Tag Archives: Higgs

Of Cows and Razors

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Last week’s post came up on Reddit, where a commenter made a good point. I said that one of the mysteries of neutrinos is that they might not get their mass from the Higgs boson. This is true, but the commenter rightly points out it’s true of other particles too: electrons might not get their mass from the Higgs. We aren’t sure. The lighter quarks might not get their mass from the Higgs either.

When talking physics with the public, we usually say that electrons and quarks all get their mass from the Higgs. That’s how it works in our Standard Model, after all. But even though we’ve found the Higgs boson, we can’t be 100% sure that it functions the way our model says. That’s because there are aspects of the Higgs we haven’t been able to measure directly. We’ve measured how it affects the heaviest quark, the top quark, but measuring its interactions with other particles will require a bigger collider. Until we have those measurements, the possibility remains open that electrons and quarks get their mass another way. It would be a more complicated way: we know the Higgs does a lot of what the model says, so if it deviates in another way we’d have to add more details, maybe even more undiscovered particles. But it’s possible.

If I wanted to defend the idea that neutrinos are special here, I would point out that neutrino masses, unlike electron masses, are not part of the Standard Model. For electrons, we have a clear “default” way for them to get mass, and that default is in a meaningful way simpler than the alternatives. For neutrinos, every alternative is complicated in some fashion: either adding undiscovered particles, or unusual properties. If we were to invoke Occam’s Razor, the principle that we should always choose the simplest explanation, then for electrons and quarks there is a clear winner. Not so for neutrinos.

I’m not actually going to make this argument. That’s because I’m a bit wary of using Occam’s Razor when it comes to questions of fundamental physics. Occam’s Razor is a good principle to use, if you have a good idea of what’s “normal”. In physics, you don’t.

To illustrate, I’ll tell an old joke about cows and trains. Here’s the version from The Curious Incident of the Dog in the Night-Time:

There are three men on a train. One of them is an economist and one of them is a logician and one of them is a mathematician. And they have just crossed the border into Scotland (I don’t know why they are going to Scotland) and they see a brown cow standing in a field from the window of the train (and the cow is standing parallel to the train). And the economist says, ‘Look, the cows in Scotland are brown.’ And the logician says, ‘No. There are cows in Scotland of which at least one is brown.’ And the mathematician says, ‘No. There is at least one cow in Scotland, of which one side appears to be brown.’

One side of this cow appears to be very fluffy.

If we want to be as careful as possible, the mathematician’s answer is best. But we expect not to have to be so careful. Maybe the economist’s answer, that Scottish cows are brown, is too broad. But we could imagine an agronomist who states “There is a breed of cows in Scotland that is brown”. And I suggest we should find that pretty reasonable. Essentially, we’re using Occam’s Razor: if we want to explain seeing a brown half-cow from a train, the simplest explanation would be that it’s a member of a breed of cows that are brown. It would be less simple if the cow were unique, a brown mutant in a breed of black and white cows. It would be even less simple if only one side of the cow were brown, and the other were another color.

When we use Occam’s Razor in this way, we’re drawing from our experience of cows. Most of the cows we meet are members of some breed or other, with similar characteristics. We don’t meet many mutant cows, or half-colored cows, so we think of those options as less simple, and less likely.

But what kind of experience tells us which option is simpler for electrons, or neutrinos?

The Standard Model is a type of theory called a Quantum Field Theory. We have experience with other Quantum Field Theories: we use them to describe materials, metals and fluids and so forth. Still, it seems a bit odd to say that if something is typical of these materials, it should also be typical of the universe. As another physicists in my sub-field, Nima Arkani-Hamed, likes to say, “the universe is not a crappy metal!”

We could also draw on our experience from other theories in physics. This is a bit more productive, but has other problems. Our other theories are invariably incomplete, that’s why we come up with new theories in the first place…and with so few theories, compared to breeds of cows, it’s unclear that we really have a good basis for experience.

Physicists like to brag that we study the most fundamental laws of nature. Ordinarily, this doesn’t matter as much as we pretend: there’s a lot to discover in the rest of science too, after all. But here, it really makes a difference. Unlike other fields, we don’t know what’s “normal”, so we can’t really tell which theories are “simpler” than others. We can make aesthetic judgements, on the simplicity of the math or the number of fields or the quality of the stories we can tell. If we want to be principled and forego all of that, then we’re left on an abyss, a world of bare observations and parameter soup.

