More Accurate Treatment of the Low-Energy Potential in the Strip Approximation

Both the tractability and the reliability of the Reggeized strip approximation are increased by normalizing the long-range part of the potential at zero energy. We make the decomposition $V(t, s)=V(t, 0)+V^R(t, s)$, where $s$ is the energy squared and $t$ is the negative square of momentum transfer, and then calculate the first part $V(t, 0)$ from a low-$t$ partial-wave expansion in the $t$ reaction. The other part, $V^R(t, s)$, which vanishes at $s=0\mathrm{}$ but which dominates at large $s$, continues to be calculated by the method of Chew and Jones from the leading crossed-reaction Regge trajectories. The normalization eliminates from the bootstrap calculation the need for accurate treatment of secondary trajectories that fail to reach $J=0\mathrm{}$; normalization, in fact, amounts to a sum over all poles, and is useful because poles that remain in the left half of the $J$ plane may have an important influence on the low-energy potential while being negligible at higher energies.