Y and ψ leptonic decays as probes of solutions to the R(D(*)) puzzle

Experimental measurements of the ratios RD∗≡ΓB→D*τvΓB→D*ℓvℓ=e,μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ R\left({D}^{\left(\ast \right)}\right)\equiv \frac{\Gamma \left(B\to {D}^{\left(*\right)}\tau v\right)}{\Gamma \left(B\to {D}^{\left(*\right)}\ell v\right)}\left(\ell =e,\mu \right) $$\end{document} show a 3.9σ deviation from the Standard Model prediction. In the absence of light right-handed neutrinos, a new physics contribution to b → cτν decays necessarily modifies also bb¯→τ+τ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ b\overline{b}\to {\tau}^{+}{\tau}^{-} $$\end{document} and/or cc¯→τ+τ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ c\overline{c}\to {\tau}^{+}{\tau}^{-} $$\end{document} transitions. These contributions lead to violation of lepton flavor universality in, respectively, Y and ψ leptonic decays. We analyze the constraints resulting from measurements of the leptonic vector-meson decays on solutions to the R(D(*)) puzzle. Available data from BaBar and Belle can already disfavor some of the new physics explanations of this anomaly. Further discrimination can be made by measuring Y(1S, 2S, 3S) → ττ in the upcoming Belle II experiment.


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This R(D ( * ) ) puzzle received a lot of attention in the literature in recent years. Various works analyze the deviation in terms of effective field theory (EFT), construct explicit models of new physics (NP), and introduce new observables and kinematical distributions which can shed light on the flavor and Lorentz structure of the underlying theory .
Given the SM particle content and, in particular, assuming that the light neutrinos are purely left-handed, NP contributions to the b → cτ ν transition imply that NP contributions to bb → τ + τ − and/or cc → τ + τ − transitions are unavoidable. This point was discussed in ref. [47] which analyze the high P T distribution of the τ + τ − signature at the LHC, arising from sea bb and/or cc annihilation. In this work we study new observables which are governed by the cc → τ τ and bb → τ τ transitions: non-universality in leptonic decays of ψ and Υ quarkonia. Specifically, we study the ratios , (V = ψ, Υ; = e, µ), (1.4) making two assumptions: • The deviation of R(D ( * ) ) from the SM prediction is generated by gauge invariant effective operators of dimension six.
• R(D ( * ) ) is modified by new physics that affects only the B → D ( * ) τ ν decays (and not the B → D ( * ) ν decays). Correspondingly, R V τ / is modified because V → τ τ is affected (and not V → ).
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 modifying the leptonic decays of Υ and ψ is that all processes occur at the same energy scale. Therefore once we fix the Wilson coefficients at the low scale to give the measured best fit values of R(D ( * ) ), no RGE effects and mixing among other operators affect our predictions for R V τ / . Once an anomaly is found in these leptonic vector meson decays, a full UV model should be scrutinized, including a proper UV matching and RGE mixing, as well as its compatibility with other relevant observables.
Tests of LFU have been carried out for Υ(1S), Υ(2S) and Υ(3S) [60,61]. We collect these results in table 1. For ψ(2S) the leptonic branching fractions are measured to be [62]: BR(ψ(2S) → µ + µ − ) = (7.9 ± 0.9) × 10 −3 , BR(ψ(2S) → e + e − ) = (7.89 ± 0.17) × 10 −3 . (1.5) The corresponding ratio is presented in table 1. We do not consider ψ(1S) whose mass is below the τ + τ − threshold. We also do not consider ψ(3770) and Υ(4S) which have negligible branching fractions into leptons because their masses are above the DD and BB threshold, respectively. Within the SM, the QED partial decay widths of a vector quarkonium into charged lepton pairs obey [63] R V 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 dominant corrections to this factor are at the level of 0.006x 2 τ , and arise from QED vertex corrections. A full discussion on the SM decay rate is presented in appendix A. The relevant masses are known with a great accuracy [62]: (1.7) The non-universality predicted in the SM agrees very well with the experimental results, as is evident from table 1. LFU in Upsilon decays was discussed in the literature in the context of light pseudoscalar (see refs. [64,65] and references within). In this scenario, the radiative Υ → γη b decay is followed by a mixing between the η b state and a CP-odd scalar A, for which the leptonic couplings are non-universal. Lepton flavor changing decays of heavy vector-mesons were discussed in refs. [48,66]. While such decays are not directly relevant to our study, the formalism is similar.
The plan of this paper is as follows. The formalism for V → decays is introduced in section 2. The effective field theory that is relevant to R(D ( * ) ) and to R V τ / is introduced in section 3. In section 4 we analyze a series of simplified models, where we add to the SM a single new boson. For each model, we find the numerical range for the Wilson coefficients that explain R(D ( * ) ), and obtain the resulting predictions for R V τ / . In section 5 we compare these predictions to present measurements and discuss the prospects for improving the experimental accuracy in the future. Our conclusions are summarized in section 6.

