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 ${\displaystyle T}$ or ${\displaystyle I}$ with the third component written as ${\displaystyle T_{\mathrm {z} }}$, ${\displaystyle T_{3}}$, ${\displaystyle I_{\mathrm {z} }}$, or ${\displaystyle I_{3}}$.[lower-alpha 1] It can be understood as the eigenvalue of a charge operator.

The weak isospin conservation law relates to the conservation of ${\displaystyle T_{3}}$; all weak interactions must conserve ${\displaystyle T_{3}}$. 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 (and weak hypercharge). Only a specific combination of them, ${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }}$ (electric charge), is conserved. ${\displaystyle T_{3}}$ 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 ${\displaystyle T={\tfrac {1}{2}}}$ and can be grouped into doublets with ${\displaystyle T_{3}=\pm {\tfrac {1}{2}}}$ that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned ${\displaystyle T_{3}}$ with the same sign as their electric charge.[lower-alpha 2] For example, up-type quarks (u, c, t) have ${\displaystyle T_{3}=+{\tfrac {1}{2}}}$ and always transform into down-type quarks (d, s, b), which have ${\displaystyle T_{3}=-{\tfrac {1}{2}}}$, and vice versa. On the other hand, a quark never decays weakly into a quark of the same ${\displaystyle T_{3}}$. Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (
e⁻
,
μ⁻
,
τ⁻
) with ${\displaystyle T_{3}=-{\tfrac {1}{2}}}$ and a neutrino (
ν
_e,
ν
_μ,
ν
_τ) with ${\displaystyle T_{3}=+{\tfrac {1}{2}}}$. In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign ${\displaystyle T_{3}}$.

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have ${\displaystyle T=T_{3}=0}$ and form singlets that do not undergo weak interactions.

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

${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }}$.

Left-handed fermions in the Standard Model[1]

Generation 1			Generation 2			Generation 3
Fermion	Symbol	Weak isospin	Fermion	Symbol	Weak isospin	Fermion	Symbol	Weak isospin
Electron neutrino	${\displaystyle \nu _{e}\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Muon neutrino	${\displaystyle \nu _{\mu }\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Tau neutrino	${\displaystyle \nu _{\tau }\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$
Electron	${\displaystyle e^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Muon	${\displaystyle \mu ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Tau	${\displaystyle \tau ^{-}\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$
Up quark	${\displaystyle u\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Charm quark	${\displaystyle c\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$	Top quark	${\displaystyle t\,}$	${\displaystyle +{\tfrac {1}{2}}\,}$
Down quark	${\displaystyle d\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Strange quark	${\displaystyle s\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$	Bottom quark	${\displaystyle b\,}$	${\displaystyle -{\tfrac {1}{2}}\,}$
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 antiparticles 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 ${\displaystyle T=1}$ (
W⁺
,
W⁻
and
W⁰
) to mediate transformations between fermions with half-integer weak isospin charges. ${\displaystyle T=1}$ implies that
W
bosons have three different values of ${\displaystyle T_{3}}$:

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

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; the resulting
Z⁰
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 ${\displaystyle Y_{\text{W}}=0}$, so unlike gluons of the color force, the electroweak bosons are unaffected by the force they mediate.

See also

• Weak hypercharge
• Weak charge

Footnotes

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

References

1. Baez, John C.; Huerta, John (2009). "The Algebra of Grand Unified Theories". Bull. Am. Math. Soc. 0904: 483–552. arXiv:0904.1556. Bibcode:2009arXiv0904.1556B. doi:10.1090/s0273-0979-10-01294-2. Retrieved 15 October 2013.