High-scale validity of a two-Higgs doublet scenario: a study including LHC data

We consider the conditions for the validity of a two-Higgs doublet model at high energy scales, together with all other low- and high-energy constraints. The constraints on the parameter space at low energy, including the measured value of the Higgs mass and the signal strengths in channels are juxtaposed with the conditions of vacuum stability, perturbativity and unitarity at various scales. We find that a scenario with an exact $\mathbb{Z}_2$ symmetry in the potential cannot be valid beyond about 10 TeV without the intervention of additional physics. On the other hand, when the $\mathbb{Z}_2$ symmetry is broken, the theory can be valid even up to the Planck scale without any new physics coming in. The interesting feature we point out is that such high-scale validity is irrespective of the uncertainty in the top quark mass as well as $\alpha_{s}(M_Z)$, in contrast with the standard model with a single Higgs doublet. It is also shown that the presence of a CP-violating phase is allowed when the $\mathbb{Z}_2$ symmetry is relaxed. The allowed regions in the parameter space are presented for each case. The results are illustrated in the context of a Type-II scenario.


Introduction
The Higgs sector of the standard electroweak model (SM) continues to appear enigmatic from several angles. The existence of such a sector, comprising at least one scalar doublet, and driving the spontaneous symmetry breakdown SU (2) L × U (1) Y −→ U (1) EM is almost impossible to deny now. It is also widely agreed that the Large Hadron Collider (LHC) has found [1,2] a neutral boson with mass around 125 GeV, which is almost certainly of spin zero [3] and dominantly a CP-even field [4][5][6][7]. However, despite the properties of the boson being consistent with that of the SM Higgs, rather persistent enquiries are on, to find out whether the electroweak symmetry breaking sector also contains some signature of physics beyond the standard model. The LHC data till date leaves room for such new physics.
Two sets of standpoints are noticed in such enquiries. First of all, with spin-1/2 fermions showing family replication, it is not obvious why the part of the matter sector containing spin-zero particles should also not have similar repetition. With this in view, multi-doublet scenarios are under regular scrutiny, the most widely investigated being models with two Higgs doublets. An extended electroweak symmetry breaking sector entails a rich phenomenology, including additional sources of CP violation [8]. Of course, scalars belonging to higher representations of SU (2) have also attracted attention, especially triplets which can play a role in the so-called Type-II mechanism of neutrino mass generation [9]. Secondly, even with just one doublet (leading to a single physical scalar), the Higgs mass is not stable under quadratically divergent radiative corrections, and it is somewhat artificial (or 'finetuned') to have a 125-GeV Higgs if the cut-off for the SM is much higher than a TeV or so.
Furthermore, it is also not clear that the SM scalar potential retains a finite and stable minimum at high scales. But for the yet uncertain measurement of the top quark mass, which is crucial in governing the evolution of the Higgs self-coupling via Yukawa interactions, we may be doomed to live in an unstable or metastable vacuum new physics intervenes at a scale no greater than 10 8−10 GeV [10][11][12][13]. Therefore, the ultraviolet incompleteness of the current scheme of electroweak symmetry breaking looms up as a distinct possibility, even if one disregards the somewhat philosophical issue of naturalness.
In this paper, we follow these two standpoints in tandem. We take up a two-Higgs doublet scenario as the minimal extension of the standard electroweak theory, assessing its viability as well sufficiency modulo all available constraints. The intuitive motivations for the study is that the proportionality constant between the top quark mass and its coupling to the 125 GeV scalar is different from its SM value when more doublets are around. Consequently, the dependence of the vacuum stability limit on the top quark mass is expected to be different.
