Chirality (physics) - Chiral Theories

Chiral Theories

Only left-handed fermions interact with the weak interaction. In most circumstances, two left-handed fermions interact more strongly than right-handed or opposite-handed fermions implying that the universe has a preference for left-handed chirality, which violates a symmetry of the other forces of nature.

Chirality for a Dirac fermion ψ is defined by the operator γ5, which has eigenvalues ±1. Any Dirac field can therefore be projected into its left- or right-handed component by the operation of the projection operator (1–γ5)/2 or (1+γ5)/2 acting on ψ. The coupling of the weak interaction to fermions is proportional to such a projection operator, which is responsible for its parity symmetry violation.

A common source of confusion is due to conflating this operator with the helicity operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is that the chirality operator is equivalent to helicity for massless fields only, for which helicity is not frame-dependent. For massive particles, chirality is not the same as helicity so there is no frame dependence of the weak interaction: a particle that interacts with the weak force does so in every frame.

A theory that is asymmetric between chiralities is called a chiral theory, while a parity symmetric theory is sometimes called a vector theory. Most pieces of the Standard Model of physics are non-chiral, which may be due to problems of anomaly cancellation in chiral theories. Quantum chromodynamics is an example of a vector theory since both chiralities of all quarks appear in the theory, and couple the same way.

The electroweak theory developed in the mid 20th century is an example of a chiral theory. Originally, it assumed that neutrinos were massless, and only assumed the existence of left-handed neutrinos (along with their complementary right-handed antineutrinos). After the observation of neutrino oscillations, which imply that neutrinos are massive like all other fermions, the revised theories of the electroweak interaction now include both right- and left-handed neutrinos. However, it is still a chiral theory, as it does not respect parity symmetry.

The exact nature of the neutrino is still unsettled and so the electroweak theories that have been proposed are different, but most accommodate the chirality of neutrinos in the same way as was already done for all other fermions.