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

TL;DR

This work assesses how to match NLO-like matrix elements with exact phase-space coverage in QED bremsstrahlung decays, focusing on $W \to l\nu(\gamma)$ and $\gamma^* \to \pi^+\pi^-(\gamma)$. By decomposing spin amplitudes into gauge-invariant pieces and isolating a Born × eikonal factor, the authors test PHOTOS' multiphoton emission kernel against exact first-order results within an exclusive-exponentiation framework. They identify regimes where the exact matrix element can be incorporated into PHOTOS (notably the $W$ case) and where large compensating weights arise (notably $\gamma^* \to \pi^+\pi^-$), providing numerical guidance and confirming the approach's relevance for precision accelerator phenomenology. The study demonstrates good agreement with benchmark calculations and offers practical prescriptions for kernel construction and weight choices to improve QED radiative corrections in decays. These insights advance reliable, high-precision predictions for collider experiments.

Abstract

The PHOTOS Monte Carlo is often used for simulation of QED effects in decay of intermediate particles and resonances. Momenta are 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 an exact multiphoton phase space. Crude matrix element is obtained by iteration of a universal multidimensional kernel. It ensures exact distribution in the soft photon region. Algorithm is compatible with exclusive exponentiation. To evaluate the program's precision, it is necessary to control the kernel with the help of perturbative results. If available, kernel is constructed from the exact first order matrix element. This ensures that all terms necessary for non-leading logarithms are taken into account. In the present paper we will focus on the W -> l nu and gamma^* -> pi^+ pi^- decays. The Born level cross sections for both processes approach zero in some points of the phase space. A process dependent compensating weight is constructed to incorporate the exact matrix element, but is recommended for use in tests only. 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 some terms appearing in QCD. The present paper can be understood either as a contribution to discussion on how to match two collinear emission chains resulting from charged sources in a way compatible with the exact and complete phase space, exclusive exponentiation and the first order matrix element of QED (scalar QED), or as the practical study of predictions for accelerator experiments.

Figures (14)

• Figure 1: Distributions of invariant masses normalized to center of mass energy and squared ($M^{\;2}/S$) for $e^+e^- \to \pi^+\pi^- (\gamma)$ at 2 GeV center of mass energy. Results from PHOTOS with matrix element taken from Nanava:2006vv are given in red (or darker grey) colour. If matrix element $\sum_{\lambda,\epsilon}|M_{I}^\prime|^2$ or $\sum_{\lambda,\epsilon} |M_{I'}^\prime|^2$ is used (the two options are effectively identical) results are given in green colour. Logarithmic scale is used, but for the ratio (black line) linear scale is used instead. Fraction of events with photons above 50 MeV is respectively 4.2279 $\pm$ 0.0021 % and 4.2269 $\pm$ 0.0021% for the two programs.
• Figure 2: Distributions of invariant masses normalized to center of mass energy and squared ($M^{\;2}/S$) for $e^+e^- \to \pi^+\pi^- (\gamma)$ at 2 GeV center of mass energy. Results from PHOTOS with matrix element taken from Nanava:2006vv are given in red colour. If matrix element $|M_{I}|^2$ is used results are given in green colour. Logarithmic scale is used, but for the ratio (black line) linear scale is used instead. Fraction of events with photons above 50 MeV is respectively 4.2279 $\pm$ 0.0021 % and 3.4435 $\pm$ 0.0019 % for the two programs.
• Figure 3: Distributions of invariant masses normalized to center of mass energy and squared ($M^{\;2}/S$) for $e^+e^- \to \pi^+\pi^- (\gamma)$ at 2 GeV center of mass energy. Results from PHOTOS with matrix element taken from Nanava:2006vv are given in red colour. If matrix element $\sum_{\lambda,\epsilon} |M_{I'}|^2$ is used results are given in green colour. Logarithmic scale is used, but for the ratio (black line) linear scale is used instead. Fraction of events with photons above 50 MeV is respectively 4.2279 $\pm$ 0.0021 % and 3.8329 $\pm$ 0.0020 % for the two programs.
• Figure 4: Distributions of invariant masses normalized to center of mass energy and squared ($M^{\;2}/S$) for $e^+e^- \to \pi^+\pi^- (\gamma)$ at 2 GeV center of mass energy. Results from PHOTOS with matrix element taken from Nanava:2006vv are given in red colour. If complete matrix element is used results are given in green colour. Logarithmic scale is used, but for the ratio (black line) linear scale is used instead. Fraction of events with photons above 50 MeV is respectively 4.2279 $\pm$ 0.0021 % and 4.4320 $\pm$ 0.0021% for the two programs.
• Figure 5: Distributions of invariant masses normalized to center of mass energy and squared ($M^{\;2}/S$) for $e^+e^- \to \pi^+\pi^- (\gamma)$ at 2 GeV center of mass energy. Results from PHOTOS with matrix element taken from Nanava:2006vv are given in red colour. If exponentiation is activated as well results are given in green colour. Logarithmic scale is used, but for the ratio (black line) linear scale is used instead. Fraction of events with at least one photon above 50 MeV is respectively 4.2279 $\pm$ 0.0021 % and 4.1377 $\pm$ 0.0020% for the two cases.