Revisiting the B -physics anomalies in R -parity violating MSSM
RRevisiting the B -physics anomalies in R -parity violating MSSM Quan-Yi Hu, ∗ Ya-Dong Yang, † and Min-Di Zheng ‡ School of Physics and Electrical Engineering, Anyang Normal University, Anyang, Henan 455000, China Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),Central China Normal University, Wuhan, Hubei 430079, China
In recent years, several deviations from the Standard Model predictions in semileptonic decays of B -mesonmight suggest the existence of new physics which would break the lepton-flavour universality. In this work,we have explored the possibility of using muon sneutrinos and right-handed sbottoms to solve these B -physicsanomalies simultaneously in R -parity violating minimal supersymmetric standard model. We find that thephotonic penguin induced by exchanging sneutrino can provide sizable lepton flavour universal contributiondue to the existence of logarithmic enhancement for the first time. This prompts us to use the two-parameterscenario ( C V9 , C U9 ) to explain b → s(cid:96) + (cid:96) − anomaly. Finally, the numerical analyses show that the muonsneutrinos and right-handed sbottoms can explain b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies simultaneously, andsatisfy the constraints of other related processes, such as B → K ( ∗ ) ν ¯ ν decays, B s − ¯ B s mixing, Z decays, aswell as D → µ + µ − , τ → µρ , B → τ ν , D s → τ ν , τ → Kν , τ → µγ , and τ → µµµ decays. I. INTRODUCTION
Recently, several flavour anomalies in semileptonic B -decays have been reported, which have been attracting greatinterest. Among them, the observables R K ( ∗ ) = B ( B → K ( ∗ ) µ + µ − ) / B ( B → K ( ∗ ) e + e − ) in flavour-changing neu-tral current b → s(cid:96) + (cid:96) − ( (cid:96) = e, µ ) transition and the ob-servables R ( D ( ∗ ) ) = B ( B → D ( ∗ ) τ ν ) / B ( B → D ( ∗ ) (cid:96)ν ) in flavour-changing charged current b → cτ ν transition areparticularly striking. The advantage of considering the ratios R K ( ∗ ) and R ( D ( ∗ ) ) instead of the branching fractions them-selves is that, apart from the significant reduction of the ex-perimental systematic uncertainties, the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements cancel out and the depen-dence on the transition form factors become much weaker.These observables can be good probes to test the lepton-flavour universality (LFU) held in the Standard Model (SM).The latest measurement of R K by LHCb collaborationgives [1, 2] R K = 0 . +0 . . − . − . , . < q < , (1)but the SM prediction is around 1 with O (1%) uncertainty [3],there is . σ discrepancy. Moreover, the measurement of R K ∗ by LHCb at low and high q are [4] R K ∗ = (cid:40) . +0 . − . ± . , . < q < . . +0 . − . ± . , . < q < . , (2)while the SM predictions are R [0 . , . K ∗ = 0 . ± . and R [1 . , . K ∗ = 1 . ± . [3]. The measurements show . σ discrepancy in the low q region and . σ discrepancyin the high q region, respectively. The Belle collaborationalso reported their measurements of R K ( ∗ ) [5, 6], which are ∗ [email protected] (corresponding author) † [email protected] ‡ [email protected] consistent with the SM predictions within their quite large er-ror bars. In addition to R K ( ∗ ) , there are also some other devia-tions in b → sµ + µ − transition, such as the angular observable P (cid:48) [7–9] of B → K ∗ µ + µ − decay with . σ discrepancy [10–15] and the differential branching fraction of B s → φµ + µ − decay with . σ discrepancy [16, 17].These deviations indicate the possible existence of newphysics (NP) beyond the SM in b → s(cid:96) + (cid:96) − transition. ThisNP may break LFU. Many recent model-independent analy-ses [18–25] show that some scenarios can explain the b → s(cid:96) + (cid:96) − anomaly well. To express the fit results, we con-sider the low-energy effective weak Lagrangian governing the b → s(cid:96) + (cid:96) − transition L eff = 4 G F √ η t (cid:88) i C i O i + H . c ., (3)where