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 or with the third component written as,,, or. It can be understood as the eigenvalue of a charge operator.
The weak isospin conservation law relates to the conservation of ; all weak interactions must conserve. It is also conserved by the electromagnetic and strong interactions. However, one of the interactions is with the Higgs field. Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. This changes their weak isospin. Only a specific combination of them, , is conserved. is more important than and often the term "weak isospin" refers to the "3rd component of weak isospin".

Relation with chirality

s with negative chirality have and can be grouped into doublets with that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned with the same sign as their electric charge. For example, up-type quarks have and always transform into down-type quarks, which have, and vice versa. On the other hand, a quark never decays weakly into a quark of the same. Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton with and a neutrino with. In all cases, the corresponding anti-fermion has reversed chirality and reversed sign.
Fermions with positive chirality and anti-fermions with negative chirality have and form singlets that do not undergo weak interactions.
The electric charge,, is related to weak isospin,, and weak hypercharge,, by

Weak isospin and the W bosons

The symmetry associated with weak isospin is SU and requires gauge bosons with to mediate transformations between fermions with half-integer weak isospin charges. implies that bosons have three different values of :
Under electroweak unification, the boson mixes with the weak hypercharge gauge boson, resulting in the observed boson and the photon of quantum electrodynamics; the resulting 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, so unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.

Footnotes