# Asymptotic freedom in \(\mathbb {Z}_2\)-Yukawa-QCD models

- 41 Downloads

## Abstract

\(\mathbb {Z}_2\)-Yukawa-QCD models are a minimalistic model class with a Yukawa and a QCD-like gauge sector that exhibits a regime with asymptotic freedom in all its marginal couplings in standard perturbation theory. We discover the existence of further asymptotically free trajectories for these models by exploiting generalized boundary conditions. We construct such trajectories as quasi-fixed points for the Higgs potential within different approximation schemes. We substantiate our findings first in an effective-field-theory approach, and obtain a comprehensive picture using the functional renormalization group. We infer the existence of scaling solutions also by means of a weak-Yukawa-coupling expansion in the ultraviolet. In the same regime, we discuss the stability of the quasi-fixed point solutions for large field amplitudes. We provide further evidence for such asymptotically free theories by numerical studies using pseudo-spectral and shooting methods.

## 1 Introduction

Gauged Yukawa models form the backbone of our description of elementary particle physics: they provide mechanisms for mass generation of gauge bosons as well as for chiral fermions via the Brout–Englert–Higgs mechanism. Many suggestions of even more fundamental theories beyond the standard model, such as grand unification, models of dark matter, supersymmetric models, etc., also involve the structures of gauged Yukawa systems. A comprehensive understanding of such systems is thus clearly indispensable.

Despite their fundamental relevance, gauged Yukawa systems can also exhibit a genuine conceptual deficiency. Many generic models develop Landau-pole singularities in their perturbative renormalization group (RG) flow towards high energies, indicating that these models may not be ultraviolet (UV) complete. If so, such models do not constitute quantum field theories which are fully consistent at any energy scale. Insisting on UV completeness by enforcing a UV cutoff to be sent to infinity typically requires to send the renormalized coupling to zero. This problem is also called *triviality*.

An important class of UV-complete nontrivial theories are those featuring asymptotic freedom [1, 2] which allow to send the cutoff to infinity at the expense of a vanishing bare coupling while keeping the renormalized coupling at a finite value. In fact, a conventional perturbative analysis [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] is capable of revealing the existence of asymptotically free gauged Yukawa models, and allows a classification in terms of their matter content and corresponding representations. Recent studies of aspects of such models [14, 15, 16] and constructions of phenomenologically acceptable models [17, 18, 19, 20, 21] have been performed; however, a unique route to an unequivocal model appears not obvious. Phenomenological constraints on the gauge and matter side typically require an appropriately designed scalar sector, as UV Landau poles often show up in the Higgs self-coupling.

The standard model is, in fact, not asymptotically free because of the perturbative Landau pole singularity in the U(1) gauge sector. Still, all other gauge couplings as well as the dominant top-Yukawa coupling and the Higgs self-coupling decrease towards higher energies. In fact, the value of the Higgs boson mass and the top quark mass are *near-critical* [22] in the sense that the perturbative potential approaches flatness towards the UV. Whereas a substantial amount of effort has been devoted to clarify whether the potential is exactly critical or overcritical (metastable and long-lived) in recent years [22, 23, 24, 25, 26, 27], a conclusive answer depends on the precise value of the strong coupling and the top Yukawa coupling [28, 29] as well as on the details of the microscopic higher-order interactions [30, 31, 32, 33, 34, 35, 36, 37, 38]. In summary, we interpret the present data as being compatible with the critical case of the Higgs interaction potential approaching flatness towards the UV. This viewpoint is also a common ground for the search for conformal extensions of the standard model [39, 40, 41, 42].

For the present work, this viewpoint serves as a strong motivation to study asymptotically free gauged Yukawa systems. Whereas perturbation theory seems ideally suited for this, conventionally made implicit assumptions may reduce the set of asymptotically free RG trajectories visible to perturbation theory. In fact, new asymptotically free trajectories in gauged-Higgs models have been discovered with the aid of generalized boundary conditions imposed on the renormalized action [43, 44]. This result has also been astonishing as it was obtained in a class of models which does not exhibit asymptotic freedom in naive perturbation theory. Still, the existence of these new trajectories has been confirmed by weak-coupling approximations, effective-field-theory approaches, large-*N* methods, as well as more comprehensively with the functional RG [44].

As such dramatic conclusions about the existence of new UV-complete theories requires substantiation and confirmation, the purpose of this work is to study the emergence of these new RG trajectories in a model that also exhibits asymptotic freedom already in standard perturbation theory. This allows to understand the novel features of the RG trajectories in greater detail. For this, we use the simplest gauged Yukawa system that exhibits asymptotic freedom perturbatively, it consists of a QCD-like matter sector with nonabelian SU(\(N_{\mathrm {c}}\)) gauge symmetry Yukawa-coupled to a single real scalar field. This \(\mathbb {Z}_2\)-Yukawa-QCD model can be viewed as a subset of the standard model [34, 45], with the Yukawa sector representing the Higgs boson and the top quark. In this model, the existence of asymptotically free trajectories has already been known since the seminal work of Cheng, Eichten, and Li [4] based on standard perturbation theory.

In the present work, we discover the existence of new asymptotically free trajectories in addition to the standard perturbative solution. For this, we follow the strategy of [43, 44] using effective-field-theory methods and the functional RG in order to get a handle on the global properties of the Higgs potential. We generalize the approach to an inclusion of a fermionic sector and also identify a new approximation technique (\(\phi ^4\)-dominance) that allows to get deeper analytical insight into the functional flow equations.

While the existence of new asymptotically free trajectories as well as some of their properties are reminiscent to the conclusions already found for the gauged-Higgs models [43, 44], we also find some interesting differences. Again, the class of new solutions has free parameters, such as a field- or coupling-rescaling exponent and the location of the (rescaled) minimum of the potential during the approach to the UV. For the present \(\mathbb {Z}_2\)-Yukawa-QCD model, we find that the exponent is more tightly constraint by the requirement of a globally stable potential. Also the rescaled potential minimum has to remain nonzero towards the UV, exemplifying the fact that the model develops a non-trivial UV structure which is not visible in the deep Euclidean region (DER). The present work thus pays special attention to the difference between working in the DER, as is often implicitly done in standard perturbation theory, and a more general analysis.

As our methods can address the global behavior of the potential, our work also adds new knowledge to the results known from standard perturbation theory: for the asymptotically free Cheng–Eichten–Li solution, we demonstrate that the potential is and remains globally stable when running the RG towards the UV; an analytic approximation of the potential can be given in terms of hypergeometric functions.

In Sect. 2, we review the standard analysis of asymptotic freedom for perturbatively renormalizable \(\mathbb {Z}_2\)-Yukawa-QCD models, for a generic number of colors and fermion flavors. We then specify our analysis to three colors and six flavors, to get closer to the standard model and only in Sect. 7, while summarizing most of our findings, we will generalize them to an arbitrary number of colors. In Sect. 3, we present the functional renormalization group (FRG) approach by which we derive the RG flow equations for our model. In Sect. 4 and Sect. 5, we generalize the treatment of Sect. 2 and include perturbatively nonrenormalizable Higgs self-interactions by polynomially truncating the FRG equations, as in effective field theory (EFT) approaches, within and beyond the deep Euclidean region. In the subsequent sections we then address the task of solving the FRG equation for a generic scalar potential. In Sect. 6, we construct functional approximations of asymptotically free solutions by inspecting a regime where the scalar fluctuations are dominated by a quartic interaction. Another description is then obtained from the expansion in powers of the weak Yukawa coupling in Sect. 7. Finally in Sect. 8, we substantiate our analytical results by using numerical tools, in particular pseudo-spectral and shooting methods. Conclusions are presented in Sect. 9.

