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A Model for ¯ Flavour Anomalies Z b sℓℓ ′ → Cheng-Wei Chiang1,2,3a, Xiao-Gang He4,3,5b, German Valencia6c 1Department of Physics and Center for Mathematics and Theoretical Physics, National Central University, Chungli, Taiwan 32001 2Institute of Physics, Academia Sinica, Taipei, Taiwan 11529 6 1 3Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan 30013 0 2 4INPAC,Department of Physics and Astronomy, r a Shanghai Jiao Tong University, Shanghai M 5Department of Physics, National Taiwan University, Taipei, 0 1 6School of Physics and Astronomy, ] h Monash University, 3800 Melbourne Australia.d p - Abstract p e h We study the implications of flavour-changing neutral currents (FCNC’s) in a model with the [ 2 SU(2)l SU(2)h U(1)Y electroweak gauge symmetry for several anomalies appearing in b sℓℓ¯ × × → v 8 inducedB decaysinLHCbdata. Inthismodel,SU(2)l andSU(2)h governtheleft-handedfermions 2 3 in the first two generations and the third generation, respectively. The physical Z and Z′ generate 7 0 the b s transition at tree level, leading to additional contributions to the b s semileptonic . → → 1 0 operators 9,10. WefindthatalthoughBs-B¯s mixingconstrainstheparametersseverely, themodel O 6 1 can produce values of NP in the range determined by Descotes-Genon et. al. in Ref. [1] for this : C9,10 v scenario to improve the global fit of observables in decays induced by the b sµµ¯ transition. i X → r The Z′ boson in this model also generates tree-level FCNC’s for the leptonic interactions that can a accommodate the experimental central value of R = (B Kµµ¯)/ (B Kee¯) = 0.75. In this K B → B → case, the model predicts sizeable branching ratios for B Keτ¯, B Kτe¯, and an enhancement → → of B Kττ¯ with respect to its SM value. → a Electronic address: [email protected] b Electronic address: [email protected] c Electronic address: [email protected] d On leave from Department of Physics, Iowa State University, Ames, IA 50011. 1 I. INTRODUCTION Experimental data have hinted at several anomalies in B decays induced by the flavour- ¯ changingneutralcurrent(FCNC)processb sℓℓ. In2013,LHCbmeasuredfourobservables → related to the angular distribution of B K∗µ+µ− in six bins of dimuon invariant mass → squared, q2, and found a deviation at the 3.7σ level from the standard model (SM) in one of them [2]. LHCb also measured the rates for the B K(∗)µ+µ− decay [3], finding → values slightly below the SM expectations. Recently, with finer binning, LHCb confirmed their earlier anomaly in the angular distribution of B K∗µ+µ− decay [4]. In addition, → LHCb has studied other modes induced by the b sµ+µ− transition as well, namely, the → B φµ+µ− decay [5] and also b se+e− in the B K∗e+e− decay [6] with results s → → → consistent with the SM. A particularly interesting discrepancy between experiment and the SM is in the ratio R of the branching fraction of B+ K+µ+µ− to that of B+ K+e+e−. Lepton- K → → universality in the SM predicts R to be very close to 1. Yet LHCb found R (B K K ≡ B → Kµµ¯)/ (B Kee¯) = 0.745+0.090 0.036 [7] for the dilepton invariant mass squared range B → −0.074 ± of 1 6 GeV2. This