Υ and ψ leptonic decays as probes of solutions to the R_D^{(*)} puzzle
ΥΥ and ψ leptonic decays as probes of solutions to the R (cid:0) D ( ∗ ) (cid:1) puzzle Daniel Aloni a , Aielet Efrati a , Yuval Grossman b and Yosef Nir a Department of Particle Physics and Astrophysics,Weizmann Institute of Science, Rehovot, Israel 7610001 Department of Physics, LEPP, Cornell University, Ithaca, NY 14853 ∗ Experimental measurements of the ratios R ( D ( ∗ ) ) ≡ Γ( B → D ( ∗ ) τν )Γ( B → D ( ∗ ) (cid:96)ν ) ( (cid:96) = e, µ ) show a 3 . σ deviationfrom the Standard Model prediction. In the absence of light right-handed neutrinos, a new physicscontribution to b → cτ ν decays necessarily modifies also b ¯ b → τ + τ − and/or c ¯ c → τ + τ − transitions.These contributions lead to violation of lepton flavor universality in, respectively, Υ and ψ leptonicdecays. We analyze the constraints resulting from measurements of the leptonic vector-meson decayson solutions to the R ( D ( ∗ ) ) puzzle. Available data from BaBar and Belle can already disfavor someof the new physics explanations of this anomaly. Further discrimination can be made by measuringΥ(1 S, S, S ) → τ τ in the upcoming Belle II experiment. I. INTRODUCTION
The Standard Model (SM) predicts that the electroweak interactions respect lepton flavor universality (LFU).Violation of LFU, beyond the small effects of the Yukawa interactions (or, equivalently, of the charged lepton masses),will constitute clear evidence for physics beyond the SM. There is growing experimental evidence that LFU is brokenin the ratios R ( D ( ∗ ) ) ≡ Γ( B → D ( ∗ ) τ ν )Γ( B → D ( ∗ ) (cid:96)ν ) , ( (cid:96) = e, µ ) . (1)The combined results of BaBar [1, 2], Belle [3–6] and LHCb [7] read [8] R ( D ) = 0 . ± . , R ( D ∗ ) = 0 . ± . , ρ = − . , (2)where ρ is the experimental correlation between R ( D ) and R ( D ∗ ). (The updated Belle result, R ( D ∗ ) = 0 . ± .
044 [6], is not included in the HFAG average [8].) The SM predictions are [9, 10] R ( D ) = 0 . ± . , R ( D ∗ ) = 0 . ± . . (3)Combined, these results show a deviation from the SM prediction at a level of 3 . σ [8].This R ( D ( ∗ ) ) puzzle received a lot of attention in the literature in recent years. Various works analyze the deviationin terms of effective field theory (EFT), construct explicit models of new physics (NP), and introduce new observablesand kinematical distributions which can shed light on the flavor and Lorentz structure of the underlying theory [11–59].Given the SM particle content and, in particular, assuming that the light neutrinos are purely left-handed, NPcontributions to the b → cτ ν transition imply that NP contributions to b ¯ b → τ + τ − and/or c ¯ c → τ + τ − transitionsare unavoidable. This point was discussed in Ref. [47] which analyze the high P T distribution of the τ + τ − signatureat the LHC, arising from sea b ¯ b and/or c ¯ c annihilation. In this work we study new observables which are governedby the c ¯ c → τ τ and b ¯ b → τ τ transitions: non-universality in leptonic decays of ψ and Υ quarkonia. Specifically, westudy the ratios R Vτ/(cid:96) ≡ Γ ( V → τ + τ − )Γ ( V → (cid:96) + (cid:96) − ) , ( V = ψ, Υ; (cid:96) = e, µ ) , (4)making two assumptions: • The deviation of R ( D ( ∗ ) ) from the SM prediction is generated by gauge invariant effective operators of dimensionsix. ∗ Electronic address: a daniel.aloni, aielet.efrati, [email protected], b [email protected] a r X i v : . [ h e p - ph ] J un • R ( D ( ∗ ) ) is modified by new physics that affects only the B → D ( ∗ ) τ ν decays (and not the B → D ( ∗ ) (cid:96)ν decays).Correspondingly, R Vτ/(cid:96) is modified because V → τ τ is affected (and not V → (cid:96)(cid:96) ).We compare our results to the current and future sensitivity for LFU violation in leptonic decays of Υ and ψ vector-mesons.One advantage of using the relation between operators responsible for the R ( D ( ∗ ) ) anomaly and those modifyingthe leptonic decays of Υ and ψ is that all processes occur at the same energy scale. Therefore once we fix the Wilsoncoefficients at the low scale to give the measured best fit values of R ( D ( ∗ ) ), no RGE effects and mixing among otheroperators affect our predictions for R Vτ/(cid:96) . Once an anomaly is found in these leptonic vector meson decays, a fullUV model should be scrutinized, including a proper UV matching and RGE mixing, as well as its compatibility withother relevant observables.Tests of LFU have been carried out for Υ(1 S ), Υ(2 S ) and Υ(3 S ) [60, 61]. We collect these results in Table I. For ψ (2 S ) the leptonic branching fractions are measured to be [62]: BR ( ψ (2 S ) → τ + τ − ) = (3 . ± . × − ,BR ( ψ (2 S ) → µ + µ − ) = (7 . ± . × − ,BR ( ψ (2 S ) → e + e − ) = (7 . ± . × − . (5)The corresponding ratio is presented in Table I. We do not consider ψ (1 S ) whose mass is below the τ + τ − threshold.We also do not consider ψ (3770) and Υ(4 S ) which have negligible branching fractions into leptons because theirmasses are above the D ¯ D and B ¯ B threshold, respectively.TABLE I: Experimental results and SM predictions for R Vτ/(cid:96) . V ( nS ) SM prediction Exp. value ± σ stat ± σ syst Υ(1 S ) 0 . ± O (10 − ) 1 . ± . ± . S ) 0 . ± O (10 − ) 1 . ± . ± . S ) 0 . ± O (10 − ) 1 . ± . ± . ψ (2 S ) 0 . ± O (10 − ) 0 . ± . Within the SM, the QED partial decay widths of a vector quarkonium into charged lepton pairs obey [63] R Vτ/(cid:96) (cid:39) (1 + 2 x τ )(1 − x τ ) / (6)where x τ = m τ /m V . This approximation neglects the electron and muon masses, one-loop corrections and weak-current effects. This phase space factor is the leading source of flavor non-universality in the SM. The dominantcorrections to this factor are at the level of 0 . x τ , and arise from QED vertex corrections. A full discussion on theSM decay rate is presented in App. A. The relevant masses are known with a great accuracy [62]: m Υ(1 S ) = 9 . ± . ,m Υ(2 S ) = 10 . ± . ,m Υ(3 S ) = 10 . ± . ,m ψ (2 S ) = 3 . ± . ,m τ = 1 . ± . . (7)The non-universality predicted in the SM agrees very well with the experimental results, as is evident from Table I.LFU in Upsilon decays was discussed in the literature in the context of light pseudo-scalar (see Refs. [64, 65] andreferences within). In this scenario, the radiative Υ → γη b decay is followed by a mixing between the η b state and aCP-odd scalar A , for which the leptonic couplings are non-universal. Lepton flavor changing decays of heavy vector-mesons were discussed in Ref. [48, 66]. While such decays are not directly relevant to our study, the formalism issimilar.The plan of this paper is as follows. The formalism for V → (cid:96)(cid:96) decays is introduced in Section II. The effectivefield theory that is relevant to R ( D ( ∗ ) ) and to R Vτ/(cid:96) is introduced in Section III. In Section IV we analyze a seriesof simplified models, where we add to the SM a single new boson. For each model, we find the numerical range forthe Wilson coefficients that explain R ( D ( ∗ ) ), and obtain the resulting predictions for R Vτ/(cid:96) . In Section V we comparethese predictions to present measurements and discuss the prospects for improving the experimental accuracy in thefuture. Our conclusions are summarized in Section VI.
