# Factorization of fermionic scattering integral in 2d momentum rep

the scattering integrals for fermions involves both momentum ($k$) and energy ($k^2$) conservation and a nonlinear phase space factor of a distribution function $f(k)$.

$$\begin{multline}I(k) = \sum_{k_1, k_2, k_3} \delta(k^2+k_1^2+k_2^2+k_3^2) \delta(k+k_1+k_2+k_3)\times \\ \Bigl[f(k)f(k_1)\bigl(1-f(k_2)\bigr)\bigl(1-f(k_3)\bigr) - f(k_2)f(k_3)\bigl(1-f(k)\bigr)\bigl(1-f(k_1)\bigr)\Bigr]\end{multline}$$

In energy space, energy conservation is linear and powerful factorizations are possible to compute the integrals fast. In momentum space, the nonlinear energy conservation constraint complicates factorizations.

Trivially, in 2 dim one can work in polar coordinates separating off the radial part. But has anybody seen a more efficient reduction of numerical complexity (e.g. mapping to FFT,...)

Thanks for any hint!

-
Hi Michael, and welcome to Physics Stack Exchange! I reformatted your equation a bit for readability. Also, this seems like a QFT question; is there a particular reason you didn't put the quantum-field-theory tag on it? –  David Z Mar 6 '12 at 20:41