Enhanced $B_q \to \ell\ell^\prime$ and $B \to (K, \pi) \ell\ell^\prime$ in light of $(g-2)_\mu$

We study lepton flavor violating (LFV) $B$ decays in a general two Higgs doublet model with sub-TeV exotic scalars. Two different parameter spaces are explored: one dominated by extra top Yukawa coupling $\rho_{tt}$, the other by LFV couplings relevant for the muon $g-2$ anomaly. In the first case, flavor constraints such as $\ell \to \ell^\prime\gamma$, $h \to \ell\ell^\prime$ imply LFV $B$ decays are far below experimental sensitivities. The second case needs to be close to the alignment limit, but $B_q \to \tau\mu$ and $B \to (K, \pi) \tau\mu$ rates can lie within the sensitivities of Belle II and LHCb Upgrade II. Neutral $B$ meson mixings and $B,\, K,\, \pi \to \ell \nu$ decays provide important flavor constraints on parameter space. $B$ decays involving $e$-$\tau$ violation are constrained by $\mu\to e$ processes.


I. INTRODUCTION
Flavor changing neutral couplings (FCNCs) in the Standard Model (SM) occur only beyond tree-level and the corresponding rates are small due to suppression from GIM mechanism [1] and vanishing neutrino masses. But new physics (NP) beyond SM could have interactions that allow for sizable FCNC processes. Therefore, the precise measurements of rare FCNC decays serve as powerful probes of physics beyond SM. In this context, rare B decays offer excellent opportunities as, in addition to loop factors, these are suppressed further by small CKM factors, while the b quark mass is sufficiently large so long-and short-distance effects can be separated and extracted with reasonable precision.
Rare leptonic decays of B mesons are advantageous for study, as all hadronic effects are contained in the decay constant, calculable by lattice QCD [2], hence the decay rates can be predicted with great precision. On the experimental front, there has been excellent progress with ever increasing precision. One example is the helicitysuppressed rare B s → µ + µ − decay, which provides one of the most sensitive probes of scalar NP interactions.
Based on full Run 1 and Run 2 data with 9 fb −1 luminosity, LHCb [3,4] reported the branching ratio,  [6,7]; but within errors, both measurements agree with SM, thereby provide strong bounds on NP interactions. The CKM suppressed B d → µ + µ − is not measured yet. The current 95% C.L. upper limit B(B d → µ + µ − ) < 1.9 × 10 −10 [5] is still above the SM prediction at B(B d → µ + µ − ) SM = (1.03 ± 0.05) × 10 −10 [7]. A more exquisite probe to hunt for NP is provided by lepton flavor violating (LFV) decays of B mesons. Since LFV phenomena is practically absent in SM, any experimental detection will be unambiguous signals for NP. BaBar, Belle, and more recently LHCb have searched for LFV B decays; no evidence so far has been observed, which provide stringent limits. In Table I we list current bounds, together with projected sensitivities in the near future on promising LFV B decays. * In this article, we explore the possible size of LFV B decays associated with b → q (q = s, d) in one of the simplest extensions of SM, the general two Higgs doublet model (g2HDM) [21] (for a review of 2HDMs, see [22]), sometimes denoted as 2HDM Type III [23], where the Lagrangian itself contains flavor changing neutral couplings of exotic scalar bosons, denoted as ρ ij (which are defined in Eq. (3) later). In our study, we investigate two very different parameter space choices motivated by different phenomenological reasons. In the first case, the NP Yukawa matrices are somewhat SM-like in strength; the largest coupling, just as in SM, is the top-related diagonal coupling ρ tt λ t , where λ t = √ 2m t /v is the SM top Yukawa coupling. Another assumption, supported by experiment [24], is of small but finite mixing, denoted as c γ (≡ cos γ), between CP even scalars in the model (alignment). It was shown [25] that lepton-related NP couplings ρ are constrained to be small, due to bounds from h → decay and µ → e, τ → µ LFV processes. Driven by ρ tt (or ρ tc ), this scenario can realize electroweak baryogenesis (EWBG) for explaining the baryon asymmetry of the Universe(BAU) [26,27], providing strong motivation for experimental exploration.

