Helicity and Chirality

For most particles, the distinction between helicity and chirality can be ignored. For neutrinos — which are very light and interact only through the weak force — it is central.

Two different projections of spin

Helicity is the projection of a particle’s spin onto its momentum: $h=\hat{\mathbf{s}}\cdot \hat{\mathbf{p}}$ A spin-½ particle has $h=+1/2$ (right-handed) or $h=-1/2$ (left-handed). For a massive particle, helicity is frame-dependent: a faster observer can overtake the particle and see its momentum reversed. Helicity is conserved in the absence of interactions but is not a Lorentz invariant.

Chirality is an intrinsic property defined by the action of the ${\gamma }^5$ matrix in the Dirac equation. Left- and right-chiral projections are ${\psi }_L=\frac{1-{\gamma }^5}{2}\psi ,{\psi }_R=\frac{1+{\gamma }^5}{2}\psi$ Chirality is Lorentz-invariant but is not conserved by the mass term in the Dirac Lagrangian: $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ A massive particle propagating through space continually mixes its left- and right-chiral components.

For a massless fermion, the two concepts coincide. For a massive fermion, a boost can flip helicity but not chirality.

Weak interactions couple only to left-chiral fermions

The weak force is uniquely chiral: the charged-current interaction involves only the left-chiral components of fermions and the right-chiral components of antifermions. The interaction term has the form ${\mathcal{L}}_{CC}\propto {\bar{\nu }}_L{\gamma }^{\mu }{e}_L{W}_{\mu }^{+}+\text{h.c.}$ This is the famous V − A structure of the weak interaction, established by Madame Chien-Shiung Wu’s 1956 experiment on parity violation in ${}^{60}$Co decay.

Because neutrinos in the Standard Model interact only weakly, only the left-chiral component ${\nu }_L$ has observable effects. The right-chiral component, if it exists, is sterile with respect to all known gauge interactions.

The Goldhaber experiment

In 1958, Maurice Goldhaber, Lee Grodzins, and Andrew Sunyar measured the helicity of the neutrino directly using electron capture in ${}^{152m}$Eu. The resulting ${}^{152}$Sm nucleus emits a photon, whose circular polarization carries away the neutrino’s helicity information. The measurement found neutrinos to be entirely left-helical within experimental error — one of the cleanest demonstrations of maximal parity violation in the weak sector.

Where neutrino mass enters

If neutrinos have mass, they cannot be purely left-chiral in the mathematical sense. A massive particle propagating at less than the speed of light has both chirality components. The small right-chiral component is however suppressed by a factor $m_{\nu }/E$. At typical neutrino energies of MeV–GeV and masses of sub-eV, this suppression is of order $10^{-6}$ or smaller — negligible for practical purposes. What experiments detect is effectively left-handed.

The tiny right-chiral content is not just a mathematical curiosity. If neutrinos are Majorana particles, the right-chiral component is the charge-conjugate of the left-chiral one, and the mass term mixes the two into a single self-conjugate state. Neutrinoless double beta decay would then be possible: a virtual right-chiral neutrino, produced at one vertex, can propagate, flip helicity through the mass insertion, and be absorbed as a left-chiral antineutrino at a second vertex. The amplitude is therefore suppressed by $m_{\nu }/Q$ where $Q$ is the energy transfer.

Summary

At the energies of neutrino experiments, the distinction between a left-chiral and a left-helical neutrino is irrelevant. At the level of fundamental theory — particularly for questions of Dirac vs. Majorana nature and neutrinoless double beta decay — the distinction is essential.

Frequently asked

Is helicity the same as chirality?

Only in the massless limit. Helicity is the projection of spin onto momentum (frame-dependent). Chirality is the eigenvalue of the γ₅ operator and is Lorentz-invariant. For a massless fermion the two coincide; for a massive fermion they do not.