Weak isospin

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 $T_{\mathrm {z} }$ , $T_{3}$ , $I_{\mathrm {z} }$ or $I_{3}$ . It can be understood as the eigenvalue of a charge operator.

The weak isospin conservation law relates to the conservation of $T_{3}$ ; all weak interactions must conserve $T_{3}$ . 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, $Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }$ (electric charge), is conserved. $T_{3}$ 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 $T={\tfrac {1}{2}}$ and can be grouped into doublets with $T_{3}=\pm {\tfrac {1}{2}}$ that behave the same way under the weak interaction. For example, up-type quarks (u, c, t) have $T_{3}=+{\tfrac {1}{2}}$ and always transform into down-type quarks (d, s, b), which have $T_{3}=-{\tfrac {1}{2}}$ , and vice versa. On the other hand, a quark never decays weakly into a quark of the same $T_{3}$ . Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (
e
,
μ
,
τ
) with $T_{3}=-{\tfrac {1}{2}}$ and a neutrino (
ν
e
,
ν
μ
,
ν
τ
) with $T_{3}=+{\tfrac {1}{2}}$ . In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and sign reversed $T_{3}$ .

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have $T=T_{3}=0$ and form singlets that do not undergo weak interactions.

The electric charge, $Q$ , is related to weak isospin, $T_{3}$ , and weak hypercharge, $Y_{\mathrm {W} }$ , by

$Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }$ .
Left-handed fermions in the Standard Model
Generation 1 Generation 2 Generation 3
Fermion Symbol Weak
isospin
Fermion Symbol Weak
isospin
Fermion Symbol Weak
isospin
Electron neutrino $\nu _{e}\,$ $+{\tfrac {1}{2}}\,$ Muon neutrino $\nu _{\mu }\,$ $+{\tfrac {1}{2}}\,$ Tau neutrino $\nu _{\tau }\,$ $+{\tfrac {1}{2}}\,$ Electron $e^{-}\,$ $-{\tfrac {1}{2}}\,$ Muon $\mu ^{-}\,$ $-{\tfrac {1}{2}}\,$ Tau $\tau ^{-}\,$ $-{\tfrac {1}{2}}\,$ Up quark $u\,$ $+{\tfrac {1}{2}}\,$ Charm quark $c\,$ $+{\tfrac {1}{2}}\,$ Top quark $t\,$ $+{\tfrac {1}{2}}\,$ Down quark $d\,$ $-{\tfrac {1}{2}}\,$ Strange quark $s\,$ $-{\tfrac {1}{2}}\,$ Bottom quark $b\,$ $-{\tfrac {1}{2}}\,$ All of the above left-handed (regular) particles have corresponding
right-handed anti-particles with equal and opposite weak isospin.
All right-handed (regular) particles and left-handed antiparticles have weak isospin of 0.

Weak isospin and the W bosons

The symmetry associated with weak isospin is SU(2) and requires gauge bosons with $T=1$ (
W+
,
W
and
W0
) to mediate transformations between fermions with half-integer weak isospin charges. $T=1$ implies that
W
bosons have three different values of $T_{3}$ :

• W+
boson $(T_{3}=+1)$ is emitted in transitions $\left(T_{3}=+{\tfrac {1}{2}}\right)$ $\left(T_{3}=-{\tfrac {1}{2}}\right)$ .

• W0
boson $(T_{3}=0)$ would be emitted in weak interactions where $T_{3}$ does not change, such as neutrino scattering.

• W
boson $(T_{3}=-1)$ is emitted in transitions $\left(T_{3}=-{\tfrac {1}{2}}\right)$ $\left(T_{3}=+{\tfrac {1}{2}}\right)$ .

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; the resulting
Z0
and the photon both have weak isospin = 0.

The sum of isospin and +charge is zero for each of the bosons, consequently, all the electroweak bosons have weak hypercharge $Y_{\text{W}}=0$ , so unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.