However, one can make precise and quantitative statements on the matter only when one takes cognizance of the exact scenario, and includes the complete set of renormalisation group equations appropriate for it. This is precisely what we aim to do here, using a two-Higgs doublet scenario at various levels of generality.
The desired suppression of flavour-changing Yukawa interactions is best implemented by imposing a discrete symmetry on such models, thus preventing both the doublets from coupling with T 3 = +1/2 and −1/2 fermions simultaneously. It is possble to go beyond such imposition and examine two Higgs doublets in a 'basis-independent' formulation [14][15][16].
However, we feel that our central issue, namely, the evolution of the Higgs self-interaction(s), is amenable to a more transparent study if one adheres to a specific Yukawa scheme. With this in view, we adopt the so-called Type-II scenario for our study [17], though our broad conclusions do not depend on this choice.
We begin by examining the situation when the discrete symmetry is exact, and derive the constraints on the low-energy values of the parameters of this scenario. The lighter neutral scalar mass being around 125 GeV is of course the prime requirement here, and constraints from rare processes such as b → sγ are also included. In addition, the constraints from perturbativity of all scalar quartic couplings are considered, together with those from vacuum stability. The parameter space thus validated is further examined in the light of the perturbativity and vacuum stability conditions at high scales. Thus we identify the parameter regions that keep a two-Higgs doublet scenario valid upto different levels of high scales-an exercise that reveals rather severe limits. The same investigation is carried out for cases where the discrete symmetry is broken by soft (dimension-2) and hard (dimension-4) terms in turn, with the Yukawa coupling assignment remaining (for simplicity) the same as in the case with unbroken symmetry. The effect of a CP-violating phase is also demonstrated.
Finally, the regions found to be allowed from all the above considerations, at both low-and high-scales, are pitted against the existing data from the LHC in different channels. Thus we identify parameter regions that are consistent with the measured signal strengths in different channels. This entire study is aimed at indicating how far a two-Higgs doublet model can remain valid, not only at the LHC energy but also upto various high scales without further intervention of new physics.
Although a number of recent studies have addressed some similar questions, the present study has gone beyond them on the following points: • Our study reveals that the precise value of the top quark mass is rather unimportant in deciding the high-scale validity of the theory. Regions in the parameter space are identified, for which the theory has no cut-off till the Planck scale, even though the top quark mass can be at the upper edge of the allowed band.
• We find that it is rather difficult to retain the validity of a two-Higgs doublet scenario well above a TeV with the discrete symmnetry intact. Also, large values of tan β, the ratio of the vacuum expectation values (vev) of the two doublets, are mostly disfavoured in this case.
• With the discrete symmetry broken, there is a correlation between allowed tan β and the extent of symmetry breaking, when it comes to validity upto the Planck scale.
• We examine the constraints on the model including a CP-violating phase [18][19][20][21]. In fact, since the existence of a phase is a natural consequence of relaxing the discrete symmetry, the high-scale validity of a two Higgs doublet model may be argued to be contingent on the possibility of CP-violation in the scalar potential.
• We have performed a detailed examination of the validity of the scenario at both low-and high scales, including dimension-4 discrete symmetry breaking terms in our analysis. The LHC constraints are also imposed in this situation.
We remind the reader of the broad features of a two-Higgs doublet scenario in section 2.
In section 3, we list and explain all the constraints that the scenario is subjected to, at both the low and high scales. Sections 4, 5 and 6 contain, in turn, the results of our analysis, with the discrete symmetry intact, softly broken and broken by hard terms, respectively. We summarise and conclude in section 7.
2 The two-Higgs-doublet scenario and the scalar po-

