Improved description of dilepton production in τ − → ντP− decays

Improved description of dilepton production in τ − →ντP−decays Adolfo Guevara ,1,2 Gabriel López Castro,3and Pablo Roig 3 1Departamento de Física Atómica, Molecular y Nuclear and Instituto Carlos I de Física Teórica y Computacional Universidad de Granada, E-18071 Granada, Spain 2Departament de Física Teòrica, IFIC, Universitat de Val`encia—CSIC, Apt. Correus 22085, E-46071 Val`encia, Spain 3Departamento de Física, Centro de Investigación y de Estudios Avanzados del Instituto Polit´ecnico Nacional. AP 14-740, 07000, Ciudad de M´exico, M´exico (Received 29 November 2021; accepted 7 March 2022; published 12 April 2022) Recently, the Belle Collaboration reported the first measurements of the τ−→ντπ−eþe−branching fraction and the spectrum of the pion-dielectron system. In an analysis previous to Belle’s results, we evaluated this branching fraction which turned out to be compatible with that reported by Belle, although with a large uncertainty. This is the motivation to seek for improvement on our previous evaluation of τ−→ντπ−lþl−decays (l¼e,μ). In this paper we improve our calculation of the WP−γvertex by including flavor-symmetry breaking effects in the framework of the resonance chiral theory. We impose QCD short-distance behavior to constrain most parameters and data on the π−eþe−spectrum reported by the Belle Collaboration to fix the remaining free ones. As a result, improved predictions for the branching ratios and hadronic/leptonic spectra are reported, which are in good agreement with observations. Analogous calculations for the strangeness-changing τ−→ντK−lþl−transitions are reported for the first time. Albeit one expects the mπμþμ−spectrum to be measured in Belle-II and the observables with l¼ecan be improved, it is rather unlikely that the Kchannels can be measured due to the suppression factor jVud=Vusj2¼0.05. DOI: 10.1103/PhysRevD.105.076007 I. INTRODUCTION The search for signals of physics beyond the Standard Model (SM) requires a good understanding of SM processes, either to discard possible backgrounds coming from it such as large radiative corrections [1–3],ortohave hadronic contamination under control in precision tests of the SM [4]. In addition to offering a clean laboratory to test the hadronization of the weak currents, some semileptonic τ lepton decays, such as τ→ντPðγÞfor P¼π,K, provide a good example where SM effects can be reliably calculated to disentangle possible new-physics signals hidden in precision observables. In Ref. [5] we reported the first prediction of Bðτ→ ντπl¯ lÞand the corresponding dilepton spectrum, where l¼e,μ(this can be viewed as the crossed channels of lepton pairs produced in πl2decays [6] in a larger kinematical domain); later on the Belle Collaboration [7] announced the first searches of these decays. Recently, some of the authors have also reported similar studies of τ−→ντπ−π0l¯ ldecays [8]. Together with the five lepton decays of tau leptons [9], they provide a better description of possible backgrounds in lepton-number or lepton-flavor violation searches in τdecays. Motivated by the Belle Collaboration studies [7], in this work we revisit our predictions for τ→ντπl¯ ldecays with the aim of improving the theoretical description of structure-dependent effects and to get reduced uncertainties. In addition, we make an analogous analysis of the strangeness-changing processes τ→ντKl¯ lfor the first time. In these phenomena, the Wγ⋆Pvertex plays a central role and its description is necessary to understand the radiative corrections to the τ−→ντP−decays [10]. This vertex also involves parameters which are needed to describe the pion transition form factor (TFF), which is required to compute the dominant piece (the pion pole) of the hadronic light-bylight contribution to the anomalous magnetic moment of the μlepton, aμ; the TFF can be obtained by our vector form factor (see Sec. III B) by considering Bose symmetry. Although knowledge on these parameters could, in principle, help reduce the uncertainty on the hadronic part of aμ [11], the τ−→ντπ−eþe−data does not (and is not foreseeable to) have the necessary precision to improve actual predictions on the π-pole contribution to aμ. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3. PHYSICAL REVIEW D 105, 076007 (2022) 2470-0010=2022=105(7)=076007(15) 076007-1 Published by the American Physical Society The problem with the description of these effective vertices arises when one tries to describe them in terms of the fundamental fields of the Standard Model, since at energies below the mτscale, one can not give a proper perturbative description of color interactions. However, the decay amplitude involving these vertices can be, for the sake of convenience, split into a part where the hadronic current h0j¯ uγμð1−γ5Þdjπ−i¼−iffiffiffi 2 pfπpμand the electromagnetic interactions are computed using scalar QED (sQED), which we call structure independent, and a part where more involved hadronic interactions are computed using an effective field theory, called structure dependent. Thus, we try to surpass the difficulties of calculating the structure-dependent part using resonance chiral theory (RχT) [12,13], which is an extension of chiral perturbation theory (χPT) [14–16] that includes resonances as active degrees of freedom. χPT relies on the chiral symmetry group G¼Uð3ÞL⊗Uð3ÞRof the massless QCD Lagrangian. After it gets spontaneously broken,G→Uð3ÞV,theremaining symmetry gets explicitly broken when the masses of the light quarks are considered to be nonvanishing. The Bðτ−→ντπ−l¯ lÞand dilepton spectrum were computedpreviouslyinRef.[5] using such techniques; however, the novelty in the present treatment is that we include the effects of finite different light-quark masses as done for the transition form factor of the pseudo-Goldstone bosons for the hadronic light-by-light part of the aμin Ref. [17] (over [18], where these were neglected). We also give a more thorough treatment of the uncertainties than those in Ref. [5], thus obtaining consistent results comparing with the corresponding form factors given in Ref. [19]. Furthermore, the recent measurement of the branching fraction with a lower limit in the invariant mass of the pion and dilepton pair [7],mπe−eþ, motivates this reanalysis further, since in the mπe−eþ≥1.05 GeV region the branching fraction gets saturated by the structure-dependent contribution. While most of the parameters of the model can be constrained by means of the high-energy behavior of QCD, some of them remain loose. We fit these to the measured invariant mass mπ−e−eþspectra and also to the measurement of the branching fraction Bðmπ−e−eþ≥ 1.05 GeVÞ¼ð5.90 1.01Þ×10−6[7]. Despite the access to the invariant mass spectra data for the τþ→ντπþ¯ ll decay,wewillonlymakeuseofthedatafortheτ−decay.The reason not to use both sets is that the spectra have incompatibilities in several bins; also, when fitting individually the πþdata set leads to unphysical conditions (see discussion in Sec. IV B). As a result, we improve our predictions, with correspondingly reduced uncertainties. The outline of the paper is as follows. In Sec. II the different contributions to the matrix element are collected. In Sec. III we introduce the Lagrangian used for computing the structure-dependent corrections, calculate the corresponding form factors (including flavor-breaking corrections to our previous results) and derive the shortdistance constraints among resonance couplings. In Sec. IV we carry out our phenomenological analysis, including a fit to Belle τ−→π−eþe−ντdata and predicting the partner ðπ↔K;e ↔μÞmodes, yet to be discovered. We give our conclusions in Sec. V. II. AMPLITUDES For convenience, we take three kinds of contributions to the decay amplitude: the first called inner bremsstrahlung (IB) or structure independent (SI). The other two are the structure dependent (SD) ones, namely the polar (V) and axial-vector (A) parts of the left-handed weak charged current. The IB amplitude