RK and RK* in an Aligned 2HDM with Right-Handed Neutrinos

PHYSICAL REVIEW D 101, 115009 (2020)

RK and RK in an aligned 2HDM with right-handed neutrinos

† ‡ Luigi Delle Rose,1,2,* Shaaban Khalil,3, Simon J. D. King ,2,4, and Stefano Moretti2,5,§ 1INFN, Sezione di Firenze, and Dipartimento di Fisica ed Astronomia, Universit`a di Firenze, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy 2School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom 3Center for Fundamental Physics, Zewail City of Science and Technology, 6 October City, Giza 12588, Egypt 4INFN, Sezione di Padova, and Dipartimento di Fisica ed Astronomia G. Galilei, Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy 5Particle Physics Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom

(Received 3 September 2019; accepted 21 May 2020; published 4 June 2020)

We consider the possibility of explaining the recent RK and RK anomalies in a two-Higgs doublet model, known as aligned, combined with a low-scale seesaw mechanism generating light neutrino masses and mixings. In this class of models, a large Yukawa coupling allows for significant nonuniversal leptonic contributions, through box diagrams mediated by charged Higgs bosons and right-handed neutrinos, to the þ − b → sl l transition that can then account for both RK and RK anomalies.

DOI: 10.1103/PhysRevD.101.115009

I. INTRODUCTION Therefore, the RK and RK anomalies attracted the attention of theoretical particle physicists, and several new physics Recently, the LHCb Collaboration announced intriguing þ → þμþμ− scenarios have been proposed to accommodate these results [1,2] for the ratios RK ¼ BRðB K Þ= – þ þ þ − 0 0 þ − results; see Refs. [4 120]. B → K e e R B → K μ μ = BRð Þ and K ¼ BRð Þ A nontrivial flavor structure in the lepton sector is 0 → 0 þ − BRðB K e e Þ. In fact, it was reported that for two already required, beyond the SM, in order to explain the dilepton invariant mass-squared bins, RK and RK , are observation of neutrino oscillations. Therefore, it is plau- given by sible and very attractive if the mechanism behind the B anomalies can be related to the same physics responsible 0 846þ0.060þ0.016 1 1 2 ≤ 2 ≤ 6 2 RK ¼ . −0.054−0.014 for . GeV q GeV ; for the nonzero neutrino masses and oscillations. In view of 0 66þ0.11 0 03 0 045 2 ≤ 2 ≤ 1 1 2 this, one then ought to explore extensions of the SM able to . −0.07 . for . GeV q . GeV R ¼ address both phenomena. K 0 69þ0.11 0 05 1 1 2 ≤ 2 ≤ 6 2 . −0.07 . for . GeV q GeV : One of the possible scenarios for generating the observed ð1Þ lepton nonuniversality is to allow for large Yukawa couplings of the right-handed neutrinos with Higgs fields These measurements contradict the standard model (SM) and charged leptons, as in a low-scale seesaw mechanism, SM ≃ SM ≃ 1 ≈2 5σ with the inverse seesaw being one of the most notorious expectations, RK RK [3],bya . deviation. Hence, they are considered as important hints of new examples, for generating light neutrino masses. In this case, a nontrivial neutrino Yukawa matrix may lead to different physics that violates lepton universality. − − Any signal of possible lepton nonuniversality would be results for BRðB → Keþe Þ and BRðB → Kμþμ Þ. This striking evidence for physics beyond the SM (BSM). framework has been recently considered in the super- symmetric (SUSY) B − L extension of the SM, where it was shown that the box diagram mediated by a right- *[email protected] † [email protected] handed sneutrino, Higgsino-like chargino, and light stop ‡ [email protected] can account simultaneously for both RK and RK [121].In §[email protected] non-SUSY models, a similar box diagram can be obtained through a charged Higgs, instead of a chargino, and a right- Published by the American Physical Society under the terms of handed neutrino, instead of a right-handed sneutrino. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Therefore, a rather minimal model that can account for the author(s) and the published article’s title, journal citation, these discrepancies is an extension of the SM with two- and DOI. Funded by SCOAP3. Higgs doublets (so that we can have a physical charged

2470-0010=2020=101(11)=115009(14) 115009-1 Published by the American Physical Society ROSE, KHALIL, KING, and MORETTI PHYS. REV. D 101, 115009 (2020)

