7Quantum chromodynamics (QCD)
III The Standard Model
7.1 QCD Lagrangian
The modern description of the strong interaction of quarks is quantum chromo-
dynamics, QCD. This is a gauge theory with a
SU
(3)
C
gauge group. The strong
force is mediated by gauge bosons known as gluons. This gauge symmetry is
exact, and the gluons are massless.
In QCD, each flavour of quark comes in three “copies” of different colour. It
is conventional to call these colours red, green and blue, even though they have
nothing to do with actual colours. For a flavour
f
, we can write these as
q
red
f
,
q
green
f
and q
blue
f
. We can put these into an triplet:
q
f
=
q
red
f
q
green
f
q
blue
f
.
Then QCD says this has an
SU
(3) gauge symmetry, where the triplet transforms
under the fundamental representation. Since this symmetry is exact, quarks
of all three colours behave exactly the same, and except when we are actually
doing QCD, it doesn’t matter that there are three flavours.
We do this for each quark individually, and then the QCD Lagrangian is
given by
L
QCD
= −
1
4
F
aµν
F
a
µν
+
X
f
¯q
f
(i
/
D − m
f
)q
f
,
where, as usual,
D
µ
= ∂
µ
+ igA
a
µ
T
a
F
a
µν
= ∂
µ
A
a
ν
− ∂
ν
A
a
µ
− gf
abc
A
b
µ
A
c
ν
.
Here T
a
for a = 1, ··· , 8 are generators of su(3), and, as usual, satisfies
[T
a
, T
b
] = if
abc
T
c
.
One possible choice of generators is
T
a
=
1
2
λ
a
,
where the
λ
a
are the Gell-Mann matrices. The fact that we have 8 independent
generators means we have 8 gluons.
One thing that is very different about QCD is that it has interactions between
gauge bosons. If we expand the Lagrangian, and think about the tree level
interactions that take place, we naturally have interactions that look like
but we also have three and four-gluon interactions
Mathematically, this is due to the non-abelian nature of
SU
(2), and physically,
we can think of this as saying the gluon themselves have colour charge.