11.6. The O(e
2 ) vertex correction, and Z 1 = Z 2
345
is once again
1
(11.64)
[2]
q 2 (1 − Π ¯ γ (q 2 ))
[2]
2
2
but now q > 4m μ , which is greater than 4m
2 so that Π ¯ γ (q
2 ) in (11.64) has
an imaginary part. There is a good physical reason for this, which has to do
with unitarity. This was introduced in section 6.2.2 in terms of the relation
SS
† = I for the S-matrix. The invariant amplitude M is related to S by
S fi = 1 + i(2π)
4 δ
4 (p i − p f )M fi (cf (6.102)). Inserting this into SS
† = I leads
to an equation of the form (for help see Peskin and Schroeder (1995, section
7.3))
(
)
∑
∑
2ImM fi =
M kf
∗
M ki (2π)
4 δ p i −
q k
(11.65)
k
∑
where ‘
’ stands for the phase space integral involving momenta q 1 , q 2 , . . .
k
over the states allowed by energy–momentum conservation. This implies that
as the energy crosses each threshold for production of a newly allowed state,
there will be a new contribution to the imaginary part of M. This is exactly
what we are seeing here, at the e
+ e
− threshold.
It is interesting, incidentally, that (11.65) can be used to derive the relativistic generalization of the optical theorem given in appendix H (note that
the right-hand side of (11.65) is clearly related to the total cross section for
i → k, if i = f).
[2]
As regards the real part of Π ¯ γ (q
2 ) in the time-like region, it will be given
2
by (11.57) with Q
2 replaced by q , or s, for large values of q
2 . Again, measurements have verified the predicted variation of α(q
2 ) in the time-like region
(Miyabayashi et al. 1995, Ackerstaff et al. 1998, Abbiendi et al. 1999, 2000).
There is one more ‘elementary’ loop that we must analyse – the vertex
correction shown in figure 11.8, which we now discuss. We will see how the
important relation Z 1 = Z 2 emerges, and introduce some of the physics contained in the renormalized vertex.
11.6 The O(e
2 ) vertex correction, and Z 1 = Z 2
The amplitude corresponding to figure 11.8 is
∫
i
′ )Γ
[2]
−ig λν
−ieu ¯(p μ (p, p
′ )u(p) = u ¯(p
′ ) (−ieγ
ν ) k 2
′
/
p − k / − m
i
d
4 k
× (−ieγ μ )
(−ieγ
λ )
u(p) (11.66)
p − k / − m
(2π) 4
/
2 ) vertex correction, and Z 1 = Z 2
345
is once again
1
(11.64)
[2]
q 2 (1 − Π ¯ γ (q 2 ))
[2]
2
2
but now q > 4m μ , which is greater than 4m
2 so that Π ¯ γ (q
2 ) in (11.64) has
an imaginary part. There is a good physical reason for this, which has to do
with unitarity. This was introduced in section 6.2.2 in terms of the relation
SS
† = I for the S-matrix. The invariant amplitude M is related to S by
S fi = 1 + i(2π)
4 δ
4 (p i − p f )M fi (cf (6.102)). Inserting this into SS
† = I leads
to an equation of the form (for help see Peskin and Schroeder (1995, section
7.3))
(
)
∑
∑
2ImM fi =
M kf
∗
M ki (2π)
4 δ p i −
q k
(11.65)
k
∑
where ‘
’ stands for the phase space integral involving momenta q 1 , q 2 , . . .
k
over the states allowed by energy–momentum conservation. This implies that
as the energy crosses each threshold for production of a newly allowed state,
there will be a new contribution to the imaginary part of M. This is exactly
what we are seeing here, at the e
+ e
− threshold.
It is interesting, incidentally, that (11.65) can be used to derive the relativistic generalization of the optical theorem given in appendix H (note that
the right-hand side of (11.65) is clearly related to the total cross section for
i → k, if i = f).
[2]
As regards the real part of Π ¯ γ (q
2 ) in the time-like region, it will be given
2
by (11.57) with Q
2 replaced by q , or s, for large values of q
2 . Again, measurements have verified the predicted variation of α(q
2 ) in the time-like region
(Miyabayashi et al. 1995, Ackerstaff et al. 1998, Abbiendi et al. 1999, 2000).
There is one more ‘elementary’ loop that we must analyse – the vertex
correction shown in figure 11.8, which we now discuss. We will see how the
important relation Z 1 = Z 2 emerges, and introduce some of the physics contained in the renormalized vertex.
11.6 The O(e
2 ) vertex correction, and Z 1 = Z 2
The amplitude corresponding to figure 11.8 is
∫
i
′ )Γ
[2]
−ig λν
−ieu ¯(p μ (p, p
′ )u(p) = u ¯(p
′ ) (−ieγ
ν ) k 2
′
/
p − k / − m
i
d
4 k
× (−ieγ μ )
(−ieγ
λ )
u(p) (11.66)
p − k / − m
(2π) 4
/