tential: basic features
In the present work, we consider the most general renormalizable scalar potential for two doublets Φ 1 and Φ 2 , each having hypercharge (+1).
The parameters m 12 , λ 5 , λ 6 and λ 7 could be complex in general, although the phase in one of them can be removed by redefinition of of the relative phase between Φ 1 and Φ 2 . Thus this scenario in general has the possibility of CP-violation in the scalar sector.
In a general two-Higgs-doublet model (2HDM), a particular fermion can couple to both Φ 1 and Φ 2 . However this would lead to the flavor changing neutral currents (FCNC) at the tree level [22][23][24][25]. One way to avoid such FCNC is to impose a Z 2 symmetry, such as one that demands invariance under Φ 1 → −Φ 1 and Φ 2 → Φ 2 . This type of symmetry puts restrictions on the scalar potential. The Z 2 symmetry is exact as long as m 12 , λ 6 and λ 7 vanish, when the scalar sector also becomes CP-conserving. The symmetry is said to be broken softly if it is violated in the quadratic terms only, i.e., in the limit where λ 6 and λ 7 vanish but m 12 does not. Finally, a hard breaking of the Z 2 symmetry is realized when it is broken in the quartic terms as well. Thus in this case, m 12 , λ 6 and λ 7 all are non-vanishing in general.
As mentioned in the introduction, we focus on a specific scheme of coupling fermions to the doublets. This scheme is referred to in the literature as the Type-II 2HDM, where the down type quarks and the charged leptons couple to Φ 1 and the up type quarks, to Φ 2 [26].
This can be ensured through the discrete symmetry Φ 1 → −Φ 1 and ψ i R → −ψ i R , where ψ is charged leptons or down type quarks and i represents the generation index. It has been already mentioned that this choice is purely for illustrative purpose; our general results are largely independent of it. Although we start by analysing the high-scale validity of the model with m 12 = λ 6 = λ 7 = 0, we subsequently include the effects of both soft and hard breaking of Z 2 in turn, which bring back these parameters. The two simplifications that we still make are as follows: (a) the phases of λ 6 and λ 7 are neglected though that of m 12 is considered, and (b) the Yukawa coupling assignments of Φ 1 and Φ 2 are left unchanged.
Minimization of the scalar potential in Eq. 2.1 yields where the vacuum expectation values (vev) are often expressed in terms of the m Z and the ratio tan We parametrize the doublets in the following fashion, In the absence of CP-violation, the squared masses of these physical scalars and the mixing angle α can be expressed as [27], where we have defined, Furthermore, the interactions of the various charged and neutral scalars to the up-and downtype fermions are functions of α and β. Their detailed forms in different 2HDM scenarios, including the Type-II model adopted here for illustration, can be found in the literature [28].

Theoretical and experimental constraints
Next, we subject the Type-II 2HDM using various theoretical and experimental constraints (though the most binding ones are often irrespective of the specific type of 2HDM). It should be remembered at the outset that the most general Z 2 violating 2HDM has seven quartic couplings, namely, λ i (i = 1, . . . , 7), in addition to tan β and m 12 , totalling to nine free parameters. Though such a nine-dimensional parameter is prima facie large enough to accommodate any phenomenology, the set of constraints under consideration below can ultimately become quite restrictive.
We discuss the theoretical constraints in subsection 3.1, and take up the experimental/phenomenological ones in the subsequent subsections. It should be noted that the parameter space is being constrained in two distinct ways. Subsections 3.2 -3.4 list essentially low-energy constraints which apply at the energy scale of the subprocesses leading to Higgs production. The various masses and couplings get restricted by the requirement of satisfying them. However, while such a strategy is valid for the discussion of subsection 3.1 as well, we additionally require the conditions laid down there to hold at various high scales, too.
This not only restricts the low-energy parameters more severely, but also answers the main question asked in this paper, namely, to what extent the 2HDM can be deemed 'ultraviolet complete'.

