Quantum Diaries

Hi everyone—it’s time that I wrap up some old posts about the Higgs boson. Last December’s tantalizing results may end up being the first signals of the real deal and the physics community is eagerly awaiting the combined results to be announce at the Rencontres de Moriond conference next month. So now would be a great time to remind ourselves of why we’re making such a big deal out of the Higgs.

Review of the story so far

Since it’s been a while since I’ve posted (sorry about that!), let’s review the main points that we’ve developed so far. See the linked posts for a reminder of the ideas behind the words and pictures.

There’s not only one, but four particles associated with the Higgs. Three of these particles “eaten” by the W and Z bosons to become massive; they form the “longitudinal polarization” of those massive particles. The fourth particle—the one we really mean when we refer to The Higgs boson—is responsible for electroweak symmetry breaking. A cartoon picture would look something like this:

The solid line is a one-dimensional version of the Higgs potential. The x-axis represents the Higgs ”vacuum expectation value,” or vev. For any value other than zero, this means that the Higgs field is “on” at every point in spacetime, allowing fermions to bounce off of it and hence become massive. The y-axis is the potential energy cost of the Higgs taking a particular vacuum value—we see that to minimize this energy, the Higgs wants to roll down to a non-zero vev.

Actually, because the Higgs vev can be any complex number, a more realistic picture is to plot the Higgs potential over the complex plane:

Now the minimum of the potential is a circle and the Higgs can pick any value. Higgs particles are quantum excitations—or ripples—of the Higgs field. Quantum excitations which push along this circle are called Goldstone bosons, and these represent the parts of the Higgs which are eaten by the gauge bosons. Here’s an example:

Of course, in the Standard Model we know there are three Goldstone bosons (one each for the W+, W-, and Z), so there must be three “flat directions” in the Higgs potential. Unfortunately, I cannot fit this many dimensions into a 2D picture. The remaining Higgs particle is the excitation in the not-flat direction:

Usually all of this is said rather glibly:

The Higgs boson is the particle which is responsible for giving mass.

A better reason for why we need the Higgs

The above story is nice, but you would be perfectly justified if you thought it sounded like a bit of overkill. Why do we need all of this fancy machinery with Goldstone bosons and these funny “Mexican hat” potentials? Couldn’t we have just had a theory that started out with massive gauge bosons without needing any of this fancy “electroweak symmetry breaking” footwork?

It turns out that this is the main reason why we need the Higgs-or-something-like it. It turns out that if we tried to build the Standard Model without it, then something very nefarious happens. To see what happens, we’ll appeal to some Feynman diagrams, which you may want to review if you’re rusty.

Suppose you wanted to study the scattering of two W bosons off of one another. In the Standard Model you would draw the following diagrams:

There are other diagrams, but these two will be sufficient for our purposes. You can draw the rest of the diagrams for homework, there should be three more that have at most one virtual particle. In the first diagram, the two W bosons annihilate into a virtual Z boson or a photon (γ) which subsequently decay back into two W bosons. In the second diagram it’s the same story, only now the W bosons annihilate into a virtual Higgs particle.

Recall that these diagrams are shorthand for mathematical expressions for the probability that the W bosons to scatter off of one another. If you always include the sum of the virtual Z/photon diagrams with the virtual Higgs diagram, then everything is well behaved. On the other hand, if you ignored the Higgs and only included the Z/photon diagram, then the mathematical expressions do not behave.

By this I mean that the probability keeps growing and growing with energy like the monsters that fight the Power Rangers. If you smash the two W bosons together at higher and higher energies, the number associated with this diagram gets bigger and bigger. If  these numbers get too big, then it would seem that probability isn’t conserved—we’d get probabilities larger than 100%, a mathematical inconsistency. That’s a problem that not even the Power Rangers could handle.

Mathematics doesn’t actually break down in this scenario—what really happens in our “no Higgs” theory is something more subtle but also disturbing: the theory becomes non-perturbative (or “strongly coupled”). In other words, the theory enters a regime where Feynman diagrams fail. The simple diagram above no longer accurately represents the W scattering process because of large corrections from additional diagrams which are more “quantum,” i.e. they have more unobserved internal virtual particles. For example:

In addition to this diagram we would also have even more involved diagrams with even more virtual particles which also give big corrections:

And so forth until you have more diagrams than you can calculate in a lifetime (even with a computer!). Usually these “very quantum” diagrams are negligible compared to the simpler diagrams, but in the non-perturbative regime each successive diagram is almost as important as the previous. Our usual tools fail us. Our “no Higgs theory” avoids mathematical inconsistency, but at the steep cost of losing predictivity.

So if we took our “no Higgs” theory seriously, we’d be in an uncomfortable situation. The theory at high energies would become “strongly coupled” and non-perturbative just like QCD at low energies. It turns out that for W boson scattering, this happens at around the TeV scale, which means that we should be seeing hints of the substructure of the Standard Model electroweak gauge bosons—which we do not. (Incidentally, the signatures of such a scenario would likely involve something that behaves somewhat like the Standard Model Higgs.)

On the other hand, if we had the Higgs and we proposed the “electroweak symmetry breaking” story above, then this is never a problem. The probability for W boson scattering doesn’t grow uncontrollably and the theory remains well behaved and perturbative.

Goldstone Liberation at High Energies

The way that the Higgs mechanism saves us is somewhat technical and falls under the name of the Goldstone Boson Equivalence Theorem. The main point is that our massive gauge bosons—the ones which misbehave if there were no Higgs—are actually a pair of particles: a massless gauge boson and a massless Higgs/Goldstone particle which was “eaten” so that the combined particle is massive. One cute way of showing this is to show the W boson eating Gold[stone]fish:

Indeed, at low energies the combined “massless W plus Goldstone” particle behaves just like a massive W. A good question right now is “low compared to what?” The answer is the Higgs vacuum expectation value (vev), i.e. the energy scale at which electroweak symmetry is broken.

However, at very high energies compared to the Higgs vev, we should expect these two particles to behave independently again. This is a very intuitive statement: it would be very disruptive if your cell phone rang at a “low energy” classical music concert and people would be very affected by this; they would shake their heads at you disapprovingly. However, at a “high energy” heavy metal concert, nobody would even hear your cell phone ring.

Thus at high energies, the “massless W plus Goldstone” system really behaves like two different particles. In a sense, the Goldstone is being liberated from the massive gauge boson:

Now it turns out that the massless W is perfectly well behaved so that at high energies. Further, the set of all four Higgses together (the three Goldstones that were eaten and the Higgs) are also perfectly well behaved. However, if you separate the four Higgses, then each individual piece behaves poorly. This is fine, since the the four Higgses come as a package deal when we write our theory.

What electroweak symmetry breaking really does is that it mixes up these Higgses with the massless gauge bosons. Since this is just a reshuffling of the same particles into different combinations, the entire combined theory is still well behaved. This good behavior, though, hinges on the fact that even though we’ve separated the four Higgses, all four of them are still in the theory.

This is why the Higgs (the one we’re looking for) is so important: the good behavior of the Standard Model depends on it. In fact, it turns out that any well behaved theory with massive gauge bosons must have come from some kind of Higgs-like mechanism. In jargon, we say that the Higgs unitarizes longitudinal gauge boson scattering.

Caveat: Higgs versus Higgs-like

I want to make one important caveat: all that I’ve argued here is that we need something to play the role of the Higgs in order to “restore” the “four well behaved Higgses.” While the Standard Model gives a simple candidate for this, there are other theories beyond the Standard Model that give alternate candidates. For example, the Higgs itself might be a “meson” formed out of some strongly coupled new physics. There are even “Higgsless” theories in which this “unitarization” occurs due to the exchange of new gauge bosons. But the point is that there needs to be something that plays the role of the Higgs in the above story.

Visit from the Higgs

by Flip Tanedo (during a long post-dinner research break)

With apologies to Clement Moore, author of “A Visit from St. Nicholas,” aka “[Twas] The Night Before Christmas”. These verses contain little/no scientific content (no rumors, just rhymes) and are here just for some timely holiday silliness. For those who are confused about what’s going on, see Aidan’s liveblog.

‘Twas the night before Higgsmas, when all through the lab,
not a student was stirring—except some undergrad.
The data were analyzed with lots of great care
in hopes that the Higgs boson soon would be there.

The press corps were nestled all snug in their beds,
while visions of exclusion plots danced in their heads.
And theorists in the US, Asia, and Europe
dug up the models that they were so sure of.

