Vacuum Stability of Asymptotically Safe Two Higgs Doublet Models

We study different types of Two Higgs Doublet Models (2HDMs) under the assumption that all quartic couplings' beta functions vanish simultaneously at the Planck scale. The Standard Model seems to display this property almost accidentally, because the Higgs boson mass is close to 125 GeV. This also ties closely into the question of whether the theory is stable or metastable. We investigate if such"fixed points"can exist in various $\mathbb{Z}_2$-symmetric 2HDM subclasses, and if the theories that meet these conditions are phenomenologically viable, as well as vacuum stable. We find that the fixed point condition drastically reduces the parameter space of 2HDM theories, but can be met. Fixed points can only exist in type II and type Y models, in regions of large tan$\beta$, and they are only compatible with all existing experimental bounds if the $\mathbb{Z}_2$-symmetry is at least softly broken, with a soft breaking parameter of at least $M_{12}$ $>$ 70 GeV (380 GeV) for type Y (type II) models. The allowed region falls into the alignment limit, with the mixing angle combination $|\alpha - \beta| \approx\frac{\pi}{2}$. While there are both vacuum-stable and vacuum-unstable solutions, only the vacuum-unstable ones really agree with Standard-Model-like CP-even Higgs boson mass values of 125 GeV. The vacuum-stable solutions favour slightly higher values. While scenarios of asymptotically safe 2HDM exist, they cannot improve over the Standard Model regarding the question of vacuum stability.


Introduction
The discovery and the mass measurement of a Standard Model (SM)-like Higgs boson by ATLAS and CMS in 2012 [1,2] so far rank among the most impactful events in this century's particle physics. It is an interesting situation that the Higgs mass of m H = (125.5 ± 0.5) GeV lies right at the edge of the so-called stability bound [3][4][5]. Extrapolating from the Higgs mass value to very short distances shows that the LHC result seems to hint at a quartic coupling of λ = 0 at Planck scale-like energies, and also the renormalisation group (RG) beta function β λ (m P l ) ∼ 0.
The argument also works in reverse: Before the LHC experiments had discovered a Higgs boson, calculations were performed to show that initial conditions of λ = 0 and β λ = 0 at high scales naturally point to Higgs mass values around 125 GeV [6], as do the combination of a vanishing beta function at high scales and the experimental measurement of the top quark mass, or of a vanishing quartic coupling at high scales and the top quark mass [7]. The idea of vanishing beta functions suggests a link to the field of Asymptotic Safety [8][9][10], in which RG flow fixed points play a critical role. Originally a concept for quantum gravity, Asymptotic Safety has in recent years become a point of interest in SM extensions, as a tool for UV completion or generalised renormalisability [11][12][13][14][15][16][17]. A relation between vanishing quartic couplings and vanishing beta functions at high scales and the measured Higgs mass at the LHC may be coincidental. On the other hand, the question of vacuum stability remains. Experimental results suggest that the SM vacuum is metastable, although agreement on how strong a statement can be made has not yet been reached [18]. Here, we study how models with an enlarged scalar sector behave in this regard. To this end, we look at Two Higgs Doublet Models (2HDMs) and investigate if the same properties can be found, and what ramifications the existence of fixed points can have for these models. Specifically, we examine 2HDMs that exhibit simultaneously vanishing quartic coupling beta functions β λ i (µ) at the Planck scale m P l = 1.2 · 10 19 GeV.
This paper is organized as follows: In Section 2, general properties of 2HDMs are reviewed. A detailed outline of how the analyses are performed are then given in Section 3. Sections 4 and 5 subsequently treat different types of 2HDMs, including the complete softlybroken Z 2 -symmetric model. An Appendix contains the complete two-loop beta functions of all 2HDM couplings used in this work.

2HDM
In this section, we briefly review the features of the general 2HDM, before reviewing the current state of bounds on the model from different sources.