II. NEW YUKAWA INTERACTIONS
Adding a second Higgs doublet to the SM gives four new Higgs bosons, the neutral scalars H, A, and charged Higgs boson H ± in mass basis. In the limit of CP conserving scalar potential, the H (A) boson is CP -even (odd). Due to absence of discrete Z 2 symmetry on Yukawa sector, it is not possible to diagonalize simultaneously the Yukawa matrices associated with the two Higgs doublets. As a result, the Yukawa Lagrangian of g2HDM contains Higgs FCNCs, giving rise to flavor violation at tree-level. Working in the so-called Higgs basis [30][31][32], the Yukawa Lagrangian in g2HDM is given by [33,34], where indices i, j denote the generation of fermion f , Q f the corresponding electric charge, and R(L) = (1 ± γ 5 )/2 are chiral projections. Note that NP Yukawa matrices ρ f are in general not Hermitian, hence elements ρ ij can have arbitrary complex phases.
The presence of tree-level Higgs FCNCs lead to potentially dangerous flavor violating decays of SM Higgs boson, h → f i f j (i = j), which are severely bound by experiments. To evade such constraints, usually some Z 2 symmetry is imposed on NP Yukawa sector to enforce the natural flavor conservation condition [35]. However, the vertex hf i f j in g2HDM is proportional to the mixing angle c γ , therefore with suppression due to sufficiently small c γ , as hinted by current Higgs data [24], the mere existence of Higgs FCNCs in g2HDM is not directly a cause of concern. But, of course, the strength of Higgs FCNC couplings will be determined by data.
The scalar potential can be found, e.g., in Ref. [36]. For our study, besides Eq. (3), we only need physical H, A, and H + masses as benchmarks. We focus on sub-TeV masses in range of [300, 500] GeV and take m A = m H + , usually adopted † to evade constraint from T parameter (constraints from S and U parameters are easily satisfied [42,43]), where the formula for T in g2HDM is given in appendix A 1. In addition to oblique parameters, the parameter space considered in Sec. III and IV satisfy [29,44] perturbativity, unitarity, and positivity.

III. CASE I: TOP YUKAWA DOMINANCE
We assume that NP top coupling ρ tt ∼ λ t is the largest coupling. This assumption finds support also from the Cheng-Sher ansatz [45], ρ ij ∝ √ m i m j /v, frequently employed to control tree-level Higgs FCNC. But as mentioned in the preceding section, small c γ can tackle the issue of Higgs FCNC, so we do not quite follow the ansatz. We take c γ ∼ 0.1 as sample value, but note that due to ρ tt being the dominant quark coupling, the main g2HDM contribution to b → q ( ) processes are induced by H + interactions that do not depend on c γ . The c γ value is relevant for constraints from h → . The leading flavor constraints on ρ tt are from B physics discussed later in detail. For LFV B decays, we also need to determine the strength of lepton couplings ρ . We have discussed previously [25] the allowed strength of ρ for large ρ tt , so let us give a brief summary. For finite c γ , h → provide important constraints on ρ , independent of ρ tt . For example, the current upper limit on h → τ µ from CMS, based on full Run 2 data [46], can be evaded, µ-τ couplings together with sizable ρ tt unavoidably generate τ → µγ at two-loop via Barr-Zee diagrams [47]. The recently updated bound of τ → µγ < 4.2 × 10 −8 [48] from Belle again gives ρ τ µ ∼ O(λ τ ) for ρ tt ∼ λ t and m H,A ∼ 300 GeV.
With discussion as delineated above, we take the following structure for NP lepton Yukawa matrix ρ [25], which will be our working assumption in estimating rates of LFV B decays in Case I. The NP contribution to b → q ( ) due to ρ tt arise from one-loop diagrams shown in Fig. 1, where the Z-penguin diagram dominates. The diagrams generate the following effective Hamiltonian, contributing to both axial and vector coefficients, while γ-penguin contributes only to vector coefficient, where x t = m 2 t /m 2 H + , s W is Weinberg angle and G γ,Z (x) are t-H + loop functions given in appendix A 5. Note that contributions in Eq. (7) and (8) are universal to all lepton flavors due to SM vertex on lepton end, and therefore only affect lepton flavor conserving B decays.
