Editing Chirality (physics) (section)

=== An application in particle physics ===
In [[theoretical physics]], the [[electroweak]] model breaks [[parity (physics)|parity]] maximally. All its [[fermion]]s are chiral [[Weyl fermion]]s, which means that the charged [[W and Z bosons|weak gauge bosons W{{sup|+}} and W{{sup|−}}]] only couple to left-handed quarks and leptons.{{efn|Unlike the W{{sup|+}} and W{{sup|−}} bosons, the neutral electroweak [[W and Z bosons|Z{{sup|0}}&nbsp;boson]] couples to both left ''and'' right-handed fermions, although not equally.}}

Some theorists found this objectionable, and so conjectured a [[Grand unification theory|GUT]] extension of the [[weak force]] which has new, high energy [[W′ and Z′ bosons]], which ''do'' couple with right handed quarks and leptons:
: <math>\frac{ \mathrm{SU}(2)_\text{W}\times \mathrm{U}(1)_Y }{ \mathbb{Z}_2 }</math>
to
: <math>\frac{ \mathrm{SU}(2)_\text{L}\times \mathrm{SU}(2)_\text{R}\times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_2 }.</math>

Here, {{math|SU(2){{sub|L}}}} (pronounced "{{math|SU(2)}} left") is {{math|SU(2){{sub|W}}}} from above, while {{math|''[[B−L]]''}} is the [[baryon number]] minus the [[lepton number]]. The electric charge formula in this model is given by
: <math>Q = T_{\rm 3L} + T_{\rm 3R} + \frac{B-L}{2}\,;</math>
where <math>\ T_{\rm 3L}\ </math> and <math>\ T_{\rm 3R}\ </math> are the left and right [[weak isospin]] values of the fields in the theory.

There is also the [[chromodynamic]] {{math|SU(3){{sub|C}}}}. The idea was to restore parity by introducing a '''left-right symmetry'''. This is a [[group extension]] of <math> \mathbb{Z}_2 </math> (the left-right symmetry) by
: <math>\frac{ \mathrm{SU}(3)_\text{C}\times \mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_6}</math>
to the [[semidirect product]]
: <math>\frac{ \mathrm{SU}(3)_\text{C} \times \mathrm{SU}(2)_\text{L} \times \mathrm{SU}(2)_\text{R} \times \mathrm{U}(1)_{B-L} }{ \mathbb{Z}_6 } \rtimes \mathbb{Z}_2\ .</math>

This has two [[connected space|connected component]]s where <math> \mathbb{Z}_2 </math> acts as an [[automorphism]], which is the composition of an [[Involution (mathematics)|involutive]] [[outer automorphism]] of {{math|SU(3){{sub|C}}}} with the interchange of the left and right copies of {{math|SU(2)}} with the reversal of {{math|U(1){{sub|''B−L''}}}}. It was shown by [[Rabindra Mohapatra|Mohapatra]] & [[Goran Senjanovic|Senjanovic]] (1975)<ref>{{cite journal |author1-link=Goran Senjanovic |first1=Goran |last1=Senjanovic |author2-link=Rabindra Mohapatra |first2=Rabindra N. |last2=Mohapatra |year=1975 |title=Exact left-right symmetry and spontaneous violation of parity |journal=[[Physical Review D]] |volume=12 |issue=5 |page=1502 |doi=10.1103/PhysRevD.12.1502 |bibcode=1975PhRvD..12.1502S }}</ref> that [[left-right symmetry]] can be [[spontaneous symmetry breaking|spontaneously broken]] to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam, and also connects the small observed neutrino masses to the breaking of left-right symmetry via the [[seesaw mechanism]].

In this setting, the chiral [[quark]]s
: <math>(3,2,1)_{+{1 \over 3}}</math>
and
: <math>\left(\bar{3},1,2\right)_{-{1 \over 3}}</math>
are unified into an [[irreducible representation]] ("irrep")
: <math>(3,2,1)_{+{1 \over 3}} \oplus \left(\bar{3},1,2\right)_{-{1 \over 3}}\ .</math>

The [[lepton]]s are also unified into an [[irreducible representation]]
: <math>(1,2,1)_{-1} \oplus (1,1,2)_{+1}\ .</math>

The [[Higgs boson]]s needed to implement the breaking of left-right symmetry down to the Standard Model are
: <math>(1,3,1)_2 \oplus (1,1,3)_2\ .</math>

This then provides three [[sterile neutrino]]s which are perfectly consistent with {{As of|2005|alt=current}} [[neutrino oscillation]] data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.

Because the left–right symmetry is spontaneously broken, left–right models predict [[Domain wall (string theory)|domain wall]]s. This left-right symmetry idea first appeared in the [[Pati–Salam model]] (1974)<ref>{{cite journal |last1=Pati |first1=Jogesh C. |last2=Salam |first2=Abdus |date=1 June 1974 |title=Lepton number as the fourth "color" |journal=[[Physical Review D]] |volume=10 |issue=1 |pages=275–289 |doi=10.1103/physrevd.10.275 |bibcode=1974PhRvD..10..275P }}</ref> and Mohapatra–Pati models (1975).<ref>{{cite journal |last1 = Mohapatra |first1 = R.N. |last2 = Pati |first2 = J.C. |year = 1975 |title = 'Natural' left-right symmetry |journal = Physical Review D |volume = 11 |issue = 9 |pages = 2558–2561 |doi = 10.1103/PhysRevD.11.2558 |bibcode = 1975PhRvD..11.2558M}}</ref>