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”.

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.

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.

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.

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!

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.)

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.]

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.

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

Helicity

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

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

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

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

To be clear, we can also draw the right-handed particle moving in the opposite direction (to the left):

Note that the direction of the spin (the red arrow) also had to change. You can confirm that if you point your thumb in the opposite direction, your fingers will also wrap in the opposite direction.

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

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

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

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

Helicity, Relativity, and Mass

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

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

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

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

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

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

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

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

Chirality

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

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

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

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

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

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

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

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

Chiral theories

[6/20/11: the following 2 paragraphs were edited and augmented slightly for better clarity. Thanks to Bjorn and Jack C. for comments.]

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

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

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

Electrons (left-chiral) and positrons (right-chiral) are two completely different particles, as evidenced by the positron's mustache.

How different are these particles? Well, the electron has electric charge -1, while the positron has electric charge +1. Further, the electron can couple to a neutrino through the W-boson, while the positron cannot. Why does the W only talk to the (left-chiral) electron? That’s just the way the Standard Model is constructed; the left-chiral electron is charged under the weak force whereas the right-chiral positron is not. Note that at this stage, even the electron and the anti-positron are not the same particle! Even though they both have the same charge and chirality, the electron can talk to a W, whereas the anti-positron cannot.

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

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

Masses mix different particles!

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

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

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

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

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

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

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

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

What this all has to do with the Higgs

We have now learned that masses are responsible for mixing between different types of particles. The mass terms combine two a priori particles (electron and anti-positron) into a single particle (physical electron). [See a very old post where I tried---I think unsuccessfully---to convey similar ideas.] The reason why we’ve gone through this entire rigmarole is to say that ordinarily, two unrelated particles don’t want to be mixed up into one another.

The reason for this is that particles can only mix if they carry the same quantum properties. You’ll note, for example, that the electron and the anti-positron both had the same electric charge (-1). It would have been impossible for the electron and anti-electron to mix because they have different electric charges. However, the electron carries a weak charge because it couples to the W boson, whereas the anti-positron carries no weak charge. Thus these two particles should not be able to mix. In highfalutin language, one might say that this mass term is prohibited by “gauge invariance,” where the word “gauge” refers to the W as a gauge boson. This is a consequence of the Standard Model being a chiral theory.

The reason why this unlikely mixing is allowed is because of the Higgs vev. The Higgs carries weak charge. When it obtains a vacuum expectation value, it “breaks” the conservation of weak charge and allows the electron to mix with the anti-positron, even though they have different weak charges. Or, in other words, the vacuum expectation value of the Higgs “soaks up” the difference in weak charge between the electron and anti-positron.

So now the mystery of the Higgs boson continues. First we said that the Higgs somehow gives particle masses. We then said that these masses are generated by the Higgs vacuum expectation value. In this post we took a detour to explain what this mass really does and got a glimpse of why the Higgs vev was necessary to allow this mass. The next step is to finally address how this Higgs managed to obtain a vacuum expectation value, and what it means that it “breaks” weak charge. This phenomenon is called electroweak symmetry breaking, and is one of the primary motivations for theories of new physics beyond the Standard Model.

Addendum: Majorana masses

Okay, this is somewhat outside of our main discussion, but I feel obligated to mention it. The kind of fermion mass that we discussed above is called a Dirac mass. This is a type of mass that connects two different particles (electron and anti-positron). It is also possible to have a mass that connects two of the same kind of particle, this is called a Majorana mass. This type of mass is forbidden for particles that have any type of charge. For example, an electron and an anti-electron cannot mix because they have opposite electric charge, as we discussed above. There is, however, one type of matter particle in the Standard Model which does not carry any charge: the neutrino! (Neutrinos do carry weak charge, but this is “soaked up” by the Higgs vev.)

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

This allowed us to draw diagrams with two fermions or two gauge bosons attached to a terminated Higgs:

We made the bold claim that these diagrams should be interpreted as masses for the particles attached to the Higgs. We’ll explore this interpretation in a later post, but for now let’s better understand why we should have this odd Feynman rule and what it means.

