Published for SISSA by Springer Received: April 28, 2014 Accepted: May 13, 2014 Published: June 4, 2014 Xin-Qiang Li,a,b Jie Luc and Antonio Pichc a Institute of Particle Physics and Key Laboratory of Quark & Lepton Physics (MOE), Central China Normal University, Wuhan, Hubei 430079, P.R. China b State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China c IFIC, Universitat de València – CSIC, Apt. Correus 22085, E-46071 València, Spain E-mail: xqli@itp.ac.cn, lu.jie@ific.uv.es, pich@ific.uv.es 0 Abstract: The rare decays Bs,d → ℓ+ ℓ− are analyzed within the general framework of the aligned two-Higgs doublet model. We present a complete one-loop calculation of the relevant short-distance Wilson coefficients, giving a detailed technical summary of our results and comparing them with previous calculations performed in particular limits or approximations. We investigate the impact of various model parameters on the branching ratios and study the phenomenological constraints imposed by present data. Keywords: Higgs Physics, Rare Decays, Beyond Standard Model, B-Physics ArXiv ePrint: 1404.5865 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP06(2014)022 JHEP06(2014)022 0 Bs,d → ℓ+ℓ− decays in the aligned two-Higgs-doublet model Contents 1 Introduction 1 2 The aligned two-Higgs doublet model 2.1 Yukawa sector 4 5 7 7 8 10 13 15 19 4 Numerical results 4.1 Input parameters 4.2 SM predictions 4.3 Results in the A2HDM 4.3.1 Choice of model parameters 4.3.2 Small ςd,ℓ 4.3.3 Large ςd,ℓ 4.4 Z2 symmetric models 20 20 21 22 22 23 24 25 5 Conclusions 27 A Scalar-penguin results within the A2HDM 28 1 Introduction The recent discovery of a Higgs-like boson [1, 2], with properties compatible with the Standard Model (SM) expectations [3–6], is one of the greatest achievements in the past decades in particle physics and represents a major confirmation of our present theoretical paradigm. The LHC data suggest that the electroweak symmetry breaking (EWSB) is probably realized in the most elegant and simple way, i.e., via the Higgs mechanism implemented through one scalar SU(2)L doublet. An obvious question we are now facing is whether the discovered 126 GeV state corresponds to the unique Higgs boson incorporated in the SM, or it is just the first signal of a much richer scenario of EWSB. None of the fundamental principles of the SM forbids the possibility of an enlarged scalar sector associated with the EWSB. –1– JHEP06(2014)022 0 → ℓ+ ℓ− ) 3 Calculation of B(Bs,d 3.1 Effective Hamiltonian 3.2 Computational method 3.3 Wilson coefficients in the SM 3.4 Wilson coefficients in the A2HDM 3.4.1 Neutral scalar exchange 0 → ℓ+ ℓ− branching ratio 3.5 Bs,d –2– JHEP06(2014)022 Among the many possible scenarios for new physics (NP) beyond the SM, the twoHiggs doublet model (2HDM) [7] provides a minimal extension of the scalar sector that naturally accommodates the electroweak (EW) precision tests, giving rise at the same time to a large variety of interesting phenomenological effects [8, 9]. The scalar spectrum of the model consists of two charged fields, H ± , and three neutral ones, h, H and A, one of which is to be identified with the Higgs-like boson found at the LHC. The direct search for these additional scalar states at high-energy collisions, or through indirect constraints via precision flavour experiments, is an important task for the next years. This will also be helpful to gain further insights into the scalar sector of supersymmetry (SUSY) and other models with similar scalar contents. Within the SM, flavour-changing neutral current (FCNC) interactions are forbidden at tree level, and highly suppressed at higher orders, due to the Glashow-IliopoulosMaiani (GIM) mechanism [10]. In a generic 2HDM, however, tree-level FCNC interactions generally exist, through non-diagonal couplings of neutral scalars to fermions. The unwanted FCNCs can be eliminated, imposing on the Lagrangian an ad-hoc discrete Z2 symmetry; depending on the different possible Z2 charge assignments, this results in four types of 2HDMs (I, II, X and Y) [8, 9], all satisfying the hypothesis of natural flavour conservation (NFC) [11]. A more general alternative is to assume the alignment in flavour space of the Yukawa matrices for each type of right-handed fermions [12]. The so-called aligned two-Higgs doublet model (A2HDM) results in a very specific structure, with all fermion-scalar interactions being proportional to the corresponding fermion masses. It also contains as particular cases the different versions of the 2HDM with NFC, while at the same time introduces new sources of CP violation beyond the Cabibbo-KobayashiMaskawa (CKM) phase [13, 14]. These features make the A2HDM a very interesting theoretical framework, which leads to a rich and viable phenomenology, both in high-energy collider experiments [15–21], as well as in low-energy flavour physics [22–27]. 0 → ℓ+ ℓ− , with In the field of rare B-meson decays, the purely leptonic processes Bs,d ℓ = e, µ or τ , play an outstanding role in testing the SM and probing physics beyond it, because they are very sensitive to the mechanism of quark-flavour mixing. Within the SM, the FCNC transition is mediated by a one-loop amplitude, suffers from a helicitysuppression factor mℓ /mb , and is characterized by a purely leptonic final state. The first two features result in a double suppression mechanism, responsible for the extremely rare nature of these decays. The third feature implies that these processes are theoretically very clean, with the only hadronic uncertainty coming from the B-meson decay constants 0 → ℓ+ ℓ− a formidable fBs,d . All these considerations make the rare leptonic decays Bs,d probe of physics beyond the SM, especially of models with a non-standard Higgs sector like multi-Higgs doublet models [28–36] as well as various SUSY scenarios [29, 30, 36–49]. As far as the experimental side is concerned, the decay modes with ℓ = µ are especially interesting because the corresponding final state can be easily tagged. Over the last decade the upper bounds for the branching ratios of these decays have been improving continuously, thanks to the CDF and DØ collaborations at the Tevatron and, more recently, the ATLAS, CMS and LHCb experiments at the LHC [50]. In November 2012, the LHCb experiment reported the first evidence of the decay Bs0 → µ+ µ− , at the 3.5 σ level [51]. The signal significance has been raised, respectively, to 4.0 σ and 4.3 σ by LHCb and CMS, after analyzing the currently available data set, with the averaged time-integrated branching ratio given by 
 2.9 +1.1 (stat.) +0.3 (syst.) × 10−9 −1.0 −0.1 B(Bs0 → µ+ µ− ) =
 −9 3.0 +1.0 −0.9 × 10 LHCb [52] , (1.1) CMS [53] B(Bs0 → µ+ µ− )exp. = (2.9 ± 0.7) × 10−9 . (1.2) At the same time, the branching fraction of Bd0 → µ+ µ− has also been determined with a signal significance of 2 σ by the two experiments: 
 3.7 +2.4 (stat.) +0.6 (syst.) × 10−10 −2.1 −0.4 B(Bd0 → µ+ µ− ) =
 −10 3.5 +2.1 −1.8 × 10 LHCb [52] . (1.3) CMS [53] The corresponding combined result reads [54]
−10 . B(Bd0 → µ+ µ− )exp. = 3.6 +1.6 −1.4 × 10 (1.4) These measurements are in remarkable agreement with the latest updated predictions within the SM [55]: B(Bs0 → µ+ µ− ) = (3.65 ± 0.23) × 10−9 , B(Bd0 → µ+ µ− ) = (1.06 ± 0.09) × 10−10 , (1.5) where the next-to-leading order (NLO) corrections of EW origin [56], as well as the QCD corrections up to the next-to-next-to-leading order (NNLO) [57], have been taken into account. Although the experimental uncertainties are still quite large, they are expected to get significantly reduced within the next few years [58]. All these experimental and theoretical progresses will lead to new stringent constraints on physics beyond the SM. Motivated by the above considerations, in this work we shall perform a study of the 0 → ℓ+ ℓ− within the A2HDM. Our paper is organized as follows. rare leptonic decays Bs,d In section 2 we give a brief overview of the A2HDM Lagrangian, especially of its Yukawa and scalar sectors. In section 3 we summarize the SM results and describe the full oneloop calculation of the relevant Feynman diagrams in the A2HDM. We have performed the calculation in two different gauges, Feynman (ξ = 1) and unitary (ξ = ∞), in order to check the gauge-independence of our results. In section 4 we discuss the impact of the model parameters on the branching ratios of these decays, taking into account the latest implications from the LHC Higgs data. Our conclusions are made in section 5. Finally, the appendix contains the explicit results for the individual Higgs-penguin diagrams. –3– JHEP06(2014)022 where the CMS uncertainty includes both the statistical and systematic components, but is dominated by the statistical uncertainties. The two measurements lead to the weighted world average [54] 2 The aligned two-Higgs doublet model The Hermiticity of the potential requires all parameters to be real except µ3 , λ5 , λ6 and λ7 ; thus, there are 14 real parameters. The minimization conditions h0|ΦT1 (x)|0i = √ (0, v/ 2) and h0|ΦT2 (x)|0i = (0, 0) impose the relations µ1 = −λ1 v 2 and µ3 = − 21 λ6 v 2 , which allow us to trade the parameters µ1 and µ3 by v and λ6 , respectively. The freedom to rephase the field Φ2 implies, moreover, that only the relative phases among λ5 , λ6 and λ7 are physical. Therefore, we can fully characterize the potential with 11 parameters: v, µ2 , λ1,2,3,4 , |λ5,6,7 |, arg(λ5 λ∗6 ) and arg(λ5 λ∗7 ). Four of these parameters can be determined through the physical scalar masses. Inserting eq. (2.1) into eq. (2.2), expanding out the resulting expression and imposing the minimization conditions, one can decompose the potential into a quadratic mass term plus cubic and quartic interactions (up to an irrelevant constant). The mass term takes the form: ! S1 1 + − 2 V2 = MH ± H H + (S1 , S2 , S3 ) M S2 2 S3 = 2 MH ± 3
