Weak isospin

Weak isospin
In particle physics, weak isospin is a quantum number relating to the electrically charged part of the weak interaction: Particles with half-integer weak isospin can interact with the bosons; particles with zero weak isospin do not.
Weak isospin is a construct parallel to 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.
is more important than ; typically "weak isospin" is used as short form of the proper term "3rd component of weak isospin".
The weak isospin conservation law relates to the conservation of weak interactions conserve . It is also conserved by the electromagnetic and strong interactions. However, interaction with the Higgs field does not conserve , 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, (electric charge), is conserved.
Relation with chirality
Fermions with negative chirality (also called "left-handed" fermions) 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 (u, c, t) have and always transform into down-type quarks (d, s, b), 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 ("right-handed" antifermion) and reversed sign
Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have and form singlets that do not undergo charged 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(2) and requires gauge bosons with (and ) to mediate transformations between fermions with half-integer weak isospin charges. implies that bosons have three different values of
boson is emitted in transitions →
boson would be emitted in weak interactions where does not change, such as neutrino scattering.
boson is emitted in transitions → .
Under electroweak unification, the boson mixes with the weak hypercharge gauge boson ; both have This results in the observed boson and the photon of quantum electrodynamics; the resulting and likewise have zero weak isospin.
The sum of negative isospin and positive 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.
See also
Weak hypercharge
Weak charge
Mathematical formulation of the Standard Model
Footnotes
References
Standard Model
Flavour (particle physics)
Electroweak theory
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