General properties of the 2HDM
A 2HDM contains two SU(2) doublets Φ 1 , Φ 2 [19]. The most general scalar potential takes the form: (2.1) In this notation, following [20], m 11 , m 22 , and λ 1 to λ 4 are real-valued, whereas M 12 , λ 5 , λ 6 , and λ 7 are complex parameters. Of these 14 degrees of freedom, only eleven are physical. The rest can be absorbed by making use of the freedom of choice of bases for the SU(2) doublets Φ i . For spontaneous symmetry breaking (SSB), both fields Φ 1 and Φ 2 are assigned a vacuum expectation value (VEV): The SU(2) doublets contain eight physical fields , three of which are absorbed during SSB. The remaining physical Higgs bosons after rotating into mass eigenstates are the charged Higgs H ± , a pseudoscalar Higgs A and two CP-even scalar Higgs h, H. The rotation angle diagonalizing the CP-even scalar mass matrix is conventionally called α, the angle diagonalizing the charged and CP-odd bosons is called β. The latter angle β also appears in the ratio of v 2 v 1 ≡ tanβ. In general, 2HDMs permit tree-level FCNCs. According to the Paschos-Glashow-Weinberg theorem [21,22], a necessary and sufficient condition for their absence is to I -1  1  1  1  1  1  type II -1  1  1  -1 -1  1  type X -1  1  1 1 -1 1 type Y -1 1 1 -1 1 1  [23]. For our purpose the leptons only contribute minor corrections when compared to the quarks, so the primary computational focus will be on type I and type II models. The Yukawa Lagrangian for type I and type II 2HDMs are hence given by: where Y u,d,l are the Yukawa matrices for up-, down-, and lepton type particles, Q l , L l , u R , d R , and l R are left-and right-handed quark and lepton fields respectively, and i, j denote the generations in flavour space. In our calculations, only the dominant y 33 entries generated by the top quark, the bottom quark, and the tau lepton respectively, will be considered. Thus, the Yukawa matrices are assumed to have the simplified structures Under the Φ 1 → −Φ 1 Z 2 symmetry mentioned above, it follows that λ 6 = λ 7 = 0, which leads to a mass matrix for the CP-even neutral scalars of the form: with λ 345 = λ 3 +λ 4 +Re(λ 5 ). The terms m 11 and m 22 can be eliminated using the minimum conditions from SSB, that is ∂V ∂v i = 0: The charged and the pseudoscalar Higgs mass matrices are given by: (2.8) Both have one zero eigenvalue, corresponding to the charged and the pseudoscalar Goldstone boson, respectively. The pseudoscalar mass vanishes for M 12 = λ 5 = 0, because of an additional accidental spontaneously broken U(1)-symmetry. In 2HDMs, to be vacuum-stable the potential needs to be bounded from below in all directions. This is the case if and only if the following set of inequalities is met [24]: Unlike the SM, theories with more than one Higgs doublet can display a range of different vacuum configurations [25]: Not only can there be more than one minimum at the same time, but the minima can also be of CP breaking type, when the VEVs have a relative complex phase, or of charge breaking type, with one VEV carrying an electric charge. It has however been shown that minima of different types (i.e. CP-breaking, charge-breaking, or normal) cannot exist simultaneously within the same model [26][27][28]. By requiring the model to fulfil the minimum conditions for normal-type minima given by Eqs. (2.5) and (2.6), it is therefore assured that the absolute minimum of the theory is also normal. It only remains to be checked if the minimum at v = 246 GeV is global, or if there is another, deeper one.

Limits on 2HDM Parameter Space
While the 2HDM is a relatively simple SM extension, it still contains up to eleven new free parameters (six in the type II models studied below). On the other hand, the model's high popularity means that its parameter space has been comprehensively explored and constrained from both the theoretical and the experimental side, and in particular by recent LHC data [29][30][31]. At this point, we briefly review current bounds, more thorough discussions of different aspects can be found for example in [32][33][34][35][36][37][38].
