Analytical Approximations for Beamstrahlung at Very High Energy Electron-Positron Colliders

TL;DR

This work develops an analytical framework for beamstrahlung in very high energy $e^+e^-$ colliders, showing that at large quantum parameter $Υ$ the electron and photon energy spectra can be described by universal functions $h_e(x,τ)$ and $h_g(x,τ)$. The key advancement is the asymptotic Yokoya-Chen (AYC) model, which yields a parameter-free, scalable description of beamstrahlung through a time-like variable $τ$ and a convolution-based construction of luminosity spectra that agrees with detailed PIC simulations. By deriving simple, accurate approximations to the AYC equations and implementing a tractable collision model, the paper provides practical tools to predict luminosity distributions and the evolution of $e^+e^-$, $e\gamma$, and $\gamma\gamma$ channels in multi-TeV machines. The findings enable more reliable design studies for next-generation linear colliders while highlighting limitations (LCFA, pair production, and nonperturbative regimes) and avenues for future refinement with high-performance simulations.

Abstract

Among the many effects that occur in beam-beam electron-positron collisions at TeV energies, emission of hard synchrotron radiation, or beamstrahlung, has special importance. Beamstrahlung determines the energy spectrum of the most energetic electrons, positrons, and photons and supplies the initial condition for the calculation of all other QED processes. In this paper, we show that the description of beamstrahlung simplifies in the limit of large quantum parameter $Υ$, which is realized in 10 TeV collider designs. The beamstrahlung spectra for electrons and photons are given in terms of universal functions. We supply approximations to these functions that will be useful for more general studies of the beam-beam interaction at very high energies.

Figures (11)

• Figure 1: Luminosity distributions $d{\cal L}/dz$ from simulation, comparing designs at 3, 10, and 30 TeV with beam parameters from Table \ref{['tab:params']}. Top: Un-normalized distributions plotted on a log scale. Bottom: Normalized distributions plotted on a linear scale. Left: flat beam parameters; Right: round beam parameters.
• Figure 2: Comparison of the numerical solution of the AYC equation for $h_e(x,\tau)$, in blue, to the formula \ref{['yeformmain']}, in red.
• Figure 3: Comparison of the numerical solution of the AYC equation for $h_g(x,\tau)$, in blue, to the formula \ref{['yeformmain']}, in red.
• Figure 4: Schematic drawing of the collision process in our simple model, defining the coordinates $T$ and $Z$.
• Figure 5: Comparison of GUINEA-PIG simulation data for the electron energy distribution in $e^+e^-$ collisions with the formula \ref{['feval']} for $f_e(x)$, for a CM energy of 10 TeV. Left: flat beams; Right: round beams. In the analytic formulae, we use the nominal values $N_\gamma$ = 1.5 for flat beams and 5.7 for round beams.