Physical Foundations of Cosmology

4.2 Quantum chromodynamics and quark–gluon plasma 141

Problem 4.5Verify that color–anticolor and three-quark combinations are really
colorless, that is, they do not change under gauge transformation. (HintThe differ-
ent anticolors can be thought of as three element rows,r ̄=( 1 , 0 , 0 ),b ̄=( 0 , 1 , 0 ),
g ̄=( 0 , 0 , 1 )and different colors as corresponding columns.)

Confinement should, in principle, be derivable from the fundamental Lagrangian
(4.19) but until now this has not been achieved. Nevertheless, there are strong
experimental and theoretical indications that this hypothesis is valid. In particular,
the increase of the strong interaction strength at low energy strongly supports the
idea of confinement. The energy dependence of the strong coupling constant has
another important feature: the interaction strength vanishes in the limit of very high
energies (or, correspondingly, at very small distances). This is calledasymptotic
freedom. As a consequence, this allows us to use perturbation theory to calculate
strong interaction processes with highly energetic hadrons. We explain below why
the coupling constant should be scale-dependent and then calculate how it “runs.” To
introduce the concept of a running coupling, we begin with familiar electromagnetic
interactions, and then derive the results for a general renormalizable field theory
and apply them to quantum chromodynamics.

4.2.1 Running coupling constant and asymptotic freedom

Let us consider two electrically charged particles. According to quantum electro-
dynamics, their interaction via photon exchange can be represented by a set of
diagrams, some of which are shown in Figure 4.3. The contribution of a particular
diagram to the total interaction strength is proportional to the number of primitive
vertices in the diagram. Each vertex brings a factor ofe.Becausee 1 ,the largest
contribution to the interaction (∝e^2 ) comes from the simplest (tree-level) diagram
with only two vertices. The next order (one-loop) diagrams have four vertices and
their contribution is proportional toe^4 .Hence, the diagrams in Figure 4.3 are sim-
ply a graphical representation of the perturbative expansion in powers of the fine
structure constant

α≡

e^2

4 π



1

137

.

The interaction strength depends not only on the charge but also on the distance
between the particles, characterized by the 4-momentum transfer

qμ=pμ 2 −p 1 μ. (4.21)

Note that for virtual photonsq^2 ≡|qμqμ| =0, that is, they do not lie on their mass
shell.