Abstract
We calculate the ratios Rτ/P≡ Γ (τ → Pντ [γ]) /Γ (P → μνμ [γ]) (P = π, K) at one loop following a large-NC expansion where Chiral Perturbation Theory is enlarged by including the lightest resonances and respecting the short-distance behavior dictated by QCD. We find δRτ/π = (0.18 ± 0.57)% and δRτ/K = (0.97 ± 0.58)%, where the uncertainties are induced fundamentally by the counterterms. We test the lepton universality, obtaining |gτ/gμ|π = 0.9964 ± 0.0038 and |gτ/gμ|K = 0.9857 ± 0.0078, and analyze the CKM unitarity, getting results at 2.1σ and 1.5σ from unitarity via |Vus/Vud| and |Vus|, respectively. We also update the search for non-standard interactions in τ decays. As a by-product, we report the theoretical radiative corrections to the τ → Pντ [γ] decay rates: δτπ = −(0.24 ± 0.56)% and δτK = −(0.15 ± 0.57)%.
Highlights
Where gμ = gτ according to LU, δRτ/P denotes the radiative corrections, and Rτ(0/)P is the leading order contribution given by
There are important reasons to address this analysis again: the hadronic form factors modeled in refs. [7,8,9,10] are different for real- and virtual-photon corrections, do not satisfy the correct QCD short-distance behavior, violate unitarity, analyticity, and the chiral limit at leading non-trivial orders; besides, a cutoff scheme was used to regulate the loop integrals, separating unphysically long- and short-distance corrections
The uncertainties given in refs. [7,8,9,10] are unrealistic, being of the order of an expected purely O(e2p2) Chiral Perturbation Theory (ChPT) result
Summary
Let us start by considering the description of the real-photon corrections, for which, we follow refs. [25,26,27]. Where MIB = MIBτ + MIBP comprises the model-independent (structure-independent) inner bremsstrahlung (IB) given by the radiation off the τ − and off the P − meson This part can be derived from the QCD low-energy theorems assuming an elementary point-like meson field with electromagnetic interaction dictated by the scalar QED Lagrangian [7,8,9,10]. In this case, the contribution of the seagull diagram (diagram (c) in figure 1) is fixed by gauge invariance, in such a way that the sum of the diagrams (a), (b), and (c) in figure 1 is given by MIB = −iGF VuDeFP Mτ Γμu(q)(1 + γ5).
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