II. V → (cid:96)(cid:96) DECAY RATE
The most general V → (cid:96) + (cid:96) − decay amplitude can be written as M ( V → (cid:96) + (cid:96) − ) = (cid:18) f V m V (cid:19) ¯ u ( p , s ) (cid:34) A q(cid:96)V γ µ + B q(cid:96)V γ µ γ + C q(cid:96)V m V ( p − p ) µ + iD q(cid:96)V m V ( p − p ) µ γ (cid:35) v ( p , s ) (cid:15) µ ( p ) , (8)where A q(cid:96)V , B q(cid:96)V , C q(cid:96)V , D q(cid:96)V are dimensionless parameters which depend on the Wilson coefficients of the operatorscontrolling the V → (cid:96) + (cid:96) − decays at the perturbative level, and on meson-to-vacuum matrix elements at the non-perturbative level. The form-factors of the Lorentz structures ( p + p ) µ and ( p + p ) µ γ do not contribute to therate. The decay width and R Vτ/(cid:96) are given byΓ[ V → (cid:96)(cid:96) ] = f V πm V (cid:113) − x (cid:96) (cid:34) | A q(cid:96)V | (cid:0) x (cid:96) (cid:1) + | B q(cid:96)V | (cid:0) − x (cid:96) (cid:1) + | C q(cid:96)V | (cid:0) − x (cid:96) (cid:1) + | D q(cid:96)V | (cid:0) − x (cid:96) (cid:1) +2Re (cid:104) A q(cid:96)V C ∗ q(cid:96)V (cid:105) x (cid:96) (cid:0) − x (cid:96) (cid:1)(cid:105) ,R Vτ/(cid:96) (cid:39) (cid:112) − x τ | A (cid:96), SM V | (cid:20) | A qτV | (cid:0) x τ (cid:1) + | B qτV | (cid:0) − x τ (cid:1) + | C qτV | (cid:0) − x τ (cid:1) + | D qτV | (cid:0) − x τ (cid:1) +2Re (cid:2) A qτV C ∗ qτV (cid:3) x τ (cid:0) − x τ (cid:1)(cid:3) . (9)Within the SM, A (cid:96), SM V (cid:39) − παQ q and B (cid:96), SM V , C (cid:96), SM V , D (cid:96), SM V (cid:39)
0. For the SM calculations we include QED one-loopcorrection which is further discussed in App. A.In order to calculate the V → (cid:96) + (cid:96) − decay rate in a specific UV or EFT model, one needs to find the relationbetween A V , B V , C V and D V and the Lagrangian parameters. Including only terms which are relevant for a leptonicmeson decay, we consider the following effective Lagrangian: L (cid:96)q = C q(cid:96)V RR ¯ e R γ µ e R ¯ q R γ µ q R + C q(cid:96)V RL ¯ e R γ µ e R ¯ q L γ µ q L + C q(cid:96)V LR ¯ e L γ µ e L ¯ q R γ µ q R + C q(cid:96)V LL ¯ e L γ µ e L ¯ q L γ µ q L + (cid:104) C q(cid:96)T ¯ e L σ µν e R ¯ qσ µν q + C q(cid:96)SL ¯ e R e L ¯ q R q L + C q(cid:96)SR ¯ e R e L ¯ q L q R + h . c . (cid:105) . (10)We find: A q(cid:96)V = − παQ q + m V (cid:20)(cid:16) C q(cid:96)V LL + C q(cid:96)V RR + C q(cid:96)V LR + C q(cid:96)V RL (cid:17) + 16 x (cid:96) f TV f V Re (cid:104) C q(cid:96)T (cid:105)(cid:21) ,B q(cid:96)V = m V (cid:16) C q(cid:96)V RR + C q(cid:96)V RL − C q(cid:96)V LR − C q(cid:96)V LL (cid:17) ,C q(cid:96)V = 2 m V f TV f V Re (cid:104) C q(cid:96)T (cid:105) ,D q(cid:96)V = 2 m V f TV f V Im (cid:104) C q(cid:96)T (cid:105) . (11)We introduced the form factors, f V and f TV , that are defined via the standard parametrization: (cid:104) | ¯ qγ µ q | V ( p ) (cid:105) = f V m V (cid:15) µ ( p ) , (cid:104) | ¯ qσ µν q | V ( p ) (cid:105) = if TV [ (cid:15) µ ( p ) p ν − (cid:15) ν ( p ) p µ ] , (12)with σ µν = i [ γ µ , γ ν ] /
2, and (cid:104) | ¯ qq | V ( p ) (cid:105) = (cid:104) | ¯ qγ q | V ( p ) (cid:105) = (cid:104) | ¯ qγ µ γ q | V ( p ) (cid:105) = 0. The relevant ratio f TV /f V shouldbe determined from measurements or lattice calculations. In the heavy quark limit f V = f TV . This is an excellentapproximation for the Υ meson. For the ψ (2 S ) state, however, relativistic effects correct this relation by a fewpercent [67]. We checked that this is a sub-leading effect, therefore in the following we neglect this correction. III. THE EFFECTIVE FIELD THEORY
Assuming that the NP contributions are related to heavy degrees of freedom, their effects can be presented bynon-renormalizable terms in the Lagrangian. There are eight combinations of two lepton and two quark fields thatcan be contracted into SU (3) C × SU (2) L × U (1) Y and Lorentz invariant operators:¯ LL ¯ QQ, ¯ eL ¯ uQ, ¯ eL ¯ Qd, ¯ LL ¯ uu, ¯ LL ¯ dd, ¯ ee ¯ uu, ¯ ee ¯ dd, ¯ ee ¯ QQ, (13)where L and e are the SU (2) L doublet and singlet lepton fields, and Q , u and d are the SU (2) L doublet, up-singlet anddown-singlet quark fields. Only the first three combinations can introduce the charged-current interaction needed fora b → c transition. Specifying the Lorentz and SU (2) L contractions, we write the complete set of (linearly dependent)gauge-invariant operators in Table II. Wherever possible, we follow the notations of Ref. [27].Given that the experimental central values of R ( D ( ∗ ) ) deviate by order 30% from the SM predictions, and giventhat the SM amplitude is tree-level and only mildly CKM-suppressed, it is likely that the NP contribution whichaccounts for the deviation is also tree-level. It is instructive, therefore, to consider simplified UV models, each with asingle new (scalar or vector) boson. Following Ref. [27], we specify the NP field content which generates the operatorsin Table II. Our convention is such that ψ c = − iγ ψ ∗ .TABLE II: Dimension six four-fermion Operators
Field content Operator Fierz identities NP rep, s = 0 , LL ¯ QQ (cid:0) ¯ Lγ µ L (cid:1) (cid:0) ¯ Qγ µ Q (cid:1) O V L B (cid:48) µ ∼ (1 , (cid:0) ¯ Lγ µ τ a L (cid:1) (cid:0) ¯ Qγ µ τ a Q (cid:1) O V L W (cid:48) µ ∼ (1 , (cid:0) ¯ Lγ µ Q (cid:1) (cid:0) ¯ Qγ µ L (cid:1) O V L + 2 O V L U µ ∼ (3 , +2 / (cid:0) ¯ Lγ µ τ a Q (cid:1) (cid:0) ¯ Qγ µ τ a L (cid:1) O V L − O V L X µ ∼ (3 , +2 / (cid:0) ¯ L(cid:15) T Q c (cid:1) (cid:0) Q c (cid:15)L (cid:1) O V L − O V L S ∼ (3 , − / (cid:0) ¯ Lτ a (cid:15) T Q c (cid:1) (cid:0) Q c (cid:15)τ a L (cid:1) O V L + O V L T ∼ (3 , − / (cid:0) ¯ L(cid:15) T σ µν Q c (cid:1) (cid:0) Q c (cid:15)σ µν L (cid:1) (cid:0) ¯ Lτ a (cid:15) T σ µν Q c (cid:1) (cid:0) Q c (cid:15)τ a σ µν L (cid:1) eL ¯ uQ (¯ eL ) (cid:15) (¯ uQ ) O S L φ ∼ (1 , +1 / (¯ uL ) (cid:15) (¯ eQ ) − O S L − O T D ∼ (3 , +7 / (cid:0) L c (cid:15)Q (cid:1) (¯ eu c ) − O S L + O T S ∼ (3 , − / (¯ eσ µν L ) (cid:15) (¯ uσ µν Q ) O T –(¯ uσ µν L ) (cid:15) (¯ eσ µν Q ) − O S L + O T – (cid:0) L c σ µν (cid:15)Q (cid:1) (¯ eσ µν u c ) 6 O S L + O T –¯ eL ¯ Qd (cid:0) ¯ Qd (cid:1) (¯ eL ) O S R φ ∼ (1 , +1 / (cid:0) ¯ Qγ µ L (cid:1) (¯ eγ µ d ) − O S R U µ ∼ (3 , +2 / (cid:0) ¯ Qγ µ e c (cid:1) (cid:0) d c γ µ L (cid:1) O S R V µ ∼ (3 , − / (cid:0) ¯ Qσ µν d (cid:1) (¯ eσ µν L ) 0 –¯ LL ¯ uu (cid:0) ¯ Lγ µ L (cid:1) (¯ uγ µ u ) O V L B (cid:48) µ ∼ (1 , (cid:0) ¯ Lγ µ u c (cid:1) (cid:0) L c γ µ u (cid:1) O V L Y µ ∼ (3 , +1 / (cid:0) ¯ Lu (cid:1) (¯ uL ) − O V L D ∼ (3 , +7 / (cid:0) ¯ Lσ µν u (cid:1) (¯ uσ µν L ) 0 –¯ LL ¯ dd (cid:0) ¯ Lγ µ L (cid:1) (cid:0) ¯ dγ µ d (cid:1) O V L B (cid:48) µ ∼ (1 , (cid:0) ¯ Lγ µ d c (cid:1) (cid:0) L c γ µ d (cid:1) O V L V µ ∼ (3 , − / (cid:0) ¯ Ld (cid:1) (cid:0) ¯ dL (cid:1) − O V L D (cid:48) ∼ (3 , +1 / (cid:0) ¯ Lσ µν d (cid:1) (cid:0) ¯ dσ µν L (cid:1) ee ¯ QQ (¯ eγ µ e ) (cid:0) ¯ Qγ µ Q (cid:1) O V R B (cid:48) µ ∼ (1 , (cid:0) ¯ Qγ µ e c (cid:1) (cid:0) Q c γ µ e (cid:1) O V R V µ ∼ (3 , − / (cid:0) ¯ Qe (cid:1) (¯ eQ ) − O V R D ∼ (3 , +7 / (cid:0) ¯ Qσ µν e (cid:1) (¯ eσ µν Q ) 0 –¯ ee ¯ uu (¯ eγ µ e ) (¯ uγ µ u ) O V R B (cid:48) µ ∼ (1 , (¯ eγ µ u ) (¯ uγ µ e ) O V R Z µ ∼ (3 , +5 / (¯ eu c ) ( u c e ) O V R S ∼ (3 , − / (¯ eσ µν u c ) ( u c σ µν e ) 0 –¯ ee ¯ dd (¯ eγ µ e ) (cid:0) ¯ dγ µ d (cid:1) O V R B (cid:48) µ ∼ (1 , (¯ eγ µ d ) (cid:0) ¯ dγ µ e (cid:1) O V R U µ ∼ (3 , +2 / (¯ ed c ) (cid:0) d c e (cid:1) O V R S (cid:48) ∼ (3 , − / (¯ eσ µν d c ) (cid:0) d c σ µν e (cid:1) As a linearly independent set, we choose the following four-fermion operators (written in the quark interactionbasis): O V L = (¯ ν L γ µ ν L ) (¯ u L γ µ u L ) + (¯ ν L γ µ ν L ) (cid:0) ¯ d L γ µ d L (cid:1) + (¯ e L γ µ e L ) (¯ u L γ µ u L ) + (¯ e L γ µ e L ) (cid:0) ¯ d L γ µ d L (cid:1) , O V L = 14 (cid:2) (¯ ν L γ µ ν L ) (¯ u L γ µ u L ) − (¯ ν L γ µ ν L ) (cid:0) ¯ d L γ µ d L (cid:1) − (¯ e L γ µ e L ) (¯ u L γ µ u L ) + (¯ e L γ µ e L ) (cid:0) ¯ d L γ µ d L (cid:1)(cid:3) + 12 (cid:2) (¯ ν L γ µ e L ) (cid:0) ¯ d L γ µ u L (cid:1) + (¯ e L γ µ ν L ) (¯ u L γ µ d L ) (cid:3) , O V L = (¯ ν L γ µ ν L ) (¯ u R γ µ u R ) + (¯ e L γ µ e L ) (¯ u R γ µ u R ) , O V L = (¯ ν L γ µ ν L ) (cid:0) ¯ d R γ µ d R (cid:1) + (¯ e L γ µ e L ) (cid:0) ¯ d R γ µ d R (cid:1) , O V R = (¯ e R γ µ e R ) (¯ u L γ µ u L ) + (¯ e R γ µ e R ) (cid:0) ¯ d L γ µ d L (cid:1) , O V R = (¯ e R γ µ e R ) (¯ u R γ µ u R ) , O V R = (¯ e R γ µ e R ) (cid:0) ¯ d R γ µ d R (cid:1) , O S L = − (¯ e R e L ) (¯ u R u L ) + (¯ e R ν L ) (¯ u R d L ) , O S R = (¯ e R ν L ) (¯ u L d R ) + (¯ e R e L ) (cid:0) ¯ d L d R (cid:1) , O T = (¯ e R σ µν ν L ) (¯ u R σ µν d L ) − (¯ e R σ µν e L ) (¯ u R σ µν u L ) . (14)A comment is in order regarding other dimension six operators which may be generated by integrating outheavy particles. Two sets of operators relevant to the study of R Vτ/(cid:96) are operators of the form O Hl L,R = i (cid:16) H † ←→ D µ H (cid:17) (cid:0) ¯ l L,R γ µ l L,R (cid:1) and the dipole operators O D = H ¯ Lσ µν e R F µν .The O Hl operators generate non-universal Z -mediated quarkonia decays, and further modify the Z → τ τ decaywidth. These two effects are related: The leading effect in R Vτ/(cid:96) arising from this operator is given by R Vτ/(cid:96) (cid:39) R V, SM τ/(cid:96) (cid:18) − m V v g qV παQ q (cid:2) δg ZτL + δg ZτR (cid:3)(cid:19) , (15)where v = 246 GeV is the electroweak vacuum expectation value. The LEP measurements of Z -pole observables [68]constrain this effect to be smaller than 10 − . For more details see App. B 1. These operators can be formed at theUV matching scale, in which case they are typically controlled by additional free parameters. They can be furthergenerated by the mixing with four fermion operators, in which case fine-tuning cancelation may be needed to ensurethat the Z -pole constraints are not violated.Dipole operators might be generated at the UV by one loop processes or by RGE mixing with the four-fermi operator O T . Typically, the resulting contribution to the vector meson decay rate is O ( αv/m V ) compared to the contributionarising from the tensor operator. Yet, the Wilson coefficients generated in the mixing are y c V cb suppressed [69, 70].Numerically, we find that this effect on R Υ(1 S ) τ/(cid:96) is always below the per-mil level and can be neglected. The dipoleoperators further modify the electric and magnetic dipole moments of the τ lepton. For completeness we write theleading contributions of the dipole operators to R Vτ/(cid:96) and the relation to the taonic dipole moments in App. B 2.In general RGE effects generate also four-quark and four-lepton operators. Such effects were studied in e.g.
Ref. [42]and we do not study their possible indirect implications on our analysis.
IV. NUMERICAL RESULTS
As explained above, it is useful to explore simplified UV models which generate the required four-fermion operatorsat tree-level. In the following subsections we examine various sets of effective operators formed by the integration outof heavy scalar and vector bosons which have the right quantum numbers to modify the b → cτ ν transition. For eachcase, we present the relevant UV couplings and obtain the resulting CC and NC operators (neglecting, as explainedabove, RGE effects). We find within 95% C.L. the numerical values for the Wilson coefficients which minimize theobserved anomaly in R ( D ( ∗ ) ), and the corresponding predictions for R Vτ/(cid:96) for the Υ and ψ (2 S ) states. Some of thesimplified models we study cannot ease the tension between the theory and the R ( D ( ∗ ) ) measurements. Clearly, inthese models the best fit point of the new couplings is zero, giving back the SM. Nevertheless, for completeness wealso consider these models and present our results for them. We present our main results in Table III, where onlymodels which ease the tension below χ = 9 (corresponding to 3 σ with one degree of freedom) are included. Note thatin some cases only one of the decay modes, i.e. Υ → τ τ or ψ → τ τ , is modified, due to the specific SU (2) structure ofthe effective operators. Furthermore, in the 2HDM case (cid:0) φ ∼ (1 , +1 / (cid:1) neither of these decays is modified becauseof the vector structure of the q ¯ q mesons.TABLE III: The simplified (single boson) models and the predicted range for R Vτ/(cid:96) for V = Υ(1 S ) , ψ (2 S ) . Theachievable and projected uncertainties are our estimations, see the text for more details. UV field content R Υ(1 S ) τ/(cid:96) R ψ (2 S ) τ/(cid:96) Predicted modification to R Υ(1 S ) τ/(cid:96) W (cid:48) µ ∼ (1 , . − . U µ ∼ (3 , +2 / . − . S ∼ (3 , − / SM 0.389-0.390 – V µ ∼ (3 , − / . − . .
992 0 . . ± .
