$b\to s\ell^+\ell^-$ Transitions in Two-Higgs-Doublet Models

In this article we study $b\to s\mu^+\mu^-$ transitions and possible correlations with the anomalous magnetic moment of the muon ($a_\mu$) within two-Higgs-doublet models with generic Yukawa couplings, including the possibility of right-handed neutrinos. We perform the matching on the relevant effective Hamiltonian and calculate the leading one-loop effects for $b\to s\ell\ell^{(\prime)}$, $b\to s\gamma$, $\Delta B=\Delta S=2$, $b\to s\nu\bar\nu$ and $\ell\to\ell^\prime\gamma$ transitions in a general $R_\xi$ gauge. Concerning the phenomenology, we find that an explanation of the hints for new physics in $b\to s\mu^+\mu^-$ data is possible once right-handed neutrinos are included. If lepton flavour violating couplings are allowed, one can account for the discrepancy in $a_\mu$ as well. However, only a small portion of parameter space gives a good fit to $b\to s\mu^+\mu^-$ data and the current bound on $h\to\tau\mu$ requires the mixing between the neutral Higgses to be very small if one aims at an explanation of $a_\mu$.


Introduction
Two-Higgs-doublet models (2HDMs) [1] have been under intensive investigation for a long time (see e.g. Ref. [2] for an introduction or Ref. [3] for a review article). There are several reasons for this intense interest: First of all, 2HDMs are extremely simple extensions of the Standard Model (SM) obtained by adding a single scalar SU (2) L doublet to the SM particle content. Furthermore, motivation for 2HDMs comes from axion models [4] because a possible CP violating QCD-theta term can be absorbed [5] if the Lagrangian possesses a global U (1) symmetry. This is only possible if the SM is extended by at least one Higgs doublet. Also the baryon asymmetry of the universe can be generated within 2HDMs while the amount of CP violation in the SM alone is too small to achieve this [6]. Finally, the Minimal Supersymmetric Standard Model predicts the presence of a second Higgs doublet [7], due to the holomorphicity of the superpotential. The effective theory obtained after integrating out the superpartners of the SM particles (sfermions, gaugions and higgsinos) is a 2HDM (with the addition of higher dimensional operators involving two Higgs doublets [8]).
2HDMs possess three additional physical scalars with respect to the single Higgs boson of the SM; a neutral CP-even H 0 , a CP-odd scalar A 0 and a charged scalar H ± (under the assumption of CP conservation). These new particles are not only interesting with respect to direct searches at the LHC (see e.g. Ref. [9,10,11,12,13,14,15,16,17] for recent reports). In addition, they give rise to important effects in low-energy precision flavour observables, providing a complementary window to physics beyond the SM. In this respect, decays of neutral mesons to charged lepton pairs (e.g. B s(d) → µ + µ − , D → µ + µ − and K L → µ + µ − ) are very interesting because they are especially sensitive to scalar operators which possess enhanced matrix elements with respect to vector operators. For this reason, B s → µ + µ − (which can be calculated more precisely than D → µ + µ − or K L → µ + µ − and has a larger branching fraction than B d → µ + µ − ) has been studied frequently in the context of 2HDMs.
(1. 2) This puts stringent constraints on 2HDMs with scalar operators contributing to b → sµ + µ − transitions. Furthermore, LHCb found significant hints for new physics in b → s + − data, showing a coherent pattern of deviations from the SM predictions with a significance of more than 4-5 σ [38,39] 1 . However, in order to explain these anomalies, vector operators, in particular O 9 , are necessary while an explanation of the anomalies with scalar operators alone is not possible.
Within 2HDMs, vector operators at the dimension 6 level can only be generated via loop effects. However, contributions to other loop-induced processes such as b → sγ (for which the SM prediction [47] is in very well agreement with the experimental average [31]), b → sνν, (where the experimental upper bound [48,49] approaches the SM prediction [50]) or B s −B s mixing [31] unavoidably arise and their constraints must be taken into account. Therefore, an explanation of b → s + − data in the context of multi-Higgs-doublet models might require the introduction of right-handed neutrinos [51,52]. Furthermore, any model with sizeable couplings to muons could potentially address the long-lasting discrepancy between experiment [53] and the SM For definiteness, and in order to be conservative, we choose a value at the lower end. In the case of lepton flavour violation, a µ is intrinsically correlated to lepton flavour violating decays such as τ → µγ whose bound must be taken into account. Furthermore, in 2HDMs also h → τ µ gives relevant bounds due to the mixing between the neutral CP-even Higgses.
In this article we want to investigate b → sµ + µ − transitions within 2HDMs in the light of the corresponding hints for new physics and its correlations with other b → s transitions and a µ . For this purpose, we will consider a 2HDM with a CP conserving Higgs potential but with generic sources of flavour violation and the possible addition of right-handed neutrinos. After establishing our conventions in Sec. 2, we will use 1 Including only R(K) and R(K * ), the significance is at the 4 σ level [40,41,42,43,44,45,46]. 2 The SM prediction of a µ is currently re-evaluated in a community-wide effort prompted by upcoming improved measurements at Fermilab [54] and J-PARC [55] (see also [56]). With electroweak [57,58,59] and QED [60] contributions under good control, recent advances in the evaluation of the hadronic part include: hadronic vacuum polarization [61,62,63,64,65,66,67], hadronic light-by-light scattering [68,69,70,71,72,73], and higher-order hadronic corrections [74,75]. this setup to calculate the tree-level matching on the effective Hamiltonian governing b → s transitions and the leading one-loop effects in Sec. 3. Section 4 is devoted to the calculation of the matching on the ∆B = ∆S = 2 Hamiltonian, to a µ , h → τ µ and b → sνν. In our phenomenological analysis in Sec. 5 we will address the question if the hints for new physics in b → sµ + µ − transitions can be explained within 2HDMs without violating the bounds from other processes, before we conclude in Sec. 6.

