Tag Archives: Higgs

When you learn physics in school, you learn it in terms of building blocks.

First, you learn about atoms. Indivisible elements, as the Greeks foretold…until you learn that they aren’t indivisible. You learn that atoms are made of electrons, protons, and neutrons. Then you learn that protons and neutrons aren’t indivisible either, they’re made of quarks. They’re what physicists call composite particles, particles made of other particles stuck together.

Hearing this story, you notice a pattern. Each time physicists find a more fundamental theory, they find that what they thought were indivisible particles are actually composite. So when you hear physicists talking about the next, more fundamental theory, you might guess it has to work the same way. If quarks are made of, for example, strings, then each quark is made of many strings, right?

Nope! As it turns out, there are two different things physicists can mean when they say a particle is “made of” a more fundamental particle. Sometimes they mean the particle is composite, like the proton is made of quarks. But sometimes, like when they say particles are “made of strings”, they mean something different.

To understand what this “something different” is, let’s go back to quarks for a moment. You might have heard there are six types, or flavors, of quarks: up and down, strange and charm, top and bottom. The different types have different mass and electric charge. You might have also heard that quarks come in different colors, red green and blue. You might wonder then, aren’t there really eighteen types of quark? Red up quarks, green top quarks, and so forth?

Physicists don’t think about it that way. Unlike the different flavors, the different colors of quark have a more unified mathematical description. Changing the color of a quark doesn’t change its mass or electric charge. All it changes is how the quark interacts with other particles via the strong nuclear force. Know how one color works, and you know how the other colors work. Different colors can also “mix” together, similarly to how different situations can mix together in quantum mechanics: just as Schrodinger’s cat can be both alive and dead, a quark can be both red and green.

This same kind of thing is involved in another example, electroweak unification. You might have heard that electromagnetism and the weak nuclear force are secretly the same thing. Each force has corresponding particles: the familiar photon for electromagnetism, and W and Z bosons for the weak nuclear force. Unlike the different colors of quarks, photons and W and Z bosons have different masses from each other. It turns out, though, that they still come from a unified mathematical description: they’re “mixtures” (in the same Schrodinger’s cat-esque sense) of the particles from two more fundamental forces (sometimes called “weak isospin” and “weak hypercharge”). The reason they have different masses isn’t their own fault, but the fault of the Higgs: the Higgs field we have in our universe interacts with different parts of this unified force differently, so the corresponding particles end up with different masses.

A physicist might say that electromagnetism and the weak force are “made of” weak isospin and weak hypercharge. And it’s that kind of thing that physicists mean when they say that quarks might be made of strings, or the like: not that quarks are composite, but that quarks and other particles might have a unified mathematical description, and look different only because they’re interacting differently with something else.

This isn’t to say that quarks and electrons can’t be composite as well. They might be, we don’t know for sure. If they are, the forces binding them together must be very strong, strong enough that our most powerful colliders can’t make them wiggle even a little out of shape. The tricky part is that composite particles get mass from the energy holding them together. A particle held together by very powerful forces would normally be very massive, if you want it to end up lighter you have to construct your theory carefully to do that. So while occasionally people will suggest theories where quarks or electrons are composite, these theories aren’t common. Most of the time, if a physicist says that quarks or electrons are “made of ” something else, they mean something more like “particles are made of strings” than like “protons are made of quarks”.

Assumptions for Naturalness

Why did physicists expect to see something new at the LHC, more than just the Higgs boson? Mostly, because of something called naturalness.

Naturalness, broadly speaking, is the idea that there shouldn’t be coincidences in physics. If two numbers that appear in your theory cancel out almost perfectly, there should be a reason that they cancel. Put another way, if your theory has a dimensionless constant in it, that constant should be close to one.

(To see why these two concepts are the same, think about a theory where two large numbers miraculously almost cancel, leaving just a small difference. Take the ratio of one of those large numbers to the difference, and you get a very large dimensionless number.)

You might have heard it said that the mass of the Higgs boson is “unnatural”. There are many different physical processes that affect what we measure as the mass of the Higgs. We don’t know exactly how big these effects are, but we do know that they grow with the scale of “new physics” (aka the mass of any new particles we might have discovered), and that they have to cancel to give the Higgs mass we observe. If we don’t see any new particles, the Higgs mass starts looking more and more unnatural, driving some physicists to the idea of a “multiverse”.

