# Higgs mass bounds from renormalization flow for a Higgs–top–bottom model

## Abstract

We study a chiral Yukawa model mimicking the Higgs–top–bottom sector of the standard model. We re-analyze the conventional arguments that relate a lower bound for the Higgs mass with vacuum stability in the light of exact results for the regularized fermion determinant as well as in the framework of the functional renormalization group. In both cases, we find no indication for vacuum instability nor meta-stability induced by top fluctuations if the cutoff is kept finite but arbitrary. A lower bound for the Higgs mass arises for the class of standard bare potentials of \(\phi ^4\) type from the requirement of a well-defined functional integral (i.e., stability of the bare potential). This consistency bound can, however, be relaxed considerably by more general forms of the bare potential without necessarily introducing new metastable minima.

## 1 Introduction

Long before the recent discovery of a comparatively light standard-model Higgs boson [1, 2], estimates and bounds on this mass parameter have been derived from renormalization arguments [3, 4, 5, 6, 7, 8, 9]. Assuming the validity of the standard model over a wide range of scales up to an ultraviolet (UV) cutoff scale \(\Lambda \), together with mild assumptions on the microscopic action at the scale \(\Lambda \), typically leads to a finite range of possible low-energy values for the Higgs mass, the so-called infrared (IR) window [6, 10]. Similar arguments can also be applied to models beyond the standard model of particle physics [11, 12, 13, 14, 15, 16, 17]. It has been suggested that Higgs masses below the lower bound necessarily require the effective potential of the standard model to develop a further minimum beyond the electroweak minimum [18, 19, 20, 21, 22, 23, 24]. Since the measured value of the Higgs mass near \(m_{\text {H}}=125\) GeV appears to be near if not below the lower bound, the standard-model vacuum could be unstable or at least metastable. In the latter case, the metastability has to be sufficiently long-lived compared to the age of the universe to allow for our existence [25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

While the occurrence of the vacuum instability is often attributed to the fluctuation of the top quark (and therefore sensitively depends on the top mass), the conventional perturbative analysis of determining the instability has been questioned by non-perturbative methods. Within the toy model of a top–Higgs–Yukawa system with discrete symmetry, lattice simulations have revealed that the full effective potential in this model with the cutoff kept finite does not develop an instability [35, 36, 37]. By contrast, the perturbative treatment of the same model in the limit \(\Lambda \rightarrow \infty \) exhibits an instability in disagreement with the simulation results. Within the same model and using functional methods, the occurrence of the erroneous instability has been traced back to an implicit renormalization condition that contradicts the underlying assumption of a well-defined functional integral [38, 39, 40].

In a series of recent lattice simulations using chiral Higgs–Yukawa models, imposing the criterion of a stable bare potential (typically of \(\phi ^4\) type) has lead to a number of quantitative results for the lower bound on the Higgs mass [37, 41, 42, 43, 44] without the need to require low-energy stability. The same line of argument can in fact be used to put strong constraints on the existence of a fourth generation of flavor in the light of the Higgs boson mass measurement [45, 46, 47, 48, 49]. These results have also been substantiated by conventional analytical methods [50].

In a recent work, we have been able to show that the sole consideration of bare potentials of \(\phi ^4\) type is actually too restrictive [51]. In fact, if the standard model is viewed as a low-energy effective theory, there is no reason to exclude higher-dimensional operators from the bare potential. Their occurrence is actually expected. Whereas Wilsonian renormalization group (RG) arguments of course suggest that low-energy observables remain almost completely unaffected by the higher-dimensional operators, we have demonstrated that Higgs-mass bounds can in fact exhibit a significant dependence on the bare potential. This may seem counter-intuitive at first sight, since the Higgs mass is clearly an IR observable. However, a Higgs-mass bound formulated in terms of a function of the cutoff, \(m_{\text {H}}{}_{,\text {bound}}(\Lambda )\), can be strongly influenced by a non-trivial RG running of the couplings near the cutoff. In [51], we have identified a simple RG mechanism that leads to a lowering of the conventional lower bound for the case of a \(\mathbb {Z}_2\)-symmetric Yukawa toy model.

Our findings of a “lowering” of the lower bound have been confirmed in the chiral Yukawa models studied on the lattice [52] as well as in a model involving an additional dark-matter scalar [53].

In view of the latent controversy about the (non-)existence of a top-fluctuation induced vacuum in-/meta- stability of the electroweak vacuum, the purpose of this work is twofold: first, we demonstrate on the basis of exact results for the fermion determinant in the presence of a scalar vacuum expectation value that the interaction part contribution to the effective potential is positive if the UV cutoff is kept finite but arbitrary. By contrast, a removal of the cutoff by taking the naive limit \(\Lambda \rightarrow \infty \) makes the fermionic contribution to the effective potential unstable. Second, we generalize our results of [51] to the chiral Yukawa model, which is also used in lattice simulations. Using the functional RG, we demonstrate that the conventional lower bound can be relaxed considerably by more general forms of the bare potential without necessarily introducing new metastable minima.

