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### Helicity, Chirality, Mass, and the Higgs

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

1. is equivalent to helicity in the massless limit
2. 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:

What happens when you rotate a left- vs right-chiral fermion 360 degree about its direction of motion. Both particles pick up a -1, but the left-chiral fermion goes one way around the complex plane, while the right-chiral fermion goes the other way. The circle on the right represents the complex phase of the particle’s quantum state; as we rotate a particle, the value of the phase moves along the circle. Rotating the particle 360 degrees only brings you halfway around the circle in a direction that depends on the chirality of the fermion.

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:

Rotating a fermion shifts its quantum wavefunction. Left- and right-chiral fermions are shifted in opposite directions. This is a purely quantum phenomenon.

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.

Electrons (left-chiral) and anti-positrons (right-chiral) have the same electric charge but are two completely different particles, as evidenced by the positron’s mustache.

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):

The electron, anti-electron, anti-positron, and positron.

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:

1. Electron: left-chiral, charge -1, can interact with the W
2. Anti-electron: right-chiral, charge +1, can interact with the W
3. Positron: left-chiral, charge +1, cannot interact with the W
4. 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:

An “electron” propagating in space and interacting with the Higgs field. Note that the Higgs-induced mass term connects an electron with an anti-positron. This means that the two types of particles are exhibiting quantum mixing.

[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:

The “physical electron” (what most people mean when they say “electron”) is a combination of an electron and an anti-positron. Note that the electron and the anti-positron have different interactions (e.g. the electron can interact with the W boson); the physical electron inherits the interactions of both particles.

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:

1. The “electron” (interacts with the W) is called eL, or the left-chiral electron
2. The “anti-positron” (does not interact with the W) is called eR, 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 eL and eR 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!
3. 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.

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.

• Hey Flip! Great post. Did you make the illustrations?

• Hi Bee! Thanks for the compliment. Yeah, I made the illustrations. 🙂

• Björn

“In particular, the W bosons will only talk to left-chiral fermions and refuse to talk to right-chiral fermions”

“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 positron also has a different antiparticle, (an “anti-positron”) which 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):”

Hi this i find very hard to understand you state above that the W only interact with left-chirality particals but still it interact with anti-electron which is a right-chirality praticel. How is this possible. And why does it not interact with the anti-positron which was a left-chirality particel ?

• Hi Bjorn, I knew somebody was going to catch me on this statement. 🙂

The correct statement is that the W will interact with an electron or anti-electron but will not interact with a positron or anti-positron.

When I originally wrote that line I meant to say that the W interacts with left-chiral fermions (meaning the electron) and, therefore, right-chiral anti-fermions (the anti-electron); but I thought that saying in so many words wouldn’t be more confusing.

The chiral nature of the Standard Model allows us to say that the W interacts with electrons but not positrons. Matter-antimatter symmetry in our theory then requires that the anti-electron also interacts with the W, while the anti-positron does not. (Nevermind that the universe is not matter-antimatter symmetric, that’s likely due to quantum fluctuations—but this is also a subtle topic.)

Thanks for the question!
F

• Splendid explanation! Also see Physics Today 64(5) 40 (2011), “The neutrino’s elusive helicity reversal.”

Physics postulates a mirror-symmetric universe. We see no vacuum photon refraction, dispersion, dichroism, or gyrotropy over billion lightyear pathlengths (arxiv:0912.5057, 0905.1929, 0706.2031; 1106.1068). Is the vacuum is isotropic toward massed fermions? Teleparallel gravitation embedded in Weitzenböck spacetime has spacetime torsion that is chiral, like Lorentz force.

Do opposite shoes violate the Equivalence Principle? “Opposite shoes” is a metaphor. Enantiomorphic space group single crystals are chemically and macroscopically identical, opposite geometric parity atomic mass distributions (one chiral lattice is the universal coordinate sign inversion of the other). A vacuum left foot is invisible to socks and left shoes. A right shoe is diagnostic. The equipment already exists,

http://www.mazepath.com/uncleal/erotor1.jpg
Two geometric parity Eotvos experiments

If a parity Equivalence Principle test succeeds, theory will be chiral at the postulate level. Symmetry breakings will be intrinsic. The universe will be matter by default (as opposed to being 50% antimatter by conservation laws). Somebody should look.

• A very nice post. I just thought I would point out a small typo: just above the heading “Masses Mix Different Particles”, item 4, the charge of the “anti-positron” should be negative.

