Tension between $e^+e^-\to\eta\pi^-\pi^+$ and $\tau^-\to\eta\pi^-\pi^0\nu_\tau$ data and non-standard interactions

We show the discrepancy between the isospin-rotated $e^+e^-\to\eta\pi^-\pi^+$ cross-section -- measured by various collaborations -- and the Belle $\tau^-\to\eta\pi^-\pi^0\nu_\tau$ spectrum, which cannot be explained by heavy new physics non-standard interactions. We give for the first time the framework needed to study these beyond the standard model contributions in three-meson tau decays.


Introduction
In the very accurate isospin symmetry limit, the e + e − → hadrons cross-section is related to the spectral function of semileptonic tau decays (see e.g., ref. [1]).Beyond tests of this property, based on the conservation of the vector current (CVC), it gave rise to tau-based evaluations of the leading-order hadronic vacuum polarization (HV P, LO) contributions to the muon g-2 [2,3,4,5,6,7,8,9], a µ = (g µ − 2)/2.However, uncertainties associated to isospin-breaking effects relating both observables are currently too large to make this determination competitive with the one using hadronic e + e − cross-section data [10,11,12,13,14,15,16,17]. Notwithstanding, checking the consistency between σ(e + e − → hadrons) and exclusive hadron tau decay data is still motivated by the tension exhibited by lattice QCD evaluations [18,19,20] of a HV P,LO µ and the e + e −based data-driven extraction [10].Depending on which number is compared to the experimental average of the recent FNAL measurement [21] and the final result from the BNL experiment [22], new physics significance varies sizeably, between barely one and slightly more than four standard deviations.
In this work, we study the discrepancy -which goes beyond isospin breaking effects-between both sets of data (e + e − and τ ) for the exclusive ηππ channels, and show that it cannot be explained by heavy new physics.
The isovector component (I = 1) of the e + e − → ηπ + π − cross section data can be converted to the decay distribution in τ − → ηπ − π 0 ν τ decays using the approximate conservation of the vector current (CVC), which becomes exact in the isospin symmetry limit [1,23,24,25,26]: with Q 2 = m 2 ηππ the invariant mass squared of the ηππ system and S EW = 1.0201(3) [94] the short-distance electroweak radiative correction.We note that V ud differs from V ud by possible non-standard effects (see section 2).
Using eq. ( 1), Belle [27] data on τ − → ηπ − π 0 ν τ decays are seen to be incompatible with e + e − → ηπ − π + measurements published by DM2 [28], ND [29], CMD2 [30], BaBar [31], SND [32,33,34] and CMD3 [35].We use the best fits obtained in Refs.[36,37] to the e + e − → (η/π 0 )π + π − data for the Standard Model prediction.Since possible heavy new physics effects are negligible compared to the photon exchange driving these processes, a possible discrepancy between the isospin-rotated e + e − → ηπ + π − cross-section and the τ − → ηπ − π 0 ν τ decay rate [1,23,24,25] (besides small isospin breaking effects) could in principle be due to non-standard interactions (NSI) modifying the latter.Thanks to the limits set on possible NSI in semileptonic tau decays [38,39,40,41,42,43,44,45,46] we will show that this seeming CVC violation is incompatible with other hadron tau decays data.Belle-II will improve the measurement of this tau decay channel [47], as understanding semileptonic tau decays with eta mesons is required to search for second-class currents and heavy new physics through the discovery of the τ − → π − η ( ) ν τ decays [38,48,49].The rest of the paper is structured as follows: in section 2 we briefly recall the formalism encoding non-standard interactions in semileptonic tau decays.In section 3 we derive the τ − → ηπ − π 0 ν τ decays amplitude, in the Standard Model and including the NSI (involving new hadron contributions, which we account for).In section 4 we study the possible effects of NSI on the observables of interest, and show that the discrepancy between e + e − and τ data cannot be explained by heavy new physics, according to the NSI bounds.We conclude in section 5.The appendix summarizes the setting in which structure-dependent contributions were evaluated.
For later convenience we introduce V := L + R and A := − L + R , so that the relevant Lagragian at dimension six is where [60].We neglect higher-dimensional operators, suppressed by powers of M τ /Λ, since current limits on the i coefficients [38,39,40,41,42,43,44,45,46] correspond to Λ ∼ O(TeV) (under the weakcoupling hypothesis).As we only compute CP-conserving observables2 , the i coefficients are taken real.They are translated straightforwardly [50,57] into the SMEFT [65,66] couplings.For vanishing i , the SM is recovered.We will work in the M S scheme, at a scale µ = 2 GeV.