When out from Geneve there arose such a clatter,
We sprung from our desks to see what was the matter.
Away to the webcast—I must install Flash,
Reloaded the webpage, I hope it didn’t crash.

The introduction recapped the latest CERN run,
and gave the impression of more fun to come.
When, what to my wondering eyes should I see,
but a miniature bump… in Higgs to ZZ?

And with all of the press and media bigwigs
I knew in a moment that it must be the Higgs.
From ATLAS and CMS the results were the same,
and we whistled, and shouted, and called them by name:

Now Higgs! Now Englert! Now Guralnik and Hagen!
On Kibble! On Brout! On, Goldstone and Anderson!
To Stockholm in December, the Nobel prize,
But a prize that only three could realize.

We wondered about the “look elsewhere effect,”
But somewhere, someone just won their Higgs bet.
Not so fast, of course, it was only three sigma.
That’s okay—it could be a ‘discovery’ by summer.

Not so fine tuned, in fact still quite natural,
in spite of electroweak precision observables,
at least in the supersymmetric Standard Model.
There’s room for new physics, we can be hopeful!

The Higgs mass? A hint? A whisper, a whim?
Theory papers will fill arXiv up to its brim.
And with a white Santa-like beard, who is this?
Oh my, straight from CERN-TH—it’s really John Ellis!

His eyes — how they twinkled! His dimples—how merry!
He spoke many great things about supersymmetry.
I tried to refrain myself from asking if he knew
That he was still off by a factor of two.

But I really shouldn’t write that here on this blog
For soon I’ll be applying to be a postdoc.
I digress. The matter we should focus on
is what’s next in the search for the Higgs boson.

It is now up to ATLAS and CMS
To combine their data in a way that makes sense.
In maybe a month, maybe early next year,
We will have new significances to hear.

We gave up our breaks and went straight to our work,
Life as a grad student! But it sure has its perks.
What’s more exciting than the science frontier?
And by reading this blog, you can also be there!

We sprang to our desks, we downed our espressos,
All in the search for what new physics might show.
And John Ellis exclaimed, to the OPERA bambinos,
“Happy Higgsmas to all, and forget those neutrinos”.

Here’s another dispatch from the intensity frontier—that is, the ongoing Fundamental Physics of the Intensity Frontier workshop in Rockville, MD… and Twitter. This one ties in to our exploration of Feynman diagrams, too.

Today’s “charged lepton working group” had an excellent experimental overview talk by Chris Polly of Fermilab on the experimental prospects for “muon g-2″ (“g minus 2″) experiments. You can find the pdf here and the rest of the charged lepton agenda here. After explaining what the g-2 is, I’d like to discuss one of Chris’ especially nice slides where he summarized the history of the heroic g-2 calculation.

Unless otherwise noted, all images from this post are from Chris’ talk (and see further references therein!) with his tacit permission.

You may remember from high school physics that moving electric charges (i.e. currents) generate a magnetic field. Further, recall that electrons spin. Even though we think of electrons as point-like and even though this “spin” is completely quantum mechanical, this also generates a magnetic field. This means that fundamental charged particles like electrons are kind of like little bar magnets. More importantly, electrons in a magnetic field will wobble (“precess”) just like a gyroscope. So while my image of an electron is this:

Chris wants us to think about it more like this:

Don’t worry about the details; the point is that this is an object with quantum spin that behaves precisely as one would expect classically. Now the big question: so what?

The response of our “electron gyroscope” to a uniform external magnetic field is for it to wobble. The technical term is precession. The sensitivity of the electron to a magnetic field is given by something called the g-factor (related to its ‘magnetic moment’), which just happens to have the distinction of being the most accurately verified prediction in the history of physics. Just for fun, the number is something like,

g = 2.0023193043617.

Look at all those significant figures! That’s the Standard Model showing off. I suspect g stands for gyromagnetic.

Precession in a magnetic field is exhibited by all spinning charged particles, including the electron’s heavier sibling, the muon. The g-factor of the muon is slightly different from that of the electron due to quantum effects. The experimentally measured value for the muon g is

g = 2.00233184178

This observed muon g-factor happens to match the theoretical calculation up to ten orders of magnitude. Chris had a very nice slide in which he dissected the history of the heroic calculation effort leading to these ten decimal points of theory prediction.

The first part of the story comes from Dirac, one of the fathers of quantum mechanics, who predicted the leading factor of 2. This value is “almost classical,” and the subsequent two zeros after the decimal point represents the smallness of quantum corrections.

After this brief “desert” of corrections from Dirac’s prediction, the first corrections from quantum electrodynamics comes from Schwinger, who calculated the corrections from quantum field theory. This is represented by a Feynman diagram which is a correction to the usual electron-electron-photon vertex:

(Pop quiz! You should have expected that the relevant diagram has something to do with the photon coupling to the charged leptons since the photon is the force particle for the electromagnetic field.)

The next advance comes from Tom Kinoshita, who calculated higher order corrections within quantum electrodynamics. In fact, he continues to work on such calculations at tenth order in the electric coupling—at this level there are 12,672 different Feynman diagrams contributing!

The real difficulties come from quantum corrections which involve intermediate hadrons. Such diagrams come from fluctuations in which a virtual photon emits a quark/anti-quark virtual pair, which may then turn into a meson/anti-meson pair before annihilating back into a virtual photon. Recall that at low energies, the theory of quarks and gluons is very non-perturbative; thus the contribution from these virtual hadrons is actually the main source of theoretical uncertainty in the calculation of this quantity.

Finally, the next correction to this value comes from the exchange of virtual heavy gauge bosons. Because these particles are much heavier than the characteristic energy of the process (the muon mass), their quantum effects are highly suppressed.

Okay, great. This is a very well calculated object. So what? Here’s the exciting part. If we rewrite this in terms of  a quantity a (which contains the same information as g), we find:

The Standard Model prediction does not agree with the experimental observation. Of course, the relevant question is: by how much? The answer turns out to be around 3.6 standard deviations! In other words, if you’re the type of person to get excited about things quickly, then this is something which seems very intriguing. This has been a well known result for some time and people would like to continue to check this with even more precise experimental measurements and theoretical calculations—if it continues to disagree then this starts to look like a very strong hint for new physics from the intensity (low energy) frontier!

Chris opened his talk with the questions he was asked by his family over Thanksgiving:

1. So what are y’all doing up at that lab [Fermilab]?
2. Why would ya do that?

That’s exactly how Chris phrased it in his talk, mentioning his Missouri heritage. Being an excellent science communicator as well as an excellent scientist, Chris explained that his collaboration is working on a more precise measurement of the (g-2) value of the muon, which is related to its gyromagnetic ratio. When he explains, however, that this is already the most accurately physical quantity of all time, his family would again scratch their heads and wonder why this is worth measuring once again.

This really gets to the heart of the intensity frontier: by measuring very precisely known quantities down to the level of their theoretical precision, we can look for the quantum (virtual!) effects of new physics in very accurately measured observables. The point isn’t that we’re pushing from ten to eleven decimal points of precision, but rather that the next decimal point will go a long way to confirm (or refute) that the observed discrepancy is indeed a signal of new physics.

My thanks to Chris Polly for sharing his slides and for an excellent talk. All credit for the information herein goes to him… except for any mistakes, which are solely the fault of the blogger.

Hi everyone! I’m currently at the “Fundamental Physics at the Intensity Frontier” workshop in Rockville, Maryland. There are about 500 high energy physicists here who have gathered to discuss the future of “intensity frontier” physics in the United States. You can find a nice summary on Symmetry Breaking and can follow along on Twitter (#intensityfrontier). For those interested in checking out some of the slides, you can find the agenda here.

In short, the “intensity frontier” is shorthand the exploration of fundamental physics from high luminosity, that is looking for very rare processes that probe the quantum effects of new physics. (I may have to revise this personal definition after attending the workshop!) This should be contrasted with the “energy frontier,” which is what we usually discuss on this blog with the direct production of new physics at the LHC.

I’ll whet your appetite , here’s a teaser image from Nima Arkani-Hamed‘s opening talk in which he plots the “coolness” and “importance” of intensity frontier physics with respect to time:

Fermilab has now passed the “energy frontier” torch to the LHC and is restructuring towards a particle physics lab dedicated to pushing the forefront of the intensity frontier. The workshop is a very unique and very special opportunity for theoretical and experimental physicists to get together and discuss the future of particle physics in the United States. There are over 500 high energy physicists here for the next three days, which perhaps makes this the center of particle physics this side of CERN.

