New $\tau$-based evaluation of the hadronic contribution to the vacuum polarization piece of the muon anomalous magnetic moment

We revisit the isospin-breaking and electromagnetic corrections to the decay $\tau^-\to\pi^-\pi^0\nu_\tau$, which allow its use as input in the two-pion contribution to the (leading order) hadronic vacuum polarization part of the muon anomalous magnetic moment. We extend a previous resonance chiral Lagrangian analysis, which included those operators saturating the next-to-leading order chiral low energy constants, by including the contributions of the next subleading terms. As a result, we improve agreement between the two-pion tau decay and $e^+e^-$ data and reduce the discrepancy between experiment and the SM prediction of $a_\mu$ (using $\tau$ input).


Introduction
The anomalous magnetic moment of the (first electron, and then) muon (a µ ≡ (g µ − 2)/2) has been crucial for the development of quantum field theory and the understanding of radiative corrections within it. Over the years, it has validated those computed in QED at increasing precision and (in the muon case) started probing the other Standard Model sectors, electroweak and QCD, setting also -and more interestingly-stringent constraints on new physics contributions. In the absence of any direct hint for heavy new particles or interactions at the LHC, clean observables both from experiment and theory -among which a µ stands out-are reinforced as a promising gate for the eagerly awaited further (indirect) discoveries in high-energy physics.
With the forthcoming measurement of a µ at FNAL [1] we will finally have an experimental update on the long-standing discrepancy (at 3 to 4 sigmas) between the SM prediction of this observable (recently refined in [2]) 1 and its most accurate measurement, at BNL [41]. On the theory side, a tremendous effort driven by the Muon g-2 Theory Initiative 2 has been reducing (and making more robust) the SM errors during the last few years, in order to profit maximally from the new data. In the near future, both the FNAL [1] and the J-PARC [42] experiments will shrink the current experimental uncertainty (63 · 10 −11 ) by a factor four. A commensurate improvement on the theory error is essential in maximizing the reach on new physics of these measurements.
The SM uncertainty on a µ (43 · 10 −11 ) is saturated by that of the hadronic contributions, where the error of the dominant hadronic vacuum polarization (HVP,LO) part has been reduced to 40 · 10 −11 , versus 17 · 10 −11 of the light-by-light piece [2]. In turn, the HVP,LO contribution is dominated by the ππ cut (yielding ∼ 73% of the overall value), where good-quality data of the corresponding e + e − hadronic cross-sections [43][44][45][46][47][48][49][50][51][52] enables its computation by dispersive methods [53,54]. Alternatively, one can also use isospinrotated τ → ππν τ measurements with that purpose, as was put forward in LEP times [55]. Although one may claim the former procedure is theoretically cleaner, it is nevertheless a convenient test checking the consistency of both extractions of a HV P,LOππ µ , especially in light of the tensions between different sets of e + e − → π + π − data that has not been resolved so far [2]. In addition to the previous data-based determinations of a HV P,LO µ , lattice QCD is also achieving computations with reduced errors, although not yet competitive with the e + e − evaluations [2]. One notable exception to this being the recent very accurate result (45 · 10 −11 error) of the BMW Coll. [56], according to which the difference with respect to the SM prediction is at the one sigma level.
Concerning the tau based determination, Refs. [57,58] computed the required isospin violating and electromagnetic corrections using Resonance Chiral Theory (RχT ) [59,60] and refs. [61,62] using Vector Meson Dominance (VMD). These series of articles were employed by ref. [63] (updated in Refs. [64,65]) which, remarkably, found that the discrepancy of the SM prediction with the measurement is reduced substantially when tau data is employed 3 . Notwithstanding, as precise measurements of σ(e + e − → hadrons) became available in the last fifteen years, the e + e − based evaluation gained preference over using tau data. Indeed, ref. [2] concludes that 'at the required precision to match the e + e − data, the present understanding of the IB (isospin breaking) corrections to τ data is unfortunately not yet at a level allowing their use for the HVP dispersion integrals', despite ref. [66] claiming that (the model-dependent) ρ − γ mixing in the neutral channel makes it agree with the results in the charged current. It is the purpose of this work 4 to extend previous analyses [57,58] of the required IB corrections to di-pion tau decays so that they can again be useful, when combined with σ(e + e − → π + π − (γ)), to increase the accuracy of the SM prediction of a HV P,LO µ .
Within the global effort of the Muon g-2 theory initiative, we revisit in this paper the RχT computations including operators that -in the chiral limit-start to contribute at O(p 6 ). This is possible by the knowledge acquired after the analyses of Cirigliano et al. [57,58], through a series of works studying OPE restrictions on RχT couplings on several relevant Green functions (and related form factors) [31, 5 . This procedure will also allow us to compute a robust error for the O(p 4 ) prediction, which is one of the main outcomes of this work, together with the new O(p 6 ) results.
