20
G. Altarelli and S. Forte
and
F μν = ∂ μ A ν − ∂ ν A μ
(2.27)
Note that in QED one usually takes the e − to be the particle, so that Q = −1 and
the covariant derivative is D μ = ∂ μ −ieA μ when acting on the electron field. In this
case, the F μν tensor is linear in the gauge field V μ so that in the absence of matter
fields the theory is free. On the other hand, in the non abelian case the F A
μν tensor
contains both linear and quadratic terms in V A
μ , so that the theory is non-trivial even
in the absence of matter fields.
2.5 Application to QCD
According to the formalism of the previous section, the statement that QCD is a
renormalizable gauge theory based on the group SU (3) with colour triplet quark
matter fields 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
(2.28)
Here q j are the quark fields (of n f different flavours) with mass m j and D μ is the
covariant derivative:
D μ = ∂ μ + ie s g μ ;
(2.29)
e s is the gauge coupling and later we will mostly use, in analogy with QED
α s =
e 2
s
4π
.
(2.30)
Also, 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
(2.31)
Chapter 4 is devoted to a detailed description of the QCD as the theory of
strong interactions. The physical vertices in QCD include the gluon-quark-antiquark
vertex, analogous to the QED photon-fermion-antifermion coupling, but also the 3gluon and 4-gluon vertices, of order e s and e 2
s respectively, which have no analogue
G. Altarelli and S. Forte
and
F μν = ∂ μ A ν − ∂ ν A μ
(2.27)
Note that in QED one usually takes the e − to be the particle, so that Q = −1 and
the covariant derivative is D μ = ∂ μ −ieA μ when acting on the electron field. In this
case, the F μν tensor is linear in the gauge field V μ so that in the absence of matter
fields the theory is free. On the other hand, in the non abelian case the F A
μν tensor
contains both linear and quadratic terms in V A
μ , so that the theory is non-trivial even
in the absence of matter fields.
2.5 Application to QCD
According to the formalism of the previous section, the statement that QCD is a
renormalizable gauge theory based on the group SU (3) with colour triplet quark
matter fields 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
(2.28)
Here q j are the quark fields (of n f different flavours) with mass m j and D μ is the
covariant derivative:
D μ = ∂ μ + ie s g μ ;
(2.29)
e s is the gauge coupling and later we will mostly use, in analogy with QED
α s =
e 2
s
4π
.
(2.30)
Also, 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
(2.31)
Chapter 4 is devoted to a detailed description of the QCD as the theory of
strong interactions. The physical vertices in QCD include the gluon-quark-antiquark
vertex, analogous to the QED photon-fermion-antifermion coupling, but also the 3gluon and 4-gluon vertices, of order e s and e 2
s respectively, which have no analogue