## 2 Asymptotic freedom within perturbative renormalizability

Let us first review the standard analysis of this model at one loop, considering only the perturbatively renormalizable couplings [4]. The latter are the scalar mass \(\bar{m}\), the Higgs self-interaction \(\bar{\lambda }\), the Yukawa coupling \(\bar{h}\) and the strong gauge coupling \(\bar{g}_{\mathrm {s}}\). In particular, we address the UV behavior of this model, and look for totally AF trajectories. To this end, one focuses on the RG equations for the renormalized dimensionless couplings \(g_\mathrm {s}\), *h*, *m*, and \(\lambda \). Their definition in terms of the bare couplings and wave function renormalizations is the usual one, which we postpone to Sect. 3 for the moment.

*quasi-fixed point*(QFP) in Ref. [44]. It is a defining condition for AF scaling solutions and a useful tool to search for such trajectories [3, 7, 13, 14].

From another perspective, the AF trajectory defined by Eq. (8) can be viewed as an upper bound on the ratio of the Yukawa coupling and the gauge coupling at some initializing scale. For \(\hat{h}^{2}>\hat{h}_{*}^{2}\), asymptotic freedom is lost and the Yukawa coupling hits a Landau pole at a finite RG time towards the UV. The Yukawa coupling becomes AF only for \(\hat{h}^{2} \le \hat{h}_{*}^{2}\). Throughout the main text of this work, we will concentrate on the implications of the RG flow for the particular ratio defined by this upper bound where the flow of the Yukawa coupling is locked to the running of \(g_\mathrm {s}\). For \(\hat{h}^{2} < \hat{h}_{*}^{2}\), the Yukawa coupling is driven faster than the gauge coupling towards the Gaußian fixed point for high energies. These scaling solutions are sketched in “Appendix A”.

*deep Euclidean region*(DER) here, where all the masses are negligible compared to the RG scale. This implies in particular that any threshold effect given by the mass parameter

*m*of the scalar field is neglected. In case the system is in the symmetry-broken regime, effects from a nonvanishing vacuum expectation value on the properties of the top quark are also ignored for the moment, as they would alter the beta functions for the Yukawa coupling and the gauge coupling as well.

*P*is determined by the requirement that \(\hat{\lambda }_{2}\) achieves a finite positive value in the UV. The flow equation for this rescaled Higgs coupling then receives contributions from the \(\beta \) function of \(h^2\). As already stated, we focus on the AF trajectories with \(\hat{h}^2=\hat{h}_{*}^2\). In this case it is convenient to define an anomalous dimension for the Yukawa coupling by

In the remaining part of this paper, we restrict ourselves to an asymptotic UV running of the Yukawa coupling described by Eqs. (4) and (5). We will refer to the AF solution described by Eq. (5) and by the positive root in Eq. (16) as the *Cheng–Eichten–Li* (CEL) solution, since it was first described in Ref. [4]. The further AF solutions with \(\hat{h}^2<\hat{h}_{*}^2\) have also already been discussed in Ref. [4] as well as in later analyses [7, 14]; for completeness, we review them in “Appendix A”. For the remainder of the paper, we consider the asymptotic UV running of the Yukawa coupling of Eqs. (4) and (5), because it is most predictive: whereas classically the gauge coupling \(g_\mathrm {s}\), the Yukawa coupling *h* and the scalar self-interaction \(\lambda \) are independent, our AF trajectory locks the running of *h* and \(\lambda \) to that of \(g_\mathrm {s}\). Physically, this implies that the mass of the fermion (top quark) as well as that of the Higgs boson will be determined in terms of the initial conditions for the gauge sector and the scalar mass-like parameter, i.e., the Fermi scale. This maximally predictive point in theory space is also called the Pendleton-Ross point [52]. Let us finally emphasize that we focus exclusively on the UV behavior of our model class in the present work. The low-energy behavior will be characterized by possible top-mass generation from \(\mathbb {Z}_2\) symmetry breaking and a QCD-like low-energy sector for the remaining fermion flavors and gauge degrees of freedom. In models with a gauged Higgs field, a distinction of Higgs- and QCD-like phases as well as details of the particle spectrum might be much more intricate [53, 54, 55, 56].

The question that is left open by the preceding standard perturbative UV analysis is as to whether the CEL solution is the only possible AF model with the same field content and symmetries of Eq. (1). More specifically, can there be more AF solutions outside the family of perturbatively renormalizable models? To address this possibility, we take inspiration from the discovery that new AF trajectories can be constructed in nonabelian Higgs models, if functional RG equations are used to explore the space of theories including also couplings with negative mass dimension [43, 44]. Therefore, as a first step of our investigation, we now turn to the computation of such functional RG equations for \(\mathbb {Z}_2\)-Yukawa-QCD models.

## 3 Functional renormalization group

*k*. Then the full (inverse) two-point function \(\Gamma _k^{(2)}\) at this scale enters the one-loop computation, supplemented by the regulator \(R_k\). Differentiating with respect to the scale

*k*leads to the Wetterich equation [59, 60, 61, 62, 63]

Equation (17) can be projected onto the RG flow of a specific coupling constant. In addition, it is also well suited to study functional parametrizations of the dynamics, such as a general scalar effective potential. These functional flow equations can then be used also outside the regime of small field amplitudes, to address problems such as the existence of a nontrivial minimum or the global stability of the theory.

*U*which exhibits a discrete \(\mathbb {Z}_{2}\) symmetry and the wave function renormalizations \(Z_{\{\phi ,\psi ,A,\eta \}}\) are scale dependent, as well as the Yukawa coupling \(h^2\) and the strong coupling \(g_\mathrm {s}\). Let us introduce a dimensionless renormalized scalar field in order to fix the usual RG invariance of field rescalings

*h*extracted from a projection onto a field-dependent two-point function \(\Gamma ^{(2)}_{\bar{\psi }\psi }(\phi )\) shows better convergence upon the inclusion of higher-dimensional Yukawa interactions than the projection onto the three-point function \(\Gamma ^{(3)}_{\phi \bar{\psi }\psi }\) in case the system is in the SSB regime [76, 80]. The flow equation for the Yukawa coupling extracted from \(\Gamma ^{(3)}_{\phi \bar{\psi }\psi }\) can be obtained from Eq. (25) by taking a derivative with respect to \(\rho \) before evaluating at \(\rho = \kappa \) which coincides with flow equation \(\partial _t h^2\) derived in [30].

In principle, functional flow equations can also be obtained for the gauge sector of the model. Nevertheless as we are interested in the properties of the flow equations far above the QCD scale where \(g_\mathrm {s}\) is small, it is legitimate to treat the running of the gauge sector in a standard way. Therefore we will use the one-loop beta function for \(g_\mathrm {s}\) as shown in Eq. (2).