CKM factor η t ≡ V tb V ∗ ts . We mainly concern thesemileptonic operators O = e π (¯ sγ µ P L b )(¯ (cid:96)γ µ (cid:96) ) , (4) O = e π (¯ sγ µ P L b )(¯ (cid:96)γ µ γ (cid:96) ) , (5)where P L = (1 − γ ) / is the left-handed chirality projec-tor. The Wilson coefficients C , = C SM9 , + C NP9 , . In thiswork, we try to explain the anomaly through a two-parameterscenario where the total NP effects are given by [26] C NP9 ,µ = C V9 + C U9 , C NP10 ,µ = − C V9 , (6) C NP9 ,e = C U9 , C NP10 ,e =0 . (7)The global analyses show that this scenario has the largestpull-value. The best-fit point performed by Ref. [20] is ( C V9 , C U9 ) = ( − . , − . , with the σ range being − . < C V9 < − . , − . < C U9 < − . . (8)As we will see in the following discussion, this scenario canbe implemented naturally in the R -parity violating minimalsupersymmetric standard model (MSSM) [27]. a r X i v : . [ h e p - ph ] M a y The combined measurements of R ( D ∗ ) and R ( D ) are fromBaBar [28, 29] and Belle [30, 31], and Belle [32, 33] andLHCb [34–36] only give the measurements of R ( D ∗ ) . Af-ter being averaged by the Heavy Flavor Averaging Group(HFLAV) [37], they give the results as follows [38] R ( D ) avg =0 . ± . ± . , (9) R ( D ∗ ) avg =0 . ± . ± . , (10)with a correlation of − . . Comparing these with the arith-metic average of the SM predictions [38–42], R ( D ) SM = 0 . ± . , R ( D ∗ ) SM = 0 . ± . , (11)one can see that the difference between experiment and the-ory is at about . σ , implying the existence of LFU violatingNP in the charged-current B -decays. Global analyses [43–47] show that the NP contributing to the left-handed operator (¯ cγ µ P L b )(¯ τ γ µ P L ν ) can solve the R ( D ( ∗ ) ) anomaly. Suchoperator can be generated in R -parity violating MSSM by ex-changing the right-handed down type squarks at tree level.There have been attempts to explain the b → s(cid:96) + (cid:96) − anomaly [48–52] or R ( D ( ∗ ) ) anomaly [53–57] or both ofthem [58–60] by R -parity violating interactions in the super-symmetric (SUSY) models. For example, based on the inspi-ration from the paper by Bauer and Neubert [61], the authorsin Ref. [58] investigated the possibility of using right-handeddown type squarks to explain the b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies simultaneously, and found that this was impossibledue to the severe constraints from B → K ( ∗ ) ν ¯ ν decays. Con-sidering that the parameter space obtained by using squarksto explain b → s(cid:96) + (cid:96) − anomaly is very small [49, 50, 58] dueto the strict constraints from other related processes, such as B → K ( ∗ ) ν ¯ ν decays and B s − ¯ B s mixing, the authors inRef. [52] used sneutrinos to explain it and found that it is al-most unconstrained by other related processes. Based on thisknowledge, in this work, we will explore the possibility of us-ing muon sneutrinos ˜ ν µ and right-handed sbottoms ˜ b R to ex-plain the b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies simultaneouslywithin the context of R -parity violating MSSM.Our paper is organized as follows. In Sec. II, we scruti-nize all the one-loop contributions of terms λ (cid:48) ijk L i Q j D ck to b → s(cid:96) + (cid:96) − processes in the framework of R -parity violatingMSSM, and then give our scenario to explain the b → s(cid:96) + (cid:96) − anomaly. Discussions of R ( D ( ∗ ) ) anomaly and other relatedprocesses are included in Sec. III. The numerical analyses andresults are shown in Sec. IV. Our conclusions are finally madein Sec. V. II. b → s(cid:96) + (cid:96) − PROCESSES IN R -PARITY VIOLATINGMSSM The superpotential terms violating R -parity in the MSSMare [27] W RPV = µ i L i H u + 12 λ ijk L i L j E ck + λ (cid:48) ijk L i Q j D ck + 12 λ (cid:48)(cid:48) ijk U ci D cj D ck , (12)where the generation indices are denoted by i, j, k = 1 , , and the colour indices are suppressed. All repeated indicesare assumed to be summed over throughout this paper un-less otherwise