2 1 X 2 H H + Mϕ0 ϕ0i , i 2 + − (2.3) i=1 2 = µ + 1 λ v 2 and with MH ± 2 2 3   R 2 λI 2λ1 v 2 v2 λ −v 6 6     2 R 2 + v 2 λ 4 + λR   M v λ −v 2 λI5 ± M= 6 5 2 H  ,  2 + v 2 λ 4 − λR −v 2 λI6 −v 2 λI5 MH ± 5 2 (2.4) I where λR i ≡ Re(λi ) and λi ≡ Im(λi ). The symmetric mass matrix M is diagonalized by an orthogonal matrix R, which defines the neutral mass eigenstates:
2 R M RT = diag Mh2 , MH , MA2 , ϕ0i = Rij Sj . (2.5) –4– JHEP06(2014)022 The 2HDM extends the SM with the addition of a second scalar doublet of hypercharge Y = 21 [7]. In the so-called “Higgs basis”, in which only one doublet gets a nonzero vacuum expectation value, the two doublets can be parametrized as " # " # G+ H+ Φ1 = 1 , Φ2 = 1 , (2.1) √ (v + S1 + iG0 ) √ (S2 + iS3 ) 2 2 √ where G± and G0 denote the Goldstone fields, and v = ( 2GF )−1/2 ≃ 246 GeV. The five physical scalar degrees of freedom are given by the two charged fields H ± (x) and three neutral scalars ϕ0i (x) = {h(x), H(x), A(x)}. The latter are related with the Si fields through an orthogonal transformation, which is fixed by the scalar potential:     h    i V = µ1 Φ†1 Φ1 + µ2 Φ†2 Φ2 + µ3 Φ†1 Φ2 + µ∗3 Φ†2 Φ1  2  2       +λ1 Φ†1 Φ1 + λ2 Φ†2 Φ2 + λ3 Φ†1 Φ1 Φ†2 Φ2 + λ4 Φ†1 Φ2 Φ†2 Φ1 h   i + λ5 Φ†1 Φ2 + λ6 Φ†1 Φ1 + λ7 Φ†2 Φ2 Φ†1 Φ2 + h.c. . (2.2) In a generic case, the three mass-eigenstates ϕ0i (x) do not have definite CP quantum numbers. In the CP-conserving limit, λI5 = λI6 = λI7 = 0 and S3 does not mix with the other two neutral fields. The scalar spectrum contains then a CP-odd field A = S3 and two CP-even scalars h and H which mix through the two-dimensional rotation matrix:1 ! " # ! h cos α̃ sin α̃ S1 = . (2.6) H − sin α̃ cos α̃ S2 where 2 2 Σ = MH ± + v s ∆=  2 λ1 + 2 + v2 MH ±  λ4 + λR 5 2 −2 λ1 +  , λ4 + λR 5 2 (2.8) 2 2 + 4v 4 (λR 6) = − 2v 2 λR 6 , sin (2α̃) (2.9) and the mixing angle is determined through tan α̃ = Mh2 − 2λ1 v 2 v 2 λR 6 = 2 . R 2 2 2λ1 v − MH v λ6 (2.10) The cubic and quartic self-couplings among the physical scalars and their interactions with the gauge bosons can be derived straightforwardly. Their explicit form could be found, for example, in refs. [8, 9, 20, 21, 59–62]. 2.1 Yukawa sector In the Higgs basis, the most generic Yukawa Lagrangian of the 2HDM is given by √ h i 2 Q̄′L (Md′ Φ1 + Yd′ Φ2 )d′R + Q̄′L (Mu′ Φ̃1 + Yu′ Φ̃2 )u′R + L̄′L (Mℓ′ Φ1 + Yℓ′ Φ2 )ℓ′R + h.c. , LY = − v (2.11) ∗ where Φ̃i (x) = iτ2 Φi (x) are the charge-conjugated scalar doublets with hypercharge Y = − 21 , Q′L and L′L denote the SM left-handed quark and lepton doublets, respectively, and u′R , d′R and ℓ′R are the corresponding right-handed singlets, in the weak interaction basis. All fermionic fields are written as 3-vectors in flavour space and, accordingly, the couplings Mf′ and Yf′ (f = u, d, ℓ) are 3 × 3 complex matrices. In general, the Yukawa matrices Mf′ and Yf′ cannot be simultaneously diagonalized in flavour space. Thus, in the fermion mass-eigenstate basis with diagonal mass matrices Mf , the corresponding Yukawa matrices Yf remain non-diagonal, giving rise to tree-level 1 The scalar mixing is often parametrized in terms of α′ = α̃ + π/2, so that the SM limit corresponds to α = π/2 [8, 9]. We prefer to describe small deviations from the SM limit with α̃ ≃ 0. ′ –5– JHEP06(2014)022 We shall adopt the conventions Mh ≤ MH and 0 ≤ α̃ ≤ π, so that sin α̃ is always positive. The masses of the three physical neutral scalars are given in this case by   1 λ4 1 2 2 2 R , (2.7) + v MH = (Σ + ∆) , MA2 = MH − λ Mh2 = (Σ − ∆) , ± 5 2 2 2 Model Type I Type II Type X (lepton-specific) Type Y (flipped) Inert ςd cot β − tan β cot β − tan β 0 ςu cot β cot β cot β cot β 0 ςl cot β − tan β − tan β cot β 0 Table 1. The one-to-one correspondence between different specific choices of the couplings ςf and the 2HDMs based on discrete Z2 symmetries. Yu = ςu∗ Mu , Yd,ℓ = ςd,ℓ Md,ℓ , (2.12) where the three proportionality parameters ςf (f = d, u, ℓ) are arbitrary complex numbers and introduce new sources of CP violation. The Yukawa interactions of the physical scalars with the fermion mass-eigenstate fields then read [12] √ i o 2 +n h † H ū ςd V Md PR − ςu Mu V PL d + ςℓ ν̄Mℓ PR ℓ LY = − v X   1 ϕ0 − yf i ϕ0i f¯Mf PR f + h.c. , (2.13) v 0 ϕi ,f 5 are the right-handed and left-handed chirality projectors, Mf the where PR,L ≡ 1±γ 2 diagonal fermion mass matrices, and V the CKM quark-mixing matrix [13, 14]. The couplings of the neutral scalar fields to fermion pairs are given by ϕ0 yd,ℓi = Ri1 + (Ri2 + i Ri3 ) ςd,ℓ , ϕ0 yu i = Ri1 + (Ri2 − i Ri3 ) ςu∗ . (2.14) In the A2HDM, all fermionic couplings to scalars are proportional to the corresponding fermion masses, and the only source of flavour-changing interactions is the CKM quarkmixing matrix V , while all leptonic couplings and the quark neutral-current interactions are diagonal in flavour. All possible freedom allowed by the alignment conditions is encoded by the three family-universal complex parameters ςf , which provide new sources of CP violation without tree-level FCNCs [12]. The usual models with NFC, based on discrete Z2 symmetries, are recovered for particular values of the couplings ςf , as indicated in table 1. Explicit examples of symmetry-protected underlying theories leading to a lowenergy A2HDM structure have been discussed in ref. [63–65]. The alignment conditions in eq. (2.12) presumably hold at some high-energy scale ΛA and are spoiled by radiative corrections. These higher-order contributions induce a misalignment of the Yukawa matrices, generating small FCNC effects suppressed by the corresponding loop factors [12, 22, 66–70]. However, the flavour symmetries of the A2HDM tightly constrain the possible FCNC structures, keeping their effects well below the present –6– JHEP06(2014)022 FCNC interactions. In the A2HDM, the tree-level FCNCs are eliminated by requiring the alignment in flavour space of the two Yukawa matrices coupling to a given type of right-handed fermions [12] experimental bounds [22–27]. Using the renormalization-group equations (RGEs) [67–70], one can check that the only FCNC local structures induced at one loop take the form [22, 66] h i X n C ϕ0i (Ri2 + i Ri3 ) (ςd − ςu ) d¯L V † Mu Mu† V Md dR LFCNC = 2 3 (1 + ςu∗ ςd ) 4π v i io h ∗ + h.c. , (2.15) − (Ri2 − i Ri3 ) (ςd − ςu∗ ) ūL V Md Md† V † Mu uR where D is the space-time dimension. Thus, the renormalized coupling satisfies CR (µ) = CR (µ0 ) − ln (µ/µ0 ) . (2.17) Assuming the alignment to be exact at the scale ΛA , i.e., CR (ΛA ) = 0, this implies CR (µ) = ln (ΛA /µ). 3 3.1 0 Calculation of B(Bs,d → ℓ+ ℓ− ) Effective Hamiltonian 0 → ℓ+ ℓ− decays proceed through loop diagrams in both the SM The rare leptonic Bs,d and the A2HDM. After decoupling the heavy degrees of freedom, including the top quark, the weak gauge bosons, as well as the charged and neutral Higgs bosons, these decays are described by a low-energy effective Hamiltonian [71–74] " # 10,S,P X
