  We review Buchler and Colangelo's result that leading divergences at any loop
order can be calculated using only one-loop calculations and we provide an
alternative proof. We then use this method to calculate the leading divergences
of and thus the leading logarithmic corrections to the meson mass in the
massive O(N) nonlinear sigma model to five-loop order. We also calculate the
all-loop result to leading order in the large $N$ expansion by showing that
only cactus diagrams contribute and by summing these via a generalized gap
equation.

High Energy Physics - Phenomenology;High Energy Physics - Theory