In essence, constraints on the 2HDM parameter space can be sorted into three categories: Theory bounds are generated by requiring the model to possess certain features, commonly referred to as positivity (the Higgs potential must be bounded from below, cf. Eq. (2.9)), perturbativity (quartic couplings must not be large), and unitarity (of the S-matrix of 2→2 scattering amplitudes) [39,40]. Secondly, there are mass bounds on the physical Higgs bosons from signal strength data by the ATLAS and CMS collaborations. These searches have confirmed the existence of a 125 GeV CP-even scalar eigenstate, and they also show that this boson couples to vector bosons and fermions in a very SM-Higgs-like fashion [41][42][43][44]. Furthermore, the absence of heavier resonances so far translates to mass bounds for the other Higgs eigenstates. Lastly, there are implications for the 2HDM from flavour physics [45]. Most notably, B(b → sγ) measurements exclude charged Higgs masses smaller than m H + = 580 GeV [46] in type II/type Y models, lower bounds on tanβ can be extracted from B s mass differences and leptonic decays [47].
Together, these bounds can be combined to make a number of statements: The masses of the three additional Higgs bosons all must be large, the mass differences between them, however, small. The rotation angles must fulfil |β − α| ≈ π 2 , ensuring that the mass basis of the CP-even scalar states aligns with the SM gauge eigenbasis. These features are thus usually referred to as alignment limit [48][49][50][51]. It should be noted that some studies have used fine-tuning arguments to impose stronger bounds on tanβ, and successively to the heavy Higgs boson masses [33]. Since the RG methods employed in this work contain a certain degree of tuning by design, they offer an alternative as to why these large tanβ regions may yet be phenomenologically viable. As a consequence, our mass bounds are slightly more conservative than some.

Solving the Fixed Point Equations
We pursue the question whether 2HDMs support "fixed points" at the Planck scale in the same way the SM does, and if the resulting models are vacuum-stable. The fixed point condition reads: i.e., the beta functions of all quartic couplings λ i present in the scalar potential are to have a root at the Planck mass m P l . Because of contributions from gauge and Yukawa couplings, the condition of β λ i = 0 is not technically sufficient to define a fixed point, nor does it necessarily lead to an asymptotically safe theory by itself; for recent progress in BSM model building see e.g. [12]. Still, because of similarities to the SM case and for convenience, the terms fixed point and fixed point condition will be used in this context, effectively interpreting effects disturbing the equilibrium into the realm of beyond the Planck scale physics. While in the SM there is only one quartic Higgs coupling λ, the 2HDM potential with a Z 2 symmetry protecting flavour conservation can contain up to five quartic terms (one of which may be complex). This means that compared to the SM, the fixed point condition has a much higher impact in terms of limiting the parameter space of the theory.
The search for fixed points comes down to solving the system of differential equations given by the beta functions of the running couplings of the theory. It involves the gauge couplings g 1 , g 2 , g 3 , the quartic scalar couplings λ i and the Yukawa couplings λ t , λ b , λ τ . The complete two-loop expressions for the most general beta functions used are calculated with the Mathematica package SARAH [52,53], and listed in Appendix A. Since the coefficients m 11 and m 22 of the dimension-two-operators do not appear directly in the beta functions of any other couplings, m ii can be ignored at this point and determined with help of the minimum conditions at the electroweak scale, see Eqs. (2.5) and (2.6). The soft breaking parameter M 12 also does not appear in the beta functions of quartic, gauge, or Yukawa couplings, and will be treated as a free parameter.
