In particle physics, the electroweak interaction
is the unified description of two of the four
known fundamental interactions of nature:
electromagnetism and the weak interaction.
Although these two forces appear very different
at everyday low energies, the theory models
them as two different aspects of the same
force. Above the unification energy, on the
order of 246 GeV, they would merge into a
single electroweak force. Thus, if the universe
is hot enough (approximately 1015 K, a temperature
not exceeded since shortly after the Big Bang),
then the electromagnetic force and weak force
merge into a combined electroweak force. During
the quark epoch, the electroweak force split
into the electromagnetic and weak force.
Sheldon Glashow, Abdus Salam, and Steven Weinberg
were awarded the 1979 Nobel Prize in Physics
for their contributions to the unification
of the weak and electromagnetic interaction
between elementary particles. The existence
of the electroweak interactions was experimentally
established in two stages, the first being
the discovery of neutral currents in neutrino
scattering by the Gargamelle collaboration
in 1973, and the second in 1983 by the UA1
and the UA2 collaborations that involved the
discovery of the W and Z gauge bosons in proton–antiproton
collisions at the converted Super Proton Synchrotron.
In 1999, Gerardus 't Hooft and Martinus Veltman
were awarded the Nobel prize for showing that
the electroweak theory is renormalizable.
== Formulation ==
Mathematically, the unification is accomplished
under an SU(2) × U(1) gauge group. The corresponding
gauge bosons are the three W bosons of weak
isospin from SU(2) (W1, W2, and W3), and the
B boson of weak hypercharge from U(1), respectively,
all of which are massless.
In the Standard Model, the W± and Z0 bosons,
and the photon, are produced by the spontaneous
symmetry breaking of the electroweak symmetry
from SU(2) × U(1)Y to U(1)em, caused by the
Higgs mechanism (see also Higgs boson). U(1)Y
and U(1)em are different copies of U(1); the
generator of U(1)em is given by Q = Y/2 +
T3, where Y is the generator of U(1)Y (called
the weak hypercharge), and T3 is one of the
SU(2) generators (a component of weak isospin).
The spontaneous symmetry breaking makes the
W3 and B bosons coalesce into two different
bosons – the Z0 boson, and the photon (γ),
(
γ
Z
0
)
=
(
cos
⁡
θ
W
sin
⁡
θ
W
−
sin
⁡
θ
W
cos
⁡
θ
W
)
(
B
W
3
)
{\displaystyle {\begin{pmatrix}\gamma \\Z^{0}\end{pmatrix}}={\begin{pmatrix}\cos
\theta _{W}&\sin \theta _{W}\\-\sin \theta
_{W}&\cos \theta _{W}\end{pmatrix}}{\begin{pmatrix}B\\W_{3}\end{pmatrix}}}
where θW is the weak mixing angle. The axes
representing the particles have essentially
just been rotated, in the (W3, B) plane, by
the angle θW. This also introduces a mismatch
between the mass of the Z0 and the mass of
the W± particles (denoted as MZ and MW, respectively),
M
Z
=
M
W
cos
⁡
θ
W
.
{\displaystyle M_{Z}={\frac {M_{W}}{\cos \theta
_{W}}}.}
The W1 and W2 bosons, in turn, combine to
give massive charged bosons
W
±
=
1
2
(
W
1
∓
i
W
2
)
.
{\displaystyle W^{\pm }={\frac {1}{\sqrt {2}}}(W_{1}\mp
iW_{2}).}
The distinction between electromagnetism and
the weak force arises because there is a (nontrivial)
linear combination of Y and T3 that vanishes
for the Higgs boson (it is an eigenstate of
both Y and T3, so the coefficients may be
taken as −T3 and Y): U(1)em is defined to
be the group generated by this linear combination,
and is unbroken because it does not interact
with the Higgs.
== Lagrangian ==
=== Before electroweak symmetry breaking ===
The Lagrangian for the electroweak interactions
is divided into four parts before electroweak
symmetry breaking becomes manifest,
L
EW
=
L
g
+
L
f
+
L
h
+
L
y
.
