Implications of the R_K and R_{K^*} anomalies
aa r X i v : . [ h e p - ph ] N ov Implications of the R K and R K ∗ anomalies Wei Wang and Shuai Zhao ∗ INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology,School of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240, China
We discuss the implications of the recently reported R K and R K ∗ anomalies, the leptonflavor non-universality in the B → Kℓ + ℓ − and B → K ∗ ℓ + ℓ − . Using two sets of hadronicinputs of form factors, we perform a fit of the new physics to the R K and R K ∗ data, andsignificant new physics contributions are found. We suggest to study the lepton flavor uni-versality in a number of related rare B, B s , B c and Λ b decay channels, and in particular wegive the predictions for the µ -to- e ratios of decay widths with different polarizations of thefinal state particles, and of the b → dℓ + ℓ − processes which are presumably more sensitive tothe structure of the underlying new physics. With the new physics contributions embeddedin Wilson coefficients, we present theoretical predictions for lepton flavor non-universality inthese processes. I. INTRODUCTION
The standard model (SM) of particle physics is now completed by the discovery of Higgs boson.Thus the focus in particle physics has been gradually switched to the search for new physics(NP) beyond the SM. This can proceed in two distinct ways. One is the direct search at thehigh energy frontier, in which new particles beyond the SM are produced and detected directly.The other is called indirect search, which is at the high intensity frontier. The new particles willpresumably manifest themselves as intermediate loop effects, and might be detectable by low-energyexperiments with high precision.In flavor physics, the b → sℓ + ℓ − process is a flavor changing neutral current (FCNC) transition.This process is of special interest since it is induced by loop effects in the SM, which leads totiny branching fractions. Many extensions of the SM can generate sizable effects that can beexperimentally validated. In particular, the B → K ∗ ( → Kπ ) µ + µ − decay offers a large numberof observables to test the SM, ranging from the differential decay widths, polarizations, to a fullanalysis of angular distributions of the final state particles, for an incomplete list one can refer toRefs. [1–21] and many references therein.In the past few years, quite a few observables in the channels mediated by b → sℓ + ℓ − transitionhave exhibited deviations from the SM expectations. The LHCb experiment has first observedthe so-called P ′ anomaly, a sizeable discrepancy at 3.7 σ between the measurement and the SMprediction in one bin for the angular observable P ′ [22]. This discrepancy was reproduced in a laterLHCb analysis for the two adjacent bins at large K ∗ recoil [23]. To accommodate this discrepancy, ∗ [email protected] considerable attentions have been paid to explore new physics contributions (see Refs. [24–31] andreferences therein), while at the same time, this has also triggered the thoughts to revisit thehadronic uncertainties [32, 33].More strikingly, the LHCb measurement of the ratio [34]: R K [ q , q ] ≡ R q q dq d Γ( B + → K + µ + µ − ) /dq R q q dq d Γ( B + → K + e + e − ) /dq , (1)gives a hint for the lepton flavour universality violation (LFUV). A plausible speculation is thatdeviations from the SM are present in b → sµ + µ − transitions instead in b → se + e − ones. Veryrecently the LHCb collaboration has found sizable differences between B → K ∗ e + e − and B → K ∗ µ + µ − at both low q region and central q region [35]. Results for ratios R K ∗ [ q , q ] ≡ R q q dq d Γ( B → K ∗ µ + µ − ) /dq R q q dq d Γ( B → K ∗ e + e − ) /dq , (2)are given in Tab. I, from which we can see the data showed significant deviations from unity. Theseinteresting results have subsequently attracted many theoretical attentions [36–59]. TABLE I: Ratios of decay widths with a pair of muons and electrons in B → Kℓ + ℓ − and B → K ∗ ℓ + ℓ − .Observable SM results Experimental data R K : q = [1 , . ± .
01 [60] 0 . +0 . − . ± .
036 [34] R low K ∗ : q = [0 . , .
1] GeV . +0 . − . [39] 0 . +0 . − . ± .
03 [35] R central K ∗ : q = [1 . ,
6] GeV . ± .
002 [39] 0 . +0 . − . ± .
