Radiative corrections to $\tau \to \pi (K) \nu_\tau [\gamma]$: a reliable new physics test

The ratios $R_{\tau/P}\equiv \Gamma(\tau \to P \nu_\tau [\gamma]) / \Gamma(P \to\mu \nu_\mu[\gamma])$ ($P=\pi, K$) provide sensitive tests of lepton universality $\left|g_\tau/g_\mu\right|=1$ and are a useful tool for new physics searches. The radiative corrections to $R_{\tau/P}$ are computed following a large-$N_C$ expansion to deal with hadronic effects: Chiral Perturbation Theory is enlarged by including the lightest multiplets of spin-one heavy states such that the relevant Green functions are well-behaved at high energies. We find $\delta R_{\tau/\pi}=(0.18\pm 0.57 )\%$ and $\delta R_{\tau/K}=(0.97\pm 0.58 )\%$, which imply $\left|g_\tau/g_\mu\right|_\pi=0.9964\pm 0.0038$ and $\left|g_\tau/g_\mu\right|_K=0.9857\pm 0.0078$, compatible with and at $1.8\sigma$ of lepton universality, respectively. We test unitarity and bind non-standard effective interactions with the $\tau \to P \nu_\tau [\gamma]$ decays.

Introduction. -Lepton universality (LU) is a basic tenet of the Standard Model of particles interactions. A large diversity of weak interaction processes are compatible with the fact that lepton doublets have identical couplings g to the W -boson. A few anomalies observed in semileptonic B meson decays [1] seem to challenge this principle or require new non-universal weak interactions. Lower energy observables where very precise comparison of theory and experiments can be done, currently provide the most precise test of LU [2].

2.
Theory. First of all, the hadronic form factors modeled in Ref. [4] are different for real-and virtual-photon corrections. Furthermore, they do not satisfy the correct QCD short-distance behavior, violate unitarity, analyticity and the chiral limit at leading non-trivial orders, and use a cutoff to regulate the loop integrals, separating unphysically long-and short-distance corrections. Finally, the uncertainties quoted in Ref. [4] are unrealistic, since they are of the order of a purely Chiral Perturbation Theory result, that is, a computation which cannot include the τ . Thus, a new analysis of δR τ /P overcoming these problems is pressing from a theoretical point of view.
Moreover, and as a by-product, an updated analysis of δR τ /P would be useful to revisit the CKM unitarity [6] and improve the constraints on possible non-standard interactions affecting this ratio [10,11].
-The inclusive P µ2[γ] decay rate can be analyzed unambiguously within the Standard Model (Chiral Perturbation Theory), being the estimation of the local counterterms the only model dependence. We follow the notation proposed in Ref. [3]   from [15]). The uncertainties correspond to the input values L r 9 (µ = mρ) = (6.9±0.7)·10 −3 , γ ≡ F π A (0, 0)/F π V (0, 0) = 0.465±0.005, and to the estimation of the counterterms (m, from matching), affecting only c reported in Refs. [12,13]: where the first bracketed term is the universal shortdistance electroweak correction (which cancels in the ratio R τ /P ), the second bracketed term is the universal long-distance correction (point-like approximation, originally calculated in Ref. [14] and to be given later), the third bracketed term includes the structure dependent contributions and Γ (0) Pµ2 is the rate in absence of radiative corrections (F π ∼ 92 MeV), being D = d, s for P = π, K, respectively. The numerical values for c (P ) n are reported in Table I [12,13]. Note that the most important uncertainties come from the estimations of the local counterterms, which were computed with a large-N C expansion of QCD where Chiral Perturbation Theory is enlarged by including the lightest multiplets of spin-one heavy states such that the relevant Green functions are well-behaved at high energies [16].
τ decays must be scrutinized by using an effective approach encoding the hadronization of the QCD currents and we consider here the same large-N C expansion of QCD used in Refs. [12,13] to estimate the counterterms of P µ2[γ] , quoted previously [16]. Similarly to (3), the decay rate can be organized as where again the point-like long-distance correction will be reported later, the structure dependent contributions have been split into the real-photon (rSD) and virtualphoton (vSD) corrections and Γ (0) τ P 2 is the rate in absence of radiative corrections, being D = d, s for P = π, K, respectively.
The matrix element of the real-photon correction reads where W 2 = (p τ − q) 2 = (p + k) 2 and Γ µ = − * µ (k) for an on-shell photon. In the first line the structureindependent contribution is shown [4], whereas in the second and third lines we give the structure-dependent contributions in terms of the relevant form factors, which encapsulate the hadronization of the related QCD currents.
At leading order in the chiral expansion, the form factors A P 2 and A P 4 are not independent and can be written in terms of a single form factor B (only depending on k 2 and identical for P = π, K at this order): . For virtual-photon corrections, we will focus on the Feynman diagram corresponding to the structure dependent contributions to τ → P ντ decays. The gray shaded box stands for the form factors.
In the case of P µ2 the structure-dependent contribution with virtual photons (vSD) can be extracted directly from (3) and the numerical values for c (P ) n of Table I: δ πµ vSD = (0.54 ± 0.12)% and δ Kµ vSD = (0.43 ± 0.12)%. The new calculation we need to perform from scratch for the analysis of R τ /P is the structure-dependent part with virtual photons for τ → P ν τ , corresponding to the Feynman diagram of figure 1. Inserting the form factors of (10) into (8) -a tedious calculation whose technical details will be explained deeper in a forthcoming longer article [19]-yields our results δ τ π vSD = −(0.48 ± 0.56)%, δ τ K vSD = −(0.45 ± 0.57)%.
(15) A reliable estimation of the uncertainties in (14) is fundamental, since it is the most important source of error in δR τ /P . Keep in mind the great difference with the P µ2 decays: (a) in P decays the calculation is performed within Chiral Perturbation Theory (ChPT), so the unknown local counterterms can be determined by matching ChPT with the effective approach at higher energies, the large-N C extension including the first resonances we have quoted previously; (b) in τ decays, and due to the energy scale at hand, the calculation is done directly with the large-N C extension of ChPT, so the matching procedure to estimate the unknown counterterms is not possible anymore. Bearing in mind this handicap, we have estimated the uncertainties of δ τ P vSD by considering two ingredients. First of all, and in order to assess the model-dependence of the effective approach, we have also calculated δ τ P vSD with a less general scenario where only well-behaved two-point Green functions and a reduced resonance Lagrangian is used; consequently, the form factors of (10) are different [17,18] and we take as a first source of error in (14) one half of the deviation in δ τ P vSD between the two scenarios, resulting in ±0.22% for the pion and ±0.24% for the kaon. Secondly, and in order to estimate the unknown local counterterms in δ τ P vSD , whose dependences on the renormalization scale are known from our calculation, we have considered as the second source of uncertainty in (14) one half of the running of the counterterms between 0.5 and 1.0 GeV, giving ±0.52% 3 . Adding quadratically these two uncertainties yields the errors of (14): ±0.56% and ±0.57% for the pion and the kaon case, respectively. Table II the different contributions to 3 We follow a conservative estimate of the local counterterms in (14), as we justify next. Seeing that the first resonances are included in the theoretical framework for τ decays, their counterterms are expected to be smaller than in P µ2 . However, with the effect of the running we consider here, the counterterms in the P µ2 case affecting δ P µ vSD imply similar corrections to the estimation we consider in δ τ P vSD . This can be seen as a check, a posteriori, that further running of the counterterms is not physically motivated.

