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# 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}, T_{3}, I_{z} or I_{3}.[1] Weak isospin is a complement of the weak hypercharge, which unifies weak interactions with electromagnetic interactions.

The weak isospin conservation law relates the conservation of T_{3}; all weak interactions must preserve T_{3}. It is also conserved by the other interactions and is therefore a conserved quantity in general. For this reason T_{3} 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 T_{3} = ±1⁄2 that behave the same way under the weak interaction. For example, up-type quarks (u, c, t) have T_{3} = +1⁄2 and always transform into down-type quarks (d, s, b), which have T_{3} = −1⁄2, and vice-versa. On the other hand, a quark never decays weakly into a quark of the same T_{3}. Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (e^{−} , μ^{−} , τ^{−} ) with T_{3} = −1⁄2 and a neutrino (ν_{e}, ν_{μ}, ν_{τ}) with T_{3} = 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, T_{3}, and weak hypercharge, YW, 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 W^{0}

. This implies that W bosons have a T = 1, with three different values of T3.

W^{+} boson (T3 = +1) is emitted in transitions {(T_{3} = +1⁄2) → (T3 = −1⁄2)},

W^{−} boson (T3 = −1) is emitted in transitions {(T_{3} = −1⁄2) → (T3 = +1⁄2)}.

W^{0} boson (T3 = 0) would be emitted in reactions where T_{3} 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.

Sources and footnotes

^ Ambiguities: I is also used as sign for the 'normal' isospin, same for the third component I3 aka Iz. T is also used as the sign for Topness. This article uses T and T_{3}.

See also

Field theoretical formulation of standard model

Weak hypercharge

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