The paper is organized as follows: in Sect. 2, we introduce our model and all relevant notation. Section 3 is devoted to an analysis of exact properties of the fermion determinant in order to explore the origin of the apparent instability of the standard-model vacuum. The non-perturbative RG flow equations of the present model are summarized in Sect. 4. These are used in Sect. 5 to compute the Higgs-mass bounds of the model non-perturbatively for various bare potentials. Conclusions are presented in Sect. 6.

## 2 Chiral Higgs–top–bottom model

^{1}in terms of this minimum and the renormalized Yukawa couplings \(h_{\text {t}}\) and \(h_{\text {b}}\),

Apart from the missing further matter and flavor content, our model also ignores the gauge sectors of the standard model. This avoids not only technical complications and subtle issues arising from the gauge-Higgs interplay [54, 55, 56] in the standard model. But it will, of course, also lead to decisive differences to standard-model properties, which will be commented on in the course of this work. Nevertheless, the gauge sector is less important for the main points of the present work. In order to make closer contact with the standard-model language, we fix \(v=246\) GeV, \(m_{\text {t}}=173\) GeV and \(m_{\text {b}}=4.2\) GeV for illustrative purposes, but leave the Higgs mass as a free parameter for the moment.

For the following discussion it is important to note that the standard model in its conventional form (as well as the present model) may not be extendible to arbitrarily high momentum scales. The problem of triviality [57, 58, 59, 60, 61, 62, 63]–where substantial evidence has been accumulated for \(\phi ^4\)-type theories–is likely to extend to the full chiral-Yukawa sector as well. If so, the definition of the model unavoidably requires a UV cutoff \(\Lambda \) which physically plays the role of the scale of maximum UV extent up to which a quantum field theory description is appropriate. If the cutoff scale is sufficiently large, Wilsonian renormalization guarantees that the IR physics essentially depends only on a finite number of relevant and marginal parameters, rendering the theory predictive (in spite of our ignorance about the physics beyond \(\Lambda \)).

In fact, the strategy of *perturbative* renormalization manifestly allows one to implicitly or explicitly take the limit \(\Lambda \rightarrow \infty \) for certain physical observables. For the general definition of the theory, it is, however, important to accept the fact that the cutoff \(\Lambda \) may unavoidably have to be kept finite.^{2}

Fixing physical parameters such as \(v,m_{\text {t}}\), and \(m_{\text {b}}\) is technically implemented by renormalization conditions. Phenomenologically, it is useful to fix these conditions at observational IR scales. Conceptually, however, it is equally well possible to impose suitable renormalization conditions at the UV cutoff \(\Lambda \). A perturbative study of possible Higgs-mass values for partly randomized UV initial conditions at the Planck scale has, for instance, been performed for the standard model in [64]. For the present model, we can fix \(h_{\text {t}}\), \(h_{\text {b}}\), and the scalar potential \(U\) in terms of their bare quantities at the cutoff, \(h_{\text {t}}{}_{\Lambda }=\bar{h}_{\text {t}}\), \(h_{\text {b}}{}_{\Lambda }=\bar{h}_{\text {b}}\), and \(U_{\Lambda }\). In practice, the fixing can be done such that the constraints set by the physical values of \(v,m_{\text {t}}\), and \(m_{\text {b}}\) are satisfied.

## 3 Fermion determinant and (in-)stability

^{3}

*mean field*\(\phi \). Deviations from this mean field contribute to the full effective potential only in terms of fluctuations at higher-loop order. Therefore, we concentrate on the fermion-fluctuation induced contribution to the effective potential

### 3.1 Sharp cutoff

Most importantly: whereas the contribution to the mass term is negative, as it should be, since fermion fluctuations tend to induce chiral symmetry breaking, the whole interaction part in square brackets is strictly positive for all \(\rho >0\). This follows immediately from the inequality \(\ln (1+x)<x\) (for \(x>0\)) applied to the last term. Similarly, it can be shown that also the derivative of the interacting part with respect to \(\rho \) is strictly positive for any finite value of \(\rho \), \(h_a\), and \(\Lambda \).

We conclude that the fermion determinant–apart from its contribution to the scalar mass term–is strictly positive and monotonically increasing in its interacting part. Therefore, once the scalar mass term has been fixed by a renormalization condition, the remaining contributions from the top fluctuations to the interacting part of the bosonic potential are strictly positive. This excludes the possibility that an instability beyond the electroweak vacuum is induced by fermionic fluctuations. This can also be phrased in terms of a more rigorous statement: if the potential of the purely bosonic part \(S_\text {B}\) of the action in Eq. (7) is bounded from below by a function of the form \(U_{\text {B}}(\rho )> c_1 + c_2 \rho ^{1+\epsilon }\) with an arbitrary finite constant \(c_1\) and finite positive constants \(c_2,\epsilon >0\), then also the full potential including the fermionic fluctuations is bounded from below.