025 0 . ± . ± . ± . L Υ(3 S ) = 1 / ab in Belle II) ± .
004 –
One clear advantage of analyzing such simplified models is that each scenario predicts distinct relations between thevarious EFT operators. These relations can, however, be modified due to SU (2) L breaking effects. Let us explain whywe ignore these effects in our analysis. Electroweak breaking effects split the spectrum of the charged and neutral NPfields, which, in turn, changes the relations between the Wilson coefficients of the CC and NC effective operators by O (∆ M/M ). This modifies our predictions for R Vτ/(cid:96) . The unavoidable loop-induced splitting is smaller than a GeV,and therefore negligible. Tree-level splitting is typically of order ∆
M/M (cid:39) v /M and might change our final resultsby a few percent. This splitting is, however, generated by a free parameter in the scalar potential which we take tobe zero (up to small loop-induced effects). Once a specific UV model is considered in full detail, this assumption canbe modified, and other consequences of it (such as corrections to the oblique T parameter) should be considered.As concerns the flavor structure of the fundamental couplings, we impose a global U (2) Q symmetry, under which thelight left-handed Q , quarks transform as a doublet. This choice is taken to avoid large production rate at the LHCand dangerous FCNC transitions in the first and second generation . To determine uniquely the flavor structure ofthe NP couplings one should further specify the mass basis alignment of the UV operators. In what follows, we alwaysconsider alignment to the down mass basis, with Q = (cid:0) V ∗ u i b u L i , b L (cid:1) . For the ¯ L L ¯ Q Q and ¯ e L ¯ Q d operators thischoice is essential to ensure that b → c transition is modified, once U (2) Q is preserved. For the ¯ L e ¯ Q u operators(generated by the S and D fields) one could choose alignment to the up-mass basis. In this case neither c L c R → τ + τ − nor b L b R → τ + τ − transitions are generated. Nevertheless, other NC operators which result in b L b L → τ + τ − and c R c R → τ + τ − transitions are formed, with the same CKM suppression (since V tb (cid:39)
1) and the same Wilson coefficientas in the down-aligned scenarios. We therefore find no change in our results once up-alignment is taken.We assume no significant mixing between the NP and SM fields (this is crucial for the W (cid:48) scenario), and take thequartic | X NP | | H | couplings to be negligible. To keep our models in the perturbative regime, we take all parametersto be smaller than 4 π at the TeV scale. We stress again that, once an anomaly is found in LFU of Υ or ψ decays,these assumptions should be reconsidered and a complete UV theory should be studied in full detail. Since we donot study the 2HDM case, condensation of the NP fields is absent due to Lorentz and/or SU (3) C symmetries. In thefollowing, we take all the couplings to be real and denote X ij ≡ X i X j .To find the best fit values of the Wilson coefficients we minimize the χ function. We use the experimental resultsquoted in Eq. (2). The theoretical uncertainties on the form factors which affect R ( D ( ∗ ) ) in the SM are small comparedto the experimental errors, as evident from the accurate SM predictions. (See also Figs. [1-2] in Ref [50].) As for theother form factor ( F T in the notations of Ref. [50]), we have explicitly checked that varying it within 20% error doesnot alter our results for R Vτ/(cid:96) . When considering a model, we are agnostic about how plausible it is from a modelbuilding point of view, and only compare it to the SM point. The latter gives χ (cid:39) q distribution of Γ [ B → D ∗ τ ν ] is hardly modified in all of the scenar- This typically suppresses the non-universality effect in ψ (2 S ) decays to the permil level, and might not be necessary in some cases [71]. ios. We comment below on the q distributions of Γ [ B → Dτ ν ], which is, in general, consistent within the currentuncertainties. Interference corrections, as analysed in Ref. [49] are expected to be small. A. W (cid:48) µ ∼ (1 , We introduce a vector-boson, color-singlet, SU (2) L -triplet W (cid:48) µ ∼ (1 , with the following interaction Lagrangian: L W = g ¯ Q τ a /W a Q + g ¯ L τ a /W a L . (16)Integrating out W (cid:48) µ , we obtain the following EFT Lagrangian: L EFT W = − g g M W (cid:48) (cid:0) ¯ Q τ a γ µ Q (cid:1) (cid:0) ¯ L τ a γ µ L (cid:1) = − g g M W (cid:48) O V L . (17)The relevant CC interactions are given by L CC = − g g V cb M W (cid:48) (¯ τ L γ µ ν L ) (¯ c L γ µ b L ) + h . c .. (18)The best-fit-point (BFP) and 95% C.L. intervals are given by g BFP12 = 7 . (cid:18) M W (cid:48) TeV (cid:19) with χ = 0 . ,g = [4 . , . (cid:18) M W (cid:48) TeV (cid:19) @ 95% C . L .. (19)where g ≡ g g . The q distribution is identical to the SM one, since the new CC operator has the same Lorentzstructure as in the SM.The relevant NC interactions are given by L NC = V cb g g M W (cid:48) (¯ τ L γ µ τ L ) (¯ c L γ µ c L ) − g g M W (¯ τ L γ µ τ L ) (cid:0) ¯ b L γ µ b L (cid:1) . (20)They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . − . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . ,R ψ (2 S ) τ/(cid:96) = 0 . . (21) B. U µ ∼ (3 , +2 / We introduce a vector-boson, color-triplet, SU (2) L -singlet U µ ∼ (3 , +2 / with the following interaction La-grangian: L U = g ¯ Q /U L + g ¯ d /U e + h . c .. (22)Integrating out U µ , we obtain the following EFT Lagrangian: L EFT U = − | g | M U (cid:0) ¯ Q γ µ L (cid:1) (cid:0) ¯ L γ µ Q (cid:1) − | g | M U (¯ e γ µ d ) (cid:0) ¯ d γ µ e (cid:1) − (cid:20) g g ∗ M U (cid:0) ¯ Q γ µ L (cid:1) (¯ e γ µ d ) + h . c . (cid:21) = − | g | M U O V L − | g | M U O V L − | g | M U O V R + (cid:20) g g ∗ M U O S R + h . c . (cid:21) . (23)The relevant CC interactions are given by L CC = − | g | V cb M U (¯ τ L γ µ ν L ) (¯ c L γ µ b L ) + 2 V cb g g ∗ M U (¯ τ R ν L ) (¯ c L b R ) + h . c .. (24)The BFP which explains R ( D ( ∗ ) ) is given by g g < (cid:0) | g | , | g | (cid:1) = (3 . , . (cid:18) M U TeV (cid:19) with χ = 0 . (25)The 95% C.L. intervals are presented in Figure 1. The q distribution of Γ [ B → Dτ ν ] is modified compared to theSM one. Yet, as is evident from Figure. 3b, this change is not very significant given the current uncertainties. | g | g FIG. 1: U µ ∼ (3 , / couplings at C.L. for M U = 1 TeV. Colored lines are contours of R Υ(1 S ) τ/(cid:96) . The relevant NC interactions are given by L NC = − | g | M U (¯ τ L γ µ τ L ) (cid:0) ¯ b L γ µ b L (cid:1) − | g | M U (¯ τ R γ µ τ R ) (cid:0) ¯ b R γ µ b R (cid:1) + (cid:20) g g ∗ M U (¯ τ R τ L ) (cid:0) ¯ b L b R (cid:1) + h . c . (cid:21) . (26)They induce only Υ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . − . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . . (27) C. X µ ∼ (3 , +2 / We introduce a vector-boson, color-triplet, SU (2) L -triplet X µ ∼ (3 , +2 / with the following interaction La-grangian: L X = g ¯ Q τ a /X a L + h . c .. (28)Integrating out X µ , we obtain the following EFT Lagrangian: L EFT X = − | g | M X (cid:0) ¯ Q τ a γ µ L (cid:1) (cid:0) ¯ L τ a γ µ Q (cid:1) = − | g | M X O V L + | g | M X O V L . (29)The relevant CC interactions are given by L CC = V cb | g | M X (¯ τ L γ µ ν L ) (¯ c L γ µ b L ) + h . c .. (30)The BFP and 95% C.L. interval are given by | g | = 0 (cid:18) M X TeV (cid:19) with χ = 20 . , | g | = [0 , . (cid:18) M X TeV (cid:19) @ 95% C . L .. (31)The q distribution is identical to the SM one, since the new CC operator has the same Lorentz structure as in theSM.The relevant NC interactions are given by L NC = − V cb | g | M U (¯ τ L γ µ τ L ) (¯ c L γ µ c L ) − | g | M U (¯ τ L γ µ τ L ) (cid:0) ¯ b L γ µ b L (cid:1) . (32)They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . ,R ψ (2 S ) τ/(cid:96) = 0 . . (33) D. S ∼ (3 , − / We introduce a scalar-boson, color-triplet, SU (2) L -singlet S ∼ (3 , − / with the following interaction Lagrangian: L S = λ S ¯ L (cid:15) T Q c + λ S ¯ e u c + h . c .. (34)We impose a global 3 B − L symmetry, which prevent an additional Yukawa couplings of the form Sdu and
SQQ .Integrating out S , we obtain the following EFT Lagrangian: L EFT S = | λ | M S | ¯ L (cid:15) T Q c | + | λ | M S | ¯ e u c | − (cid:20) λ ∗ λ M S (cid:0) L c (cid:15)Q (cid:1) (¯ e u c ) + h . c . (cid:21) = | λ | M S O V L − | λ | M S O V L + | λ | M S O V R + (cid:20) λ ∗ λ M S O S L − λ ∗ λ M S O T + h . c . (cid:21) . (35)The relevant CC interactions are given by L CC = − | λ | V cb M S (¯ τ L γ µ ν L ) (¯ c L γ µ b L ) + λ ∗ λ M S (¯ τ R ν L ) (¯ c R b L ) − λ ∗ λ M S (¯ τ R σ µν ν L ) (¯ c R σ µν b L ) + h . c .. (36)The BFP which explains R ( D ( ∗ ) ) is given by λ λ < (cid:0) | λ | , | λ | (cid:1) = (0 . , . (cid:18) M S TeV (cid:19) with χ = 0 . (37)The 95% C.L. intervals are presented in Figure 2. The q distribution of Γ [ B → Dτ ν ] is modified compared to theSM one. Yet, as is evident from Figure. 3c, this change is not very significant given the current uncertainties.The relevant NC interactions are given by L NC = | λ | M S V cb (¯ τ L γ µ τ L ) (¯ c L γ µ c L ) + | λ | M S (¯ τ R γ µ τ R ) (¯ c R γ µ c R )+ (cid:20) − λ ∗ λ M S V cb (¯ τ R τ L ) (¯ c R c L ) + λ ∗ λ M S V cb (¯ τ R σ µν τ L ) (¯ c R σ µν c L ) + h . c . (cid:21) . (38)They induce only ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R ψ (2 S ) τ/(cid:96) = 0 . − . . (39)0 | λ | λ FIG. 2: S ∼ (3 , − / couplings at C.L. for M S = 1 TeV. Red region corresponds to λ λ < and blue regioncorresponds to λ λ > . Colored lines are contours of R ψ (2 S ) τ/(cid:96) E. T ∼ (3 , − / We introduce a scalar-boson, color-triplet, SU (2) L -triplet S ∼ (3 , − / with the following interaction Lagrangian: L T = λT a ¯ L τ a (cid:15) T Q c + λ ∗ T ∗ a Q c (cid:15)τ a L . (40)We impose global 3 B − L symmetry to forbid T QQ terms. Integrating out T , we obtain the following EFT Lagrangian: L EFT T = 1 M T | λ | (cid:0) ¯ L τ a (cid:15) T Q c (cid:1) (cid:0) Q c (cid:15)τ a L (cid:1) = 3 | λ | M T O V L + | λ | M T O V L . (41)The relevant CC interactions are given by L CC = | λ | V cb M T (¯ τ L γ µ ν L ) (¯ c L γ µ b L ) . (42)The BFP and 95% C.L. interval are given by | λ | = 0 (cid:18) M T TeV (cid:19) with χ = 20 . , | λ | = [0 , . (cid:18) M T TeV (cid:19) @ 95% C . L .. (43)The q distribution is identical to the SM one, since the new CC operator has the same Lorentz structure as in theSM.The relevant NC interactions are given by L NC = | λ | V cb M T (¯ τ L γ µ τ L ) (¯ c L γ µ c L ) + | λ | M T (¯ τ L γ µ τ L ) (cid:0) ¯ b L γ µ b L (cid:1) . (44)They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . − . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . ,R ψ (2 S ) τ/(cid:96) = 0 . . (45)1 F. φ ∼ (1 , +1 / A scalar-boson, color-singlet, SU (2) L -doublet φ ∼ (1 , +1 / does not affect V → (cid:96)(cid:96) as it generates only scalarcouplings. G. D ∼ (3 , +7 / We introduce a scalar-boson, color-triplet, SU (2) L -doublet D ∼ (3 , +7 / with the following interaction Lagrangian: L D = λ D ¯ Q e + λ D(cid:15) ¯ u L + h . c .. (46)Integrating out D , we obtain the following EFT Lagrangian: L EFT D = | λ | M D | ¯ Q e | − | λ | M D | ¯ u L | − (cid:20) λ ∗ λ M D (¯ u L ) (cid:15) (¯ e Q ) (cid:21) = − | λ | M D O V R + | λ | M D O V L + (cid:20) λ ∗ λ M D (cid:18) O S L + 14 O T (cid:19) + h . c . (cid:21) . (47)The relevant CC interactions are given by L CC = λ ∗ λ M D (¯ τ R ν L ) (¯ c R b L ) + λ ∗ λ M D (¯ τ R σ µν ν L ) (¯ c R σ µν b L ) + h . c .. (48)The BFP and 95% C.L. interval are given by λ BFP12 = 0 . (cid:18) M D TeV (cid:19) with χ = 17 ,λ = [0 , . (cid:18) M D TeV (cid:19) @ 95% C . L .. (49)The q distribution of Γ [ B → Dτ ν ] is modified compared to the SM one. Yet, as is evident from Figure. 3d, thischange is not very significant given the current uncertainties.The relevant NC interactions are given by L NC = − | λ | M D V cb (¯ τ R γ µ τ R ) (¯ c L γ µ c L ) − | λ | M D (¯ τ R γ µ τ R ) (cid:0) ¯ b L γ µ b L (cid:1) + | λ | M D (¯ τ L γ µ τ L ) (¯ c R γ µ c R ) − (cid:20) λ ∗ λ M D V cb (¯ τ R τ L ) (¯ c R c L ) + λ ∗ λ M D V cb (¯ τ R σ µν τ L ) (¯ c R σ µν c L ) + h . c . (cid:21) . (50)They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . − . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . ,R ψ (2 S ) τ/(cid:96) = 0 . − . . (51) H. V µ ∼ (3 , − / We introduce a vector-boson, color-triplet, SU (2) L -doublet V µ ∼ (3 , − / with the following interaction La-grangian: L V = g ¯ Q /V e c + g ¯ L /V d c + h . c .. (52)2Integrating out V µ , we obtain the following EFT Lagrangian: L EFT V = − | g | M V (cid:0) ¯ Q γ µ e c (cid:1) (¯ e c γ µ Q ) − | g | M V (cid:0) ¯ L γ µ d c (cid:1) (cid:0) ¯ d c γ µ L (cid:1) − (cid:20) g g ∗ M V (cid:0) ¯ Q γ µ e c (cid:1) (cid:0) ¯ d c γ µ L (cid:1) + h . c . (cid:21) = − | g | M V O V R − | g | M V O V L − (cid:20) g g ∗ M V O S R + h . c . (cid:21) . (53)The relevant CC interactions are given by L CC = − V cb g g ∗ M V (¯ τ R ν L ) (¯ c L b R ) + h . c .. (54)The BFP and 95% C.L. interval are given by g BFP12 = 4 . (cid:18) M V TeV (cid:19) with χ = 8 . ,g = [1 . , . (cid:18) M V TeV (cid:19) @ 95% C . L .. (55)The q distribution of Γ [ B → Dτ ν ] is modified compared to the SM one. Yet, as is evident from Figure. 3e, thischange is not very significant given the current uncertainties.The relevant NC interactions are given by L NC = − | g | M V (¯ τ L γ µ τ L ) (cid:0) ¯ b L γ µ b L (cid:1) − | g | M V (¯ τ R γ µ τ R ) (cid:0) ¯ b R γ µ b R (cid:1) − (cid:20) g g ∗ M V (¯ τ R τ L ) (cid:0) ¯ b L b R (cid:1) + h . c . (cid:21) . (56)They induce only Υ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities R Υ(1 S ) τ/(cid:96) = 0 . − . ,R Υ(2 S ) τ/(cid:96) = 0 . − . ,R Υ(3 S ) τ/(cid:96) = 0 . − . . (57) V. DISCUSSION AND FUTURE PROSPECTS
The allowed ranges for R Vτ/(cid:96) in simplified models that account for the deviations of R ( D ( ∗ ) ) are presented in TableIII. As concerns R ψ (2 S ) τ/(cid:96) , it is modified by no more than three permil, while the present experimental accuracy isof order thirteen percent. Thus, the maximal modification is about a factor of fifty below current sensitivity. Weconclude that R ψ (2 S ) τ/(cid:96) does not probe at present models that solve the R ( D ( ∗ ) ) puzzle. (Removing the imposed U (2) Q symmetry might lead to a much larger modification of R ψ (2 S ) τ/(cid:96) [71].)As concerns R Υ(1 S ) τ/(cid:96) , in all simplified models, except the T model, it is smaller than the SM. The lowest valueis 4% below the SM value. The current experimental accuracy is 2 . . σ ) then the SM prediction. We conclude that R Υ(1 S ) τ/(cid:96) is startingto probe relevant models, disfavoring parts of the parameter space in some of the models.Non-universality in leptonic Υ decays was tested by CLEO [60] for the 1 S, S and 3 S states, and by BaBar [61] forthe 1 S state. These measurements read R Υ(1 S ) , BaBar τ/µ = 1 . ± . stat ± . syst ,R Υ(1 S ) , CLEO τ/µ = 1 . ± . stat ± . syst ,R Υ(2 S ) , CLEO τ/µ = 1 . ± . stat ± . syst ,R Υ(3 S ) , CLEO τ/µ = 1 . ± . stat ± . syst . (58)3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● SM q [ GeV ] ( / Γ ) d Γ / dq [ G e V - ] (a) SM ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● U μ q [ GeV ] ( / Γ ) d Γ / dq [ G e V - ] (b) U µ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● S q [ GeV ] ( / Γ ) d Γ / dq [ G e V - ] (c) S ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● D q [ GeV ] ( / Γ ) d Γ / dq [ G e V - ] (d) D ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● V μ q [ GeV ] ( / Γ ) d Γ / dq [ G e V - ] (e) V µ FIG. 3:
Normalized q distribution of Γ[ B → Dτ ν ] . Data points and error bars are taken from Ref. [2]. CLEO’s data used in this analysis includes both on-resonance and off-resonance subsamples, which correspond ap-proximately to 21 ,
10, and 6 million events of the 1 S, S and 3 S states, respectively. The Υ(2 S ) on-resonance samplecollected by BaBar (Belle) is about ∼
10 (16) times larger than CLEO’s sample. The Υ(3 S ) on-resonance samplecollected by BaBar (Belle) is about ∼
