Effective-field theory analysis of the $\tau^-\to \pi^-\pi^0\nu_\tau$ decays

We perform an effective field theory analysis of the $\tau^- \to \pi^- \pi^0 \nu_\tau$ decays, that includes the most general interactions between Standard Model fields up to dimension six, assuming left-handed neutrinos. We constrain as much as possible the necessary Standard Model hadronic input using chiral symmetry, dispersion relations, data and asymptotic QCD properties. As a result, we set precise (competitive with low-energy and LHC measurements) bounds on (non-standard) charged current tensor interactions, finding a very small preference for their presence, according to Belle data. Belle-II near future measurements can thus be very useful in either confirming or further restricting new physics tensor current contributions to these decays. For this, the spectrum in the di-pion invariant mass turns out to be particularly promising. Distributions in the angle defined by the $\tau^-$ and $\pi^-$ momenta can also be helpful if measured with less than $10\%$ accuracy, both for non-standard scalar and tensor interactions.


INTRODUCTION
Early studies of nuclear beta decays and, particularly, the problem of apparent nonconservation of energy and violation of the spin-statistics theorem lead to Pauli's postulation of the neutrino. Soon after, Fermi proposed a theory [1] describing these decays which was inspired by QED's vector current interaction which, however, was of a local current-current type. This was the first step towards establishing the V-A nature of the weak force and understanding its maximal parity violation. Now the original Fermi theory is regarded as one of the possible contributions of dimension six effective operators to these decays and it constitutes the basis for effective field theories. In this spirit, not only nuclear beta decays, but also purely leptonic lepton decays, pion decays into a lepton and its corresponding neutrino and also strangeness-changing meson and baryon decays involving a lepton charged current can be studied in a coherent and comprehensive way with direct connection to the underlying theory at some TeVs [2][3][4][5][6][7][8][9][10]. Thus, it is possible to obtain bounds on non-standard charged current interactions from either of these processes that can be compared among them (assuming lepton universality if necessary). As a result, quite generic New Physics (NP) is restricted in absence of deviations from the Standard Model (SM) predictions. In the event of any such departures appearing, one would expect them to point to the underlying new dynamics, as (nuclear) beta and muon decays did with the W mass value (provided the coupling intensity can be estimated from some symmetry argument) and its left-handed couplings.
In ref. [11] we put forward that semileptonic tau decays are also an interesting scenario in this respect. Particularly, our study of the τ − → π − (η/η )ν τ decays [11] showed that they could be competitive with superallowed nuclear beta decays in restricting scalar nonstandard interactions. Our aim in this paper is to extend our previous analysis to the τ − → π − π 0 ν τ decays, which should not be sensitive to NP charged current scalar interactions (as generally, they are very suppressed by the small isospin breaking effects giving rise to them in this decay channel [12]) but could instead be very competitive restricting charged-current tensor interactions.
Only if the SM input (and particularly the hadronization) to the considered decays is well under control one can actually set bounds on NP effective couplings. This is the case for the vector and -to a lesser extent-the scalar interactions (where we will follow the treatment in refs. [13] and [14], respectively) but only a theory-driven approach is possible for the tensor form factor (where we will complement our previous work [11] guided by the recent letter [57]). In all cases it is desirable to fulfill the requirements imposed by the approximate chiral symmetry of QCD, which are automatically enforced in its low-energy effective field theory, Chiral Perturbation Theory (χP T ) [15][16][17]. If possible, it is also convenient to use dispersion relations to warrant analyticity and comply with unitarity, at least in the elastic region (for the ππ system it amounts to ∼ 1 GeV). Within this formalism, known short-distance QCD constraints [18,19] can also be satisfied. In the absence of data (as it the case for the tensor form factor) enlarging the domain of applicability of χP T coupled to tensor sources [20] by including resonances as explicit degrees of freedom [21,22] could seem useful, although we will show in the appendix of this paper that it is not the case.
This work is organized as follows: in section 2 we present the basics for an effective field theory treatment of the considered decays. In section 3 the different contributions to the matrix element are identified and the participant meson form factors defined. These are the subject of section 4, with a special focus on the tensor form factor. With all SM contributions fixed, we perform a phenomenological study in search for NP signatures, examining the hadron spectrum and branching ratio, the Dalitz plot distributions and the forward-backward asymmetry in section 5. The conclusions of this research are summarized in section 6.