{\displaystyle {\mathcal {L}}_{\text{EW}}={\mathcal
{L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal
{L}}_{y}~.}
The
L
g
{\displaystyle {\mathcal {L}}_{g}}
term describes the interaction between the
three W vector bosons and the B vector boson,
L
g
=
−
1
4
W
a
μ
ν
W
μ
ν
a
−
1
4
B
μ
ν
B
μ
ν
{\displaystyle {\mathcal {L}}_{g}=-{\tfrac
{1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\tfrac
{1}{4}}B^{\mu \nu }B_{\mu \nu }}
,where
W
a
μ
ν
{\displaystyle W^{a\mu \nu }}
(
a
=
1
,
2
,
3
{\displaystyle a=1,2,3}
) and
B
μ
ν
{\displaystyle B^{\mu \nu }}
are the field strength tensors for the weak
isospin and weak hypercharge gauge fields.
L
f
{\displaystyle {\mathcal {L}}_{f}}
is the kinetic term for the Standard Model
fermions. The interaction of the gauge bosons
and the fermions are through the gauge covariant
derivative,
L
f
=
Q
¯
i
i
D
/
Q
i
+
u
¯
i
i
D
/
u
i
+
d
¯
i
i
D
/
d
i
+
L
¯
i
i
D
/
L
i
+
e
¯
i
i
D
/
e
i
{\displaystyle {\mathcal {L}}_{f}={\overline
{Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}iD\!\!\!\!/\;u_{i}+{\overline
{d}}_{i}iD\!\!\!\!/\;d_{i}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline
{e}}_{i}iD\!\!\!\!/\;e_{i}}
,where the subscript i runs over the three
generations of fermions; Q, u, and d are the
left-handed doublet, right-handed singlet
up, and right handed singlet down quark fields;
and L and e are the left-handed doublet and
right-handed singlet electron fields.
The h term describes the Higgs field and its
interactions with itself and the gauge bosons,
L
h
=
|
D
μ
h
|
2
−
λ
(
|
h
|
2
−
v
2
2
)
2
{\displaystyle {\mathcal {L}}_{h}=|D_{\mu
}h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}}
The y term displays the Yukawa interaction
with the fermions,
L
y
=
−
y
u
i
j
ϵ
a
b
h
b
†
Q
¯
i
a
u
j
c
−
y
d
i
j
h
Q
¯
i
d
j
c
−
y
e
i
j
h
L
¯
i
e
j
c
+
h
.
c
.
,
{\displaystyle {\mathcal {L}}_{y}=-y_{u\,ij}\epsilon
^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline
{Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline
{L}}_{i}e_{j}^{c}+h.c.~,}
and generates their masses, manifest when
the Higgs field acquires a nonzero vacuum
expectation value, discussed next.
=== After electroweak symmetry breaking ===
The Lagrangian reorganizes itself as the Higgs
boson acquires a non-vanishing vacuum expectation
value dictated by the potential of the previous
section. As a result of this rewriting, the
symmetry breaking becomes manifest.
Due to its complexity, this Lagrangian is
best described by breaking it up into several
parts as follows.
L
EW
=
L
K
+
L
N
+
L
C
+
L
H
+
L
HV
+
L
WWV
+
L
WWVV
+
L
Y
.
{\displaystyle {\mathcal {L}}_{\text{EW}}={\mathcal
{L}}_{\text{K}}+{\mathcal {L}}_{\text{N}}+{\mathcal
{L}}_{\text{C}}+{\mathcal {L}}_{\text{H}}+{\mathcal
{L}}_{\text{HV}}+{\mathcal {L}}_{\text{WWV}}+{\mathcal
{L}}_{\text{WWVV}}+{\mathcal {L}}_{\text{Y}}.}
The kinetic term
L
K
{\displaystyle {\mathcal {L}}_{K}}
contains all the quadratic terms of the Lagrangian,
