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 T₃ or I₃.^[a] It can be understood as the eigenvalue of a charge operator.

T₃ is more important than T and typically the term "weak isospin" may refer to the "3rd component of weak isospin".

The weak isospin conservation law relates to the conservation of ${\displaystyle T_{3}}$; weak interactions conserve T₃. It is also conserved by the electromagnetic and strong interactions. However, interaction with the Higgs field does not conserve T₃, as directly seen by propagation of fermions, mixing chiralities by dint of their mass terms resulting from their Higgs couplings. Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. Interaction with the Higgs field changes particles' weak isospin (and weak hypercharge). Only a specific combination of them, ${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }}$ (electric charge), is conserved.

Relation with chirality[]

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.^[b] 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. (They do not interact with W^± bosons; however, they do all interact with the Z⁰ boson.^[c])

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 anti-particles 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 ${\displaystyle \,T=1\,}$ (
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
B⁰
, resulting in the observed
Z⁰
boson and the photon of quantum electrodynamics; the resulting
Z⁰
and the
γ⁰
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.

See also[]

• Weak hypercharge
• Weak charge

Footnotes[]

References[]