Hadronization: Standard Model and beyond
In the Standard Model, the τ − → ηπ − π 0 ν τ decay amplitude is where H µ encodes the hadronization into the three final-state mesons (h 1 = η, h 2 = π − , h 3 = π 0 in our case and with our conventions).Lorentz invariance determines the most general decomposition of H µ to be where the chosen set of independent Lorentz structures is and the relevant form factors (F i , i = 1, ..., 4) are driven by either vector or axial-vector currents (as indicated by their superscript, V /A) and carry quantum numbers of pseudoscalar (F 4 ), vector (F 3 ) or axial-vector (F 1,2 ) degrees of freedom.Very approximate G-parity conservation by the strong interactions5 produces vanishing axial-vector form factors in this channel, in such a way that -to an excellent accuracy-the dynamics of the considered decays are driven solely by the vector form factor, F 3 , which will be taken from the best fits of Refs.[36,37].As explained, F A 1,2,4 vanish in the limit of G-parity conservation.We will however compute the isospin-breaking contributions to these form factors given by scalar resonance exchanges.Our motivation to include these subleading effects only for the scalar mesons contributions is two-folded: on the one hand isospin-violating f 0 − a 0 mixing is enhanced with respect to other isospin breaking effects by the approximate degeneracy of these states and their comparable value to the kaon-antikaon thresholds [69].On the other hand, Belle-II shall measure the di-meson mass spectra6 in τ − → ηπ − π 0 ν τ decays and a theoretically-motivated parametrization of scalar meson exchanges in these processes will benefit their analysis.In this way, we will construct the hadronic input needed for NSI contributions to the considered decays.
The Γ a0 (s i ) energy-dependent width is given by [49] with ∆ P Q = m 2 P − m 2 Q and φ ηη = (41.4± 0.5) • [82].Mass and on-shell width of the a 0 resonance will be taken from the PDG [77].Scalar contributions in eq. ( 6) can be written in terms of the F A 1,2,4 form factors using 7 T α s 2 has been obtained assuming, for simplicity, that f 0 (980) is a pure octet state.If it comes from the mixing of the octet and singlet f states, the corresponding mixing coefficient can be absorbed in the constant a 0 f 0 , that we will fix to unity for definiteness.This and other ambiguities present in the description of the scalar mesons (like possible tetraquark components [77] and more complicated mixing pattern [78]) prevent us from attempting to derive the real part of the meson-meson loops which should be present in the a 0 propagators in eq. ( 6) to fulfill analyticity.
For consistency -as scalar resonance contributions are included-axial-vector current contributions induced from τ − → π − π 0 π 0 decays coming from π 0 − η mixing need to be accounted for as well.This is done following references [83,84,85] (including also the KKπ cuts [86] into the energy-dependent Γ a1 ).The overall factor 2 πη suppresses strongly this contribution, which does not introduce any additional free parameter.
Beyond the Standard Model, the vector and axial-vector matrix elements (corresponding to the dγ µ u and dγ µ γ 5 u quark currents) can be written in terms of the {F i } i=1,...,4 form factors (we omit their dependence on Q 2 , s 1 , s 2 below) which are defined by the currents We can relate the previous terms with the hadronization in eq. ( 3) with the relation where H L contains all SM interactions and H µ V /A ∈ H NSI .The (pseudo)scalar matrix elements can be related to the former using Dirac equation.This shows the vanishing of the hadron matrix element of the scalar current, while the pseudoscalar one (for the dγ 5 u quark current) can be related to H µ A , which is defined as yielding We will finally address the hadronization of the tensor current ( ηπ − π 0 | dσ µν u|0 ) for which we will employ Chiral Perturbation Theory [87] with tensor sources [88].The leading contribution in the chiral counting is given in terms of a single coupling constant, Λ 2 , which can be determined from the lattice [89] to be Λ 2 = (11.1 ± 0.4) MeV [42].In terms of it, the hadron matrix element for the tensor current is
(18) Using Dirac equation, L µ Q µ = M τ L is obtained.This, together with eq. ( 15), allows the convenient rewriting A L µ H µ A − P LH P = A L µ H µ A , where which in turn allows to recast eq. ( 17) as that we have used to compute the observables presented in the following section.We provide an ancillary file with the analytic results for the different contributions to |M| 2 , for which we used FeynCalc [90,91,92].
4 CVC prediction of the τ − → ηπ − π 0 ν τ decay rate and NSI For our isospin-rotated prediction of the τ − → ηπ − π 0 ν τ decays in absence of new physics we will use the CVC relation (see eq. ( 1)), with e + e − → ηπ + π − given by the best fit solutions of refs.[36,37] .Specifically, Fit 4 in ref. [36] and Fit II in ref. [37], respectively.The amplitudes were calculated using RχT [72] and confronted with the latest high statistics experimental measurements of e + e − → ηπ + π − cross sections up to 2.3 GeV, including those of Babar [31], SND [32,34], and CMD3 [35].By isospin rotation, the prediction of the invariant mass spectrum of τ − → ηπ − π 0 ν τ decays is given in Fig. 1.It can be seen that the prediction from e + e − → ηπ + π − is quite different from that of the Belle data [27], especially in the region of 0.9-1.4GeV.The τ → ηπ − π 0 ν τ branching ratio is (1.71±0.13)×10−3 , using the Fit II in Ref. [37], and (1.55±0.18)×10−3 from Fit 4 of Ref. [36].The PDG quotes (1.39 ± 0.07) × 10 −3 instead, from which our previous numbers are 2.2 and 0.8 σ away, respectively.Meanwhile, e + e − → ηπ + π − data are considered much more accurate and trustworthy.Hence, it would be rather important for Belle-II to improve the measurement of this decay channel in the future.The effects of NSI are constrained thanks to the most recent determination (in agreement with previous ones) of these couplings [46], yielding with the correlation matrix M hpp (GeV)   Figure 1: The prediction of τ − → ηπ − π 0 ν τ from e + e − → ηπ + π − amplitudes.The black solid line is calculated from Fit II of Ref. [37] and the red dashed line is from Fit 4 of Ref. [36].The cyan band describes the uncertainty obtained by a combined statistics of the error band of Fit II in Ref. [37] and the difference between the red and black lines.Belle data are represented by purple dots.Figure 2: Invariant mass spectrum of the τ − → ηπ − π 0 ν τ transition.The purple stripe is the error band obtained from the Gaussian variation of parameters at each bin, the solid purple line represents the mean of the distribution.
In our numerical analysis we used 2500 points9 generated randomly following a Gaussian distribution using the parameters and errors in eq. ( 21) and the correlation matrix of eq. ( 22).The vector form factor in ref. [25] was used in the following.
We also computed the invariant mass m ηππ spectrum for which we again used a Gaussian variation of the parameters, generating 2500 points at each bin of the spectrum, shown in figure 2.
Figure 3: Comparison between the spectra obtained using the complete amplitude (purple line) with the NSI turned off (green line) and the Belle data (pale blue dots).The difference between both lines can be slightly appreciated only near the peak.
When comparing the result of the total differential decay width to that obtained only from the SM contribution to the amplitude and the Belle spectrum, shown in Figure 3, we confirm that the possible NSI contribution is undetectable with current data.In Fig. 4 we show a close up of the region where both curves of Fig. 3 differ a bit more.
We also obtained the contributions to the decay width from the different terms in the squared amplitude, this is, pure V , A, P , T , SM terms or only one of the interference terms among them, in turn.This is shown in Figure 5 in a logarithmic scale.In Fig. 6 we show all such contributions, except for the pure SM one, in a normal scale.For most of the phase space the interference of the SM with the vector non-SM interaction dominates.At low invariant masses there is a small window where the SM-tensor and SM-axial interferences overcome it slightly.It is also seen that the SM-tensor interference dominates near the endpoint.It must be noted, however, that the tensor effects at high invariant masses may be smaller than depicted, as we are using for this form

