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What is “Model Building”?

Thursday, August 18th, 2016

Hi everyone! It’s been a while since I’ve posted on Quantum Diaries. This post is cross-posted from ParticleBites.

One thing that makes physics, and especially particle physics, is unique in the sciences is the split between theory and experiment. The role of experimentalists is clear: they build and conduct experiments, take data and analyze it using mathematical, statistical, and numerical techniques to separate signal from background. In short, they seem to do all of the real science!

So what is it that theorists do, besides sipping espresso and scribbling on chalk boards? In this post we describe one type of theoretical work called model building. This usually falls under the umbrella of phenomenology, which in physics refers to making connections between mathematically defined theories (or models) of nature and actual experimental observations of nature.

One common scenario is that one experiment observes something unusual: an anomaly. Two things immediately happen:

  1. Other experiments find ways to cross-check to see if they can confirm the anomaly.
  2. Theorists start figure out the broader implications if the anomaly is real.

#1 is the key step in the scientific method, but in this post we’ll illuminate what #2 actually entails. The scenario looks a little like this:

An unusual experimental result (anomaly) is observed. One thing we would like to know is whether it is consistent with other experimental observations, but these other observations may not be simply related to the anomaly.

An unusual experimental result (anomaly) is observed. One thing we would like to know is whether it is consistent with other experimental observations, but these other observations may not be simply related to the anomaly.

Theorists, who have spent plenty of time mulling over the open questions in physics, are ready to apply their favorite models of new physics to see if they fit. These are the models that they know lead to elegant mathematical results, like grand unification or a solution to the Hierarchy problem. Sometimes theorists are more utilitarian, and start with “do it all” Swiss army knife theories called effective theories (or simplified models) and see if they can explain the anomaly in the context of existing constraints.

Here’s what usually happens:

Usually the nicest models of new physics don't fit! In the explicit example, the minimal supersymmetric Standard Model doesn't include a good candidate to explain the 750 GeV diphoton bump.

Usually the nicest models of new physics don’t fit! In the explicit example, the minimal supersymmetric Standard Model doesn’t include a good candidate to explain the 750 GeV diphoton bump.

Indeed, usually one needs to get creative and modify the nice-and-elegant theory to make sure it can explain the anomaly while avoiding other experimental constraints. This makes the theory a little less elegant, but sometimes nature isn’t elegant.

Candidate theory extended with a module (in this case, an additional particle). This additional model is "bolted on" to the theory to make it fit the experimental observations.

Candidate theory extended with a module (in this case, an additional particle). This additional model is “bolted on” to the theory to make it fit the experimental observations.

Now we’re feeling pretty good about ourselves. It can take quite a bit of work to hack the well-motivated original theory in a way that both explains the anomaly and avoids all other known experimental observations. A good theory can do a couple of other things:

  1. It points the way to future experiments that can test it.
  2. It can use the additional structure to explain other anomalies.

The picture for #2 is as follows:

A good hack to a theory can explain multiple anomalies. Sometimes that makes the hack a little more cumbersome. Physicists often develop their own sense of 'taste' for when a module is elegant enough.

A good hack to a theory can explain multiple anomalies. Sometimes that makes the hack a little more cumbersome. Physicists often develop their own sense of ‘taste’ for when a module is elegant enough.

Even at this stage, there can be a lot of really neat physics to be learned. Model-builders can develop a reputation for particularly clever, minimal, or inspired modules. If a module is really successful, then people will start to think about it as part of a pre-packaged deal:

A really successful hack may eventually be thought of as it's own variant of the original theory.

A really successful hack may eventually be thought of as it’s own variant of the original theory.

Model-smithing is a craft that blends together a lot of the fun of understanding how physics works—which bits of common wisdom can be bent or broken to accommodate an unexpected experimental result? Is it possible to find a simpler theory that can explain more observations? Are the observations pointing to an even deeper guiding principle?

Of course—we should also say that sometimes, while theorists are having fun developing their favorite models, other experimentalists have gone on to refute the original anomaly.


Sometimes anomalies go away and the models built to explain them don’t hold together.


But here’s the mark of a really, really good model: even if the anomaly goes away and the particular model falls out of favor, a good model will have taught other physicists something really neat about what can be done within the a given theoretical framework. Physicists get a feel for the kinds of modules that are out in the market (like an app store) and they develop a library of tricks to attack future anomalies. And if one is really fortunate, these insights can point the way to even bigger connections between physical principles.

I cannot help but end this post without one of my favorite physics jokes, courtesy of T. Tait:

 A theorist and an experimentalist are having coffee. The theorist is really excited, she tells the experimentalist, “I’ve got it—it’s a model that’s elegant, explains everything, and it’s completely predictive.”The experimentalist listens to her colleague’s idea and realizes how to test those predictions. She writes several grant applications, hires a team of postdocs and graduate students, trains them,  and builds the new experiment. After years of design, labor, and testing, the machine is ready to take data. They run for several months, and the experimentalist pores over the results.

The experimentalist knocks on the theorist’s door the next day and says, “I’m sorry—the experiment doesn’t find what you were predicting. The theory is dead.”

The theorist frowns a bit: “What a shame. Did you know I spent three whole weeks of my life writing that paper?”


