Weak Isospin
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In
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
, weak isospin is a
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
relating to the electrically charged part of the
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
: Particles with half-integer weak isospin can interact with the bosons; particles with zero weak isospin do not. Weak isospin is a construct parallel to the idea of
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
under the
strong interaction In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
. Weak isospin is usually given the symbol or , with the third component written as or is more important than ; typically "weak isospin" is used as short form of the proper term "3rd component of weak isospin". It can be understood as the
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of a charge operator.


Notation

This article uses and for weak isospin and its projection. Regarding ambiguous notation, is also used to represent the 'normal' (strong force)
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
, same for its third component a.k.a. or  . Aggravating the confusion, 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\ ; weak interactions conserve . It is also conserved by the electromagnetic and strong interactions. However, interaction with the
Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
does ''not'' conserve , 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 In quantum field theory, the vacuum expectation value (VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. One of the most widely used exa ...
is nonzero, particles interact with this field all the time, even in vacuum. Interaction with the
Higgs field The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
changes particles' weak isospin (and weak hypercharge). Only a specific combination of electric charge is conserved. The electric charge, \ Q\ , is related to weak isospin, \ T_3\ , and
weak hypercharge In the Standard Model (mathematical formulation), Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently deno ...
, \ Y_\mathrm\ , by :Q = T_3 +\tfracY_\mathrm ~. In 1961 Sheldon Glashow proposed this relation by analogy to the Gell-Mann–Nishijima formula for charge to
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
.


Relation with chirality

Fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s with negative
chirality Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable fro ...
(also called "left-handed" fermions) have \ T = \tfrac\ and can be grouped into doublets with T_3 = \plusmn \tfrac that behave the same way under the
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
. By convention, electrically charged fermions are assigned T_3 with the same sign as their electric charge. For example, up-type
quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
( u, c, t) have \ T_3 = +\tfrac\ and always transform into down-type quarks ( d, s, b), which have \ T_3 = -\tfrac\ , 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
lepton In particle physics, a lepton is an elementary particle of half-integer spin (Spin (physics), spin ) that does not undergo strong interactions. Two main classes of leptons exist: electric charge, charged leptons (also known as the electron-li ...
s, which exist as doublets containing a charged lepton (, , ) with \ T_3 = -\tfrac\ and a
neutrino A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
(, , ) with \ T_3 = +\tfrac ~. In all cases, the corresponding ''anti''-fermion has reversed chirality ("right-handed" antifermion) and reversed sign \ T_3 ~.
Fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s 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. Particles with \ T_3 = 0\ do not interact with ; however, they do all interact with the .


Neutrinos

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\ , so those neutrinos have \ T_3 = +\tfrac ~. Since right-handed antineutrinos are paired with positively charged right-handed anti-leptons with \ T_3 = +\tfrac\ , those antineutrinos are assigned \ T_3 = -\tfrac ~. The same result follows from particle-antiparticle charge & parity reversal, between left-handed neutrinos (\ T_3 = +\tfrac\ ) and right-handed antineutrinos (\ T_3 = -\tfrac\ ).


Weak isospin and the W bosons

The symmetry associated with weak isospin is
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
and requires gauge
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...
s with \, T = 1 \, (, , and ) to mediate transformations between fermions with half-integer weak isospin charges. ''An introduction to quantum field theory'', by M.E. Peskin and D.V. Schroeder (HarperCollins, 1995) ; ''Gauge theory of elementary particle physics'', by T.P. Cheng and L.F. Li (Oxford University Press, 1982) ; ''The quantum theory of fields'' (vol 2), by S. Weinberg (Cambridge University Press, 1996) .\ T = 1\ implies that bosons have three different values of \ T_3 \ : * boson (\,T_3 = +1\,) is emitted in transitions \left(\,T_3 = +\tfrac\,\right)\left(\,T_3 = -\tfrac\,\right)~. * boson (\, T_3 =\, 0 \,) would be emitted in weak interactions where \, T_3 \, does not change, such as
neutrino A neutrino ( ; denoted by the Greek letter ) is an elementary particle that interacts via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass is so small ('' -ino'') that i ...
scattering. * boson (\, T_3 = -1 \,) is emitted in transitions \left(\, T_3 = -\tfrac \,\right)\left(\, T_3 = +\tfrac \,\right). Under electroweak unification, the boson mixes with the
weak hypercharge In the Standard Model (mathematical formulation), Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently deno ...
gauge boson ; both have This results in the observed boson and the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
; the resulting and likewise have zero weak isospin.


See also

*
Weak hypercharge In the Standard Model (mathematical formulation), Standard Model of electroweak interactions of particle physics, the weak hypercharge is a quantum number relating the electric charge and the third component of weak isospin. It is frequently deno ...
* Weak charge * Mathematical formulation of the Standard Model


Footnotes


References

{{Standard model of physics Standard Model Flavour (particle physics) Electroweak theory he:איזוספין חלש