# 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.

Advertisements

# 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.

# Hooray for Neutrinos!

Congratulations to Takaaki Kajita and Arthur McDonald, winners of this year’s Nobel Prize in Physics, as well as to the Super-Kamiokande and SNOLAB teams that made their work possible.

Congratulations!

Unlike last year’s Nobel, this is one I’ve been anticipating for quite some time. Kajita and McDonald discovered that neutrinos have mass, and that discovery remains our best hint that there is something out there beyond the Standard Model.

But I’m getting a bit ahead of myself.

Neutrinos are the lightest of the fundamental particles, and for a long time they were thought to be completely massless. Their name means “little neutral one”, and it’s probably the last time physicists used “-ino” to mean “little”. Neutrinos are “neutral” because they have no electrical charge. They also don’t interact with the strong nuclear force. Only the weak nuclear force has any effect on them. (Well, gravity does too, but very weakly.)

This makes it very difficult to detect neutrinos: you have to catch them interacting via the weak force, which is, well, weak. Originally, that meant they had to be inferred by their absence: missing energy in nuclear reactions carried away by “something”. Now, they can be detected, but it requires massive tanks of fluid, carefully watched for the telltale light of the rare interactions between neutrinos and ordinary matter. You wouldn’t notice if billions of neutrinos passed through you every second, like an unstoppable army of ghosts. And in fact, that’s exactly what happens!

Visualization of neutrinos from a popular documentary

In the 60’s, scientists began to use these giant tanks of fluid to detect neutrinos coming from the sun. An enormous amount of effort goes in to understanding the sun, and these days our models of it are pretty accurate, so it came as quite a shock when researchers observed only half the neutrinos they expected. It wasn’t until the work of Super-Kamiokande in 1998, and SNOLAB in 2001, that we knew the reason why.

As it turns out, neutrinos oscillate. Neutrinos are produced in what are called flavor states, which match up with the different types of leptons. There are electron-neutrinos, muon-neutrinos, and tau-neutrinos.

Radioactive processes usually produce electron-neutrinos, so those are the type that the sun produces. But on their way from the sun to the earth, these neutrinos “oscillate”: they switch between electron neutrinos and the other types! The older detectors, focused only on electron-neutrinos, couldn’t see this. SNOLAB’s big advantage was that it could detect the other types of neutrinos as well, and tell the difference between them, which allowed it to see that the “missing” neutrinos were really just turning into other flavors! Meanwhile, Super-Kamiokande measured neutrinos coming not from the sun, but from cosmic rays reacting with the upper atmosphere. Some of these neutrinos came from the sky above the detector, while others traveled all the way through the earth below it, from the atmosphere on the other side. By observing “missing” neutrinos coming from below but not from above, Super-Kamiokande confirmed that it wasn’t the sun’s fault that we were missing solar neutrinos, neutrinos just oscillate!

What does this oscillation have to do with neutrinos having mass, though?

Here things get a bit trickier. I’ve laid some of the groundwork in older posts. I’ve told you to think about mass as “energy we haven’t met yet”, as the energy something has when we leave it alone to itself. I’ve also mentioned that conservation laws come from symmetries of nature, that energy conservation is a result of symmetry in time.

This should make it a little more plausible when I say that when something has a specific mass, it doesn’t change. It can decay into other particles, or interact with other forces, but left alone, by itself, it won’t turn into something else. To be more specific, it doesn’t oscillate. A state with a fixed mass is symmetric in time.

The only way neutrinos can oscillate between flavor states, then, is if one flavor state is actually a combination (in quantum terms, a superposition) of different masses. The components with different masses move at different speeds, so at any point along their path you can be more or less likely to see certain masses of neutrinos. As the mix of masses changes, the flavor state changes, so neutrinos end up oscillating from electron-neutrino, to muon-neutrino, to tau-neutrino.

So because of neutrino oscillation, neutrinos have to have mass. But this presented a problem. Most fundamental particles get their mass from interacting with the Higgs field. But, as it turns out, neutrinos can’t interact with the Higgs field. This has to do with the fact that neutrinos are “chiral”, and only come in a “left-handed” orientation. Only if they had both types of “handedness” could they get their mass from the Higgs.

As-is, they have to get their mass another way, and that way has yet to be definitively shown. Whatever it ends up being, it will be beyond the current Standard Model. Maybe there actually are right-handed neutrinos, but they’re too massive, or interact too weakly, for them to have been discovered. Maybe neutrinos are Majorana particles, getting mass in a novel way that hasn’t been seen yet in the Standard Model.

Whatever we discover, neutrinos are currently our best evidence that something lies beyond the Standard Model. Naturalness may have philosophical problems, dark matter may be explained away by modified gravity…but if neutrinos have mass, there’s something we still have yet to discover. And that definitely seems worthy of a Nobel to me!

# Want to Make Something New? Just Turn on the Lights.

Isn’t it weird that you can collide two protons, and get something else?

It wouldn’t be so weird if you collided two protons, and out popped a quark. After all, protons are made of quarks. But how, if you collide two protons together, do you get a tau, or the Higgs boson: things that not only aren’t “part of” protons, but are more massive than a proton by themselves?

It seems weird…but in a way, it’s not. When a particle releases another particle that wasn’t inside it to begin with, it’s actually not doing anything more special than an everyday light bulb.

Eureka!

How does a light bulb work?

You probably know the basics: when an electrical current enters the bulb, the electrons in the filament start to move. They heat the filament up, releasing light.

That probably seems perfectly ordinary. But ask yourself for a moment: where did the light come from?

Light is made up of photons, elementary particles in their own right. When you flip a light switch, where do the photons come from? Were they stored in the light bulb?

Silly question, right? You don’t need to “store” light in a light bulb: light bulbs transform one type of energy (electrical, or the movement of electrons) into another type of energy (light, or photons).

Here’s the thing, though: mass is just another type of energy.

I like to describe mass as “energy we haven’t met yet”. Einstein’s equation, $E=mc^2$, relates a particle’s mass to its “rest energy”, the energy it would have if it stopped moving around and sit still. Even when a particle seems to be sitting still from the outside, there’s still a lot going on, though. “Composite” particles like protons have powerful forces between their internal quarks, while particles like electrons interact with the Higgs field. These processes give the particle energy, even when it’s not moving, so from our perspective on the outside they’re giving the particle mass.

What does that mean for the protons at the LHC?

The protons at the LHC have a lot of kinetic energy: they’re going 99.9999991% of the speed of light! When they collide, all that energy has to go somewhere. Just like in a light bulb, the fast-moving particles will release their energy in another form. And while that some of that energy will add to the speed of the fragments, much of it will go into the mass and energy of new particles. Some of these particles will be photons, some will be tau leptons, or Higgs bosons…pretty much anything that the protons have enough energy to create.

So if you want to understand how to create new particles, you don’t need a deep understanding of the mysteries of quantum field theory. Just turn on the lights.