### Spinors, Chirality, and Majorana Mass

04Jan08

When I started learning field theory out of the canonical texts, I used to be very confused about the origin of spinors. [1] It wasn’t until I took a course in representation theory for physicists that I was able to appreciate the representations of the Poincare group. [2]

Representations of the Lorentz Group

In a few words, the point is that particles are irreducible representations of the Poincare group—the group of Lorentz transformations, rotations, and translations under which nature is symmetric. In a nutshell, a representation‘ is a set of particles that transform into one another under a symmetry. [3]

These representations of the Poincare group are classified by spin in the usual half-integer denominations: 0, 1/2, 1, etc. The meat of a representation theory course involves quantifying the transformation properties of these objects by constructing isomorphisms between this group and the special unitary groups to quantify these representations, but I’ll leave this to the reader to explore. (See references in [2].)

Spinors, in particular, are the spin-1/2 representation of the Poincare (or Lorentz) group. There are two types: the fundamental (left-handed) and the anti-fundamental (right-handed). This is no surprise to students of field theory, even those without any representation theory background; we understand that a 4-component Dirac spinor is just a direct sum of left- and right-handed 2-component Weyl spinors.

In this sense they are representations of the Lorentz group in the same way that isodoublets’ are representations of the gauge group SU(2). A spin-up fermion transforms into a spin-down fermion under the Lorentz group in the same way that a proton transforms into a neutron under the [archaic] SU(2) isospin group.

Chirality and Helicity: slightly different

Now here’s the point I wanted to get to, and it’s a subtle one. The Standard Model is a chiral theory. This means that different representations of the Lorentz group transform differently not only under the Lorentz group itself, but also with respect to the Standard Model SU(3)$\times$SU(2)$\times$U(1) gauge group.

This is why we denote left-handed (fundamental Lorentz rep) quarks by $q$ and the right-handed (anti-fundamental Lorentz rep) quarks by $u$ and $d$. The left-handed up and down quarks are in a SU(2) doublet, $q$. Meanwhile, their right-handed counterparts are stuck in lonely SU(2) singlets $u$ and $d$.

This is the very subtle difference between chirality and helicity. The chirality of a spin-1/2 particle refers to whether it is in the fundamental of anti-fundamental representation of the Lorentz group. Meanwhile, the helicity refers to the projection of the spin onto the direction of motion (e.g. the z-axis).

In the massless case, these are the same thing. A free massless spin-1/2 particle is either left- or right- handed and stays that way.

Dirac Mass Terms

Things become more interesting in the massive case. The standard Diract mass term looks like $m^2 \bar \psi \psi$, where $\psi$ is a 4-component Dirac spinor. Some foresight leads us to work with Dirac rather than Weyl spinors. We expect the left- and right- handed Weyl spinor representations of the Lorentz group to mix under the mass term. Indeed, if $\psi$ is a left-handed particle, then $\bar \psi$ is a right-handed particle. [4] Hence the irreducible representation’ with the mass term is a product of the left- and right- handed Weyl representations.

Intuitively this makes perfect sense. A massless particle travels at the speed of light, c. One can never boost into a frame where the particle is either massless or spinning in the opposite direction. On the other hand, a massive particle travels at some speed less than c. This means an observer can boost into a frame where the particle is travelling in the opposite direction with the opposite helicity, or is even at rest. [5]

The big question, then, is whether this particle also has the opposite chirality. The answer is yes, and this is clear if we take another perspective. Think of the Dirac mass term as an interaction term, i.e. part of the interaction Lagrangian rather than the free Lagrangian. Then, as we showed above, the corresponding interaction vertex has an incoming particle of left(right)-handed chriality and an outgoing particle of right(left)-handed chirality. So indeed, the Dirac mass means that chirality and helicity are intimately linked. (If you’re apprehensious about this argument, then that’s good! Keep reading.)

Majorana Mass Terms

Dirac masses aren’t the only sort of mass terms. A two-component Weyl spinor $\chi$ can have a Majorana mass term of the form $m^2 \chi \chi$. [6] This is often accompanied by an explanation that the particle is it’s own antiparticle. We can go through the same arguments as above — but this time the chirality of the particle is fixed: a left-handed Majorana fermion remains a left-handed (chirally) Majorana fermion even if it has a right-handed helicity.

That’s sounds like a weird idea, but the weirdness is just in our naming. Imagine a massive Majorana fermion travelling in the positive z-direction with left-handed helicity and left-handed chirality. If we boost into a frame where the particle is moving in the negative z-direction, thenit will still have left-handed chirality, but the helicity will now be right-handed.

To reinforce the point: the helicity is a physical property of the particle, while the chirality is a label associated with a representation of the Lorentz group.

