What about small momentum divergences?

[LAHLH]

In Srednicki (ch14 say), he looks at integrals of the form $$\int\, \frac{d^{d}l}{l^4}...$$. This is of course diveregent at large l if d>=4, which is easily seen by looking at the integrals measure in hyperspherical coords.

However, what about small l divergences, surely these occur if d<4, e.g. if d=3 we have something that goes as $$\frac{1}{l^2}$$, which diverges for small l. Why are we not concerned about this and only large l divergences?

Thanks

 

[strangerep]

Why are we not concerned about this and only large l divergences?

Because in the example you mentioned, Srednicki is dealing with a massive field.
I.e., there's a nonzero mass term in the denominator.

However, for QED involving the massless photon field, one does indeed encounter
so-called infrared divergences, which require special handling.

 

[LAHLH]

ah I see, thanks a lot.

 

[humanino]

Those small momentum divergences may be real : for instance, there is really an infinity of zero angle zero energy photons along an electron's line. One has to integrate the cross section over finite angle and finite energy to get a finite result. This kind of divergence is also at the root of the concept of "jet" in QCD.