with sufficient accuracy we proceed, with a shooting method, to the numerical integrationfrom the asymptotic region towards the origin. The properties of the solutions which reach theorigin depend on the free parameters in the asymptotic expansions. By requiring the solutions to15ransform correctly under Z , one can uniquely fix the latter parameters to their FP values [38]The leading term of the asymptotic expansion for both v and h is determined, in the LPA withvanishing anomalous dimensions, by the classical scaling. Here we report the first correction toit. Denoting α = 2 / ( d − v asympt ( φ ) (cid:39) A φ α +2 + φ − α C d ( B − AX f ( α + 1)(2 α + 1))2 AB ( α + 1)(2 α + 1)( d + 2) + · · · (V.1) h ( φ ) (cid:39) B φ α + φ − − α C d α (4 α (2 α + 1) A + B )2 A ( α + 1)(2 α + 1) ( d + 2) + · · · and depends on two real parameters A and B . In our analysis we have computed and usedasymptotic expansions with eight terms for each potential. Starting the numerical evolutionfrom some large value for φ = φ max , we have then investigated v (cid:48) (0) and h (0) as functionsof A and B . Computing numerically the gradient of these two functions, we were able toemploy a kind of Newton-Raphson method to determine their zeros, i.e. the values of A and B corresponding to Z -symmetric scaling solutions. In Fig. 4 we present two examples of globalsolutions for the cases X f = 1 and X f = 2. The former is in the broken regime, since the Z symmetric scalar potential has a non trivial minimum, while the latter is in the symmetricregime. Any solution ( v , h ) is characterized by two parameters, such as for example A and B , or v (cid:48)(cid:48) (0) and h (cid:48) (0), which indeed fix completely the Cauchy problem once they are complementedby the symmetry conditions. In Fig. 5 we show the FP values of the integration constants A and B as defined by Eq. (V.1). The locus of the FP solutions in the plane ( v (cid:48)(cid:48) (0) , h (cid:48) (0)) as a functionof X f ∈ [10 − ,
3] is instead presented in Fig. 6. Notice that as X f approaches zero, in the lowerleft end of the curve, h (cid:48) (0) attains a finite value, which is situated around 3.3. It is evident thatthe two regimes, broken and symmetric, are realized in two complementary intervals of X f . Thetransition between the two occurs at X f (cid:39) .
64 for the LPA. In the next Section we will see thatthis value is slightly modified in the LPA (cid:48) , and becomes X f (cid:39) .
62. The vacuum expectationvalue φ and the value of h (cid:48) ( φ ) as functions of X f are presented in Fig. 7.The critical exponents of these scaling solutions and the corresponding eigenperturbationsare an important piece of information. This is obtained by studying the evolution of the smallperturbations around the FPs. Therefore the linearized flow equations are the main tool tostudy such a problem. They are constructed, taking advantage of the separation of variables in φ and k , by substituting into the flow equations v k ( φ ) = v ∗ ( φ ) + (cid:15)δv ( φ ) e λt , y k ( φ ) = y ∗ ( φ ) + (cid:15)δy ( φ ) e λt (V.2)16 IG. 4: The potentials v and h at the global scaling solution, computed numerically within the LPA.The case X f = 1, which is in the broken regime, appears in the first two panels (top), while X f = 2, inthe symmetric regime, is shown in the last two panels (bottom). X f A X f B FIG. 5: The values of the asymptotic parameters ( A , B ) defined by Eq. (V.1) at the scaling solutions,varying X f in the range 10 − < X f < and then keeping the first term in (cid:15) , for (cid:15) (cid:28)
1. Such a procedure leads to the following eigenvalue17
IG. 6: The values of ( v (cid:48)(cid:48) (0), h (cid:48) (0)) from the numerical global scaling solutions, varying X f in the range10 − < X f <
3. One can notice the transition from the broken to the symmetric regime, which occursat X f (cid:39) .
64 for the present LPA.FIG. 7: The vacuum expectation value φ ( X f ) from the numerical global scaling solutions is shown inthe left panel, while in the right panel we plot the corresponding value of h (cid:48) ( φ )( X f ), both in the LPA. problem 0 = ( λ − d ) δv + 12 ( d − φ δv (cid:48) + C d (cid:18) X f (1 + y ) δy − v (cid:48)(cid:48) ) δv (cid:48)(cid:48) (cid:19) (V.3)18nd 0 = ( λ − δy + (cid:18) d − (cid:19) φ δy (cid:48) + C d (cid:34) δv (cid:48)(cid:48) (cid:0) y ( y + 1) y (cid:48)(cid:48) − ( y (cid:48) ) (cid:0) y ( v (cid:48)(cid:48) + 5) + 3 y + 1 (cid:1)(cid:1) y (1 + y ) (1 + v (cid:48)(cid:48) ) − δy ( y (cid:48) ) (cid:32) y ) ( v (cid:48)(cid:48) + 1) + (cid:0) y + 2 y + 1 (cid:1) y (1 + y ) (1 + v (cid:48)(cid:48) ) (cid:33) + δy (cid:48) y (cid:48) (cid:18) y ) (1 + v (cid:48)(cid:48) ) + (3 y + 1) y (1 + y ) (1 + v (cid:48)(cid:48) ) (cid:19) − δy (cid:48)(cid:48) (1 + v (cid:48)(cid:48) ) (cid:35) (V.4)where for simplicity we have renamed v ∗ and y ∗ as v and y . This system is of the form (cid:16) ˆ O − λ (cid:17) δf = 0 . (V.5)if δf is the vector of perturbations, δf T = ( δv, δy ), and ˆ O is the corresponding differentialoperator. We have considered two different ways to solve this eigenvalue problem.The first approach is a direct generalization of the one we have already discussed for scalingsolutions, in this case applied to the full set of equations: FP plus linearized flow. The asymptoticbehavior of the eigenperturbations is computed by solving the asymptotic form of the linearizedequations for large field, which is obtained using the known asymptotic expansion for v and y at the FP, given in Eq. (V.1). In d = 3 one finds δv asympt = φ − λ + φ − λ − (cid:0) A βX f + B (cid:0) − λ + 11 λ − (cid:1)(cid:1) π A B + O (cid:16) φ − − λ (cid:17) (V.6) δy asympt = βφ − λ − φ − λ − (cid:32) (cid:0) λ − λ + 15 (cid:1) (20 A + B )16875 π A + β (cid:0) Aλ + B (cid:0) λ + 5 λ − (cid:1)(cid:1) π A B (cid:33) + O (cid:16) φ − − λ (cid:17) In practice we used an asymptotic expansion with up to three terms per perturbation. We notethat in a linear homogeneous problem the overall normalization of the eigenvector δf plays norole. Therefore the asymptotic form of δf depends only on a relative real parameter β , whichwe choose to be a constant multiplying the leading term of δy . One more free parameter isneeded for tuning the behavior of the solutions at the origin, such that they fulfill the symmetryrequirements δv (cid:48) (0) = 0 and δy (0) = 0. This can be interpreted as the eigenvalue λ itself. As aconsequence, one expects a discrete spectrum of allowed values for λ and β . Unfortunately, dueto numerical uncertainties, with this method we have been able only to restrict the eigenvaluesto an interval described by a continuous function λ ( β ). Indeed one has to remember that theglobal numerical solutions have been constructed on some bounded neighborhood of the origin,19ven if the latter overlaps with the region were the large field asymptotic behavior becomesdominant. Moreover, the linearized equations depend on derivatives of the numerical global FPsolutions, for which the accuracy is reduced.The second approach we considered consists in inserting the known numerical FP solutions inthe linearized equations, computing a numerical expression for all the φ -dependent coefficientsof this eigenvalue problem, and then solving them by means of a pseudo-spectral method basedon Chebyshev polynomials. Also in this case some uncertainties remain, for the same reasonsmentioned above. As an example, for X f = 1 the leading critical exponent we find is θ = − λ =1 . θ = 3, since it is related toan additive constant in the potential and it is unphysical in flat space). All the other eigenvalues λ i are positive and associated to irrelevant operators, for instance θ = − λ = − . θ = − λ = − . δf = ( δv, δh )shown in Fig 8. Notice the fact that the relevant eigenpertubation has δh ( φ ) (cid:54) = 0 unlike in thelarge- X f analysis, where the only relevant perturbation compatible with symmetry requirementsis δv ( φ ) = δc v φ , which corresponds to θ = 1. Even if X f = 1 is quite away from thislimit, it is know that in this case the FP theory is a N = 1 Wess-Zumino model [17, 23], andthat the supersymmetry-preserving relevant perturbation is a change in the mass of the scalarfield [17, 39], which therefore leaves the Yukawa sector unchanged. Hence δh (cid:54) = 0 is probably aconsequence of the explicit breaking of supersymmetry introduced by our regularization scheme.We do not push further here the spectral analysis of the critical exponents and associatedperturbations as a function of X f , leaving it for a future study based on algorithms giving bettercontrol on the numerical errors. In the present work, these global numerical computations at X f = 1 will serve as a reference for the development of a different, local, approximation method,based on polynomial truncations of the functions v ( φ ) and h ( φ ). The latter will be discussed inthe next Section, and will be also used for a more reliable discussion of the dependence of thecritical exponents on the number of fermion degrees of freedom. VI. POLYNOMIAL ANALYSIS IN d = 3 In this Section we are going to discuss the use of polynomial parameterizations and consequenttruncations of the functions v ( φ ) and h ( φ ). Though for definiteness we will address the specificcase of the unique d = 3 nontrivial critical Yukawa theory, similar techniques can be applied20 IG. 8: Case d = 3 and X f = 1: the components δv and δh of the relevant eigenperturbation, from theglobal numerical analysis of the LPA. to the other scaling solutions in 2 < d <
3, presumably with the same degree of success.Sect. VI A will present results obtained within the LPA, which can be directly compared to thefull functional analysis developed in the previous Section. This will make us confident about theeffectiveness and soundness of polynomial truncations, as well as of the necessity to go beyond asimple linear Yukawa coupling for an accurate description of critical properties of the theory. Onthese grounds, Sect. VI B will push forward the analysis to a self-consistent inclusion of the wavefunction renormalization of the fields, which is essential for quantitative estimates of the criticalexponents, which will be compared with some literature for several values of X f . Polynomialtruncations will be also used in Sect. VII for some comments on the four-dimensional model.Let us start by presenting the truncation schemes we are going to analyze. Since we restrictourselves to d = 3, we will demand v ( φ ) and h ( φ ) to be even and odd respectively. We will usethe common notation ρ = φ /