Model and Conventions
As outlined in the introduction, we supplement the SM by a second scalar doublet with the same hypercharge as the first one. For the calculation of flavour observables it is convenient to work in the Higgs basis [76,77,78] where only one Higgs doublet acquires a vacuum expectation value and therefore the generation of the fermions and gauge boson masses is separated from the couplings to fermions. Using the notation of Ref. [79], we have with v 246 GeV. G + and G 0 are the Goldstone bosons and A 0 denotes the physical CP-odd scalar, assuming that CP is conserved in the Higgs potential. The CP-even mass eigenstates are where we defined the mixing angle as β − α for easier comparison with the wellknown type-I/II/X/Y 2HDMs. In the following, we will abbreviate s βα ≡ sin(β − α) and c βα ≡ cos(β − α) and assume that h 0 is the SM-like Higgs boson with a mass of around 125 GeV. We require c βα to be small (at most O(0.1)) such that its properties are compatible with experiments [80,81]. With these conventions the couplings of the scalar bosons to fermions are given by We also allowed for the presence of right-handed neutrinos N with a Majorana mass term −1/2MN c N . This manifests itself in Eq. (2.3) through the terms m ν and ε ν which otherwise would be absent. Note that m ν corresponds to the Dirac mass term of the neutrinos which is related to the physical neutrino mass via the see-saw mechanism. Assuming a mass scale of the right-handed neutrinos at the TeV scale requires m ν to be at most around 10 MeV. Thus we can safely neglect its effect on the Higgs couplings to fermions and focus on ε ν which is decoupled from the neutrino masses and thus unconstrained.
We do not need to discuss the Higgs potential in detail since, in addition to the physical masses and mixing angles, only the two Higgs self-couplings enter in our calculation in the case of CP conservation. We will simply parametrize these couplings as λ h 0 H + H − and λ H 0 H + H − and refer the interested reader to Eq. currents. However, the parameters ε F ij in the Higgs basis can be related to the ones within the four 2HDMs with natural flavour conservation (type-I/II/X/Y) as Theε F ij are the flavour changing entries in the new basis, i.e. the corrections to natural flavour conservation. The coefficients c f y and c F ε are given in Table 1. In this basis, the termsε F ij break the Z 2 symmetry and lead to deviations from natural flavour conservation.

b → s + − Processes
We define the effective Hamiltonian giving direct effects in b → s ( ) and b → sγ transitions as with the operators plus their primed counterparts which are obtained by exchanging P L and P R . We did not include tensor operators here since they are not generated at the dim-6 level.
In addition, we include four-quark operators which are generated by charged Higgs exchange (analogous to O 2 in the SM) which can contribute to b → s + − processes at the loop-level. The operators are defined as with A, B = L, R and the colour indices are contracted within the bilinears.