If a physicist looks out a train window, will they say that all the electrons they see get their mass from the Higgs? Maybe, still. But they should be careful about it.

Redefining Fields for Fun and Profit

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When we study subatomic particles, particle physicists use a theory called Quantum Field Theory. But what is a quantum field?

Some people will describe a field in vague terms, and say it’s like a fluid that fills all of space, or a vibrating rubber sheet. These are all metaphors, and while they can be helpful, they can also be confusing. So let me avoid metaphors, and say something that may be just as confusing: a field is the answer to a question.

Suppose you’re interested in a particle, like an electron. There is an electron field that tells you, at each point, your chance of detecting one of those particles spinning in a particular way. Suppose you’re trying to measure a force, say electricity or magnetism. There is an electromagnetic field that tells you, at each point, what force you will measure.

Sometimes the question you’re asking has a very simple answer: just a single number, for each point and each time. An example of a question like that is the temperature: pick a city, pick a date, and the temperature there and then is just a number. In particle physics, the Higgs field answers a question like that: at each point, and each time, how “Higgs-y” is it there and then? You might have heard that the Higgs field gives other particles their mass: what this means is that the more “Higgs-y” it is somewhere, the higher these particles’ mass will be. The Higgs field is almost constant, because it’s very difficult to get it to change. That’s in some sense what the Large Hadron Collider did when they discovered the Higgs boson: pushed hard enough to cause a tiny, short-lived ripple in the Higgs field, a small area that was briefly more “Higgs-y” than average.

We like to think of some fields as fundamental, and others as composite. A proton is composite: it’s made up of quarks and gluons. Quarks and gluons, as far as we know, are fundamental: they’re not made up of anything else. More generally, since we’re thinking about fields as answers to questions, we can just as well ask more complicated, “composite” questions. For example, instead of “what is the temperature?”, we can ask “what is the temperature squared?” or “what is the temperature times the Higgs-y-ness?”.

But this raises a troubling point. When we single out a specific field, like the Higgs field, why are we sure that that field is the fundamental one? Why didn’t we start with “Higgs squared” instead? Or “Higgs plus Higgs squared”? Or something even weirder?

The inventor of the Higgs-squared field, Peter Higgs-squared

That kind of swap, from Higgs to Higgs squared, is called a field redefinition. In the math of quantum field theory, it’s something you’re perfectly allowed to do. Sometimes, it’s even a good idea. Other times, it can make your life quite complicated.

The reason why is that some fields are much simpler than others. Some are what we call free fields. Free fields don’t interact with anything else. They just move, rippling along in easy-to-calculate waves.

Redefine a free field, swapping it for some more complicated function, and you can easily screw up, and make it into an interacting field. An interacting field might interact with another field, like how electromagnetic fields move (and are moved by) electrons. It might also just interact with itself, a kind of feedback effect that makes any calculation we’d like to do much more difficult.

If we persevere with this perverse choice, and do the calculation anyway, we find a surprise. The final results we calculate, the real measurements people can do, are the same in both theories. The field redefinition changed how the theory appeared, quite dramatically…but it didn’t change the physics.

You might think the moral of the story is that you must always choose the right fundamental field. You might want to, but you can’t: not every field is secretly free. Some will be interacting fields, whatever you do. In that case, you can make one choice or another to simplify your life…but you can also just refuse to make a choice.

That’s something quite a few physicists do. Instead of looking at a theory and calling some fields fundamental and others composite, they treat every one of these fields, every different question they could ask, on the same footing. They then ask, for these fields, what one can measure about them. They can ask which fields travel at the speed of light, and which ones go slower, or which fields interact with which other fields, and how much. Field redefinitions will shuffle the fields around, but the patterns in the measurements will remain. So those, and not the fields, can be used to specify the theory. Instead of describing the world in terms of a few fundamental fields, they think about the world as a kind of field soup, characterized by how it shifts when you stir it with a spoon.

It’s not a perspective everyone takes. If you overhear physicists, sometimes they will talk about a theory with only a few fields, sometimes they will talk about many, and you might be hard-pressed to tell what they’re talking about. But if you keep in mind these two perspectives: either a few fundamental fields, or a “field soup”, you’ll understand them a little better.

Discovering the Rules, Discovering the Consequences

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Two big physics experiments consistently make the news. The Large Hadron Collider, or LHC, and the Laser Interferometer Gravitational-Wave Observatory, or LIGO. One collides protons, the other watches colliding black holes and neutron stars. But while this may make the experiments sound quite similar, their goals couldn’t be more different.

