I posted this question before in the Physics stack exchange, but it was recommended to post it better here.

While reading a paper I saw the following integral equation.

$\frac{1}{g} = \int_{-\pi}^{\pi} (\prod_{\sigma}\frac{\mathrm{d}p_{\sigma}}{2\pi}) \frac{\delta(\Sigma_{\sigma} p_{\sigma})}{(\prod_{\sigma} (1+\frac{2 \mathrm{sin}(p_{\sigma}/2)}{m}) -1)} \, ,\, \mathrm{with}\, \sigma = {1,2,3,4} $

But there was no solution given for it. To learn more about this integral equation, I would like to solve it. Sadly I don't even know how to begin/deal with this equation or more precisely with the integral itself. Hopefully someone can give me a hint or two.


  • $\begingroup$ i guess you already have the starting point that the integral only has a non-zero component when $\sum_\sigma p_\sigma = 0$? so I guess you dealing with conservative (momenta? in different dimensions) $\endgroup$ – Chinny84 Apr 29 '15 at 14:16
  • $\begingroup$ $p_1, ..., p_4$ are one dimensional momenta, which are conserved, indicated by the delta-function. should have mentioned that, sorry ... $\endgroup$ – nerdizzle Apr 29 '15 at 14:38
  • $\begingroup$ can you solve the case of $\sigma=1$? it's quite straightforward $\endgroup$ – tired Apr 29 '15 at 14:40
  • $\begingroup$ I tried to solve the trivial case $\sigma = 1$, but I did not succeed.. $\endgroup$ – nerdizzle Apr 29 '15 at 14:49
  • $\begingroup$ the integral reads in this case $\int_{\pi}^{-\pi}\frac{dp}{2 \pi}\frac{\delta(p)}{2\sin(p/2)/m}$ what is the effect of the delta function? $\endgroup$ – tired Apr 29 '15 at 14:52

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