Quantum Fields Forever

Before we get into that, though, we need to get back to one of the fundamental ideas of quantum physics: wave-particle duality. Douglas Hofstadter’s famous ambrigram summarizes the well-known statement in pop-science:

Amigram by Douglas Hofstadter, image used according to the Creative Commons Attribution-Share Alike 3.0 Unported license.

Wave-particle duality is one of those well-known buzz words of quantum mechanics. Is light a particle or a wave? Is an electron a particle or a wave? If these things are waves, then what are they waves of?

In high energy physics we’re usually interested in small things that move very quickly, so we use the framework of quantum field theory (QFT) which is the marriage of quantum mechanics (which describes “small” things) and special relativity (which describes “fast” things). The quantum ‘waves’ are waves in the quantum field associated with a particle. The loose interpretation of the field is the probability that you might find a particle there.

So the quantum field is a mathematical object which we construct which tells us how likely it is that there’s a particle at each point in space and time. Most of the time the quantum field is pretty much zero: the vacuum of space is more-or-less empty. We can imagine a particle as a ripple in the quantum field, which former US LHC blogger Sue Ann Koay very nicely depicted thusly:

Sue Ann Koay's depiction of a quantum field. Ripples in the field should be interpreted as particles. Here we have two particles interacting. (For experts: Sue pointed out the ISR in the image.)

Sometimes ripples can excite others ripples (perhaps in other quantum fields), this is precisely what’s happening when we draw a Feynman diagram that describes the interaction of different particles.

The vacuum and the ‘Higgs phase’

Now we get to the idea of the vacuum—space when there isn’t any stuff in it. Usually when you think of the vacuum of empty space you’re supposed to think of nothingness. It turns out that the vacuum is a rather busy place on small scales because of quantum fluctuations: there are virtual particle–anti-particle pairs that keep popping into existence and then annihilating. Further still, vacuum is also filled with cosmic microwave background radiation at 2.725 Kelvin. But for now we’re going to ignore both of these effects. It turns out that there’s something much more surprising about the vacuum:

It’s full of Higgs bosons.

The quantum field for normal particle species like electrons or quarks is zero everywhere except where there are particles moving around. Particles are wiggles on top of this zero value. The Higgs is different because the value of its quantum field in the vacuum is not zero. We say that it has a vacuum expectation value, or “vev” for short. It is precisely this Higgs vev which is represented by the crossed out Higgs line in our Feynman rules.

A loose interpretation for the Higgs vev is a background probability for there to be a Higgs boson at any given point in spacetime. These “background” Higgs bosons carry no momentum, but they can interact with other particles as we saw above:

The cross means that instead of a ‘physical’ Higgs particle, the dashed line corresponds to an interaction with one of these background Higgses. In this sense, we are swimming in a sea of Higgs. Our interactions with the Higgs are what give us mass, though this statement will perhaps only make sense after we spend some time in a later post understanding what mass really is.

A good question to ask is why the Higgs has a vacuum expectation value. This is the result of something called electroweak symmetry breaking and is related to the unification of the electromagnetic force and the weak force, i.e. somehow the Higgs is part of a broader story about unification of the fundamental forces.

Often people will say that the universe is in a ‘Higgs phase,’ a phrase which draws on very elegant connections between the quantum field theory of particles and the statistical field theory of condensed matter systems. Just as we can discuss phase transitions between liquid and gas states (or more complicated phases), we can also discuss how the universe underwent an electroweak phase transition which led to the Higgs vev that lends masses to our favorite particles.

Next time…

When we continue our story of the Higgs, we’ll start to better understand the relation of the Higgs vev with the mass of the other Standard Model particles and will learn more about electroweak symmetry breaking.

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.

• 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.