disagreement occurs only at the 2.6σ level, but would be extremely − interesting if confirmed. ¯ As expected, the anomalies in the b sℓℓ measurements have received considerable → attention in the literature [8] and several models have been put forward as possible new physics explanations[9]. Ithasalsobeenarguedthatmorecarefultreatment oflongdistance physics would eliminate most of these anomalies, as done most recently in Ref. [10]. Models havealsobeenputforthattemptingtoexplaintheapparentleptonnon-universality observed in R [11]. A recent analysis of these experimental results is that of Ref. [1], where global K fits of the observables in terms of new physics parametrised by deviations from the SM values of certain Wilson coefficients are presented. This model-independent analysis and its results are the starting point of our discussions. Inthis paper we will focus our discussion around the scenario in which new physics affects primarily the and Wilson coefficients, which has been found in Ref. [1] to significantly 9 10 C C improve the agreement between the measurements and the theoretical predictions. We recall ¯ that these coefficients appear in the low-energy effective Hamiltonian responsible for b sℓℓ → 2 transitions as follows: 4G = FV V∗ ℓℓ + ℓℓ , Heff − √2 tb ts C9 O9 C10O10 e2 (cid:0) (cid:1) e2 = (s¯γ P b) ℓ¯γµℓ , = (s¯γ P b) ℓ¯γµγ ℓ , (1) O9 16π2 µ L O10 16π2 µ L 5 (cid:0) (cid:1) (cid:0) (cid:1) where P = (1 γ )/2 and, in the absence of flavour universality, ℓℓ can have different L − 5 C9,10 values for different lepton flavours. Within the SM, ℓℓ are approximately the same for all leptons with SM 4.1, and C9,10 C9 ≈ SM 4.1. To reduce the tension in the global fit associated with the b sµµ¯ anomalies, C10 ≈ − → the new physics contribution NP,µµ is required to be of order 1.0 and for scenarios where C9 − NP,µµ is also not zero, the best fit occurs for NP,µµ 0.3 [1]. To address the anomaly in C10 C10 ∼ the value of R , the absolute value of SM + NP,ee is required to be larger than that of K C9,10 C9,10 SM + NP,µµ. C9,10 C9,10 When going beyond the SM, additional operators with different chiral structures that contribute to b sℓℓ¯can also be generated, such as ′ = (e2/16π2)(s¯γ P b) ℓ¯γµℓ and → O9 µ R O1′0 = (e2/16π2)(s¯γµPRb) ℓ¯γµγ5ℓ , where PR = (1+γ5)/2. For the remaining of(cid:0)this p(cid:1)aper, we will neglect this possibi(cid:0)lity and(cid:1)concentrate on a scenario with modified 9,10 only, corre- C sponding to a particular Z′ interpretation of the anomalies. This particular interpretation is motivated by the possibility of lepton non-universality hinted at by R , and its occurrence K in non-universal Z′ models that single out the third generation. AcommonextensionoftheSMthatproducestree-levelFCNC’sisaZ′ boson, particularly whenitisnon-universalingenerations. Thisnewinteractioncanhavedifferenttypesofchiral structures in both quark and lepton sectors. A model that singles out the third generation with an additional right-handed interaction [12] leads to tree-level FCNC’s for ′ which, O9,10 accordingtotheglobalfitsofRef.