Weak isospin

Conservation law

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 in propagating fermions, which mix their chiralities by the mass terms that result 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 electric charge is conserved. The electric charge, ${\displaystyle Q,}$ is related to weak isospin, ${\displaystyle T_3,}$ and weak hypercharge, ${\displaystyle Y_{\mathrm{W}},}$ by

${\displaystyle Q=T_3+\frac{1}{2}{Y}_{\mathrm{W}}.}$

In 1961 Sheldon Glashow proposed this relation by analogy to the Gell-Mann–Nishijima formula for charge to isospin.

Relation with chirality

Fermions with negative chirality (also called "left-handed" fermions) have ${\displaystyle T=\frac{1}{2}}$ and can be grouped into doublets with ${\displaystyle T_3=\pm \frac{1}{2}}$ 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. For example, up-type quarks (u, c, t) have ${\displaystyle T_3=+\frac{1}{2}}$ and always transform into down-type quarks (d, s, b), which have ${\displaystyle T_3=-\frac{1}{2},}$ 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 ${\displaystyle T_3=-\frac{1}{2}}$ and a neutrino (ν
_e, ν
_μ, ν
_τ) with ${\displaystyle T_3=+\frac{1}{2}.}$ 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. Particles with ${\displaystyle T_3=0}$ do not interact with W^±
bosons; however, they do all interact with the Z⁰
boson.

Neutrinos

Lacking any distinguishing electric charge, neutrinos and antineutrinos are assigned the ${\displaystyle T_3}$ opposite their corresponding charged lepton; hence, all left-handed neutrinos are paired with negatively charged left-handed leptons with ${\displaystyle T_3=-\frac{1}{2},}$ so those neutrinos have ${\displaystyle T_3=+\frac{1}{2}.}$ Since right-handed antineutrinos are paired with positively charged right-handed anti-leptons with ${\displaystyle T_3=+\frac{1}{2},}$ those antineutrinos are assigned ${\displaystyle T_3=-\frac{1}{2}.}$ The same result follows from particle-antiparticle charge & parity reversal, between left-handed neutrinos (${\displaystyle T_3=+\frac{1}{2}}$) and right-handed antineutrinos (${\displaystyle T_3=-\frac{1}{2}}$).

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 ${\displaystyle \left(T_3=+\frac{1}{2}\right)}$ → ${\displaystyle \left(T_3=-\frac{1}{2}\right).}$
• 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 ${\displaystyle \left(T_3=-\frac{1}{2}\right)}$ → ${\displaystyle \left(T_3=+\frac{1}{2}\right)}$.

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.