2, and we will adopt only one name for one and the same quantity,regarless of whether it is considered as a function of φ or as a function of ρ . In the symmetricregime, the physically meaningful parameterization of the scalar potential is a Taylor expansionaround vanishing field v ( ρ ) = N v (cid:88) n =0 λ n n ! ρ n . (VI.1)Regarding the Yukawa potential, we are interested in two possible Taylor expansions, one for21 ( φ ), already adopted in [31], and one for y ( ρ ) = [ h ( φ )] . In the symmetric regime they read h ( φ ) = φ N h − (cid:88) n =0 h n n ! ρ n (VI.2) y ( ρ ) = N h (cid:88) n =1 y n n ! ρ n . (VI.3)In the regime of spontaneous symmetry breaking (SSB) the potential v ( ρ ) develops a nontrivialminimum κ = φ /
2, which becomes the preferred reference point for a different Taylor expansion v ( ρ ) = λ + N v (cid:88) n ≥ λ n n ! ( ρ − κ ) n . (VI.4)Though, in general, κ is no special point for the function h ( φ ), it still enters in the definition ofthe vertex functions, from which one extracts the physical multi-meson Yukawa couplings. Asa consequence, in this regime it is necessary to change also the parameterizations of h ( φ ) and y ( ρ ), as follows h ( φ ) = φ N h − (cid:88) n =0 h n n ! ( ρ − κ ) n (VI.5) y ( ρ ) = N h (cid:88) n =1 y n n ! [( ρ − κ ) n − ( − κ ) n ] . (VI.6)The pair ( N v , N h ), or more generally an ordering of the polynomial couplings by priority ofinclusion in the truncations, can be chosen by relying on naive dimensional counting, as in aneffective field theory setup, or on the knowledge of the dynamics at a deeper level, e.g. a globalnumerical solution for the FP functionals and the critical exponents. In the latter strategy onewould sort the critical exponents in order of relevance and would try to accurately describe thecorresponding perturbations. Alternatively, and maybe less efficiently, one could scan over theresults produced by different pairs ( N v , N h ) and select them on the base of a comparison tothe global numerical solution. In the former strategy instead, since the dimension of a scalarself-interaction φ n is n , and the one of a multi-meson Yukawa coupling ¯ ψφ n +1 ψ is 5 / n ,we would expect that the pairs ( N v = D, N h = D − h ( φ ) given in Eqs.(VI.2,VI.5), correspond to including operators up to dimension D . However, since by truncatingat level N h = D − D +1 /
2, if we want to beslightly more accurate we could include the latter and consider the pairs ( N v = D, N h = D − N v , N h ), andwe found that the two strategies nicely agree, so that ( N v = D, N h = D −
1) is a very good22ystematic choice for polynomial truncations. For similar reasons, as well as for the sake ofcomparison, we made the same choice also for the truncation of y ( ρ ) given in Eqs. (VI.3,VI.6).It is necessary to stress that, in both the parameterizations given above, even at lowest orderin the truncation for the Yukawa coupling, the beta-functions for h or y are different from theclassic result [26] illustrated in the reviews [33] and used for the present d = 3 critical theoryfor instance in [14–16, 18, 19]. This happens because ∂ t h ( φ ), which comes from the projectionof the r.h.s. of the flow equation on the term i ¯ ψψ , is a nonlinear function of φ , independentlyof the parameterization of h ( φ ), be it linear in φ or not. Hence, in order to define the runningof a linear Yukawa coupling, a further projection is needed. The prescription adopted by theabove-mentioned studies is to identify the beta function of the linear Yukawa coupling withthe first φ -derivative of ∂ t h ( φ ) at the minimum of the potential. For the truncations underconsideration in this work instead, ∂ t h comes from the zeroth order φ -derivative of ∂ t h ( φ ) /φ ,while ∂ t y is defined as the first order ρ -derivative of ∂ t y ( ρ ) = 2 h ( φ ) ∂ t h ( φ ), always evaluated atthe minimum of the potential. Simplicity is our main motivation for choosing a parameterizationof the running Yukawa sector which does not include the traditional Yukawa beta-function, aswe are now going to explain.The traditional projection has the structure of a Taylor expansion of ∂ t h ( φ ) about φ = φ ( φ being the minimum of v ( φ )). The choice of such an expansion for the parameterization of h ( φ ) would entail an explicit breaking of Z symmetry, which requires this function to be odd.Ideally, one would need to match two Taylor expansions, one about φ = φ and another oneabout φ = − φ , by imposing suitable conditions at the origin. These are just provided by Z symmetry. The result of this construction however is not a simple Taylor expansion any more h ( φ ) = 12 N h (cid:88) n =1 g n n ! (cid:2) ( φ − φ ) n + ( − n +1 ( φ + φ ) n (cid:3) (VI.7)and the projection rule on the generic coupling g n is more involved than simply taking the n -th φ -derivative and evaluating it at φ = φ . Yet, it is true that the latter projection works for the N h -th coupling, such that this truncation does include the traditional beta-function of the linearYukawa coupling as the N h = 1 case. In this work we preferred to consider and compare onlythe two truncation schemes presented in Eqs. (VI.2,VI.5) and Eqs. (VI.3,VI.6), leaving the onein Eq. (VI.7) aside. In the next Sections we are going to show that both polynomial truncationsconverge to the same results for large enough N v and N h , an observation that clearly shouldapply to all possible parameterizations. Furthermore, in both polynomial truncations simply23y setting N h = 1 one gets estimates that are significantly different from the full truncation-independent results. That the latter statement also applies to the truncation in Eq. (VI.7), canbe assessed by comparison to the literature, which the reader can find in Sect. VI B. A. LPA
In Sect. V we looked for the d = 3 nontrivial critical theories at varying X f within the LPA, bymeans of numerical solvers for the ODEs defining the FP potentials. Here we repeat this analysiswith the different method of polynomial truncations and we compare the results with the ones wepreviously found. The FPs emerge from the solution of a system of coupled nonlinear algebraicequations for the couplings. The critical exponents are defined by (minus) the eigenvalues ofthe stability matrix at the FP, i.e. the matrix of derivatives of the beta-functions with respectto the couplings [33]. The anomalous dimensions are computed in a non-self-consistent way, byneglecting them in the FP equations descending from Eqs. (II.3,II.4), and then by evaluatingthe flow equations for the wave function renormalizations Eqs. (II.5,II.6) at this FP position.Let us start from the standard way of describing the Yukawa models, that is by approximatingthe Yukawa potential h ( φ ) with a single linear coupling. On the grounds of the results of thefull functional analysis presented in Sect. V, one could expect that this approximation performswell, since far enough from the large-field region the FP function h ( φ ) does not strongly deviatefrom a straight line, see Fig. 4. For a linear Yukawa function, the expansions around the originof h ( φ ) and y ( ρ ) give results which are identical order by order in N v , both in the shape ofthe FP functions (in the sense that y = 2 h at the FP) and in the critical exponents. As aconsequence we can present them in a single table for the former parameterization, the latterproviding the same results. This is Tab. I, where we set e.g. X f = 1. The first two criticalexponents form a complex conjugate pair, which is clearly unsatisfactory. This is producedby the expansion around a trivial minimum of v ( φ ), that for X f = 1 is not justified. Oncewe turn to the SSB parameterization of h ( φ ), which is given on the left panel of Tab. II, theybecome real. However, things become cumbersome for the single-coupling SSB parameterizationof y ( ρ ), since we were not able to find any FP at all (which might nevertheless exist). Let usrecall that, even in the case of a single Yukawa coupling, the beta functions descending fromthe two different polynomial truncations of h ( φ ) and y ( ρ ) are different, hence one cannot simplytranslate the FP position from one parameterization to the other. As soon as we add y the FP24 N v ,N h ) (2,1) (3,1) (4,1) (5,1) (6,1) (7,1) (8,1) (9,1) (10,1) λ − − − − − − − − − λ λ — 84.22 121.9 134.7 136.7 134.5 132.7 132.4 132.8 h θ θ − − − i − i − i − i − i − i i θ — − − − i − − i − − i − − i − − i − − i − − i η ψ η φ TABLE I: Case d = 3 and X f = 1, polynomial expansion of h ( φ ) around a trivial vacuum of the potential,with a fixed linear Yukawa function (standard Yukawa interaction), in the LPA. can be easily found. This then stimulates to consider the general effect of allowing for higherpolynomial Yukawa couplings.The immediate observation is that their inclusion significantly alters the position of theFP and the critical exponents. Some degree of convergence is observed in several systematicstrategies for the increase of N v and/or N h , but this can be convergence to the wrong results,i.e. to FP functions that do not agree with the numerical global solution. The linear Yukawatruncations provide one example of this fact. This is visible by comparing the two panels ofTab. II, where on the r.h.s. we show the results provided by the ( N v = D, N h = D − D, D − y ( ρ ).Comparing the two panels one can see how the critical exponents can be computed by largepolynomial truncations independently of whether these are around the origin or a nontrivialvacuum. Furthermore, comparing the right panels of Tab. III and Tab. II it can be observedhow both the FP potentials and the critical exponents converge to values that are independentof the chosen parameterization. That these values are the ones corresponding to the full globalsolution provided in Sect. V, is shown in the right panel of Tab. III. Notice however that thereis a 0.6% difference between the relevant exponent computed with the polynomial truncationsand the one obtained by the global numerical analysis. Even if we feel that we have the formermethod under a better control, we cannot give our preference to any of these estimates.In Fig. 9 we plot different kinds of polynomial solutions, all in a ( N v = 9 , N h = 8) truncation,against the numerical global FP functions, still for X f = 1. For the potential v we show onlythe domain φ ≥ .