EFFECTIVE THEORY ANALYSIS OF
For low-energy charged current processes, the effective Lagrangian with SU (2) ⊗ U (1) invariant dimension six operators 1 reads [2,3] L If we particularize it for the O(1 GeV) semileptonic strangeness and lepton-flavor conserving 2 charged current transitions involving any lepton ( = e, µ, τ ) and only left-handed neutrino fields, the following Lagrangian is obtained (where subscripts L(R) stand for lefthandedness (right-handedness)) In the previous equation G F is the tree-level definition of the Fermi constant and Heavy degrees of freedom (H, W ± and Z bosons plus c, b and t quarks) have been integrated out to obtain eq. (2.2). The effective couplings v L,R , s L,R and t L generated by the NP can be taken real since we are only interested in CP conserving observables 3 .
Although observables are renormalization scale and scheme independent, this scale independence comes after the cancellation of the scale dependence of the effective couplings (v L,R , s L,R and t L ) by the corresponding scale dependence of the hadronic matrix elements. These encode the amplitude for the quark current to produce/annihilate the measured hadrons. As it conventional, we select µ = 2 GeV as the renormalization scale.
It is advantageous to shift our basis for the spin-zero currents so that the new ones have defined parity. This is achieved by means of introducing S = s L + s R and P = s L − s R .
Although the other elements in the basis of currents remain unmodified, we also rename them to avoid any confusion between both bases: R,L = v L,R and T = t L .
One can proceed with = e, µ, τ in full generality (which may be profitable if lepton universality is an approximate symmetry). We, however, focus now on the tau case, in such a way that the corresponding semileptonic effective Lagrangian is: whereˆ i ≡ i /(1 + L + R ) for i = R, S, P, T . From this expression it is easily seen that, working at linear order in theˆ i , one is insensitive to non-standard spin-one charged current interactions because the overall dependence on L + R cannot be isolated, as it is subsumed in the determination of G F . That is, conveniently normalized rates cancel the overall factor (1 + L + R ) in the previous equation. We note that, at linear order in thê i 's, these agree with ref. [3].

SEMILEPTONIC τ DECAY AMPLITUDE
From now on, we will study the semileptonic τ − → π − (P π − ) π 0 (P π 0 ) ν τ (P ) decays, where pions parity determines that only scalar, vector and tensor currents contribute. The decay amplitude reads 4 where the following lepton currents were introduced: The scalar (H), vector (H µ ) and tensor (H µν ) hadron matrix elements entering eq. (3.1) can be decomposed using Lorentz invariance and discrete QCD symmetries in terms of a number of allowed Lorentz structures times the corresponding form factors, which are scalar functions encoding the hadronization procedure. Specifically, these are In the previous equations, the momentum of the meson system is q µ = (P π − + P π 0 ) µ , with s = q 2 . We also introduced Q µ = (P π − − P π 0 ) µ + (∆ π 0 π − /s)q µ , and ∆ π 0 π − = m 2 π 0 − m 2 π − . Clebsch-Gordan flavor coefficients are C S = C V = √ 2 for this decay channel. The F S (s) and F 0 (s) form factors can be related by taking the divergence of the vector current via As in ref. [11], the scalar contribution can be absorbed into the vector current amplitude. This can achieved by replacing , (3.5) in eq. (3.3b).
Obtaining the F 0 (s), F + (s) and F T (s) form factors is discussed in the following section.

HADRONIZATION OF THE SCALAR, VECTOR AND TENSOR CURRENTS
Lorentz invariance, together with the discrete symmetries of the strong interactions, determine eqs. (3.3a) to (3.3c). QCD dynamics is encoded in these hadron matrix elements, although it is not possible to determine them using the Lagrangian of the underlying theory unambiguously. Nevertheless, QCD properties are useful in restricting this hadronic input. On the one hand, it is desirable to keep the properties derived from the (very approximate) chiral symmetry of low-energy QCD and from asymptotic strong interactions, where known. On the other, using dispersion relations is ideal to warrant the correct analytic structure of the amplitudes and to comply with unitarity (at least in the elastic region). These properties will be exploited in what follows, as we will briefly review.