The problem of this line of argument becomes obvious once we go back to the cutoff-dependent leading order terms in Eq. (13). It is straightforward to work out that also these leading-order terms seem to have an instability: the interaction part of the potential in square brackets first develops a maximum and then eventually turns negative for large fields \(\rho \). However, the location of the maximum is in fact at \(h_a^2\rho = \Lambda ^2\). In other words, these seeming instability features appear precisely at those field values, where the expansion in terms of the parameter \(\frac{h_a^2\rho }{\Lambda ^2}\ll 1\) breaks down. We conclude that the instability “discovered” in Eq. (14) is an artifact of having tried to send the cutoff to infinity \(\Lambda \rightarrow \infty \) together with a problematic choice of the renormalization conditions. In fact, it has been shown in [38, 39, 40] for the \(\mathbb {Z}_2\)-Yukawa model that the renormalization conditions needed to arrive at Eq. (14) require an unstable bare bosonic potential with negative bare \(\phi ^4\) coupling, \(\lambda (\Lambda )<0\).

- (1)
Our conclusions are identical to those of [35, 36], where essentially the same results have been found for the \(\mathbb {Z}_2\)-Yukawa model. In these works, fully non-perturbative lattice simulations have been compared with the one-loop effective potential with a cutoff kept finite, matching the lattice data almost perfectly. By contrast, the effective potential with the cutoff removed à la Eq. (14) shows an artificial instability in strong disagreement with the non-perturbative simulation. This work has been criticized [68, 70] also because it is generically difficult on the lattice to bridge wide ranges of scales, in particular to separate the cutoff from the long-range mass scales by many orders of magnitude. As is clear from the above discussion, this problem does not exist for the present line of argument; the cutoff can be arbitrarily large in the above discussion of the fermion determinant. As long as it is finite, the interaction part of the determinant does not induce any instability.

- (2)
For the above discussion and the comparison to the standard line of arguments at one-loop order, it has been sufficient to evaluate the determinant for a homogeneous mean field. Though this does not interfere with our argument, one might ask whether the determinant behaves qualitatively differently for non-homogeneous fields. Some exact results are known for \(d=1+1\) dimensional determinants, where the Peierls instability at a finite chemical potential can lead to inhomogeneous ground states with lower free energy [71, 72]. However, the vacuum ground state is generically homogeneous as no mechanism exists that can “pay” for the higher cost in kinetic energy. Absolute lower and upper bounds for fermion determinants have been found, e.g., for QED [73].

- (3)
The fact that the interaction part of the fermion contribution to the scalar potential is positive does not imply that the full theory cannot have further potentially (meta-) stable vacua. The conclusion rather is that such further vacua have to be provided by the bosonic sector. In particular, the bare bosonic potential \(U_{\text {B}}\) can in principle be chosen such that it has several vacua.

^{4}As a further special case, it is even possible to construct somewhat special examples such that the bare bosonic potential has one minimum, but the sum of \(U_{\text {B}}\) and \(U_{\text {F}}\) has two minima. This is still very different from the perturbative reasoning which for the present model seems to suggest a global instability due to the fermionic fluctuations, whereas a global instability of \(U_{\text {B}}+U_{\text {F}}\) in our analysis would have to be seeded from the choice of \(U_{\text {B}}\). The fact a second global minimum of the effective potential could be generated by physics in the UV has also been emphasized in [70]. To summarize, our arguments do not exclude that our electroweak vacuum is unstable, but they suggest that such an in/meta-stability would have to be provided by the microscopic underlying theory; see [74] for a specific example from string phenomenology. In this case, however, the Higgs-mass bounds from metastability as well as the life-time estimates of the electroweak vacuum would be very different from the conventional estimates; see e.g. [75]. - (4)
As mentioned above, the result for the fermion determinant is regulator dependent, as long as the cutoff is kept finite. The preceding results have been derived for a sharp cutoff in momentum space. These results in fact generalize to arbitrary smooth cutoff shape functions in momentum space as they can be implemented straightforwardly within the functional RG framework; see below and Appendix B. Though this is not an issue for the present model, one might be concerned about the fact that such regularizations are not gauge invariant. Hence a gauge-invariant regularization is studied in the remainder of this section.

### 3.2 \(\zeta \) function regularization

Whereas for the sharp cutoff, this was an obvious artifact of the \(\Lambda \rightarrow \infty \) limit, the failure is less obvious here. Nevertheless, as we have derived this misleading result of a negative contribution from a strictly positive expression given in Eq. (18), it is clear that the standard strategies of dimensional regularization fail to describe the global behavior of the fermion determinant properly. The reason is that dimensional regularization is not only a regularization but at the same time a projection solely onto the logarithmic divergencies. It has in fact long been known that the use of dimensional regularization in the presence of large fields can become delicate; procedures to deal with this problem typically suggest to go back to the dimensionally continued proper-time/\(\zeta \)-function representation that we started out with [76, 77].

