Physics:Weak isospin

Short description: Quantum number related to the weak interaction

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 W^± 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 T or I, with the third component written as T₃ or I₃. It can be understood as the eigenvalue of a charge operator.

T₃ is more important than T; 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 [math]\displaystyle{ \ T_3\ ; }[/math] 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, [math]\displaystyle{ \ Q = T_3 + \tfrac{1}{2}Y_\mathrm{W}\ }[/math] (electric charge), is conserved.

Relation with chirality

Fermions with negative chirality (also called "left-handed" fermions) have [math]\displaystyle{ \ T = \tfrac{1}{2}\ }[/math] and can be grouped into doublets with [math]\displaystyle{ T_3 = \plusmn \tfrac{1}{2} }[/math] that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned [math]\displaystyle{ T_3 }[/math] with the same sign as their electric charge. For example, up-type quarks (u, c, t) have [math]\displaystyle{ \ T_3 = +\tfrac{1}{2}\ }[/math] and always transform into down-type quarks (d, s, b), which have [math]\displaystyle{ \ T_3 = -\tfrac{1}{2}\ , }[/math] and vice versa. On the other hand, a quark never decays weakly into a quark of the same [math]\displaystyle{ \ T_3 ~. }[/math] Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (Electron-, Muon-, Tau-) with [math]\displaystyle{ \ T_3 = -\tfrac{1}{2}\ }[/math] and a neutrino (Electron neutrino, Muon neutrino, Tau neutrino) with [math]\displaystyle{ \ T_3 = +\tfrac{1}{2} ~. }[/math] In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign [math]\displaystyle{ \ T_3 ~. }[/math]

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have [math]\displaystyle{ \ T = T_3 = 0\ }[/math] and form singlets that do not undergo charged weak interactions.

The electric charge, [math]\displaystyle{ \ Q\ , }[/math] is related to weak isospin, [math]\displaystyle{ \ T_3\ , }[/math] and weak hypercharge, [math]\displaystyle{ \ Y_\mathrm{W}\ , }[/math] by

[math]\displaystyle{ Q = T_3 +\tfrac{1}{2}Y_\mathrm{W} ~. }[/math]

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	[math]\displaystyle{ \nu_\mathrm{e}\ }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]	Muon neutrino	[math]\displaystyle{ \nu_\mu\ }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]	Tau neutrino	[math]\displaystyle{ \nu_\tau\, }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]
Electron	[math]\displaystyle{ \mathrm e^-\, }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]	Muon	[math]\displaystyle{ \mu^-\, }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]	Tauon	[math]\displaystyle{ \tau^-\, }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]
Up quark	[math]\displaystyle{ \mathrm{u}\ }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]	Charm quark	[math]\displaystyle{ \mathrm{c}\ }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]	Top quark	[math]\displaystyle{ \mathrm{t}\ }[/math]	[math]\displaystyle{ +\tfrac{1}{2}\ }[/math]
Down quark	[math]\displaystyle{ \mathrm{d}\ }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]	Strange quark	[math]\displaystyle{ \mathrm{s}\ }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]	Bottom quark	[math]\displaystyle{ \mathrm{b}\ }[/math]	[math]\displaystyle{ -\tfrac{1}{2}\ }[/math]
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 [math]\displaystyle{ \, T = 1 \, }[/math] ( W⁺ , W⁻ , and W⁰ ) to mediate transformations between fermions with half-integer weak isospin charges. [math]\displaystyle{ \ T = 1\ }[/math] implies that W bosons have three different values of [math]\displaystyle{ \ T_3 \ : }[/math]

• W⁺ boson [math]\displaystyle{ (\,T_3 = +1\,) }[/math] is emitted in transitions [math]\displaystyle{ \left(\,T_3 = +\tfrac{1}{2}\,\right) }[/math] → [math]\displaystyle{ \left(\,T_3 = -\tfrac{1}{2}\,\right)~. }[/math]
• W⁰ boson [math]\displaystyle{ (\, T_3 =\, 0 \,) }[/math] would be emitted in weak interactions where [math]\displaystyle{ \, T_3 \, }[/math] does not change, such as neutrino scattering.
• W⁻ boson [math]\displaystyle{ (\, T_3 = -1 \,) }[/math] is emitted in transitions [math]\displaystyle{ \left(\, T_3 = -\tfrac{1}{2} \,\right) }[/math] → [math]\displaystyle{ \left(\, T_3 = +\tfrac{1}{2} \,\right) }[/math].

Under electroweak unification, the W⁰ boson mixes with the weak hypercharge gauge boson B⁰ ; both have weak isospin = 0 . This results in the observed Z boson0 boson and the photon of quantum electrodynamics; the resulting Z boson0 and photon0 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 [math]\displaystyle{ \ Y_\text{W} = 0\ , }[/math] 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

References

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