V → decay rate
The most general V → + − decay amplitude can be written as

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where A q V , B q V , C q V , D q V are dimensionless parameters which depend on the Wilson coefficients of the operators controlling the V → + − 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 2 + p 1 ) µ and (p 2 + p 1 ) µ γ 5 do not contribute to the rate. The decay width and R V τ / are given by For the SM calculations we include QED one-loop correction which is further discussed in appendix A.
In order to calculate the V → + − decay rate in a specific UV or EFT model, one needs to find the relation between A V , B V , C V and D V and the Lagrangian parameters. Including only terms which are relevant for a leptonic meson decay, we consider the following effective Lagrangian: Tē L σ µν e Rq σ µν q + C q SLē R e LqR q L + C q SRē R e LqL q R + h.c. .

(2.3)
We find: We introduced the form factors, f V and f T V , that are defined via the standard parametrization: with σ µν = i[γ µ , γ ν ]/2, and 0|qq |V (p) = 0|qγ 5 q |V (p) = 0|qγ µ γ 5 q |V (p) = 0. The relevant ratio f T V /f V should be determined from measurements or lattice calculations. In the heavy quark limit f V = f T V . This is an excellent approximation for the Υ meson. For the ψ(2S) state, however, relativistic effects correct this relation by a few percent [67]. We checked that this is a sub-leading effect, therefore in the following we neglect this correction.

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3 The effective field theory Assuming that the NP contributions are related to heavy degrees of freedom, their effects can be presented by non-renormalizable terms in the Lagrangian. There are eight combinations of two lepton and two quark fields that can be contracted into SU(3) C ×SU(2) L ×U(1) Y and Lorentz invariant operators: LLQQ,ēLūQ,ēLQd,LLūu,LLdd,ēeūu,ēedd,ēeQQ, (3.1) 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 and down-singlet quark fields. Only the first three combinations can introduce the charged-current interaction needed for a b → c transition. Specifying the Lorentz and SU(2) L contractions, we write the complete set of (linearly dependent) gauge-invariant operators in table 2. 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 given that the SM amplitude is tree-level and only mildly CKMsuppressed, it is likely that the NP contribution which accounts for the deviation is also tree-level. It is instructive, therefore, to consider simplified UV models, each with a single new (scalar or vector) boson. Following ref. [27], we specify the NP field content which generates the operators in table 2. Our convention is such that ψ c = −iγ 2 ψ * .
As a linearly independent set, we choose the following four-fermion operators (written in the quark interaction basis): ,

(3.2)
A comment is in order regarding other dimension six operators which may be generated by integrating out heavy particles. Two sets of operators relevant to the study of R V τ / are operators of the form O Hl L,R = i H † ← → D µ H l L,R γ µ l L,R and the dipole operators The O Hl operators generate non-universal Z-mediated quarkonia decays, and further modify the Z → τ τ decay width. These two effects are related: the leading effect in R V τ /

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arising from this operator is given by 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 −5 . For more details see appendix B.1. These operators can be formed at the UV matching scale, in which case they are typically controlled by additional free parameters. They can be further generated by the mixing with four fermion operators, in which case fine-tuning cancelation may be needed to ensure that 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 contribution arising 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 is always below the per-mil level and can be neglected. The dipole operators further modify the electric and magnetic dipole moments of the τ lepton. For completeness we write the leading contributions of the dipole operators to R V τ / and the relation to the taonic dipole moments in appendix 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.

