Unified resolution of the R(D) and R(D^*) anomalies and the lepton flavor violating decay h\toμτ
Debajyoti Choudhury, Anirban Kundu, Soumitra Nandi, Sunando Kumar Patra
UUnified resolution of the R ( D ) and R ( D ∗ ) anomalies and the leptonflavor violating decay h → µτ Debajyoti Choudhury ∗ , Anirban Kundu † , Soumitra Nandi ‡ and Sunando Kumar Patra § 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 Department of Physics, Indian Institute of Technology, North Guwahati, Guwahati 781039, Assam, India
Abstract
Taking advantage of the fact that the flavor of the neutrino in semileptonic B decays B → D ( ∗ ) τ ν is not known, we show how a minimal set of higher-dimensional lepton flavor violating(LFV) operators can explain the R ( D ( ∗ ) ) anomalies, and as a spin-off, can give rise to the LFVdecay of the Higgs boson, h → µτ . We also show how none but the minimal set of operators survivethe present data. PACS no.: 12.60.Fr, 13.20.He, 14.80.Bn
The search for signals of lepton flavor violation (LFV) has been a long and varied quest, for it isbelieved to not only constitute a smoking gun for new physics (NP) beyond the Standard Model (SM),but also shed light on a variety of issues ill-understood within the SM, such as the origin of flavor onthe one-hand and the generation of non-zero lepton and baryon number in the universe, on the other.While the SM can incorporate LFV, as seen, e.g., in neutrino oscillations, by the mere inclusion ofright-handed neutrino fields and consequent Dirac masses, the corresponding LFV amplitudes wouldbe too small to be manifested in processes involving charged leptons . Even the proposed upgrades,or new experiments, are expected to improve the limits on LFV processes by at most one order ofmagnitude, except for µ → e and µ - e conversion [1]. Indeed, if decays such as µ → eγ or τ → µ are seen in experiments currently in operation or due to start in the near future, the correspondingamplitudes would be too large to be supported by such trivial extensions of the SM.It is in this context that the recently reported [2] hint, from the CMS experiment, of the Higgs bosondecay h → µτ is to be viewed. If this is not a mere background fluctuation but an actual signal, one hasto entertain the possibility that such LFV decays are flavor-specific, as neither CMS nor ATLAS hasseen any LFV in channels like h → eτ or h → eµ [3]. This, however, is not unnatural, simply becausesuch a decay is quite likely to be generated from Yukawa couplings, and the latter are believed to be ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] It should also be noted that total lepton number conservation is an accidental symmetry within the SM, and that theinclusion of right-handed neutrino fields would allow for unsuppressed Majorana masses as well (unless a global U (1) L isimposed), thereby further enriching the neutrino mass sector. With the Majorana/Dirac masses suffering only logarithmiccorrections, ascribing appropriate (small) values to these is technically natural. a r X i v : . [ h e p - ph ] F e b ypically stronger for the higher generations, even in extensions of the SM. While the results from theATLAS experiment on h → µτ are more or less consistent with zero, these too can allow for a nontrivialbranching ratio (BR) for this channel. The measurements have yielded [2, 4]BR( h → µτ ) = 0 . +0 . − . % (CMS) , . ± .
