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Flip Tanedo | USLHC | USA

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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.

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Tools of the trade: MC4BSM 2012

Sunday, April 1st, 2012

Last week Cornell hosted the sixth “Monte Carlo Tools for Beyond the Standard Model” mini-workshop and I thought it was a terrific success. Here “Monte Carlo” refers to the computer simulation techniques used to solve difficult problems such as the behavior of high energy particles at the LHC. The name is a reference to the famous casino in Monaco since these methods are based on random sampling.

 

MC4BSM poster: if CMS reminds you of a roulette wheel, you may have a gambling problem.

 

Playing Dice with the LHC?

It’s not what it sounds like. At first glance talking about ‘random sampling’ might make it sound like someone doesn’t know what they’re doing. It’s actually quite the opposite. The theory of hadron colliders (which is mostly quantum chromodynamics) is well established, but actual calculation requires compromise.

I won’t go into details, but the rough sketch is that a high energy collision at the LHC does not look like the nice Feynman diagrams that we’ve been drawing (and that we can calculate easily):

Nope. In fact, the events look much, much more complicated (from S. Hoeche’s talk):

Needless to say, this is very difficult to calculate using pen and paper. In fact, the situation is even more difficult than it looks: many of the steps in this calculation require systematic approximations and are non-perturbative (hopeless to calculate in the usual method). There’s more: the above picture is just what happens when there’s a high energy collision in vacuum. We also have to model how all of that interacts with the detector to give a picture more like this:

It is practically impossible to calculate a closed form expression for the Standard Model prediction for the distribution of detector signatures. On the other hand, what we can do is imagine actually simulate particle production and decay at each step of the process so that the random evolution of the initial collision into the final detector signature follows the probability distribution of the “closed form expression” that we can’t write down. By doing this many times, we can determine the probability distribution simply by looking at the distribution of the Monte Carlo simulation.

This probably still sounds a little abstract—but it’s the analog of determining the interference pattern of the double slit system by actually doing the experiment with electrons and looking at the distribution of electron hits on the screen. Another nice example is to determine the area of a circle (or the value of π) by Monte Carlo.

 

Tools of the Trade

Suffice it to say that Monte Carlo is a very important tool in high energy physics. For example, the results of Monte Carlo studies are used to determine what sorts of events we should be looking at to find the cleanest signals for a Higgs or for new physics—this sort of thing is especially relevant since the rate of high energy events at the LHC is actually much larger than our bandwidth for recording data, so we need to be able to trigger on particular events that we think are worth a closer look.

On the more theoretical side, Monte Carlo gives us a handle for mapping models of new physics to experimental signatures. For example, if we see a definitive signal of a new particle outside of the Standard Model, how can we begin to determine whether it is a supersymmetric partner, an extra dimensional resonance, or something else?

In between theory and experiment, there’s a lot of hard work done ‘in the trenches’ to develop better tools (both theoretical and computational) to model quantum chromodynamics. Often times this is under appreciated in the field since the work is not glamorous enough to land in one of Dennis Overbye’s New York Times articles, but recently three of the leaders of this field received the 2012 Sakurai prize—congrats to AltarelliSjostrand, and Webber!

Theorists will wax poetic about espresso machines and long nights at a chalkboard, experimentalists will tell you what it’s like to jump into the world’s largest scientific apparatus (armed with a vacuum cleaner), but the truth of the matter is that we spent a lot of time running computer simulations. We rely on the subset of the community that develops and maintains these tools, and occasionally we hold workshops (such as MC4BSM) to learn the latest and greatest tools.

MC4BSM 2012

Probably the first mystery of the MC4BSM series of workshops is the strange logo:

Apparently the illustration was done by a professional artist who is a friend of one of the organizers. The interpretation still isn’t clear to me—though it’s been suggested that it represents the “elephant in the room” associated with the lack of training opportunities to learn Monte Carlo techniques. Alternately, it was also pointed out that it’s a different kind of “pink elephant.”