Higgs boson) and right-handed neutrinos along with a low- II. A2HDM scale seesaw mechanism (to guarantee large neutrino The 2HDM is characterized by two Higgs doublets with Yukawa couplings). For earlier works in the context of hypercharge Y ¼ 1=2, which, in the Higgs basis, can be two-Higgs doublet models with or without extra neutral parametrized as leptons, see, for instance, Refs. [101,122–125]. Note that there have been several attempts at explaining the above Gþ Hþ results through the penguin diagram as well, with a Φ1 ¼ 1 0 0 ; Φ2 ¼ 1 0 0 ; ð2Þ pffiffi ϕ pffiffi ϕ ϕ nonuniversal Z0 (see the previous list for examples, 2ðvþ 1 þiG Þ 2ð 2 þi 3Þ Refs. [4–120]) and also through tree-level mediation of a flavor-violating Z0 or leptoquark that induces a nonuniver- where v is the electroweak (EW) vacuum expectation value 0 sal b → slþl− transition. and Gþ and G denote the Goldstone bosons. The two The two-Higgs doublet model (2HDM), which is moti- doublets describe five physical scalar degrees of freedom vated by SUSY and grand unified theories, is the simplest which are given by the two components of the charged φ0 model that includes charged Higgs bosons. According to Higgs H and three neutral states i ¼fh; H; Ag, the latter ϕ0 the types of couplings of the two-Higgs doublets to the SM obtained from the rotation of the i fields into the mass fermions doublets and singlets, we may have a different M2 eigenstate basis. The scalar squared mass matrix S is type of 2HDMs. For example, if only one Higgs doublet determined by the structure of the 2HDM scalar potential, couples to the SM fermions, one obtains the type I 2HDM, see, for instance, Refs. [126–128], and diagonalized by the while in the case of one Higgs doublet coupling to the up orthogonal matrix R, where quarks and the second Higgs doublet coupling to the down quarks and charged leptons, one obtains the type II 2HDM. RM2RT 2 2 2 φ0 R ϕ0 S ¼ diagðMh;MH;MAÞ; i ¼ ij i : ð3Þ Also, we may have a type III or IV 2HDM if both Higgs doublets couple to both up and down quarks as well as φ0 In general, the three mass eigenstates i do not have charged leptons. However, severe constraints are imposed definite CP transformation properties, but in the CP- on 2HDMs due to their large contributions to flavor 0 conserving scenario, ϕ3 does not mix with the other two changing neutral currents (FCNCs) that contradict the neutral states, and the scalar spectrum consists of a CP-odd current experimental limits. Therefore, assumptions on 0 field A ¼ ϕ3 and two CP-even fields h and H that are the Yukawa couplings generated by different Higgs dou- defined from the interaction eigenstates through the two- blets are usually imposed. One of these assumptions is the dimensional orthogonal matrix alignment between the Yukawa couplings generated by Φ1 and Φ2, the two Higgs doublet fields. This class of models 0 h cos α sin α ϕ1 is called the aligned 2HDM (A2HDM). It is worth ¼ : ð4Þ − α α ϕ0 mentioning that, as discussed below, other 2HDMs cannot H sin cos 2 account for the LHCb results of lepton nonuniversality. In this paper, we emphasize that in the A2HDM, The most general Yukawa Lagrangian of the 2HDM can extended by (heavy) right-handed neutrinos in order to be written in the Higgs basis as generate the (light) neutrino masses through a seesaw −L ¯ 0 0 Φ 0 Φ 0 ¯ 0 0 Φ˜ 0 Φ˜ 0 mechanism, interesting results can be obtained for several Y ¼ QL ðY1d 1 þ Y2d 2ÞdR þ QL ðY1u 1 þ Y2u 2ÞuR flavor observables, such as μ → eγ; B → μμ; and, indeed, ¯ 0 0 Φ 0 Φ l0 ¯ 0 0 Φ˜ 0 Φ˜ ν0 s þ LLðY1l 1 þ Y2l 2Þ R þ LLðY1ν 1 þ Y2ν 2Þ R R ðÞ . In particular, one can account simultaneously for K þ H:c:; ð5Þ both aforementioned results on RK and RK , through a box diagram mediated by a right-handed neutrino, top quark, 0 0 0 0 l0 ν0 and charged Higgs boson. where the quark QL;uR;dR and lepton LR; R; R fields are The paper is organized as follows. In Sec. II,we defined in the weak interaction basis and we also included the couplings of the left-handed lepton doublets with introduce the main features of our A2HDM focusing on Φ the neutrino sector and its interplay with the Higgs the right-handed neutrinos. The 1;2 fields are the two structures. (Here, we also briefly review some particular Higgs doublets in the Higgs basis, and, as customary, Φ˜ σ2Φ 0 0 realizations of low-scale seesaw models for generating light i ¼ i i . The Yukawa couplings Y1j and Y2j, with 0 0 neutrino masses.) In Sec. III, we discuss the most relevant j ¼ u; d; l, are 3 × 3 complex matrices, while Y1ν and Y2ν constraints from flavor physics that affect our A2HDM are 3 × nR matrices, with nR being the number of right- 0 0 parameter space. The calculation of the A2HDM contri- handed neutrinos. In general, the Yukawas Y1 and Y2 butions to b → slþl− transitions mediated by charged cannot be simultaneously diagonalized in flavor space, so 0 Higgs bosons and right-handed neutrinos is given in while the quark and the charged-lepton Y1 can be recast into a diagonal form in the fermion mass eigenstate basis, Sec. IV. Our numerical results are presented in Sec. V. pffiffiffi Finally, our conclusions and remarks are given in Sec. IV. namely, Y1 ¼ 2=vM, with M being the fermion mass

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matrix, Y2 would remain nondiagonal and thus give rise to TABLE I. Relation between the ζf couplings of the A2HDM potentially dangerous tree-level FCNCs. This problem is and the ones of the Z2 symmetric scenarios. usually solved by enforcing that only one of the two Higgs Aligned Type I Type II Type III Type IV doublets couple to a given right-handed field. This require- ζ β β β β ment is satisfied by implementing a discrete Z2 symmetry u cot cot cot cot ζ β − β − β β acting on the Higgs and fermion fields. There are four d cot tan tan cot ζ β − β β − β nonequivalent choices: types I, II, III, and IV (as previously l cot tan cot tan intimated). Another general way to avoid tree-level FCNCs in the Higgs sector is to require the alignment, in flavor 0 space, of the two Yukawa matrices that couple to the same supplemented by a Majorana mass term MR for the right- right-handed fermion [129], namely, handed ones,

Y2 ¼ ζ Y1 ≡ ζ Y ;Y2 ¼ ζ Y1 ≡ ζ Y ; 1 ;d d ;d d d ;u u ;u u u −L ¼ ν0 TCM0 ν0 þ H:c:; ð8Þ MR 2 R R R Y2;l ¼ ζlY1;l ≡ ζlYl; ð6Þ where C is the charge-conjugation operator. In particular, where the proportionality constants ζf are arbitrary family universal complex parameters. This scenario is dubbed by exploiting a biunitary transformation in the charged- A2HDM. The allowed sources of FCNCs at quantum level lepton sector and a unitary transformation on the right- 0 l0 l l are highly constrained, and the resulting structures are handed neutrinos LL ¼ ULLL, R ¼ UR R, and ν0 ν ν functions of the mass matrices and Cabibbo-Kobayashi- R ¼ UR R, it is always possible to diagonalize (with real Maskawa matrix elements, so this model provides an eigenvalues) the charged-lepton and Majorana mass matri- explicit example of the popular minimal flavor violation ces at the same time, scenario [130]. pffiffiffi Even though the alignment of the Yukawa matrices is 2 † 0 e U Y U ¼ Yl ≡ diagðm ;mμ;mτÞ; strictly required, from observations, only in the quark and L l R v e charged-lepton sectors, we assume that the same mecha- ν T 0 ν U M U ¼ M ≡ diagðM1; …M Þ; ð9Þ nism which guarantees the aligned structure in the SM R R R R nR flavor space also holds in the neutrino sector and leads to † 0 ν while Yν ¼ ULYνUR remains nondiagonal. In this basis, Y2;ν ¼ ζνY1;ν ≡ ζνYν: ð7Þ the neutrino mass matrix can be written as

