Do Total Cross Sections for Scattering of Off-Shell Particles Grow Like a Power of the Energy?

Abstract

Let ${\sigma }_P(Q^2, s)$ be the total cross section for scattering of meson $P$ on some target, with $-Q^2$ the off-shell mass of $P$. It is shown that, when $-Q^2\ne {m_P}^2$ (the physical $P$ mass), quantum chromodynamics suggests that, as the c.m. energy squared $s$ approaches $\infty$, ${\sigma }_P(Q^2, s)\sim C_{1}f\left(Q^2\right)s^{\lambda }+\dots , \lambda >0,$ where $f\left(Q^2\right)$ is calculable for large $Q^2$. This is compatible with the Froissart bound, but only for on-shell particles, provided $f(-{m_P}^2)=0\mathrm{}$. It is shown that there is experimental evidence supporting such behavior.