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 $T_{\mathrm {z} }$ , $T_{3}$ , $I_{\mathrm {z} }$ or $I_{3}$ .^[1] It can be understood as the eigenvalue of a charge operator.

The weak isospin conservation law relates the conservation of $T_{3}$ ; all weak interactions must preserve $T_{3}$ . 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, $Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }$ (electric charge), is conserved. $T_{3}$ 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 $T={\tfrac {1}{2}}$ and can be grouped into doublets with $T_{3}=\pm {\tfrac {1}{2}}$ that behave the same way under the weak interaction. 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 sign reversed $T_{3}$ .

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 weak interactions.

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} }

.

Weak isospin and the W bosons

The symmetry associated with weak isospin is SU(2) and requires gauge bosons with integral $T_{3}$ (
W⁺
,
W⁻
and
W⁰
) to mediate transformations between fermions with half-integer weak isospin charges. This implies that
W
bosons must have $T = 1$ , with three different values of $T_{3}$ :

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

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. However, the resulting
Z⁰
and the
γ
both have weak isospin 0. As a consequence of their weak isospin values and charges, all the electroweak bosons have weak hypercharge $Y_{\text{w}}=0$ , so unlike gluons and the color force, the electroweak bosons are unaffected by the force they mediate.

References

↑ Ambiguities: $I$ is also used as sign for the ‘normal’ isospin, same for the third component $T_{3}$ aka $I_{\mathrm {z} }$ . $T$ is also used as the sign for Topness. This article uses $T$ and $T_{3}$ .

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[1] Ambiguities: $I$ is also used as sign for the ‘normal’ isospin, same for the third component $T_{3}$ aka $I_{\mathrm {z} }$ . $T$ is also used as the sign for Topness. This article uses $T$ and $T_{3}$ .

Flavour in particle physics
Flavour quantum numbers
Isospin: I or I₃ Charm: C Strangeness: S Topness: T Bottomness: B′
Related quantum numbers
Baryon number: B Lepton number: L Weak isospin: T or T₃ Electric charge: Q X-charge: X
Combinations
Hypercharge: Y Y = (B + S + C + B′ + T) Y = 2 (Q − I₃) Weak hypercharge: Y_W Y_W = 2 (Q − T₃) X + 2Y_W = 5 (B − L)
Flavour mixing
CKM matrix PMNS matrix Flavour complementarity

Weak isospin

Relation with chirality

Weak isospin and the W bosons

See also

References