138 The very early universe
with detU= 1 .Therefore we can writeU(N)=U( 1 )×SU(N),and consider the
localSU(N)gauge groups separately. TheSU(N)group hasN^2 −1 independent
generators and hence we need at least that number of independent compensating
fieldsACμ.The Hermitian matrixAμcan then be written as
Aμ=ACμTC. (4.18)ForSU( 2 )andSU( 3 )groups it is convenient to use as the basis matricesσC/ 2
andλC/2 respectively, whereσCare three familiar Pauli matrices andλCare eight
Gell-Mann matrices,the explicit form of which will not be needed here.
4.2 Quantum chromodynamics and quark–gluon plasma
The strong force is responsible for binding neutrons and protons within nuclei.
The particles participating in strong interactions are called hadrons. They can be
either fermions or bosons. The fermions have half-integer spin and they are called
baryons, while the bosons have integer spin and are called mesons. The hadron
family is extremely large. To date, several hundred hadrons have been discovered.
It would be a nightmare if all these particles were elementary. Fortunately, they
are composite and built out of fermions of spin 1/2 called quarks. This is similar
to the way all chemical elements are made of protons and neutrons. In contrast to
the chemical elements, each of which has its own name, only the lightest and most
important hadrons have names reflecting their “individuality.” To classify hadrons
(or, in other words, put them in their own “Periodic Table”) we need five different
kinds (flavors) of quarks, which are accompanied by appropriate antiquarks. The
sixth quark, needed for cancellation of anomalies in the Standard Model, was also
discovered experimentally. The quarks have different masses and electric charges.
Three of them, namelyu(up),c(charm), andt(top) quarks, have a positive electric
charge, which is+ 2 /3 of the elementary charge. The other three quarks,d(down),
s(strange), andb(bottom), have a negative electric charge equal to− 1 / 3.
Strong interactions of quarks are described by anSU( 3 )gauge theory called
quantum chromodynamics.According to this theory, every quark of a given flavor
comes in three different colors: “red” (r), “blue” (b), and “green” (g). The colors are
simply names for the charges of theSU( 3 )gauge group, which acts on triplets of
spinor fields of the same flavor but different colors. The gauge-invariant quantum
chromodynamics Lagrangian is
L=∑
f,C(
iψ ̄fγμ∂μψf−mfψ ̄fψf−1
2
gs( ̄
ψfγμλCψf)
ACμ)
−
1
2
tr(
FμνFμν