11
Loops and Renormalization II: QED
The present electrodynamics is certainly incomplete, but is no longer certainly incorrect.
—F. J. Dyson (1949b)
We now turn to the analysis of loop corrections in QED. As we might expect,
a theory with fermionic and gauge fields proves to be a tougher opponent than
one with only spinless particles, even though we restrict ourselves to one-loop
diagrams only.
At the outset we must make one important disclaimer. In QED many
loop diagrams diverge not only as the loop momentum goes to infinity (‘ultraviolet divergence’) but also as it goes to zero (‘infrared divergence’). This
phenomenon can only arise when there are massless particles in the theory –
for otherwise the propagator factors ≈(k
2
− M
2 )
−1 will always prevent any
infinity at low k. Of course, in a gauge theory we do have just such massless quanta. Our main purpose here is to demonstrate how the ultraviolet
divergences can be tamed and we must refer the reader to Weinberg (1995,
chapter 13), or to Peskin and Schroeder (1995, section 6.5), for instruction in
dealing with the infrared problem. The remedy lies, essentially, in a careful
consideration of the contribution, to physical cross sections, of amplitudes involving the real emission of very low frequency photons, along with infrared
divergent virtual photon processes. It is a ‘technical’ problem, having to do
with massless particles (of which there are not that many), whereas ultraviolet
divergences are generic.
11.1 Counter terms
We shall consider the simplest case of a single fermion of bare mass m 0 and
bare charge e 0 (e 0 > 0) interacting with the Maxwell field, for which the bare
(i.e. actual!) Lagrangian is
1
1
L ˆ = ψ
¯ ˆ
0 (i∂ / − m 0 )ψ ˆ 0 − e 0 ψ
¯ ˆ
0 γ
μ ψ ˆ 0 A ˆ 0μ − F ˆ 0μν F ˆ
0
μν −
(∂ · A ˆ 0 )
2
(11.1)
4
2ξ 0
327
Loops and Renormalization II: QED
The present electrodynamics is certainly incomplete, but is no longer certainly incorrect.
—F. J. Dyson (1949b)
We now turn to the analysis of loop corrections in QED. As we might expect,
a theory with fermionic and gauge fields proves to be a tougher opponent than
one with only spinless particles, even though we restrict ourselves to one-loop
diagrams only.
At the outset we must make one important disclaimer. In QED many
loop diagrams diverge not only as the loop momentum goes to infinity (‘ultraviolet divergence’) but also as it goes to zero (‘infrared divergence’). This
phenomenon can only arise when there are massless particles in the theory –
for otherwise the propagator factors ≈(k
2
− M
2 )
−1 will always prevent any
infinity at low k. Of course, in a gauge theory we do have just such massless quanta. Our main purpose here is to demonstrate how the ultraviolet
divergences can be tamed and we must refer the reader to Weinberg (1995,
chapter 13), or to Peskin and Schroeder (1995, section 6.5), for instruction in
dealing with the infrared problem. The remedy lies, essentially, in a careful
consideration of the contribution, to physical cross sections, of amplitudes involving the real emission of very low frequency photons, along with infrared
divergent virtual photon processes. It is a ‘technical’ problem, having to do
with massless particles (of which there are not that many), whereas ultraviolet
divergences are generic.
11.1 Counter terms
We shall consider the simplest case of a single fermion of bare mass m 0 and
bare charge e 0 (e 0 > 0) interacting with the Maxwell field, for which the bare
(i.e. actual!) Lagrangian is
1
1
L ˆ = ψ
¯ ˆ
0 (i∂ / − m 0 )ψ ˆ 0 − e 0 ψ
¯ ˆ
0 γ
μ ψ ˆ 0 A ˆ 0μ − F ˆ 0μν F ˆ
0
μν −
(∂ · A ˆ 0 )
2
(11.1)
4
2ξ 0
327