In all sectors, the alignment is fixed to be exact at some 1 T μ −LM ¼ N CMN þ H:c: specified scale 0 and subsequently will misalign due to ν 2 L L radiative corrections, as discussed in Refs. [131,132]. 1 0 MD νL However, the flavor structure of the model constrains the ¼ ðνT νcTÞC ; ð10Þ 2 L R T νc nature of the new sources of FCNCs induced by renorm- MD MR R alization group effects. Quantitatively, in the quark sector, vﬃﬃ these FCNC contributions are suppressed by mass hier- with MD ¼ p Yν being the neutrino Dirac mass. This can 2 3 2 archies mqm 0 =v and provide negligible effects [131,132]. q be diagonalized with the unitary ð3 þ nRÞ × ð3 þ nRÞ We will not consider the impact of the misalignment in matrix U, this work. Interestingly, ζf can provide new sources of CP viola- νL νl ULl ULh νl tion, but in this work, we will consider only real values. ¼ U ≡ ; ð11Þ νc ν ν Notice also that the usual 2HDMs in which tree-level R h URcl URch h FCNCs are removed by exploiting the discussed Z2 discrete T symmetry, namely, the types I, II, III, and IV, can be such that Mν ¼ U MU provides the masses of the three recovered for particular values of the proportionality light active neutrinos νl and of the remaining nR heavy ζ constants f as shown in Table I. sterile neutrinos νh. The Yukawa Lagrangian in Eq. (5) generates a Dirac The Yukawa interactions of the physical scalars with the mass matrix for the standard neutrinos and can also be mass eigenstate fermions are then described by

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pffiffiffi 2 † † −L ¼ ½u¯ð−ζ m V P þ ζ V m P Þd þ ν¯ ð−ζνmν U P þ ζlU mlP Þl Y v u u ud L d ud d R l l Ll L Ll R ν¯ −ζ † ζ † l þ þ hð νmνh ULhPL þ lULhmlPRÞ H þ H:c: 1 X X 1 X i 0 ¯ i 0 † † c c þ ξ φ fm P f þ ξνφ ðν¯ U þ ν¯ U ÞP ðU mν ν þ U mν ν ÞþH:c:; ð12Þ v f i f R v i l Ll h Lh R Ll l l Lh h h i f¼u;d;l i where the couplings of the neutral Higgs states to the structure. To match the nomenclature usually employed in fermions are given by the literature, we split the set of right-handed neutrinos defined above into two classes, one of nS SM-singlet i i ξu;ν ¼ Ri1 þðRi2 − iRi3Þζu; fermionic fields S and another one of fermionic states i that we continue to call right-handed neutrinos. The ξ l ¼ R 1 þðR 2 þ iR 3Þζ l: ð13Þ d; i i i d; differences between the two will be clear in a moment. ν νc T Because of the alignment of the Yukawa matrices, all the In the basis NL ¼ð L; R;SÞ , the mass matrix can be couplings of the scalar fields to fermions are proportional to parametrized as the corresponding mass matrices. Finally, the weak neutral 0 1 0 and charged interactions of the neutrinos are mD mS B C M T ¼ @ mD mN mR A; ð15Þ g † † μ L ¼ ðν¯ U þ ν¯ U Þγ ðU ν þ U ν ÞZμ; T T Z l Ll h Lh Ll l Lh h m m μS 2 cos θW S R g † † μ þ L ¼ − pffiffiffi ½ðν¯ U þ ν¯ U Þγ P lWμ þ H:c: ð14Þ W 2 l Ll h Lh L which has the same structure as the one given in Eq. (10) provided that In this paper, rather than presenting a complete model in the neutrino sector by specifying the structure and the mN mR M ≡ ðm ;m Þ;M≡ : ð16Þ hierarchies of the neutrino mass matrices, we work in a D D S R T m μS simplified framework that captures the interesting phenom- R enology while preserving a significant degree of model The mD and mS are, respectively, 3 × nR and 3 × nS mass independence. In particular, we consider a single extra matrices mediating the interactions between the charged heavy neutrino despite the usual requirement of additional leptons and the right-handed and sterile neutrinos, while sterile states to fully accommodate the observed pattern of m , m , and μ are n × n , n × n , and n × n mass the light neutrino masses and mixing angles. Indeed, low- N R S R R R S S S matrices, respectively. As both right-handed ν and sterile scale right-handed neutrinos with sizeable mixings with the R S neutrinos can be assigned lepton number L ¼ 1, the mass SM left-handed neutrino states, such that they may provide terms m , m , and μ violate lepton number by two units. visible effects in physical observables at the EW scale, N S S Two commonly studied mass patterns are the inverse and usually affect the light neutrino masses with inadmissible linear seesaws, which are characterized by m ¼ m ¼ 0 large contributions. This issue is nicely solved in extended S N and μ ¼ m ¼ 0, respectively. In these cases, the vanish- seesaw models [133–141], as, for example, the linear or S N ing of the Majorana mass μ or m would restore lepton inverse seesaw mechanism, in which extra sterile neutrino S S number conservation and, as such, would increase the states are introduced to allow for large mixings while symmetry of the model. This feature makes the two masses correctly reproducing the observed smallness of the light naturally small accordingly to the ’t Hooft naturalness neutrino masses. For the sake of definiteness, in the following section, we principle. Following the standard seesaw calculation and by briefly present some specific setup that could be employed μ ≪ to realize the phenomenological scenario described above. assuming the hierarchy SðmSÞ mD;mR for the inverse (linear) seesaw scenario, the 3 × 3 light neutrino mass A. Some explicit examples of low-scale matrix is seesaw mechanism m ðmT Þ−1μ m−1mT inverseseesaw As stated above, in order to achieve a sizeable mixing m ≃ D R S R D 17 light −1 T T −1 T ð Þ with the heavy sterile neutrino while avoiding, at the same mSmR mD þmDðmRÞ mS linear seesaw; time, large contributions to the light neutrino masses, it is necessary to require the neutrino Majorana mass matrix and which is diagonalized by the Pontecorvo-Maki-Nakagawa- the Dirac neutrino Yukawa coupling to realize a particular Sakata (PMNS) matrix UPMNS, namely,

