Personal Sandbox for Chirality article

A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality). The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity. The action of parity acting on a Dirac fermion is called chiral symmetry.

An experiment on the weak decay of cobalt in 1956 showed that parity is not a symmetry of the universe.

Absolute and Relative Chirality

The chirality of a particle is Right-handed if the direction of its spin is the same as the direction of its motion. It is Left-handed if the directions of spin and motion are opposite. By convention for rotation, a standard clock, tossed with its face directed forwards, has Left-handed chirality. Mathematically, chirality is the sign of the projection of the spin vector onto the momentum vector: Left is negative, Right is positive.

Massless particles — such as the photon, the gluon, and the (hypothetical) graviton — have absolute chirality: (better known as helicity) a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

Particles which do have mass — such as electrons, quarks, and neutrinos — have relative chirality, which depends on the observer’s reference frame. In the case of these particles, it is possible for an observer to change to a reference frame that overtakes the spinning particle, in which case the particle will then appear to move backwards, and its apparent chirality will reverse.

A massless particle moves with the speed of light, so a real observer (who must always travel at less than the speed of light) cannot be in any reference frame where the particle appears to reverse its relative direction, meaning that all real observers see the same chirality. Because of this, the direction of spin of massless particles is not affected by a Lorentz boost (change of viewpoint) in the direction of motion of the particle, and the sign of the projection (chirality) is fixed for all reference frames: the chirality is absolute.

With the discovery of neutrino oscillations, which imply that neutrinos have mass, the only observed massless particle is the photon. The gluon also is expected to be massless (although its mass has not been measured). Hence, these are the only two particles now known with absolute chirality. All other observed particles have mass and so can only have relative chirality. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have absolute chirality like the photon. It is also not known for certain that the gluon is actually massless, it is only supposed; all that is certain from measurement is that if it is not zero then its mass must be very small. And because of confinement, observation of gluons is complicated and difficult; it may be that they cannot exist as a free particle and only come in bound states called a glueball.

Helicity

In particle physics, helicity is the projection of the angular momentum to the direction of motion:

h={\vec {J}}\cdot {\hat {p}},\qquad {\hat {p}}={\vec {p}}/|{\vec {p}}|

Because the angular momentum with respect to an axis has discrete values, helicity is discrete, too. For spin-1/2 particles such as the electron, the helicity can either be positive ( $+\hbar /2$ ) - the particle is then "right-handed" - or negative ( $-\hbar /2$ ) - the particle is then "left-handed".

For massless (or extremely light) spin-1/2 particles, helicity is equivalent to the operator of chirality multiplied by $\hbar /2$ .

Chiral Theories

A so-far perplexing property observed for the weak interaction has been that the strength of the interaction is different for left- and right-handed fermions, even though chirality for these particles is not an absolute or universal symmetry. In most circumstances, two left-handed fermions will interact more strongly than right-handed or different-handed fermions. Experiments which show this effect imply that the universe has an otherwise unexplained preference for left-handed chirality. In order to conform to the observed interaction rates and cross-sections, all theories of the electroweak interaction treat left- and right-handed chiralities unequally.

Chirality for a Dirac field, represented by the wave function ψ, is defined to be one of the two eigenvalues of the operator γ⁵: either +1 or –1. Any Dirac field can therefore be projected into its left- or right-handed component by the operation of the composite operator (1–γ⁵)/2 or (1+γ⁵)/2 acting on ψ, so combinations of the operator γ⁵ frequently are introduced into the formulas of particle theories to create chiral bias in the results of the calculations.

Both chiralities of a particle may appear in a theory, in which case the theory is called a vector theory. If only one chirality appears in a theory, then it is called a chiral theory.

Quantum chromodynamics is an example of a vector theory since both chiralities of all quarks appear in the theory.

The version of the electroweak theory developed in the mid twentieth century was an example of a chiral theory. It assumed that neutrinos were massless, hence had absolute chirality, and only accepted the existence of left-handed neutrinos (along with their complementary right-handed anti-neutrinos). After the observation of neutrino oscillations, which strongly implies that neutrinos have some mass and have relative chirality like all other fermions, the revised theories of the electroweak interaction now include both right- and left-handed neutrinos, and none is any longer a chiral theory.

The exact nature of the neutrino is still unsettled and so the many 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. The consequence of these changes to adapt to the discovery of neutrino mass is that no current theory of particle physics is a "chiral theory" in the sense used above.

Chiral symmetry

Vector gauge theories with massless Dirac fermion fields $\psi$ exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:

\psi _{L}\rightarrow e^{i\theta _{L}}\psi _{L}

and

\psi _{R}\rightarrow \psi _{R}

or

\psi _{L}\rightarrow \psi _{L}

and

\psi _{R}\rightarrow e^{i\theta _{R}}\psi _{R}

.

With N flavors, we have unitary rotations instead: SU(N)_L×SU(N)_R.

Massive fermions do not exhibit chiral symmetry. One also says that the mass term in the Lagrangian, $m{\overline {\psi }}\psi$ breaks chiral symmetry explicitly. Spontaneous chiral symmetry breaking may also occur in some theories, most notably in quantum chromodynamics.