The Post-Higgs Hangover: where’s the new physics?

Thursday, July 19th, 2012

Now that the good people at CERN have finished their Higgs-discovery champagne, many of us have found ourselves drawn to harder drinks. While the Higgs is the finishing touch on the elegant edifice of the Standard Model, it’s the culmination of theoretical physics from the 1960s. Where’s all the exciting new physics that we’d been expecting “just around the corner” at the terascale?

My generation of particle physicists entered graduate school expecting a cornucopia of supersymmetry and extra dimensions at the TeV scale just waiting for us to join the party—unfortunately those hopes and dreams have yet come up short. While the book has yet to be written on whether or not the Higgs branching ratios are Standard Model-like, two recent experimental updates in collider and dark matter physics have also turned up empty.

No Z’ at 1 TeV

The first is the search for Z’ (“Z prime”) resonances, these are “smoking gun” signatures of a new particle which behaves like a heavy copy of the Z boson. Such particles are predicted by several models of new physics. There was some very cautious excitement after the 2011 data showed a 2σ bump in the dilepton channel around 1 TeV (both at CMS and ATLAS):

The horizontal axis is the mass of the hypothetical particle (measured by the momenta of the two leptons it supposedly decays to) in GeV, while the vertical axis is the rate at which these two lepton events are seen. (The other lines are examples for what one would expect for a Z’ from different models, for our purposes we can ignore them.) A bump would be indicative of a new particle causing a resonance: an increased rate in the observation of two leptons with a given energy. You can see something that is beginning to “kinda-sorta” look like a bump around 1 TeV. Of course, 2σ signals come and go with statistics—and this is indeed what happened with this year’s data [CMS EXO-12-015]:

Bummer. (Again, one really doesn’t have much right to be disappointed—that’s just the way the statistics works.)

Still no WIMP dark matter

Another area where we have good reason to expect new physics is dark matter. Astrophysical observations have given very strong evidence that the dark matter that gravitationally seeds our galaxies is composed of some new particle that is not described by the Standard Model. One nice feature is that astrophysical and cosmological data tell us the dark matter density in our galaxy, from which we can deduce a relation between the dark matter mass and its interaction strength.

Physicists observed that one particularly interesting scenario is when the dark matter particle interacts via the weak force—the sector of our the Standard Model that gets tied up with electroweak symmetry breaking and the Higgs. In this case, the dark matter mass should be right around a few hundred GeV, right in the ballpark of the LHC. To some, this is very suggestive evidence that dark matter may be related to electroweak physics. This class of models got a cute name: WIMPs, for weakly interacting massive particles. There are other types of dark matter, but until fairly recently WIMPs were king because they fit so nicely with models of new physics that were already modifying the electroweak scale.

Unfortunately, the flagship dark matter detector, XENON, recently released a sobering summary of its latest data at the Dark Attack conference in Switzerland. Yes, that’s really the conference name. XENON is a wonderful piece of detector technology that any particle physicist would be proud of. Their latest data-taking run found only two events (what’s expected from background). The result is the following plot:

How to read this plot: the horizontal axis is the mass of the WIMP particle. You get to pick this (or your model of new physics predicts this).  The vertical axis is the cross section, which measures the number of dark matter–detector interactions that such a WIMP is expected to undergo. The large boomerang-shaped lines are the limits set by the experiment—as the red text says, for a mass of around 55 GeV, it rules out cross sections that are above a certain number. For “garden variety” interaction channels, this number is already much smaller than the ball park estimate for the weak force.

The blob at the bottom right is some fairly arbitrary slice of the supersymmetry parameter space, but this is really just there for illustrative purposes and shouldn’t be interpreted as any kind of exclusion of supersymmetry. The other lines are other past experiments. The circles at the top left are slightly controversial ‘signals’ that have been ruled out within the WIMP paradigm by the last few direct detection experiments (XENON and CDMS).

The story is not necessarily as dour as the plot seems to indicate. There are many clever ways to get dark matter, not all of them WIMP-like. In fact, even the above plot is limited to the “spin-independent” coupling—an assumption about the particular way that dark matter interacts with nuclei. But these WIMP searches will eventually hit a brick wall around 2017: that’s when the XENON 1T (“one ton”) experiment will be sensitive to cross sections that are three orders of magnitude smaller than the current bounds. At that level of sensitivity, you end up with a lot of background noise from cosmic neutrinos which, as far as the detector is concerned, behave very much like dark matter. (They’re not.) Looking for a dark matter signal against this background is like looking for a needle in a stack of needles.

Where do we stand?

Between the infamous magnet quench of 2008 to the sobering exclusion plots of the last couple of years, an entire generation of graduate students and young postdocs is internalizing the idea that finding new physics will not be as simple as turning on the LHC as some of us had believed as undergrads. Despite our youthful naivete, the LHC is also still in its infancy with a 14 TeV run coming after its year-long shutdown. The above results are sobering, but they just mean that there wasn’t any low-hanging fruit for us to gobble up right away.


More Post-Higgs silliness

Friday, July 6th, 2012

I recently got to eavesdrop on a delightful and silly e-mail exchange between US LHC’s very own Burton and Aidan, both ATLAS physicists, after I pointed out that Wikipedia now mentions the ATLAS Higgs talk as a “notable use” of the infamous font Comic Sans. The quotes below are lifted directly from their e-mail exchange (with their permission), as illustrated by yours truly.

