Minimal unified resolution to R_{K^{(*)}} and R(D^{(*)}) anomalies with lepton mixing
Debajyoti Choudhury, Anirban Kundu, Rusa Mandal, Rahul Sinha
MMinimal unified resolution to R K ( ∗ ) and R ( D ( ∗ ) ) anomalies with lepton mixing Debajyoti Choudhury , Anirban Kundu , Rusa Mandal and Rahul Sinha Department of Physics and Astrophysics, University of Delhi, Delhi 110007, India Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India Institute of Mathematical Sciences, HBNI, Taramani, Chennai 600113, India
It is a challenging task to explain, in terms of a simple and compelling new physics scenario,the intriguing discrepancies between the standard model expectations and the data for the neutral-current observables R K and R K ∗ , as well as the charged-current observables R ( D ) and R ( D ∗ ).We show that this can be achieved in an effective theory with only two unknown parameters. Inaddition, this class of models predicts some interesting signatures in the context of both B decaysas well as high-energy collisions. Introduction and the data – Several recent hints of dis-crepancies in a few charged- as well as neutral-currentsemileptonic decays of B -mesons have intrigued the com-munity. Unlike the case for fully hadronic decay modesthat suffer from large (and, in some cases, not-so-well un-derstood) strong interaction corrections, the theoreticaluncertainties in semileptonic decays are much better con-trolled. Even these uncertainties are removed to a greatextent in ratios of similar observables. While, individu-ally, none of the observables, militate against the stan-dard model (SM), viewed together, they strongly suggestthat some new physics (NP) is lurking around the cor-ner [1, 2]. The pattern also argues convincingly for theviolation of lepton-flavor universality.With the ratios of partial widths being particularlyclean probes of physics beyond the SM, on account ofthe cancellation of the leading uncertainties, let us focuson R ( D ) and R ( D ∗ ) defined as R ( D ( ∗ ) ) ≡ BR( B → D ( ∗ ) τ ν )BR( B → D ( ∗ ) (cid:96)ν ) , (cid:96) ∈ { e, µ } (1)and analogous ratios for the neutral-current sector R K ( ∗ ) ≡ BR( B → K ( ∗ ) µµ )BR( B → K ( ∗ ) ee ) . (2)With the major source of uncertainty in the individualmodes being the form factors, they largely cancel out in ratios like R ( D ( ∗ ) ) or R K ( ∗ ) , and the SM estimatesfor these ratios are rather robust. Several measurementsof R ( D ) and R ( D ∗ ) by the B A B AR [3], Belle [4, 5], andLHCb [6, 7] Collaborations indicated an upward devia-tion from the SM expectations. Combining the individ-ual results, namely, R ( D ) = 0 . ± . ± .
024 and R ( D ∗ ) = 0 . ± . ± . ∼ . σ and ∼ . σ respectively. On the inclusion of thecorrelation between the data, the combined significanceis at the ∼ . σ level [8] from the SM predictions [9]. The cancellation works best for relatively large momentum trans-fers (where the leptonic mass effects are negligible), the regionwith the best data.
The data on R K and R K ∗ , on the other hand, lie sys-tematically below the SM expectations [10, 11]: R K = 0 . +0 . − . ± . q ∈ [1 : 6] GeV ,R low K ∗ = 0 . +0 . − . ± . q ∈ [0 .
045 : 1 .