51% (ATLAS) , (1)so that the 95% CL upper limits on the BR are 1.51% (CMS) and 1.41% (ATLAS) respectively.While the CMS measurement per se. does not call for new physics right away, it is interesting tojuxtapose it against another long-standing anomaly, albeit in a completely different sector. The ratiosof the partial widths of B mesons, R ( D ) and R ( D ∗ ), defined as R ( D ( ∗ ) ) = Γ( B → D ( ∗ ) τ ν )Γ( B → D ( ∗ ) (cid:96)ν ) , (2)(with (cid:96) = e, µ ) are particularly clean probes of physics beyond the SM, on account of the cancellationof the leading uncertainties inherent in individual BR predictions. The values of R ( D ) and R ( D ∗ )as measured by B A B AR [5], when taken together, exceed SM expectations by more than 3 σ , whichgenerated interest in the first place. Furthermore, the Belle measurements for the same observableslie in between the SM expectations and the B A B AR measurements and are consistent with both [6].Recently, Belle has published their new result on R ( D ∗ ) [7] with τ decaying semileptonically, and thisagrees with the SM expectations only at the 1 . σ level, while the first measurement by LHCb [8] is also2 . σ above the SM prediction. Taking all the results together, including the correlations, the tensionbetween data and SM is at the level of 3 . σ . On the other hand, the recent results on the measurementof τ -polarization for the decay B → D ∗ τ ν in Belle [9] are consistent with the SM predictions, albeitwith only a large uncertainty.While the “anomalies” in either of R ( D ) and R ( D ∗ ) do not call for LFV, clearly they seem to beassociated with a loss of lepton universality, and involving the very same fermions as the anomalousdecay. It is therefore conceivable that the individual excesses, intriguing in their own right but notcalling out for a rejection of the SM, are, together, indicative of some new physics. A combinedapproach to treat both these anomalies together within the scope of a particular model may be foundin Ref. [10] . At this point, we may refer the reader to Refs. [12, 13], and the references therein, fora detailed analysis of the NP operators. In this paper, we investigate this more closely, coupled withthe LFV Higgs decays. In particular, if anomalous Higgs interactions are indeed called for, we showthat the difference between the chiral structure of the ensuing four-fermi operators and that of the SMoperator could possibly explain why the experimental discrepancies are seen only in certain channels.The generation of such LFV decays of the Higgs is relatively simple if the scalar sector is enlarged,as in a Type-III two-Higgs doublet model wherein the 125 GeV scalar has a tiny component of thefield responsible for the LFV decays [14]. A variation is afforded by scenarios [15] wherein thereare two or more nearly degenerate scalars with one of them being SM-like and the other(s) havingexplicitly LFV couplings. On the other hand, lepton flavor non-universality can appear in many aguise, whether it be through Higgs couplings or through gauge couplings in a theory with extendedsymmetry or even through the exchange of other non-standard particles such as superpartners in asupersymmetric extension of the SM, or leptoquarks. Hence, rather than adopt any particular scenario,we investigate the structure of the minimal alteration to the SM that can satisfactorily explain the There have been numerous attempts to relate the R ( D ( ∗ ) ) anomaly with some other anomalous observables, see, e.g. ,Ref. [11]. h → µτ and B → D ( ∗ ) τ ν can be simultaneously affected by a single four-fermion operator, keeping the scalar sector tobe completely SM-like at the electroweak scale. There are at least two points worth emphasizing, solet us note them down here. • If the scalar sector is completely SM-like at all energies, i.e. , if the mass matrix and the Yukawamatrix are proportional, there can be no flavor-changing coupling of the Higgs boson of the form hf i f j with i (cid:54) = j , even at the one-loop level. This is in contradiction to what has been claimedin, for example, Refs. [16, 17]. The reason is not difficult to understand: as soon as one generatesan off-diagonal Yukawa coupling h ij , an analogous term m ij = vh ij is also generated in the massmatrix, where v is the vacuum expectation value (VEV) for the CP-even neutral component ofthe SM Higgs field Φ. Thus, one needs to redefine the stationary basis for the fermions again,and in that new basis, such off-diagonal effective Yukawa couplings no longer exist. However,there are possible ways out [18, 19], and we will later show, with a toy model, how to achievethis. In this sense, we demonstrate how to generate the LFV decay of the Higgs boson withoutintroducing any low-energy extension of the scalar sector. • NP has to be there in some form or other at some high scale, but if the low-energy sector isSM-like, then any new state can exist only at a scale Λ ∼ > O (1 TeV), the natural scale for NP. It ispossible, though, that NP can appear at several (well-separated) scales, with the aforementionedΛ being the lowest of them all.Here, we will focus on some possible dimension-6 four-fermion operators to explain both the anoma-lies, relating the charged current operator b → cτ ν with the neutral current operator, that produces τ µ in the final state, through SU(2) L . We will take advantage of two facts: first, the quark mixing in theright-chiral sector is essentially unconstrained, and second, the flavor of the neutrino that comes outin semileptonic B decays is not determined. While a similar exercise using higher dimensional effectiveoperators has been performed [20], it was restricted only to the B -sector observables. The novelty, inour approach, lies in that we do not consider any extension of the SM scalar sector, and the Yukawacouplings remain unchanged. As we will show, the new operators that we consider produce an effective hµτ vertex, which we illustrate with the help of a toy model. Showing how experimental constraintsalready rule out most of the possible operators, we identify the minimal set of operators necessary toexplain the anomalies.The paper is arranged as follows. In Section II, we will first describe a toy model to generate flavor-changing Higgs couplings with lowest dimensional effective operators, and then elaborate our model.In Section III, we show how it affects the LFV Higgs decay h → µτ , and semileptonic B decays aretreated in Section IV. We summarize and conclude in Section V. Assuming that the (low-energy) scalar sector is just as in the SM, the only way to explain a LFV decayof the Higgs boson h (such as the one under discussion) would be to postulate a term (cid:104) − y ij (cid:96) iL (cid:96) jR h + h . c . (cid:105) i (cid:54) = j ), in the Lagrangian, keeping in abeyance, for the time being, any discussion of the sourceof this term. Written in full, the relevant term is − y µτ ( µ L τ R + τ R µ L ) h − y τµ ( τ L µ R + µ R τ L ) h , (3)and the corresponding branching fraction is given byBR( h → µτ ) = m h π Γ h (cid:0) | y τµ | + | y µτ | (cid:1) , (4)where y µτ and y τµ are effective Yukawa couplings, which need not be equal, or even of the samemagnitude. If h → µτ (and other possible new decay channels) have only a small BR, one can assumeΓ h ≈ Γ SM h ≈ .
07 MeV for m h ≈
125 GeV.If the scalar sector (both the field content and interactions) is restricted to being exactly as in theSM, clearly, terms as in Eq. (3) cannot occur at the tree-level. They may appear as quantum correctionsthough, and the required size clearly does not preclude this. However, for even this to work, either thefield content of the theory has to be enlarged or non-renormalizable interactions introduced or both.
As was discussed earlier, one cannot simply postulate such an off-diagonal coupling for the Yukawaand the mass matrices often turn out to be proportional to each other (not only at the tree level, butto any given order in perturbation theory). To circumvent this argument, let us consider a toy model.Suppose the Lagrangian contains dimension-5 terms like1Λ (cid:104) a t t R Q L ˜Φ X + a l τ R L L Φ X ∗ (cid:105) + H . c . (5)where Φ is the SM doublet ( ˜Φ = iσ Φ ∗ ), and X is a complex SU(2) L triplet with hypercharge Y = 2. Wewill assume that the mass-squared term for X is positive and O (TeV ). Consequently, the componentsof X receive no vacuum expectation value, thereby trivially satisfying the constraints from the ρ -parameter. A further consequence is that they are almost degenerate in mass, which allows the scenarioto evade the remaining constraints from electroweak