Matching NLO parton shower matrix element with exact phase space: case of $W\to l\nu(\gamma)$ and $\gamma^* \to \pi^{+}\pi^{-}(\gamma)$

In practical applications PHOTOS Monte Carlo is often used for simulation of QED effects in decay of intermediate particles and resonances. Generated in such a way that samples of events cover the whole bremsstrahlung phase space. With the help of selection cuts, experimental acceptance can be then taken into account. The program is based on exact multiphoton phase space. To evaluate the program precision it is necessary to control its matrix element. Generally it is obtained using iteration of the universal multidimensional kernel. In some cases it is however obtained from the exact first order matrix element. Then, as a consequence, all terms necessary for non-leading logarithms are taken into account. In the present paper we will focus on the decays W -> l nu and gamma^* -> pi^+ pi^-. The Born level cross sections for both processes approach zero in some points of the phase space. Process dependent, compensating weight is constructed to implement exact matrix element, but it will be recommended for use only for tests. In the hard photon region, where scalar QED is not expected to be reliable, the compensating weight for gamma^* decay can be large. With respect to the total rate, the effect remains at the permille level. It is nonetheless of interest. The terms leading to the effect are analogous to the ones appearing in QCD. Present paper can be understood either as a contribution to the general discussion on how to match two collinear emission chains resulting from charged sources in a way compatible with the exact and complete phase space and the first order matrix element of QED (scalar QED) or as the practical study of predictions for accelerator experiments.