# Left–right symmetry

Left–right symmetry is a general principle in physics which holds that valid physical laws must not produce a different result for a motion that is left-handed than motion that is right-handed. The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.

The general principle is often referred to by the name chiral symmetry. The rule is absolutely valid in the classical mechanics of Newton and Einstein, but results from quantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiral subatomic particles.

## Particle Physics

In theoretical physics, the electroweak model breaks parity maximally. All its fermions are chiral Weyl fermions, which means that the charged weak gauge bosons only couple to left-handed quarks and leptons. (Note that the neutral electroweak Z boson already couples to left and right-handed fermions.) Some theorists found this objectionable, and so proposed a GUT extension of the weak force which has new, high energy W' and Z' bosons which couple with right handed quarks and leptons.

${[SU(2)_W\times U(1)_Y]\over \mathbb{Z}_2}$

to

${SU(2)_L\times SU(2)_R\times U(1)_{B-L}\over \mathbb{Z}_2}.$

Here, SU(2)L (pronounced SU(2) left) is none other than SU(2)W and B−L is the baryon number minus the lepton number. An advantage of this model over the Standard Model is that the electric charge formula in this model is given by

$Q= I_{3L}+I_{3R}+\frac{B-L}{2}$;

where $\! I_{3L,R}$ are the weak isospin values of the fields in the theory.

There is also the chromodynamic SU(3)C. The idea was to restore parity by introducing a left-right symmetry. This is a group extension of Z2 (the left-right symmetry) by

${SU(3)_C\times SU(2)_L \times SU(2)_R \times U(1)_{B-L}\over \mathbb{Z}_6}$

to the semidirect product

${SU(3)_C\times SU(2)_L \times SU(2)_R \times U(1)_{B-L}\over \mathbb{Z}_6} \rtimes \mathbb{Z}_2.$

This has two connected components where Z2 acts as an automorphism, which is the composition of an involutive outer automorphism of SU(3)C with the interchange of the left and right copies of SU(2) with the reversal of U(1)B−L. It was shown by Rabindra N. Mohapatra and Goran Senjanovic in 1975 that left-right symmetry can be spontaneously broken to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg and Salam and it also connects the small observed neutrino masses to the breaking of left-right symmetry via the seesaw mechanism.

In this setting the chiral quarks

(3,2,1)1/3

and

$(\bar{3},1,2)_{-{1\over 3}}$

are unified into an irrep

$(3,2,1)_{1\over 3}\oplus(\bar{3},1,2)_{-{1\over 3}}.$

The leptons are also unified into an irrep

$(1,2,1)_{-1}\oplus(1,1,2)_1.$

The Higgs bosons needed to implement the breaking of left-right symmetry down to the Standard Model are

$(1,3,1)_{2}\oplus(1,1,3)_{2}.$

This then predicts three sterile neutrinos, which is perfectly consistent with 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 walls.

This left-right symmetry idea first appeared in the Pati–Salam model (1974), Mohapatra–Pati models (1975) and later in trinification (1984).