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T ≡ 1. Neutrino mass squared differences UPMNS mlightUPMNS ¼ mν diagðmν1 ;mν2 ;mν3 Þ: ð18Þ The 3σ confidence-level (CL) ranges on the mass Differently from the standard type I seesaw case, in squared differences ∼ 2 ≪ which mν mD=mR with mD mR, the lightness of the active neutrino masses is ensured in these low-scale seesaw 2 −5 2 Δm21 ¼ð6.80 → 8.02Þ× 10 eV scenarios by the smallness of the μS ðmSÞ parameters. This 2.399 → 2.593 × 10−3 eV2 for l 1 NO feature prevents mD from being extremely suppressed with 2 ð Þ ð ¼ Þ Δm3 ¼ respect to the Majorana mass and, as such, may allow for l ð−2.562 → −2.369Þ ×10−3 eV2 ðfor l ¼ 2 IOÞ non-negligible couplings between the heavy neutrinos and ð21Þ the SM gauge bosons, which are set by the mixing ULh. To understand the dependence of the mixing ULh of the left-handed SM neutrinos with the extra sterile states, it is where the first and second possibility refer to the instructive to study the mixing matrix U in the limit of assumption of normal (NO) and inverted ordering (IO) in the light neutrino masses, respectively. negligible μSðmSÞ and small mD=mR. While the first requirement is necessary to reproduce the lightness of the active neutrino states, the latter is used here only to 2. Leptonic mixing matrix simplify the structure of U, which reads as The 3σ CL ranges on the elements of the leptonic mixing matrix UPMNS 0 1 −1 −1 1 1 pffiffi piffiffi 2 mDmR 2 mDmR 2 2 B 1 C m sin θ12 ¼ð0.272 → 0.346Þ 0 pffiffi − piffiffi O D U ¼ @ 2 2 A þ 2 : m ð0.418 → 0.613Þ N:O: −1 1 R 2 −m mT pffiffi piffiffi sin θ23 ¼ R D 2 2 ð0.435 → 0.616Þ I:O: ð19Þ 0 01981 → 0 02436 2 ð . . Þ N:O: sin θ13 ¼ ð22Þ ð0.02006 → 0.02452Þ I:O: Notice that the PMNS matrix has been set to the unit 1, consistently with the approximation mν ≃ 0. From i Notice also that mD and mR cannot be chosen freely since Eq. (19), one can immediately realize that ULh is set, as the ULh block of the mixing matrix that they define is naively expected from dimensional arguments, by the constrained by the unitarity requirement that directly affects m =m ratio D R. the analysis presented in this work. The corresponding Once some specific inputs are provided for m and m , D R bound is discussed in Sec. III together with all the other the corresponding μ m matrix that ensures the agree- Sð SÞ relevant constraints. ment with the light neutrino mass splittings and mixing angles can always be reconstructed from Eq. (17).For instance, in the inverse seesaw case, we find III. RELEVANT PARAMETER SPACE AND CONSTRAINTS μ T −1 † T −1 S ¼ mRmD UPMNSmνUPMNSðmDÞ mR; ð20Þ A. Unitarity bounds on the neutrino mixing matrix 3 3 where mν and U are chosen in agreement with the The × block of the mixing matrix U corresponds to a PMNS ˜ bounds from the low-energy neutrino data, which we report nonunitary UPMNS matrix. The bounds on the deviation ˜ below for the sake of completeness. In particular, one from unitarity of UPMNS have been obtained in should enforce the following constraints from the latest Refs. [143,144] using an effective field theory approach results of the Vfit group [142] extracted from the Vfit 3.2 in which the masses of the heavy neutrinos lie above the (2018) data. EW scale. This constraint can be recast as follows:

XnR XnR ϵ ≡ ≡ δ − ˜ † ˜ αβ UαiUβi ðULhÞαiðULhÞβi ¼j αβ ðUPMNSUPMNSÞαβj; 1 0 i i¼ 1 ð0.9979 → 0.9998Þ <10−5 <0.0021 B C ˜ ˜ † @ 10−5 0 9996 → 1 0 0 0008 A jUPMNSUPMNSj¼ < ð . . Þ < . : ð23Þ <0.0021 <0.0008 ð0.9947 → 1.0Þ