As stated by Henry Weerts in his welcome talk, the workshop has four goals:

1. Produce a single coherent document that explains the science opportunities at the intensity frontier.
2. Identify the experiments and facilities needed to explore the intensity frontier.
3. Demonstrate the importance of the intensity frontier to the physics and broader community.
4. Educate the community.

The last item was particularly directed to the broader community, not just physicists but also to congress and—by extension—to the general public which ultimately supports research into fundamental science. To that end, it’s a busy workshop, but I’ll do my best to provide some updates about what’s going on.

In recent posts we’ve seen how the Higgs gives a mass to matter particles and force particles. While this is nice, it is hardly a requirement there must be a Higgs boson—maybe particles just happen to have mass and there’s no “deeper” origin of that mass. In fact, there’s a different reason why particle physicists are obsessed about finding the Higgs (or something like it)—that’s called electroweak symmetry breaking.

The statement that we’d like to understand is the following:

The Higgs boson breaks electroweak symmetry spontaneously.

That’s pretty heady stuff, but we’ll take it one piece at a time. Write it down and use it to impress your friends. Just be sure that you read the rest of this post so you can explain it to them afterward. (There’s a second part to the statement that we’ll examine in a follow up post.)

Electroweak symmetry

You might be familiar with the idea that electricity and magnetism are two manifestations of the same fundamental force. This is manifested in Maxwell’s equations and is often seen written on t-shirts worn by physics undergraduates. (If you happen to own such a t-shirt, I refer you to this article.) Electroweak symmetry is, in a sense, the next step in this progression, by which the electromagnetic force is unified with the weak force. This unification into an ‘electroweak’ theory and the theory’s subsequent ‘breaking’ into separate electromagnetic and weak forces led to the 1979 Nobel Prize in Physics.

So what’s going on here? We know that the force particle for electromagnetism is the photon, and we know that the force particles for the weak force are the W+, W-, and Z bosons. Permit me to make the a priori bold claim that the “unified” set of particles are actually the following: three W bosons and something we’ll call a B boson.

What? Now there are three W particles? And what’s this funny B boson; we never drew any diagrams with that weirdo in our guide to Feynman diagrams! Don’t worry, we’ll see shortly that because of the Higgs, these particles all mix up into the usual gauge bosons that we know and love. This should at least be plausible, since there are four particles above which we know must give us the four electroweak particles that we know: the W+, W-, photon, and Z.

Note that this new “unified” batch of gauge bosons don’t really look very unified: The Ws look completely different from the B. This illustration reflects an actual physical difference: the Ws mediate one type of force while the B mediates a different force. In this sense, the “unified” electroweak symmetry isn’t actually so unified!

Electroweak symmetry  is broken

In everyday phenomena, we observe electricity and magnetism as distinct phenomena. The same thing happens for electromagnetism and the weak force: instead of seeing three massless Ws and a massless B, we see two massive charged weak bosons (W+ and W-), a massive neutral weak boson (Z) and a massless photon. We say that electroweak symmetry is broken down to electromagnetism.

Now that masses have come up you should suspect that the Higgs has something to do with this. Now is a good time to remember that there are, in fact, four Higgs bosons: three of which are “eaten” by the weak gauge bosons to allow them to become massive. It turns out that this “eating” does more that that: it combines the ‘unified’ electroweak bosons into their ‘not-unified’ combinations!

The first two are easy; the W1 and W2 combine into the W+ and W- by “eating” the charged Higgs bosons. (Technically we should now call them “Goldstone” bosons.)

We’ll say a bit more about why eating a Higgs/Goldstone can cause the W1 and W2 particles to combine into, say, a W+. For now, note that the number of “degrees of freedom” match. Recall that ‘degree of freedom’ roughly translates in to the number of distinct particle states. In the electroweak theory we have two massless gauge bosons (2 × 2 polarizations = 4 degrees of freedom) and two charged Higgses (2 degrees of freedom) for a total of six degrees of freedom. In the broken theory, we have two massive gauge bosons (2 × 3 polarizations) which again total to six degrees of freedom.

A similar story goes through for the W3, B, and H⁰ (recall that this is not the same as the Higgs boson, which we write with a lowercase h). The W3 and B combine and eat the neutral Higgs/Goldstone to form the massive Z boson. Meanwhile, the photon is the leftover combination of the W3 and B. There are no more Higgses to eat, so the photon remains massless.

It’s worth noting that the Ws didn’t combine into charged Ws until electroweak symmetry breaking. This is because [electric] charge isn’t even well-defined until the electroweak theory has broken to electromagnetic theory. It’s only after this breaking that we have a photon that mediates the force that defines electric charge.

Electroweak symmetry is broken spontaneously

Alright, we have some sense of what it means that “electroweak symmetry” is broken. What does it mean that it’s broken spontaneously, and what does this whole story have to do with the Higgs? Now we start getting into the thick of things.

The punchline is this: the Higgs vacuum expectation value (“vev” for short) is what breaks electroweak symmetry. You might want t quickly review this post where we first introduced the Higgs vev in the context of particle mass. For those who like hearing fancy physics-jargon, you can use the following line:

The Higgs vev is the order parameter for electroweak symmetry breaking.

First, let’s see why the Higgs obtains a vacuum expectation value at all. We can draw nice pictures since the vev is a classical quantity. The potential is a function that tells you the energy of a particular configuration. You might recall problems in high school physics where you had to find the minimum of an electric potential, or determine the gravitation potential energy of a rock being held at some height. This is pretty much the same thing: we would like to draw the potential of the Higgs field. (To be technically clear: this is the potential for the combined bunch of four Higgses.)

Let’s start with what a “normal” potential looks like. Here on the x and y axes we’ve plotted the real and imaginary parts of a field ϕ; all that’s important is that a point on the x-y plane corresponds to a particular field configuration. If the particle is sitting at the origin (in the middle) then it has no vacuum expectation value, otherwise, it does obtain a vacuum expectation value.

On the z axis we draw the potential V(ϕ). The particle wants to roll to the minimum of the potential, so in the cartoon above—the “normal” case—the particle obtains no vacuum expectation value. I’ll mention in passing that concave of the potential is related to the particle’s mass.

Now let’s examine what the Higgs potential looks like. Physicists refer to this as the “Mexican hat” potential (These images are based on an illustration that is often used in physics talks. Unfortunately I am unable to find the original source of this graphic and ended up re-drawing it.):

What we observe is that the origin is no longer a minimum of the potential. In other words, the Higgs wants to roll down the hill where it can have lower potential energy. I’m not telling you why the potential is shaped this way (there are a few plausible guesses), and within the Standard Model this is an assumption about the Higgs.

So the Higgs must roll off of its hill into the ravine of minimum potential energy. This happens at every point in spacetime, meaning that the Higgs vev is “on” everywhere and matter particles can bounce off it to obtain mass. There’s something even more important though: this vev breaks electroweak symmetry.

In the cartoons above, there’s something special about the origin. If the particle sits at the origin, you can do a rotation about the x-y plane and the configuration doesn’t change. On the other hand, if the particle is off of the origin, then doing a rotation will send the particle around along a circular trajectory (shown as a solid green line). In other words, the rotational symmetry is broken because the physical configuration changes.

The case of electroweak symmetry is the same, though it requires more dimensions than we can comfortably draw. The point is that there are originally four Higgses which are all parts of a single “Higgs.” In the unified theory where electroweak symmetry is unbroken, these four Higgses can be rotated into one another and the physics doesn’t change. However, when we include the Mexican hat potential, the system rolls into the bottom of the Mexican hat: one of the Higgses obtains a vev while the others do not. Performing a “rotation” then moves the vev from one Higgs to the others and the symmetry is broken—the four Higgses are no longer being treated equally.

Now to whet your appetite for my next post: you can see that once electroweak symmetry is broken, there is a “flat direction” in the potential (the green circle). Remember when I said that the concave of the potential has to do with the particle’s mass? The fact that there is a flat direction means that there are massless particles. In fact, for the Higgs, there are three flat directions that correspond to—you guessed it—the three massless Higgs/Goldstone particles which are eaten by the weak gauge bosons: the H+, H-, and H⁰. The fourth Higgs—the particle that we usually call the Higgs—corresponds to an excitation in the radial direction where there is a concave, so the Higgs boson has mass.

Do we really need a Higgs?

Okay, so if you’ve followed so far, you have an idea of how electroweak symmetry breaking explains how the massless W and B bosons combine with the Higgses to form the usual W+, W-, Z, and photon. We’ve also reviewed how matter particles get mass (by bumping into the resulting vev) and how some of those gauge bosons got mass (by eating some of the Higgses). But was all of this necessary, or did we just cook it all up because we liked the idea of electroweak unification?