3, the agreement among all the curves being perfect for smaller values. Theexpansion around the origin has a smaller domain of validity as expected. Regarding the two25 N v , N h ) (5,1) (6,1) (7,1) (8,1) (9,1) κ λ λ h θ θ − − − − − θ − − − − − η ψ η φ ( N v , N h ) (5,4) (6,5) (7,6) (8,7) (9,8) κ λ λ h h θ θ − − − − − θ − − − − − η ψ η φ TABLE II: Case d = 3 and X f = 1, polynomial expansion of h ( φ ) around a non trivial vacuum for boththe potential and the Yukawa function, in the LPA, with or without the inclusion of multiple-meson-exchange interactions (right and left panel respectively). ( N v , N h ) (4,3) (5,4) (6,5) (8,7) (9,8) λ − − − − − λ λ y y θ θ − − − − − θ − − − − − η ψ η φ N v , N h ) (5,4) (6,5) (7,6) (8,7) (9,8) ( ∞ , ∞ ) κ λ λ y y θ θ − − − − − − θ − − − − − − η ψ η φ TABLE III: Case d = 3 and X f = 1, polynomial expansion of y ( ρ ) in the LPA. Left panel: expansionaround the origin, for which the global numerical solution provides λ = − . y = 28 .
47, andunstable higher couplings. Right panel: expansion around a nontrivial vacuum and, in the last column,the corresponding couplings extracted from the global numerical solution. set of expansions around a non trivial vacuum, the scalar potentials for the two cases are almostindistinguishable, while for the Yukawa function we obtain a slightly better result employingthe one of Eq. (VI.6), as it is shown in the right panel of the figure. The same kind of plots canbe obtained for the polynomial truncations based on a single Yukawa coupling, correspondingto a linear Yukawa function. These are shown in Fig. 10, were we consider both polynomialexpansions, around the origin and the non trivial minimum, for N v = 9. The left panel isespecially interesting since it shows how, if one forces a linear Yukawa function, even with theSSB expansion, the shape of the potential is poorly reproduced.Having observed that in the LPA the ( D, D − X f = 1, we assume that this is always the case, and make use of themfor addressing how the FP and the critical exponents depend on X f within the LPA. In Sect. IIIwe have argued that when X f is not small, there is no reason to trust the LPA for the d = 3critical theory, since η φ should approch unity as X f increases. This is what the global numerical26 IG. 9: Comparison of the X f = 1 global numerical solution in the LPA (blue, continuous) with thecorresponding ( N v = 9 , N h = 8) polynomial solutions, around the origin as in Eqs. (VI.1)-(VI.3) (red,dotted), around a non trivial vacuum as in Eqs. (VI.4)-(VI.6) (brown, dashed) and in Eqs. (VI.4)-(VI.5)(green, dot-dashed), for the potential v ( φ ) (left panel) and the Yukawa function y ( φ ) = h ( φ ) (rightpanel).FIG. 10: Comparison of the X f = 1 global numerical solution in the LPA (blue, continuous) with thecorresponding ( N v = 9 , N h = 1) polynomial solutions, around the origin as in Eqs. (VI.1)-(VI.3) (red,dotted) and around a non trivial vacuum as in Eqs. (VI.4)-(VI.5) (green, dot-dashed), for the potential v ( φ ) (left panel) and the Yukawa function h ( φ ) (right panel). analysis also indicates. Indeed in Sect. V we found that the constants A and B wildly growfrom X f = 3 on, in practice making the construnction of FP potentials harder and harder. Thisproblem is easily addressed by means of the polynomial expansions. The results obtained witha (9 , h ( φ ) and y ( ρ ), are shown in Tab. IV and Tab. V.As expected, the anomalous dimensions show a very different X f -dependence. Starting with27 f . κ .
311 10 − .
704 10 − .
173 10 − .
845 10 − .
219 10 − .
126 10 − λ λ h h θ θ − − − − − − θ − − − − − − η ψ η φ X f λ − .
267 10 − λ λ .
313 10 .
090 10 h h θ θ − − − − − θ − − . ± . − . ± . − . ± . − . ± . η ψ η φ TABLE IV: Case d = 3 and varying X f , polynomial expansion of h ( φ ) around the non-trivial (left panel)or trivial (right panel) minimum for both the potential and the Yukawa function, with N h = 8 and N v = 9in the LPA. X f . κ .
310 10 − .
705 10 − .
174 10 − .
846 10 − .
187 10 − .
115 10 − λ λ y y θ θ − − − − − − θ − − − − − − η ψ η φ X f λ − .
085 10 − λ λ .
301 10 .
089 10 y y .
990 10 .
310 10 θ θ − − − − − θ − − − . ± . − . ± . − . ± . η ψ − − η φ TABLE V: Case d = 3 and varying X f , polynomial expansion of y ( ρ ) around the non-trivial (left panel)or trivial (right panel) minimum for both the potential and the Yukawa function, with N h = 8 and N v = 9in the LPA. X f log y FIG. 11: Behavior of the coupling y in a N h = 5, N v = 6 polynomial truncation of y ( ρ ) around a trivialvacuum, within the LPA. The curve is a fit of data from X f = 3 . X f = 4 − − . η ψ > η φ for very small X f , the former decreases and the latter increases as X f is increased.Still for X f around one, the two are small enough for qualitatively trusting the LPA, though forestimates of the critical exponents the LPA (cid:48) provides different and more accurate results. Thepolynomial truncations agree with the global analysis and locate around X f = 1 .
64 the transition28rom the SSB to the SYM regime for the FP potential. Around this value η φ reaches unity thussignalling the inconsistent use of the LPA. Yet if we insist on using this approximation for largervalues of X f , the breakdown of the approach is signalled by different phenomena. First of allthe critical exponents become complex, from about X f = 2 on. Then the anomalous dimensions η φ and η ψ , which are determined in a somehow un-legitimate way, become much bigger thanunity and negative respectively. At the same time the couplings at the FP increase very rapidly,similarly to what was observed in Fig. 5. Actually in LPA it is easier than in the global numericalanalysis to understand how quickly they grow. The result of a (6 , y ( ρ ) around a trivial minimum is shown in Fig. 11. It is quite accurate to fit the behavior ofthe coupling y close to X f = 4 with a simple pole y ≈ . / (3 . − X f ). Also the remainingcouplings have a rate of growth that is compatible to a divergence at a finite value of X f , butthese values would lie beyond the pole of y .Also the comparison between the polynomial truncations and the global numerical resultsillustrates the appearance of severe problems as X f increases. Moving to larger values of X f and entering the symmetric regime one sees, again comparing against the numerical solution ofthe ODEs, that the polynomial approximation has a smaller radius of convergence and thereforeleads to a less trustworthy estimate of the LPA results. As an example we present the case X f = 2 . φ < .
18, both for v ( φ ) and y ( φ ), while at X f = 1 the same grade of agreement was found for φ < .
28. Again the strongestrestriction is imposed by the Yukawa function. Instead of interpreting these problems as a signof the generic weakness of the polynomial truncations for large- X f , we take the point of viewthat they are the way in which these truncations manifest the failure of the LPA for X f roughlybigger than 1 .
6. We think that the results of the next Section support this interpretation.
B. LPA (cid:48)
In the LPA (cid:48) the anomalous dimensions are consistently determined by solving the FP equa-tions together with the flow equations for the wave function renormalizations. In the previousSections we have shown that this is necessary for a correct qualitative description of the dy-namics of the model, roughly above X f ≈ .
6. The expectation is that thanks to the wavefunctions renormalizations the system should gradually move towards the large- X f limit, as itwas already checked for truncations with a linear Yukawa function [14–17]. In this Section we29 IG. 12: Comparison of the numerical solution in the LPA (blue, continuous) with the corresponding( N v = 9 , N h = 8)-polynomial solutions, for X f = 2, around the origin as in Eqs. (VI.1)-(VI.3) (red,dotted), around a non trivial vacuum as in Eqs. (VI.4)-(VI.6) (brown, dashed) and in Eqs. (VI.4)-(VI.5)(green, dot-dashed), for the potential v (left panel) and the Yukawa function y ( φ ) = h ( φ ) (right panel). ( N v , N h ) (5,1) (6,1) (7,1) (8,1) (9,1) κ .
250 10 − λ λ h θ θ − − − − − θ − − − − − η ψ η φ ( N v , N h ) (5,4) (6,5) (7,6) (8,7) (9,8) κ λ λ h h θ θ − − − − − θ − − − − − η ψ η φ TABLE VI: Case d = 3 and X f = 1, polynomial expansion of h ( φ ) around a non trivial vacuum for boththe potential and the Yukawa function, in the LPA (cid:48) , with or without the inclusion of multiple-meson-exchange interactions (right and left panel respectively). want also to understand how big are the effects of the wave function renormalizations on thecritical exponents, already for small X f .As in the previous Section, let us start our discussion with the X f = 1 model. Tab. VI is theLPA (cid:48) version of Tab. II, which considers the truncation of h ( φ ) with or without higher Yukawacouplings. If the effect of the inclusion of multi-meson exchange on the relevant exponent θ wasof the 8% in the LPA, it gets reduced to the 7% in the LPA (cid:48) . However, in the truncation of y ( ρ )the effect is of the 20%, see Tab. VII Also, the convergence of the polynomial truncations seemsquicker in the LPA (cid:48) . A comparison between the left panels of Tab. VI and Tab. VII illustrateshow the predictions of the FRG can be made independent of the truncation scheme, here inthe form of a different definition of Yukawa couplings, only by including full functions of field30 N v ,N h ) (5 ,
1) (6,1) (7,1) (8,1) (9,1) κ .
208 10 − .
210 10 − .
212 10 − .
213 10 − .
