Weak isospin

Fermions with negative chirality (also called "left-handed" fermions) have ${\displaystyle \ T={\tfrac {1}{2}}\ }$ and can be grouped into doublets with ${\displaystyle T_{3}=\pm {\tfrac {1}{2}}}$ that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned ${\displaystyle T_{3}}$ with the same sign as their electric charge.[lower-alpha 3] For example, up-type quarks (u, c, t) have ${\displaystyle \ T_{3}=+{\tfrac {1}{2}}\ }$ and always transform into down-type quarks (d, s, b), which have ${\displaystyle \ T_{3}=-{\tfrac {1}{2}}\ ,}$ and vice versa. On the other hand, a quark never decays weakly into a quark of the same ${\displaystyle \ T_{3}~.}$ Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (
e⁻
,
μ⁻
,
τ⁻
) with ${\displaystyle \ T_{3}=-{\tfrac {1}{2}}\ }$ and a neutrino (
ν
_e,
ν
_μ,
ν
_τ) with ${\displaystyle \ T_{3}=+{\tfrac {1}{2}}~.}$ In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign ${\displaystyle \ T_{3}~.}$

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have ${\displaystyle \ T=T_{3}=0\ }$ and form singlets that do not undergo charged weak interactions.[lower-alpha 4]

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

${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }~.}$

Left-handed fermions in the Standard Model[1]

Generation 1			Generation 2			Generation 3
Fermion	Symbol	Weak isospin	Fermion	Symbol	Weak isospin	Fermion	Symbol	Weak isospin
Electron neutrino	${\displaystyle \nu _{\mathrm {e} }\ }$	${\displaystyle +{\tfrac {1}{2}}\ }$	Muon neutrino	${\displaystyle \nu _{\mu }\ }$	${\displaystyle +{\tfrac {1}{2}}\ }$	Tau neutrino	${\displaystyle \nu _{\tau }\,}$	${\displaystyle +{\tfrac {1}{2}}\ }$
Electron	${\displaystyle \mathrm {e} ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\ }$	Muon	${\displaystyle \mu ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\ }$	Tauon	${\displaystyle \tau ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\ }$
Up quark	${\displaystyle \mathrm {u} \ }$	${\displaystyle +{\tfrac {1}{2}}\ }$	Charm quark	${\displaystyle \mathrm {c} \ }$	${\displaystyle +{\tfrac {1}{2}}\ }$	Top quark	${\displaystyle \mathrm {t} \ }$	${\displaystyle +{\tfrac {1}{2}}\ }$
Down quark	${\displaystyle \mathrm {d} \ }$	${\displaystyle -{\tfrac {1}{2}}\ }$	Strange quark	${\displaystyle \mathrm {s} \ }$	${\displaystyle -{\tfrac {1}{2}}\ }$	Bottom quark	${\displaystyle \mathrm {b} \ }$	${\displaystyle -{\tfrac {1}{2}}\ }$
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 anti-particles have weak isospin of 0.

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

• 
  W⁺
  boson ${\displaystyle (\,T_{3}=+1\,)}$ is emitted in transitions ${\displaystyle \left(\,T_{3}=+{\tfrac {1}{2}}\,\right)}$ → ${\displaystyle \left(\,T_{3}=-{\tfrac {1}{2}}\,\right)~.}$
• 
  W⁰
  boson ${\displaystyle (\,T_{3}=\,0\,)}$ would be emitted in weak interactions where ${\displaystyle \,T_{3}\,}$ does not change, such as neutrino scattering.
• 
  W⁻
  boson ${\displaystyle (\,T_{3}=-1\,)}$ is emitted in transitions ${\displaystyle \left(\,T_{3}=-{\tfrac {1}{2}}\,\right)}$ → ${\displaystyle \left(\,T_{3}=+{\tfrac {1}{2}}\,\right)}$.

Under electroweak unification, the
W⁰
boson mixes with the weak hypercharge gauge boson
B⁰
; both have weak isospin = 0 . This results in the observed
Z⁰
boson and the photon of quantum electrodynamics; the resulting
Z⁰
and
γ⁰
likewise have zero weak isospin.

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

• Weak hypercharge
• Weak charge
• Mathematical formulation of the Standard Model