Hadronic Junk

Let’s review what we know about quantum chromodynamics (QCD). Protons and neutrons are composite objects built out of quarks which are bound together by gluons. Like electrons and photons in quantum electrodynamics (QED), quarks and gluons are assumed to be “fundamental” particles. Unlike electrons and photons, however, we do not observe individual quarks or gluons in isolation. You can pull an electron off of a Hydrogen atom without much ado, but you cannot pull a quark out of a proton without shattering the proton into a bunch of other very different looking things (things like pions).

The reason is that QCD is very nonperturbative at low energies. QCD hates to have color-charged particles floating around, it wants them to immediately bind into color-neutral composite objects, even if that means producing new particles out of the quantum vacuum to make everything neutral. These color-neutral composite objects are called hadrons. Unfortunately, usually the process of hadronizing a quark involves radiating off other quarks of gluons which themselves hadronize. This process continues until you end up with a messy spray of particles in place of the original colored object. This spray is called a jet. (Every time I write about jets I feel like I have to reference West Side Story.)

Simulated event from ATLAS Experiment © 2011 CERN

As one can see in the image above, the problem is that the nice Feynman diagrams that we know how to calculate do not directly correspond to the actual mess of particles that form the jets which the LHC experiments measure. And it really is mess. One cannot effectively measure every single particle within each jet and even if one could, it is impractically difficult to calculate Feynman diagrams for very large numbers of particles.

Thus we’re stuck having to work with the jets themselves. High energy jets usually correspond to the production of a single high-energy colored particle, so it makes sense to talk about jets as “single objects” even though they’re really a spray of hadrons.

Identifying Jets

So we’ve accepted the following fact of life for QCD at a particle collider:

Even though our high energy collisions produce ‘fundamental’ particles like quarks and gluons, the only thing we get to observe are jets: messy sprays of hadrons.

Thus one very important task is trying to make the correspondence between the ‘fundamental’ particles in our Feynman diagrams and the hadronic slop that we actually measure. In fact, it’s already very hard to provide a technical definition of a jet. Our detectors can identify most of the “hadronic slop,” but how do we go from this to a measurement of some number of jets?

This process is called clustering and involves developing algorithms to divide hadrons into groups which are each likely to have come from a single high energy colored particle (quarks or gluons). For example, for the simple picture above, one could develop a set of rules that cluster hadrons together by drawing narrow cones around the most energetic directions and defining everything within the cone to be part of the jet:

Simulated event from ATLAS Experiment © 2011 CERN

One can then measure the energy contained within the cone and say that this must equal the energy of the initial particle which produced the jets, and hence we learn something about fundamental object. I’ll note that this kind of “cone algorithms” for jet clustering can be a little crude and there are more sophisticated techniques on the market (“sequential recombination”).

Boosted Jets

Even though the above cartoon was very nice, you can imagine how things can become complicated. For example, what if the two cones started to approach each other? How would you know if there was one big jet or two narrow jets right next to each other? In fact, this is precisely what happens when you have a highly boosted object decaying into jets.

By “boosted” I mean that the decaying particle has a lot of kinetic energy. This means that even though the particle decays into two colored objects—i.e. two jets—the jets don’t have much time to separate from one another before hitting the detector. Thus instead of two well-separated jets as we saw in the example above, we end up with two jets that overlap:

Collimation of two jets into a single jet as the decaying particle is boosted. Image from D. Krohn.

Now things become very tricky. Here’s a concrete example. At the LHC we expect to produce a lot of top/anti-top pairs (t–t-bar). Each of these tops immediately decays into a b-quark and a W. Thus we have

t, t-bar → b, b-bar, W W

(As an exercise, you can draw a Feynman diagram for top pair production and the subsequent decay.) These Ws are also fairly massive particles and can each decay into either a charged lepton and a neutrino, or a pair of quarks. Leptons are not colored objects and so they do not form jets; thus the charged lepton (typically a muon) is a very nice signal. One promising channel to look for top pair production, then, is the case where one of the Ws decays into a lepton and neutrino and the other decays into two quarks:

t, t-bar → b, b-bar, W W → b, b-bar, q, q-bar, lepton, ν

The neutrino is not detected, and all of the quarks (including the bottoms) turn into jets. We thus can search for top pair production by counting the number of four jet events with a high energy lepton. For this discussion we won’t worry about background events, but suffice it to say that one of the reasons why we require a lepton is to help discriminate against background.