212 10 − λ λ y θ θ − − − − − θ − − − − − η ψ η φ ( N v , N h ) (5 ,
4) (6,5) (7,6) (8,7) (9,8) κ λ λ y y θ θ − − − − − θ − − − − − η ψ η φ TABLE VII: Case d = 3 and X f = 1, polynomial expansion of y ( ρ ) around a non trivial vacuum for boththe potential and the Yukawa function, in the LPA (cid:48) , with or without the inclusion of multiple-meson-exchange interactions (right and left panel respectively). X f . κ .
377 10 − .
793 10 − .
253 10 − .
316 10 − .
171 10 − .
164 10 − λ λ h h θ θ − − − − − − θ − − − − − − η ψ η φ X f λ − .
622 10 − .
135 10 − λ λ h h θ θ − − − − − − θ − − − − − − η ψ .
347 10 − .
939 10 − .
341 10 − .
073 10 − η φ TABLE VIII: Case d = 3 and various X f , polynomial expansion of h ( φ ) around the non-trivial (leftpanel) or trivial (right panel) minimum for both the potential and the Yukawa function, with N h = 8and N v = 9 in the LPA (cid:48) . amplitudes, that is by allowing for higher polynomial couplings.Once we turn to the dependence of the results on X f , which is shown in Tab. VIII andTab. IX, it becomes visible how the difference between the LPA and the LPA (cid:48) can be negligibleonly for unphysical very small values of X f . For θ , it is the 7% at X f = 0 .
3, and the 14% X f . κ .
377 10 − .
793 10 − .
253 10 − .
315 10 − .
169 10 − .
125 10 − λ λ y y θ θ − − − − − − θ − − − − − − η ψ η φ X f λ − .
366 10 − .
137 10 − λ λ y y θ θ − − − − − − θ − − − − − − η ψ .
347 10 − .
939 10 − .
341 10 − .
073 10 − η φ TABLE IX: Case d = 3 and various X f , polynomial expansion of y ( ρ ) around the non-trivial (left panel)or trivial (right panel) minimum for both the potential and the Yukawa function, with N h = 8 and N v = 9in the LPA (cid:48) . X f = 1. On the contrary, as we will see later in this Section by comparing our resultsto the literature, the effect of the inclusion of higher Yukawa couplings decreases with incresing X f . The transition between the SSB and the symmetric regime for the FP potential in theLPA (cid:48) is around X f = 1 .
62, while it occurs at X f = 2 .
31 for truncations with a linear Yukawafunction [19]. From these tables it also seems reasonable to expect that in the X f → θ and η φ is compatible with an approach to thecorresponding Ising values, thus further supporting the discussion at the end of Sect. II. As faras the X f → ∞ limit is concerned instead, the smooth transition to the large- X f exponents isevident in the right panels of Tab. VIII and Tab. IX.Let’s now come to the comparison of our results with the literature. The classic methods forthe investigation of the critical properties of the Gross-Neveu and Yukawa models are the (cid:15) - andthe 1 /N f -expansions [2–8]. The former can be of great utility since both expansions around theupper and the lower critical dimensions give comparable results, such that d = 3 does not seem atoo wild extrapolation. Yet, some treatment for these asymptotic series is needed. Resummationis unfortunately out of reach since they are known only up to the second or third order [3, 5],apart for the anomalous dimensions for which the computations have been pushed up to thefourth order [6]. Polynomial interpolations of the two different (cid:15) -expansions have been studiedin [18] for the case X f = 8, and we report their results borrowing their notations, such that P i,j denotes a polynomial which is i -loop exact near the lower critical dimension, and j -loop exactnear the upper. We also report the crude extrapolations that are obtained by simply setting (cid:15) = 1 in the expansions of θ = ν − , η φ and η ψ . Also the 1 /N f expansion clearly needs somecare, since we are interested in low number of fermions. Actually we are going to refer to thismethod only for X f = 8 and X f = 4, corresponding to N f = 2 and N f = 1 respectively. Againonly the second or third order is known [7, 8]. For the correlation-length exponent θ = ν − weadopt the Pad´e approximant used in [18], while for the anomalous dimensions we refer to thePad´e - Borel treatment reported in [15].The available FRG literature is rich and it offers a precious background on which we canmeasure the effects of the enlargement of the truncation discussed in this work. Essentially allthe past studies considered the LPA (cid:48) , including a scalar potential and a simple linear Yukawa We made use of the formulas reported in [3], with typos corrected according to the observations of [18]. X f = 1 case, which retained a full superpotential [39, 41, 42], thus includingmulti-meson exchange in the Yukawa sector, and sometimes was pushed to the next-to-next-to-leading order of the (supercovariant) derivative expansion. Also the choice of regulators isdiverse, comprehending the linear, the sharp and the exponential ones (which in the tableswe abbreviate with lin, sha, exp). In some studies the scalar potential was approximated bypolynomial truncations in the symmetric regime, for which we provide the corresponding N v ( N w in case of truncations of the superpotential for supersymmetric flows). In others, that welabel by N v = ∞ (or N w = ∞ ), the differential equations for the FP and the perturbationsaround it were solved by numerical methods, which are different from paper to paper. Ourresults are labeled by N h > X f Yukawa models. For X f = 8 two lattice calculations of the critical exponents are available. One based on staggeredfermions [10], though ignoring a sign problem, provides results which are in good agreementwith continuum methods, as it appears from Tab. X. An independent work applying the fermionbag approach [12], that is free from the sign problem, is instead offering very different results: ν = 0 . η φ = 0 . η ψ = 0 . . Recently, another sign-problem-freesimulation adopting the continuous time quantum Monte-Carlo method for a model of spinlessfermions on a honeycomb lattice, provides estimates of the critical exponents of the chiral Isinguniversality class for X f = 4, i.e. a single Dirac field [13]. These results are compared to thoseemerging from the continuum methods in Tab. XI. Surprisingly they are much closer to ourestimates for the case X f = 2, see Tab. XII.Regarding the latter case, notice that the results from [15] are affected by the absence of someterms in the flow equations that, being proportional to the vev of the scalar, become importantfor X f ≤ . Their effect significantly reduces the value of ν . Since upon inclusion of multi-meson exchange the transition from the symmetric to the SSB regime occurs at lower values of We are grateful to H. Gies for informing us about these discussions. See the discussion in [19]. θ η φ η ψ FRG ( N v = 9 , N h = 8) lin (this work) 1.004 0.996 0.789 0.031FRG ( N v = 3 , N h = 1) exp [15] 1.016 0.984 0.786 0.028FRG ( N v = 6 , N h = 1) sha [18] 1.022 0.978 0.767 0.033FRG ( N v = 11 , N h = 1) lin [16] 1.018 0.982 0.760 0.032FRG ( N v = ∞ , N h = 1) lin [15] 1.018 0.982 0.756 0.032FRG ( N v = ∞ , N h = 1) lin [19] 1.018 0.982 0.760 0.032Monte-Carlo [10] 1.00(4) 1.00(4) 0.754(8) —1 /N f (cid:15) ) 3rd order [5] 1.309 0.764 0.602 0.081(4 − (cid:15) ) 2nd order [3] 0.948 1.055 0.695 0.065 P , interpolated (cid:15) -expansion [18] 1.005 0.995 0.753 0.034 P , interpolated (cid:15) -expansion [18] 1.054 0.949 0.716 0.041 TABLE X: Critical exponents for X f = 8. For a short description of the approximations involved in eachmethod, see the main text. ν θ η φ η ψ FRG ( N v = 9 , N h = 8) lin (this work) 0.929 1.077 0.602 0.069FRG ( N v = 3 , N h = 1) exp [15] 0.962 1.040 0.554 0.067FRG ( N v = ∞ , N h = 1) lin [15, 19] 0.927 1.079 0.525 0.071Monte-Carlo [13] 0.80(3) 1.25(3) 0.302(7) —1 /N f − (cid:15) ) 2nd order [3] 0.862 1.160 0.502 0.110 TABLE XI: Critical exponents for X f = 4. For a short description of the approximations involved ineach method, see the main text. X f , our computations are still in the symmetric regime. This might qualitatively explain thedrastic departure from the results of [19].Also the comparison for X f = 1, which is presented in Tab. XIII, requires some comments.Let us recall that for this field-content the system at criticality is described by a N = 1 Wess-Zumino model [17, 23]. Hence, if the regularization does not break supersymmetry, the criticalanomalous dimensions of the scalar and of the spinor should be equal. Furthermore, a superscal-ing relation ν − = ( d − η ) /
2, which was first observed in [40] and later proved to hold at any orderin the supercovariant derivative expansion in [41], is expected to hold. This is what happens for ν θ η φ η ψ FRG ( N v = 9 , N h = 8) lin (this work) 0.814 1.229 0.372 0.131FRG ( N v = 3 , N h = 1) exp [15] 0.633 1.580 0.319 0.113FRG ( N v = 3 , N h = 1) lin [15] 0.623 1.605 0.308 0.112FRG ( N v = ∞ , N h = 1) exp [15] 0.640 1.563 0.319 0.114FRG ( N v = ∞ , N h = 1) lin [15] 0.621 1.610 0.308 0.112FRG ( N v = ∞ , N h = 1) lin [19] 0.4836 2.068 0.3227 0.1204(4 − (cid:15) ) 2nd order [3] 0.773 1.293 0.317 0.154 TABLE XII: Critical exponents for X f = 2. For a short description of the approximations involved ineach method, see the main text. θ θ η φ η ψ θ FRG ( N v = 9 , N h = 8) lin (this work) 0.693 1.443 − N w = ∞ ) opt n = 2 NLO [41] 0.711 1 . − .
771 0.186 0.186 0.186SUSY FRG ( N w = ∞ ) opt n = 2 NNLO [41] 0.710 1 . − .