Numerical results
As explained above, it is useful to explore simplified UV models which generate the required four-fermion operators at tree-level. In the following subsections we examine various sets of effective operators formed by the integration out of heavy scalar and vector bosons which have the right quantum numbers to modify the b → cτ ν transition. For each case, we present the relevant UV couplings and obtain the resulting CC and NC operators (neglecting, as explained above, RGE effects). We find within 95% C.L. the numerical values for the Wilson coefficients which minimize the observed anomaly in R(D ( * ) ), and the corresponding predictions for R V τ / for the Υ and ψ(2S) states. Some of the simplified models we study cannot ease the tension between the theory and the R(D ( * ) ) measurements. Clearly, in these models the best fit point of the new couplings is zero, giving back the SM. Nevertheless, for completeness we also consider these models and present our results for them. We present our main results in table 3, where only models which ease the tension below χ 2 = 9 (corresponding to 3σ with one degree of freedom) are included. Note that in 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 of the effective operators. Furthermore, in the 2HDM case φ ∼ (1, 2) +1/2 neither of these decays is modified because of the vector structure of the qq mesons.
One clear advantage of analyzing such simplified models is that each scenario predicts distinct relations between the various EFT operators. These relations can, however, be modified due to SU(2) L breaking effects. Let us explain why we ignore these effects in Projected uncertainty (L Υ(3S) = 1/ab in Belle II) ±0.004 - Table 3.
The simplified (single boson) models and the predicted range for R V τ / for V = Υ(1S), ψ(2S). The achievable and projected uncertainties are our estimations, see the text for more details. our analysis. Electroweak breaking effects split the spectrum of the charged and neutral NP fields, 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 V τ / . The unavoidable loop-induced splitting is smaller than a GeV, and therefore negligible. Treelevel splitting is typically of order ∆M/M v 2 /M 2 and might change our final results by a few percent. This splitting is, however, generated by a free parameter in the scalar potential which we take to be zero (up to small loop-induced effects). Once a specific UV model is considered in full detail, this assumption can be 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 the light left-handed Q 1,2 quarks transform as a doublet. This choice is taken to avoid large production rate at the LHC and dangerous FCNC transitions in the first and second generation. 1 To determine uniquely the flavor structure of the NP couplings one should further specify the mass basis alignment of the UV operators. In what follows, we always consider alignment to the down mass basis, with operators this choice is essential to ensure that b → c transition is modified, once U(2) Q is preserved. For theL 3 e 3Q3 u 2 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 1) and the same Wilson coefficient as 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 scenario), and take the quartic |X NP | 2 |H| 2 couplings to be negligible. To keep our models in the perturbative regime, we take all parameters to be smaller than 4π at the TeV scale. We stress again that, once an anomaly is found in LFU of Υ or ψ decays, these JHEP06(2017)019 assumptions should be reconsidered and a complete UV theory should be studied in full detail. Since we do not study the 2HDM case, condensation of the NP fields is absent due to Lorentz and/or SU(3) C symmetries. In the following, 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 2 function. We use the experimental results quoted in eq. (1.2). The theoretical uncertainties on the form factors which affect R(D ( * ) ) in the SM are small compared to the experimental errors, as evident from the accurate SM predictions. (See also figures 1-2 in ref. [50].) As for the other form factor (F T in the notations of ref. [50]), we have explicitly checked that varying it within 20% error does not alter our results for R V τ / . When considering a model, we are agnostic about how plausible it is from a model building point of view, and only compare it to the SM point. The latter gives χ 2 20.
As concerns kinematical observables, the q 2 distribution of Γ [B → D * τ ν] is hardly modified in all of the scenarios. We comment below on the q 2 distributions of Γ [B → Dτ ν], which is, in general, consistent within the current uncertainties. Interference corrections, as analysed in ref. [49] are expected to be small.
We introduce a vector-boson, color-singlet, SU(2) L -triplet W µ ∼ (1, 3) 0 with the following interaction Lagrangian: Integrating out W µ , we obtain the following EFT Lagrangian: The relevant CC interactions are given by The best-fit-point (BFP) and 95% C.L. intervals are given by where g 12 ≡ g 1 g 2 . The q 2 distribution is identical to the SM one, since the new CC operator has the same Lorentz structure as in the SM. The relevant NC interactions are given by

X µ ∼ (3, 3) +2/3
We introduce a vector-boson, color-triplet, SU(2) L -triplet X µ ∼ (3, 3) +2/3 with the following interaction Lagrangian: Integrating out X µ , we obtain the following EFT Lagrangian: (4.14) The relevant CC interactions are given by The q 2 distribution is identical to the SM one, since the new CC operator has the same Lorentz structure as in the SM. The relevant NC interactions are given by