20 (2) times larger than CLEO’s sample. Analyzing these existing data setswill allow to reduce each of the statistical uncertainties to roughly 1 − τ and µ totalefficiencies and event shapes. The larger statistics can realistically lead to a reduction of this uncertainty by a factorof two. Altogether, we estimate that a total uncertainty of about 1% can be obtained by analyzing the existing data.Future measurement in Belle II can further reduce this uncertainty. Assuming that the important systematical error isgoverned by the limited statistics, we estimate that reaching a σ sys = 0 .
4% for R Υ(1 S ) , Belle II τ/µ would require integratedluminosity of
L ∼ / ab at the Υ(3 S ) energy. Thus, our study gives additional motivation to the proposal of Ref. [72]to study the Υ(3 S ) resonance at the early physics program of the Belle II experiment.One important point to emphasize is that theoretically any model that affects an Υ state affects them all. Thus,it is wise to combine the experimental results of Υ(1 S ), Υ(2 S ), and Υ(3 S ) into one test of universality. Eq. (6) as a4function of m Υ( nS ) , R τ/(cid:96) ( m Υ( nS ) ) = (cid:34) x τ, S (cid:18) m Υ(1 S ) m Υ( nS ) (cid:19) (cid:35) (cid:34) − x τ, S (cid:18) m Υ(1 S ) m Υ( nS ) (cid:19) (cid:35) / , (59)where x τ, S ≡ m τ /m Υ(1 S ) = 0 . ψ (2 S ) → τ + τ − ) = (3 . ± . stat ± . sys ) × − using 14M ψ (2 S ) events.BESII already collected 106M events which will reduce the statistical error by a factor of ∼
3. (KEDR also mea-sures this tauonic branching ratio [74] but it is not used by the PDG fit.) The relative systematic uncertainty onBR( ψ (2 S ) → µ + µ − ) (as measured by BaBar) is 10% [75], while the relative systematic error on BR( ψ (2 S ) → e + e − )(as measured by BESII) is approximately 4% [76]. It is then not very likely that the ratio R ψ (2 S ) τ/(cid:96) will be measuredto an accuracy better than 4%. We conclude that at least an order of magnitude improvement in this uncertainty isneeded to be achieved in Bess III to start probing the relevant parameter space. VI. SUMMARY AND CONCLUSIONS
There is a 3 . σ evidence that the ratio R ( D ( ∗ ) ) ≡ Γ( B → D ( ∗ ) τ ν ) / Γ( B → D ( ∗ ) (cid:96)ν ), where (cid:96) = µ, e , is considerablyenhanced compared to the Standard Model value. To explain a large enhancement of a SM tree-level process, therequired new physics is likely to include new bosons which mediate the B → D ( ∗ ) τ ν decay at tree level. There areseven such candidates, and none is a SM singlet. Thus, it is highly likely that their mass is well above m B , whichallows one to examine their effects in an EFT language. Specifically, they should generate dimension-six, four-fermi(two quark and two lepton fields) operators.In the absence of light right-handed neutrinos, the four-fermi operators include one or two SU (2) L -doublet quarkfields. Consequently, a variety of processes, in addition to B → D ( ∗ ) τ ν , are affected. Some of these, such as t → cτ + τ − decay, B c → τ ν decay [24, 41], Λ b → Λ c τ ¯ ν τ decay [51, 59], and b ¯ b /c ¯ c → τ + τ − scattering [47], have been previouslystudied in the literature, and probe the proposed models. Here, we suggest another class of observables: lepton non-universality in leptonic decays of the Υ and ψ vector-mesons, parameterized by the ratio R Vτ/(cid:96) ≡ Γ( V → τ τ ) / Γ( V → (cid:96)(cid:96) ).We find that, once a U (2) Q symmetry is assumed, current measurements of ψ decays do not have the accuracyrequired to probe the models in a significant way. On the other hand, for Υ(1 S ) decays, the current experimentalaccuracy and the predicted deviations are of the same order of magnitude. A modest improvement in the experimentalaccuracy is capable of favoring some models and disfavoring others. If, for example, it is established that R Υ(1 S ) τ/(cid:96) islarger than the SM value, than all the simplified models will be disfavored. (The only model which enhances R Υ(1 S ) τ/(cid:96) is the T model, which gives, however, a very small effect.) If experiments reach high accuracy in the leptonic vectormeson decays and observe no signal, then the models that allow negligible deviations will be favored. Anotherpossibility is that the light neutrino has a significant ν R component, in which case R ( D ( ∗ ) ) could be explained byoperators which do not affect V → τ τ .We discussed the prospects of such improvement in the experimental accuracy. Current data samples should alreadyallow to reduce the statistical error considerably and reach an accuracy of about 1.5%. If Belle II operates below theΥ(4 S ) resonance, it can contribute significantly, via the measurements of R Υ( nS ) τ/(cid:96) , to our understanding of the R ( D ( ∗ ) )puzzle. Acknowledgments
We thank Avital Dery, Admir Greljo, Avner Soffer, Alexander Penin and Nadav Priel for useful discussions. We arethankful to Diptimoy Ghosh for his great help with the numerical evaluation of R ( D ( ∗ ) ). This research is supportedby the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel (grant number 2014230), and bythe I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number1937/12). YG is supported in part by the NSF grant PHY-1316222. YN is the Amos de-Shalit chair of theoreticalphysics. YN is supported by grants from the Israel Science Foundation (grant number 394/16) and from the MinervaFoundation.5 Appendix A: The leptonic width in the SM
The leading SM decay rate is given by the Van Royen-Weisskopf formula [63],Γ [ V → (cid:96)(cid:96) ] = 4 πα (cid:18) f V m V (cid:19) Q q (1 − x (cid:96) ) / (cid:0) x (cid:96) (cid:1) . (A1)The dominant corrections arise from tree-level Z exchange and quantum QED and QCD corrections,Γ [ V → (cid:96)(cid:96) ] (cid:39) Γ [ V → (cid:96)(cid:96) ] (cid:0) δ tree Z + δ QCD + δ EM (cid:1) . (A2)The tree-level Z mediated correction is δ tree Z = (cid:18) (cid:15)(cid:15) − (cid:19) (cid:18) g qV Q q c W s W (cid:19) (cid:0) − s W (cid:1) + (cid:18) (cid:15)(cid:15) − (cid:19) (cid:18) g qV Q q c W s W (cid:19) (cid:20)(cid:0) − s W (cid:1) + (cid:18) − x (cid:96) x (cid:96) (cid:19)(cid:21) . (A3)Here (cid:15) = m V /m Z and g qV = T q − Q q s W . This is an O (10 − ) correction, well below the experimental sensitivity.The resulting change in the non-universality relation (Eq. (6)) is, however, only O (10 − ), as the interference withthe photon projects out the vectorial part of the Z coupling. QCD corrections have been studied extensively in theliterature (see, for example, [77] and references therein). A crucial point is that these short- and long-distance QCDeffects do not depend on m (cid:96) , and cancel in the ratio R Vτ/(cid:96) . These can be absorbed in the definition of the vector mesonform factor, f V .The leading QED corrections which are not included in the definition of f V arise from corrections to the photontwo-point function, the photon-lepton vertex corrections, and the irreducible real photon emission (FSR). Correctionsdue to two-photon exchange are absent: The Landau-Yang theorem [78, 79] implies M ( V → γγ ) = 0 for massivevector bosons. Therefore exploiting the optical theorem and dispersion relation for real analytic functions, as wellas QED and QCD CP invariance, forbids M ( V → (cid:96)(cid:96) ) via two photon exchange. The corrections to Π γγ are takeninto account by using the running couplings, namely evaluating α ( µ ) at µ = m V . Once again these are universalcorrections that do not affect R Vτ/(cid:96) .As for the leptonic-vertex corrections, these exhibit IR singularities which are regulated by the real emission ofsoft photons. Clearly, the latter depends on the experimental resolution and should be determined (and unfolded)by the experiments using state-of-the-art detector simulation in MC analysis. Following the LEP report for Z poleobservables [68], we quote here the inclusive Γ[ V → (cid:96) + (cid:96) − + γ ] at one-loop order in QED, which does not depend onthe experimental setup. It reads [80] δ EM = α π (cid:20) − − x (cid:96) ] (cid:0) x (cid:96) ] (cid:1) − [1 − x (cid:96) ] − x (cid:96) (2 + 3 x (cid:96) ) + 16 x (cid:96) (cid:0) x (cid:96) (cid:1) log 4 + 4 π (cid:21) (cid:39) α π (cid:2) x (cid:96) (2 − log 4) (cid:3) (cid:39) .