While the quartic couplings are already fixed implicitly by (3.1), the remaining initial conditions are given explicitly at low scales: Both gauge couplings and Yukawa couplings can be determined from experimental measurements. The MS gauge coupling initial conditions for g 1 and g 2 are calculated using the fine structure constant α −1 (M Z ) = 127.95 ± 0.017 and the weak mixing angle sin 2 θ W = 0.23129 ± 5 · 10 −5 [54,55] to: Uncertainties on gauge coupling initial values are small enough to be inconsequential. The relations between Yukawa couplings and quark masses are model-dependent. In a type II model, the Yukawa couplings are related to the quark masses by the tree-level relations: while in the type I model the bottom quark Yukawa coupling is instead determined by the other VEV: GeV, the τ -lepton mass is given by m τ (m τ ) = 1.78 GeV [55]. For the purpose of this paper, it is assumed that all non-SM effects only affect the running from the electroweak scale onwards. In other words, the Yukawa couplings are run up to M Z under SM-like conditions, at which point the tanβ-enhancement is switched on. The effective initial values used are thus: The so defined initial value problem is solved numerically. With the full RG flow of all couplings known, their low scale values are used to determine the mass spectrum of the theory using the matrices given in Section 2. The results depend on the parameters treated as free (in these cases tanβ and later M 12 ) and on the experimentally determined coupling initial conditions, but beyond this are a direct consequence of the theory itself and the fixed point assumption.
With all couplings known at all scales, the question of vacuum stability can also be answered: For a solution to be vacuum-stable, the quartic couplings must fulfil the inequalities given by Eq. (2.9) at all energy scales up to µ = m P l .

The CP-conserving 2HDM with Z 2 Symmetry
We study a 2HDM with a discrete Z 2 symmetry (Φ 1 → −Φ 1 , Φ 2 → Φ 2 ) introduced in order to ensure CP-conservation in the scalar sector. The scalar potential reads: with real-valued mass parameters m ii and quartic couplings λ i . The restriction Im(λ 5 ) = 0 does not follow immediately from the Z 2 symmetry, but can be assumed in this case without loss of generality due to the structure of the beta functions: λ 5 always appears in the other quartic couplings' beta functions in the form of the norm squared, |λ 5 | 2 , and the function β λ 5 is proportional to λ 5 , with the remainder being comprised by real-valued terms only. It follows that the function β λ 5 must have a constant phase. It is also easy to see that in these circumstances for every function λ 5 (µ) that solves the system of differential equations, a phase-shifted e iθ · λ 5 (µ) is also a solution for every constant phase θ. Therefore, it suffices to look at the real values for λ 5 when searching for fixed points. The different Yukawa Lagrangians in type I and type II models lead to differences in the beta functions, as the Yukawa terms will appear in the running of different couplings. As an example, in the type I model one-loop beta function of quartic coupling λ 1 can be written as: (4. 2) The right hand side is expressed as a sum of perfect squares. Furthermore, it does not contain any Yukawa terms, as in type I models, Φ 1 does not couple to any fermions. As was also discussed in [11], Yukawa couplings are often times indispensable for enabling fixed points, as they can be the only terms with a negative sign in the beta functions. The consequence of this is that the β 1l λ 1 above will never vanish, and type I models are excluded as viable candidates, when looking for 2HDMs with fixed points. As a corollary, models like the one discussed in [56] extending the SM by a singlet can also not support fixed points in their new quartic coupling.
The full two-loop beta functions for the type II Z 2 -symmetric 2HDM are given in Appendix A. In this model, tanβ is a free parameter that fixes the exact starting conditions. For fixed points to exist, Yukawa contributions to both Higgs field have to be big enough. This translates into a lower bound on tanβ, as the bottom-type Yukawa couplings in a type II model are (sin β) −1 -enhanced compared to the SM (cf. Eq. (4.2)). On the other hand, tanβ cannot be too large either, or the running Yukawa couplings will become divergent below the Planck scale. Accordingly, there is a tanβ-interval in which fixed points can exist. In this two-loop framework, fixed points can be found for:  value for stability in the lower branch is marked by a red vertical line. An estimate for the theoretical uncertainty is given by the difference between one-loop and two-loop results.