The box diagram in Fig. 1, however, does depend on lepton flavor and will contribute to LFV B decays (box diagram with W + -H + in the loop depends on the downtype couplings and therefore does not contribute). The corresponding Wilson coefficients are given by [58], Before discussing numerical results, let us discuss briefly constraints on ρ tt from the flavor conserving decays B q → µµ and neutral B q mixings, which will help us determine the upper limit of ρ tt from data for a given value of m H + . For numerical analysis, we use the open-source packages Flavio [59] and Wilson [60] for calculating flavor observables and QCD running of Wilson coefficients from NP scale to physical low-energy processes.
In Fig. 2, we give the region ruled out at 95% C.L. in ρ tt -m H + plane for both LHCb [3,4] (orange) and CMS [5] (green) measurements of B s → µµ. It is interesting to note that, though the central value of the latest CMS measurement is closer to the SM prediction of (3.66 ± 0.14) × 10 −9 , the resulting constraint on ρ tt is weaker compared to that from LHCb. This is because only the axial vector coefficient C Z A modifies B s → µµ (see appendix A 3) with the following correction, with G Z (x) < 0. Since there is no sensitivity to arg ρ tt , B s → µµ rate can only be enhanced, so the central value of CMS being on the higher side of SM allows for relaxed constraint on ρ tt . Note also that, since Eq. (10) 200 400 600 800 does not depend on CKM elements, the result holds true for B d → as well. The current measurements for B q → ee, τ τ [24] are rather poor, therefore no improved constraint can be obtained. For ρ tt ∼ 0.5 and m H + = 300 GeV, Eq. (10) implies that the rates of all B q → get enhanced by ∼ 10% over their SM value, which fits the rising experimental trend.
A better constraint can be obtained from neutral B q mixings. The ∆B = ∆S = 2 transitions arise from H + box diagrams of Fig. 3, which generate [58], where C HH and with y = m 2 W /m 2 H + , and loop functions f (x), g(x) are given in appendix A 5. Similar expressions for B 0 mixing are obtained by replacing s → d in Eqs. (12) and (13).
The current values of mass differences ∆M q are [24], replacing external fermion lines {bs} → {sd} in Fig. 3, these box diagrams will contribute to neutral kaon mixing and modify mixing parameters ∆M K and ε K [24], but the resulting constraints [62] on ρ tt are not competitive with B s mixing.
We find that ρ tt ∼ 0.5 is a reasonable choice with scalar mass spectrum in [300, 500] GeV range. With lepton couplings from Eq. (5), we can now estimate various LFV B branching ratios. As discussed, both Zand γ-penguins preserve lepton flavor, and only box diagrams of Fig. (1) contribute. But this contribution is rather suppressed by small ρ . For ρ tt = 0.5 and m H + = 300 GeV, we find, with B q → eτ, eµ further suppressed due to smaller electron Yukawa couplings. We therefore find that LFV B decays will be far below future sensitivities in Table I in g2HDM for Case I, with semileptonic decays B → (K, π) analogously suppressed.

IV. CASE II: µ-τ YUKAWA DOMINANCE
The weakness of ρ in Eq. (5), together with GIM suppression, do not allow large LFV effects in Case I, but instead constrains ρ tt through B s → µµ and B s mixing. Now we explore a scenario where the LFV µ-τ couplings can be sizable. One motivation for such parameter space is to address the disagreement between SM prediction and experimental measurement of the anomalous magnetic moment of the muon, a µ = (g − 2) µ /2.