The workshops are geared toward an audience of theorists who don’t necessarily have a background in Monte Carlo methods. The “big idea” is connecting our models of new physics to experimental data (image from M. Perelstein’s slides):

The key to doing this efficiently has been to develop a pipeline of Monte Carlo tools which interface with one another and take a theorist’s model to something that can be compared to real data; one example of such a pipeline is (image from C. Duhr’s talk):

The ovals are different stages of computational tools—the first two or three stages can usually be done by hand by a careful graduate student. From there on out we really rely on the Monte Carlo tools available to us. The red text highlights common programs used to connect each step, while the green-ish text are common languages that are used to provide a standardized language for program to communicate with one another.

All of these programs are open source (though some depend on commercial software like Mathematica) and are developed by high energy physicists for high energy physicists.

Tutorial

The real highlight of the workshop were the two tutorial sessions where attendees had a chance to play with various programs in a hands-on environment. The whole point of the meeting, after all, is to learn how to use these tools. The tutorial session allow attendees to ask questions directly to the program developers and to build their own templates by solving a simple toy problem.

Unlike previous MC4BSM workshops, the organizers adopted a novel format for the tutorial sessions which I thought worked very well. Each participant brought their own laptop and had a choice choice of which chain of programs they would use to solve the toy problem:

Instead of having representatives from each program give a short talk about how to install and run their code, users were left to themselves to jump in head first with their colleagues and then flag own experts as needed. (The night before the workshop there was also a group installation session where people could work out kinks in getting specific programs to compile on specific operating systems.)

Several of the graduate students there got their first taste in going through the entire series of programs, while more senior researchers learned how to use alternate tools than the ones they’re used to.

The tutorial information is all available online for anyone who wants to follow along on their own. The material will eventually be made available as proceedings for the workshop; I think it will be a valuable resource for anyone interested in learning to use these tools.

Human vs. Machine

One of the running jokes at the workshop was that eventually we’d be able to select a few options in a smart phone app to cook up a model of new physics and then send it to a computing cluster to work out the detailed phenomenology—perhaps obviating the need for graduate students. However, one thing that computers cannot yet replace is the value of having face-to-face interactions with one’s colleagues.

I’ve said many times that physics is a social activity and the field progresses from the collaborative efforts of the entire community. Meetings like MC4BSM are more than just ways to learn new tools, but are also ways to catch up with friends and colleagues and bounce new ideas off one another.

One bright idea that was promptly shot down was a request to hold the next MC4BSM meeting at the Monte Carlo casino in Monaco.

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Dissecting the Penguin

Monday, March 19th, 2012

No animals were harmed in the writing of this blog post.

One of the amusing tales in particle physics is the story of how the “penguin diagram” got its name. We won’t go into that here, instead, we’ll make use of some of the tools we’ve developed with Feynman diagrams to understand the physics behind these ‘penguin’ diagrams. In doing so, we’ll have a nice playground to really make use of what we’ve learned so far about Feynman rules. (Feel free to review the series if you need a refresher!)

Caveat: we’ll try to squeeze as much physics as we can out of our diagrams, and occasionally I’ll lapse into some more technical details. Feel free to skip these if you just want a ‘big picture,’ the main idea is independent of the details.

Drawing the penguin diagram

In a nutshell, a penguin (in the Feynman diagram sense) is a process where one flavor of matter particle changes into another flavor while emitting a photon or gluon: for example, a bottom (b) quark converting into a strange (s) quark and a photon. These processes are rare in the Standard Model and are interesting because they can get large enhancements from new physics

So let’s jump right in. If I told you that I wanted to study a process where a bottom quark turns into a photon and a strange quark, the simplest diagram to draw would be something like this:

However, this is wrong. The problem is that this is not an allowed Feynman rule in the Standard Model—the photon does not connect particles of different flavors. It can talk to a bottom and anti-bottom, or a strange and an anti-strange, but not to a bottom and an anti-strange. (Recall that an anti-strange quark going into the vertex is the same as a strange quark coming out.)