715 0.180 0.180 0.180SUSY FRG ( N w = ∞ ) opt n = 1 [42] 0.708 1.413 − N w = ∞ ) opt n = 2 [42] 0.706 1.417 − N v = 2 , N h = 1) 1-loop [17] 0.72 1.39 − − (cid:15) ) 1st order [23] — — — 0.143 0.143 —(4 − (cid:15) ) 2nd order [3] 0.710 1.408 — 0.184 0.184 0.184Conformal Bootstrap [43] — — — 0.13 0.13 — TABLE XIII: Critical exponents for X f = 1. About the FRG results, the schemes, the regulators, andthe approximations are very different, see the main text. example in the (cid:15) -expansions or in the SUSY FRG. Since the scheme adopted in the present workexplicitely breaks supersymmetry, we expect and we observe violations of these properties. Alsoin [17] supersymmetry is broken by regularization, and these violations are present, but theycould be partially reduced or canceled by tuning the regulator. This tuning gives the resultsreported in Tab. XIII. A similar analysis of the regulator dependence of universal quantitiesand of the consequent breaking of supersymmetry could be performed in future studies for thepresent family of truncations. Yet, even by explicitly breaking the FP supersymmetry, we getexponents which are not very far from the ones produced by the above mentioned methods. Letus add few details on the SUSY FRG results shown in Tab. XIII. They are obtained by settingone of the regulators to zero, and choosing a shape similar to the linear regulator for the other,with an exponent n that differentiates between the conventional linear regulator (opt n = 2) anda slight variant (opt n = 1). Also the truncation scheme is different from the one discussed inthe present paper, since it is related to an expansion in powers of the supercovariant derivative,that has been considered at the level of the LPA (cid:48) [39, 42], at next-to-leading order (NLO) or atnext-to-next-to-leading order (NNLO) [41]. For the case X f = 1 we can also compare with apioneering study based on the conformal bootstrap [43]. In Tab. XIII we included the one-loopcomputations of [17, 23], even if two-loop results are on the market [3], on the base of the naiveobservation that for Yukawa systems with complex scalars and spinors, whose FP should effec-tively show N = 2 supersymmetry [24], the anomalous dimensions obtained from the first-orderof the (4 − (cid:15) ) expansion, η φ = η ψ = 1 /
3, agree with the available exact results [44].35 h Φ c FIG. 13: Spike plots for X f = 0, v ( φ ) = 0 and d ∈ { . , . , . , . , . } from red (upper) to blue(lower) in the LPA (cid:48) . VII. D=4
From the leading order of the 1 /X f -expansion one expects that for large enough X f the chiralIsing FP merges with the Gaußian FP in the d → X f = 0, for which we knowfrom the discussion at the end of Sect. II that only mirrored images of the purely scalar FPs canexist, one can observe that the latter merge with the Gaußian FP for d →
4, compatibly withthe presumed triviality of four-dimensional scalar theory. This is illustrated in Fig. 13, whichis produced as Fig. 3 but integrating only the FP equation for h ( φ ) at v ( φ ) = 0 and X f = 0in the LPA (cid:48) . Yet, it remains to be shown what happens for a small non-vanishing number offermions. Dimensional analysis indicates d = 4 as the upper critical dimension for any X f . Thiscan be checked by means of the FRG, either by numerical integration of the FP equation, asit was shown for example in Sect. IV for X f = 1, or by the polynomial truncations discussedin the last Sections. Indeed, the latter have already been used in the past, precisely to addressthis question.In fact, an exploratory study of what happens to the d → Z -symmetric Yukawamodel with very small X f was performed in [27], in order to test a mechanism for the generationof nontrivial FPs in fermion-boson models, that has subsequently found in chiral-Yukawa modelssome natural candidates [28]. That analysis pointed out that within a ( N v = 2 , N h = 1)polynomial truncation, according to the scheme of Eq. (VI.7), the FRG detects nontrivial FPsalso in d = 4, for unphysical small values of X f . This holds both in the LPA and in the LPA (cid:48) .However, the fact that the FP position and the critical exponents are significantly different inthe two approximations was interpreted as a signal of the need to include further boson-fermion36nteractions in the truncation, in order to understand if these FPs are physical or merely anartifact of the approximations. This Section reports on the changes brought by the differenttreatment of the Yukawa sector presented in this work.At the level of the LPA we generated three-dimensional plots similar to the ones illustratedin Fig. 1, second panel, by shooting from the origin with random values of ( v (cid:48)(cid:48) (0) , h (cid:48) (0)), forseveral values of X f <
1, and we looked for spikes signaling possible FPs, but we have not foundany of them. We were also not able to produce any global solution studying numerically theCauchy problem from the asymptotic region, along the lines of Sect. V. We then re-consideredthe analysis at the level of polynomial truncations. Already trying to reproduce the resultsof [27] in other truncations with N v = 2 and N h = 1, can be a nontrivial test, because of thedifferent beta-function of the Yukawa coupling, associated to different projection rules. We havealready argued that a change of the results depending on the parameterization employed signalsthe presence of errors induced by the use of inconsistent truncations. We first concentrated onthe LPA, which at least for d < X f ≤ X f = 0 . κ = 0 . , λ = 27 . , g = 81 .
13 (VII.1)with two relevant directions θ = 2 . , θ = 0 . , θ = − . . (VII.2)We observed that in a polynomial truncation of y ( ρ ) as in Eq. (VI.6), the FP position is different κ = 0 . , λ = 54 . , y = 494 . θ = 1 . , θ = 0 . , θ = − . . (VII.4)Still, the changes are not dramatic. On the other hand, we could not find any real FP within thesame order of the truncation of h ( φ ) given in Eq. (VI.5). We tried to circumvent this problemas in d = 3, by following the FP found in one parameterization to higher orders, and thentranslating back to the other parameterization. Yet, we were not able to reveal the FP for y ( ρ )for bigger values of N v and N h , nor to find it by chance in different orders of the truncation of h ( φ ). 37oping that the inclusion of the wave function renormalizations could stabilize the polynomialtruncations and help us in the search for FPs, we then considered LPA (cid:48) , using the resultsof [27] as a guide for the localization of the interesting region in the space of couplings. Whilethe FP is present in the first order of the truncation of Eq. (VI.7), we could not find it inthe parameterizations considered in this paper. Let us once more stress that this does notcompletely exclude that it can be found by other methods, even if we consider this very unlikely.Nevertheless, for the LPA (cid:48) we have not tried a numerical shooting at nonvanishing X f as inthe LPA. Hence, a more careful numerical analysis is needed, to exclude with a higher levelof confidence the presence of low- X f FPs in the theory space described by the truncation inEq. (II.2). A even better test would be to consider the full next-to-leading order of the derivativeexpansion.
VIII. CONCLUSIONS
A proper quantitative control of the quantum dynamics of the Z -symmetric Yukawa model,beyond the domain of applicability of perturbative methods, is important not only from ageneric field-theoretical point of view, but also for phenomenological reasons, since the latter isvery useful as a toy-model of numerous condensed matter systems, as well as of specific sectors ofmodern particle theory, see Sect. I for more details. The functional renormalization group (FRG)is a simple analytic nonperturbative method that can provide a detailed description of stronglycoupled systems, under approximations that are testable and improvable in several systematicways. Furthermore, these results can be produced, almost simultaneously, in a continuousnumber of spacetime dimensions d and fermionic degrees of freedom X f , thus allowing for aquick analysis of the dependence of the dynamics on the latter parameters.In this work we focused on the critical behavior of the Z -symmetric Yukawa model at zerotemperature and density. Our principal aim was to test the impact of multi-meson-exchange,encoded in a Yukawa coupling which is a full function of the scalar field, on the FRG descriptionof the latter behavior, a question that to our knowledge has never been considered before.Nevertheless, our analysis is relevant not only for the FRG community. For instance, in Sect. IIIwe discussed the leading order of the 1 /X f -expansion, whose results can be directly exported outof the FRG framework in which we produced them, and recovered also by different methods. Thisstudy illustrated how, by allowing for multi-meson-exchange, one can describe the generation of38ulti-critical conformal Yukawa models as d is lowered from d = 3 towards d = 2, across thecorresponding upper critical dimensions d n = 2 + 2 /n , with n a positive integer. We also showedhow the large- X f limit quantizes the corresponding critical anomalous dimension η φ = d n − d .In Sect. IV we checked that this pattern of generation of critical theories as a function of d holds also at X f = 1, and presumably at any other finite X f . This would imply that in the d → X f <