*d*from the threshold functions, as we work with \(d=4\) and \(d_\gamma =4\) from now on.

The freedom to choose different regularization schemes is parametrized by the threshold functions \(l,m,\dots \). This includes general mass-dependent schemes as well as mass-independent schemes as a particular limiting case. Using an EFT-like analysis, we investigate in the following whether the results in the more general mass-dependent schemes are sensitive to the assumption of working in the DER as a special case. It turns out below that the restriction to the DER is severe and legitimate only for the CEL solution. A more general class of asymptotically free solutions requires to take threshold effects into account.

## 4 Effective field theory analysis in the deep Euclidean region

In the present section and Sect. 5, we discuss a generalization of the construction outlined in Sect. 2, by including perturbatively nonrenormalizable interactions. In adding higher-dimensional operators to Eq. (1), we follow the EFT paradigm, but we do so only for momentum-independent scalar self-interactions. In fact, as will be explained in the next sections, a justification of the consistency of the new AF solutions we construct requires an infinite number of higher-dimensional operators, which cannot be generally dealt with, unless further restrictions are imposed. The focus on point-like scalar self-interactions is one such additional specification, and it will be extensively discussed in the following.

Regardless of our choice to depart from a standard EFT setup, the AF solutions can be studied also within the latter. The goal of the present section and of Sect. 5 is precisely to explain how to reveal these solutions and to properly account for some of their properties in a parameterization where a finite number of couplings with higher dimension is included. These steps can be followed also when *all* interactions up to some given dimensionality are included in the effective Lagrangian. Still, the crucial ingredient in the construction is a treatment of the \(\beta \) functions of these operators that slightly differs from the standard EFT one. Namely, one has to treat the scale dependence of one coupling or Wilson coefficient in the EFT expansion as free. Finally, we will show in the next sections that this additional freedom has to be present in any rigorous definition of the RG flow of the model, due to the infinite dimensionality of the theory space, and plays the role of a boundary condition in a functional representation of the quantum dynamics.

*P*of Eq. (11), it will become clear soon that the only possibility in the DER is \(P=1/2\). In fact, since \(\hat{\lambda }_{3}\) and \(\hat{\lambda }_{4}\) contribute to the \(\beta \) function of \(\hat{\lambda }_{2}\),

*P*cannot be fixed without fixing simultaneously all the other powers \(P_n\) with \(n>2\). To simplify the discussion, we already start with the ansatz \(P=1/2\) and look for the corresponding values of \(P_n\) and \(\hat{\lambda }_{n}\). The flow equation for \(\hat{\lambda }_{3}\) then reads

*n*quartic self-interaction vertices and of a fermion loop with 2

*n*Yukawa vertices, respectively. This can be drawn diagrammatically as in Fig. 3. Thus, among all possible scalar self-interactions, the \(\hat{\lambda }_{2}\) coupling plays a dominant role in the UV. This \(\phi ^4\)-dominance regime can be studied by specifying a pure \(\phi ^4\) interaction in the bosonic threshold function that appears in the RG flow equation for \(u(\rho )\). This means

*z*as defined in Eq. (36). Therefore we can identify an outer region where \(z\gg 1\) and an inner region where \(z\ll 1\). In “Appendix B” we address in more detail this combined limit and show that it exists and is the same in both asymptotic regions, such that Eq. (40) does give a definite answer concerning the stability of the potential \(u(\rho )\) for an arbitrarily small value of \(h^2\). In factThis proves that the CEL solution corresponds to a bounded potential in the DER.

## 5 Effective field theory analysis including thresholds

*Q*is a priori arbitrary.

Let us denote by \(\beta _n\) the beta function of \(\hat{\lambda }_{n}\), \(\beta _n=\partial _t\hat{\lambda }_{n}\). In order to construct polynomial solutions of the QFP equations for the couplings \(\hat{\lambda }_{n}\) and \(\hat{\kappa }\), we set up the following recursive problem: we solve the equation \(\beta _{\hat{\kappa }}=0\) for \(\hat{\lambda }_{2}\), and \(\beta _n=0\) for \(\hat{\lambda }_{n+1}\). Upon truncating the series of equations at some \(\beta _{N_\mathrm {p}}\), this can be achieved only if one more coupling \(\hat{\lambda }_{N_\mathrm {p}+1}\) is retained. The result of this construction is a set of QFPs for \(\hat{\lambda }_{n}\) as functions of the couplings \(h^2\) and \(\hat{\kappa }\). Also, some of the parameters *P*, \(P_n\) and *Q* might remain unconstrained. A defining requirement for a viable QFP solution to represent an AF trajectory is that the couplings \(\hat{\lambda }_{n}\) and \(\hat{\kappa }\) approach constants for \(h^2\rightarrow 0\).

Clearly, there is some freedom in the search for scaling solutions and particularly in the recursive procedure we have described. Of course, it is likewise possible to treat another scalar coupling as a “free” parameter and to solve for \(\hat{\kappa }\) in terms of some \(\hat{\lambda }_{n}\). The question which coupling should meaningfully be treated as free parameter cannot be answered a priori and depends again on the precise details of the model. We choose \(\hat{\lambda }_{N_\mathrm {p}+1}\) here to start with. For definiteness, we concentrate in this work on solutions exhibiting the property that \(\hat{\lambda }_{2}\ne 0\) at the QFP (though this might be a scheme-dependent statement).

### 5.1 \(P\in \left( 0,1/2\right) \)

*P*being equal or smaller then 1 / 2. Thus, we immediately make the ansatz \(Q=0\), which turns out to be the correct solution. Indeed the leading orders in \(h^2\) in the flow equations of the rescaled couplings are

While the first class of solutions in Eqs. (46) and (47) had already been discovered in Refs. [43, 44], the second one given by Eqs. (48) and (49) is new. These solutions were not observed in Refs. [43, 44] because of simplifying approximations in the analysis of the RG equations. In particular, only linear insertions of the coupling \(\lambda _3\) into the beta functions of lower-dimensional parameters were considered.

### 5.2 \(P=1/2\)

If \(1<P_3<2\), the contribution coming from \(\hat{\lambda }_{3}\) plays the dominant role in the RG flow of \(\hat{\lambda }_{2}\) but is subleading for \(\kappa \). The solution of the corresponding QFP equations is \(\kappa =5/(64\pi ^2)\) and \(\hat{\lambda }_{2}=-6\), implying that the expansion point is a nontrivial maximum. As we have assumed in our analysis that the expansion point of the Taylor series is a minimum of the potential, we reject this solution albeit it might lead to further interesting solutions if an appropriate expansion scheme is used. Thus, the only two new solutions correspond to \(P_3=1\) and \(P_3=2\).

