Expanding in the gradient at weak coupling.

Abstract

A new method for producing the weak-coupling expansion in nonlinear $\ensuremath{\sigma}$ models and Yang-Mills theories is developed. It assumes only the smallness of the gradient of the field, rather than of the field itself, as the coupling constant $\frac{1}{\ensuremath{\beta}}\ensuremath{\rightarrow}0$. The new method reproduces all the known exact results. It agrees with ordinary perturbation theory only in Abelian problems or when the number of space-time dimensions $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Notably the new scheme reveals that the $\mathrm{O}(N)$ nonlinear $\ensuremath{\sigma}$ models in $d=2$ are not asymptotically free; it also indicates that the leading term of the $\ensuremath{\beta}$ function of QCD in four dimensions is likely different from its presently accepted value.