A couple of weeks ago we met the Higgs boson and discussed its Feynman rules.

I had forgotten to put up the obligatory Particle Zoo plush Higgs picture in my last post, but US LHC readers will know that Burton has the best photos of the [plushy] Higgs. (It seems that the Higgs has changed color over that the Particle Zoo.)

We learned that the Higgs is a different kind of particle from the usual **gauge boson** “force” particles or the **fermion** “matter” particles: it’s a **scalar** particle which, for those who want to be sophisticated, means that it carries no intrinsic quantum mechanical spin. Practically for these posts, it means that we ended up drawing the Higgs as a dashed line. For the most part, however, the Feynman rules that we presented in the previous post were pretty boring…

Recall the big picture for how to draw Feynman diagrams:

- Different particles are represented by lines. We now have three kinds: fermions (solid lines with arrows), gauge bosons (wiggly lines), and scalars (dashed lines).
- When these particles interact, their lines intersect. The “rules” above tell us what kinds of intersections are allowed.
- If we want to figure out whether a process is possible, we have to decide whether or not we can use the rules to convert the initial set of particles into the final set of particles.

If you’ve been following our posts on Feynman diagrams, then you might already be bored of this process. We could see how electrons could turn into muons, or even how the Higgs boson might be produced at the LHC; but now we’ve arrived at the Higgs boson—one of the main goals of the LHC—where is the pizzazz? What makes it *special*, and how do we see it in our Feynman rules?

## The Higgs *is* special

It turns out that the Higgs has a trick up it’s sleeve that the other particles in the Standard Model do not. In the language of Feynman diagrams, a Higgs line can terminate:

The “x” means that the line just ends; there are no other particles coming out. Very peculiar! We know that ordinary particles don’t do this… we don’t see matter particles disappearing into nothing, nor do we see force particles disappearing without being absorbed by other particles. We can think about what happens when matter and anti-matter annihilate, but there we usually release energy in the form of force particles (usually photons). The above rule tells us that a single Higgs line—happily doing its own thing—can be suddenly be cut off. It shouldn’t be read as an initial state or final state particle. It’s just some intermediate line which happens to stop.

We’ll discuss the physical meaning of this in upcoming posts. Sometimes when people try to explain the physical meaning they can get caught up in their own analogies. Instead, let us use the Feynman diagrams as a crutch to see the effects of this weird Feynman rule. Recall that in the previous post we introduced a four-point Higgs self-interaction (“four-point” means four Higgs lines intersecting):

If we take one of the lines and terminate it, we end up with a ** three-**point Higgs self interaction:

In fact, since the crossed out line isn’t doing anything, we might as well say that there is a new Feynman rule of the form

Now that’s somewhat interesting. We could have forgotten about the “crossed out Higgs line” rule and just postulated a three-point vertex. In fact, usually this is the way people write out Feynman rules (this is why our method has been “idiosyncratic“); however, for our particular purposes it’s important to emphasize that what people *really *mean is that there is implicitly a “crossed out Higgs line.” The significance is closely tied up to what makes the Higgs so special.

We could play this game again and cross one one of these three lines. This would lead us to a ** two-point** Higgs interaction.

Once again, we could just as well chop off the two terminated lines and say that there is a ‘new’ two-point Higgs Feynman rule. But this is really just a line, and we already knew that we could draw lines as part of our Feynman rules. In fact, we know that that lines just mean that a particle moves from one place to another. So it seems like this interaction with two crossed out lines doesn’t give us anything news.

… except there’s more to it, and this is where we start to get a hint of the magic associated with the Higgs. Let me make the following statement without motivation:

Claim:the above Feynman rule is a contribution to the Higgs mass.

At this point, you should say something incredulous like, “*Whaaaaaat?*” Until now, we’ve said that particles have some particular mass. The number never really mattered that much, some particles are lighter than others, some particles have zero mass. Mass is just another property that each particle seems to have. Now, however, we’ve made a rather deep statement that puts us at the tip of a rather large iceberg: we’re now relating a particular Feynman rule to the mass of the particle, which we had previously assumed was just some number that we had to specify with our theory.

We’ll have to wait until my next post to really get into *why* such a relation should exist and really what we even mean by mass, but this should at least start to lend credence to the idea that the Higgs boson can give masses to particles. At this point this should still feel very mysterious and somewhat unsatisfying—that’s okay! We’ll get there. For now, I just want you to feel comfortable with the following string of ideas:

- The Higgs boson has a special Feynman rule where a line can terminate.
- This means we can take any interaction and effectively remove the Higgs line by terminating it immediately after the vertex.
- In particular, this means that we generate a vertex with just two lines.
- This vertex with two lines should—for reasons which are presently mysterious—be identified with mass.