### 5.3 \(P\in \left( 1/2,1\right) \)

*Q*is positive. Choosing \(Q=2 P-1\) as in the gauged-Higgs model turns out to be the correct scaling also for the present system. However, we prefer to be more general and consider

*Q*as an undetermined positive power in the first place. It is possible to verify that, under the assumptions that \(Q>0\), \(P_3>0\), and \(P>1/2\), the only terms that can contribute to the leading parts in the RG flow for \(\hat{\lambda }_{2}\) and \(\hat{\kappa }\) are

*Q*, \(P_3\) and

*P*, one has to take care that the two powers of \(h^2\) in the denominators, i.e., \(2P-Q\) and \(1-Q\), give different contributions to the \(\beta \) functions depending on whether they are positive or negative. Moreover, we have to keep in mind that – by definition of the finite ratios – \(\hat{\lambda }_{2}\) and \(\hat{\kappa }\) have to approach their QFP values in the UV limit up to subleading corrections in some positive power of \(h^2\). Among the set of all possible configurations there are only two QFP solutions. One of these corresponds to the case where the contribution arising from \(\hat{\lambda }_3\) is subleading in Eq. (57):

### 5.4 \(P=1\)

*Q*. One solution is

In “Appendix C” we complete the EFT analysis of the present section, by discussing \(P>1\). Also in this case we conclude that all the QFP solutions we observe have either \(\hat{\lambda }_{2}\) or \(\hat{\kappa }\) negative.

## 6 Full effective potential in the \(\phi ^4\)-dominance approximation

*P*is chosen to be that of Eq. (11) so that \(\xi _2=\hat{\lambda }_{2}\), because we specifically look for QFPs where \(\hat{\lambda }_{2}\ne 0\). It might happen that at a QFP \(x_0\ne \hat{\kappa }\), and \(\xi _n\ne \hat{\lambda }_{n}\) for \(n>2\), such that solutions of the equation \(\partial _t f(x)=0\) might differ from the actual scaling solutions. Thus, the rescaling of Eq. (66) is expected to be useful as long as the quartic scalar coupling is the leading term in the approach of the scalar potential towards flatness.

*x*includes also the anomalous dimension of the Yukawa coupling \(\eta _{h^2}\) defined in Eq. (12). All anomalous dimensions are also evaluated by neglecting the possible appearance of higher-dimensional couplings, as well as contributions from the scalar mass term. Thus, in the following, we use Eqs. (2), (3), and the one-loop value for the anomalous dimension of the scalar field given in Eq. (10).

*P*and its QFP solution is

For \(C_{\mathrm {f}}=0\) we can straightforwardly impose the consistency condition \(f^{\prime \prime }(0)=\xi _2\). Instead, for any nonvanishing \(C_{\mathrm {f}}\), the QFP potential behaves as a nonrational power of *x* at the origin. Its second order derivative is not defined at the origin as long as \(\eta _{x}>0\) which is generically the case for a potential in the symmetric regime. This problem might be avoided if there is at least one nontrivial minimum \(x_0\), in the spirit of the Coleman-Weinberg mechanism [81]. In fact, we can impose \(f''(x_0)=\xi _2\) for this xcase.

As a first analysis, we want to understand the asymptotic properties of the full \(h^2\)-dependent solution *f*(*x*). Specifically, we want to identify parameter ranges for \(C_{\mathrm {f}}\) and \(\xi _2\) for which the potential is bounded from below. To this end, we focus on the asymptotic behavior of the solution, \(x\rightarrow +\infty \). In particular, we are interested in the UV regime where \(h^2\rightarrow 0\). Since the QFP potential *f*(*x*) for given \(C_{\mathrm {f}}\), which might also depend on \(h^2\), is a function of the two variables *x* and \(h^2\), we have to take the limit process with care to investigate the asymptotic behavior of *f* in the deep UV.

*h*-dependent boundary \(x_{1}(h)\) of an inner interval \(x \in [0,x_{1})\) by requiring \(z_{{\mathrm {B}}}=1\) and the boundary \(x_{2}(h)\) of an outer interval \((x_{2},\infty )\) by \(z_{{\mathrm {F}}}=1\) for fixed

*P*and \(\xi _{2}\). For \(z_{\mathrm {F}} > z_{\mathrm {B}}\), the requirement \(z_{{\mathrm {B}}}=1\) and \(z_{{\mathrm {F}}}=1\) will define \(x_{2}\) and \(x_{1}\), respectively. In case \(P<1\), the two boundaries \(x_{1}\) and \(x_{2}\) grow towards larger values and always fulfill \(x_{2}>x_{1}\) when we send \(h \rightarrow 0\).

Approximating the hypergeometric functions for small but fixed arguments \(z_{{\mathrm {B}}/{\mathrm {F}}} \ll 1\), we obtain a valid approximation of the potential in the first interval as this also implies \(x \ll x_{1}\). Thus, we are able to reliably check the asymptotic behavior by first performing the limit \(h \rightarrow 0\) and afterwards \(x \rightarrow \infty \) in this region. In case the hypergeometric functions shall be investigated for large arguments, we have to perform first the limit \(x \rightarrow \infty \) before sending \(h \rightarrow 0\) to investigate the asymptotic behavior such that one stays in the outer interval as only there the results can be trusted for the used approximations. Further details can be found in “Appendix B2”. The rescaled potential *f*(*x*) turns out to be stable in the deep UV for both regimes, and the two asymptotic behaviors are in agreement.

### 6.1 Large-field behavior

### 6.2 Small-field behavior and the CEL solution

*f*(

*x*) for small arguments \(x \ll 1\). This is relevant to address both the \(x\rightarrow 0\) limit at fixed \(h^2\), and also to inspect the large field asymptotics for \(P<1\) in the limit where \(h^2\rightarrow 0\) and \(x\rightarrow +\infty \) at \(z_{{\mathrm {B}}/{\mathrm {F}}}\ll 1\). For this purpose, we start from the expansion of the QFP potential

*f*(

*x*) for small

*x*, which can be found in “Appendix B2”, cf. Eq. (B13). The Gauß hypergeometric functions are analytical for small

*x*, but the scaling term is not, due to the nonrational power of

*x*. The first derivative at the origin is

For \(P=1/2\), the first derivative at the origin changes sign at \(\xi _2=4\). In this case, we find that the two lines \(C_{\mathrm {f},\infty }=0\) and \(\xi _2=4\) divide the (\(C_{\mathrm {f},\infty },\xi _{2}\)) plane in four regions with different qualitative behavior for *f*(*x*), as represented in the right panel of Fig. 6 with solid black line and dashed blue line respectively. In region II the QFP potential is bounded from below and has a nontrivial stable minimum. In region IV the potential has a nontrivial maximum but is unbounded from below. Instead in regions I and III the function *f*(*x*) is monotonically increasing towards \(+\infty \) and decreasing to \(-\infty \), respectively. For \(P<1/2\), there are only regions of type II and III.

*f*(

*x*) includes all the information about \(u(\rho )\) plus a linear term that was discarded in Sect. 4 by the definition of the DER. Furthermore, Eqs. (72) and (74) apply to all values of \(C_{\mathrm {f}}\), thus by choosing \(P=1/2\) and \(C_{\mathrm {f}}=0\) in these equations, and specifying the QFP value of \(\xi _2\), we deduce that the asymptotic behavior for the CEL potential isThus, we conclude that the CEL solution is stable for arbitrary small values of the Yukawa coupling.

