## Science & Nature

# Minding Matter

In classical physics, mass is taken for granted; in the quantum world, it’s a puzzle

Jun 16, 2016

26 Min read time

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The quantum puzzle of mass.

*Nobel Prize–winning theoretical physicist Peter Higgs visits the Compact Muon Solenoid at CERN. Photo: Marc Buehler*

*Editors' Note: This is the fourth article in a series on new experiments at CERN.*

The discovery of the Higgs boson on July 4th, 2012, at CERN was one of the great achievements in the history of particle physics, the capstone of over four decades of work by tens of thousands of researchers from all over the world. Though finding the Higgs was exciting, its discovery did not come as a complete surprise: it was predicted by the Standard Model, a theory embraced by all particle physicists (at least as a starting point), which attempts to organize into a single precise framework all the known matter and forces in the Universe.

In the first article of this series, I described something exciting and *un*expected: an anomaly announced at CERN in December 2015. Reporting on data accumulated over the previous year, physicists described a surprisingly large number of two-photon events at a very high invariant mass (750 GeV). The finding was anomalous because it was *not *predicted by the Standard Model. If it holds up—and we should know whether it does by late this summer—then we physicists will need to move beyond the Standard Model, and no one knows what that will look like. That uncertainty is hugely exciting, but in a different way from the Higgs search.

The Standard Model is a *quantum field theory *(QFT). In my second article, I took a step back to explain what that means. In a nutshell, the QFT idea is that the world is composed ultimately of fields, which are assignments of values to every point in spacetime. Moreover, in QFT, those fields are quantized. Quantized fields, unlike wind fields or temperature fields, can only take on certain values: the possible values of the fields are lumpy, not smooth. According to QFT, there are lots of quantum fields, including one for every fundamental particle: an electron field, a photon field, many quark fields, a Higgs field, and so on.

In my third article, I introduced three key elements of the Standard Model:

**Fermions**are excitations of certain quantum fields. They provide the stuff (the matter particles) the Universe is made of, including quarks, electrons, and various flavors of neutrino.**Forces**are the ways that the matter particles interact. The Standard Model includes three forces or types of interaction: electromagnetism, the strong nuclear force, and the weak nuclear force.**Gauge bosons**are excitations of the force fields, and are exchanged in interactions between stuff. The bosons include photons (associated with the electromagnetic force), gluons (associated with the strong force), and Ws and Zs (associated with the weak force).

One more component is needed to make the Standard Model all work: the Higgs boson. By the end of this article, you should have a sense of how and why.

**• • •**

The Standard Model was constructed in the 1970s. As it was being created, it was already obvious to its architects that it demanded a new, undiscovered, quantum field. This field needed to have unique properties. These properties had already been identified by three independent sets of theorists a decade prior (Peter Higgs; Robert Brout and François Englert; and Gerald Guralnik, C.R. Hagen, and Tom Kibble).

Like all quantum fields, this new field would have an associated quantum excitation—a particle (a boson it turns out, not a fermion). Only Peter Higgs’s work, published in *Physical Review Letters *in 1964, explicitly mentioned this particle—and it did so only in one sentence added in a revision after the paper was initially rejected. In the intervening years the field, the particle, and the mechanism by which this new field interacted with other Standard Model particles were referred to as the “Higgs field,” the “Higgs particle” (more accurately, the Higgs boson), and the “Higgs mechanism.”

In the quantum world, the property of “having mass” is something that needs to be understood.

If you’ve read anything about the Higgs boson or Higgs field, you probably read that it “gives mass” to the other massive fundamental particles. This is a puzzling concept: “mass” is such an integral property of “stuff” that the idea that mass is something that could be “given” probably seems completely crazy. And in the world of classical, non-QFT physics, that view is the correct one. Mass in such a world just *is*, it needs no explanation. But in the quantum world, that changes and the property of “having mass” is something that needs to be understood.

Mass is pretty important: without mass, electrons could not remain in bound states around atomic nuclei, which means that we would not have chemistry and life would be impossible. So if you care about understanding the world at the fundamental level, then you want to know how the Higgs does its critical, mass-giving work. But before I get to *how* it gives mass, we should ask a much simpler question: What does this statement even mean? What is mass and what does it mean to “get a mass”?