If you find parts of this argument hokey, you’re not alone. Critics of naturalness point out that we don’t really have a good reason to favor “numbers close to one”, nor do we have any way to quantify how “bad” a number far from one is (we don’t know the probability distribution, in other words). They critique theories that do preserve naturalness, like supersymmetry, for being increasingly complicated and unwieldy, violating Occam’s razor. And in some cases they act baffled by the assumption that there should be any “new physics” at all.

Some of these criticisms are reasonable, but some are distracting and off the mark. The problem is that the popular argument for naturalness leaves out some important assumptions. These assumptions are usually kept in mind by the people arguing for naturalness (at least the more careful people), but aren’t often made explicit. I’d like to state some of these assumptions. I’ll be framing the naturalness argument in a bit of an unusual (if not unprecedented) way. My goal is to show that some criticisms of naturalness don’t really work, while others still make sense.

I’d like to state the naturalness argument as follows:

1. The universe should be ultimately described by a theory with no free dimensionless parameters at all. (For the experts: the theory should also be UV-finite.)
2. We are reasonably familiar with theories of the sort described in 1., we know roughly what they can look like.
3. If we look at such a theory at low energies, it will appear to have dimensionless parameters again, based on the energy where we “cut off” our description. We understand this process well enough to know what kinds of values these parameters can take, starting from 2.
4. Point 3. can only be consistent with the observed mass of the Higgs if there is some “new physics” at around the scales the LHC can measure. That is, there is no known way to start with a theory like those of 2. and get the observed Higgs mass without new particles.

Point 1. is often not explicitly stated. It’s an assumption, one that sits in the back of a lot of physicists’ minds and guides their reasoning. I’m really not sure if I can fully justify it, it seems like it should be a consequence of what a final theory is.

(For the experts: you’re probably wondering why I’m insisting on a theory with no free parameters, when usually this argument just demands UV-finiteness. I demand this here because I think this is the core reason why we worry about coincidences: free parameters of any intermediate theory must eventually be explained in a theory where those parameters are fixed, and “unnatural” coincidences are those we don’t expect to be able to fix in this way.)

Point 2. may sound like a stretch, but it’s less of one than you might think. We do know of a number of theories that have few or no dimensionless parameters (and that are UV-finite), they just don’t describe the real world. Treating these theories as toy models, we can hopefully get some idea of how theories like this should look. We also have a candidate theory of this kind that could potentially describe the real world, M theory, but it’s not fleshed out enough to answer these kinds of questions definitively at this point. At best it’s another source of toy models.

Point 3. is where most of the technical arguments show up. If someone talking about naturalness starts talking about effective field theory and the renormalization group, they’re probably hashing out the details of point 3. Parts of this point are quite solid, but once again there are some assumptions that go into it, and I don’t think we can say that this point is entirely certain.

Once you’ve accepted the arguments behind points 1.-3., point 4. follows. The Higgs is unnatural, and you end up expecting new physics.

Framed in this way, arguments about the probability distribution of parameters are missing the point, as are arguments from Occam’s razor.

The point is not that the Standard Model has unlikely parameters, or that some in-between theory has unlikely parameters. The point is that there is no known way to start with the kind of theory that could be an ultimate description of the universe and end up with something like the observed Higgs and no detectable new physics. Such a theory isn’t merely unlikely, if you take this argument seriously it’s impossible. If your theory gets around this argument, it can be as cumbersome and Occam’s razor-violating as it wants, it’s still a better shot than no possible theory at all.

In general, the smarter critics of naturalness are aware of this kind of argument, and don’t just talk probabilities. Instead, they reject some combination of point 2. and point 3.

This is more reasonable, because point 2. and point 3. are, on some level, arguments from ignorance. We don’t know of a theory with no dimensionless parameters that can give something like the Higgs with no detectable new physics, but maybe we’re just not trying hard enough. Given how murky our understanding of M theory is, maybe we just don’t know enough to make this kind of argument yet, and the whole thing is premature. This is where probability can sneak back in, not as some sort of probability distribution on the parameters of physics but just as an estimate of our own ability to come up with new theories. We have to guess what kinds of theories can make sense, and we may well just not know enough to make that guess.

One thing I’d like to know is how many critics of naturalness reject point 1. Because point 1. isn’t usually stated explicitly, it isn’t often responded to explicitly either. The way some critics of naturalness talk makes me suspect that they reject point 1., that they honestly believe that the final theory might simply have some unexplained dimensionless numbers in it that we can only fix through measurement. I’m curious whether they actually think this, or whether I’m misreading them.