GF α Heff = − √ Vtb Vtq∗ Ci Oi + Ci′ Oi′ + h.c. , (3.1) 2πs2W i where GF is the Fermi coupling constant, α = e2 /4π the QED fine-structure constant, and sW = sin θW the sine of the weak angle. The effective four-fermion operators are given, respectively, as O10 = (q̄γµ PL b) (ℓ̄γ µ γ5 ℓ) , mℓ mb (q̄PR b) (ℓ̄ℓ) , OS = 2 MW mℓ mb OP = (q̄PR b) (ℓ̄γ5 ℓ) , 2 MW ′ O10 = (q̄γµ PR b) (ℓ̄γ µ γ5 ℓ) , mℓ mb OS′ = (q̄PL b) (ℓ̄ℓ) , 2 MW mℓ mb OP′ = (q̄PL b) (ℓ̄γ5 ℓ) , 2 MW (3.2) where ℓ = e, µ, τ ; q = d, s, and mb = mb (µ) denotes the b-quark running mass in the modified minimal subtraction (MS) scheme. In this paper, we shall neglect the operators Oi′ , –7– JHEP06(2014)022 which vanishes identically when ςd = ςu (Z2 models of types I, X and inert) or ςd = −1/ςu∗ (types II and Y), as it should be. Although the numerical effect of the local term in eq. (2.15) is suppressed by mq m2q′ /v 3 and quark-mixing factors, its tree-level contribution is needed to render finite the contri0 → ℓ+ ℓ− , as will be detailed later. bution from one-loop Higgs-penguin diagrams to Bs,d The renormalization of the coupling constant C is determined to be   1 2µD−4 + γE − ln (4π) , (2.16) C = CR (µ) + 2 D−4 3.2 Computational method The standard way to find the Wilson coefficients is to require equality of one-particle irreducible amputated Green functions calculated in the full and in the effective theory [75, 76]. The former requires the calculation of various box, penguin and self-energy diagrams. We firstly use the program FeynArts [77], with the model files provided by the package FeynRules [78–80], to generate all the Feynman diagrams contributing to the 0 → ℓ+ ℓ− , as well as the corresponding amplitudes, which can then be evaluated decays Bs,d straightforwardly. Throughout the whole calculation, we set the light-quark masses md,s to zero; while for mb , we keep it up to linear order. As the external momenta are much smaller than the masses of internal top-quark, gauge bosons, as well as charged and neutral scalars, the Feynman integrands are expanded in external momenta before performing the loop integration [81, 82]   1 l2 + 2(k · l) 1 4(k · l)2 1− 2 + O(l4 /M 4 ) , (3.3) = 2 + 2 (k + l)2 − M 2 k − M2 k − M2 (k − M 2 )2 where k denotes the loop momentum, M a heavy mass and l an arbitrary external momentum. In addition, we employ the naive dimensional regularization scheme with an –8– JHEP06(2014)022 because they only give contributions proportional to the light-quark mass mq . Operators 0 → ℓ+ ℓ− because the conserved involving the vector current ℓ̄γ µ ℓ do not contribute to Bs,d vector current vanishes when contracted with the Bq0 momentum. Since the matrix element h0|q̄σµν b|B̄q0 (p)i = 0, there is also no contribution from the tensor operators. Thus, only the operators O10 , OS and OP survive in our approximation. 0 → ℓ+ ℓ− , short-distance As there are highly separated mass scales in the decays Bs,d QCD corrections can contain large logarithms like ln (µb /MW ) with µb ∼ O(mb ), which must be summed up to all orders in perturbation theory with the help of renormalizationgroup techniques. The evolution of the Wilson coefficients from the scale O(MW ) down to O(µb ) requires the solution of the RGEs of the corresponding operators O10 , OS and OP . However, the operator O10 has zero anomalous dimension due to the conservation of the (V − A) quark current in the limit of vanishing quark masses. The operators OS and OP have also zero anomalous dimension, because the anomalous dimensions of the b-quark mass mb (µ) and the scalar current (q̄PR b)(µ) cancel each other. Thus, with the operators defined by eq. (3.2), the corresponding Wilson coefficients do not receive additional renormalization due to QCD corrections. In the SM, the contributions from the scalar and pseudoscalar operators are quite suppressed and, therefore, are usually neglected in phenomenological analyses. However, they can be much more sizeable in models with enlarged Higgs sectors, such as the A2HDM, especially when the Yukawa and/or scalar-potential couplings are large. Therefore, the 0 → ℓ+ ℓ− data provide useful constraints on the model parameters. To get the theoretBs,d 0 → ℓ+ ℓ− ), the main task is then to calculate the three Wilson ical predictions for B(Bs,d coefficients C10,S,P in both the SM and the A2HDM, details of which will be presented in the next few subsections. anti-commuting γ5 to regularize the divergences appearing in Feynman integrals. After the Taylor expansion and factorizing out the external momenta, the integrals remain dependent only on the loop momentum and the heavy masses M . Subsequently, we apply the partial fraction decomposition [83]   1 1 1 1 − , (3.4) = 2 (q 2 − m21 )(q 2 − m22 ) m1 − m22 q 2 − m21 q 2 − m22 with an arbitrary integer power n and with m 6= 0. The computational procedure has also been checked through an independent analytic calculation of the Feynman diagrams, using more standard techniques such as the Feynman parametrization to combine propagators. We found full agreement between the results obtained with these two methods. It should be noted that, in deriving the effective Hamiltonian in eq. (3.1), the limit mu,c → 0 and the unitarity of the CKM matrix, ∗ Vuq Vub + Vcq∗ Vcb + Vtq∗ Vtb = 0 , (3.6) have been implicitly exploited. In general, the Wilson coefficients Ci are functions of the internal up-type quark masses, together with the corresponding CKM factors [75, 76]: X Ci = Vjq∗ Vjb Fi (xj ) , (3.7) j=u,c,t 2 , m2j /MW where xj = and Fi (xj ) denote the loop functions. The unitarity relation in eq. (3.6) implies vanishing coefficients Ci if the internal quark masses are set to be equal, i.e., xu = xc = xt . For this reason, we need only to calculate explicitly the contributions from internal top quarks, while those from up and charm quarks are taken into account by means of simply omitting the mass-independent terms in the basic functions Fi (xt ). For simplicity, we also introduce the following mass ratios: m2 xt = 2t , MW xH + M2 ± = H2 , MW xϕ0 = i Mϕ20 i 2 MW xhSM = , Mh2SM 2 MW , (3.8) where mt = mt (µ) is the top-quark running mass in the MS scheme, and hSM the SM Higgs boson. In order to make a detailed presentation of our results, we shall split the different contributions to the Wilson coefficients into the form: Z penguin, A2HDM SM C10 = C10 + C10 , (3.9) ϕ0i , A2HDM CS = CSbox, SM + CSbox, A2HDM + CS CP = , CPbox, SM + CPZ penguin, SM + CPGB penguin, SM + CPbox, A2HDM ϕ0 , A2HDM + CPZ penguin, A2HDM + CPGB penguin, A2HDM + CP i . –9– (3.10) (3.11) JHEP06(2014)022 which allows a reduction of all the Feynman integrals to those in which only a single mass parameter occurs in the propagator denominators. Finally, after reduction of tensor integrals to scalar ones, the only non-vanishing one-loop integrals take the form [84]   Z (−1)n i Γ(n − D/2) dD k 1 1 n−D/2 = , (3.5) (2π)D (k 2 − m2 )n Γ(n) m2 (4π)D/2 ℓ b W± νℓ t s W± (1.1) ℓ b G± νℓ t ℓ s G± (1.2) ℓ b G± νℓ t ℓ s W± (1.3) ℓ b W± νℓ t ℓ s G± (1.4) ℓ Figure 1. SM W -box diagrams contributing to B̄s0 → ℓ+ ℓ− . Diagrams involving Goldstone bosons G± are absent in the unitary gauge. 0 → ℓ+ ℓ− the external momenta are small compared to the EW scale M . One In Bs,d W can then set all external momenta to zero when evaluating C10 . However, the external momenta must be taken into account to evaluate the scalar Wilson coefficients CS and CP , otherwise some contributions would be missed. 3.3 Wilson coefficients in the SM 0 → ℓ+ ℓ− come from the W -box In the SM, the dominant contributions to the decays Bs,d and Z-penguin diagrams shown in figures 1 and 2, respectively, which generate the Wilson coefficient: SM C10 = −ηYEW ηYQCD Y0 (xt ) , (3.12)   3xt xt xt − 4 + ln xt Y0 (xt ) = 8 xt − 1 (xt − 1)2 (3.13) where is the one-loop function that was calculated for the first time in ref. [87]. The factor ηYEW accounts for both the NLO EW matching corrections [56], as well as the logarithmically enhanced QED corrections that originate from the renormalization group evolution [55, 57], while the coefficient ηYQCD stands for the NLO [88–91] and NNLO [57] QCD corrections. When the small external momenta are taken into account, the SM W -box and Zpenguin diagrams also generate contributions to the Wilson coefficients CS and CP . The contribution from diagram 1.2 can be neglected, because it contains two leptonic Goldstone 2 . The scalar contribution from the couplings which generate a suppression factor m2ℓ /MW – 10 – JHEP06(2014)022 The pieces labeled with “SM” only involve SM fields (without the Higgs), while those denoted by “A2HDM” contain the scalar contributions. We have calculated all the individual diagrams in both the Feynman (ξ = 1) and the unitary (ξ = ∞) gauges. Goldstone boson (GB) contributions are of course absent in the unitary gauge. While the contributions of the box and penguin diagrams to the Wilson coefficients are separately gauge dependent, their sum is indeed independent of the EW gauge fixing [85, 86]. Note that photonic penguin diagrams, in both the SM and the A2HDM, do not contribute to the decays 0 → ℓ+ ℓ− because of the pure vector nature of the electromagnetic leptonic coupling. Bs,d b ℓ b ℓ t s b (2.2) b t W± ℓ s ℓ b t ℓ s (2.6) b G± t Z (2.9) ℓ (2.4) b s s Z Z ℓ s (2.7) ℓ W± t ℓ ℓ s (2.8) ℓ ℓ ℓ b b ℓ s (2.3) G± G± Z (2.5) G± Z ℓ s b W± Z (2.10) ℓ Figure 2. SM Z-penguin diagrams contributing to B̄s0 → ℓ+ ℓ− . Diagrams involving Goldstone bosons G± are absent in the unitary gauge. remaining box diagrams is given by: (xt − 2)(3xt − 1) xt (xt − 2) + ln xt , 12(xt − 1)2 24(xt − 1)3 xt (xt + 1) (xt − 2)(3x2t − 3xt + 1) =− − ln xt , 48(xt − 1)2 24(xt − 1)3 box, SM CS, Feynman = − box, SM CS,Unitary (3.14) (3.15) where the two different expressions correspond to the results obtained in the Feynman and unitary gauges, respectively. In the SM there is an additional contribution to the scalar Wilson coefficient CS from the Higgs-penguin diagrams shown in figure 3, which is by itself gauge dependent [86, 92, 93] and should cancel the gauge dependence of the W -box contribution. We find the result: h penguin, SM CS, Feynman h penguin, SM CS, Unitary   xt − 3 xt (xt − 2) 3 xt − + ln xt , =− 8 xhSM 2(xt − 1)2 (xt − 1)3 3xt =− . 