In particle physics, weak isospin is a quantum
number relating to the weak interaction, and
parallels the idea of isospin under the strong
interaction. Weak isospin is usually given
the symbol T or I with the third component
written as
T
z
{\displaystyle T_{\mathrm {z} }}
,
T
3
{\displaystyle T_{3}}
,
I
z
{\displaystyle I_{\mathrm {z} }}
or
I
3
{\displaystyle I_{3}}
. It can be understood as the eigenvalue of
a charge operator.
The weak isospin conservation law relates
the conservation of
T
3
{\displaystyle T_{3}}
; all weak interactions must preserve
T
3
{\displaystyle T_{3}}
. It is also conserved by the electromagnetic,
and strong interactions. However, one of the
interactions is with the Higgs field. Since
the Higgs field vacuum expectation value is
nonzero, particles interact with this field
all the time even in vacuum. This changes
their weak isospin (and weak hypercharge).
Only a specific combination of them,
Q
=
T
3
+
1
2
Y
W
{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm
{W} }}
(electric charge), is conserved.
T
3
{\displaystyle T_{3}}
is more important than T and often the term
"weak isospin" refers to the "3rd component
of weak isospin".
== Relation with chirality ==
Fermions with negative chirality (also called
“left-handed” fermions) have
T
=
1
2
{\displaystyle T={\tfrac {1}{2}}}
and can be grouped into doublets with
T
3
=
±
1
2
{\displaystyle T_{3}=\pm {\tfrac {1}{2}}}
that behave the same way under the weak interaction.
For example, up-type quarks (u, c, t) have
T
3
=
+
1
2
{\displaystyle T_{3}=+{\tfrac {1}{2}}}
and always transform into down-type quarks
(d, s, b), which have
T
3
=
−
1
2
{\displaystyle T_{3}=-{\tfrac {1}{2}}}
, and vice versa. On the other hand, a quark
never decays weakly into a quark of the same
T
3
{\displaystyle T_{3}}
. Something similar happens with left-handed
leptons, which exist as doublets containing
a charged lepton (e−, μ−, τ−) with
T
3
=
−
1
2
{\displaystyle T_{3}=-{\tfrac {1}{2}}}
and a neutrino (νe, νμ, ντ) with
T
3
=
+
1
2
{\displaystyle T_{3}=+{\tfrac {1}{2}}}
. In all cases, the corresponding anti-fermion
has reversed chirality (“right-handed”
antifermion) and sign reversed
T
3
{\displaystyle T_{3}}
.
Fermions with positive chirality (“right-handed”
fermions) and anti-fermions with negative
chirality (“left-handed” anti-fermions)
have
T
=
T
3
=
0
{\displaystyle T=T_{3}=0}
and form singlets that do not undergo weak
interactions.
Electric charge,
Q
{\displaystyle Q}
, is related to weak isospin,
T
3
{\displaystyle T_{3}}
, and weak hypercharge,
Y
W
{\displaystyle Y_{\mathrm {W} }}
, by
Q
=
T
3
+
1
2
Y
W
{\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm
{W} }}
.
== Weak isospin and the W bosons ==
The symmetry associated with weak isospin
is SU(2) and requires gauge bosons with integral
T
3
{\displaystyle T_{3}}
(W+, W− and W0) to mediate transformations
between fermions with half-integer weak isospin
charges. This implies that W bosons must have
T
=
1
{\displaystyle T=1}
, with three different values of
T
3
{\displaystyle T_{3}}
:
W+ boson
(
T
3
=
+
1
)
{\displaystyle (T_{3}=+1)}
is emitted in transitions
(
T
3
=
+
1
2
)
{\displaystyle (T_{3}=+{\tfrac {1}{2}})}
→
(
T
3
=
−
1
2
)
{\displaystyle (T_{3}=-{\tfrac {1}{2}})}
.
W0 boson
(
T
3
=
0
)
{\displaystyle (T_{3}=0)}
would be emitted in weak interactions where
T
3
{\displaystyle T_{3}}
does not change, such as neutrino scattering.
W− boson
(
T
3
=
−
1
)
{\displaystyle (T_{3}=-1)}
is emitted in transitions
(
T
3
=
−
1
2
)
{\displaystyle (T_{3}=-{\tfrac {1}{2}})}
→
(
T
3
=
+
1
2
)
{\displaystyle (T_{3}=+{\tfrac {1}{2}})}
.Under electroweak unification, the W0 boson
mixes with the weak hypercharge gauge boson
B, resulting in the observed Z0 boson and
the photon of Quantum Electrodynamics. However,
the resulting Z0 and the γ both have weak
isospin 0. As a consequence of their weak
isospin values and charges, all the electroweak
bosons have weak hypercharge
Y
w
=
0
{\displaystyle Y_{\text{w}}=0}
, so unlike gluons and the color force, the
electroweak bosons are unaffected by the force
they mediate.
== See also ==
Field theoretical formulation of standard
model
Weak hypercharge