Perturbativity, unitarity and vacuum stability
For the 2HDM to behave as a perturbative quantum field theory at any given scale, one must impose the conditions |λ i | ≤ 4π (i = 1, . . . , 7) and |y i | ≤ √ 4π (i = t, b, τ ) at that scale 1 . On applying such conditions, one implies upper bounds on the values of the couplings at low as well as high scales.
Next, we impose the more stringent condition of unitarity on the tree-level scattering amplitudes involving the scalar degrees of freedom. In a model with an extended scalar sector, the scattering amplitudes are taken between various two-particle states constituted out of the fields w ± i , h i and z i corresponding to the parametrization of Eq. 2.4. Maintaining this, there will be neutral two-particle states (e.g., as well as singly charged two-particle states (e.g., w ± i h j , w ± i z j ). The various two particle initial and final states give rise to a 2 → 2 scattering matrix whose elements are the lowest order partial wave expansion coefficients in the corresponding amplitiudes. The method used by Lee, Quigg and Thacker (LQT) [29] prompts us to consider the eigenvalues of this two-particle scattering matrix [30][31][32]. These eigenvalues, labelled as a i , should satisfy the condition Re[a i ] < 1/2.
Again, these conditions apply to high scales as well, if we expect perturbativity to hold.
When the quartic part of the scalar potential preserves CP [33,34] and Z 2 symmetries, the LQT eigenvalues are discussed in [35][36][37]. For λ 6 , λ 7 = 0, we follow the procedure described 1 The conditions are slightly different for the two types of couplings. The reason becomes clear if we note that the perturbative expansion parameter for 2 → 2 processes driven by the quartic couplings is λ i . The corresponding parameter for Yukawa-driven scattering processes is |y i | 2 in [35]. The general formulas including λ 6 , λ 7 , is given in Appendix B.
The condition to be taken up next is that of vacuum stability. For the scalar potential of a theory to be stable, it must be bounded from below in all directions. This condition is threatened if the quartic part of the scalar potential, which is responsible for its behaviour at large field values, turns negative. Avoiding such a possibility upto any given scale ensures vacuum stability upto that scale. The issue of vacuum stability in context of a 2HDM has been discussed in detail in [38][39][40][41] The 2HDM potential has eight real scalar fields. By studying the behaviour of the quartic part of its scalar potential along different field directions, one arrives at the following conditions [28,42], vsc1 : The couplings in the general Z 2 violating Type-II 2HDM evolve from a low scale to a high scale according to a set of renormalization group (RG) equations listed in the Appendix A. If one proposes the UV cut-off scale of the model to be some Λ U V , it might so happen that the couplings grow with the energy scale and hit the Landau pole before Λ U V . A second, still unacceptable, possibility is that of the LQT eigenvalues crossing their perturbativity/unitarity limits. The RG evolution of the 2HDM couplings has been recently studied in [43,44]. Finally, the stability conditions can get violated below Λ U V , making the scalar potential unbounded from below. All these problems are avoided if one postulates that all of the conditions laid down above are valid upto Λ U V , which marks the maximum energy upto which the 2HDM can be valid without the intervention of any additional physics.

Higgs mass constraints
The spectrum of a generic 2HDM consists of a charged scalar, a CP-odd neutral scalar and two CP-even neutral scalars. Since the LHC has observed a CP-even neutral boson around 125 GeV, we allow only those regions in the parameter space for which h, the lighter neutral scalar, lies in the mass range 123-127 GeV (keeping an error bar). In addition, the charged scalar is required to have a mass greater than 315 GeV due to low energy constraints, coming mainly from b → sγ [45,46].

Oblique parameter constraints
The presence of an additional SU (2) doublet having a hypercharge Y = 1 modifies the electroweak oblique parameters [47]. It is to be noted that since the couplings of the fermions to gauge bosons remain unaltered even after the introduction of the second doublet, all the additional contributions come from the scalar sector of the 2HDM. The oblique parameters can be decomposed as, in [16,[48][49][50]. The corresponding bounds we have used are |∆S| < 0.10 and |∆T | < 0.12 [51].
While ∆S hardly imposes any constraint on this scenario, the splitting amongst the scalar masses affects the T parameter. Typically for m 12 = 0, T prevents large mutual splitting among states other than the lightest neutral scalar. For m 12 = 0, the scalars other than the light neutral one have masses ∼ m 12 . As m 12 is increased, the masses approach the decoupling limit, and in that case, the T parameter constraint ceases to play a significant role. The oblique parameters constraints turn out to be redundant in our analysis in such a case. The consistency with these parameters has nevertheless been explicitly ensured at each allowed point of the parameter space.