8xhSM (3.16) (3.17) The sum of the two contributions to CS is indeed gauge independent: box, SM h penguin, SM box, SM h penguin, SM CSSM = CS, = CS, Feynman + CS,Feynman Unitary + CS, Unitary =− (xt − 2)(3x2t − 3xt + 1) xt (xt + 1) 3xt − ln xt . − 8xhSM 48(xt − 1)2 24(xt − 1)3 – 11 – (3.18) JHEP06(2014)022 s t W± Z t W± Z s Z t ℓ s ℓ G± ℓ G± t Z (2.1) b W± G± W± t ℓ b t t b ℓ b ℓ s b b ℓ G± (3.2) W± t ℓ s ℓ b t ℓ s t b s hSM ℓ s (3.7) ℓ (3.8) ℓ t b hSM W± ℓ s (3.9) ℓ W± t s ℓ s (3.6) hSM G± b ℓ ℓ (3.4) hSM ℓ b b s G± G± hSM (3.5) ℓ s ℓ (3.10) Figure 3. SM Higgs-penguin diagrams contributing to B̄s0 → ℓ+ ℓ− . Contributions with Goldstone bosons G± are absent in the unitary gauge. b b ℓ ℓ ℓ b t t 0 t s G b t s ℓ (4.1) G0 (4.2) b ℓ s G± t b G0 (4.5) ℓ s W± ℓ s G± G0 ℓ b ℓ (4.3) ℓ G± s ℓ s (4.7) ℓ W± t s ℓ (4.4) b G0 G0 (4.6) t W ± G0 t t b W± t G± W± ℓ b G± s G0 ℓ s (4.8) ℓ Figure 4. SM Goldstone-penguin diagrams contributing to B̄s0 → ℓ+ ℓ− . These contributions are absent in the unitary gauge. The contribution from the SM W -box diagrams (figure 1) to the pseudoscalar Wilson coefficient CP is given by: xt (35x2t − 82xt − 1) 9x3t − 28x2t + xt + 2 − ln xt , 72(xt − 1)3 24(xt − 1)4 xt (71x2t − 172xt − 19) x4t − 12x3t + 34x2t − xt − 2 = + ln xt . 144(xt − 1)3 24(xt − 1)4 box, SM CP, Feynman = box, SM CP, Unitary (3.19) (3.20) Additional contributions to CP are generated by the Z- and Goldstone-penguin diagrams shown in figures 2 and 4, respectively. The contributions from diagrams 4.6, 4.7 – 12 – JHEP06(2014)022 s G± hSM (3.3) t W ± hSM t W ± hSM hSM t ℓ s (3.1) G± t hSM t ℓ W± G± W± b ℓ b t t and 4.8 are proportional to the light-quark mass and can be therefore neglected. We find: xt (5x2t + 16xt + 3) x4t + x3t + 18x2t − 12xt + 4 − ln xt (3.21) 48(xt − 1)3 24(xt − 1)4   xt (5x2t + 40xt − 21) 3x4t − 3x3t + 36x2t − 32xt + 8 2 − ln xt , − sW 72(xt − 1)3 36(xt − 1)4  
xt xt − 6 3xt + 2 (3.22) + ln xt , = 1 − s2W 4 xt − 1 (xt − 1)2 Z penguin, SM CP, = Feynman GB penguin, SM CP, Feynman and Using the above results, one can easily check that the SM contribution to CP is also gauge independent: box, SM Z penguin, SM GB penguin, SM box, SM Z penguin, SM CPSM = CP, + CP, = CP,Unitary + CP, (3.24) Feynman + CP,Feynman Feynman Unitary   1 xt (36x3t − 203x2t + 352xt − 209) 17x4t − 34x3t + 4x2t + 23xt − 6 = + ln xt 24 6(xt − 1)3 (xt − 1)4   s2W xt (18x3t − 139x2t + 274xt − 129) 24x4t − 33x3t − 45x2t + 50xt − 8 + ln xt . − 36 2(xt − 1)3 (xt − 1)4 The GIM mechanism has eliminated those contributions which are independent of the virtual top-quark mass. However, the ln xt terms in the Wilson coefficients CSSM and 1 ln xt and CPSM ∼ CPSM do not vanish in the massless limit: at xt ≪ 1, CSSM ∼ − 12
− 41 1 − 89 s2W ln xt . These infrared-sensitive terms arise from diagrams 1.1 and 2.1 in both gauges. The corresponding contributions from virtual up and charm quarks cancel in the matching process with the low-energy effective theory, which has the same infrared behaviour.2 3.4 Wilson coefficients in the A2HDM In the A2HDM, the only new contribution to C10 comes from the Z-penguin diagrams shown in figure 5. The result is gauge independent and given by   2 xH + 1 Z penguin, A2HDM 2 xt A2HDM = C10 = |ςu | C10 + (ln xt − ln xH + ) . (3.25) 8 xH + − xt (xH + −xt )2 In the particular case of the type-II 2HDM (or MSSM), ςu = 1/ tan β, this result agrees with the one calculated in ref. [36]. The box diagrams shown in figure 6 involve charged scalar exchanges and contribute to the Wilson coefficients CSA2HDM and CPA2HDM . The contributions from diagrams 6.3 and 6.4 2 In the low-energy effective theory the same ln xc (ln xu ) terms appear from analogous diagrams with a c ν̄ℓ (u ν̄ℓ ) or c c̄ (u ū) loop connecting two four-fermion operators. – 13 – JHEP06(2014)022   1 xt (18x3t − 137x2t + 262xt − 95) 8x4t − 11x3t − 15x2t + 12xt − 2 = + ln xt 12 6(xt − 1)3 (xt − 1)4   s2 xt (18x3t −139x2t +274xt −129) 24x4t −33x3t −45x2t +50xt −8 − W + ln x t . (3.23) 36 2(xt − 1)3 (xt − 1)4 Z penguin, SM CP, Unitary b ℓ b ℓ H± Z ℓ (5.1) t s t b Z ℓ (5.2) b ℓ H± t s b t H± s H± H± t ℓ s Z Z ℓ (5.3) s ℓ (5.4) Figure 5. Z-penguin diagrams involving H ± exchanges in the A2HDM. ℓ b H± s W± (6.1) W± νℓ t ℓ s H± (6.2) ℓ b νℓ t ℓ b H± s H± (6.3) νℓ t ℓ s ℓ H ± /G± G± /H ± (6.4) ℓ Figure 6. Box diagrams involving H ± exchanges in the A2HDM. Diagrams with Goldstone bosons are absent in the unitary gauge. 2 . For the scalar coefficients we can be neglected, since they are proportional to m2ℓ /MW find the results: (   xH + xt − xH + xt xt box, A2HDM ∗ ςℓ ςu ln xt − + ln xH + CS, Feynman = 8(xH + −xt ) (xH + −1)(xt −1) (xt −1)2 (xH + −1)2   xH + xH + (2xH + − xt −1) 1 ∗ + ln xt − ln xH + − ςu ςℓ xH + −1 (xH + −xt )(xt −1) (xH + −xt )(xH + −1)2 )   1 1 ∗ + 2 ςd ςℓ , (3.26) ln xt ln xH + − xH + − 1 xt − 1 (   xt xH + xt box, A2HDM ∗ CS, Unitary = ln xt − ln xH + ςℓ ςu 8(xH + − xt ) xt − 1 xH + − 1   xH + (xt − 1) xH + − x2t ∗ ln xt − ln xH + + ςu ςℓ 1 − (xH + − xt )(xt − 1) (xH + − xt )(xH + − 1) ) h i , (3.27) + 2 ςd ςℓ∗ ln xt − ln xH + while the pseudoscalar contributions are given by: box, A2HDM box, A2HDM CP, = −CS, Feynman Feynman box, A2HDM box, A2HDM CP, = −CS, Unitary Unitary ςℓ ςu∗ →−ςℓ ςu∗ ςℓ ςu∗ →−ςℓ ςu∗ , (3.28) . (3.29) Most of the previous calculations in the literature focused on the type-II 2HDM in the large tan β limit; i.e., only those contributions proportional to tan2 β were kept, which correspond to the ςd ςℓ∗ terms in eqs. (3.26)–(3.29). For this specific case, our results agree with ref. [28]. – 14 – JHEP06(2014)022 νℓ t ℓ b b ℓ b b ℓ t t H± s t t G0 ℓ s (7.1) b s G0 H± (7.2) ℓ H± G0 ℓ s (7.3) ℓ Figure 7. Goldstone-boson penguin diagrams involving H ± exchanges in the A2HDM. These contributions are absent in the unitary gauge. The gauge dependence of these two contributions compensates each other. Since there is no contribution from Goldstone-penguin topologies in the unitary gauge, the Z-penguin result should satisfy in this case: Z penguin, A2HDM Z penguin, A2HDM GB penguin, A2HDM CP, = CP, + CP, . Unitay Feynman Feynman (3.32) This relation has been validated by the actual calculation. 3.4.1 Neutral scalar exchange The Wilson coefficients CSA2HDM and CPA2HDM receive a direct tree-level contribution from the scalar-exchange diagram shown in figure 8, where the FCNC vertex ϕ0i s̄b is generated by the local operator in eq. (2.15). This contribution must be combined together with the scalar penguin diagrams shown in figure 9. The structure of the common ϕ0i ℓ̄ℓ vertex relates the resulting scalar and pseudoscalar Wilson coefficients, which take the form: X X 0 0 ϕ0 ϕ0 ϕ0 , A2HDM ϕ0 , A2HDM Im(yℓ i ) Ĉ ϕi . (3.33) Re(yℓ i ) Ĉ ϕi , CP i = i CS i = ϕ0i ϕ0i – 15 – JHEP06(2014)022 Similarly to the SM case, the coefficient CPA2HDM receives additional contributions from Z- and Goldstone-penguin diagrams shown in figures 5 and 7, respectively. They are given by: (   xt xt xH + xt + xH + Z penguin, A2HDM ∗ CP, Feynman = + (ln xH + − ln xt ) ςd ςu − 4(xH + − xt )2 2 xH + − xt  2 ) 2 2 (3x x −8x + xt −17xt x +x ) 1 + + H t H H + t (ln xH + −ln xt ) + |ςu |2 6(xH + −xt ) 6 xH + − xt (   x xH + (2xH + − 3xt ) t 2 ∗ 5xt − 3xH + + sW + (ln xH + − ln xt ) ςd ςu 6(xH + − xt )2 2 xH + − xt  3 4xH + − 12x2H + xt + 9xH + x2t + 3x3t 1 2 (ln xH + − ln xt ) − |ςu | 6(xH + − xt ) xH + − xt )  17x2H + − 64xH + xt + 71x2t − , (3.30) 6 h i x2t GB penguin, A2HDM 2 2 CP, = |ς | (1−s ) x + (ln xH + −ln xt ) + xt − xH + . (3.31) u H W Feynman 4(xH + −xt )2 ℓ b ϕ0i s ℓ Figure 8. Tree-level FCNC diagram mediated by the neutral scalars ϕ0i = {h, H, A}. ℓ t t ϕ0i t ℓ (9.1) b ϕ0i t ℓ (9.5) b ℓ W± s ℓ W± ℓ b W± s ϕ0i ϕ0i (9.7) ℓ s ℓ b (9.8) ℓ ℓ t ℓ 0 W ± ϕi s ℓ (9.10) ℓ b b b ℓ G± t t ϕ0i ℓ s (9.11) ℓ b t 0 G± ϕi ℓ ℓ b s t (9.15) ℓ s s ϕ0i ϕ0i G± ℓ s (9.13) b (9.14) G± W± t 0 G± ϕi (9.16) ℓ s ℓ ℓ b ℓ H± 0 H ± ϕi b ℓ G± t (9.12) G± s ℓ (9.4) ℓ b t ϕ0i b 0 H ± ϕi (9.9) t s ℓ (9.3) H± s s ϕ0i t (9.6) W± t s s ϕ0i H± b ℓ t G± b W± W± t ℓ (9.2) b ℓ t 0 H ± ϕi ℓ H± t s t s b H± H± s ℓ b ℓ b t 0 G± ϕi (9.17) ℓ 0 W ± ϕi s (9.18) ℓ Figure 9. Scalar penguin diagrams in the A2HDM, where ϕ0i = {h, H, A}. Diagrams 9.11 to 9.18 are absent in the unitary gauge. The contributions from diagrams 9.4, 9.7, 9.8 and 9.14 are proportional to the light-quark mass mq and, therefore, vanish in our massless approximation. Diagrams 9.1, 9.3, 9.11 and 9.13 in Feynman gauge and diagrams 9.1, 9.3, 9.5, 9.6, 9.9 and 9.10 in unitary gauge generate a divergent contribution, which is not eliminated by the GIM – 16 – JHEP06(2014)022 b Ĉ ϕ0i = xt v 2 ϕ0 1 (ςu −ςd )(1+ςu∗ ςd ) (Ri2 + iRi3 )CR (MW ) + 2 λHi+ H − g0 (xt , xH + , ςu , ςd ) 2xϕ0 Mϕ 0 i i   3 X 1 (a) (b) + Rij ξj g (xt , xH + , ςu , ςd ) + gj (xt , xH + , ςu , ςd ) , (3.34) 2xϕ0 j  j=1 i ϕ0 I where λHi+ H − = λ3 Ri1 + λR 7 Ri2 − λ7 Ri3 , ξ1 = ξ2 = 1 and ξ3 = i. The functions (a) (b) g0 (xt , xH + , ςu , ςd ), gj (xt , xH + , ςu , ςd ) and gj (xt , xH + , ςu , ςd ) are given in the appendix, both in the Feynman and unitary gauges, together with the separate contributions from each diagram in figure 9. In the limit ςu,d → 0, xH,A → ∞, xh → xhSM , Ri2,i3 → 0, R11 → 1, this result reduces to the SM expression in eqs. (3.16) and (3.17). The orthogonality relation [20, 21] 3 X i=1 ϕ0 yℓ i Rij = δj1 + (δj2 + i δj3 ) ςℓ (3.35) (b) allows us to separate the total contribution from the functions gj (xt , xH + , ςu , ςd ), which does not depend on the neutral scalar masses: h i ϕ0 , A2HDM (b) (b) (b) = x g + Re(ς ) g − i Im(ς ) g , (3.36) CS i t ℓ ℓ 1 2 3 g (b) h i ϕ0 , A2HDM (b) (b) CP i = xt i Im(ςℓ ) g2 − Re(ςℓ ) g3 . (3.37) (b) g (b) It is also noted that the functions gj (xt , xH + , ςu , ςd ) only receive contributions in the Feynman gauge, because they arise from the scalar penguin diagrams involving the Goldstone bosons. Actually, the gauge dependent pieces from the box diagrams shown in figures 1 and 6 are exactly cancelled by these terms: (b) box, SM box, SM CS, Unitary − CS, Feynman = xt g1 , h i (b) (b) box, A2HDM box, A2HDM CS, − C = x Re(ς ) g − i Im(ς ) g , t ℓ ℓ 2 3 Unitary S, Feynman h i (b) (b) box, A2HDM box, A2HDM CP, − CP, = xt i Im(ςℓ ) g2 − Re(ςℓ ) g3 . Unitary Feynman – 17 – (3.38) (3.39) (3.40) JHEP06(2014)022 mechanism; i.e., it remains even after summing over contributions of the three virtual uptype quarks. This divergence matches exactly the expected behaviour predicted through the RGEs, which originated in the local term LFCNC . Thus, the one-loop divergence is cancelled by the renormalization of the coupling C in eq. (2.16) which, moreover, reabsorbs the µ dependence of the loops into the combination CR (MW ) = CR (µ) − ln (MW /µ). The scalar penguin diagrams 9.2, 9.12, 9.15 and 9.16 involve the cubic couplings 0 ϕi H + H − , ϕ0i G+ G− , ϕ0i H + G− and ϕ0i G+ H − , respectively, which are functions of the scalar-potential parameters. Since the last three couplings can be fully determined in terms of the vacuum expectation value v and the scalar masses and mixings, we can express the total scalar-exchange (tree-level plus one-loop penguin) contribution in the form: We shall perform our phenomenological analyses in the CP-conserving limit, with real potential and alignment parameters, where A = S3 is a CP-odd state while H and h are two CP-even states defined by the rotation in eq. (2.6). The 1/xϕ0 contributions take then i the form:  xt ϕ0i , A2HDM CP con. (cα̃ + sα̃ ςℓ ) sα̃ (ςu − ςd ) (1 + ςu ςd ) CR (MW ) = CS 2xh C+g0 +g (a)  2v 2 (a) (a) + (cα̃ λ3 + sα̃ λ7 ) 2 g0 + cα̃ g1 + sα̃ g2 MW  xt + (cα̃ ςℓ − sα̃ ) cα̃ (ςu − ςd ) (1 + ςu ςd ) CR (MW ) (3.43) 2xH  2v 2 (a) (a) , − (sα̃ λ3 − cα̃ λ7 ) 2 g0 − sα̃ g1 + cα̃ g2 MW i xt h ϕ0 , A2HDM CP con. (a) (ς − ς ) (1 + ς ς ) C (M ) + g , (3.44) CP i = −ς u u d R W d ℓ 3 2xA C+g0 +g (a) where cα̃ = cos α̃ and sα̃ = sin α̃. For degenerate neutral scalars, this reproduces the results in eqs. (3.41) and (3.42) (in the CP-conserving limit). The terms proportional to CR (MW ) in eqs. (3.43) and (3.44) are absent in Z2 symmetric models, because the alignment conditions are protected by the Z2 symmetry at any scale. In the particular case of the type-II 2HDM at large tan β, the only terms (a) (a) enhanced by a factor tan2 β originate from the ςℓ g2 (for CS ) and ςℓ g3 (for CP ) contri(a) (a) butions, due to the factors ςd2 ςu∗ and ςd in the definitions for g2 and g3 (see eqs. (A.29) and (A.30)). In this specific case, our results agree with the ones calculated in ref. [28]. Especially, we confirmed the observation that the dependence on the masses of the neutral Higgs bosons from the penguin and fermion self-energy diagrams drops out in their sum without invoking any relation between the mixing angle and the Higgs masses [28]. – 18 – JHEP06(2014)022 The remaining contributions in eq. (3.34), which are all proportional to 1/Mϕ20 , are i gauge independent but are sensitive to the scalar mixing parameters. Nevertheless, a naive mixing-independent estimate can be obtained in the limit of degenerate neutral-scalar masses:  h i xt ϕ0i , A2HDM xh =xH =xA ∗ = CS (ς − ς ) (1 + ς ς ) C (M ) Re(ς ) − i Im(ς ) u R W d ℓ ℓ u d 2xh C+g0 +g (a) i h 2v 2 I Im(ς ) Re(ς ) + λ + 2 g 0 λ3 + λ R ℓ ℓ 7 7 MW  (a) (a) (a) +g1 + Re(ςℓ ) g2 − i Im(ςℓ ) g3 , (3.41)  h i xt ϕ0 , A2HDM xh =xH =xA ∗ CP i = (ς − ς ) (1 + ς ς ) C (M ) i Im(ς ) − Re(ς ) u R W d ℓ ℓ u d 2xh C+g0 +g (a) i h 2v 2 I + 2 g 0 i λR Re(ς ) Im(ς ) − λ ℓ ℓ 7 7 MW  (a) (a) . (3.42) +i Im(ςℓ ) g2 − Re(ςℓ ) g3 3.5 0 → ℓ+ ℓ− branching ratio Bs,d Due to the pseudoscalar nature of the Bq meson, only the following two hadronic matrix 0 → ℓ+ ℓ− decays: elements are involved in Bs,d 0|q̄ γµ γ5 b|B̄q (p) 0|q̄ γ5 b|B̄q (p) = ifBq pµ , = −ifBq MB2 q mb + mq , (3.45) τB G 4 M 4 SM 2 2 fBq MBq m2ℓ B(Bq0 → ℓ+ ℓ− ) = q F5 W Vtb Vtq∗ C10 8π h i = B(Bq0 → ℓ+ ℓ− )SM |P |2 + |S|2 , s 1− i 4m2ℓ h 2 2 |P | + |S| , MB2 q (3.46) where τBq is the Bq -meson mean lifetime, and P and S are defined, respectively, as [71–73]   MB2 q CP − CPSM mb C10 P ≡ SM + ≡ |P | eiφP , 2 SM mb + mq 2MW C10 C10 s  2  4m2ℓ MBq CS − CSSM mb S ≡ 1− 2 ≡ |S| eiφS . 2 SM mb + mq MBq 2MW C10 (3.47) (3.48) We have approximated the negligibly small (and usually neglected) SM scalar/pseudoscalar 2 . In the SM, P = 1 and S = 0. In contributions3 CSSM and CPSM to first order in MB2 q /MW a generic case, however, P and S can carry nontrivial CP-violating phases φP and φS . It is also noted that, even in models with comparable Wilson coefficients, the contributions 2 with respect to that from O . from OS and OP are suppressed by a factor MB2 q /MW 10 Therefore, unless there were large enhancements for CS and CP , the coefficient C10 still provides the dominant contribution to the branching ratio. In order to compare with the experimental measurement, the effect of Bq0 − B̄q0 oscillations should be taken into account, and the resulting averaged time-integrated branching ratio is given by [71–73]   1 + Aℓℓ 0 + − ∆Γ yq B(Bq → ℓ ℓ ) = B(Bq0 → ℓ+ ℓ− ) , (3.49) 1 − yq2 where Aℓℓ ∆Γ is a time-dependent observable introduced firstly in ref. [72, 73], and yq is related to the decay width difference ∆Γq between the two Bq -meson mass eigenstates, ΓqL − ΓqH ∆Γq yq ≡ q , q = 2Γq ΓL + ΓH 3 Here, CSSM and CPSM denote the full SM contribution, including the Higgs-penguin terms. – 19 – (3.50) JHEP06(2014)022 where fBq and MBq are the Bq -meson decay constant and mass, respectively. The second equation follows from the first one by using the QCD equation of motion for the quark fields. Starting with eq. (3.1) and using eq. (3.45), we can express the branching ratio of 0 Bs,d → ℓ+ ℓ− decays as the average with ΓqH(L) denoting the heavier (lighter) eigenstate decay width and Γq = τB−1 q Bq -meson width. Within the SM, Aℓℓ ∆Γ = 1 and the averaged time-integrated branching ratio is given by B(Bq0 → ℓ+ ℓ− )SM = 1 B(Bq0 → ℓ+ ℓ− )SM , 1 − yq G4 M 4 SM = F5 qW Vtb Vtq∗ C10 8π ΓH 2 fB2 q MBq m2ℓ s 1− 4m2ℓ . MB2 q (3.51) B(Bq0 h i 1 + Aℓℓ ∆Γ yq |P |2 + |S|2 B(Bq0 → ℓ+ ℓ− )SM , →ℓ ℓ ) = 1 + yq     ∆Γq 2 0 + − 2 = ˙ B(Bq → ℓ ℓ )SM |P | + 1 − q |S| , ΓL + −  (3.52) where the second line is valid only in the absence of