### 6.3 New solutions with a nontrivial minimum

*P*lead to different leading behaviors in \(h^2\) for the latter expression. These can be summarized in the following way

*x*,Also for \(P<1/2\) it is possible to find a parametrization \(C_{\mathrm {f}}(\xi _2)\) for the QFP solutions with a nontrivial minimum satisfying the consistency condition in \(x_0\). Its leading order contribution in \(h^2\) reads

*x*which reads

*f*(

*x*) and the finite ratio \(\hat{\lambda }_{3}\) is

Finally, let us summarize once more the results of the fixed-point potential analysis for *f*(*x*) and for general \(P<1\). Starting from a pure quartic scalar interaction for the potential given by \(\lambda _2\rho ^2/2=\xi _2 x^2/{2}\) with a trivial minimum at the origin, we obtain a QFP potential of the same type and with the required property \(f^{\prime \prime }(0)=\xi _2\) only for the particular choice for the parameters \(\{P,C_{\mathrm {f}},\xi _2\}=\{1/2,0,\hat{\lambda }_{2}^\pm \}\). This is the CEL solution. We argued that it is stable with a well defined asymptotic behavior in the combined limit \(x\rightarrow \infty \) and \(h^2\rightarrow 0\). In addition for \(P\le 1/2\), we discovered in the \((\hat{C}_{\mathrm {f}},\xi _2)\) plane the existence of a one-parameter family of *new* solutions. Despite the presence of a log-type singularity at the origin, these solutions have a nontrivial minimum \(x_0\) which satisfies the consistency condition \(f^{\prime \prime }(x_0)=\xi _2\). For \(P=1/2\) these new solutions are stable and present the same quadratic asymptotic behavior as for the CEL solution. For \(P<1/2\), the QFP potential becomes asymptotically flat in the combined limit \(x\rightarrow \infty \) and \(h^2\rightarrow 0\), because \(\hat{C}_{\mathrm {f},\infty }=0\).

## 7 Full effective potential in the weak-coupling expansion

*f*(

*x*) in powers of \(h^2\). The one-loop flow equation for

*f*(

*x*) takes the form

*x*. The arguments \(\omega _f\) and \(\omega _{1f}\), defined as

*x*and

*f*, yields the following expression

*f*(

*x*), its derivatives, and its minimum \(x_0\) are assumed to be \(h^2\)-independent in the expansion of Eq. (94). Therefore, a consistency check of the validity of the expansion has to be performed after the analytic QFP solution for

*f*(

*x*) is computed. This applies, for instance, to the contributions coming from the anomalous dimensions which, according to Eqs. (25), (26) and (27), depend on the properties of the potential at its minimum.

### 7.1 \(P\in (0,1/2)\)

*c*is an integration constant. The second solution grows exponentially for large field amplitudes. However, we are only interested in solutions that obey power-like scaling for \(x\rightarrow \infty \), since in this case a scalar product can be defined on the space of eigenperturbations of these solutions [83, 84, 85]. Thus, we set the second integration constant to zero.

### 7.2 \(P=1/2\)

*x*by setting the corresponding integration constant to zero. The remaining solution is a quadratic polynomial

### 7.3 \(P\in (1/2,1)\)

For all values of \(P<1\) in the present approximation, we have obtained QFP solutions which are analytic in *x*. In Sect. 5, this was implemented by construction, since we have projected the functional flow equation onto a polynomial ansatz. In the present analysis, this happens because the contributions to \(\beta _f\) producing non-analyticities are accompanied by subleading powers of \(h^2\) for \(P<1\). Indeed, both the anomalous dimension of *x* and contributions from the loops proportional to \(x^2\) would produce a logarithmic singularity of \(f''(x)\) at \(x=0\) for any \(h^2\ne 0\), as discussed in Sect. 6.3, see also below. Knowing about the presence of this singularity for any *P* for \(h^2\ne 0\), we can accept the previous solutions only if \(x_0>0\), which appears to be impossible for \(P\in (1/2,1)\).

### 7.4 \(P=1\)

*c*allows to construct physical QFP solutions, that avoid the divergence at small fields by developing a nontrivial minimum. The defining equation \(f'(x_0)=0\) for this minimum, where

*f*is given by Eq. (106), can straightforwardly be solved for

*c*in terms of \(x_0\). From the point of view where the latter is the free parameter labeling the QFP solutions, the natural question then is as to whether it can be chosen such that \(f''(x_0)=\xi _2\) is positive and finite for \(h^2\rightarrow 0\). The answer is negative, since the piecewise linear regulator gives

## 8 Numerical solutions

*f*(

*x*) as in Eq. (92), where we have computed the threshold functions \(l_0^{{\mathrm {B}}/{\mathrm {F}}}\) in Eq. (23) by choosing the piece-wise linear regulator. We make a further approximation evaluating the anomalous dimensions \(\eta _\phi \), \(\eta _\psi \) and \(\eta _{h^2}\) in the DER, leading to the expressions in Eqs. (10), (13) and (28). We are moreover interested in the \(P=1/2\) case characterized by the existence of the CEL solution, regular at the origin, and a family of new QFP potentials, singular in \(x=0\) but featuring a nontrivial minimum \(x_0\ne 0\). To address this numerical issue we exploit two different methods. First, we study the global behavior of the CEL solution using pseudo-spectral methods. And second, we corroborate the existence of the new QFP family of solutions using the shooting method. s

### 8.1 Pseudo-spectral methods

Pesudo-spectral methods provide for a powerful tool to numerically solve functional RG equations, provided the desired solution can be spanned by a suitable set of basis functions. Here, we are interested in a numerical construction of global properties of the QFP function *f*(*x*). We follow the method presented in [86], as this approach has proven to be suited for this purpose, see [74, 87, 88, 89] for a variety of applications, and [90] for earlier FRG implementations; a more general account of pseudo-spectral methods can be found in [91, 92, 93, 94].

*f*(

*x*) into two series of Chebyshev polynomials. The first series is defined over some domain \([0,x_\mathrm {M}]\) and is spanned in terms of Chebyshev polynomials of the first kind \(T_i(z)\). The second series is defined over the remaining infinite domain \([x_\mathrm {M},+\infty )\) and expressed in terms of rational Chebyshev polynomials \(R_i(z)\). Moreover, to capture the correct asymptotic behavior of

*f*(

*x*), the latter series is multiplied by the leading asymptotic term \(x^{d/d_x}\), which is in fact the solution of the homogeneous scaling part of Eq. (92). Finally the ansatz reads

*f*(

*x*) and \(f^\prime (x)\) must be taken into account. The solutions presented in the following are obtained by choosing \(x_\mathrm {M}=2\). We have further examined that the results do not change once \(x_\mathrm {M}\) is varied.

In Fig. 7, we compare the first derivative \(f^\prime (x)\) obtained from this pseudo-spectral method and the analytical solution derived from the \(\phi ^4\)-dominance EFT approximation, see Eq. (70), for a fixed value of \(h^2=10^{-4}\) and \(\xi _2=\hat{\lambda }_{2}^+\). The two solutions lie perfectly on top of each other within the numerical error. Moreover, the coefficients \(a_i\) and \(b_i\) exhibit an exponential decay with increasing \(N_a\) and \(N_b\) – and thus indicate an exponentially small error of the numerical solution – until the algorithm hits machine precision.

The pseudo-spectral method thus allows us to provide clear numerical evidence for the global existence of the CEL solution within the full non-linear flow equation in the one-loop approximation. To our knowledge, this is the first time that results about global stability have been obtained for the scalar potential of this model.

