7Quantum chromodynamics (QCD)
III The Standard Model
7 Quantum chromodynamics (QCD)
In the early days of particle physics, we didn’t really know what we were doing.
So we just smashed particles into each other and see what happened. Our initial
particle accelerators weren’t very good, so we mostly observed some low energy
particles.
Of course, we found electrons, but the more interesting discoveries were in
the hadrons. We had protons and neutrons,
n
and
p
, as well as pions
π
+
, π
0
and
π
0
. We found that
n
and
p
behaved rather similarly, with similar interaction
properties and masses. On the other hand, the three pions behaved like each
other as well. Of course, they had different charges, so this is not a genuine
symmetry.
Nevertheless, we decided to assign numbers to these things, called isospin.
We say
n
and
p
have isospin
I
=
1
2
, while the pions have isospin
I
= 1. The
idea was that if a particle has spin
1
2
, then it has two independent spin states;
If it has spin 1, then it has three independent spin states. This, we view
n, p
as different “spin states” of the same object, and similarly
π
±
, π
0
are the three
“spin states” of the same object. Of course, isospin has nothing to do with actual
spin.
As in the case of spin, we have spin projections
I
3
. So for example,
p
has
I
3
=
+
1
2
and
n
has
I
3
=
−
1
2
. Similarly,
π
+
, π
0
and
π
0
have
I
3
= +1
,
0
, −
1 respectively.
Mathematically, we can think of these particles as living in representations of
su
(2). Each “group”
{n, p}
or
{π
+
, π
0
, π}
corresponded to a representation
of
su
(2), and the isospin labelled the representation they belonged to. The
eigenvectors corresponded to the individual particle states, and the isospin
projection I
3
referred to this eigenvalue.
That might have seemed like a stretch to invoke representation theory. We
then built better particle accelerators, and found more particles. These new
particles were quite strange, so we assigned a number called strangeness to
measure how strange they are. Four of these particles behaved quite like the
pions, and we called them Kaons. Physicists then got bored and plotted out
these particles according to isospin and strangeness:
I
3
S
π
+
K
+
¯
K
0
π
−
K
0
K
−
η
π
0
Remarkably, the diagonal lines join together particles of the same charge!
Something must be going on here. It turns out if we include these “strange”
particles into the picture, then instead of a representation of
su
(2), we now have
representations of su(3). Indeed, this just looks like a weight diagram of su(3).
Ultimately, we figured that things are made out of quarks. We now know
that there are 6 quarks, but that’s too many for us to handle. The last three
quarks are very heavy. They weren’t very good at forming hadrons, and their
large mass means the particles they form no longer “look alike”. So we only
focus on the first three.
At first, we only discovered things made up of up quarks and down quarks.
We can think of these quarks as living in the fundamental representation
V
1
of
su(2), with
u =
1
0
, d =
0
1
.
These are eigenvectors of the Cartan generator
H
, with weights +
1
2
and
−
1
2
(using
the “physicist’s” way of numbering). The idea is that physics is approximately
invariant under the action of
su
(2) that mixes
u
and
d
. Thus, different hadrons
made out of
u
and
d
might look alike. Nowadays, we know that the QCD part
of the Lagrangian is exactly invariant under the
SU
(2) action, while the other
parts are not.
The anti-quarks lived in the anti-fundamental representation (which is also
the fundamental representation). A meson is made of two quarks. So they live
in the tensor product
V
1
⊗ V
1
= V
0
⊕ V
2
.
The
V
2
was the pions we found previously. Similarly, the protons and neutrons
consist of three quarks, and live in
V
1
⊗ V
1
⊗ V
1
= (V
0
⊕ V
2
) ⊗V
1
= V
1
⊕ V
1
⊕ V
3
.
One of the V
1
’s contains the protons and neutrons.
The “strange” hadrons contain what is known as the strange quark,
s
. This
is significantly more massive than the
u
and
d
quarks, but are not too far off,
so we still get a reasonable approximate symmetry. This time, we have three
quarks, and they fall into an su(3) representation,
u =
1
0
0
, d =
0
1
0
, s =
0
0
1
.
This is the fundamental
3
representation, while the anti-quarks live in the
anti-fundamental
¯
3. These decompose as
3 ⊗
¯
3 = 1 ⊕
¯
8
3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10.
The quantum numbers correspond to the weights of the eigenvectors, and hence
when we plot the particles according to quantum numbers, they fall in such a
nice lattice.
There are a few mysteries left to solve. Experimentally, we found a baryon
∆
++
=
uuu
with spin
3
2
. The wavefunction appears to be symmetric, but this
would violate Fermi statistics. This caused theorists to go and scratch their
heads again and see what they can do to understand this. Also, we couldn’t
explain why we only had particles coming from
3 ⊗
¯
3
and
3 ⊗3 ⊗3
, and nothing
else.
The resolution is that we need an extra quantum number. This quantum
number is called colour . This resolved the problem of Fermi statistics, but also,
we postulated that any bound state must have no “net colour”. Effectively,
this meant we needed to have groups of three quarks or three anti-quarks, or
a quark-antiquark pair. This leads to the idea of confinement. This principle
predicted the Ω
−
baryon sss with spin
3
2
, and was subsequently observed.
Nowadays, we understand this is all due to a
SU
(3) gauge symmetry, which
is not the
SU
(3) we encountered just now. This is what we are going to study
in this chapter.