precision observables. Λ above is a cutoff scale,with Λ (cid:29) m X so as to validate the effective Lagrangian approach.Written in full, with X = ( x ++ , x + , x ), the relevant terms look like L ⊃ √ (cid:20) a t (cid:18) t R t L φ ∗ x − √ t R b L φ ∗ x + (cid:19) + a l (cid:18) τ R µ L φ x ∗ − √ τ R ν µL φ x − (cid:19)(cid:21) + H.c. . (6)Integrating out the X fields yields a dimension-8 term in the Lagrangian of the form − a t a l m X | φ | t L t R µ L τ R + h.c., (7)valid at scales well below m X . Here, analogous terms involving the putative Goldstones have beensuppressed. On the breaking of the electroweak symmetry, one may write φ = ( h + v ) / √
2, with h being the physical Higgs field. This yields not only a four-Fermi term of the form L = a t a l v m X (cid:0) t L t R (cid:1) ( τ R µ L ) + H . c ., (8)4ut also couplings of the same set of fields with both a single higgs and a pair of higgses, or, in otherwords, a five-field and a six-field vertex each. Of immediate concern are the first two of these terms.Clearly the (2 vh ) t L t R µ L τ R term, on contracting the top-fields, would lead to an effective LFV coupling hµ L τ R . Similarly, the term in Eq. (8) would contribute to an off-diagonal mass term connecting themuon and the tau. Importantly, these one loop contributions to the Yukawa and the mass matricesbear a relation different from the tree-level terms, viz. δy µτ = 2 δm µτ /v . The extra factor of 2 destroysthe overall proportionality of the Yukawa and the mass matrices, thereby allowing for a LFV Higgscoupling when the fermions are rotated into the stationary basis.The evaluation of the loop contributions is quite straightforward. While they are, formally, quadrat-ically divergent, it needs to be realized that the effective theory under consideration has a natural cutoffat m X . The leading term, apart from the overall coupling, is thus − N c m X m t / π , where the minussign comes from the fermion loop and m t from the chirality flip. Thus, the effective LFV Yukawacoupling is given by 12 × a t a l v N c π m t µ L τ R h . (9)The factor of half needs explaining. As mentioned above, the term proportional to v generates anoff-diagonal term in the mass matrix and, consequently, an extra rotation is needed to get back to thenew mass basis. This absorbs half of the effect (which is why a coupling proportional to ( h + v ) cannotlead to flavor-changing Yukawa couplings), leaving us with the remaining half.It should be noted that much the same low-energy phenomenology could have been obtained, hadwe started with an Y = 0 triplet instead, with the Lagrangian now being1Λ (cid:104) a t t R Q L Φ X + a l τ R L L ˜Φ X ∗ (cid:105) + H . c . . Similarly, had we started with a scalar leptoquark field, coupling to both a t - τ and a t - µ current, theensuing effective Lagrangian, on Fierz-rearrangement, would yield terms analogous to those above, butwith (axial-)vector couplings instead. Having argued that it is indeed possible to generate flavor-changing Higgs couplings (for a theory with asingle scalar doublet) within the stationary basis, and that this may be achieved quite naturally withinthe paradigm of an effective theory, we now turn to the other anomalies at hand, namely R ( D ( ∗ ) ). Tothis end, we augment the SM by postulating at most a couple of effective dimension-6 operators obeyingthe full symmetry of the SM. These operators will be shown to generate an effective hµτ vertex, by amechanism similar to that outlined above, which is of the right magnitude. While a similar approachwas adopted in Ref. [21] to explain h → µτ alone, we go much beyond and relate the operators to theanomalies in R ( D ) and R ( D ∗ ).Following Refs. [22, 23], let us consider an effective charged-current Hamiltonian of the form H eff = 4 G F √ V cb [ O SM + C S O S + C S O S + C T O T ] , (10)5here O SM = ( c L γ λ b L )( τ L γ λ ν τL ) ,O S = ( c L b R )( τ R ν µL ) ,O S = ( c R b L )( τ R ν µL ) ,O T = ( c R σ µλ b L )( τ R σ µλ ν µL ) , (11)and the fermion fields are weak-eigenstates, as befits operators in an effective theory defined abovethe electroweak scale. While O S and O S might result from the mechanism discussed in the previoussubsection (albeit with different fermionic fields), the generation of O T is more non-trivial, and theultraviolet