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I know electrons have helicity but can particles which are not fundamental i.e.protons have helicity?

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Helicity of a particle is defined to be the projection of its spin vector $\mathbf{s}$ along the direction of its impulse $\mathbf{p}$:

$$H = \frac{\mathbf{s}\cdot\mathbf{p}}{||\mathbf{s}\cdot \mathbf{p}||}$$

  • For a massless particle, the helicity is equivalent to the chirality;
  • If the mass of the particle is not zero, then helicity is not a Lorentz invariant;
  • Neutrino helicity is $-1$

So every particle with spin $\neq 0$ and/or $m \neq 0$ has helicity.

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Yes. Helicity is nothing but the projection of the spin of the particle along the direction of the momentum. Protons have spin as well as the other composite particles.

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