Having mass bestows a number of important properties on things that have it. Here are three:

- Massive particles—by which I mean, simply, particles with mass—have
*inertia*, which means they resist acceleration. The larger the mass, the greater the force needed to achieve a specified acceleration. - As I explained in the second article, particles are excitations or oscillations in quantum fields. The mass of a particle is the energy (using E = mc
^{2}) required to make the excitation self-sustaining. Thus, each field—say, the electron field or the top quark field—has a property called “mass.” If you excite the field but the energy and momentum are smaller than this mass-property, the excitation will die away very quickly. So every excitation that lasts long enough to have its mass measured will always have mc^{2}of energy (in the case of the electron, that means 0.511 MeV). When we say, then, that the Higgs makes a particle “gain mass,” we are saying that the Higgs somehow explains the energy levels needed for self-sustaining excitations in fields. - Massive particles move more slowly than light. Massless particles, like photons, always move at the speed of light. No matter how much observers speed up or slow down, they always see the photon moving at this speed. Said differently, the speed of light is a constant of Nature, and everyone measures it to be the same. You can never run fast enough to catch a photon. With a massive particle, you can imagine running fast enough to catch up to it, no matter how fast the particle is moving to start with.

The Higgs field has some important and complicated work to do if it is going to explain all this. Before I dive in and explain how it works, I need to add one final clarification. While the Higgs field gives mass to *fundamental* particles, it is *not* responsible for most of the mass in the universe. Consider your own body. Almost all its mass comes from protons and neutrons, both of which contain massive quarks. But the reason protons and neutrons are massive is not principally because the quarks are massive (the quark mass comes from the Higgs). Most of the mass of protons and neutrons exists because the strong nuclear force—which operates between quarks and gluons—is a strongly interacting force (as I described in the third article). This means that a collection of quarks disturbs all of the other fields (both quarks and gluons) that feel the strong nuclear force. This requires energy: so a collection of quarks, like a proton or a neutron, requires a minimum amount of energy, and these particles, therefore, have mass. In fact, you can account for about 95 percent of the mass of protons and neutrons—and thus about 95 percent of your mass—this way. So even if quarks were massless, most of the mass in the Universe would remain.

Still, the Higgs accounts for the mass of fundamental particles. So let's see how.

**• • •**

One thing we mean when we say the Higgs field is responsible for giving mass to fundamental particles is that without the Higgs, those particles would be massless—which in turn means that they would all travel at the speed of light, and that the minimum energy needed for all the self-sustaining excitations would be zero (as it is with photons).

How would things look if the fundamental particles were all massless?

Let’s start with the Standard Model matter fields. As I explained in my last article, excitations in the matter fields are called *fermions*, which includes both quarks and leptons. They all have a spin, or intrinsic angular momentum, of 1/2 of the fundamental unit of angular momentum (Planck’s constant, ℏ). I’m going to want you to picture how these spinning fermions are oriented in space.

Physicists like to represent particles in terms of vectors (arrows) because they make our mathematical constructions simpler. But how can we associate a vector to a spinning object? Here is the convention physicists came up with: if you have a spinning object, take your *right* hand, and curve your fingers around the spinning edge, in the direction of the rotation. The direction your thumb points is the direction we have decided the spin vector points in. So if you’re facing a spinning wheel, the spin vector would point *away* from you if the wheel were spinning clockwise but *toward* you if the wheel were spinning counterclockwise.

Fundamental particles have spin, too, but a pointlike fundamental particle isn’t an extended object like a wheel: there is no spinning edge you can run your right hand along to define the spin vector. Nevertheless—and this is one of the puzzles of the quantum world—these particle spin and so we can talk about the direction of the spin vector for fundamental particles. Picture a spinning fermion, then, as a little point (the particle) with an arrow (the spin) associated with it.

Why does this matter?

Consider an electron, which is one type of fermion, flying through space in some direction. I can measure its spin. When I do, one of the quirks of the quantum world ensures that the spin vector will point in only one of two directions: in the direction that the electron is moving, or in the opposite direction. The laws of quantum mechanics tell us that no other directions are possible. As a matter of convention we call the electrons whose spins point in the direction of travel *right-handed electrons*, while those with spins pointing opposite their direction of motion are called *left-handed electrons*. The distinction between left- and right-handedness is known as the *helicity *of a particle.