which include the dynamic terms (the partial
derivatives) and the mass terms (conspicuously
absent from the Lagrangian before symmetry
breaking)
L
K
=
∑
f
f
¯
(
i
∂
/
−
m
f
)
f
−
1
4
A
μ
ν
A
μ
ν
−
1
2
W
μ
ν
+
W
−
μ
ν
+
m
W
2
W
μ
+
W
−
μ
−
1
4
Z
μ
ν
Z
μ
ν
+
1
2
m
Z
2
Z
μ
Z
μ
+
1
2
(
∂
μ
H
)
(
∂
μ
H
)
−
1
2
m
H
2
H
2
,
{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{K}}=\sum
_{f}{\overline {f}}(i\partial \!\!\!/\!\;-m_{f})f-{\frac
{1}{4}}A_{\mu \nu }A^{\mu \nu }-{\frac {1}{2}}W_{\mu
\nu }^{+}W^{-\mu \nu }+m_{W}^{2}W_{\mu }^{+}W^{-\mu
}\\\qquad -{\frac {1}{4}}Z_{\mu \nu }Z^{\mu
\nu }+{\frac {1}{2}}m_{Z}^{2}Z_{\mu }Z^{\mu
}+{\frac {1}{2}}(\partial ^{\mu }H)(\partial
_{\mu }H)-{\frac {1}{2}}m_{H}^{2}H^{2}~,\end{aligned}}}
where the sum runs over all the fermions of
the theory (quarks and leptons), and the fields
A
μ
ν
{\displaystyle A_{\mu \nu }^{}}
,
Z
μ
ν
{\displaystyle Z_{\mu \nu }^{}}
,
W
μ
ν
−
{\displaystyle W_{\mu \nu }^{-}}
, and
W
μ
ν
+
≡
(
W
μ
ν
−
)
†
{\displaystyle W_{\mu \nu }^{+}\equiv (W_{\mu
\nu }^{-})^{\dagger }}
are given as
X
μ
ν
a
=
∂
μ
X
ν
a
−
∂
ν
X
μ
a
+
g
f
a
b
c
X
μ
b
X
ν
c
,
{\displaystyle X_{\mu \nu }^{a}=\partial _{\mu
}X_{\nu }^{a}-\partial _{\nu }X_{\mu }^{a}+gf^{abc}X_{\mu
}^{b}X_{\nu }^{c}~,}
with X to be replaced by the relevant field,
and f abc by the structure constants of the
appropriate gauge group.
The neutral current
L
N
{\displaystyle {\mathcal {L}}_{\text{N}}}
and charged current
L
C
{\displaystyle {\mathcal {L}}_{\text{C}}}
components of the Lagrangian contain the interactions
between the fermions and gauge bosons,
L
N
=
e
J
μ
em
A
μ
+
g
cos
⁡
θ
W
(
J
μ
3
−
sin
2
⁡
θ
W
J
μ
em
)
Z
μ
{\displaystyle {\mathcal {L}}_{\text{N}}=eJ_{\mu
}^{\text{em}}A^{\mu }+{\frac {g}{\cos \theta
_{W}}}(J_{\mu }^{3}-\sin ^{2}\theta _{W}J_{\mu
}^{\text{em}})Z^{\mu }}
,where e= g sin θW= g' cos θW; while the
electromagnetic current
J
μ
em
{\displaystyle J_{\mu }^{\text{em}}}
and the neutral weak current
J
μ
3
{\displaystyle J_{\mu }^{3}}
are
J
μ
em
=
∑
f
q
f
f
¯
γ
μ
f
{\displaystyle J_{\mu }^{\text{em}}=\sum _{f}q_{f}{\overline
{f}}\gamma _{\mu }f}
,and
J
μ
3
=
∑
f
I
f
3
f
¯
γ
μ
1
−
γ
5
2
f
{\displaystyle J_{\mu }^{3}=\sum _{f}I_{f}^{3}{\overline
{f}}\gamma _{\mu }{\frac {1-\gamma ^{5}}{2}}f}
where
q
f
{\displaystyle q_{f}^{}}
and
I
f
3
{\displaystyle I_{f}^{3}}
are the fermions' electric charges and weak
isospin.
The charged current part of the Lagrangian
is given by
L
C
=
−
g
2
[
u
¯
i
γ
μ
1
−
γ
5
2
M
i
j
CKM
d
j
+
ν
¯
i
γ
μ
1
−
γ
5
2
e
i
]
W
μ
+
+
h.c.
,
{\displaystyle {\mathcal {L}}_{\text{C}}=-{\frac
{g}{\sqrt {2}}}\left[{\overline {u}}_{i}\gamma
^{\mu }{\frac {1-\gamma ^{5}}{2}}M_{ij}^{\text{CKM}}d_{j}+{\overline
{\nu }}_{i}\gamma ^{\mu }{\frac {1-\gamma
^{5}}{2}}e_{i}\right]W_{\mu }^{+}+{\text{h.c.}}~,}
where
L
H
{\displaystyle {\mathcal {L}}_{\text{H}}}
contains the Higgs three-point and four-point
self interaction terms,
L
H
=
−
g
m
H
2
4
m
W
H
3
−
g
2
m
H
2
32
m
W
2
H
4
.