stated (For example, repeated indices in bothnumerator and denominator are not automatically summed). H u , L and Q are SU (2) doublet chiral superfields while E c , D c and U c are SU (2) singlet chiral superfields.In this work, we are mainly interested in the terms λ (cid:48) ijk L i Q j D ck which related to both quarks and leptons. Thischoice can also alleviate the constraint of sneutrino masses onthe collider, because the lower limit of sneutrino masses willbe as high as TeV scale [62–65] when there are non-zero λ and λ (cid:48) at the same time. The corresponding Lagrangian can be ob-tained by the chiral superfields composing of the fermions andsfermions as follows L = λ (cid:48) ijk (cid:0) ˜ ν Li ¯ d Rk d Lj + ˜ d Lj ¯ d Rk ν Li + ˜ d ∗ Rk ¯ ν cLi d Lj − ˜ l Li ¯ d Rk u Lj − ˜ u Lj ¯ d Rk l Li − ˜ d ∗ Rk ¯ l cLi u Lj (cid:1) + H . c ., (13)where the sparticles are denoted by “ ˜ ”, and “c” indicatescharge conjugated fields. Working in the mass eigenstatesfor the down type quarks and assuming sfermions are in theirmass eigenstates, one replaces u Lj by ( V † u L ) j in Eq. (13).These R -parity violating interactions can induce b → s(cid:96) + (cid:96) − processes by exchanging left-handed up squarks ˜ u Lj at tree level, but resulting in the operators with right-handedquark current, which are unable to explain the b → s(cid:96) + (cid:96) − anomaly. This unwanted effect can be eliminated by assumingthat the masses of ˜ u Lj are very large or/and by assuming that λ (cid:48) ij = 0 . Assuming that λ (cid:48) ij = 0 also forbids the exchangeof ˜ l Li or/and ˜ d Lj in one loop level to affect the b → s(cid:96) + (cid:96) − processes . In the following discussion, we should assumethat λ (cid:48) ij = λ (cid:48) ij = 0 .Next, we will show the contributions of R -parity violatingMSSM to b → s(cid:96) + (cid:96) − processes. All the Feynman diagramsinclude four ˜ W − b box diagrams (Fig. 1a), five W − ˜ b R box diagrams (one of which is Goldstone − ˜ b R box diagram)(Fig. 1b), one H ± − ˜ b R box diagram (Fig. 1c), two λ (cid:48) boxdiagrams (Fig. 1d) and two γ -penguin diagrams (Fig. 2). Mostof these results can be found in Refs [49, 50, 52, 58], however,to our knowledge, the results of the diagram induced by ex-changing charged Higgs H ± and right-handed sbottom ˜ b R inloop are the first to be given in this paper. The photonic pen-guin diagrams, which have been neglected in previous work,play an important role in our discussion, as we will explain inmore detail later. We do not find sizable Z -penguin contribu-tions to b → s(cid:96) + (cid:96) − processes. In this work, the contributionsof γ/Z -penguin diagrams always include their supersymmet-ric counterparts unless otherwise specified. For convenience, In this work, we don’t consider contributions only from R -parity conserv-ing MSSM, because these contributions can be ignored numerically [66]. the following Passarino-Veltman functions [67] D and D are defined as D [ m , m , m , m ] ≡ (cid:90) d k (2 π ) k − m )( k − m )( k − m )( k − m )= − i π (cid:20) m log( m )( m − m )( m − m )( m − m )+ ( m ↔ m ) + ( m ↔ m ) + ( m ↔ m ) (cid:21) , (14) D [ m , m , m , m ] ≡ (cid:90) d k (2 π ) k ( k − m )( k − m )( k − m )( k − m )= − i π (cid:20) m log( m )( m − m )( m − m )( m − m )+ ( m ↔ m ) + ( m ↔ m ) + ( m ↔ m ) (cid:21) . (15) b sµ µ ˜ ν b ˜ ν ˜ W (a) µ µb su ˜ b R uW (b) µ µb su ˜ b R uH + (c) b µs µ ˜ b R ˜ b R ν u (d)1 FIG. 1. Box diagrams for b → sµ + µ − transition in our scenario.Fig. 1a shows an example ˜ W − b box diagram, Fig. 1b shows anexample W − ˜ b R box diagram, Fig. 1c shows the H ± − ˜ b R boxdiagram, and Fig. 1d shows an example λ (cid:48) box diagram. The contributions of box diagram are listed below. Weeliminate the contributions of all box diagrams to b → se + e − processes by assuming λ (cid:48) j = 0 . • The contributions of ˜ W − b box diagram to b → sµ + µ − processes are given by C V( ˜ W )9 = − iπ √ G F sin θ W η t × (cid:16) λ (cid:48) i