In particle physics, weak isospin is a quantum number relating to the electrically charged part of the weak interaction. Particles with nonzero weak isospin can interact with the W± bosons, while particles with zero weak isospin do not. Weak isospin is a concept 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 . T3 is more important than T; typically "weak isospin" is used as short form of the proper term "3rd component of weak isospin". It can be understood as the eigenvalue of a charge operator.
Notation This article uses T and T3 for weak isospin and its projection. Regarding ambiguous notation, I is also used to represent the 'normal' (strong force) isospin, same for its third component I3 a.k.a. T3 or Tz . Aggravating the confusion, T is also used as the symbol for the Topness quantum number.
Conservation law The weak isospin conservation law relates to the conservation of T 3 ; {\displaystyle \ 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 in propagating fermions, which mix their chiralities by the mass terms that result 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 electric charge is conserved. The electric charge, Q , {\displaystyle \ Q\ ,} is related to weak isospin, T 3 , {\displaystyle \ T_{3}\ ,} and weak hypercharge, Y W , {\displaystyle \ Y_{\mathrm {W} }\ ,} by
Q = T 3 + 1 2 Y W . {\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{\mathrm {W} }~.}
In 1961 Sheldon Glashow proposed this relation by analogy to the Gell-Mann–Nishijima formula for charge to isospin.
Relation with chirality Fermions with negative chirality (also called "left-handed" fermions) have T = 1 2 {\displaystyle \ T={\tfrac {1}{2}}\ } and can be grouped into doublets with T 3 = ± 1 2 {\displaystyle T_{3}=\pm {\tfrac {1}{2}}} that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned T 3 {\displaystyle T_{3}} with the same sign as their electric charge. For example, up-type quarks (u, c, t) have T 3 = + 1 2 {\displaystyle \ T_{3}=+{\tfrac {1}{2}}\ } and always transform into down-type quarks (d, s, b), which have T 3 = − 1 2 , {\displaystyle \ 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 . {\displaystyle \ T_{3}~.} Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton (e−, μ−, τ−) with T 3 = − 1 2 {\displaystyle \ T_{3}=-{\tfrac {1}{2}}\ } and a neutrino (νe, νμ, ντ) with T 3 = + 1 2 . {\displaystyle \ T_{3}=+{\tfrac {1}{2}}~.} In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign T 3 . {\displaystyle \ T_{3}~.}
Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have T = T 3 = 0 {\displaystyle \ T=T_{3}=0\ } and form singlets that do not undergo charged weak interactions. Particles with T 3 = 0 {\displaystyle \ T_{3}=0\ } do not interact with W± bosons; however, they do all interact with the Z0 boson.
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