Feynman Path Integral Formulation

3.7 Running of α(μ) in Gauge Theories 99
with
for small r, and
e
4π r →
e Q(r) (3.137)
4π r
Q(r) =1 + α
3π ln 1
m 2 + ... mr ≪ 1 (3.138)
r2 Q(r) =1 +
α
4 √ π (mr) 3/2 e−2mr + ... mr ≫ 1 , (3.139)
for large r. Here the normalization is such that the potential at infinity has Q(∞)=
1. 3 The reason we have belabored this example is to show that the screening vacuum
polarization contribution would have dramatic effects in QED if for some reason
the particle running through the fermion loop diagram had a much smaller (or even
close to zero) mass. There are two interesting aspects of the (one-loop) result of
Eqs. (3.138) and (3.139). The first one is that the exponentially small size of the
correction at large r is linked with the fact that the electron mass m e is not too small:
the range of the correction term is ξ = 2¯h/mc = 0.78 × 10 −10 cm, but would have
been much larger if the electron mass had been a lot smaller.
In QCD (and related Yang-Mills theories) radiative corrections are also known to
alter significantly the behavior of the static potential at short distances. The changes
in the potential are best expressed in terms of the running strong coupling constant
α S (μ), whose scale dependence is determined by the celebrated beta function of
SU(3) QCD with n f light fermion flavors
μ ∂α S
∂μ = 2β(α S)=− β 0
2π α2 S − β 1
4π 2 α3 S − β 2
64π 3 α4 S − ... (3.140)
with β 0 = 11− 2 3 n f , β 1 = 51− 19
3 n f , and β 2 = 2857− 5033
9 n f + 325
27 n2 f . The solution
of the renormalization group equation of Eq. (3.140) then gives for the running of
α S (μ)
[
4π
α S (μ) =
β 0 ln μ 2 /Λ 2 1 − 2β 1
ln[ln μ 2 /Λ 2 MS ]
]
β 2 MS 0
ln μ 2 /Λ 2 + ... , (3.141)
MS
(see Fig. 3.8). The non-perturbative scale Λ MS
appears as an integration constant
of the renormalization group equations, and is therefore - by construction - scale
independent. The physical value of Λ MS
cannot be fixed from perturbation theory
alone, and needs to be determined by experiment, giving Λ MS
≃ 220MeV.
In principle one can solve for Λ MS
in terms of the coupling at any scale, and in
particular at the cutoff scale Λ, obtaining
3 The running of the fine structure constant has recently been verified experimentally at LEP. The
scale dependence of the vacuum polarization effects gives a fine structure constant changing from
α(0) ∼ 1/137.036 at atomic distances to about α(m Z0 ) ∼ 1/128.978 at energies comparable to the
Z 0 boson mass, in good agreement with the theoretical renormalization group prediction.