can be obtained using the sQED Lagrangian, where the photon is either radiated by the τ lepton, off the pseudo-Goldstone boson (πor K) or by the longitudinal propagation mode of the W−boson, a contribution which is needed to achieve gauge invariance of the total IB amplitude. The total IB contribution is shown in Eq. (1), along with the parametrization of the SD parts as given in Ref. [5]. The momenta definition is given in Fig. 1. The different contributions to the matrix element are (D¼d,sfor P¼π,K) MIB ¼−iGFVuDfπmτ e2 k2Jν l¯ uντð1þγ5Þ ×2pν 2p·kþk2þ2pτν− = kγν −2pτ·kþk2uτ;ð1aÞ MV¼−GFVuD e2 k2Jν lJμ τFVðW2;k 2Þεμναβkαpβ;ð1bÞ FIG. 1. Feynman diagrams of the SI contributions (only scalar QED is used for the radiation off the P−meson) to the τ−ðpτÞ→ ντðqÞP−ðpÞlðp−Þ¯ lðpþÞdecay amplitude. The diamond vertex is an effective vertex meaning the Wboson has been integrated out. GUEVARA, LÓPEZ CASTRO, and ROIG PHYS. REV. D 105, 076007 (2022) 076007-2 MA¼iGFVuD e2 k2Jν lJμ τfFAðW2;k 2Þ½ðW2þk2−m2 πÞgμν −2kμpν−A2ðW2;k 2Þk2gμν þA4ðW2;k 2Þk2ðpþkÞμpνg:ð1cÞ Here, Jν l¼¯ uðp−ÞγνvðpþÞand Jμ τ¼¯ uðqÞð1þγ5ÞγμuðpτÞ are the lepton electromagnetic and τweak charged currents, respectively. We use W2≡ðpτ−qÞ2and k2≡ðp−þpþÞ2 as the two independent Lorentz-invariants upon which the form factors (FV;FA;A 2;A 4) depend. In Ref. [5] the axial amplitude was given only in terms of three form factors (FV,FA, and a combination of A2and A4called B), since at chiral order p4, the A2and A4form factors are linearly dependent and can be written in terms of the pseudoGoldstone electromagnetic form factor FP Vðk2Þ[6]. Here, A2and A4cannot be recast in terms of FP Vðk2Þ, since we are considering contributions of chiral order p6. Furthermore, including the complete set of leading-order chiral symmetry breaking contributions will change the pion pole for the massive pion propagator. As a result, the A2and A4 form factors become linearly independent and the axial-vector part of the left hadronic current cannot be expressed in terms of the two form factors FðW2;k 2;p 2Þ and GðW2;k 2;p 2Þof Refs. [19–21] [see discussion after Eq. (24)]. III. STRUCTURE DEPENDENT FORM FACTORS A. The relevant operators In this section we will present, for the sake of simplicity, only the relevant operators in the RχT Lagrangian needed to compute the form factors, which are given in the next subsection. We will be concise here, for a more extended discussion see e.g., Ref. [17].RχT extends the domain of applicability of chiral perturbation theory [14–16] (χPT) by adding the light-flavored resonances as active degrees of freedom. We start with operators involving no resonances, these being1 L0Res ¼f2 4huμuμþχþiþLWZW þCW 7OW 7 þCW 11OW 11 þCW 22OW 22;ð2Þ where the first term is given by the leading χPT Lagrangian operators of chiral order p2[14–16], the second one is the anomalous Wess-Zumino-Witten Lagrangian of Oðp4Þ [25,26] and the last three operators belong to the subleading odd-intrinsic parity sector Oðp6ÞLagrangian [27].We neglect operators not included in this Lagrangian. Congruently with Refs. [19,21,28], we will not consider any Oðp8Þcontribution whatsoever. In the first term, fis the decay constant in the chiral limit, which we will set to f¼fπ∼92 MeV, uμand χþare chiral tensors [29], the former containing derivatives of the π=K fields and external spin-one currents and the latter scalar currents involving the previous fields’masses squared, m2 π=K, times even powers of such fields. The equations of motion of the resonances give their classical fields in terms of series of chiral tensors of different order. The resonances are said to be integrated out (tree-level integration) when the classic fields are substituted in favor of chiral tensors in the resonant Lagrangian. Integrating the resonances out using the leading-order terms of the equations of motion very approximately saturates the Oðp4Þ[and leading Oðp6Þ] contributions in the even-intrinsic parity sector [12,13,19]; therefore, we will not use the nonresonant