Conclusions
We have studied whether the discrepancy between isospin-rotated σ(e + e − → ηπ + π − ) and dΓ(τ − → ηπ + π 0 ν τ )/dm ηππ data can be explained by heavy new physics beyond the SM.Within an effective field theory approach for the NSI (assuming left-handed neutrinos), and using the bounds obtained previously on the corresponding new physics couplings, we have shown that it is impossible to explain this tension between e + e − and τ data by heavy new physics.Future measurement of the τ decay channel at Belle-II will shed light on the origin of this controversy.As a by-product of our analysis, we have developed the formalism needed to study NSI in three-meson tau decays (see ancillary file), which can be useful for other decay channels where hadronization is more complicated.

Appendix: Brief overview of Resonance Chiral Theory
In this appendix we recapitulate briefly the framework in which the modeldependent contributions have been evaluated [36,37,25], Resonance Chiral Theory (RχT) [72].See, for instance ref. [95] for further details.
Resonances are added as explicit degrees of freedom to the χPT Lagrangian, which is enlarged by terms including them 10 , where the χPT chiral tensors also appear.The symmetries determining the Lagrangian operators are the chiral one for the lightest pseudoscalar mesons (which are pseudoGoldstone bosons) and unitary symmetry for the resonances, SU (3) L ⊗SU (3) R → SU (3) V and U (3) V , respectively, for the three lightest quark flavors.The expansion parameter of RχT is the inverse of the number of colors [96], where the leading order corresponds to tree level diagrams with an infinite tower of mesons per quantum number [96,97] (the most important subleading correction comes from finite resonance widths).
Symmetries do not restrict the coupling values, so these should in principle be determined phenomenologically.However, assuming that the theory with resonances can interpolate between the chiral and parton regimes, Green functions in RχT need to comply with the known (from the corresponding operator product expansion) QCD short-distance behaviour.This determines or relates some of the couplings, increasing the predictivity of RχT.At the same time, this requirement tightly constrains contributions from operators with high-order chiral tensors.Complementary, the number of resonance fields is limited by the process at hand (via the number of initial and final state mesons, to which exchanged resonances couple).Altogether, this restricts, in practice, the number of operators of the RχT Lagrangians in the large-N C limit.The minimal interactions with (pseudo)scalar and (axial)vector resonances are given by [72] see ref. [72] for further details.

54 Figure 4 : 8 Figure 5 : 8 Figure 6 :
Figure 4: Close up of Figure 3 in the small region where they differ.
Cruz work on this topic.Useful discussions on this subject with Gabriel López Castro and Antonio Rodríguez Sánchez are also acknowledged.P. R. was partly funded by Conacyt's project within 'Paradigmas y Controversias de la Ciencia 2022', number 319395, and by Cátedra Marcos Moshinsky (Fundación Marcos Moshinsky) 2020, whose support is also acknowledged by A. G.