330
11. Loops and Renormalization II: QED
regularization to make (11.9) ultraviolet finite. We shall not be specific (as
yet) about what choice we are making, since whatever it may be the outcome
[2]
will be qualitatively similar to the Π case.
C
There is, however, one interesting new feature in this (fermion) case. As
previously indicated, power-counting in the integral of (11.9) might lead us to
expect that – if we adopt a simple cut-off – the leading ultraviolet divergence
of Σ
[2] would be proportional to Λ rather than ln Λ. This is because we
have that one extra power of k in the numerator and Σ
[2] has dimensions
of mass. However, this is not so. The leading p-independent divergence is,
in fact, proportional to m ln(Λ/m). The reason for this is important and
it has interesting generalizations. Suppose that m in (11.4) were set equal
to zero. Then, as we saw in problem 9.4, the two helicity components ψ ˆ L
and ψ ˆ R of the electron field will not be coupled by the QED interaction.
¯ ˆ
¯ ˆ
ˆ
ˆ
It follows that no terms of the form ψ L ψ R or ψ R ψ L can be generated, and
hence no perturbatively induced mass term, if m = 0. The perturbative mass
shift must be proportional to m and therefore, on dimensional grounds, only
logarithmically divergent.
There is also a p-dependent divergence of the self-energy, of which warning
was given in section 10.3.2. As in the scalar case, this will be associated with
the field strength renormalization factor Z 2 . It is proportional to / p ln(Λ/m)
(Z 2 is the coefficient of ∂ / in (11.8), which leads to p / in momentum space). The
upshot is that the fermion propagator, including the one-loop renormalized
self-energy, is given by
i
(11.10)
/ p − m − Σ ¯ [2] (p)
where (cf (10.74))
dΣ
[2]
¯
Σ
[2] (p) = Σ
[2] (p) − Σ
[2] ( / p = m) − ( / p − m)
.
(11.11)
p
|
|
|
|
p=m
d / /
Σ
[2]
Whatever form of regularization is used, the twice-subtracted ¯ will be
finite and independent of the regulator when it is removed. In terms of the
‘compensating’ quantities Z 2 and m 0 − m, we find (problem 11.1, cf (10.70))
dΣ
[2]
−Z
−1 Σ
[2] ( /
Z 2 = 1 +
m 0 − m =
p = m).
(11.12)
2
p
|
|
|
|
p=m
d / /
Note that, as in the case of Π ¯ [2] , the definition (11.11) of Σ ¯ [2] implies that
C
propagator corrections vanish for external (on-shell) fermions. The quantities
Z 2 and m 0 determined by (11.12) now carry a superscript ‘[2]’ to indicate that
they are correct at O(e
2 ).
We must now remind the reader that, although we have indeed eliminated
the ultraviolet divergences in Σ ¯ [2] by the subtractions of (11.11), there remains
an untreated infrared divergence in dΣ
[2] /d / p. To show how this is dealt with
11. Loops and Renormalization II: QED
regularization to make (11.9) ultraviolet finite. We shall not be specific (as
yet) about what choice we are making, since whatever it may be the outcome
[2]
will be qualitatively similar to the Π case.
C
There is, however, one interesting new feature in this (fermion) case. As
previously indicated, power-counting in the integral of (11.9) might lead us to
expect that – if we adopt a simple cut-off – the leading ultraviolet divergence
of Σ
[2] would be proportional to Λ rather than ln Λ. This is because we
have that one extra power of k in the numerator and Σ
[2] has dimensions
of mass. However, this is not so. The leading p-independent divergence is,
in fact, proportional to m ln(Λ/m). The reason for this is important and
it has interesting generalizations. Suppose that m in (11.4) were set equal
to zero. Then, as we saw in problem 9.4, the two helicity components ψ ˆ L
and ψ ˆ R of the electron field will not be coupled by the QED interaction.
¯ ˆ
¯ ˆ
ˆ
ˆ
It follows that no terms of the form ψ L ψ R or ψ R ψ L can be generated, and
hence no perturbatively induced mass term, if m = 0. The perturbative mass
shift must be proportional to m and therefore, on dimensional grounds, only
logarithmically divergent.
There is also a p-dependent divergence of the self-energy, of which warning
was given in section 10.3.2. As in the scalar case, this will be associated with
the field strength renormalization factor Z 2 . It is proportional to / p ln(Λ/m)
(Z 2 is the coefficient of ∂ / in (11.8), which leads to p / in momentum space). The
upshot is that the fermion propagator, including the one-loop renormalized
self-energy, is given by
i
(11.10)
/ p − m − Σ ¯ [2] (p)
where (cf (10.74))
dΣ
[2]
¯
Σ
[2] (p) = Σ
[2] (p) − Σ
[2] ( / p = m) − ( / p − m)
.
(11.11)
p
|
|
|
|
p=m
d / /
Σ
[2]
Whatever form of regularization is used, the twice-subtracted ¯ will be
finite and independent of the regulator when it is removed. In terms of the
‘compensating’ quantities Z 2 and m 0 − m, we find (problem 11.1, cf (10.70))
dΣ
[2]
−Z
−1 Σ
[2] ( /
Z 2 = 1 +
m 0 − m =
p = m).
(11.12)
2
p
|
|
|
|
p=m
d / /
Note that, as in the case of Π ¯ [2] , the definition (11.11) of Σ ¯ [2] implies that
C
propagator corrections vanish for external (on-shell) fermions. The quantities
Z 2 and m 0 determined by (11.12) now carry a superscript ‘[2]’ to indicate that
they are correct at O(e
2 ).
We must now remind the reader that, although we have indeed eliminated
the ultraviolet divergences in Σ ¯ [2] by the subtractions of (11.11), there remains
an untreated infrared divergence in dΣ
[2] /d / p. To show how this is dealt with