Recently, the Fermilab Muon g-2 experiment [28] reported its first measurement of a µ . Combined with the previous result of Brookhaven [63], the result of a Exp µ = 116592061(41) × 10 −11 [28] compared to the theory consensus value of a SM µ = 116591810(43)×10 −11  is larger by more than 4σ [28] (17) † The SM prediction of (g − 2)µ based on the recent lattice results [86][87][88][89][90] is closer to the experimental value. However, the low energy data on σ(e + e − → hadrons) [91][92][93] show tension with these lattice results, which calls for further investigation. In this paper we will take Eq. (17) as evidence of NP. The difference can be explained in g2HDM via one-loop diagram ‡ given in Fig. 4, which in the limit of c γ → 0 gives the following NP correction [43,96,97], for each φ = H, A. The total contribution is ∆a µ = (∆a µ ) H − (∆a µ ) A as H and A effects are opposite in sign. Therefore, to obtain a finite ∆a µ , H and A must be nondegenerate: ∆m = m A − m H = 0. To present our numerical results, we follow Ref. [29] and assume H to be lighter, setting m H = 300 GeV. For ∆m, we take two choices for illustration: 40 and 200 GeV. The small ∆m = 40 GeV implies large cancellation between H and A contributions, and therefore a larger value of ρ τ µ ∼ 30λ τ (we implicitly assume ρ τ µ = ρ µτ ) is required to account for difference in Eq. (17) within 1σ solution. For larger ∆m case, cancellation between H and A becomes mute since effect of heavy A starts to decouple, and one only needs a smaller ρ τ µ = ρ µτ ∼ 20λ τ .
With strength of ρ τ µ more than an order larger than Case I (compare Eq. (5)), the experimental bound of Eq. (4) implies c γ ∼ 0.005 or smaller. We therefore set c γ = 0 to simplify (which would demand some yet unknown symmetry). Another important implication of large ρ τ µ is smallness of ρ tt [29], because of bound from τ → µγ [48]. In fact, a more stringent constraint on ρ tt can be set [29] by the collider search for gg → H, A → τ µ [98]. Therefore, if muon g − 2 arises from one-loop in g2HDM, ρ tt is unavoidably small.
One possibility for enhancing LFV B decays in g2HDM is to allow the ρ d Yukawa matrix to be nondiagonal. Explicitly, if one allows for finite ρ bq and ρ qb for q = s, d, b → q ( ) is at tree level while B q → do not suffer helicity suppression. The effective Hamiltonian is, where O S = (sRb)(¯ ), O P = (sRb)(¯ γ 5 ), and O S,P are obtained by exchanging L ↔ R. The scalar Wilson ‡ In Case I, ρtt together with ρµµ can contribute to aµ at two-loop, but the contribution to ∆aµ is small [29], due to constraint from gg → H/A → µµ [94,95] direct search.
coefficients at NP scale in alignment limit are given by, Note that under ρ = ρ condition, for each C ( ) S,P the H and A contributions are not simultaneously present.
That down-type couplings ρ bq , ρ qb can be finite and allowed from various flavor and collider constraints has been discussed [99] for h → bq decays in g2HDM. The most important constraints on ρ bq couplings come from neutral B q mixings, which are now induced at treelevel. Ref. [99] pointed out an effective mechanism where if one imposes the conditions ρ bq ρ qb = 0 and NP effects in B q mixing can be easily evaded. However, note that the latter condition in the alignment limit implies m A = m H , which would rule out the possibility to explain the muon g − 2 anomaly [Eq. (17)]. Therefore, one must confront B q mixing constraints in scenarios with ∆m = 0.