Best,
Steve

• Excellent animations! I didn’t know they were yours, cool. So I linked to your nice article now, too.

• Many thanks for catching this sign error, Steve! 🙂

• Jack Frillman

I have always read that an electron is a fundamental particle (i.e. it can’t be broken down into other particles) that has no size or radius. If this true then how can a electron be a combination an electron and an anti-positron?

• Thanks Lubos! For those with a more technical background in field theory, I recommend Lubos’ post (http://motls.blogspot.com/2011/06/particles-of-different-spins-and-their.html) to go even deeper into this rich subject.

Jack: This is a great question. The difference is that the physical electron is a “mixture” of an electron and anti-positron, but it is not a composite or bound state. The Hydrogen atom, for example, is a “bound state” that is made up of an electron and a proton *simultaneously.* You can look at the Hydrogen atom and ‘see’ that it is composed of both particles. On the other hand, a physical electron is a quantum mixture of an electron and anti-positron: when you observe the physical electron, you see *either* the electron or anti-positron, but never both simultaneously.

For example, I can have a physical electron that interacts with a W boson. At the moment it interacts with the W boson, I know that it is 100% an electron and not an anti-positron. But later on I might observe the same particle to be an anti-positron.

In other words: there is no sense in which a physical electron “contains” both the electron and the anti-positron at the same time. Rather, it’s like a shape-shifter which is constantly changing between the two.

Great question!
Thanks,
F

• ParoXoN

Very cool post! Thanks for that 🙂

I’m a little confused though:
You say towards the end:
“The “anti-positron” (does not interact with the W) is called eR (or sometimes eRc or eR-bar), the right-chiral electron.”

But I thought the anti-positron was left-chiral?

• jal

If more that two people link to your article, does that make it viral?
Great job.
jal

• Tony

Very nice post Flip! Love the googly-eyed electron too.

• Ah, good catch ParoXoN. The positron is right-chiral, and the anti-positron is left-chiral. I’ve corrected the post (and given credit to you for pointing this out).

Tony and jal: thanks! 🙂

• Björn

Hi now is start making more and more sense,

But i wonder what the “right-chiral electron” is in your picture.

The “anti-positron” (does not interact with the W) is called eRc or eR-bar, is a left-chiral particle which is the anti-partner of right-chiral electron (eR) [6/21: corrected, thanks to ParoXoN.]

The positron is right-chiral and the anti-electron is right chiral, but i dont see any right chiral electron,

• It seems that helicity is a property of motion and chirality a property of shape (where, in the case of an elementary particle, this might be represented by something like the shape of a level surface of its wave function). The language chosen by physicists is unfortunate as a helix is an object with fixed chirality but the chirality of the path of a “helical” motion depends on the relative motion of the medium in which it is traced.

• Electron: left-chiral, charge -1, cannot interact with the W
Anti-electron: right-chiral, charge +1, cannot interact with the W
Positron: right-chiral, charge +1, can interact with the W
Anti-positron: left-chiral, charge -1, can interact with the W this is right or not?

• Brian

Very nice post! I’m glad to see an explanation of chirality that’s understandable to a non-specialist like myself.

There is a connection between this post and the branching ratios for charged pion decay, and reading this helps clarify things on that front. A negative (say) pion decays by the weak force into a (virtual) W-, and then into either an electron/neutrino pair, or a muon/neutrino pair. The first decay path is highly relativistic, but the second decay path is not (due to the muon’s large rest mass, almost as large as the pion). As you say, the mixing angle between the electron and anti-positron is energy dependent, and for the highly relativistic “electron” decay of the pion, the emitted electron is almost all anti-positron, with barely any electron in its wave function to interact with the weak force. (The neutrino must be emitted with the correct chirality = helicity, which means the helicity of the emitted electron or muon is “wrong”, so the “wrong”-chirality component dominates its wave function.) This strongly inhibits the electron decay, but not the muon decay, where the mixing angle is close to the 45 degree non-relativistic limit.

• Hi everyone, thanks for the comments over the past couple of days. My apologies for the delayed response.

@ Bjorn: Sorry, the term “right chiral electron” is what particle physicists usually call what I am here calling the “anti-positron.”

@ Magneton: the way it is written now should be correct. Only the non-mustached particles interact with W bosons, i.e. only the electron and anti-electron.

@ Brian: yes! Excellent example!

