Two-Higgs-doublet models and enhanced rates for a 125 GeV Higgs

We examine the level of enhancement that can be achieved in the ZZ and γγ channels for a two-Higgs-doublet model Higgs boson (either the light h or the heavy H) with mass near 125 GeV after imposing all constraints from LEP data, B physics, precision electroweak data, vacuum stability, unitarity and perturbativity. The latter constraints restrict substantially the possibilities for enhancing the gg → h → γγ or gg → H → γγ signal relative to that for the SM Higgs, hSM. Further, we find that a large enhancement of the gg → h → γγ or gg → H → γγ signal in Type II models is possible only if the gg → h → ZZ or gg → H → ZZ mode is even more enhanced, a situation disfavored by current data. In contrast, in the Type I model one can achieve enhanced rates in the γγ final state for the h while having the ZZ mode at or below the SM rate — the largest [gg → h → γγ]/[gg → hSMγγ] ratio found is of order ~ 1.3 when the two Higgs doublet vacuum expectation ratio is tan β = 4 or 20 and the charged Higgs boson has its minimal LEP-allowed value of mH± = 90 GeV.


Introduction
The original data from the ATLAS and CMS collaborations [1,2] provided an essentially 5σ signal for a Higgs-like resonance with mass of order 123-128 GeV. The updates from Moriond 2013 include those for the γγ channel from ATLAS [3] and CMS [4]. The earlier ATLAS and CMS gluon fusion induced rates were significantly enhanced relative to the Standard Model (SM) prediction. The Moriond ATLAS data still shows substantial enhancement for the γγ channel while the CMS MVA analysis finds a roughly SM-like rate in the γγ channel. Here, we consider the extent to which an enhanced γγ rate is possible in various 2HDM models once all relevant theoretical and experimental constraints are imposed It is known that enhancements with respect to the SM in the γγ channel are generically possible in two-Higgs-doublet models (2HDM) of Type-I and Type-II as explored in [6][7][8][9][10]. However, these papers do not make clear what level of enhancement is possible after all constraints from B physics and LEP data (B/LEP), precision electroweak data, unitarity and perturbativity are imposed. In this paper, we impose all such constraints and determine the maximum possible enhancement. We employ a full 1-loop amplitude for Higgs→ γγ without neglecting any contributions from possible states in the loop. We examine correlations with other channels. We also consider cases of degenerate scalar masses at ∼ 125 GeV [11,12].

2HDM models
The general Higgs sector potential employed is where, to avoid explicit CP violation in the Higgs sector, all λ i and m 2 12 are assumed to be real. We choose a basis in which where v = ( √ 2G F ) −1/2 ≈ 246 GeV. By convention 0 ≤ β ≤ π/2 is chosen. For real parameters, the phase ξ could still be non-zero if the vacuum breaks CP spontaneously. We avoid parameter choices for which this happens and take ξ = 0. Then, we define with v 1 = v cos β and v 2 = v sin β. The neutral Goldstone boson is G 0 = η 1 cos β + η 2 sin β while the physical pseudoscalar state is 3) The physical scalars are: h = −ρ 1 sin α + ρ 2 cos α, H = ρ 1 cos α + ρ 2 sin α . (2.4) Without loss of generality, one can assume that the mixing angle α varies between −π/2 and π/2. We choose our independent variables to be tan β and sin α, which are single valued in the allowed ranges. We adopt the code 2HDMC [13,14] for numerical calculations. All relevant contributions to loop induced processes are taken into account, in particular those with heavy quarks (t, b and c), W ± and H ± . A number of different input sets can be used in the 2HDMC context. We have chosen to use the "physical basis" in which the inputs are the physical Higgs masses (m H , m h , m A , m H ± ), the vacuum expectation value ratio (tan β), and the CP-even Higgs mixing angle, α, supplemented by m 2 12 . The additional parameters λ 6 and λ 7 are assumed to be zero as a result of a Z 2 symmetry being imposed on the dim 4 operators under which H 1 → H 1 and H 2 → −H 2 . m 2 12 = 0 is still allowed as a "soft" breaking of the Z 2 symmetry. With the above inputs, λ 1,2,3,4,5 as well as m 2 11 and m 2 22 are determined (the latter two via the minimization conditions for a minimum of the vacuum) [15].
In this paper we discuss the Type I and Type II 2HDM models, that are defined by the fermion coupling patterns as specified in table 1 -for more details see [16].