The CP-even neutral scalar mass eigenstates are split apart wide due to the large parameter tanβ: Without significant mixing, one of the eigenstates is of the order of v 1 , the other of the order v 2 . The heavier CP-even eigenstate takes on mass values of around 125-130 GeV for the (mostly) vacuum-unstable branch, and 135-140 GeV for the stable one. The lighter CP-even eigenstate lies in the O(1) GeV region, which is ruled out experimentally. This will be addressed in Section 5. The heavier CP-even eigenstate in the vacuum-unstable case has roughly the correct mass to be considered as a candidate for a SM-like Higgs. However, while it is theoretically possible that the observed Higgs boson at 125 GeV is the heavier of the two CP-even eigenstates, this configuration is heavily disfavoured by experimental observations, due to strong bounds from the H → hh decay [57]. It is therefore usually assumed in 2HDMs that the 125 GeV Higgs is the lighter of the two CP-even neutral eigenstates. The charged Higgs boson mass is below 200 GeV, which falls into the regions excluded bȳ B → X s γ measurements, as mentioned in Section 2.2. The pseudoscalar mass is not shown, because it vanishes completely: As it turns out, all fixed point solutions contain λ 5 ≡ 0. In this case (with M 12 forbidden by the Z 2 -symmetry), this means that the model displays an accidental U(1)-symmetry which forces the pseudoscalar into the role of a pseudo-Goldstone boson, and hence to become massless. The mixing angle α is close to zero in all cases.
Together with the large tanβ values, this ensures that the type II alignment limit condition of |β − α| ∼ π 2 (cf. Section 2.2) is always met, as is shown in Fig. 4.
To summarise, the mass spectrum produced by the vacuum-stable solutions to the fixed point equations in the Z 2 -symmetric 2HDM exhibits an SM-Higgs candidate in the vacuum-unstable branch, but is excluded by experimental observations because of the remaining boson spectrum. We fix this problem in the next section. Performing the same analysis in a type-Y 2HDM leads to generally analogous results at slightly elevated tanβvalues. To allow for larger, phenomenologically viable masses for m h , m H + , and m A , it is necessary to go beyond the Z 2 -symmetric model. The least invasive way to generate heavier masses is to include the so-called softly-broken Z 2 -symmetric 2HDMs. The assumption of a Z 2symmetry under the transformation φ 2 → −φ 2 is not completely dropped, but a mass term M 2 12 (Φ † 1 Φ 2 + Φ † 2 Φ 1 ) mixing between the two Higgs field is allowed. Like the other mass parameters, M 12 does not appear in any quartic, gauge, or Yukawa beta function. It can instead be treated as a free parameter. For this reason, M 12 does not influence the fixed point search itself. Phenomenologically, on the other hand, the mixing parameter can have a big impact, especially in the case of λ 5 = 0 observed in our models. The additional global U(1) symmetry is now broken by non-vanishing M 12 -terms, which allows the pseudoscalar Higgs boson to acquire mass. Additionally, three of the other four bosons grow approximately linear with M 12 , which allows them to evade experimental constraints. The influence of M 12 on the different boson masses is illustrated in Fig. 5. As can be seen in the plot, the CP-even neutral scalar eigenvalues (blue/violet) depend on M 12 in different ways (cf. Eq.(2.4)). The SM-like eigenstate (originally m H ) only shows a minor dependence, and hardly changes even for very large values of M 12 . For the originally smaller eigenstate however, M 12 can easily become the dominating contributor. While the original mass of this state was mainly generated by a small VEV v 2 , it soon starts to grow almost linearly with M 12 , surpassing the mass of the former heavier eigenstate in a level crossing at values of roughly M 12 ≈ 20 GeV.
Both the charged Higgs (yellow) and the pseudoscalar Higgs (green) also grow together with M 12 , and adopt an asymptotically linear behaviour as M 12 becomes large. For the pseudoscalar, the linear dependence is actually exact as long as λ 5 = 0, as it is the case  Figure 7 shows the corresponding plots for m H + (left) and m A (right). The SM-like Higgs in Fig. 6 is the only eigenstate for which there remains a significant difference in stable (top) and unstable (bottom) branch.