λ (cid:48)∗ V ib D [ m W , m u Li , m ν µ , m b ] − λ (cid:48) i λ (cid:48)∗ j V ib V ∗ js D [ m W , m u Li , m u Lj , m b ]+ λ (cid:48) λ (cid:48)∗ j V ∗ js D [ m W , m u Lj , m ν µ , m b ] − λ (cid:48) λ (cid:48)∗ D [ m W , m ν µ , m ν µ , m b ] (cid:17) , (16)where the winos engage these interactions with left-hand up type squarks and muon sneutrinos. The lastterm plays an important role in numerical analysis [52]. • The contributions of W − ˜ b R box diagram to b → sµ + µ − processes are given by C V( W )9 = − iπ √ G F sin θ W η t × (cid:16) ˜ λ (cid:48) i λ (cid:48)∗ V ib D [ m b R , m u i , m W , − ˜ λ (cid:48) i ˜ λ (cid:48)∗ j V ib V ∗ js D [ m b R , m u i , m u j , m W ]+ λ (cid:48) ˜ λ (cid:48)∗ j V ∗ js D [ m b R , m u j , m W , − λ (cid:48) λ (cid:48)∗ D [ m b R , m W , , λ (cid:48) i ˜ λ (cid:48)∗ j V ib V ∗ js m u i m u j m W × D [ m b R , m u i , m u j , m W ] (cid:17) , (17)where ˜ λ (cid:48) ijk ≡ λ (cid:48) ilk V ∗ jl . The right-hand sbottom ˜ b R isthe only NP particle here. In the limit m ˜ b R (cid:29) m t , onehas C V( W )9 = m t παm bR | λ (cid:48) | [49, 50, 61] which isobviously positive. • The contributions of H ± − ˜ b R box diagram to b → sµ + µ − processes are given by C V( H ± )9 = − iπ V ib V ∗ js ˜ λ (cid:48) i ˜ λ (cid:48)∗ j √ G F sin θ W tan βη t m u i m u j m W × D [ m H ± , m u i , m u j , m b R ] , (18)which should be considered in the following numeri-cal analysis. The tan β = v u /v d where v u and v d arethe vacuum expectation values of two Higgs doubletsrespectively. • The contributions of λ (cid:48) box diagram to b → sµ + µ − processes are given by C V(4 λ (cid:48) )9 = − iπλ (cid:48) i λ (cid:48)∗ i √ G F αη t (cid:16) | ˜ λ (cid:48) j | D [ m b R , m b R , m u j , | λ (cid:48) j | D [ m u Lj , m ν i , m b , m b ] (cid:17) . (19) bs ‘‘bb ˜ ν γ (a) bs ‘‘ ˜ b R ˜ b R ν γ (b)1 FIG. 2. Photonic penguin diagrams studied in our scenario.
The contributions of photonic penguin diagrams are leptonflavour universal which naturally gives us a nonzero C U9 C U9 = √ λ (cid:48) i λ (cid:48)∗ i G F η t (cid:20) m b R − (cid:18)
43 + log m b m ν i (cid:19) m ν i (cid:21) . (20)As stated in Ref. [52], this result is consistent with that inRef. [68], but it has a negative sign different from that inRef. [50]. The first term in Eq. (20) comes from the contri-bution of Fig. 2b, like the photonic penguin induced by scalarleptoquark. We find this term gives a negligible contribution,which is in agreement with Refs. [61, 69]. However the sec-ond term in Eq. (20) has a significant contribution because ofthe logarithmic enhancement, which has never been addressedbefore. These photonic penguins also contribute new electro-magnetic dipole operator O = m b e (¯ sσ αβ P R b ) F αβ , which isstrictly constrained by B → X s γ decay [9]. Fortunately, wefind that the corresponding contribution can be ignored nu-merically because there such logarithmic enhancement is ab-sent [50, 52, 68].We will discuss the possibility of using muon sneutrinos ˜ ν µ and right-handed sbottoms ˜ b R to explain b → s(cid:96) + (cid:96) − anomaly, for which we set the mass of tau sneutrinos ˜ ν τ andthree left-handed up type squarks ˜ u Lj sufficiently large thatthe contributions of the loop diagrams containing them areignored . The contribution from H ± − ˜ b R box diagram isusually positive, and we find that it is numerically negligiblewhen tan β > . Thus, the contributions to only muon chan-nel are C V9 = − √ λ (cid:48) λ (cid:48)∗ f ( x ˜ ν µ )32 G F sin θ W η t m ν µ + | λ (cid:48) | x ˜ b R πα (21) − λ (cid:48) i λ (cid:48)∗ i (cid:104) | ˜ λ (cid:48) | + | ˜ λ (cid:48) | − | ˜ λ (cid:48) | f (cid:0) /x ˜ b R (cid:1)(cid:105) √ πG F αη t m b R , where x ˜ ν µ ≡ m ν µ /m W , x ˜ b R ≡ m t /m b R , and the loop func-tion f ( x ) ≡ x (1 − x +log x )(1 − x ) . III. R ( D ( ∗ ) ) ANOMALY AND OTHER CONSTRAINTS
