Large $h\to b s$ in generic two-Higgs-doublet models

We investigate the possible size of $h\to bs$ in two-Higgs-doublet models with generic Yukawa couplings. Even though the corresponding rates are in general expected to be small due to the indirect constraints from $B_s\to\mu^{+}\mu^{-}$ and $B_s$--$\overline{B}_s$ mixing, we find regions in parameter space where $h\to bs$ can have a sizable branching ratio well above 10\%. This requires a tuning of the neutral scalar masses and their couplings to muons, but then all additional constraints such as $B\to X_s \gamma$, $(g-2)_\mu$, and $h\to\mu^{+}\mu^{-}$ are satisfied. In this case, $h\to bs$ can be a relevant background in $h\to b\bar{b}$ searches and vice versa due to the imperfect $b$-tagging purity. Furthermore, if $h\to bs$ is sizeable, one expects two more scalar resonances in the proximity of $m_h$. We briefly comment on other flavour violating Higgs decays and on the 95 GeV $\gamma\gamma$ resonance within generic two-Higgs-doublet models.


I. INTRODUCTION
The possibility of flavour-changing decays of the Brout-Englert-Higgs boson h (Higgs for short in the following) has been discussed for a long time as a possible signal for physics beyond the Standard Model (SM) [1][2][3][4][5][6][7]. Indirect constraints on these couplings come from flavour-changing neutral-current observables. In many analyses one follows an effective-field-theory approach in which one assumes that only the couplings of the SM-like Higgs to fermions are modified and derives constraints on these couplings from low-energy processes [6,7]. This leads one to conclude that no flavour-changing Higgs decays can be observable at the LHC, with the possible exception of h → τ e and h → τ µ [6,7]. This is a dangerous conclusion because the very existence of flavour-changing Higgs couplings in a renormalizable SM extension implies additional states which posses flavour-changing couplings as well. The indirect constraints from flavourchanging neutral currents and rare decays are thus inherently model-dependent and can be decoupled from Higgs decays. This generically involves finetuning of the mass spectrum and couplings of the additional states, but opens the way for some new channels to look for physics beyond the SM.
In this article we will study the arguably simplest SM extension that can lead to flavour-changing couplings of the SM-like Higgs: the two-Higgs-Doublet Model (2HDM) with generic Yukawa couplings, i.e. type III. 1 After computing the effects in B s -B s mixing, B s → µ + µ − and b → sγ, we identify regions of parameter space that can lead to sizable decay rates of h → bs (upwards of 10%) which are potentially observable at the LHC, hopefully motivating dedicated searches. This is particularly relevant now that the largest Higgs decay mode, h → bb, has finally been observed [18,19], rendering it background for h → bs. While not the focus of our work, we stress that the additional neutral states (H or A) can easily have even larger flavour-violating branching ratios, so general resonance searches for bs final states are encouraged as well.
The rest of this article is structured as follows: in Sec. II we set up our 2HDM notation. In Sec. III we discuss the main observables that could invalidate large h → bs rates and identify ways to circumvent their constraints. Sec. IV deals with direct searches for the new scalars at colliders, pointing out their main production and decay channels. We comment on different choices of bases for the 2HDM in Sec. V. Finally, we conclude in Sec. VI and provide an outlook for other rare Higgs decays. Appendix A provides one-loop formulae relevant for b → sγ.