[1], donothelpmuchinaddressingtheobservedanomalies. At one-loop level, it is possible to produce the pattern NP = NP which is disfavoured by C9 C10 the data on B µµ. In this context, a model that more naturally fits the scenario is s 9,10 → C one where the SU(2) gauge group in the SM is extended to be generation-dependent [13], L an example of which has been dubbed ‘top-flavour’ before [14]. The model has the SU(2) SU(2) U(1) gauge symmetry, where SU(2) governs the l h Y l × × left-handed fermions in the first two light generations and SU(2) governs those in the third h heavy generation. This model has been studied before by two of us in Ref. [15]. It affects the ¯ b sℓℓ process at tree level mostly through modifications to . The relevant parameters 9,10 → C 3 are severely constrained by B -B¯ mixing. Nevertheless, the model can still produce values s s of NP in the right ranges to improve the global fits as described in Ref. [1]. In addition, the C9,10 model can break lepton universality and lepton number, accommodating R and predicting K sizeable branching ratios for B Keτ¯, B Kτe¯ and an enhancement of B Kττ¯ with → → → respect to its SM value. This paper is organized as follows. In Section II, we review the tree-level FCNC’s induced by the Z and Z′ exchanges in the model, deriving the basis for the latter analyses. In Section III, we compute the corrections to the Wilson coefficients NP occurring in this C9,10 model. In Section IV, we update the global fit to the electroweak precision data and the B -B¯ mixing constraint, thereby obtaining preferred ranges of the theory parameters. The s s results are then used to evaluate NP numerically and to check against the preferred values C9,10 presented in Ref. [1]. Taking a step further, we make predictions for R and the decay K branching ratios of B Keτ¯, B Kτe¯, and B Kττ¯. Section V summarizes our → → → findings. II. TREE-LEVEL FCNC’S DUE TO Z AND Z′ IN THE MODEL With the gauge group extended from SU(3) SU(2) U(1) to SU(3) SU(2) C L Y C l × × × × SU(2) U(1) , there are additional gauge bosons: a pair of W′± bosons and a Z′ boson. h × Y µ With an appropriate Higgs sector, the SU(2) SU(2) symmetry is broken down to SU(2) l h L × attheTeVscale, leavingtheSMgaugegroupfollowedbythestandardelectroweaksymmetry breakdown [15] . The Z and Z′ FCNC’s relevant to the b sℓℓ¯transitions are caused by → the neutral gauge boson interactions with fermions. The left-handed quark doublets Q , the right-handed quark singlets U and D , the L R R left-handed lepton doublets L , and the right-handed charged leptons E transform under L R the original gauge group as Q1,2 : (3,2,1,1/3) , Q3 : (3,1,2,1/3) , U1,2,3 : (3,1,1,4/3) , D1,2,3 : (3,1,1, 2/3) , L L R R − L1,2 : (1,2,1, 1) , L3 : (1,1,2, 1) , E1,2,3 : (1,1,1, 2) , (2) L − L − R − where the numbers in each bracket are the quantum numbers of the corresponding field under SU(3) , SU(2) , SU(2) and U(1) , respectively. The superscript on each field labels C l h Y the generation of the fermion. 