Weak isospin

This article uses {{mvar|T}} and {{mvar|T}}{{sub|3}} for weak isospin and its projection.

Regarding ambiguous notation, {{mvar|I}} is also used to represent the 'normal' (strong force) isospin, same for its third component {{mvar|I}}{{sub|3}} a.k.a. {{mvar|T}}{{sub|3}} or {{mvar|T}}{{sub|z}} . Aggravating the confusion, {{mvar|T}} is also used as the symbol for the Topness quantum number.

The weak isospin conservation law relates to the conservation of $\backslash \; T\_3\backslash \; ;$ weak interactions conserve {{mvar|T}}{{sub|3}}. It is also conserved by the electromagnetic and strong interactions. However, interaction with the Higgs field does not conserve {{mvar|T}}{{sub|3}}, 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, $\backslash \; Q\backslash \; ,$ is related to weak isospin, $\backslash \; T\_3\backslash \; ,$ and weak hypercharge, $\backslash \; Y\_\backslash mathrm\{W\}\backslash \; ,$ by

:$Q\; =\; T\_3\; +\backslash tfrac\{1\}\{2\}Y\_\backslash mathrm\{W\}\; ~.$

In 1961 Sheldon Glashow proposed this relation by analogy to the Gell-Mann–Nishijima formula for charge to isospin.{{Cite journal |last=Glashow |first=Sheldon L. |date=1961-02-01 |title=Partial-symmetries of weak interactions |url=https://dx.doi.org/10.1016/0029-5582%2861%2990469-2 |journal=Nuclear Physics |volume=22 |issue=4 |pages=579–588 |doi=10.1016/0029-5582(61)90469-2 |bibcode=1961NucPh..22..579G |issn=0029-5582|url-access=subscription }}{{Cite book |last1=Greiner |first1=Walter |title=Gauge theory of weak interactions |last2=Müller |first2=Berndt |last3=Greiner |first3=Walter |date=1996 |publisher=Springer |isbn=978-3-540-60227-9 |edition=2 |location=Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo}}{{rp|152}}

Fermions with negative chirality (also called "left-handed" fermions) have $\backslash \; T\; =\; \backslash tfrac\{1\}\{2\}\backslash$ and can be grouped into doublets with $T\_3\; =\; \backslash plusmn\; \backslash tfrac\{1\}\{2\}$ that behave the same way under the weak interaction. By convention, electrically charged fermions are assigned $T\_3$ with the same sign as their electric charge.

For example, up-type quarks (u, c, t) have $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash$ and always transform into down-type quarks (d, s, b), which have $\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash \; ,$ and vice versa. On the other hand, a quark never decays weakly into a quark of the same $\backslash \; T\_3\; ~.$ Something similar happens with left-handed leptons, which exist as doublets containing a charged lepton ({{math|{{SubatomicParticle|link=yes|Electron-}}}}, {{math|{{SubatomicParticle|link=yes|Muon-}}}}, {{math|{{SubatomicParticle|link=yes|Tau-}}}}) with $\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash$ and a neutrino ({{math|{{SubatomicParticle|link=yes|Electron neutrino}}}}, {{math|{{SubatomicParticle|link=yes|Muon neutrino}}}}, {{math|{{SubatomicParticle|link=yes|Tau neutrino}}}}) with $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\; ~.$ In all cases, the corresponding anti-fermion has reversed chirality ("right-handed" antifermion) and reversed sign $\backslash \; T\_3\; ~.$

Fermions with positive chirality ("right-handed" fermions) and anti-fermions with negative chirality ("left-handed" anti-fermions) have $\backslash \; T\; =\; T\_3\; =\; 0\backslash$ and form singlets that do not undergo charged weak interactions.

Particles with $\backslash \; T\_3\; =\; 0\backslash$ do not interact with {{nobr|{{math|{{SubatomicParticle|link=yes|W boson+-}}}} bosons}}; however, they do all interact with the {{nobr|{{math|{{SubatomicParticle|link=yes|Z boson0}}}} boson}}.

Lacking any distinguishing electric charge, neutrinos and antineutrinos are assigned the $\backslash \; T\_3\backslash$ opposite their corresponding charged lepton; hence, all left-handed neutrinos are paired with negatively charged left-handed leptons with $\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash \; ,$ so those neutrinos have $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\; ~.$ Since right-handed antineutrinos are paired with positively charged right-handed anti-leptons with $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash \; ,$ those antineutrinos are assigned $\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\; ~.$ The same result follows from particle-antiparticle charge & parity reversal, between left-handed neutrinos ($\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash$) and right-handed antineutrinos ($\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash$).

The symmetry associated with weak isospin is SU(2) and requires gauge bosons with $\backslash ,\; T\; =\; 1\; \backslash ,$ ({{math| {{SubatomicParticle|W boson+|link=yes}} }}, {{math| {{SubatomicParticle|W boson-|link=yes}} }}, and {{math| {{SubatomicParticle|W boson0|link=yes}} }}) 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) {{ISBN|0-201-50397-2}};

Gauge theory of elementary particle physics, by T.P. Cheng and L.F. Li (Oxford University Press, 1982) {{ISBN|0-19-851961-3}};

The quantum theory of fields (vol 2), by S. Weinberg (Cambridge University Press, 1996) {{ISBN|0-521-55002-5}}.$\backslash \; T\; =\; 1\backslash$ implies that {{math| {{SubatomicParticle|W boson}} }} bosons have three different values of $\backslash \; T\_3\; \backslash \; :$

• {{math| {{SubatomicParticle|W boson+|link=yes}} }} boson $(\backslash ,T\_3\; =\; +1\backslash ,)$ is emitted in transitions $\backslash left(\backslash ,T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash ,\backslash right)$ → $\backslash left(\backslash ,T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash ,\backslash right)~.$
• {{math| {{SubatomicParticle|W boson0|link=yes}} }} boson $(\backslash ,\; T\_3\; =\backslash ,\; 0\; \backslash ,)$ would be emitted in weak interactions where $\backslash ,\; T\_3\; \backslash ,$ does not change, such as neutrino scattering.
• {{math| {{SubatomicParticle|W boson-|link=yes}} }} boson $(\backslash ,\; T\_3\; =\; -1\; \backslash ,)$ is emitted in transitions $\backslash left(\backslash ,\; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\; \backslash ,\backslash right)$ → $\backslash left(\backslash ,\; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\; \backslash ,\backslash right)$.

Under electroweak unification, the {{math| {{SubatomicParticle|W boson0|link=yes}} }} boson mixes with the weak hypercharge gauge boson {{math| {{SubatomicParticle|B boson0}} }}; both have {{nobr|weak isospin {{=}} 0 .}} This results in the observed {{math| {{SubatomicParticle|Z boson0|link=yes}} }} boson and the photon of quantum electrodynamics; the resulting {{math| {{SubatomicParticle|Z boson0|link=yes}} }} and {{math| {{SubatomicParticle|photon0|link=yes}}}} likewise have zero weak isospin.

• Weak hypercharge
• Weak charge
• Mathematical formulation of the Standard Model

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