We will see in one of my follow up posts that in fact, electroweak symmetry breaking is almost necessary for our theory to make sense. (I’ll quantify the “almost” when we get there, but the technical phrase will be “perturbative unitarity.”) Note that I said that electroweak symmetry breaking is the important thing. Throughout this entire post you could have replaced the Higgs boson with “something like it.” There are plenty of theories out there with multiple Higgs bosons, no Higgs bosons, or generically Higgsy-things-but-not-quite-the-Higgs. That’s fine—in all of these theories, the “Higgsy-thing” always breaks electroweak symmetry. In doing so, you always end up with Goldstone bosons that are eaten by the W+, W-, and Z. And you always end up with some kind of particle like the Higgs that we expect to find at the LHC.

One last request: vote to support this blog

Hi everyone, if you liked this post (or any of my other posts, e.g. the Feynman diagram series) I’d like to ask you to vote for me (Philip Tanedo) for the 2011 Blogging Scholarship. The voting goes on for about another week and you can vote once per day. If you re-blog any of my posts, it would mean a lot if you could encourage your readers/friends/Facebook friends, etc. to also vote for me. For the past two years I’ve been able to blog due to support from the National Science Foundation and the Paul and Daisy Soros foundation, but without additional support like the Blogging Scholarship for next year I would be unable to continue with US LHC / Quantum Diaries.

While one of the priorities of the LHC is to find the Higgs boson (also see Aidan’s rebuttal), it should also be pointed out that we have already discovered three quarters of the Standard Model Higgs. Just don’t expect to hear about this in the New York Times; this isn’t breaking news—we’ve known about this “three quarters” of a Higgs for nearly two decades now. In fact, these three quarters of a Higgs live inside the belly of two beasts: the Z and W bosons!

What the heck do I mean by all this? What is “three quarters” of a particle? What does the Higgs have to do with the Z and the W? And to what extent have we or haven’t we discovered the Higgs boson? These are all part a subtle piece of the Standard Model story that we are now in an excellent position to decipher.

What we will find is that there’s not one, but four Higgs bosons in the Standard Model. Three of them are absorbed—or eaten—by the Z and W bosons when they become massive. (This is very different from the way matter particles obtain mass!) In this sense the discovery of massive Z and W bosons was also a discovery of these three Higgs bosons. The fourth Higgs is what we call the Higgs boson and its discovery (or non-discovery) will reveal crucial details about the limits of the Standard Model.

The difference between massless and massive vectors

In the not-so-recent past we delved into some of the nitty-gritty of vector bosons such as the force particles of the Standard Model. We saw that relativity forces us to describe these particles with four-component mathematical objects. But alas, such objects are redundant because they encode more polarization states than are physically present. For example, a photon can’t spin in the direction of motion (longitudinal polarization) since this would mean part of the field is traveling faster than the speed of light.

Now, what do we mean by polarization anyway? We’d previously seen that polarizations are different ways a quantum particle can spin. In fact, each polarization state can be thought of as an independent particle, or an independent “degree of freedom.” In this sense there are two photons: one which has a left-handed polarization and one with a right-handed polarization.

Because massive particles (which travel slower than light) can have a longitudinal polarization, they have an extra degree of freedom compared to massless particles. So repeat after me:

The difference between massless force particles (like the photon and gluon) and massive force particles (like the W and Z) is the longitudinal degree of freedom.

Since a “degree of freedom” is something like an independent particle, what we’re really saying is that the W and Z seem to have an “extra particle’s worth of particle” in them compared to the photon and gluon. We will see that this poetic language is also technically correct.

The mass of a force particle is important for large scale physics: the reason why Maxwell was able to write down a classical theory of electromagnetism in the 19th century is because the photon has no mass and hence can create macroscopic fields. The W and Z on the other hand, are heavy and can only mediate short-range forces—it costs energy for particles to exchange heavy force particles.

Massive vectors are a problem

The fact that the W and Z are massless is also important for the following reason:

In the early days of quantum field theory, massive vector particles didn’t seem to make any sense!

The details don’t matter, but the punchline is that the very mathematical consistency of a typical theory with massive vector particles breaks down at high energies. You can ask a well-posed physical question—what’s the probability of Ws to scatter off one another—and it is as if the theory itself realizes that something isn’t right and gives up halfway through, leaving your calculations in tatters. It seemed like massive vector particles just weren’t allowed.

If that’s the case, then how can the W and Z bosons be massive? Contrary to lyrics to a popular Lady Gaga song, the W and Z bosons were not “born this way.” Force particles naturally appear in theories as massless particles. From our arguments above, we now know that the difference between a massless and a massive particle is a single, extra longitudinal degree of freedom. Somehow we need to find extra longitudinal degrees of freedom to lend to the W and Z.

Technical remark & update (10 Oct): As a commenter has pointed out below, I should be more careful in how I phrase this. Theories of massive vectors (essentially nonlinear sigma models) only become non-unitary at tree-level so that we say they lose “perturbative unitarity.” This on its own is not a problem and certainly doesn’t mean that the they is “mathematically inconsistent” since they become strongly coupled and get large corrections from higher order terms. What we do lose is calculability and one has to wonder if there’s a better description of the physics at those scales. Many thanks to the ‘anonymous’ commenter for calling me out on this.

Let them eat Goldstone bosons

Jeffrey Goldstone, image from his MIT webpage

Where can this extra degree of freedom come from? One very nice resolution to this puzzle is called the Higgs mechanism. The main idea is that vector particles can simply annex another particle to make up the “extra particle’s worth of particle” it needs to become massive. We’ll see how this works below, but what’s really fantastic is that this is one of the very few known ways to obtain a mathematically consistent theory of massive vector particles.

So what are these extra particles?

Since particles with spin carry at least two degrees of freedom, this “extra longitudinal degree of freedom” can only come from a spin-less (or scalar) particle. Such a particle has to somehow be connected to the force particles that want to absorb it, so it should be charged under the weak force. (For example, neutrinos are uncharged under electromagnetism since they don’t talk to photons, but they are charged under the weak force since they talk to the W and Z bosons.)

Further, this particle has to obtain a vacuum expectation value (“vev”). Those of you who have been following along with our series on Feynman diagrams will already be familiar with this, though we’re now approaching the topic from a different direction.

In general, particles that can be combined with massless force particles to form massive force particles are called Goldstone bosons (or Nambu-Goldstone bosons including one of the 2008 Nobel prize winners) after Jeffrey Goldstone, pictured to the right. The Goldstone theorem states that

A theory with spontaneous symmetry breaking has massless scalar particle in its spectrum.

For now don’t worry about any of these words other than the fact that this gives a condition for which there must be scalar particles in a theory. We’ll get back to the details below and we’ll see that these scalar particles, the Goldstone bosons, are precisely the scalars which massless force particles can absorb in order to become massive.

So now we arrive at another aphorism in physics:

Force particles can eat Goldstone bosons to become massive.

In light of this terminology, perhaps a more appropriate cartoon of this is to draw the Goldstone particle as a popular type of fish-shaped cracker…

A W eating a Goldstonefish cracker... get it? (I hope we don't get sued for this.)

Four Higgses: A different kind of redundancy

Okay, so we have three massive gauge bosons: the W⁺, W^-, and Z. Each one of these has two transverse polarizations (right- and left-handed) in addition to a longitudinal polarization. This means we need three Goldstone bosons to feed them. Where do these particles come from? The answer should be no surprise, the Higgs.

Indeed, you might think I’m selling you the Standard Model like an informercial:

If you buy now, the Standard Model comes with not one, not two, not even three, but four—count them, four—Higgs bosons!

Four Higgs bosons?! That’s an awful lot of Higgs. But it turns out this is exactly what we have: we call them the H⁺, H^-, H⁰, and h. As you can see, two of them are charged (you can guess these will be eaten by the Ws), two are uncharged. Here’s they are:

"The Four Higgses of the Standard Model," biblical pun intended

Where did all of these Higgses come from? And why did our theory just happen to have enough of them? These four Higgses are all manifestations of a different kind of redundancy called gauge symmetry. The name is related to gauge bosons, the name we give to force particles.

When we described vector particles, we said that our mathematical structure was redundant: our four-component objects have too many degrees of freedom than the physical objects they represented. One redundancy came from the restriction that massless particles can have no longitudinal polarization. This brings us down from 4 degrees of freedom to 3. However, we know that massless particles only have two polarizations—we have to remove one more polarization. (Similarly for massive particles, which have 3, not 4, degrees of freedom.) This left-over redundancy is precisely what we mean by gauge symmetry.