As shown in ref. [13], the scalar form factor F 0 (s) can be determined in an essentially model-independent way in the low-energy region, though it does not involve resonance contributions to first order in isospin breaking. The S-wave π − π 0 system must have isospin I = 2. Watson's final-state interactions theorem [33] ensures that -in the elastic regionthe phase of the di-meson form factor with definite angular momentum (L) and isospin (I) coincides with the corresponding meson-meson scattering phase shift having the same L and I values (L = 0 and I = 2 in our case, so this phase shift is δ 2 0 (s) according to the usual notation). Neglecting inelastic effects (that is a good approximation up to s ∼ 1 GeV 2 in this case), the required di-pion scalar form factor can be obtained [13] by means of a phase dispersive representation (F 0 (0) = 1 has been used) since the phase shift δ 2 0 (s) has been measured [34,35]. |F 0 (s)| and δ 2 0 (s) are plotted in the upper panel of Fig. 12 in ref. [13]. As expected, there is no hint of resonance dynamics in F 0 (s).
Theoretically, F + (s) is well-constrained at low-energies by χP T [15][16][17] and in the asymptotic regime by short-distance QCD results [18,19]. In the intermediate energy (O(1) GeV) region, resonance dynamics is needed to interpolate between the two former limits. An adequate tool to connect all energy ranges taking advantage of analyticity and unitarity constraints on F + (s) are the dispersion relations, which have been employed widely in this context (see i. e. ref. [14] and references therein). We will not discuss at length the procedure here, but only recall that an excellent description of the data can be achieved with three subtractions (one is used to set F + (0) = 1) being α 1,2 the remaining subtraction constants, to be fitted to low-energy data, and δ 1 1 (s) the relevant phase shift. In ref. [14], δ 1 1 (s) is given in terms of the ρ(770) pole position and the pion decay constant, F π . We will follow this description (using their best fit results) in what follows. The modulus and phase of F + (s) are plotted and compared to data in Figs. 1 and 2 in ref. [14].
Although it is difficult to constrain the hadronization of the tensor current, eq. (3.3c), from first principles, this would be desirable as it turns out that the τ − → π − π 0 ν τ decays have the potential to set competitive bounds on (non-standard) charged current tensor interactions. This is in contrast with the τ − → π − η ( ) ν τ decays explored in ref. [11], which are competitive for new scalar contributions but not for tensor ones, which justified using leading-order χP T results for eq. (3.3c) in that analysis. Unfortunately, there is no experimental data that can guide us in building F T (s), so will rely only on theory to accomplish this task.
Since s can vary from the two-pion threshold up to M 2 τ , light resonances contribution (giving the energy dependence of the form factor) should be included in a refined analysis, as we intend. We show in the appendix that, for F T (s), it is not convenient to extend the energy range of applicability of χP T by including the resonances as explicit degrees of freedom, in the so-called Resonance Chiral Theory [21]. Instead, it will be more appropriate to use a dispersive construction of F T (s) taking advantage of unitarity constraints on its phase [57]. F T (0) will be fixed from χP T in the following.
The lowest-order χP T Lagrangian with tensor sources, which is O(p 4 ) in the chiral counting [20], includes only four operators. Among them, only the one with coefficient Λ 2 contributes to the studied decays: In the preceding equation, t µν + = u † t µν u † + ut µν † u and · · · means a flavor space trace. Operators in eq. (4.3) are built with chiral tensors [51], with three of them entering the displayed operators:

which includes the left-and right-handed
sources, µ and r µ .
• The chiral tensor sources t µν and its adjoint, and including the left-and right-handed field-strength tensors, F µν L and F µν R , given in terms of µ and r µ .
Let us recall the non-linear representation of the pseudo Goldstone bosons, given by F being the pion decay constant in the chiral limit, F ∼ F π ∼ 92 MeV. All resonance multiplets considered below have analogous flavor structure to eq. (4.4). The tensor source (t µν ) is related to its chiral projections (t µν and t µν † ) by means of [20] t µν = P µνλρ Lt λρ , 4P µνλρ L = (g µλ g νρ − g µρ g νλ + i µνλρ ), (4.5) whereΨσ µνt µν Ψ is the tensor quark current.
