What's the reason for this epsilon? (about Schwinger effect)

[cello]

In Schwinger's classic http://dx.doi.org/10.1103/PhysRev.82.664" on pair production, he inserted an infinitesimal to get the desired pair production rate. (On page 13, he said: "We shall now simply remark that, to extend our results to pair-production fields, it is merely necessary to add an infinitesimal negative imaginary constant to the denominator of Eq. (6.30) ")

Also I found a pedagogical http://dx.doi.org/10.1119/1.19313" in which the author said "Defining the integration contour by infinitesimal semicircular deviations into the upper half plane from the straight line path (in order to guarantee exponential decay)." It seems that this is a choice made by hand.

Can anyone explain in detail the justification for this epsilon? I'm aware of the epsilon we added to the time (or the mass) when doing path integral, but I can't see if the epsilon in Schwinger's paper is of the same nature.

You can download the two papers I mentioned http://ifile.it/u9smtwa", if you don't have the access.

 

[LightHero]

In Schwinger's paper, the infinitesimal is added to the denominator of a specific equation (6.30). The purpose of this infinitesimal is to make sure that the denominator does not become zero, which would lead to an undefined result. By introducing the infinitesimal, the denominator is always non-zero and the resulting equation is well-defined.In the second paper, the infinitesimal is added to the integration contour. This is done to guarantee exponential decay of the integrand. Without adding the infinitesimal, the integral could potentially diverge. By adding the infinitesimal, the integral converges and the result is well-defined.