For more substantial physics discussion, check out Aidan and Seth’s Higgs postgame video and Anna’s ongoing posts from ICHEP.

Update [7/08]: the “4.9 sigma” comment below is a mistake, the actual “global significance” includes the ‘look elsewhere effect’ and is lower than this.



Photoshop the Higgs

Thursday, July 5th, 2012

Symmetry Breaking has a fun contest going on to photoshop the Higgs into interesting photos… by the way this is not how ATLAS and CMS do their data analysis.

Here are a few examples featuring familiar faces from the US LHC blog:

Many thanks to Aidan for his Higgs liveblog. He's now a certified Higgs-buster.



Shout out to Katie Yurkewicz, Fermilab office of communications director and former US LHC communicator


And a very special congrats to Kathryn Jepsen, US LHC communicator, who got married earlier this year!




What to look for: the Higgs-to-gamma-gamma branching ratio

Tuesday, July 3rd, 2012

There’s a lot of press building up to the Higgs announcement at CERN in just a few hours, and you’ll have Aidan’s live-blog for the play-by-play commentary. I just wanted to squeeze in more chatter about what to look for in the talks besides the usual “oh look how many sigmas we have.”

[caveat: the above cereal guy meme is purely hypothetical!]

Since we’re all friends here, I’ll be candid and say that many physicists have taken the existence of a 125 GeV-ish Higgs-like particle as a foregone conclusion—in large part because any alternative would be even more dramatic. (Recall: the Standard Model is begging for there to be a Higgs.) Whether the evidence for the Higgs is just above or just below the magic 5-sigma “discovery” threshold won’t change anything other than how much champagne Aidan will be drinking.

But that shouldn’t deter you from tuning into the 3am EST webcast. Besides getting a chance to see some famous faces in the audience, the thing to look for are hints that there’s actually more to the Higgs than the Standard Model. As described very nicely at Resonaances, the 2011 LHC data presented last December suggested that the Higgs (if it’s there) decays into photons slightly more often than the Standard Model predicts. Could this be a hint that there’s exciting (and unexpected) new physics right around the corner?

Let’s back up a little bit. Before we can talk about how the Higgs decays, we have to talk about how it’s produced at the LHC. The two main mechanisms are called gluon fusion and vector boson fusion (where the vector boson V can be a Z or W):

The gluon fusion diagram dominates at the LHC since there are plenty of high energy gluons in a multi-TeV proton beam. Note that the loop of virtual top quarks is required since the Higgs has no direct coupling to gluons (it’s not colored); the top is a good choice since it has a large coupling to the Higgs (which is why the top is so heavy). As an exercise, use the Standard Model Feynman rules to draw other Higgs production diagrams.

Once you have a Higgs, you can look at the different ways it can decay. The photon-photon final state is very rare, but particularly intriguing because the Higgs doesn’t have electric charge  and photons don’t have mass—so these particles don’t tend to talk to each other. In fact, such a Higgs-photon-photon interaction only occurs when mediated by virtual particles like the top and W:

Why these diagrams? They’re heavy enough to have a large coupling to the Higgs and also charged so they can emit photons. (Exercise: draw the other W boson diagram contributing to h to γγ.) In fact, the W diagram is about 5 times larger than the top diagram.

The great thing about loop diagrams is that any particle (with electric charge and coupling to the Higgs) can run in the loop. You can convince yourself that other Standard Model particles don’t make big contributions to hγγ, but—and here’s the good part—if there are new particles beyond the Standard Model, they could potentially push the h → γγ rate larger than the Standard Model prediction. This is what we’re hoping.

What to look for: keep an eye out for a measurement of the h → γγ cross section (a measurement of the rate). Cross sections are usually denoted by σ. Because we don’t care so much about the actual number but rather its difference from the Standard Model, what is usually presented is a ratio of the observed cross section to the Standard Model cross section: σ/σ(SM). If this ratio is one within uncertainty, then things look like the Standard Model, but otherwise (and hopefully) things are much more interesting.

The outlook on the eve of ‘the announcement’

[I thank my colleagues Jack, Mathieu, and Javi for sharing their insights on this.]

Given the assumption that there indeed is a particle at 125-ish GeV that does all the great things that the Standard Model Higgs should do, we would like to ask whether or not this is really the Standard Model (SM) Higgs, or whether it is some other Higgs-like state that may have different properties. In particular, is it possible that this particle talks to the rest of the Standard Model with slightly different strengths than the SM Higgs? And maybe, if we really want to push our luck, could this more exotic Higgs-like particle push the h → γγ rate to be larger than expected?

To answer this question, we don’t want to restrict ourselves to any one specific model of new physics, we’d rather be as general as possible. One way to do this is to use an “effective theory” that parameterizes all of the possible couplings of the “Higgs” to Standard Model particles. Here’s what one such effective theory looks like in sloppy handwriting:

Don’t worry, you don’t have to know what these all mean, but just for fun you can compare to this famous expression. The parameters here are the variables labelled a, b, c, and d. Of these, the two important ones to consider are a, which controls the Higgs coupling to two W bosons, and c, which controls the Higgs coupling to fermions (like top quarks). The Standard Model corresponds to a = c = 1.