1, at least within the ansatz of Eq. (II.2). Let us remark that, as far as we know,the observation of multi-critical conformal Yukawa models at finite X f and in continuous fractaldimensions 2 < d < X f results, they indicate that in several cases the effect of multi-meson-exchange cannot be neglected, either quantitatively or even qualitatively. We argued that thesehigher Yukawa interactions are required by consistency of the truncation, otherwise the solutionsof the system of differential equations defining the flow of the scalar potential v ( φ ) and of theYukawa “potential” h ( φ ) would depend on the chosen parametrization of these functions. Forinstance, the same FP solutions should be reproduced using any polynomial truncation of thesefunctions, at least within a certain domain. On these grounds we believe that general FRGstudies of Yukawa models should at least consider the inclusion of these interactions, and possiblycheck when they can actually be neglected.On the quantitative side, in Sect. V we explicitly numerically constructed these global FPsolutions for d = 3 and several values of X f . These results include the Gross-Neveu univer-sality class for X f >
1, and the superconformal N = 1 Wess-Zumino model for X f = 1. At X f = 1, we also numerically computed the critical exponent ν , and the corresponding linearperturbation around the FP. In Sect. VI we showed how the results of the global analysis can beeasily reproduced by two different kinds of high-order polynomial truncations. However, thesestudies were performed in the local-potential approximation (LPA), that is by neglecting therenormalization of the fields. Taking into account the anomalous dimensions (LPA (cid:48) ) was crucialto obtain a more accurate picture, especially for X f >
1, so that in Sect. VI B we developed aLPA (cid:48) analysis, based on the same polynomial truncations which were proved to be trustworthyin the LPA studies.This allowed us to to produce estimates of the critical exponents ν , η φ and η ψ , in d = 3 andvarious X f , and to compare them with some of the existing literature. We concentrated on the39specially interesting cases of two and one massless Dirac ( X f = 8 and 4), of a Weyl ( X f = 2),and of a Majorana spinor ( X f = 1). They can be found in Tab. X, XI, XII, XIII. Often, therestill exists some significant mismatch among the available estimates, such that more studies byall kinds of methods, including Monte-Carlo simulations and higher-order (cid:15) - or 1 /N f -expansions,are welcome. As far as the FRG is concerned, the results seem stable for X f ≥
4, while forlower number of fermions there is still room for debate, and probably larger truncations areneeded. The supersymmetric case X f = 1 is an exception also in this sense, since it enjoys agood agreement among the results produced with different methods.Larger truncations, such as a next-to-leading order of the derivative expansion, are anywayneeded for a quantitative analysis of multi-critical models in 2 < d <
3, as we argued inApp. III in the large- X f limit. Still within the LPA (cid:48) , the next natural step is to produce globalnumerical studies similar to the ones presented for the LPA in Sect. IV and V. Regarding thepossible applications of the present analysis to different models, one possibility is to enlarge thesymmetry group from Z to O (N). The N= 3 three-dimensional chiral Heisenberg universalityclass, for instance, can be interesting for the physics of electrons in graphene [18]. With anenlarged symmetry, the effect of different representations would become a natural case-studyand would further widen the class of physical applications of these studies [45]. The same kindof truncation can also be used in the context of a Yukawa model interacting with gravity, alongthe lines of [46], to investigate first the asymptotic safety properties of the model, and then toconstruct global flows from the UV to the IR. Some scenarios could be of particular interest forcosmology. Acknowledgments
We would like to thank Alessandro Codello, Stefan Rechenberger, Michael Scherer and Ren´eSondenheimer for inspiring discussions, and Julia Borchardt, Tobias Hellwig, Benjamin Knorrand Omar Zanusso for providing us some of their results, which can be found in Sect. VI B,as well as for valuable comments. We are grateful to Holger Gies for several explanations andsuggestions and for a critical reading of the manuscript.L. Z. acknowledges support by the DFG under grant GRK1523/2.40 ppendix A: Regulators and threshold functions
We have to evaluate the r.h.s of Eq. (II.1), for which we need the Γ (2) k matrix. Consideringthe field ψ as a column and ¯ ψ as a row vector, let us denote by Φ T ( q ) the row vector withcomponents φ ( q ), ψ T ( q ), ¯ ψ ( − q ), and by Φ( p ) the column vector given by its transposition.Then Γ (2) k is obtained by the formulaΓ (2) k = −→ δδ Φ T ( − p ) Γ k ←− δδ Φ( q ) . This inverse propagator is regularized by addition of the following regulator R k ( q, p ) = δ ( p − q ) R B ( p ) 00 R F ( p ) , where R B ( p ) = Z φ p r B ( p ) ,R F ( p ) = − δ ij p/ T δ ij p/ Z ψ r F ( p ) , is a 2 d γ N f × d γ N f matrix. In principle one can have different regulators for the scalar (B) andfor the spinors (F). A compact way to rewrite the flow equation is ∂ t Γ k = 12 ˜ ∂ t STr log(Γ (2) k + R k ) , where ˜ ∂ t ≡ ∂ t ( Z φ r B ) Z φ · δδr B + ∂ t ( Z ψ r F ) Z ψ · δδr F and · denotes multiplication as well as integration over the common argument of the shapefunctions of the two factors. Then the regularized kinetic (or squared kinetic) terms are givenby P B / F ( x ) = x (1 + r B / F ( x )), and the loop momentum integrals appearing on the r.h.s. of theflow equation give rise to corresponding regulator dependent threshold functions. Introducing41he abbreviation (cid:82) p ≡ (cid:82) d d p (2 π ) d these threshold functions read l (B / F) d ( ω ) = k − d v d (cid:90) p ˜ ∂ t log (cid:0) P B / F + ωk (cid:1) l (B / F) d ( ω ) = − k − d v d (cid:90) p ˜ ∂ t P B / F + ωk l (FB) dn ,n ( ω , ω ) = − k n + n ) − d v d (cid:90) p ˜ ∂ t P F + ω k ) n ( P B + ω k ) n m (F) d ( ω ) = − k − d v d (cid:90) p p ˜ ∂ t (cid:18) ∂∂p P F + ωk (cid:19) m (F) d ( ω ) = − k − d v d (cid:90) p p ˜ ∂ t (cid:18) ∂∂p r F P F + ωk (cid:19) m (B) d ( ω ) = − k − d v d (cid:90) p p ˜ ∂ t (cid:32) ∂∂p P B ( P B + ω k ) (cid:33) m (FB) d , ( ω , ω ) = − k − d v d (cid:90) p p ˜ ∂ t (cid:32) r F P F + ω k ∂∂p P B ( P B + ω k ) (cid:33) . In this work we adopted the linear regulator xr B ( x ) = (1 − x ) θ (1 − x ), where x = q /k . Forspinors this corresponds to a shape function r F such that x (1 + r B ( x )) = x (1 + r F ( x )) . For it,the threshold functions can be computed analytically, and give l (B) d ( ω ) = 2 d − η φ d +2 ω ,l (F) d ( ω ) = 2 d − η ψ d +1 ω ,l (B / F) d ( ω ) = − ∂∂ω l (B / F) d ( ω ) ,l (FB) dn ,n ( ω , ω ) = 2 d (cid:34) n − η ψ d +1 (1 + ω ) n (1 + ω ) n + n − η φ d +2 (1 + ω ) n (1 + ω ) n (cid:35) ,m (F) d ( ω ) = 1(1 + ω ) ,m (F) d ( ω ) = 1(1 + ω ) + 1 − η ψ ( d − ω ) − (cid:18) − η ψ d − (cid:19) ω ) ,m (B) d ( ω ) = 1(1 + ω ) ,m (FB) d , ( ω , ω ) = 1 − η φ d +1 (1 + ω )(1 + ω ) . ppendix B: Properties of the large- X f solutions In both versions of the LPA, with or without η φ = 0, and also in the LPA (cid:48) , Eq. (III.8) enablesus to write the potentials in the form h ( φ ) = c h φ n , v ( φ ) = c v φ dn − v d d F (cid:18) , − d − d − c h φ n (cid:19) . (B.1)The behavior of v for φ → ±∞ is v asympt ( φ ) (cid:39) (cid:18) sgn( φ ) dn c v + Γ( − d/ d +1 π d/ | c h | d (cid:19) | φ | dn (B.2)and, since we are assuming 2 < d <
4, the gamma function in front of | c h | d is positive. If c v (cid:54) = 0,the scalar potential can be real only if ( − dn has a real branch, that is if d = mn j , j ∈ { , , , . . . } , m ∈ N , nj < m < nj . (B.3)Its stability further requires | c h | d ≥ d +1 π d/ Γ( − d/
2) max {− c v , ( − dn c v } = c dh, crit (B.4)and for special values of c v and c h , namely when | c h | = | c h, crit | , it can become asymptoticallyflat (possibly only on one side) instead of growing like φ dn .In order to understand the physical properties of the large- X f FP’s, we need to considerthe RG flow in vicinity of the corresponding critical points. In particular we consider thelinearization of the flow, by looking at small fluctuations of the potentials v = v + δv , h = h + δh and for eigenvalue solutions ˙ δv = − θδv , ˙ δh = − θδh . These equations at large- X f are extremely simple and, for the linearized regulator, they read − θδv = − dδv + 1 n φδv (cid:48) + δη φ φv (cid:48) + 4 v d d hδh (1 + h ) (B.5) − θδh = − δh + 1 n φδh (cid:48) + δη φ φh (cid:48) . (B.6)In this appendix we want to sketch a study of the properties of these FPs as well as of thelinearized flow around them. We believe it can be instructive to consider separately the resultsobtained with or without the inclusion of the flow equation for η φ . This will make evident thatlarger truncations, out of the reach of the present work, are necessary to get a complete pictureof the large- X f multicritical Yukawa theories. 43 - Φ - - - v H Φ L - - Φ v H Φ L FIG. 14: The d = 3, n = 2, FP scalar potential at nonvanishing c v . Left panel: c v = − c h ∈{ c h, crit + 10 − , c h, crit , c h, crit − − } , from bounded (green) to unbounded (blue). Right panel: c v = 1and c h ∈ { c h, crit + 2 , c h, crit , c h, crit − } , from steeper (green) to broader (blue). - - Φ - - v H Φ L - - Φ v H Φ L FIG. 15: The d = 8 / n = 3, FP scalar potential at nonvanishing c v . Left panel: c v = − c h ∈ { c h, crit + 10 − , c h, crit , c h, crit − − } , from bounded (green) to unbounded (blue). Right panel: c v = 1 and c h ∈ { c h, crit + 2 , c h, crit , c h, crit − } , from steeper (green) to broader (blue).
