Physical Foundations of Cosmology

142 The very early universe

pμ

qμ

1 p

μ

2

Fig. 4.3.

The diagrams containing closed loops are generically divergent. Fortunately, in
so-called renormalizable theories, the divergences can be “isolated and combined”
with bare coupling constants, bare masses, etc. What we measure in experiment is
not the value of the bare parameter, but only the finite outcome of “its combination
with infinities.” For instance, given a distance characterized by the momentum
transferq^2 =μ^2 (called the normalization point), we can measure the interaction
force and thus determine the renormalized coupling constantα(μ^2 ) which becomes
the actual parameter of the perturbative expansion.After removing and absorbing
the infinities, there remain finiteq^2 -dependent loop contributions to the interaction
force (the vacuum polarization effect). They too can be absorbed by redefining the
coupling constant which becomesq^2 -dependent, or in other words, begins to run. In
the limit of vanishing masses (or forq^2 m^2 ), the expansion of thedimensionless
running coupling “constant”α(q^2 ) in powers of the renormalized coupling constant
α(μ^2 ) can be expressed on dimensional grounds as

α(q^2 )=α(μ^2 )+α^2 (μ^2 )f 1

(

q^2

μ^2

)

+··· =

∑∞

n= 0

αn+^1 (μ^2 )fn

(

q^2

μ^2

)

, (4.22)

where f 0 =1 and the other functions fnare determined by appropriaten-loop
diagrams. Sinceα(q^2 )=α(μ^2 )atq^2 =μ^2 ,we have

fn( 1 )= 0

forn≥ 1.
If we consider a process withq-momentum transfer, we can use the running
constantα(q^2 ) instead ofα(μ^2 ) as a small expansion parameter in the remaining
finite diagrams.This corresponds to the resummation of finite contributions from
divergent diagrams. However, to take advantage of this resummation, we have to