Now we can start playing an interesting game:

  1. If we increase the coupling a of the Higgs to W bosons, then we increase the rate for h → γγ via the W loop above.
  2. If, on the other hand, we increase the coupling c of the Higgs to the top quark, then we increase the rate of h → γγ via the top quark loop above.

Thus the observation of a larger-than-expected rate for h → γγ could point to either a or c >1 (or both). How would we distinguish between these? Well, note that (see the production diagrams above):

  1. If the a (Higgs to W) coupling were enhanced, then we would also expect an enhancement in the “vector boson fusion” rate for Higgs production. When the Higgs is produced this way, you can [with some efficiency] tag the quark remnants and say that the Higgs was produced through vector boson fusion.
  2. On the other hand, if the c (Higgs to top) coupling were enhanced, then we would also expect an enhancement in the “gluon fusion” rate for Higgs production.

Thus we have some handle for where we could fit new physics to explain a possible h → γγ excess. (Again, by “excess” we mean relative to the expected production in the Standard Model.)

Here’s a quick plot of where we stand currently, including recent results from Moriond, from 1202.3697, I refer experts to that paper for further details and plots:

(There are many similar plots out there—some by good friends of mine—I apologize for not providing a more complete reference list… the Higgs seminar is only a few hours away!) The green/yellow/gray blobs are the 1,2,3 sigma confidence regions for the parameters a and c above. The red and blue lines are ATLAS and CMS exclusions. The reason why there are two green blobs is that there is a choice for the sign of c, this corresponds to whether the Higgs-top loops interfere constructively or destructively with the Higgs-W loops. For more details, see this Resonaances post.

The plot above includes the latest LHC data (Moriond, pre-ICHEP) as well as the so-called “electroweak precision observables” which tightly constrain the effects of virtual particles on the Standard Model gauge bosons. These are the blobs to keep an eye on—the lines indicate the Standard Model point a=c=1. If the blob continues to creep away from this point, then there will be good reason to expect exciting new physics beyond the Higgs… and that’s what makes it worth tuning in at 3am.


Tim Tait: “Why look for the Higgs?”

Tuesday, July 3rd, 2012

For those of you who are itching to learn more about the Higgs in anticipation of the Higgs announcement and Aidan’s liveblog, I encourage you to check out Tim Tait’s recent colloquium at SLAC titled, “Why look for the Higgs?” It’s an hour-long talk aimed at a non-physics audience (Tim says “engineers and programmers”).

Tim is a professor at UC Irvine whose enthusiasm and natural ability to explain physics carries through in his talk.

Last summer Tim was a co-director for the “Theoretical Advanced Study Institute in Elementary Particle Physics” summer school for graduate students. I heard that the students tried to get Tim’s portrait immortalized on the official school t-shirt.

For more SLAC colloquia and public lectures, see their channel on YouTube.


The Hierarchy Problem: why the Higgs has a snowball’s chance in hell

Sunday, July 1st, 2012

The Higgs boson plays a key role in the Standard Model: it is related to the unification of the electromagnetic and weak forces, explains the origin of elementary particle masses, and provides a weakly coupled way to unitarize longitudinal vector boson scattering.

As particle physics community eagerly awaits CERN’s special seminar on a possible Higgs discovery (see Aidan’s liveblog), it’s a good time to review why Higgs—the last piece of the Standard Model—is also one of the big reasons why we expect even more exciting physics beyond the Standard Model.

The main reason is called the Hierarchy problem. This is often ‘explained’ by saying that quantum corrections want to make the Higgs much heavier than we need it to be… say, 125-ish GeV. Before explaining what that means, let me put it in plain language:

The Higgs has a snowball’s chance in hell of having a mass in that ballpark.

This statement works as an analogy, not just an idiom. (This analogy is adapted from one originally by R. Rattazzi involving a low energy particle passing through a thermal bath. Edit: I’m told this analogy was by G. Giudice, thanks Duccio.)

If you put a glass of water in a really hot place—you expect it to also become really hot, maybe even to off into steam.  It would be really surprising if we put an ice cube in a hot oven and 10 minutes later it had not melted. This is because the ambient thermal energy is expected to be transferred to the ice cube by the energetic air molecules bouncing off it. Sure, it is possible that the air molecules just happen to bounce in a way that doesn’t impart much thermal energy—but that would be ridiculously improbable, as we learn in thermodynamics.

The Higgs is very similar: we expect its mass to be around 125 GeV (not too far from W and Z masses), but ambient quantum energy wants to make its mass much larger through interactions with virtual particles. While it is possible that the Higgs stays light without any additional help, it’s ridiculously improbable, as we learn from quantum physics.

Remark: the relation between thermal/statistical uncertainty and quantum uncertainty is actually one that is deeply woven into their mathematical descriptions and is the reason why quantum (or statistical) field theory is the common language of both particle physics and condensed matter physics.

