Are the B decay anomalies related to neutrino oscillations?
AAre the B decay anomalies related to neutrino oscillations?
Sofiane M. Boucenna, ∗ José W.F. Valle, † and Avelino Vicente
2, 3, ‡ INFN, Laboratori Nazionali di Frascati, C.P. 13, 100044 Frascati, Italy. Instituto de Física Corpuscular (CSIC-Universitat de València), Apdo. 22085, E-46071 Valencia, Spain. IFPA, Dep. AGO, Université Liège, Bat B5, Sart-Tilman B-4000, Liège 1, Belgium
Neutrino oscillations are solidly established, with a hint of CP violation just emerging. Similarly,there are hints of lepton universality violation in b → s transitions at the level of . σ . By assumingthat the unitary transformation between weak and mass charged leptons equals the leptonic mixingmatrix measured in neutrino oscillation experiments, we predict several lepton flavor violating (LFV)B meson decays. We are led to the tantalizing possibility that some LFV branching ratios for Bdecays correlate with the leptonic CP phase δ characterizing neutrino oscillations. Moreover, wealso consider implications for (cid:96) i → (cid:96) j (cid:96) k (cid:96) k decays. PACS numbers:
Introduction
The historical discovery of the Higgs boson [1, 2] wouldhave completed our picture of particle physics, were itnot for the solid evidence we now have that neutrino fla-vors interconvert [3]. Apart from neutrino oscillationsand cosmology, no other signs of new physics (NP) havebeen established. However, some indirect signs mighthave been found by the LHCb collaboration. In 2013,they have published the results of the measurement ofa variety of observables in b → s transitions. In somecases, the experimental result was found to be in cleartension with the Standard Model (SM) prediction. Theseinclude angular observables [4–7] in B → K ∗ µ + µ − [8],as well as a sizable suppression of several branching ra-tios [9, 10]. Recently, the LHCb announced new resultsbased on the complete LHC Run I dataset [11]. The in-clusion of new data has confirmed the anomalies, whichare currently at the ∼ σ level. Furthermore, in 2014,the LHCb collaboration found an intriguing indication oflepton universality violation in the ratio [12] R K = BR ( B → Kµ + µ − ) BR ( B → Ke + e − ) = 0 . +0 . − . ± . . (1)This experimental measurement, obtained in the lowdilepton invariant mass regime, is . σ away from theSM result R SMK = 1 . ± . [13]. Although thestatistical significance of this discrepancy is not enoughto claim a discovery, it is highly suggestive that severalindependent global fits [14–18] have shown that this hintcan be explained by the same type of new physics con-tributions as the previous b → s anomalies.The violation of lepton universality usually comes to-gether with the violation of lepton flavor. Based on sym-metry arguments, Glashow, Guadagnoli and Lane [19] ∗ Electronic address: [email protected] † Electronic address: valle@ific.uv.es ‡ Electronic address: [email protected] recently argued that the observation of universality vio-lation in the lepton flavor conserving (LFC) B → K(cid:96) + i (cid:96) − i decays implies the existence of the lepton flavor violating(LFV) processes B → K(cid:96) + i (cid:96) − j (with i (cid:54) = j ). The ideaof LFV in B meson decays has been further exploredin [20–24].Here we take a step further in this direction. Sincewe lack a theory a flavor, we can not make definite pre-dictions for the LFV rates using the LFC ones as in-put. Hence we make the simplest alternative assump-tion, namely, that the unitary transformation betweenweak and mass charged lepton states is given by the lep-tonic mixing matrix measured in neutrino oscillation ex-periments. Under this assumption we make numericalpredictions for several LFV observables in the B system.We emphasize that this assumption is not completely adhoc . It will actually be a prediction in models where theleptonic mixing arises from the charged lepton sector.We refer to [25] for a general discussion and an examplemodel. General aspects of the b → s anomalies The effective hamiltonian describing b → s transitionscan be expressed as: H eff = − G F √ V tb V ∗ ts e π (cid:88) i ( C i O i + C (cid:48) i O (cid:48) i ) + h.c. (2)Here G F is the Fermi constant, e the electric charge and V the CKM matrix. The Wilson coefficients C i and C (cid:48) i encode the different (SM and NP) contributions to theeffective operators O i and O (cid:48) i . The analysis of the avail-able experimental data on b → s transitions reveals thatthe effective operators relevant for the resolution of the b → s anomalies are: O ≡ O µµ = (¯ sγ α P L b ) (¯ µγ α µ ) , (3) O ≡ O µµ = (¯ sγ α P L b ) (¯ µγ α γ µ ) , (4)where P L = (1 − γ ) is the left-chirality projector. Sev-eral independent global fits [15–18] find a significant ten- a r X i v : . [ h e p - ph ] O c t sion between the SM results for the Wilson coefficientsof these operators and the experimental data. This canbe clearly alleviated in the presence of NP contributions.According to the global fit [18], the C µµ coefficient is thekey to improve the fits. More precisely, one finds a rea-sonable agreement with data when NP provides a nega-tive contribution to O µµ , with C µµ, NP ∼ − × C µµ, SM .Similar improvements are found when NP enters in the SU (2) L invariant direction C µµ, NP = − C µµ, NP , with C µµ, NP ∼ − × C µµ, SM . Predicting lepton flavor violation in B meson decays Here we raise the following question: can the leptonicmixing matrix provide the required lepton flavor struc-ture in O and O ? And if so, what are the predictionsfor lepton flavor violation in the B sector? As suggestedby global fits, let us assume that the relevant NP opera-tor contains a left-handed leptonic current. In this case,this operator can be generally written in the mass basisas: O ij = 1Λ J dα J α(cid:96) ij , (5)where J dα = C Qbs ¯ bγ α P L s , (6) J α(cid:96) ij = C Lij ¯ (cid:96) i γ α P L (cid:96) j , (7)and Λ is the energy scale of the NP inducing this oper-ator. The i, j indices denote the lepton flavor combina-tion characterizing the operator in eq. (7). The × matrices C Q and C L completely determine the relationsamong the Wilson coefficients for different flavor choices.On the other hand, in the interaction (gauge) basis, O takes the same form, but the quark and lepton currentsare written in terms of gauge eigenstates d (cid:48) and (cid:96) (cid:48) as J dα = ˜ C Qmn ¯ d (cid:48) m γ α P L d (cid:48) n , (8) J α(cid:96) ij = ˜ C Lij ¯ (cid:96) (cid:48) i γ α P L (cid:96) (cid:48) j . (9)We now focus on the leptons. By combining eqs. (7)and (9) one finds the relation between C L and ˜ C L , C L = U † (cid:96) ˜ C L U (cid:96) , (10)where U (cid:96) is the unitary matrix which relates the left-handed charged lepton gauge and mass eigenstates as (cid:96) (cid:48) = U (cid:96) (cid:96) . Similarly, the left-handed neutrino gauge andmass eigenstates are connected by another matrix, U ν , as ν (cid:48) = U ν ν . The product of these two matrices determinesthe leptonic charged current weak interaction, L cc = − g √ (cid:2) W − µ ¯ (cid:96) (cid:48) γ µ P L ν (cid:48) + h.c. (cid:3) = − g √ (cid:2) W − µ ¯ (cid:96)γ µ KP L ν + h.c. (cid:3) , (11) where K = U † (cid:96) U ν is the leptonic mixing matrix measuredin neutrino oscillation experiments. If U ν = I , the left-handed neutrino gauge and mass eigenstates are the sameand all the mixing is in the left-handed charged leptons.In this case K = U † (cid:96) and eq. (10) leads to C L = K ˜ C L K † . (12)We do not attempt to give any model prediction for ˜ C L . Instead, we will assume that it is diagonal but withnon-universal entries. In that case one can determinethe required ˜ C L which, after using eq. (12), leadsto a C L matrix compatible with the observations in b → s transitions. In particular, the resulting C L musthave a strong hierarchy between the ee and µµ entries, C Lee (cid:28) C Lµµ , in order to induce a sizable correction to B → K ( ∗ ) µ + µ − and a negligible one to B → K ( ∗ ) e + e − . “Deriving” C L from neutrino oscillations Barring tuning of the parameters, we find two generic ˜ C L matrices in the gauge basis that lead to valid C L matrices in the mass basis. Their forms define our twoscenarios: • Scenario A : ˜ C L = diag (0 , (cid:15), • Scenario B : ˜ C L = diag ( (cid:15), , Here (cid:15) (cid:28) is a small parameter (interestingly enough,note that Ref. [19] considered ˜ C L = diag (0 , , , whichcorresponds to any of our scenarios in the limit (cid:15) = 0 ) .Using the standard parameterization for the leptonicmixing matrix K , one finds that in order to suppressthe contributions to the ee Wilson coefficients, (cid:15) must beclose to (cid:15) A = − tan θ sin θ in scenario A , (13) (cid:15) B = − tan θ cos θ in scenario B . (14)Taking σ ranges for the mixing angles from the latestglobal fit to neutrino oscillation data [26], one finds theranges [ − . , − . for scenario A and [ − . , − . for scenario B, irrespective of the neutrino mass spec-trum; normal and inverted hierarchies giving basicallythe same results. Interestingly, θ (cid:54) = 0 implies (cid:15) (cid:54) = 0 ,indicating a suggestive connection between quarks andleptons.We can now obtain C L for both scenarios. Let us firstconsider case A. Assuming (cid:15) = (cid:15) A and taking the best-fitvalues from [26], we find: C L = − .