Here’s what such an event might look like:

Simulated event from ATLAS Experiment © 2011 CERN

Here “pT” refers to the energy (momentum perpendicular to the beam) of the top quarks. In the above event the tops have a modest kinetic energy. On the other hand, it might be the case that the tops are highly boosted—for example, they might have come from the decay of a very heavy particle which thus gives them a lot of kinetic energy. In the following simulated event display, the tops have a pT that is ten times larger than the previous event:

Simulated event from ATLAS Experiment © 2011 CERN

Now things are tricky! Instead of four clean jets, it looks like two slightly fat jets. Even though this simulated event actually had the “b, b-bar, q, q-bar, lepton, ν” signal we were looking for, we probably wouldn’t have counted this event because the jets are collimated.

There are other ways that jets tend to be miscounted. For example, if a jet (or anything really) is pointed in the direction of the beam, then it is not detected. This is why it’s something of an art to identify the kinds of signals that one should look for at a hadron collider. One will often find searches where the event selection criteria requires “at least” some number of jets (rather than a fixed number) with some restriction on the minimum jet energy.

Jet substructure

One thing you might say is that even though the boosted top pair seemed to only produce two jets, shouldn’t there be some relic that they’re actually two small jets rather than one big jet? There has been a lot of recent progress in this field.

Distinguishing jets from a boosted heavy particle (two collimated jets) from a "normal" QCD jet with no substructure. The plot is a cylindrical cross section of the detector---imagine wrapping it around a toilet paper roll aligned with the beam. Image from D. Krohn.

The main point is that one can hope to use the “internal radiation distribution” to determine whether a “spray of hadrons” contains a single jet or more than one jets. As you can see from the plots above, this is an art that is similar to reading tea leaves. (… and I only say that with the slightest hint of sarcasm!)

[For experts: the reason why the QCD jets look so different are the Alterelli-Parisi splitting functions: quarks and gluons really want to emit soft, collinear stuff.]

There’s now a bit of an industry for developing ways to quantify the likelihood that a jet is really a jet (rather than two jets). This process is called jet substructure. Typically one defines an algorithm that takes detector data and spits out a number called a jet shape variable that tells you something about the internal distribution of hadrons within the jet. The hope is that some of these variables will be reliable and efficient enough to help us squeeze as much useful information as we can out of each of our events. There also seems to be a rule in physics that the longer you let theorists play with an idea, the more likely it is that they’ll give it a silly name. One recent example is the “N-subjettiness” variable.

Jet superstructure

In addition to substructure, there has also been recent progress in the field of jet superstructure, where one looks at correlations between two or more jets. The basic idea boils down to something very intuitive. We know that the Hydrogen atom is composed of a proton and an electron. As a whole, the Hydrogen atom is electrically neutral so it doesn’t emit an electric field. (Of course, this isn’t quite true; there is a dipole field which comes from the fact that the atom is actually composed of smaller things which are charged.) The point, however, is that far away from the atom, it looks like a neutral object so we wouldn’t expect it to emit an electric field.

We can say the same thing about color-charged particles. We already know that quarks and gluons want to recombine into color-neutral objects. Before this happens, however, we have high energy collisions with quarks flying all over the place trying to figure out how to become color neutral. Focusing on this time scale, we can imagine that certain intermediate configurations of quarks might already be color neutral and hence would be less likely to emit gluons (since gluons are the color-field). On the other hand, other intermediate configurations might be color-charged, and so would be more likely to emit gluons. This ends up changing the distribution of jet slop.

