this question is for experts in asymptotic analysis, in particular expansions of integrals.
Consider the following type of integral
$$ I(\ell) = \ell^2\int_{[0,2\pi]^2}\frac{dk dq}{(2\pi)^2} \frac{\operatorname{sinc}^2((k-q)\ell/2)}{\operatorname{sinc}^2((k-q)/2)}\left[e^{ \beta (\epsilon(k)-\epsilon(q))}-1\right]\frac{n(k)(1-n(q))}{\beta(\epsilon(k) -\epsilon(q))} $$
where
$$ \operatorname{sinc}(x) = \frac{\sin(x)}{x} $$
is the cardinal sine and
$$ n(k) = \frac{1}{e^{-\beta(\cos k + \mu)}+1} $$
is a kind of Fermi-Dirac distribution and
$$ \epsilon(k) = -\cos k - \mu\,. $$
I am interested in understanding the behavior of $I(\ell)$ when $\mathbb{N}\ni\ell\to+\infty$. In particular I consider the case where $\beta>0$ and $\mu\in[-1,1]$ are fixed and do not scale with $\ell$. I am familiar with standard methods like stationary phase, Laplace and steepest descent for one dimensional integrals but it looks like the story for higher dimensions is much more complicated. Googling around I found these integrals appear a lot in scattering theory of light where even higher dimensional analogs are needed.
The leading order in $\ell$ is just
$$ I(\ell) \sim \ell\int_{[0,2\pi]}\frac{dk}{2\pi} n(k) (1-n(k)) $$
because
$$ \lim_{\ell \to +\infty}\ell\,\operatorname{sinc}^2(\ell x/2) = 2\pi \delta(x) $$
in the sense of distributions. So the leading order is linear in $\ell$. What I would like to know is whether there is a way to go further. On physical grounds I expect that
$$ I(\ell) = a_1 \ell + a_2 \log\ell + a_3 + O(\ell^{-1}) $$
where $a_i$ are finite constants. I would already be happy to understand how to systematically get $a_2$ and maybe that will shed light on how to go further and how to approach the general case for other types of integrals.
Thank you very much for the help.
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