002 + 0 . x (cid:96) . (A4)We further estimate the two-loop non-universality effect to be δ − loop = O (cid:0) α x τ (cid:1) (cid:39) − . (A5)We therefore consider, for all practical purposes,Γ [ V → (cid:96)(cid:96) ] = 4 πα (cid:18) f V m V (cid:19) Q q (1 − x l ) / (cid:0) x l (cid:1) (cid:20) α π + 4 αx (cid:96) π (2 − log 4) (cid:21) , (A6)and R Vτ/(cid:96) = (1 − x τ ) / (cid:0) x τ (cid:1) (cid:20) αx τ π (2 − log[4]) (cid:21) . (A7) Appendix B: Other EFT operators1. Z mediated operators Here we consider the following set of dimension six operators L Hl = iC H(cid:96) (cid:16) H † ←→ D µ H (cid:17) (cid:0) ¯ Lγ µ L (cid:1) + iC H(cid:96) (cid:16) H † ←→ D µ τ a H (cid:17) (cid:0) ¯ Lγ µ τ a L (cid:1) + iC He (cid:16) H † ←→ D µ H (cid:17) (¯ e R γ µ e R ) , (B1)6which affect R Vτ/(cid:96) by modifying the
Zτ τ vertex. Their contributions to the ψ and Υ leptonic decays are given by A q(cid:96)V = − παQ q − m V g qV (cid:0) C H(cid:96) + C H(cid:96) / C He (cid:1) (cid:18) − (cid:15) (cid:19) (cid:39) − παQ q + m V g qV v (cid:0) δg ZτL + δg ZτR (cid:1) ,B q(cid:96)V = m V g qV (cid:0) C H(cid:96) + C H(cid:96) / − C He (cid:1) (cid:18) − (cid:15) (cid:19) (cid:39) − m V g qV v (cid:0) δg ZτL − δg ZτR (cid:1) , (B2)where, as before, (cid:15) = m V /m Z and g qv = T q − Q q s W , and for the Zτ τ vertex corrections we follow the definitionsof [81]. The consistency of LEP data with the SM prediction requires δg ZτL,R ≤ − at 2 σ , which, in turn, can modify R Vτ/(cid:96) by, at most, 10 − − − . This effect is negligible given the current and future experimental sensitivity.
2. Dipole operator
Here we consider the dimension six dipole operator L D = √ παC (cid:96)D ¯ Lσ µν e R HF µν + h . c .. (B3)Its contribution to the ψ and Υ leptonic decays is given by A q(cid:96)V = − παQ q + 16 παQ q vm (cid:96) √ (cid:2) C (cid:96)D (cid:3) = − παQ q − παQ q ∆ a (cid:96) ,C q(cid:96)V = 8 παQ q vm V √ (cid:2) C (cid:96)D (cid:3) = − παQ q ∆ a (cid:96) (cid:18) m V m (cid:96) (cid:19) ,D (cid:96)V = 8 παQ q vm V √ (cid:2) C (cid:96)D (cid:3) = 4 παQ q m V (cid:18) d (cid:96) e (cid:19) , (B4)where ∆ a (cid:96) and d (cid:96) are the leptonic magnetic and electric moments, respectively. The τ constraints read − . ≤ ∆ a τ ≤ . , − . − ≤ (cid:18) d τ e (cid:19) ≤ . − . (B5)These bounds are, however, not strong enough to suppress the dipole contribution to R Vτ/(cid:96) . [1] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. Lett. , 101802 (2012) [arXiv:1205.5442 [hep-ex]].[2] J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D , 072012 (2013) [arXiv:1303.0571 [hep-ex]].[3] M. Huschle et al. [Belle Collaboration], Phys. Rev. D , 072014 (2015) [arXiv:1507.03233 [hep-ex]].[4] Y. Sato et al. [Belle Collaboration], Phys. Rev. D , 072007 (2016) [arXiv:1607.07923 [hep-ex]].[5] A. Abdesselam et al. [Belle Collaboration], arXiv:1608.06391 [hep-ex].[6] S. Hirose et al. [Belle Collaboration], arXiv:1612.00529 [hep-ex].[7] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. , 111803 (2015) Addendum: [Phys. Rev. Lett. , 159901(2015)] [arXiv:1506.08614 [hep-ex]].[8] Y. Amhis et al. et al. , arXiv:1607.00299 [hep-lat].[10] S. Fajfer, J. F. Kamenik and I. Nisandzic, Phys. Rev. D , 094025 (2012) [arXiv:1203.2654 [hep-ph]].[11] Y. Sakaki and H. Tanaka, Phys. Rev. D , 054002 (2013) [arXiv:1205.4908 [hep-ph]].[12] S. Fajfer, J. F. Kamenik, I. Nisandzic and J. Zupan, Phys. Rev. Lett. , 161801 (2012) [arXiv:1206.1872 [hep-ph]].[13] A. Crivellin, C. Greub and A. Kokulu, Phys. Rev. D , 054014 (2012) [arXiv:1206.2634 [hep-ph]].[14] A. Datta, M. Duraisamy and D. Ghosh, Phys. Rev. D , 034027 (2012) [arXiv:1206.3760 [hep-ph]].[15] D. Choudhury, D. K. Ghosh and A. Kundu, Phys. Rev. D , 114037 (2012) [arXiv:1210.5076 [hep-ph]].[16] A. Celis, M. Jung, X. Q. Li and A. Pich, JHEP , 054 (2013) [arXiv:1210.8443 [hep-ph]].[17] M. Tanaka and R. Watanabe, Phys. Rev. D , 034028 (2013) [arXiv:1212.1878 [hep-ph]].[18] P. Biancofiore, P. Colangelo and F. De Fazio, Phys. Rev. D , 074010 (2013) [arXiv:1302.1042 [hep-ph]].[19] M. Duraisamy and A. Datta, JHEP , 059 (2013) [arXiv:1302.7031 [hep-ph]].[20] I. Dorner, S. Fajfer, N. Koˇsnik and I. Niandic, JHEP , 084 (2013) [arXiv:1306.6493 [hep-ph]]. [21] Y. Sakaki, M. Tanaka, A. Tayduganov and R. Watanabe, Phys. Rev. D , 094012 (2013) [arXiv:1309.0301 [hep-ph]].[22] M. Duraisamy, P. Sharma and A. Datta, Phys. Rev. D , 074013 (2014) [arXiv:1405.3719 [hep-ph]].[23] B. Bhattacharya, A. Datta, D. London and S. Shivashankara, Phys. Lett. B , 370 (2015) [arXiv:1412.7164 [hep-ph]].[24] R. Alonso, B. Grinstein and J. Martin Camalich, JHEP , 184 (2015) [arXiv:1505.05164 [hep-ph]].[25] A. Greljo, G. Isidori and D. Marzocca, JHEP , 142 (2015) [arXiv:1506.01705 [hep-ph]].[26] L. Calibbi, A. Crivellin and T. Ota, Phys. Rev. Lett. , 181801 (2015) [arXiv:1506.02661 [hep-ph]].[27] M. Freytsis, Z. Ligeti and J. T. Ruderman, Phys. Rev. D , 054018 (2015) [arXiv:1506.08896 [hep-ph]].