completion of the same would, typically, require the introduction of exotic fields , such as adoublet scalar leptoquark with a hypercharge of . Note that this set is not exactly identical to thatgiven in Ref. [23]. For one, the new operators contain ν µ instead of ν τ . With the neutrinos in a decaybeing unidentified, this does not affect the analysis of R ( D ) and R ( D ∗ ) except for the fact that, now,no interference between the SM operator O SM and the new operators would exist. Furthermore, wehave dropped some operators, involving (axial-)vector currents, as they (to be demonstrated shortly)not only do not lead to h → µτ , but, in addition, cause disagreements with other observables. Lateron, we will show that C S should be of the order of unity to produce a good fit with the data, and itis almost trivial to show that this leads to an unacceptably large contribution to the decay B s → µτ ,which is yet to be observed. Thus, even the operator O S falls out of favor, but we will keep this in ouranalysis for the time being.The origin of the specific set of operators is, of course, uncertain. Given that the family numberis conserved, it is quite conceivable, for example, that these arise on account of flavor dynamics. Wedo not, however, attempt to answer such questions, but only offer the argument that this leads us tothe minimal set of new operators required to explain the data. To further reduce the number of freeparameters, we shall consider an additional simplification and consider two reduced sets, namely • Model 1: C S , C S (cid:54) = 0, C T = 0; • Model 2: C S , C T (cid:54) = 0, C S = 0.In other words, only two new Wilson coefficients are introduced in each case.The new operators also imply the existence of their SU(2) conjugates, with identical Wilson coeffi-cients, namely O (cid:48) S = ( s L b R )( τ R µ L ) ,O (cid:48) S = ( c R t L )( τ R µ L ) ,O (cid:48) T = ( c R σ µλ t L )( τ R σ µλ µ L ) . (12)This, immediately, puts into perspective our earlier assertion about O S being highly constrained, for O (cid:48) S would readily generate semileptonic LFV decays like B → K ( ∗ ) τ µ and the purely leptonic decay B s → τ µ . In fact, if the corresponding Wilson coefficient C S is of order unity, the BR of B s → τ µ becomes so large ( ∼ O (0 . C S is of theorder of at least 10 − , it is hard, but not entirely impossible, to entertain O S (and hence Model I asmentioned before) as a possible candidate for the minimal set of operators. It should be noted, though, that such a rendition would require the simultaneous introduction of other operators aswell. × U L,R , D L,R ) in play, one each for the (left-) right-handed (up-) down-quarks. Thanks tothe right-handed fields being SU (2) L singlets and universality of the gauge-structure across generations,within the SM, two of these matrices ( U R and D R ) play no dynamic role, and only the combination U † L D L is manifested physically (as the Cabibbo-Kobayashi-Maskawa matrix). In the presence of thesenew operators, this would no longer be the case. In particular, both of U R and D R would now playa nontrivial role. Once again, rather than consider the most general case, we simplify the analysis byretaining only the most important term, namely c R = cos α c (cid:48) R + sin α t (cid:48) R , t R = − sin α c (cid:48) R + cos α t (cid:48) R , (13)where the primed fields are in the mass basis. This immediately leads to O S = cos α ( c (cid:48) R b L )( τ R ν µL ) + · · · ,O T = cos α ( c (cid:48) R σ µλ b L )( τ R σ µλ ν µL ) + · · · ,O (cid:48) S = sin α ( t (cid:48) R t L )( τ R µ L ) + · · · ,O (cid:48) T = sin α ( t (cid:48) R σ µλ t L )( τ R σ µλ µ L ) + · · · . (14)The left-chiral quark fields are also rotated to the mass basis as per the Cabibbo-Kobayashi-Maskawaparadigm. These rotations have important physical consequences. For example, even if the mixing isconfined to the down quark sector alone, O (cid:48) S , after field rotation, can lead to Υ → µτ , which, withinthe SM, is highly suppressed compared to the electromagnetic decay Υ → (cid:96) + (cid:96) − . This particular mode,though, is not very restrictive once the aforementioned constraints from B s → τ µ are satisfied. Similarly,if the mixing is for the up-type quarks, O (cid:48) S and O (cid:48) T can lead to LFV