In fact, we know that the only source of flavor-changing in the Standard Model comes from the W boson. Thus we conclude that whatever diagram mediates this process, which we succinctly write as b → sγ (“bottom quark to strange quark and photon”), must include a W boson somewhere. Here’s a simple diagram one could draw:

Great! This actually works. A couple of things to notice right off the bat:

  1. This is a “loop” diagram, to be distinguished from the “tree” diagram we tried to draw above. This demonstrates a principle in the Standard Model: there are no tree-level “flavor-changing neutral currents” (FCNC). That is to say, the neutral gauge bosons (photon, gluon, Z) do not have direct interactions to fermions which change flavor. The b → sγ transition evades this principle by being a loop-level (and hence “more quantum”) process.
  2. The top quark (t) in this diagram could have been replaced by any of the up-type quarks: up, charm, top. In fact, since these internal quarks are virtual—they’re not directly observed—quantum mechanically all three quarks simultaneously mediate this process. (Mathematically this means that there are three complex numbers that add together.)
  3. In fact, there are several other diagrams one could draw. Since we can learn a lot just by looking at this diagram, I’ll leave it to you to draw these diagrams as homework. (Think of different places to put the photon, and if you’re really slick, think of other bosons that could replace the W.)
Technical comments: For those with some more mathematical background, the loop diagrams are higher order in a Taylor expansion. In fact, the whole point of drawing Feynman diagrams is to have a succinct way of writing out a Taylor expansion for a process. The expansion parameters are the coupling constants, g, associated with each vertex: note that a loop diagram contains two extra vertices compared to a  tree-level diagram for the same process. It turns out that calculating the loop also involves a 4D integral from including all possible momentum configurations for the virtual particles, and this ends up giving a factor of 1/16π2 so that the expansion parameter is really g2/16π2. For the electromagnetic coupling we know that e2/4π is 1/137, so that this is indeed a small parameter to expand about.

Drawing the diagram using chiral notation

Edit, 19 March: an earlier version of this post had incorrectly suggested that angular momentum restricts the penguin to be a dipole operator. Thanks to Jack Collins for pointing this out.

Instead of pushing this diagram for all it’s worth, let’s try to make use of the “chiral nature” of the Standard Model that tells us that left-handed particles are rather different from right-handed particles.

We propose using different notation which makes the “chirality” (handedness) of the matter particles manifest; in other words, we explicitly say if a particles is right- or left-hande, e.g. bR for the right-handed bottom. This notation will lead to more complicated diagrams, but it won’t require us to do any math to understand physics that is hidden in the simpler notation.

Angular momentum plays an important role here. What we really mean by a penguin process is one that connects a right-handed fermion with a left-handed fermion, for example bR → sL γ. This may sound little abstract, so here’s a cartoon picture, where the red arrows are carefully drawn to indicate handedness. The green numbers represent the angular momentum in the bottom quark’s direction of motion: right-handed fermions carry half a unit of angular momentum, photons carry 1 unit of angular momentum. (The left-handed strange quark has negative angular momentum since it’s spinning in the opposite direction.)

Okay: so now we’ve learned that penguins are transitions in which a right-handed fermion decays into a photon and left-handed fermion of a different flavor. Of course you can also have left-handed fermions decaying into a photon and a right-handed fermion. Similarly, you can replace the photon with a gluon (“gluonic penguin”). Photonic penguins in which the flavors don’t change have their own fancy name, “electric dipole moments.”

Technical comment, for physics students: The identification of the penguin with a “dipole” should make sense from the chirality of the operator. These right-to-left transitions are mediated by a σμν . Alternately (and slightly more technically), you can appeal to gauge invariance and observe that the only gauge-invariant operator you can write with one photon that is not a kinetic term must contain a field strength Fμν. The antisymmetry of the Lorentz indices requires that it contracts with a σμν which necessarily requires a chirality-flip. 

You might wonder why we don’t consider same-chirality transitions of the form bL → sL γ. Loop diagrams contribute to this are field-strength renormalizations, they correct the tree-level kinetic term and lead to a redefinition of flavors with respect to the tree-level basis.

Instead of dictating the chiral Feynman rules to you, let’s discover them together. We’ll be sketchy since for our purposes the details won’t buy us much. Let’s naively try to draw a chiral Feynman diagram by just copying the non-chiral diagram above.

We know that the chiral transition must be of the form bRsL γ. First let’s make the observation that we’ve drawn the bR leg with an arrow going in the opposite direction. This is one of the new conventions of the chiral Feynman diagrams:

Arrows no longer correspond to particle or anti-particle, instead, they correspond to the chirality of the particle.