In this section, we discuss the interpretation of R ( D ( ∗ ) ) anomaly and consider the constraints imposed by other relatedprocesses from B, D, K, τ , and Z decays. III.1. R ( D ( ∗ ) ) anomaly In R -parity violating MSSM, the charged current processes d j → u n l l ν i are induced by exchanging ˜ b R at tree level. Theeffective Lagrangian of these processes are given by L eff = − G F √ V nj ( δ li + C njli )¯ u n γ µ P L d j ¯ l l γ µ P L ν i + H . c ., (22) In our numerical analysis, we find that the contribution of the loop diagramscontaining ˜ ν τ is numerically negligible when the mass of ˜ ν τ is a few TeVor larger. The same conclusion is true for ˜ u L where the mass of ˜ u L isa few 10TeV or larger. Here, we consider that m ˜ ν µ < m ˜ ν τ , which canbe achieved, for example, by setting the hierarchy of neutrino Yukawas Y ν < Y ν in the µν SSM [70]. where the Wilson coefficient C njli is C njli = λ (cid:48) ij ˜ λ (cid:48)∗ ln √ G F V nj m b R . (23)Because taking λ (cid:48) j = 0 to eliminate the contributions ofbox diagrams to b → se + e − processes , we have C nj i = C njl = 0 . It is useful to define the ratio R njl ≡ B ( d j → u n l l ν ) B ( d j → u n l l ν ) SM = (cid:88) i =1 | δ li + C njli | , (24)and we have R ( D ) R ( D ) SM = R ( D ∗ ) R ( D ∗ ) SM = 2 R R + 1 . (25)To obtain the allowed parameter region, we use the followingbest fit value in the R -parity violating scenario R ( D ) R ( D ) SM = R ( D ∗ ) R ( D ∗ ) SM = 1 . ± . . (26) III.2. Constraints from the tree-level processes
In the scenario we set up, some other processes receivetree level R -parity violating contributions. Here we mainlydiscuss the constraints from neutral current processes B → K ( ∗ ) ν ¯ ν , B → πν ¯ ν , K → πν ¯ ν , D → µ + µ − and τ → µρ ,as well as charged current processes B → τ ν , D s → τ ν and τ → Kν . These decays relate to λ (cid:48) ij λ (cid:48)∗ lm m b R ¯ d m γ µ P L d j ¯ ν l γ µ P L ν i , (27) ˜ λ (cid:48) ij ˜ λ (cid:48)∗ lm m b R ¯ u m γ µ P L u j ¯ l l γ µ P L l i , (28) − λ (cid:48) ij ˜ λ (cid:48)∗ lm m b R ¯ u m γ µ P L d j ¯ l l γ µ P L ν i . (29)The effective Lagrangian for B → K ( ∗ ) ν ¯ ν , B → πν ¯ ν and K → πν ¯ ν decays are defined by L eff = ( C SM mj δ li + C ν l ¯ ν i mj )( ¯ d m γ µ P L d j )(¯ ν l γ µ P L ν i ) + H . c ., (30)where [71] C SM mj = − √ G F αX ( x t ) π sin θ W V tj V ∗ tm , (31) In fact, by combining the assumptions λ (cid:48) j = 0 and λ (cid:48) ij = λ (cid:48) ij = 0 ,we can get λ (cid:48) jk = 0 , which implies that the contribution of box diagramsof NP to the first generation leptons and sleptons is zero, because we onlyconsider the terms λ (cid:48) ijk L i Q j D ck . is the SM one. The loop function X ( x t ) ≡ x t ( x t +2)8( x t − + x t ( x t − x t − log( x t ) with x t ≡ m t /m W . The R -parity violatingcontributions are given by C ν l ¯ ν i mj = λ (cid:48) ij λ (cid:48)∗ lm m b R . (32)It is useful to define the ratio R ν ¯ νmj ≡ B ( d j → d m ν ¯ ν ) B ( d j → d m ν ¯ ν ) SM = (cid:80) i =1 (cid:12)(cid:12) C SM mj + C ν i ¯ ν i mj (cid:12)(cid:12) + (cid:80) i (cid:54) = l (cid:12)(cid:12) C ν l ¯ ν i mj (cid:12)(cid:12) (cid:12)(cid:12) C SM mj (cid:12)(cid:12) . (33)The upper limit of B → K ( ∗ ) ν ¯ ν decay corresponds to R ν ¯ ν < . [71–73] at 90% confidence level (CL), and the upper limitof B → πν ¯ ν decay is related to R ν ¯ ν < . [74, 75] at 90%CL. By combining the SM prediction B ( K + → π + ν ¯ ν ) SM =(9 . ± . × − [76] with experimental measurement B ( K + → π + ν ¯ ν ) exp = (1 . ± . × − [77], we obtain astringent constraint from K → πν ¯ ν decay that makes | λ (cid:48) i λ (cid:48)∗ l | < . × − ( m ˜ b R / . (34)Therefore, we will assume λ (cid:48) i k = 0 to satisfy this constraint.At the same time, under this assumption, B → πν ¯ ν decay isunaffected by the NP.The branching fraction for D → µ + µ − decay is givenby [58] B ( D → µ + µ − ) = τ D f D m D m µ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ λ (cid:48) ˜ λ (cid:48)∗ m b R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:115) − m µ m D , (35)where decay constant of D is f D = 209 . ± . MeV [78].The mean life τ D = 410 . ± . fs [77] and the upper limitof branching fraction of D → µ + µ − decay is . × − at 90% CL [77]. The corresponding constraint is | λ (cid:48) | < . m ˜ b R / .The branching fraction for τ → µρ decay is given by [79] B ( τ → µρ ) = τ τ f ρ m τ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ λ (cid:48) ˜ λ (cid:48)∗ m b R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:32) − m ρ m τ (cid:33) × (cid:32) m ρ m τ − m ρ m τ (cid:33) , (36)where τ τ = 290 . ± . fs and the decay constant f ρ =153 MeV [50]. The current experimental upper limit onthe branching fraction for this process is B ( τ → µρ ) < . × − at 90% CL [77]. The corresponding constraintis | λ (cid:48) λ (cid:48)∗ | < . m ˜ b R / .The formulas for charged current processes are given, re-spectively, by B ( B → τ ν ) B ( B → τ ν ) SM = R , (37) B ( D s → τ ν ) B ( D s → τ ν ) SM = R , (38) B ( τ → Kν ) B ( τ → Kν ) SM = R . (39)The corresponding experimental and theoretical values arelisted, respectively, as follows: B ( B → τ ν ) exp = (1 . ± . × − [77], B ( B → τ ν ) SM = (9 . ± . × − [80]; B ( D s → τ ν ) exp = (5 . ± . [77], B ( D s → τ ν ) SM = (5 . ± . ; B ( τ → Kν ) exp = (6 . ± . × − [77], B ( τ → Kν ) SM = (7 . ± . × − [56]. III.3. Constraints from the loop-level processes
First of all, the most important one-loop constraint comesfrom B s − ¯ B s mixing, which is governed by L eff = ( C SM B s + C NP B s )(¯ sγ µ P L b )(¯ sγ µ P L b ) + H . c ., (40)where the SM and NP Wilson coefficients are given respec-tively by C SM B s = − π G F m W η t S ( x t ) , (41) C NP B s = − π (cid:20) ( λ (cid:48) i λ (cid:48)∗ i ) m b R + ( λ (cid:48) λ (cid:48)∗ ) m ν µ (cid:21) , (42)where loop function S ( x t ) = x t (4 − x t + x t )4( x t − + x t log( x t )2( x t − . At σ level, the UT f it collaboration [81] gives the bound . < | C NP B s /C SM B s | < . .Next, we investigate a series of Z decaying to two chargedleptons with the same flavour like Z → µµ ( τ τ ) and thedifferent one like Z → µτ . The amplitude of these dia-grams is i M = i g π cos θ W B ij (cid:15) α ¯ u (cid:96) i γ α P L v (cid:96) j [50], where B ij = B ij + B ij and [50, 82] B ij = (cid:88) l =1 ˜ λ (cid:48) jl ˜ λ (cid:48)∗ il m Z m b R (cid:20)(cid:18) −
43 sin θ W (cid:19) × (cid:18) log m Z m b R − iπ − (cid:19) + sin θ W (cid:21) , (43) B ij = 3˜ λ (cid:48) j ˜ λ (cid:48)∗ i (cid:26) − x ˜ b R (1 + log x ˜ b R )+ m Z m b R (cid:20) (11 −
10 sin θ W ) + (6 − θ W ) log x ˜ b R + 110 ( − θ W ) m Z m t (cid:21)(cid:27) , (44)here B ij is the contribution from the diagram induced by ex-changing ˜ b R − u − u or ˜ b R − c − c in triangular loop and B ij is the contribution from the diagram induced by exchang-ing ˜ b R − t − t in triangular loop. As shown in Ref. [50],for Z → µµ ( τ τ ) , demanding the interference term in thepartial width between the SM tree-level contribution and theNP one-loop level ones is less than twice the experimentaluncertainty on the partial width [77], there are the bounds |(cid:60) ( B ) | < . and |(cid:60) ( B ) | < . [50]. And the ex-perimental upper limit B ( Z → µτ ) < . × − [77] makesthe bound (cid:112) | B | + | B | < . [50].Finally, we discuss the lepton-flavour violating decay of τ lepton, including τ → µγ and τ → µµµ . In the limit m µ /m τ → , the branching fraction for τ → µγ is givenby [68, 83, 84] B ( τ → µγ ) = τ τ αm τ | A L | + | A R | ) , (45)where the effective couplings A L,R come from on shell pho-ton penguin diagrams [68], A L = − λ (cid:48) j λ (cid:48)∗ j π m b R , A R = 0 . (46)The current experimental upper limit is B ( τ → µγ ) < . × − at 90% CL [77].In general, the effective Lagrangian leading to τ → µµµ decay is given by [83, 84] L eff = − B τ γ ν P L µ )(¯ µγ ν P R µ ) − B τ γ ν P R µ )(¯ µγ ν P L µ )+ C (¯ τ P R µ )(¯ µP R µ ) + C (¯ τ P L µ )(¯ µP L µ )+ G (¯ τ γ ν P R µ )(¯ µγ ν P R µ ) + G (¯ τ γ ν P L µ )(¯ µγ ν P L µ ) − A R (¯ τ [ γ µ , γ ν ] q ν q P R µ )(¯ µγ µ µ ) − A L (¯ τ [ γ µ , γ ν ] q ν q P L µ )(¯ µγ µ µ ) + H . c .. (47)This Lagrangian leads to [83, 84] B ( τ → µ ) = τ τ m τ π (cid:20) | B | + | B | + 8( | G | + | G | )+ | C | + | C | (cid:18) m τ m µ − (cid:19) | A R | + | A L | m τ − (cid:60) ( A L G ∗ + A R G ∗ ) m τ + 32 (cid:60) ( A L B ∗ + A R B ∗ ) m τ (cid:21) . (48)In our scenario, there are three different types of contributions,the photonic and Z penguins as well as box diagrams withfour λ (cid:48) couplings, that can contribute to τ → µµµ decay. Thenonzero Wilson coefficients are [50, 68] B = − (cid:0) παA L + sin θ W B (cid:48) (cid:1) , (49) G =4 παA L + (cid:18) −
12 + sin θ W (cid:19) B (cid:48) + C τ , (50) A L =2 παm τ A L , (51)where B (cid:48) = − α ˜ λ (cid:48) ˜ λ (cid:48)∗ x ˜ b R (1 + log x ˜ b R )8 π cos θ W sin θ W m Z , (52) C τ = i λ (cid:48) i ˜ λ (cid:48) ∗ i ˜ λ (cid:48) j ˜ λ (cid:48) ∗ j D [ m b R , m b R , m u i , m u j ] , (53)and the off-shell effective coupling A L is [68] A L = λ (cid:48) j λ (cid:48)∗ j π m b R (cid:20) − (cid:18)
43 + log m u j m b R (cid:19)(cid:21) . (54)The current experimental upper limit on the branching frac-tion for this decay is B ( τ → µµµ ) < . × − at 90%CL [77]. IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we discuss how to interpret both b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies and satisfy all these potential con-straints simultaneously. The relevant model parameters in ourscenario are the wino mass m ˜ W , the mass of muon sneu-trino m ˜ ν µ , the mass of right-handed sbottom m ˜ b R , as wellas four nonzero couplings λ (cid:48) , λ (cid:48) , λ (cid:48) , and λ (cid:48) . We set m ˜ W = 250 GeV. It can be seen from Ref. [52] that a pos-itive product λ (cid:48) λ (cid:48)∗ is needed to explain the b → s(cid:96) + (cid:96) − anomaly mainly through muon sneutrinos (the C V9 part). Both λ (cid:48) and λ (cid:48) are positive to help solve R ( D ( ∗ ) ) anomalyby exchanging ˜ b R at tree level [56]. The combination of thechoice of above couplings will naturally produce a negative C U9 , which is in line with the conclusion of the global anal-ysis [20]. Our numerical results are shown in Fig. 3. Theseresults show that it is possible to explain b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies simultaneously at σ level . The regionsof NP parameters that can solve B -physics anomalies are mostconstrained by B → K ( ∗ ) ν ¯ ν decays, B s − ¯ B s mixing and Z decays. In addition, the τ → µµµ decay can provide aweak constraint. We find that other related processes, such as D → µ + µ − , τ → µρ , B → τ ν , D s → τ ν , τ → Kν , and τ → µγ decays, do not provide available constraints.We show in Fig. 3a and Fig. 3b the allowed regions in theplanes of coupling parameters ( λ (cid:48) , λ (cid:48) ) and ( λ (cid:48) , λ (cid:48) ) respectively when other parameters are fixed. These two sub-figures show that in order to explain the B -physics anomalies,the coupling parameters need to satisfy the relation λ (cid:48) >λ (cid:48) > λ (cid:48) (cid:39) λ (cid:48) , and the required λ (cid:48) and λ (cid:48) are verysmall. Therefore, the next four subfigures in Fig. 3 mainly dis-cuss the relationships between the coupling parameters λ (cid:48) and λ (cid:48) and the masses m ˜ b R and m ˜ ν µ . From Fig. 3a, we cansee that λ (cid:48) is more constrained by R ( D ( ∗ ) ) , B → K ( ∗ ) ν ¯ ν and Z decays, but less affected by b → s(cid:96) + (cid:96) − processes and B s − ¯ B s mixing. On the contrary, λ (cid:48) is greatly constrainedby b → s(cid:96) + (cid:96) − processes and B s − ¯ B s mixing, but has lit-tle influence on R ( D ( ∗ ) ) , B → K ( ∗ ) ν ¯ ν and Z decays. Asshown in Fig. 3c, after the variable parameter m ˜ b R is added, In order to consider the constraints from B → K ( ∗ ) ν ¯ ν , τ → µγ and τ → µµµ decays at σ level, we get the experimental bounds (assuming theuncertainties follow the Gaussian distribution [85]) R ν ¯ ν < . , B ( τ → µγ ) < . × − and B ( τ → µµµ ) < . × − , respectively.
