Asymptotic Behavior of Trajectory Functions and Size of Classical Orbits

The strong-coupling limits of several ladder-graph models are explored. A correlation is found between the power of the coupling constant appearing in the leading Regge trajectory $\alpha \mathrm{}(t=0\mathrm{})\mathrm{}$ and the size of classical orbits described by the coordinate-space Bethe-Salpeter wave function. Specifically: (i) For the ${\phi }^3$ theory with exchanged mass $m\ne 0$, the orbit radius $r_0$ approaches a fixed value and $\alpha \mathrm{}(t=0\mathrm{})\mathrm{}\sim g^{\frac{1}{2}}$. (ii) For ${\phi }^3$ with $m=0\mathrm{}$, $r_0$ grows linearly with $g$ and $\alpha \mathrm{}(t=0\mathrm{})\mathrm{}\sim g$. (iii) For ${\phi }^4$, $r_0\to 0$ and the leading singularity is a fixed cut in $t$. Expansions about classical orbits are possible in the first two cases, and lead in lowest order to a harmonic-oscillator equation from which corrections to the classical result may be derived.