The fact that we refer to chiralities as being left- or right- handed is, I assume, a relic of the Dirac mass case where the chiralities indeed match with the helicities.

The Standard Model does not contain any Majorana mass terms. It is possible, however, that neutrinos have a Majorana mass. Further, in supersymmetry, the gauginos (superpartners of gauge bosons) are Majorana fermions.

Relativist versus Particle Physicist

Now let’s get back to something we’d swept under the rug. We were passing between two points of view. First, we took a relativist’ point of view when we boosted from one frame to another to show that helicity will swap. Secondly, we took a particle physicist’ point of view where we used perturbation theory with respect to the mass term in the Lagrangian.

One should pause to think about whether or these correspond to the same thing. Does it make sense that these mass insertions are related to the boosts?

There’s a nice heuristic argument we can make to this point. We shall compare the particle’s mass (a Lorentz invariant) with its momentum (a Lorentz covariant). The mass-insertion (particle physicist’) perturbative approach only works when the mass is small compared to the energy, $E = \sqrt{p^2 + m^2}$, i.e. when the 3-momentum is large.

Let us start by boosting into the opposite regime, where the 3-momentum is small. In this regime we are near the particle’s rest-frame. A small boost (of our frame) in either direction would cause us to see the particle in either helicity. Further, perturbation theory with mass-insertions would be meaningless since this regime would be strongly coupled and hence one would have to do a nonperturbative analysis. Of course, since the mass term is quadratic, such an analysis is possible. [7] Anyway, the point is that there’s not much to say in this frame.

Now let’s boost to a frame where the particle has a large 3-momentum in the positive z-direction. In this regime, the particle has a large energy relative to its mass. From the relativist’s point of view we’ve boosted into a frame where the particle’s helicity is heavily biased to be in a certain directon (depending on its chirality). From the particle physicist’s point of view, this is a regime where the mass is small compared to the energy and hence perturbation theory with mass-insertions is valid.

Thus the frame in which the helicity of the particle is pushed a certain way are the same regimes in which $p \gg m$ and hence where perturbation theory with the mass insertions is valid.

Proper representation theory for physicists’ courses seem to be out of fashion in my limited familiarity with American particle physics graduate courses, even though Lie algebras were the bread and butter of any particle physicist in the past generation. I know of only one university with a dedicated semester-long course to the subject. (On the other hand, both Cambridge and Durham stress group theory as a foundational course in their respective theoretical physics courses.)

I’m not sure why this is the case, but many of the QFT texts written in the past generation seem to assume that the reader has a 1960’s-style background in representation theory (especially of SU(N) and the Poincare group). Even more recent books (Zee and Srednicki) leave the meat of this subject to other resources. The material I’ve suggested in [2] is very helpful for understanding the objects one works with in quantum field theory, and is especially important for supersymmetry and grand unification.

As usual, the conventional wisdom is to learn representation theory from a physicist rather than a mathematician, for much the same reason why engineers have special engineering physics’ and engineering mathematics’ courses: the applications of interest have different focus from the more formal theory.

Notes

[1] Sure, one learns how to calculate with spinors — but it took me a while before I properly understood what a spinor is. If I were a bit more clever at the time, I would also have been very confused about the vector representation. But courses in relativity had fooled me into complacency.

[2] I refer the reader to an excellent set of Part III notes, or the QFT texts Weinberg Vol I and Ramond. There’s also a cute book my Hladik and Cole, but there are plenty of small typos and it takes a while to get the the point. Preferable background knowledge: a bit about group theory, a bit more about representations of the special unitary groups. See note [3] below.

[3] For more details, start with Bob Cahn’s book or go to all the way to the beginning with a text on abstract algebra.

[4] This is apparent if you insert projection operators in between the two spinors and write everything out explicitly as matrices.

[5] Activity! Make a fist with your right hand with your thumb sticking out. Your tumb is pointing in the direction of a right-handed particle’s spin. Your fingers are curled in the direction of the right-handed particle’s spin. Now boost’ into a frame where the particle is moving in the opposite direction: squeeze your right thumb under your fingers so that it’s pointing in the opposite direction from before. The thumb-and-finger orientation should now look just like a left-handed particle. Check this against your left hand if you’re not convinced. 🙂

[6] The form of the Majorana mass term restricts the gauge group representation of the particle since all gauge group indices must be fully contracted.

[7] A QFT student should already be familiar with this argument, it’s exactly the same as the resummation of the two-point 1PI diagrams to extract the full propagator.

#### 3 Responses to “Spinors, Chirality, and Majorana Mass”

1. Good post flip. i remember being quite confused between what chirality, helicity and spin. If I recall correctly it was you who explained it to me (in the Burrell’s Field Common Room was it?)

🙂

Evan