Expanding in the Gradient at Weak Coupling

A new method for producing the weak-coupling expansion in nonlinear $\sigma$ 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 $\frac{1}{\beta }\to 0$. 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 $d\to \infty$. Notably the new scheme reveals that the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ nonlinear $\sigma$ models in $d=2\mathrm{}$ are not asymptotically free; it also indicates that the leading term of the $\beta$ function of QCD in four dimensions is likely different from its presently accepted value.