Setup of the analysis
The 2HDMC code implements precision electroweak constraints (denoted STU) and limits coming from requiring vacuum stability, unitarity and coupling-constant perturbativity (denoted jointly as SUP). We note that it is sufficient to consider the SUP constraints at tree level as usually done in the literature. Evolution to higher energies would make these constraints, outlined below, stronger and would not be appropriate when considering the 2HDM as an effective low energy theory. In more detail, the vacuum stability condition requires that the scalar potential be positive in all directions in the limit of growing field

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Type I Type II Higgs up quarks down quarks leptons up quarks down quarks leptons h cos α/ sin β cos α/ sin β cos α/ sin β cos α/ sin β −sin α/ cos β −sin α/ cos β H sin α/ sin β sin α/ sin β sin α/ sin β sin α/ sin β cos α/ cos β cos α/ cos β strength [17][18][19]. Tree-level necessary and sufficient conditions for unitarity are formulated in terms of eigenvalues of the S-matrix in the manner specified in [20] for the most general 2HDM -the criterion is that the multi-channel Higgs scattering matrix must have a largest eigenvalue below the unitarity limit. Coupling constant perturbativity is defined as in 2HDMC by the requirement that all self-couplings among the Higgs-boson mass eigenstates be smaller than 4π. For the scenarios we consider, this becomes an important constraint on λ 1 . The SUP constraints are particularly crucial in limiting the level of enhancement of the gg → h → γγ channel, which is our main focus. For all our scans, we have supplemented the 2HDMC code by including the B/LEP constraints. For the LEP data we adopt upper limits on σ(e + e − → Z h/H) and σ(e + e − → A h/H) from [21] and [22], respectively. 1 Regarding B physics, the constraints imposed are those from BR(B s → X s γ), R b , ∆M Bs , K , BR(B + → τ + ν τ ) and BR(B + → Dτ + ν τ ). The most important implications of these results are to place a lower bound on m H ± as a function of tan β as shown in figure 15 of [23] in the case of the Type II model and to place a lower bound on tan β as a function of m H ± as shown in figure 18 of [23] in the case of the Type I model.
While looking for an enhancement of the signal in the γγ channel we also computed the extra Higgs-sector contributions to the anomalous magnetic moment of the muon, a µ = (g µ − 2)/2. Since the experimentally measured value, a µ = (1165920.80 ± 0.63) × 10 −9 [24], differs by ∼ 3σ from its SM value it is important to check correlations between δa µ ≡ a µ − a SM µ and the signal in the γγ channel. Given the B/LEP, STU and SUP constraints, it turns out that one-loop contributions within the 2HDM are small and negligible, and the leading contribution is that known as the Barr-Zee diagram [25] which emerges at the two-loop level. For completeness we include also sub-leading contributions, see [13,14]. Since the overall ∼ 3σ discrepancy between the experimental and theoretical SM values could still be due to fluctuations (the world average is based mainly on the E821 result [26] with uncertainties dominated by statistics) or underestimates of the theoretical uncertainties, we do not use the a µ measurement as an experimental constraint on the models we discuss. However, in tables presented hereafter we do show (in the very last column in units of 10 −11 ) δa µ , the judgment as to whether δa µ is acceptable being left to the reader. In fact, for all parameter choices yielding an enhanced Higgs to two-photon rate the extra contributions to a µ are very small and the a µ discrepancy is not resolved. induced Higgs cross section times the Higgs branching ratio to a given final state, X, relative to the corresponding value for the SM Higgs boson as follows: given that kinematic factors cancel out of all these ratios and that these ratios are computed in a self-consistent manner (that is, treating radiative corrections for the SM Higgs boson in the same manner as for the 2HDM Higgs bosons). When considering cases where more than one h i has mass of ∼ 125 GeV [11], we sum the different R h i for the production/decay channel of interest. This is justified by the fact that we always choose masses (as indicated in table 2) that are separated by at least 100 MeV -in this case interference effects are negligible (given that the Higgs widths are substantially smaller than 100 MeV in the range of Higgs masses we consider).
We have performed five scans over the parameter space with the range of variation specified in  Figure 1. The top two plots show the maximum R h gg (γγ) values in the Type I (left) and Type II (right) models for m h = 125 GeV as a function of tan β after imposing various constraints -see figure legend. Corresponding R h gg (ZZ) and R h gg (τ τ ) are shown in the middle and lower panels. Disappearance of a point after imposing a given constraint set means that the point did not satisfy that set of constraints. In the case of boxes and circles, if a given point satisfies subsequent constraints then the resulting color is chosen according to the color ordering shown in the legend. This same pattern is adopted in the remaining plots.