There is another perspective that explains why the standard perturbative argument of integrating the \(\beta \) function of the \(\phi ^4\) theory is misleading as far as vacuum stability is concerned: the \(\beta \) functions are typically derived in mass-independent regularization schemes (though mass-dependent schemes have recently also been studied [78]), and it is implicitly assumed that the discussion can be performed in the deep Euclidean region where all mass scales are much smaller than any of the involved momentum scales of the fluctuations. The latter assumption is in fact not valid, as both scales the value of the field as well as the cutoff \(\Lambda \) can interfere non-trivially with each other. This is illustrated rather explicitly in the sharp-cutoff calculation given above.

## 4 Renormalization flow

Independently of the validity of the perturbative arguments about vacuum stability, the comparatively small mass of the observed Higgs boson [1, 2] poses a challenge: the fermionic fluctuations (dominated by top loops) contribute to the curvature of the effective potential which determines the Higgs boson mass. Even in the absence of any bosonic interactions, this appears to lead to a lower bound on the value of the Higgs mass which are in tension with the measured value. This line of argument has been used in quantitative lattice studies [37, 41, 42, 43, 44]. For rendering simulations on a Euclidean lattice well defined, the bosonic action has to be bounded from below; in practice, lower bounds on the Higgs mass thus arise in the limit of the bare \(\phi ^4\) coupling approaching zero, \(\bar{\lambda }\rightarrow 0_+\).

While it is debatable whether this criterion could be relaxed for a Minkowskian functional integral in the continuum, we have already provided first examples in the \(\mathbb {Z}_2\)-symmetric Yukawa toy model that these conventional lower bounds can be relaxed by allowing for more general forms of the bare potential [51]. These results have recently been confirmed in lattice simulations [52]. In particular, no state of meta-/instability is required for relaxing the lower bound. In the following, we generalize these results to the chiral top–bottom–Higgs Yukawa model.

^{5}The reliability of the derivative expansion can be monitored with the aid of the anomalous dimension, providing a rough measure for the importance of the higher-derivative terms. In practice, we study the convergence of the derivative expansion by comparing leading-order results (\(\eta _i=0\)) to the full NLO calculation; see below.

## 5 Non-perturbative Higgs-mass bounds

The flow equations listed above enable us to take a fresh look at Higgs boson mass bounds possibly arising from the RG flow of the model. For this, it is useful to think of the RG flow as a mapping from a microscopic theory defined at some high scale \(\Lambda \) onto the effective long-range theory governing the physics observed in collider experiments. For this mapping, we use the standard-model-type parameters \(v=246\) GeV, \(m_{\text {t}}=173\) GeV and \(m_{\text {b}}=4.2\) GeV as constraints. The range of all possible Higgs boson masses resulting from the remaining UV parameters for a given cutoff then defines the IR window and correspondingly puts bounds on the Higgs boson mass as a function of the cutoff scale \(\Lambda \).

In full generality, constructing this mapping is a complex problem, since the microscopic theory at scale \(\Lambda \) is a priori unconstrained to a large extent. At first sight, it seems natural to allow only the renormalizable terms in the bare action. For the present model, this has been successfully implemented in extensive lattice simulations [37, 41, 42, 43, 44, 45, 46] yielding quantitative results for the Higgs-mass bounds. In particular, the lower bound arises from the lowest possible value for the Higgs selfinteraction \(\bar{\lambda }\phi ^4\), i.e., \(\bar{\lambda }\rightarrow 0\), for which the lattice theory remains well defined. The resulting bounds should therefore not be viewed as vacuum stability bounds, but as consistency bounds arising from the requirement that the underlying lattice partition function is well defined.

However, there is no need at all to confine the bare theory to just the renormalizable operators. On the contrary, generic underlying theories (UV completions) are expected to produce all terms allowed by the symmetries, such that the search for Higgs-mass bounds corresponds to finding extrema of a function (the Higgs mass) depending on infinitely many variables (the bare action). Formulated in terms of this generality, it is actually unclear whether these extrema and thus a universal consistency bound would exist at all. We therefore confine our study to a much simpler question: given the lower mass bound arising within the conventional class of \(\phi ^4\) potentials, can we find more general bare potentials, which (a) lower the mass bounds and (b) do not show an instability towards a different vacuum neither in the UV nor in the IR? While (a) is obviously inspired by the fact that the measured Higgs boson mass seems to lie below the conventional lower bound, an answer to (b) can serve as an illustration that no meta-stability is required in order to relax the lower bound.

While these two questions have been answered in the affirmative for the \(\mathbb {Z}_2\)-Yukawa model in [51] with functional RG methods as well as for the present chiral model with lattice simulations up to cutoff scales on the order of several TeV [52], the present flow equation study can elucidate the underlying RG mechanisms in more detail and can bridge a wide range of scales. Furthermore, it is straightforward to deal with two distinct quark masses, \(h_{\text {t}}\ne h_{\text {b}}\), in our functional approach, whereas simulations with the physical ratio \(m_{\text {t}}/m_{\text {b}}\simeq 40\) would be rather expensive on the lattice.

