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Taming forward scattering singularities in partial waves

Marta Fuentes Zamoro, Benjamín Grinstein, Pablo Quílez

Abstract

Perturbative partial-wave amplitudes diverge in cases with a massless exchanged particle in the $t$-channel. We argue that the divergence is an artifact of perturbation theory and give a prescription for the all-orders correction factor that renders the partial waves finite. As an example, we apply this to longitudinal $W^+W^-$ elastic scattering, for which there is a photon exchange $t$-channel contribution, and derive improved quasi-perturbative unitarity bounds on the mass of the Higgs. The method is also useful for $t$-channel exchange of very light particles.

Taming forward scattering singularities in partial waves

Abstract

Perturbative partial-wave amplitudes diverge in cases with a massless exchanged particle in the -channel. We argue that the divergence is an artifact of perturbation theory and give a prescription for the all-orders correction factor that renders the partial waves finite. As an example, we apply this to longitudinal elastic scattering, for which there is a photon exchange -channel contribution, and derive improved quasi-perturbative unitarity bounds on the mass of the Higgs. The method is also useful for -channel exchange of very light particles.

Paper Structure

This paper contains 1 section, 20 equations, 2 figures.

Table of Contents

  1. Acknowledgements

Figures (2)

  • Figure 1: In the non-relativistic limit, the phase $e^{iW}$, see \ref{['Eq:SchrodingerPhase']}, arises from the resummation of the ladder diagrams (top line); the crossed diagrams do not contribute in the non-relativistic limit (bottom line).
  • Figure 2: Comparison of the perturbative unitarity bound from Lee, Quigg and Thacker (LQT) Lee:1977eg (dashed) with our improved quasi-perturbative unitarity bound (solid) as a function of $s/m_H^2$. Following Lee:1977eg, we show the $W_L^+W_L^- \to W_L^+W_L^-$, $J=0$ partial-wave amplitude $a_0$ for $m_H=300$ GeV (blue) and $m_H^2=4\sqrt2\pi/G_F$ (green). As opposed to LQT we include all contributions to the tree level scattering amplitude, and tame the forward singularity by including the effect of the multiple photon resummation in \ref{['eq:T-corrected', 'eq:phase-coef-W']}.