05 [35]
The statistics significance in the data is low at this stage, about 3 σ level. In order to obtainmore conclusive results, one should measure the muon-versus-electron ratios in the B → Kℓ + ℓ − and B → K ∗ ℓ + ℓ − more precisely, meanwhile one should also investigate more channels with bettersensitivities to the structures of new physics contributions. In this paper, we will focus on thelatter. To do so, we will first discuss the implications of the R K and R K ∗ anomalies in a model-independent way, where the new particle contributions are parameterized in terms of effectiveoperators. Since there is lack of enough data, we analyze their impact on the Wilson coefficients ofSM operators O , . We then propose to study the lepton flavor universality in a number of rare B, B s , B c and Λ b decay channels. Incorporating the new physics contributions, we will present thepredictions for the muon-versus-electron ratios in these channels, making use of various updates ofform factors [61–66]. We will demonstrate that the measurements of lepton flavor non-universalitywith different polarizations of the final state hadron, and in the b → dℓ + ℓ − processes are of greatvalue to decode the structure of the underlying new physics.The rest of this paper is organized as follows. In the next section, we will use a model-independent approach and quantify the new physics effects in terms of the short-distance Wilsoncoefficients. In Section III, we will study the LFUV in various FCNC channels. Our conclusion isgiven in the last section. II. IMPLICATIONS FROM THE R K AND R K ∗ In this section, we will first study the impact of the R K and R K ∗ data. In the SM, the effectiveHamiltonian for the transition b → sℓ + ℓ − H eff = − G F √ V tb V ∗ ts X i =1 C i ( µ ) O i ( µ )involves the four-quark and the magnetic penguin operators O i . Here C i ( µ ) are the Wilson co-efficients for these local operators O i . G F is the Fermi constant, V tb and V ts are CKM matrixelements. The dominant contributions to b → sℓ + ℓ − come from the following operators: O = em b π ¯ sσ µν (1 + γ ) bF µν + em s π ¯ sσ µν (1 − γ ) bF µν ,O = α em π (¯ lγ µ l )¯ sγ µ (1 − γ ) b, O = α em π (¯ lγ µ γ l )¯ sγ µ (1 − γ ) b. (3)The above effective Hamiltonian gives the B → Kℓ + ℓ − decay width as: d Γ( B → Kℓ + ℓ − ) dq = G F √ λα em β l m B π | V tb V ∗ ts | × " λ (1 + 2 ˆ m l ) (cid:12)(cid:12)(cid:12)(cid:12) C f + ( q ) + C m b f T ( q ) m B + m K (cid:12)(cid:12)(cid:12)(cid:12) + λβ l | C | f ( q ) + 6 ˆ m l | C | ( m B − m K ) f ( q ) i , (4)where ˆ m l = m l / p q , β l = q − ˆ m l , λ = ( m B − m K − q ) − m K q , and f + , f and f T are the B → K form factors. In the above expression, we have neglected the non-factorizable contributionswhich are expected to be negligible for R K .The decay width for B → K ∗ ℓ + ℓ − can be derived in terms of the helicity amplitude [67–71].The differential decay width is given as d Γ( B → K ∗ ℓ + ℓ − ) dq = 34 (cid:16) I c + 2 I s (cid:17) − (cid:16) I c + 2 I s (cid:17) , (5)with I c = ( | A L | + | A R | ) + 8 ˆ m l Re[ A L A ∗ R ] + 4 ˆ m l | A t | ,I s = (cid:0) / − ˆ m l (cid:1) [ | A L ⊥ | + | A L || | + | A R ⊥ | + | A R || | ] + 4 ˆ m l Re[ A L ⊥ A ∗ R ⊥ + A L || A ∗ R || ] ,I c = − β l ( | A L | + | A R | ) ,I s = 14 β l ( | A L ⊥ | + | A L || | + | A R ⊥ | + | A R || | ) . (6)The handedness label L or R corresponds to the chirality of the di-lepton system. Functions A L/Ri can be expressed in terms of B → K ∗ form factors A t = 2 q N K ∗ J N C √ λ p q A ( q ) , (7) A L = N p N K ∗ J m K ∗ J p q (cid:20) ( C − C )[( m B − m K ∗ − q )( m B + m K ∗ ) A − λm B + m K ∗ A ]+2 m b C [( m B + 3 m K ∗ − q ) T − λm B − m K ∗ T ] (cid:21) , (8) A