Short and Long Distance Effects in the Decay $\tau \to \pi {\nu }_{\tau }(\gamma )$

Abstract

We calculate the radiative corrections to the decays $\tau \to M{\nu }_{\tau }$ and $\pi \to l{\nu }_{\text{l}}$, where the meson $M$ is $M=\pi$ or $K$ and the lepton $l$ is $l=e$ or $\mu$. We perform a complete calculation, which includes internal bremsstrahlung and structure dependent radiation in the radiative decays and point meson, hadronic structure dependent and short distance contributions in the virtual corrections. Our result for the radiative correction to the ratio $\Gamma (\tau \to \pi {\nu }_{\tau }(\gamma ))/\Gamma (\pi \to \mu {\nu }_{\mu }(\gamma ))$ is $\delta R_{\tau /\pi }=\left(0.16_{\text{T}-0.14}^{\text{T}+0.09}\right)\%$. For the ratio $\Gamma (\tau \to K{\nu }_{\tau }(\gamma ))/\Gamma (K\to \mu {\nu }_{\mu }(\gamma ))$, we obtain $\delta R_{\tau /K}=\left(0.90_{\text{T}-0.26}^{\text{T}+0.17}\right)\%$. For completeness we have also calculated the ratio of the electronic and muonic decay modes of the pion.

Comments: 39 pages, Latex [19 figures, not included]: A complete postscript file, including figures, is available via anonymous ftp at ttpux2.physik.uni-karlsruhe.de (129.13.102.139) as /pub/ttp94-05/ttp94-05.ps, Local preprint# TTP94-5. Revised version: In the previous version of this paper we used a relative sign $s=-1$ between internal bremsstrahlung and the structure dependent radiation. However, as explained in TTP93-1A (hep-ph/940538), we now believe that $s=+1$ is the physical choice