A finite M 12 has a number of implications on the validity of the theory. Most importantly, it has the anticipated effect of allowing the model to produce phenomenologically viable mass spectra by opening up a way to drive m h , m A and m H + to higher values. The experimental bounds on the physical Higgs bosons can be translated to a lower bound on M 12 . From the bound of m H + > 580 GeV [46] in type-Y models it follows that: with the exact value depending on tanβ. Re-translated, this condition implies in terms of other boson masses: Because of the mixing angle α being close to zero, the SM-like Higgs mass stays almost unchanged. This means that in terms of vacuum stability, the situation also remains consistent with the Z 2 -symmetric case: While there are vacuum-stable solutions to the fixed point equations, only the vacuum-unstable ones include masses around 125 GeV. The SMlike Higgs is thus in a unique position among the 2HDM bosons, in that its mass cannot be heavily adjusted in this model. Whereas the SM-like CP-even scalar eigenstate is independent of M 12 , the opposite is true for all other bosons: Even at its minimum, M 12 ∼ 70 GeV is already large enough to make it the controlling factor in generating the masses of the three bosons H, H + and A.
For larger values of M 12 , the degeneracy in masses becomes even stronger. Therefore, most of the parameter space of viable asymptotically safe 2HDMs falls into the decoupling limit [48], with one SM-like and three heavy bosons with m H ≈ m H + ≈ m A ∝ M 12 .
The high tanβ-values necessary to find fixed points mean that in type II models specifically, bounds from B s → µµ decays are much more restrictive [47,58]. They demand heavy boson masses upwards of

Stability Analysis
In order to understand the characteristics of a given fixed point, we study the linearized RG flow around the fixed point, described by the stability matrix given by: In this case, g j includes gauge, Yukawa and quartic couplings. The number of negative eigenvalues of M ij corresponds to the dimension of the critical surface from which trajectories run into the fixed point. However, it has less significance here: While it is important to confirm that the fixed points are indeed UV-attractive (which they are), both the exact fixed point scale and the low scale initial conditions give additional constraints that intersect non-trivially with the critical surface. The solution to Eq. (3.1) is always a single trajectory in parameter space. On the other hand, by construction our method of finding fixed points ensures that the solutions found connect to the critical surface. It is therefore necessary to examine which of the initial conditions used is subject to uncertainties, and how these translate to changes in fixed point solutions and thus in Higgs boson mass spectra.

Uncertainty Estimates
There are several factors that influence the fixed point analysis. Below, we look at changes in scale where the fixed point condition is applied, followed by a discussion about low scale top and bottom quark mass uncertainties. Unless stated differently, values for the SM-like CP-even scalar eigenstate (here h) will be evaluated using M 12,Min = 380 GeV. The pseudoscalar A and M 12 -dependent CP-even scalar eigenstate H are entirely or almost entirely generated by the free parameter M 12 and therefore have negligible uncertainties. In general, the models are studied with the condition that the quartic coupling beta functions become zero at m P l = 1.2 · 10 19 GeV. The mass spectrum shows a minor dependence on where exactly the fixed points are assumed to occur. The masses m h are shown in Figure  8 for different fixed point scales. We see that lower fixed point scales correspond to a larger difference between vacuum-stable (upper) and vacuum-unstable (lower) branches, and notably bring down the lower branch mass values. Also, the tanβ-range in which fixed points can be found changes with the fixed point scale: When the fixed points scale is chosen at higher values than m P l , the divergence of large Yukawa couplings, especially y b becomes an even more pronounced problem. The mass spectra depend on the initial values chosen for the gauge and Yukawa couplings. The dependency on the top quark mass turns out to be especially strong. In much smaller in the first plot of Figure 10 compared to the second. Figure 11 shows a typical set of running