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 2] 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 weak interactions.

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	${\displaystyle \nu _{e}\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Muon neutrino	${\displaystyle \nu _{\mu }\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Tau neutrino	${\displaystyle \nu _{\tau }\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$
Electron	${\displaystyle e^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Muon	${\displaystyle \mu ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Tau	${\displaystyle \tau ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$
Up quark	${\displaystyle u\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Charm quark	${\displaystyle c\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Top quark	${\displaystyle t\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$
Down quark	${\displaystyle d\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Strange quark	${\displaystyle s\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Bottom quark	${\displaystyle b\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$
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 antiparticles 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. ${\displaystyle T=1}$ 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
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 ${\displaystyle Y_{\text{W}}=0}$, so unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.

• Weak hypercharge
• Weak charge