Physical Foundations of Cosmology

4.2 Quantum chromodynamics and quark–gluon plasma 145

Fig. 4.4.

diagrams with virtualgluonbubbles (Figure 4.4). Their contribution to f 1 ′( 1 )is
negativeand the number of possible diagrams of this kind is proportional to the
number of colors. For non-Abelian gauge theory with f massless flavors andn
colors,

f 1 ′( 1 )=

1

12 π

(2f− 11 n). (4.30)

Problem 4.8Why does the fermion contribution tof 1 ′( 1 )not depend on the number
of colors? Why is the gluon contribution proportional to the number of colors, but
not to the number of different gluons? Why do one-loop gluon diagrams, which are
due to the couplingA^4 ,not contribute to f 1 ′( 1 )?

The formula for the running coupling constant was derived in the limit when
fermion masses are negligible compared toq.It turns out that the contribution of
fermions with massmtof 1 ′( 1 )becomes significant only whenq^2 becomes larger
thanm^2 .Hence theβfunction coefficients change by discrete amounts as quark
masses are crossed. In quantum chromodynamics,f=6 for energies larger than
the top quark mass (∼170 GeV) andf=5 in the range 5 GeVq170 GeV. On
the other hand, since the number of colors isn= 3 ,f 1 ′( 1 )is always less than zero.
This has far-reaching consequences. As follows from (4.29), the running “strong
fine structure constant”,αs(q^2 )≡gs^2 / 4 π,decreases asq^2 increases. This is opposite
to the situation in electrodynamics. The strength of strong interactions decreases at
very high energies (small distances), so thatαsbecomes much smaller than unity
and we can use perturbation theory to calculate highly energetic hadron processes.
The approximation we used to deriveαs(q^2 ) becomes more and more reliable as
q^2 grows and in the limit thatq^2 →∞,interactions disappear. This property of