Collider constraints
Apart from the theoretical constraints discussed above, we also strive to find the region of parameter space of a 2HDM allowed by the recent Higgs data. The ATLAS [52][53][54] and CMS [55] collaborations have measured the production cross section for a ∼125 GeV Higgs multiplied by its branching ratios to various possible channels. In our case, since the underlying theory is a 2HDM, all the cross sections and decay widths get modified compared to the corresponding SM values. For example, the production cross section of the light neutral Higgs through gluon fusion will get rescaled in the case of a 2HDM due to the fact that the fermionic couplings of the 125 GeV Higgs are now changed with respect to the SM values by appropriate multiplicative factors. Similarly, the loop induced decay h → γγ will now draw an additional contribution from the charged scalars. Some recent investigations in this area, can be found in [56][57][58][59][60][61][62][63][64][65][66]. Also, model-independent analysis of the data, which impose constraints on non-SM couplings of the scalar discovered, have to allow such contributions [67][68][69][70][71][72]. In order to check the consistency of a 2HDM with the measured rates in various channels, We theoretically compute the signal strength µ i for the i th channel using the relation: Here R prod , R i decay and R width denote respectively the ratios of the theoretically calculated production cross section, the decay rate to the i th channel and the total decay width for a ∼125 GeV Higgs to their corresponding SM counterparts. Thus, our analysis strategy is to generate a region in parameter space allowed by the constraints coming from vacuum stability, perturbative unitarity and electroweak precision data. We subsequently compute µ i for each point in that allowed region and compare them to the experimentally measured signal strengths,μ i , supplied by the LHC. This exercise carves out a sub-region, which is allowed by the recent Higgs data, from the previously obtained parameter space.
For our numerical analysis, we have taken gluon fusion to be the dominant production mode for the SM-like Higgs. As for the subsequent decays of h, we have considered all the decay channels mentioned in Table 1. Unless otherwise stated we use 1σ allowed ranges of µ i .

Results with exact discrete symmetry
In this section, we set out to obtain the allowed parameter space of a Type-II 2HDM having an exact Z 2 symmetry consistent with the various theoretical and collider constraints described above. In this particular case, one naturally has m 12 = 0, λ 6 , λ 7 = 0. Thus, we scan over the quartic couplings λ i (i = 1, . . . , 5) within their perturbative limits    We thus see that the uncertainity in the top quark mass measurement has no bearing on the allowed region of the parameter space. This result is a precursor to a much stronger one This leads to the observation that although λ 1 remains within the perturbative limit upto 10 TeV, the LQT eigenvalue |a + | crosses its unitarity limit of 8π beyond that scale. Thus, this example illustrates the interplay among perturbativity and unitarity in determining the UV fate of this scenario and it appears that unitarity often proves stronger as a constraint than perturbativity. Further, it follows from Eq. 2.5 that relatively larger values of tan β are difficult to accommodate in this case, since one has to comply with the measured value of m h . Also, one can generally conclude that in order to push the UV limit of 2HDM to higher scales, one must look beyond the exact Z 2 symmetric case. Having tan β higher than in the previous section is possible in this case, so long as m 12 is correspondingly large, thus generating an acceptable m h .

