High-energy amplitudes of Yang-Mills theory in arbitrary perturbative orders

Abstract

In this paper, we give a summary of our work on perturbative calculations of high-energy amplitudes in Yang-Mills theory. Using the model of SU(2) with an isodoublet of Higgs bosons, we calculate the leading real part and the leading imaginary part of the vector-meson-vector-meson scattering amplitude in the 2nd through 10th orders. We also calculate two sets of diagrams to all perturbative orders: the ladder diagrams and the multi-meson-exchange diagrams. The leading terms of the amplitude of $I=1$ (one unit of isospin exchanged) come from the ladder diagrams only and are shown to add up to a Regge-pole term corresponding to the Reggeization of the vector meson. The above results are shown to generalize easily to other non-Abelian gauge field theories. The extension to the process of fermion-fermion scattering is also straightforward, and we give a proof that these amplitudes are asymptotically proportional to the corresponding ones in vector-meson-vector-meson scattering. The 2nd- through 10th-order calculations show the following features: (i) All factors of $\mathrm{ln}s$ come from integration over longitudinal momenta. (ii) All divergent integrals over the transverse momenta cance. (This has been explicitly verified up to the 8th order only.) The 2nd- through 10th-order results suggest to us a recursion formula which determines to all perturbative orders the leading terms of the scattering amplitudes. Summing up these leading terms, we found the following: (a) For the amplitude of $I=0$ (no exchange of isospin), the sum of the leading terms exceeds the Froissart bound, representing a fixed branch point at ${J}_{0}=1+[\frac{(2\mathrm{ln}2)}{{\ensuremath{\pi}}^{2}}]{g}^{2}$ in the plane of the angular momentum. (b) For the amplitude of $I=2$ (two units of isospin exchanged), the sum of leading terms at $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}=0$ has a branch point at ${J}_{2}=2{\ensuremath{\alpha}}_{1}(0)\ensuremath{-}1$, where ${\ensuremath{\alpha}}_{1}$ is the Regge trajectory on which the vector meson lies. (c) From (a) and (b) we have $\ensuremath{\infty}g{J}_{0}g1g{\ensuremath{\alpha}}_{1}(0)g{J}_{2}$. The qualitative features of high-energy scattering in Yang-Mills theories are therefore exactly the same as those in QED. In particular, the violation of the Froissart bound by summing leading terms in all these cases indicates the necessity of a calculational program to go beyond summing leading terms.