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Prepared for submission to JHEP τ dipole moments via radiative leptonic τ decays S. Eidelmana,b D. Epifanova,b,c M. Faeld L. Mercollie M. Passeraf 6 1 aBudker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russian Federation 0 bNovosibirsk State University, Novosibirsk 630090, Russian Federation 2 cThe University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113-0033, Japan n a dAlbert Einstein Center for Fundamental Physics, J Institute for Theoretical Physics, University of Bern, CH-3012 Bern, Switzerland 9 eFederal Office of Public Health FOPH, CH-3003 Bern, Switzerland 2 fIstituto Nazionale Fisica Nucleare, Sezione di Padova, I-35131 Padova, Italy ] h E-mail: [email protected], [email protected], p [email protected], [email protected], [email protected] - p e Abstract: We propose a new method to probe the magnetic and electric dipole moments h [ of the τ lepton using precise measurements of the differential rates of radiative leptonic 1 τ decays at high-luminosity B factories. Possible deviations of these moments from the v Standard Model values are analyzed in an effective Lagrangian approach, thus providing 7 8 model-independent results. Analytic expressions for the relevant non-standard contribu- 9 tions to the differential decay rates are presented. Earlier proposals to probe the τ dipole 7 0 moments are examined. A detailed feasibility study of our method is performed in the . 1 conditions of the Belle and Belle II experiments at the KEKB and Super-KEKB colliders, 0 respectively. This study shows that our approach, applied to the planned full set of Belle II 6 1 data for radiative leptonic τ decays, has the potential to improve the present experimental : v bound on the τ anomalous magnetic moment. On the contrary, its foreseen sensitivity is i X not expected to lower the current experimental limit on the τ electric dipole moment. r a Contents 1 Introduction 1 2 The τ lepton electromagnetic form factors 2 3 Status of the τ lepton g-2 and EDM 4 4 Radiative τ leptonic decays: theoretical framework 8 5 Feasibility study at Belle and Belle II 9 6 Conclusions 14 1 Introduction The very short lifetime of the τ lepton (2.9 10−13s) makes it very difficult to measure × its electric and magnetic dipole moments. While the Standard Model (SM) prediction of the τ anomalous magnetic moment a = (g 2) /2 is known with a tiny uncertainty of τ τ − 5 10−8 [1], thisshortlifetimehassofarpreventedthedeterminationofa measuringtheτ τ × spinprecessioninamagneticfield, likeintheelectronandmuong 2experiments. Instead, − experiments focused on various high-precision measurements of τ pair production in high- energy processes, comparing the measured cross sections with the SM predictions. As these processes involve off-shell photons or taus in the ττ¯γ vertices, the measured quantity is not directly a . The present resolution on a obtained by these experiments is only of τ τ O(10−2) [2], more than an order of magnitude larger than its leading SM contribution α 0.001 [3]. 2π (cid:39) The electron and muon g 2, a and a , have been measured with the remarkable e µ − precision of 0.24 ppb [4] and 540 ppb [5], respectively. While a perfectly agrees with the e SMprediction[6],a ,whichismuchmoresensitivethana tostrongandweakinteractions, µ e shows a long-standing puzzling discrepancy of about 3–4σ and provides a powerful test of physics beyond the SM [7–11]. A precise measurement of a would offer a new excellent τ opportunity to unveil new physics effects. Indeed, in a large class of theories beyond the SM, new contributions to the anomalous magnetic moment of a lepton l of mass m scale l with m2. Therefore, given the large factor m2/m2 283, the g 2 of the τ is much l τ µ ∼ − more sensitive than the muon one to electroweak