Editing Chirality (physics) (section)

== Chiral theories ==
Particle physicists have only observed or inferred left-chiral [[fermion]]s and right-chiral antifermions engaging in the [[weak force|charged weak interaction]].<ref>{{cite book |author1=Povh, Bogdan |author2=Rith, Klaus |author3=Scholz, Christoph |author4=Zetsche, Frank |title=Particles and Nuclei: An introduction to the physical concepts |publisher=Springer |year=2006 |page=145 |isbn=978-3-540-36683-6}}</ref> In the case of the weak interaction, which can in principle engage with both left- and right-chiral fermions, only two left-handed [[fermion]]s interact. Interactions involving right-handed or opposite-handed fermions have not been shown to occur, implying that the universe has a preference for left-handed chirality. This preferential treatment of one chiral realization over another violates parity, as first noted by [[Chien Shiung Wu]] in her famous experiment known as the [[Wu experiment]]. This is a striking observation, since parity is a symmetry that holds for all other [[fundamental interaction]]s.

Chirality for a [[Fermionic field#Dirac fields|Dirac fermion]] {{mvar|ψ}} is defined through the [[Gamma matrices#The fifth "gamma" matrix, γ5|operator {{math|''γ''<sup>5</sup>}}]], which has [[eigenvalue, eigenvector, and eigenspace|eigenvalue]]s ±1; the eigenvalue's sign is equal to the particle's chirality: +1 for right-handed, −1 for left-handed. Any Dirac field can thus be projected into its left- or right-handed component by acting with the [[Projection (linear algebra)|projection operators]] {{math|{{sfrac|1|2}}(1 − ''γ''<sup>5</sup>)}} or {{math|{{sfrac|1|2}}(1 + ''γ''<sup>5</sup>)}} on {{mvar|ψ}}.

The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's [[parity (physics)|parity symmetry]] violation.

A common source of confusion is due to conflating the {{math|''γ''<sup>5</sup>}}, chirality operator with the [[helicity (particle physics)|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 {{Em|the chirality operator is equivalent to helicity for massless fields only}}, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, or, alternatively,  helicity is not Lorentz invariant, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.

A theory that is asymmetric with respect to chiralities is called a '''chiral theory''', while a non-chiral (i.e., parity-symmetric) theory is sometimes called a '''vector theory'''. Many pieces of the [[Standard Model]] of physics are non-chiral, which is traceable to  [[Anomaly (physics)|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 to gluons in the same way.

The [[electroweak theory]], developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed that [[neutrino#Mass|neutrinos were massless]], and assumed the existence of only left-handed [[neutrino]]s and right-handed antineutrinos. After the observation of [[neutrino oscillation]]s, which implies that no fewer than two of the three [[neutrino#Mass|neutrinos are massive]], the revised [[electroweak theory|theories of the electroweak interaction]] now include both right- and left-handed [[neutrino]]s. 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 theory|electroweak theories]] that have been proposed are somewhat different, but most accommodate the chirality of [[neutrino]]s in the same way as was already done for all other [[fermions]].