Chirality (physics)

A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity. Invariance under parity by a Dirac fermion is called chiral symmetry.

An experiment on the weak decay of cobalt-60 nuclei carried out by Chien-Shiung Wu and collaborators in 1957 demonstrated that parity is not a symmetry of the universe.

Chirality and helicity

See also: Helicity (particle physics)

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, with its spin vector defined by the rotation of its hands, 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 massive particles—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 of massless particles is a relativistic invariant (i.e. a quantity whose value is the same in all inertial reference frames).

With the discovery of neutrino oscillation, which implies that neutrinos have mass, the only observed massless particle is the photon. The gluon is also expected to be massless, although the assumption that it is has not been conclusively tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one of them 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.

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 through the operator γ⁵, which has eigenvalues ±1. Any Dirac field can thus be projected into its left- or right-handed component by acting with the projection operators (1−γ⁵)/2 or (1+γ⁵)/2 on 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 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 false paradox is that 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, so there is no frame dependence of the weak interaction: a particle that couples 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 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 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 somewhat different, but most accommodate the chirality of neutrinos in the same way as was already done for all other fermions.

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:

$\psi_L\rightarrow e^{i\theta_L}\psi_L$  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: U(N)_L×U(N)_R.

More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are

$P_R = \frac{1 + \gamma^5}{2}$

and

$P_L = \frac{1 - \gamma^5}{2}$

Massive fermions do not exhibit chiral symmetry, as the mass term in the Lagrangian, m —ψψ, breaks chiral symmetry explicitly.

Spontaneous chiral symmetry breaking may also occur in some theories, as it most notably does in quantum chromodynamics.

The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry. A scalar field model encoding chiral symmetry and its breaking is the sigma model.

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.

Example: u and d quarks in QCD

Consider quantum chromodynamics (QCD) with two massless quarks u and d (massive fermions do not exhibit chiral symmetry). The Lagrangian reads

$\mathcal{L} = \overline{u}\,i\displaystyle{\not}D \,u + \overline{d}\,i\displaystyle{\not}D\, d + \mathcal{L}_\mathrm{gluons}~.$

In terms of left-handed and right-handed spinors, it reads

$\mathcal{L} = \overline{u}_L\,i\displaystyle{\not}D \,u_L + \overline{u}_R\,i\displaystyle{\not}D \,u_R + \overline{d}_L\,i\displaystyle{\not}D \,d_L + \overline{d}_R\,i\displaystyle{\not}D \,d_R + \mathcal{L}_\mathrm{gluons} ~.$

(Here, i is the imaginary unit and $\displaystyle{\not}D$ the Dirac operator.)

Defining

$q = \begin{bmatrix} u \\ d \end{bmatrix} ,$

it can be written as

$\mathcal{L} = \overline{q}_L\,i\displaystyle{\not}D \,q_L + \overline{q}_R\,i\displaystyle{\not}D\, q_R + \mathcal{L}_\mathrm{gluons} ~.$

The Lagrangian is unchanged under a rotation of q_L by any 2 x 2 unitary matrix L, and q_R by any 2 x 2 unitary matrix R.

This symmetry of the Lagrangian is called flavor chiral symmetry, and denoted as U(2)_L×U(2)_R. It decomposes into

$SU(2)_L \times SU(2)_R \times U(1)_V \times U(1)_A ~.$

The singlet vector symmetry, U(1)_V, acts as

$q_L \rightarrow e^{i\theta} q_L \qquad q_R \rightarrow e^{i\theta} q_R ~,$

and corresponds to baryon number conservation.

The singlet axial group U(1)_A acts as

$q_L \rightarrow e^{i\theta} q_L \qquad q_R \rightarrow e^{-i\theta} q_R ~,$

and it does not correspond to a conserved quantity, because it is explicitly violated due to a quantum anomaly.

The remaining chiral symmetry SU(2)_L×SU(2)_R turns out to be spontaneously broken by a quark condensate $\langle \bar{q}^a_R q^b_L \rangle = v \delta^{ab}$ formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup SU(2)_V known as isospin. The Goldstone bosons corresponding to the three broken generators are the three pions. As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this chiral symmetry breaking induces the bulk of hadron masses, such as those for the nucleons−−in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks, SU(2)_L×SU(2)_R is only an approximate symmetry to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.

More Flavors

For more "light" quark species, N flavors in general, the corresponding chiral symmetries are U(N)_L×U(N)_R, decomposing into

$SU(N)_L \times SU(N)_R \times U(1)_V \times U(1)_A ~,$

and exhibiting a very analogous chiral symmetry breaking pattern.

Most usually, N=3 is taken, the u, d, and s quarks taken to be light (the Eightfold way (physics)), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.

An application in 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 conjectured a GUT extension of the weak force which has new, high energy W' and Z' bosons, which now 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 the above SU(2)_W, while B−L is the baryon number minus the lepton number. 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).

See also

• Electroweak theory
• Chirality (chemistry)
• Chirality (mathematics)
• Chiral symmetry breaking
• Handedness
• Spinors and Dirac fields

External links

• To see a summary of the differences and similarities between chirality and helicity (those covered here and more) in chart form, one may go to Pedagogic Aids to Quantum Field Theory and click on the link near the bottom of the page entitled "Chirality and Helicity Summary". To see an in depth discussion of the two with examples, which also shows how chirality and helicity approach the same thing as speed approaches that of light, click the link entitled "Chirality and Helicity in Depth" on the same page.
• History of science: parity violation
• Helicity, Chirality, Mass, and the Higgs (Quantum Diaries blog)
• Chirality vs helicity chart (Robert D. Klauber)