323
10.4. Bare and renormalized perturbation theory
2
2
2
2
except that g and m are replaced by g and m . From (10.55) it then
i
ph
ph,i
follows that, as Λ → ∞,
2
2
2
∫ 1
[2]
2
ph
ph
ph
Π
(q , Λ
2 ) = −
g
ln Λ −
g
(ln 2 − 1) +
g
dx ln Δ(x, q
2 ), (10.73)
ph,C
8π 2
8π 2
16π 2
0
and hence
(
)
∫
2
1
g
2 )
[2]
2
[2]
2
ph
Δ(x, q
Π
(q , Λ
2 ) − Π
(m ph,C , Λ
2 ) =
dx ln
(10.74)
ph,C
ph,C
2
16π 2
Δ(x, m
)
0
ph,C
[2]
2
which is finite as Λ → ∞. It is also clear from (10.73) that dΠ
/dq is
ph,C
finite as Λ → ∞. Thus the quantity Π
[2]
2 ) is finite as Λ → ∞, and
ph,C (q
is understood to be evaluated in that limit; the subtraction in (10.74) has
removed the infinity. The additional subtraction in (10.72) would in fact
have removed a logarithmic divergence in Z C , had there been one. Note that
the form of (10.72) guarantees that the leading behaviour of Π
[2]
2 ) near
ph,C (q
2
2
2
q = m
is (q
2
− m
)
2 , so that the behaviour of (10.71) near the massph,C
ph,C
2
shell point is indeed i/(q
2
− m ph,C ) as desired.
A succinct way of summarizing our final renormalized result (10.71), with
the definition (10.72), is to say that the C propagator may be defined by
[2]
2
(10.71) where the O(g ) renormalized self-energy Π
satisfies the renorph
ph,C
malization conditions
[2]
2
2
d [2]
Π
= m ph,C ) = 0
Π
2 )
= 0.
(10.75)
ph,C (q
ph,C (q
|
|
|
|
dq 2
2
2
ph,C
q =m
Relations analogous to (10.75) clearly hold for the A and B self-energies also.
In this definition, the explicit introduction and cancellation of large-Λ terms
has disappeared from sight, and all that remains is the importation of one
2
constant from experiment, m ph,C , and a (hidden) rescaling of the fields. It is
useful to bear this viewpoint in mind when considering more general theories,
including ones that are ‘non-renormalizable’ (see section 11.8 of the following
chapter).
There is a lot of good physics in the expression (10.71), which we shall elucidate in the realistic case of QED in the next chapter. For the moment, we
just whet the reader’s appetite by pointing out that (10.71) must amount to
the prediction of a finite, calculable correction to the Yukawa 1 − C exchange
potential, which after all is given by the Fourier transform of the (static form
of) the propagator, as we learned long ago. In the case of QED, this will
amount to a calculable correction to Coulomb’s law, due to radiative corrections, as we shall discuss in section 11.5.1.
There is an important technical implication we may draw from (10.75).
Consider the Feynman diagram of figure 10.12 in which a propagator correc4
tion has been inserted in an external line. This diagram is of order g , and
ph
10.4. Bare and renormalized perturbation theory
2
2
2
2
except that g and m are replaced by g and m . From (10.55) it then
i
ph
ph,i
follows that, as Λ → ∞,
2
2
2
∫ 1
[2]
2
ph
ph
ph
Π
(q , Λ
2 ) = −
g
ln Λ −
g
(ln 2 − 1) +
g
dx ln Δ(x, q
2 ), (10.73)
ph,C
8π 2
8π 2
16π 2
0
and hence
(
)
∫
2
1
g
2 )
[2]
2
[2]
2
ph
Δ(x, q
Π
(q , Λ
2 ) − Π
(m ph,C , Λ
2 ) =
dx ln
(10.74)
ph,C
ph,C
2
16π 2
Δ(x, m
)
0
ph,C
[2]
2
which is finite as Λ → ∞. It is also clear from (10.73) that dΠ
/dq is
ph,C
finite as Λ → ∞. Thus the quantity Π
[2]
2 ) is finite as Λ → ∞, and
ph,C (q
is understood to be evaluated in that limit; the subtraction in (10.74) has
removed the infinity. The additional subtraction in (10.72) would in fact
have removed a logarithmic divergence in Z C , had there been one. Note that
the form of (10.72) guarantees that the leading behaviour of Π
[2]
2 ) near
ph,C (q
2
2
2
q = m
is (q
2
− m
)
2 , so that the behaviour of (10.71) near the massph,C
ph,C
2
shell point is indeed i/(q
2
− m ph,C ) as desired.
A succinct way of summarizing our final renormalized result (10.71), with
the definition (10.72), is to say that the C propagator may be defined by
[2]
2
(10.71) where the O(g ) renormalized self-energy Π
satisfies the renorph
ph,C
malization conditions
[2]
2
2
d [2]
Π
= m ph,C ) = 0
Π
2 )
= 0.
(10.75)
ph,C (q
ph,C (q
|
|
|
|
dq 2
2
2
ph,C
q =m
Relations analogous to (10.75) clearly hold for the A and B self-energies also.
In this definition, the explicit introduction and cancellation of large-Λ terms
has disappeared from sight, and all that remains is the importation of one
2
constant from experiment, m ph,C , and a (hidden) rescaling of the fields. It is
useful to bear this viewpoint in mind when considering more general theories,
including ones that are ‘non-renormalizable’ (see section 11.8 of the following
chapter).
There is a lot of good physics in the expression (10.71), which we shall elucidate in the realistic case of QED in the next chapter. For the moment, we
just whet the reader’s appetite by pointing out that (10.71) must amount to
the prediction of a finite, calculable correction to the Yukawa 1 − C exchange
potential, which after all is given by the Fourier transform of the (static form
of) the propagator, as we learned long ago. In the case of QED, this will
amount to a calculable correction to Coulomb’s law, due to radiative corrections, as we shall discuss in section 11.5.1.
There is an important technical implication we may draw from (10.75).
Consider the Feynman diagram of figure 10.12 in which a propagator correc4
tion has been inserted in an external line. This diagram is of order g , and
ph