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 T3 or I3.[a] It can be understood as the eigenvalue of a charge operator.

T3 is more important than T and typically the term "weak isospin" may refer to the "3rd component of weak isospin".

The weak isospin conservation law relates to the conservation of $T_{3}$ ; 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 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, $Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }$ (electric charge), is conserved.

Relation with chirality

Fermions with negative chirality (also called "left-handed" fermions) have $T={\tfrac {1}{2}}$  and can be grouped into doublets with $T_{3}=\pm {\tfrac {1}{2}}$  that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned $T_{3}$  with the same sign as their electric charge.[b] For example, up-type quarks (u, c, t) have $T_{3}=+{\tfrac {1}{2}}$  and always transform into down-type quarks (d, s, b), which have $T_{3}=-{\tfrac {1}{2}}$ , 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 $T_{3}=-{\tfrac {1}{2}}$  and a neutrino (
ν
e
,
ν
μ
,
ν
τ
) with $T_{3}=+{\tfrac {1}{2}}$ . In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign $T_{3}$ .

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have $T=T_{3}=0$  and form singlets that do not undergo charged weak interactions. (They do not interact with W± bosons; however, they do all interact with the Z0 boson.[c])

The electric charge, $Q$ , is related to weak isospin, $T_{3}$ , and weak hypercharge, $Y_{\mathrm {W} }$ , by

$Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }$ .
Left-handed fermions in the Standard Model
Generation 1 Generation 2 Generation 3
Fermion Symbol Weak
isospin
Fermion Symbol Weak
isospin
Fermion Symbol Weak
isospin
Electron neutrino $\nu _{e}\,$  $+{\tfrac {1}{2}}\,$  Muon neutrino $\nu _{\mu }\,$  $+{\tfrac {1}{2}}\,$  Tau neutrino $\nu _{\tau }\,$  $+{\tfrac {1}{2}}\,$
Electron $e^{-}\,$  $-{\tfrac {1}{2}}\,$  Muon $\mu ^{-}\,$  $-{\tfrac {1}{2}}\,$  Tau $\tau ^{-}\,$  $-{\tfrac {1}{2}}\,$
Up quark $u\,$  $+{\tfrac {1}{2}}\,$  Charm quark $c\,$  $+{\tfrac {1}{2}}\,$  Top quark $t\,$  $+{\tfrac {1}{2}}\,$
Down quark $d\,$  $-{\tfrac {1}{2}}\,$  Strange quark $s\,$  $-{\tfrac {1}{2}}\,$  Bottom quark $b\,$  $-{\tfrac {1}{2}}\,$
All of the above left-handed (regular) particles have corresponding
right-handed anti-particles with equal and opposite weak isospin.
All right-handed (regular) particles and left-handed anti-particles have weak isospin of 0.

Weak isospin and the W bosons

The symmetry associated with weak isospin is SU(2) and requires gauge bosons with $\,T=1\,$  (, , and ) to mediate transformations between fermions with half-integer weak isospin charges. $\,T=1\,$  implies that
W
bosons have three different values of $\,T_{3}\,:$

• boson $(\,T_{3}=+1\,)$  is emitted in transitions $\left(\,T_{3}=+{\tfrac {1}{2}}\,\right)$ $\left(\,T_{3}=-{\tfrac {1}{2}}\,\right)~.$

• W0
boson $(\,T_{3}=\,0\,)$  would be emitted in weak interactions where $\,T_{3}\,$  does not change, such as neutrino scattering.
• boson $(\,T_{3}=-1\,)$  is emitted in transitions $\left(\,T_{3}=-{\tfrac {1}{2}}\,\right)$ $\left(\,T_{3}=+{\tfrac {1}{2}}\,\right)$ .

Under electroweak unification, the boson mixes with the weak hypercharge gauge boson
B0
, resulting in the observed boson and the photon of quantum electrodynamics; the resulting and the 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 $\,Y_{\text{W}}=0\;,$  so unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.

1. ^ Regarding ambiguous notation, $I$  is also used to represent the 'normal' (strong force) isospin, same for its third component $T_{3}$  a.k.a. $I_{\mathrm {z} }$  or $I_{3}$ . $T$  is also used as the symbol for the Topness quantum number. This article uses $T$  and $T_{3}$  for weak isospin and its projection.
2. ^ Lacking any distinguishing electric charge, neutrinos and antineutrinos are assigned the $T_{3}$  opposite their corresponding charged lepton; hence, all left-handed neutrinos are paired with negatively charged left-handed leptons with $T_{3}=-{\tfrac {1}{2}},$  so those neutrinos have $T_{3}=+{\tfrac {1}{2}}.$  Since right-handed antineutrinos are paired with positively charged right-handed anti-leptons with $T_{3}=+{\tfrac {1}{2}},$  those antineutrinos are assigned $T_{3}=-{\tfrac {1}{2}}.$  The same result follows from particle-antiparticle charge & parity reversal, between left-handed neutrinos ($T_{3}=+{\tfrac {1}{2}}$ ) and right-handed antineutrinos ($T_{3}=-{\tfrac {1}{2}}$ ).