From eq. (4.3) it can be shown [11] that, in the limit of isospin symmetry 5 , We show in the appendix that it is not convenient to include the energy-dependence of the tensor form factor by extending χP T [15][16][17] including resonances [21,22]. We will follow the procedure in ref. [11] for the estimation of Λ 2 . Although we are not aware any QCD restriction on it, short-distance QCD properties [56] do predict Λ 1 : which corresponds to Λ 1 4πF = 0.028 ± 0.002. Since both operators displayed in eq. (4.3) have the same chiral counting order, we assume Λ 2 ∼ Λ 1 and take |Λ 2 | 4πF ≤ 0.05 as a conservative estimate. In our numerics we will consider both Λ 2 4πF = 0.05, and half this value. We will use Λ max 2 and Λ min 2 for them, respectively. We will follow ref. [57] and obtain F T (s) using again a phase dispersive representation. As shown in ref. [57] (see also the appendix of this article), the tensor form factor phase equals the vector form factor phase, δ T (s) = δ + (s), in the elastic region. We will use the previous equation also above the onset of inelasticities in our dispersion relation and fix F T (0) = √ 2Λ 2 F 2 according to the leading-order χP T result. We will consider Λ 2 = Λ min 2 and Λ 2 = Λ max 2 in our numerical evaluations. We plot in figure 1 the modulus and phase of F T (s) obtained using eq. (4.8). The different curves on the left panel are obtained for s max = M 2 τ , 4 and 9 GeV 2 6 and we will take this range for F T (s) as an estimate of our corresponding error (our plots will be given for s max = 4 GeV 2 in the following). We neglect the uncertainty associated to our ignorance on the inelasticities affecting δ T (s) (see the related discussion in ref. [57]), which are small below √ s = 1.3 GeV.
In order to study possible NP effects in these decays, one should use not only the hadronic spectrum and branching ratio, but also Dalitz plot distributions and the measurable forwardbackward asymmetry. In this section, we focus in the study of the possible effects of the non-standard effective couplings described in section 2 in these τ − → π − π 0 ν τ decay observables. We will start with the Dalitz plots (which should contain more dynamical information, as no integration over any of the two independent kinematical variables has been performed) and move later on to (partially) integrated observables: differential decay rate as function of the di-meson invariant mass, forward-backward asymmetry and, finally, branching ratio.

Dalitz plot
Including possible non-standard weak charged current interactions, the unpolarized spinaveraged squared amplitude yields 7 where the scalar, vector and tensor squared amplitudes are M 00 , M ++ and M T T , respectively. Their corresponding interferences are denoted M 0+ , M T + , M T 0 . All these read where the familiar definitions ∆ π − π 0 = m 2 π − −m 2 π 0 and Σ π − π 0 = m 2 π − +m 2 π 0 were employed. Noteworthy, the scalar form factor is always suppressed by ∆ π − π 0 , which is tiny, in the previous equations for M 00 , M T 0 and M 0+ . This makes its effect negligible even for |ˆ S | ∼ 1 (radiative pion decay limits |ˆ S | 0.01 and, under the reasonable assumption of lepton flavor universality, this limit should also apply for the tau flavor considered here), but for the special case of the forward-backward asymmetry, as we will discuss. We now turn to analyze possible NP signatures in Dalitz plots distributions. The left panel of figure 2 shows the squared the matrix element |M| 2 00 in the (s,t) plane, which is obtained using the SM predictions for τ − → π − π 0 ν τ form factors [13,14]. The ρ(770) meson dominance of the dynamics is clearly seen in this plot. In the left panel of figures 3 and 4, the same quantity is shown for two representative values of the set of (ˆ S ,ˆ T ) parameters that are consistent with the known information on the BR(τ − → π − π 0 ν τ ) decays (obtaining these limits will be discussed in subsection 5.5). Comparing these graphs yields to the conclusion that Dalitz plot distributions are not sensitive to the effects of nonstandard tensor and scalar interactions in the considered decays.

Angular distribution
The hadronic mass and angular distributions are also modified by the generic new effective interactions that we are studying and can have different sensitivity toˆ S andˆ T . The rest  frame of the hadronic system is convenient for this analysis. It is defined by p π − + p π 0 = p τ − p ν = 0. In this frame, the charged particle energies are given by E τ = (s+M 2 τ )/2 √ s and E π − = (s + m 2 π − − m 2 π 0 )/2 √ s. The measurable angle θ between these two particles can be obtained from the invariant t variable by means of t = m 2 The Dalitz decay distribution in the (s, cos θ) variables, for generalˆ S andˆ T reads which coincides with the SM result when these two effective NP couplings are set to zero.