Here’s a nice example from one of the first papers in this line of work. Consider the production of a Higgs boson through “quark fusion,” i.e. a quark and an antiquark combining into a Higgs boson. We already started to discuss the Higgs in a recent post, where we made two important points: (1) once we produce a Higgs, it is important to figure out how it decays, and (2) once we identify a decay channel, we also have to account for the background (non-Higgs events that contribute to that signal).

One nice decay channel for the Higgs is b b-bar. The reason is that bottom quark jets have a distinct signature—you can often see that the b quark traveled a small distance in the detector before it started showering into more quarks and gluons. Thus the signal we’re looking for is two b-jets. There’s a background for this: instead of qq-bar → Higgs → b-jets, you could also have qq-bar → gluon → b-jets.

The gluon-mediated background is typically very large, so we would like to find a clever way to remove these background events from our data. It turns out that jet superstructure may be able to help out. The difference between the Higgs → b-jets decay versus the gluon → b-jets decay is that the gluon is color-charged. Thus when the gluon decays, the two b-quarks are also color-charged. On the other hand, the Higgs is color-neutral, so that the two b-quarks are also color neutral.

One can draw this heuristically as “color lines” which represent which quarks have the same color charge. In the image below, the first diagram represents the case where an intermediate Higgs is produced, while the second diagram represents an intermediate gluon.

Color lines for qq-bar → Higgs → b-jets and qq-bar → gluon → b-jets. Image from 1001.5027

For the intermediate Higgs, the two b-jets must have the same color (one is red, the other is anti-red) so that the combined object is color neutral. For the intermediate gluon, the color lines of the two b-jets are tied up to the remnants of the protons (the thick lines at the top and bottom). The result is that the hadronic spray that makes up the jets tend to be pulled together for the Higgs decays, while pushed apart for the gluon decays. This is shown heuristically below, where again we should understand the plot as being a cylindrical cross section of the detector:

Higgs decays into two b-jets (signal) versus gluon decays (background). Image from 1001.5027

One can thus define a jet superstructure variable (called ‘pull‘) to quantify how much two jets are pulled together or pushed apart. The hope is that this variable can be used to discriminate between signal and background and give us better statistics for our searches for new particles.

Anyway, that’s just a sample of the types of neat things that people have been working on to improve the amount of information we can get out of each event at hadron colliders like the LHC. I’d like to thank David Krohn, once again, for a great talk and very fun discussions. For experts, let me make one more plug for his workshop next month: Boost 2011.

Update: for those who read this in time, there will be a seminar on this result broadcast online through the Fermilab webpage. (The talk will be at the level of physicists working in the field.) This result was also mentioned in the NY Times today.

Even though its running days are numbered, the Tevatron reminds us that it can still muster up an interesting hint of new physics. This is a quick post on a brand new result from the CDF collaboration, “Invariant Mass Distribution of Jet Pairs Produced in Association with a W boson in p-pbar Collisions at Sqrt(s) = 1.96 TeV” [arXiv:1104.0699]. Just to whet your appetite, here’s one of the plots that we’d like to understand (hot off the press!):

Unpacking the title

Before diving in, let’s first understand what the title means. It’s somewhat cumbersome, but that’s because it encodes a lot of good physics. It turns out that it’s easier to start at the end of the title and work our way backwards.