Before we turn to a quantitative analysis of the RG flow, we have to cure a deficiency of the chiral Yukawa model in comparison with the full standard-model top–bottom–Higgs sector. Since chiral symmetry breaking in the present model breaks a global symmetry, our present model has massless Goldstone bosons in the physical spectrum. This is different from the standard model where the would-be Goldstone bosons due to their interplay with the gauge sector are ultimately absent from the physical spectrum, the latter finally containing massive vector excitations. In order to make contact with the standard-model physics, we therefore have to modify our chiral Yukawa model, otherwise the massless Goldstone modes have the potential to induce an IR behavior which is very different from that of the standard model. This modification of the model is not unique and could be done in various ways. For instance on the lattice, the influence of the unwanted Goldstone bosons is identified by their strong finite volume effects and subtracted accordingly [41, 42, 43, 44, 45, 46]. Similarly, we could study the onset of Goldstone dynamics in the limit \(k\rightarrow 0\) and perform a similar subtraction.

As a result, all fluctuations acquire a mass in the regime of spontaneous symmetry breaking (SSB) and the whole flow freezes out, similarly to the \(\mathbb {Z}_2\) Yukawa model and as expected in the full standard model.

Now that we have amended our model with a dynamical removal of the unwanted Goldstone bosons, the RG flow of the model is technically similar to the simpler \(\mathbb {Z}_2\) invariant Yukawa model extensively studied in [51]. In the following, we therefore focus on the new features induced by the additional degrees of freedom of the chiral model such as the bottom quark and the three additional real scalar fields. Further technical details follow those of [51].

### 5.1 Bare potentials of \(\phi ^4\) type

By contrast: for large \(\lambda _{2,\Lambda }\), the theory already starts in the SSB regime with a small value for \(\kappa _{\Lambda }\). The flow still runs over many scales, depending on the initial conditions, until \(\kappa \) eventually grows large near the electroweak scale. As a result, all modes decouple and we can read off the long-range observables.

The flow equation provide us with a map of the UV parameters to physical parameters such as the mass of the Higgs, the top or the bottom quark. In the following, we fine tune either \(\lambda _{1,\Lambda }\) if we start in the SYM regime or \(\kappa _{\Lambda }\) in the SSB regime, in order to arrive at a vev of \(v_{k\rightarrow 0}=246\) GeV in the IR. Further, we vary the bare top \(h_{t,\Lambda }\) and bottom \(h_{b,\Lambda }\) Yukawa coupling such that we obtain the desired top and bottom quark mass, \(m_{\text {t}}\simeq 173\) GeV and \(m_{\text {b}}\simeq 4.2\) GeV. For this reduced class of bare \(\phi ^4\) potentials, the Higgs mass is only a function of the bare quartic coupling \(\lambda _{2,\Lambda }\) for a fixed cutoff. In order to start with a well-defined theory in the UV, \(\lambda _{2,\Lambda }\) must be strictly nonnegative.

We find that the Higgs mass is a monotonically increasing function of the bare quartic coupling, which can be seen in Fig. 1. Here, the Higgs mass \(m_{\text {H}}\) is plotted as a function of the bare quartic coupling \(\lambda _{2,\Lambda }\) for a fixed cutoff \(\Lambda =10^7\) GeV. The lower bound is approached for \(\lambda _{2,\Lambda }\rightarrow 0\), where the Higgs mass becomes rather independent of \(\lambda _{2,\Lambda }\). This was also shown in lattice simulations for the \(\mathbb {Z}_2\) model [35] as well as for a chiral Yukawa theory [41, 42, 43, 44]. For large bare quartic couplings the Higgs mass reaches a region of saturation.

To test the convergence of our expansion and truncation, we plotted the Higgs mass in various approximations in Fig. 1. The derivative expansion is tested by comparing leading-order (LO) (dashed lines) to NLO results (solid lines). At LO, we drop the running of the kinetic terms in Eq. (22), achieved by setting the anomalous dimensions to zero in the flow equation (23) and (24). These differ by at most \(12\,\%\) for small as well as for large couplings. The difference for small couplings is somewhat larger than in the \(\mathbb {Z}_2\)-symmetric Yukawa model because of the larger number of fluctuating scalar components.

Furthermore, we varied \(N_{\text {p}}\) to check the convergence of the polynomial expansion of the potential. The simplest non-trivial order is given by \(N_{\text {p}}=2\) and plotted as red lines with squares in Fig. 1. For \(N_{\text {p}}=4\) (blue lines with circles) there are only small deviations for small \(\lambda _{2,\Lambda }\) (\(\sim 2\) GeV) and deviations of \(5\,\%\) for large \(\lambda _{2,\Lambda }\) compared to \(N_{\text {p}}=2\). Beyond this, we find no deviations between the Higgs masses for \(N_{\text {p}}=4,5,6,8,10\) within our numerical accuracy, demonstrating the remarkable convergence of the polynomial truncation for the present purpose.