1] GeV ,R cntr K ∗ = 0 . +0 . − . ± . q ∈ [1 . . (3)For both R K and R cntr K ∗ , the SM predictions are virtuallyindistinguishable from unity [12], whereas for R low K ∗ it is ∼ m µ ). Except for R low K ∗ , the theoreti-cal uncertainties have been subsumed in the experimentalones. Thus the measurements of R K , R low K ∗ and R cntr K ∗ , re-spectively, correspond to 2 . σ , 2 . σ and 2 . σ shortfallsfrom the SM expectations.For the K ∗ mode, a discrepancy is visible not only inthe ratios of binned differential distribution for muon andelectron modes but also in some angular distributions,like the celebrated P (cid:48) [13] anomaly for the decay B → K ∗ µµ [14], at more than 3 σ . Restricting ourselves toonly the low and medium- q region, namely, q ≤ (as the high- q region can be affected by a different kindof physics [15]), we do not include this anomaly in ouranalysis. However, we see later that our fitted Wilsoncoefficients can explain this discrepancy as pointed outin global fits [1].A similar suppression (at a level of approximately 3 σ )is seen in the observable Φ ≡ d BR( B s → φµµ ) /dm µµ inthe analogous bin ( m µµ ∈ [1 : 6] GeV ) [16–18], namely,Φ = (cid:40)(cid:0) . +0 . − . ± . ± . (cid:1) × − GeV − (exp . )(4 . ± . × − GeV − (SM) . (4)With low theoretical error, this bin is virtually the sameas that for R K and R cntr K ∗ . This suggests strongly thatthe discrepancies in the latter have been caused by a de-pletion of the b → sµµ channel, rather than an enhance-ment in b → see , a surmise further vindicated by the P (cid:48) anomaly. Note that P (cid:48) is dominated by the vector oper-ator O , while the two-body decay B s → µµ is controlledby the axial vector operator O , both of them definedlater. a r X i v : . [ h e p - ph ] O c t With possible corrections from large ∆Γ s , as well asnext-to-leading-order (NLO) electroweak and next-to-next-to-leading-order QCD corrections calculated, theSM prediction is quite robust with only small uncer-tainties accruing from the Cabibbo-Kobayashi-Mashkawa(CKM) matrix elements and the decay constant of B s .The LHCb measurement at a significance of 7 . σ [19, 20]shows an excellent agreement between the data and theSM:BR( B s → µµ ) = (cid:40)(cid:0) . ± . +0 . − . (cid:1) × − (exp . ) , (3 . ± . × − (SM) , (5)and hence puts very strong constraints on NP models, inparticular on those incorporating (pseudo)scalar or axial-vector currents [21]. However, note that the central valuecan accommodate a ∼
20% suppression. Thus, one isnaturally led to models that preferentially alter O ratherthan O .Similarly, neither the radiative decay B → X s γ northe mass difference ∆ M s and mixing phase φ s measure-ments for the B s system show any appreciable discrep-ancy with the SM expectations. The pattern of devi-ations is thus a complicated one and, naively at least,does not appear to show a definite direction towards anywell-motivated NP model. Consequently, most efforts atexplaining the anomalies consider only a subset, either R K and/or R ( D ( ∗ ) ) data [22, 23], or R K ( ∗ ) and b → s(cid:96)(cid:96) data [24]. Those that do attempt a more complete treat-ment either invoke very complicated models, or resultin fits that are not very good. In addition, they are li-able to result in other unacceptable phenomenologicalconsequences. Analyses within specific models, like lep-toquarks, are available in the literature [25].In view of this, we adopt a very phenomenological ap-proach, rather than advocate a particular model. Assum-ing an effective Lagrangian, with the minimal number ofnew parameters, in the guise of the unknown Wilson co-efficients (WCs), we seek the best fit. While not an en-tirely new idea, our analysis takes into account not onlythe anomalous channels but also the existing limits onseveral other channels; as we will show, they provide thetightest constraints on the parameter space. This ap-proach hopefully will pave the way to unravelling the asyet unknown flavor dynamics. Models – Within the SM, the b → cτ ν τ transition pro-ceeds through a tree-level W exchange. If the NP addscoherently to the SM, one can write the effective Hamil-tonian as H eff = 4 G F √ V cb (cid:0) C NP (cid:1) [( c, b )( τ, ν τ )] , (6)where the NP contribution is parametrized by C NP van-ishes in the SM limit and we have introduced the short-hand notation ( x, y ) ≡ x L γ µ y L ∀ x, y . To explain the data, one thus needs either small positive or large nega-tive values of C NP .The flavor-changing neutral-current decays B → K ( ∗ ) µµ and φµµ are occasioned by the b → sµµ transi-tion proceeding, within the SM, primarily through a com-bination of the penguin and the box diagrams (driven, es-sentially, by the top quark). Parametrizing the ensuingeffective Hamiltonian as H eff = − G F √ V tb V ∗ ts (cid:88) i C i ( µ ) O i ( µ ) , (7)where the relevant operators are O = ( α em ( m b ) m b / π ) ( sσ µν P R b ) F µν , O = ( α em ( m b ) / π ) ( sγ µ P L b ) ( µγ µ µ ) , O = ( α em ( m b ) / π ) ( sγ µ P L b ) ( µγ µ γ µ ) . (8)The WCs, matched with the full theory at m W andthen run down to m b at the next-to-next-to-leading log-arithmic accuracy [23], are given in the SM as C = − . , C = 4 .
