I would like to calculate the 1-loop 1-PI correction to the propagator for $\phi^3$ scalar theory in 2 dimensions, where the integral is finite. Performing the usual procedure (Feynman trick, Wick rotation into Euclidean space momenta) I reduce the 1-loop integral :

$$\Pi(p^2)=\frac{g^2}{2}\int\frac{d^2q}{(2\pi)^2}\frac{1}{[(p-q)^2-m^2+i\epsilon][q^2-m^2+i\epsilon]}=$$ $$...=\frac{g^2}{4\pi}\int\limits_{0}^{1} dx \int\limits_{0}^{\infty}dq_E \frac{q_E}{(q_E^2+m^2+(x^2-x)p^2)^2}.$$

The momentum integral is easy to do and the result is:

$$\frac{g^2}{8\pi}\int\limits_{0}^{1} dx \frac{1}{m^2+(x^2-x)p^2}.$$

However, integrating over the Feynman parameter is interesting, because the integrand has a pole above threshold, but is regular below threshold, while what I would have expected would be the exact opposite. Am I missing some physics in this calculation or is there a problem with regularization of the integral?


You are getting poles when $p^2\geq (2m)^2$. This is exactly what you expect since then there is enough energy flowing through the diagram to put the two internal lines on shell.


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