Gauge symmetry doesn’t just explain the redundancy in the vector particles, but it also imposes a redundancy in any matter particles that are charged under the associated force. In particular, the gauge symmetry associated with the weak force requires that the Higgs is described by a two component complex-valued object. Since a complex number contains two real numbers, this means the Higgs is really composed of four distinct particles—the four particles we met above.

Now let’s get back to the statement of Goldstone’s theorem that we gave above:

A theory with spontaneous symmetry breaking has massless scalar particle in its spectrum.

We’re already happy with the implications of having a scalar. Let’s unpack the rest of this sentence. The hefty phrase is “spontaneous symmetry breaking.” This is a big idea that deserves its own blog post, but in our present case (the Standard Model) we’ll be “breaking” the gauge symmetry associated with the W and Z bosons.

What happens is that one of the Higges (in fact, this is “the Higgs,” the one called h) gets a vacuum expectation value. This means that everywhere in spacetime there Higgs field is “on.” However, the Higgs carries weak charge—so if it is “on” everywhere, then something must be ‘broken’ with this gauge symmetry… the universe is no longer symmetric since there’s a preferred weak charge (the charge of the Higgs, h).

For reasons that we’ll postpone for another time, Goldstone’s theorem then implies that the other Higgses serve as Goldstone bosons. That is, the H⁺, H^-, and H⁰ can be eaten by the W⁺, W^-, and Z respectively, thus providing the third polarization required for a massive vector particle (and doing so in a way that is mathematically consistent at high energies).

Three of the four Higgses are Goldstones and are eaten by the W and Z.

Epilogue

There are still a few things that I haven’t told you. I haven’t explained why there was exactly one Goldstone particle for each heavy force particle. Further, I haven’t explained why it turned out that each Goldstone particle had the same electric charge as the force particle that ate it. And while we’re at it, I haven’t said anything about why the photon should be massless while the W and Z bosons gain mass—they’re close cousins and you may wonder why the photon couldn’t have just gone off and eaten the h.

Alas, all of these things will have to wait for a future post on what we really mean by electroweak symmetry breaking.

What we have done is shown how gauge symmetry and the Higgs are related to the mass of force particles. We’ve seen that the Higgs gives masses to vector bosons in a way that is very different from the way it gives masses to fermions. Fermions never “ate” any part of the Higgs but bounced off its vacuum expectation value, while the weak gauge bosons feasted on three-fourths of the Higgs! This difference is related to the way that relativity restricts the behavior of spin-one particles versus spin–one-half particles.

Finally, while we’ve shown that we’ve indeed discovered “3/4th of the Standard Model Higgs,” that there is a reason why the remaining Higgs is special and called the Higgs—it’s the specific degree of freedom which obtains the vacuum expectation value which breaks the gauge symmetry (allowing its siblings to be eaten). The discovery of the Higgs would shed light on the physics that induces this so-called electroweak symmetry breaking, while a non-discovery of the Higgs would lead us to consider alternate explanations for what resolves the mathematical inconsistencies in WW scattering at high energies.

The raison d’être of this blog is to tell continuing story and science of the LHC, and this story would not be complete without a glimpse of the human infrastructure behind the world’s largest science experiment. To this end, I’d like to share a few vignettes from this past summer conference season. (For those who want more technical posts about the Standard Model, don’t worry, I’ve got a long queue of those to write.)

Particle physics: as much about people as it is about particles

There are a lot of scientists working on the LHC: the general purpose detectors (ATLAS and CMS) have thousands of experimentalists, LHCb and ALICE add another thousand or two, and there’s a slew of theorists who are not officially affiliated with any particular experiment. (And this isn’t even accounting for the accelerator physicists who manage and improve the beam quality (and work on R&D for future colliders), the support staff, and partners in industry.)

A photo mural at CMS showing a small subset of its members.

What’s even more impressive, though, is that the vast majority of the several thousand of scientists behind the LHC are specialists. A given person might be an expert on a particular analysis, a particular part of a particular detector, or a particular class of theories and how they might be manifested in data.

Ultimately, however, everyone relies  on one another to make progress. New data spurs new models, which in turn suggest new search strategies. (Recent examples of this include many of the recent anomalies at the Tevatron.)

It is crucial in such a large, co-dependent community that researchers are able to communicate with one another. This is the reason why flocks of physicists migrate over the summer to conferences and workshops around the world to discuss the latest experimental and theoretical results. Just as new particles are produced when you collide protons at the LHC, new ideas are generated when you bring together researchers in one place to discuss experimental and theoretical developments in the field.

No, it’s not summer vacation…

After bumping into each other an LHC workshop in Santa Barbara, Robin and I found ourselves discussing how difficult it is to convince friends and family that spending part of the summer somewhere with nice weather is actually a crucial part of your research and is not a vacation. To be fair, we were a the Kavli Institute for Theoretical Physics, which is so close to the beach that you could walk down there to catch a breath of fresh air (and work out some kinks in a calculation):

This is actually a physics pun: "penguins on the beach."

Of course, no matter how nice the scenery, it’s difficult to call it a “vacation” if you’re busy busting your butt trying to tie up lose ends in your research in preparation for an upcoming talk while simultaneously bouncing ideas off colleagues for new ideas to play with over the next year.

The people colliders

H. Haber's opening slide at SUSY 2011 in Fermilab. Images from the Particle Zoo were very popular this year... as were jokes about the Higgs boson being available for $9.75.

So once several physicists have gathered, what actually goes on during summer conferences? A glance at one of the official programs will show a schedule that is packed with presentations. This is one way that researchers can promote new results and get a broad view of what’s going on in the field. It’s a great chance to see where your own work fits into the “big picture” and what the next steps forward ought to be.

Some conferences, such as this year’s Lepton-Photon and EPS-HEP, are also venues where big experiments announce results their latest  data. With the LHC now entering the regime where it has enough data to seriously search for the Higgs and new physics, each of these talks are big events.

Given all the programming dedicated to presentations, one might think that this is the whole point of summer meetings. In fact, it’s quite the opposite. The real value of these events occurs in the time between talks: the question and answer sessions after a tantalizing plot, the scribbles over napkins, the discussions at the espresso machine, and the sleepless nights working out ideas that were generated during the morning sessions—that is where the magic happens.

Outside the conference rooms you’ll find people huddled over their laptops, forming little communities around power outlets. They’re putting finishing touches on talks, Skyping with colleagues to let them know about an interesting result, or perhaps checking up on an analysis that they’re running on a computer cluster at their home institution. There are also the groups of old friends who became buddies as grad students attending the same summer school and spent much of the rest of their careers collaborating with one another. In between catching up with their latest ideas, one might overhear them planning an excursion into a local dive or (depending on their age) reminiscing about past exploits. There are the debates after talks about the merits of one idea versus another, the grumblings over coffee about how early the morning sessions are, the grad students trying to make a good impression as they nervously count the months before graduation.

Overall, there’s a lot of physics—but there’s also a lot of personal interactions. Despite the size of the high energy physics community, it is still small enough that you regularly bump into old friends. Amidst the debates about whether or not the Higgs might have a low mass or whether or not SUSY is in trouble, people will ask about each others’ families, share humorous stories about colleagues who couldn’t make it, and will implore one another to come visit their local institution to give a talk so that they might properly catch up.

The people behind the US LHC blog

Speaking of the personal aspect of summer conferences, I was especially delighted to have the opportunity to bump into a few US LHC blog members and alumni. Since readers of this blog mostly know us from our words rather than our faces, I thought I’d share a few photos with familiar bloggers.

With US LHC blog alum Mike in Madison, Wisconsin for the Phenomenology ("Pheno") conference	A mini bloggers meeting at a pizza place near CERN; Lauren R. (US LHC intern), Burton, Aidan, and Anna. Not pictured: Matt, Kathryn (US LHC Communication), and me.
With Katie Yurkewicz, formerly US LHC editor and now Director of the Office of Communication in Fermilab. (I was at the SUSY 2011 conference and swing by to visit.)	With Robin at the Adler Planetarium during part of the SUSY 2011 conference. Not pictured: Robin's youngest child, who's been to more physics conferences than most grad students.

Judging from this pictures, it seems that I only have one nice shirt for conferences.

