336
11. Loops and Renormalization II: QED
FIGURE 11.4
The contribution of a massless particle to the photon self-energy.
have no contribution of the form A
2 /q
2 . If such a contribution were present,
(11.35) shows that it would result in a photon propagator having the form
−ig μν
(11.37)
q 2 − A 2
which is, of course, that of a massive particle. Thus, provided no such contribution is present, the photon mass will remain zero through all radiative
corrections. It is important to note, though, that gauge invariance is fully satisfied by the general form (11.36) relating Π ¯ μν to Π ¯ γ ; it does not prevent the
occurrence of such an ‘A
2 /q
2 ’ piece in Π ¯ γ . Remarkably, therefore, it seems
possible, after all, to have a massive photon while respecting gauge invariance! This loophole in the argument ‘gauge invariance implies m γ = 0’ was
first pointed out by Schwinger (1962).
Such a 1/q
2 contribution in Π ¯ γ must, of course, correspond to a massless single particle intermediate state, via a diagram of the form shown in
figure 11.4. Thus if the theory contains a massless particle, not the photon
(since 1–γ states are omitted from Π ¯ μν ) but coupling to it, the photon can
acquire mass. This is one way of understanding the ‘Higgs mechanism’ for
generating a mass for a gauge-field quantum while still respecting the gauge
symmetry (Englert and Brout 1964, Higgs 1964, Guralnik et al. 1964). The
massless particle involved is called a ‘Goldstone boson’. As we shall see in
volume 2, just such a photon mass is generated in a superconductor, and a
similar mechanism is invoked in the Standard Model to give masses to the
W
± and Z
0 gauge bosons, which mediate the weak interactions.
Π
[2]
11.5 The physics of ¯ γ (q
2 )
We now consider some immediate physical consequences of the formulae (11.32)
and (11.34).
11.5.1 Modified Coulomb’s law
In section 1.3.3 we saw how, in the static limit, a propagator of the form
2
2
2
−g (q + m )
−1 could be interpreted (via a Fourier transform) in terms of a
N
U
11. Loops and Renormalization II: QED
FIGURE 11.4
The contribution of a massless particle to the photon self-energy.
have no contribution of the form A
2 /q
2 . If such a contribution were present,
(11.35) shows that it would result in a photon propagator having the form
−ig μν
(11.37)
q 2 − A 2
which is, of course, that of a massive particle. Thus, provided no such contribution is present, the photon mass will remain zero through all radiative
corrections. It is important to note, though, that gauge invariance is fully satisfied by the general form (11.36) relating Π ¯ μν to Π ¯ γ ; it does not prevent the
occurrence of such an ‘A
2 /q
2 ’ piece in Π ¯ γ . Remarkably, therefore, it seems
possible, after all, to have a massive photon while respecting gauge invariance! This loophole in the argument ‘gauge invariance implies m γ = 0’ was
first pointed out by Schwinger (1962).
Such a 1/q
2 contribution in Π ¯ γ must, of course, correspond to a massless single particle intermediate state, via a diagram of the form shown in
figure 11.4. Thus if the theory contains a massless particle, not the photon
(since 1–γ states are omitted from Π ¯ μν ) but coupling to it, the photon can
acquire mass. This is one way of understanding the ‘Higgs mechanism’ for
generating a mass for a gauge-field quantum while still respecting the gauge
symmetry (Englert and Brout 1964, Higgs 1964, Guralnik et al. 1964). The
massless particle involved is called a ‘Goldstone boson’. As we shall see in
volume 2, just such a photon mass is generated in a superconductor, and a
similar mechanism is invoked in the Standard Model to give masses to the
W
± and Z
0 gauge bosons, which mediate the weak interactions.
Π
[2]
11.5 The physics of ¯ γ (q
2 )
We now consider some immediate physical consequences of the formulae (11.32)
and (11.34).
11.5.1 Modified Coulomb’s law
In section 1.3.3 we saw how, in the static limit, a propagator of the form
2
2
2
−g (q + m )
−1 could be interpreted (via a Fourier transform) in terms of a
N
U