¯ [2] 2 )
11.5. The physics of Π γ (q
339
Referring now to the discussion of form factors for charged spin1 particles in
2
section 8.8, we can share the correction (11.46) equally between the e
− and
the μ
− vertices and write
Π
[2]
Π ¯ [2]
eu ¯ k ' γ μ u k → eu ¯ k ' γ μ u k (1 + ¯ (q
2 ))
1/2
≈ eu ¯ k ' γ μ u k (1 +
1
(q
2 )) (11.47)
γ
2 γ
for the electron, and similarly for the muon. From (8.208) this means that our
‘radiative correction’ has generated some effective extension of the charge, as
1 [2]
¯
given by a charge form factor F 1 (q
2 ) = 1 + Π γ (q
2 ). Note that the condition
2
[2]
F 1 (0) = 1 is satisfied since Π ¯ γ (0) = 0.
In the static case, or for scattering of equal mass particles in the CM
2
2
system, we have q = −q and we may consider the Fourier transform of
the function F 1 (−q
2 ), to obtain the charge distribution. The integral is discussed in Weinberg (1995, section 10.2) and in Peskin and Schroeder (1995,
section 7.5). The latter authors show that the approximate radial distribution of charge is ∼e
−2mr /(mr)
3/2 , indicating that it has a range ∼
1 . This is
2m
precisely the mass of the fermion–antifermion intermediate state in the loop
[2]
which yields Π ¯ γ , so this result represents a plausible qualitative extension of
Yukawa’s relationship (1.20) to the case of two-particle exchange. In any case,
[2]
the range represented by Π ¯ γ is of order of the fermion Compton wavelength
1/m, which is an important insight; this is why we need to do better than the
point-like approximation (11.42) in the case of muonic atoms.
11.5.3 The running coupling constant
There is yet another way of interpreting (11.38). Referring to (11.46), we may
regard
Π ¯ [2]
e
2 (q
2 ) = e
2 [1 +
(q
2 )]
(11.48)
γ
2
as a ‘q
2 -dependent effective charge’. In fact, it is usually written as a ‘q
dependent fine structure constant’
Π ¯ [2]
α(q
2 ) = α[1 +
(q
2 )].
(11.49)
γ
The concept of a q
2 -dependent charge may be startling but the related one of
a spatially dependent charge is, in fact, familiar from the theory of dielectrics.
Consider a test charge q in a polarizable dielectric medium, such as water.
If we introduce another test charge −q into the medium, the electric field
between the two test charges will line up the water molecules (which have a
permanent electric dipole moment) as shown in figure 11.5. There will be an
induced dipole moment P per unit volume, and the effect of P on the resultant
field is (from elementary electrostatics) the same as that produced by a volume
charge equal to − div P . If, as is usual, P is taken to be proportional to E,
so that P = χ∈ 0 E, Gauss’ law will be modified from
div E = ρ free /∈ 0
(11.50)
11.5. The physics of Π γ (q
339
Referring now to the discussion of form factors for charged spin1 particles in
2
section 8.8, we can share the correction (11.46) equally between the e
− and
the μ
− vertices and write
Π
[2]
Π ¯ [2]
eu ¯ k ' γ μ u k → eu ¯ k ' γ μ u k (1 + ¯ (q
2 ))
1/2
≈ eu ¯ k ' γ μ u k (1 +
1
(q
2 )) (11.47)
γ
2 γ
for the electron, and similarly for the muon. From (8.208) this means that our
‘radiative correction’ has generated some effective extension of the charge, as
1 [2]
¯
given by a charge form factor F 1 (q
2 ) = 1 + Π γ (q
2 ). Note that the condition
2
[2]
F 1 (0) = 1 is satisfied since Π ¯ γ (0) = 0.
In the static case, or for scattering of equal mass particles in the CM
2
2
system, we have q = −q and we may consider the Fourier transform of
the function F 1 (−q
2 ), to obtain the charge distribution. The integral is discussed in Weinberg (1995, section 10.2) and in Peskin and Schroeder (1995,
section 7.5). The latter authors show that the approximate radial distribution of charge is ∼e
−2mr /(mr)
3/2 , indicating that it has a range ∼
1 . This is
2m
precisely the mass of the fermion–antifermion intermediate state in the loop
[2]
which yields Π ¯ γ , so this result represents a plausible qualitative extension of
Yukawa’s relationship (1.20) to the case of two-particle exchange. In any case,
[2]
the range represented by Π ¯ γ is of order of the fermion Compton wavelength
1/m, which is an important insight; this is why we need to do better than the
point-like approximation (11.42) in the case of muonic atoms.
11.5.3 The running coupling constant
There is yet another way of interpreting (11.38). Referring to (11.46), we may
regard
Π ¯ [2]
e
2 (q
2 ) = e
2 [1 +
(q
2 )]
(11.48)
γ
2
as a ‘q
2 -dependent effective charge’. In fact, it is usually written as a ‘q
dependent fine structure constant’
Π ¯ [2]
α(q
2 ) = α[1 +
(q
2 )].
(11.49)
γ
The concept of a q
2 -dependent charge may be startling but the related one of
a spatially dependent charge is, in fact, familiar from the theory of dielectrics.
Consider a test charge q in a polarizable dielectric medium, such as water.
If we introduce another test charge −q into the medium, the electric field
between the two test charges will line up the water molecules (which have a
permanent electric dipole moment) as shown in figure 11.5. There will be an
induced dipole moment P per unit volume, and the effect of P on the resultant
field is (from elementary electrostatics) the same as that produced by a volume
charge equal to − div P . If, as is usual, P is taken to be proportional to E,
so that P = χ∈ 0 E, Gauss’ law will be modified from
div E = ρ free /∈ 0
(11.50)