In our Universe, with the Higgs mechanism playing its mass-giving role, an electron is a massive particle (0.511 MeV). Therefore, it moves more slowly than the speed of light. Thus, no matter how fast the electron is moving relative to you at the start, you can imagine running fast enough to catch up and pass it. What happens to the spin vector of the electron if you pass it?

To see the answer, you can do an experiment. Take one finger and trace out a circle clockwise in the air in front of you. Keep moving your finger in that circle. Now move your head to look at your moving finger from the other side, opposite to your original vantage point. What happened? Your finger now appears to be moving counterclockwise. For another example, consider again a rotating wheel. If you look at a wheel that is spinning clockwise, the spin vector will point away from you. But if you walk around the wheel and look at it from the other side, the wheel will appear to be spinning counterclockwise, so the spin vector will point toward you.

Let us apply these insights to the electron. Imagine you chase an electron and overtake it. Before you catch up to the electron, it has a spin vector pointing in one direction, but after you pass the electron, that spin vector will appear to change directions. For example, if the electron initially looks right-handed, with a spin vector pointing in its direction of motion, when you pass it the spin vector will point opposite to the direction of motion, so the electron will then appear left-handed. Nothing has changed with the particle; what has changed is your frame of reference. (Keep in mind that quantum field theory marries quantum mechanics and special relativity, in which lots of properties are dependent on reference frames. In turning the description of quantum fields into a more easily grasped analogy, I’ve had to sweep relativity under the rug.)

In the language of quantum fields, we would say that the electron is an excitation of two quantum fields: one for the left-handed orientations, one for the right-handed. As we imagine running faster or slower than a massive particle, the wave that makes up that particle would appear to switch the field it resides in, from left to right or vice versa. If a wave can jump from one field to another, then the two fields are coupled. Thus, for fermions, *mass* is just the coupling between the two hands or helicities of the fermion fields, each of which requires the same energy for a self-sustaining excitation.

Because a massive fermion can be chased down and passed, it must be capable of switching right-handed and left-handed spins, which means that two different fields must be coupled. By comparison, you can’t catch up and pass a massless particle because it would always be moving at the speed of light. So for massless particles, left-handed stays left-handed and right-handed stays right. No mass means no coupling between the left- and right-handed fields. In fact, you can have a massless particle that comes only in left-handed or right-handed versions.

All well and good, you might say. But left- and right-handedness are just a matter of which way a little spinning top is pointing. Why make a big deal about it?

Because—and this is *incredibly* counterintuitive—in our Universe, the weak nuclear force distinguishes between particles with left-handed spin and right-handed spin. They are treated differently, which means that *the left-handed and right-handed fermions are not the same particle.* We call this a *parity violation*. Specifically, the W bosons (weak force bosons with positive and negative charge) *only *interact with the left-handed fermions. The (electrically neutral) Z boson interacts with both, but with a different strength to left and right.

Thus, if this were the end of the story, the fermions could not have mass. If a fermion has mass, then it must switch between left- and right-handed. But we cannot have a single (massive) particle switching between left-handed and right-handed and subject to different forces depending on the helicity. In contrast, if left-handed and right-handed electrons are simply two different massless particles, the fact that they interact with different forces creates no problem. We just treat them as two different quantum fields, just like we treat quarks and electrons as different fields.

However, we know that the electrons (and the rest of the matter fields) *do *have mass. What is going on?

**• • •**

What is going on is the Higgs mechanism. The mechanism is somewhat complicated, as you’ll see. To focus the discussion, I will keep talking about electrons, but everything I say applies to all fermions, not just electrons. The story will proceed in three steps.

**Step One: assume there is no Higgs**. The Standard Model says that we have two different particles, which I’ll call left-electrons and right-electrons. Left-electrons always have spin 1/2 pointing opposite their direction of travel, while right-electrons have spins pointed in the direction of travel. Neither particle has mass, which means that they travel at the speed of light and that we never have to worry about seeing a left-electron as a right-electron, or vice versa.

