Editing Renormalization (section)

=== A loop divergence ===
[[Image:Loop-diagram.png|thumb|upright=1.1|Figure 2. A diagram contributing to electron–electron scattering in QED. The loop has an ultraviolet divergence.]]
The diagram in Figure 2 shows one of the several one-loop contributions to electron–electron scattering in QED. The electron on the left side of the diagram, represented by the solid line, starts out with 4-momentum {{math|''p<sup>μ</sup>''}} and ends up with 4-momentum {{math|''r<sup>μ</sup>''}}. It emits a virtual photon carrying {{math|''r<sup>μ</sup>'' − ''p<sup>μ</sup>''}} to transfer energy and momentum to the other electron.  But in this diagram, before that happens, it emits another virtual photon carrying 4-momentum {{math|''q<sup>μ</sup>''}}, and it reabsorbs this one after emitting the other virtual photon. Energy and momentum conservation do not determine the 4-momentum {{math|''q<sup>μ</sup>''}} uniquely, so all possibilities contribute equally and we must integrate.

This diagram's amplitude ends up with, among other things, a factor from the loop of
<math display="block">-ie^3 \int \frac{d^4 q}{(2\pi)^4} \gamma^\mu \frac{i (\gamma^\alpha (r - q)_\alpha + m)}{(r - q)^2 - m^2 + i \epsilon} \gamma^\rho \frac{i (\gamma^\beta (p - q)_\beta + m)}{(p - q)^2 - m^2 + i \epsilon} \gamma^\nu \frac{-i g_{\mu\nu}}{q^2 + i\epsilon}.</math>

The various {{math|''γ<sup>μ</sup>''}} factors in this expression are [[gamma matrices]] as in the covariant formulation of the [[Dirac equation]]; they have to do with the spin of the electron. The factors of {{mvar|e}} are the electric coupling constant, while the <math>i\epsilon</math> provide a heuristic definition of the contour of integration around the poles in the space of momenta. The important part for our purposes is the dependency on {{math|''q<sup>μ</sup>''}} of the three big factors in the integrand, which are from the [[propagator]]s of the two electron lines and the photon line in the loop.

This has a piece with two powers of {{math|''q<sup>μ</sup>''}} on top that dominates at large values of {{math|''q<sup>μ</sup>''}} (Pokorski 1987, p.&nbsp;122):
<math display="block">e^3 \gamma^\mu \gamma^\alpha \gamma^\rho \gamma^\beta \gamma_\mu \int \frac{d^4 q}{(2\pi)^4} \frac{q_\alpha q_\beta}{(r - q)^2 (p - q)^2 q^2}.</math>

This integral is divergent and infinite, unless we cut it off at finite energy and momentum in some way.

Similar loop divergences occur in other quantum field theories.