5Electroweak theory
III The Standard Model
5.5 Summary of electroweak theory
We can do a quick summary of the electroweak theory. We start with the picture
before spontaneous symmetry breaking.
The gauge group of this theory is
SU
(2)
L
×
U(1)
Y
, with gauge fields
W
µ
∈
su
(2) and
B
µ
∈ u
(1). The coupling of U(1) with the particles is specified by a
hypercharge
Y
, and the
SU
(2) couplings will always be trivial or fundamental.
The covariant derivative is given by
D
µ
= ∂
µ
+ igW
a
µ
τ
a
+
1
2
g
0
Y B
µ
,
where
τ
a
are the canonical generators of
su
(2). The field strengths are given by
F
W,a
µν
= ∂
µ
W
a
ν
− ∂
ν
W
a
µ
− gε
abc
W
b
µ
W
c
ν
F
B
µν
= ∂
µ
B
ν
− ∂
ν
V
µ
.
The theory contains the scalar Higgs field
φ ∈ C
2
, which has hypercharge
Y
=
1
2
and the fundamental
SU
(2) representation. We also have three generations of
matter, given by
Type G1 G2 G2 Q Y
L
Y
R
Positive quarks u c t +
2
3
+
1
6
+
2
3
Negative quarks d s b −
1
3
+
1
6
−
1
3
Leptons (e, µ, τ) e µ τ −1 −
1
2
−1
Leptons (neutrinos) ν
e
ν
µ
ν
τ
0 −
1
2
???
Here G1, G2, G3 are the three generations,
Q
is the charge,
Y
L
is the hypercharge
of the left-handed version, and
Y
R
is the hypercharge of the right-handed version.
Each of these matter fields is a spinor field.
From now on, we describe the theory in the case of a massless neutrino,
because we don’t really know what the neutrinos are. We group the matter fields
as
L =
ν
e
L
e
L
ν
µ
L
µ
L
ν
τ
L
τ
L
R = ( e
R
µ
R
τ
R
)
u
R
= ( u
R
c
R
t
R
)
d
R
= ( d
R
s
R
b
R
)
Q
L
=
u
L
d
L
c
L
s
L
t
L
b
L
The Lagrangian has several components:
– The kinetic term of the gauge field is given by
L
gauge
= −
1
2
Tr(F
W
µν
F
W,µν
) −
1
4
F
B
µν
F
B,µν
,
– The Higgs field couples with the gauge fields by
L
gauge,φ
= (D
µ
φ)
†
(D
µ
φ) − µ
2
|φ|
2
− λ|φ|
4
,
After spontaneous symmetry breaking, this gives rise to massive
W
±
, Z
and Higgs bosons. This also gives us
W
±
, Z
-Higgs interactions, as well as
gives us Higgs-Higgs interactions.
– The leptons interact with the Higgs field by
L
lept,φ
= −
√
2(λ
ij
¯
L
i
φR
j
+ h.c.).
This gives us lepton masses and lepton-Higgs interactions. Of course, this
piece has to be modified after we figure out what neutrinos actually are.
– The gauge coupling of the leptons induce
L
EW
lept
=
¯
Li
/
DL +
¯
Ri
/
DR,
with an implicit sum over all generations. This gives us lepton interactions
with
W
±
, Z, γ
. Once we introduce neutrino masses, this is described by
the PMNS matrix, and gives us neutrino oscillations and (possibly) CP
violation.
– Higgs-quark interactions are given by
L
quark,φ
= −
√
2(λ
ij
d
¯
Q
i
L
φd
i
R
+ λ
ij
u
¯
Q
i
L
φ
c
u
j
R
+ h.c.).
which gives rise to quark masses.
– Finally, the gauge coupling of the quarks is given by
L
EW
quark
=
¯
Q
L
i
/
DQ
L
+ ¯u
R
i
/
Du
R
+
¯
d
R
i
/
Dd
R
,
which gives us quark interactions with
W
±
, Z, γ
. The interactions are
described by the CKM matrix. This gives us quark flavour and CP
violation.
We now have all of the standard model that involves the electroweak part
and the matter. Apart from QCD, which we will describe quite a bit later, we’ve
got everything in the standard model.