### 8.2 Shooting method

## 9 Conclusions

Models that feature the existence of asymptotically free RG trajectories represent “perfect” quantum field theories in the sense that they could be valid and consistent models at any energy and distance scale. Identifying such RG trajectories hence provides information that can be crucial for our attempt at constructing fundamental models of particle physics. Based on the observation that part of the standard model including the Higgs-top sector exhibits a behavior reminiscent to an asymptotically free trajectory, we have taken a fresh look at asymptotic freedom in a gauged-Yukawa model from a perspective that supersedes conventional studies within standard perturbation theory.

Gauged-Yukawa models form the backbone not only of the standard model, but also of many models of new physics. Our study concentrates on the simplest model, a \(\mathbb {Z}_2\)-Yukawa-QCD model, that features asymptotically free trajectories already in standard perturbation theory as first found in Ref. [4]. Using effective-field-theory methods as well as various approximations based on the functional RG, we discover additional asymptotically free trajectories. One key ingredient for this discovery is a careful analysis of boundary conditions on the correlation functions of the theory, manifested by the asymptotic behavior of the Higgs potential in field space in our study. Whereas standard perturbation theory corresponds to an implicit choice of these boundary conditions, generalizing this choice explicitly yields a further two-parameter family of asymptotically free trajectories.

Our findings in this work generalize the strategy developed in Refs. [43, 44] for gauged-Higgs models to systems including a fermionic matter sector. The new solutions also share the property that the quasi-fixed-point potentials, i.e., the solution to the fixed-point equation for a given small value of the gauge coupling, exhibit a logarithmic singularity at the origin in field space. Nevertheless, standard criteria (polynomial boundedness of perturbations, finiteness of the potential and its first derivative, global stability) are still satisfied. Moreover, since the quasi-fixed-point potential exhibits a nonzero minimum at any scale, correlation functions defined in terms of derivatives at this minimum remain well-defined to any order. Hence, we conclude that our solutions satisfy all standard criteria that are known to be crucial for selecting physical solutions in statistical-physics models [83, 84, 85].

The occurrence of a nontrivial minimum in the quasi-fixed-point solutions also indicates that standard arguments based on *asymptotic symmetry* [99] are sidestepped: conventional perturbation theory often focuses on the deep Euclidean region (DER), thereby implicitly assuming the irrelevance of nonzero minima or running masses for the RG analysis. In fact, all our new solutions demonstrate that the inclusion of a nonzero minimum is mandatory to reveal their existence. In this sense, the CEL solution found in standard perturbation theory is just a special case that features the additional property of asymptotic symmetry.

Our analysis is capable of extracting information about the global shape of the quasi-fixed-point potential. In fact, the requirement of global stability leads to constraints in the two-parameter family of solutions. The scaling exponent is confined to the values \(P\le 1/2\). This constraint is new in the present model in comparison with gauged-Higgs models [43, 44], and may be indicative for the fact that further structures in the matter sector may lead to further constraints. The CEL solution is a special solution with \(P=1/2\), such that our results provide direct evidence for the first time that the CEL solution indeed features a globally stable potential.

In our work, we so far concentrated on the flow of the effective potential \(u(\rho )\) (or *f*(*x*)). This does, of course, not exhaust all possible structures that may be relevant for identifying asymptotically free trajectories. A natural further step would be a study of a full Yukawa coupling potential \(h(\rho )\). This would generalize the single Yukawa coupling *h* which corresponds to the coupling defined at the minimum \(h(\rho =\kappa )\). In fact, the functional RG methods are readily available to also deal with this additional layer of complexity [72, 73, 76, 80, 100, 101, 102, 103]. As further boundary conditions have to be specified, it is an interesting open question as to whether the set of asymptotically free trajectories becomes more diverse or even more constrained.

In view of the standard model with its triviality problem arising from the U(1) hypercharge sector, it also remains to be seen if our construction principle can be applied to this U(1) sector. We believe that the construction of UV complete quantum field theories with a U(1) factor as part of the fundamental gauge-group structure should be a valuable ingredient in contemporary model building.

## Notes

### Acknowledgements

We thank J. Borchardt and B. Knorr for insightful discussions especially concerning the pseudo-spectral methods. Interesting discussion with C. Kohlf\(\ddot{\text {u}}\)rst and R. Martini are acknowledged. This work received funding support by the DFG under Grants No. GRK1523/2 and No. Gi328/9-1. RS and AU acknowledge support by the Carl-Zeiss foundation.