The paper is organized as follows. In section 1 we review the main features of the τ − → π − π 0 ν τ γ decays and split the model-independent part from the hadron form factors, computed in RχT including new terms, subleading in the chiral expansion. We then recall the short-distance (SD) QCD constraints on the Lagrangian couplings, their phenomenological determinations and explain our estimation of the remaining free couplings, based on chiral counting. After that, in section 2 we recap the radiative corrections needed for the tau-based calculation of a HV P,LO µ and predict several observables for the processes where the real photon is detected together with the pion pair. Then, in section 3 we evaluate a HV P,LO|ππ µ using tau data, which is the main result of this article. Finally, our conclusions are presented in section 4. Several appendices complement the main material, giving a full account of the kinematics and the complete expressions for the structure-dependent (axial-)vector form factors of the τ − → π − π 0 ν τ γ decays.

Amplitude
For the radiative decay τ − (P ) → π − (p − ) π 0 (p 0 ) ν τ (q) γ (k), we can split the contribution due to the bremsstrahlung off the initial tau lepton from the one coming from the hadronic part.

Theoretical framework
We will present in the following the model-dependent contributions to the V µν and A µν tensors. We will closely follow ref. [58], extending it to include subleading terms in the chiral expansion. In this reference, a large-N C [92][93][94] inspired computation was carried out. Specifically, it was restricted to the dominant (for N C → ∞) tree level diagrams, although the relevant loop corrections for the τ − → π − π 0 ν τ γ decays -giving the ρ (and a 1 , for completeness) off-shell width 6 -were taken into account 7 . Also, given the limited phase space of tau decays and the fact that the region E M ρ + Γ ρ is the most important one for the IB corrections needed for a HV P,LOππ µ [58], the contribution of the ρ(1450) and other heavier resonances was neglected in this reference (despite the fact that, in the large-N C limit, there is an infinite tower of resonances per channel), as we will also do 8 . Within this setting, our computation will include all RχT operators contributing -in the chiral limit-up to O(p 6 ) in the chiral expansion. Our results agree with those in ref. [58] up to the O(p 4 ) included there, providing the new contributions at O(p 6 ) (where possible, our computations at this order have been checked against the results in ref. [89]).
As explained in ref. [58], this procedure warrants the correct low-energy limit (as given by Chiral Perturbation Theory) and includes consistently the most general pion and photon interactions with the lightest resonances. Demanding the known QCD short-distance constraints results in relations among the Lagrangian couplings, and chiral counting can be employed to estimate those still unconstrained after using phenomenological information. It should then provide an accurate description of the τ − → π − π 0 ν τ γ decays for s 1 GeV 2 , which gives ∼ 99.8% of the whole a HV P,LO|ππ µ contribution.

Vector Form Factors
Within RχT [59,60], the diagrams contributing to the vector form factors of the τ − → π − π 0 γν τ decays at O p 6 in Chiral Perturbation Theory [106][107][108][109][110] are shown in Figs. 1, 2 and 3 9 . The first three diagrams in Fig. 1 and the first diagram in Fig. 2 contribute to the pion vector form factor entering the structure-independent (SI) piece (1.6) The contribution of both the last diagram in Fig. 1 and the last diagram in Fig. 2 vanishes for a real photon, as the corresponding (f + (0) = 1 part) contribution is already in the SI piece. We note we are using F ∼ 92 MeV for the pion decay constant and that QCD operator product expansion (OPE) constraints λ 21 = λ 22 = 0 [71]. Figure 1. One-resonance exchange contributions from the RχT to the vector form factors of the τ − → π − π 0 γν τ decays. Figure 2. Two-resonance exchange contributions from the RχT to the vector form factors of the τ − → π − π 0 γν τ decays. Figure 3. Three-resonance exchange contributions from the RχT to the vector form factors of the τ − → π − π 0 γν τ decays.
For the vector form factors, we get where v 0 i is the contribution at O p 4 as in ref. [58] (D −1 R stands for the inverse resonance and v R+RR GIi 10 correspond to contributions including O p 6 vertices. Due to their length, the expressions for these form factors are in App. B. In writing the contributions to v i at O p 6 , the basis given in ref. [71] has been used for the even-intrinsic parity operators (with couplings λ X i ) and the basis given in ref. [76] has been employed for the odd-intrinsic parity operators (κ X i couplings). Both sets of λ X i and κ X i couplings have dimensions of inverse energy.
At leading chiral order in the low-energy limit of RχT , only the contribution from the exchange of ρ and a 1 resonances on the vector form factor appeared [58], but at the next order we also have a contribution from the ω exchange coming from the odd-intrinsic parity sector for both vector and axial-vector form factors. Apparently, such contribution was responsible for the larger effect of the IB corrections obtained in [61,62] with respect to refs. [57,58]. As a result, ref. [63] (and later evaluations by this group) ascribed an error to these corrections covering both contradictory evaluations. As we include (among others) contributions with an ω −ρ−π vertex in this work, closer agreement with the VMD evaluation should, in principle, be expected.
We have verified that all diagrams including scalar mesons vanish in the isospin symmetry limit. We point out that all contributions involving pseudoscalar mesons can be obtained from those with an axial-vector resonance by replacing it by a pseudoscalar resonance. Then, at leading chiral order, the saturation of the LECs by spin-one mesons [59] shows that diagrams including pseudoscalar resonances are suppressed. If we assume that this feature also holds at the next chiral order, then pseudoscalar resonance exchanges could be safely neglected 11 . 10 In general, diagrams are gauge-invariant by themselves. Those giving the contribution v R+RR GIi need to be summed to achieve gauge invariance. These are the first three diagrams in Fig. 1 and the first diagram in Fig. 2. 11 Since contributions from scalar and pseudoscalar resonances are suppressed, we will neglect them for

Axial-Vector Form Factors
The axial form factors at O p 4 get contibutions from the Wess-Zumino-Witten functional [111,112]: (1.8) The diagrams that receive contributions due to the anomaly are shown in Fig. 4 12 .
One-resonance exchange contributions from the RχT to the axial-vector form factors of the τ − → π − π 0 γν τ decays.
the axial form factors in the next section. 12 The first diagram, when coupled to a vector current, contributes to the SI piece in V µν . Figure 6. Two-resonance exchange contributions from the RχT to the axial-vector form factors of the τ − → π − π 0 γν τ decays. Figure 7. Three-resonance exchange contributions from the RχT to the axial-vector form factors of the τ − → π − π 0 γν τ decays.
For the axial form factors, we get and a RRR i include O p 6 vertices. Due to their length, the expressions for these form factors appear in App. C.

SD constraints
Including the contributions up to O p 6 , we have now so many parameters (see Table  1) allowed by the discrete symmetries of QCD and chiral symmetry that, in practice, prevent making phenomenological predictions. It is possible to find relations between these couplings by means of SD properties of QCD and its OPE. We summarize these results in this section.
For the parameters contributing to the O p 4 chiral low-energy constants (LECs), the constraints [59,60,[113][114][115][116][117]: (1.10) are set by the known asymptotic behaviour of: the pion vector form factor, the V − A correlator (yielding the Weinberg sum rules), the scalar form factor and the S−P correlator. We note that the vanishing of the axial pion form factor (giving the π-to-γ matrix element) at infinite momentum transfer demands -if only the original RχT Lagrangian [60] (including operators linear in resonance fields) is used- This, together with the two first eqs. in (1.10), determine (1.11) all in terms of the pion decay constant. These relations were employed in ref. [58]. We emphasize that -once operators with more than one resonance field (which start contributing at O p 6 ) are considered-, the relations (1.11) no longer hold true (see e. g. Ref. [71]). We will come back to discussing the actual values of the F V , G V , F A couplings before closing this section, as they are essential to assess the error associated to the predictions up to O p 4 in the chiral expansion. Now, we consider the RχT operators contributing to the O p 6 chiral LECs. For the even intrinsic parity sector [71,86]: 12) using these SD constraints in eq. (1.6) and the Brodsky-Lepage behaviour [118,119] of f + (s), we get: The study of the V AP and SP P Green functions yield the following restrictions on the resonance couplings [69][70][71]: (1.14) For the odd-intrinsic parity sector [76]: (1.15) -10 - The analysis of the V AS Green's function yields [76]: (1. 16) and through the study of the V V A Green's function in ref. [31]: (1. 17) A comparison between two basis for the odd-intrinsic operators [68,76] was given in ref. [82], which is consistent with those in eq. (1.15) 13 For the even-and odd-intrinsic parity sectors, there are 115 (EIP)+67 (OIP)=182 operators at O p 6 but only a few of them contribute to a given process. The form factors of the τ − → π − π 0 γν τ decays at O p 6 are given by 32 (EIP)+23(OIP)=55 operators ( Table  1). Taking into account the relations in eqs. (1.12)-(1.18) we get 24 (EIP)+17 (OIP)=41 undetermined couplings.