There’s a general point to be made here about framing. Suppose that tomorrow someone figures out a way to start with a theory with no dimensionless parameters and plausibly end up with a theory that describes our world, matching all existing experiments. (People have certainly been trying.) Does this mean naturalness was never a problem after all? Or does that mean that this person solved the naturalness problem?

Those sound like very different statements, but it should be clear at this point that they’re not. In principle, nothing distinguishes them. In practice, people will probably frame the result one way or another based on how interesting the solution is.

If it turns out we were missing something obvious, or if we were extremely premature in our argument, then in some sense naturalness was never a real problem. But if we were missing something subtle, something deep that teaches us something important about the world, then it should be fair to describe it as a real solution to a real problem, to cite “solving naturalness” as one of the advantages of the new theory.

If you ask for my opinion? You probably shouldn’t, I’m quite far from an expert in this corner of physics, not being a phenomenologist. But if you insist on asking anyway, I suspect there probably is something wrong with the naturalness argument. That said, I expect that whatever we’re missing, it will be something subtle and interesting, that naturalness is a real problem that needs to really be solved.

Calabi-Yaus for Higgs Phenomenology

less joking title:

You Didn’t Think We’d Stop at Elliptics, Did You?

When calculating scattering amplitudes, I like to work with polylogarithms. They’re a very well-understood type of mathematical function, and thus pretty easy to work with.

Even for our favorite theory of N=4 super Yang-Mills, though, they’re not the whole story. You need other types of functions to represent amplitudes, elliptic polylogarithms that are only just beginning to be properly understood. We had our own modest contribution to that topic last year.

You can think of the difference between these functions in terms of more and more complicated curves. Polylogarithms just need circles or spheres, elliptic polylogarithms can be described with a torus.

A torus is far from the most complicated curve you can think of, though.

String theorists have done a lot of research into complicated curves, in particular ones with a property called Calabi-Yau. They were looking for ways to curl up six or seven extra dimensions, to get down to the four we experience. They wanted to find ways of curling that preserved some supersymmetry, in the hope that they could use it to predict new particles, and it turned out that Calabi-Yau was the condition they needed.

That hope, for the most part, didn’t pan out. There were too many Calabi-Yaus to check, and the LHC hasn’t seen any supersymmetric particles. Today, “string phenomenologists”, who try to use string theory to predict new particles, are a relatively small branch of the field.

This research did, however, have lasting impact: due to string theorists’ interest, there are huge databases of Calabi-Yau curves, and fruitful dialogues with mathematicians about classifying them.

This has proven quite convenient for us, as we happen to have some Calabi-Yaus to classify.

Our midnight train going anywhere…in the space of Calabi-Yaus

We call Feynman diagrams like the one above “traintrack integrals”. With two loops, it’s the elliptic integral we calculated last year. With three, though, you need a type of Calabi-Yau curve called a K3. With four loops, it looks like you start needing Calabi-Yau three-folds, the type of space used to compactify string theory to four dimensions.

“We” in this case is myself, Jacob Bourjaily, Andrew McLeod, Matthias Wilhelm, and Yang-Hui He, a Calabi-Yau expert we brought on to help us classify these things. Our new paper investigates these integrals, and the more and more complicated curves needed to compute them.

Calabi-Yaus had been seen in amplitudes before, in diagrams called “sunrise” or “banana” integrals. Our example shows that they should occur much more broadly. “Traintrack” integrals appear in our favorite N=4 super Yang-Mills theory, but they also appear in theories involving just scalar fields, like the Higgs boson. For enough loops and particles, we’re going to need more and more complicated functions, not just the polylogarithms and elliptic polylogarithms that people understand.

(And to be clear, no, nobody needs to do this calculation for Higgs bosons in practice. This diagram would calculate the result of two Higgs bosons colliding and producing ten or more Higgs bosons, all at energies so high you can ignore their mass, which is…not exactly relevant for current collider phenomenology. Still, the title proved too tempting to resist.)

Is there a way to understand traintrack integrals like we understand polylogarithms? What kinds of Calabi-Yaus do they pick out, in the vast space of these curves? We’d love to find out. For the moment, we just wanted to remind all the people excited about elliptic polylogarithms that there’s quite a bit more strangeness to find, even if we don’t leave the tracks.