and new physics loop effects that give contributions proportional to m2. In these scenarios, the present discrepancy in the muon l g 2 suggests a new-physics effect in a of (10−6); several theories exist where this naive τ − O scaling is violated and much larger effects are expected [12]. The SM prediction of a lepton electric dipole moment (EDM) is extremely small and far below present experimental capabilities. Therefore, a measurement of a non-zero value – 1 – would be direct evidence of new physics. Moreover, models for physics beyond the SM generally induce large contributions to lepton EDMs so that, although there has been no experimental evidence for an EDM so far, we hope that this kind of experiments will soon shed new light on the nature of CP violation. In this article we study the possibility to determine the electromagnetic dipole moments of the τ via the radiative leptonic decays τ lγνν¯, with l = µ,e, comparing the → theoretical prediction for the differential decay rates with precise data from high-luminosity B factories [13, 14]. In particular, we present the results of a feasibility study performed in the conditions of the Belle [15–18] and Belle II [19] experiments at the KEKB [20] and SuperKEKB [21, 22] colliders, respectively. Following the strategy of the authors of refs. [23, 24], deviations of the τ dipole moments from the SM values are analyzed in an effective Lagrangian approach, thus avoiding the interpretation of off-shell form factors. We also examine the feasibility of earlier proposals; in particular, one based on the study of the Pauli form factor of the τ via τ+τ− production in e+e− collisions at the Υ resonances [25, 26], and another relying on the analysis of the radiation zero which occurs in radiative leptonic τ decays [27]. In section 2 we establish our conventions for the τ electromagnetic form factors and introduce an effective Lagrangian to study the τ dipole moments. In section 3 we review the present theoretical and experimental status on the τ g 2 and EDM. The theoretical − frameworktoanalyzeradiativeleptonicτ decaysispresentedinsection4, whereweprovide explicit analytic expressions for the relevant non-standard contributions to the differential decay rates. In section 5 we outline our method to determine the τ dipole moments and report the results of our feasibility study for the sensitivities that may be reached at the Belle and upcoming Belle II experiments. Conclusions are drawn in sec. 6. 2 The τ lepton electromagnetic form factors Let us consider the structure of the ff¯γ coupling. The most general vertex function de- scribing the interaction between a photon and the initial and final states of an arbitrary on-shell spin 1/2 fermion f, with four-momenta p and p(cid:48), respectively, can be written in the form (cid:26) σµνq (cid:104) (cid:105) (cid:16) 2qµm (cid:17) (cid:27) Γµ(q2) = ieQ γµF (q2)+ ν iF (q2)+F (q2)γ + γµ f γ F (q2) , − f 1 2m 2 3 5 − q2 5 4 f (2.1) where e > 0 is the positron charge, m is the mass of the fermion, σ = i/2[γ ,γ ], and f µν µ ν q = p(cid:48) p is the ingoing four-momentum of the off-shell photon. Equation (2.1), when − sandwiched in u(p)Γ (q2)u(p(cid:48)), is the most general expression that satisfies Lorentz and µ QED gauge invariance. The functions F (q2) and F (q2) are called the Dirac and Pauli 1 2 form factors, respectively. In general, they are not physical quantities (for example, they can contain infrared divergences [28, 29]), but in the limit q2 0 they are measurable and → related to the static quantities 2m f F (0) = 1, F (0) = a , F (0) = d , (2.2) 1 2 f 3 f eQ f – 2 – where eQ is the charge of the fermion, a its anomalous magnetic moment, and d its f f f EDM. The electric dipole contribution F (q2) violates the discrete symmetries P (parity) 3 and T (time