quartic couplings each for the vacuum-stable and unstable branch and how different top quark mass initial values influence λ i . For most couplings, the m t -induced relative uncertainty becomes smaller as the scale decreases, even if the coupling itself becomes larger. This is most notable for λ 1 (blue) and λ 2 (violet). The quartic coupling λ 5 is not shown in these graphs, as λ 5 = 0 is an exact solution of the fixed point equations regardless of initial conditions. A last important detail to take note of is the range of λ 2 in the right graph: Depending on the exact initial conditions, λ 2 may become negative (and thereby break the vacuum stability conditions of Eq. (2.9)) as early   Figure 11: Running quartic couplings with 1σ uncertainty intervals from low scale top quark mass initial value for the vacuum-stable (left) and vacuum-unstable (right) fixed point branch. The colours correspond to: λ 1 (blue), λ 2 (violet, dashed), λ 3 (yellow, dotted) and λ 4 (green, dot-dashed). as 10 8 GeV, or not at all. To further analyse the influence of the exact starting parameters on the Higgs mass, all SM-like Higgs mass values generated by a fixed point solution can be shown against the corresponding top quark Yukawa initial value. Figure 12 shows the Higgs mass m h for the 1σ regions of m b and m t for two different values of tanβ. The vertical spread in points is generated by shifting the bottom quark initial value. Once again, all vacuum-stable points are coloured blue, the vacuum-instable ones violet. The right hand graph in particular illustrates well the two branches of fixed point solutions: The branch corresponding to higher Higgs boson masses is fully vacuum-stable, whereas the lower mass branch can only be vacuum-stable for top quark mass values on the lower end of its 1σ band. Comparing both graphs also shows how tanβ needs to be of certain size to facilitate the existence of fixed points. For the central mass values of m b and m t , there are no fixed points at tanβ = 60. However, as shown in the left graph, fixed points can be found there if either or both initial values are slightly larger.

Summary
Proposing simultaneously vanishing quartic coupling beta functions at the Planck scale severely constrains the 2HDM parameter space, but is possible in a way similar to the SM. As such, the 2HDM likewise supports the idea of being extended to high scales through means of asymptotic safety.
The parameter tanβ needs to be large, as both the up-type and the down-type Yukawa couplings have to be large in order to keep the positive contributions in quartic couplings beta functions in check. For the same reason, only type II and type Y models are viable, while type I and type X models are not. In the type II/type Y models studied, there always exists a tanβ-interval in which fixed points can be found, see Eq. (4.3). The most minimal model that also agrees with all experimental bounds is the softly-broken Z 2 -symmetric 2HDM, cf. Figs. 5 -7.
The allowed parameter region defined by the fixed point assumption meets the characteristics of the decoupled alignment limit, with three heavy Higgs bosons m H ≈ m A ≈ m H + ∝ M 12 , and |β − α| ≈ π 2 . To be consistent with experimental constraints, a lower bound is given on M 12 > 70 GeV (380 Gev) for type Y (type II) models, corresponding to the charged Higgs limits for type Y. This implies lower limits on m A and m H , see Eqs. 5.3 and 5.4.
Similar to the SM, both the existence of fixed points and the vacuum stability depend strongly on the low scale initial values, most notably the exact top quark mass. As illustrated in Fig. 12, the central values of Higgs and top quark mass indicate an instable vacuum, but are rather close to the criticality border. Fixed point solutions with stability can be possible by having the MS top quark mass values lower than 160 GeV [55] or the fixed point scale set below m P l = 1.2 · 10 19 GeV. A more precise determination of m t and m h by future experiments will allow a more definite statement.
As it stands, the 2HDM does not solve the stability problem of the SM. Instead, the situation is mirrored, or even worse. Note added: During the final phase of this project, an analysis of 2HDM fixed points using slightly different methodology appeared [59]. While our approach differs in details, we agree with the general conclusion that fixed points in type II 2HDMs are possible.