L ⊥ = − q N K ∗ J N h ( C − C ) √ λVm B + m K ∗ + 2 m b C q √ λT i , (9) A L || = q N K ∗ J N h ( C − C )( m B + m K ∗ ) A + 2 m b C q ( m B − m K ∗ ) T i , (10)with N = iG F √ α em π V tb V ∗ ts , N K ∗ J = 8 / √ λq β l / (256 π m B ) and λ ≡ ( m B − m K ∗ − q ) − m K ∗ q .The right-handed decay amplitudes are obtained by reversing the sign of C : A Ri = A Li | C →− C . (11)Within the SM, one can easily find that results for R K and R K ∗ are extremely close to 1 andthus deviate from the experimental data. If new physics is indeed present, it can be in b → sµ + µ − and/or b → se + e − transitions. In order to explain the R K and R K ∗ data, one can enhance thepartial width for the electronic mode or reduce the one for the muonic mode. It seems that the SMresult for the B → Ke + e − is consistent with the data, and thus here we will adopt the strategythat the muonic decay width is reduced by new physics.After integrating out the high scale intermediate states the new physics contributions can beincorporated into the effective operators. As there is lack of enough data that shows significantdeviations with SM, we will assume that NP contributions can be incorporated into Wilson coeffi-cients C and C . For this purpose, we define δC µ = C µ − C SM9 , δC µ = C µ − C SM10 . (12)The O contribution to b → sℓ + ℓ − arises from the coupling of a photon with the lepton pair. Onone hand, this coupling is highly constrained from the b → sγ data. On the other hand, thiscoefficient is flavor blinded and thus even if NP affect C , the µ -to- e will not be affected.For the analysis, we adopt three scenarios,1. Only C is affected with δC µ = 0.2. Only C is affected with δC µ = 0.3. Both C and C are affected in the form: δC µ = − δC µ = 0.Using the R K and R K ∗ data, we show our results in FIG. 1. The left panel corresponds to scenario1, and the middle panel corresponds to the constraint on δC µ , the last one corresponds to the - - - - - ∆ C Μ Χ ∆ C Μ Χ - - - - - - - - ∆ C Μ -∆ C Μ Χ FIG. 1: Impact of R K and R K ∗ data on the δC µ (left panel), δC µ (central panel) or δC µ − δC µ (rightpanel). The dependence of the total χ for all data in Tab. I on Wilson coefficients is shown as the solid(red) and dashed (blue) curves, which correspond to the form factors from LQCD [65, 75] and LCSR [72, 73],respectively. Removing the low- q data for B → K ∗ ℓ + ℓ − , the results are shown as dotted (black) and anddot-dashed (green) curves. scenario 3 with a nonzero δC µ − δC µ . In this analysis, we have used two sets of B → K and B → K ∗ form factors. One is from the light-cone sum rules (LCSR) [72–74], corresponding tothe dashed curves. The other is from Lattice QCD (LQCD) [65, 75], which gives the solid curves.As one can see clearly from the figure, the results are not sensitive to the form factors, and thisalso partly validate the neglect of other hadronic uncertainties like non-factorizable contributions.Using the LQCD set of form factors [65, 75] and the data in Tab.I, we found the best-fitted centralvalue and the 1 σ range for δC µ in scenario 1 as δC µ = − . , − . < δC µ < − . . (13)For scenario 2, we have δC µ = 1 . , . < δC µ < . , (14)while for the δC µ = − δC µ , we obtain δC µ − δC µ = − . , − . < δC µ − δC µ < − . . (15)A few remarks are given in order. • Since the Wilson coefficient in the electron channel is unchanged, the δC µ and δC µ couldbe viewed as the difference between the Wilson coefficients for the lepton and muon case. • We have found the largest deviation between the fitted results and the data comes from thelow- q region. Removing this data, we show the χ in FIG. 1 as dotted and dot-dashed ℓ − ℓ + B K ( K ∗ ) b s ( c )( d ) ℓ − ℓ + B