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B. Lepton flavor–violating processes delineating the allowed parameter space spanned by the ζ l couplings and the charged Higgs mass M . We consider the lepton flavor–violating decays lα → u;d; H lβγ induced at one-loop order by the sterile neutrinos and check their compatibility with the experimental upper 1. Neutral meson mixing bounds at 90% CL [145], The meson mixing observables ΔMs, ΔMd, and jϵKj constrain large values of the ζu parameter and smaller BRðμ → eγÞ ≤ 4.2 × 10−13; BRðτ → eγÞ ≤ 3.3 × 10−8; values of mH , due to the box diagram with two charged ∓ BRðτ → μγÞ ≤ 4.4 × 10−8: ð24Þ Higgses, or HW . In turn, they do not significantly affect ζd, the dependence of which dependence is suppressed by 2 2 The branching ratios (BRs) of the aforementioned decay mb=MW in the first two observables, while it does not rates are given by appear at all in the third one. Explicit formulas for the corrections to the SM predictions of these observables in BRðlα → lβγÞ the 2HDM are lengthy and can be found in Ref. [151]. 0 n 2 2 2 Here, we present only the results for ΔM since the B − XR mν mν q q C G hi G hi ¯ 0 ¼ ðULhÞαiðULhÞβi W 2 þ H 2 Bq mixing is the one mostly affecting our parameter space, i¼1 MW MH G2 ð25Þ Δ F 2 2 2 ˆ η ˆ STηST Mq ¼ 2 M MB jVtqV j f ½BB B CV þB CST; 24π W q tb Bq q q Bq Bq with ð28Þ 3 2 4 ˆ ˆ ST α s ml ml where B and B are the bag parameters, while η and C ¼ W W α α ; ð26Þ Bq Bq Bq 256π2 Γ ST MW lα η account for the running of the QCD corrections. Their Bq values can be found in Ref. [151]. The Wilson coefficients where Γl is the total decay width of the lepton lα and the α CV and CST arise, respectively, from the vector operator and loop functions are from a combination of the scalar and tensor operators. Their leading contributions can be parametrized as − 6 2 − 3 3 − 2 4 6 3 G x þ x x x þ x logx W ðxÞ¼ 4 ; 4ðx −1Þ CV ¼ xt½AWW þ 2xtAWH þ xtAHH; 2 2 2 ST ST ζν x −1 x − 2xlogx 4 G G ζ ζ ð þ Þ CST ¼ xbxt ½AWH þ AHH; ð29Þ H ðxÞ¼ 3 W ðxÞþ ν l 2 − 1 3 ; ð27Þ ðx Þ 2 2 with xq ¼ Mq=MW. The term AWW originates from the SM, where we have neglected the mass of the lepton in the final while all the others are induced by the exchange of one or state (for a study of lepton flavor–violating processes in two charged Higgses in the box diagrams. The SM also the context of low-scale seesaw models, see, for instance, contributes to the Wilson coefficient CST but suffers from a Refs. [146–149]). large suppression by the bottom quark mass. In contrast, G G The H can offer large contributions, larger than W , this suppression can be alleviated in the 2HDM by large ζ for sizeable values of the couplings ζν, ζl, which are, as factors of u;d. In our scenario, the main corrections to the such, strongly constrained by lepton flavor–violating proc- SM prediction appear in CV and, in particular, in AWH since þ esses. These can be tamed by controlling the size of the AHH is suppressed by an additional H propagator in the mixing matrix elements ðULhÞαi, which should be highly box diagram. The relevant coefficients are suppressed in order to avoid any large contribution to these 9 6 log x sensitive processes. A ¼ 1 þ − − 6x2 t ; WW 1 − x ð1 − x Þ2 t ð1 − x Þ3 t t t C. Flavor constraints from meson processes x ðx − 4Þ x ðx − 4x Þ log x A ¼ ζ2 H t þ H t H H Here, we briefly mention the relevant constraints from WH u x ðx − 1Þðx − 1Þ x ðx − 1Þ2ðx − x Þ t H t t H H t measurement of flavor observables in meson mixing and 3x log x decays. These have been studied in the context of general þ H t ; ðx − 1Þ2ðx − x Þ 2HDMs, and the majority of the bounds extracted from t H t 1 2 2 these can be straightforwardly applied in our case since the ζ4 xHðxH þ Þ − xH log xH AHH ¼ u 2 3 : ð30Þ presence of the sterile neutrinos does not add any signifi- xtðxH − 1Þ xtðxH − 1Þ cant contribution at leading order. In particular, the 2HDM with the alignment in the flavor sector has been scrutinized We require that our benchmark scenario, detailed below, in Refs. [131,132,150,151], to which we refer for maximizes the impact on RK and RK while complying with

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0 ¯ 0 SM the observed Bq − Bq mixing at the level of 2σ. A lighter respect to C10 as, for instance, in a Z2 symmetric model β charged Higgs boson or a larger ζu would spoil the with large tan . In the flavor aligned 2HDM and in the ζ ≃ 1 constraint from ΔMs. Obviously, there is no dependence parameter space in which we are interested, namely, u on ζl and ζν. and ζd ≃ ζl ≃ 0, the bound from the measurement of the 0 þ − Bs → μ μ transition is usually weaker than the one from → γ 2. Radiative Bs Xs decay the Bs → Xsγ decay. Nevertheless, due to the contribution The B → X γ decay rate represents one of the best from the heavy sterile neutrinos which is proportional to the s s ζ measured observables, and it is employed to constrain new parameter ν, we recompute the corresponding con- several new physics scenarios. The contribution of the straint by employing the FLAVIO package [152]. charged Higgs boson is encoded in the Wilson coefficients C7 and C8. At leading order, the corresponding new physics 4. Leptonic decay of the charged mesons M → τν corrections are sensitive to ζ and ζ , and are given by u d The M → τν decay occurs at tree level through 2 2 2 charged current processes, and the corresponding BR is ðζ Þ M M C ¼ u G1 t þ ζ ζ G2 t ð31Þ i 3 i 2 u d i 2 M þ M BRðM → τ νÞ H H 2 2 2 2 τ G m mτ mτ with M F M 1 − 2 2 1 2 ¼ 8π 2 jVudj fMj þ CHj ; ð35Þ mM 2 2 1 yð7 − 5y − 8y Þ y ð3y − 2Þ G7ðxÞ¼ þ log x; where f and τ are the decay constant and the lifetime, 24ðy − 1Þ3 4ðy − 1Þ4 M M respectively, and the contribution of the charged Higgs yð3 − 5yÞ yð3y − 2Þ G2 x x; boson is encoded in 7ð Þ¼12 − 1 2 þ 6 − 1 3 log ðy Þ ðy Þ 2 ζ ζlm − ζ ζlm m 2 5 − 2 3 2 C ¼ u u d d M : ð36Þ 1 yð þ y y Þ − y H 2 G8ðxÞ¼ log x; mu þ md M 8ðy − 1Þ3 4ðy − 1Þ4 H → τν 2 yð3 − yÞ y The most constraining decay mode is found to be B , G8ðxÞ¼ − log x: ð32Þ 4ðy − 1Þ2 2ðy − 1Þ3 which is, however, only relevant for light charged Higgs masses. As for the new physics contributions to the meson mixing observables, the values of the ζl and ζν parameters are → γ IV. CONTRIBUTIONS TO completely irrelevant in the determination of the b s b → sl + l − PROCESSES transition. The effective Hamiltonian for the b → slþl− transitions 0 + − 3. Leptonic decay of the neutral mesons Bq → μ μ is given by 0 → μþμ− X The BR of the Bq meson decay is given by 4G α H ¼ − pffiffiffiF V V C O þ H:c:; ð37Þ eff 2 4π ts tb i i 0 → μþμ− 0 → μþμ− 2 2 i BRðBq Þ¼BRSMðBq ÞðjPj þjSj Þð33Þ where the relevant operators for the analysis of the RK and with RK anomalies are 2 m 0 ¯ μ ¯ ¯ μ ¯ C10 B m C O9 ¼ðbγ P sÞðlγμlÞ; O10 ¼ðbγ P sÞðlγμγ5lÞ: ð38Þ P ¼ þ q b P ; L L CSM 2M2 m þ m CSM v10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiW b q 10 The new physics effects in the corresponding Wilson u 2 2 0 coefficients can be recast as u 4mμ mB m C S ¼ t1 − q b S : ð34Þ 2 2 2 SM Δ Δ m 0 MW mb þ mq C10 Δ ð ZÞ ð ZÞ Bq Ci ¼ Ci ðH ÞþCi ðNRÞ Z penguin Δ Δ ð γ Þ ð γ Þ γ The leading new physics contribution appears in the Wilson þ Ci ðH ÞþCi ðNRÞ penguin coefficient C10, and it is mediated by the charged Higgs in □ □ □ þ Cð ÞðHÞþCð ÞðN ÞþCð ÞðN ;HÞ box; 2HDMs. Interestingly, the same coefficient is also corrected i i R i R by the presence of the heavy sterile neutrinos, and it is ð39Þ sensitive to, besides ζu;d;l, the coupling ζν of the charged where Δ , Δγ, and □ denote the contributions from the Z Higgs to the sterile neutrino states. The impact of the CP Z penguins, the photon penguins, and the box diagrams, and CS coefficients is, in contrast, suppressed by the 2 2 ð•Þ m 0 =M factor unless the former can be enhanced with respectively. Moreover, C ðH Þ represents the charged Bq W i