Oðp4Þset of operators (for the sake of simplicity), since they are considered to yield negligible contributions. Since we will only consider leading-order terms in the resonances equations of motion, the Oðp6Þchiral low-energy constants in the odd-intrinsic parity sector cannot be saturated upon resonance exchange [28]; therefore, we have to include the three contributing CW iOW iterms [27], OW 7¼iϵμναβhχ−fμν þfαβ þi; OW 11 ¼iϵμναβhχþ½fμν þ;fαβ −i; OW 22 ¼iϵμναβhuμf∇ρfρν þ;fαβ þgi;ð3Þ where the following chiral tensors [29] enter; χ−gives odd powers of the π=K fields with factors involving m2 πor m2 K, ∇μis the covariant derivative and includes spin-one left and vector external currents through the connection, and fμν  yields the field-strength tensors of the charged-weak or electromagnetic fields. We turn next to those operators with one resonance field, in either intrinsic parity sector, L1Res ¼Leven 1Res þLodd 1Res:ð4Þ In turn, the first piece can be further divided according to the quantum numbers of this resonance Leven 1Res ¼X Ri¼V;A;P Leven 1Ri:ð5Þ The contributions with one vector resonance read [12,19]2 1Although these terms also appear in the χPT Lagrangian, their couplings get shifted in the presence of resonance contributions (see for instance [22–24]). 2Vμν (analogously Aμν for axial resonances below) is a matrix in flavor (u,d,s) space and we use the antisymmetric tensor formalism for spin-one fields for convenience [12,13]. IMPROVED DESCRIPTION OF DILEPTON PRODUCTION IN …PHYS. REV. D 105, 076007 (2022) 076007-3 Leven 1V¼FV 2ffiffiffi 2 phVμνfμν þiþ i 2ffiffiffi 2 pGVhVμν½uμ;u νi þλV ffiffiffi 2 phVμνffμν þ;χþgi;ð6Þ where the Vfield (we assume ideal mixing of neutral mesons) has an analogous flavor structure as the pseudoGoldstone field ϕ, namely Vμν ¼0 B B @ 1 ffiffi2 pðρ0 μν þωμνÞρþ μν K⋆þ μν ρ− μν 1 ffiffi2 pð−ρ0 μν þωμνÞK⋆0 μν K⋆− μν K⋆0 μν ϕμν 1 C C A :ð7Þ In Eq. (6), the first two operators give the contribution from the coupling of vector resonances to external fields in the chiral limit and the last term gives the flavor-breaking corrections to such couplings. Our λV¼ffiffiffi 2 pλV 6, using the notation in Ref. [19]. This last operator is the only one included from the full basis of Oðp4Þeven-intrinsic parity operators in Ref. [19] since it is the single one that can contribute to the Uð3ÞVbreaking in the V−γcoupling. There are, however, two reason to disregard basis of operators in the even-intrinsic parity sector: The operators that are relevant to the process can be dismissed on the basis of resonance field redefinitions;3if we, however, keep such operators, they will only give subleading contributions to those from the first two operators in Eq. (6) with no contribution to Uð3ÞV-breaking vertices. The axial resonance operators present a similar feature and an analogous flavor-space structure to that of the vector mesons. This is, the Oðp4Þone-resonance even-intrinsic parity operators for axial resonances in Ref. [19] can be absorbed through field redefinitions. We will therefore disregard any contribution from this part of the Lagrangian, including the Uð3ÞVbreaking terms to the axial-vector resonance coupling to external currents; namely, the JA vertex. The remaining contributions with one resonance field are [12] Leven 1A=P ¼FA 2ffiffiffi 2 phAμνfμν −iþidmhPχ−i;ð8Þ with Pa matrix in three-flavor space containing the lightest pseudoscalar resonances. The inclusion of the pseudoscalar resonance is necessary in order to obtain consistent shortdistance constraints in hVAPiand hVJPiGreen’s functions [19,28,30,31]. All Feynman diagrams involving these resonances will give Uð3Þbreaking contributions to the amplitude due to the last term in Eq. (8). We have neglected other spin-zero resonance contributions (scalar and heavier pseudoscalar resonances [5]), which are not needed for theoretical consistency and are irrelevant phenomenologically.Theodd-intrinsicparity contributionsto L1Res are [32]4 Lodd 1Res ¼X 7 j¼1 cj MV Oj VþεμναβhκP 5ffμν þ;fαβ þgPi;ð9Þ with the operators O1 V¼εμνρσhfVμν;fρα þg∇αuσi; O2 V¼εμνρσhfVμα;fρσ þg∇αuνi; O3 V¼iεμνρσhfVμν;fρσ þgχ−i; O4 V¼iεμνρσhVμν½fρσ −;χþi; O5 V¼εμνρσhf∇αVμν;fρα þguσi; O6 V¼εμνρσhf∇αVμα;fρσ þguνi; O7 V¼εμνρσhf∇σVμν;fρα þguαi:ð10Þ In the following, we quote those terms bilinear in resonance fields (we do not display the kinetic terms for the resonances, which can be found in Ref. [12], as they do not contribute to the effective vertices). L2Res ¼Leven 2Res þLodd 2Res;ð11Þ with [19,33–35]5 Leven 2Res ¼−eV MhVμνVμνχþiþλPV 1OPV 1þλPV 2OPV 2 þλPA 1OPA 1þX 5 i¼1 λVA iOVA i;ð12Þ and [28,32] Lodd 2Res ¼X 3 i¼1 diOVV iþκPV 3OPV 3:ð13Þ The operators appearing in the two previous equations are (hμν ¼∇μuνþ∇νuμ) 3Through the redefinition of the vector resonance field V→VþgfV;χþgit is possible to cancel the λVoperator [19]; however, we keep it in order to show the full basis of possible Uð3ÞVbreaking operators since we do not consider the full even-intrinsic parity basis of Ref. [19]. We will show later that this is consistent, since the short-distance constraints give λV¼0. 4Since we are only considering operators with one π=K field, these constitute a basis. In the general case, the basis is given in Ref. [28]. The translation between them can be read in Ref. [30]. 5The operator with coefficient eV Mallows to account for Uð3Þ breaking effects in the vector resonance masses, in agreement with phenomenology. GUEVARA, LÓPEZ CASTRO, and ROIG PHYS. REV. D 105, 076007 (2022) 076007-4 OPV 1¼ih½∇μP; Vμνuνi; OPV 2¼ih½P; Vμνfμν −i; OPA 1¼ih½P; Aμνfμν þi; OVA 1¼h½Vμν;A μνχ−i; OVA 2¼ih½Vμν;A ναhα μi; OVA 3¼ih½∇μVμν;A ναuαi; OVA 4¼ih∇αVμν;A ανuμi; OVA 5¼ih½∇αVμν;A μνuαi; OVV 1¼εμνρσhfVμν;Vραg∇αuσi; OVV 2¼iεμνρσhfVμν;Vρσgχ−i; OVV 3¼εμνρσhf∇αVμν;Vραguσi; OPV 3¼εμναβhfVμν;fαβ þgPi:ð14Þ There is only one relevant operator with three resonance fields in either parity sector, L3Res ¼iλVAPh½Vμν;A μνPiþκPVV εμνρσhVμνVαβPi:ð15Þ Operators with a higher number of resonant fields will not be included, since then one has to include subleading diagrams with loops where some of the internal lines are given by resonances. The present analysis is restricted to tree-level diagrams, which should already capture the leading effects associated with resonance exchange. Oneloop diagrams with resonances are expected to be a numerically small correction since these would be subleading in the 1=NCexpansion [4]. Such corrections will be neglected due to the already sizeable number of parameters involved in the tree-level analysis and the current precision of the experimental data. B. Form factors In this section we quote our results for the different contributions to the FV,FA,A2, and A4form factors, for P¼π,K. All resonance propagators are to be understood as provided with an energy-dependent width (M2 R−x→M2 R−x−iMRΓRðxÞ;x¼W2;k 2) computed within RχT, using those in Refs. [36] [ρð770Þ], [37] [Kð892Þ], and [38,39] [a1ð1260Þ, including the KKπ cuts [40]. A constant width will suffice for the very narrow ωð782Þand ϕð1020Þmesons (their Particle Data Group [PDG] [41] values will be taken). For the K1ð1270=1400Þ states we will follow [42]. The Feynman diagrams contributing to the vector form factors are shown in Fig. 2, these form factors are (NC¼3 in QCD). FðπÞ VðW2;k 2Þ¼ 1 3f−NC 8π2þ64m2 πCW⋆ 7−8CW 22ðW2þk2Þþ 4Fud V 2 M2 ρ−W2 d3ðW2þk2Þþd⋆ 123m2 π M2 ω−k2 þ2ffiffiffi 2 pFud V MV c1256W2−c⋆ 1235m2 π−c125k2 M2 ρ−W2þ2ffiffiffi 2 pFud V MV c1256k2−c⋆ 1235m2 π−c125W2 M2 ω−k2; ð16Þ FðKÞ VðW2;k 2Þ¼1 f−NC 24π2þ64 3m2 KCW⋆ 7þ32CW 11Δ2 Kπ−8 3CW 22ðW2þk2Þþ2Fus V½d3ðW2þk2Þþd⋆ 123m2 K M2 K⋆−W2 ×Fud V M2 ρ−k2þ1 3 Fud V M2 ω−k2−2 3 Fss V M2 ϕ−k2þ2ffiffiffi 2 pFus V 3MV c1256W2−c⋆ 1235m2 K−c125k2þ24c4Δ2 Kπ M2 K⋆−W2 þffiffiffi 2 pðc1256k2−c⋆ 1235m2 K−c125W2Þ MVFud V M2 ρ−k2þ1 3 Fud V M2 ω−k2−2 3 Fss V M2 ϕ−k2;ð17Þ FIG. 2. Feynman diagrams contributing to the vector part of the left hadronic current. The circled cross vertex indicates vector current. The resonance P⋆is the pseudoscalar resonance corresponding to πð1300Þ≡π0[Kð1460Þ≡K0] for