A theorem on resumming leading logarithms in non-Abelian gauge theories at short distances

Given a power series expansion in two parameters alpha and t (with, for instance, alpha =O(1/10) and t>>>1), the sum of the leading asymptotic part in t of each individual term in the series does not in general give the correct asymptotic behaviour of the whole sum of the series. In fact, when such a procedure works, it is the exception rather than the rule. In QCD at short distances a similar procedure known as the 'leading logarithm approximation' is nevertheless used. The author presents a theorem justifying this procedure. The underlying reason is the invariance with respect to the renormalization group (RG) of physical quantities and the covariance of Green functions and vertex amplitudes under this group. Whereas the individual terms of straightforward perturbation expansion for these quantities are not invariant (covariant) with respect to RG, the quantities themselves are invariant (covariant). In order to maintain these invariance properties, the usual perturbation expansion must be recast in a different form, the leading term of which (in the short distance region) is precisely the sum of the leading logarithms. (9 refs).