Results with softly broken discrete symmetry
Some recent studies have put constraints on the 2HDM parameters using the current Higgs data. We find that for the benchmark point tan β = 2 and m 12 = 200 GeV, a substantial region of the parameter space allowed by the theoretical constraints also gets allowed by the Higgs data even at the 1σ confidence limit. For other benchmark points, the Higgs data allow the entire region allowed by the theoretical constraints. Hence in those cases, the regions allowed by the Higgs data are not shown separately.
We note here that though m 12   recent Higgs data. An inspection of the results so obtained shows that as Λ U V is pushed towards higher scales, the allowed parameter shrinks, and finally at the Planck scale, it is most constrained. Though the bound obtained on one quartic coupling is correlated to the bounds on the other quartic couplings, one can still make a few remarks on the obtained results. As Fig. 5 shows, for the benchmark point tan β = 2, m 12 = 200GeV, the theory is not UV complete beyond 10 16 GeV. However, higher tan β allow for an UV completion upto the Planck scale.
It is worth commenting further on the bounds on the λ i that the plots in Fig. 5 show. to turn negative at some scale destroying the vacuum stability of the theory in the process.
To illustrate the point better, we choose an initial condition for the quartic couplings at tan β = 2 and m 12 = 1000 GeV, out of the allowed set of couplings which obey all the imposed constraints upto the Λ U V = 10 19 GeV, and display the RG running of the λ i , the stability conditions and the LQT eigenvalues in Fig. 7.  We took λ 2 (M t ) = 0.39 in Fig. 8. It is seen that the RG running is not significantly altered even by a 3σ variation of α s (M z ). Hence, for any value of α s (M z ) within this band, the parameter spaces will not change in a major fashion, and whatever constraints apply to λ 2 (M t ) will continue to be valid rather insensitively to α s (M z ).
The implications of having a complex m 12 in the scalar potential [33,34] is also investigated here. We rewrite the quadratic part of the scalar potential as, The quartic couplings are kept real as in the previous case. The presence of an arbitrary phase α in m 2 12 , leads to a charged scalar H + , three neutral scalars H 1 , H 2 and H 3 which are not eigenstates of CP, and of course the charged and neutral Goldstone bosons. The masses of the neutral scalars can not be obtained in closed form in this case, rather, the corresponding mass matrix has to be diagonalized numerically. In the process, we choose the lightest neutral scalar, say H 3 to be around 125 GeV and the charged scalar to be have a mass higher than 300 GeV. The quartic couplings satisfying these conditions are selected and are further constrained by the imposition of the theoretical constraints under RG.  Our conclusion therefore is that the regions in the parameter space of a 2HDM, consistent In the case where Λ U V = 10 3 GeV, we show the subregions in the parameter spaces which are also allowed by the recent Higgs data. The major constraint, however, comes from the signal strength corresponding to h → γγ. It is clearly seen in Fig. 11 that m 12 = 1000 GeV allows for a bigger region in the parameter space that is allowed by the Higgs data, compared to what m 12 = 200 GeV does. This is obviously expected, given the fact that a high value of m 12 takes the theory towards the decoupling limit and hence, the 125 GeV Higgs becomes more SM like in the process.
We demonstrate the UV completion of the hard Z 2 violating case by showing the RG evolution of the various quartic couplings and stability conditions upto Λ U V = 10 19 GeV. We choose the following initial condition for the quartic couplings at tan β = 2 and m 12 = 1000 GeV.   Table 3: λ 1 and λ 2 bounds at Λ U V = 10 11 and 10 19 GeV As shown in Fig. 13, λ 3 increases most sharply whereas λ 2 first plunges down due to the top quark effect and then starts increasing. Choosing same initial conditions for λ 6 and λ 7 causes their evolutions to become fairly similar. In this section, it should be noted that the allowed parameter spaces found are not expected to be exhaustive as we have not scanned over all λ i (M t ) independently, rather, have put λ 1 (M t ) = 0.02 and λ 6 (M t ) = λ 7 (M t ) while doing so. However, given the similar structure of the 1-loop beta functions of λ 6 and λ 7 , the bounds obtained on them would have not substantially changed even if an independent scanning would have been allowed.

Summary and Conclusions
We set out to investigate the high-scale behaviour of a 2HDM. The results are illustrated in the context of a Type-II scenario. We have subjected the parameter space of this model to the theoretical constraints based on perturbativity, unitarity and vacuum stability. The relatively less stringent constraints from oblique parameters, and also the LHC constraints on the signal strength in each decay channel of a Higgs around 125 GeV have also been taken into account.