II. TYPE-III 2HDM
Our starting point is the 2HDM with generic couplings to fermions (type III) and a CP conserving scalar potential [20]. In the Higgs basis [21][22][23] in which only one doublet acquires a vacuum expectation value (using notation close to Ref. [24]) we have with v 246 GeV, the Goldstone bosons G 0,+ , and the physical CP-odd scalar A. Assuming that CP is conserved in the scalar potential, the CP-even mass eigenstates are where we defined the mixing angle as β − α for easier comparison with the well-known type-I/II/X/Y 2HDM. We will abbreviate s βα ≡ sin(β − α), c βα ≡ cos(β − α), and t βα ≡ tan(β − α) below.
In the physical basis with diagonal fermion mass matrices the Yukawa couplings are given by where η d, = 1 = −η u and V is the Cabibbo-Kobayashi-Maskawa (CKM) matrix. (y f ) ij = δ ij m f j /v are the standard (diagonal) SM Yukawa couplings, while ε u,d, are arbitrary complex 3 × 3 matrices in flavour space. Offdiagonal elements in ε f lead to flavour-changing Higgs couplings. For our channel of interest, h → bs + bs, we have Note that this expression is valid at tree level; next-toleading order QCD corrections might increase the decay rate by 10-20% [9]. However, since we are interested in an order of magnitude estimate, such corrections are not of particular importance here. The resulting branching ratio is then with Γ SM 4.1 MeV and assuming all ε u,d, to be zero, except of course those for h → bs. Note that a branching ratio of h → bs of 1% (10%) requires ε d 23,32 couplings of order 0.02 (0.06), assuming c βα = 0.1.
So far no searches for h → bs have been performed, making it difficult to assess the sensitivity. The channel h → bb, which has a large SM branching ratio of 58%, has only recently been observed [18,19] despite its better b-tagging possibilities compared to h → bs. Nevertheless, we can obtain a model-independent limit on Γ(h → bs) of 1.1 GeV [25], corresponding roughly to the CMS energy resolution. This is still almost three orders of magnitude above the SM value Γ SM , and thus still allows for BR(h → bs) ∼ 1. A more intricate upper limit on the Higgs width can be obtained by comparing onand off-shell cross sections, as proposed in Ref. [26]. A recent CMS analysis of run-1 data along these lines obtains Γ h < 13 MeV [27]. While it cannot be claimed to be a model-independent limit [28], it should hold true in our scenario with c βα 1, seeing as h becomes arbitrarily SM-like. Naively applying Γ h < 13 MeV on our model and using c βα ≤ 0.55 as a very conservative bound (see below), this implies BR(h → bs) 78%; for c βα 1 the limit is BR(h → bs) 68%. This is obviously still very large and can most likely be improved by a direct search for h → bs. We will use this as a conservative limit in the following.
Stronger limits can be obtained from global fits to observed Higgs production and decay channels, seeing as a large Γ(h → bs) would reduce all measured Higgs branching ratios and hence require a larger production cross section to obtain the same rates. An analysis of this type with LHC run-1 data was performed in Ref. [29] and lead to the 95% C.L. limit BR(h → new) < 34% on any new decay channels, including bs. This is a factor of two stronger than the limit from the Higgs width, in part because it is based on a combination of ATLAS and CMS data and makes use of more search channels. We will also show this limit in the following, but stress that it should be taken with a grain of salt; global-fit limits are very indirect and depend strongly on the assumptions one puts in. With the many parameters available in a type-III 2HDM, it is conceivable that the limit could be weakened by increasing some parameters relevant to Higgs production. A dedicated search for h → bs will yield far more direct constraints and should always be preferred to global-fit limits.
The goal of our article is to show that a sizable branching ratio for h → bs is possible, even up to the conservative limit of 68%. To simplify the analysis we will set as many entries of ε f to zero as possible, i.e. ε u,d, ij = 0 is the starting point of our investigation. In this limit, we can obtain bounds on the masses and on the mixing angle β − α by comparison with the type-I 2HDM (in the limit tan(β) → ∞, i.e. β → π/2, identifying our c βα with the type-I sin(α) = cos(π/2 − α)). This gives the rather weak bound |c βα | 0.55 from LHC run-1 Higgs measurements [30,31]. In the limit c βα → 0, the new scalars become completely fermiophobic and the model resembles the Inert Higgs Doublet (IDM), with a Z 2 symmetry that only allows the new scalars to be produced in pairs. This Z 2 is of course broken in the scalar potential and by c βα = 0, but it allows us to use well-known limits on IDM. In particular, LEP constraints on the Z and W widths approximately require while LEP-II excludes m H + < 70 GeV and also restricts the m A -m H parameter space [32]. Additional bounds come from LHC searches, which most importantly constrain the masses below m h /2 [33,34]. The Peskin-Takeuchi parameters S and T also provide constraints, unless the mass spectrum satisfies m A m H + (for ∆T 0) and m A m H m H + (for ∆S 0) [35][36][37]. All in all, the fermiophobic limit still allows for newscalar masses around 100 GeV, depending on the hierarchy. Turning on the mixing angle β − α will significantly affect the limits on m H as it opens up gluon fusion, diphoton decay, etc., to be discussed below.