Thanks!
F

• Alejandro

AWESOME explanation!!!!!… I really enjoy your blog, one of the best in Quantum Diaries!!!

• Stephen Brooks

–[Sorry, the term “right chiral electron” is what particle physicists usually call what I am here calling the “anti-positron.”]–

• Lennart

Hello Flip Tanedo,

Your last change has made things a bit foggy for me. In the summary you changed the chirality of the positron to left chiral while throughout the rest of the post it is still called right chiral.

Also the reason why the W boson doesn’t talk with positrons is a bit unclear. The way I understand it RH chiral particles do not carry electroweak charge so W bosons only interact with LH chiral particles. Is this correct? If so shouldn’t the summary become:

1. LH Chiral electron, W interaction, negative charge
2. LH Chiral positron, W interaction, positive charge
3. RH Chiral anti-electron, no W interaction, positive charge
4. RH Chiral anti-positron, no W interaction, negative charge

Your posts are wonderful and a true delight to read. If you would ever decide to write a book about physics I would love to buy it.

Regards,

Lennart

• David George

Dear Flip,

Thank you for a fascinating introduction to helicity and chirality. However, I do not believe the process you describe represents the workings of nature. In other words, I have no faith in the particle model as a description of nature. I have no formal physics training and so you would probably class me as a crackpot. Knowing from my experience that physicists do not like to talk to me I must nevertheless propose an alternative process (based on an alternative atomic model). If this process does not provide a way to match exactly the quantum mechanical predictions, it should be easily demolished. My proposal is that an electron (and proton) is a rotating sphere occupying the entire atomic orbital (i.e. it resembles a ‘smeared out’ electric charge). There are three possible orthogonal axes, combinations of which can describe the sphere’s rotation (i.e. the electron ‘superposition’ of two electron states may be resolved into a rotation on any two of the three axes). Firstly, as an expert on chirality and helicity, which are intimately related to rotation, I wonder if you have ever dealt with combinations of potentially triaxial rotations (say, two of three axes). Secondly, it appears that the electron-antipositron mixture is due to the measurement uncertainty principle +h/2 pi versus -h/2 pi. So in particle physics the up-versus-down becomes electron-versus-antipositron. Is that correct?
My belief in triaxial rotation is quite firm, since simple equations lead to a formula that gives the electron magnetic moment as 9.284 765 18 e-24 joules per tesla, without any spin-g factor (however, a combination of the spin-g factor and a factor which appeared in my calculations regains the conventional mass values of the electron and proton). In this model the system electron and proton have different radii than their free versions (which are the measured versions I believe).
I should note that a comment I made on the Physics 2.0 website — to the effect that the q.m. probabilities for a measurement of entangled electrons (or from a superposition state) can be reproduced by manipulating the two-out-of-three possibilities for a triaxial rotation — was deleted from the comment section altogether. I may be appear to be overly hostile sometimes (being a non-believer in the particle model) but there was no hostility in that post. The only reason I can come up with is that the comment was removed (rather than simply dismissed with a chuckle, etc.) because it posed some kind of threat to the original poster. So having found your very interesting post (and having been unaware of the four particle scenario) I wonder if you can comment on the possible outcomes of rotational combinations of spheres of rotating space, given the existence of three axes to choose from.
I am submitting this because you appear to be sincerely interested in revealing details of quantum mechanics as applied to fields. As I said earlier, I do not believe any such process as the electron zig-zag (with accompanying singularity, I think) exists in nature. It is an ingenious mathematical contrivance to deal with a measurement phenomenon but does not represent a natural process (and so also for the particle models which have grown out of it).

• Lennart

Ay there’s the rub. I just learned that the W boson interacts with left handed particles and right handed anti particles. So if you call the LH particle state a electron and the right handed particle state a positron and do the same vice versa for the anti particles I get:

1. LH electron, W interaction, negative charge
2. RH positron, no W interaction, negative charge
3. RH anti-electron, W interaction, positive charge
4. LH anti-positron, no W interaction, positive charge

Which is the same as yours save for the names of the particles:

1. Electron: left-chiral, charge -1, can interact with the W
2. Anti-electron: right-chiral, charge +1, can interact with the W
3. Positron: left-chiral, charge +1, cannot interact with the W
4. Anti-positron: right-chiral, charge -1, cannot interact with the W.

