Weak decays of the $\tau$ lepton

For the $\tau$ lepton, an explicit list of decay channels has been put together, which includes channels with up to five final-state particles, some of which may be unstable and subsequently decay to produce even larger total multiplicities.

The leptonic decays $\tau^- \to \nu_{\tau} \ell^- \overline{\nu}_{\ell}$, where $\ell$ is $\mathrm{e}$ or $\mu$, are distributed according to the standard $V-A$ matrix element

$$\vert{\cal M}\vert^2 = (p_{\tau} p_{\overline{\nu}_{\ell}}) (p_{\ell} p_{\nu_{\tau}}) ~.$$ (262)

(The corresponding matrix element is also used in $\mu$ decays, but normally the $\mu$ is assumed stable.)

In $\tau$ decays to hadrons, the hadrons and the $\nu_{\tau}$ are distributed according to phase space times the factor $x_{\nu} \, (3 - x_{\nu})$, where $x_{\nu} = 2E_{\nu}/m_{\tau}$ in the rest frame of the $\tau$. The latter factor is the $\nu_{\tau}$ spectrum predicted by the parton level $V-A$ matrix element, and therefore represents an attempt to take into account that the $\nu_{\tau}$ should take a larger momentum fraction than given by phase space alone.

The probably largest shortcoming of the $\tau$ decay treatment is that no polarization effects are included, i.e. the $\tau$ is always assumed to decay isotropically. Usually this is not correct, since a $\tau$ is produced polarized in $\mathrm{Z}^0$ and $\mathrm{W}^{\pm}$ decays. The PYTAUD routine provides a generic interface to an external $\tau$ decay library, such as TAUOLA [Jad91], where such effects could be handled (see also MSTJ(28)).