1. LPA
If we set by hand η φ = 0, regardless of c v or c h Eq. (III.8) leaves a discrete set of dimensionsas the only possibility, the ones in Eq. (III.11). As a consequence dn = 2( n + 1) and the scalarpotential is real and even also in case c v (cid:54) = 0. The stability properties, depending on c h and c v according to Eq. (B.4), are illustrated in the plots of Fig. 14 and Fig. 15. The special case c v = 0 is shown in Fig. 16.Let us now turn to the linear perturbations of these FP’s. By definition in the LPA oneneglects a possible change of anomalous dimension. Thus, setting δη φ = 0, the solution for the44 - Φ v H Φ L - - Φ v H Φ L FIG. 16: The even FP scalar potentials for c v = 0. For illustration the value of c h has been chosenaccording to Eq. (B.12), even if this is mandatory only for n = 1 in the LPA (cid:48) . Left panel: n = 1 and d ∈ { . , , . } , from steeper (green) to broader (blue). Right panel: n ∈ { , , , } , in the correspondingdimension d = 2 + 2 /n , from steeper (green) to broader (blue). perturbations reads δh ( φ ) = δc h φ N (B.7) δv ( φ ) = δc v φ ( d − n + N − v d d c h δc h φ n + N (cid:20)
11 + c h φ n − dd − F (cid:18) , − d − d − c h φ n (cid:19)(cid:21) Here we restricted our analysis to the perturbations with δc h (cid:54) = 0, and required their smoothnessby setting (1 − θ ) n = N ∈ N . For the special case δc h = 0 the solution is simply δv ( φ ) = δc v φ M with critical exponent θ M = d − M/n , and will not be discussed any further. Notice that theseeigenfunctions are independent of c v , which is due to the suppression of scalar loops in thelarge- X f limit. They are regular at the origin, since the leading behavior is δv ( φ ) ∼ φ → v d d ( d − c h δc h φ n + N . (B.8)Recall that the FP potential had, as leading small field dependence, φ n ; as a consequence,the relevant perturbations with N < n change the behavior of the potential at the origin, themarginal ones only change the coefficient in front of φ n , and the irrelevant ones leave the leadingterm unaltered. For large value of the field δv ( φ ) ∼ φ →∞ (cid:18) δc v + sgn( φ ) dn d Γ( − d/ d +1 π d/ | c h | d − c h δc h (cid:19) φ dn + N − n (B.9)while the FP potential behaves like | φ | dn at infinity. As a consequence, the irrelevant perturba-tions with N > n completely change the asymptotic behavior of the potential for large fields,the marginal ones with N = n only change the coefficient in front of the leading power, and the45 - - - Φ - - v H Φ L - - Φ - - v H Φ L FIG. 17: The d = 3 and n = 1 FP scalar potential at nonvanishing c v . Left panel: c v ∈{ c v, crit − , c v, crit , c v, crit + 1 } , from bounded (green) to unbounded (blue). Right panel: c v ∈ {− c v, crit − , − c v, crit , − c v, crit + 1 } , from bounded (green) to unbounded (blue). Notice that the value of the potentialat the origin is arbitrary, while its behavior for large fields is not. relevant ones only change the sub-leading terms. Clearly this is not the case for those potentials,with special values of c v , that are asymptotically flat.Let’s now discuss the symmetry properties of the perturbations. Trivially, the symmetry ofYukawa potential under Z is preserved or violated depending on n and N . We now want tounderstand what this entails for the scalar potential. Recall that in the LPA dn = 2( n + 1)and the FP v is always even. Then, the fluctuations behave as φ n + N +2 , and whenever N + n is odd, the Z symmetry of both h and v at the FP is spoiled by the perturbations. Amongthese symmetry breaking perturbations, the irrelevant ones, with N > n , give rise to unstablepotentials. Notice that the relevant perturbations, even if spoiling symmetry, do not directlycause instabilities (though they might induce them indirectly, i.e. beyond linearization). Thepossibility to have stable theories with no definite Z symmetry emanating from symmetric FPsin the UV or IR is in any case a question that requires a global study of the RG flow, and it isbeyond the scope of this work.
2. LPA (cid:48)
So far we have not used the flow equation for η φ . In order to do so, we first have to analyzethe possible presence of a nontrivial minimum for v . The general expectation is that, since onlyfermion loops survive in the leading order of the 1 /X f expansion, the potential is always in thesymmetric regime. This is suggested by the expansion of the potential around the origin, basedon Eq. (III.7). We assume that this is always the case for the time being, as it is indeed for46 - Φ - - v H Φ L - Φ - - v H Φ L FIG. 18: The d = 7 / n = 1 FP scalar potential at nonvanishing c v . Left panel: c v ∈{ c v, crit − , c v, crit , c v, crit + 1 } , from bounded (green) to unbounded (blue). Right panel: c v ∈ {− c v, crit − , − c v, crit , − c v, crit + 1 } , from bounded (green) to unbounded (blue). every specific example we have considered. Under this assumption, we need to take the φ → η φ , which is proportional to h (cid:48) ( φ ) , i.e. to φ n − . Therefore, onlyfor n = 1 such a limit can be nonvanishing. This shows how LPA (cid:48) is an improvement of LPAonly for the n = 1 critical theory. For the remaining values of n , one finds again η φ = 0, whichartificially forces the dimension d to its critical value. We expect this condition to be lifted bymore general truncations, and a nontrivial η φ should emerge for any n .Let’s then discuss the change brought by LPA (cid:48) in the description of the large- X f n = 1 FP.As argued in Sect. III, the nontrivial η φ allows for the existence of the non Gaußian FP in any d <
4, as long as η φ = 4 − d , n = 1 . (B.10)Actually this is the case only for the Z symmetric solution with c v = 0. As soon as c v (cid:54) = 0 thereality of the potential requires d = mj , j ∈ { , , , . . . } , m ∈ N , j < m < j . (B.11)Regardless of c v , by using Eq. (B.10) the flow equation for Z φ can be solved for c h as a functionof d , giving [16] c h = d (4 − d )( d − v d (6 d − . (B.12)Then, the stability condition Eq. (B.4) for the nonvanishing c v FP’s is best phrased as a boundon c v c v ≥ − c v, crit , c v, crit = Γ( − d/ d +1 π d/ (cid:20) d (4 − d )( d − v d (6 d − (cid:21) d/ (B.13)47nd additionally, only for odd m , c v ≤ c v, crit . The scalar FP potential with c v (cid:54) = 0 is an evenfunction if and only if m is even.Let us then turn to perturbations, and allow for a nontrivial δη φ . We postpone for a whilethe task of solving the linearized equation for η φ , which provides us the first correction δη φ tothe anomalous dimension, as a function of the FP h and δh . This is because such an equationinvolves the variation δφ in the location of the minimum of the potential, which in turn can becomputed from the variation of the potential by the formula δφ = − δv (cid:48) (0) v (cid:48)(cid:48) (0) (B.14)where we stuck to our assumption that the minimum of the FP potential is always trivial. As aconsequence we first solve for δv and δh as parametric functions of δη φ , and then plug Eq. (B.14)into the linearized equation for η φ , to compute the actual δη φ . Solving for δh is again trivial,and it immediately allows us to extract the eigenvalues of the linearized flow. When θ (cid:54) = 0 thesolution for δh is δh ( φ ) = δc h φ N − δη φ n n − N c h φ n , N ∈ N , N (cid:54) = n (B.15)where again we focused on δc h (cid:54) = 0 and set N = (1 − θ ) n ∈ N . For θ = 0 instead δh ( φ ) = δc h φ n − δη φ n c h φ n log( φ ) (B.16)Notice that the second term in the last equation is simply the first order in the expansion of c h φ / ( d − η φ ) , which is the exactly marginal h , around the n -th FP. As a consequence, theapparent instability that can come from the second term in Eq. (B.16) is actually a fake oflinearization, as long as δη φ > − /n . On the other hand, a logarithmic singularity at theorigin appears even beyond linearization, and we believe this to be a pathology produced bythe leading order in 1 /X f . The solution to this pathology will come soon, in the form of theconstraint δη φ = 0 for these perturbations.The equation for δv is much more involved in the LPA (cid:48) than in the LPA, since it now dependson the FP potential. Yet, its solutions for generic δη φ can be easily given analytically. It is notnecessary to show them here. It suffices to report that quite in general they have the property δv (cid:48) (0) = 0, as it could be expected by the argument that fermion loops are generally associatedwith scalar potentials with a trivial minimum . As a consequence the scalar potential stays in For δc h (cid:54) = 0 only the n = 1, N = 0 case gives rise to a nonvanishing δv (cid:48) (0). For δc h = 0 only the M = 1 case.In what follows we discard these cases. δv ( φ ) is also in the symmetricregime.With this piece of information, one can work out the linearized δη φ , by varying the r.h.s. ofthe flow equation for Z φ with respect to h and v (whose fluctuations still depend parametri-cally on δη φ itself) and η φ , while keeping φ fixed, and then taking φ →
0. The latter limitmakes the r.h.s. vanishing unless n = 1, in which case it reaches a d -dependent constant times c h δη φ . Hence, for general n and N we find δη φ = 0, which boils the analysis of the linearizedperturbations down to the one sketched in the last Section within the LPA. d = 4 The expression in Eq. (B.1), cannot be used in d = 4 nor in d = 2, since the hypergeometricfunction in v has simple poles at these values. The d → d → X f system of flow equations one can find the following FP solutions h ( φ ) = c h φ n , v ( φ ) = c v φ n + 164 π (cid:0) c h φ n − c h φ n log( c h + φ − n ) (cid:1) (B.17)where we already demanded the Yukawa potential to be smooth, according to Eq. (III.8). Thecrucial fact is again that the minimum of v is always trivial. This allow us to take the φ → η φ . For n = 1 this leaves us with the equation c h = η φ = 0, thusboiling every feature of the critical theory down to the classical counting. For n ≥ η φ = 0, which is inconsistent with Eq. (III.8) and therefore eliminates these solutions. [1] D. J. Gross and A. Neveu, Phys. Rev. D (1974) 3235.[2] B. Rosenstein, B. J. Warr and S. H. Park, Phys. Rev. D (1989) 3088; A. Kovner, B. Rosensteinand G. Gat, Helv. Phys. Acta (1992) 411.[3] B. Rosenstein, H. L. Yu and A. Kovner, Phys. Lett. B (1993) 381.[4] J. Zinn-Justin, Nucl. Phys. B (1991) 105; J. Zinn-Justin, “Phase transitions and renormalizationgroup”, Oxford, UK: Oxford Univ. Pr. (2007) 452 p Such a constant is actually infinite for the marginal perturbation, the r.h.s. inheriting a logarithmic singularityat the origin from δh . Yet the simple way to cure this pathology and get a self-consistent answer is to set δη φ = 0.