So right- and left-handed particles have opposite arrows.

Now there’s a clear problem in the diagram above, which we’ve marked with a question mark (?). We know that the W boson is biased: it only talks to left-handed particles! Thus we are not allowed to have the coupling of a bR to a tL. We need to convert the right-handed bottom quark to a left-handed bottom quark (and these are totally different particles!).

Fortunately, we can do that! A right-chiral particle can convert into its a left-chiral sibling by bouncing off the Higgs vacuum expectation value. (Once again, electroweak symmetry breaking plays a key role!) We draw the “bounce off of the Higgs vacuum expectation value” as a cross on the fermion that changes the direction of the arrow. We call these crosses “mass insertions” because they are proportional to the mass of the particle.

Technical details: for our new “chiral” Feynman diagrams, the arrows no longer represent particle/anti-particle flow. So how do you know if a line corresponds to a particle or anti-particle? Usually it’s clear from angular-momentum conservation. For example, we should have been more precise and said that in the above diagram, the right-chiral bR is converted into the anti-bL since that’s the guy with same angular momentum as the original bR.

Why cross on the bR (the “bounce of the Higgs vacuum expectation”) physically represent a mass? This makes sense: a massive particle is one which can come in both chiralities since you can always boost into a frame where it’s spinning in the opposite direction.

For students who want to a reference for chiral diagrams, I refer to the encyclopedic “two-component spinor bible.” The results are completely equivalent to what one would obtain using four-component Dirac spinors, but the main benefit is that you hardly have to do any Dirac algebra to see the chiral structure of the amplitude. Further, things like Majorana fermions can be very difficult in Dirac notation but are straightforward using the two-component Weyl spinors.

There! Now we have a diagram that appears to work. Except it doesn’t. The reason is a a bit technical, but it has to do with the fact that not only are there tops (tL) running in the loop, but there are also charms (cL) and ups (uL). When you sum over all of these contributions, it turns out that the final result is zero! This result is known as the GIM mechanism.

Technical detail: What is the origin of this GIM mechanism cancellation? The W boson coupling is actually a 3×3 matrix corresponding to which down-type flavor is being converted into which up-type flavor. These matrices are unitary, they encode a change in flavor basis, and the diagram above is proportional to:

This is basically the relation UU = 1 for a unitary matrix U.

The lesson from this is that the previous Feynman diagram is too simple—it needs more internal structure to avoid the GIM cancellation. The particular structure that it needs is something that differentiates the top/charm/up type diagrams. Fortunately, there’s a way to do this: we just add more mass insertions: since the top, charm, and up each have different masses, the sum of the following diagrams (with t replaced by c and u in other diagrams) will not vanish:

Of course, the question mark is our way of pointing out that this still doesn’t work. The mass insertion in the loop converts the left-handed top quark into a right-handed top quark. However, the W only couples to left-handed particles, so the WtRsL is not allowed. This means that we need another internal mass insertion to convert the right-handed top back into a left-handed top:

I drew the mass insertion after the photon, but there are other places it could have gone. As an exercise you can draw the other diagrams that contribute to bR → sL γ. These extra mass insertions come at a price: they tend to reduce the size of the diagram.

What have we learned?

  1. One reason that these penguin decays are rare is that there is no tree-level diagram in the Standard Model. It’s a loop-level process which makes it “more quantum.”
  2. Another reason why the penguin is rare is the “GIM mechanism,” which requires that the diagram picks up additional mass insertions. In order to avoid this, we need additional internal mass insertions which come in pairs and typically suppresses the process. (This also tells us that in the limit where all of the up-type quarks have the same mass, the probability of this process must vanish.)
  3. The chiral structure of the Standard Model tells us a lot about what’s actually happening in a penguin! We learned that penguins are left-right “dipole” transitions and that (in the Standard Model) they require that the fermions bounce off the Higgs vacuum expectation value a few times due to (a) angular momentum conservation and (2) the coupling of the W to only left-handed fermions.

