We’ve been discussing the Higgs (its interactions, its role in particle mass, and its vacuum expectation value) as part of our ongoing series on understanding the Standard Model with Feynman diagrams. Now I’d like to take a post to discuss a very subtle feature of the Standard Model: its chiral structure and the meaning of “mass.” This post is a little bit different in character from the others, but it goes over some very subtle features of particle physics and I would really like to explain them carefully because they’re important for understanding the entire scaffolding of the Standard Model.

My goal is to explain the sense in which the Standard Model is “chiral” and what that means. In order to do this, we’ll first learn about a related idea, helicity, which is related to a particle’s spin. We’ll then use this as an intuitive step to understanding the more abstract notion of chirality, and then see how masses affect chiral theories and what this all has to do with the Higgs.

Helicity

Fact: every matter particle (electrons, quarks, etc.) is spinning, i.e. each matter particle carries some intrinsic angular momentum.

Let me make the caveat that this spin is an inherently quantum mechanical property of fundamental particles! There’s really no classical sense in which there’s a little sphere spinning like a top. Nevertheless, this turns out to be a useful cartoon picture of what’s going on:

This is our spinning particle. The red arrow indicates the direction of the particle’s spin. The gray arrow indicates the direction that the particle is moving. I’ve drawn a face on the particle just to show it spinning.

The red arrow (indicating spin) and the gray arrow (indicating direction of motion) defines an orientation, or a handedness. The particular particle above is “right-handed” because it’s the same orientation as your right hand: if your thumb points in the direction of the gray arrow, then your fingers wrap in the direction of the red arrow. Physicists call this “handedness” the helicity of a particle.

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. You can confirm that if you point your thumb in the opposite direction, your fingers will also wrap in the opposite direction.

Sounds good? Okay, now we can also imagine a particle that is left-handed (or “left helicity”). For reference here’s a depiction of a left-handed particle moving in each direction; to help distinguish between left- and right-handed spins, I’ve given left-handed particles a blue arrow:

[Confirm that these two particles are different from the red-arrowed particles!]

An observation: note that if you only flip the direction of the gray arrow, you end up with a particle with the opposite handedness. This is precisely the reason why the person staring back at you in the mirror is left-handed (if you are right-handed)!

Thus far we’re restricting ourselves to matter particles (fermions). There’s a similar story for force particles (gauge bosons), but there’s an additional twist that will deserve special attention. The Higgs boson is another special case since it doesn’t have spin, but this actually ties into the gauge boson story.

Once we specify that we have a particular type of fermion, say an electron, we automatically have a left-helicity and a right-helicity version.

Helicity, Relativity, and Mass

Now let’s start to think about the meaning of mass. There are a lot of ways to think about mass. For example, it is perhaps most intuitive to associate mass with how ‘heavy’ a particle is. We’ll take a different point of view that is inspired by special relativity.

A massless particle (like the photon) travels at the speed of light and you can never catch up to it. There is no “rest frame” in which a massless particle is at rest. The analogy for this is driving on the freeway: if you are driving at the same speed as the car in the lane next to you, then it appears as if the car next to you is not moving (relative to you). Just replace the car with a particle.

On the other hand, a massive particle travels at less than the speed of light so that you can (in principle) match its velocity so that the particle is at rest relative to you. In fact, you can move faster than a massive particle so that it looks like the particle is traveling in the opposite direction (this flips the direction of the gray arrow). Note that the direction of its spin (the red arrow) does not change! However, we already noted that flipping only the particle’s direction—and not its spin—changes the particle’s helicity:

Here we’ve drawn the particle with a blue arrow because it has gone from being right-handed to left-handed. Clearly this is the same particle: all that we’ve done is gone to a different reference frame and principles of special relativity say that any reference frame is valid.

Okay, so here’s the point so far: mass is a something that tells us whether or not helicity is an “intrinsic” property of the particle. If a particle is massless, then its helicity has a fixed value in all reference frames. On the other hand, if a particle has any mass, then helicity is not an intrinsic property since different observers (in valid reference frames) can measure different values for the helicity (left- or right-helicity). So even though helicity is something which is easy to visualize, it is not a “fundamental” property of most particles.