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• Δ Δ □ ð Þ ð ZÞ ð γ Þ 0 ð Þ Higgs contribution typical of the 2HDM, Ci ðNRÞ denotes Ci ðNRÞ¼Ci ðNRÞ¼ . Moreover, Ci ðH Þ¼ Δ the loop corrections from heavy sterile neutrinos and W ð γ Þ 0 Ci ðH Þ¼ since in the 2HDM the charged Higgs bosons that is present in the seesaw extensions of the SM contributes only to the Z penguin diagram. Finally, ð•Þ ð□Þ without extra Higgses, and Ci ðNR;H Þ represents a Ci ðNR;H Þ includes the contributions of the heavy combined contribution from diagrams with both sterile neutrinos exchange mediated by a charged Higgs current neutrinos and charged Higgs. and, as such, is peculiar of the model considered in this As the sterile neutrinos are not charged under the (color) paper. The analytic expressions of the new physics con- 3 SUð Þ gauge group, their contribution to the penguin tributions to the Wilson coefficients for the penguin diagrams is identically zero at leading order, namely, diagrams are as follows:

Penguins—contributions from the charged Higgs boson.— 2 Δ −1 4 − 1 − ð ZÞ 2 þ sW xHxtðxH log xHÞ C9 ðH Þ¼−ζ ; u 8s2 ðx − 1Þ2 W H Δ 1 x x ðx − 1 − log x Þ ð ZÞ −ζ2 H t H H C10 ðH Þ¼ u 8 2 − 1 2 ; sW ðxH Þ Δ x ð γ Þ ζ2 H 6 3 3 − 6 4 − − 1 47 − 79 38 C9 ðH Þ¼ u 4 ½ ð xH xH þ Þ log xH ðxH ÞðxHð xH Þþ Þ; 108ðxH − 1Þ Δ ð γ Þ C10 ðH Þ¼0: ð40Þ Penguins—contributions from the heavy neutrinos.—

Δ Δ ð ZÞ ð γ Þ 0 Ci ðNRÞ¼Ci ðNRÞ¼ : ð41Þ

The analytic expressions of the new physics contributions to the Wilson coefficients for the box diagrams are as follows: Box—contributions from the charged Higgs boson.—

ð□Þ ð□Þ C9 ðH Þ¼C10 ðH Þ¼0: ð42Þ

Box—contributions from the heavy neutrinos.—

XnR ð□Þ 2 xt C9 ðN Þ¼ jðU Þl j ½ðx − 1Þððx − 1Þðx − x Þð7x − 4x Þ R Lh i 16s2 x − 1 x − 1 2 x − x 2 Ni t t Ni t Ni i¼1 wð Ni Þð t Þ ð Ni tÞ − ð4x2 − x x ð3x þ 8Þþx2ð6x þ 1ÞÞ log x Þ − ðx − 1Þ2ð4x2 − 8x x þ x2Þ log x ; Ni Ni t t t t t t Ni Ni t t Ni ð□Þ ð□Þ C10 ðNRÞ¼−C9 ðNRÞ: ð43Þ Box—contributions from the charged Higgs boson and the heavy neutrinos.— XnR ζ2ζ2 ð□Þ 2 u νxHxt C ðN ;H Þ¼ jðU Þl j 9 R Lh i 16s2 x − 1 2 x − 1 x − x 2 i¼1 Wð H Þ ð Ni Þð H Ni Þ − − 1 2 − − 1 − 1 − −2 1 × ½ xHðxH Þ log xNi ðxNi ÞððxH ÞðxH xNi ÞþxHð xH þ xNi þ Þ log xHÞ

ζuζνxHxt þ 8 2 − 1 − 1 − 1 − − − sWðxH ÞðxNi Þðxt ÞðxH xNi ÞðxH xtÞðxNi xtÞ × ½ðx − 1Þðx − 1Þð4x − x Þ log x ðx − x Þ Ni t H t H Ni t − 1 − 1 − − 4 3 − 1 − þðxH Þððxt ÞðxH xtÞðxt xNi Þ log xNi þ ðxNi ÞxtðxH xNi Þ log xtÞ ;

ð□Þ ð□Þ C10 ðNR;H Þ¼−C9 ðNR;H Þ: ð44Þ

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(a) (b) (c)

FIG. 1. The heavy neutrino contributions to the one-loop box diagrams of the b → slþl− transition, with (a) two W bosons, (b) one W and one charged Higgs, and (c) two charged Higgs.

In the previous equations, we used the following mass the Yukawa couplings is in place in both the quark and ratios: lepton sectors.