Regards,

Lennart

• Hello David G. I believe one quick answer is that the shell structure of molecules has been measured experimentally (I believe? You should cross check this with someone more knowledgeable), which provides a very non-trivial affirmation of quantum theory. -F

• David George

Dear Flip,

I see no problem with quantum mechanics so long as its statistical foundation (and lack of explanation for its 100% certain predictions) is acknowledged. It is the “fundamental” particle-and-field model that seems contrived. Studying collision debris may be like breaking clocks and studying their parts, but it may also be like breaking billiard balls (rotating, that is) thinking them to be clocks. (And I seem to be having a hard time finding someone — anyone — to analyze that triaxial rotation “math problem”. If I could do it I would, but I haven’t a clue how to do it and my brain has a hard time learning new tricks.)

But I must thank you for your response — I appreciate it, usually there is just silence.

David G.

• r binwal

A beutiful post. You really excel in making great concepts simple. We believe that world urgently needs people like you who make abstruce topics appear so appealing and inspiring. I had the same doubts raised about L and R chirality but now you have resolved these in the forgoing posts. Thanks.

• All right then! So you are prepared to believe the MINOS results then, which clearly indicate that the so called antineutrino is NOT the antiparticle of the neutrino, so that CPT may be conserved. That is, the neutrino mixes a left and right Bilson-Thompson braid pair to form one (kind of Majorana) particle, while the antineutrino mixes the other pair, so that these two ‘physical’ particles may have distinct masses. As observed. No Lagrangians required. Who thought mass ever had anything to do with Lagrangians anyway?

• Hi Kea, I do not know which MINOS results you are referring to, but to the best of my limited knowledge in that field, there has not been definitive evidence for any specific model of the right-handed neutrino sector other than the fact that neutrino indeed oscillate (and, most recently, that theta_13 >0).

• The MINOS experiment measures the two delta m squares. Whether or not theta_13 is greater than zero is a separate issue (maybe, that is).

• Hello Flip Tanedo,

Mu name is Manel Rosa Martins and I would like to ask for your kind permission to translate into Portuguese (European)and publish this excellent article of yours in the Astronomy Blog Astro PT. I am Particle Physics student and volunteer writer for the Astro Pt. We’re aiming to enhance functional literacy amongst our readers and to try to reach them with accessible and accurate writing. I will respect attribution, publish your bio and send you a copy upon completion, if your authorization is granted , that is. Thanks
Manel R Martins

• DK

Great post! But I’m a little confused about something. In the “CHIRAL THEORIES” section you say “the W bosons will only talk to electrons (left-chiral electrons and right-chiral anti-electrons) and refuses to talk to positrons (right-chiral positrons and left-chiral anti-positrons)” and a little later you say “The electron is left-chiral, while the positron is right-chiral. They’re two completely different particles.”

But then this is the opposite of what’s written in the “Summary Section”:

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.

I think the summary section is correct while the earlier statements (regarding the positrons) is not. If the positron was right-chiral, the antipositron would have to be left-chiral. But then the electron mass term would have a left-chiral electron and a left-chiral antipositron, which is not correct.

Do you agree that there is a conflict between the statements or am I misinterpreting it somehow?

cheers,
D

• Old Wolf

Hi Flip. This is probably a silly question, but why can the electron only be spinning ‘up’ or ‘down’ ? Could it not be spinning sideways?

What happens if you drive in your car and match speeds with the electron, so it is not moving at all relative to you? Then which direction is its direction of motion? If you then moved perpendicular to how you were moving before, then have you caused the electron to change its spin direction? (it was previously spinning north, now it is spinning east, even though it didn’t change its velocity, only you did) ?

• Hi DK, it’s entirely possible that I was sloppy with conventions, I’ll try to go over it again when I get a chance… but there’s a possibility that I won’t get the chance in the near future. -F

• Pingback: Who ate the Higgs? | My Blog()

• beefy

thank you very much for this simple explanation of the basic concepts of spin and angular momentum , it was worth watching

• Ken

You can never know the “true” direction of an electron’s spin in quantum mechanics. This is similar to how you can’t know the exact position and momentum of a particle. In QM parlance, the operators for spin around different axes don’t commute.

What you can do is measure the spin around one particular axis. And when you do, you can get one of two values: spin up or spin down. If you later measure along a different axis, this new measurement is only statistically correlated to the previous one (and is uncorrelated if the new axis is 90 degrees from the old one). If you measure the z-axis spin, then later measure the y-axis spin, then go back and measure the z-axis spin again, the spin could be different! Measuring the y-axis spin destroyed any information about z-axis spin in the system (through “wavefunction collapse”). This is the uncertainty principle at work.