The right panel of figure 2 shows eq. (5.5) for π − π 0 in the SM case. In the right panel of figures 3 and 4 the (s, cos θ) distributions are plotted, for the same representative values of (ˆ S ,ˆ T ) used in order to obtain the left panel of these figures. A comparison between figures 2, 3 and 4 shows that SM and plausible NP distributions cannot be differentiated using the (s, t) or the (s, cos θ) Dalitz plot analyses.

Decay rate
The di-pion invariant mass distributions is obtained integrating upon the t variable in eq. (5.1) where Again, the SM limit is recovered with L = R =ˆ S =ˆ T = 0. Figure 5 plots the invariant mass distribution of the di-pion system for τ − → π − π 0 ν τ decays. It is almost impossible to distinguish the case of tensor interactions from the SM curve and, although some departure is seen for non-standard scalar interactions, it goes away when realistic values on |ˆ S | ∼ 10 −2 [2,3,11] are considered.

Forward-backward asymmetry
The forward-backward asymmetry is defined [13] by We can obtain it for τ − → π − π 0 ν τ decays plugging in eq. (5.5) into eq. (5.8) and integrating upon the cos θ variable, where, again, the SM forward-backward asymmetry is recovered for R = L =ˆ S =ˆ T = 0. This reference case is plotted in figure 6, which agrees with the prediction in ref. [13] (this asymmetry was first studied in ref. [63]). In figure 7 we plot this forward-backward asymmetry for the same representatives values of (ˆ S ,ˆ T ) parameters used to obtain figs. 3 and 4. In theˆ S = 0 case one might get excited because of the notable (clearly measurable) difference with the SM case. However, as commented before, τ − → π − π 0 ν τ decays are not competitive setting bounds on non-standard scalar interactions. According to known information from light quark beta decays [2,3] and rare tau decays [11], |ˆ S | must be some two orders of magnitude smaller than in figure 7. As a result, the difference with the SM case goes mostly away and only a fine measurement could reveal this type of NP, as shown in figure 8. It is interesting to note, that while using the limits on |ˆ S | from refs. [2,3] requires assuming lepton universality, this is not the case for the limits in ref. [11] so the conclusions drawn from figure 8 hold irrespective of this assumption. The tensor interaction cannot be distinguished from the SM case in any region of the plots in figures 7 and 8.  Figure 6. The forward-backward asymmetry in the τ − → π − π 0 ν τ decay as a function of the ππ energy for the SM case.
As advanced before, A ππ (s) in eq. (5.9) is a good observable for finding non-standard scalar interactions: despite its numerator is suppressed by the small value of ∆ π − π 0 , its denominator is further suppressed by the dependence of X S 2 on ∆ 2 π − π 0 , which enhances the sensitivity of this forward-backward asymmetry to scalar contributions. However, as just observed, if the strict limits on |ˆ S | obtained in other low-energy processes are applied, even A ππ (s) happens to be unable of evidencing this kind of NP contributions.  Figure 9. ∆ as a function ofˆ S forˆ T = 0 (left-hand) andˆ T forˆ S = 0 (right-hand) for τ − → π − π 0 ν τ decays. Horizontal lines represent the values of ∆ according to the current measurement (at three standard deviations) of the branching ratio (dashed line) and the hypothetical case of this value being measured by Belle-II with three times reduced error (dotted line).
The τ − → π − π 0 ν τ decay width can be obtained integrating the invariant mass dis-tribution, using the expressions for the form factors [13,14]. Since the total decay width depends on the effective couplings, this process branching ratio sets bounds onˆ S andˆ T 9 .
