Radiative Corrections to π l 2 and K l 2 Decays

We reexamine radiative corrections to πl2 and Kl2 decays. We perform a matching calculation, including vector and axial vector resonances as explicit degrees of freedom in the long distance part. By considering the dependence on the matching scale and on the hadronic parameters, and by comparing with model independent estimates, we scrutinize the model dependence of the results. For the pseudoscalar meson decay constants, we extract the values fπ = (92.1±0.3)MeV and fK = (112.4±0.9)MeV. For the ratios Rπ and RK of the electronic and muonic decay modes, we predict Rπ = (1.2354± 0.0002) · 10 and RK = (2.472± 0.001) · 10. PACS numbers: 13.40.Ks, 13.20.Cz, 13.20.Eb 1. Radiative corrections to πl2 and Kl2 decays are interesting for two separate reasons. On the one hand, measurements of the decay rates for Γ(π → μνμ) and Γ(K → μνμ) are used to extract the decay constants fπ and fK , which are important input parameters for chiral perturbation theory [1]. Therefore it is important to understand how radiative corrections affect these parameters [2]. On the other hand, in the ratios Rπ = Γ(π → eνe)/Γ(π → μνμ) and RK , strong interaction uncertainties cancel to a large degree. Therefore they can be predicted very precisely [3, 4, 5, 6, 7, 8] and allow for low energy precision tests of the standard model. The most recent discussion of radiative corrections to πl2 decays can be found in [8]. These authors seperate the radiative corrections into a long and a short distance part, matched at a scale mρ. In the long distance part, only pions are considered as active degrees of freedom. QED corrections to the decay of a pointlike pion [4] give rise to the leading, model independent contribution. Hadronic structure effects of the order ml /m 2 ρ ln(m 2 ρ/m 2 l ) are also model independent [5] and have been included. For the remaining hadronic structure effects, order-of-magnitude estimates are given. Regarding the short distance part, the running of the effective semileptonic four fermion interaction is used to evolve GF down from mZ to mρ. This matching procedure is somewhat simplified, because in the long distance part there are missing hadronic structure effects, which become important for energy scales approaching mρ, and in the short distance part, the effective quark-antiquarklepton-neutrino operator has been evolved down to the rather low scale mρ. We will use a different approach. We include vector and axial vector resonances as explicit degrees of freedom in the long distance part, which allows us to use a larger matching scale. We show that this leads to a drastic reduction of the matching scale dependence of the radiative correction. The inclusion of the hadronic resonances, however, unavoidably introduces model dependence. Our main goal therefore will be to study the size and the uncertainties of the model dependent contributions in detail, replacing the order-of-magnitude estimates in [8]. 2. We seperate the loop integration over the Euklideanized momentum k of the virtual photon into long distances k = 0 · · ·μcut and short distances k = μcut · · ·mZ . To calculate the long distance part, we construct an effective model by starting with the low energy theorems of QCD and adding resonance degrees of freedom along the lines of vector meson dominance. The amplitude for the radiative decay π → lνlγ consists of the model independent internal bremsstrahlung part (IB) and the hadronic structure dependent (SD) part [9, 10]. The latter is parametrized by two form factors FV (s) and FA(s). FV (0) and FA(0) can be determined from chiral perturbation theory [1, 10] F (π) V (0) = mπ 4 √ 2πfπ