Neutrinos, however, have some kind of inherent helicity called chirality. But they can have either helicity. How is chirality different from helicity? At first glance, chirality and helicity seem to have no relationship to each other. Helicity, as you said, is whether the spin is aligned or anti aligned with the momentum. Chirality is like your left hand versus your right hand. Its just a property that makes them different than each other, but in a way that is reversed through a mirror imaging - your left hand looks just like your right hand if you look at it in a mirror and vice-versa.
If you do out the math though, you find out that they are linked. Helicity is not an inherent property of a particle because of relativity. Suppose you have some massive particle with spin. In one frame the momentum could be aligned with the spin, but you could just boost to a frame where the momentum was pointing the other direction boost meaning looking from a frame moving with respect to the original frame.
But if the particle is massless, it will travel at the speed of light, and so you can't boost past it. So you can't flip its helicity by changing frames. In this case, if it is "chiral right-handed", it will have right-handed helicity. If it is "chiral left-handed", it will have left-handed helicity. So chirality in the end has something to do with the natural helicity in the massless limit. Note that chirality is not just a property of neutrinos. It is important for neutrinos because it is not known whether both chiralities exist.
It is possible that only left-handed neutrinos and only right-handed antineutrinos exist. The helicity of a particle is the normalized projection of the spin on the direction of momentum. If the spin is more along the same direction of the momentum than against it, then the helicity is positive; otherwise it is negative. Chirality is to do with the way the particle's properties transform when they are described with respect to one inertial reference frame or another.
The difference between right-handed and left-handed is like the difference between contravariant and covariant 4-vectors, but now we are talking about spinors. For a massless spin half particle, the spin and momentum can both be extracted from a single spinor. When one transforms from one frame to another, one should use the ordinary Lorentz transformation for a right-handed spinor, and the inverse Lorentz transformation for a left-handed spinor.
Thus chirality is an intrinsic property of such a particle, but one whose influence is only revealed in this subtle way. It influences how the spinor enters into the Weyl equation, for example. Massive spin-half particles such as electrons have their spin and momentum described by Dirac spinors which are made of two Weyl spinors, one of each chirality.
The helicity of a particle is the normalized projection of the spin on the direction of momentum. If the spin is more along the same direction of the momentum than against it, then the helicity is positive; otherwise it is negative. Chirality is to do with the way the particle's properties transform when they are described with respect to one inertial reference frame or another. The difference between right-handed and left-handed is like the difference between contravariant and covariant 4-vectors, but now we are talking about spinors.
For a massless spin half particle, the spin and momentum can both be extracted from a single spinor. When one transforms from one frame to another, one should use the ordinary Lorentz transformation for a right-handed spinor, and the inverse Lorentz transformation for a left-handed spinor.
Thus chirality is an intrinsic property of such a particle, but one whose influence is only revealed in this subtle way. It influences how the spinor enters into the Weyl equation, for example.
Massive spin-half particles such as electrons have their spin and momentum described by Dirac spinors which are made of two Weyl spinors, one of each chirality. What distinguishes a neutrino treated here as massless from an anti-neutrino is primarily its chirality. But whenever just a single 2-component spinor describes both the momentum and the spin, one finds that the helicity for such a particle can only take one value and for the antiparticle it takes the opposite value.
Thus the helicity and the chirality then have the same value, but it does not mean they are the same thing. When a given type of particle can only have one helicity, one has a situation that does not respect parity mirror-reflection symmetry. This is at the heart of the breaking of parity invarience by the weak force. For those that want to learn the two-component formalism in gory detail, I strongly recommend the recent review by Dreiner, Haber, and Martin. We have now learned that masses are responsible for mixing between different types of particles.
The mass terms combine two a priori particles electron and anti-positron into a single particle physical electron. The reason for this is that particles can only mix if they carry the same quantum properties. It would have been impossible for the electron and anti-electron to mix because they have different electric charges.
However, the electron carries a weak charge because it couples to the W boson, whereas the anti-positron carries no weak charge. Thus these two particles should not be able to mix. This is a consequence of the Standard Model being a chiral theory. The reason why this unlikely mixing is allowed is because of the Higgs vev.
The Higgs carries weak charge. So now the mystery of the Higgs boson continues. First we said that the Higgs somehow gives particle masses.
We then said that these masses are generated by the Higgs vacuum expectation value. In this post we took a detour to explain what this mass really does and got a glimpse of why the Higgs vev was necessary to allow this mass.
This phenomenon is called electroweak symmetry breaking , and is one of the primary motivations for theories of new physics beyond the Standard Model. Okay, this is somewhat outside of our main discussion, but I feel obligated to mention it. The kind of fermion mass that we discussed above is called a Dirac mass.
This is a type of mass that connects two different particles electron and anti-positron. It is also possible to have a mass that connects two of the same kind of particle, this is called a Majorana mass. This type of mass is forbidden for particles that have any type of charge.
For example, an electron and an anti-electron cannot mix because they have opposite electric charge, as we discussed above. There is, however, one type of matter particle in the Standard Model which does not carry any charge: the neutrino!
Within the Standard Model, Majorana masses are special for neutrinos. It is also possible for the neutrino to have both a Majorana and a Dirac mass. This would have some interesting consequences. Majorana masses, on the other hand, do not cause any charge non-conservation and can be arbitrarily large. Quantum Diaries Thoughts on work and life from particle physicists from around the world. Ricky Nathvani. Andrea Signori Nikhef Netherlands.
Richard Ruiz Univ. Steven Goldfarb University of Michigan. Mandeep Gill. Alex Millar University of Melbourne Australia. Fermilab budget woes continue ».
Helicity Fact: every matter particle electrons, quarks, etc. To be clear, we can also draw the right-handed particle moving in the opposite direction to the left : Note that the direction of the spin the red arrow also had to change. This is precisely the reason why the person staring back at you in the mirror is left-handed if you are right-handed! Some technical remarks : The broad procedure being outlined in the last two sections can be understood in terms of group theory.
Technical remark: the difference between chirality and helicity is one of the very subtle points when one is first learning field theory. The mathematical difference can be seen just by looking at the form of the helicity and chirality operators. Intuitively, helicity is something which can be directly measured by looking at angular momentum whereas chirality is associated with the transformation under the Lorentz group e.
In order to conserve angular momentum, the helicities of the e L and e R have to match. This means that one of these particles has opposite helicity and chirality—and this is the whole point of distinguishing helicity from chirality!
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