reversal) [30–32], and therefore CP, because of the CPT theorem. F (q2) is 4 called the anapole form factor and violates P. In the limit q2 0, the dipole interactions → in eq. (2.1) can be cast in the form C σ qνP +C σ qνP , (2.3) L µν L R µν R where P = (1 γ5)/2. Hermiticity of this expression requires that C = C∗ = c , with L,R ∓ R L f eQ c = a f id , a ,d R. (2.4) f f f f f 2m − ∈ f Deviations of the τ dipole moments from the SM values can be analyzed in the framework of an effective field theory description where the SM Lagrangian is extended by a set of gauge-invariant higher-dimensional operators, built with the SM fields, suppressed by powers of the scale of new physics Λ [33]. We will consider only dimension-six operators, which are the lowest dimensional ones relevant for our analysis. Out of the complete set of 59independentdimension-sixoperatorsinref.[34],onlytwoofthemcandirectlycontribute to the τ lepton g 2 and EDM at tree level (i.e., not through loop effects): − Q33 = (cid:0)¯l σµντ (cid:1)σIϕWI , (2.5) lW τ R µν Q33 = (cid:0)¯l σµντ (cid:1)ϕB , (2.6) lB τ R µν whereϕandl = (ν ,τ )aretheHiggsandtheleft-handedSU(2)doublets,σI arethePauli τ τ L matrices, and WI and B are the gauge field strength tensors. The leading non-standard µν µν effects will therefore arise from the effective Lagrangian = 1 (cid:2)C33 Q33 +C33Q33 +h.c.(cid:3). (2.7) Leff Λ2 lW lW lB lB After the electroweak symmetry breaking, these two operators mix and give additional, beyond the SM, contributions to the τ anomalous magnetic moment and EDM: a˜ = 2mτ √2v Re(cid:2)cosθ C33 sinθ C33 (cid:3), (2.8) τ e Λ2 W lB − W lW d˜ = √2v Im(cid:2)cosθ C33 sinθ C33 (cid:3), (2.9) τ Λ2 W lB − W lW where v = 246 GeV and sinθ is the weak mixing angle. Moreover, through the coupling W to the Z boson, the effective Lagrangian (2.7) also gives non-standard contributions to the neutral weak dipole moments: a˜W = 2mτ √2v Re(cid:2)sinθ C33 +cosθ C33 (cid:3), (2.10) τ e Λ2 W lB W lW d˜W = √2v Im(cid:2)sinθ C33 +cosθ C33 (cid:3). (2.11) τ − Λ2 W lB W lW – 3 – The operator Q33 in (2.5) also generates an additional chirality-flipping coupling be- lW tween the τ and the W boson, and a four-point vertex that couples the τ and the W to the photonortheZ (otherfour-andfive-pointvertices, involvingthephysicalHiggsboson, will not be considered since they do not contribute to the τ dipole moments nor to the decays τ lνν¯(γ)). These additional τ-W couplings are proportional to the complex parame- → ter C33 and, therefore, to the real combinations ˜b = (2m /e)(√2v/Λ2)sinθ ReC33 = lW τ − τ W lW sin2θ a˜ sinθ cosθ a˜W andc˜ = (√2v/Λ2)sinθ ImC33 = sin2θ d˜ +sinθ cosθ d˜W. W τ− W W τ τ − W lW W τ W W τ The dynamics of radiative leptonic τ decays is modified both by non-standard terms proportional to a˜ and d˜ (see section 4), as well as by contributions generated by these new τ τ couplings between the τ and the W boson, which are proportional to ˜b and c˜ . However, τ τ as these new τ-W couplings also affect the ordinary (inclusive) leptonic τ decays τ lνν¯, → we will assume that future bounds on˜b and c˜ will be more stringent than those on a˜ and τ τ τ d˜ obtained via radiative leptonic decays. The present limits on ˜b and c˜ are of (10−3); τ τ τ O should future bounds on a˜ and d˜ reach the sensitivity of ˜b and c˜ , then a combined τ τ τ τ analysis of ordinary and radiative leptonic τ decays for τ dipole moments and Bouchiat- Michel-Kinoshita-Sirlin parameters [35–38] will become necessary. For the time being, we will neglect these new τ-W couplings. 