charmonium decays, which arealso yet to be observed. While eq. (14) lists all the operators relevant for our study, it is instructive,at this stage, to examine the ramifications thereof. Clearly, engendering the flavor-changing Yukawacoupling hµτ by Wick-contracting the top-fields is possible only for O (cid:48) S . Thus, only this operator (andits sibling, O S ) are relevant for this aspect. On the other hand, O ( (cid:48) ) S and O ( (cid:48) ) T appear at the same orderin the effective theory and, like O ( (cid:48) ) S , can contribute to both R ( D ) and R ( D ∗ ). Thus, the inclusion ofat least two operators is necessary to maintain agreement for these decays.Before we end this section, we would like to point out out that, in obtaining the operators in Eq. (12)from those in Eq. (11) through basis transformations, we would also generate many other operators,designated by the ellipses in Eq. (12). These would have their own consequences, such as the FCNCtop decay t → cµτ . We have checked that, for the sizes of the Wilson coefficients ( C S accompanied byone of C S and C T ) that we would need, such effects are negligible. The presence of an operator such as ( f Γ a f ) ( τ Γ a µ ), where f is a SM fermion and Γ a a Dirac matrix,denotes the violation of both N τ and N µ while preserving their difference. Clearly, this can result in h → µτ , at least at the loop-level. Fig. 1 shows two typical diagrams, in the context of the toy modeldiscussed before, that contributes to such a process.7t is easy to see that O T cannot contribute to this amplitude, for, to obtain a hµτ vertex, we wouldneed to contract the leptonic current with two external momenta which, of course, is not possible.For (axial-)vector operators (not listed in Eq. (11)), on the other hand, only one such contraction isneeded and, consequently, the amplitude is proportional to the lepton mass. Furthermore, the verystructure of the operator ensures that the loop integral is logarithmically divergent and scales only as m − X ln( m h /m X ) at the most. While this suppression is not necessarily an overwhelming one (provided m X is not too large), it should be realized that corresponding diagrams exist where the Higgs field isreplaced by the Z . The latter would lead to an unsuppressed contribution [24] to the decay Z → τ µ ,well beyond the experimental limits, unless the Wilson coefficient for the four-fermion interaction issuppressed enough. This, though, would imply that the operator has a negligibly small effect in Higgsdecays.Figure 1: Typical contributions to the decay h → µ + τ − initiated by the new operators. Diagrams forthe conjugate process would be analogous.This leaves us with the (pseudo-)scalar operators O S and O S . Let us concentrate on the latter,and take our toy model as a concrete example. This gives4 G F √ V cb C S sin α = a t a l v m X , (15)and hence, the first diagram of Fig.1 yields y µτ = G F √ π m X V cb N c m t v C S sin α ≈ . (cid:16) m X (cid:17) C S sin α , (16)where N c = 3(1 + α s /π ) ≈ .
11 is the effective number of colors, and h t ≈ m t = 175 GeV, G F = 1 . × − GeV − , and | V cb | = (41 . ± . × − .The contribution of the second diagram of Fig. 1 is further suppressed by a factor of ∼ v/m X . Thisgives BR( h → µτ ) ≈ . (cid:16) m X (cid:17) [ C S sin α ] < . ⇒ C S sin α < . × − (cid:18) m X (cid:19) . (17)Thus, if | C S | is of order unity, one needs a small mixing in the t R - c R sector, namely, tan α ∼ − , toexplain the LFV Higgs decay. Note that while the estimation has been done for a particular toy model,the essence is model-independent. 8 The B -decay anomalies In terms of the differential distributions d Γ /dq for the decay B → X(cid:96)ν , where q µ ≡ ( p B − p X ) µ is themomentum transfer, the ratios R ( D ) and R ( D ∗ ) are defined as R ( D ( ∗ ) ) = (cid:34)(cid:90) q max m τ d Γ (cid:0) B → D ( ∗ ) τ ν τ (cid:1) dq dq (cid:35) (cid:34)(cid:90) q max m (cid:96) d Γ (cid:0) B → D ( ∗ ) (cid:96)ν (cid:96) (cid:1) dq dq (cid:35) − (18)with q max = ( m B − m D ( ∗ ) ) , and (cid:96) = e or µ . In each case, both isospin channels are taken into account.Using the effective Hamiltonian in Eq. (10), the expressions for these distributions are given as d Γ (cid:0) B → Dτ ν τ (cid:1) dq = G F | V cb | π m B q (cid:112) λ D ( q ) (cid:18) − m τ q (cid:19) × (cid:26)(cid:20)(cid:18) m τ q (cid:19) H sV, + 32 m τ q H sV,t (cid:21) + 32 | C S + C S | H sS + 8 | C T | (cid:18) m τ q (cid:19) H sT (cid:27) , (19)and d Γ (cid:0) B → D ∗ τ ν τ (cid:1) dq = G F | V cb | π m B q (cid:112) λ D ∗ ( q ) (cid:18) − m τ q (cid:19) × (cid:26)(cid:20)(cid:18) m τ q (cid:19) (cid:0) H V, + + H V, − + H V, (cid:1) + 32 m τ q H V,t (cid:21) + 32 | C S − C S | H S + 8 | C T | (cid:18) m τ q (cid:19) (cid:0) H T, + + H T, − + H T, (cid:1)(cid:27) , (20)with λ X ( q ) ≡ m B + m X + q − m B m X − m B q − m X q . Here, H i s are the respective form factorsas defined within the Heavy Quark Effective Theory [25], and we use the values determined by theHeavy Flavor Averaging Group (HFAG) [26]. For more details, we refer the reader to Ref. [22]. Whilethe results for the lighter leptons are obtained by substituting m τ → m (cid:96) ≈
0, putting all the C i s equalto zero would yield the SM results. R ( D ) and R ( D ∗ ) Let us first focus on R ( D ) and R ( D ∗ ). Several experiments have measured these ratios, and the currentstatus is summarized in Fig. 2 as well as in Table 1. However, while Table 1 includes the latest Belleresult [9] on R ( D ∗ ), Fig. 2 takes into account only the Belle update till August 2016. Though thechange is quite small and can easily be neglected, we have used the updated result [9] in our analysis.While the two scenarios ( C S = 0 vs. C T = 0) are identical as far as h → µτ is concerned, theireffects are quite markedly different on R ( D ( ∗ ) ). We perform a χ goodness-of-fit analysis to fit the newphysics Wilson coefficients through their effects as summarized in Eqs. 19 and 20. In our analysis, weuse the q -integrated data on R ( D ) and R ( D ∗ ), given in Tables 1 and 2 for different isospin channels( i.e. , both B + and B decays) with appropriate correlations wherever the data is available. However,we have not used the isospin-constrained data measured by B A B AR (given in Table 1) as an input inour analysis as those are not independent data-points. Our analysis involves 11 data-points: 4 fromRef. [5], 2 from Ref. [6], 2 from Ref. [30], and 1 each from Refs. [7], [8], and [9]. Ref. [30] supplies the9 (D) R ( D * ) BaBar, PRL109,101802(2012)Belle, PRD92,072014(2015)LHCb, PRL115,111803(2015)Belle, PRD94,072007(2016)Belle, arXiv:1608.06391Average
SM Predictions = 1.0 contours cD R(D)=0.300(8) HPQCD (2015)R(D)=0.299(11) FNAL/MILC (2015)R(D*)=0.252(3) S. Fajfer et al. (2012)
HFAG
Summer 2016 ) = 70% c P( HFAG
Summer 2016
Figure 2: Current experimental status in the measurements of R ( D ) and R ( D ∗ ) [27]. R ( D ) R ( D ∗ )SM prediction 0 . ± .
008 [28] 0 . ± .
003 [29] B A B AR (Isospin constrained) 0 . ± . ± .
042 0 . ± . ± .
018 [5]Belle (2015) 0 . ± . ± .
026 0 . ± . ± .
015 [6]Belle (2016) - 0 . ± . ± .
011 [7]Belle (2016, Full Dataset) - 0 . ± . +0 . − . [9]LHCb - 0 . ± . ± .
030 [8]Table 1: The SM predictions for and the data on R ( D ) and R ( D ∗ ). While B A B AR considers bothcharged and neutral B decay channels, LHCb and Belle results, as quoted here, are based only on theanalysis of neutral B modes.Experiment Channel R ( D ( ∗ ) ) B − → D τ − ν τ . ± . ± . B A B AR [5] B → D + τ − ν τ . ± . ± . B − → D ∗ τ − ν τ . ± . ± . B → D ∗ + τ − ν τ . ± . ± . B − → D τ − ν τ . ± . B − → D ∗ τ − ν τ . ± . R ( D ∗ ) in different isospin channels. Only Belle 2010 and not the laterBelle papers gives the isospin break-up.data in the form of branching fractions. We have converted them to R ( D ( ∗ ) ) by normalizing them withBR( B → D ( ∗ ) (cid:96)ν ) [31] while propagating the errors.An important point to note is that the expressions depend only on | C S | and | C S | (or | C T | and | C S | ) and hence there is a fourfold ambiguity on the position of the minimum. This is best understood10 - - - Re ( C S ) R e ( C S ) - - - - - - Re ( C S ) R e ( C T ) Figure 3: The χ contours for Model 1 (left) and Model 2 (right). The 1 σ (68.27%), 2 σ (95.45%), and4 σ (99.99%) confidence levels are shown by red, orange, and blue lines respectively.from the χ contours shown in Fig. 3. For example, the best fit points areModel 1 : C S cos α = ± (1 . ± . , C S cos α = − sgn( C S cos α )(1 . ± . , or C S cos α = ± (1 . ± . , C S cos α = − sgn( C S cos α )(1 . ± . , Correlation coefficient = − .
