The integral I want to calculate is defined as
$$
P(s)=\int\_{-\infty}^{\infty}{\rm d}x\int\_{-\infty}^{\infty}{\rm d}y\ \delta\left(\frac{(x+y)^2+4x^2y^2}{(x+y)^2+(x+y)^4}-s\right)e^{-\left(x^2+y^2\right)/2a^2}
$$
This integral seems no explicit solution (?) since the quartic terms in the $\delta$ function. So I just want to find the asymptotic behavior for small and positive $s$, i.e $P(s)$ for $s\to0^+$. I am trying to find the answer by the changing of variables as $x=p+q,y=p-q$, then I have
$$
P(s)=2\int\_{-\infty}^{\infty}{\rm d}p\int\_{-\infty}^{\infty}{\rm d}q\ \delta\left(\frac{p^2+(p^2-q^2)^2}{p^2+4p^4}-s\right)e^{-\left(p^2+q^2\right)/a^2}.
$$
To get small $s$, the variable $q$ should closed to $\pm p$, and $p$ should be as large as possible. So I directly let $q=p$, neglect the integral over $q$, and I have
$$
P(s)\propto\int\_{-\infty}^{\infty}{\rm d}p\ \delta\left(\frac{1}{4p^2}-s\right)e^{-p^2/a^2},
$$
where I used the assumption $p\gg1$ so that $1+4p^2\approx 4p^2$. Using the properties of $\delta$ function, I finally obtain
$$
P(s)\propto s^{-2}e^{-1/(2a^2s)}.
$$
I am probably wrong, especially for the step that I neglect the integration over $q$. But this is all I can get. Any suggestion will be helpful.



Asymptotic behavior of the integral that contains $\delta$ function
$$P(s)=2\int\_{-\infty}^{\infty}{\rm d}p\int\_{-\infty}^{\infty}{\rm d}q\ \delta\left(\frac{p^2+(p^2-q^2)^2}{p^2+4p^4}-s\right)e^{-\left(p^2+q^2\right)/a^2}$$
$$\qquad=2\int\_{0}^{\infty}\frac{{\rm d}x}{\sqrt{x}}(x+4x^2)\int\_{0}^{\infty}\frac{{\rm d}y}{\sqrt{y}}\ \delta\left(x+(x-y)^2-s(x+4x^2)\right)e^{-\left(x+y\right)/a^2}$$
$$\qquad=2\int\_{0}^{\infty}\frac{{\rm d}x}{\sqrt{x}}e^{-x/a^2}\frac{x+4x^2}{z\_+-z\_-}\left[\frac{\theta(z\_+)}{\sqrt{z\_+}}e^{-z\_+/a^2}+\frac{\theta(z\_-)}{\sqrt{z\_-}}e^{-z\_-/a^2}\right]\theta\bigl(x-(1-s)/4s\bigr)$$
$$\qquad=\int\_{(1-s)/4s}^{\infty}dx\,e^{-x/a^2}\frac{1+4x}{\sqrt{4 s x+s-1}}\left[\frac{1}{\sqrt{z\_+}}e^{-z\_+/a^2}+\frac{1}{\sqrt{z\_-}}e^{-z\_-/a^2}\right],$$
with $z\_\pm=x\pm\sqrt{x (4 s x+s-1)}$ and $\theta(z)$ the unit step function. In the final equality I assumed $0<s<1/4$.


For $s\ll 1$ this approximates to
$$P(s)\approx\frac{4}{\sqrt{s}}\int\_{1/4s}^{\infty}dx\,e^{-2x/a^2}\frac{1}{\sqrt{4 s x-1}}=\sqrt{2\pi}(a/s)\exp\left(-\frac{1}{2a^2s}\right).$$


