The Mysteries of Mass

Physicists are hunting for an elusive particle that would reveal the presence of a new kind of field that permeates all of reality. Finding that Higgs field will give us a more complete understanding about how the universe works.

Most people think they know what mass is, but they understand only part of the story. For instance, an elephant is clearly bulkier and weighs more than an ant. Even in the absence of gravity, the elephant would have greater mass--it would be harder to push and set in motion. Obviously the elephant is more massive because it is made of many more atoms than the ant is, but what determines the masses of the individual atoms? What about the elementary particles that make up the atoms--what determines their masses? Indeed, why do they even have mass?

We see that the problem of mass has two independent aspects. First, we need to learn how mass arises at all. It turns out mass results from at least three different mechanisms, which I will describe below. A key player in physicists' tentative theories about mass is a new kind of field that permeates all of reality, called the Higgs field. Elementary particle masses are thought to come about from the interaction with the Higgs field. If the Higgs field exists, theory demands that it have an associated particle, the Higgs boson. Using particle accelerators, scientists are now hunting for the Higgs.

The second aspect is that scientists want to know why different species of elementary particles have their specific quantities of mass. Their intrinsic masses span at least 11 orders of magnitude, but we do not yet know why that should be so. For comparison, an elephant and the smallest of ants differ by about 11 orders of magnitude of mass.

Isaac newton presented the earliest scientific definition of mass in 1687 in his landmark Principia: "The quantity of matter is the measure of the same, arising from its density and bulk conjointly." That very basic definition was good enough for Newton and other scientists for more than 200 years. They understood that science should proceed first by describing how things work and later by understanding why. In recent years, however, the why of mass has become a research topic in physics. Understanding the meaning and origins of mass will complete and extend the Standard Model of particle physics, the well-established theory that describes the known elementary particles and their interactions. It will also resolve mysteries such as dark matter, which makes up about 25 percent of the universe.

The foundation of our modern understanding of mass is far more intricate than Newton's definition and is based on the Standard Model. At the heart of the Standard Model is a mathematical function called a Lagrangian, which represents how the various particles interact. From that function, by following rules known as relativistic quantum theory, physicists can calculate the behavior of the elementary particles, including how they come together to form compound particles, such as protons. For both the elementary particles and the compound ones, we can then calculate how they will respond to forces, and for a force F, we can write Newton's equation F = ma, which relates the force, the mass and the resulting acceleration. The Lagrangian tells us what to use for m here, and that is what is meant by the mass of the particle.

But mass, as we ordinarily understand it, shows up in more than just F = ma. For example, Einstein's special relativity theory predicts that massless particles in a vacuum travel at the speed of light and that particles with mass travel more slowly, in a way that can be calculated if we know their mass. The laws of gravity predict that gravity acts on mass and energy as well, in a precise manner. The quantity m deduced from the Lagrangian for each particle behaves correctly in all those ways, just as we expect for a given mass.

Fundamental particles have an intrinsic mass known as their rest mass (those with zero rest mass are called massless). For a compound particle, the constituents' rest mass and also their kinetic energy of motion and potential energy of interactions contribute to the particle's total mass. Energy and mass are related, as described by Einstein's famous equation, E = mc² (energy equals mass times the speed of light squared).

An example of energy contributing to mass occurs in the most familiar kind of matter in the universe--the protons and neutrons that make up atomic nuclei in stars, planets, people and all that we see. These particles amount to 4 to 5 percent of the mass-energy of the universe. The Standard Model tells us that protons and neutrons are composed of elementary particles called quarks that are bound together by massless particles called gluons. Although the constituents are whirling around inside each proton, from outside we see a proton as a coherent object with an intrinsic mass, which is given by adding up the masses and energies of its constituents.

The Standard Model lets us calculate that nearly all the mass of protons and neutrons is from the kinetic energy of their constituent quarks and gluons (the remainder is from the quarks' rest mass). Thus, about 4 to 5 percent of the entire universe--almost all the familiar matter around us--comes from the energy of motion of quarks and gluons in protons and neutrons.

Unlike protons and neutrons, truly elementary particles--such as quarks and electrons--are not made up of smaller pieces. The explanation of how they acquire their rest masses gets to the very heart of the problem of the origin of mass. As I noted above, the account proposed by contemporary theoretical physics is that fundamental particle masses arise from interactions with the Higgs field. But why is the Higgs field present throughout the universe? Why isn't its strength essentially zero on cosmic scales, like the electromagnetic field? What is the Higgs field?

