Regge-pole behavior from perturbative scalar-field theories

Abstract

In the planar approximation, we consider the large-energy-$s$, fixed-transfer-$t$ limit of the four-point vertex function. In ${\ensuremath{\varphi}}^{3}$ theory (6 space-time dimensions) and in ${\ensuremath{\varphi}}^{4}$ theory (4 space-time dimensions), for any essentially and crossed planar graph, we analytically calculate the coefficients of all powers of logarithms of $s$ for the leading power of $s$. After summing the series in logarithms obtained when the four-point function is considered, we discuss the existence of Regge trajectories from a Riccati-type differential equation. In $g{({\ensuremath{\varphi}}^{3})}_{6}$ theory we find a discrete family of Regge-pole trajectories with $g$-dependent intercepts accumulating at $\ensuremath{\alpha}=\ensuremath{-}1$. In $g{({\ensuremath{\varphi}}^{4})}_{4}$ theory the solution of the Riccati equation may be easily found if there exists a fixed point $g={g}^{*}$; the result then is a fixed $g$-independent cut and an infinite number of pole trajectories (the square-root branch point is above the intercepts at $g={g}^{*}$).