These two particles are very different. The massless *right-electron* does not feel the weak nuclear force. But it does feel a force called “hypercharge.” (Without the Higgs, there is no force called “electromagnetism,” rather it is replaced with a very similar force called “hypercharge.” Confusing I know, but bear with me.) The right-electron has a charge of -1 under this hypercharge force. Now, in the previous article, I described the connection between forces, local (gauge) symmetries, and bosons. For example, the strong nuclear force is deeply connected to a local symmetry that lets us rearrange the “colorcharges” of quarks, and this symmetry in turn requires the existence of gluons, a type of boson that carries the strong force. Hypercharge, too, is due to a local, gauge symmetry. And there is a gauge boson—with the unexciting name “B”—that carries the hypercharge force.

The massless *left-electron* also interacts with the hypercharge force, though with a charge of -1/2. Unlike the right electron, however, it *does* feel the weak nuclear force. The weak nuclear force (like the strong force and hypercharge) is also due to a local, gauge symmetry. This symmetry allows us to interchange left-electrons with electron-neutrinos. Without the Higgs, the left-electron and electron-neutrino are indistinguishable: both have -1/2 hypercharge, and we could freely rearrange and recombine the fields that make up these two particles, and nothing would change. Once more, this gauge symmetry requires gauge bosons. Specifically, the weak nuclear force requires three of these bosons, massless as usual, and they again get exciting names: the W^{1}, W^{2}, and W^{3}.

To review Step 1: if we assume no Higgs field, then the Standard Model tells us:

- We have two distinct particles, right and left electrons.
- There is a force called hypercharge, associated with a local symmetry and with the B boson.
- The right electron has hypercharge of -1, while the left has hypercharge -1/2.
- There is a weak force, which reflects a local symmetry that permits the transposition of left-electrons and electron neutrinos.
- The weak force is associated with three bosons: W
^{1}, W^{2}, and W^{3}. - The left-electron feels the weak force, whereas the right does not.

**Step Two: add in the Higgs field. **We now add in the Higgs field. The particles (excitations) associated with this field carry no spin (spin zero), so we call them bosons—Higgs bosons. This field has very particular properties. Like the left-electron, it feels the hypercharge force with a charge of -1/2. And like the right-electron, it has two components that can be relabeled under the symmetry that is associated with the weak nuclear force.

As you may have just noticed, the Higgs field fits very nicely with both the right- and left-electrons. It has just the correct hypercharge so that, if a right-electron came wandering along (with hypercharge -1), it could turn into a left-electron (hypercharge -1/2) and an excitation of this new Higgs field (hypercharge -1/2), as -1/2-1/2 = -1 (thus conserving hypercharge). What’s more, since the left-electron and this new Higgs field both can be rearranged under the weak nuclear force, that symmetry too would be respected. So it is possible that the left-electron, the right-electron, and the Higgs field can all be “coupled” together: an excitation in one (a particle, the right-electron) can jump its energy over to the other two, creating two new particles, left-electron and Higgs.

The stage is set for the emergence of mass, but we are not quite there yet. We do have a Higgs field, but the left-electron and the right-electron remain different particles, neither of which has or could have mass. And we do not yet have a force called “electromagnetism.”

**Step Three: energize the Higgs field. **Remember that you can think of quantum fields sort of like the surfaces of lakes, with particles as waves traveling across them. Most quantum fields are content in the absence of particles to relax down to some resting state, an unexcited state with no waves, with the field set at “zero,” containing no energy at all.

The Higgs field is different (unique, as far as we know): it doesn’t sit at “zero.” Everywhere in the Universe, this field, even in the absence of particles or excitations, contains a non-zero amount of energy. *Even in the vacuum*, with no particles and no radiation, the Higgs field is energized. This non-zero “vacuum expectation value” has enormous consequences.

To be more precise, the Higgs field *can* have zero value everywhere. But the zero value is extremely unstable. The slightest perturbation at any point will send the field there moving to some non-zero value. That one point will drag the rest of the field—everywhere in the Universe—along with it.