3 Status of the τ lepton g-2 and EDM In this section we discuss the present status of the SM prediction and experimental determination of the anomalous magnetic moment and EDM of the τ lepton. The SM prediction for a is given by the sum of QED, electroweak (EW) and hadronic τ terms. The QED contribution has been computed up to three loops: aQED = 117324(2) τ × 10−8 [39–42], where the uncertainty π2ln2(m /m )(α/π)4 2 10−8 has been assigned τ e ∼ × for uncalculated four-loop contributions. The errors due to the uncertainties of the (α2) O and (α3) terms, as well as that induced by the uncertainty of α, are negligible. The O sum of the one- and two-loop EW contributions is aEW = 47.4(5) 10−8 [1, 43, 44]. The τ × uncertainty encompasses the estimated errors induced by hadronic loop effects, neglected two-loop bosonic terms and the missing three-loop contribution. It also includes the tiny errors due to the uncertainties in m and m . top τ Similarly to the case of the muon g 2, the leading-order hadronic contribution to a is τ − obtainedviaadispersionintegralofthetotalhadroniccrosssectionofthee+e− annihilation (theroleoflowenergiesisveryimportant,althoughnotasmuchasfora ). Theresultofthe µ latest evaluation, using experimental data below 12 GeV, is aHLO = 337.5(3.7) 10−8 [1]. τ × The hadronic higher-order (α3) contribution aHHO can be divided into two parts: aHHO = τ τ aHHO(vp)+aHHO(lbl).Thefirstone, the (α3)contributionofdiagramscontaininghadronic τ τ O self-energy insertions in the photon propagators, is aHHO(vp) = 7.6(2) 10−8 [45]. Note τ × that naively rescaling the corresponding muon g 2 result by a factor m2/m2 leads to the τ µ − incorrect estimate aHHO(vp) 28 10−8 (even the sign is wrong!). Estimates of the light- τ ∼ − × by-light contribution aHHO(lbl) obtained rescaling the corresponding one for the muon g 2 τ − by a factor m2/m2 fall short of what is needed – this scaling is not justified. The parton- τ µ level estimate of [1] is aHHO(lbl) = 5(3) 10−8, a value much lower than those obtained by τ × – 4 – naive rescaling. Adding up the above contributions one obtains the SM prediction [1] aSM = aQED+aEW +aHLO+aHHO = 117721(5) 10−8. (3.1) τ τ τ τ τ × Errors were added in quadrature. The EDM interaction violates the discrete CP symmetry. In the SM with massless neutrinos, the only source of CP violation is the CKM-phase (and a possible θ-term in the QCDsector). Inrefs.[46,47]itwasshownthatallCP-violatingamplitudesareproportional to the Jarlskog invariant J, defined as Im(cid:2)V V V∗V∗(cid:3) = J(cid:88)ε ε , (3.2) ij kl il kj ikm jln m,n whereV aretheCKMmatrixelements. Therefore,theleptonEDMmustarisefromvirtual ij quarkslinkedtotheleptonthroughtheW boson,thusbeingsensitivetotheimaginarypart oftheCKMmatrixelements. Theleadingcontributionisnaivelyexpectedatthethree-loop level, since two-loop diagrams are proportional to V 2. The problem was first analyzed ij | | in some detail in [48], but it was subsequently shown that also three-loop diagrams yield a zero EDM contribution in the absence of gluonic corrections to the quark lines [49]. For this reason, lepton EDMs are predicted to be extremely small in the SM, of the (10−38 O − 10−35)e cm [32], far below the present (10−17)e cm experimental reach on the τ EDM. · O · Evenfortheelectron, thefantasticexperimentalupperbounddEXP < 0.87 10−28 e cm[50] e × · is still much larger than the SM prediction dSM (10−38)e cm and it is hard to imagine e ∼ O · improvements in the sensitivity by ten orders of magnitude! However, new EDM effects could arise at the one- or two-loop level from new physics that violates P and T, and be much larger than the tiny SM value, even if they arise from high mass scales. The present experimental resolution on the τ anomalous magnetic moment is only of (10−2) [2], more than an order of magnitude larger than its SM prediction in Eq. (3.1). O In fact, while the SM value of a is known with a tiny uncertainty of 5 10−8, the τ short τ × lifetime has so far prevented the determination of a by measuring the τ spin precession in τ a magnetic field, like