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Let us begin by discussing the case in which the h has mass m h = 125 GeV while scanning over the masses of the other Higgs eigenstates (cases where two Higgs are approximaely degenerate are discussed below). The upper plots of figure 1 show the maximum value achieved for the ratio R h gg (γγ) as a function of tan β after scanning over all other input parameters (as specified earlier), in particular sin α. These maximum values are plotted both prior to imposing any constraints and after imposing various combinations of the constraints outlined earlier with point notation as specified in the figure legend. We observe that for most values of tan β the B/LEP and STU precision electroweak constraints, both individually and in combination, leave the maximum R h gg (γγ) unchanged relative to a full scan over all of parameter space. In contrast, the SUP constraints greatly reduce the maximum value of R h gg (γγ) that can be achieved and that value is left unchanged when B/LEP and STU constraints are imposed in addition. Remarkably, in the Type I model maximum R h gg (γγ) values much above 1.3 are not possible, with values close to 1 being more typical for most tan β values. In contrast, maximum R h gg (γγ) values in the range of 2−3 are possible for 2 ≤ tan β ≤ 7 and tan β = 20 in the Type II model. In figure 1 we also show the values of R h gg (ZZ) and R h gg (τ τ ) (middle and bottom plots, respectively) found for those parameter choices giving the maximum R h gg (γγ) values appearing in the upper plots. One can get a feeling for how the different constraints impact R h gg (γγ) by plotting this quantity as a function of sin α at fixed tan β for different constraint combinations and a selection of different other input parameters. As shown in figure 2, R h gg (γγ) typically has a maximum as sin α is varied but the height of this maximum depends very much on the constraints imposed as there is also variation with the other input parameters.
Tables 3 and 4 display the full set of input parameters corresponding to the maximal R h gg (γγ) values at each tan β for models of Type I and Type II, respectively. It is important to notice that in the Type II model, the value of R h gg (ZZ) corresponding to the parameters that maximize R h gg (γγ) is typically large, ∼ 3. In fact, as discussed shortly, R h gg (ZZ) > R h gg (γγ) whenever R h gg (γγ) is even modestly enhanced. The current experimental situation is confused. The Moriond 2013 ATLAS data [3] shows central values of R h gg (γγ) ∼ 1.6 and R h gg (ZZ) ∼ 1.5. In the Type II model case, the former would imply R h gg (ZZ) > 2, somewhat inconsistent with the observed central value. However, the data uncertainties are significant and so it is too early to conclude that the Type II model cannot describe the ATLAS data. The Moriond 2013 CMS data has central values of R h gg (γγ) < 1 and R h gg (ZZ) ∼ 1, a situation completely consistent with the Type II model predictions.
As an aside, we note that R h gg (γγ)/R h gg (ZZ) 1 when R h gg (γγ) > 1 is fairly typical of the MSSM model (which has a Type II Higgs sector), especially with full or partial GUT scale unification for the soft-SUSY-breaking parameters, see for example [27]. In such scenarios the primary modification to the γγ rate relative to the SM is due to the light stop loop contribution to the hγγ coupling (which enters with the same sign as the W loop and has a color factor enhancement) which enhances BR(h → γγ). Note that the stop loop contribution to the hgg production coupling is the same for both the ZZ and γγ final states. In the absence of GUT scale unification, there are many other potentially significant loops contributing to an increase in the hγγ coupling, the most important being the light chargino loop and the light stau loop, as studied for example in [28].       Corresponding results for the H are presented for the Type I and Type II models in tables 5 and 6, respectively. In the case of the Type I model, an enhanced gluon fusion rate in the γγ final state does not seem to be possible after imposing the SUP constraints, whereas maximal enhancements of order   For the decay mode s → ZZ * , the tree level amplitude is present and dominant so that the denominator simply reduces to (C s ZZ ) 2 . For the decay mode s → γγ, there is no tree level contribution -the sγγ coupling first arises at the one-loop level with the t-loop, W -loop and H ± -loop being the important contributions. As a result, the numerator can be written as where C s tt and C s W W are the stt and sW W couplings normalized to those of the h SM , while A SM W and A SM t are the W -loop and t-loop amplitudes, respectively, for the h SM . Finally, A H ± is the H ± -loop amplitude in the 2HDM; since it is very small in the Type II model, it can be neglected. Thus, where C s ZZ = C s W W in any doublets+singlets models. Note that when the t-loop contribution is negligible then r s → 1. It is easy to see that r s < 1 if the following inequality is JHEP05(2013)072 In the case s = h, R h gg (γγ) is maximized by suppressing the h total width, which corresponds to chosing α so as to minimize the hbb coupling, i.e. α ∼ 0, resulting in C h tt /C h W W ∼ 1/ sin 2 β > 1 (and < 5 for tan β > 0.5). Consequently r h < 1, as observed in table 4. The argument is similar in the case of the H: this time the Hbb coupling is chosen to be small (equivalent to α ∼ ±π/2) in order to minimize the H total width and therefore maximize R H gg (γγ), with the result that once again C H tt /C H W W ∼ 1/ sin 2 β > 1, yielding r H < 1. These analytic results explain why large R s gg (γγ) is correlated with even larger R s gg (ZZ) in Type II 2HDMs.