beyond-SM sources of CP violation, which will be assumed in the following.4 4 Numerical results 4.1 Input parameters To evaluate numerically the branching ratios in eqs. (3.51) and (3.52), we need several input parameters collected in table 2. For the matching scale µ0 ∼ O(MW ) and the low-energy scale µb ∼ O(mb ), we fix them to µ0 = 160 GeV and µb = 5 GeV [55]. In addition, the on-shell scheme is adopted for the EW parameters, which means that the Zboson and top-quark masses coincide with their pole masses, and the weak angle is given 2 /M 2 , where M by s2W ≡ 1 − MW W = 80.359 ± 0.012 GeV is the W -boson on-shell mass Z obtained according to the fit formulae in eqs. (6) and (9) of ref. [101]. For the top-quark mass, we assume that the combined measurement of Tevatron and LHC [95] corresponds to the pole mass, but increase its systematic error by 1 GeV to account for the intrinsic ambiguity in the mt definition; i.e. we shall take Mt = (173.34 ± 0.27 ± 1.71) GeV. With the aid of the Mathematica package RunDec [102], four-loop QCD RGEs are applied to evolve the strong coupling αs (µ) as well as the MS renormalized masses mt (µ) and mb,s (µ) between different scales, and a three-loop relation has been used to convert the pole mass Mt to the scale-invariant mass mt (mt ), which gives mt (mt ) ≃ 163.30 GeV. The decay constants fBq are taken from the updated FLAG [96] average of Nf = 2 + 1 lattice determinations, which are consistent with the naive weighted average of Nf = 2 + 1 [103–105] and Nf = 2 + 1 + 1 [106, 107] results. For the Bq -meson lifetimes, while a sizable decay width difference ∆Γs has been established [97], the approximation 1/ΓdH ≃ 1/ΓdL ≡ τBd can be safely set, given the tiny SM expectation for ∆Γd /Γd [108]. 4 The explicit formulae in a generic case with new CP-violating phases could be found in refs. [57, 71–73]. – 20 – JHEP06(2014)022 By exploiting eqs. (3.46) and (3.51), we can rewrite eq. (3.49) as GF = 1.1663787 × 10−5 GeV−2 [94] αs (MZ ) = 0.1185 ± 0.0006 [94] MZ = 91.1876 ± 0.0021 GeV [94] MhSM = 125.9 ± 0.4 GeV [94] ∆αhadr (MZ ) = 0.02772 ± 0.00010 [94] Mt = 173.34 ± 0.27 ± 1.71 GeV MBs = 5366.77 ± 0.24 MeV MBd = 5279.58 ± 0.17 MeV ms (2 GeV) = 95 ± 5 MeV mµ = 105.65837 MeV [94] [94] [94] [94] [94] [96] fBd = 190.5 ± 4.2 MeV [96] τBs = 1.516 ± 0.011 ps [97] τBd = 1.519 ± 0.007 ps [97] 1/ΓsH = 1.615 ± 0.021 ps 1/ΓsL = 1.428 ± 0.013 ps ∆Γs = 0.081 ± 0.011 ps−1 |Vcb | = (42.42 ± 0.86) × [97] [97] [97] 10−3 |Vtb∗ Vts /Vcb | = 0.980 ± 0.001 |Vtb∗ Vtd | = 0.0088 ± 0.0003 [98] [99, 100] [99, 100] Table 2. Relevant input parameters used in our numerical analysis. For the CKM matrix element |Vcb |, we adopt the recent inclusive fit performed by taking into account both the semileptonic data and the precise quark mass determinations from flavour-conserving processes [98]. However, one should be aware of the present disagreement between inclusive and exclusive determinations [96]. With |Vcb | fixed in this way, the needed CKM factors are then obtained (within the SM) from the accurately known ratio |Vtb∗ Vts /Vcb | [99, 100]. 4.2 SM predictions SM is relevant and, using the fitting formula Within the SM, only the Wilson coefficient C10 in eq. (4) of ref. [55] (which has been transformed to our convention for the effective Hamiltonian),  1.52   0.89    Mt αs (MZ ) −0.09 αs (MZ ) −0.09 Mt SM C10 = −0.9604 + 0.0224 , 173.1 GeV 0.1184 173.1 GeV 0.1184 1.53    αs (MZ ) −0.09 Mt . (4.1) = −0.9380 173.1 GeV 0.1184 The EW and QCD factors introduced in eq. (3.12) are extracted as: ηYQCD = 1.010 . ηYEW = 0.977 , (4.2) With the input parameters collected in table 2, the SM predictions for the branching 0 → ℓ+ ℓ− decays are: ratios of Bs,d B(Bs → e+ e− ) = (8.58 ± 0.59) × 10−14 , B(Bs → µ+ µ− ) = (3.67 ± 0.25) × 10−9 , B(Bs → τ + τ − ) = (7.77 ± 0.53) × 10−7 , B(Bd → e+ e− ) = (2.49 ± 0.22) × 10−15 , B(Bd → µ+ µ− ) = (1.06 ± 0.10) × 10−10 , B(Bd → τ + τ − ) = (2.23 ± 0.20) × 10−8 , – 21 – (4.3) JHEP06(2014)022 mb (mb ) = 4.18 ± 0.03 GeV [95] fBs = 227.7 ± 4.5 MeV where a 1.5% nonparametric uncertainty has been set to the branching ratios, and the main parametric uncertainties come from fBq and the CKM matrix elements [55]. The small differences with respect to the results given in ref. [55] are due to our slightly different (more conservative) input value for the top-quark mass Mt . In order to explore constraints on the model parameters, it is convenient to introduce the ratio [71–73]     B(Bq0 → ℓ+ ℓ− ) ∆Γq 2 2 , (4.4) |S| Rqℓ ≡ = |P | + 1 − ΓqL B(Bq0 → ℓ+ ℓ− )SM Rdµ = 3.38+1.53 −1.35 , Rsµ = 0.79 ± 0.20 , SM (4.5) SM to be compared with the SM expectation Rsµ = Rdµ = 1. Since only the Bs → µ+ µ− branching ratio is currently measured with a signal significance of ∼ 4.0σ [54], we shall investigate the allowed parameter space of the A2HDM under the constraint from Rsµ given in eq. (4.5). Although the experimental uncertainty is still quite large, it has already started to put stringent constraints on many models beyond the SM [71]. Notice that, in addition to modifying the ratios Rqℓ , the scalar contributions to Bq0 –B̄q0 mixings also change the fitted values of the relevant CKM parameters and, therefore, the normalization B(Bq0 → ℓ+ ℓ− )SM . This should be taken into account, once more precise Bq0 → ℓ+ ℓ− data becomes available, through a combined global fit. 4.3 Results in the A2HDM 4.3.1 Choice of model parameters In the following we assume that the Lagrangian of the scalar sector preserves the CP symmetry i.e., that the only source of CP violation is still due to the CKM matrix. This makes all the alignment and scalar-potential parameters real. Assuming further that the lightest CP-even scalar h corresponds to the observed neutral boson with Mh ≃ 126 GeV, there are ten free parameters in our calculation: three alignment parameters ςf , three scalar masses (MH , MA , MH ± ), one mixing angle α̃, two scalar-potential couplings (λ3 , λ7 ), and the misalignment parameter CR (MW ). 0 → ℓ+ ℓ− decays, it In order to gain insight into the parameter space allowed by Bs,d is necessary to take into account information about the h(126) collider data and flavour physics constraints, as well as EW precision observables, which will be crucial for making simplifying assumptions and reducing the number of relevant variables. Explicitly, the following constraints and assumptions on the model parameters are taken into account: • Firstly, the mixing angle α̃ is constrained at | cos α̃| > 0.90 (68% CL) through a global fit to the latest LHC and Tevatron data for the h(126) boson [20, 21], which is very close to the SM limit; i.e., the lightest CP-even scalar h behaves like the SM Higgs boson. – 22 – JHEP06(2014)022 where the hadronic factors and CKM matrix elements cancel out. Combining the theoretical SM predictions in eq. (4.3) with the experimental results in eqs. (1.2) and (1.4), we get • To assure the validity of perturbative unitarity in the scalar-scalar scattering amplitudes, upper bounds on the quartic Higgs self-couplings are usually imposed by requiring them to be smaller than 8π [8, 9]; i.e., |λ3,7 | . 8π. • With our convention, the lower bound on the heavier CP-even scalar mass is MH ≥ Mh ≃ 126 GeV. Much lower values of MA are still allowed experimentally. There are, however, no stringent upper limits on these masses. Here we limit them at MH ∈ [130, 500] GeV and MA ∈ [80, 500] GeV. • The alignment parameters ςd and ςℓ are only mildly constrained through phenomenological requirements involving other model parameters. As in our previous works, we restrict them at |ςd,ℓ | ≤ 50 [23–27]. • At present, there are no useful constraints on the misalignment parameter CR (MW ). For simplicity, it is assumed to be zero. Numerically, it is found that the ratio Rsµ is less sensitive to the scalar-potential couplings λ3 and λ7 than to the other model parameters, especially when the alignment parameters are small and/or the neutral scalar masses are large. The mixing angle α̃, when constrained in the range cos α̃ ∈ [0.9, 1], is also found to have only a marginal impact on Rsµ . Thus, for simplicity, we shall assign the following values to these parameters: λ3 = λ7 = 1, cos α̃ = 0.95 . (4.6) As can be seen from eqs. (3.47) and (3.48), the Wilson coefficients CS and CP are 2 compared to C . The always accompanied with the power-suppressed factor MB2 q /MW 10 2 NP contribution to C10 is, however, proportional to |ςu | and depends only on the chargedscalar mass. It is, therefore, interesting to discuss the following two special cases with respect to the choice of the alignment parameters: the first one is when |ςd,ℓ | . |ςu | ≤ 2, where the NP contribution is dominated by C10 while CS and CP are negligible. The second one is when |ςd,ℓ | ≫ |ςu |, which means that CS and CP play a significant role. 4.3.2 Small ςd,ℓ When the alignment parameters ςd,ℓ are of the same size as (or smaller than) ςu , the NP contributions from CS and CP are negligible. In this case, we need only to focus on the SM and the charged-Higgs Wilson coefficient C10 , which is the sum of the SM contribution C10 A2HDM due to Z-penguin diagrams shown in figure 5. The latter involves contribution C10 only two free parameters, ςu and MH ± , and goes to zero when ςu → 0 and/or MH ± → ∞. The dependence of R̄sµ on the alignment parameter ςu with three typical charged-Higgs masses (80, 200 and 500 GeV) is shown in figure 10. One can see that, with the contributions from CS and CP ignored, the observable R̄sµ puts a strong constraint on the parameter ςu . – 23 – JHEP06(2014)022 • The charged Higgs mass is assumed to lie in the range MH ± ∈ [80, 500] GeV, which would require |ςu | ≤ 2 to be compatible with the present data on loop-induced pro0 − B̄ 0 mixing, as well as the h(126) cesses, such as Z → b̄b, b → sγ and Bs,d s,d decays [20, 21, 23–27]. For MH ± = 80 (500) GeV, a 95% CL upper bound |ςu | ≤ 0.49 (0.97) is obtained, with the assumption |ςd,ℓ | . |ςu |, which is stronger than the constraint from Rb [22]. Since A2HDM ∼ |ς |2 , this constraint is independent of any assumption about CP and, therefore, C10 u applies in the most general case.5 For larger charged-Higgs masses, the constraint becomes A2HDM = 0. weaker as the NP effect starts to decouple, reflected by lim C10 xH + →∞ 4.3.3 Large ςd,ℓ When ςd and ςℓ are large, the scalar and pseudoscalar operators can induce a significant enhancement of the branching ratio. To see this explicitly, we vary ςd and ςℓ within the range [−50, 50], and choose three representative values of ςu , ςu = 0, ±1. We also take three different representative sets of scalar masses: Mass1 : MH ± = MA = 80 GeV, MH = 130 GeV , Mass2 : MH ± = MA = MH = 200 GeV , Mass3 : MH ± = MA = MH = 500 GeV , (4.7) which cover the lower, intermediate, and upper range, respectively, of the allowed scalar spectrum. With the above specification, we show in figure 11 the allowed regions in the ςd –ςℓ plane under the constraint from R̄sµ . One can see that, irrespective of the scalar masses, regions with large ςd and ςℓ are already excluded, especially when they have the same sign. The 5 Actually, the explicit correction factor given at the end of eq. (3.52) is valid only in the absence of new sources of CP violation beyond the SM. Taking the correct general relation into account, the upper  1/4 bounded parameter is |ςu | 1 + ys cos (2φP − φNP ≈ |ςu |, where the phase φNP denotes the s ) /(1 + ys ) s 0 0 CP-violating NP contribution to Bs –B̄s mixing. – 24 – JHEP06(2014)022 Figure 10. Dependence of R̄sµ on ςu (left), for |ςd,ℓ | . |ςu | ≤ 2 and MH ± = 80, 200 and 500 GeV (upper, middle and lower curves, respectively). The shaded horizontal bands denote the allowed experimental region at 1σ (dark green), 2σ (green), and 3σ (light green), respectively. The right panel shows the resulting upper bounds on ςu , as function of MH ± . JHEP06(2014)022 Figure 11. Allowed regions (at 1σ, 2σ and 3σ) in the ςd –ςℓ plane under the constraint from R̄sµ , with three different assignments of the scalar masses and ςu = 0, ±1. impact of ςu , even when varied within the small range [−1, 1], is found to be significant: a nonzero ςu will exclude most of the regions allowed in the case with ςu = 0, and changing the sign of ςu will also flip that of ςℓ . This is mainly due to the factors ςd2 ςu∗ appearing (a) (a) in the functions g2 and g3 defined, respectively, by eqs. (A.29) and (A.30). It is also observed that the allowed regions expand with increasing scalar masses, as expected, since larger scalar masses make the NP contributions gradually decouple from the SM. 4.4 Z2 symmetric models The five types of Z2 -symmetric models listed in Table 1 are particular cases of the CPconserving A2HDM, with the three alignment factors ςf reduced to a single parameter tan β = v2 /v1 ≥ 0. In the particular scalar basis where the discrete Z2 symmetry is – 25 – implemented, the scalar-potential couplings µ′i and λ′i must be real, and µ′3 = λ′6 = λ′7 = 0; however, the rotation into the Higgs basis generates non-zero values of µ3 = − 12 λ6 v 2 and λ7 . Furthermore, the alignment condition is protected by the Z2 symmetry at any energy scale, which means that the misalignment parameter CR (MW ) does not contribute and the Higgs-penguin diagrams are free of divergences. Thus, for Z2 -symmetric models, the ratio R̄sµ only involves seven free parameters: MH ± , MH , MA , λ3 , λ7 , cos α̃, and tan β. A much more constrained case is the inert 2HDM, where the Z2 symmetry is imposed in the Higgs basis: all SM fields and Φ1 are even while Φ2 → −Φ2 under the Z2 transformation. This implies that there is no mixing between the CP-even neutral states h and H, and the scalars H, A and H ± decouple from the fermions: cos α̃ = 1, λ6 = λ7 = 0, ςf = 0. Moreover, the couplings of h to fermions and vector bosons are identical to the SM ones. inert = 1. Therefore, in the inert model R̄sµ For the other four types of Z2 -symmetric models, we continue to use the assignments cos α̃ = 0.95 and λ3 = λ7 = 1. One can easily check that the effects of MH and MA on R̄sµ are tiny, unless tan β is extremely small which is excluded by the flavour constraint |ςu | ≤ 2. For simplicity, we fix them to be MH = MA = 500 GeV in the following analysis. Figure 12 shows the dependence of R̄sµ on the parameter tan β, for three representative values of the charged-Higgs mass: MH ± = 80, 200 and 500 GeV. The four different panels correspond to the Z2 -symmetric models of types I, II, X and Y, respectively. A lower bound tan β > 1.6 is obtained at 95% CL under the constraint from the current experimental data on R̄sµ . This implies ςu = cot β < 0.63, which is stronger than the bounds obtained previously from other sources [22–27]. – 26 – JHEP06(2014)022 Figure 12. Dependence of R̄sµ on tan β for the 2HDMs of types I, II, X and Y. The upper, middle and lower curves correspond to MH ± = 80, 200 and 500 GeV, respectively. The horizontal bands denote the allowed experimental region at 1σ (dark green), 2σ (green), and 3σ (light green), respectively. 5 Conclusions 0 → ℓ+ ℓ− within In this paper, we have performed a detailed analysis of the rare decays Bs,d the general framework of the A2HDM. Firstly, we presented a complete one-loop calculation of the short-distance Wilson coefficients C10 , CS and CP , which arise from various box and penguin diagrams, and made a detailed technical summary of our results and a comparison with previous calculations performed in particular limits or approximations. In order to make sure our results are gauge independent, the calculations were carried out in both the Feynman and the unitary gauges. With the current data on B(Bs0 → µ+ µ− ) taken into account, we have also investigated the impact of various model parameters on the branching ratios and studied the phenomenological constraints imposed by present data. The resulting information about the model parameters will be crucial for the model building and is complementary to the collider searches for new scalar resonances in the near future. When |ςd,ℓ | . |ςu |, the contributions to B(Bs0 → µ+ µ− ) from the scalar and pseudoscalar operators are negligible compared to the leading Wilson coefficient C10 . Since A2HDM ∼ |ς |2 , the measured B(B 0 → µ+ µ− ) branching ratio implies then an upper C10 u s bound on the up-family alignment parameter, which only depends on the charged Higgs mass. At 95% CL, we obtain: |ςu | ≤ 0.49 (0.97) , for MH ± = 80 (500) GeV and |ςd,ℓ | . |ςu | . (5.1) This bound is stronger than the constraints obtained previously from other sources [22–27]. The role of the scalar and pseudoscalar operators becomes much more important for large values of |ςd,ℓ |. This region of parameter space was previously explored within the context of the type-II 2HDM, where these contributions are enhanced by a factor tan2 β. Our analysis agrees with previous results in the type-II case and shows, moreover, that this tan2 β enhancement is absent in the Z2 -symmetric models of types I, X and Y, which approach the SM prediction for large values of tan β. From the current experimental data on R̄sµ , we derive the 95% CL bound: tan β > 1.6 , for 2HDMs of types I, II, X and Y. (5.2) This implies ςu = cot β < 0.63, which is also stronger than the