K ( K ∗ ) b s ( b ) ℓ − ℓ + B K ( K ∗ ) b sℓ − ℓ + B K ( K ∗ ) b s ( a ) ℓ − ℓ + B K ( K ∗ ) b s K ( K ∗ )( e ) ℓ − ℓ + B K ( K ∗ ) b s ( g ) B ℓ − ℓ + b s ( f ) FIG. 2: The electromagnetic corrections to B → Kℓ − ℓ + and B → K ∗ ℓ − ℓ + . curves, where the χ has been greatly reduced. The reason is that in low- q region, thedominant contribution to R K ∗ arises from the transverse polarization of K ∗ . From Eq. (9)and (10), one can see this contribution is dominated by O and less sensitive to O , . Alight mediator that only couples to the µ + µ − is explored for instance in Refs. [47, 52, 54]. • For the R K and R K ∗ predictions in Refs. [39, 60], theoretical errors are typically lessthan one percent, while Ref. [76] gives the prediction with even smaller uncertainty R K =1 . ± . α/π ln( q /m ℓ ). The difference between the double logarithms for the electron and muonmode is about 3%. A complete analysis requests the detailed calculation of all diagramsin Fig. 2 and analyses can be found in Ref. [77]. The nonfactorizable corrections to theamplitude can be found in Ref. [78]. • It is necessary to point out that there are a number of observables in B → Kµ + µ − and B → K ∗ µ + µ − that have been experimentally measured. These observables are of greatvalues to provide very stringent constraints on the Wilson coefficients in the factorizationapproach. On the other hand, most of these observables in B → Kµ + µ − and B → K ∗ µ + µ − are not sensitive to the flavor non-universality coupling since only the mu lepton is involved.The exploration of the µ -to- e ratios will be able to detect the difference in the new physicscouplings to fermions. It is always meaningful to conduct a comprehensive global analysisand incorporate as many observables as possible. At this stage, the study of flavor non-universality in flavor physics is at the beginning, and we believe measuring more µ to e ratios (for instance the ones in Table II shown in the following section) will be helpful. • For a more comprehensive analysis, one may combine various experimental data on theflavor changing neutral current processes for instance in Refs. [36–40]. We quote the resultsin scenario I in Ref. [36], δC µ = − . ± . , δC e = − . ± . , (16)from which we can see that the results are close to our scenario 1. This implies that for thedetermination of flavor dependent Wilson coefficient, the R K and R K ∗ are dominant. Froma practical viewpoint, since the main purpose of this paper is to explore the implications ofthe large lepton flavor non-universality, we will use our fitted results to predict the leptonflavor non-universality for a number of other channels.Explicit models which can realize these scenarios include the flavor non-universal Z ′ model,leptoquark model and vector-like models, see, e.g., Refs. [79–108] and many references therein.Their generic contributions are shown in FIG. 3. Taking the Z ′ model as an example, the SM canbe extended by including an additional U (1) ′ symmetry, which can leads to the Lagrangian of Z ′ ¯ bs couplings L Z ′ FCNC = − g ′ ( B Lsb ¯ s L γ µ b L + B Rsb ¯ s R γ µ b R ) Z ′ µ + h . c . . (17)It contributes to the b → sℓ + ℓ − decay at tree level H Z ′ eff = 8 G F √ ρ Lsb ¯ s L γ µ b L + ρ Rsb ¯ s R γ µ b R )( ρ Lll ¯ ℓ L γ µ ℓ L + ρ Rll ¯ ℓ R γ µ ℓ R ) , (18)where the coupling is ρ L,Rff ′ ≡ g ′ M Z gM Z ′ B L,Rff ′ (19)where the g standard model SU (2) L coupling. For simplicity, one can assume that the FCNCcouplings of the Z ′ and quarks only occur in the left-handed sector: ρ Rsb = 0. Thus in this case theeffects of the Z ′ will modify the Wilson coefficients C and C : C Z ′ = C − πα em ρ Lsb ( ρ Lll + ρ Rll ) V tb V ∗ ts , C Z ′ = C + 