P¼πðKÞ. The resonance V0 means ωfor P¼πand ρ0;ω;ϕ, for P¼K. IMPROVED DESCRIPTION OF DILEPTON PRODUCTION IN …PHYS. REV. D 105, 076007 (2022) 076007-5 where Δ2 Kπ¼m2 K−m2 πand we have used the combinations of coupling constants [43] c125 ¼c1−c2þc5; c1256 ¼c1−c2−c5þ2c6; c1235 ¼c1þc2þ8c3−c5; d123 ¼d1þ8d2−d3:ð18Þ FuD;ss Vand starred coefficients absorb Uð3Þbreaking contributions induced by λVin Eq. (6) and pseudoscalar resonances, respectively. Their expressions are given after Eq. (24). The Feynman diagrams contributing to the axial form factors are shown in Fig. 3, these form factors are6 FðπÞ AðW2;k2Þ¼Fud V 2f Fud V−2GV−m2 π 4ffiffi2 pdm M2 π0ðλPV 1þ2λPV 2Þ M2 ρ−k2 −FA 2f FA−2m2 π 4ffiffi2 pdm M2 π0λPA 1 M2 a1−W2 þffiffiffi 2 p f FAFud V M2 a1−W2 λ⋆ 0m2 π−λ0k2−λ00W2 M2 ρ−k2;ð19Þ FðKÞ AðW2;k 2Þ¼−FA 2f FA−2m2 K 4ffiffi2 pdm M2 K0λPA 1 M2 K1−W2þ2 6 4ffiffiffi 2 pFA 2f λ⋆ 0m2 K−λ0k2−λ00W2 M2 K1−W2þ Fus VðFus V−2GVþm2 K 4ffiffi2 pdm M2 K0ðλPV 1þ2λPV 2ÞÞ 4f3 7 5 ×Fud V M2 ρ−k2þ1 3 Fud V M2 ω−k2þ2 3 Fss V M2 ϕ−k2;ð20Þ AðπÞ 2ðW2;k 2Þ¼2 fGVþ2ffiffiffi 2 pm2 πdm M2 π0 λPV 1þffiffiffi 2 pFA M2 a1−W2W2ðλ0þλ00ÞFud V M2 ρ−k2;ð21Þ AðKÞ 2ðW2;k 2Þ¼GV fþ2ffiffiffi 2 pm2 Kdm M2 K0 λPV 1 fþffiffiffi 2 pFA M2 K1−W2 W2ðλ0þλ00Þ f Fud V M2 ρ−k2þ1 3 Fud V M2 ω−k2þ2 3 Fss V M2 ϕ−k2;ð22Þ AðπÞ 4ðW2;k 2Þ¼2 f Fud V M2 ρ−k2GV W2−m2 πþ2ffiffiffi 2 pdmm2 πλPV 1 M2 π0ðW2−m2 πÞþffiffiffi 2 pFAðλ0þλ00Þ M2 a1−W2;ð23Þ AðKÞ 4ðW2;k 2Þ¼1 fGV W2−m2 Kþ2ffiffiffi 2 pdmm2 KλPV 1 M2 K0ðW2−m2 πÞþffiffiffi 2 pFAðλ0þλ00Þ M2 K1−W2 Fud V M2 ρ−k2þ1 3 Fud V M2 ω−k2þ2 3 Fss V M2 ϕ−k2:ð24Þ It is worth noting that by replacing the Ppropagator in AðPÞ 2and AðPÞ 4with the massless pole propagator, one recovers the linear dependence between both form factors, thus getting a congruent expression with those in Ref. [19]. Therefore, the short-distance constraints obtained in this reference can be used as shown there if the Weinberg’s sum rules are imposed. We, however, do not make use of these sum rules, as FV=A are fitted to data (see discussion in Secs. III C and IV B). FIG. 3. Feynman diagrams contributing to the axial part of the left hadronic current. The circled cross vertex indicates axial current. Conventions for P⋆is the same as in the previous figure, the resonance V0means ρ0for P¼πand ρ0;ω;ϕ, for P¼K. 6We note two mistakes in writing FðπÞ Ain ref. [5], see the Appendix. The result written here agrees with the one in Ref. [44] for k2→0. GUEVARA, LÓPEZ CASTRO, and ROIG PHYS. REV. D 105, 076007 (2022) 076007-6 We introduced the short-hand notation Fud V≡FVþ8m2 πλV; Fus V≡FVþ8m2 KλV; Fss V≡FVþ8ð2m2 K−m2 πÞλV;ð25Þ for the shifts appearing also in [17].7 We also used [33] −ffiffiffi 2 pλ⋆ 0¼4λ⋆ 1þλ2þλ4 2þλ5; ffiffiffi 2 pλ0¼λ2−λ3þλ4 2þλ5;ð26Þ and ffiffiffi 2 pλ00 ¼λ2−λ4 2−λ5: We employed several starred coefficients including Uð3Þ breaking contributions, λ⋆ 1¼λ1−λVAPdm M2 P ; CW⋆ 7¼CW 7þκP 5dm M2 P ; c⋆ 3¼c3þκPV 3dmMV M2 P ;ð27Þ implying c⋆ 1235 ¼c1þc2þ8c⋆ 3−c5;and ð28Þ d⋆ 2¼d2þκVVPdm 2M2 P ;ð29Þ yielding d⋆ 123 ¼d1þ8d⋆ 2−d3:ð30Þ We have first shown here the correction to λ1appearing in λ⋆ 1, while the remaining starred couplings were already introduced in Ref. [17]. We will follow the scheme explained in Ref. [42] to account for the mixing between the K1ð1270Þ¼K1L and the K1ð1400Þ¼K1Hstates. This amounts to replacing, in eq. (20),ðM2 K1−W2Þ−1→cos2θAðM2 K1H−W2Þ−1þ sin2θAðM2 K1L−W2Þ−1, with mixing angle θA∈½37;58°. C. Short-distance constraints We will demand that the different form factors have an asymptotic behavior in agreement with QCD [45,46]. Specifically, we will require their vanishing for large λ in the limλ→∞FVðλW2;0Þand limλ→∞FVðλW2;λk2Þcases. We will do this first in the chiral limit and then at Oðm2 PÞ,8 paralleling the discussion in Ref. [17] for the neutral pseudoscalar transition form factors. In this way, we find the following relations: (a) FðπÞ VðW2;k 2Þ,Oðm0 PÞ: CW 22 ¼0;ð31Þ c125 ¼0;ð32Þ c1256 ¼−NCMV 32 ffiffiffi 2 pπ2FV ;ð33Þ d3¼−NCM2 V 64π2F2 V :ð34Þ (b) FðπÞ VðW2;k 2Þ,Oðm2 PÞ: λV¼−64π2FV NC CW⋆ 7;ð35Þ c⋆ 1235 ¼NCMVeV m 8ffiffiffi 2 pπ2FVþNCM3 VλV 4ffiffiffi 2 pπ2F2 V :ð36Þ (c) FðKÞ VðW2;k 2Þ,Oðm0 PÞ: Same constraints as for the π case, since both form factors9are identical in the Uð3Þ symmetry limit. (d) FðKÞ VðW2;k 2Þ,Oðm2 PÞ: CW 11 ¼NCλV 64π2FV :ð37Þ For the sake of predictability and in order to further constrain the parameters in the form factor, we use the Vector-Vector-Pseudoscalar (VVP) Green’s function, ΠVVPðr2;p 2;q 2Þ, constraints [28] obtained from the high-energy behavior when r2→∞,p2→∞,q2→∞, and matching to the operator product expansion (OPE) leading terms in the chiral and large-NClimits. These give 7As mentioned in Sec. III A, a similar shift can be introduced in FA, however, the operator responsible for such shift can be absorbed through axial resonance field redefinitions [19]. 8Since we are considering a complete basis of chiral symmetry breaking operators at order m2 P, we neglect higher-order chiral corrections. 9FP Aand AP 2;4form factors are also identical in this limit for P¼πor K, obviously. IMPROVED DESCRIPTION OF DILEPTON PRODUCTION IN …PHYS. REV. D 105, 076007 (2022) 076007-7 c125 ¼c1235 ¼0;c 1256 ¼−NCMV 32 ffiffiffi 2 pπ2FV ; κP 5¼0;d 3¼−NCM2 V 64π2F2 VþF2 8F2 Vþ4ffiffiffi 2 pdmκPV 3 FV ; CW 7¼CW 22 ¼0;d 123 ¼F2 8F2 V :ð38Þ Notice that these constraints coincide with our expressions in Eqs. (31)–(34) and that they imply CW 11 ¼λV¼CW⋆ 7¼0; d123 ¼F2 8F2 V ; dmκPV 3¼NCM2 π0eV m 64 ffiffiffi 2 pπ2FV :ð39Þ One can see that from the definition of c⋆ 1235 [Eqs. (27) and (28)] combined with the last expression and the shortdistance constraints Eqs. (34),(36), and (38) would imply a relation of eV min terms of Fand Mπ0, namely eV m¼−2π2F2 NCM2 π0 :ð40Þ However, we do not rely on this relation since comparison with previous phenomenology [34,35] shows that the absolute value of Eq. (40) obtained for f≈92 MeV and Mπ0¼1.3GeV is an order of magnitude smaller than required by phenomenology. On the other hand, no relation among parameters of the axial form factors can be obtained by taking the infinite virtualities limit, since it already has the right asymptotic behavior. Instead, we will rely on the relations obtained using the VAP Green function10 ΠVAPðp2;q 2;ðpþqÞ2Þ [21] in an analogous manner to that done for the ΠVVPðr2;p 2;q 2ÞGreen function. We recall that the simultaneous analysis of the scalar form factor [49,50] and the SS-PP sum rules [51] yields dm¼f=ð2ffiffiffi 2 pÞ. Additionally, notice that AðPÞ 2;4depend on λPV 1. In turn, FðPÞ Adepends on λPA 1and λPV 1þ2λPV 2. The appropriate short-distance behavior of the VAP Green function [21] fixes all of them but λ⋆ 0or, in other words λVAP, as noted in Ref. [19] λ0¼f2 4ffiffiffi 2 pFVFA ;λ0¼f2þF2 A 2ffiffiffi 2 pFVFA ; λ00 ¼−f2þF2 A−2FVGV 2ffiffiffi 2 pFVFA ;d mλPV 1¼−f2 4ffiffiffi 2 pFV ; dmλPV 2¼3f2þ2F2 A−2F2 V 16 ffiffiffi 2 pFV ;d mλPA 1¼f2 16 ffiffiffi 2 pFA :ð41Þ Despite the relation for dmfrom the scalar form factor and the SS-PP Green’s function, notice that there is no need for one since dmalways appears multiplied by one of the other parameters to be constrained. We will also make use of the constraint [12] FVGV¼f2:ð42Þ In order to gain predictability, we will use the values of d⋆ 123,MV, and eV mgiven for the best fit of Ref. [17], namely (their correlations are given in the quoted reference) d⋆ 123 ¼−ð2.31.5Þ×10−1; MV¼ð791 6ÞMeV; eV m¼−ð0.36 0.10Þ:ð43Þ IV. PHENOMENOLOGICAL ANALYSIS A. Phase space In order to compare our results with those of Ref. [5] we use the same phase space configuration. We recall that the variables in Ref. [5] are the invariant mass squared of the pseudo-Goldstone and the neutrino, s12 ¼m2 Pντ, the invariant mass squared of the charged lepton pair, s34 ¼m2 l¯ l,two polar angles θ1,θ3, and one azimuthal angle ϕ3, with the integration limits given by ðm3þm4Þ2≤s34 ≤ðM−m1−m2Þ2;ð44aÞ ðm1þm2Þ2≤s12 ≤ðM−ffiffiffiffiffiffi s34 pÞ2;ð44bÞ 0≤θ1;3≤π;0≤ϕ3≤2π:ð44cÞ If we identify the particle with tag 1 with ντ, the invariant mass of the weak gauge boson can be related to