III. OBSERVABLES
Since we are interested in h → bs we will use the ansatz where in addition to ε d 23,32 we also allow for non-zero values of ε µµ because this entry is important for B s → µ + µ − . In addition to B s → µ + µ − , the most relevant constraints originate from B s -B s mixing and B → X s γ.
These channels were also discussed in the MSSM (i.e. type-II 2HDM), where the h → bs branching ratio was found to be tiny [9,10]. Here it is important to discuss the difference of our analysis to the MSSM. Even though at the loop-level non-decoupling effects in the MSSM induce non-holomorphic Higgs couplings [38][39][40][41][42][43][44][45] (making it a type-III 2HDM), these effects are only corrections to the type-II structure. Therefore, the strong bounds from direct LHC searches for additional Higgs bosons as well as the stringent bounds from b → sγ on the charged Higgs mass of around 570 GeV apply [46]. Furthermore, in the MSSM the angle α is directly related to m A m H + , rendering it small and further suppressing h → bs.

A. Bs-Bs mixing
The ∆F = 2 process B s -B s mixing is unavoidably modified already at tree-level if h → bs has a non- GeV and c βα = 0.1, requiring that the 2HDM contribution to Bs-Bs mixing should not exceed 10% compared to the SM which is of the order of the uncertainty in the lattice calculation of the matrix elements. Here we scanned over mA from 100 to 200 GeV. Note that the dependence on c βα is very weak. As one can see, in order to get potentially large effects in h → bs, either ε d 23 or ε d 32 must be very small.
vanishing rate. To describe this process we use the effective Hamiltonian (see for example [47]) where non-vanishing Wilson coefficients are generated for the three operators with A and B being colour indices. At tree level, we obtain the Wilson coefficients [48] Computing the B s -B s mass difference by inserting the matrix elements together with the corresponding bag factor and taking into account the renormalization group evolution [47], we show the result in Fig. 1 The ε d 23,32 couplings necessary for h → bs also induce a modification of B s → µ + µ − at tree level, because by construction all three neutral scalars couple to bs, and at least two scalars also couple to µ + µ − . The effective Hamiltonian takes the form [48] H Bs→µµ with O X are obtained from O X by replacing P L with P R . The branching ratio then reads [48] BR experimentally determined to be 2.8 +0.7 −0.6 × 10 −9 [50]. The SM yields only one non-zero Wilson coefficient, First of all note that one cannot avoid effects here by setting ε d 32 = ±ε d 23 due to the constraints from B s -B s mixing. Adjusting ε µµ allows one to eliminate the muon coupling of at most one of the neutral scalars, leaving the other two contributing to B s → µ + µ − at tree level. Setting for example ε µµ = 0 gives C P − C P = 0 and , which can only be made small for m H ∼ m h with our ansatz from Eq. (15). As can be seen in Fig. 2 (left), this is already sufficient to obtain BR(h → bs) = O(10%) while satisfying the experimental B s → µ + µ − result within 2σ.
An even better ansatz is to choose a (real) ε µµ such that C S − C S = 0, as this allows for new-physics contributions interfering with the SM Wilson coefficient C A . The required coupling for C S − C S = 0 is 2 which gives, using also Eq. (15), The most obvious way to eliminate the new-physics effect here is to choose m H = m h , which also implies m A = m h with Eq. (15). Another possibility is to pick the phase of ε d 23,32 in such a way that it induces destructive interference with the SM contribution C A , which will soften the limits and allow for larger h → bs, see Fig. 2 (right). The largest possible h → bs values arise when C P − C P destructively interferes with C SM A , while keeping BR(B s → µ + µ − ) close to its SM value. Indeed, if we impose the condition then all observables in B s → µ + µ − will remain exactly at their SM values, as we are effectively just flipping the sign of the SM contribution, which is unphysical (see, for example, Refs. [52,53]  where m b should now be evaluated at the scale m H to take the running of C P − C P into account [48]. To reiterate, choosing masses and couplings according to Eqs. the neutral scalars we find that C 8 = −3C 7 depends only on m H but not on the mixing angle. We can thus predict the size of C 7 as a function of m H (or BR(h → bs) with the help of Fig. 3). For m H in the region of interest for a large h → bs, we find a tiny |C 7 | 2 × 10 −3 (2 × 10 −6 ) for ε d 32 = 0 (ε d 23 = 0), far below the current limit [55]. B → X s γ is hence trivially compatible with a large h → bs in the region of parameter space under study here.
The decay B s → µ + µ − is sensitive to the difference of the Wilson coefficients C P,S − C P,S , whereas B → Kµ + µ − depends on their sum C P,S + C P,S [56][57][58]. With our ansatz from Eqs. (15), (23), (26), we have C S = C S = 0 and either C P = 0 or C P = 0, depending on which ε d 23,32 we set to zero. We thus unavoidably modify B → Kµ + µ − at tree level. Using the results of Ref. [57], we checked that this effect is very small, keeping B → Kµ + µ − close to the SM value. This means that our model cannot address the observed deviations from the SM prediction in current global fits to b → sµ + µ − observables [59][60][61].
Similarly, one can expect an effect in B → Kτ + τ − or B s → τ + τ − . Even though the effect is enhanced by m τ /m µ , the very weak experimental bounds on the branching ratio [62,63] (several times 10 −3 ) do not pose relevant constraints on our parameter space.