The Higgs field is a quantum field. That may sound mysterious, but the fact is that all elementary particles arise as quanta of a corresponding quantum field. The electromagnetic field is also a quantum field (its corresponding elementary particle is the photon). So in this respect, the Higgs field is no more enigmatic than electrons and light. The Higgs field does, however, differ from all other quantum fields in three crucial ways.

The first difference is somewhat technical. All fields have a property called spin, an intrinsic quantity of angular momentum that is carried by each of their particles. Particles such as electrons have spin 1/2 and most particles associated with a force, such as the photon, have spin 1. The Higgs boson (the particle of the Higgs field) has spin 0. Having 0 spin enables the Higgs field to appear in the Lagrangian in different ways than the other particles do, which in turn allows--and leads to--its other two distinguishing features.

The second unique property of the Higgs field explains how and why it has nonzero strength throughout the universe. Any system, including a universe, will tumble into its lowest energy state, like a ball bouncing down to the bottom of a valley. For the familiar fields, such as the electromagnetic fields that give us radio broadcasts, the lowest energy state is the one in which the fields have zero value (that is, the fields vanish)--if any nonzero field is introduced, the energy stored in the fields increases the net energy of the system. But for the Higgs field, the energy of the universe is lower if the field is not zero but instead has a constant nonzero value. In terms of the valley metaphor, for ordinary fields the valley floor is at the location of zero field; for the Higgs, the valley has a hillock at its center (at zero field) and the lowest point of the valley forms a circle around the hillock. The universe, like a ball, comes to rest somewhere on this circular trench, which corresponds to a nonzero value of the field. That is, in its natural, lowest energy state, the universe is permeated throughout by a nonzero Higgs field.

The final distinguishing characteristic of the Higgs field is the form of its interactions with the other particles. Particles that interact with the Higgs field behave as if they have mass, proportional to the strength of the field times the strength of the interaction. The masses arise from the terms in the Lagrangian that have the particles interacting with the Higgs field.

Our understanding of all this is not yet complete, however, and we are not sure how many kinds of Higgs fields there are. Although the Standard Model requires only one Higgs field to generate all the elementary particle masses, physicists know that the Standard Model must be superseded by a more complete theory. Leading contenders are extensions of the Standard Model known as Supersymmetric Standard Models (SSMs). In these models, each Standard Model particle has a so-called superpartner (as yet undetected) with closely related properties [see "The Dawn of Physics beyond the Standard Model," by Gordon Kane; Scientific American, June 2003]. With the Supersymmetric Standard Model, at least two different kinds of Higgs fields are needed. Interactions with those two fields give mass to the Standard Model particles. They also give some (but not all) mass to the superpartners. The two Higgs fields give rise to five species of Higgs boson: three that are electrically neutral and two that are charged. The masses of particles called neutrinos, which are tiny compared with other particle masses, could arise rather indirectly from these interactions or from yet a third kind of Higgs field.

Theorists have several reasons for expecting the SSM picture of the Higgs interaction to be correct. First, without the Higgs mechanism, the W and Z bosons that mediate the weak force would be massless, just like the photon (which they are related to), and the weak interaction would be as strong as the electromagnetic one. Theory holds that the Higgs mechanism confers mass to the W and Z in a very special manner. Predictions of that approach (such as the ratio of the W and Z masses) have been confirmed experimentally.

Second, essentially all other aspects of the Standard Model have been well tested, and with such a detailed, interlocking theory it is difficult to change one part (such as the Higgs) without affecting the rest. For example, the analysis of precision measurements of W and Z boson properties led to the accurate prediction of the top quark mass before the top quark had been directly produced. Changing the Higgs mechanism would spoil that and other successful predictions.

Third, the Standard Model Higgs mechanism works very well for giving mass to all the Standard Model particles, W and Z bosons, as well as quarks and leptons; the alternative proposals usually do not. Next, unlike the other theories, the SSM provides a framework to unify our understanding of the forces of nature. Finally, the SSM can explain why the energy "valley" for the universe has the shape needed by the Higgs mechanism. In the basic Standard Model the shape of the valley has to be put in as a postulate, but in the SSM that shape can be derived mathematically.