Quantum corrections: the analogy of the point electron

The phrase “quantum corrections” is somewhat daunting, so let’s appeal to a slightly more familiar problem (from H. Murayama) and draw some pictures. The analog of the Hierarchy problem in classical physics is the question of the electron self energy:

The electron has charge but is nearly point-like. It must have a very large charge density and thus have a very large self-energy (mass).

Self-energy here just means the contribution to the electron mass coming from repulsive electrostatic energy of one part of the electron from another. The problem thus reduces to: how can the electron mass be so small when we expect it to be large due to electrostatic energy? Yet another way to pose the question is to say that the electron mass has contributions from some ‘inherent mass’ (or ‘bare mass’) m0 and the electrostatic energy, ΔE:

mmeasured = m0ΔE

Since mmeasured is small while ΔE is large, then it seems that m0 must be very specifically chosen to cancel out most of ΔE but still leave the correct tiny leftover value for the electron mass. In other words, the ‘bare mass’ m0 must be chosen to uncomfortably high precision.

I walk through the numbers in a previous post (see also the last few pages of these lectures to undergraduates [pdf] from here), but here’s the main idea: the reason why there isn’t a huge electrostatic contribution to the electron mass is that virtual electron–positron pairs smear out the electric charge over a radius larger than the size of the electron:

In other words: current experimental bounds tell me that the electron is smaller than 10-17 cm and the “electron hierarchy problem” arises when I calculate the energy associated with packing in one unit of electric charge into that radius. The resolution is that even though the electron may be tiny, at a certain length scale quantum mechanics becomes relevant and you start seeing virtual electron–positrion pairs which interact with the physical electron to smear out the charge over a larger distance (this is called vacuum polarization).

The distance at which this smearing takes place is predicted by quantum mechanics—it’s the distance where the virtual particles have enough energy to become real—and when you plug in the numbers, it’s precisely where it needs to be to prevent a large electrostatic contribution to the electron mass. Since we’re now experts with Feynman diagrams, here’s what such a process looks like in that language:

Higgs: the petulant child of the Standard Model

The Hierarchy problem for the Higgs is the quantum version of the above problem. “Classically” the Higgs has a mass that comes from the following diagram (note the Higgs vev):

This diagram is perfectly well behaved. The problem occurs from contributions that include loops of virtual particles—these play the role of the electrostatic contribution to the electron mass in the above analogy:

As an exercise, use the Higgs Feynman rules to draw other contributions to the Higgs mass which contain a single loop; for our present purposes the one above is sufficient. Recall, further, that  one of our rules for drawing diagrams was that momentum is conserved. In the above diagram, the incoming Higgs has some momentum (which has to be the same as the outgoing Higgs), but the virtual particle momenta (k) can be anything. What this means is that we have to sum over an infinite number of diagrams, each with a different momentum k running through the loop.

We’ll ignore the mathematical expression that’s actually being summed, but suffice it to say that it is divergent—infinity. This is a good place for you to say, what?! the Higgs mass isn’t infinity… that doesn’t even make sense! That’s right—so instead of summing up to diagrams with infinite loop momentum, we should stop where we expect our model to break down. But without any yet undiscovered physics, the only energy scale at which we know our description must break down is the gravitational scale: MPlanck ~ 1018 GeV. And thus, as a rough estimate, these loop diagrams want to push the Higgs mass up to 1018 GeV… which is way heavier than we could ever hope to discover from a 14 TeV (= 14,000 GeV) LHC. (Recall that these virtual contributions to the Higgs mass are what were analogous to thermal energy in our “snowball in Hell” analogy.)

But here’s the real problem: the Standard Model really, really wants the Higgs to be around the 100 GeV scale. This is because it needs something to “unitarize longitudinal vector boson scattering.” It needs to have some Higgs-like state accessible at low energies to explain why certain observed particle interactions are well behaved. Thus if the Higgs indeed has a mass around 125 GeV, then the only way to make sense of the 1018 GeV mass contribution from the loop diagram above is if the “classical” (or “tree”) diagram has a value which precisely cancels that huge number to leave only a 125 GeV mass. This is the analog of choosing m0 in the electron analogy above.

Unlike the electron analogy above, we don’t know what kind of physics can explain this 1016 ‘fine-tuning’ of our Standard Model parameters. For this reason, we expect there to be some kind of new physics accessible at TeV energies to explain why the Higgs should be right around that scale rather than being at the Planck mass.

Outlook on the Hierarchy

The Hierarchy problem has been the main motivation for new physics at the TeV scale for over two decades. There are a few obvious questions that you may ask.

1. Is it really a problem? Maybe some number just has to be specified very precisely.

Indeed—it is possible that the Higgs mass is 125 GeV due to some miraculous almost-cancellation that set it to be in just the right ballpark to unitarize longitudinal vector boson scattering. But such miracles are rare in physics without any a priori explanation. The electron mass is an excellent example. There are some apparent (and somewhat controversial) counter-examples: the cosmological constant problem is a much more severe ‘fine-tuning’ problem which may be explained anthropically rather than through more fundamental principles.

2. I can draw loop diagrams for all of the Standard Model particles… why don’t they all have Hierarchy problems?

If you’ve asked this question, then you get an A+. Indeed, based on the arguments in this post, it seems like any diagram with a loop gives a divergence when you sum over the possible intermediate momenta so that we would expect all Standard Model particles to have Planck-scale masses due to quantum corrections. However, the important point was that we never wrote out the mathematical form of the thing that we’re summing.