4 πα em ρ Lsb ( ρ Lll − ρ Rll ) V tb V ∗ ts . (20)From this expression, we can see that the δC µ and δC µ are not entirely correlated. This corre-sponds to the scenario 1 and 2 in our previous analysis.The impact in a leptoquark model has been discussed for instance in Ref. [43], where the NPcontribution satisfies δC LQ,µ = − δC LQ,µ . (21)This corresponds to the scenario 3. b sZ ′ ℓ − ℓ + b ℓ − ∆ (2 / s ℓ + b ℓ − ∆ (2 / s ℓ + ( a ) ( b ) ( c ) bW ∆ (5 / u ℓ − ℓ + u ′ s bW ∆ (5 / u ℓ − ℓ + u ′ s ( d ) ( e ) FIG. 3: New physics scenarios that can contribute to b → sµ + µ − . The panel (a) shows a Z ′ , and in theother four diagrams ∆ denotes a leptoquark with different spins and charges. III. LEPTON FLAVOR UNIVERSALITY IN FCNC CHANNELS
In this section, we will study the µ -to- e ratios of decay widths in various FCNC channels. Sincethe three scenarios considered in the last section describe the data equally well, we will choose thefirst one for illustration in the following. We follow a similar definition R B,M [ q , q ] ≡ R q q dq d Γ( B → M µ + µ − ) /dq R q q dq d Γ( B → M e + e − ) /dq , (22)where B denotes a heavy particle and M denotes a final state. The channels to be studied include B → K ∗ , (1430) ℓ + ℓ − , B s → f (980) ℓ + ℓ − , B → K (1270) ℓ + ℓ − , B s → f (1525) ℓ + ℓ − , B s → φℓ + ℓ − , B c → D s ℓ + ℓ − , B c → D ∗ s ℓ + ℓ − . The expressions for their decay widths have been given in the lastsection. In addition, we will also analyze on the R ratio for the baryonic decay Λ b → Λ ℓ + ℓ − . Thedifferential decay width for Λ b → Λ ℓ + ℓ − is given as [109]dΓd q [Λ b → Λ ℓ + ℓ − ] = 2 K ss + K cc , (23)where K ss ( q ) = 14 h | A R ⊥ | + | A R k | + 2 | A R ⊥ | + 2 | A R k | + ( R ↔ L ) i ,K cc ( q ) = 12 h | A R ⊥ | + | A R k | + ( R ↔ L ) i . (24)The functions A are defined as A L ( R ) ⊥ = √ N (cid:20) ( C ∓ C ) H V + − m b C q H T + (cid:21) , A L ( R ) k = −√ N (cid:20) ( C ∓ C ) H A + + 2 m b C q H T (cid:21) ,A L ( R ) ⊥ = √ N (cid:20) ( C ∓ C ) H V − m b C q H T (cid:21) , A L ( R ) k = −√ N (cid:20) ( C ∓ C ) H A + 2 m b C q H T (cid:21) , (25)where the normalization factor N is N = G F V tb V ∗ ts α em vuut q q λ ( m b , m , q )3 · m b π . (26)The helicity amplitudes are given by H V = f V ( q ) m Λ b + m Λ p q √ s − , H V + = − f V ⊥ ( q ) p s − ,H A = f A ( q ) m Λ b − m Λ p q √ s + , H A + = − f A ⊥ ( q ) p s + H T = − f T ( q ) p q √ s − , H T + = f T ⊥ ( q ) ( m Λ b + m Λ ) p s − ,H T = f T ( q ) p q √ s + , H T = − f T ⊥ ( q ) ( m Λ b − m Λ ) p s + , (27)where s ± ≡ ( m Λ b ± m Λ ) − q . The f i / ⊥ with i = V, A, T, T b → Λ form factors.The B s → φℓ + ℓ − and Λ b → Λ form factors are used from LQCD calculation in Refs. [65, 110],respectively. The B → K ∗ (1430) and B s → f (980) form factors are taken from Ref. [61, 111].The B → K (1270) form factors are calculated in the perturbative QCD approach [63], andthe mixing angle between K (1 ++ ) and K (1 + − ) is set to be approximately 45 ◦ . In this casethe B → K (1400) ℓ + ℓ − is greatly suppresed [112]. The B → K and B s → f (1525) formfactors are taken from Ref. [64]. The B c → D s /D ∗ s form factors are provided in light-front quarkmodel [62], and in this work we have calculated the previously-missing tensor form factors. Usingthe Wilson coefficient δC µ in Eq. (13), we present our numerical results for R B,M in TABLE II.Three kinematics regions are chosen in the analysis: low q with [0.045, 1] GeV , central q with[1, 6] GeV and high q region with [14 GeV , q = ( m B − m M ) ]. For a vector final state, thelongitudinal and transverse polarizations are separated and labeled as L and T , respectively. ForΛ b → Λ ℓ + ℓ − , a similar decomposition is used, in