Naturally, physicists want to carry out direct tests of the idea that mass arises from the interactions with the different Higgs fields. We can test three key features. First, we can look for the signature particles called Higgs bosons. These quanta must exist, or else the explanation is not right. Physicists are currently looking for Higgs bosons at the Tevatron Collider at Fermi National Accelerator Laboratory in Batavia, Ill.

Second, once they are detected we can observe how Higgs bosons interact with other particles. The very same terms in the Lagrangian that determine the masses of the particles also fix the properties of such interactions. So we can conduct experiments to test quantitatively the presence of interaction terms of that type. The strength of the interaction and the amount of particle mass are uniquely connected.

Third, different sets of Higgs fields, as occur in the Standard Model or in the various SSMs, imply different sets of Higgs bosons with various properties, so tests can distinguish these alternatives, too. All that we need to carry out the tests are appropriate particle colliders--ones that have sufficient energy to produce the different Higgs bosons, sufficient intensity to make enough of them and very good detectors to analyze what is produced.

A practical problem with performing such tests is that we do not yet understand the theories well enough to calculate what masses the Higgs bosons themselves should have, which makes searching for them more difficult because one must examine a range of masses. A combination of theoretical reasoning and data from experiments guides us about roughly what masses to expect.

The Large Electron-Positron Collider (LEP) at CERN, the European laboratory for particle physics near Geneva, operated over a mass range that had a significant chance of including a Higgs boson. It did not find one--although there was tantalizing evidence for one just at the limits of the collider's energy and intensity--before it was shut down in 2000 to make room for constructing a newer facility, CERN's Large Hadron Collider (LHC). The Higgs must therefore be heavier than about 120 proton masses. Nevertheless, LEP did produce indirect evidence that a Higgs boson exists: experimenters at LEP made a number of precise measurements, which can be combined with similar measurements from the Tevatron and the collider at the Stanford Linear Accelerator Center. The entire set of data agrees well with theory only if certain interactions of particles with the lightest Higgs boson are included and only if the lightest Higgs boson is not heavier than about 200 proton masses. That provides researchers with an upper limit for the mass of the Higgs boson, which helps focus the search.

For the next few years, the only collider that could produce direct evidence for Higgs bosons will be the Tevatron. Its energy is sufficient to discover a Higgs boson in the range of masses implied by the indirect LEP evidence, if it can consistently achieve the beam intensity it was expected to have, which so far has not been possible. In 2007 the LHC, which is seven times more energetic and is designed to have far more intensity than the Tevatron, is scheduled to begin taking data. It will be a factory for Higgs bosons (meaning it will produce many of the particles a day). Assuming the LHC functions as planned, gathering the relevant data and learning how to interpret it should take one to two years. Carrying out the complete tests that show in detail that the interactions with Higgs fields are providing the mass will require a new electron-positron collider in addition to the LHC (which collides protons) and the Tevatron (which collides protons and antiprotons).

What is discovered about Higgs bosons will not only test whether the Higgs mechanism is indeed providing mass, it will also point the way to how the Standard Model can be extended to solve problems such as the origin of dark matter.

With regard to dark matter, a key particle of the SSM is the lightest superpartner (LSP). Among the superpartners of the known Standard Model particles predicted by the SSM, the LSP is the one with the lowest mass. Most superpartners decay promptly to lower-mass superpartners, a chain of decays that ends with the LSP, which is stable because it has no lighter particle that it can decay into. (When a superpartner decays, at least one of the decay products should be another superpartner; it should not decay entirely into Standard Model particles.) Superpartner particles would have been created early in the big bang but then promptly decayed into LSPs. The LSP is the leading candidate particle for dark matter.

The Higgs bosons may also directly affect the amount of dark matter in the universe. We know that the amount of LSPs today should be less than the amount shortly after the big bang, because some would have collided and annihilated into quarks and leptons and photons, and the annihilation rate may be dominated by LSPs interacting with Higgs bosons.

As mentioned earlier, the two basic SSM Higgs fields give mass to the Standard Model particles and some mass to the superpartners, such as the LSP. The superpartners acquire more mass via additional interactions, which may be with still further Higgs fields or with fields similar to the Higgs. We have theoretical models of how these processes can happen, but until we have data on the superpartners themselves we will not know how they work in detail. Such data are expected from the LHC or perhaps even from the Tevatron.