In Fig. 2, the resulting Higgs masses are plotted as a function of the UV cutoff for different bare quartic couplings for a wide range of cutoff values \(\Lambda =10^3,\ldots ,10^9\) GeV. The lower black line is derived for \(\lambda _{2,\Lambda }=0\) and indicates a lower bound for \(m_{\text {H}}\) within the \(\phi ^4\) type bare potentials. Incidentally, it agrees comparatively well with the results of a simple mean-field (“large-\(N_{\mathrm {f}}\)”) calculation sketched in the appendix. Dashed lines depict upper Higgs-mass bounds if one restricts the bare quartic coupling to \(\lambda _{2,\Lambda }\le 1,10,100\) (from bottom to top). Artificially restricting the coupling \(\lambda _{2,\Lambda }\) to the perturbatively accessible domain, say, \(\lambda _{2,\Lambda }\lesssim 1\), the upper bound is obviously significantly underestimated.

Finally, we should emphasize once more that the use of standard-model-like parameters is only for the purpose of illustration. The quantitative difference becomes obvious, e.g., from Fig. 2 where the “channel” of Higgs mass values that allow for a large cutoff is centered near \(m_{\text {H}}\simeq 200\) GeV. The same channel-like behavior in the full standard model occurs near \(m_{\text {H}}\simeq 130\) GeV. This quantitative difference is mainly due to the influence of the gauge sectors, in particular the strong interactions. But also the electroweak gauge sector can take a conceptually (if not quantitatively) important influence on mass bounds: e.g., recent non-perturbative lattice simulations of the Yang–Mills–Higgs system suggest that the Higgs mass has to be larger than the weak gauge-boson masses in certain parameter regimes, otherwise the electroweak sector would rather be in a QCD-like domain [94].

### 5.2 Generalized bare potentials

Motivated by previous continuum calculations in the \(\mathbb {Z}_2\) model [51] and by lattice studies in the chiral version [52], we study whether more general bare potentials can modify the phenomenologically relevant lower Higgs-mass bound. The main purpose of this study is to demonstrate that the lower bound can be relaxed without the occurrence of an in- or metastability of the potential. An analysis of all conceivable bare potentials is a numerically challenging problem and beyond the scope of this work.

Similarly to the \(\mathbb {Z}_2\) invariant model, this phenomenon of a relaxed bound as a consequence of a modified bare theory can be understood by the RG flow itself [51]. First note that the parameters for the generalized bare potential are chosen in such a way that the potential is initially in the SYM regime; this is also true for the lower bound within \(\phi ^4\) theory. In the present case of the generalized bare potential, the negative quartic coupling \(\lambda _2\) flows quickly to positive values whereas \(\lambda _3\) becomes small as expected in the vicinity of the Gaußian fixed point. Therefore, the system essentially flows back into the class of \(\phi ^4\)-type potentials. In other words, the system defined for a fixed cutoff \(\Lambda \) with \(\lambda _{2,\Lambda }<0\) and \(\lambda _{3,\Lambda }>0\) can be mapped to a system with \(\lambda _{2,\Lambda }>0\) and \(\lambda _{3,\Lambda }\approx 0\) for an effectively smaller cutoff \(\tilde{\Lambda }<\Lambda \). Roughly speaking some RG time is required to flow from the beyond-\(\phi ^4\)-type potentials back to the class of standard \(\phi ^4\)-type potentials. Thereby, the red dashed line can be interpreted as a horizontally shifted variant of the Higgs mass curve derived from \(\phi ^4\) potentials for effectively larger cutoffs.

We emphasize that the effective potential is stable at all scales with one well-defined minimum for the present choice of parameters.^{6} Finally, we would like to point out that the non-perturbatively computed effective potential for a finite cutoff is of course regularization scheme dependent much in the same way as the fermion determinant presented above. Choosing different regulator shape functions would also lead to (typically slightly) different Higgs mass curves in Figs. 1, 2, 3 and 4. We expect that this scheme change could be compensated for by a corresponding change of the bare action. Hence, it suffices to vary only the bare action for a fixed regulator in order to illustrate our main points.

## 6 Conclusions

### 6.1 Summary

We have analyzed a chiral Yukawa model featuring the interactions of a scalar SU(2) Higgs field with a chiral top-bottom quark sector similar to the Higgs sector of the standard model. We have critically re-examined conventional perturbative arguments that relate a lower bound for the Higgs mass with the stability of the effective potential. Based on exact results for the regularized fermion determinant, we have shown that the interacting part of the fermion determinant contributes strictly positively to the effective scalar potential for any finite field value – as long as the UV cutoff \(\Lambda \) is kept finite. We have shown that this result holds for a variety of regularization schemes including the sharp momentum cutoff as well as (gauge-invariant) \(\zeta \)-function/proper-time regularization schemes.

Furthermore, we have shown that the conventional perturbative conclusion of a vacuum in-/metastability of the effective potential due to top fluctuations can be rediscovered if the cutoff is forced to approach infinity together with standard ad hoc recipes to project onto the finite parts. For the example of the sharp cutoff, we have shown explicitly that this corresponds to an illegitimate order of limits, as the resulting instability occurs at scalar field values where the supposedly small expansion parameter of the \(\Lambda \rightarrow \infty \) limit is actually of order 1. A similar failure occurs for dimensional regularization where the standard procedures of projecting onto the finite parts violate the positivity properties of the interacting part of the effective potential. Our findings corroborate earlier results from non-perturbative lattice simulations [35, 36], but in addition allow for a large separation of the UV cutoff from the Fermi scale and an analytic control of the corresponding limits.

