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Дата индексирования: Mon Oct 1 22:42:33 2012
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Breit-Wigner propagator

The propagator of virtual particle has a pole at $p^2 = m^2$:

$$\frac{1}{p^2 -m^2}\;\;.$$

If the pole is situated inside the phase space volume it leads to a non-integrable singularity. The general solution of this problem is an account of special set of high order corrections [Bjorken&Drell]. They transform the propagator to the Breit-Wigner form

$$\frac{1}{p^2 -m^2 -i \cdot \Gamma(p^2) \cdot m}\;\;,$$

where the value $\Gamma(m^2)$ is the particle width (reversed mean life time).

First problem which appears in a way of implementation of this expression is a choice of $\Gamma(p^2)$ dependence. The $\Gamma$ value is essential near the pole point $p^2 = m^2$. Thus, for the first approximation we can put $\Gamma(p^2)=\Gamma(m^2)=const$. It corresponds to the position OFF of 'S dependence' switch. In some papers it is declared that $\Gamma(p^2)=\Gamma(m^2) \cdot \sqrt{p^2}/m$ describes the pole shape better. This choice corresponds to the position ON of 'S dependence' switch.

The second and even more important problem is a gauge symmetry breaking. Generally we have this symmetry in any order of perturbation theory but the intervention of a part of higher order terms to the lowest order expression via the Breit-Wigner propagator can break it.

The gauge symmetry is responsible for some cancellation of diagram contributions (see Section ), and its violation, in turn, prevents the cancellations and can lead to a completely wrong result. The user could solve this problem by setting the Gauge invariance menu switch to the position ON. In this case the contribution of a diagram which does not contain the Breit-Wigner propagator is multiplied by factor [Baur-1992,Kurihara-1995]

$$\frac{(p^2-m^2)^2}{(p^2 -m^2)^2 + (width(p^2) \cdot m)^2}\;\;.$$ (1)

This trick corresponds to the symbolic summation of all diagram contributions at a common denominator expression and to a subsequent substitution of the width term into the factored denominator. The trick allows to keep all gauge-motivated cancellations. As a defect of the trick it must be mentioned that the factor (1) kills a contribution of non-resonant diagrams in the resonance point [Boos-1996]. If the particle width is very small such an approximation is reasonable, but in some cases it can also lead to an error.