One of the nice things about the chiral Feynman diagrams is that they’re easier to read when trying to estimate the size of the diagram without doing the nitty gritty details of the calculation. Each mass insertion gives a factor of the fermion mass (or the mass splitting in the case of GIM cancellation) and then we can fudge additional factors by dimensional analysis. This is beyond the scope of this post, but it’s worth explaining why these slightly-more-complicated diagrams are worth their complications. For the above diagram, one can see that the bR → sL γ penguin is proportional to the mass of the bottom quark and the difference in the squared masses of the internal up-type quarks.

Bonus: Leptonic penguins

As a final example, let’s quickly go over the story of the leptonic penguin. The prime example is the decay of a muon into a photon and an electron, μR → eL γ. Usually the relation between up/down quarks is analogous to that of electrons and neutrinos. This leads us to guess the following diagram:

Here we’ve taken the liberty of introducing a right-handed neutrino into the theory to account for the experimental observation that neutrinos have a very tiny, but non-zero mass. Unfortunately, the above diagram does not work since the neutrinos are not electrically charged and so they do not interact with the photon. We need to look for other diagrams. In particular, the photon must not come off of the fermion leg, but perhaps from the W leg.

One thus ends up with diagrams of the form:

What is that dashed line? That’s a charged Higgs—one of the Goldstone bosons that was eaten by the W. I’ve drawn it here just to show off a little: we can draw diagrams in which we make the interactions with the different components of the W manifest. Here we know that the charged Higgs is really the “longitudinal polarization” of the W, but we’re drawing it as an independent particle. (We could have “picked a gauge” in which this diagram is absent, but let’s allow ourselves to show off for pedagogical purposes.) Let us use this opportunity to highlight another aspect of the chiral notation:

  1. Interactions with vector particles (spin 1) preserve fermion chirality. We saw this with the W boson above: a left-handed particle stayed left-handed after interacting via a W boson. This was also true with the photon coupling, and turns out to be true for all of the gauge bosons. It has nothing to do with the chiral structure of the Standard Model, rather it has to do with conservation of angular momentum. (The chiral structure of the Standard Model shows up when we say the W only talks to left-handed particles; compare this to the photon which will talk to pairs of left-handed particles or pairs of right-handed particles, but never a left-handed and a right-handed particle in the same vertex.)
  2. On the other hand, interactions with scalars (spin 0), such as the Higgs vacuum expectation value or any of its components, do change chirality. This is just a feature of scalar interactions versus vector interactions.

Now some remarks about the leptonic penguins:

  1. Note that because there is no photon–neutrino coupling, the set of diagrams for the leptonic penguins are different from those of the quark penguins! (As an exercise, try drawing all the diagrams for the quark–quark–gluon penguins, there are even more since gluons can couple to other gluons.)
  2. In the diagram above, the charged Higgs coupling to a right-handed muon and a left-handed electron is proportional to the mass of the muon—thus one still picks up a factor of the initial fermion mass.
  3. We could also draw the reversed diagram where we pick up the mass of the electron. (Exercise: draw this diagram and label all internal states with their chiralities.) However, since the electron is so much lighter than the muon, we can ignore this contribution.
  4. [technical detail] For those who know a little field theory, it should be straight forward to do a dimensional analysis on this to determine the dependence of the branching ratio on the internal neutrino masses and the external muon mass. (Let the W mass be the dominant mass scale in the diagram, though kinematic factors have to be made up with the mass scale of the process, which is the muon mass.)
  5. The muon mass insertions are really small. We know that these mass insertions are really mass splittings (i.e. proportional to the differences in mass). Since the neutrino mass splittings are experimentally known to be very small, the μR → eL γ penguin is very rare!

Punchline: looking for new physics

While this post has been somewhat technical, for the most part we’ve managed to avoid doing any mathematics while still being able to make some fairly quantitative statements about the penguin process. We could, for example, talk about how the bR → sL γ penguin vanishes if the internal quarks have the same mass, or even guess the dependence of the quantum mechanical amplitude on the masses of the internal and external particles. If we were to do the calculation using the “standard” (rather than chiral) Feynman diagram, these properties would require a little bit of mathematical work to see explicitly.