For that, we compare the decay rate (Γ) for τ − → π − π 0 ν τ in the presence of non-vanishing NP effective couplings with respect to the one (Γ 0 ) obtained by neglecting them (SM case). Integrating eq. (5.6) we get the shift produced by NP contributions as follows for whose coefficients we get: α = 3.5 × 10 −4 , β = 15, γ = 2.2 × 10 −2 and δ = 105 (for Λ 2 = Λ min 2 we obtain β = 8 and δ = 26). Eq. (5.10) is a quadratic function of the effective scalar and tensor couplings, which can be used to explore the sensitivity of τ − → π − π 0 ν τ decays to non-standard scalar and tensor interactions. We will do this in two steps. Firstly, we can make the analysis for one vanishing and one non-vanishing coupling. This is shown in figure 9 where we represent with horizontal lines the current experimental limits on ∆ (at three standard deviations) and use eq. (5.10) to translate this information into bounds forˆ S andˆ T . According to this procedure, we get the following constraint −0.944 ≤ˆ S ≤ 0.928 withˆ T = 0 and −0.16 ≤ˆ T ≤ 1.3 × 10 −3 withˆ S = 0. The upper limits of the previous inequalities were used to estimate the values ofˆ S andˆ T which were employed in the preceding subsections:ˆ S ∼ 0.93 andˆ T ∼ 0.001. The dotted lines illustrate how the limits would evolve for an error reduced by a factor three, which could be achieved at Belle-II. Interestingly, in this optimistic case, one could start excluding the SM case (although the statistical significance shown in the plot must be taken with a grain of salt: all precision analysis of F + (s) should agree on this conclusion and our estimated value of Λ 2 needs to be confirmed).
Then, we can also fix joint constraints on the scalar and tensor effective interactions assuming bothˆ S andˆ T non-vanishing and using again eq. (5.10) as before. This result is shown in figure 10, where the limits on the scalar and tensor couplings are contained inside an ellipse in theˆ S −ˆ T plane. As a rough estimate of the possible impact of Belle-II data we repeat the exercise of assuming a factor three reduction with respect to the Belle-I error.
As indicated by the dashed lines of the figure 10 (right panel), this would create a sizable tension with the SM case (with the restrictions on systematic theoretical uncertainties just pointed out). Table 1 summarizes the constraints on the scalar and tensor effective couplings that can be obtained from the Belle measurement of the branching ratio for τ − → π − π 0 ν τ decays. Moreover, table 1 shows the results that could be reached from the future measurements of these decays by Belle-II with an error reduced to a third of the current one. Table 2 shows the analogous results to table 1, but now using Λ 2 = Λ min 2 . Next we consider fits to the data reported by Belle [36] for the normalized spectrum 9 Obviously, the results forˆ T will differ depending on whether we use Λ2 = Λ max 2 or Λ2 = Λ min 2 . Both are considered, as limiting cases, in the following.  Figure 10. Constraints on the scalar and tensor couplings obtained from ∆(τ − → π − π 0 ν τ ) using the Belle measurement (at three standard deviations) of the branching ratio. The left-hand plot shows the constraints obtained from current data. On the right-hand plot we show a magnification of the top part of this ellipse, where the solid line represents the upper limit onˆ S andˆ T , while the dashed lines represent the upper and lower limits on these couplings that could be obtained if the current branching ratio error is reduced by a factor three in the Belle-II experiment. Table 1. Constraints on the scalar and tensor couplings obtained through the limits on the current branching ratio measurements and the hypothetical case where this value be measured by Belle II with a three times smaller error using Λ 2 = Λ max  (1/N ππ )(dN ππ /ds) and integrated branching ratio using the function 10 In this way, we found the following constraints: |ˆ S | < 0.52 for the scalar coupling (at one-sigma), andˆ T = −1.9 +1.3 −0.8 × 10 −2 for the tensor coupling (the uncertainty on Λ 2 was taken into account), which shows a moderate 1.5 sigma discrepancy with respect to the SM, due to charged current tensor interactions that should be checked with more precise measurements of these decays and scrutinizing F + (s), hopefully with improved knowledge on Λ 2 . As expected, the constraints on the scalar coupling are not competitive with refs. [2,3,11], because of the suppression of the scalar form factor by ∆ π − π 0 . Considering this last feature, we can perform more realistic fits restricting |ˆ S | < 0.8 × 10 −2 [2,3] and fitting onlyˆ T . In this case we getˆ T = (−0.42 ± 0.32) × 10 −3 for Λ 2 = Λ max 2 and T = (−0.85 ± 0.64) × 10 −3 for Λ 2 = Λ min 2 . In either case, the deviation from the SM case is slightly reduced to ∼ 1.3σ, which could be combined intoˆ T = −0.64 +0. 54 −0.85 × 10 −3 11 . A caveat is, of course, in order: although chiral symmetry (at low energies) and the use of dispersion relations together with precise measurements (especially useful outside the χP T regime of applicability) makes us confident on our knowledge of the vector two-pion form factor, F + (s), one should be very cautious before claiming evidence for NP from this type of analysis 12 . Provided a hint for an anomaly appears, different investigations should be performed to test it: it may be worth considering a dispersive coupled-channel analysis of the two-pion and two-kaon vector form factors [59][60][61][62], one should analyze along these lines the compatibility between the F + (s) form factor measured by Belle and the L = 1 = I ππ scattering amplitude, a lattice evaluation of Λ 2 can (dis)prove our estimate and give a reliable error...