It turns out that the Hierarchy problem is unique for scalar particles like the Higgs. Loop contributions to fermion masses are not so sensitive to the ‘cutoff’ scale where the theory breaks down. This is manifested in the mathematical expression for the fermion mass and is ultimately due to the chiral structure of fermions in four dimensions. Gauge boson masses are also protected, but from a different mechanism: gauge invariance. More generally, particles that carry spin are very picky about whether they’re massive or massless, whereas scalar particles like the Higgs are not, which makes the Higgs susceptible to large quantum corrections to its mass.

3. What are the possible ways to solve the Hierarchy problem?

There are two main directions that most people consider:

  1. Supersymmetry. Recall in our electron analogy that the solution to the ‘electron mass hierarchy problem’ was that quantum mechanics doubled the number of particles: in addition to the electron, there was also a positron. The virtual electron–positron contributions solved the problem by smearing out the electric charge. Supersymmetry is an analogous idea where once again the set of particles is doubled, and in doing so the loop contributions of one particle to the Higgs are cancelled by the loop contributions of its super-partner. Supersymmetry has deep connections to an extension of space-time symmetry since it relates matter particles to force particles.
  2. Compositeness/extra dimensions. The other solution is that maybe our description of physics breaks down much sooner than the Planck scale. In particular, maybe at the TeV scale the Higgs no longer behaves like a scalar particles, but rather as a bound state of two fermions. This is precisely what happens with the mesons: even though the pion is a scalar, there is no pion ‘hierarchy problem’ because as you probe smaller distances, you realize the pion is actually a bound state of two quarks and it starts behaving as such. One of the beautiful developments of theoretical physics in the 1990s and early 2000s was the realization that this is precisely what is being described by theories of extra dimensions through the so-called holographic principle.

So there you have it—while you’re celebrating the [anticipated] Higgs discovery with fireworks on July 4th, also take a moment to appreciate that this isn’t the end of a journey culminating in the Standard Model, but the beginning of an expedition for exciting new physics at the terascale.


An experiment: Feynman Diagrams for Undergrads

Thursday, May 31st, 2012

The past couple of weeks I’ve been busy juggling research with an opportunity I couldn’t pass up: the chance to give lectures about the Standard Model to Cornell’s undergraduate summer students working on CMS.

The local group here has a fantastic program which draws motivated undergrads from the freshman honors physics sequence. The students take a one credit “research in particle physics course” and spend the summer learning programming and analysis tools to eventually do CMS projects. Since the students are all local, some subset of them stay on and continue to work with CMS during their entire undergraduate careers. Needless to say, those students end up with fantastic training in physics and are on a trajectory to be superstar graduate students.

Anyway, I spent some time adapting my Feynman diagram blog posts into a series of lectures. In case anyone is interested, I’m posting them publicly here, along with some really nice references at the appropriate level.

There are no formal prerequisites except for familiarity with particle physics at the popular science/Wikipedia level, though they’re geared towards enthusiastic students who have been doing a lot of outside [pop-sci level] reading and have some sophistication with freshman level math and physics ideas.

The whole thing is an experiment for me, but the first lecture earlier today seems to have gone well.


Name these brands/plants? Name these particles!

Tuesday, April 17th, 2012

I don’t know the original source, but there’s an image that has gone semi-viral over the past year which challenges the reader to identify several brand names based on their logos versus plant names based on their leaves. (Here’s a version at Adbusters.) The point is to contrast consumerism to the outdoors-y/science-y education that kids would get if they just played outside.

This isn’t the place to discuss consumerism, but I don’t agree with idea that the ability to identify plant names carries any actual educational value. Here’s my revision to the image:

Adapted from the original “Name these brands/plants” image (original source unknown).

On the right we’ve encoded all of the particles in the Standard Model in a notation based on representation theory. In fact, this is almost all of the information you need to know to write down all of the Feynman rules in the Standard Model (more on this below).

Tables that the one above are a compact way to describe the particle content of a model because the information in the table specifies all of the properties of each particle. And that’s the point: whether we name a particle the “truth quark” or the “top quark” doesn’t matter—what matters is the physics behind these names, and that’s captured succinctly in the table. Science isn’t about classification, it’s about understanding. I leave you with this quote from Feynman (which you can watch in his own words here):

You can know the name of a bird in all the languages of the world, but when you’re finished, you’ll know absolutely nothing whatever about the bird… So let’s look at the bird and see what it’s doing — that’s what counts. I learned very early the difference between knowing the name of something and knowing something.