Now that we’ve really beaten ourselves over the head with these penguins, let me just close by explaining that these penguins are interesting primarily because they are loop level processes where any allowed particle may run in the loop, including new particles that aren’t in the Standard Model. This is because such internal particles are virtual and don’t need to be on-shell, that is to say that they don’t need to have enough energy to actually exist for long periods of time. In popular books this is often explained with Heisenberg’s uncertainty principle: the internal particles can violate energy conservation for a very short period of time, as long as they decay into states which do respect energy conservation relative to the initial particle. Thus the inside of the penguin can include contributions from exotic new particles. Since the Standard Model contribution is suppressed, there’s a chance that the effect of the exotic new particles might be seen in an enhanced decay rate.

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Particle Paparazzi: the private lives of the Standard Model particles (summary)

Sunday, March 4th, 2012

I wanted to take a break from our ongoing discussion of Feynman diagrams and the Standard Model to highlight what we’ve learned the true identities of the particles we now know and love.

The “fake” Particle Periodic Table

These days most people who read Scientific American or Dennis Overbye’s articles in the NY Times can list off the Standard Model particles off the top of their heads. Their list would look something like this (which we posted earlier):

These are, of course, the matter particles. Savvy science fans will also list off the photon, W, Z, and gluon as the Standard Model force particles.  With these particles you can explain a whole swath of nuclear physics and chemistry. But the chart above doesn’t shed light on the electroweak (Higgs!) physics that leads to this structure.

For this reason, these aren’t the particles that physicists use when describing the theoretical structure of the Standard Model and its extensions. Instead, we distinguish between things like the left-handed electron” and “right-handed anti-positron,” or the “transverse W polarizations” and versus the longitudinal polarization which is really a Goldstone boson stolen from a group of [four!] Higgses.

Instead of a lengthy recap of previous blog posts, let’s try to summarize with the right cartoon pictures. In doing so, we’ll get to learn something about the true hidden identities of the Standard Model particles and how they really interact with one another.

The true identities of the Standard Model particles

Let’s start with the electron. We learned that the electron is really a mixture of two chiral particles: a massless “left-handed electron” and a massless “right-handed anti-positron.” These two particles bounce off of the Higgs vacuum expectation value (vev) an convert into one another, leading to the massive particle that we just called the “electron” above.


(We denote the right-handed matter particle with a mustache to highlight that it’s really a totally different object.) The same story is true for the electrons two siblings, the muon and tau, and also for each of their antiparticles.

What about the neutrino? In the Standard Model, the neutrino is massless. We now know they have very small mass, but we’ll stick to the vanilla version of the theory. For this reason, the neutrino doesn’t have to bounce off of the Higgs vev and should be identified with only a left-handed neutrino.

I drew speed lines on the neutrino because the Standard Model neutrino travels at the speed of light due to being massless. I mention this to remind you that the reason why the massive electron could swap between being a “left-handed electron” (yellow particle above) and a “right-handed anti-positron” (green particle with a mustache) is that we could imagine being in a reference frame where the electron spins in the opposite direction, say by speeding past the electron along its trajectory. We can’t do that with the massless neutrino because it’s going at the speed of light; thus all neutrinos that we observe are left-handed.

The Standard Model quarks are all massive and behave just like the electron: they are a mixture of “left-handed quarks” (yellow) and right-handed anti-quarks (green with mustache). Also shown below are the different “color charge” that each quark comes in: red, blue, and green. This has nothing to do with actual colors in the visible spectrum, rather it’s a way of describing objects with three types of charge rather than just two (positive or negative).

There are two types of quarks: up-type (up, charm, top) and down-type (down, strange, bottom); they differ by their electric charge. Up-type and down-type quarks interact with one another through the massive W boson. Part of the theoretical structure of the Standard Model is that the W boson only talks to the left-handed particles (the yellow guys with no mustache)—we say that the Standard Model is chiral (compare this to the definition in chemistry and biology).

The W comes along with its also-massive cousin, the Z boson. We now know that massive gauge bosons come about when previously-massless gauge bosons “eat” a Goldstone boson. These ‘eaten’ Goldstone bosons are three of the four parts of the Standard Model Higgs, leaving one neutral particle which we call the Higgs. This entire process by which force particles have become massive has a fancy name, electroweak symmetry breaking.

