144 The very early universe
the expression on the right hand side of these equations,
β(α)=f 1 ′( 1 )α^2 +f 2 ′( 1 )α^3 +···, (4.28)is called theβfunction.
The results obtained are generic and valid in any renormalizable quantum field
theory. The only input we need from concrete theory is the numerical values of
the coefficients fn′( 1 ).For instance, to determine f 1 ′( 1 ),one has to calculate the
appropriate one-loop diagrams. Other coefficients require the calculations of higher-
order diagrams.
Let us assume thatα(q^2 )1 forq^2 interest of (this assumption should be
checkeda posteriori). In this case we may retain in theβfunction only the leading
one-loop termf 1 ′( 1 )α^2 ,and neglect all higher order contributions. Equation (4.27)
is then easily integrated, with the result
α(
q^2)
=
α(μ^2 )
1 −f 1 ′(1)α(μ^2 ) ln(q^2 /μ^2 ), (4.29)
whereα(μ^2 ) reappears as an integration constant. The expression obtained corre-
sponds to the partial resummation of series (4.22). As is clear from (4.26), this
resummation takes into account only the leading(lnx)ncontributions of alln-loop
diagrams. Knowledge of theβ-function to two loops, combined with the Gell-
Mann–Low equations, would allow us to resum next-to-leading logarithms.
Problem 4.7Find the behavior of the running coupling constant in the two-loop ap-
proximation. (The coefficientf 1 ′( 1 )does not depend on the renormalization scheme,
whilef 2 ′( 1 ), f 3 ′( 1 )etc. can be scheme-dependent).
In quantum electrodynamics the coefficientf 1 ′( 1 )is positive and equals 1/ 3 π.
In this case the couplingα(q^2 ) increases as the charges get closer together (q^2
increases). It is a straightforward consequence of vacuum polarization. In fact, the
vacuum can be thought of as a kind of “dielectric media” where negative charge
attracts positive charges and repel negative ones. As a result, the charge is sur-
rounded by a polarized “halo”, which screens it. Therefore, a negative charge,
observed from far away (smallq^2 ), will be reduced by the charge of the surround-
ing halo. At higherq^2 we approach the charge more closely, penetrating inside the
halo, and see a diminished screening of the charge.
In quantum chromodynamics we have one-loop diagrams of the kind shown in
Figure 4.3, where quarks and gluons should be substituted for the electrons and
photons respectively. They also give a positive contribution tof 1 ′( 1 ),proportional
to the number of possible diagrams of this kind and hence to the number of quark
flavors. As previously noted, gluons unlike photons, are charged. Hence, in quantum
chromodynamics, in addition to the diagrams in Figure 4.3 there are also one-loop