If you change your frame of reference relative to the particle, the spin you measure won’t change along the same axis (the helicity might). If you measure along different axes you certainly could see how much the electron was spinning “sideways” — but as above, you lose any information of how it was spinning “up” or “down.”

• Ken

Argh this was so confusing! The “summary” contradicts all the text and diagrams before it! I certainly hope the “summary” part was right, because that’s what I’m gonna stick in my brain.

Just the resource I was looking for, and at an engaging level. Thanks!

• Richard

There seem to be two ways to take combinations of the e_L and e_R states, e.g. when the electrons interact with the Higgs field, what choses the relative phases of the e_R and e_L components?

e_S = (e_L + e_R) / sqrt(2)
e_A = (e_L + e-R) / sqrt(2)

Could I pick a symmetric and antisymmetric combination of these particles? etc. Do the different linear combinations have different masses? Does the full picture take the electron muon and tau chiral states, and mix them to produce real particles?

• Thomas Walsh

“As we said above, for massless particles the chirality and helicity are the same. A massless left-chiral particle also has left-helicity”. The photon is massless; can it be left-chiral/helicity and/or right-chiral/helicity. Are there then two form of the photon? If the photon carries the electro-magnetic force, does this mean both forms carry the electro-magnetic force or just one form? If just one form, then the other form carries…?

• Hi everyone—I think some of the notation I use here is unnecessarily cumbersome, I apologize. For an overview using simpler notation, see my particle physics lecture notes here: http://www.lepp.cornell.edu/~pt267/undergradparticles.html

@Richard: great question. The Higgs will interact with an eR and an eL. It cannot talk to two eRs at the same time, nor to two eLs at the same time. This is imposed by the “gauge” structure of the theory: the Higgs and eL are sensitive to the weak force, while the eR is not. The particular interaction of a Higgs-eR-eL is what is required to conserve weak charge (and also hypercharge, which is related to the weak and electric charges).

The particular combinations that have well defined masses are given by the Higgs interactions. You end up with three massive particles with well defined masses, after starting with six massless particles. (Recall one massive fermion is equivalent to two massless fermions.)

@Thomas: photons gauge bosons and have spin-1, the discussion here focused on fermions which are spin-1/2. But indeed, photons can be left- or right-handed—we know this more commonly as being left or right circularly polarized. It’s the same photon, but just spinning in different directions—it both polarizations carry the same electromagnetic force.

• Rene Kail

Hi Flip

For your convenience, I endeavor to correct and reformulate your section “Chiral Theories”, providing your bloggers with a more easy reading. The following text is based on the asumption that the now published Table “Important Summary” related to the electron an positron is correct and that all particle icons are drawn correctly.
You may check the new text ocasionally and reply if you agree.

Kind greetings from Switzerland

Rene

Chiral theories

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! The weak force shows a selective behavior, violating Parity symmetry. (Of course, biologists are already familiar with this from ‘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 the helicity of a particle are the same. The W will couple to both left- and right-helicity particles: the electron and the anti-electron respectively. However, it still ignores the positron and anti-positron. In other words, the W will couple to an electric charge -1 left-handed particle (the electron), but does not couple to an electric charge -1 right-handed particle (the anti-positron). This is a very subtle point!

Technical Remark:
etc…

In order to really drive this point home, let me reintroduce two particles to you: the electron and the positron. You already know that the positron is the anti-partner of the electron… but for now, pretend you didn’t know that. The electron shown below has left-handed helicity or spin, while the positron has right-handed helicity or spin. Both are chiral left-handed. They’re two completely different particles.

Figure

The electron (with left-handed spin) and the positron (with right-handed spin) are two completely different particles, as evidenced by the positron’s moustache. Both are left-chiral.

How different are these particles? Well, the electron has electric charge -1, while the positron has electric charge +1. Further the electron can couple to a neutrino through the W-boson, while the 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 (“carries weak charge”), whereas the left-chiral positron is not. Note that at this stage, even the electron and the anti-positron are NOT the same particle! Even though they both have the same electric charge and the same helicity, the electron can talk to a W, whereas the antipositron cannot.