Tree-Level
At tree-level, in the approximation of vanishing external momenta, we only get contributions to semi-leptonic scalar and pseudoscalar operators from neutral Higgs exchange (see Fig. 1). They are given by where we defined and In addition, we define for future convenience the squared mass ratios for heavy Majorana neutrino, up-type quark and the W boson with respect to the charged Higgs We derived Eq. (3.5) by working at leading order in the external momenta (which we will also do for all following results). This corresponds to an expansion in m b,s and m over the Higgs masses which we assume to be at least at the EW scale. For consistency, one has to take into account all masses m b,s and m in this expansion, also the ones entering via Higgs couplings 3 . Equation (3.5) contains terms linear in light fermion masses which therefore correspond to dim-7 contributions. However, since from the expansion in the external momenta no dim-7 terms arise (the next non-vanishing order is dim-8), it is consistent to keep these terms even though in the loop effects, to be studied later, we only consider dim-6 terms.
The Wilson coefficients of the four-quark operators in Eq. (3.3) due to tree-level charged Higgs exchange read (3.9)

b → sγ
Here (and for all loop effects to be calculated) we do not consider multiple flavour changes which are phenomenologically known to be small. Regarding the (numerically) leading contributions due to the charged Higgs (see Fig. 2) exchange we therefore only have to distinguish the top contribution (for which all particles in 3 Note that it is a convenient feature of the Higgs basis that only the couplings which are related to EW symmetry breaking contain fermion masses (unlike in type-I/II/X/Y). Thus one can directly expand in these parameters without taking into account factors of sin α, tan β, etc.
the loop are heavy) from the charm contribution (where we set the mass equal to zero). For the first case the result is given by to each other. Since there is no constant term in Eq. (3.12) the inclusion of the soft contribution just leads to a replacement of µ by µ b in Eq. (3.11).
While an explicit splitting into the hard matching contribution and the effect from the four-quark operators is necessary if one aims at including α s corrections, this is not necessary at leading order and one can just add both contributions. In fact, since the neutral Higgs contribution is phenomenologically small, a leading order estimate is sufficient and we give here the sum of the soft and the hard contribution It is straightforward to use the NLO QCD corrections calculated in Ref. [82] (for our prediction with a top-quark in the loop), where QCD corrections in a generic 2HDM with a discrete symmetry were considered. The Wilson coefficients C 7 and C 8 can be included by simply setting the couplings X and Y defined in Ref. [82] to (3.14) The primed operators can be treated in an analogous way taking into account that C 2 = 0.

One-Loop Effects in b → s ( )
We will now calculate the "leading" one-loop matching contributions to the op- and C ( ) 10 . We will perform this calculation in a general R ξ gauge expanding all diagrams up to the first non-vanishing order in the external momenta, corresponding to dim-6 operators. In addition, we neglect all quark masses, except for the top-quark and integrate out all Higgses, W , Z and the top at a common scale m EW .
By "leading" one-loop effects we also mean that we will only calculate the loop corrections to a Wilson coefficient if there is no corresponding tree-level effect. In addition, we will neglect small effects originating from multiple flavour changes, i.e.
3 → 1 → 2. Thus, since the tree-level contribution involve ε d 23,32 , we will assume these couplings to be zero when calculating the loop correction. Therefore, flavour violation in the quark sector can either originate from the CKM matrix multiplying a diagonal ε d ii or from the term ε u * jf V ji P L which contributes both for diagonal and also off-diagonal elements ε u * jf . Note that the latter terms only enter via charged Higgs couplings to quarks. Hence, we just need to calculate diagrams with a charged Higgs and/or W boson together with the corresponding charged Goldstones. Finally, we obtain gauge-invariant results.