which the superscript 0 means the Λ b and Λ havethe same polarization while 1 corresponds to different polarizations. The SM predictions for theseratios are listed in Tab. III.A few remarks are given in order. • From the decay widths for B → K ∗ ℓ + ℓ − , we can see that in the transverse polarization, thecontribution from O is enhanced at low q , and thus the R TB,M is less sensitive to the NPin O , . Measurements of the µ -to- e ratio in the transverse polarization of B → V ℓ + ℓ − atlow q can tell whether the NP is from the q independent contribution in C , or the q dependent contribution in C .0 TABLE II: Theoretical results for the µ -to- e ratio R B,M of decay widths as defined in Eq. (22) in various b → sℓ + ℓ − channels. Three kinematics regions are chosen: low, central and high q regions. Wilsoncoefficient C is used as in Eq. (13) based on the analysis of R K and R K ∗ . For a vector final state,the longitudinal and transverse polarizations are separated and labeled as L and T , respectively. ForΛ b → Λ ℓ + ℓ − , a similar decomposition is used: the superscript 0 means that the Λ b and Λ have the samepolarization, while 1 corresponds to different polarizations.Observable Low q : [0 . , Central q : [1 , High q : [14GeV , q ] R B,K ∗ (1430) . +0 . − . . +0 . − . . +0 . − . R B s ,f (980) . +0 . − . . +0 . − . . +0 . − . R B c ,D s . +0 . − . . +0 . − . . +0 . − . R B s ,φ . +0 . − . . +0 . − . . +0 . − . R LB s ,φ . +0 . − . . +0 . − . . +0 . − . R TB s ,φ . − . . . − . . . +0 . − . R B c ,D ∗ s . − . . . − . . . +0 . − . R LB c ,D ∗ s . +0 . − . . +0 . − . . +0 . − . R TB c ,D ∗ s . − . . . − . . . +0 . − . R B,K ∗ . +0 . − . . +0 . − . . +0 . − . R LB,K ∗ . +0 . − . . +0 . − . . +0 . − . R TB,K ∗ . − . . . − . . . +0 . − . R B s ,f . +0 . − . . +0 . − . . +0 . − . R LB s ,f . +0 . − . . +0 . − . . +0 . − . R TB s ,f . − . . . − . . . +0 . − . R B,K (1270) . +0 . − . . +0 . − . . +0 . − . R LB,K (1270) . +0 . − . . +0 . − . . +0 . − . R TB,K (1270) . − . . . − . . . +0 . − . R Λ b , Λ . +0 . − . . +0 . − . . +0 . − . R b , Λ . +0 . − . . +0 . − . . +0 . − . R b , Λ . − . . . − . . . +0 . − . • In the central q region, the operators O and O , will contribute destructively to the trans-verse polarization of B → V ℓ + ℓ − . Reducing C with δC µ < • Results for Λ b → Λ with different polarizations are similar, but it should be pointed out thatdifferential decay widths in Eq. (23) have neglected the kinematic lepton mass corrections.Thus the results in the low q region are not accurate. • For the B → K , (1430) ℓ + ℓ − and B c → D ∗ s , the high q region has a limited kinematics,and thus the results are difficult to be measured.1 TABLE III: Theoretical results for the µ -to- e ratio R B,M of decay widths as defined in Eq. (22) in various b → sℓ + ℓ − channels in the SM. Three kinematics regions are chosen: low, central and high q regions. Fora vector final state, the longitudinal and transverse polarizations are separated and labeled as L and T ,respectively. We do not present the results Λ b → Λ ℓ + ℓ − since the lepton mass effects are not included inEq. (25). Observable Low q : [0 . , Central q : [1 , High q : [14GeV , q ] R B,K ∗ (1430) .
980 1 .
001 1 . R B s ,f (980) .
980 1 .
000 1 . R B c ,D s .
981 1 .
001 1 . R B s ,φ .
937 0 .
998 0 . R LB s ,φ .
991 1 .
001 0 . R TB s ,φ .
902 0 .
985 0 . R B c ,D ∗ s .
917 0 .
995 0 . R LB c ,D ∗ s .
978 0 .
997 0 . R TB c ,D ∗ s .
908 0 .
990 0 . R B,K ∗ .
932 0 .
996 0 . R LB,K ∗ .
971 0 .
998 0 . R TB,K ∗ .
902 0 .
985 0 . R B s ,f .
930 0 .
995 0 . R LB s ,f .
971 0 .
998 0 . R TB s ,f .
902 0 .
985 0 . R B,K (1270) .
950 1 .
015 0 . R LB,K (1270) .
064 1 .
039 0 . R TB,K (1270) .
901 0 .