• “…in p-pbar Collisions at Sqrt(s) = 1.96 TeV“
  This is telling us about the general Tevatron experiment: they collide a proton (p) with an anti-proton (p-bar) at an energy of about 2 TeV. A TeV is roughly the kinetic energy of a flying mosquito. In Feynman diagram calculations it’s often useful to use the parameter s, which is the square of the energy, rather than the energy itself, hence the weird “Sqrt(s) = 1.96 TeV.” For comparison, recall that the LHC is now operating at 7 TeV and will eventually go up to 14 TeV.
• “…in association with a W boson …”
  This should sound a lot like the “associated production” diagram that we drew for the Higgs boson in a previous post:Reading from left to right, we have a quark and an antiquark producing a gauge boson (in this case the W boson) along with a Higgs. I’ll tell you right now: the CDF result does not appear to be the Higgs, but if there is a new particle responsible for it, it is produced from the same diagram (with the dashed line representing the new particle). [Correction, 6 Apr: a few people have correctly pointed out that the new state could also come from a "t-channel" diagram with an intermediate fermion. (I'll leave the actual diagram as an exercise for those who have been reading my Feynman diagram posts).] By the way, the W decays before it reaches the detector. Two signals are W?e? or W???, which appear in the detector as a charged lepton and “missing energy” (the neutrino).
• “… Jet Pairs Produced …“
  We also know that the Higgs decays into other stuff before it reaches our detectors. Thus we have to tell our detectors to keep an eye out not for the exotic particle, but for the ordinary stuff that it decays into. If it decays into quarks or gluons, then we know that we end up with jets. Thus this paper is looking for a W boson (which itself becomes a lepton + neutrino/missing energy) and a pair of jets. This is the criterion for picking out ‘interesting’ events relevant for this analysis.
• “Invariant mass distribution of…“
  Now that we’ve picked out interesting events, we want to plot them in a way that can tell us if there’s a new particle hidden in the data. The easiest way to do this is to go “bump hunting,” which is what we discussed for the detection of the Z boson. The key idea is this: if we sum the energy and momentum of the two jets, we should get the mass of the intermediate particle that produced them (if they were produced by the same particle). This sum is called the invariant mass, and by plotting the number of interesting events based on the invariant mass, we can look for bumps that are characteristic of new particles.

Phew—that was quite a lot packed into a title. But now we’ve established most of the physics to understand the plot and see what people are [cautiously] excited about!

Results

Here’s the plot, once again:

On the horizontal axis is the invariant mass of the two jets, which is roughly the sum of the jet energies. The vertical axis is the number of events in the data set with the given invariant mass. If there were a particle which produced the two jets, then there should be a bump in the number of events with an invariant mass around the mass of the new particle.

So… what’s all of that colorful mess in the plot? It’s the thing that makes experiments hard: background. We have identified a particular experimental signal (2 jets, a lepton, and missing energy) which could come from a new particle. What we still have to account for are “boring” processes which could lead to the same signal. By “boring” I mean Standard Model processes that we already understand. Here’s the inventory from the plot above:

• The red contribution to the histogram are events where a pair of Ws or a WZ pair are produced. The second W (or the Z) then decays into a pair of jets.
• The big green contribution comes from various processes where a single W is produced and the two jets are separately produced independently of the W.
• The white (with a pink border) sliver are events where a top-anti-top pair are produced. These tops each immediately decay into b (or anti-b) and a W boson. One of these W bosons decays to lepton + neutrino and is tagged by the experiment, while the other one might decay into jets. Now we have four jets (2 from the b quarks, two from the decay of a W), but it is possible for two of those jets to get ‘lost’ because they don’t fulfill the detector criteria for identifying jets. (This is a notoriously subtle thing.) There is also a contribution from the production of a single top.
• The blue sliver is the production of a single Z boson with two jets. The Z decays into two leptons One of the leptons can be ‘hidden’ because the particular search only looks at particles which are fairly perpendicular to the beam direction, or misidentified as a jet.
• Finally, the shaded sliver is QCD background: these are gluon mediated processes that can give two jets and a lepton.

Once we take all of these background effects into account—and this is a very nontrivial thing to do—we can subtract these from the actual number of observed events per invariant mass bin. This “background subtracted data” is plotted below:

Now things look rather interesting. First note that not all of the background is subtracted: they leave in the WW and WZ background because these will also produce a characteristic bump (red line) because the 2 jets come from a single particle (indeed the bump is around the W and Z mass). The other backgrounds have broad, smooth profiles and can be reliably subtracted—bumps are harder to subtract so we keep them in. (Update: I’m told that this may also be partly included for comparison reasons: the W/Z bump is really well understood, so it helps to be able to use it as a measuring stick.)