211 and C = − . . The differ-ential widths for the B → K ( ∗ ) µµ decay are obtained interms of algebraic functions of these. NP contributionsto H eff can be parametrized by C i → C i + C NP i .Similarly, the b → sνν transition (which governs the B → K ( ∗ ) νν decays) proceeds through the Z penguinsand box diagrams. Unless right-handed neutrino fieldsare introduced, the low energy effective Hamiltonian canbe parametrized by [26] H eff = 2 G F √ V tb V ∗ ts α em π C SM L (cid:0) C NP ν (cid:1) ( s, b )( ν, ν ) , (9)where C NP ν denotes the NP contribution. Including theNLO QCD correction and the two-loop electroweak con-tribution, the SM WC is given by C SM L = − X t /s w wherethe Inami-Lim function X t = 1 . ± .
017 [26, 27].While it may seem trivial to write down extra four-fermi operators that would produce just the right contri-butions, care must be taken to see that this does notintroduce unwelcome consequences. For one, a largeenhancement of C could lead an unacceptably largeBR( B s → µµ ), with O being the leading contributorto this decay. Similarly, the said four-fermi operatorsneed to be invariant under the SM gauge group (assumingthat the NP appears only above the electroweak scale).A non-zero C NP (see Eq. (6)) would, potentially, lead toan analogue of C NP10 for the tau-channel. This, in turn,would lead to an enhancement of B s → τ τ , where thechirality suppression is less operative than in the muoniccase. Indeed, the LHCb Collaboration [28] has obtaineda 95% C.L. upper limit of 6 . × − on the branch-ing fraction for this mode , with the SM value being It should be noted, though, that this analysis does not actuallyreconstruct the τ s, but employs neural networks. Hence, it ispossible that future measurements would point to a value higherthan the limits quoted. (7 . ± . × − [20]. Similarly, none of the threeoperators ( b, s ) ( ν i , ν i ) may receive large corrections lestthe SM expectations, namely [26]BR( B + → K + νν ) SM = (3 . ± . ± . × − , BR( B → K ∗ νν ) SM = (9 . ± . ± . × − , (10)be augmented to levels beyond the 90% C.L upperbounds (summed over all three neutrinos) as obtainedby the Belle Collaboration [29], viz. BR( B → K ( ∗ ) νν ) < . . × − . (11)In view of the aforementioned constraints, we consideronly a combination of two four-fermi operators, charac-terized by a single WC (assumed to be real to avoid newsources of CP violation). Since we do not claim to obtainthe ultraviolet completion thereof, we do not speculateon the (flavor) symmetry that would have led to such astructure, which could have arisen from a plethora of NPscenarios, such as models of (gauged) flavor, leptoquarks(or, within the supersymmetric paradigm, a breaking of R parity) etc. To wit, we propose a model involving twofour-fermi operators in terms of the second- and third-generation (weak-eigenstate) fields H NP = A ( Q L γ µ L L ) ( L L γ µ Q L )+ A ( Q L γ µ Q L ) ( τ R γ µ τ R ) (12)where the overall Clebsch-Gordan coefficients have beensubsumed and we demand A = A .This operator, seemingly, contributes to R ( D ( ∗ ) ) butnot to the other anomalous processes. This, though, istrue only above the electroweak scale. Below this scale,the Hamiltonian needs to be rediagonalized In the quarksector, this is determined by the quark masses and thesmall non-alignment due to A , can be neglected. Inthe leptonic sector, though, the extreme smallness of theneutrino masses implies that the nonuniversal term H NP plays a major role [30]. To this end, we consider thesimplest of field rotations for the left-handed leptons fromthe unprimed (flavor) to the primed (mass) basis, namely τ = cos θ τ (cid:48) + sin θ µ (cid:48) , ν τ = cos θ ν (cid:48) τ + sin θ ν (cid:48) µ . (13)This, immediately, generates a term with the potentialto explain the b → sµµ anomalies. Results — The scenario is, thus, characterized by twoparameters, namely A and sin θ . The best fit values Note that the neutrino flavors need not be identical for the NP. With NP only modifying the Wilson coefficients of certain SMoperators to a small extent, the QCD corrections (as well ashadronic uncertainties) are analogous. Additional effects due tooperator mixings are too small to be of any concern. for these can be obtained by effecting a χ -test definedthrough χ = (cid:88) i =1 (cid:0) O exp i − O th i (cid:1) (∆ O exp i ) + (cid:0) ∆ O th i (cid:1) (14)where O exp i ( O th i ) denote the experimental (theoreti-cal) mean and ∆ O exp i (∆ O th i ) the corresponding 1 σ un-certainty, with the theoretical values depending on themodel parameters. We include a total of seven measure-ments for the evaluation of χ , namely, R ( D ), R ( D ∗ ), R K , R low K ∗ , R cntr K ∗ , Φ, and BR( B s → µµ ) (while not af-fected by the NP interactions in Eq. (12), this is relevantfor the scenario considered later). Only for the last twoobservables, do ∆ O th i need to be considered explicitly ,while, for the rest, they have been subsumed within theexperimental results. For our numerical analysis, we use V cb = 0 . V tb V ∗ ts = − . , and find, for the SM, χ (cid:39) χ (cid:39) C NP9 = − . C NP = − .
12. Interms of the model parameters, this corresponds to (notethat there is a θ → − θ degeneracy) A (= A ) = − .
92 TeV − , sin θ = ± . , (15)Even this low value of χ is largely dominated by asingle measurement, namely, R low K ∗ . This is not unex-pected, as an agreement to this experimental value tobetter than 1 σ is not possible if the NP contributioncan be expressed just as a modification of the SM WCs,rather than through the introduction of a new and smalldynamical scale (such a change could be tuned so as tomanifest itself primarily only in the low- q region, but islikely to have other ramifications). Note that the smallvalue of sin θ can only partially explain the atmosphericneutrino oscillation, while the full explanation needs ad-ditional dynamics. - - - - - - - - - - - A [ TeV - ] s i n θ FIG. 1. The light and dark blue regions denote 95% and99% C.L. bands, respectively, around the best-fit points. Thered shaded region is allowed by bounds from BR( B + → K + µ − τ + ). More importantly, in effecting the field rotation of Eq.(13) in H NP , we generate terms of the form ( s, b )( µ, τ ),leading to potential lepton-flavor violating (LFV) decays.The current limits on the relevant ones are [31]BR( B + → K + µ ± τ ∓ ) < . . × − . (16)In Fig. 1, we display the constraints from this particularmode. While the best-fit point is summarily ruled out,clearly solutions can be found if a slight worsening of the χ (to (cid:39)
15) is acceptable. This would still represent amuch better agreement than is possible within the SM.The corresponding values of the observables are: R K =0 . R cntr K ∗ = 0 . R low K ∗ = 0 . R (cid:0) D ( ∗ ) (cid:1) = 1 . × R SM (cid:0) D ( ∗ ) (cid:1) , and Φ = 4 . × − GeV − , representingquite a reasonable fit to all but R low K ∗ . It should be notedhere that the θ → − θ degeneracy is broken by the LFVconstraint, with θ > R K ( ∗ ) requires the intro-duction of a small bit of C NP10 . Postponing the discussionof B s → τ τ , this is most easily achieved if we chooseto destroy, to a small degree, the relation A = A . Asan illustrative example, we consider A = 4 A /
5. Theconsequent best fit values for A and sin θ remain virtu-ally the same but, now, χ = 7 with NP contributionsbeing C NP9 = − . C NP10 = 0 .