The goal of the LHC, put simply, is to discover the rules that govern reality. Should the LHC find a new fundamental particle, it will tell us something we didn’t know about the laws of physics, a newly discovered fact that holds true everywhere in the universe. So far, it has discovered the Higgs boson, and while that particular rule was expected we didn’t know the details until they were tested. Now physicists hope to find something more, a deviation from the Standard Model that hints at a new law of nature altogether.

LIGO, in contrast, isn’t really for discovering the rules of the universe. Instead, it discovers the consequences of those rules, on a grand scale. Even if we knew the laws of physics completely, we can’t calculate everything from those first principles. We can simulate some things, and approximate others, but we need experiments to tweak those simulations and test those approximations. LIGO fills that role. We can try to estimate how common black holes are, and how large, but LIGO’s results were still a surprise, suggesting medium-sized black holes are more common than researchers expected. In the future, gravitational wave telescopes might discover more of these kinds of consequences, from the shape of neutron stars to the aftermath of cosmic inflation.

There are a few exceptions for both experiments. The LHC can also discover the consequences of the laws of physics, especially when those consequences are very difficult to calculate, finding complicated arrangements of known particles, like pentaquarks and glueballs. And it’s possible, though perhaps not likely, that LIGO could discover something about quantum gravity. Quantum gravity’s effects are expected to be so small that these experiments won’t see them, but some have speculated that an unusually large effect could be detected by a gravitational wave telescope.

As scientists, we want to know everything we can about everything we find. We want to know the basic laws that govern the universe, but we also want to know the consequences of those laws, the story of how our particular universe came to be the way it is today. And luckily, we have experiments for both.

Unification That Does Something

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I’ve got unification on the brain.

Recently, a commenter asked me what physicists mean when they say two forces unify. While typing up a response, I came across this passage, in a science fiction short story by Ted Chiang.

Physics admits of a lovely unification, not just at the level of fundamental forces, but when considering its extent and implications. Classifications like ‘optics’ or ‘thermodynamics’ are just straitjackets, preventing physicists from seeing countless intersections.

This passage sounds nice enough, but I feel like there’s a misunderstanding behind it. When physicists seek after unification, we’re talking about something quite specific. It’s not merely a matter of two topics intersecting, or describing them with the same math. We already plumb intersections between fields, including optics and thermodynamics. When we hope to find a unified theory, we do so because it does something. A real unified theory doesn’t just aid our calculations, it gives us new ways to alter the world.

To show you what I mean, let me start with something physicists already know: electroweak unification.

There’s a nice series of posts on the old Quantum Diaries blog that explains electroweak unification in detail. I’ll be a bit vaguer here.

You might have heard of four fundamental forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force. You might have also heard that two of these forces are unified: the electromagnetic force and the weak nuclear force form something called the electroweak force.

What does it mean that these forces are unified? How does it work?

Zoom in far enough, and you don’t see the electromagnetic force and the weak force anymore. Instead you see two different forces, I’ll call them “W” and “B”. You’ll also see the Higgs field. And crucially, you’ll see the “W” and “B” forces interact with the Higgs.

The Higgs field is special because it has what’s called a “vacuum” value. Even in otherwise empty space, there’s some amount of “Higgsness” in the background, like the color of a piece of construction paper. This background Higgs-ness is in some sense an accident, just one stable way the universe happens to sit. In particular, it picks out an arbitrary kind of direction: parts of the “W” and “B” forces happen to interact with it, and parts don’t.

Now let’s zoom back out. We could, if we wanted, keep our eyes on the “W” and “B” forces. But that gets increasingly silly. As we zoom out we won’t be able to see the Higgs field anymore. Instead, we’ll just see different parts of the “W” and “B” behaving in drastically different ways, depending on whether or not they interact with the Higgs. It will make more sense to talk about mixes of the “W” and “B” fields, to distinguish the parts that are “lined up” with the background Higgs and the parts that aren’t. It’s like using “aft” and “starboard” on a boat. You could use “north” and “south”, but that would get confusing pretty fast.

My cabin is on the west side of the ship…unless we’re sailing east….

What are those “mixes” of the “W” and “B” forces? Why, they’re the weak nuclear force and the electromagnetic force!