2 2 2 Mt Mt Mt V. RESULTS xt ¼ 2 ;xH ¼ 2 ;xNi ¼ 2 : ð45Þ M M mν W H hi Among the different lepton flavor–violating processes, μ → eγ is the most constraining one, we use the Mu We now discuss some interesting features concerning the to E Gamma experiment (MEG) bound of BRðμ → eγÞ ≤ structure of these coefficients. First, one can see for the box 5.7 × 10−13, from Ref. [153]. In Fig. 2, we show the diagram contribution, shown in Fig. 1, that the heavy corresponding BR generated by a single heavy neutrino as neutrinos give C9 ¼ −C10. One can also see that the SM a function of the squared mixing angle of the same heavy box contributionP with the light neutrinos, not shown above, neutrino with the electron one. The other parameters 3 ˜ li 2 2 is rescaled by 1 jU j ≃ 1 − η , where l ¼ e, μ, have been fixed as M ¼ 700 GeV, mν ¼ 500 GeV, i¼ PMNS H hi η2 2 −3 and controlling the departure from unitarity of the the ζl ¼ 0, and jðU Þμ j ¼ 0.4 × 10 . The latter corre- η2 Lh i PMNS matrix. (Since is expected to be small and no sponds to the maximum allowed value from unitarity new physics enhancement factors are present in the SM constraints; see Eq. (23). Since in a model with a single diagram, we can safely neglect this correction.) Concerning Higgs doublet we would obtain BR=U4 ∼ ð6–7Þ × 10−4 in the box diagrams, there is also a new contribution the heavy neutrino mass range of 500–1000 GeV, it is clear ð□Þ Ci ðNRÞ with the heavyP neutrinos and virtual W s. that the diagrams with the charged Higgs give the dominant nR 2 This is proportional to i¼1 jðULhÞlij , and we do not and a very large contribution to the BR. Using the largest expect, as confirmed by the numerical analysis, that this values for the mixing angles allowed by unitarity, we obtain contribution can provide large effects to flavor observables. a suppression of the BR of ∼10−7, which cannot accom- Indeed, the coupling of the heavy neutrinos to the leptons modate the strong bound on μ → eγ from the MEG mediated by the W boson is fixed by the gauge invariance experiment. This suggests that, if a low-scale seesaw is and proportional to the SUð2Þ weak gauge coupling. To embedded into a 2HDM framework, a given heavy neutrino allow for more freedom, one has to rely on an extra charged (or, in a scenario with a large hierarchy between heavy degree of freedom with the simplest possibility being the neutrino states, the lightest one) may have a non-negligible charged scalar of a 2HDM extension. These contributions ð□Þ are encoded into Ci ðNR;H Þ. The Z2 symmetric scenarios of the 2HDM are among the simplest ones but barely produce significant effects in the C9;10 Wilson coefficients. This can be understood from Table I, because the correc- 2 2 2 tions to C9;10 would be proportional to ζu ¼ ζν ¼ cot β, independently from the specific realization, and thus relevant only for tan β < 1, which is severely constrained by b → sγ. The A2HDM allows us to disentangle ζu from ζν, such that, while the former is still bound from b → sγ, the latter can be varied freely, thus providing significant contributions to the Wilson coefficients in some region of the parameter space. We recall again that the alignment in

μ → γ 700 500 the neutrino sector is not strictly required by the flavor FIG. 2. BRð e Þ for MH ¼ GeV, mνh ¼ GeV, 2 0 4 10−3 ζ 0 i physics but we, nevertheless, impose it by assuming that jðULhÞμij ¼ . × , and l ¼ . The dashed line corresponds the same mechanism ensuring the proportionality between to the MEG exclusion bound.

115009-9 ROSE, KHALIL, KING, and MORETTI PHYS. REV. D 101, 115009 (2020) mixing only with SM neutrinos of a given flavor eigenstate; Other choices are obviously acceptable but not consid- otherwise, the charged Higgs boson would induce unac- ered here. ceptably large effects on lepton flavor–violating processes. The main result of the analysis is presented in Fig. 3. ≠ 0 ≃ 0 The two realizations would be ðULhÞμi ; ðULhÞei or The numerical values of the flavor observables RK and RK ≃ 0 ≠ 0 have been obtained from the evaluation of the Wilson ðULhÞμi ; ðULhÞei . As we will see below, the first coefficients computed above and from the FLAVIO package possibility can be also used to explain the deviation of R K [152]. In the interesting region of the parameter space, the and R from the SM prediction. These two conditions can K predictions for the two ratios in the A2HDM are the same, be achieved by suitably choosing the mass matrices mD and and, for the sake of simplicity, we will only discuss RK . mR. Another possibility, allowing one to control the large μ → γ The latterpffiffiffi is presented as a function of the parameter effects in BRð e Þ, which, anyway, we will not explore ζ ≡ 2ζ 2 νYν νðULhÞμiðmν =vÞ that mostly controls the in this work, is to assume a ζl ≠ 0 and tune it against the ζν hi G C9;10 Wilson coefficients. In particular, the blue points term in Eq. (27) to reduce the H form factor with respect G correspond to the configuration in which the heavy to W . With the analytic expressions of the Wilson coefficients, neutrino couples to the SM muon sector and has negligible mixing with the first family. As anticipated above, this we may turn to finding realistic benchmark scenarios. We setup allows us to reproduce the measured reduction in the explore the parameter space spanned by the three parameters: 2 RK ratio, which is represented in the plot with a dashed ζν, mν , and ðU Þμ considering the impact of a single heavy i Lh ;i horizontal line, together with 1σ and 2σ (green) bands. The neutrino, thus assuming, for the sake of simplicity, a red points, instead, are representative of the scenario in hierarchy in the neutrino mass spectrum. The general case which the heavy neutrino has a mixing with the electrons. can be obtained straightforwardly and does not add much to In this case, the predicted RK is above 1 and contradicts the the present discussion. In particular, one finds that if all the LHCb observations. The extent of the reduction is mainly Pneutrino masses are almost degenerate the combination controlled by the parameter ζνYν. Interestingly, the same 6 2 i¼1ðULhÞμ;i can be factored out from the Wilson coef- parameter defines the strength of the coupling among the ficients and can be treated as an independent parameter charged Higgs, heavy neutrino, and lepton and, as such, is leading to the same conclusions of the single neutrino case. subject to the perturbativity bound of Ref. [154]. The upper This combination of squared mixing angles is bound from the limit for the coupling is usually extracted from naive nonunitarity test of the PMNS matrix to be less than scaling arguments but also depends on the loop functions ∼0 4 10−3 of the involved processes. . × . Notice that we considered the scenario ζ ≠ 0 ≃ 0 Besides perturbativity, the parameter νYν is restricted by ðULhÞμi ; ðULhÞei in linewith the previous discussion – the requirement that the total widths of the charged Higgs on lepton flavor violating processes. The performed scan boson and heavy neutrinos are not larger than their masses, −80 ζ 80 10−5 2 10−3 takes the ranges < ν < , < ðULhÞμ;i < , since it controls, in particular, the partial decay widths and 200 GeV

þ − FIG. 3. (a) RK in the central bin and (b) BRðBs → μ μ Þ as a function of the combination of parameters that mostly controls the observable. The green bands are the 1σ and 2σ bands for the RK measurement. Blue (red) points correspond to the scenario in which the heavy neutrino has a non-negligible coupling only to the muon (electron) flavor eigenstates.

