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_z, T₃, I_z or I₃.^[1] It can be understood as the eigenvalue of a charge operator.

The weak isospin conservation law relates the conservation of T₃; all weak interactions must preserve T₃. It is also conserved by the other interactions. However, one of the interactions is with Higgs field. Since 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=T3+Y/2 (electric charge), is conserved. T₃ 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 =  ¹⁄₂ and can be grouped into doublets with T₃ = ± ¹⁄₂ that behave the same way under the weak interaction. For example, up-type quarks (u, c, t) have T₃ = + ¹⁄₂ and always transform into down-type quarks (d, s, b), which have T₃ = − ¹⁄₂, and vice versa. On the other hand, a quark never decays weakly into a quark of the same T₃. Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (
e−
,
μ−
,
τ−
) with T₃ = − ¹⁄₂ and a neutrino (
ν
e,
ν
μ,
ν
τ) with T₃ =  ¹⁄₂.

Fermions with positive chirality (also called right-handed fermions) have T = 0 and form singlets that do not undergo weak interactions.

Electric charge, Q, is related to weak isospin, T₃, and weak hypercharge, Y_W, by

$Q = T_3 + \frac{Y_\mathrm{W}}{2}.$

Weak isospin and the W bosons

The symmetry associated with spin is SU(2). This requires gauge bosons to transform between weak isospin charges: bosons
W+
,
W−
and
W0
. This implies that
W
bosons have a T = 1, with three different values of T₃.

• 
  W+
  boson (T₃ = +1) is emitted in transitions {(T₃ = + ¹⁄₂) → (T₃ = − ¹⁄₂)},
• 
  W−
  boson (T₃ = −1) is emitted in transitions {(T₃ = − ¹⁄₂) → (T₃ = + ¹⁄₂)}.
• 
  W0
  boson (T₃ = 0) would be emitted in reactions where T₃ does not change. However, 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.

See also

• Field theoretical formulation of standard model
• Weak hypercharge

References

1. Ambiguities: I is also used as sign for the 'normal' isospin, same for the third component I₃ aka I_z. T is also used as the sign for Topness. This article uses T and T₃.