The couplings ρ bq , ρ qb via tree-level H/A exchange generate the effective Hamiltonian, where the Wilson coefficients at NP scale in alignment limit are given by, with C 2 obtained after substituting ρ * bs → ρ sb in C 2 . In Fig. 5, we show the 95% C.L. allowed region by ∆M s measurement [Eq. (14)] for ∆m = 40 GeV (blue) and 200 GeV (orange). One sees that, so long the product ρ bs ρ sb is very small (but finite), constraints on individual couplings ρ bs or ρ sb can be evaded (similar results follow for ρ bd , ρ db from B d mixing). Note, however, that if either ρ bs or ρ sb is zero, i.e. with ρ bs ρ sb exactly zero, then the size of the other coupling is severely constrained and cannot be larger than O(10 −3 ). Another important probe for Case II comes from H +induced processes. With lepton couplings fixed by 1σ solution to muon g − 2, ρ qb and ρ bq couplings contribute to leptonic decays such as M + → + ν via tree-level H + exchange, where M = B, K, π, described by the effective Hamiltonian, which modifies the branching ratios as follows [100], where m M is the mass of meson M , and quark masses are evaluated at NP scale to account for renormalization group running. In Eq. (25), neutrino species are summed over, since neutrino flavor is not detected by experiment (the earlier work of Ref. [101] contains an error here). Decays B → µν and B → τ ν provide important constraints on the coupling products ρ qb ρ τ µ and ρ qb ρ µτ , respectively. Adapting Eq. (25) for B → µν, one notes that the SM-NP interference term (for = µ) involves the coupling ρ µµ , which is strongly constrained by τ → µµµ [51]. Similarly, in case of B → τ ν, the SMinterference term involves the coupling ρ τ τ , which gets constrained by τ → µγ [51]. We therefore ignore the SM-NP interference term and focus on contributions of the coupling product ρ qb ρ τ µ (ρ qb ρ µτ ), which contribute through the incoherent term in B → µν (B → τ ν).
It is worth mentioning that the ratio R µτ B in 2HDM Type-II (such as in minimal supersymmetric models) is lepton flavor independent and therefore remains the same as in SM. Therefore, the ratio R µτ B is one of the most important observables to probe genuine NP effects of g2HDM couplings [100]. We mention in passing that constraints from other decays such as K, D → µν and τ → (K, π)ν do not impose any significant bounds. Before presenting our results, we mention few important collider probes of Scenario II. As noted in Ref. [99], if quark coupling ρ bs is large then pseudoscalar A produced via strange-quark sea, i.e., sg → bA, followed by A → bs is one of the best channel to search for. However, ρ bs ∼ O(10 −3 ) is very small in our setup. But lepton couplings ρ τ µ = ρ µτ are quite large. Then exotic scalars H, A can be probed with 4-lepton final state (especially, the same-sign dimuon and same-sign ditau) via electroweak scalar pair production: qq → AH → µ ± µ ± τ ∓ τ ∓ , as pointed out in Ref. [103]. If the scalar pair is HH + , AH + (or H + H − ) then 3-lepton plus neutrino (2-lepton plus 2 neutrinos) are also channels to search for (see Ref. [103] for detail). Another potential channel could be bs → H, A → µτ . Due to very small ρ bs the production cross-section of H, A at LHC is expected to be small, but given that strange quark is involved, and that we also have sizable ρ τ µ = ρ µτ , it is not clear if this constraint can be ignored. We leave a detailed analysis of collider signatures of scenario II as future work.
In Fig. 6, we present various LFV B decays as functions of ρ τ µ for a range of ρ bq values, while setting ρ qb = 0. The upper (lower) row shows results for b → sτ µ (b → dτ µ) related decays. Note that decays B q → depend on the difference C i −C i , i = S, P scalar Wilson coefficients, while semileptonic decays B → (K, π) depend on the sum C i + C i (see appendix A 3). But since C S,P vanish because ρ qb = 0, only C S,P contribute. Thus, B q → and B → (K, π) rates are correlated, as reflected in Fig. 6, where both modes are indicated for the y axis. The dark green band in each plot corresponds to the region ruled out by current leptonic bounds. The light green region will be probed in the near future, according to Table I. * We have not shown experimental sensitivity of semileptonic decays as leptonic modes appear to be the leading probe. The plots show that, for ρ bq ∼ O(10 −3 ) and ρ τ µ ∼ O(20)λ τ as motivated by the muon g − 2 anomaly, large rates of LFV B decays are possible and within reach of future searches. One also notes from Fig. 6 that scenarios with smaller mass splitting ∆m < 200 GeV have better prospects for discovery, although the needed ρ τ µ value is larger, hence somewhat less attractive.
