Weak isospin

 ( Standard Model ) 

In particle physics, weak isospin is a quantum number relating to the electrically charged part of the weak interaction: 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 under the strong interaction. 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 of a charge operator.

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Notation This article uses and for weak isospin and its projection. Regarding ambiguous notation, is also used to represent the 'normal' (strong force) isospin, same for its third component a.k.a. or  . Aggravating the confusion, is also used as the symbol for the Topness quantum number.

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Conservation law The weak isospin conservation law relates to the conservation of $\backslash \; T\_3\backslash \; ;$ weak interactions conservation law . It is also conserved by the electromagnetic and strong interactions. However, interaction with the Higgs field 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 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.

(1996). 9783540602279, Springer. ISBN 9783540602279

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Relation with chirality 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 (up quark, charm quark, top quark) have $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash$ and always transform into down-type quarks (down quark, strange quark, bottom quark), 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 , which exist as doublets containing a charged lepton (, , ) with $\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash$ and a neutrino (, , ) with $\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\; ~.$ In all cases, the corresponding antiparticle has reversed chirality ("right-handed" antifermion) and reversed sign $\backslash \; T\_3\; ~.$

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 ; however, they do all interact with the .

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Neutrinos 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 CPT symmetry, between left-handed neutrinos ($\backslash \; T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash$) and right-handed antineutrinos ($\backslash \; T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash$).

+ Left-handed fermions in the Standard Model
Electron	$\backslash \; -\backslash !1~$	$\backslash quad\; \backslash mathrm\{e\}^-\backslash$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$	Muon	$\backslash \; -\backslash !1~$	$\backslash quad\; \backslash mathrm\{\backslash mu\}^-\backslash$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$	Tauon	$\backslash \; -\backslash !1~$	$\backslash quad\; \backslash mathrm\{\backslash tau\}^-~$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$
Up quark	$\backslash \; +\backslash !\backslash tfrac\{2\}\{3\}~$	$\backslash \; \backslash mathrm\{u\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$	Charm quark	$\backslash \; +\backslash !\backslash tfrac\{2\}\{3\}~$	$\backslash \; \backslash mathrm\{c\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$	Top quark	$\backslash \; +\backslash !\backslash tfrac\{2\}\{3\}~$	$\backslash \; \backslash mathrm\{t\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$
Down quark	$\backslash \; -\backslash !\backslash tfrac\{1\}\{3\}~$	$\backslash \; \backslash mathrm\{d\}\backslash$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$	Strange quark	$\backslash \; -\backslash !\backslash tfrac\{1\}\{3\}~$	$\backslash \; \backslash mathrm\{s\}\backslash$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$	Bottom quark	$\backslash \; -\backslash !\backslash tfrac\{1\}\{3\}~$	$\backslash \; \backslash mathrm\{b\}\backslash$	$\backslash \; -\backslash !\backslash tfrac\{1\}\{2\}~$
Electron neutrino	$\backslash \; \backslash quad\; 0~$	$\backslash \; ~\backslash nu\_\backslash mathrm\{e\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$	Muon neutrino	$\backslash \; \backslash quad\; 0~$	$\backslash \; ~\backslash nu\_\backslash mathrm\{\backslash mu\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$	Tau neutrino	$\backslash \; \backslash quad\; 0~$	$\backslash \; ~\backslash nu\_\backslash mathrm\{\backslash tau\}\backslash$	$\backslash \; +\backslash !\backslash tfrac\{1\}\{2\}~$

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Weak isospin and the W bosons The symmetry associated with weak isospin is SU(2) and requires gauge with $\backslash ,\; T\; =\; 1\; \backslash ,$ (, , 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) .$\backslash \; T\; =\; 1\backslash$ implies that bosons have three different values of $\backslash \; T\_3\; \backslash \; :$

• boson $(\backslash ,T\_3\; =\; +1\backslash ,)$ is emitted in transitions $\backslash left(\backslash ,T\_3\; =\; +\backslash tfrac\{1\}\{2\}\backslash ,\backslash right)$ → $\backslash left(\backslash ,T\_3\; =\; -\backslash tfrac\{1\}\{2\}\backslash ,\backslash right)~.$
• 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.
• 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 boson mixes with the weak hypercharge gauge boson ; both have This results in the observed boson and the photon of quantum electrodynamics; the resulting and likewise have zero weak isospin.

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See also

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

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Footnotes

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