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In figure 5 we plot contours of r s and of R s gg (γγ) in the Type II model. It is seen from the left panel that if tan β is large then only small α's will maximize R s gg (γγ). And, in that region, r h is always less than 1. Note that the R s gg (γγ) > 1 region shrinks for large tan β, so the the values of α preferred for large R h gg (γγ) converge to 0 when β → π/2. For the case of s = H, the right panel shows that when tan β is large then only vertical bands of α corresponding to values close to ±π/2 are allowed if R s gg (γγ) > 1. From the plots, we see that R s gg (γγ) > 1 could be consistent with r s > 1 only if tan β 1, which explains the pattern observed in tables 4 and 6 and figure 3. Note, however, that small tan β is disfavored by B-physics as it enhances the H +t b coupling too much, see for example [23].
Once again, we emphasize that a substantial enhancement of the γγ rate is possible for the h in Type I models without enhancing the ZZ rate. In particular, from table 3 we see that the enhancement in the γγ channel is ∼ 1.3 (for both gg fusion and VBF) for tan β = 4 and 20 while other final states, in particular ZZ, have close to SM rates. The table also shows that this maximum is achieved for sin α ∼ 0. Thus, β ∼ π/2 and cos α ∼ 1 yielding SM-like coupling of the h to quarks (see table 1) and vector bosons. It turns out that in these cases the total enhancement, ∼ 30%, is provided by the charged Higgs boson loop contribution to the γγ-coupling. In these same cases, the mass of the heavier Higgs boson is m H = 225 GeV. As such a mass is within the reach of the LHC, it is important to make sure that the H cannot be detected (at least with the current data set). It is easy to see that indeed this is the case. Since g HZZ ∝ cos(β −α) and g Hbb,Htt ∝ sin α one finds that the H decouples from both vector bosons and fermions given that α ∼ 0 and β ∼ π/2. The A will also be difficult to detect since it has no tree-level W W, ZZ coupling and the Abb, Att couplings, being proportional to cot β, will be quite suppressed, especially at tan β = 20. From table 3, we observe that for tan β = 4 and 20 the corresponding charged Higgs is light, m H ± = 90 GeV, i.e. as small as allowed by LEP2 direct searches in e + e − → H + H − . Searches for a light H ± are underway at the LHC along the lines described in [29]. The most promising H ± production and decay process is pp → tt → H ± bW ∓b → τ νbbq q. According to figure 3 of [29], for the Type I model, the region of tan β 6−7 for m H ± ∼ 90 GeV could be efficiently explored at the 14 TeV LHC by ATLAS even at the integrated luminosity of 10 fb −1 -for more details see [29]. The existing LHC bounds on BR(t → H + b) obtained assuming BR(H ± → τ ± ν τ ) = 1 are only moderately restrictive: 5%−1% [30] (4%−2%) [31] for masses of the charged Higgs boson m H ± = 90(80) − 160 GeV in the case of ATLAS (CMS), respectively. These bounds are weakened in the Type I model where BR(H ± → τ ± ν τ ) 0.7. Since BR(t → H + b) ∼ 1/ tan 2 β, large tan β suppresses BR(t → H + b). Indeed, it is easy to verify that for m H ± = 90 GeV BR(t → H + b) is ∼ 3.8% and ∼ 0.15% for tan β = 4 and tan β = 20, respectively. So, a charged Higgs yielding enhanced h → γγ rates in gg fusion and VBF is still completely consistent with current data.     For the Type I model, we see from table 7 that R h gg (γγ) is significantly enhanced only for the same tan β = 4 and tan β = 20 values as in the case of having (only) m h = 125 GeV and that the pseudoscalar contribution R A gg (γγ) turns out to be tiny. However, the contribution to the bb final state from the A can be substantial. Given that the top loop dominates both the Agg and hgg coupling one finds (C A gg /C h gg ) 2 ∼ (3/2) 2 (cos β/ cos α) 2 , where we used C A tt /C h tt = cos β/ cos α from table 1 and the m h,A 2m t fermionic loop ratio of A/h = 3/2. As a result, the A can contribute even more to the bb final state rate than the h if tan β is small. This (unwanted) contribution to the bb final state from A production is apparent from the results for R h+A gg (bb) in table 7 for tan β = 2−4. In the end, only tan β = 20 yields both an enhanced γγ rate, R h+A gg max (γγ) = 1.31, and SM-like rates for the ZZ and bb final states, R h+A gg (ZZ) = R h+A gg (bb) = 1. For this case β π/2 and α = 0 implying that the h couples to fermions and gauge bosons like a SM Higgs boson and the enhancement of R h+A gg max (γγ) is due exclusively to the charged Higgs loop contribution to the γγ couplings.