Self-Energies and Renormalization
Here we will discuss the renormalization which can be solely derived from expressions for the self-energies. The reason is that in our setup (with ε d 23,32 = 0) ultraviolet divergences only arise in (pseudo)scalar operators originating from Higgs penguins and Higgs couplings are intrinsically related to chirality changing self-energies (see Ref. [84]). We will also use this opportunity to illustrate the cancellation of the gauge dependence in the renormalization of the quark masses. We performed the calculation in a general R ξ gauge. We begin by defining the self-energies as b and we obtain the following expressions for b → s transitions with ξ denoting the gauge parameter.
Let us now consider the general effect of self-energies on kinetic terms and quark masses (see e.g. Ref. [85]). First of all, one has to render the kinetic terms canonical, leading to the shifts in the quark fields These shifts then enter not only in all couplings but also in quark masses. Since the quark mass terms receive contributions from the chirality changing self-energies as well, we have The eigenvalues of this matrix after renormalization in the M S scheme are identified with the physical quark masses, extracted from data according to the SM prescription. Note that at first order in perturbation theory (i.e. linear in Σ), the eigenvalues just correspond to the diagonal terms where the dependence on ξ drops out and thus rendering the renormalized parameter gauge-independent, as required for a physical quantity. The rotations that diagonalize the mass matrix as read at leading order (considering only the s-b sector) These rotations, together with the shifts in Eq. (3.20) result iñ This agrees with the diagrammatical approach of Ref. [86] and confirms the statements of Ref. [22] that diagrams involving flavour changing self-energies can be treated as one-particle irreducible. Thus, we apply Eq. (3.24) to the couplings ε d ij and take into account all self-energy contributions.
Let us now turn to the renormalization. As stated above, it can be determined solely from the expressions for the self-energies. Unlike in the SM or in 2HDMs with natural flavour conservation, our results for b → s + − will be divergent for generic couplings ε u ij . The reason for this is that once ε u ij does not correspond to a special case of the four 2HDMs with natural flavour conservation (see Table 1), the Z 2 symmetry in the Yukawa sector is broken and no symmetry protects ε d ij from being flavour changing. In fact, counterterms to off-diagonal elements of ε d ij are required to render the result finite. Since all divergences originate from Higgs penguin diagrams, we can determine the 1/ structure of our results from the self-energies. For this, we start with the interaction basis in which the Yukawa Lagrangian is given by where for simplicity we considered the neutral current part only. Assuming (3.25) is already in the basis with diagonal mass matrices, the masses then are given by Since the chirality flip on the fermion line in Σ LR 23 always originates from an up-quark mass, we can define We keep only the relevant divergent part and we obtain We invert the relations in Table 1 to go to the Higgs basis and set for consistency reasons the quark masses to zero. Then we apply the rotations in Eq. (3.24) and find where the definition for the bare couplings ε d(0) 23,32 = ε d 23,32 + δε d 23,32 was used. Again, note that these counterterms are independent of the gauge parameter ξ. As we will see later, these counterterms, inserted into the tree-level expressions for B s → + − (see Eq. (3.5)), will render the results finite.

Z and γ Penguins
The Wilson coefficients originating from Z penguins and involving the charged Higgs (see Fig. 3), are only relevant for top exchange and are given by where the loop function I 1 (x) is defined in the appendix. Note that I 1 (0) − 1 = 0 justifying that we only consider the top quark here.
For the off-shell photon penguin, also shown in Fig. 3, we obtain for the top quark (3.31) Concerning light-quarks, the hard matching contributions get amended by the mixing of the four-quark operators in Eq. (3.9) into C 9 and C 9 . We obtain (3.32) The same result can be obtained by expanding Eq. (3.31) in m t and then replacing m t in the logarithm by the B meson scale µ b . Once more, note that at LO adding the soft to the hard matching contribution is justified.

Higgs Penguin and W -Higgs Boxes
Here, contributions originating from flavour changing self-energies appear that are parametrically enhanced by for i = 3. Using these definitions, the neutral Higgs penguin contributions involving a top quarks and a H ± in the loop, (see Fig. 4) read The charm contribution is obtained in the limit z → 0 and is explicitly given in the appendix. The top quark contributions of diagrams including both W ± and H ± , i.e. mixed boxes and Higgs penguins with a W in the loop (see Fig. 5) yield the result

H ± Boxes
The expressions for the box diagrams involving two charged Higgses (see Fig. 6) are given by

Processes and Observables
For b → sµ + µ − transitions it is helpful to distinguish three regimes, the one of scalar operators (C  However, the effect of vector operators cannot be neglected here, since they have different parametric dependences, notably contributions independent of ε d ij . Concerning magnetic operators, the inclusive b → sγ decay is most sensitive. The SM prediction [47,88] Br[B → X s γ] SM = (3.36 ± 0.23) × 10 −4 (3.40) has to be compared to the experimental value [31] Br Note that the contributions in Eqs. (3.11, 3.13), which would require the addition of the four Fermion operators in Eq. (3.9) are all proportional to ε d , which we set to zero in our analysis. Finally, semi-leptonic decays are important to constrain vector operators since their dependence on scalar ones is very weak [90]. However, many processes and observables have been measured and one therefore should use a global fit to constrain C ( )µµ 9,10 (taking also into account B s → µ + µ − if one assumes the absence of scalar operators). The scenario with a lepton flavour conserving C 10 effect (C U 10 ) and a contribution to C 9 = −C 10 with muons only (C V 9 = −C V 10 ) (following the conventions of Ref. [91]) is phenomenologically the most important scenario for us.
We will discuss this in the next section.