5] J. A. Gracey, Nucl. Phys. B (1990) 403; Nucl. Phys. B (1991) 657; C. Luperini and P. Rossi,Annals Phys. (1991) 371.[6] A. N. Vasiliev and M. I. Vyazovsky, Theor. Math. Phys. (1997) 1277 [Teor. Mat. Fiz. (1997)85]; J. A. Gracey, Nucl. Phys. B (2008) 330 [arXiv:0804.1241 [hep-th]].[7] J. A. Gracey, Int. J. Mod. Phys. A (1994) 567 [hep-th/9306106].[8] J. A. Gracey, Int. J. Mod. Phys. A (1994) 727 [hep-th/9306107]; A. N. Vasiliev, S. E. Derkachov,N. A. Kivel and A. S. Stepanenko, Theor. Math. Phys. (1993) 127 [Teor. Mat. Fiz. (1993)179].[9] S. Hands, A. Kocic and J. B. Kogut, Annals Phys. (1993) 29 [hep-lat/9208022].[10] L. Karkkainen, R. Lacaze, P. Lacock and B. Petersson, Nucl. Phys. B (1994) 781 [Erratum-ibid.B (1995) 650] [hep-lat/9310020].[11] E. Focht, J. Jersak and J. Paul, Nucl. Phys. Proc. Suppl. (1996) 709 [hep-lat/9509040]; S. Christofiand C. Strouthos, JHEP (2007) 088 [hep-lat/0612031].[12] S. Chandrasekharan and A. Li, Phys. Rev. D (2013) 021701 [arXiv:1304.7761 [hep-lat]].[13] L. Wang, P. Corboz and M. Troyer, New J. Phys. (2014) 10, 103008 [arXiv:1407.0029 [cond-mat.str-el]].[14] L. Rosa, P. Vitale and C. Wetterich, Phys. Rev. Lett. (2001) 958 [hep-th/0007093].[15] F. Hofling, C. Nowak and C. Wetterich, Phys. Rev. B (2002) 205111 [cond-mat/0203588].[16] J. Braun, H. Gies and D. D. Scherer, Phys. Rev. D (2011) 085012 [arXiv:1011.1456 [hep-th]]; J.Phys. A (2013) 285002 [arXiv:1212.4624 [hep-ph]].[17] H. Sonoda, Prog. Theor. Phys. (2011) 57 [arXiv:1102.3974 [hep-th]].[18] L. Janssen and I. F. Herbut, Phys. Rev. B (2014) 20, 205403 [arXiv:1402.6277 [cond-mat.str-el]].[19] J. Borchardt and B. Knorr, arXiv:1502.07511 [hep-th].[20] N. Dorey and N. E. Mavromatos, Phys. Lett. B (1990) 107; I. J. R. Aitchison and N. E. Mavro-matos, Phys. Rev. B (1996) 9321 [hep-th/9510058]; N. E. Mavromatos and J. Papavassiliou, Re-cent Res. Devel. Phys. (2004) 369 [cond-mat/0311421]. M. Franz and Z. Tesanovic, Phys. Rev. Lett. (2001) 257003; I. F. Herbut, Phys. Rev. Lett. (2002) 047006 [cond-mat/0110188]; Phys. Rev.B (2002) 094504 [cond-mat/0202491]; Phys. Rev. Lett. (2005) 237001 [cond-mat/0410557].[21] G. W. Semenoff, Phys. Rev. Lett. (1984) 2449; I. F. Herbut, Phys. Rev. Lett. (2006)146401 [cond-mat/0606195]; I. F. Herbut, V. Juricic and B. Roy, Phys. Rev. B (2009) 085116[arXiv:0811.0610 [cond-mat.str-el]]; V. P. Gusynin, S. G. Sharapov and J. P. Carbotte, Int. J. Mod.Phys. B (2007) 4611 [arXiv:0706.3016 [cond-mat.mes-hall]]; S. Hands and C. Strouthos, Phys.Rev. B (2008) 165423 [arXiv:0806.4877 [cond-mat.str-el]]; J. E. Drut and T. A. Lahde, Phys.Rev. Lett. (2009) 026802 [arXiv:0807.0834 [cond-mat.str-el]]; Y. Araki and T. Hatsuda, Phys.Rev. B (2010) 121403 [arXiv:1003.1769 [cond-mat.str-el]].[22] H. Gies and L. Janssen, Phys. Rev. D (2010) 085018 [arXiv:1006.3747 [hep-th]]; L. Janssen and . Gies, Phys. Rev. D (2012) 105007 [arXiv:1208.3327 [hep-th]].[23] T. Grover, D. N. Sheng and A. Vishwanath, Science (2014) 6181, 280 [arXiv:1301.7449 [cond-mat.str-el]].[24] L. Balents, M. P. A. Fisher and C. Nayak, Int. J. Mod. Phys. B (1998) 1033 [cond-mat/9803086[cond-mat.supr-con]], S. S. Lee, Phys. Rev. B (2007) 075103 [cond-mat/0611658].[25] Y. Nambu and G. Jona-Lasinio, Phys. Rev. (1961) 345; J. B. Kogut, M. A. Stephanov andC. G. Strouthos, Phys. Rev. D (1998) 096001 [hep-lat/9805023]. S. Chandrasekharan, J. Cox,K. Holland and U. J. Wiese, Nucl. Phys. B (2000) 481 [hep-lat/9906021]; S. Hands, B. Luciniand S. Morrison, Phys. Rev. D (2002) 036004 [hep-lat/0109001].[26] D. U. Jungnickel and C. Wetterich, Phys. Rev. D (1996) 5142 [hep-ph/9505267].[27] H. Gies and M. M. Scherer, Eur. Phys. J. C (2010) 387 [arXiv:0901.2459 [hep-th]].[28] H. Gies, S. Rechenberger and M. M. Scherer, Eur. Phys. J. C (2010) 403 [arXiv:0907.0327 [hep-th]]; H. Gies, L. Janssen, S. Rechenberger and M. M. Scherer, Phys. Rev. D (2010) 025009[arXiv:0910.0764 [hep-th]]; H. Gies, S. Rechenberger, M. M. Scherer and L. Zambelli, Eur. Phys. J.C (2013) 2652 [arXiv:1306.6508 [hep-th]].[29] Y. Nambu, in: Z. Ajduk, S. Pokorski, A. Trautman ed. (World Scientific Pub., ISBN 9971-50-691-2, 1988, p. 1-10) 11. Warsaw symposium on elementary particle physics: new theories in physics,Kazimierz (Poland), 1988; V. A. Miransky, M. Tanabashi and K. Yamawaki, Phys. Lett. B (1989) 177; Mod. Phys. Lett. A (1989) 1043; H. Gies, J. Jaeckel and C. Wetterich, Phys. Rev. D (2004) 105008 [hep-ph/0312034].[30] O. Zanusso, L. Zambelli, G. P. Vacca and R. Percacci, Phys. Lett. B (2010) 90 [arXiv:0904.0938[hep-th]]; G. P. Vacca and O. Zanusso, Phys. Rev. Lett. , 231601 (2010) [arXiv:1009.1735 [hep-th]].[31] J. M. Pawlowski and F. Rennecke, arXiv:1403.1179 [hep-ph]; J. Braun, L. Fister, J. M. Pawlowskiand F. Rennecke, arXiv:1412.1045 [hep-ph].[32] C. Wetterich, Phys. Lett. B , 90 (1993).[33] K. Aoki, Int. J. Mod. Phys. B , 1249 (2000); J. Braun, J. Phys. G , 033001 (2012)[arXiv:1108.4449]; J. Berges, N. Tetradis and C. Wetterich, Phys. Rept. , 223 (2002) [hep-ph/0005122]; B. Delamotte, Lect. Notes Phys. , 49 (2012) [cond-mat/0702365 [COND-MAT]];P. Kopietz, L. Bartosch and F. Schutz, Lect. Notes Phys. , 1 (2010); T. R. Morris, Prog.Theor. Phys. Suppl. (1998) 395 [hep-th/9802039]; O. J. Rosten, Phys. Rept. (2012) 177[arXiv:1003.1366 [hep-th]].[34] G. Felder, Commun. Math. Phys. 111, 101-121 (1987).[35] A. Codello, J. Phys. A (2012) 465006 [arXiv:1204.3877 [hep-th]]; A. Codello and G. D’Odorico,Phys. Rev. Lett. (2013) 141601 [arXiv:1210.4037 [hep-th]]; A. Codello, N. Defenu andG. D’Odorico, arXiv:1410.3308 [hep-th].
36] T. R. Morris, Phys. Lett. B (1994) 355 [hep-th/9405190].[37] T. R. Morris, Phys. Lett. B (1995) 139 [hep-th/9410141].[38] T. R. Morris, Phys. Lett. B (1994) 241 [hep-ph/9403340]; T. R. Morris and M. D. Turner, Nucl.Phys. B (1998) 637 [hep-th/9704202].[39] F. Synatschke, J. Braun and A. Wipf, Phys. Rev. D (2010) 125001 [arXiv:1001.2399 [hep-th]].[40] H. Gies, F. Synatschke and A. Wipf, Phys. Rev. D (2009) 101701 [arXiv:0906.5492 [hep-th]].[41] M. Heilmann, T. Hellwig, B. Knorr, M. Ansorg and A. Wipf, JHEP (2015) 109 [arXiv:1409.5650[hep-th]].[42] T. Hellwig, A. Wipf, and O. Zanusso, in preparation.[43] D. Bashkirov, arXiv:1310.8255 [hep-th].[44] O. Aharony, A. Hanany, K. A. Intriligator, N. Seiberg and M. J. Strassler, Nucl. Phys. B (1997)67 [hep-th/9703110].[45] D. Mesterhazy, J. Berges and L. von Smekal, Phys. Rev. B (2012) 245431 [arXiv:1207.4054[cond-mat.str-el]].[46] R. Percacci and G. P. Vacca, arXiv:1501.00888 [hep-th].(2012) 245431 [arXiv:1207.4054[cond-mat.str-el]].[46] R. Percacci and G. P. Vacca, arXiv:1501.00888 [hep-th].