# How can neutrinos have both mass and helicity?

If a neutrino has mass it must travel at less than the speed of light. So how can it possess helicity, which can change depending on relative velocity?

• Have a look at this physics.stackexchange.com/q/1111 – anna v Mar 1 '16 at 10:38
• Good question. If there is a nonvanishing mass, there is the rest frame and therein the helicity does vanish! I think an answer could be related to the fact that the neutrino mass is not defined in quantum sense (its operator does not commute with the Hamiltonian). – Valter Moretti Mar 1 '16 at 12:55
• @annav The link you provide does not actually explain, except to say spin and helicity are related – user56903 Mar 1 '16 at 13:00
• look up chirality en.wikipedia.org/wiki/Chirality_%28physics%29 – anna v Mar 1 '16 at 15:02
• @anna v Yes, but I cannot see the answer to the question there. How can helicity be defined (independent form the reference frame) if the particle has mass? – Valter Moretti Mar 1 '16 at 15:21

Helicity is well-defined for both massive and massless particles, as far as we keep the velocity $v>0$. See, M. Jacob and G. C. Wick, Annals of Physics 281, (2000), 774-799
• The helicity can be defined as $\vec{S}\cdot \vec{P}/P^0$ in both cases. For massive particles it depends on the reference frame and can be reversed. For massless particles it cannot be reversed and is a intrinsic property of the particle. The question is what is the meaning of statements like "the helicity of a neutrino is such" (see,e.g. here en.wikipedia.org/wiki/Neutrino#Mass at the item Chirality) when it is not an intrinsic property. The spin should be the relevant intrinsic property. Otherwise we could speak about helicity of electrons as an intrinsic property of them. – Valter Moretti Mar 1 '16 at 15:29
• The weak interaction of electrons does depend on the helicity. See, for instance, the results of the $G^0$ experiment. The reason that we don't talk about it much is that the weak interactions are dominated by electromagnetic ones in most settings, but neutrinos only interact weakly so their leading interaction shows the oddities of the weak force. – dmckee Mar 1 '16 at 15:31
• @Wen Chern Thanks. The point is that, if I understand well what you are saying, I can always fix a reference frame where $v<0$ reversing the helicity. On the other hand, if the particle is massless, I cannot. In this sense the helicity is intrinsic for massless particles. – Valter Moretti Mar 1 '16 at 15:44