In fact, in the picture above we show a very important feature of electroweak symmetry breaking: in addition to H0, the Z boson is also a mixture of a third W boson (called W3 above) and something we called the B boson. Like its charged siblings, the W3 only talks to left-handed particles, though the B is more sociable and will happily talk to both left- and right-handed particles. Note, however, that even here the Standard Model is biased: the “B charge” of a left-handed electron (yellow, no mustache) is different from the “B charge” of a right-handed electron!

What happened to the other leftover parts of the W3 and B? They form another force particle, the photon! The interesting thing about the photon is that it’s the electroweak force particle that didn’t eat any Goldstones—it’s the combination of gauge bosons that survived “electroweak symmetry breaking” without putting on weight. We say that “electroweak symmetry has broken to electromagnetism.” Unlike the full electroweak theory, electromagnetism is left/right democratic in how it talks to matter particles.

Since the photon has not eaten any Goldstone boson, it remains massless and travels at the speed of light. This sounds silly since it is light, so I should say that it travels at the “universal speed of all things which have no mass.” Speaking of which, there’s one more force particle that has nothing to do with electroweak symmetry and also is massless, the gluon:

 

There are actually eight gluons coming from different color and anti-color combinations, i.e. corresponding to different combinations of quarks and anti-quark colors that may interact with the gluon. (Astute readers will wonder why there aren’t nine combinations… this is due to a subtlety in group theory!)

What does this all buy us?

So we see that the Standard Model is actually a bit more complicated than the “fake” version that we showed at the top of the page. Even though it might not seem like it, the theory of this “more complicated” Standard Model is actually rather elegant and minimal. I should also say that calling the simple table a “fake” is too harsh: that is indeed an accurate description of the Standard Model after “electroweak symmetry breaking,” but it doesn’t illuminate the rich structure of interactions included in the theory.

For example, one would have never understood why nuclear beta decays (mediated by the W) always produce left-handed neutrinos. It also wouldn’t have been clear how “longitudinal vector boson scattering is rendered unitary” at high energies—in other words, the description of certain types of gauge boson scattering breaks down without something like a Higgs to keep the calculations under control.

The first example is a real matching of observed phenomenon to a theoretical framework. The second example shows how our theory is prediction about missing pieces that it needs to make sense. While it may sound dry, these are important points—this is part of the Scientific Method: we use observations of natural phenomena and build hypotheses (theories) that make predictions. We can then go and build and LHC (… as well as LEP and the Tevatron) to confirm or refute these predictions—the data from these experiments then feed back in to revise (or overthrow) our theories.

So now that we’re more or less up to speed with the moving parts of the Standard Model, we can push forward to figure out why we believe it should still break down at TeV-scale energies and give some hint of even more fundamental organizing principles. (This is the really exciting part!)

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Why do we expect a Higgs boson? Part II: Unitarization of Vector Boson Scattering

Tuesday, February 14th, 2012

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

Review of the story so far

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

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

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

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

 

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

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

Usually all of this is said rather glibly:

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

A better reason for why we need the Higgs

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

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

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

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

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

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

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

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

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

Now let me be totally clear: there’s nothing “wrong” with this scenario… nature may very well have chosen this path. In fact, we know at least one example where it has: the theory of quarks and gluons (QCD) at low energies is non-perturbative. But this is just telling us that the “particles” that we see at those energies aren’t quarks and gluons since they’re too tightly bound together: the relevant particles at those energies are mesons and baryons (e.g.pions and protons). Even though QCD—a theory of quarks and gluons—breaks down as a calculational tool, nature allowed us to describe physics in terms of perfectly well behaved (perturbative) “bound state” objects like mesons in aneffective theory of QCD. The old adage is true: when nature closes a door, it opens a window.

