Uniform control of the low-temperature pp-wave expansion

Establish uniform control of the low-temperature expansion of the critical large-N O(N) model on the pp-wave geometry in the simultaneous limit of vanishing positive null momentum k and large transverse coordinate |y|.

Background

The paper derives the low-temperature expansion of the semi-universal residue by expanding around the zero-temperature critical saddle on the pp-wave geometry. The calculation involves thermal sources localized in transverse position and an integral over positive null momentum k, with the asymptotic expansion organized in sectors of q=e^{-β/2}.

Although the relevant integrals are finite and the expansion is computed through several exponential sectors, the authors explicitly state that they have not established uniform control when k approaches zero while |y| becomes large. Proving such control would clarify the validity and error estimates of the low-temperature asymptotic expansion in this potentially nonuniform region.