Pion bremsstrahlung in the splitting function formalism and the dark photon production

TL;DR

The paper addresses dark-photon production in $\pi^-p$ collisions via two mechanisms: inelastic pion bremsstrahlung and a Drell–Yan–like process, across $m_{\gamma'}$ from $0.4$ to $3.5$ GeV. It argues that LO ChPT is not reliable at the energies of interest and develops a factorization-based splitting-function framework for bremsstrahlung, alongside a perturbative QCD Drell–Yan calculation with a Breit–Wigner propagator to compare contributions. The study finds that inelastic pion bremsstrahlung dominates for $m_{\gamma'}$ in $0.4$–$1.3$ GeV, while the Drell–Yan mechanism takes over for $m_{\gamma'}$ in $1.3$–$3.5$ GeV; the resulting dark-photon energy spectra and mean energies offer practical discriminants for experiments like NA64h and future beam-dump setups. These results have direct implications for interpreting dark-photon searches in fixed-target environments and guiding strategies to distinguish production channels through energy measurements.

Abstract

We study the production of hypothetical vector portal mediators, dark photons $γ^\prime$, with masses in the range 0.4-3.5 GeV in the negatively charged pion-proton collisions $π^-p\rightarrow γ^\prime X$ via inelastic pion bremsstrahlung and QCD Drell-Yan-like process. In both cases we estimate the value of total dark photon production cross section and obtain the energy distribution for dark photons that could be produced in the NA64h experiment. We also present the mean energies of dark photons produced in the same way by the secondary pions with momenta typical for T2K, DUNE and SHiP experiments.

Figures (6)

• Figure 1: Feynman diagrams for elastic $\pi^-p$-scattering in the leading order of Heavy Baryon ChPT: (a) $s$-channel, (b) $t$-channel, (c) 4-point. Figures are made with the help of the package TikZ-FeynmanEllis:2016jkw.
• Figure 2: Differential cross section of elastic pion-proton scattering $\pi^-p\rightarrow\pi^-p$ as the function of $\cos\theta$, where $\theta$ is the proton scattering angle in the center-of-mass frame and the momentum of the incident pion in the lab frame $P=50\text{\,GeV}$. The results of the LO ChPT calculation are shown in blue. The fit of Fermilab experimental data FermilabSingleArmSpectrometerGroup:1976krf for the pion beam of the same momentum is presented in red for comparison.
• Figure 3: Feynman diagram for the dark photon production in the initial state radiation of inelastic pion bremsstrahlung.
• Figure 4: (a) Differential cross section of inelastic pion bremsstrahlung \ref{['eq:dphspectrum']} for a set of dark photon masses, $m_{\gamma^\prime}$, listed in the legend, as a function of dark photon energy $E_{\gamma^\prime}$. The incident pion momentum $P=50\text{\,GeV}$ corresponds to NA64h experiment. (b) Mean energy of dark photons \ref{['eq:meanE-brem']} produced via inelastic pion bremsstrahlung as a function of dark photon mass for the incident pion momenta \ref{['eq:Peff']} typical for T2K, DUNE and SHiP experiments, see main text for details.
• Figure 5: In green: the dependence of full inelastic pion bremsstrahlung cross section \ref{['eq:fullcrsec']} on dark photon mass $m_{\gamma^\prime}$ calculated with the pion electromagnetic form factor $F^\pi_\text{em}(m^2_{\gamma^\prime})$RuizArriola:2024gwb (dark green line) and with the same form factor put equal to unity (light green line). In blue: full cross section of dark photon production via Drell--Yan process in $\pi^-p\rightarrow\gamma^\prime X$ calculated within perturbative QCD. Light blue line shows the LO result, whereas dark blue line presents the sum of LO and NLO cross sections. In all cases the calculations are made for the negatively charged pion beam of the NA64h experiment with $P=50\text{\,GeV}$.