1Overview
III The Standard Model
1 Overview
We begin with a quick overview of the things that exist in the standard model.
This description will mostly be words, and the actual theory will have to come
quite some time later.
– Forces are mediated by spin 1 gauge bosons. These include
◦
The electromagnetic field (EM ), which is mediated by the photon.
This is described by quantum electrodynamics (QED);
◦
The weak interaction, which is mediated by the
W
±
and
Z
bosons;
and
◦
The strong interaction, which is mediated by gluons
g
. This is
described by quantum chromodynamics (QCD).
While the electromagnetic field and weak interaction seem very different,
we will see that at high energies, they merge together, and can be described
by a single gauge group.
–
Matter is described by spin
1
2
fermions. These are described by Dirac
spinors. Roughly, we can classify them into 4 “types”, and each type comes
in 3 generations, which will be denoted G1, G2, G3 in the following table,
which lists which forces they interact with, alongside with their charge.
Type G1 G2 G3 Charge EM Weak Strong
Charged leptons e µ τ −1 3 3 7
Neutrinos ν
e
ν
µ
ν
τ
0 7 3 7
Positive quarks u c t +
2
3
3 3 3
Negative quarks d s b −
1
3
3 3 3
The first two types are known as leptons, while the latter two are known
as quarks. We do not know why there are three generations. Note that a
particle interacts with the electromagnetic field iff it has non-zero charge,
since the charge by definition measures the strength of the interaction with
the electromagnetic field.
–
There is the Higgs boson, which has spin 0. This is responsible for giving
mass to the
W
±
, Z
bosons and fermions. This was just discovered in 2012
in CERN, and subsequently more properties have been discovered, e.g. its
spin.
As one would expect from the name, the gauge bosons are manifestations of
local gauge symmetries. The gauge group in the Standard Model is
SU(3)
C
× SU(2)
L
× U(1)
Y
.
We can talk a bit more about each component. The subscripts indicate what
things the group is responsible for. The subscript on
SU
(3)
C
means “colour”,
and this is responsible for the strong force. We will not talk about this much,
because it is complicated.
The remaining
SU
(2)
L
×
U(1)
Y
bit collectively gives the electroweak inter-
action. This is a unified description of electromagnetism and the weak force.
These are a bit funny. The
SU
(2)
L
is a chiral interaction. It only couples to
left-handed particles, hence the
L
. The U(1)
Y
is something we haven’t heard of
(probably), namely the hypercharge, which is conventionally denoted
Y
. Note
that while electromagnetism also has a U(1) gauge group, it is different from
this U(1) we see.
Types of symmetry
One key principle guiding our study of the standard model is symmetry. Sym-
metries can manifest themselves in a number of ways.
(i)
We can have an intact symmetry, or exact symmetry. In other words,
this is an actual symmetry. For example, U(1)
EM
and
SU
(3)
C
are exact
symmetries in the standard model.
(ii)
Symmetries can be broken by an anomaly. This is a symmetry that exists
in the classical theory, but goes away when we quantize. Examples include
global axial symmetry for massless spinor fields in the standard model.
(iii)
Symmetry is explicitly broken by some terms in the Lagrangian. This is
not a symmetry, but if those annoying terms are small (intentionally left
vague), then we have an approximate symmetry, and it may also be useful
to consider these.
For example, in the standard model, the up and down quarks are very
close in mass, but not exactly the same. This gives to the (global) isospin
symmetry.
(iv)
The symmetry is respected by the Lagrangian
L
, but not by the vacuum.
This is a “hidden symmetry”.
(a)
We can have a spontaneously broken symmetry: we have a vacuum
expectation value for one or more scalar fields, e.g. the breaking of
SU(2)
L
× U(1)
Y
into U(1)
EM
.
(b)
Even without scalar fields, we can get dynamical symmetry breaking
from quantum effects. An example of this in the standard model is
the SU(2)
L
× SU(2)
R
global symmetry in the strong interaction.
One can argue that (i) is the only case where we actually have a symmetry, but
the others are useful to consider as well, and we will study them.