This, broadly speaking, is the kind of unification physicists look for. It doesn’t have to be a “mix” of two different forces: most of the models physicists imagine start with a single force. But the basic ideas are the same: that if you “zoom in” enough you see a simpler model, but that model is interacting with something that “by accident” picks a particular direction, so that as we zoom out different parts of the model behave in different ways. In that way, you could get from a single force to all the different forces we observe.

That “by accident” is important here, because that accident can be changed. That’s why I said earlier that real unification lets us alter the world.

To be clear, we can’t change the background Higgs field with current technology. The biggest collider we have can just make a tiny, temporary fluctuation (that’s what the Higgs boson is). But one implication of electroweak unification is that, with enough technology, we could. Because those two forces are unified, and because that unification is physical, with a physical cause, it’s possible to alter that cause, to change the mix and change the balance. This is why this kind of unification is such a big deal, why it’s not the sort of thing you can just chalk up to “interpretation” and ignore: when two forces are unified in this way, it lets us do new things.

Mathematical unification is valuable. It’s great when we can look at different things and describe them in the same language, or use ideas from one to understand the other. But it’s not the same thing as physical unification. When two forces really unify, it’s an undeniable physical fact about the world. When two forces unify, it does something.

How the Higgs Is, and Is Not, Like an Eel

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In the past, what did we know about eel reproduction? What do we know now?

The answer to both questions is, surprisingly little! For those who don’t know the story, I recommend this New Yorker article. Eels turn out to have a quite complicated life cycle, and can only reproduce in the very last stage. Different kinds of eels from all over Europe and the Americas spawn in just one place: the Sargasso Sea. But while researchers have been able to find newborn eels in those waters, and more recently track a few mature adults on their migration back, no-one has yet observed an eel in the act. Biologists may be able to infer quite a bit, but with no direct evidence yet the truth may be even more surprising than they expect. The details of eel reproduction are an ongoing mystery, the “eel question” one of the field’s most enduring.

But of course this isn’t an eel blog. I’m here to answer a different question.

In the past, what did we know about the Higgs boson? What do we know now?

Ask some physicists, and they’ll say that even before the LHC everyone knew the Higgs existed. While this isn’t quite true, it is certainly true that something like the Higgs boson had to exist. Observations of other particles, the W and Z bosons in particular, gave good evidence for some kind of “Higgs mechanism”, that gives other particles mass in a “Higgs-like-way”. A Higgs boson was in some sense the simplest option, but there could have been more than one, or a different sort of process instead. Some of these alternatives may have been sensible, others as silly as believing that eels come from horses’ tails. Until 2012, when the Higgs boson was observed, we really didn’t know.

We also didn’t know one other piece of information: the Higgs boson’s mass. That tells us, among other things, how much energy we need to make one. Physicists were pretty sure the LHC was capable of producing a Higgs boson, but they weren’t sure where or how they’d find it, or how much energy would ultimately be involved.

Now thanks to the LHC, we know the mass of the Higgs boson, and we can rule out some of the “alternative” theories. But there’s still quite a bit we haven’t observed. In particular, we haven’t observed many of the Higgs boson’s couplings.

The couplings of a quantum field are how it interacts, both with other quantum fields and with itself. In the case of the Higgs, interacting with other particles gives those particles mass, while interacting with itself is how it itself gains mass. Since we know the masses of these particles, we can infer what these couplings should be, at least for the simplest model. But, like the eels, the truth may yet surprise us. Nothing guarantees that the simplest model is the right one: what we call simplicity is a judgement based on aesthetics, on how we happen to write models down. Nature may well choose differently. All we can honestly do is parametrize our ignorance.

In the case of the eels, each failure to observe their reproduction deepens the mystery. What are they doing that is so elusive, so impossible to discover? In this, eels are different from the Higgs boson. We know why we haven’t observed the Higgs boson coupling to itself, at least according to our simplest models: we’d need a higher-energy collider, more powerful than the LHC, to see it. That’s an expensive proposition, much more expensive than using satellites to follow eels around the ocean. Because our failure to observe the Higgs self-coupling is itself no mystery, our simplest models could still be correct: as theorists, we probably have it easier than the biologists. But if we want to verify our models in the real world, we have it much harder.

Made of Quarks Versus Made of Strings

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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

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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

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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.

983px-calabi_yau_formatted-svgString 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.

traintrackpic

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

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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

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How can colliding two protons give rise to more massive particles? Why do vibrations of a string have mass? And how does the Higgs work anyway?

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=mc² 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.