in the electron and muon g 2 experiments. The present PDG limit − on the τ g 2 was derived in 2004 by the DELPHI collaboration from e+e− e+e−τ+τ− − → total cross section measurements at √s between 183 and 208 GeV at LEP2 (the study of a τ via this channel was proposed in [51]). The measured values of the cross-sections were used to extract limits on the τ g 2 by comparing them to the SM values, assuming that possible − deviations were due to non-standard contributions a˜ . The obtained limit at 95% CL is [2] τ 0.052 < a˜ < 0.013, (3.3) τ − which can be also expressed in the form of central value and error as [2] a˜ = 0.018(17). (3.4) τ − The present PDG limit on the EDM of the τ lepton at 95% CL is 2.2 < Re(d ) < 4.5 (10−17 e cm), τ − · (3.5) 2.5 < Im(d ) < 0.8 (10−17 e cm); τ − · – 5 – it was obtained by the Belle collaboration [52] following the analysis of ref. [24] for the impact of an effective operator for the τ EDM in the process e+e− τ+τ−. → The reanalysis of ref. [23] of various LEP and SLD measurements – mainly of the e+e− τ+τ− cross sections – allowed the authors to set the indirect 2σ confidence interval → 0.007 < a˜ < 0.005, (3.6) τ − a bound stronger than that in Eq. (3.3). This analysis assumed d˜ = 0. We updated τ this analysis using more recent data [53, 54] obtaining the almost identical 2σ confidence interval 0.007 < a˜ < 0.004. τ − AttheLHC,boundsontheτ dipolemomentsareexpectedtobesetinτ pairproduction via Drell-Yan [55, 56] or double photon scattering processes [57]. The best limits achievable in pp τ+τ− + X are estimated to be comparable to present existing ones if the total → cross section for τ pair production is assumed to be measured at the 14% level [55]. Earlier proposals to set bounds on the τ dipole moments can be found in [58–61]. Yetanothermethodtodeterminea˜ wouldusethechannelingofpolarizedτ leptonsina τ bent crystal similarly to the suggestion for the measurement of magnetic moments of short- living baryons [62]. This approach has been successfully tested by the E761 collaboration at Fermilab, which measured the magnetic moment of the Σ+ hyperon [63]. The challenge of this method is to produce a polarized beam of τ leptons. One could use the decay B+ τ+ν , which would produce polarized τ leptons [64]; however this particular decay τ → of the B has a very tiny branching ratio of (10−4). In 1991, when this proposal was O published, the idea seemed completely unlikely. Nonetheless, in the era of B factories, when the decay B+ τ+ν is already observed [54], the realization of this idea in a τ → dedicated experiment is definitively not excluded. The Belle II experiment at the upcoming high-luminosity B factory SuperKEKB will offer new opportunities to improve the determination of the τ electromagnetic properties. The authors of ref. [25, 26] proposed to determine the Pauli form factor F (q2) of the τ via 2 τ+τ− production in e+e− collisions at the Υ resonances (Υ(1S), Υ(2S) and Υ(3S)) with a sensitivity of (10−5) or even better (of course, the center-of-mass energy at super B facto- O riesis√s M 10GeV,sothattheformfactorF (q2)isnottheanomalousmagnetic Υ(4S) 2 ∼ ≈ moment). When attempting to extract the value of F (q2) from scattering experiments (as 2 opposed to using a background magnetic field) one encounters additional complications due to the contributions of various other Feynman diagrams not related to the magnetic form factor. In particular, in the e+e− τ+τ− case, contributions to the cross section arise → not only from the usual s-channel one-loop vertex corrections, but also from box diagrams, which should be somehow subtracted out. The strategy proposed in [25, 26] to eliminate their contamination is to measure the observables on top of the Υ resonances, where the non-resonant box diagrams should be numerically