331
11.3. The O(e
2 ) photon self-energy
would take us beyond our intended scope, as explained at the start of the
chapter. Suffice it to say that by the introduction of a ‘regulating’ photon
mass μ
2 , and consideration of relevant real photon processes along with virtual
ones, these infrared problems can be controlled (Weinberg 1995, Peskin and
Schroeder 1995).
11.3 The O(e
2 ) photon self-energy
[2]
The amplitude corresponding to figure 11.2(b) is iΠ μν (q) where
∫ d
4 k
i
i
iΠ
[2]
μν (q) = (−1)(−ie)
2 Tr
γ μ
γ ν (11.13)
(2π) 4 / q + k / − m k / − m
∫ d
4 k Tr[( / q + k / + m)γ μ (k / + m)γ ν ]
2
= −e
.
(11.14)
(2π) 4 [(q + k) 2 − m 2 ][k 2 − m 2 ]
Once again, this photon self-energy is analogous to the scalar particle selfenergy of chapter 10. There are two new features to be commented on in
(11.14). The first is the overall ‘−1’ factor, which occurs whenever there is a
closed fermion loop. The keen reader may like to pursue this via problem 11.2.
The second feature is the appearance of the trace symbol ‘Tr’: this is plausible
as the amplitude is basically a 1γ → 1γ one with no spinor indices, but again
the reader can follow that through in problem 11.3.
[2]
We now want to go some way into the calculation of Π μν because it will,
in the end, contain important physics – for example, corrections to Coulomb’s
law. The first step is to evaluate the numerator trace factor using the theorems
of section 8.2.3. We find (problem 11.4)
Tr[( / q + k / + m)γ μ (k / + m)γ ν ] = 4{(q μ + k μ )k ν + (q ν + k ν )k μ
− g μν ((q · k) + k
2
− m
2 )}. (11.15)
We then use the Feynman identity (10.40) to combine the denominators, yielding
∫ 1
1
1
=
dx
(11.16)
[(q + k) 2 − m 2 ][k 2 − m 2 ]
[k ′2 − Δ γ + i∈] 2
0
2
where k
′ = k + xq, Δ γ = −x(1 − x)q + m
2 (note that Δ γ is precisely the same
as Δ of (10.43) with m A = m B = m) and we have reinstated the implied ‘i∈’.
Making the shift to the variable k
′ in the numerator factor (11.15) produces
a revised numerator which is
4{2k μ
′ k ν
′
−g μν (k
′2
−Δ γ )−2x(1−x)(q μ q ν −g μν q
2 )+terms linear in k
′
} (11.17)
where the terms linear in k
′ will vanish by symmetry when integrated over k
′
in (11.14). Our result so far is therefore
11.3. The O(e
2 ) photon self-energy
would take us beyond our intended scope, as explained at the start of the
chapter. Suffice it to say that by the introduction of a ‘regulating’ photon
mass μ
2 , and consideration of relevant real photon processes along with virtual
ones, these infrared problems can be controlled (Weinberg 1995, Peskin and
Schroeder 1995).
11.3 The O(e
2 ) photon self-energy
[2]
The amplitude corresponding to figure 11.2(b) is iΠ μν (q) where
∫ d
4 k
i
i
iΠ
[2]
μν (q) = (−1)(−ie)
2 Tr
γ μ
γ ν (11.13)
(2π) 4 / q + k / − m k / − m
∫ d
4 k Tr[( / q + k / + m)γ μ (k / + m)γ ν ]
2
= −e
.
(11.14)
(2π) 4 [(q + k) 2 − m 2 ][k 2 − m 2 ]
Once again, this photon self-energy is analogous to the scalar particle selfenergy of chapter 10. There are two new features to be commented on in
(11.14). The first is the overall ‘−1’ factor, which occurs whenever there is a
closed fermion loop. The keen reader may like to pursue this via problem 11.2.
The second feature is the appearance of the trace symbol ‘Tr’: this is plausible
as the amplitude is basically a 1γ → 1γ one with no spinor indices, but again
the reader can follow that through in problem 11.3.
[2]
We now want to go some way into the calculation of Π μν because it will,
in the end, contain important physics – for example, corrections to Coulomb’s
law. The first step is to evaluate the numerator trace factor using the theorems
of section 8.2.3. We find (problem 11.4)
Tr[( / q + k / + m)γ μ (k / + m)γ ν ] = 4{(q μ + k μ )k ν + (q ν + k ν )k μ
− g μν ((q · k) + k
2
− m
2 )}. (11.15)
We then use the Feynman identity (10.40) to combine the denominators, yielding
∫ 1
1
1
=
dx
(11.16)
[(q + k) 2 − m 2 ][k 2 − m 2 ]
[k ′2 − Δ γ + i∈] 2
0
2
where k
′ = k + xq, Δ γ = −x(1 − x)q + m
2 (note that Δ γ is precisely the same
as Δ of (10.43) with m A = m B = m) and we have reinstated the implied ‘i∈’.
Making the shift to the variable k
′ in the numerator factor (11.15) produces
a revised numerator which is
4{2k μ
′ k ν
′
−g μν (k
′2
−Δ γ )−2x(1−x)(q μ q ν −g μν q
2 )+terms linear in k
′
} (11.17)
where the terms linear in k
′ will vanish by symmetry when integrated over k
′
in (11.14). Our result so far is therefore