## References

- 1.D.J. Gross, F. Wilczek, Phys. Rev. Lett.
**30**, 1343 (1973)ADSCrossRefGoogle Scholar - 2.H.D. Politzer, Phys. Rev. Lett.
**30**, 1346 (1973)ADSCrossRefGoogle Scholar - 3.D.J. Gross, F. Wilczek, Phys. Rev. D
**8**, 3633 (1973)ADSCrossRefGoogle Scholar - 4.T.P. Cheng, E. Eichten, L.-F. Li, Phys. Rev. D
**9**, 2259 (1974)ADSCrossRefGoogle Scholar - 5.D.J. Gross, F. Wilczek, Phys. Rev. D
**9**, 980 (1974)ADSCrossRefGoogle Scholar - 6.H.D. Politzer, Phys. Rept.
**14**, 129 (1974)ADSCrossRefGoogle Scholar - 7.N.-P. Chang, Phys. Rev. D
**10**, 2706 (1974)ADSCrossRefGoogle Scholar - 8.N.-P. Chang, J. Perez-Mercader, Phys. Rev. D
**18**, 4721 (1978). [Erratum: Phys. Rev.D19,2515(1979)]Google Scholar - 9.E.S. Fradkin, O.K. Kalashnikov, J. Phys. A
**8**, 1814 (1975)ADSCrossRefGoogle Scholar - 10.A. Salam, J.A. Strathdee, Phys. Rev. D
**18**, 4713 (1978)ADSCrossRefGoogle Scholar - 11.F.A. Bais, H.A. Weldon, Phys. Rev. D
**18**, 1199 (1978)ADSCrossRefGoogle Scholar - 12.A. Salam, V. Elias, Phys. Rev. D
**22**, 1469 (1980)ADSMathSciNetCrossRefGoogle Scholar - 13.D.J.E. Callaway, Phys. Rept.
**167**, 241 (1988)ADSCrossRefGoogle Scholar - 14.G.F. Giudice, G. Isidori, A. Salvio, A. Strumia, JHEP
**02**, 137 (2015). arXiv:1412.2769 [hep-ph]ADSCrossRefGoogle Scholar - 15.
- 16.F.F. Hansen, T. Janowski, K. Langaeble, R.B. Mann, F. Sannino, T.G. Steele, Z.-W. Wang (2017). arXiv:1706.06402 [hep-ph]
- 17.J. Hetzel, B. Stech, Phys. Rev. D
**91**, 055026 (2015). arXiv:1502.00919 [hep-ph]ADSCrossRefGoogle Scholar - 18.G.M. Pelaggi, A. Strumia, S. Vignali, JHEP
**08**, 130 (2015). arXiv:1507.06848 [hep-ph]CrossRefGoogle Scholar - 19.C. Pica, T.A. Ryttov, F. Sannino (2016). arXiv:1605.04712 [hep-th]
- 20.E. Molgaard, F. Sannino (2016). arXiv:1610.03130 [hep-ph]
- 21.M. Heikinheimo, K. Kannike, F. Lyonnet, M. Raidal, K. Tuominen, H. Veermäe, JHEP
**10**, 014 (2017). arXiv:1707.08980 [hep-ph]ADSCrossRefGoogle Scholar - 22.D. Buttazzo, G. Degrassi, P.P. Giardino, G.F. Giudice, F. Sala, A. Salvio, A. Strumia, JHEP
**12**, 089 (2013). arXiv:1307.3536 [hep-ph]ADSCrossRefGoogle Scholar - 23.A.V. Bednyakov, B.A. Kniehl, A.F. Pikelner, O.L. Veretin, Phys. Rev. Lett.
**115**, 201802 (2015). arXiv:1507.08833 [hep-ph]ADSCrossRefGoogle Scholar - 24.L. Di Luzio, G. Isidori, G. Ridolfi (2015). arXiv:1509.05028 [hep-ph]
- 25.A. Andreassen, W. Frost, M.D. Schwartz (2017). arXiv:1707.08124 [hep-ph]
- 26.S. Chigusa, T. Moroi, Y. Shoji, Phys. Rev. Lett.
**119**, 211801 (2017). arXiv:1707.09301 [hep-ph]ADSCrossRefGoogle Scholar - 27.S. Chigusa, T. Moroi, Y. Shoji (2018). arXiv:1803.03902 [hep-ph]
- 28.S. Alekhin, A. Djouadi, S. Moch, Phys. Lett. B
**716**, 214 (2012). arXiv:1207.0980 [hep-ph]ADSCrossRefGoogle Scholar - 29.F. Bezrukov, M. Shaposhnikov, J. Exp. Theor. Phys.
**120**, 335 (2015). arXiv:1411.1923 [hep-ph]ADSCrossRefGoogle Scholar - 30.H. Gies, C. Gneiting, R. Sondenheimer, Phys. Rev. D
**89**, 045012 (2014). arXiv:1308.5075 [hep-ph]ADSCrossRefGoogle Scholar - 31.V. Branchina, E. Messina, Phys. Rev. Lett.
**111**, 241801 (2013). arXiv:1307.5193 [hep-ph]ADSCrossRefGoogle Scholar - 32.P. Hegde, K. Jansen, C.J.D. Lin, A. Nagy, Proceedings, 31st International Symposium on Lattice Field Theory (Lattice 2013), PoS
**LATTICE2013**, 058 (2014). arXiv:1310.6260 [hep-lat] - 33.H. Gies, R. Sondenheimer, Eur. Phys. J. C
**75**, 68 (2015). arXiv:1407.8124 [hep-ph]ADSCrossRefGoogle Scholar - 34.A. Eichhorn, H. Gies, J. Jaeckel, T. Plehn, M.M. Scherer, R. Sondenheimer, JHEP
**04**, 022 (2015). arXiv:1501.02812 [hep-ph]ADSCrossRefGoogle Scholar - 35.D.Y.J. Chu, K. Jansen, B. Knippschild, C.J.D. Lin, A. Nagy, Phys. Lett. B
**744**, 146 (2015). arXiv:1501.05440 [hep-lat]ADSCrossRefGoogle Scholar - 36.D.Y.J. Chu, K. Jansen, B. Knippschild, C.J.D. Lin, K.-I. Nagai, A. Nagy, Proceedings, 32nd International Symposium on Lattice Field Theory (Lattice 2014), PoS
**LATTICE2014**, 278 (2014). arXiv:1501.00306 [hep-lat] - 37.O. Akerlund, P. de Forcrand, Phys. Rev. D
**93**, 035015 (2016). arXiv:1508.07959 [hep-lat]ADSCrossRefGoogle Scholar - 38.R. Sondenheimer (2017). arXiv:1711.00065 [hep-ph]
- 39.M. Holthausen, J. Kubo, K.S. Lim, M. Lindner, JHEP
**12**, 076 (2013). arXiv:1310.4423 [hep-ph]ADSCrossRefGoogle Scholar - 40.A.J. Helmboldt, P. Humbert, M. Lindner, J. Smirnov (2016). arXiv:1603.03603 [hep-ph]
- 41.A. Ahriche, A. Manning, K.L. McDonald, S. Nasri, Phys. Rev. D
**94**, 053005 (2016). arXiv:1604.05995 [hep-ph]ADSCrossRefGoogle Scholar - 42.M. Shaposhnikov, A. Shkerin (2018). arXiv:1803.08907 [hep-th]
- 43.H. Gies, L. Zambelli, Phys. Rev. D
**92**, 025016 (2015). arXiv:1502.05907 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 44.H. Gies, L. Zambelli, Phys. Rev. D
**96**, 025003 (2017). arXiv:1611.09147 [hep-ph]ADSMathSciNetCrossRefGoogle Scholar - 45.M. Reichert, A. Eichhorn, H. Gies, J.M. Pawlowski, T. Plehn, M.M. Scherer, Phys. Rev. D
**97**, 075008 (2018). arXiv:1711.00019 [hep-ph]ADSCrossRefGoogle Scholar - 46.U. Ellwanger, M. Hirsch, A. Weber, Z. Phys. C
**69**, 687 (1996). arXiv:hep-th/9506019 [hep-th]MathSciNetCrossRefGoogle Scholar - 47.D.F. Litim, J.M. Pawlowski, Phys. Lett. B
**435**, 181 (1998). arXiv:hep-th/9802064 [hep-th]ADSCrossRefGoogle Scholar - 48.
- 49.A.D. Bond, D.F. Litim (2016). arXiv:1608.00519 [hep-th]
- 50.A. Codello, K. Langæble, D.F. Litim, F. Sannino, JHEP
**07**, 118 (2016). arXiv:1603.03462 [hep-th]ADSCrossRefGoogle Scholar - 51.B. Bajc, F. Sannino (2016). arXiv:1610.09681 [hep-th]
- 52.B. Pendleton, G.G. Ross, Phys. Lett. B
