Simulation of QED Radiation in Particle decays using the YFS Formalism

TL;DR

This work introduces SOPHTY, a module integrated into the HERWIG++ event generator that applies YFS soft-photon resummation to QED radiation in particle decays, aiming for universality across decay modes. It builds two building-block dipole algorithms (final-final and initial-final) to dress decays with photons while preserving correct soft and collinear limits, and adds higher-order corrections through real-emission β̄1^1 and virtual β̄0^1 terms using leading-log and dipole-splitting formalisms. The approach allows decays to be simulated with all-orders soft radiation and leading collinear logs in a process-independent way, with exact corrections incorporated where available via the C residuals. Validation across Z, W, and various meson decays shows stable, physically reasonable photon spectra and good agreement with other established tools, demonstrating the method's broad applicability and setting the stage for future inclusion of full O(α) corrections in more processes.

Abstract

In this paper we describe a program (SOPHTY) implementing QED corrections to decays in the HERWIG++ event generator. In order to resum the dominant soft emissions to all orders, the program is based on the YFS formalism. In addition, universal large collinear logarithms are included and the approach can be systematically extended to incorporate exact, process specific, higher order corrections to decays. Due to the large number of possible decay modes the program is designed to operate, as far as possible, independently of the decay matrix elements.

Figures (7)

• Figure 1: The total energy $\left(K_{0}\right)$ of the photons radiated in $\mathrm{Z}$ boson decays to leptons: (a) shows the $K_{0}$ spectrum for the case that no infrared residuals are considered $\left(\mathcal{C}=1\right)$; (b) shows the effect of including the collinear approximation for the $\mathcal{O}\left(\alpha\right)$ residual $\bar{\beta}_{1}^{1}$.
• Figure 2: The total energy $\left(K_{0}\right)$ of the photons radiated in $\mathrm{W}$ boson decays to leptons; (a) shows the $K_{0}$ spectrum for the case that no infrared residuals are considered $\left(\mathcal{C}=1\right)$; (b) shows the effect of including the collinear approximation for the $\mathcal{O}\left(\alpha\right)$ residual $\bar{\beta}_{1}^{1}$.
• Figure 3: The total energy $\left(K_{0}\right)$ of the photons radiated in $\mathrm{W}^{\pm}\rightarrow\mathrm{e}^{\pm}\nu_{e}/\bar{\nu_{e}}$ decays. In figure (a) the red histogram was generated using the WINHACPlaczek:2003zg simulation, including exact $\mathcal{O}\left(\alpha\right)$ real emission corrections to the $\mathrm{W}^{\pm}$ decay, while the black line was generated using the SOPHTY module for QED radiation in HERWIG++. In figure (b) we show the difference between the spectra shown in (a) divided by the associated statistical error. The discrepancy in the region beyond 40 GeV is exclusively comprised of events with at least two non-soft photons, which neither program is designed to model well.
• Figure 4: The total energy $\left(K_{0}\right)$ of the photons radiated in the decays of neutral vector mesons to pseudoscalar mesons for a number of different decays: (a) $\rho\rightarrow\pi^{+}\pi^{-}$; (b) $\phi\rightarrow\pi^{+}\pi^{-}$ and $\phi\rightarrow{\textrm{K}}^{+}{\textrm{K}}^{-}$; (c) ${\textrm{K}}^{*0}\rightarrow{\textrm{K}}^{\pm}\pi^{\mp}$; (d) ${\textrm{J}}/\psi\rightarrow\pi^{+}\pi^{-}$ and ${\textrm{J}}/\psi\rightarrow{\textrm{K}}^{+}{\textrm{K}}^{-}$. In addition to using the real physical masses of the decay products we have included the effect of varying the masses of the decay products.
• Figure 5: The total energy $\left(K_{0}\right)$ of the photons radiated in the decays of charged vector mesons to pseudoscalar mesons for the decays: (a) $\rho^{\pm}\rightarrow\pi^{\pm}\pi^{0}$; (b) ${\textrm{K}}^{*\pm}\rightarrow{\textrm{K}}^{\pm}\pi^{0}$ and ${\textrm{K}}^{*\pm}\rightarrow{\textrm{K}}^{0}\pi^{\pm}$. In addition to using the real physical masses of the decay products we have included the effect of varying the masses of the decay products.