What’s particularly neat, however, is that there seems to be a second bump with a peak right around 150 GeV. This is what is shown in the blue line. The significance of this bump is around 3.2 standard deviations, which roughly means that we can be 99.7% sure that this is not a statistical fluctuation.

It’s not the (standard) Higgs

If this bump really does come from a particle with mass around 150 GeV, then the first thing one might think is that this is the first hint of the Higgs boson. Indeed, we even showed above that the production of the Higgs includes diagrams that would give this particular signal when the Higgs decays into two quarks. However, one very, very interesting part of the analysis is that it does not seem like this bump could come from the standard Higgs boson!

The reason is simple: we understand the standard Higgs well enough to know that if it had a mass of 150 GeV, then we would expect an effect (a bump) that would be about three hundred times smaller. In the parlance of the field, the observed bump corresponds to a particle with a 4 picobarn dijet cross section, while a 150 GeV Higgs is expected to have a 12 femtobarn dijet cross section.

Further, CDF has already done a closely related analysis: WH ?l? b b-bar. This is basically the same analysis as the present paper, except that they are able to identify jets that come from b-quarks (this is called b-tagging). The analysis with b-tagged jets showed that there was no significant excess in the range 100 — 150 GeV.

What this means is that if this bump is indeed coming from a new particle, then it must not be a particle which decays into b-quarks, at least not very often. We know, however, that the standard Higgs does decay into b-quarks, so this hypothetical new particle could not be the usual Higgs.

This is actually much more interesting, since this could either suggest a non-standard Higgs sector or it could be a sign of completely different new physics.

I should note that there is one sentence whose significance is a little unclear to me:

We compare the fraction of events with at least one b-jet in the excess region (120-160 GeV) to that in the sideband regions (100-120 and 160-180 GeV) and find them to be compatible with each other.

Basically, they look at how many b-quark jets were in the bump versus those that weren’t in the bump, and they find that the number is roughly the same. This seems to imply that whatever is causing the bump is not decaying into b quarks, but I’m not an expert on this and might be misreading it.

Where to go from here

Don’t get too excited, though. Nobody is breaking out champagne bottles yet. Three standard deviation effects have been known to come and go—i.e. it is possible that it is just an unlucky statistical/systematic fluctuation. For example, it might be a mis-modeling of the background that had to be subtracted. (The three standard deviation significance assumes that one “knows how to estimate what one doesn’t know,” as one person explained it to me.) All the same, I expect that there will be plenty of model-building papers by eager theorists in the next few weeks. [By the way, CDF has known about this effect for some time; the current excitement comes from breaking the 3 standard deviation significance and their subsequent publication of the result.]

There are a few things to look out for:

1. More data! We measure data in “inverse femtobarns” (1/fb). The current paper is based on the analysis of 4.3/fb. My [outsider's] understanding is that CDF should have around 10/fb by the end of the year, so the collaboration should be able to say something with more significance if this is a real effect.
2. What about D0? Fermilab’s other collaboration should be able to corroborate (or refute) this effect.
3. I do not believe that the LHC has enough data to say much about this at the moment, though I understand that we could be looking at 1/fb of data by summer time, and maybe a few inverse femtobarns through 2012. If the signal is real, there might be some hope to see the effect before the long shutdown at the end of 2012.

There are a lot of people who are cautiously optimistic about this. It’s almost certain that many theorists will jump on this to see if their favorite models can be tweaked to give a 150 GeV particle decaying to jets (but visible in the b-jet analysis), and that’s part of the fun. I look forward to seeing how things develop (and perhaps jumping in if the opportunity presents itself)!

Acknowledgements: I would like to thank my experimental colleagues, SP and DP for many helpful conversations. Any errors in this post are purely due to my own misunderstanding. I challenged some hep-ex grad students to foosball to try to squeeze info out of them before the paper was published… but I lost and they didn’t spill any beans. [They're also strictly prohibited from such gossip... especially to theorists.]