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They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities We introduce a scalar-boson, color-triplet, SU(2) L -singlet S ∼ (3, 1) −1/3 with the following interaction Lagrangian: We impose a global 3B − L symmetry, which prevent an additional Yukawa couplings of the form Sdu and SQQ. Integrating out S, we obtain the following EFT Lagrangian: The relevant CC interactions are given by

T ∼ (3, 3) −1/3
We introduce a scalar-boson, color-triplet, SU(2) L -triplet S ∼ (3, 3) −1/3 with the following interaction Lagrangian: We impose global 3B − L symmetry to forbid T QQ terms. Integrating out T , we obtain the following EFT Lagrangian: The relevant CC interactions are given by The q 2 distribution is identical to the SM one, since the new CC operator has the same Lorentz structure as in the SM. The relevant NC interactions are given by

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They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities We introduce a scalar-boson, color-triplet, SU(2) L -doublet D ∼ (3, 2) +7/6 with the following interaction Lagrangian: Integrating out D, we obtain the following EFT Lagrangian: The relevant CC interactions are given by The q 2 distribution of Γ [B → Dτ ν] is modified compared to the SM one. Yet, as is evident from figure 3d, this change is not very significant given the current uncertainties. The relevant NC interactions are given by

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They induce both Υ → τ τ and ψ → τ τ . Given the 95% C.L. intervals quoted above, we find the following non-universalities We introduce a vector-boson, color-triplet, SU(2) L -doublet V µ ∼ (3, 2) −5/6 with the following interaction Lagrangian: Integrating out V µ , we obtain the following EFT Lagrangian: The relevant CC interactions are given by The q 2 distribution of Γ [B → Dτ ν] is modified compared to the SM one. Yet, as is evident from figure 3e, this change is not very significant given the current uncertainties. The relevant NC interactions are given by

Discussion and future prospects
The allowed ranges for R V τ / in simplified models that account for the deviations of R(D ( * ) ) are presented in table 3. As concerns R ψ(2S) τ / , it is modified by no more than three permil, while the present experimental accuracy is of order thirteen percent. Thus, the maximal modification is about a factor of fifty below current sensitivity. We conclude that R ψ(2S) τ / 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(2S) τ / [71].)

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As concerns R Υ(1S) τ / , in all simplified models, except the T model, it is smaller than the SM. The lowest value is 4% below the SM value. The current experimental accuracy is 2.5%, comparable to the predicted deviations. Furthermore, the central value is higher (by about 0.5σ) then the SM prediction. We conclude that R Υ(1S) τ / is starting to 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 1S, 2S and 3S states, and by BaBar [61] for the 1S state. These measurements read  (2) times larger than CLEO's sample. Analyzing these existing data sets will allow to reduce each of the statistical uncertainties to roughly 1 − 2 percent. Additional improvement can be achieved by using the Υ cascade chains, which will render this error negligible. The systematic error in BaBar's analysis is currently controlled by the uncertainties on the different τ and µ total efficiencies and event shapes. The larger statistics can realistically lead to a reduction of this uncertainty by a factor of 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 is governed by the limited statistics, we estimate that reaching a σ sys = 0.4% for R Υ(1S), Belle II τ /µ would require integrated luminosity of L ∼ 1/ab at the Υ(3S) energy. Thus, our study gives additional motivation to the proposal of ref. [72] to study the Υ(3S) 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(1S), Υ(2S), and Υ(3S) into one test of universality. Eq. (1.6) as a function of m Υ(nS) , where x τ,1S ≡ m τ /m Υ(1S) = 0.187823, can be used to test the SM, and a violation of it would constitute a signal for NP. The correlation of the systematic uncertainties between the different Υ states is probably large (if not maximal), not allowing to further reduce this part of the error with a combined analysis.

