- #1
geoduck
- 258
- 2
If you have a momentum integral over the product of propagators of the form [itex]\frac{1}{k_o^2-E_k^2+i\epsilon} [/itex], why are there divergences associated with setting m=0?
Factoring you get: [itex]\frac{1}{k_o^2-E_k^2+i\epsilon}=\frac{1}{(k_o-E_k+i\epsilon)
(k_o+E_k-i\epsilon)} [/itex]. This expression has simple poles at [itex]k_0=\pm E_k [/itex]. These two poles do merge to form a pole of order 2 when m=0 at the special value of [itex]\vec{k}=0 [/itex]. But this special value of [itex]\vec{k}=0 [/itex] is only one point in the integration region, and the value of an integral doesn't depend on the behavior at a single point. Everywhere else besides this single point the integrand only has simple poles of order 1 and hence should be convergent around these points.
So it seems to me that the only divergences should be UV divergences and not IR divergences.
Factoring you get: [itex]\frac{1}{k_o^2-E_k^2+i\epsilon}=\frac{1}{(k_o-E_k+i\epsilon)
(k_o+E_k-i\epsilon)} [/itex]. This expression has simple poles at [itex]k_0=\pm E_k [/itex]. These two poles do merge to form a pole of order 2 when m=0 at the special value of [itex]\vec{k}=0 [/itex]. But this special value of [itex]\vec{k}=0 [/itex] is only one point in the integration region, and the value of an integral doesn't depend on the behavior at a single point. Everywhere else besides this single point the integrand only has simple poles of order 1 and hence should be convergent around these points.
So it seems to me that the only divergences should be UV divergences and not IR divergences.