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Widths

A width is calculated perturbatively for those resonances which appear in the PYTHIA hard process generation machinery. The width is used to select masses in hard processes according to a relativistic Breit-Wigner shape. In many processes the width is allowed to be $\hat{s}$-dependent, see section [*].

Other particle masses, as discussed so far, have been fixed at their nominal value. We now have to consider the mass broadening for short-lived particles such as $\rho$, $\mathrm{K}^*$ or $\Delta$. Compared to the $\mathrm{Z}^0$, it is much more difficult to describe the $\rho$ resonance shape, since nonperturbative and threshold effects act to distort the naïve shape. Thus the $\rho$ mass is limited from below by its decay $\rho \to \pi\pi$, but also from above, e.g. in the decay $\phi \to \rho \pi$. Normally thus the allowed mass range is set by the most constraining decay chains. Some rare decay modes, specifically $\rho^0 \to \eta \gamma$ and $a_2 \to \eta' \pi$, are not allowed to have full impact, however. Instead one accepts an imperfect rendering of the branching ratio, as some low-mass $\rho^0/a_2$ decays of the above kind are rejected in favour of other decay channels. In some decay chains, several mass choices are coupled, like in $\mathrm{a}_2 \to \rho \pi$, where also the $\mathrm{a}_2$ has a non-negligible width. Finally, there are some extreme cases, like the $\mathrm{f}_0$, which has a nominal mass below the $\mathrm{K}\mathrm{K}$ threshold, but a tail extending beyond that threshold, and therefore a non-negligible branching ratio to the $\mathrm{K}\mathrm{K}$ channel.

In view of examples like these, no attempt is made to provide a full description. Instead a simplified description is used, which should be enough to give the general smearing of events due to mass broadening, but maybe not sufficient for detailed studies of a specific resonance. By default, hadrons are therefore given a mass distribution according to a non-relativistic Breit-Wigner

\begin{displaymath}
{\cal P}(m) \, \d m \propto \frac{1}{(m - m_0)^2 + \Gamma^2/4}
\, \d m ~.
\end{displaymath} (264)

Leptons and resonances not taken care of by the hard process machinery are distributed according to a relativistic Breit-Wigner
\begin{displaymath}
{\cal P}(m^2) \, \d m^2 \propto \frac{1}{(m^2 - m_0^2)^2 +
m_0^2 \Gamma^2} \, \d m^2 ~.
\end{displaymath} (265)

Here $m_0$ and $\Gamma$ are the nominal mass and width of the particle. The Breit-Wigner shape is truncated symmetrically, $\vert m - m_0\vert < \delta$, with $\delta$ arbitrarily chosen for each particle so that no problems are encountered in the decay chains. It is possible to switch off the mass broadening, or to use either a non-relativistic or a relativistic Breit-Wigners everywhere.

The $\mathrm{f}_0$ problem has been `solved' by shifting the $\mathrm{f}_0$ mass to be slightly above the $\mathrm{K}\mathrm{K}$ threshold and have vanishing width. Then kinematics in decays $\mathrm{f}_0 \to \mathrm{K}\mathrm{K}$ is reasonably well modelled. The $\mathrm{f}_0$ mass is too large in the $\mathrm{f}_0 \to \pi\pi$ channel, but this does not really matter, since one anyway is far above threshold here.


next up previous contents
Next: Lifetimes Up: Masses, Widths and Lifetimes Previous: Masses   Contents
Stephen Mrenna 2005-07-11