8Lie groups in nature

III Symmetries, Fields and Particles

8.3 Internal symmetries and the eightfold way

Finally, we get to the notion of internal symmetries. Recall that for a complex

scalar field, our field was invariant under a global U(1) action given by phase

change. More generally, fields can carry some representation of a Lie group

G

.

In general, if this representation has dimension greater than 1, this means that

we have multiple different “particles” which are related under this symmetry

transformation. Then by symmetry, all these particles will have the same mass,

and we get a degeneracy in the mass spectrum.

When we have such degeneracies, we would want to distinguish the different

particles of the same mass. These can be done by looking at the weights of the

representation. It turns out the different weights correspond to the different

quantum numbers of the particles.

Often, we do not have an exact internal symmetry. Instead, we have an

approximate symmetry. This is the case when the dominant terms in the

Lagrangian are invariant under the symmetry, while some of the lesser terms are

not. In particular, the different particles usually have different (but very similar)

masses.

These internal symmetries are famously present in the study of hadrons, i.e.

things made out of quarks. The basic examples we all know are the nucleons,

namely protons and neutrons. We can list them as

Charge (Q) Mass (M)

p +1 938 MeV

n 0 940 MeV

Note that they have very similar masses. Later, we found, amongst many other

things, the pions:

Charge (Q) Mass (M)

π

+

+1 139 MeV

π

0

0 135 MeV

π

−

-1 139 MeV

Again, these have very similar masses. We might expect that there is some

approximate internal symmetry going on. This would imply that there is some

conserved quantity corresponding to the weights of the representation. Indeed,

we later found one, known as isospin:

Charge (Q) Isospin (J) Mass (M)

p +1 +

1

2

938 MeV

n 0 −

1

2

940 MeV

π

+

+1 +1 139 MeV

π

0

0 0 135 MeV

π

−

-1 -1 139 MeV

Isospin comes from an approximate

SU

(2)

I

symmetry, with a generator given by

H = 2J.

The nucleons then have the fundamental ρ

1

representation, and the pions have

the ρ

2

representation.

Eventually, we got smarter, and discovered an extra conserved quantum

number known as hypercharge. We can plot out the values of the isospin and

the hypercharge for our pions and some other particles we discovered, and we

found a pattern:

π

+

K

0

K

+

π

−

K

−

¯

K

0

η

π

0

At firsts, physicists were confused by the appearance of this pattern, and tried

very hard to figure out generalizations. Of course, now that we know about Lie

algebras, we know this is the weight diagram of a representation of su(3).

However, the word

SU

(3) hasn’t resolved all the mystery. In reality, we only

observed representations of dimension 1

,

8 and 10. We did not see anything

else. So physicists hypothesized that there are substructures known as quarks.

Each quark (

q

) carry a

3

representation (flavour), and antiquarks (

¯q

) carry a

¯

3

representation.

The mesons correspond to

q¯q

particles, and the representation decompose as

3 ⊗

¯

3 = 1 ⊕ 8.

Bosons correspond to qqq particles, and these decompose as

3 ⊗ 3 ⊗ 3 = 1 ⊕ 8 ⊕ 8 ⊕ 10.

Of course, we now have to explain why only

q¯q

and

qqq

appear in nature,

and why we don’t see quarks appearing isolated in nature. To do so, we have

to go very deep down in QCD. This theory says that quarks have a

SU

(3)

gauge symmetry (which is a different

SU

(3)) and the quark again carries the

fundamental representation

3

. More details can be found in the Lent Standard

Model course.