Eq. (1.14) leaves two λ V A i couplings undetermined, the numerical values of the restricted combinations are: (1. 19) Since the same linear combination of λ 4 and λ 5 is in all couplings in eq. (1.19), we choose λ 4 as independent. By similar reasons we take λ 2 as the other independent coupling. Based on eq. (1.19), we conservatively estimate |λ 2 | ∼ |λ 4 | ≤ 0.4. According to ref. [71] the λ X i couplings can be estimated from low energy couplings C R i of the O p 6 χP T Lagrangian as 14 20) where we take the relation There is a sign ambiguity on the determination of c 3 from τ → ηπ − π 0 ν τ decays [77]. We will take c 3 = 0.007 +0.020 −0.012 according to the determinations by Y. H. Chen et al. in Refs. [78,83,120] (which is also in agreement with the most elaborated e + e − → (η/π 0 )π + π − fit [80]). Although c 4 was first evaluated by studying σ(e + e − → KKπ) in ref. [74], this yielded an inconsistent result for τ − → K − γν τ branching ratio [75], so we will use c 4 = −0.0024 ± 0.0006 [83] as the most reliable estimation. In view of all these results, we will take |c i | 0.015 as a reasonable estimate, which is translated to |κ V i | 0.025 GeV −1 . Since there is not enough information on κ A i , we will take |κ A i | ∼ |κ V i | 0.025 GeV −1 15 . For the remaining couplings, we will employ d 2 = 0.08 ± 0.08, which has been determined simultaneously with c 3 according to the quoted references. For d 4 we will assume |d 4 | < 0.15, or in terms of κ V V i , we get |κ V V i | 0.1. Again we will adopt |κ V A i | ∼ |κ V V i | 0.1, which agrees with the prediction κ V A 5 ∼ −0.14 in eq. (1.17). At O p 4 it is clear that we must use the relations (1.11). However, at O p 6 , these relations are no longer fulfilled. In particular, F V = √ 3F , which implies (via (1.10)) Therefore, we will also be showing our O p 4 results with the latter set of constraints (inconsistent at this chiral order) so that the impact of the change of F V , F A , G V from O p 4 to O p 6 is appreciated. We will refer to the original [58] constraints (1.11) as 'F V = √ 2F ' and by 'F V = √ 3F ' to their consistent set of values At this order, the consistent set of SD constraints in both parity sectors [69,71,76,82] determines the F V = √ 3F relations (among many others, reviewed in this section).
14 Couplings of operators with two resonance fields are dimensionless [71,76]. 15 We will see in the following sections that the observables that we consider and the IB corrections for a HV P,LO|ππ µ depend mostly on the κ V i couplings, among those contributing at O(p 6 ) -in either parity sectorin the chiral counting.

Radiative corrections for hadronic vacuum polarization
The four-body differential decay width is given by [58] ds du dx and integrating over the three-momentum of the photon and neutrino 17 , we get working at leading order in the Low expansion and in the isospin limit m u = m d , we have is the amplitude at leading order for the non-radiative decay that includes the SD electroweak radiative corrections (S EW ). At O k −1 , the amplitude for the radiative decay is proportional to the amplitude of the non-radiative decay according to the Low's theorem [91].
The unpolarized spin-averaged squared amplitude is given by using the relation γ * µ (k) ν (k) = −g µν and massive photons (k µ k µ = M 2 γ ). The sum over photon polarizations should include the longitudinal part, since our photon has mass and the amplitude is no longer gauge invariant. We do not take into account this contribution because it will vanish in the limit M γ → 0.
Thus, eq. (2.4) becomes (2.5) 16 Although the analytical results in this section were presented in the quoted reference, we include them here given their importance in the evaluation of the relevant IB corrections, and take advantage to add a few explanations to previous discussions of this subject [58,62]. 17 The kinematics for these decays are in App. A. where with D(s, u) = 1 2 m 2 τ m 2 τ − s + 2m 4 π − 2u(m 2 τ − s + 2m 2 π ) + 2u 2 . eq. (2.5) does not have contributions at O k −1 , these terms are canceled out by the terms in eq. (2.6) according to the Burnett-Kroll theorem [121].
Replacing eqs. (2.5) and (2.6) in eq. (2.2), we get the I mn (s, u, x) is defined as performing an integration over x, we can split the decay width according to the integration region and and (2.14) Eq. (2.8) is an invariant, so we can evaluate it in any reference frame in order to simplify the integration, working in the γ − ν τ center of mass, we have Integrating this equation over x in D IV /III and D III , as in Refs. [58,122] we get , where the expressions in eq. (2.16) are given by Experimentally, it is impossible to measure the full photon spectrum because of acceptances, efficiencies and cuts. For this reason, we need to calculate the inclusive decay width, since we can not distinguish the radiative decay from the non-radiative decay for low-energy (or collinear) photons. For the non-radiative decay, we have that includes isospin violation and photonic corrections according to ref. [57], where f elm loop (u, M γ ) is given by in terms of the variables and of the dilogarithm Thus, the inclusive decay width is In the previous expression we neglected the quadratic term for f elm loop (u, M γ ), and Integrating eq. (2.25) over u, and using for this we follow the same notation as in ref. [58], We can split the electromagnetic correction factor (G EM (s)) in two parts, G EM (s) and G rest EM (s), the first one corresponds to taking g rest (s, u) → 0 and the second one is the remainder of G EM (s), In eq. (2.31a), the term 2f elm (2.32) In this limit, we have The leading Low approximation for G Fig. 8. This function has two poles, one at s = 4m 2 π and the other at s = m 2 τ . We will use the same conventions as ref. [58], so we denote as 'complete Bremsstrahlung' the amplitude where the SD part vanishes, i.