71 (21)Model 2 : | C S cos α | = 0 . ± . , | C T cos α | = 0 . ± . , Correlation coefficient = − .
29 (22)with almost identical χ / d . o . f ≈ . /
9, whereas the SM has χ = 33 .
05. From the smallness of α , it isclear that Model 1, with the operator O S , is almost ruled out from the non-observation of B s → µτ .For the best fit points, the values of R ( D ) and R ( D ∗ ) are given in Table 3. We also show, in Fig.4, how the 1 σ contours in the C S - C T plane translate to the R ( D )- R ( D ∗ ) plane. The plot is for Model2, but it would have been the same for Model 1 if it were not disfavored, as the goodness-of-fit is thesame in both cases. While the operator O S can lead to the chirally unsuppressed decay through weakannihilation B c → τ ν , whose partial width is bounded from the lifetime of the B c meson [32], it is easyto check that the Wilson coefficient C S is not so large as to put that bound in jeopardy. In this paper, we have tried to explain, with the introduction of a minimal set of operators, two ap-parently uncorrelated anomalies. The first one is that of the normalized B → D ( ∗ ) τ ν decay widths,denoted as R ( D ) and R ( D ∗ ), for which almost all the experiments find a nontrivial pull from the SMexpectations. The second one is the hint of the LFV decay h → µτ as seen by the CMS collaboration.While none of them immediately calls for a beyond-SM explanation right now, it is nevertheless inter-esting to see whether one can relate these two sets of data following the principle of Occam’s razor, i.e. R ( D ) R ( D ∗ )From B + . ± .
072 0 . ± . . ± .
040 0 . ± . B . ± .
064 0 . ± . . ± .
036 0 . ± . R ( D ( ∗ ) ) with the fitted Wilson coefficients as given in Eq.(22). BaBar,PRL109,101802 ( ) Belle,PRD92,072014 ( ) LHCb,PRL115,111803 ( ) Belle,PRD94,072007 ( ) Belle,arXiv:1612.00529
Our FitWorld Avg. ( HFAG ) SM ( D ) R ( D * ) Figure 4: The 1 σ contour in the R ( D )- R ( D ∗ ) plane with the best fit points for Model 2. The currentexperimental results and the world average are also shown for comparison.by the introduction of a minimal set of higher-dimensional operators.We find that this is indeed possible. However, not all operators invoked in the literature to explainthe R ( D ( ∗ ) ) can do the job. The situation apparently becomes even more complicated from the fact thatno LFV Higgs coupling can survive if the scalar sector is SM-like. However, this can be circumventedby postulating the existence of new degrees of freedom at a higher scale while the low-energy scalarsector remains completely SM-like. This also leads to new four-fermion operators which can possiblycontribute to b → cτ ν decays. Arguing that the undetermined nature of the neutrino flavor allows forthe anomaly to be explained in terms of the muon-neutrino, we relate it, through the SU (2) L symmetryto the τ µ final state. While many Lorentz structures, per se., could explain the anomaly(ies), only somesurvive the stringent limits imposed by the Z and B s decays.We find that it is indeed possible to find a parameter space where both the anomalies can besuccessfully explained, with the fit showing a very marked improvement over the SM. This region isalso physically meaningful in the sense that all the Wilson coefficients for the new operators are of theorder of unity.This scenario can be tested in a number of ways. First, the τ polarization, P τ , can be measured withmuch improved precision in future B factories. The SM τ s are all left-chiral, while our model predictsa large number of right-chiral τ s as well. The second way is to investigate the LFV couplings of the12iggs boson in future electron-positron colliders. As has been shown in Ref. [33], the InternationalLinear Collider can have a reach one order of magnitude better than the LHC. As for which modelscan produce such effective operators, we leave that to the model builders. Acknowledgements – D.C. acknowledges partial support from the European Union’s Horizon 2020research and innovation program under Marie Sk(cid:32)lodowska-Curie grant No 674896. A.K. acknowledgesthe Council for Scientific and Industrial Research, Government of India, for a research grant. Healso thanks the Physics Department of IIT, Guwahati, for hospitality where a part of this work wascompleted.
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