Now a good question to ask is: Is there some property of a particle related to the helicity which is intrinsic to the particle? In other words, is there some property which

is equivalent to helicity in the massless limit is something which all observers in valid reference frames would measure to be the same for a given particle.

The good news is that such a property exists, it is called chirality. The bad news is that it’s a bit more abstract. However, this is where a lot of the subtlety of the Standard Model lives, and I think it’s best to just go through it carefully.

Chirality

Chirality and helicity are very closely related ideas. Just as we say that a particle can have left- or right-handed helicity, we also say that a particle can have left- or right-handed chirality. As we said above, for massless particles the chirality and helicity are the same. A massless left-chiral particle also has left-helicity.

However, a massive particle has a specific chirality. A massive left-chiral particle may have either left- or right-helicity depending on your reference frame relative to the particle. In all reference frames the particle will still be left-chiral, no matter what helicity it is.

Unfortunately, chirality is a bit trickier to define. It is an inherently quantum mechanical sense in which a particle is left- or right-handed. For now let us focus on fermions, which are “spin one-half.” Recall that this means that if you rotate an electron by 360 degrees, you don’t get the same quantum mechanical state: you get the same state up to a minus sign! This minus sign is related to quantum interference. A fermion’s chirality tells you how it gets to this minus sign in terms of a complex number:

The physical meaning of this is the phase of the particle’s wavefunction. When you rotate a fermion, its quantum wavefunction is shifted in a way that depends on the fermion’s chirality:

We don’t have to worry too much about the meaning of this quantum mechanical phase shift. The point is that chirality is related in a “deep” way to the particle’s inherent quantum properties. We’ll see below that this notion of chirality has more dramatic effects when we introduce mass.

Some technical remarks: The broad procedure being outlined in the last two sections can be understood in terms of group theory. What we claim is that massive and massless particles transform under different [unitary] representations of the Poincaré group. The notion of fermion chirality refers to the two types of spin-1/2 representations of the Poincaré group. In the brief discussion above, I tried to explain the difference by looking at the effect of a rotation about the z-axis, which is generated by ±σ3/2.

The take home message here is that particles with different chiralities are really different particles. If we have a particle with left-handed helicity, then we know that there should also be a version of the particle with right-handed helicity. On the other hand, a particle with left-handed chirality needn’t have a right-chiral partner. (But it will certainly furnish both helicities either way.) Bear with me on this, because this is really where the magic of the Higgs shows up in the Standard Model.

Chiral theories

[6/20/11: the following 2 paragraphs were edited and augmented slightly for better clarity. Thanks to Bjorn and Jack C. for comments. 4/8/17: corrected “right-chiral positron” to “left-chiral positron” and analogously for anti-positrons; further clarification to text and images; thanks to Martha Lindeman, Ph.D.]

One of the funny features of the Standard Model is that it is a chiral theory, which means that left-chiral and right-chiral particles behave differently. In particular, the W bosons will only talk to electrons (left-chiral electrons and right-chiral anti-electrons) and refuses to talk to positrons (left-chiral positrons and right-chiral anti-positrons). You should stop and think about this for a moment: nature discriminates between left- and right-chiral particles! (Of course, biologists are already very familiar with this from the ‘chirality’ of amino acids.)

Note that Nature is still, in some sense, symmetric with respect to left- and right-helicity. In the case where everything is massless, the chirality and helicity of a particle are the same. The W will couple to both a left- and right-helicity particles: the electron and anti-electron. However, it still ignores the positrons. In other words, the W will couple to a charge -1 left-handed particle (the electron), but does not couple to a charge -1 right-handed particle (the anti-positron). This is a very subtle point!

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.g. the quantum mechanical phase under a rotation).