We will refer as O p 4 and O p 6 to the contributions from RχT including operators that contribute up to O p 4 and O p 6 in its low energy limit, respectively.
In G EM (s), the difference between using the F V = √ 2F or F V = √ 3F constraints at O p 4 is only appreciated for s 0.35 GeV 2 , with the latter set producing the largest deviation with respect to the SI result (Fig. 8). It is important to note that -as put forward in ref. [58]-with F V = √ 2F constraints (those consistent at O(p 4 )) the impact of the SD corrections on G EM (s) is negligible and the evaluation with SI gives already an excellent approximation. On the contrary, we find that using the F V = √ 3F set this is no longer true, which will increase the G EM (s) correction in a HV P,LO|ππ µ using τ data (even before adding the O(p 6 ) contributions).
In Fig. 8 , represented by dashed (dotted) lines. Compared to previous results [57,58,61,62], we note the appearance of a bump near the end of the phase space on G EM (s) due to the inclusion of the ρ(1450) and the ρ(1700) resonances in the dispersive representation of the vector form factor [104,105]. The blue band in Fig. 8 shows the uncertainty of the O(p 6 ) contribution, evaluated according to that on the couplings which were determined phenomenologically or estimated from chiral counting in section 1.5 18 . While the central values of the O p 6 corrections change mildly the results obtained at O p 4 19 , their huge uncertainties suggest that our estimate of the RχT couplings which start contributing at O p 6 was too conservative. Lacking a better way for this estimation, we consider this uncertainty band as an extremely conservative upper limit on the corresponding uncertainties and by no means a realistic estimation of them. Therefore, our errors at O p 6 should be regarded accordingly in the following. On the contrary, the small modification induced by those O p 6 couplings fixed by SD constraints (with all remaining ones vanishing) with respect to the O p 4 [58] results, suggests that the difference between those is a realistic estimate of the missing higherorder terms in the O p 4 [58] evaluation 20 and will be given as such in the remainder of the paper.
The blue shaded region is the full O p 6 contribution, including (overestimated) uncertainties. The left-hand side plot corresponds to the dispersive parametrization [104] while the right-hand side corresponds to the Guerrero-Pich parametrization [123] of the form factor (the latter was used in ref. [58]).

Radiative decay
The differential decay width [122] is given by where |M| 2 is the unpolarized spin-averaged squared amplitude that corresponds to the τ − → π − π 0 γ ν τ decays, and E γ is the photon energy in the τ rest frame. It is not worth to quote here the full analytical expression for |M| 2 . For these decays, we have the following integration region

38)
20 These were not estimated in ref. [58] as SI was already an excellent approximation to the result up to O p 4 (using the FV = √ 2F set).
with boundaries or interchanging the last two limits, There are other ways to write these, (2.43) (2.44) We recall that this amplitude has IR divergences due to soft photons, i.e. E γ → 0, which is the same problem with M γ → 0 outlined in the previous section. Correspondingly, the experiment is not able to measure photons with energies smaller than some E cut γ (which is related with the experimental resolution).
Concerning the O p 6 contributions, once we employ the relations obtained from the SD behaviour of QCD and its OPE, it is seen that observables are basically insensitive (at the percent level of precision) to O(1) changes of all the couplings but κ V i (the ρ − ω − π vertex is described by these couplings), which will saturate the (overestimated) uncertainty of our predictions at this order.
If we integrate eq. (2.37) using the limits in eq. (2.44) and the dispersive vector form factor [104,105], we get the π − π 0 invariant mass distribution, the photon energy distribution and the branching ratios as a function of E cut γ , shown in Figs. 10, 11, 12, 13 and 14 and summarized in Table 2. In these figures, the dotdashed red line corresponds to taking the limit where all the couplings at O p 6 vanish except for those constrained by SD and the band overestimates the corresponding uncertainties. Table 2. Branching ratios Br(τ − → π − π 0 γν τ ) for different values of E cut γ . The second column corresponds to the complete Bremsstrahlung and the third and fourth to the O p 4 contributions.
