84
G. Altarelli and S. Forte
QCD is a renormalisable gauge theory based on the group SU (3) with colour triplet
quark matter fields [7] fixes the QCD lagrangian density to be:
L = −
1
4
8
A=1
F
Aμν F
A
μν +
n f
j =1
¯
q j (iD / − m j )q j
(4.1)
Here: q j are the quark fields (of n f different flavours) with mass m j ; D / = D μ γ μ ,
where γ μ are the Dirac matrices and D μ is the covariant derivative:
D μ = ∂ μ + ie s g μ ;
(4.2)
e s is the gauge coupling, later we will mostly use, in analogy with QED
α s =
e 2
s
4π
;
(4.3)
g μ =
A t A g A
μ where g A
μ , A = 1, 8, are the gluon fields and t A are the SU (3)
group generators in the triplet representation of quarks (i.e. t A are 3 × 3 matrices
acting on q); the generators obey the commutation relations [t A , t B ] = iC ABC t C
where C ABC are the complete antisymmetric structure constants of SU (3) (the
normalisation of C ABC and of e s is specified by T r[t A t B ] = δ AB /2);
F
A
μν = ∂ μ g
A
ν − ∂ ν g
A
μ − e s C ABC g
B
μ g
C
ν
(4.4)
For quantisation the classical Lagrangian in Eq. (4.1) must be enlarged to contain
gauge fixing and ghost terms, as described in Chap. 2. The Feynman rules of
QCD are listed in Fig. 4.1. The physical vertices in QCD include the gluon-quarkantiquark vertex, analogous to the QED photon-fermion-antifermion coupling, but
also the 3-gluon and 4-gluon vertices, of order e s and e 2
s respectively, which have
no analogue in an abelian theory like QED.
The QCD lagrangian in Eq. (4.1) has a simple structure but a very rich dynamical
content. It gives rise to a complex spectrum of hadrons, it implies the striking
properties of confinement and asymptotic freedom, is endowed with an approximate
chiral symmetry which is spontaneously broken, has a highly non trivial topological
vacuum structure (instantons, U(1) A symmetry breaking, strong CP violation
(which is a problematic item in QCD possibly connected with new physics, like
axions), . . . ), an intriguing phase transition diagram (colour deconfinement, quarkgluon plasma, chiral symmetry restoration, colour superconductivity, . . . ).
Confinement is the property that no isolated coloured charge can exist but only
colour singlet particles. For example, the potential between a quark and an antiquark
has been studied on the lattice. It has a Coulomb part at short distances and a linearly
rising term at long distances:
V q ¯
q ≈ C F [
α s (r)
r
+ . . . . + σ r]
(4.5)
G. Altarelli and S. Forte
QCD is a renormalisable gauge theory based on the group SU (3) with colour triplet
quark matter fields [7] fixes the QCD lagrangian density to be:
L = −
1
4
8
A=1
F
Aμν F
A
μν +
n f
j =1
¯
q j (iD / − m j )q j
(4.1)
Here: q j are the quark fields (of n f different flavours) with mass m j ; D / = D μ γ μ ,
where γ μ are the Dirac matrices and D μ is the covariant derivative:
D μ = ∂ μ + ie s g μ ;
(4.2)
e s is the gauge coupling, later we will mostly use, in analogy with QED
α s =
e 2
s
4π
;
(4.3)
g μ =
A t A g A
μ where g A
μ , A = 1, 8, are the gluon fields and t A are the SU (3)
group generators in the triplet representation of quarks (i.e. t A are 3 × 3 matrices
acting on q); the generators obey the commutation relations [t A , t B ] = iC ABC t C
where C ABC are the complete antisymmetric structure constants of SU (3) (the
normalisation of C ABC and of e s is specified by T r[t A t B ] = δ AB /2);
F
A
μν = ∂ μ g
A
ν − ∂ ν g
A
μ − e s C ABC g
B
μ g
C
ν
(4.4)
For quantisation the classical Lagrangian in Eq. (4.1) must be enlarged to contain
gauge fixing and ghost terms, as described in Chap. 2. The Feynman rules of
QCD are listed in Fig. 4.1. The physical vertices in QCD include the gluon-quarkantiquark vertex, analogous to the QED photon-fermion-antifermion coupling, but
also the 3-gluon and 4-gluon vertices, of order e s and e 2
s respectively, which have
no analogue in an abelian theory like QED.
The QCD lagrangian in Eq. (4.1) has a simple structure but a very rich dynamical
content. It gives rise to a complex spectrum of hadrons, it implies the striking
properties of confinement and asymptotic freedom, is endowed with an approximate
chiral symmetry which is spontaneously broken, has a highly non trivial topological
vacuum structure (instantons, U(1) A symmetry breaking, strong CP violation
(which is a problematic item in QCD possibly connected with new physics, like
axions), . . . ), an intriguing phase transition diagram (colour deconfinement, quarkgluon plasma, chiral symmetry restoration, colour superconductivity, . . . ).
Confinement is the property that no isolated coloured charge can exist but only
colour singlet particles. For example, the potential between a quark and an antiquark
has been studied on the lattice. It has a Coulomb part at short distances and a linearly
rising term at long distances:
V q ¯
q ≈ C F [
α s (r)
r
+ . . . . + σ r]
(4.5)