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I am learning P&S's Introduction of quantum field theory. My teacher said that if the mass of neutrino is exactly 0, then we should not observe any right-handed neutrinos and left-handed anti-neutrinos according to Weyl's theory. It is because $$ \left( i\gamma^\mu \partial_\mu - m \right) \psi = \left( \begin{array}{cc} -m & i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right) \\ i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right) & -m \end{array} \right) \left( \begin{array}{c} \psi_L \\ \psi_R \end{array} \right) =0. $$

If we set m=0, then we will have:

\begin{align} i\left(\partial_0 - \vec\sigma \cdot \vec\nabla\right) \psi_L &= 0; \\ i\left(\partial_0 + \vec\sigma \cdot \vec\nabla\right) \psi_R &= 0. \end{align}

And if these formulas can describe neutrinos we must conclude that there are two types of neutrinos.

My confusions are:

  1. It is known that the mass of neutrino is not 0, therefore there are chances (depending on the mass) to find right-handed neutrinos and left-handed anti-neutrinos. But we don't.

  2. If I accept that there are only these two types of neutrinos (right-handed anti-neutrinos and left-handed neutrinos), this property of neutrino causes the parity violation, so parity violation must relate to angular momentum and therefore spinor space. Am I write?

  3. Are there any reasons why neutrino is so weird that is has mass but still has helicity + Or - 1. This goes against with what I have learned that only when a particle moves at the speed of light can its spin aligned with its momentum.

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  • $\begingroup$ I am sure you are aware that the general concensus amoung the physics community is that neutrinos are now believed to have mass. en.wikipedia.org/wiki/Neutrino#Mass $\endgroup$ – user108787 Nov 1 '16 at 2:54
  • $\begingroup$ Yeah. But the mass is really small compared to the scale of neutrons. Therefore Weyl's theory is still a good approximation. If we can do experiments at scale of neutrino and the rest mass of neutrino is indeed 0, then, we may find that parity is conserved. $\endgroup$ – ZHANG Juenjie Nov 1 '16 at 3:11
  • $\begingroup$ Parity isn't conserved in the weak interaction, that was clearly shown 60 years ago by Chien-Shiung Wu and her team. And as CountTo10's link says, we have very good evidence that neutrino flavor mixing occurs, and such mixing would be impossible if neutrinos had zero rest mass. Sure, it's a bit disturbing to learn that P isn't conserved, but thankfully CPT symmetry appears to be fundamental. :) $\endgroup$ – PM 2Ring Nov 1 '16 at 5:37
  • $\begingroup$ @PM 2Ring I have restated my questions. Would please take a look? $\endgroup$ – ZHANG Juenjie Nov 1 '16 at 12:37
  • $\begingroup$ Related: physics.stackexchange.com/questions/267035/… $\endgroup$ – tparker Nov 5 '16 at 7:38
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About your questions:

"It is known that the mass of the neutrino is not 0, therefore there are chances (depending on the mass) to find right-handed neutrinos and left-handed anti-neutrinos. But we don't."

Actually no. The way you wrote it down assumes that the mass is a so-called Dirac mass. In that case the left-handed and right-handed neutrinos would be connected via the mass term. Therefore one would need both left- and right-handed neutrinos. However, one can also have a different kind of mass term called a Majorana mass, which does not require the right-handed neutrino.

"If I accept that there are only these two types of neutrinos (right-handed anti-neutrinos and left-handed neutrinos), this property of neutrinos causes the parity violation, so parity violation must relate to angular momentum and therefore spinor space. Am I [right]?"

Parity, as you probably know, is the transformation where something is replaced by its mirror image. So, under parity, a left-handed particle is replaced by a right-handed particle and visa verse. Parity violation implies that nature does not behave the same as its mirror image. However, this does not have anything to do with angular momentum. In other words, the violation of parity does not mean that angular momentum is not conserved.

For the weak interaction, it was found that parity is maximally violated. This is incorporated into the Standard Model in that only the left-handed neutrino couples to the weak force. So the left-handed neutrino, together with the left-handed electron, form a weak doublet, while the right-handed electron is a singlet under the weak force. So if there is a right-handed neutrino, then it is believed that it should be a singlet, otherwise we should have seen it being produced via the weak force in high energy experiments.

Because of this difference in the transformations of left- and right-handed neutrinos under the weak interaction, one cannot represent the mass of the neutrinos as a Dirac mass, because such a Dirac mass term would explicitly break the gauge symmetry associated with the weak interaction. Therefore one needs to use the Majorana mass term to represent the mass of the neutrino in the Standard Model.

"Are there any reasons why neutrinos are so weird that they have mass, but still has helicity + Or - 1. This goes against with what I have learned that only when a particle moves at the speed of light can its spin aligned with its momentum."

Firstly, what is helicity? It is the component of the spin along the direction of the momentum vector, regardless of the mass of the particle. It can either be parallel or anti-parallel to the momentum, which are represented by $+1$ and $-1$, respectively. So a particle with mass can have a helicity of either $+1$ and $-1$, provided it is not a scalar.