Consider an analogy. Imagine a flexible sheet of magnet (like a large, thin refrigerator magnet). You place one magnet above the sheet and one below it so that everything is perfectly balanced and the flexible magnet in the middle is attracted equally in both directions and the forces exactly cancel. Even a tiny wiggle in that central magnetic sheet upsets the delicate balance. At some random point, one part of the sheet would get a little closer to either the magnet above or below, and shoot towards it. That would drag the sheet around that first point closer to the same magnet (either above or below), and soon the entire sheet has moved to its new, stable location. Before the delicate equilibrium is disturbed, there was no difference between above and below the sheet. After the sheet moves, the symmetry is broken, and the sheet everywhere has picked the same direction to move in. Similarly, the Higgs field, delicately balanced at zero, is ready to move toward a more stable configuration. Like the sheet, it has more than one option for which “direction” to move in, but once one part of the field settles into its new, stable configuration, the field everywhere will follow.

The magnetic sheet in our analogy had only two choices of stable states: move toward the upper magnet or the lower. The Higgs has more choices. Imagine trying to balance a sharpened pencil on its tip: that state is unstable, and the pencil can fall not in just two directions, but in any direction in a circle. Before the pencil falls over, no direction around is “special.” We physicists would say there is a symmetry here: rotating the pencil on its tip (assuming it continues to be balanced) doesn’t change the setup. In falling over, the pencil moves a particular direction. This breaks the symmetry.

The analogies with the magnetic sheet and the pencil are helpful but imperfect. Helpful because they illustrate a type of instability that breaks spontaneously; imperfect because the “direction” that describes how the Higgs falls over to a non-zero value is not a direction in space. Rather, it tells us which particular combination of the multiple fields that make up the Higgs have gotten non-zero values. Remember that the Higgs field is really a combination of two fields that can be rearranged under the weak nuclear force. When the Higgs field is in the unstable “zero energy” configuration, no combination of these two fields is special, as both have zero values. That is, there is a symmetry between all the combinations of the fields.

However, when the Higgs field “falls over” to the stable, nonzero value, it has to pick a “direction,” selecting some particular combination of special fields. The symmetry responsible for the weak nuclear force is *broken*. In addition, remember that the Higgs field itself has hypercharge (of charge -1/2); since this field is now nonzero everywhere, empty space now has a hypercharge, thanks to the Higgs mechanism.

This process of breaking the symmetry between the fields that make up the Higgs, called electroweak symmetry breaking, also picks out a special combination of the left-electron and neutrino. This is *how* we decide what a left-electron and a neutrino are: they are the bits of these fields that are combined in the same way as the Higgs field, once symmetry is broken.

**• • •**

We have been on a long trip, but now have all the pieces needed to give fermions their mass. The left-electron and right-electron could not pair up—and thus had to have zero mass—because the two fields had different hypercharges, and because the left-handed electrons had a weak nuclear force symmetry and the right-handed did not. Those problems are now solved: with symmetry broken, the Universe no longer respects the symmetries responsible for the weak nuclear force and hypercharge.

Before electroweak symmetry breaking, a left-electron could turn into a right-electron and a Higgs particle: the excitation in the left-electron field can set up waves in these other two fields. Now, imagine the left-electron traveling through space. It is still coupled to the right-electron field and the Higgs field, but now it doesn’t need to set up a wave in *both *the right-electron field and Higgs field. Rather, the wave can just jump over to the right-electron field, with the non-zero Higgs field “absorbing” the extra hypercharge. A left-handed fermion entered, and a right-handed one exited. That’s *exactly* what needs to happen if a particle is to have mass.

Furthermore, the stronger the connection between the Higgs field and left- and right-handed fermions, the easier this switch between left- and right- becomes, which translates to a higher mass for the fermion. This is a very important prediction: the Higgs field should couple more to the heavier matter particles. As a result, the Higgs boson (the excitation of the field) should interact most strongly to the heaviest fermions around. If it doesn’t, then it’s not really the Higgs, or at least, the Higgs mechanism is more complicated than the already complex tale I’m telling you here. Testing this prediction is part of the LHC’s continuing mission over the next years and decades.

The Higgs field has lots more work to do: it must give mass to the W and Z gauge bosons, and also needs to give us a force called electromagnetism.

When the Higgs picked a “direction” as it broke the electroweak symmetry, it selected a combination of left-handed fermion fields for us to call “the left-up quark,” “the left-down quark” and so on. It also selected a particular combination of the W^{1}, W^{2}, and W^{3} gauge bosons as special. Along with the B gauge boson of hypercharge, these particular combinations of gauge bosons now find that moving through a Universe filled with non-zero Higgs field is difficult. It used to require no energy at all to set up a wave in each of these quantum fields: thus, no mass. Now these fields are coupled to the Higgs field, which is resistant to moving from its new, stable, non-zero value. This makes the W and B fields “stiffer” in some sense, more resistant to waves moving through them. After electroweak symmetry breaking, without some minimum energy, a wave in the W or B fields waves dies away quickly. That is: these gauge bosons used to be massless, but now they have a mass.