Tutoring at GGI

I’m still at the Galileo Galilei Institute this week, tutoring at the winter school.

At GGI’s winter school, each week is featuring a pair of lecturers. This week, the lectures alternate between Lance Dixon covering the basics of amplitudeology and Csaba Csaki, discussing ways in which the Higgs could be a composite made up of new fundamental particles.

Most of the students at this school are phenomenologists, physicists who make predictions for particle physics. I’m an amplitudeologist, I study the calculation tools behind those predictions. You’d think these would be very close areas, but it’s been interesting seeing how different our approaches really are.

Some of the difference is apparent just from watching the board. In Csaki’s lectures, the equations that show up are short, a few terms long at most. When amplitudes show up, it’s for their general properties: how many factors of the coupling constant, or the multipliers that show up with loops. There aren’t any long technical calculations, and in general they aren’t needed: he’s arguing about the kinds of physics that can show up, not the specifics of how they give rise to precise numbers.

In contrast, Lance’s board filled up with longer calculations, each with many moving parts. Even things that seem simple from our perspective take a decent amount of board space to derive, and involve no small amount of technical symbol-shuffling. For most of the students, working out an amplitude this complicated was an unfamiliar experience. There are a few applications for which you need the kind of power that amplitudeology provides, and a few students were working on them. For the rest, it was a bit like learning about a foreign culture, an exercise in understanding what other people are doing rather than picking up a new skill themselves. Still, they made a strong go at it, and it was enlightening to see the pieces that ended up mattering to them, and to hear the kinds of questions they asked.

Mass Is Just Energy You Haven’t Met Yet

There is one central misunderstanding that makes each of these topics confusing. It’s something I’ve brought up before, but it really deserves its own post. It’s people not realizing that mass is just energy you haven’t met yet.

It’s quite intuitive to think of mass as some sort of “stuff” that things can be made out of. In our everyday experience, that’s how it works: combine this mass of flour and this mass of sugar, and get this mass of cake. Historically, it was the dominant view in physics for quite some time. However, once you get to particle physics it starts to break down.

It’s probably most obvious for protons. A proton has a mass of 938 MeV/c², or 1.6×10⁻²⁷ kg in less physicist-specific units. Protons are each made of three quarks, two up quarks and a down quark. Naively, you’d think that the quarks would have to be around 300 MeV/c². They’re not, though: up and down quarks both have masses less than 10 MeV/c². Those three quarks account for less than a fiftieth of a proton’s mass.

The “extra” mass is because a proton is not just three quarks. It’s three quarks interacting. The forces between those quarks, the strong nuclear force that binds them together, involves a heck of a lot of energy. And from a distance, that energy ends up looking like mass.

This isn’t unique to protons. In some sense, it’s just what mass is.

The quarks themselves get their mass from the Higgs field. Far enough away, this looks like the quarks having a mass. However, zoom in and it’s energy again, the energy of interaction between quarks and the Higgs. In string theory, mass comes from the energy of vibrating strings. And so on. Every time we run into something that looks like a fundamental mass, it ends up being just another energy of interaction.

If mass is just energy, what about gravity?

When you’re taught about gravity, the story is all about mass. Mass attracts mass. Mass bends space-time. What gets left out, until you actually learn the details of General Relativity, is that energy gravitates too.

Normally you don’t notice this, because mass contributes so much more to energy than anything else. That’s really what E=m is really about: it’s a unit conversion formula. It tells you that if you want to know how much energy a given mass “really is”, you multiply it by the speed of light squared. And that’s a large enough number that most of the time, when you notice energy gravitating, it’s because that energy looks like a big chunk of mass. (It’s also why physicists like silly units like MeV/c² for mass: we can just multiply by c² and get an energy!)

It’s really tempting to think about mass as a substance, of mass as always conserved, of mass as fundamental. But in physics we often have to toss aside our everyday intuitions, and this is no exception. Mass really is just energy. It’s just energy that we’ve “zoomed out” enough not to notice.

A Collider’s Eye View

When it detected the Higgs, what did the LHC see, exactly?

What do you see with your detector-eyes, CMS?

The first problem is that the Higgs, like most particles produced in particle colliders, is unstable. In a very short amount of time the Higgs transforms into two or more lighter particles. Often, these particles will decay in turn, possibly many more times.  So when the LHC sees a Higgs boson, it doesn’t really “see the Higgs”.