Addendum: naming those particles

For those who want to know, the particles in the table are, from top down:

  1. The left-handed quark doublet, containing the left-handed up quark and left-handed down quark
  2. The anti-right-handed-up quark
  3. The anti-right-handed-down quark
  4. The left-handed lepton doublet, containing the left-handed electron and left-handed neutrino
  5. The anti-right-handed electron (a.k.a the right-handed positron)
  6. The anti-right-handed neutrino
  7. The Standard Model Higgs

SU(3), SU(2), and U(1) refer to the strong force, weak force, and hypercharge. Upon electroweak symmetry breaking, the weak force and hypercharge combine into electromagnetism and the heavy W and Z bosons. Here’s how to read the funny notation:

  1. Under SU(3): particles with a box come in three colors (red, green, blue). Particles with a barred box come in three anti-colors (anti-red, anti-green, anti-blue). Particles with a ‘1’ are not colored.
  2. Under SU(2): particles with a box have two components, an upper and a lower component. That is to say, a box means that there are actually two particles being represented. More on this below. Particles with a ‘1’ do not carry weak charge and do not talk to the W boson.
  3. Under U(1): this is the “hypercharge” of the particle.
  4. The electric charge of a particle is given by adding to the hypercharge +1/2 if it’s the upper component of an SU(2) box, -1/2 if it’s the lower component of an SU(2) box, or 0 if it is not an SU(2) box (just ‘1’).

As a consistency check, you can convince yourself that both the left- and right-handed neutrinos carry zero electric charge. Note, also, the fact that we’ve written out left-handed and right-handed particles differently. This is a reflection of the fact that the Standard Model is a chiral theory.

Finally, I said above that the table of particles almost specifies the structure of the Standard Model completely, the additional pieces of information required are:

  1. Which of the above particles are fermions and which are scalars (the gauge bosons are implied)
  2. Write down the most general ‘renormalizable’ theory (we write only the simplest interaction vertices)
  3. Specify the pattern of electroweak symmetry breaking (the Higgs)
  4. Specify the flavor symmetries (three of each type of matter  particle)

From this one can write the complete mathematical expressions for the Standard Model. One then just has to fill in the observed numerical values to be able to calculate concrete predictions for actual processes.


Grad School Confidential

Wednesday, April 11th, 2012

Later this week another generation of academics will finalize their decisions about which graduate programs to attend next year—many congratulations to all of you soon-to-be grad students who will join us in the trenches at the frontier of human knowledge.

Unlike undergraduate life which has a well-known idealization in Animal House (or the TV series Greek), grad school doesn’t get much publicity other than the sardonic (and delightful) PhD comics. I wanted to take a moment to share some observations of what graduate school is actually like, with the usual caveat that this is just my personal perspective—each person has their own experience. (Grads and former grads: feel free to add to the discussion in the comments section.)  Without further ado, here are five observations about grad school.

[All illustrations are my own and brought back memories of my failed first-year aspirations of becoming a chalkboard Banksy.]

1. Grad school: more like Zelda than Mario

College is a lot like a Super Mario Bros. video game. You wake up, go to class, do the homework that’s assigned, and study the chapters you were told to, and rock the exam after practicing on past exams. Sure, sometimes you have to try a few times before you can make that jump right at the end of the level, but at each step it was clear what you had to do.

You may have to rethink your measures of success and re-evaluate the tools you need to get there.

Grad school is different. You can’t just wake up in the morning and do all the stuff that you know you have to do—because research is precisely about figuring out what to do when it is not clear at all what the next step is. In this respect grad school is more like playing a Legend of Zelda video game.

Unlike coursework, research is about open questions. Usually these questions are still open for a good reason: they’re hard! You won’t have an answer key in the back of the book or a TA’s office hours to show you the trick. There’s no road map; you need to carve out your own path and figure out what tools you need to develop to move forward. Sometimes there will be dead ends and you’ll have to back-track, but in the end this can be a rewarding experience. You don’t remember Mario Bros. for how hard it was to stomp on Bowser’s head, but you do remember all the time you spent trying to figure out the puzzle to break into that one dungeon so you could rescue Zelda.

2. Get paid to do what you love—just not very much

One of the perks of grad school that often surprises non-academics is that yes, you get paid to do science! (Usually this is associated with doing some teaching.) In a difficult economy and with undergraduate student loans soaring, this is a welcome respite from large tuition bills and reliance on parental support. On the other hand, don’t expect to be drowning in disposable income.

One trick to stretch out your stipend money: lower your standards when treating yourself to something nice.

Just be careful not to fall into the trap of comparing your income to your college friends who got ‘real’ jobs. That being said, you’ll have health insurance, be able afford an apartment and food, and most importantly, you’ll have the freedom to work on what you want and how (and when) you want to.

3. Somewhere between being a kid and a grown up

Maybe it’s just me (and I really hope not), but part of being a grad student is living precipitously on the edge of growing up. My personal experience has been full of office pranks, jokes, and the “child-ish” silliness that sometimes comes with the “child-like” curiosity that is at the heart of being a scientist. At the same time, one has to balance one’s aforementioned budget, keep pushing the less-fun parts of projects, and be responsible for the direction and content of one’s research.

It’s worth mentioning that sometimes it can feel like the rest of the world is growing up way faster than you are. In addition to earning much more than you, your old high school friends will be getting married and starting families—the latter of which is something which can be difficult (though not impossible) as a young academic.

High school reunion can be a reminder that everyone else has "grown up" while you're still in school.

In a larger sense, grad students are fledgling scientists, apprentices to professors who train their academic offspring. And just like biological offspring, it’s often the case that the apple doesn’t fall far from the tree—after all, your grad school mentors are the ones who teach you how to think about your science, how to grapple with hard problems, and (very important) how to interact with other scientists.

4. Sometimes the next step is a step back

Whatever discipline you’re in, and no matter how smooth things seem to be going for the other students, graduate school is hard. (So are professional schools and real grown up life, for that matter.) Sure, most people are prepared to spend their PhD working on hard research questions. What people don’t usually expect is that often it’s actually everything else that makes a PhD hard: balancing your work with the rest of your life.

“Rest of your life?” It’s a cliche that grad students don’t have lives outside of their labs, and it’s completely wrong. The most successful students—both in undergrad and grad school—are often the ones who have something else that they’re passionate about and that is totally unrelated to their work. Maybe music or art, maybe a particular sport, or a social activity (blogging?)… something to dive into and keep you sane when work isn’t going well—and there will be times when work is not going well.

In many ways the defining moments in graduate school aren’t when research is going well, but rather those times when it feels like everything is crumbling beneath you. Those moments when you feel like you should chain yourself to your desk until everything works? Those are usually the times when the best thing you can do is to take a step back for a bit and relax.

It’s crucially important to recognize that things will not go as smoothly as you plan. Consider the following very-scientific graph of happiness over time.

Actually, the pointy curves come from something I've been working on (with different labels).

Naively, one might imagine that grad school is a period where you just keep learning more and more about something you enjoy until you steadily become the world expert on something really important. What actually happens is that you spend most of your time grappling with the frustrating problems that prevented other people from doing this research before you. Then, with some luck, there are brief moments of ecstatic clarity where you make progress: you’ll remember why you’re doing a PhD and all will be right in the world… for maybe a day or two, at which point you’ll come up to the next hurdle that you’ll have to struggle with.

This perpetual struggle at the heart of research can be hard to swallow, especially for those to whom undergraduate coursework came fairly naturally. The feelings of self-doubt that often arise are so common that it even has a name, impostor syndrome, wherein people feel like their struggles indicate that they are not ‘good enough’ to be a PhD student and their university made a big mistake accepting them to such a program. Just remember: all this is normal! (See Zelda analogy above.)

Footnote: Not every PhD becomes an academic!

I wanted to address something related to this: not every grad student goes on to become an academic, and that this is okay. Somehow it’s almost taboo to talk about going off into industry after grad school instead of continuing to become a postdoc and then a faculty member somewhere—even though there are clearly fewer postdoc positions than grad students, and fewer still faculty hires. (I think it’s great that Burton’s mentioned this in recent posts.)

While there is something special about spending your life pursuing fundamental science, but that doesn’t mean it’s the right path for everyone. And this is not to say that some people “aren’t cut out” for research or that their PhD was not well spent: I’ve seen some truly special and talented individuals with bright academic futures decide that they would be happier applying the skills they developed on something else. And that’s great—one of the reasons why our country invests in fundamental research is to support a highly skilled workforce doing exciting things outside of the ivory tower.

I’ve had difficult conversations with multiple young academics who have struggled to weigh their passion for science against pragmatism: what if they can’t find a job sufficiently close to their spouse? What if they want to settle down and start a family rather than having to bounce between temporary grad and postdoc positions? What if they need to take care of ailing parents and cannot hold off until the indeterminate future to secure that kind of financial stability?

Fortunately, a PhD is something which generally translates into marketable skills “in the real world,” and I think it’s important for those in academia to recognize that sometimes good people will leave the field for good reasons.

5. How to be a good graduate student

I’d like to wrap up by once again addressing the next generation of grad students with some unsolicited advice from someone crawling towards the light at the end of his own PhD tunnel.

1. Find good mentors. Your adviser will have a big impact on your PhD and career, but you should also make a point to find mentors in the form of other faculty, postdocs, and graduate students. Learn as much as you can from the people around you, especially when they can offer advice that they had to learn the hard way.

2. Persistence and enthusiasm goes a long way. You can expect to run into setbacks and roadblocks. One of the most useful things you can develop is an enthusiasm for your work and the persistence to keep pushing even when things feel futile. Persistence and enthusiasm can make up for a lot of things: lost sleep, raw intelligence (when you feel like everyone else is smarter than you), gaps in your problem-solving toolbox, etc.

3. Learn how to communicate. One of the cornerstones of science is being able to effectively communicate your work to others. Learn how to effectively read and write papers, and learn how to give good talks about your research.

4. Use your freedom wisely. For the most part, people won’t tell you how to spend your time. It’ll be up to you to work on what you want, when you want to, and however you think will best solve the problem. Just be careful that you’re not using all of this extra rope to hang yourself. Find the right balance of work and play that works for you.

5. Science is social. There is synergy in academia. People wonder what theorists do all day long since it seems like all we do is to think up silly ideas—we spend most of the day talking to each other. Ideas are meant to be bounced off of one another: revised, refined, and re-assessed. Don’t fall into the bad habit of hiding in a hole in the ground until you find the answer—make use of the community around you!

Science is a team sport, it helps to figure this out earlier rather than later.

While we’re on this note—take time to be part of the science community in your field. There are some scientists who develop their best ideas while hiking with friends or at a pub after a conference.

6. Let it be fun. Despite all the things one has to struggle with from research to personal life and everything in-between, grad school is a special time in your life; enjoy it.