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FIG. 4. (a) Correlation between RK in the central and low bins. (b) Correlation between RK in the central bin and the predicted þ − BRðBs → μ μ Þ. Here, only the 1σ bounds have been considered in both figures.

∓ as well as an analogous result for Γðνh → H μ Þ with the muon, RðDÞ and RðD Þ; nevertheless, the explanation ↔ þ of these anomalies lies outside the motivation of this paper. mνh MH and an extra factor of 2 to account for the charged-conjugated final states. Ignoring the kinematic Figure 4(a) displays the 1σ bounds for both the R low pffiffiffi K factor, we obtain the constraint ζνYν ≲ 4 π, which lies and central bins. One can see that parameter configurations between the two commonly adopted upper bounds that can satisfy either the low or central bins separately, but not can be used to estimate the region of tree-level perturba- simultaneously both. In Fig. 4(b), we consider the effects → μþμ− tivity and that are shown in the plot. One should keep in on the BRðBs Þ. One can see that for several points both predictions are simultaneously compatible with the mind that,ffiffiffiffiffiffi by adopting the most conservative bound, p experimental measurements. ζνYν ≲ 4π, it is possible to explain the present RK →μþμ− 2 8þ1.4 10−9 measurement and the BRðBs Þ¼ð . −1.2 Þ× [155] only at the 2σ level. VI. CONCLUSIONS ζ → 0 While the limit νYν switches off the nonuniversal The LHCb experiment at CERN has recently reported leptonic contributions, lepton universal effects beyond the the existence of some anomalies in their data, with respect SM predictions still survive. This is particularly clear in 0 to the predictions of the SM. Specifically, the measured Fig. 3(b). The latter shows that the BRðB → μμ¯Þ is þ þ þ − s values of the observables RK ¼ BRðB → K μ μ Þ= characterized by a correction of ∼20% with respect to þ þ þ − 0 0 þ − BRðB → K e e Þ and RK ¼ BRðB → K μ μ Þ= the SM value for the chosen benchmark point. These effects BRðB0 → K0eþe−Þ revealed a ≈2.5σ deviation when are intimately linked to the 2HDM nature of the BSM compared to the SM rates, which are essentially 1. In scenario discussed here and, in particular, to the presence of addition, the discrepancies occur in two dilepton invariance a charged Higgs state in the scalar spectrum. The lepton mass bins. Therefore, these results must be taken as a universal contributions arise from the penguin diagrams serious hint of possible BSM physics. and affect both C9 and C10 as detailed in Eq. (40). The Herein, we have considered the possibility of explaining corrections to C10 are larger (∼10%) than those to C9 in such RK and RK anomalies in an A2HDM, wherein an which case the Z-penguin and photon-penguin diagrams alignment is present between the Yukawa couplings gen- almost cancel each other. This feature breaks the C9 ¼ erated by the two Higgs doublet fields, combined with a − C10 effect induced by the heavy neutrinos. The impact of low-scale seesaw mechanism generating light neutrino the charged Higgs boson in other flavor observables and the masses and mixings in compliance with current experi- corresponding constraints have been discussed in mental measurements. Such a scenario allows for signifi- Refs. [132,151]. As stated above, we have ensured that cant nonuniversal leptonic contributions, through box the benchmark point considered complies with all the diagrams mediated by H and νR states, which in turn relevant bounds. The scalar charged current mainly affects → lþl− 0 ¯ 0 0 ¯ 0 alter the yield of the partonic decay b s entering neutral meson mixings (Bq − Bq and K − K , with ΔMs the definition of both the RK and RK observables. To providing the strongest constraint), leptonic decays of render our explanation phenomenologically viable, we 0 → μþμ− charged and neutral mesons (in particular, Bs , have made sure to comply with both theoretical (chiefly, 0 − B → μþμ , and B → τν), and the radiative decay the unitarity bounds stemming from the neutrino mixing ¯ d B → Xsγ. The 2HDM extension of the SM has also been matrix) and experimental (the strongest being those due to employed to address the anomalous magnetic moment of lepton flavor–violating processes and mesonic decay

115009-11 ROSE, KHALIL, KING, and MORETTI PHYS. REV. D 101, 115009 (2020) channels) constraints. In fact, the masses required for the measurements can be found in the envisioned A2HDM charged Higgs and right-handed neutrino states entering the plus low-scale seesaw scenario. above transition are also well beyond their current direct limits. Furthermore, to make clear that our explanation of ACKNOWLEDGMENTS the RK and RK anomalies is not particularly ad hoc,we have left the actual low-scale dynamics onsetting the The work of L. D. R. and S. M. is supported in part by the seesaw mechanism undetermined, by illustrating that this NExT Institute. S. M. also acknowledges partial financial could be realized through different scenarios, e.g., the so- contributions from the STFC Consolidated Grant No. ST/ called inverse and linear seesaw cases. Therefore, our setup L000296/1. S. J. D. K. and S. K. have received support under captures a variety of light neutrino masses and mixings that the H2020-MSCA InvisiblesPlus (RISE) Grant No. 690575 can be tuned to further experimental observation in the and Elusives (ITN) Grant No. 674896. All authors acknowl- neutrino sector while leaving predictions in the B one edge support under the H2020-MSCA NonMinimalHiggs unchanged. Finally, we have correlated our predictions for (RISE) Grant No. 645722. L. D. R. thanks Fady Bishara, RK and RK in both dilepton invariant mass bins to those for Marco Nardecchia, and Olcyr Sumensari for very fruitful þ − the highly constraining observable BRðBs → μ μ Þ, show- discussions. ing that simultaneous solutions to both sets of

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