{\displaystyle {\mathcal {L}}_{\text{H}}=-{\frac
{gm_{H}^{2}}{4m_{W}}}H^{3}-{\frac {g^{2}m_{H}^{2}}{32m_{W}^{2}}}H^{4}~.}
L
HV
{\displaystyle {\mathcal {L}}_{\text{HV}}}
contains the Higgs interactions with gauge
vector bosons,
L
HV
=
(
g
m
W
H
+
g
2
4
H
2
)
(
W
μ
+
W
−
μ
+
1
2
cos
2
⁡
θ
W
Z
μ
Z
μ
)
.
{\displaystyle {\mathcal {L}}_{\text{HV}}=\left(gm_{W}H+{\frac
{g^{2}}{4}}H^{2}\right)\left(W_{\mu }^{+}W^{-\mu
}+{\frac {1}{2\cos ^{2}\theta _{W}}}Z_{\mu
}Z^{\mu }\right).}
L
WWV
{\displaystyle {\mathcal {L}}_{\text{WWV}}}
contains the gauge three-point self interactions,
L
WWV
=
−
i
g
[
(
W
μ
ν
+
W
−
μ
−
W
+
μ
W
μ
ν
−
)
(
A
ν
sin
⁡
θ
W
−
Z
ν
cos
⁡
θ
W
)
+
W
ν
−
W
μ
+
(
A
μ
ν
sin
⁡
θ
W
−
Z
μ
ν
cos
⁡
θ
W
)
]
.
{\displaystyle {\mathcal {L}}_{\text{WWV}}=-ig[(W_{\mu
\nu }^{+}W^{-\mu }-W^{+\mu }W_{\mu \nu }^{-})(A^{\nu
}\sin \theta _{W}-Z^{\nu }\cos \theta _{W})+W_{\nu
}^{-}W_{\mu }^{+}(A^{\mu \nu }\sin \theta
_{W}-Z^{\mu \nu }\cos \theta _{W})].}
L
WWVV
{\displaystyle {\mathcal {L}}_{\text{WWVV}}}
contains the gauge four-point self interactions,
L
WWVV
=
−
g
2
4
{
[
2
W
μ
+
W
−
μ
+
(
A
μ
sin
⁡
θ
W
−
Z
μ
cos
⁡
θ
W
)
2
]
2
−
[
W
μ
+
W
ν
−
+
W
ν
+
W
μ
−
+
(
A
μ
sin
⁡
θ
W
−
Z
μ
cos
⁡
θ
W
)
(
A
ν
sin
⁡
θ
W
−
Z
ν
cos
⁡
θ
W
)
]
2
}
.
{\displaystyle {\begin{aligned}{\mathcal {L}}_{\text{WWVV}}=-{\frac
{g^{2}}{4}}{\Big \{}&[2W_{\mu }^{+}W^{-\mu
}+(A_{\mu }\sin \theta _{W}-Z_{\mu }\cos \theta
_{W})^{2}]^{2}\\&-[W_{\mu }^{+}W_{\nu }^{-}+W_{\nu
}^{+}W_{\mu }^{-}+(A_{\mu }\sin \theta _{W}-Z_{\mu
}\cos \theta _{W})(A_{\nu }\sin \theta _{W}-Z_{\nu
}\cos \theta _{W})]^{2}{\Big \}}.\end{aligned}}}
L
Y
{\displaystyle {\mathcal {L}}_{\text{Y}}}
contains the Yukawa interactions between the
fermions and the Higgs field,
L
Y
=
−
∑
f
g
m
f
2
m
W
f
¯
f
H
.
{\displaystyle {\mathcal {L}}_{\text{Y}}=-\sum
_{f}{\frac {gm_{f}}{2m_{W}}}{\overline {f}}fH.}
Note the
1
2
(
1
−
γ
5
)
{\displaystyle {\tfrac {1}{2}}(1-\gamma ^{5})}
factors in the weak couplings: these factors
project out the left handed components of
the spinor fields. This is why electroweak
theory is said to be a chiral theory.
== See also ==
Fundamental forces
History of quantum field theory
Standard Model (mathematical formulation)
Unitarity gauge
Weinberg angle
== Notes