the Lorentz invariants of Eqs. (44) via W2¼M2 234 ¼M2þm2 1−ðM2þs12 −s34Þðs12 þm2 1−m2 2Þ 2s12 −Xβ12 cosθ1;ð45Þ where βij ¼λ1=2ðsij;m 2 i;m 2 jÞ=sij and X¼λ1=2ðM2;s 12; s34Þ=2, being λða; b; cÞ¼a2þb2þc2−2ab − 2ac −2bc, the Käll´en lambda function. Equation (45) 10See, however, the discussion in Sec. 6.2 of [47] comparing these short-distance constraints to the results in Refs. [20,48]. GUEVARA, LÓPEZ CASTRO, and ROIG PHYS. REV. D 105, 076007 (2022) 076007-8 allows us to eliminate θ1in favor of M234. The importance of the phase space configuration with W2instead of θ1 relies on the need to compute the mπeþe−spectrum in order to fit unconstrained parameters to the Belle invariant mass spectrum [7]. The kinematic limits on the nonangular variables for this phase space configuration read m2þm3þm4≤M234 ≤M; ð46aÞ ðm3þm4Þ2≤s34 ≤ðM234 −m2Þ2;ð46bÞ s− 12 ≤s12 ≤sþ 12;ð46cÞ where s 12 ¼M2þm2 1þm2 2þs34 −W2þðM2−m2 1Þðm2 2−s34Þ W2 1 2W2λ1=2ðM2;W2;m 2 1Þλ1=2ðW2;s 34;m 2 2Þ:ð47Þ With this, the differential decay rate is given as dΓðτ−→ντP−l¯ lÞ ¼Xβ12β34 4ð4πÞ6m3 τjMj2ds34ds12dðcos θ1Þdðcosθ3Þdϕ3 ¼β34 4ð4πÞ6m3 τjMj2dM2 Pl¯ lds34ds12dðcosθ3Þdϕ3:ð48Þ B. Fit to data Short-distance QCD behavior [17,19] does not constrain all parameters. Thus, we fit some of the remaining unknowns (see Fig. 4) using the invariant mass spectra of the Wboson, mπ−eþe−, measured by the Belle Collaboration [7]. We start with a four-parameters fit (FV,FA,λ0, and B, the branching fraction, are floated). Despite the fact that the whole mπ−eþe−spectrum has been measured, not all the data is available for this minimization since points below mπ−eþe−<1.05 GeV were used as a control region to validate the Monte Carlo simulation, leaving the most sensitive part to SD contributions as the signal region [7]; therefore, we use for the minimization the data above the cut mπ−eþe−¼1.05 GeV. We use only the data set for the τ−→π−eþe−ντmode.11 Comparison of this with the expected signal events distribution in this reference allows us to roughly quantify the deconvolution of signal from detector, which we ignore. We have assumed this to be an energy-independent effect for simplicity, and taken it into account as a systematic uncertainty in the data. This error turns out to be comparable to the one reported by Belle for the branching fraction measured above the cut. In addition to this, the Belle Collaboration used the expressions of Ref. [5], which had typos in some of the FAðt; k2Þterms (see Appendix). Besides, trying to keep our previous analysis as simple as possible, it gave an incomplete result in the sense of VAP Green’s function analysis,12 which could lead to biased estimations of the decay observables. Both reasons motivate our choice of fitting the total branching fraction, B, as an additional parameter instead of simply computing it from the decay width expression in Eq. (48). We used the relation Zdmπ−eþe− 1 Γ dΓ dmπ−eþe−¼1¼Zdmπ−eþe− 1 N dN dmπ−eþe− ¼X bins 1 N Nbin Δmπ−eþe− ;ð49Þ where Nis the total number of events, Nbin is the number of events in a given bin, and Δmπ−eþe−is the bin width. We thus minimize the χ2given by χ2¼NΔmπ−eþe− Γεbin dΓ dmπ−eþe− −Nbin εbin 2 þB−BR εBR 2 ;ð50Þ FIG. 4. Normalized invariant mass spectra obtained with the two sets of parameters obtained from fitting to the Belle data. The purple line corresponds to the data with FAfixed, while the green one stands for that with λ⋆ 0fixed. The blue data corresponds to measurements of τ−decays, which shows the best agreement with our model. [7]. 11This is the one shown in the plots of Ref. [7]. We have checked better agreement with the Monte Carlo simulation (based on our previous paper [5], see also [52,53]) for this mode with respect to its charge-conjugated mode. 12Pseudoscalar resonance exchange and Oðp6Þoperators in the AðπÞ 2and AðπÞ 4form factors are lacking in Ref. [5]. This leads to relating both form factors to the πelectromagnetic form factor [6]. IMPROVED DESCRIPTION OF DILEPTON PRODUCTION IN …PHYS. REV. D 105, 076007 (2022) 076007-9