E. Anomalous Magnetic Moment of the Muon
The choice of ε µµ in Eq. (23) reduces the coupling of h to µµ, but enhances the one of H by a factor of few, and also couples A and H + to muons. As a result, one could expect a modification of (g − 2) µ , an observable that famously deviates from the SM value by around 3σ and can be explained in 2HDMs [64][65][66]. However, the one-loop effect is still suppressed by the small muon mass. In addition, the usually dominant Barr-Zee contributions [54] are also not important in our Higgs basis (with a minimal number of free parameters ε f ) since the couplings to heavy fermions (top, bottom or tau) are not enhanced for the heavy scalars. Furthermore, B s → µ + µ − prefers nearly degenerate masses for A and H, leading to a cancellation in the anomalous magnetic moment of the muon. Concerning H + effects, the best channel is B − → µν (assuming ε τ τ = 0), with the rate where m b is again to be evaluated at the scale m H + to take the running of the Wilson coefficients into account. The predicted SM branching ratio BR(B − → µν) SM 6 × 10 −7 is small and not observed yet, but our new contribution could reach the current upper limit BR(B − → µν) < 10 −6 [67]. From Fig. 4 we see that the limits are rather weak and automatically satisfied in the region m H + ∼ m H .
Two other indirect channels are of potential interest: D − s → µν and K − → µν, the latter of which is sup-pressed but measured with more accuracy. We have which gives much weaker bounds than B − → µν before.

IV. COLLIDER CONSTRAINTS
Having explored the indirect constraints that come with a large h → bs decay, let us briefly comment on possible collider searches.

A. Charged Scalar
The charged scalar has barely played a role in any of the processes discussed so far, thanks to our ansatz for the ε couplings in Eq. (8) together with Eq. (15). Its mass is hence a more-or-less free parameter, as long as we keep it close enough to m A,H to not induce too large S and T parameters. Let us briefly comment on the H + phenomenology beyond electroweak precision observables (see also Ref. [68] for a recent review). Aside from gauge couplings, H + only couples according to Eq. (4): (29) where ε contains only the non-zero entry ε µµ and the quark couplings are determined by the matrix Since we impose ε d 32 ε d 23 = 0 in order to satisfy limits from B s -B s mixing, the H + couples only either to b or s quarks. In particular, it does not contribute to b → sγ. ε d 23,32 is much bigger than the ε µµ given by Eq. (23) for a sizeable h → bs rate, so the dominant coupling of H + is to quarks. If H + is lighter than the top quark, it can be produced in its decays: If ε d 32 = 0, the production channel is t → bH + , suppressed by V ts , followed by H + →bc with branching ratio 1. This channel has been looked for [69], with constraints around |ε d 23 | 2 for m H + between 90 and 150 GeV. This is still compatible with a large h → bs rate as long as c βα is not too small (c βα > 0.1). For completeness, we can replace ε d 23 directly with the h → bs branching ratio BR bs h to predict If instead ε d 23 = 0, the production channel will be t → sH + with BR(H + →sc) 1. The rate can be obtained from Eq. (31) via V ts → V tb , so this channel is enhanced by |V tb /V ts | 2 580 compared to the previous one. Since this final state has only been considered with the production channel t → bH + [70,71], we cannot obtain useful limits.
For H + masses above the top mass the typical search channel is H + → tb [72] or H + → τ ν, which are suppressed or even zero in our scenario and hence not good signatures.