Because of the presumable triviality of the present model as well as the Higgs sector of the full standard model, the cutoff most likely cannot be removed from the theory – at least not within a straightforward manner. The cutoff as well as a corresponding regularization scheme should rather be viewed as part of the definition of our particle physics models that parametrize the embedding of this field-theory description into a possibly UV complete theory. Still, as long as the cutoff is large compared to the Fermi scale, Wilsonian renormalization arguments guarantee that the low-energy observables are largely insensitive to the details of this embedding. We have demonstrated that a counter-example to this generic rule is given by bounds on the mass of the Higgs boson.

In this work, we have not performed an exhaustive analysis of different bare actions or potentials, but simply focused on a constructive example that leads to Higgs boson masses below the conventional lower bound. Most importantly, this example exhibits no vacuum in-/metastability.

This together with our basic line of argument involving exact results for the fermion determinant demonstrate that there is no reason for concern arising from top-quark fluctuations as far as false vacuum decay in our universe is concerned, despite the comparatively light value of the measured Higgs mass. This does not mean that there might be no reason for concern at all. For instance, if the bare scalar potential itself features an instability induced by the underlying UV complete theory, our standard model could still live in an un- or metastable vacuum. Our arguments only exclude instabilities caused by the fluctuations of the fermionic matter fields within the standard model.

### 6.2 Vacuum stability vs. consistency bounds

Our results suggest a revision of the standard picture of Higgs mass bounds as a function of the UV cutoff. Depending on the implicit assumptions made to derive mass bounds, this revision might be more or less significant.

From our results on fermion determinants, it is clear that the conventional interpretation that top-quark fluctuations induce a vacuum instability is not tenable; this interpretation is a result of taking an inconsistent \(\Lambda \rightarrow \infty \) limit. Still, the top-quark fluctuations play, of course, a decisive role for the value of the Higgs mass. In order to reconcile these observations, we propose a UV-to-IR viewpoint: the Higgs-mass bounds should be understood as a mapping from initial conditions set at the UV cutoff given in terms of a microscopic bare action \(S_\Lambda \) onto all IR values accessible by the RG flow of the system, \(m_{\text {H}}=m_{\text {H}}[\Lambda ;S_\Lambda ]\). In this manner, Higgs-mass bounds arise from consistency conditions imposed on the bare action. For instance, in order to start from a well-defined (Euclidean) partition function, the action needs to be bounded from below.

The conventional vacuum stability bounds then are approximately equivalent to such a consistency bound arising within a restricted class of bare actions, e.g., bare potentials of \(\phi ^4\) type; here, the bare \(\phi ^4\) coupling is required to be positive for consistency of the generating functional. However, as the bare action is not at our disposal but generally a result of the underlying UV embedding, there is no reason to make such restrictive assumptions. Already for slightly more general bare actions, we have been able to show that the conventional lower mass bounds can be substantially relaxed in the present chiral Yukawa model. The reason is that a more general bare action can modify the RG flow near and below the cutoff. As a result, the consistency bound lies below the vacuum stability bound. In particular, we have given an explicit example with a Higgs boson mass below the “stability bound” but an in fact stable effective potential on all scales; our results are in agreement with lattice simulations [52] and extend them to a much wider range of scales.

Determining the consistency bound remains an open problem, the solution of which requires further assumptions. One natural but not necessary assumption could be that the effective action should feature a unique minimum on all scales. The consistency bound would then arise from a complicated extremization problem in the space of all consistent bare actions subject to the unique-minimum constraint (to be satisfied on all scales). Even in this case, it seems unclear whether the bound remains finite. Therefore, it appears reasonable to add another physical assumption: since the bare action is expected to be provided by an underlying (UV complete) theory at scale \(\Lambda \), it is natural to assume (in the absence of any concrete knowledge about the underlying theory) that the couplings of all possible operators are of order \(\mathcal {O}(1)\) if measured in terms of \(\Lambda \). For future studies, it is one of the most pressing questions to quantitatively estimate the resulting consistency bound under such a set of assumptions.

Of course, it appears equally legitimate to give up the criterion of a unique minimum at all scales, but instead allow for further minima in the bare action. If the resulting IR effective action turns out to have one unique minimum again (to be identified with the electroweak minimum), such bare actions can lead to a further relaxation of the consistency bound described above. In the general case, it should be possible to construct bare actions with multiple local minima such that the full effective action has a global minimum different from the local electroweak minimum. Since such bare actions are less constrained than those of the preceding scenarios, we expect the resulting lower Higgs mass consistency bounds to be even more relaxed to smaller values. Again, a quantitative estimate of such consistency bounds including metastable scenarios remains an urgent question.