negligible. However, because of the natural irreducible beam energy spread associated to any e+e− synchrotron, it is very difficult to resolve the narrow peaks of the Υ(1S,2S,3S) in the τ+τ− decay channel (the Υ(4S) decays almost entirely in BB¯). Indeed, the total visible cross section of these resonances is not a perfect Breit-Wigner, but the convolution of the – 6 – σmax Υ M [GeV] Γ [keV] σ [nb] ρ vis Υ Υ peak σ non−res Υ(1S) 9.46 54 101 6.2 10−3 69% × Υ(2S) 10.02 32 56 3.7 10−3 22% × Υ(3S) 10.36 20 68 2.3 10−3 17% × Υ(4S) 10.58 20 103 – – – × Table 1: Estimated visible cross section at Belle II for e+e− Υ τ+τ−. The machine → → parameters are from ref. [22]. theoretical Breit-Wigner cross section with a Gaussian spread, (cid:90) σ (s) (cid:20) (√s M )2(cid:21) ee→Υ→ττ Υ σ = exp − d√s, (3.7) vis √2πσ − 2σ2 W W where σ is the irreducible beam energy spread of the accelerator at √s = M (σ = W Υ W 5.45 MeV at the upcoming SuperKEKB collider), σ (s) is the total cross section in ee→Υ→ττ the Breit-Wigner approximation, M2Γ2 σ (s) = σ Υ Υ , (3.8) ee→Υ→ττ peak (s M2)2+M2Γ2 − Υ Υ Υ M and Γ are the masses and the widths of the Υ resonances, and the cross section at the Υ Υ peak is given by σ = 12π (Υ ee) (Υ ττ)/M2. In the limit Γ σ of narrow peak B → B → Υ Υ (cid:28) W resonances, σ (s) can be approximated by ee→Υ→ττ σ (s) σ πM Γ δ(s M2). (3.9) ee→Υ→ττ ≈ peak Υ Υ − Υ Theexpressionforthemaximumvisibleresonantcrosssectionobtainedsubstitutingeq.(3.8) into eq. (3.7) is (cid:114) π Γ σmax = ρσ , with ρ = Υ. (3.10) vis peak 8 σ W In table 1 we compare the maximum visible resonant cross sections for e+e− Υ τ+τ− → → with the non-resonant cross section σ = 0.919(3) nb at √s = M [65]. From this non−res Υ table we can conclude that, at the Belle II experiment, the τ+τ− events produced with beams at a center-of-mass energy √s M are mostly due to non-resonant contributions; Υ ∼ indeed the visible resonant cross sections are of the same order of the non-resonant ones, or smaller. Even for the multihadron events in the region of Υ(1S,2S,3S), the non-resonant cross section dominates with respect to the resonant one (see, for example, [66]). The situation at Belle was similar (the energy spread at KEKB was σ = 5.24 MeV [20]). W We therefore conclude that measuring the e+e− τ+τ− cross section at the upcoming → SuperKEKB collider on top of the Υ resonances will not eliminate the contamination of the non-resonant contributions. In the next section we will propose a new method to determine the electromagnetic dipole moments of the τ lepton via precise measurements of its radiative leptonic decays. – 7 – 4 Radiative τ leptonic decays: theoretical framework The SM prediction, at next-to-leading order (NLO), for the differential rate of the radiative leptonic decays τ− l−ν ν¯ γ, (4.1) τ l → with l = e or µ, of a polarized τ− with mass m in its rest frame is τ d6Γ(y ) αG2m5 xβ (cid:20) (cid:21) 0 = F τ l G+xβ nˆ pˆ J +ynˆ pˆ K +xyβ nˆ (pˆ pˆ )L , (4.2) dxdydΩ dΩ (4π)6 1+δ l · l · γ l · l × γ l γ W whereG = 1.1663787(6) 10−5 GeV−2 [67]istheFermiconstantdeterminedbythemuon F × lifetime and α = 1/137.035999157(33) is the fine-structure constant [6, 68]. Calling m the mass of the final charged lepton (neutrinos and antineutrinos are considered massless) we define r = m/m and r = m /M , where M is the W-boson mass; p and n = (0,nˆ) are τ W τ W W the four-momentum and polarization vector of the initial τ, with n2 = 1 and n p = 0. (cid:112) − · Also, x = 2E /m , y = 2E /m and β p(cid:126) /E = 1 4r2/x2, where p = (E ,p(cid:126) ) and l τ γ τ l l l l l l ≡ | | − p = (E ,p(cid:126) ) are the four-momenta of the final charged lepton and photon, respectively. γ γ γ ThefinalchargedleptonandphotonareemittedatsolidanglesΩ andΩ , withnormalized l γ three-momenta pˆ and pˆ , and c is the cosine of the