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For the Type II model, see table 8, the pseudoscalar contribution R A gg (γγ) is also (as for the Type I model) negligible. Thus, the enhancement of R h+A gg (γγ) is essentially the same as that for R h gg (γγ) for the case when only m h = 125 GeV, reaching maximum values of order 2−3. However, as in the pure m h = 125 GeV case, a substantial enhancement of R h+A gg (γγ) is most often associated with R h+A gg (ZZ) > R h+A gg (γγ) (contrary to the LHC observations). But this is not always the case. Among the m h ∼ m A scenarios we find 56 points in our parameter space for which R h+A gg (ZZ) < 1.3 and R h+A gg (γγ) > 1. 3. Unfortunately for all those points the τ τ signal is predicted to be too strong, R h+A gg (τ τ ) > 3.82, a result that is now excluded by the CMS analysis in the gluon fusion dominated 1-jet trigger mode which finds R h+A gg (τ τ ) < 1.8 at 95% CL. This situation is illustrated in figure 8 . As seen from the upper panels in figure 8, for tan β = 1 there exist points (blue diamonds) such that 1 GeV and we allow m H ± and m A to vary freely. Following a similar search strategy, we find that some of the tan β values previously available when only m h = 125 GeV or m H = 125 GeV was required are ruled out by the full set of constraints and that there is no gain in maximal R h+H gg (γγ) values, and often some loss, relative to the cases where only the h or only the H was required to have mass of 125 GeV.
As discusssed earlier and in [33], the charged Higgs contribution to the γγ coupling loops is sometimes relevant. Therefore, in figure 9 we show separately the fermionic loop, W loop and H ± loop contributions normalized to the total amplitude for the most interesting cases of a Type I model with m h = 125 GeV and with m h = 125 GeV, m A = 125.1 GeV (left plots). One sees that the tan β values of 4 and 20 associated with R h gg (γγ) ∼ 1.3 are associated with large A H ± /A. Indeed, in these two cases, the relative charged Higgs contribution reaches nearly ∼ 0.2 and is as large as the fermionic contribution, but of the opposite sign. In fact, although the dominant loop is the W loop, the H ± loop may contribute as much as the dominant (top quark) fermionic loop.
This should be contrasted with other cases, such as the Type II m h = 125 GeV and m h = 125 GeV, m A = 125.1 GeV cases illustrated in the right-hand plots of figure 9. One finds that the charged Higgs contributions are small when SUP constraints are imposed. In fact, the enhancement of R h gg (γγ) observed in figure 1 prior to imposing SUP is caused just by the charged Higgs loop. When SUP constraints are imposed the charged Higgs amplitude is strongly reduced by the requirement that the quartic couplings not violate the perturbativity condition. Note that the SUP constraints can be violated even though   all the mass parameters have been varied within what, a priori, appears to be a reasonable range, namely from a few GeV up to 1000 GeV. This is due to the fact that, for our input, the SUP conditions imply a strong constraint on m 2 12 that comes mainly from the requirement of keeping λ 1 small enough.

Conclusions
We have analyzed the Type I and Type II two-Higgs-doublet extensions of the Standard Model with regard to consistency with a significant enhancement of the gluon-fusioninduced γγ signal at the LHC at ∼ 125 GeV, as seen in the ATLAS data set, but possibly not in the CMS results presented at Moriond 2013. All possible theoretical and experimental constraints have been imposed. We find that vacuum stability, unitarity and perturbativity play the key role in limiting the maximal possible enhancement which, in the most interesting scenarios, is generated by the charged Higgs loop contribution to the Higgs to two photon decay amplitude. Generically, we conclude that the Type II model allows a maximal enhancement of order of 2 − 3, whereas within the Type I model the maximal enhancement is limited to 1.3. Moriond 2013 ATLAS results suggest an enhancement for gg → h → γγ of order 1.6 (but with large errors). Only Type II models can give such a large value.