The second problem is that you can’t “see” the lighter particles either. They’re much too small for that. Instead, the LHC has to measure their properties.

Does the particle have a charge? Then its path will curve in a magnetic field, and it will send electrical signals in silicon. So the LHC can “see” charge.

Can the particle be stopped, absorbed by some material? Getting absorbed releases energy, lighting up a detector. So the LHC can “see” energy, and what it takes for a particle to be absorbed.

Diagram of a collider’s “eye”

And that’s…pretty much it. When the LHC “sees” the Higgs, what it sees is a set of tracks in a magnetic field, indicating charge, and energy in its detectors, caused by absorption at different points. Everything else has to be inferred: what exactly the particles were, where they decayed, and from what. Some of it can be figured out in real-time, some is only understood later once we can add up everything and do statistics.

On the face of it, this sounds about as impossible as astrophysics. Like astrophysics, it works in part because what the colliders see is not the whole story. The strong force has to both be consistent with our observations of hadrons, and with nuclear physics. Neutrinos aren’t just mysterious missing energy that we can’t track, they’re an important part of cosmology. And so on.

So in the sense of that massive, interconnected web of ideas, the LHC sees the Higgs. It sees patterns of charges and energies, binned into histograms and analyzed with statistics and cross-checked, implicitly or explicitly, against all of the rest of physics at every scale we know. All of that, together, is the collider’s eye view of the universe.

The Higgs Solution

My grandfather is a molecular biologist. Over the holidays I had many opportunities to chat with him, and our conversations often revolved around explaining some aspect of our respective fields. While talking to him, I came up with a chemistry-themed description of the Higgs field, and how it leads to electro-weak symmetry breaking. Very few of you are likely to be chemists, but I think you still might find the metaphor worthwhile.

Picture the Higgs as a mixture of ions, dissolved in water.

In this metaphor, the Higgs field is a sort of “Higgs solution”. Overall, this solution should be uniform: if you have more ions of a certain type in one place than another, over time they will dissolve until they reach a uniform mixture again. In this metaphor, the Higgs particle detected by the LHC is like a brief disturbance in the fluid: by stirring the solution at high energy, we’ve managed to briefly get more of one type of ion in one place than the average concentration.

What determines the average concentration, though?

Essentially, it’s arbitrary. If this were really a chemistry experiment, it would depend on the initial conditions: which ions we put in to the mixture in the first place. In physics, quantum mechanics plays a role, randomly selecting one option out of the many possibilities.

Choose wisely

(Note that this metaphor doesn’t explain why there has to be a solution, why the water can’t just be “pure”. A setup that required this would probably be chemically complicated enough to confuse nearly everybody, so I’m leaving that feature out. Just trust that “no ions” isn’t one of our options.)

Up till now, the choice of mixture didn’t matter very much. But different ions interact with other chemicals in different ways, and this has some interesting implications.

Suppose we have a tube filled with our Higgs solution. We want to shoot some substance through the tube, and collect it on the other side. This other substance is going to represent a force.

If our force substance doesn’t react with the ions in our Higgs solution, it will just go through to the other side. If it does react, though, then it will be slowed down, and only some of it will get to the other side, possibly none at all.

You can think of the electro-weak force as a mixture of these sorts of substances. Normally, there is no way to tell the different substances apart. Just like the different Higgs solutions, different parts of the electro-weak force are arbitrary.

However, once we’ve chosen a Higgs solution, things change. Now, different parts of our electro-weak substance will behave differently. The parts that react with the ions in our Higgs solution will slow down, and won’t make it through the tube, while the parts that don’t interact will just flow on through.

We call the part that gets through the tube electromagnetism, and the part that doesn’t the weak nuclear force. Electromagnetism is long-range, its waves (light) can travel great distances. The weak nuclear force is short-range, and doesn’t have an effect outside of the scale of atoms.

The important thing to take away from this is that the division between electromagnetism and the weak nuclear force is totally arbitrary. Taken by themselves, they’re equivalent parts of the same, electro-weak force. It’s only because some of them interact with the Higgs, while others don’t, that we distinguish those parts from each other. If the Higgs solution were a different mixture (if the Higgs field had different charges) then a different part of the electroweak force would be long-range, and a different part would be short-range.

We wouldn’t be able to tell the difference, though. We’d see a long-range force, and a short-range force, and a Higgs field. In the end, our world would be completely the same, just based on a different, arbitrary choice.