one-loop

The full photon propagator can also in general be written separating its transversal an longitudinal parts

We have obtained, in rst order, that the vacuum polarization tensor is transversal,

This result is in fact valid to all orders of perturbation theory, a result that can be shown using the Ward-Takahashi identities. This means that the longitudinal part of the photon propagator is not renormalized,

All that we have done up to this point is formal because the function (k) diverges. The most satisfying way to solve this problem is the following. The initial lagrangian from which we started has been obtained from the classical theory and nothing tell us that it should be exactly the same in quantum theory. In fact, has we have just seen, the normalization of the wave functions is changed when we calculate one-loop corrections, and the same happens to the physical parameters of the theory, the charge and the mass. Therefore we can think that the correct lagrangian is obtained by adding corrections to the classical lagrangian, order by order in perturbation theory, so that we keep the de nitions of charge and mass as well as the normalization of the wave functions. The terms that we add to the lagrangian are called counterterms 3. The total lagrangian is then,

3This interpretation in terms of quantum corrections makes sense. In fact we can show that an expansion in powers of the coupling constant can be interpreted as an expansion in hL, where L is the number of the loops in the expansion term.

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Counterterms are de ned from the normalization conditions that we impose on the elds are other parameters of the theory. In QED we have a our disposal the normalization of the electron and photon elds and of the two physical parameters, the electric charge and the electron mass. The normalization conditions are, to a large extent, arbitrary. It is however convenient to keep the expressions as close as possible to the free eld case, that is, without radiative corrections. We de ne therefore the normalization of the photon eld as,

where GRT is the renormalized propagator (the transversal part) obtained from the la-

grangian Ltotal. The justi cation for this de nition comes from the following argument. Consider the Coulomb scattering to all orders of perturbation theory. We have then the

situation described in Fig. 5. Using the Ward-Takahashi identities one can show that the

=

Figure 5:

last three diagrams cancel in the limit q = p0 p ! 0. Then the normalization condition, Eq. (2.47), means that we have the situation described in Fig. 6, that is, the experimental value of the electric charge is determined in the limit q ! 0 of the Coulomb scattering.

The counterterm lagrangian has to have the same form as the classical lagrangian to respect the symmetries of the theory. For the photon eld it is traditional to write

corresponding to the following Feynman rule

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4Notice that the photon mass is not renormalized, that is the pole of the photon propagator still is at k2 = 0.

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and it is related to the pair production that can occur 5 for k2 > 4m2.

2.2Self-energy of the electron

The electron full propagator is given by the diagrammatic series of Fig. 7, which can be written as,

where we have identi ed

5For k2 > 4m2 there is the possibility of producing one pair e+e . Therefore on top of a virtual process (vacuum polarization) there is a real process (pair production).

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Figure 7:

k

p p+k p

Figure 8:

We conclude that it is enough to calculate (p) to all orders of perturbation theory to obtain the full electron propagator. The name self-energy given to (p) comes from the fact that, as can be seen in Eq. (2.67), it comes as an additional (momentum dependent) contribution to the mass.

In lowest order there is only the diagram of Fig. 8 contributing (p) and therefore we get,

where we have chosen the Feynman gauge ( = 1) for the photon propagator and we have introduced a small mass for the photon , in order to control the infrared divergences

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(IR) that will appear when k2 ! 0 (see below). Using dimensional regularization and the results of the Dirac algebra in dimension d,

The contribution from the loop in Fig. 8 to the electron self-energy (p) can then be

6The linear term in k vanishes.

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To continue with the renormalization program we have to introduce the counterterm lagrangian and de ne the normalization conditions. We have

L = i (Z2 1) @ (Z2 1) m + Z2 m + (Z1 1)e A (2.77)

and therefore we get for the self-energy

Contrary to the case of the photon we see that we have two constants to determine. In the on-shell renormalization scheme that is normally used in QED the two constants are obtained by requiring that the pole of the propagator corresponds to the physical mass (hence the name of on-shell renormalization), and that the residue of the pole of the renormalized electron propagator has the same value as the free led propagator. This implies,

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where in the last step in Eq. (2.80) we have taken the limit ! 0 because the integral does not diverge in that limit7. In a similar way we get for Z2,

where we have taken the ! 0 limit in all cases that was possible. It is clear that thenal result in Eq. (2.83) diverges in that limit, therefore implying that Z2 is IR divergent. This is not a problem for the theory because Z2 is not a physical parameter. We will see in section 5.2 that the IR diverges cancel for real processes. If we had taken a general gauge ( 6= 1) we will nd out that m would not be changed but that Z2 would show a gauge dependence. Again, in physical processes this should cancel in the end.

2.3The Vertex

The diagram contributing to the QED vertex at one-loop is the one shown in Fig. 9. In the Feynman gauge ( = 1) this gives a contribution,

7m is not IR divergent.

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p’

μ k

p

Figure 9:

where is related to the full vertex through the relation

i = ie ( + loop + Z1)

The integral that de nes loop(p0; p) is divergent. As before we expect to solve this problem by regularizing the integral, introducing counterterms and normalization conditions. The counterterm has the same form as the vertex and is already included in Eq. (2.85). The normalization constant is determined by requiring that in the limit q = p0 p ! 0 the vertex reproduces the tree level vertex because this is what is consistent with the de nition of the electric charge in the q ! 0 limit of the Coulomb scattering. Also this should be de ned for on-shell electrons. We have therefore that the normalization condition gives,

If we are interested only in calculating Z1 and in showing that the divergences can be removed with the normalization condition then the problem is simpler. It can be done in two ways.

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In this second method we do not rely in the Ward identity but just calculate the integrals for the vertex in Eq. (2.84). For the moment we do not put p0 = p but we will assume that the vertex form factors are to be evaluated for on-shell spinors. Then we have