For now let us assume that all 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-handed chirality which couples to the W-boson. The positron also has a different antiparticle (the “anti-positron”) which also has right-handed chirality, but does NOT couple to the W-boson. We thus have a total of four particles (plus the four with opposite helicities):

Figure

The electron, anti-electron, positron and anti-positron. (Anti-particles are drawn with a slight green tint). It is crucially important that the electron and anti-positron are two different particles.

Technical Remark:
etc….

• Peter A Lawrence

Flip,

The Standard Model is weird because “electrons” with different chirality behave differently, those with one chirality interact with W bosons, those with the other chirality do not.

here is my reaction to that:

just as photons are the bosons that allow electrically charged fermions to interact, so likewise W bosons are what allows “???”-charged fermions to interact.

therefore my question is where is the Right-Chiral “???”-charged “electron”, it sure seems like it should exist even if we have not observed it.

-Peter A. Lawrence, San Jose, CA.

• Brian

Hey Flip, I recently wrote a paper regarding chirality and the weak interaction, and thought you might be interested in looking at it. I’m not sure it’s correct – you might have some insight. Let me know if you’re interested and if so where to send it or if I can post it here, etc. Thanks.

• Jesse

Hi Flip,

I want to clarify my doubt here. Isn’t the physicsl electron the combination of left helicity left chiral electron component and right helicity left chiral anti-positron component?

left helicity left chiral electron will have left helicity right chiral positron in mirror (CPT). while left helicity left chiral electron would have right helicity right chiral anti-electron projection in morror. In case of positron, left helicity right chiral would have right helicity left chiral anti-positron.

in this scenario, left helicity left chiral electron should combine with righthelicity left chiral anti-positron and right helicity left chiral electron should combine with left helicity left chiral anti-positron.

Please think on it and resolve this doubt. I’m totally perplexed you can imigine this in writing that i can’t even describe simply. many thanks!!

• Jesse

and i more question: In electron pairing, electron up and down mean left and right helicity or left and right chirality?

• henry-couannier

Hello ,

The analogy of the chirality mixing mass term with another kind of mixing that takes place in the neutrino sector is interesting but raises a few questions for me:

Shouldnt we expect to be able to test (experimental evidence) a kind of disapearing effect if starting from a left chiral electron eigenstate of the weak interaction (emitted in a weak interaction for instance) then after some flight distance again a weak interaction only sensitive to the left chiral component of the mixing (mass mixing term due to propagation) should manifest giving only half of the interactions that would have resulted from an electron remaining pure left during its propagation. Is that the case?

Alternatively if the electron interacts electromagnetically is it possible to probe experimentally (for instance by studying the angular repartition of the tracks) wether the electron then interacted as a left or right chiral particle. The latter case would be an evidence for oscillation as for the neutrinos case.

In other words , is it possible to probe the existence of the chriral mixing term appart from the existence of mass itself? Does this question make sense at all?

best

F H-C

• Haibara

Hi.why no helicity=0 for massless particle?

• So … what is the chirality of the physical electron, at rest? It looks like it should be 0?

• Joseph

Hi Flip,

I’m not sure where the best place is to post this question, but I thought here would be good. I’m wondering what the explanation is (if there is one) for how mass increases in terms of the Higgs mechanism. The mass of the leptons comes from how frequently they interact with particles from the Higgs condensate (is that correct?…). At speeds where relativity becomes relevant, mass increases, and so does that mean that at higher speeds it becomes more likely for a particle to interact with Higgs particles? Is the answer insightful in any way?

• Quantum Diaries

Hi Joseph,
A good place for reading more about the Higgs mechanism and how it relates to the mass of the fundamental particles would be this post:
http://www.quantumdiaries.org/2013/10/07/higgs-mechanism-for-electrical-engineers/
Or this post:
http://www.quantumdiaries.org/2012/03/02/all-on-the-higgs-for-nearly-everyone/

• Skyguy93

First of all, love this blog. I’m not a physics student, but I decided to try and understand the Higgs and the only thing I don’t understand is how is the “neutral Higs” different than “The Higgs”…it’s right chiral, as opposed to spinless? What then is the difference between H+ and H-? one of them got their weak charge from an electron and one of them got their weak charge from an anti-electron? But which is which?

Also, if I understand all of this correctly, when a fermion hits “the Higgs” the higs particle is converted into the H+ H- H0 bosoms which are then promptly asborbed by the weak force bosoms. And because of this there is really only one Higs ( the other three being unstable?)