I also wanted to highlight the two Katies: Katie Yurkewicz (Director of the Office of Communications in Fermilab) and Kathryn Grim, this US LHC blog editor and US LHC communicator at CERN. If you’re a regular on this blog then you already have an idea of some of the ways that the Fermilab Office of Communication connects the US high energy physics community to the rest of the world. Besides organizing this blog, the Office of Communication puts out Symmetry magazine, interfaces with the press, coordinates with other labs, makes all sorts of multimedia available through VMS (e.g. the Fermilab Wine & Cheese seminars during big announcements), and writes all sorts of brochures/flyers/summaries of the physics going on at the LHC. They don’t get enough credit for how much they’ve served the high energy physics community and everyone who is interested in particle physics.

Speaking of the blog, over the summer regular US LHC readers noticed that we’ve moved to the revamped Quantum Diaries aggregate blog. This has been part of an experiment to aggregate a few official laboratory blogs together to try to collectively broaden our audience. We’ve all appreciated your comments and feedback during our transition—for example, comments from US LHC readers about the font size and color contrast have been implemented for the whole Quantum Diaries site (thanks to Kevin of Xeno Media and Chris of Quantum Diaries). We’ll continue to strive to provide great content from the frontiers of particle physics.

As I tried to give a glimpse of here, the mission of the LHC is not just about particles—but it’s very much about people. And it’s not just about the scientists and the staff associated with the labs, but it’s also about people like you who read blogs like this and are excited about pushing our knowledge about fundamental science. For all the kind feedback we’ve gotten about this blog, I think we—and the broader science community—appreciate you many times more.

Epilogue

There are a couple of random silly things that I can’t help myself from mentioning:

I really enjoyed meeting other grad students and young scientists over the summer. I was particularly amused/embarrassed on those occasions when someone would mention the blog (since our intended audience is more towards the general public rather than other scientists). But a special shout out goes to Sandeepan and Andrea of the CERN theory group who excitedly asked me if I was going to blog about the food at Restaurant 1 after I’d remarked how yummy it was. Well, there you have it.

I think academia is still trying to figure out what to make of blogs beyond vessels for outreach, and this past summer we’ve really seen some blogs do a lot to parse and highlight the exciting results from conferences. It will be interesting to see the evolving role of blogs in this regard. One reason why the community is still unsure about the role of blogs has been the role of blogs in spreading rumors (especially given the size of LHC collaborations). As an outreach blog supported by the particle physics community, the US LHC blog does not post rumors… but somehow I ended up in a discussion about this with a CMS experimentalist and can’t help but sharing his reply:

“Rumors? Okay, I’ve got a really good one. Okay? Listen up. Here it is. [Dramatic pause.] I heard that CMS is awesome and is way better than ATLAS.”

CERN stretches across both sides of the Switzerland–France border. As such, the CMS vending machines take Euros while the ATLAS vending machines take Swiss Francs. The main cafeteria, Restaurant 1, is on the ATLAS side but will begrudgingly take Euros. The real currency of CERN, however, are jeton (French for “token”), which are used to pay for espresso.

Many thanks to all of the friends and colleagues that I met (and re-met) over the summer!

Particles have an inherent spin. We explored the case of fermions (“spin-1/2″) in a recent post on helicity and chirality. Now we’ll extend this to the case of vector (“spin-1″) particles which describe gauge bosons—force particles.

By now regular US LHC readers are probably familiar with the idea that there are two kinds of particles in nature: fermions (matter particles) and bosons (force particles). The matter particles are the ‘nouns’ of the Standard Model. The ‘verbs’ are the bosons which mediate forces between these particles. The Standard Model bosons are the photon, gluon, W, Z, and the Higgs. The first four (the gauge bosons of the fundamental forces) are what we call vector particles because of the way they spin.

An arrow that represents spin

You might remember the usual high school definition of a vector: an object that has a direction and a magnitude. More colloquially, it’s something that you can draw as an arrow. Great. What does this have to do with force particles?

In our recent investigation of spin-1/2 fermions, the punchline was that chiral (massless) fermions either spin clockwise or counter-clockwise relative to their direction of motion. We can convert this into an arrow by identifying the spin axis. Take your right hand and wrap your fingers around the direction of rotation. The direction of your thumb is an arrow that identifies the helicity of the fermion, it is a ‘spin vector.’ In the following cartoon, the gray arrows represent the direction of motion (right) and the big colored arrows give the spin vector.

You can see that a particle has either spin up (red: spin points in the same direction as motion) or spin down (blue: spin points in the opposite direction as motion). It should not surprise you that we can write down a two-component mathematical object that describes a particle. Such an object is called a spinor, but it’s really just a special kind of vector. In can be represented this way:

ψ = ( spin up , spin down )

As you can see, there’s one slot that contains information about the particle when it is spin up and another slot that contains information about the particle when it is spin down. It’s really just a list with two entries.

Don’t panic! We’re not going to do any actual math in this post, but it will be instructive—and relatively painless—to see what the mathematical objects look like. This is the difference between taking a look at the cockpit of a jet versus actually flying it.

All you have to appreciate at this point is that we’ve described fermions (spin-1/2 particles) in terms of an arrow that determine its spin. Further, we can describe this object as a two-component ‘spinor.’

Relativity and spin

Why did we bother writing the spinor as two components? Why not just work with one component at a time: we pick up a fermion and if it’s spin up we use one object and if it’s spin down we use another.

This, however, doesn’t work. To see why, we can imagine what happens if we take the same particle but change the observer. You can imagine driving next to a spin-up particle on the freeway, and then accelerating past it. Relative to you, the particle reverses its direction of motion so that it becomes a spin-down particle.

What does this mean? In order to account for relativity (different observers see different things) we must describe the particle simultaneously in terms of being spin-up and spin-down. To describe this effect mathematically, we would perform a transformation on the spinor which changes the spin up component into the spin down component.

In fact, to a mathematician, this is the whole point of constructing vectors: they’re things which know how to transform properly when you rotate them. In this way they are intimately linked to the symmetries of spacetime: we should know how particles behave when we grab them and rotate them.

Spin-1 (vector) particles

Now that we’ve reviewed spin-1/2 (fermions), let’s move on to spin-1: these are the vector particles and include the gauge bosons of the Standard Model. Unlike the spin-1/2 particles, whose spin arrows must be parallel to the direction of motion, vector particles can have their spin point in any direction. (This is subject to some constraints that we’ll get to below.) We know how to write arrows in three dimensions: you just write down the coordinates of the arrow tip:

3D arrow = (x-component, y-component, z-component)

When we take into account special relativity, however, we must work instead in four dimensional spacetime, i.e. we need a vector with four components (sometimes called a four-vector, see Brian’s recent post). The reason for this is in addition to rotating our vector, we can also boost the observer—this is precisely what we did in the example above where we drove past a particle on the freeway—so that we need to be able to include the length contraction and time dilation effects that occur in special relativity. Heuristically, these are rotations into the time direction.

So now we’ve defined vector particles to be those whose spin can be described by an arrow pointing in four dimensions. A photon, for example, can thus be represented as:

A^μ = (A⁰, A¹, A², A³)

Here we’ve used the standard convention of labeling the x, y, and z directions by 1, 2, and 3. The A⁰ corresponds to the component of the spin in the time direction. What does this all mean? The (spin) vector associated with a spin-1 particle has a more common name: the polarization of the particle.

You’ve probably heard of polarized light: the electric (and hence also the magnetic) field is fixed to oscillate along only one axis; this is the basis for polarized sunglasses. Here’s a heuristic drawing of electromagnetic radiation from a dipole (from Wikipedia, CC-BY-SA license):

The polarization of a photon refers to the same idea. As mentioned in Brian’s post, the electric and magnetic fields are given by derivatives of the vector potential A. This vector potential is exactly the same thing that we have specified above; in a sense, a photon is a quantum of the vector potential.

Four vectors are too big

Now we get to a very important point: we’ve argued based on spacetime symmetry that we should be using these four-component vectors to describe particles like photons. Unfortunately, it turns out that four components are too many! In other words, there are some photon polarizations that we could write down which are not physical!

Here we’ll describe one reason why this is true; we will again appeal to special relativity. One of the tenets of special relativity is that you cannot travel faster than the speed of light. Further, we know that photons are massless and thus travel at exactly the speed of light. Now consider a photon with is spinning in the same direction as its motion (i.e. the spin vector is perpendicular to the page):

In this case the bottom part of the photon (blue) is moving opposite the direction of motion and so travels slightly slower than the speed of light. On the other hand, the top part of the photon is moving with the photon and thus would be moving faster than the speed of light!