In order to really drive this point home, let me reintroduce two particles to you: the electron and the “anti-positron.” We often say that the positron is the anti-partner of the electron, so shouldn’t these two particles be the same? No! The real story is actually more subtle—though some of this depends on what people mean by ‘positron,’ here we are making a useful, if unconventional, definition. The electron is a left-chiral particle while the positron is a right-chiral particle. Both have electric charge -1, but they are two completely different particles.

How different are these particles? The electron can couple to a neutrino through the W-boson, while the anti-positron cannot. Why does the W only talk to the (left-chiral) electron? That’s just the way the Standard Model is constructed; the left-chiral electron is charged under the weak force whereas the right-chiral anti-positron is not. So let us be clear: the electron and the anti-positron are not the same particle! Even though they both have the same charge, they have different chirality and the electron can talk to a W, whereas the anti-positron cannot.

For now let us assume that all of these particles are massless so that these chirality states can be identified with their helicity states. Further, at this stage, the electron has its own anti-particle (an “anti-electron”) which has right-chirality which couples to the W boson. The anti-positron also has a different antiparticle which we call the positron (the same as an “anti-anti-positron”) and has left-chirality but does not couple to the W boson. We thus have a total of four particles (plus the four with opposite helicities):

Technical remark: the left- & right-helicity electrons and left- & right-helicity anti-positrons are the four components of the Dirac spinor for the object which we normally call the electron (in the mass basis). Similarly, the left- & right-helicity anti-electrons and left- & right-helicity positrons for the conjugate Dirac spinor which represents what we normally call the positron (in the mass basis).

Important summary: [6/20/11: added this section to address some lingering confusion; thanks to David and James from CV, and Steve. 6/29: Thanks to Rainer for pointing out a mistake in 3 and 4 below (‘left’ and ‘right’ were swapped).] We’re bending the usual nomenclature for pedagogical reasons—the things which we are calling the “electron” and “positron” (and their anti-partners) are not the “physical” electron in, say, the Hydrogen atom. We will see below how these two ideas are connected. Thus far, the key point is that there are four distinct particles: Electron: left-chiral, charge -1, can interact with the W Anti-electron: right-chiral, charge +1, can interact with the W Positron: left-chiral, charge +1, cannot interact with the W Anti-positron: right-chiral, charge -1, cannot interact with the W. We’re using names “electron” and “positron” to distinguish between the particles which couple to the W and those that don’t. The conventional language in particle physics is to call these the left-handed (chirality) electron and the right-handed (chirality) electron. But I wanted to use a different notation to emphasize that these are not related to one another by parity (space inversion, or reversing angular momentum).

Masses mix different particles!

Now here’s the magical step: masses cause different particles to “mix” with one another.

Recall that we explained that mass could be understood as a particle “bumping up against the Higgs boson’s vacuum expectation value (vev).” We drew crosses in the fermion lines of Feynman diagrams to represent a particle interacting with the Higgs vev, where each cross is really a truncated Higgs line. Let us now show explicitly what particles are appearing in these diagrams:

[6/25: this paragraph added for clarity] Note that in this picture the blue arrow represents helicity (it is conserved), whereas the mustache (or non-mustache) represents chirality. The mass insertions flip chirality, but maintain helicity.

This is very important; two completely different particles (the electron and the anti-positron) are swapping back and forth. What does this mean? The physical thing which is propagating through space is a mixture of the two particles. When you observe the particle at one point, it may be an electron, but if you observe it a moment later, the very same particle might manifest itself as an anti-positron! This should sound very familiar, it’s the exact same story as neutrino mixing (or, similarly, meson mixing).