SUMMARY AND CONCLUSIONS
We have considered the τ − → π − π 0 ν τ decays in the presence of generic New Physics effective interactions up to dimension-six operators, assuming left-handed neutrinos and that the new dynamics scale is in the multi-TeV range. Within this setting, we have paid particular attention to the hadron matrix elements, which are needed SM inputs in order to set bounds on the non-standard scalar and tensor couplings,ˆ S andˆ T , respectively (we recall that it is not possible to restrict spin-one non-standard interactions in the considered processes). For this, we have employed previous results using dispersion relations for the 11 For completeness, we quote that fitting bothˆ S andˆ T the χ 2 /dof is ∼ 0.85 and it slightly grows to ∼ 1.18 when |ˆ S | < 0.8 × 10 −2 is used and onlyˆ T is fitted. Best SM fits yield χ 2 /dof ∼ 1.3. 12 In the case of τ − → π − (η/η )ντ decays [11] this would be noticeably more difficult: although the hadronization of the vector current is given again in terms of the precisely-known two-pion vector form factor, the dominant scalar contribution is subject to large uncertainties still [58]. scalar [13], vector [14] and tensor [57] form factors implementing the known chiral constraints at low energies and QCD asymptotics at short distances, according to data. For the tensor form factor, since no experimental information is available, we have pursued a purely theoretical determination of its leading chiral behaviour using Chiral Perturbation Theory. In this work, we improved over our previous treatment of the tensor form factor where only leading-order chiral predictions were considered and unitarity constraints were ignored [11], motivated here by the fact that di-pion tau decays constitute an excellent arena to set competitive limits onˆ T . Use of short-distance QCD constraints and the chiral counting allowed estimating the only leading low-energy constant of the tensor form factor, permitting a direct access toˆ T .
Within this framework, we have set bounds onˆ S andˆ T using the measured Belle branching ratio, through our observable ∆. This procedure yields quite competitive limits with the world-best bounds for the tensor case (that we have thus used in the remaining analysis), but quite poor (unrealistic assuming some reasonable approximate lepton universality holds for them) in the scalar case, which is a consequence of its suppression in all considered observables (but the forward-backward asymmetry) by the tiny difference between charged and neutral pion masses squared. Because of this feature, we have assumed S limits similar to those obtained in light quark beta and τ − → π − (η/η )ν τ decays in the remaining analysis.
As a result of our study, it turns out that Dalitz plot distributions (both in the Mandelstam variables s and t and also replacing t by the angle between the two charged particles) are not sensitive to non-zero realistic values ofˆ S andˆ T , as it also happens with the forward-backward asymmetry. Apparently, the hadronic invariant mass distribution is not sensitive either to charged-current tensor interactions. However, a fit to Belle data on this observable hints for a slight preference for non-zeroˆ T . Therefore, it is very worth measuring with extreme precision the di-pion invariant mass distribution in τ − → π − π 0 ν τ decays at Belle-II, as it will serve to further restrictˆ T and this way offer complementary information to other low-energy processes in the searches for non-standard charged current interactions. This effort would need to come together with both a tight scrutiny of the dominant vector form factor SM prediction and a lattice evaluation of the lowest-order chiral coupling of the tensor source to a pion pair in order to eventually pin down the NP contribution.

APPENDIX: F T (s) including resonances as explicit degrees of freedom
We show in this appendix that it is not convenient to build F T (s)/F T (0) including resonances as explicit degrees of freedom.
As we will see, the tensor current couples to the J P C = 1 −− and J P C = 1 +− resonances, but the contribution of the second tower of resonances is suppressed in the processes under consideration. This can be seen phenomenologically, since the b 1 (1235) resonance (which shares all quantum numbers with the ρ(770) meson but has opposed parity) is not known to couple to the two-pion system (precisely because of parity b 1 cannot decay into two pseudoscalars, though it could be exchanged in meson-meson scattering, but ππ scattering data do not show any hint for exchange of the b 1 meson). Therefore, the ρ(770) is the lightest resonance whose exchange provides an energy-dependence to F T , increasing its effect and allowing us to set more restrictive bounds onˆ T (we neglect the contributions from ρ excitations in this study).