angle between pˆ and pˆ . The term l γ l γ δ = 1.04 10−6 is the tree-level correction to muon decay induced by the W-boson W × propagator [69, 70]. Equation (4.2) includes the possible emission of an additional soft photon with normal- ized energy y(cid:48) lower than the photon detection threshold y (with y 1): y(cid:48) < y < y. 0 0 0 (cid:28) The function G(x,y,c,y ) and, analogously, J and K, are given by 0 4 (cid:104) α (cid:105) G(x,y,c,y ) = g (x,y,z)+r2 g (x,y,z)+ g (x,y,z,y ) , (4.3) 0 3yz2 0 W W π NLO 0 where z = xy(1 cβ )/2; the LO function g (x,y,z), computed in [71–74], arises from the l 0 − pure Fermi V–A interaction, whereas g (x,y,z) is the LO contribution of the W-boson W propagator derived in [70]. The NLO term g (x,y,z,y ) is the sum of the virtual and NLO 0 soft bremsstrahlung contributions calculated in [75] (see also refs. [76, 77]). The function L(x,y,z), appearing in front of the product nˆ (pˆ pˆ ), does not depend on y ; it is only l γ 0 · × induced by the loop corrections and is therefore of (α/π). In particular, L(x,y,z) is of (cid:80) O the form P (x,y,z)Im[I (x,y,z)], where P are polynomials in x,y,z and I (x,y,z) n n n n n are scalar one-loop integrals whose imaginary parts are different from zero. Tiny terms of (αm2/M2) 10−6 were neglected; they are expected to be comparable to the uncom- O τ W ∼ puted next-to-next-to-leading order (NNLO) corrections of ((α/π)2). The functions G, O J, K and L are free of UV and IR divergences. Their (lengthy) explicit expressions are provided in [75]. The corresponding formula for the radiative decay of a polarized τ+ can be simply obtained replacing J J and K K in eq. (4.2) (see table 2). If the initial → − → − τ± are not polarized, eq. (4.2) simplifies to d3Γ(y ) αG2m5 xβ 0 = F τ l 8π2G(x,y,c,y ). (4.4) dxdcdy (4π)6 1+δ 0 W – 8 – For the differential rate of leptonic τ decays in which a virtual photon is emitted and converted into a lepton pair, we refer the reader to the recent comprehensive article in [78]. The effective Lagrangian (2.7) generates additional non-standard contributions to the differential decayrate ofa polarizedτ− ineq. (4.2).1 Theycan be summarisedin the shifts: G G + a˜ G , (4.5) τ a → J J + a˜ J , (4.6) τ a → K K + a˜ K , (4.7) τ a → L L + (m /e) d˜ L , (4.8) τ τ d → where G = 4 (cid:2)r2(cid:0)y2 yz+3z2(cid:1) z(y+2z)(x+y z 1)(cid:3), (4.9) a 3z − − − − J = 2 (cid:2)3r2(cid:0)xy+y2 2z(cid:1) 2x2y 4xy2+2xyz+xy+4xz 2y3+2y2z a 3z − − − − +2y2+3yz 4z2 2z(cid:3), (4.10) − − K = 2 (cid:2)12r4y+r2(cid:0) 3x2y 3xy2 8xy 6y2+8yz+4y+6z2(cid:1)+2x3y+4x2y2 a 3yz − − − − 2x2yz x2y+2xy3 2xy2z 2xy2 xyz 4xz2 2y2z 2yz2+2yz+4z3+2z2(cid:3), − − − − − − − − (4.11) L = 4 (cid:2)3r2(cid:0)xy+y2 2z(cid:1) 2x2y 4xy2+2xyz+xy+4xz 2y3+2y2z d 3yz − − − − +2y2+3yz 4z2 2z(cid:3) (4.12) − − (we note that L = 2J /y). Tiny terms of (a˜2), (d˜2) and (a˜ d˜) were neglected. For d a τ τ τ τ O O O τ+ decays, the theoretical prediction for the differential decay rate can again be obtained from eq. (4.2), simply performing the following substitutions (see table 2): G G + a˜ G , (4.13) τ a → J J a˜ J , (4.14) τ a → − − K K a˜ K , (4.15) τ a → − − L L (m /e) d˜ L . (4.16) τ τ d → − Deviations of the τ dipole moments from the SM values can be determined comparing the SM prediction for the differential rate in eq. (4.2), modified by the terms G , J , K and a a a L , with sufficiently precise data. d 5 Feasibility study at Belle and Belle II In this section we outline our technique to estimate the sensitivity on τ dipole moments via τ leptonic radiative decays. First, however, we will discuss the possibility, suggested in ref. [27], to determine a˜ taking advantage of the radiation zero which occurs in the τ 1As discussed in section 2, we neglect non-standard τ-W couplings arising from the operator Q33 . lW – 9 –