4 The neutral gauge boson interactions with fermions are given by g s c = ψ¯γ eAµQ+ Zµ Tl +Th Qs2 +gZµ ETl ETh ψ , (3) L µ c L 3 3 − W H c 3 − s 3 (cid:20) W (cid:18) E E (cid:19)(cid:21) (cid:0) (cid:1) where ψ represents a quark or lepton field, Tl,h are the third components of the SU(2) 3 l,h generators, the electric charge Q is given by Q = T +Y/2 with T = Tl +Th, and s and 3 3 3 3 E c respectively are defined in terms of the gauge couplings g of SU(2) by E 1,2 l,h g2 g2 s2 sin2θ = 1 c2 cos2θ = 2 . (4) E ≡ E g2 +g2 E ≡ E g2 +g2 1 2 1 2 The SM couplings g and e are then given in terms of g and U(1) coupling g′ by 1,2 Y g2g2 g2g′2 g2 = 1 2 , e2 = . (5) g2 +g2 g2 +g′2 1 2 The fields A, Z , Z are defined in terms of the third components W3 of the SU(2) L H l,h l,h gauge fields and the U(1) gauge field B through the following transformation: Y Wl s c c c s Z 3 E E W E W H Wh  =  c s c s s Z  , (6) 3 − E E W E W L      B 0 s c A    W W      −        where g′2 g2 s2 = c2 = . (7) W g2 +g′2 W g2 +g′2 In general Z are not mass eigenstates. Writing them in terms of light and heavy mass L,H eigenstates Z and Z , we have l h Z = sinξZ +cosξZ , Z = cosξZ +sinξZ , (8) L h l H h l − where a rotation angle ξ is introduced. Assume that the breaking of SU(2) SU(2) to SU(2) is achieved by a bi-doublet l h L × η : (1,2,2,0) with a non-zero vacuum expectation value (VEV), u (TeV), and the ∼ O subsequent symmetry breaking isachieved by two doublets Φ : (1,2,1,1)andΦ : (1,1,2,1) 1 2 with respective VEV’s v and v with v2 +v2 = (174 GeV)2. We then have to the leading 1 2 1 2 order in ǫ v/u ≡ s c m2 s2c2 ξ E E(s2 s2)ǫ2 , Zl ǫ2 E E , (9) ≈ c β − E m2 ≈ c2 W Z′ W h 5 where s2 v2/(v2 +v2). Because of the mass hierarchy between fermions belonging to the β ≡ 1 1 2 third generation and the first two generations, s2 is expected to be small. β Now we can express the neutral gauge boson interactions with fermions in the small ǫ limit as g = f¯γ eQA + Zµ T Qs2 ǫ2c2(s2T Th) L µ µ c l 3 − W − E E 3 − 3 (cid:26) W Zµ (cid:2) s2c4 (cid:3) +g h s2T Th +ǫ2 E E(T Qs2 ) f . (10) s c E 3 − 3 c2 3 − W E E (cid:20) W (cid:21)(cid:27) Th acts only on the third generation and the terms proportional to it will induce FCNC’s 3 in the fermion mass eigenstate basis. III. b sℓℓ¯TRANSITIONS → Through the exchanges of Z and Z′ at tree level, the following effective four-fermion interactions can be induced: g2 = Z ǫ2c2 q¯∆˜qγ P q ℓ¯γµ(4s2 1+γ )ℓ Heff −8m2 E µ L W − 5 Zl (cid:16) (cid:17) g2 (cid:0) (cid:1) q¯∆˜qγ P q ℓ¯γµ(s2I ∆˜l)(1 γ )ℓ , (11) −8s2c2m2 µ L E − − 5 E E Zh (cid:16) (cid:17)(cid:16) (cid:17) where ∆˜f = T†diag(0,0,1)T with f¯ M f = f¯ S Mˆ T†f , and S and T = (Tf) are f f R f L R f f f L f f ij unitary matrices for a bi-unitary transformation to obtain the diagonal eigenmass matrix Mˆ . Here we have used the fact that the eigenvalue of T for down quarks and charged f 3 leptons is 1/2. One can further re-write the above expression as − 4G π ∆˜q = FV V∗ ǫ2c2 sb δ (4s2 1) ij + ij Heff − √2 tb tsα EV V∗ ij W − O9 O10 tb ts 4G π ∆˜q (cid:2) (cid:3) FV V∗ ǫ2 sb (s2δ ∆˜ℓ ) ij ij , (12) − √2 tb tsα V V∗ E ij − ij O9 −O10 tb ts (cid:0) (cid:1) where ∆˜q = Tq∗Tq and ∆˜ℓ = Tℓ∗Tℓ , and sb bs bb ij 3i 3j e2 e2 ij = (s¯γ P b) ℓ¯γµℓ , ij = (s¯γ P b) ℓ¯γµγ ℓ . (13) O9 16π2 µ L i j O10 16π2 µ L i 5 j From Eq. (12), we can read off(cid:0)the exp(cid:1)ressions for ij as (cid:0) (cid:1) C9,10 π ∆˜q π ∆˜q Z contribution : Z,ij = ǫ2c2 sb 4s2 1 δ , Z,ij = ǫ2c2 sb δ , C9 α EV V∗ W − ij C10 α EV V∗ ij tb ts tb ts Z′ contribution : Z′,ij = Z′,ij = π(cid:0)ǫ2 ∆˜qsb (cid:1)s2δ ∆˜ℓ . (14) C9 −C10 α V V∗ E ij − ij tb ts (cid:16) (cid:17) 6 The total new physics contributions to the Wilson coefficients are NP,ij = Z,ij + Z′,ij. C9,10 C9,10 C9,10 This implies that within this model and ∆˜ℓ = 0 for i = j, we have the relation ij 6 NP NP = C9 . (15) C10 2s2 (sec2θ +1) 1 W E − IV. NUMERICAL ANALYSIS We now explore the numerical ranges that can be obtained for NP and compare them C9,10 with those of Ref. [1] that can reduce the tension between the predictions and measurements ¯ for the observables in b sℓℓ induced B decays. For this purpose, we need to know the → constraints for the new model parameters, ǫ, c , ∆˜q and ∆˜ℓ . E sb ij The model parameters ǫ and c were constrained by the electroweak precision data in E Ref. [15]. We update this fit here using the latest data [16]. The χ2 contours on the ǫ2-c2 E plane are shown in Fig. 1. It is seen that the best fit values for ǫ2 and c2 are 0.0031 and E 0.4629, respectively. The former indicates that both the VEV of η and the Z′ mass are about 3 TeV. The 1σ and 2σ upper bounds on ǫ2 are 0.0064 and 0.0085 as marked by the vertical dashed and dotted lines, respectively. c2 can range from 0 to 1 at both 1σ and 2σ E levels. In particular, Eq. (15) allows NP to be vanishing when θ = π/4. C10 E ± The parameters ∆˜ℓ involve only leptons and are not well constrained yet. On the other ij hand, ∆˜q is severely constrained by the B -B¯ mixing. The contribution of Z′ exchange to sb s s ∆M of the B mixing system is given by Bs s G ∆MZ′ = F ǫ2(∆˜q )2 B¯ (s¯γµP b)(s¯γ P b) B ηˆ Bs √2m sb h s| L µ L | si B Bs √2G 2 = F ǫ∆˜q m f2 B ηˆ (16) 3 sb Bs Bs Bs B (cid:16)α (cid:17) = √2G V V∗m f2 B ηˆ ∆˜q NP + NP(1 2s2 ) , F6πs2 tb ts Bs Bs Bs B sb C9 C10 − W W (cid:2) (cid:3) where the last expression has been written in terms of the Wilson coefficients NP given in C9,10 Eq. (14) to emphasize the correlation. Note that NP are also linear in the flavour-changing C9,10 coupling ∆˜q . sb Numerically, f B = (216 15) MeV[17]. WehavealsoincludedtheQCDcorrection Bs Bs ± factor ηˆB 0.84 [17p] to account for the renormalization group running of the operator from ≈ theelectroweak scaletotheB scaleandneglectedasmallcorrectionfromadditionalrunning s 7 FIG. 1. χ2 contours of fit to electroweak precision data. The best-fit point, 1σ contour and 2σ contour are marked by a red cross, a blue solid curve, and a red dashed curve, respectively. The vertical dotted lines mark the 1σ and 2σ upper bounds on ǫ2. of the operator between the electroweak scale and the Z′ scale. For our numerical estimates, it is convenient to rewrite the non-perturbative factors in terms of the SM contribution: ∆MZ′ 2√2π2 ǫ∆˜q 2 Bs = sb ∆MSM G M2 S [x ] V V∗ Bs F W 0 t | tb ts|! (17) 2 ǫ∆˜q 2.29 161.8 sb . ≈ V V∗ S [x ] | tb ts|! (cid:18) 0 t (cid:19) We thus remove the main uncertainties from non-perturbative QCD factors, and ignore all but parametric uncertainties in the short-distance part due to the Z′ exchange. Experiments have determined ∆M to high precision. The latest HFAG average [18] Bs of the CDF [19] and LHCb [20] results is ∆Mexp = (17.757 0.021) ps−1. This value is Bs ± consistent with the latest SM prediction, ∆MSM = (18.3 2.7) ps−1 [17], leaving little room Bs ± fornewphysics, particularlyifitinterferesconstructively withtheSMastheterminEq.