As it can be observed from Table 2 and Fig. 14, the main contribution at O p 4 corresponds to the complete Bremsstrahlung (SI) amplitude (in agreement with ref. [58]), and the value for the branching ratio becomes smaller with larger values of E cut γ . The values in Table 2 are slightly different from those reported in ref. [58], this effect is mainly due to the parametrization of the pion vector form factor (see Fig. 9). The form factor obtained from the dispersion relation [104] is above the one obtained using the Guerrero-Pich parametrization [123] at s M 2 ρ , and also the former includes the ρ (1450) and ρ (1700) resonances.  According to our discussion on error estimation of the O p 4 result (including the uncertainty coming from missing higher-order terms from the result at O p 6 when only short-distance constraints are used), we have -for E cut The spectrum for these decays with v i = a i = 0 is plotted in Fig. 10, the dominant peak corresponds to bremsstrahlung off the π − , and the secondary receives two contributions: one from bremsstrahlung off the τ lepton and another from a resonance exchange in V µν (or E cut γ ≤ 100 MeV, these two are merged into one single peak). The rate and spectrum are dominated by the complete bremsstrahlung (SI) contribution. In Fig. 11, we show the distribution for E cut γ = 300 MeV taking into account the SI contribution (dotted line) and the O p 4 amplitude obtained using F V = √ 2F (dashed line) and F V = √ 3F (solid line), the most important contribution corresponds to the ρ resonance exchange at s ∼ 0.6 GeV 2 . The main difference between these two approaches is seen in Fig. 11, where up to s ∼ 0.4 GeV 2 the dashed line is below and the solid line is above the bremmstrahlung (SI) contribution (dotted line). The dashed line is quite similar to the distribution in Fig. 2 of ref. [58] while the solid line resembles closely the distribution in Fig. 4.6 of ref. [122] obtained from the vector meson dominance (VMD) model [124] neglecting the ω-resonance contribution. In Fig. 12 we show a comparison between the di-pion distribution at different orders. As we can see, the inclusion of the corrections at O p 6 gives a noticeable enhancement at low s. For the photon energy distribution Fig. 13, we can differentiate between the full amplitude (solid, dashed lines up to O p 4 and dotdashed red line up to O p 6 ) and the bremsstrahlung contribution (dotted line) but, as in the case of the branching fraction, the distribution decreases for high-energies. In the case of the O p 6 distribution there is an enhancement at middle and high photon energies.  Figure 13. Photon energy distribution for the τ − → π − π 0 γν τ decays normalized with the nonradiative decay width. The dotted line represents the Bremsstrahlung contribution. The solid and dashed lines represent the O p 4 corrections using F V = √ 3F and F V = √ 2F , respectively. The dotdashed red line corresponds to using only SD constraints at O p 6 (with overestimated uncertainties in the blue shaded area). According to Figs. 11 to 14, measurements of the ππ invariant mass, of the photon spectrum and the partial decay width, for a reasonable cut on E γ (at low enough energies the inner bremmstrahlung contribution hides completely any SD effect), could decrease substantially the uncertainty of the O p 6 computation. This was already emphasized in ref. [58] but remained unmeasured at BaBar and Belle. We hope these data can finally be acquired and analyzed at Belle-II.
In Fig. 15, we show the branching ratio for E cut γ = 100, 300, and 500 MeV from top to bottom. The outcomes were summarized in Table 3.

IB corrections to a HV P,LO ππ µ
We can evaluate the leading contributions to the hadronic vacuum polarization (HVP) by means of the dispersion relation [125], where K(s) is a smooth QED kernel concentrated at low energies, which increases the E M ρ contribution, and σ 0 e − e + →hadrons (s) is the bare hadronic cross section 21 . We can relate the hadronic spectral function from τ decays to the e + e − hadronic cross section by including the radiative corrections and the IB effects. For the ππ final state, we have [57,58]: where 4) and the IB corrections (3.5) The S EW term encodes the SD electroweak corrections [126][127][128][129][130][131][132][133] and F SR(s) accounts for the radiation from the final-state pions [134,135]. The G EM (s) term was already discussed at length in section 2, the β 3 π + π − /β 3 π + π 0 term is a phase space factor and the last term in R IB (s) is a ratio between the neutral (F V (s)) and the charged (f + (s)) pion form factor.
In order to study the effect of the radiative correction G EM (s) on a HV P,LO µ [ππ], we have evaluated the following expression [58] ∆a HV P,LO taking S EW = 1, The results are summarized in Table 4 using DR form factor. The results obtained for the G EM and the complete O p 4 contribution (with F V = √ 2F ) agree with those in [58], which are +16·10 −11 and −10·10 −11 , respectively (for the whole integral). In Table 5, we summarized the results obtained using the Guerrero-Pich [123] parametrization of the form factor (which only accounts for the completely dominant ρ exchange), which are in nice agreement with those found with the dispersive form factor (that also includes the ρ(1450) and ρ(1700) effects). This checks, a posteriori, that excited resonance contributions make a negligible effect in the G EM (s) corrections to a HV P,LO µ 22 .