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

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

Goldstone Liberation at High Energies

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

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

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

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

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

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

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

For advanced readers: What’s happening here is that the theory of a complex scalar Higgs doublet is perfectly well behaved. However, when we write the theory nonlinearly (e.g. chiral perturbation theory, nonlinear sigma model) to incorporate electroweak symmetry breaking, we say something like: H(x) = (v+h(x)) exp (i π(x)/v). The π’s are the Goldstone bosons. If we ignore the Higgs, h, we’re doing gross violence to the well behaved complex scalar doublet. Further, we’re left with a non-renormalizable theory with dimensionful couplings that have powers of 1/v all over the place. Just by dimensional analysis, you can see that scattering cross sections for these Goldstones (i.e. the longitudinal modes of the gauge bosons) must scale like a positive power of the energy. In this sense, the problem of “unitarizing W boson scattering” is really the same as UV completing a non-renormalizable effective theory. [I thank Javi S. for filling in this gap in my education.]

Caveat: Higgs versus Higgs-like

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

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“The Night Before Higgsmas”

Monday, December 12th, 2011

Visit from the Higgs

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

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

 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

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Dispatch from the intensity frontier: Muon (g-2)

Thursday, December 1st, 2011

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

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

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

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

Chris wants us to think about it more like this:

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

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

g = 2.0023193043617.

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

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

g = 2.00233184178

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

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

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

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

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

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

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

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

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

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

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

 

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

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

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

 

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Dispatches from the Intensity Frontier

Wednesday, November 30th, 2011

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

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

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

From Nima Arkani-Hamed's talk at the Intensity Frontier Workshop

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.

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Why do we expect a Higgs boson? Part I: Electroweak Symmetry Breaking

Monday, November 21st, 2011
Announcement: I’ve been selected as a finalist for the 2011 Blogging Scholarship. To support this blog, please vote for me (Philip Tanedo) and encourage others to do the same! See the bottom of this post for more information.

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.

Wanted: the Higgs Boson

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!

Remark: The next natural step in unifying the forces would be  to actually unify the W and B particles with one another. In fact, mathematically one can find ways to combine the B, all three Ws, and all eight gluons in what is referred to as a grand unified theory (GUT). The next step beyond this would be to unify those forces with gravity, which is referred to in popular literature as a `theory of everything.’ Unlike electroweak unification, however, there’s no reason to suspect that either of these phenomena should be accessible at the TeV scale.
Technical remark: mathematically the unification of forces falls under the representation theory of continuous groups (or rather, their algebras). The electroweak group is the product SU(2) × U(1). Note that SU(2) has three generators—this is precisely why there are three W bosons. 

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.

 

 

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Who ate the Higgs?

Monday, October 10th, 2011

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

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

Goldstone Eaten

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

Jeff Goldstone

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…

Force particles like Gold[stone]fishes because they're so delicious.

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

Technical remarks for experts: (corrected Oct 10 thanks to anon.) The “mathematical inconsistency” of a generic theory of massive vectors is the non-unitarity of tree-level WW scattering. This isn’t really an inconsistency since the theory of massive vectors has a cutoff; as one approaches the cutoff loop-level diagrams give large corrections to the amplitude and the theory becomes strongly coupled. While this isn’t a technical necessity for new physics, it is at least a very compelling reason to suspect that there is at least a better description.

In the Standard Model this is done perturbatively. The tree-level cross section for WW scattering increases with energy but is unitarized by the Higgs boson.

Saying that force particles are “born massless” is a particular viewpoint that lends itself to this UV completion by linearization of the nonlinear sigma model associated with a phenomenological theory of massive vectors. This isn’t the only game in town. For example, one can treat the ρ meson is a vector that can be understood as the massive gauge boson of a `hidden’ gauge symmetry in the chiral Lagrangian. The UV completion of such a theory is not a Higgs, but the appearance of the bound quarks that compose the ρ. The analogs of this kind of UV completion in the Standard Model are technicolorcomposite Higgs, and Higgs-less models.

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

"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.

For those with some calculus-based physics background: this is related to the fact that the electromagnetic field can be written as derivatives of a potential. This means the potential is defined up to an constant. This overall constant (more generally, a total derivative) is a gauge symmetry. To connect to the quantum picture, we previously mentioned that the vector potential is the classical analog of the 4-vector describing the photon polarization.

Technical remark: in some sense, this gauge symmetry is not a ‘symmetry’ at all but an overspecification of a physical state such that distinct 4-vectors may describe identical state. (Compare this to a symmetry where different states yield the same physics.)

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.

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