Since the Higgs field picked a particular combination of fields when it broke the electroweak symmetry, only certain combinations of the W^{1}, W^{2}, and W^{3}, and B gain these masses. One particular combination of the W^{3} and the B becomes the massive gauge boson we call the “Z.” Since the left-handed parts of our newly massive fermions interact with the W^{3} and B differently than the right-handed parts do, the Z itself has unequal couplings to the left- and right-handed components of the matter fields.

There is an elegance at the core of all of this, a beautiful economy that leaves what appears to be an orderly Universe.

However, there is a combination of W^{3} and B “perpendicular” to the direction picked out by the Higgs field. Because it is perpendicular, this combination of W^{3} and B does not interact with the non-zero Higgs field (if you think of the Higgs field’s analogy to the pencil falling over, this combination of fields would be the direction at right angles to the fallen pencil). This combination *does not* get a mass. Because it is massless, the combined particle, which we call the photon, can transmit forces over long distances. Thus, electromagnetism is a mix-up of the weak nuclear force and hypercharge.

If you look at how the precise combination of W^{3} and B that make up the photon would interact with the left- and right-handed pieces of the fermions, they are identical. Therefore, we say that both pieces have the same charge. (This is not a coincidence: regardless of which direction the Higgs field selected, the electric charges of the two helicities for fermions would work out to be equal.)

Finally, the W^{1 }and W^{2 }gain masses, like the Z did. They only interact with the left-handed pieces of fermions, just as they did before electroweak symmetry breaking. They also interact in a somewhat complicated way with the Z. And they interact with the photon exactly as if they had an electric charge. So we can identify these as the W^{+} and W^{–} bosons; the + and – signs indicate that these gauge bosons have electric charge.

The Higgs field must have a symmetry that requires interaction with the weak and hypercharge gauge bosons. If it did not, none of this story of electroweak symmetry breaking would work as it should — the W and Z bosons would not gain masses. As a result, the Higgs boson must also interact with the W and Z bosons, with precisely predicted couplings. These have been measured with some accuracy already at the LHC, and they match our predictions so far. Yet the possibility of small deviations remain, and would again signal the need for subtle changes to this “simplest’ Higgs model.

**• • •**

Think of everything that needs to happen for the Higgs mechanism to work. You need left-handed and right-handed fermions, which don’t have the same quantum charges (in particular, hypercharge) under strange local gauge symmetries. You must have a field—the Higgs field—that is very, very unstable at zero energy. Furthermore, the differences between the left-handed fields and the right-handed fermion fields have to be exactly matched by the charges of this Higgs field. This matching might seem suspicious. One would hope that there is some overarching reason behind this, rather than sheer coincidence. We physicists have long hoped and searched for evidence of such a structure, but as yet, we have not found any.

The Higgs mechanism is complicated, with many moving parts. The fermions and the gauge bosons must both gain masses, and the forces involved are not even the forces of Nature you know and love. However, despite this complexity, there is an elegance at the core of all of this, a beautiful economy where a single field can do so many things, and leave what appears to be an orderly Universe, with straightforward matter fields, interacting with a single long range force via electric charge. Only by peeking under the hood do you start seeing the difficult task that has been achieved.

Over the last two articles, I’ve explained most of the organization of the physics we understand: the physics of the Standard Model. With the discovery of the Higgs, the LHC moved into the business of looking for new physics, beyond the Standard Model. In the next article, I will explain how these experiments work, what pieces of information we get from colliding protons together, and how we use that information as physicists.

From there, I will tell you about how we actually discovered the Higgs, and then what new physics we theoretical physicists have hoped to find. By then, the LHC should have told us what is going on with the strange anomalous results I told you about in my first article, and my final article will tell you what we think is going on. I cannot spoil the ending because I have no idea what that article will say, and no one else in the world does either—which is incredibly exciting.

June 16, 2016

26 Min read time