[28] M. A. Ivanov, J. G. K¨orner and C. T. Tran, Phys. Rev. D , 114022 (2015) [arXiv:1508.02678 [hep-ph]].[29] S. Bhattacharya, S. Nandi and S. K. Patra, Phys. Rev. D , 034011 (2016) [arXiv:1509.07259 [hep-ph]].[30] M. Bauer and M. Neubert, Phys. Rev. Lett. , 141802 (2016) [arXiv:1511.01900 [hep-ph]].[31] C. Hati, G. Kumar and N. Mahajan, JHEP , 117 (2016) [arXiv:1511.03290 [hep-ph]].[32] S. Fajfer and N. Koˇsnik, Phys. Lett. B , 270 (2016) [arXiv:1511.06024 [hep-ph]].[33] R. Barbieri, G. Isidori, A. Pattori and F. Senia, Eur. Phys. J. C , 67 (2016) [arXiv:1512.01560 [hep-ph]].[34] J. M. Cline, Phys. Rev. D , 075017 (2016) [arXiv:1512.02210 [hep-ph]].[35] R. Alonso, A. Kobach and J. Martin Camalich, Phys. Rev. D , 094021 (2016) [arXiv:1602.07671 [hep-ph]].[36] I. Dorner, S. Fajfer, A. Greljo, J. F. Kamenik and N. Koˇsnik, Phys. Rept. , 1 (2016) [arXiv:1603.04993 [hep-ph]].[37] S. M. Boucenna, A. Celis, J. Fuentes-Martin, A. Vicente and J. Virto, Phys. Lett. B , 214 (2016) [arXiv:1604.03088[hep-ph]].[38] D. Buttazzo, A. Greljo, G. Isidori and D. Marzocca, JHEP , 035 (2016) [arXiv:1604.03940 [hep-ph]].[39] D. Das, C. Hati, G. Kumar and N. Mahajan, Phys. Rev. D , 055034 (2016) [arXiv:1605.06313 [hep-ph]].[40] S. Nandi, S. K. Patra and A. Soni, arXiv:1605.07191 [hep-ph].[41] X. Q. Li, Y. D. Yang and X. Zhang, JHEP , 054 (2016) [arXiv:1605.09308 [hep-ph]].[42] F. Feruglio, P. Paradisi and A. Pattori, Phys. Rev. Lett. , 011801 (2017) [arXiv:1606.00524 [hep-ph]].[43] A. K. Alok, D. Kumar, S. Kumbhakar and S. U. Sankar, arXiv:1606.03164 [hep-ph].[44] M. A. Ivanov, J. G. K¨orner and C. T. Tran, Phys. Rev. D , 094028 (2016) [arXiv:1607.02932 [hep-ph]].[45] D. Becirevic, S. Fajfer, N. Koˇsnik and O. Sumensari, Phys. Rev. D , 115021 (2016) [arXiv:1608.08501 [hep-ph]].[46] S. Sahoo, R. Mohanta and A. K. Giri, arXiv:1609.04367 [hep-ph].[47] D. A. Faroughy, A. Greljo and J. F. Kamenik, Phys. Lett. B , 126 (2017) [arXiv:1609.07138 [hep-ph]].[48] B. Bhattacharya, A. Datta, J. P. Guvin, D. London and R. Watanabe, JHEP , 015 (2017) [arXiv:1609.09078 [hep-ph]].[49] Z. Ligeti, M. Papucci and D. J. Robinson, JHEP , 083 (2017) [arXiv:1610.02045 [hep-ph]].[50] D. Bardhan, P. Byakti and D. Ghosh, JHEP , 125 (2017) [arXiv:1610.03038 [hep-ph]].[51] X. Q. Li, Y. D. Yang and X. Zhang, arXiv:1611.01635 [hep-ph].[52] R. Barbieri, C. W. Murphy and F. Senia, Eur. Phys. J. C , no. 1, 8 (2017) [arXiv:1611.04930 [hep-ph]].[53] R. Alonso, B. Grinstein and J. Martin Camalich, arXiv:1611.06676 [hep-ph].[54] D. Choudhury, A. Kundu, S. Nandi and S. K. Patra, arXiv:1612.03517 [hep-ph].[55] A. Celis, M. Jung, X. Q. Li and A. Pich, arXiv:1612.07757 [hep-ph].[56] M. A. Ivanov, J. G. K¨orner and C. T. Tran, arXiv:1701.02937 [hep-ph].[57] R. Dutta and A. Bhol, arXiv:1701.08598 [hep-ph].[58] M. Wei and Y. Chong-Xing, arXiv:1702.01255 [hep-ph].[59] A. Datta, S. Kamali, S. Meinel and A. Rashed, arXiv:1702.02243 [hep-ph].[60] D. Besson et al. [CLEO Collaboration], Phys. Rev. Lett. , 052002 (2007) [hep-ex/0607019].[61] P. del Amo Sanchez et al. [BaBar Collaboration], Phys. Rev. Lett. , 191801 (2010) [arXiv:1002.4358 [hep-ex]].[62] C. Patrignani, Chin. Phys. C , 100001 (2016).[63] R. Van Royen and V. F. Weisskopf, Nuovo Cim. A , 617 (1967) Erratum: [Nuovo Cim. A , 583 (1967)].[64] M. A. Sanchis-Lozano, J. Phys. Soc. Jap. , 044101 (2007) [hep-ph/0610046].[65] F. Domingo, U. Ellwanger and M. A. Sanchis-Lozano, Phys. Rev. Lett. , 111802 (2009) [arXiv:0907.0348 [hep-ph]].[66] D. E. Hazard and A. A. Petrov, Phys. Rev. D , 074023 (2016) [arXiv:1607.00815 [hep-ph]].[67] D. Becirevic, G. Duplancic, B. Klajn, B. Melic and F. Sanfilippo, Nucl. Phys. B , 306 (2014) [arXiv:1312.2858 [hep-ph]].[68] S. Schael et al. [ALEPH and DELPHI and L3 and OPAL and SLD Collaborations and LEP Electroweak Working Groupand SLD Electroweak Group and SLD Heavy Flavour Group], Phys. Rept. , 257 (2006) [hep-ex/0509008].[69] R. Alonso, E. E. Jenkins, A. V. Manohar and M. Trott, JHEP , 159 (2014) [arXiv:1312.2014 [hep-ph]].[70] G. M. Pruna and A. Signer, JHEP , 014 (2014) [arXiv:1408.3565 [hep-ph]].[71] Y. Nir et al , work in progress.[72] H. Ye, PoS DIS , 262 (2016) [arXiv:1607.01740 [hep-ex]].[73] M. Ablikim et al. [BES Collaboration], Phys. Rev. D , 112003 (2006).[74] V. V. Anashin et al. , JETP Lett. , 347 (2007).[75] B. Aubert et al. [BaBar Collaboration], Phys. Rev. D , 031101 (2002) [hep-ex/0109004].[76] M. Ablikim et al. [BES Collaboration], Phys. Lett. B , 74 (2008).[77] M. Beneke, Y. Kiyo, P. Marquard, A. Penin, J. Piclum, D. Seidel and M. Steinhauser, Phys. Rev. Lett. , 151801 (2014)[arXiv:1401.3005 [hep-ph]].[78] C. N. Yang, Phys. Rev. , 242 (1950).[79] L. D. Landau, Dokl. Akad. Nauk Ser. Fiz. , no. 2, 207 (1948).[80] D. Y. Bardin and G. Passarino, (International series of monographs on physics. 104)[81] A. Efrati, A. Falkowski and Y. Soreq, JHEP1507