17 and C NP = − . R K (cid:39) . R cntr K ∗ (cid:39) . R low K ∗ (cid:39) . R (cid:0) D ( ∗ ) (cid:1) (cid:39) . × R SM (cid:0) D ( ∗ ) (cid:1) , and Φ (cid:39) . × − GeV − , show-ing marked improvement in the fit to all but R low K ∗ andcorrespond to χ (cid:39)
10. While the finite contribution to C NP10 does enhance B s → τ τ , the latter (gray shaded re-gion in Fig. 2) does not have a major impact. It should berealized, though, that a stronger breaking of the A = A relation would have led to a better (worse) agreementwith the LFV ( B s → τ τ ) constraints.It is interesting to speculate on the origin of this splitbetween the A i . A naive explanation would be to at-tribute the difference to the quantum numbers of the lep-tonic fields under an as yet unidentified gauge symmetry,with the attendant anomaly cancellation being effectedby either invoking heavier fermionic fields or throughother means. Care must be taken, however, not to in-duce undesirable phenomenology. An alternative is toattribute the difference to quantum corrections, althoughthe aforementioned shift is somewhat larger than that ex-pected from a naive renormalization group flow perspec-tive, namely, ∼ ( α wk . / π ) ln(Λ /m b ), where Λ NP ∼ H eff suffer differing renor-malization group flow down to the m b scale, and the con-sequent breaking of the degeneracy is, putatively, of theright magnitude to explain the remaining discrepancies. - - - - - - - - - - - A [ TeV - ] s i n θ R K * cntr R K R D (*) B + → K + μ - τ + ( allowed ) B s →ττ ( disallowed ) FIG. 2. The fit for A = 4 A /
5, with the bands aroundthe best-fit points corresponding to 95% and 99% C.L. Alsoshown are the 1 σ bands from R K ( ∗ ) and R ( D ), and the 95%upper limits from B s → τ τ and B + → K + µ − τ + . It is worthwhile, at this stage, to explore theconsequences of introducing other operators in H NP .While operators constructed out of SU (2) L -tripletcurrents (denoted by the subscript ‘3’) such as( Q L γ µ Q L ) ( L L γ µ L L ) , ( Q L γ µ L L ) ( L L γ µ Q L ) ,etc., would also have admitted solutions to the anoma-lies, they, typically, would also result in unsuppressed b → sνν transitions. Circumventing the bounds would,then, require the introduction of multiple operators andcancellations between them. We will discuss such possi-bilities in detail in a subsequent paper.This would, typically, still leave behind too large arate for B s → τ τ (first reference of [25]) and, henceneeds the further introduction of yet another operatorsuch as the second one in H NP . Apart from enhancing B s → τ τ ( B → X s τ τ and Λ b → Λ τ τ are affected too,but bounds from these sectors are not too serious), thiswould also affect the other modes to varying degrees.Consequently, the best fit values will change. Indeed alower χ ( (cid:39) .
4) is achievable for virtually the same A ,but slightly smaller | sin θ | ( (cid:39) . B s → τ τ bound as well that in Eq. (16) areto be satisfied, the χ can be reduced to at most (cid:39) B → X s τ τ ), as well as BR(Λ b → Λ τ τ )will also be increased and should be close to observa-tion at the LHCb. However, processes like b → sγ or τ → µγ will remain under control, as we have checked.Similarly, while we do not “explain” ( g − µ , the agree-ment is marginally better than within the SM. The newoperators also generate, through renormalization grouprunning, operators involving four leptons [32], and thusmay lead to effects like τ → µ . They are, however, wellwithin control, mostly because of the small value of sin θ .In summary, we have identified the minimal modifica-tion to the SM in terms of an effective theory that can ex-plain the anomalies in both the charged- and the neutral-current decays of bottom mesons, a task that has beenchallenging on account of the seemingly contradictory re-quirements that the data demand. We circumvent thisby postulating just two four-fermi operators with WCsrelated by a symmetry and taking advantage of the pos-sibility of a small but nontrivial rotation of the chargedlepton fields that a flavor-nonuniversal operator entails.Taking all the data into account, we find that with justtwo new parameters, the χ can be reduced from 46 (inthe SM) to below 15 while being consistent with all otherdata. 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