We shall now discuss the chiral couplings of meson resonances to the pseudoscalar Goldstone fields in the presence of tensor currents. We use the antisymmetric tensor representation [21,22] in order to describe the relevant spin-one degrees of freedom. To determine the resonance exchange contributions to the τ − → π − π 0 ν τ decays (or to the effective chiral Lagrangian) we need the lowest order operators in the chiral expansion which are linear in the resonance fields. Using the P and C transformation properties of given J P C resonance fields: Table 2 in ref. [21]), and H(1 +− ) and T (2 ++ ) (see ref. [54]), we can, for the first time, construct the RχT Lagrangian linear in resonance fields and coupled to the tensor source of lowest chiral order, which has the following two pieces: In the following, we neglect the effect of the latter operator (assuming F T H negligible) because of the seemingly small b 1 ππ coupling commented above. A straightforward computation of the contribution of the former operator to the relevant hadronic matrix element yields where in which the operator iG V √ 2 V µν u µ u ν [21] was used in order to obtain the ρππ coupling. Eq. (6.3) depends on three a priori unknown couplings. Fortunately, short-distance QCD properties can shed light on their values, as we explain next. First, it is known from the analysis of two-point correlators within RχT that G V = F/ √ 2 [21] (also F V = √ 2F , which is used next). The large-N C asymptotic analysis of V V , T T and V T correlators determines F T V /F V = 1/ √ 2 [55], in such a way that only Λ 2 remains unrestricted and eq. (6.3) simplifies to Taking the upper bound for Λ 2 it is seen that the ρ meson contribution shifts the value of F T (0) by ∼ 13%.
As in the case of the vector form factor, the ρ-propagator in eq. (6.3) is modified by the inclusion of the width Γ ρ (s) (proportional to the imaginary part of the corresponding loop contributions) and also by shifting the pole mass value (according to the real part of the loop contribution), as required by analyticity. Specifically, x . The tensor form factor, F T (s), given by eq. (6.4), and using the substitution eq. (6.5), is plotted in figure 11 for Λ 2 = Λ max 2 . There, it is seen how the ρ(770) meson contribution modifies the constant χP T lowest-order result for |F T (s)|. The form factor phase, δ T (s), grows from zero to ∼ π/4 at √ s = M ρ and decreases softly to zero for larger energies. Both |F T (s)| and δ T (s) are influenced by the on-shell ρ(770) meson width as expected, according to its value of ∼ 145 MeV. The effect of varying the value of Λ 2 is illustrated in fig. 12, where Λ 2 = Λ min 2 is used. At this point unitarity arguments may convince us that this description of F T (s) cannot be complete 13 . As explained in ref. [57], the phase of F T (s) must coincide with the phase of F + (s) in the elastic region (in this paper this was shown for the tau decays into the Kπ system, but it is completely analogous to the ππ one considered here). We briefly review the argument in what follows.
The unitarity relation for F + (s) can be written mF + (s) = σ π (s)F + (s)(f 1 1 (s)) * θ(s − 4m 2 π ) , (6.7) 13 We thank Bastian Kubis for pointing this to us. where f 1 1 (s) is the the corresponding partial wave in ππ scattering. The previous equation implies that, in the elastic region, δ 1 1 (s) = δ + (s), which is again Watson's theorem. The crucial point is that an analogous unitarity relation holds for F T (s): mF T (s) = σ π (s)F T (s)(f 1 1 (s)) * θ(s − 4m 2 π ) , (6.8) from which one can immediately derive that, in the elastic region, δ T (s) = δ + (s), a feature that is not satisfied by our expression for F T (s) considered up to now (and it will not be satisfied for any value of Λ 2 ). This should not be understood as a failure of eq. (6.4) (together with eq. (6.5)), but rather as a manifestation of its incompleteness. Indeed, the contributions from the next-to-leading order χP T Lagrangian with tensor sources (O(p 6 ) in the chiral counting [20]) should provide with the needed energy-dependence to satisfy eq. (6.8). However, since the number of such operators is 75 (plus 3 contact terms) even in the SU (2) case [20], we refrain from proceeding this way as any predictability would be lost.