B. Neutral Scalars
The neutral scalars H and A have large couplings to bs, but also the far easier to detect muon coupling exists. For A, the branching ratio is however very small, especially in the region m H ∼ m h where the h → bs branching ratio BR bs h is largest. As a result, the best search channel is typically A → bs. The same is true for H in the limit c βα 1, although a sizeable c βα can lead to a large H → bb. With essentially only a large bs coupling, A can be produced at the LHC via the strangequark sea, e.g. sg → bA, followed by A → bs or A → µµ. Similar channels have been discussed in the past, see for example Refs. [73,74]. For H, the c βα -suppressed gluon or vector-boson-fusion channels become available too, allowing for a search analogous to h → bs.
A particularly interesting, albeit also c βα -suppressed, decay channel for H, A is H, A → γγ. Recent √ s = 13 TeV CMS limits for this signature can be found in Ref. [75], which also shows a small (2.9σ local, 1.5σ global) excess around m 95 GeV. This would be an interesting value for m H , as it can lead to BR(h → bs) ∼ 20% (Fig. 3). With the couplings at hand, the cross section pp → H → γγ is simply too small for realistic values of c βα . However, the discussion so far assumed that all other entries ε f ij except ε d 23,32 are zero. Introducing extra couplings, in particular ε u 33 , enhances both the gluonfusion H, A production as well as the H, A branching ratio into γγ since H, A with a mass of 95 GeV cannot decay into two top quarks. In order to keep h → γγ close to SM value, one needs c βα 1, which in turn gives m A m H due to Eq. (15). Therefore, the CMS diphoton excess would have to be interpreted as two unresolved peaks from gg → A/H → γγ. Since the total signal corresponds approximately to the expected signal strength of an SM-like Higgs boson [75] each boson should reproduce approximately half of the expected SM signal. Nevertheless, if one aims at large rates of h → bs, very large values of ε u 33 will be required to obtain the desired γγ-signal. We will leave a detailed discussion of this for future work.

V. DIFFERENT CHOICE OF BASIS
So far, we worked in the Higgs basis in which only one Higgs doublet requires a vacuum expectation value. However, this basis is not motivated by a symmetry and allows for generic large and potentially dangerous flavour violation, while the type-I, II, X and Y models posses a Z 2 symmetry ensuring natural flavour conservation (see Here,ε f ij breaks the Z2 symmetry of the four 2HDMs with natural flavour conservation and induces flavour changing neutral currents. Ref. [20] for an overview). Therefore, let us consider these models but allow for a breaking of this Z 2 symmetry such that flavour changing Higgs couplings are possible. In Tab. I we give the relation between the couplings ε f ij defined in the Higgs basis and the quantitiesε f ij which break the Z 2 symmetry of the four 2HDMs with natural flavour conservation. Our new free parameters which induce flavour-changing neutral Higgs couplings are now ε f ij instead of ε f ij . tan(β) corresponds as always to the ratio of the two vacuum expectation values.
First of all, we can rule out the type-II as well as the type-Y model since they lead to large effects in b → sγ and direct LHC searches, leading to stringent lower bounds on the masses of the additional scalars.

A. Type-I Model
Concerning B s -B s mixing the analysis remains unchanged compared to the one in the Higgs basis. For B s → µ + µ − we can setε 22 = 0, the condition to cancel C S − C S reads andε d 23,32 have to be chosen as is inherently model-dependent and thus difficult to assess in an effective-field-theory framework. In this article we have shown explicitly how the h → bs branching ratio can be enhanced to nearly arbitrary levels in a generic 2HDM while keeping other processes such as B s → µ + µ − , B → X s γ and B s -B s mixing essentially at their SM values. Of course, this requires some tuning in the mass spectrum (new neutral scalars with masses similar to the SM Higgs) and couplings of the new scalars, but illustrates the importance of flavour-changing Higgs decays as a complementary probe of new physics. We strongly encourage dedicated experimental searches for bs resonances.
Other rare or forbidden Higgs decays [6] can be analysed in a similar way within the 2HDM with generic Yukawa couplings: • h → bd: Here the analogy with h → sd is straightforward, i.e. the same conditions for the cancellations in flavour observables are required. However, the parameters must be adjusted even more precisely such that large decay rates can be possible.
• h → ds, uc: Here the experimental problem of tagging light flavour makes it very hard to distinguish such modes from h → qq or h → gg. Anyway, ε q

12,21
is stringently constrained from Kaon or D-D mixing. This bound can be avoided in the same way as the B s -B s mixing bound studied here. However, an even more precise cancellation would be required and bounds from D → µν and K → µν become relevant.
• h → τ e: Here the situation is very much like in the case of h → τ µ since the experimental bounds from τ → eγ and τ → eµµ are comparable to the corresponding τ → µ processes.
• h → eµ: Obtaining large rates for h → µe is very difficult, not only because of the stringent bounds from µ → eγ but also because of µ → e conversion, where in a 2HDM [85] an accurate cancellation among all the couplings to quarks would be required.

(A7)
Here λ t = V tb V ts and we used the couplings Γ defined as with H 0 1,2,3 = h, H, A. In order to compare with the couplings given in Eq. (4), see Tab. II.