This is a big no-no, and so we cannot have any photons polarized in this way. Our four-component vector contains more information than the physical photon. Or more accurately: being able to write down our theory in a way that manifestly respects spacetime symmetry comes at the cost of introducing extra, non-physical degrees of freedom in how we describe some of our particles.

(If we removed this degree of freedom and worked with three-component vectors, then our mathematical formalism doesn’t have enough room to describe how the particle behaves under rotations and boosts.)

Fortunately, when we put four-component photons through the machinery of quantum field theory, we automatically get rid of these unphysical polarizations. (Quantum field theory is really just quantum mechanics that knows about special relativity.)

Gauge invariance: four vectors are still too big

Now I’d like to introduce one of the key ideas of particle physics. It turns out that even after removing the unphysical ‘faster than light’ polarization of the photon, we still have too many degrees of freedom. A massless particle only has two polarizations: spin-up or spin-down. Thus our photon still has one extra degree of freedom!

The resolution to this problem is incredibly subtle: some of the polarizations that we could write down using a four-vector are physically identical. I don’t just mean that they give the same numbers when you do the math, I mean that they literally describe the same physical state. In other words, there is a redundancy in this four-vector description of particles! Just as the case of the unphysical polarization above, this redundancy is the cost of writing things in a way which manifestly respects spacetime symmetry. This redundancy is called gauge invariance.

Gauge invariance is a big topic that deserves its own post—I’m still thinking of a good way to present it—but the “gauge” refers to the same thing in term “gauge boson.” This gauge invariance (redundancy in our description of physics) is intimately linked to the fundamental forces of our theory.

For experts: at this point, I should probably mention that the mathematical object which really describes gauge bosons aren’t vectors, but rather co-vectors, or (one-)forms. One way to see this is that these are objects that get integrated over in the action. The distinction is mostly pedantic, but a lot of the power of differential geometry and topology is manifested when one treats gauge theory in its ‘natural’ language of fiber bundles. For more prosaic goals, we can write down Maxwell’s equations in an even more compact form: d*F = j. (Even more compact than Brian’s notation! )

Wigner’s classification

Let me take a step back to address the ‘big picture.’ In this post I’ve tried to give a hint of a classification of “irreducible [unitarity] representations of the Poincaré group” by Hungarian mathematical physicist Eugene Wigner in the late 1930s.

At the heart of this program is a definition of what we really mean by ‘particle.’ A particle is something with transforms in a definite way under the symmeties of spacetime, which we call the Poincaré group. Wigner developed a systematic way to write down all of the ‘representations’ of the Poincaré group that describe quantum particles; these representations are what we mean by spin-1, spin-1/2, etc.

In addition to these two examples, there are fields which do nothing under spacetime symmetries: these are the spin-0 scalar fields, such as the Higgs boson. If we treated gravity quantum mechanically, then the graviton would be a spin-2 [antisymmetric] tensor field. If nature is supersymmetric, then the graviton would also have a spin-3/2 gravitino partner. Each of these different spin fields is represented by a mathematical object with different numbers of components that mix into one another when you do a spacetime transformation (e.g. rotations, boosts).

In principle one can construct higher spin fields, e.g. spin-3, but there are good reasons to believe that such particles would not be manifested in nature. These reasons basically say that those particles wouldn’t be able to interact with any of the lower-spin particles (there’s no “conserved current” to which they may couple).

Next time: there are a few other physics (and some non-physics) topics that I’d like to blog about in the near future, but I will eventually get back to this burning question about the meaning of gauge symmetry. From there we can then talk about electroweak symmetry breaking, is the main reason why we need the Higgs boson (or something like it) in nature. (For those who have been wondering why I haven’t been writing about the Higgs—this is why! We need to go over more background to do things properly.)

`Yesterday marked the beginning of the HEP summer conference season with EPS-HEP 2011, which is particularly exciting since the LHC now has  enough luminosity (accumulated data) to start seeing hints of new physics. As Ken pointed out, the Tevatron’s new lower bound on the B_s → μμ decay rate seemed to be a harbinger of things to come (Experts can check out the official paper, the CDF public page, and the excellent summaries by Tommaso Dorigo and Jester.).

Somewhat unfortunately, the first LHCb results on this process do not confirm the CDF excess, though they are not yet mutually exclusive. Instead of delving too much into this particular result, I’d like to give some background to motivate why it’s interesting to those of us looking for new physics. This requires a lesson in “the birds and the Bs”—of course, by this I mean B mesons and the so-called ‘penguin’ diagrams.

The B_s meson: why it’s special

It's a terrible pun, I know.

A B_s meson is a bound state of a bottom anti-quark and strange quark; it’s sort of like a “molecule” of quarks. There are all sorts of mesons that one could imagine by sticking together different quarks and anti-quarks, but the B_s meson and it’s lighter cousin, the B_d meson, are particularly interesting characters in the spectrum of all possible mesons.

The reason is that both the B_s and the B_d are neutral particles, and it turns out that they mix quantum mechanically with their antiparticles, which we call the B_s and B_d. This mixing is the exact same kind of flavor phenomenon that we described when we mentioned “Neapolitan” neutrinos and is analogous to the mixing of chiralities in a massive fermion. Recall that properties like “bottom-ness” or “strangeness” are referred to as flavor. Going from a B_s to a B_s changes the “number of bottom quarks” from -1 to +1 and the “number of strange quarks” from +1 to -1, so such effects are called flavor-changing.

To help clarify things, here’s an example diagram that encodes this quantum mixing:

The ui refers to any up-type quark.

Any neutral meson can mix—or “oscillate”—into its antiparticle, but the B mesons are special because of their lifetime. Recall that mesons are unstable and decay, so unlike neutrinos, we can’t just wait for a while to see if they oscillate into something interesting. Some mesons live for too long and their oscillation phenomena get ‘washed out’ before we get to observe them. Other mesons don’t live long enough and decay before they have a chance to oscillate at all. But B mesons—oh, wonderful Goldilocks B mesons—they have a lifetime and oscillation time that are roughly  of the same magnitude. This means that by measuring their decays and relative decay rates we can learn about how these mesons mix, i.e. we can learn about the underlying flavor structure of the Standard Model.

The CDF and LHCb results: B_s → mu mu

It turns out that there are interesting flavor-changing effects even without considering meson mixing, but rather in the decay of the B meson itself. For example, we can modify the previous diagram to consider the decay of a B_s meson into a muon/anti-muon pair:

This is still a flavor-changing decay since the net strangeness (+1) and bottom-ness (-1) is not preserved; but note that the lepton flavor is conserved since the muon/anti-muon pair have no net muon number. (As an exercise: try drawing the other diagrams that contribute; the trick is that you need W bosons to change flavor.) You could also replace muons by electrons or taus, but those decays are much harder to detect experimentally. As a rule of thumb muons are really nice final state particles since they make it all the way through the detector and one has a decent shot at getting good momentum measurements.

It turns out that this decay is extremely rare. For the B_s meson, the Standard Model predicts a dimuon branching ratio of around 3 × 10^-9, which means that a B_s will only decay into two muons 0.0000003% of the time… clearly in order to accurately measure the actual rate one needs to produce a lot of B mesons.

In fact, until recently, we simply did not have enough observed B meson decays to even estimate the true dimuon decay rate. The ‘B factories’ of the past decade were only able to put upper limits on this rate. In fact, this decay is one of the main motivations for LHCb, which was designed to be the first experiment that would be sensitive enough to probe the Standard Model decay rate. (This means that if the decay rate is at least at the Standard Model rate, then LHCb will see it.)

The exciting news from CDF last week was that—for the first time—they appeared to have been able to set a lower bound on the dimuon decay rate of the B_s meson. (The B_d meson has a smaller decay rate and CDF was unable to set a lower bound.) The lower bound is still statistically consistent with the Standard Model rate, but the suggested (‘central value’) rate was 1.8 × 10^-8. If this is true, then it would be a fairly strong signal for new physics beyond the Standard Model. The 90% confidence level range from CDF is:

4.6 × 10^-9 < BR(B_s → μ⁺μ^-) < 3.9 × 10^-8.

Unfortunately, today’s new result from LHCb didn’t detect an excess with which it could set a lower bound and could only set a 90% confidence upper bound,

BR(B_s → μ⁺μ^-) < 1.3 × 10^-8.

This goes down to 1.2 × 10^-8 when including 2010 data. The bounds are not yet at odds with one another, but many people were hoping that LHCb would have been able to confirm the CDF excess in dimuon events. The analyses of the two experiments seem to be fairly similar, so there isn’t too much wiggle room to think that the different results just come from having different experiments.

More data will clarify the situation; LHCb should accumulate enough data to prove branching ratios down to the Standard Model prediction of 3 × 10^-9. Unfortunately CDF will not be able to reach that sensitivity.

New physics in loops

Now that we’re up to date with the experimental status of B_s → μμ, let’s figure out why it’s so interesting from a theoretical point of view. One thing you might have noticed from the “box” Feynman diagrams above is that they involve a closed loop. An interesting thing about closed loops in Feynman diagrams is that they can probe physics at much higher energies than one would naively expect.

The reason for this is that the particles running in the loop do not have their momenta fixed in terms of the momenta of the external particles. You can see this for yourself by assigning momenta (call them p₁, p₂, … , etc.) to each particle line and (following the usual Feynman rules) impose momentum conservation at each vertex. You’ll find that there is an unconstrained momentum that goes around the loop. Because this momentum is unspecified, the laws of quantum physics say that one must add together the contributions from all possible momenta. Thus it turns out that even though the B_s meson mass is around 5 GeV, the dimuon decay is sensitive to particles that are a hundred times heavier.

Note that unlike other processes where we study new physics by directly producing it and watching it decay, in low-energy loop diagrams one only intuits the presence of new particles through their virtual effects (quantum interference). I’ll leave the details for another time, but here are a few facts that you can assume for now:

1. Loop diagrams can be sensitive to new heavy particles through quantum interference.
2. Processes which only occur through loop diagrams are often suppressed. (This is partly why the Standard Model branching ratio for B_s → μμ is so small.)
3. In the Standard Model, all flavor-changing neutral currents (FCNC)—i.e. all flavor-changing processes whose intermediate states carry no net electric charge—only occur at loop level. (Recall that the electrically-charged W bosons can change flavor, but the electrically neutral Z bosons cannot. Similarly, note that there is no way to draw a B_s → μμ diagram in the Standard Model without including a loop.)
4. Thus, processes with a flavor-changing neutral current (such as B_s → μμ) are fruitful places to look for new physics effects that only show up at loop level. If there were a non-loop level (“tree level”) contribution from the Standard Model, then the loop-induced new physics effects would tend to be drowned out because they are only small corrections to the tree-level result. However, since there are no FCNCs in the Standard Model, the new physics contributions have a ‘fighting change’ at having a big effect relative to the Standard Model result.
5. Semi-technical remark, for experts: indeed, for B_s → μμ the Standard Model diagrams are additionally suppressed by a GIM suppression (as is the case for FCNCs) as well as helicity suppression (the B meson is a pseudoscalar, so the final states require a muon mass insertion).

So the punchline is that B_s → μμ is a really fertile place to hope to see some deviation from the Standard Model branching ratio due to new physics.

Introducing the Penguin

I would be remiss if I didn’t mention the “penguin diagram” and its role in physics. You can learn about the penguin’s silly etymology in its Wikipedia article; suffice it for me to ‘wow’ you with a picture of an autographed paper from one of the penguin’s progenitors:

A copy of the original "penguin" paper, autographed by John Ellis.

The main idea is that penguin diagrams are flavor-changing loops that involve two fermions and a neutral gauge boson. For example, the b→s penguin takes the form (no, it doesn’t look much like a penguin)

You should have guessed that in the Standard Model, the wiggly line on top has to be a W boson in order for the fermion line to change flavors. The photon could also be a Z boson, a gluon, or even a Higgs boson. If we allow the boson to decay into a pair of muons, we obtain a diagram that contributes to B_s → μμ.

Thus, in addition to the box diagrams above, there are penguin diagrams which contribute to B_s → μμ. As a nice ‘homework’ exercise, you can try drawing all of the penguins that contribute to this process in the Standard Model. (Most of the work is relabeling diagrams for different internal states.)

[Remark, 6/23: my colleague Monika points out that it's ironic that I drew the b, s, photon penguin since this penguin doesn't actually contribute to the dimuon decay! (For experts: the reason is the Ward identity.) ]

Supersymmetry and the B_s → mu mu penguin

Finally, I’d like to give an example of a new physics scenario where we would expect that penguins containing new particles give a large contribution to the B_s → μμ branching ratio. It turns out that this happens quite often in models of supersymmetry or, more generally, ‘two Higgs doublet models.’

If neither of those words mean anything to you, then all you have to know is that these models have not just one, but two independent Higgs particles which obtain separate vacuum expectation values (vevs). The punchline is that there is a free parameter in such theories called tan β which measures the ratio of the two vevs, and that for large values of tan β, the B_s → μμ branching ratio goes like (tan β)⁶ … which can be quite large and can dwarf the Standard Model contribution.

Added 6/23, because I couldn't help it: a supersymmetric penguin. Corny image from one of my talks.

[What follows is mostly for 'experts,' my apologies.]

On a slightly more technical note, it’s not often well explained why this branching ratio goes like the sixth power of tan β, so I did want to point this out for anyone who was curious. There are three sources of tan β in the amplitude; these all appear in the neutral Higgs diagram:

Each blue dot is a factor of tan β. The Yukawa couplings at each Higgs vertex goes like the fermion mass divided by the Higgs vev. For the down-type quarks and leptons, this gives a factor of m/v ~ 1/cos β ~ tan β for large tan β. An additional factor of comes from the mixing between the s and b quarks, which also goes like the Yukawa coupling. (This is the blue dot on the s quark leg.) Hence one has three powers of tan β in the amplitude, and thus six powers of tan β in the branching ratio.

Outlook

While the LHCb result was somewhat sobering, we can still cross our fingers and hope that there is still an excess to be discovered in the near future. The LHC shuts down for repairs at the end of next year; this should provide ample data for LHCb to probe all the way down to the Standard Model expectation value for this process. Meanwhile, it seems that while I’ve been writing this post there have been intriguing hints of a Higgs (also via our editor)… [edit, 6/23: Aidan put up an excellent intro to these results]

[Many thanks to the experimentalists with whom I've had useful discussions about this.]

Last month had the unique pleasure of making my first trip to CERN (more on that in a later post). I made a point to stop by the CERN gift shop to pick up a snazzy mug to show off to my colleagues back in the US, and am now the proud owner of a new vessel for my tea:

My brand new "Standard Model Lagrangian" mug from CERN.

The equation above is the Standard Model Lagrangian, which you can think of as the origin of all of the Feynman rules that I keep writing about. Each term on the right-hand side of the above equation actually encodes several Feynman rules. Roughly speaking, terms with an F or a D contain gauge fields (photon, W, Z, gluon), terms with a ψ include fermions, and terms with a ϕ include the Higgs boson. Some representative diagrams coming from each of the terms are depicted below:

Representative Feynman rules coming from each term in the Lagrangian.

But alas, there’s a bit of a problem with the design. It appears that there’s an extra term which isn’t included in the usual parametrization of the Standard Model:

This term really shouldn't be here. It's not necessarily "wrong," but it is misleading and doesn't match what is written in textbooks. Technically, it is not `canonically normalized.'

I won’t go so far as to call this a mistake because technically it’s not wrong, but I suspect that whoever designed the mug didn’t mean to write this term. Let me put it this way: if I had written the above expression, my adviser would pretend he didn’t know me. The “h.c.” means Hermitian conjugate, which is a generalization of the complex conjugate of a complex number. In terms of Feynman diagrams, this “+h.c.” term means “the same diagram with antiparticles.”

The problem is that the term above,

already includes its Hermitian conjugate. In physics-speak, we say that the kinetic term is self-conjugate (or Hermitian, or self-adjoint). This just means that there is no additional “+h.c.” necessary. In fact, including the “+h.c.” means that you are writing the same term twice and the equation is no longer “canonically normalized.” This just means that you ought to rescale some of your variables.

I was mulling over this not-quite-correct term on my mug while looking over photos from CERN when I discovered the same ‘error’ in a chalkboard display in the “Universe of Particles” exhibit:

Display at the "Universe of Particles" exhibit in The Globe of Science and Innovation at CERN.

The “+h.c.” on the top right is the same ‘error’ as printed on the CERN mug. I wonder who wrote this?

To be clear: this expression does summarize the basic structure of the Standard Model in the sense that it does give all of the correct Feynman rules. However, the extra “+h.c.” introduces a factor of two that needs to be accounted for by weird conventions elsewhere (that would not match any of the usual literature or textbooks).

Anyone who has done tedious physics calculations is familiar with the frequent agony of being off by a factor of 2. Now when people make remarks about this ‘error’ on my mug, I’m quick to tell them that the factor of 2 mistake just makes it more authentic.