The values in the last column of Tables 4 and 5 were obtained evaluating the eq. (3.6) according to the couplings discussed in section 1.5 for a sample of 200 points for each interval of integration (results were stable under increasing this number). 21 Although final state radiation would belong to HVP,NLO it is always included in HVP,LO (and not in HVP,NLO) as eliminating this radiation from the measured data is unfeasible. Thus, a final state radiation (FSR) factor is also needed in the radiative corrections discussed below. 22 By replacing The other contributions are summarized in Table 6.
• The correction due to the ratio of the form factors (Fig. 16) is harder to evaluate.
• Finally, we get (−15.9 +5.7 −16.0 ) · 10 −11 ((−79 ± 61) · 10 −11 ) for the G EM (s) correction at O(p 4 ) (O(p 6 )), versus −10 · 10 −11 in [58] and −37 · 10 −11 in [61] (from the last two results, (−19.2 ± 9.0) · 10 −11 was used in [63]). As explained before, the previous uncertainty on the O(p 6 ) can only be taken as an upper bound on it. Also interesting is the G EM (s) correction when only the couplings restricted by SD are used (with all others at this order set to zero), which allows us to estimate the effect of missing higher-order terms on the O(p 4 ) result quoted above. This O(p 4 ) result, which is our reference value, is consistent with both the earlier RχT [58] and the VMD [63] evaluations, albeit with a larger (asymmetric) error.  In Fig. 17, we show the full IB correction factor R IB (s) for the different orders of approximation in the G EM (s) factor using the DR parametrization of the form factor. As we can see, there is a difference between the contributions at O(p 4 ) and those at O(p 6 ) for energies below ∼ 0.5GeV 2 and above ∼ 0.7GeV 2 .
If we include all the couplings contributing to G EM (s) at O(p 6 ) according to section 1.5 we have an additional error associated to the EM contributions. Thus, we get where the first uncertainty is statistical and the second is systematic. Nonetheless, they are in some tension with the very precise ALEPH measurement (25.471 ± 0.097 ± 0.085)% [139]. We show in figs. 18 and 19 the prediction for the e + e − → π + π − cross section using the data reported by Belle [138] (as it is the most precise measurement of this spectrum) for the normalized spectrum (1/N ππ )(dN ππ /ds) compared to the last measurements from BaBar [50] and KLOE [140] 24 .
We recall that the e + e − → π + π − cross section obtained using τ data is given by [138] (3.14) In fig. 18 the τ -based prediction is obtained using the O(p 4 ) result for G EM (s), with the estimated uncertainty from missing higher-order corrections given by the result at O(p 6 ) (employing only the SD constraints). In fig. 18, the band shown overestimates the error at O(p 6 ).
From both figs. 18 and 19, we observe good agreement between the BaBar data and the τ decays prediction (slightly better for FF1). This consistency justifies our evaluation of the IB-corrected a HV P,LO µ [ππ, τ ] in the following 25 . The previous comparisons make us consider our evaluation with FF1 the reference one (so that its difference with FF2 will assess the size of the error induced by IB among the ρ → ππγ decay channels).  Using eq. (3.14) we evaluate the IB-corrected a HV P,LO µ [ππ, τ ] from the Belle mass spectrum. We use the PDG values [136] for m τ , V ud and B e . In addition to the experimental error shown in Table 8, we have an additional uncertainty of 1.56% due to B e and B ππ 0 .
In tables 7 (9) and 8 (10) we show IB-corrected a HV P,LO µ [ππ, τ ] in units of 10 −10 using the measured mass spectrum by Belle (ALEPH). For each dataset, the first (second) table collects the results for the different approximations to the (complete) G EM (s). We choose showing first the results with both Belle and ALEPH datasets as the first (second) one yields the most accurate spectral function (branching ratio) measurement. As in ref. [63] (and later works by the Orsay group), the contributions are split in two intervals. In the first one, √ s ∈ [2m π ± , 0.36 GeV], (the very scarce) data is not used, as this affects the precision of the integral. Instead, we use the results of the dispersive fits in ref. [105]. We proceed analogously in tables 11 (13) and 12 (14) with the CLEO [141] and OPAL [142] 26 measurements.