Let us call this propagating particle is a “physical electron.” The mass-basis-electron can either be an electron or an anti-positron when you observe it; it is a quantum mixture of both. The W boson only interacts with the “physical electron” through its electron component and does not interact with the anti-positron component. Similarly, we can define a “physical positron” which is the mixture of the positron and anti-electron. Now I need to clarify the language a bit. When people usually refer to an electron, what they really mean is the mass-basis-electron, not the “electron which interacts with W.” It’s easiest to see this as a picture:

Note that we can now say that the “physical electron” and the “physical positron” are antiparticles of one another. This is clear since the two particles which combine to make up a physical electron are the antiparticles of the two particles which combine to make up the physical positron. Further, let me pause to remark that in all of the above discussion, one could have replaced the electron and positron with any other Standard Model matter particle (except the neutrino, see below). [The electron and positron are handy examples because the positron has a name other than anti-electron, which would have introduced language ambiguities.]

Technical remarks: To match to the parlance used in particle physics: The “electron” (interacts with the W) is called e L , or the left-chiral electron The “anti-positron” (does not interact with the W) is called e R , or the right-chiral electron. [6/25: corrected and updated, thanks to those who left comments about this] Note that I very carefully said that this is a right-chiral electron, not a right-helicity electron. 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! The “physical electron” is usually just called the electron, e, or mass-basis electron The analogy to flavor mixing should be taken literally. These are different particles that can propagate into one another in exactly the same way that different flavors are different particles that propagate into one another. Note that the mixing angle is controlled by the ratio of the energy to the mass and is 45 degrees in the non-relativistic limit. [6/22: thanks to Rainer P. for correcting me on this.] Also, the “physical electron” now contains twice the physical degrees of freedom as the electron and anti-positron. This is just the observation that a Dirac mass combines two 2-component Weyl spinors into a 4-component Dirac spinor. When one first learns quantum field theory, one usually glosses over all of these details because one can work directly in the mass basis where all fermions are Dirac spinors and all mass insertions are re-summed in the propagators. However, the chiral structure of the Standard Model is telling us that the underlying theory is written in terms of two-component [chiral] Weyl spinors and the Higgs induces the mixing into Dirac spinors. For those that want to learn the two-component formalism in gory detail, I strongly recommend the recent review by Dreiner, Haber, and Martin.



What this all has to do with the Higgs

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). [See a very old post where I tried—I think unsuccessfully—to convey similar ideas.] The reason why we’ve gone through this entire rigmarole is to say that ordinarily, two unrelated particles don’t want to be mixed up into one another.

The reason for this is that particles can only mix if they carry the same quantum properties. You’ll note, for example, that the electron and the anti-positron both had the same electric charge (-1). 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. In highfalutin language, one might say that this mass term is prohibited by “gauge invariance,” where the word “gauge” refers to the W as a gauge boson. 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. When it obtains a vacuum expectation value, it “breaks” the conservation of weak charge and allows the electron to mix with the anti-positron, even though they have different weak charges. Or, in other words, the vacuum expectation value of the Higgs “soaks up” the difference in weak charge between the electron and anti-positron.

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. The next step is to finally address how this Higgs managed to obtain a vacuum expectation value, and what it means that it “breaks” weak charge. This phenomenon is called electroweak symmetry breaking, and is one of the primary motivations for theories of new physics beyond the Standard Model.

Addendum: Majorana masses

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! (Neutrinos do carry weak charge, but this is “soaked up” by the Higgs vev.)

Within the Standard Model, Majorana masses are special for neutrinos. They mix neutrinos with anti-neutrinos so that the “physical neutrino” is its own antiparticle. (In fancy language, we’d say the neutrino is a Majorana fermion, or is described by a Weyl spinor rather than a Dirac spinor.) It is also possible for the neutrino to have both a Majorana and a Dirac mass. (The latter would require additional “mustached” neutrinos to play the role of the positron.) This would have some interesting consequences. As we suggested above, the Dirac mass is associated with the non-conservation of weak charge due to the Higgs, thus Dirac masses are typically “small.” (Nature doesn’t like it when things which ought to be conserved are not.) Majorana masses, on the other hand, do not cause any charge non-conservation and can be arbitrarily large. The “see-saw” between these two masses can lead to a natural explanation for why neutrinos are so much lighter than the other Standard Model fermions, though for the moment this is a conjecture which is outside of the range of present experiments.