Chirality (physics)
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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.
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[edit] Chirality and helicity
The helicity 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 helicity. Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: Left is negative, Right is positive.
The chirality of a particle is more abstract. It is determined by whether the particle transforms in a right or left handed representation of the Poincaré group. (However, some representations, such as Dirac spinors, have both right and left handed components. In cases like this, we can define projection operators that project out either the right or left hand components and discuss the right and left handed portions of the representation.)
For massless particles — such as the photon, the gluon, and the (hypothetical) graviton — chirality is the same 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.
For particles that do have mass — such as electrons, quarks, and neutrinos — chirality and helicity must be distinguished. 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 helicity (which may be thought of as 'apparent chirality') will be reversed.
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 (helicity) is fixed for all reference frames: the helicity is a relativistic invariant.
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 the assumption that it is massless has not been tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one that has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity 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. 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 glueballs.
[edit] Chiral theories
It has been observed that 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. Experiments sensitive to this effect imply 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 twentieth 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 anti-neutrinos). 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.
[edit] Chiral symmetry
Vector gauge theories with massless Dirac fermion fields ψ 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:
and 
or
and 
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, 
breaks chiral symmetry explicitly. Spontaneous chiral symmetry breaking may also occur in some theories, most notably in quantum chromodynamics.
[edit] References and external links
- D.A. Bromley (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.
- Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.
[edit] Summary and comparison chart of chirality vs helicity
To see an explanation and summary of the differences and similarities between chirality and helicity (those covered here and more) in chart form, go to [1] and click on the appropriate link near the bottom of the page. Please note the discussion page herein, as the chart agrees with what is presented there.
