# 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.

# What Counts as a Fundamental Force?

I’m giving a presentation next Wednesday for Learning Unlimited, an organization that presents educational talks to seniors in Woodstock, Ontario. The talk introduces the fundamental forces and talks about Yang and Mills before moving on to introduce my work.

While practicing the talk today, someone from Perimeter’s outreach department pointed out a rather surprising missing element: I never mention gravity!

Most people know that there are four fundamental forces of nature. There’s Electromagnetism, there’s Gravity, there’s the Weak Nuclear Force, and there’s the Strong Nuclear Force.

Listed here by their most significant uses.

What ties these things together, though? What makes them all “fundamental forces”?

Mathematically, gravity is the odd one out here. Electromagnetism, the Weak Force, and the Strong Force all share a common description: they’re Yang-Mills forces. Gravity isn’t. While you can sort of think of it as a Yang-Mills force “squared”, it’s quite a bit more complicated than the Yang-Mills forces.

You might be objecting that the common trait of the fundamental forces is obvious: they’re forces! And indeed, you can write down a force law for gravity, and a force law for E&M, and umm…

[Mumble Mumble]

Ok, it’s not quite as bad as xkcd would have us believe. You can actually write down a force law for the weak force, if you really want to, and it’s at least sort of possible to talk about the force exerted by the strong interaction.

All that said, though, why are we thinking about this in terms of forces? Forces are a concept from classical mechanics. For a beginning physics student, they come up again and again, in free-body diagram after free-body diagram. But by the time a student learns quantum mechanics, and quantum field theory, they’ve already learned other ways of framing things where forces aren’t mentioned at all. So while forces are kind of familiar to people starting out, they don’t really match onto anything that most quantum field theorists work with, and it’s a bit weird to classify things that only really appear in quantum field theory (the Weak Nuclear Force, the Strong Nuclear Force) based on whether or not they’re forces.

Isn’t there some connection, though? After all, gravity, electromagnetism, the strong force, and the weak force may be different mathematically, but at least they all involve bosons.

Well, yes. And so does the Higgs.

The Higgs is usually left out of listings of the fundamental forces, because it’s not really a “force”. It doesn’t have a direction, instead it works equally at every point in space. But if you include spin 2 gravity and spin 1 Yang-Mills forces, why not also include the spin 0 Higgs?

Well, if you’re doing that, why not include fermions as well? People often think of fermions as “matter” and bosons as “energy”, but in fact both have energy, and neither is made of it. Electrons and quarks are just as fundamental as photons and gluons and gravitons, just as central a part of how the universe works.

I’m still trying to decide whether my presentation about Yang-Mills forces should also include gravity. On the one hand, it would make everything more familiar. On the other…pretty much this entire post.

# How to Predict the Mass of the Higgs

Did Homer Simpson predict the mass of the Higgs boson?

No, of course not.

Apart from the usual reasons, he’s off by more than a factor of six.

If you play with the numbers, it looks like Simon Singh (the popular science writer who reported the “discovery” Homer made as a throwaway joke in a 1998 Simpsons episode) made the classic physics mistake of losing track of a factor of $2\pi$. In particular, it looks like he mistakenly thought that the Planck constant, $h$, was equal to the reduced Planck constant, $\hbar$, divided by $2\pi$, when actually it’s $\hbar$ times $2\pi$. So while Singh read Homer’s prediction as 123 GeV, surprisingly close to the actual Higgs mass of 125 GeV found in 2012, in fact Homer predicted the somewhat more embarrassing value of 775 GeV.

D’Oh!

That was boring. Let’s ask a more interesting question.

Did Gordon Kane predict the mass of the Higgs boson?

I’ve talked before about how it seems impossible that string theory will ever make any testable predictions. The issue boils down to one of too many possibilities: string theory predicts different consequences for different ways that its six (or seven for M theory) extra dimensions can be curled up. Since there is an absurdly vast number of ways this can be done, anything you might want to predict (say, the mass of the electron) has an absurd number of possible values.

Gordon Kane and collaborators get around this problem by tackling a different one. Instead of trying to use string theory to predict things we already know, like the mass of the electron, they assume these things are already true. That is, they assume we live in a world with electrons that have the mass they really have, and quarks that have the mass they really have, and so on. They assume that we live in a world that obeys all of the discoveries we’ve already made, and a few we hope to make. And, they assume that this world is a consequence of string (or rather M) theory.

From that combination of assumptions, they then figure out the consequences for things that aren’t yet known. And in a 2011 paper, they predicted the Higgs mass would be between 105 and 129 GeV.

I have a lot of sympathy for this approach, because it’s essentially the same thing that non-string-theorists do. When a particle physicist wants to predict what will come out of the LHC, they don’t try to get it from first principles: they assume the world works as we have discovered, make a few mild extra assumptions, and see what new consequences come out that we haven’t observed yet. If those particle physicists can be said to make predictions from supersymmetry, or (shudder) technicolor, then Gordon Kane is certainly making predictions from string theory.

So why haven’t you heard of him? Even if you have, why, if this guy successfully predicted the mass of the Higgs boson, are people still saying that you can’t make predictions with string theory?

Trouble is, making predictions is tricky.

Part of the problem is timing. Gordon Kane’s paper went online in December of 2011. The Higgs mass was announced in July 2012, so you might think Kane got a six month head-start. But when something is announced isn’t the same as when it’s discovered. For a big experiment like the Large Hadron Collider, there’s a long road between the first time something gets noticed and the point where everyone is certain enough that they’re ready to announce it to the world. Rumors fly, and it’s not clear that Kane and his co-authors wouldn’t have heard them.

Assumptions are the other issue. Remember when I said, a couple paragraphs up, that Kane’s group assumed “that we live in a world that obeys all of the discoveries we’ve already made, and a few we hope to make“? That last part is what makes things tricky. There were a few extra assumptions Kane made, beyond those needed to reproduce the world we know. For many people, some of these extra assumptions are suspicious. They worry that the assumptions might have been chosen, not just because they made sense, but because they happened to give the right (rumored) mass of the Higgs.

If you want to predict something in physics, it’s not just a matter of getting in ahead of the announcement with the right number. For a clear prediction, you need to be early enough that the experiments haven’t yet even seen hints of what you’re looking for. Even then, you need your theory to be suitably generic, so that it’s clear that your prediction is really the result of the math and not of your choices. You can trade off aspects of this: more accuracy for a less generic theory, better timing for looser predictions. Get the formula right, and the world will laud you for your prediction. Wrong, and you’re Homer Simpson. Somewhere in between, though, and you end up in that tricky, tricky grey area.

Like Gordon Kane.