**98**, 291 (1981)ADSCrossRefGoogle Scholar - 53.
- 54.A. Maas, T. Mufti, Phys. Rev. D
**91**, 113011 (2015). arXiv:1412.6440 [hep-lat]ADSCrossRefGoogle Scholar - 55.A. Maas, R. Sondenheimer, P. Törek (2017). arXiv:1709.07477 [hep-ph]
- 56.A. Maas (2017). arXiv:1712.04721 [hep-ph]
- 57.K.G. Wilson, J.B. Kogut, Phys. Rept.
**12**, 75 (1974)ADSCrossRefGoogle Scholar - 58.F.J. Wegner, A. Houghton, Phys. Rev. A
**8**, 401 (1973)ADSCrossRefGoogle Scholar - 59.C. Wetterich, Phys. Lett. B
**301**, 90 (1993)ADSCrossRefGoogle Scholar - 60.U. Ellwanger, Proceedings, Workshop on Quantum field theoretical aspects of high energy physics: Bad Frankenhausen, Germany, September 20–24, 1993. Z. Phys. C
**62**, 503 (1994). arXiv:hep-ph/9308260 [hep-ph] - 61.U. Ellwanger, Proceedings, Workshop on Quantum field theoretical aspects of high energy physics: Bad Frankenhausen, Germany, September 20–24, 1993. Z. Phys. C
**62**, 206 (1993)Google Scholar - 62.T.R. Morris, Int. J. Mod. Phys.
**A9**, 2411 (1994). arXiv:hep-ph/9308265 [hep-ph]ADSCrossRefGoogle Scholar - 63.M. Bonini, M. D’Attanasio, G. Marchesini, Nucl. Phys. B
**409**, 441 (1993). arXiv:hep-th/9301114 [hep-th]ADSCrossRefGoogle Scholar - 64.J. Berges, N. Tetradis, C. Wetterich, Phys. Rept.
**363**, 223 (2002). arXiv:hep-ph/0005122 [hep-ph]ADSCrossRefGoogle Scholar - 65.
- 66.H. Gies, ECT* School on Renormalization Group and Effective Field Theory Approaches to Many-Body Systems Trento, Italy, February 27-March 10, 2006. Lect. Notes Phys.
**852**, 287 (2012). arXiv:hep-ph/0611146 [hep-ph] - 67.B. Delamotte, Lect. Notes Phys.
**852**, 49 (2012). arXiv:cond-mat/0702365 [cond-mat.stat-mech]ADSMathSciNetCrossRefGoogle Scholar - 68.
- 69.H. Gies, S. Rechenberger, M.M. Scherer, L. Zambelli, Eur. Phys. J. C
**73**, 2652 (2013). arXiv:1306.6508 [hep-th]ADSCrossRefGoogle Scholar - 70.A. Eichhorn, M.M. Scherer, Phys. Rev. D
**90**, 025023 (2014). arXiv:1404.5962 [hep-ph]ADSCrossRefGoogle Scholar - 71.A. Jakovac, I. Kaposvari, A. Patkos, Proceedings, Gribov-85 Memorial Workshop on Theoretical Physics of XXI Century: Chernogolovka, Russia, June 7–20, 2015. Int. J. Mod. Phys. A
**31**, 1645042 (2016). arXiv:1510.05782 [hep-th] - 72.A. Jakovac, I. Kaposvari, A. Patkos, Mod. Phys. Lett. A
**32**, 1750011 (2016). arXiv:1508.06774 [hep-th]ADSCrossRefGoogle Scholar - 73.G.P. Vacca, L. Zambelli, Phys. Rev. D
**91**, 125003 (2015). arXiv:1503.09136 [hep-th]ADSCrossRefGoogle Scholar - 74.J. Borchardt, H. Gies, R. Sondenheimer, Eur. Phys. J. C
**76**, 472 (2016). arXiv:1603.05861 [hep-ph]ADSCrossRefGoogle Scholar - 75.A. Jakovác, I. Kaposvári, A. Patkós, Phys. Rev. D
**96**, 076018 (2017). arXiv:1703.00831 [hep-ph]ADSCrossRefGoogle Scholar - 76.H. Gies, R. Sondenheimer, M. Warschinke, Eur. Phys. J. C
**77**, 743 (2017). arXiv:1707.04394 [hep-ph]ADSCrossRefGoogle Scholar - 77.H. Gies, R. Sondenheimer, in Higgs cosmology Newport Pagnell, Buckinghamshire, UK, March 27–28, 2017 (2017). arXiv:1708.04305 [hep-ph]
- 78.
- 79.
- 80.J.M. Pawlowski, F. Rennecke, Phys. Rev. D
**90**, 076002 (2014). arXiv:1403.1179 [hep-ph]ADSCrossRefGoogle Scholar - 81.S.R. Coleman, E.J. Weinberg, Phys. Rev. D
**7**, 1888 (1973)ADSCrossRefGoogle Scholar - 82.R. Jackiw, Phys. Rev. D
**9**, 1686 (1974)ADSCrossRefGoogle Scholar - 83.T.R. Morris, Phys. Rev. Lett.
**77**, 1658 (1996). arXiv:hep-th/9601128 [hep-th]ADSCrossRefGoogle Scholar - 84.J. O’Dwyer, H. Osborn, Ann. Phys.
**323**, 1859 (2008). arXiv:0708.2697 [hep-th]ADSCrossRefGoogle Scholar - 85.I.H. Bridle, T.R. Morris, (2016). arXiv:1605.06075 [hep-th]
- 86.J. Borchardt, B. Knorr, Phys. Rev. D
**91**, 105011 (2015). arXiv:1502.07511 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 87.J. Borchardt, B. Knorr, Phys. Rev. D
**94**, 025027 (2016). arXiv:1603.06726 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 88.J. Borchardt, A. Eichhorn, Phys. Rev. E
**94**, 042105 (2016). arXiv:1606.07449 [cond-mat.stat-mech]ADSCrossRefGoogle Scholar - 89.M. Heilmann, T. Hellwig, B. Knorr, M. Ansorg, A. Wipf, JHEP
**02**, 109 (2015). arXiv:1409.5650 [hep-th]ADSCrossRefGoogle Scholar - 90.
- 91.J .P. Boyd,
*Chebyshev and Fourier Spectral Methods*, 2nd edn. (Dover Publications, Dover, 2000)Google Scholar - 92.R. Robson, A. Prytz, Aust. J. Phys.
**46**(1993)Google Scholar - 93.M. Ansorg, A. Kleinwachter, R. Meinel, Astron. Astrophys.
**405**, 711 (2003). arXiv:astro-ph/0301173 [astro-ph]ADSCrossRefGoogle Scholar - 94.R.P. Macedo, M. Ansorg, J. Comput. Phys.
**276**, 357 (2014). arXiv:1402.7343 [physics.comp-ph]ADSMathSciNetCrossRefGoogle Scholar - 95.T .R. Morris, Nonperturbative QCD: structure of the QCD vacuum. Progr. Theor. Phys. Suppl.
**131**, 395 (1998). arXiv:hep-th/9802039 [hep-th]ADSCrossRefGoogle Scholar - 96.
- 97.A. Codello, J. Phys. A
**A45**, 465006 (2012). arXiv:1204.3877 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 98.A. Codello, G. D’Odorico, Phys. Rev. Lett.
**110**, 141601 (2013). arXiv:1210.4037 [hep-th]ADSCrossRefGoogle Scholar - 99.B.W. Lee, W.I. Weisberger, Phys. Rev. D
**D10**, 2530 (1974)ADSCrossRefGoogle Scholar - 100.O. Zanusso, L. Zambelli, G.P. Vacca, R. Percacci, Phys. Lett. B
**689**, 90 (2010). arXiv:0904.0938 [hep-th]ADSCrossRefGoogle Scholar - 101.G.P. Vacca, O. Zanusso, Phys. Rev. Lett.
**105**, 231601 (2010). arXiv:1009.1735 [hep-th]ADSCrossRefGoogle Scholar - 102.J. Braun, L. Fister, J. M. Pawlowski, F. Rennecke (2014). arXiv:1412.1045 [hep-ph]
- 103.B. Knorr, Phys. Rev. B
**94**, 245102 (2016). arXiv:1609.03824 [cond-mat.str-el]ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}