bounds obtained previously from other sources [22–27]. – 27 – JHEP06(2014)022 One can see that the predicted R̄sµ in the type-I, type-X and type-Y models are almost indistinguishable from each other and, in the large tan β region, approach the SM prediction, irrespective of the choices of scalar masses. For the type-II model, on the other hand, an enhancement of R̄sµ is still possible in the large tan β region. This can be understood since the Wilson coefficients in the type-II model contain the factor tan2 β arising from the product of alignment parameters ςf , while in the other three models they contain at most one power of tan β. So only the type-II model can receive a large tan β enhancement, which has been studied intensively in the literature [28–30]. It is also 0 → ℓ+ ℓ− branching interesting to note that in the type-II 2HDM with large tan β the Bs,d ratios depend only on the charged-Higgs mass and tan β [28]. Acknowledgments We are grateful to Alejandro Celis and Victor Ilisie for useful discussions on the 2HDM parameters. This work was supported in part by the National Natural Science Foundation of China (NSFC) under contract No. 11005032, the Spanish Government and ERDF funds from the EU Commission [Grants FPA2011-23778 and CSD2007-00042 (Consolider Project CPAN)] and by Generalitat Valenciana under Grant No. PROMETEOII/2013/007. X. Q. Li was also supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20104104120001) and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. A Scalar-penguin results within the A2HDM 0 The coefficients Ĉ ϕi , defined in eq. (3.33), are given by ) ( 18 2 X m 1 0 0 Ĉ ϕi = 2t C k, ϕi + (ςu − ςd ) (1 + ςu∗ ςd ) (Ri2 + iRi3 ) C , 2 Mϕ 0 i (A.1) k=1 where the last term is the tree-level contribution from the local operator in eq. (2.15). We 0 detail next the contributions C k, ϕi from the separate diagrams (k = 1, · · · , 18) shown in figure 9. The gauge-independent coefficients are: (   ϕ0i y xH + xt 0 u 1, ϕi ∗ C = 1− (ln xH + − ln xt ) ςd ςu 4 xH + − xt xH + − xt  ) 3x − x x (x − 2x ) x + + + t t t H H + H (ln xH + − ln xt ) + |ςu |2 2(xH + − xt )2 2 xH + − xt (   ϕ0 ∗ x2H + xt xt (2xH + − xt ) yu i ∗ ςd ςu Λ + − ln xH + + ln xt + 4 xH + − xt (xH + − xt )2 (xH + − xt )2  ) 2 x x 3x − x + + t t H H + |ςu |2 − (ln xH + − ln xt ) , (A.2) 2(xH + − xt )2 2 xH + − xt – 28 – JHEP06(2014)022 The enhancement of the scalar and pseudoscalar contributions at large values of |ςd,ℓ | is present in the most general A2HDM scenario and could give rise to interesting phenomenological signals. To exemplify this possibility, we have analyzed the ratio R̄sµ in the simpler CP-conserving case, showing the important impact of the A2HDM corrections whenever enhanced Yukawa couplings to leptons and down-type quarks are present. The resulting constraints on the alignment parameters are given in figure 11. It would be interesting to analyze the possible impact of the new CP-violating phases present within the A2HDM framework, at large values of |ςd,ℓ |. They could generate sizeable phases φP and φS in eqs. (3.47) and (3.48), which could manifest themselves in the timedependence of the Bs0 → µ+ µ− decay amplitude [71]. To quantify the possible size of this effect requires a more careful assessment of the allowed parameter space of the A2HDM, which we plan to further investigate in future works. C 2, ϕ0i 0 C 3, ϕi 0 (   ϕ0 s2W λHi+ H − xt ∗ ςd ςu (ln xH + − ln xt ) − 1 = 4πα(xH + − xt ) xH + − xt )  2 x − 3x x + t t (ln xH + − ln xt ) + H , + |ςu |2 2(xH + − xt )2 4(xH + − xt )   ϕ0 yd i xH + xt ∗ = ςd ςu −Λ + ln xH + − ln xt , 4 xH + − xt xH + − xt 0 0 C 4, ϕi = C 7, ϕi = C 8, ϕi = 0 . (A.3) (A.4) (A.5) 5, ϕ0i CUnitary 6, ϕ0 i CUnitary 9, ϕ0 i CUnitary 10, ϕ0 i CUnitary (  5x2t − 13xt + 2 2x3t − 6x2t + 9xt − 2 − ln xt Λ− 4(xt − 1)2 2(xt − 1)3 )  Λ 2x2t − xt − 7 x3t − 3x2t + 3xt + 2 ϕ0i , (A.6) − − ln xt + yu 2 4(xt − 1)2 2(xt − 1)2   ϕ0 λWi + W − x2t − 2xt − 11 3xt (x2t − 3xt + 4) (A.7) −3Λ + + ln xt , = 8 2(xt − 1)2 (xt − 1)3   ϕ0 λHi+ W − ∗ 1 xH + (xH + + 2) ln xH + xt (xt + 2) ln xt = ςu −Λ+ − , (A.8) 8 2 (xH + − 1)(xH + − xt ) (xt − 1)(xH + − xt ) (    ϕ0 ∗ λHi+ W − ςu xt (xH + xt −4xH + +3xt ) xH + ln xH + xt ln xt − − ln xt ςd −Λ + = 4 xH + − xt xH + −xt 2 (xt − 1)(xH + − xt )2 ) xH + xH + (xH + xt − 3xH + + 2xt ) + . (A.9) − ln xH + xH + − xt (xH + − 1)(xH + − xt )2 1 = 4 ϕ0 ∗ yu i  In the Feynman gauge the results are: 5, ϕ0i CFeynman 6, ϕ0 i CFeynman 9, ϕ0 i CFeynman 10, ϕ0 i CFeynman (    ) 2 ln xt 2(1−2xt ) ln xt 1 ϕ0i ϕ0i ∗ + yu 3 − x t − yu 3xt − 1 + , (A.10) = 8(xt −1)2 xt − 1 xt − 1   ϕ0 λWi + W − 2xt ln xt − xt − 1 , (A.11) = 4(xt − 1)2 xt − 1   ϕ0 ςu∗ λHi+ W − xH + (3xH + −2) ln xH + xt (3xt −2) ln xt xH + − xt , + − = 8(xH + − xt ) (xH + −1)(xt −1) (xH + − 1)2 (xt − 1)2 (A.12) ( 0    ϕ ∗ λHi+ W − xt ln xt xH + ln xH + xt (4xH + −3xt ) ln xt ςu xH + − ςd + + = 4(xH + −xt ) xt − 1 xH + − 1 2 xH + − 1 (xt − 1)(xH + − xt ) )  xH + (4x2H + − 3xH + xt − 3xH + + 2xt ) − ln xH + , (A.13) (xH + − 1)2 (xH + − xt ) – 29 – JHEP06(2014)022 In the unitary gauge, we find: 11, ϕ0i CFeynman 12, ϕ0 i CFeynman 13, ϕ0 i CFeynman ϕ0 ∗ yu i   (A.14) (A.15) (A.16) i CFeynman = 0, 15, ϕ0 i CFeynman = 16, ϕ0 i CFeynman = 17, ϕ0 i CFeynman = 18, ϕ0 i CFeynman = (A.17)   s2W ςu∗ λ xH + (xH + −2) xt (xt −2) xH + − xt ln xt − + ln xH + , 8πα(xH + − xt ) (xH + −1)(xt −1) (xt − 1)2 (xH + − 1)2 (A.18) ( 0    ϕ ∗ s2W λHi+ G− x2t ln xt xt xH + ln xt − ln xH + + ςu 2 ςd 8πα(xH + − xt ) xt −1 xH + −1 (xt −1)(xH + −xt ) ) x + (xH + xt + xH + − 2xt ) xH + − H ln xH + , (A.19) + xH + − 1 (xH + − 1)2 (xH + − xt )   ϕ0 λGi+ W − 5 − 7xt xt (3xt − 2) (A.20) + ln xt , 8(xt − 1)2 2 xt − 1   ϕ0 λGi+ W − 9xt − 11 xt (5xt − 6) − (A.21) ln x t . 8(xt − 1)2 2 xt − 1 ϕ0i H + G− D−4 µ Here Λ = − 2D−4 −γE +ln (4π)−ln are defined, respectively, as ϕ0  2 MW µ2  +1, and the (rescaled) cubic coupling constants ϕ0 λWi + W − = λGi+ W − = Ri1 , (A.22) λ (A.23) ϕ0i H+W − ϕ0 λHi+ H − ϕ0 = Ri2 − iRi3 , I = λ3 Ri1 + λR 7 Ri2 − λ7 Ri3 , I λGi+ G− = 2λ1 Ri1 + λR 6 Ri2 − λ6 Ri3 = ϕ0 λHi+ G− (A.24) Mϕ20 i v2 Ri1 , (A.25) 2 Mϕ20 −MH + i 1 i (Ri2 − iRi3 ) . (A.26) = λ6 Ri1 + (λ4 +2λ5 ) Ri2 − (λ4 −2λ5 ) Ri3 = 2 2 2 v (a) The functions g0 (xt , xH + , ςu , ςd ) and gj (xt , xH + , ςu , ςd ) introduced in eq. (3.34) are gauge independent. For g0 (xt , xH + , ςu , ςd ) we find 0 g0 (xt , xH + , ςu , ςd ) = – 30 – πα C 2, ϕi ϕ0 s2W λHi+ H − , (A.27) JHEP06(2014)022 14, ϕ0 ( xt (5xt − 7) xt (2x2t − 6xt + 5) Λ− − ln xt 4(xt − 1)2 2(xt − 1)3 ) ϕ0  xt yu i xt (xt − 3) , + ln xt − 2 2(xt − 1)2 (xt − 1)3   ϕ0 s2W λGi+ G− 2xt (xt − 2) = xt − 3 − ln xt , 16πα(xt − 1)2 xt − 1  ϕ0  yd i xt ln xt , −Λ + = 4 xt − 1 1 = 4 (a) while the functions gj (xt , xH + , ςu , ςd ) are given, respectively, as:   xH + xt 1− + (ln xH + − ln xt ) (A.28) 4 xH + − xt xH + − xt   xt xH + + xt xH + xt +|ςu |2 − (ln xH + − ln xt ) , 2 2(xH + − xt ) 2 xH + − xt 3 (a) g1 (xt , xH + , ςu , ςd ) = − (a) ςd ςu∗ g2 (xt , xH + , ςu , ςd ) = ςd2 ςu∗ f1 (xt , xH + ) + ςd (ςu∗ )2 f2 (xt , xH + ) 2 2 +ςd |ςu | f3 (xt , xH + ) + ςu |ςu | f4 (xt , xH + ) − (A.29) ςu∗ |ςu |2 f5 (xt , xH + ) (a) g3 (xt , xH + , ςu , ςd ) = ςd2 ςu∗ f1 (xt , xH + ) − ςd (ςu∗ )2 f2 (xt , xH + ) (A.30) +ςd |ςu |2 f3 (xt , xH + ) + ςu |ςu |2 f4 (xt , xH + ) + ςu∗ |ςu |2 f5 (xt , xH + ) +ςu f6 (xt , xH + ) + ςu∗ f7 (xt , xH + ) + ςd f1 (xt , xH + ) . (b) The functions gj (xt , xH + , ςu , ςd ) are zero in the unitary gauge, because they are all related to Goldstone boson vertices. In the Feynman gauge, they are given, respectively, as (b) g1,Feynman (xt , xH + , ςu , ςd ) (b) 1 = 8(xt − 1)2   xt − 3 xt (xt − 2) − ln xt , 2 xt − 1 (A.31) g2,Feynman (xt , xH + , ςu , ςd ) = ςd f8 (xt , xH + ) + ςu f9 (xt , xH + ) + ςu∗ f10 (xt , xH + ) , (A.32) (b) g3,Feynman (xt , xH + , ςu , ςd ) (A.33) = ςd f8 (xt , xH + ) + ςu f9 (xt , xH + ) − ςu∗ f10 (xt , xH + ) . Here the functions fj (xt , xH + ) are defined, respectively, as f1 (xt , xH + ) = f2 (xt , xH + ) = f3 (xt , xH + ) = f4 (xt , xH + ) = f5 (xt , xH + ) = f6 (xt , xH + ) = 1 [−xH + + xt + xH + ln xH + − xt ln xt ] , 2(xH + − xt )   1 xH + xt (ln xH + − ln xt ) , xt − 2(xH + − xt ) xH + − xt   x2H + ln xH + xt (2xH + − xt ) ln xt 1 , + x +− 2(xH + − xt ) H xH + − xt xH + − xt   x2H + xt 1 xt (3xH + − xt ) − (ln xH + − ln xt ) , 4(xH + − xt )2 2 xH + − xt   1 xt (xH + −3xt ) xH + xt (xH + −2xt ) − (ln xH + −ln xt ) , 4(xH + −xt )2 2 xH + − xt "
xt x2t − 3xH + xt + 9xH + − 5xt − 2 1 2(xH + − xt ) 4(xt − 1)2 x2H + (A.35) (A.36) (A.37) (A.38) (A.39) xH + (xH + xt − 3xH + + 2xt ) ln xH + 2(xH + − 1)(xH + − xt ) #

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