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Congratulations goes out to fellow US LHC Blogger Prof. Sarah Demers for just being awarded the Department of Energy’s Early Career Award.  The announcement is naturally featured prominently on the website of her home institution, Yale University Physics Department.  This award has recently replaced the long-standing DOE Outstanding Junior Investigator Award (OJI), which has awarded grants to promising junior faculty members from 1978-2008, an impressive run!  The new Early Career award and has brought the previous National Science Foundation’s Early Career Award and DOE OJI awards together to a more similar format and award level.

These awards can mean a tremendous amount to a new faculty member in particle physics. I was fortunate enough to receive an OJI from the DOE, and fellow blogger Prof. Ken Bloom was fortunate to receive a Career Award from NSF, when we were both new junior professors.  This allowed us both to support perhaps a graduate student and part of a postdoc’s salary as well as our own summer salaries while we established our research programs as new faculty members.  Now Sarah has earned a peer-reviewed grant, which is a major milestone for a new professor, and which enables her to proceed with her successful research program without relying on university start-up funds (which eventually dry up).  Here’s to Sarah’s future success!

Photo by Waldo Jaquith

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:

1. Different particles are represented by lines. We now have three kinds: fermions (solid lines with arrows), gauge bosons (wiggly lines), and scalars (dashed lines).
2. When these particles interact, their lines intersect. The “rules” above tell us what kinds of intersections are allowed.
3. 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:

1. The Higgs boson has a special Feynman rule where a line can terminate.
2. This means we can take any interaction and effectively remove the Higgs line by terminating it immediately after the vertex.
3. In particular, this means that we generate a vertex with just two lines.
4. 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 a mass for 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 why the two-point interaction from terminating a Higgs line should be interpreted as a mass. 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.
• 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 many different particles can you make from quarks? A lot. Every two years or so, the particle data group puts out a catalog of the ones we know about. I always love getting mine in the mail. It’s as big as a phone book, with thin paper like a Bible. The compilation of all the particles and their properties represents a truly massive intellectual effort. Most of the hadrons are just labeled with Greek letters, but they’re festooned with all kinds of superscripts and asterisks, and their properties have names as colorful and idiosyncratic as their discoverers. For example, the neutral Ξ or “cascade” hyperon is a doubly-strange baryon with negative half-integer isospin. To my ear, most science fiction falls flat compared to real conversations between particle physicists.

By adding energy to hadrons, they can change their nature and go into excited states called resonances. The idea is loosely analogous to exciting atoms in a laser or fluorescent lamp, except more relativistic. Their mass can change. The humble proton, for example, can be excited into something called a Δ resonance, which is around 30% more massive, because some of the absorbed energy converts to mass. They don’t hang around very long, but as you look at higher and higher masses, you see more and more of them. By the 1960’s, the number of newly discovered particles and resonances had grown rapidly in step with the energy of the accelerators that produced them. This proliferation led to questions about how to explain such large variety, and what, if any, the limitations are in the number of states. When the quantum-mechanical rules governing properties like spin, charge, angular momentum, etc. were taken into account, the number of hadronic states was found to rise exponentially with mass. This plot is a fairly recent example:

Up to a certain mass, the number of hadrons rises exponentially. The red curve includes particles that weren't plotted in earlier references, represented by the green curve.

When you see a straight line on a semi-log plot, it’s a dead giveaway for an exponential form. Why is that pattern followed? What’s even more interesting is that the number of particles rises with mass at the same rate as it falls with increasing (transverse) momentum, at least below a few GeV. Several creative ideas emerged as attempts to explain the hadron spectra, but a physicist named Rolf Hagedorn gets the credit for developing a theory using statistical mechanics. This is before the era of quarks, remember: he referred to hadrons as “fireballs”, and considered that the heavy resonances were compositions of lighter ones, which were in turn composed of still lighter ones. In one of his lively papers, he said:

His mathematical line of reasoning implied that if you were to collect a bunch of hadrons together and treat them as a gas of particles, their energy would become infinite as the temperature approached a limiting value. He seems to have been quite a character. In the same paper, he concluded:

It follows that T is the highest possible temperature—a kind of ‘boiling point of hadronic matter’ in whose vicinity particle creation becomes so vehement that the temperature cannot increase anymore, no matter how much energy is fed in.

And now we come to the point. Hagedorn’s argument implies a change in the number of fundamental degrees of freedom of the system. In other words, it has to break down to more fundamental building blocks. Instead of remaining as a gas of hadrons, a superheated system would melt into a phase with simpler constituents at a temperature near what is now known as the Hagedorn temperature. Using the best data available, he extrapolated from the known spectra to obtain a value of the critical temperature near 160 MeV, or in more familiar units, a trillion degrees Celsius.

With a more sophisticated understanding thanks to Quantum Chromodynamics (QCD), more tools have become available to check this number. It’s a tough job, because this physics lies in the so-called “non-perturbative” regime,  where pencil-and-paper solutions to the QCD equations don’t work well. But that’s what supercomputers are for. The founders of QCD devised a way to crunch out the answers by dividing space-time itself into a grid of points called the lattice, “playing” the equations forward numerically in steps. It takes a lot of CPU cycles, but the answer seems to corroborate Hagedorn’s estimate.

So nuclear matter melts if you get it hot enough. It was suggested over 40 years ago, and theoretical innovations only seem to confirm it. So what happens then? And is this temperature achievable in the lab? I’ll post again soon to follow up on these questions.

Hi everyone! Readers of this blog might enjoy some of the following recent multimedia by some well-known  particle physicists.

• First, a podcast from Jim Gates of the University of Maryland about his path in  Go Tell It on the Mountain (link to iTunes, link to mp3) from The Moth. The talk is from the 2008 World Science Festival, which will be held again this year in New York City in a month.
• Next, a very nice animated discussion with Daniel Whiteson and Jonathan Feng from UC Irvine on PhD Comics. They discuss dark matter, particle physics, and the Large Hadron Collider.
• Along the lines of dark matter and particle physics, here’s a mission briefing from NASA on AMS-2, the “particle detector in space,” featuring principal investigator (and Nobel laureate for the discovery of the J/ψ particle) Sam Ting. Matt mentioned AMS-2 in his inaugural post. A lot of particle physicists are excited about AMS due to recent anomalies in the spectrum cosmic positrons and anti-protons that may be a result of dark matter interactions.
• Finally, some time ago I had a general-public-level post about Nima Arkani-Hamed‘s (and collaborators) work in scattering amplitudes. For those with a technical background who interested in learning more, his informal lectures to the Cornell particle theory group are now posted online: part 1, part 2, part 3, part 4, part 5. For those who can’t get enough, there’s also an ongoing program at the KITP with lots of recorded talks. These links are at the level of theoretical physicists doing work in the field; for a general public version, see Nima’s messenger lectures.