90 100 110 120 130 140 1501.61.71.81.92.0
90 100 110 120 130 140 1500.30.40.50.60.7
FIG. 3. Numerical analysis in which b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies are solved and other constraints are satisfied. The masses m ˜ b R and m ˜ ν µ are given in units of GeV. The σ favored regions from the b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) measurements are shown in blue and green,respectively. The hatched areas filled with black-vertical, black-horizontal, red-horizontal, and red-vertical lines are excluded by B → K ( ∗ ) ν ¯ ν decays, B s − ¯ B s mixing, Z decays, and τ → µµµ decay, respectively. The overlaps are marked in purple. the constraints of λ (cid:48) from R ( D ( ∗ ) ) , B → K ( ∗ ) ν ¯ ν and Z decays will be relaxed a lot. The parameters λ (cid:48) and m ˜ b R are highly correlated. Because we choose a smaller mass ofmuon sneutrino, the B s − ¯ B s mixing is more sensitive to m ˜ ν µ than to m ˜ b R , which can be seen by comparing Fig. 3d withFig. 3f. All subfigures contain parameter spaces (marked inpurple) that can resolve b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies,and satisfy the constraints from other related processes simul-taneously. V. CONCLUSIONS
The recent measurements on semileptonic decays of B -meson suggest the existence of NP which breaks the LFU.Among them, the observables R K ( ∗ ) and P (cid:48) in b → s(cid:96) + (cid:96) − processes and the R ( D ( ∗ ) ) in B → D ( ∗ ) τ ν decays aremore striking. They are collectively called B -physics anoma-lies. In this work, we have explored the possibility of usingmuon sneutrinos ˜ ν µ and right-handed sbottoms ˜ b R to solvethese B -physics anomalies simultaneously in R -parity violat- ing MSSM.To explain the anomalies in b → s(cid:96) + (cid:96) − processes, weuse a two-parameter scenario, where the total Wilson coef-ficients of NP are divided into two parts, one is the C V9 (Not-ing C NP10 ,µ = − C V9 ) that only contributes the muon channeland the other is the C U9 that contributes both the electronand the muon channels. First, we scrutinize all the one-loopcontributions of the superpotential terms λ (cid:48) ijk L i Q j D ck to the b → s(cid:96) + (cid:96) − processes under the assumptions λ (cid:48) ij = λ (cid:48) ij = 0 and λ (cid:48) j = 0 . We find that the contribution from the H ± − ˜ b R box diagram (Fig. 1c) is missed in the literature, this contri-bution is usually positive, and we find that it is numericallynegligible when tan β > . The photonic penguin induced byexchanging sneutrino can provide important contribution dueto the existence of logarithmic enhancement, which has neverbeen addressed before. This contribution is lepton flavour uni-versal due to the SM photon, so it is natural to contribute anonzero C U9 .Global analyses show that the sizable magnitude of C V9 isneeded to explain b → s(cid:96) + (cid:96) − anomaly. However, C V9 inthe scenario with nonzero C U9 is smaller than the one in thescenario without C U9 . With the addition of the latest mea-surements from the Belle collaboration, the world averagesof R ( D ( ∗ ) ) are closer to the predicted values of the SM.These changes make it possible to use ˜ ν µ and ˜ b R to explain b → s(cid:96) + (cid:96) − and R ( D ( ∗ ) ) anomalies, simultaneously. We alsoconsider the constraints of other related processes in our sce-nario. The strongest constraints come from B → K ( ∗ ) ν ¯ ν decays, B s − ¯ B s mixing, and the processes of Z decays. Be- sides, τ → µµµ decay can provide a few constraints. Theother decays, such as D → µ + µ − , τ → µρ , B → τ ν , D s → τ ν , τ → Kν , and τ → µγ , do not provide availableconstraints. ACKNOWLEDGEMENTS
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