Results -In
δR τ /P are summarized, leading to our final result: δR τ /π = (0.18 ± 0.57)%, δR τ /K = (0.97 ± 0.58)%, (16) with dominant uncertainties coming from δ τ P vSD . These results should be compared with the previous ones of Refs. [4], δR τ /π = (0.16 ± 0.14)% and δR τ /K = (0.90 ± 0.22)%. Although their central values agree remarkably, this is merely a coincidence, as the one-sigma confidence intervals agree only at the 25(38)% level for the π(K) case. In our understanding uncertainties were underestimated in Refs. [4], since they have approximately the size which would be expected in a purely Chiral Perturbation Theory computation. Besides, it is important to stress again that the hadronization of the QCD currents used in that work differs for real-and virtual-photon corrections, does not satisfy the high-energy behavior dictated by QCD, violates unitarity, analyticity and the chiral limit, and a cutoff is used to regulate the loop integrals, splitting unphysically long-and short-distance regimes.
An interesting application is the unitarity test (see e. g. [20] and references therein) from the ratio where, as a result of our calculation, Using the FLAG 2+1+1 result for the meson decay constants ratio F K /F π = 1.1932 ± 0.0019 [21] and masses and branching ratios from the PDG [7], one gets V us V ud = 0.2288±0.0010 th ±0.0017 exp = 0.2288 ± 0.0020, which is 2.1σ away from unitarity 4 , according to |V ud | = 0.97373 ± 0.00031 [22].
In conclusion, our final result for δR τ /P is consistent with the previous literature [4], but with much more robust assumptions, yielding a reliable uncertainty. Extracted ratios of lepton couplings are compatible with lepton universality (pion case) and at 1.8σ (kaon case) and can also be used for testing CKM unitarity and binding effective non-standard interactions, as we have illustrated.
We wish to thank V. Cirigliano, M. González-Alonso and A. Pich for their helpful comments and for reading the manuscript. This work has been supported in part by the Spanish Government and ERDF funds from the Euro-