023 + 0 . e iδ .
026 + 0 . e iδ − .
023 + 0 . e − iδ .
005 cos δ + 0 . − .
001 cos δ + 0 . i sin δ + 0 . .
026 + 0 . e − iδ − .
001 cos δ − . i sin δ + 0 .
509 0 . − .
005 cos δ , (15)where δ is the Dirac leptonic CP violating phase. In theCP conserving case ( δ = 0 ) this matrix simplifies to C L = .
094 0 . .
094 0 .
537 0 . .
128 0 .
508 0 . . (16) Regarding case B, assuming now (cid:15) = (cid:15) B and taking thebest-fit values for the mixing angles from [26], one finds C L = .
011 + 0 . e iδ − .
012 + 0 . e iδ .
011 + 0 . e − iδ . − .
003 cos δ − . i sin δ + 0 . − .
012 + 0 . e − iδ . i sin δ + 0 .
489 0 .
003 cos δ + 0 . . (17) FIG. 1: The branching ratio of the decay B → Keµ versusthe CP violating phase δ in scenarios A and B. The bands areobtained by taking the leptonic mixing angles within their σ range w.r.t. the best-fit value (solid line) [26]. In the CP conserving case ( δ = 0 ) this matrix simplifiesto C L = .
128 0 . .
128 0 .
545 0 . .
090 0 .
489 0 . . (18)Comparing the C L matrices for our two scenarios, wefind that they are of the same of order of magnitude andthe most significant difference lies in the terms involving δ . This is what will allow us to relate B decays to theleptonic CP phase. Lepton flavor violation in the B system
The matrix C L can be used to make definite predic-tions for ratios of branching ratios in B → K(cid:96) + i (cid:96) − j decays,BR ( B → K(cid:96) ± i (cid:96) ∓ j ) = 2 ρ NP Φ ij (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C Lij C Lµµ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) BR ( B → Kµ + µ − ) . (19)Here BR ( B → K(cid:96) ± i (cid:96) ∓ j ) = BR ( B → K(cid:96) + i (cid:96) − j ) + BR ( B → K(cid:96) − i (cid:96) + j ) and BR ( B → Kµ + µ − ) = (4 . ± . × − is the LHCb result [9], measured using the fb − dataset after LHC Run I in the complete q range, where q = M µµ is the dimuon invariant mass. The factor ρ NP is the NP fraction of the B → Kµ + µ − amplitude, ρ NP = M NP / M Total [19]. Using the results of theglobal fit [18], which gives C µµ, NP ∼ − × C µµ, SM , ρ NP is found to be ρ NP ∼ − . . Finally, the Φ ij factor accounts for phase space and charged lepton masseffects. These introduce sizable corrections for finalstates including τ leptons. Using the results of Ref. [27],we find Φ µe (cid:39) and Φ τe = Φ τµ (cid:39) . . Finally, we notethat the parameterization in terms of ρ NP is only exactin the limit of vanishing non-factorizable contributions.However, we have found that these corrections arenegligible for the processes we are interested in.For δ = 0 , we obtain the following predictions for the The authors of [19] derive their value for ρ NP from the LHCb R K measurement, obtaining ρ NP ∼ − . . FIG. 2: Same as fig. (1) for the branching ratio of the decay B → Keτ . B → K LFV transitions in scenario A,BR ( B → Ke ± µ ∓ ) ∈ [4 . , . × − , (20)BR ( B → Ke ± τ ∓ ) ∈ [0 . , . × − , (21)BR ( B → Kµ ± τ ∓ ) ∈ [0 . , . × − . (22)These have been derived using the LHCb central valueand taking the leptonic mixing angles in the preferred σ ranges found by the fit [26]. The main generic predictionfrom our setup is thusBR ( B → Kµ ± τ ∓ ) (cid:29) BR ( B → Ke ± µ ∓ ) , BR ( B → Ke ± τ ∓ ) . (23)However, experimentally the decay B → Ke ± µ ∓ isthe easiest to search for and reconstruct. Indeed, elec-tron and tau final states are, O (20%) and O (80%) re-spectively, worse to reconstruct. Moreover future RUNII data will probe the region of O (10 − ) for this channelproviding a test of our scenario. Rare B decays and leptonic CP violation
One can now consider a scenario with a non-zero valueof the CP violating phase δ characterizing neutrino oscil-lations. In this case, we are led to the fascinating possi-bility that the LFV branching ratios for B meson decayswill depend upon δ . Our results can be found in figs. (1)and (2) corresponding to the decay modes B → Kµ ± e ∓ and B → Kτ ± e ∓ respectively. This would suggest analternative way of probing δ by using LFV B meson de-cays. (cid:96) i → (cid:96) j (cid:96) k (cid:96) k decays The same strategy can be extended to other LFV ob-servables if induced mainly by vectorial operators, as ineqs. (3) and (4). Assuming the same leptonic currents, the analogous operators for the purely leptonic LFV pro-cesses (cid:96) i → (cid:96) j (cid:96) k (cid:96) k are: O (cid:96) = 1Λ (cid:0) C Lij ¯ (cid:96) i γ α P L (cid:96) j (cid:1) (cid:0) C Lmn ¯ (cid:96) m γ α P L (cid:96) n (cid:1) . (24)Here we assume that the scale of the NP responsible forthe vectorial LFV operators is the same as the one rel-evant for B meson decays, eq. (5), although in full gen-erality these could be unrelated. The flavor structure of O (cid:96) ≡ O ijmn (cid:96) is given by the product C Lij C Lmn which, fol-lowing the same prescription as for the B meson decays,can be written as C Lij C Lmn = (cid:16) K ˜ C L K † (cid:17) ij (cid:16) K ˜ C L K † (cid:17) mn .The O (cid:96) operator induces several (cid:96) i → (cid:96) j (cid:96) k (cid:96) k decayprocesses: (i) (cid:96) − i → (cid:96) − j (cid:96) − k (cid:96) + k , and (ii) (cid:96) − i → (cid:96) + j (cid:96) − k (cid:96) − k (with k (cid:54) = j ). Their branching ratios can be written as [28]BR ( (cid:96) i → (cid:96) j (cid:96) k (cid:96) k ) = κ m (cid:96) i π Γ (cid:96) i | M ijk | Λ , (25)where κ = 2 / when there are two identical leptons in thefinal state, and κ = 1 / otherwise, and m (cid:96) i and Γ (cid:96) i arethe mass and decay width of the (cid:96) i lepton, respectively.The coefficient M ijk takes the form C Lij C Lkk in case (i), C Lik C Ljk in case (ii).One can now use the experimental limits on these LFVbranching ratios to derive bounds on Λ . Processes in-volving C Lee are strongly suppressed and thus they donot provide meaningful bounds. This is the case of µ − → e − e − e + , τ − → e − e − e + and τ − → µ − e − e + .In contrast, the combined LHCb+BaBar+Belle limitBR ( τ − → µ − µ − µ + ) < . × − [29] translates into Λ (cid:38) . TeV (in both scenarios, A and B). The other τ decay modes lead to slightly less stringent bounds. Fu-ture B factories are expected to improve on the searchfor τ − → µ − µ − µ + , with sensitivies to branching ratiosas low as ∼ − [30], allowing us to probe NP scales upto Λ ∼ TeV.
Conclusions and discussion
In summary, we have suggested that the universalityand flavor violating b → s anomalies may be related tothe pattern of neutrino oscillations. By assuming that theunitary transformation between weak and mass chargedlepton eigenstates is given by the leptonic mixing matrixmeasured in neutrino oscillations we predict several lep-ton flavor violating B meson decay rates. This way we areled to the thrilling possibility that some of the rare LFVB decay branching ratios correlate with the leptonic CPphase δ that characterizes neutrino oscillations. Otherlepton flavor violating processes processes such as (cid:96) i → (cid:96) j (cid:96) k (cid:96) k have been considered in a similar manner. Im-proved measurements at Belle should probe new physicsscale at the level Λ ∼ TeV. Relevant scenarios involveadditional neutral currents, such as schemes containingan extra Z (cid:48) boson with lepton universality violation in B decays [31–34], or possibly some realizations of theelectroweak symmetry SU (3) C ⊗ SU (3) L ⊗ U (1) X [35–40]. Such schemes should be taken seriously should theobserved hints in the B sector persist. Finally, we notethat in this paper we have focussed on the case wherethe violation of lepton universality is caused by NP in B → Kµµ , with negligible contributions to B → Kee .The alternative hypothesis is also plausible, though ithas a lower constraining power since the electron chan-nel is experimentally somewhat less constrained than themuonic one.
Note added
A few days ago, an update of [18] was presented in [41].While this would change slightly the value of ρ NP used inour analysis, our main point remains and the numericalresults are also left essentially unchanged. Acknowledgments
We are grateful to Javier Virto, Jorge Martin Ca-malich, and Marcin Chrząszcz for enlightening discus-sions. Work supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 (MINECO), and thegrant PROMETEOII/2014/084 from Generalitat Va-lenciana. SMB acknowledges financial support fromthe research grant “Theoretical Astroparticle Physics”number 2012CPPYP7 under the program PRIN 2012funded by the Italian “Ministero dell’Istruzione, Uni-versitá e della Ricerca” (MIUR) and from the INFN“Iniziativa Specifica” Theoretical Astroparticle Physics(TAsP-LNF). AV acknowledges partial support from theEXPL/FIS-NUC/0460/2013 project financed by the Por-tuguese FCT. [1]
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