b → sνν
For b → sνν processes the corresponding effective Hamiltonian is defined as with the operators From box diagrams with charged Higgses we obtain We follow [50] and define This allows us to write the branching ratio in terms of where κ η encapsules the dependence on the form factors. In Ref. [50] this quantity is evaluated using as input for the B → K * form factors a combined fit to lattice and LCSR results performed in [94], finding κ η = 1.34 ± 0.04. The branching ratio reads This has to be compared to the experimental limits [48] R νν K < 3.9 , R νν K * < 2.7 . (4.8)

B s −B s Mixing
The effective Hamiltonian is defined as (4.10) We obtain at tree level (see left diagram in Fig. 7) find for the pure H + boxes and for the W + -H + boxes The corresponding diagrams are shown in Fig. 7. The loop functions are given in the appendix and once more we did not distinguish between the cases of light and heavy quarks, since the contribution of the light quarks trivially follows by taking the convergent limit z i → 0.
Phenomenologically, we only need to consider the contributions to C 1 , since the other Wilson coefficients are proportional to ε d ij which we will assume to be small. The constraints on NP crucially depend on the hadronic matrix elements calculated in lattice QCD. While Ref. [95] finds a preference for destructive interference with the SM, Ref. [96] finds a preference for constructive interference. We will therefore use the ratio C NP 1 /C SM 1 , where all hadronic uncertainties drop out. We assume a conservative bound of ±30%.

→ γ and a
Since it is important for our phenomenological analysis, we generalize the formula of Ref. [29] to include right-handed neutrinos. Following the conventions of Ref. [97] we define with and For the loop diagrams with charged Higgses we obtain where we set the left-handed neutrino mass to zero. The neutral Higgs bosons give Also here, we included the hard matching contribution together with the soft contribution from the effective theory in the formula since we do not aim at calculating QED corrections [98]. For our purposes we require only the lepton flavour violating decay τ → µγ whose experimental upper limit is given by Br [τ → µγ] < 4.4 · 10 −8 [99,100].

h → τ µ
Here, we find for the decay width with Γ SM 4.1MeV. This has to be compared to the current experimental limit [101,102] Br [h → τ µ] ≤ 1.43% . Due to the suppressed SM decay width, h → τ µ will turn out to be surprisingly constraining.

Phenomenological Analysis
In our numerical analysis we want to focus on the possibility to explain the hints for NP in b → sµ + µ − transitions and a µ within 2HDMs. Concerning b → sµ + µ − data, it is well-known from global fits that a sizeable contribution to the Wilson coefficient C 9 (and possibly also C 10 ) is required to explain the data. Additional substantial effects in C 9 and C 10 are possible. However, contributions to scalar operators must be suppressed due to the strong constraints from B s → µ + µ − where they enter with an enhancement factor of m 2 b /m 2 µ . C 9 and C 10 can only be generated from γ and Z penguins (see Eqs. (3.30)-(3.32)) or from charged Higgs boxes (see Eq. (3.37)). Interestingly, all contributions to C 9 and C 10 involve ε u ij but not ε d ij while the effect in C 9 , C 10 only appears once ε d ii is unequal to zero. Furthermore, scalar operators involve both ε d ii and ε u ij . To accommodate the strong constraints on scalar operators we will assume that ε d ii is negligibly small in the following. As stated above, an effect in C 9 is mandatory to explain the anomalies. However, the Z penguin contribution to C 9 is suppressed by (1−4s 2 W ) and the off-shell photon effect is small due to the electromagnetic coupling. Hence, in the limit of ε ij = ε ν ij = 0 we are left with a lepton flavour universal C U 10 effect (following the conventions of Ref. [91]) to a good approximation. This effect is also strongly correlated to (and therefore limited by) B s −B s mixing, as shown in Fig. 8. Note that this correlation is to a good approximation independent of the Higgs masses. The bound from b → sγ in this setup turns out to be in general weaker than the ones from B s −B s mixing.
Therefore, we need in addition the charged Higgs boxes if we aim at a good fit to b → sµ + µ − data. Here, ε I2 generates C V 9 = C V 10 effect in muons only, while ε ν