# Giving mass to the other particles

Now that we see how this game works, we should immediately go back to the first two Feynman rules we wrote down:

These are the interactions of the Higgs with fermions and gauge bosons. Here’s what you should be thinking:

Hm… I know that the Higgs boson line can terminate; I can just cross out the end points of a dashed line. And I just saw that when I do this to the Higgs self-interaction vertex enough times, I end up with a two-point

interaction which Flip tells me is amassfor some weird reason. Now I these two vertexes representing the Higgs interaction with two matter particles or two force particles.Does terminating the Higgs line also give mass to these particles?

The answer is ** yes! **We end up with vertices like this:

For aesthetic reasons (and really only for aesthetic reasons) we can shrink this diagram to:

We can even drop the “x” if you want to be even more of a purist… but for clarity we’ll leave it here to distinguish this from a normal line. These diagrams indeed represent a mass contribution to fermions and gauge bosons. Again, I’m just telling you this as a mysterious fact—we’ll explain *why* this interpretation is accurate later on. We’ll need to first understand what “mass” really is… and that will require some care.

# Bumping up against the Higgs

In fact, instead of saying that particles “start out” with any masses, one can formulate our entire Feynman diagram program in terms of completely massless particles. In such a picture, particles like the top quark or *Z* boson undergo lots of the aforementioned two-point “mass” interactions and so are observed to have larger masses. Heuristically, heavy particles barrel along and have lots of these two-point interactions:

For comparison, a light particle like the electron would have fewer of these interactions. Their motion (again, *heuristically*) looks more like this:

We should remember that each of these crosses is really a terminated Higgs line. To use some fancy parlance which will come up in a later post, we say that the Higgs has a “**vacuum expectation value**” and that these particles are bumping up against it. The above pictures are just ‘cartoons’ of Feynman diagrams, but you can see how this seems to convey a sense of “inertia.” More massive particles (like the top quark) are harder to push around because they keep bumping up against the Higgs. Light particles, like the electron, don’t interact with the Higgs so much and so can be pushed more easily.

In this sense, we can think of all particles as being massless, but their interactions with the Higgs generates a two-point interaction which is *effectively* a mass. Particles which interact more strongly with the Higgs have more mass, while particles which interact weakly with the Higgs have less mass. In fact, once we assume this, we might as well drop all of the silly crosses on these lines—and then we’re left with the usual Feynman rules (with *no* terminating Higgs lines) that are usually presented.

(A small technical note: the Higgs isn’t actually responsible for *all *mass. For example, bound states get masses from their binding energy. Just look up the mass of the proton and compare it to the mass of its constituent quarks. The proton has a mass of about 1 GeV, while the up/down quarks are only one *thousandth* of this. Most of the proton mass comes from the binding energy of QCD.)

# Some closing remarks

Before letting you ponder these things a bit more, let me make a few final remarks to whet your appetite for our next discussion.

- The photon, as we know, is massless. We thus expect that the
**Higgs does not interact with the photon**, or else we could have ‘terminated’ the Higgs lines in the interaction vertex and generated a photon mass. - On the other hand, the Higgs gives the
*W*and*Z*bosons mass. This means that it costs energy to produce these guys and so the weak is only really effective over a short distance. Compare this to photons, which are massless, and so can produce a long range force. (Gluons are also massless, but they have a short range force due to their confinement.) Thus the**Higgs is responsible for the “weakness” of the weak force**. - … on that note, it’s worth noting that the “weak” force isn’t really so weak—it only appears weak at long distances due to the mass of the
*W*and*Z*. If you look at shorter distances—say on distances shorter than the distance between two Higgs crosses in the cartoon picture above—then you’d find that the weak force is actually quite potent compared to electromagnetism. Thus a more accurate statement is that the**Higgs is responsible for the short-ranged-ness of the weak force**.

There are also a few open questions that are worth pointing out at this point. We’ll try to wrap these up in the upcoming posts on this subject.

- The big elephant in the room is the question of
. We got a hint in the picture above of how “bumping off the Higgs” can at least heuristically appear to have something to do with inertia. We’d like to better understand what we really mean by mass.*why*the two-point interaction from terminating a Higgs line should be interpreted as a mass - We also very glibly talked about treating everything as massless and only generating ‘effective’ masses through such Higgs interactions. Special relativity tells us that there is a
**very big difference between a particle with exactly no mass and those with some mass**… this has to do with whether or not it is possible*in principle*to catch up to a particle. How does this mesh with our picture above that masses can come from ‘bumping off the Higgs?” - What does it mean
*physically*that the Higgs line can terminate? What do we mean by the “vacuum expectation value?” This will turn out to be related to the idea that**all of our particles are manifested as quantum fields**. What does this mean? - This whole business is related to something called
**electroweak symmetry breaking,**and*that*is the phenomenon associated with the Higgs which is really, really magical.

how does the photon have no mass when it can be bent around and absorbed by a black hole?

Good Q! Because it’s not reacting to “mass” per se; it’s that space-time itself is curved (or warped) so severely around a black hole that even something going the speed of light can’t “outrun” the curve and escape.