Summary and conclusions
There is a 3.9σ evidence that the ratio R(D ( * ) ) ≡ Γ(B → D ( * ) τ ν)/Γ(B → D ( * ) ν), where = µ, e, is considerably enhanced compared to the Standard Model value. To explain a large enhancement of a SM tree-level process, the required new physics is likely to include new bosons which mediate the B → D ( * ) τ ν decay at tree level. There are seven such candidates, and none is a SM singlet. Thus, it is highly likely that their mass is well above m B , which allows 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 quark fields. 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 τν τ decay [51,59], and bb /cc → τ + τ − scattering [47], have been previously studied 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 V τ / ≡ Γ(V → τ τ )/Γ(V → ). We find that, once a U(2) Q symmetry is assumed, current measurements of ψ decays do not have the accuracy required to probe the models in a significant way. On the other hand, for Υ(1S) decays, the current experimental accuracy and the predicted deviations are of the same order of magnitude. A modest improvement in the experimental accuracy is capable of favoring some models and disfavoring others. If, for example, it is established that R Υ(1S) τ / is larger than the SM value, than all the simplified models will be disfavored.
(The only model which enhances R Υ(1S) τ / is the T model, which gives, however, a very small effect.) If experiments reach high accuracy in the leptonic vector meson decays and observe no signal, then the models that allow negligible deviations will be favored. Another possibility is that the light neutrino has a significant ν R component, in which case R(D ( * ) ) could be explained by operators which do not affect V → τ τ .
We discussed the prospects of such improvement in the experimental accuracy. Current data samples should already allow to reduce the statistical error considerably and reach an accuracy of about 1.5%. If Belle II operates below the Υ(4S) resonance, it can contribute significantly, via the measurements of R Υ(nS) τ / , to our understanding of the R(D ( * ) ) puzzle.

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Acknowledgments We thank Avital Dery, Admir Greljo, Avner Soffer, Alexander Penin and Nadav Priel for useful discussions. We are thankful to Diptimoy Ghosh for his great help with the numerical evaluation of R(D ( * ) ). This research is supported by the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel (grant number 2014230), and by the I-CORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). YG is supported in part by the NSF grant PHY-1316222. YN is the Amos de-Shalit chair of theoretical physics. YN is supported by grants from the Israel Science Foundation (grant number 394/16) and from the Minerva Foundation.

A The leptonic width in the SM
The leading SM decay rate is given by the Van Royen-Weisskopf formula [63], The dominant corrections arise from tree-level Z exchange and quantum QED and QCD corrections, The tree-level Z mediated correction is This is an O(10 −4 ) correction, well below the experimental sensitivity. The resulting change in the non-universality relation (eq. (1.6)) is, however, only O(10 −5 ), as the interference with the photon projects out the vectorial part of the Z coupling. QCD corrections have been studied extensively in the literature (see, for example, [77] and references therein). A crucial point is that these short-and long-distance QCD effects do not depend on m , and cancel in the ratio R V τ / . These can be absorbed in the definition of the vector meson form factor, f V .
The leading QED corrections which are not included in the definition of f V arise from corrections to the photon two-point function, the photon-lepton vertex corrections, and the irreducible real photon emission (FSR). Corrections due to two-photon exchange are absent: the Landau-Yang theorem [78,79] implies M(V → γγ) = 0 for massive vector bosons. Therefore exploiting the optical theorem and dispersion relation for real analytic functions, as well as QED and QCD CP invariance, forbids M(V → ) via two photon exchange. The corrections to Π γγ are taken into account by using the running couplings, namely evaluating α(µ) at µ = m V . Once again these are universal corrections that do not affect R V τ / . As for the leptonic-vertex corrections, these exhibit IR singularities which are regulated by the real emission of soft photons. Clearly, the latter depends on the experimental resolution and should be determined (and unfolded) by the experiments using state-of-the-art JHEP06(2017)019 detector simulation in MC analysis. Following the LEP report for Z pole observables [68], we quote here the inclusive Γ[V → + − + γ] at one-loop order in QED, which does not depend on the experimental setup. It reads [80] δ which affect R V τ / by modifying the Zτ τ vertex. Their contributions to the ψ and Υ leptonic decays are given by where, as before, = m 2 V /m 2 Z and g q v = T q 3 − 2Q q s 2 W , and for the Zτ τ vertex corrections we follow the definitions of [81]. The consistency of LEP data with the SM prediction requires δg Zτ L,R ≤ 10 −3 at 2σ, which, in turn, can modify R V τ / by, at most, 10 −5 -10 −6 . This effect is negligible given the current and future experimental sensitivity.

B.2 Dipole operator
Here we consider the dimension six dipole operator L D = √ 4παC DL σ µν e R HF µν + h.c..

(B.3)
Its contribution to the ψ and Υ leptonic decays is given by These bounds are, however, not strong enough to suppress the dipole contribution to R V τ / .
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