Taking into account all di-pion tau decay data from the ALEPH [139], Belle [138], CLEO [141] and OPAL [142] Colls. (the latter yielding the largest contribution to a HV P,LO|ππ µ 26 We thank to Jorge Portolés for providing us with the OPAL data set. [ππ, τ ] in units of 10 −10 using the measured mass spectrum by Belle with B ππ = (25.24 ± 0.01 ± 0.39)%. The first errors are related to the uncertainties on the mass spectrum, and also include contributions from the τ -mass and V ud uncertainties. The second errors arise from B ππ 0 and B e . The different results correspond to the approximations to G EM (s) discussed in the caption of table 7. exceeding ∼ 10.7 · 10 −10 the mean, although with the largest errors as well), we get the [ππ, τ ] in units of 10 −10 using the measured mass spectrum by CLEO with B ππ = (25.36 ± 0.44)%. The first errors are related to the systematic uncertainties on the mass spectrum, and also include contributions from the τ -mass and V ud uncertainties. The second errors arise from B ππ 0 and B e .
The IB errors come from the uncertainty on Γ(ρ → ππγ) (FF1 vs FF2) and either from the difference between the F V = √ 2F and SD results (in eq. (3.15)) or from the difference between the 'mean' and 'min'/'max' results (in eq. (3.16)).
When eqs. (3.15) and (3.16) are supplemented with the four-pion tau decays measurements (up to 1.5 GeV) and with e + e − data at larger energies in these modes (and with [ππ, τ ] in units of 10 −10 using the measured mass spectrum by OPAL with B ππ = (25.46±0.17±0.29)%. The first errors are related to the systematic uncertainties on the mass spectrum, and also include contributions from the τ -mass and V ud uncertainties. The second errors arise from B ππ 0 and B e . e + e − data in all other channels making up the hadronic cross section), we get [7,65] 10 10 · a HV P,LO| τ data µ = 705.7 ± 2.8 spectra+BRs +1.9  (O(p 6 )). These numbers have to be compared with the error of 4.0 in ref. [2].
When all other (QED, EW and subleading hadronic) contributions are added to eqs. (3.17) and (3.18) according to ref. [2], the 3.7σ [2] deficit of the SM prediction with respect to the BN L measurement [41] is reduced to

Conclusions
In this work we have revisited the resonance chiral Lagrangian computation of the isospin and radiative corrections to the τ − → π − π 0 ν τ γ decays in ref. [58], by including the terms that start to contribute at O(p 6 ) in the chiral expansion. Our main motivation for that was to revisit the determination of a HV P,LO µ using tau decay data, so that it could -when combined with the e + e − measurements-reduce the Standard Model error on a µ , thus enhancing the sensitivity to new physics of the current BNL and future FNAL and J-PARC measurements.
Our isospin breaking corrections improve the agreement between τ and e + e − di-pion data (both in the spectrum and its integral), which endorses our evaluation of a We also provide with a detailed study of the ππ spectrum, E γ distribution and branching ratio, for different cuts on the photon energy. These τ − → π − π 0 ν τ γ decays observables have the potential to reduce drastically the error of the O(p 6 ) prediction, so we eagerly await their measurement at Belle-II.
benefitted from enriching discussions on this topic with Gabriel López Castro and Genaro Toledo Sánchez. We thank Vincenzo Cirigliano for reading our draft and making useful suggestions on the presentation of the results, Antonio Pich for helpful suggestions regarding the presentation of the form factors, Antonio Rojas, Eduard de la Cruz and Iván Heredia for their valuable help. We are indebted to Alex Keshavarzi, Bogdan Malaescu, Hisaki Hayashii and Jorge Portolés for providing us with the BaBar, Belle and OPAL data sets.

A Kinematics
In order to describe this type of decays we need five independent variables. We choose 2 , θ ν which is the angle between the direction of the π − π 0 CM frame in the τ lepton rest frame and the direction of q in the π − π 0 CM frame (see Fig. 20) and φ − , which is angle between the plane of the π − π 0 CM frame and the plane of the γν τ CM frame. We can write the invariants in terms of these variables Figure 20. The τ − → π − π 0 γν τ decay in the τ -lepton rest frame.
Working in the τ -lepton rest frame, we have where λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz is the Kallen function, andê ρ = cos φ −êx + sin φ −êy . From eq. (A.17), we get and these bounds on u and x correspond to the forward and backward direction, i.e. by taking θ − = 0, π. For the non-radiative decay, we have this region is plotted in Fig. 21 which corresponds to the projection R III onto the su-plane.
In the case of the radiative decay, we have is the value that maximizes u + s, M 2 γ . We will be working in the isospin-limit (m u = m d ), i.e. m 2 π − = m 2 π 0 and thus many of the last expressions will be simplified. We use a non-vanishing M γ in order to deal with the IR divergences, at the end these divergences are canceled out by those divergences of the non-radiative decay so we can take the limit M γ → 0. The projection R IV = R IV /III ∪ R III of the D IV is plotted in Fig.  21 for M γ → 0.  Figure 21. Projection of the kinematic region for the non-radiative decay R III (gray) and the radiative decay R IV = R IV /III ∪ R III (black and gray) onto the su−plane. R IV /III (black) is the kinematic region which is only accessible to the radiative decay.