Publication : t95/048

Abstract:
In a recent letter [Phys. Rev. Lett. {\bf 74} (1995) 1920-1923, hep-lat/9311019] Patrascioiu and Seiler argued that the results of standard perturbation theory (PT) are not valid for two dimensional models with a non-abelian global symmetry. They considered such a theory in a finite box of size $L$, with special boundary conditions (BC) called ``superinstantons" (SI) and showed that in the thermodynamic limit $L\to\infty$ the 2-loop corrections are finite, but differ from those obtained from PT with standard BC. They concluded that, when SI configurations are taken into account, renormalization group (RG) $\beta$-functions are modified and that the limit $L\to\infty$ and the weak-coupling expansion do not commute. In fact the results of Patrascioiu and Seiler do not contradict standard PT. Indeed one can show from general principles that: (1) PT with a SI BC is infra-red (IR) divergent; (2) the IR divergent terms are associated, via the short distance operator product expansion (OPE), to singular local operators, not present for classical backgrounds; (3) taking into account these operators, the perturbative RG functions are not modified.

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