Let us now briefly comment on τ -e and µ-e sectors. The B decays with τ -e flavor violation involve ρ τ e and ρ eτ couplings, but µ → eγ puts a strong bound on them. The corresponding contribution to µ → eγ is generated by a diagram similar to Fig. 4, but with outgoing fermion replaced by electron. For values of ρ τ µ , ρ µτ that explain * For Bs → τ ± µ ∓ , due to lack of public results, we have conservatively assumed that future measurements can improve current limit at least by a factor of 2. the (g − 2) µ anomaly, the MEG bound of µ → eγ < 4.2 × 10 −13 [50] would imply ρ τ e = ρ eτ O(λ e ) [51], which is quite severe. Therefore, to avoid the charged LFV constraint, we take ρ τ e = ρ eτ ∼ λ e . Then predictions with ρ bq = 10 − We find the coupling of the µ-e sector only weakly constrained in Case II by charged LFV processes. The couplings ρ µτ , ρ τ µ together with ρ µe , ρ eµ contribute to τ → eγ via diagrams similar to Fig. 4, after replacing initial and final fermions by τ and e and internal fermion by µ. But the diagram is chirally suppressed by small m µ . Taking same mass as before and ρ eµ = ρ µe , the current measurement of B(τ → eγ) = 3.3 × 10 −8 [49] sets the bound ρ τ µ ρ µe (5 × 10 5 )λ τ λ e , which is quite poor. Note that τ − → µ − e + µ − gives better constraint, as this decay is mediated by tree-level H, A exchange hence does not suffer chiral suppression. Adapting the formula of τ − → µ − µ + µ − given in Ref. [101] to τ − → µ − e + µ − , we find an order of magnitude improvement in constraint on ρ τ µ ρ µe compared to τ → eγ. Then taking ρ eµ = ρ µe ∼

V. SUMMARY
We have explored prospects of enhanced lepton flavor violation in B decays in g2HDM with sub-TeV exotic scalars. We focus on two different cases of parameter space. For Case I, we assume the top Yukawa coupling ρ tt is the dominant quark coupling and take c γ ∼ 0.1. Then charged LFV processes τ → µγ and µ → eγ constrain ρ to Eq. (5). Even with O(1) strength of ρ tt , LFV B decay rates are highly suppressed by small ρ , and are far from the sensitivities of upcoming LHCb Upgrade II and Belle II. For Case II where µ-τ flavor violating lepton couplings are motivated to be about 20-30 times larger than SM Yukawa λ τ 0.01. Contrary to Case I, one finds that ρ tt has to be small due to bounds from τ → µγ and gg → H, A → τ µ direct search at LHC. However, together with ρ bs , ρ bd as small as O(10 −3 ), we find that Case II allows substantial rates of µ-τ flavor violation in B decays, with B mixing and B → µν being the leading, but forgiving, flavor constraints.
Concerning τ -e flavor violation in B decays, we find that µ → eγ would make it difficult to have simultaneously large ρ τ e , making B q → τ e and B → (K, π)τ e rates too small to be probed at upcoming experiments. Furthermore, we find that current constraints on µ-e flavor violating coupling are not that severe. The future measurements of LFV B decays related to b → qµe will provide crucial constraint on g2HDM.
Acknowledgments This work is supported by NSTC 111-2639-M-002-002-ASP of Taiwan, and NTU grants 111L104019 and 111L894801. The parameter in g2HDM is defined as [42,43] where function F (a, b) is given by which vanishes in the limit a → b.

h → ( )
The tree-level decay rate for h → ( = ) is, The effective Hamiltonian for b → q ( ) is, where primed counterparts are obtained by L → R exchange. The full b → q operator basis can be found, e.g., in Refs. [104][105][106]. One should take note of the normalization used in H eff in Eq. (A5) when comparing with H eff defined in Eq. (6) and Eq. (19) in the main text.

b → sγ
The H + induced dipole coefficients C 7 and C 8 mediating b → sγ and b → sg in g2HDM are given by, where the loop functions F 7(8) (x) are in the notation of Ref. [109] (originally calculated in Ref. [110]) and provided in the next appendix.