Comparing the conventional stability bounds with the present consistency bounds, the overall picture seems to be qualitatively similar. The primary main difference is of quantitative nature, since the unique-minimum consistency bound lies below the stability bound. Wilsonian RG arguments, however, suggest that this difference could become small for large UV cutoffs, as is also reflected by the example of Fig. 4. Nevertheless, the size of this quantitative difference substantially depends on the assumptions imposed on the size of the couplings in the bare action. Also, the consistency bound is necessarily regularization scheme dependent. As the regularization actually should model the details of the embedding into the underlying UV completion, this dependence has a physical meaning. Even larger differences are expected between the conventional meta-stability bound and the consistency bound including metastable scenarios. The reason is that the metastable features are expected to arise from the bare action and thus are largely unknown. The size of the metastable region and corresponding life-time estimates will be even more subject to assumptions on the bare action.

### 6.3 Outlook

Independently of whether the measured value of the Higgs boson mass eventually turns out to lie slightly above or below the conventional lower bound, it is remarkable that the Higgs- and top-mass parameters appear to lie close to a region in the IR parameter space that can be connected to a bare UV effective potential that could exhibit almost vanishing scalar self-interactions. In this sense, a precise measurement of these mass parameters is relevant beyond the pure goal of precision data. These measurements can impose requirements that any UV embedding has to satisfy. The viewpoint of consistency bounds presented above provides a means to quantify these requirements. Therefore, a comprehensive quantitative exploration of these bounds appears most pressing.

One scenario appears particularly interesting: if the Higgs mass eventually turns out to be exactly compatible with a UV flat potential (apart from a possible mass term), a corresponding embedding would have to explain this rather particular feature. It is interesting to note that such scenarios exist even within purely quantum field theory approaches as, for instance, in models with asymptotically safe gravity [95, 96].

Recently, an asymptotically safe/free scenario in a gauged chiral Yukawa model has been identified [93], the UV limit of which corresponds to a flat scalar potential also allowing for comparatively light Higgs masses in the IR. In order to explore this option of a UV complete limit, we perform an RG fixed-point search also within this model along the lines of [89, 90] in Appendix E. However, in this ungauged model, we find no reliable indication for the existence of a non-Gaußian fixed point. Still, our present findings should serve as a strong motivation to further search for asymptotically free gauged chiral models with viable low-energy properties.

## Footnotes

- 1.
For the derivative expansion used below, these mass definitions already agree with the pole masses.

- 2.
In principle, perturbative predictions for \(\Lambda \rightarrow \infty \) may differ from those with a finite cutoff. However, let \(m_{\text {Obs}}\) denote the scale of a typical IR observable; then this difference is typically of order \((m_{\text {Obs}}/\Lambda )^{(2p)}\), where \(p\) is some integer. For sufficiently large \(\Lambda \), this difference hence becomes insignificant.

- 3.
In this section, we do not distinguish between bare and renormalized scalar fields, as this is not relevant at this order.

- 4.
Let us illustrate this with an extreme example: if the bare bosonic potential \(U_{\text {B}}\) with several minima is chosen such that the global minimum occurs at field values \(\phi _{\text {gm}}\gg \Lambda \) with a local curvature \(U''(\phi _{\text {gm}})\gg \Lambda ^2\), the fluctuations with momenta \(p<\Lambda \) can be expected to essentially renormalize only the inner part of the potential. The dynamics near the global minimum \(\phi _{\text {gm}}\) would completely decouple from all fluctuation physics of such a model. Hence, this global minimum of the bare potential is likely to remain the global minimum of the full potential, such that further minima at smaller field values would be metastable.

- 5.
As further cross-checks, we note that our flows also agree with those of [89, 90, 91] for \(N_{\text {L}}=2\) and \(h_{\text {b}}=0\) (apart from a missing factor of \(1/2\) in \(\eta _{\text {L,R}}\) as also noted in [92]). We also observe agreement with the flows of [93] for \(h_{\text {b}}=0\) and upon dropping the gauge sector in that work.

- 6.
As we compute the effective potential by means of a polynomial expansion about the minimum, these polynomials can seemingly develop new minima or instabilities at extremely large field values as we vary the truncation order \(N_{\text {p}}\). In [51], we have therefore carefully estimated the convergence radius of this expansion. For all examples presented here, the effective potential does not show any instability or second minimum within this radius of convergence where the polynomial expansion can be trusted. This is confirmed by numerical integrations of the flow equation for the full effective potential as performed in [40] using pseudo-spectral methods (Chebyshev expansion). For more general bare potentials possibly featuring further minima, the polynomial expansion appears insufficient and numerical precision computations for the full effective potential become mandatory.

## Notes

### Acknowledgments

We thank Gerald Dunne, Tobias Hellwig, Jörg Jäckel, Stefan Lippoldt, Axel Maas, Jan Pawlowski, Tilman Plehn, Michael Scherer, Andreas Wipf, Christof Wetterich, and Luca Zambelli for interesting discussions. We acknowledge support by the DFG under grants No. GRK1523/2, and Gi 328/5-2 (Heisenberg program).

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