2I
gives C V 9 = −C V 10 . Let us first consider the case with only ε IJ since these couplings are present also in the scenario without right-handed neutrinos. Since we aim at an explanation of a µ , we focus on the elements ε 23,32 which give an m τ /m µ enhanced effect in this observable 5 . For the numerical analysis we chose for definiteness m A 0 = 300 GeV and m H 0 = 250 GeV. Even though a detailed collider analysis is well beyond the scope of this article, note that the small values of c βα are compatible with direct LHC searches [80]. The effect in a µ is directly correlated to h → τ µ which strongly constrains c βα as shown in Fig. 9. The bounds from h → τ µ do not only depend on fewer parameters than τ → µγ but are even much stronger for ε 22,33 = 0. Concerning b → s + − , the impact with ε 23,32 = 0 is small. Since the effect in a µ is chirally enhanced, it significantly limits the product ε 23 ε 32 rendering the deviation from C V 9 = −C V 10 unimportant. In a next step, we allow for the presence of right-handed neutrinos and ε ν ij = 0 where the C V 9 = −C V 10 effect has to be added to C U 10 from the Z penguin. The result is shown in Fig. 10 where we can see that it is difficult to find points which give a good fit to b → sµ + µ − data. While the effect of ε ν IJ = 0 in a µ is always destructive, i.e. it increases the discrepancy between theory and experiment, the effect is small since it is not enhanced by m τ /m µ . It is therefore possible to tackle b → sµ + µ − fixing ε ν IJ and δa µ fixing ε IJ semi independently, while choosing the Higgs masses consistent with direct searches and taking into account the smallness of c βα , required by h → τ µ. One can see that in order to be in agreement with b → s + − data, positive effects in B s −B s mixing are preferred.

Conclusions
In this article we studied b → s transitions in 2HDMs with generic Yukawa couplings (including right-handed neutrinos) with focus on b → sµ + µ − transitions and its possible correlations with a µ . We first recalled the tree-level effects in b → s observables which involve ε d 23,32 . If these elements are zero or negligibly small, loop effects involving W bosons and charged Higgses can become numerically important. We calculated these leading one-loop corrections to b → s + − , b → sνν and ∆B = ∆S = 2 transitions in a general R ξ gauge and confirmed their correctness finding gauge invariant results. Additionally, we discuss the treatment of self-energy contributions and renormalization in detail. In addition, we provided the formula for τ → µγ and a µ including the contributions from heavy (TeV scale) right-handed neutrinos.   [100,350]. In total, we generated 10 6 points. The red regions are preferred by b → s + − data according to updated fit of Ref. [39] and includes the new LHCb [103] and Belle [104] measurement of R(K) and R(K * ), respectively. It is interesting to note that using the new fit significantly more points lie within the preferred regions.
Concerning the phenomenology, we found that without right-handed neutrinos sizeable contributions to vector operators can only be generated via photon and Z penguins. However, this does not allow for lepton flavour universality violation and the effect in C U 10 with respect to C U 9 is too big to give a good fit to data. Therefore, we included in a next step right-handed neutrinos which lead in general to a lepton flavour universality violating C V 9 = −C V 10 effect. This can provide an explanation of the anomalies especially with the recently updated b → s + − data.
If we allow for Higgs to τ µ couplings, we can explain the anomalous magnetic moment by a chirally enhanced m τ /m µ effect. This leads at the same time to nonvanishing branching ratios τ → µγ and τ → 3µ which are however compatible with the experimental limits. The effect in h → τ µ is found to be dominant, i.e. most constraining. In case of an explanation of a µ , h → τ µ requires a close alignment in the Higgs sector, i.e. very small c βα . Furthermore, a small C V 9 = +C V 10 effect is generated which does not significantly improve the goodness of the fit to data.
2HDMs have a rich flavour phenomenology since they give effects in many classes of observables. As we showed in this article, these models are in principle capable to explain the discrepancies between the SM and experiment. Once one allows for a generic flavour structure and right-handed neutrinos, this provides a possible solution to the deviations in b → s + − transitions and a µ , even though some degree of finetuning is necessary. Furthermore, also the anomalies in b → cτ ν processes [31] might be addressed by 2HDMs [29,105,106,107,108,109,110,111,112,113].
However, these solutions are under pressure from the measurement of the B c lifetime [114,115,116,117] and LHC searches [118]. Furthermore, also the / anomaly (see e.g. Ref. [119] for a review) could be explained [52,120], leaving 2HDMs still as one of the most appealing NP scenarios.

Acknowledgments
The work of A.C. and D.