- #1
maline
- 436
- 69
- TL;DR Summary
- The integral used in Dim Reg to renormalize the photon field without a mass term doesn't converge anywhere near d=4.
In the loop integral for the one-loop correction to the photon propagator in QED, the dominant term, after Wick rotation and angular averaging, has the form (omitting uninteresting factors) $$(1-2/d) e^2 \eta^{\mu\nu}\int_0^\infty \frac{p^{d+1}}{(p^2+\Delta)^2}dp,$$ where ##p## is the absolute value of the Euclidean loop momentum, ##d## is the number of spacetime dimensions, ##-e## is the electron charge, and ##\Delta=m^2+q^2x(1=x)## is a positive constant depending on the electron mass ##m##, the photon momentum ##q^\mu##, and the Feynman parameter ##x##. The integral is quadratically divergent for ##d=4##, and the dominant part is independent of the photon momentum. This would seem to require a mass correction term for the photon, breaking gauge invariance. Dimensional Regularization is presented as a solution: we treat ##d## as a continuous complex variable, and use the "well-known formula" $$\int_0^\infty \frac{p^{d+1}}{(p^2+\Delta)^2}dp=\frac12 \Delta^{d/2-1}\Gamma(1-d/2)\Gamma(1+d/2).$$ The prefactor ##(1-2/d)\propto(1-d/2)## miraculously shifts the argument of the first gamma function to ##(2-d/2)##, matching the next term of the integral result, and the ##q^2## found within ##\Delta## magically unites with a ##-q^\mu q^\nu## from the next term, so the end result has exactly the form required by gauge invariance.
But this inspiring tale has a plot hole: the "well-known formula" is false, and blatantly so. The integral on the LHS converges only for ##-2<Re(d)<2##, not anywhere in the neighborhood of ##d=4##. For purely real values of ##d\ge2## the integral goes to ##+\infty## like ##\Lambda^{d-2}## (with ##\Lambda## the UV cutoff), or like ##\log\Lambda## for ##d=2##. For complex ##d## with ##Re(d)<2##, the integral oscillates, with the amplitude diverging. The RHS, meanwhile, is finite except for simple poles at the nonzero even numbers. The discrepancy is most incredible at the (physically meaningful!?) value ##d=3##, where using the above procedure would lead to the conclusion that the loop diagram is finite, despite the clear linear divergence of the integral.
Now I can accept that if we somehow know (how??) that the diagram's value must be analytic in ##d## in a region that surrounds ##d=4## and also extends into the ##-2<Re(d)<2## region, then the convergence of the integral for ##-2<Re(d)<2## shows that the function we end up with is the only possible candidate. Also, I expect that actually formulating QED in ##(2+1)## dimensions probably requires changes more substantial than the ##d## dependence we use in Dim Reg for ##(3+1)## dimensions, so the finite value at ##d=3## need not be understood as physical.
But none of this answers the question: where did the divergent mass term go??
Is there a more convincing way to calculate this diagram, so that I can be reassured that the mass term truly does vanish somehow?
But this inspiring tale has a plot hole: the "well-known formula" is false, and blatantly so. The integral on the LHS converges only for ##-2<Re(d)<2##, not anywhere in the neighborhood of ##d=4##. For purely real values of ##d\ge2## the integral goes to ##+\infty## like ##\Lambda^{d-2}## (with ##\Lambda## the UV cutoff), or like ##\log\Lambda## for ##d=2##. For complex ##d## with ##Re(d)<2##, the integral oscillates, with the amplitude diverging. The RHS, meanwhile, is finite except for simple poles at the nonzero even numbers. The discrepancy is most incredible at the (physically meaningful!?) value ##d=3##, where using the above procedure would lead to the conclusion that the loop diagram is finite, despite the clear linear divergence of the integral.
Now I can accept that if we somehow know (how??) that the diagram's value must be analytic in ##d## in a region that surrounds ##d=4## and also extends into the ##-2<Re(d)<2## region, then the convergence of the integral for ##-2<Re(d)<2## shows that the function we end up with is the only possible candidate. Also, I expect that actually formulating QED in ##(2+1)## dimensions probably requires changes more substantial than the ##d## dependence we use in Dim Reg for ##(3+1)## dimensions, so the finite value at ##d=3## need not be understood as physical.
But none of this answers the question: where did the divergent mass term go??
Is there a more convincing way to calculate this diagram, so that I can be reassured that the mass term truly does vanish somehow?