985 0 . • Among the decay processes involved in Table II, a few of them have been experimentallyinvestigated: the branching fractions of B s → φℓ + ℓ − [113, 114], Λ b → Λ ℓ + ℓ − [115] and B s → f (980) ℓ + ℓ − [116] have been measured. So for these channels, the measurement of the µ -to-e ratio will be straightforward when enough statistical luminosity is accumulated.For the other channels, we believe most of them except the B c decay might also be experi-mentally measurable, especially at the Belle-II with the designed 50 ab − data and the highluminosity upgrade of LHC. • In FIG. 3, a new particle like Z ′ or leptoquark can contribute to the R K and R K ∗ . Thecoupling strength is unknown, and in principle it could be different from the CKM pattern.In the SM, the B → πℓ + ℓ − and B s → Kℓ + ℓ − have smaller CKM matrix elements. Thus if theNP contributions had the same magnitude as in b → sℓ + ℓ − , their impact in B → πℓ + ℓ − and B s → Kℓ + ℓ − would be much larger. But in many frameworks, the new physics in b → dℓ + ℓ − b → sℓ − ℓ − , for recent discussions see Ref. [117]. Thiscan be resolved by experiments in the future. • The weak phases from Z ′ and leptoquark can be different from that in b → sµ + µ − or b → dµ + µ − , which may induce direct CP violations. In the b → dµ + µ − process, the currentdata on B → πµ + µ − contains a large uncertainty [118] A CP ( B ± → π ± µ + µ − ) = ( − . ± . ± . . (28)This can be certainly refined in the future. It should be noticed that the SM contributionmay also contain CP violation source [119, 120] since the up-type quark loop contributionsare sizable. IV. CONCLUSIONS
Due to the small branching fractions in the SM, rare decays of heavy mesons can provide a richlaboratory to search for effects of physics beyond the SM. Up to date, quite a few quantities in B decays have exhibited moderate deviations from the SM. This happens in both tree operatorand penguin operator induced processes. The so-called R D ( D ∗ ) anomaly gives a hint that the taulepton might have a different interaction with the light leptons. The V ub and V cb puzzles referto the difference for the CKM matrix elements extracted from the exclusive and inclusive decaymodes. In the b → sℓ + ℓ − mode, the P ′ in B → K ∗ ℓ + ℓ − has received considerable attentions onboth the reliable estimates of hadronic uncertainties and new physics effects. In addition, LHCbalso observed a systematic deficit with respect to SM predictions for the branching ratios of severaldecay modes, such as B s → φµ + µ − [113, 114]. Though the statistical significance is low, all theseanomalies indicate that the NP particles could be detected in flavor physics.In this work, we have presented an analysis of the recently observed R K and R K ∗ anomalies. Interms of the effective operators, we have performed a model-independent fit to the R K and R K ∗ data. In the analysis, we have used two sets of form factors and found the results are rather stableagainst these hadronic inputs. Since the statistical significance in R K and R K ∗ is rather low, weproposed to study a number of related rare B, B s , B c and Λ b decay channels, and in particularwe have pointed out that the µ -to- e ratios of decay widths with different polarizations of thefinal state particles, and in the b → dℓ + ℓ − processes are likely more sensitive to the structure ofthe underlying new physics. After taking into account the new physics contributions, we madetheoretical predictions on lepton flavor non-universality in these processes which can stringentlyexamined by experiments in future.3 Acknowledgements
We thank Yun Jiang and Yu-Ming Wang for useful discussions. This work is supported in partby National Natural Science Foundation of China under Grant No.11575110, 11655002, 11735010,Natural Science Foundation of Shanghai under Grant No. 15DZ2272100 and No. 15ZR1423100,by the Young Thousand Talents Plan, and by Key Laboratory for Particle Physics, Astrophysicsand Cosmology, Ministry of Education.
Appendix A: Definitions of R L,T and R , For B decays to vector final state, we define the longitudinal and transverse ratios R L and R T as R L,TV [ q , q ] ≡ R q q dq d Γ L,T ( B → V µ + µ − ) /dq R q q dq d Γ L,T ( B → V e + e − ) /dq , (A1)where the longitudinal and transverse differential widths are defined by d Γ L ( B → V µ + µ − ) /dq = 34 I c − I c , (A2) d Γ T ( B → V µ + µ − ) /dq = 32 I s − I s , (A3) V denotes a vector final state. The expressions for I c,s and I c,s are given by Eq. (6).For Λ b → Λ ℓ + ℓ − decays, we define ratios with equal or different polarization as [109] R , [ q , q ] ≡ R q q dq d Γ , (Λ b → Λ µ + µ − ) /dq R q q dq d Γ , (Λ b → Λ e + e − ) /dq , (A4)the superscript 0 means that the Λ b and Λ have the same polarization, while 1 corresponds todifferent polarizations. 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