• Walter

I just want to get it clear in my head what’s happening here!
The mixing that’s happening,it’s confusing to me,the naming

For example take the electron defined as the particle “as it is now”a mix between two quanta.
If it’s a mix, what actually are the two mixed particles, one can’t surely therefore be an electron as we NOW know it (as you suggest) which puts in doubt that the other particle is an anti-positron?
The properties of the mixed particle and each individual components that makes it up must also differ ?

I’ve been trying to get clarity on this since I doing a presentation on the Higgs and this part I can’t resolve, it’s almost if you don’t mind me saying, that your making it up ?

Walter

• Walter

No need to reply to my previous message, if indeed anyone was going to.
I realise it’s a complex issue and I am following up on it using other sources etc, so hopefully it will understand it eventually.

Walter

• You wrote: “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).”

I did a double-take on this paragraph, but I believe you simply switched from talking about massless particles (that you cannot catch up with) to massive particles (that you can).

I don’t believe I know a single fact about the weak force that isn’t weird.

• gary

Excellent explanation . So my question now is what exactly is the empirical evidence for separating an electron into two chiralities…Is there experimental evidence that one chirality ‘interacts” with the W ? Or is there some other evidence?
Thanks.

• Ivan

You say here the positron has right-chiral: “Electrons (left-chiral) and positrons (right-chiral) are two completely different particles, as evidenced by the positron’s mustache.”
And you say here that it has left-chiral: “Positron: left-chiral, charge +1, cannot interact with the W”

• Jakob

Hi Flip, now with your changes your text has become impossible to understand because you change your names halfway trough. As pointed out by Ivan you start with defining the positron as right-chiral. This way you define the usual Dirac spinors as left-chiral electron and left-chiral anti-positron. In addition, you point out that the electron and the antipositron have the same charge and chirality. I think you noticed that this can’t be correct, because during propagation something left-chiral mixes with something right-chiral. With your definitions from the beginning this is impossible, because everything right-chiral has positive charge and therefore we would have a mixing of charge during propagation. Therefore you use a new definition after the “important summary”. Then, there you define the positron as left-chiral and we can have a mixing of the electron (=left-chiral) with the anti-positron (right-chiral) without violating charge conservation. Anyway, then it becomes unclear what we write conventionally into the Dirac spinor and call electron and positron. I’m pretty sure you’re aware of all this, but rewriting would mean drawing lots of pictures again. Nevertheless a short remark may be in order, because things are really confusing right now.

• Kevin

How many times per second does an electron at rest interact with the Higgs field? Is the interaction rate proportional to the rest mass of the particle? As a particle is accelerated does it interact less frequently from an inertial observer’s perspective? If so, is the change in interaction rate due just to time dialation or some other effect?

Great article! Slight typo… “Let us call this propagating particle is a “physical electron.” Change to “Let us call this propagating particle a “physical electron.”

Wow, thanks for that. Guess some people will always take things negatively. As an author I greatly appreciated all the feedback I got from others, catching my typos, letting me know which sections needed to be made clearer, etc. I really liked this article and can only assume that with all the work that went into it the author might be seeking feedback. Please try not to exercise your attacks without thinking it through. “

Study Says People Who Continually Point Out Typos Are ‘Jerks’

• koun

I think having two RH 1/2 particles which has same charge +1 and call one positron another anti-electron is making non sense. There’s nothing like the colour (‘yellow, green’) you used to distinguish them in the real universe.
You don’t have RH 1/2 particle with charge = -1 and LH 1/2 particle with charge = +1 in your particle list – among the 4 particles you drew together, the two particles on the right side should flip their chirality and you could call them positron(LH) and anti-positron(RH).
Also change as:
refuses to talk to positrons (right-chiral positrons and left-chiral anti-positrons).
->
refuses to talk to positrons (left-chiral positron and right-chiral anti-positrons).

• Topor Alexandru

None really, the pairing in orbitals is simply a pairing of magnetic moments, helicity is defined for free electrons. Electron moving around a nucleus have both orbital angular momentum and spin angular momentum, but the direction of the spin vector is completely random for a simple paramagnetic atom.

• Hugo

I did understand your explanation, thanks. The only thing is that some times doble negatives make me a little confused. For instance “do not cause any charge non-conservation” I have to translate it into “do cause charge conservation” and hope it maintains its meaning. Thanks again for that wonderful explanation.