But now what about neutrinos? If they are always left-handed, but at the same time have mass, shouldn't one be able to boost to a frame in which the neutrinos are at rest? Well, strictly speak the left-handedness refers to chirality and not to the helicity. These two are not exactly the same thing, because they don't share eigenstates. As a result, although neutrinos have left-handed chirality, they can still be a mixture of both helicities. In general, neutrinos are always observed as fast-moving (relativistic) particles, so that the difference between chirality and helicity is vanishingly small.

Hope these answers address your questions.

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  • $\begingroup$ Thanks. For your second answer: can I conclude that of we could find the same amount of right handed and left handed neutrinos then parity is conserved? For your third answer: I understand that helicity can be 1, but if it is about always to be 1, then we must need the particle to be massless, otherwise we can always boost the particle in the perpendicular direction to change its helicity. $\endgroup$ – ZHANG Juenjie Nov 4 '16 at 13:59
  • $\begingroup$ No, even if there were as many right-handed as left-handed neutrinos, parity would still be violated, because only the left-handed neutrinos feel the weak force. Yes, if we are in the rest frame of a massive particle, then obviously the helicity is not well-defined anymore. Then we just have the spin of the particle which does not require the momentum of the particle. $\endgroup$ – flippiefanus Nov 4 '16 at 14:13
  • $\begingroup$ Yeah. I understand what you said. But for your first sentence, "No, even if there were as many right-handed as left-handed neutrinos, parity would still be violated, because only the left-handed neutrinos feel the weak force. " , are there any reasons why only left-handed neutrinos feel the weak force, any references? $\endgroup$ – ZHANG Juenjie Nov 5 '16 at 4:29
  • $\begingroup$ The helicity of neutrino is always +1. Therefore, we cannot find a reference frame in which the neutrino is at rest. This contradicts with the fact that neutrino has mass. How can you explain this one? BY the way, is Majorana mass an inertia mass? $\endgroup$ – ZHANG Juenjie Nov 5 '16 at 4:34
  • $\begingroup$ I believe that is still an open question. At the moment it is simply an experimental fact, but I wouldn't know where to find a reference to this experimental evidence. There are some models, such as the Pati-Salam model that tries to explain this parity violation as the result of a symmetry breaking process. Don't if there has been recent work done on this. $\endgroup$ – flippiefanus Nov 5 '16 at 4:36
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We don't observe the right handed neutrinos directly because, to good approximation, only left handed neutrinos interact with the weak force, and the weak force is the only mechanism we have observed neutrinos to interact with at all. As far as I know, directly observed right handed neutrinos would require observing it interacting via gravity. We can barely detect really massive black holes colliding via gravity waves, so there's basically no hope there, and I don't see us being able to produce a lump of stationary neutrinos we could put on a scale or feed into a Cavendish type experiment.

That leaves us with avenues that we're already pursuing: experiments with neutrino beams (like MiniBooNE), neutrino detectors (like Super-Kamiokande, SNO, and IceCube), and 'missing energy' experiments like were used to infer the existence of neutrinos in the first place. All of these experiments interact directly only with left handed neutrinos, and infer the existence of right handed neutrinos from neutrino oscillations, and (perhaps someday) a measurement of neutrino mass beyond the current upper limits around 2 eV.

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    $\begingroup$ Don't overlook the neutrinoless double beta decay experiments. If the process were observed it amount to having seen the fluctuation of a Majorana state from one handedness to the other, but would also establish that handedness is matterness or anti-matterness for neutrinos. $\endgroup$ – dmckee Nov 1 '16 at 18:39
  • $\begingroup$ @Sean Lake From your words can I conclude that due to the oscillations we can find right handed neutrinos and hence the mass of left handed neutrino.? But what about parity violation? If there are indeed right handed neutrinos, the parity will not be strictly violated at scale of neutrinos(Even if there is little chance to find right handed neutrinos, but after all there are. ) Does that mean that parity violation is a statistic result, and may not strictly hold for single case? Could you please explain more or provide any references? $\endgroup$ – ZHANG Juenjie Nov 2 '16 at 0:50
  • $\begingroup$ I confess I'm not an expert, so answering your questions would require me to do about as much research as you would have to do. I'm not aware of any theory where left and right handed neutrinos have different masses - as far as I know the default neutrino model is Dirac since Weyl are massless and Majorana don't mix. Parity violation is nearly maximal from the fact that the weak force only interacts with left handed particles. $\endgroup$ – Sean E. Lake Nov 2 '16 at 0:58
  • $\begingroup$ Well, I see. Thank you. Maybe I should wait a bit. :) $\endgroup$ – ZHANG Juenjie Nov 2 '16 at 1:15
  • $\begingroup$ Left-handed neutrinos behave quite differently from right-handed neutrinos. That's parity violation. $\endgroup$ – Peter Shor Nov 3 '16 at 11:27

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