Excellent question, Flash! The answer is that gravity couples not only to mass, but to energy. Photons have no mass, but because they have energy, they feel the curvature of spacetime. Alternately (and equivalently), they interact with gravitons.

The bottom line is that the “charge” that gravity couples to is total energy, not just rest mass.

Cheers,

F

And at the end of the day all you have ever observed, lived and believed in is a lie. Even the cake.

Cake is a lie.

I made another observation earlier – regarding flavours, colours etc… it can be mechanical analogy of degrees of freedom in mechanics. We have axes x, y, z, a, b, c. Thus particles can be imagined and manufactured as mechanical models which will or will not couple the same as the real ones. Together – when we consider the time granularity experiment – adds a little bit to the existential anxiety: where are we? WHAT IS this place? Why is it so big? Is there a space granularity too? What does this mean? Do we really consist of only state variables in a PL/1 style matrix?

Curiously intimidating, fear inducing our material world is.

Well Flip, . . . makes me wonder about mass, energy, and gravity. Thanks for the reply;

If the Higgs doesn’t interact with photons, then why is the H -> two photons decay channel one of the ones that is talked about as a search target at the LHC? Is that because it is a 3-pt instead of 2-pt interaction, so there is no mass term, and this is the next lowest diagram?

Great question, David! The Higgs does not directly interact with photons, however, it does interact with quarks and [charged] leptons, which in turn interact with photons. A good exercise is to draw this diagram. (Hint: it’s the same diagram for Higgs to gluon coupling.) As to why the Higgs can ‘indirectly’ couple to photons and not give them mass, the reason is a bit subtle. Unfortunately I’m not sure if I can explain it at the appropriate level for the blog—which means (as Feynman once said) that perhaps I don’t understand it deeply enough. The semi-technical explanation is that the particular kind of interaction between the Higgs and two photons (also two gluons) is not the sort that gives the mass term. In fancy words it couples through a quantum anomaly (triangle diagram). I’ll try to think further if I can give a better explanation. Thanks for the great question.

Maybe I don’t understand the explanation deep enough, but for me there is one more big elephant in the room. If Higgs is interacting with some particles, it is then actually representaing a friction. Thus particles should slow down and finally stop (with respect to some weird Higgs reference frame). But the Higgs interaction should happen only when I’m trying to accelerate particles. Because only during acceleration I feel the resistance of mass. Higgs must somehow identify if a particle is accelerated or not and then start doing its job. How does this happen?

Hi Martin! My apologies for the long delay before getting to your question, I was away traveling. I don’t think it’s accurate to associate the Higgs interaction with friction or that one should think of a Higgs rest frame. For the latter point, perhaps a decent analogy would be the cosmological constant—there’s no “cosmological constant” rest frame. (Just don’t try to take this analogy much further, the point is just that there’s a background Higgs value which is—I believe—Lorentz invariant.)

The usual Higgs boson—the particle that we’d like to produce at the LHC—interacts via a Yukawa interaction, which is effectively the same way that physicists in the 50s modeled the interactions between baryons and mesons. The subtle things are the interactions which induce a particle mass. These are interactions with the Higgs vacuum expectation value (see my recent post) which do not transfer any momentum. It’s kind of a weird idea. In fact, usually when grad students first learn this there’s a bit of sleight of hand so that they don’t usually think about mass in this way.

I hope that helps—apologies if my answer is a bit vague.

-F

I am a layman.

I am interested in vorticies. Is the reason the black hole is black, because we are exceeding the speed of light???

A real big Bang can only occur, in a vortex, when super speed particles come in contact with a solid mass.

What happens when we create a vortex of higher frequency particles (bluer parts of the electromagnetic spectrum?? I only know light but know it is only a small part of the spectrum). Seems like REAL energy is to be found there.

Since I am on a roll, I thought i would say that the Big Bang occurred when a vortex of energy laden particles HIT a Super planet. Our existing planets are simply fragments of that Super planet and the stars are simply pieces that haven’t yet cooled.

what is relation between gravity and curvature of space ?

As an engineer “growing up” in the second half of the 20th century,a great deal of which was devoted to designing and building nuclear power plants – from their inception, literally, I am spellbound by your current day descriptions and explanations of particle mass and energy. In fact, I just don’t get it — yet!

But tell me, if you will, how would you expect the Einstein of e=mc square,to react to your current day representations of mass-less particles?

Thank you!

Hi Paul, thanks for the kind words. I think the following post addresses your question: E=mc2. Thanks! -F

What are non perturbative objects?

excellent

Hi Flip! Thank you for your posts! I have a question:

you have been talking about the Higgs boson, and you gave the Feynman rules for it. But when you talk about particles acquiring masses, shouldn’t you refer to the Higgs FIELD instead? Can you clarify a little bit about this?

The motion of an electron looks like the motion of a photon in a light-clock.

If this kind of motion is the reason for mass perhaps it is also the reason for time dilation at all?