(16) does. Combining these errors in quadrature, we restrict the new physics contribution to be 0 ∆MZ′ 2.7 (5.4) ps−1 at 1σ (2σ). ≤ Bs ≤ 8 InFig.2, weshowthe1σ (solidblue) and2σ (dashed blue)contoursinthe NP parameter C9,10 space, as determined by the electroweak precision data in Fig. 1 and by ∆Mexp, for the Bs particular value ∆˜q = 0.02. This value is chosen so that it allows the 2σ contour to be in sb the vicinity of the best fit for the b sℓℓ¯anomalies in the NP scenario of Ref. [1], shown → C9,10 by the red . The 1σ (solid red) and 2σ (dashed red) contours from that global fit are also × shown in the figure. Our results show that although it is not possible to reach the best-fit point within our model, there is a substantial overlap at the 2σ level between the values of NP that can be obtained in this model and those that improve the b sℓℓ¯global fit. C9,10 → exp FIG. 2. The region allowed by the electroweak precision data fit and ∆M is shown in blue (the Bs curves on the right). The region allowed by a global fit to b sℓℓ¯observables in Ref. [1] is shown → in red (the curves on the left) for comparison. So far we have assumed ∆˜ℓ = 0 for i = j in Eq. (14). Nevertheless, they can be ij 6 non-vanishing and lead to the possibilities of lepton non-universality and of lepton-flavour violation within this model. To limit the parameter space, we start with a point within the 9 2σ contour of Fig. 2 that is closest to the best-fit point; namely, ∆˜q = 0.02 , ǫ = 0.088 , cosθ = 0.63 , sb E (18) ∆MZ′ = 5.4 ps−1 , NP = 0.87 , NP = 0.32 . Bs C9 − C10 Since ∆˜ℓ = Tℓ∗Tℓ , setting ∆˜ℓ = 0 will maximize NP,µµ. This implies that for real Tℓ, it ij 3i 3j 22 C9 has the following form cosθ 0 sinθ − Tℓ =  0 1 0  . (19)   sinθ 0 cosθ       The non-zero entries for ∆˜ℓ are then: ∆˜ℓ = sin2θ, ∆˜ℓ = ∆˜ℓ = sinθcosθ and ∆˜ℓ = ij 11 13 31 33 cos2θ. Varying the value for sinθ will change our predictions for B K(ee¯,ττ¯,eτ¯,τe¯), → breaking both lepton universality and lepton flavour conservation. Inparticular, themodel can accommodate the value R = 0.745+0.090 0.036. Neglecting K −0.074± the lepton masses, we have 2 2 (B Kℓ ℓ¯) SM + NP,ij + SM + NP,ij , B → i j ∝ C9 C9 C10 C10 (cid:16) (cid:17) (cid:16) 2 (cid:17) 2 NP,µµ ∆˜ℓ (B Kee¯) SM + NP,µµ ∆˜ℓ C10 + SM + NP,µµ 1+ 11 , B → ∝ C9 C9 − 11 c2 C10 C10 c2 E ! " E !# 2 2 (20) NP,µµ ∆˜ℓ (B Kττ¯) SM + NP,µµ ∆˜ℓ C10 + SM + NP,µµ 1+ 33 , B → ∝ C9 C9 − 33 c2 C10 C10 c2 E ! " E !# 2 2 ∆˜ℓ (B Keτ¯,τe¯) 2 NP,µµ 13 . B → ∝ C10 1 2c2 (cid:16) (cid:17) − E! With the numbers given in Eq. (18), we then obtain R = 0.745 sin2θ = 0.37 , (21) K ⇒ (B Kττ¯) B → = 1.36 , (22) (B Kµµ¯) B → (B K(eτ¯,τe¯)) B → = 0.037 . (23) (B Kµµ¯) B → V. SUMMARY AND CONCLUSIONS As one intriguing feature, the model with the SU(2) SU(2) U(1) electroweak gauge l h Y × × symmetry proposed earlier [15] has flavour-changing neutral currents (FCNC’s) at tree level, 10