Neutrino masses may also arise from interactions with additional Higgs or Higgs-like fields, in a very interesting way. Neutrinos were originally assumed to be massless, but since 1979 theorists have predicted that they have small masses, and over the past decade several impressive experiments have confirmed the predictions [see "Solving the Solar Neutrino Problem," by Arthur B. McDonald, Joshua R. Klein and David L. Wark; Scientific American, April 2003]. The neutrino masses are less than a millionth the size of the next smallest mass, the electron mass. Because neutrinos are electrically neutral, the theoretical description of their masses is more subtle than for charged particles. Several processes contribute to the mass of each neutrino species, and for technical reasons the actual mass value emerges from solving an equation rather than just adding the terms.

Thus, we have understood the three ways that mass arises: The main form of mass we are familiar with--that of protons and neutrons and therefore of atoms--comes from the motion of quarks bound into protons and neutrons. The proton mass would be about what it is even without the Higgs field. The masses of the quarks themselves, however, and also the mass of the electron, are entirely caused by the Higgs field. Those masses would vanish without the Higgs. Last, but certainly not least, most of the amount of superpartner masses, and therefore the mass of the dark matter particle (if it is indeed the lightest superpartner), comes from additional interactions beyond the basic Higgs one.

Finally, we consider an issue known as the family problem. Over the past half a century physicists have shown that the world we see, from people to flowers to stars, is constructed from just six particles: three matter particles (up quarks, down quarks and electrons), two force quanta (photons and gluons), and Higgs bosons--a remarkable and surprisingly simple description. Yet there are four more quarks, two more particles similar to the electron, and three neutrinos. All are very short-lived or barely interact with the other six particles. They can be classified into three families: up, down, electron neutrino, electron; charm, strange, muon neutrino, muon; and top, bottom, tau neutrino, tau. The particles in each family have interactions identical to those of the particles in other families. They differ only in that those in the second family are heavier than those in the first, and those in the third family are heavier still. Because these masses arise from interactions with the Higgs field, the particles must have different interactions with the Higgs field.

Hence, the family problem has two parts: Why are there three families when it seems only one is needed to describe the world we see? Why do the families differ in mass and have the masses they do? Perhaps it is not obvious why physicists are astonished that nature contains three almost identical families even if one would do. It is because we want to fully understand the laws of nature and the basic particles and forces. We expect that every aspect of the basic laws is a necessary one. The goal is to have a theory in which all the particles and their mass ratios emerge inevitably, without making ad hoc assumptions about the values of the masses and without adjusting parameters. If having three families is essential, then it is a clue whose significance is currently not understood.

The standard model and the SSM can accommodate the observed family structure, but they cannot explain it. This is a strong statement. It is not that the SSM has not yet explained the family structure but that it cannot. For me, the most exciting aspect of string theory is not only that it may provide us with a quantum theory of all the forces but also that it may tell us what the elementary particles are and why there are three families. String theory seems able to address the question of why the interactions with the Higgs field differ among the families. In string theory, repeated families can occur, and they are not identical. Their differences are described by properties that do not affect the strong, weak, electromagnetic or gravitational forces but that do affect the interactions with Higgs fields, which fits with our having three families with different masses. Although string theorists have not yet fully solved the problem of having three families, the theory seems to have the right structure to provide a solution. String theory allows many different family structures, and so far no one knows why nature picks the one we observe rather than some other [see "The String Theory Landscape," by Raphael Bousso and Joseph Polchinski; Scientific American, September 2004]. Data on the quark and lepton masses and on their superpartner masses may provide major clues to teach us about string theory.

One can now understand why it took so long historically to begin to understand mass. Without the Standard Model of particle physics and the development of quantum field theory to describe particles and their interactions, physicists could not even formulate the right questions. Whereas the origins and values of mass are not yet fully understood, it is likely that the framework needed to understand them is in place. Mass could not have been comprehended before theories such as the Standard Model and its supersymmetric extension and string theory existed. Whether they indeed provide the complete answer is not yet clear, but mass is now a routine research topic in particle physics.

It is possible that the particles we see are all actually massless, their apparent masses corresponding to extra-dimensional momentum components we can't as yet detect.

Interesting, but I don't get it. (Not an uncommon event when I read about this stuff). So you have a bunch of massless particles in a 3d space, presumably moving in straight lines except when they collide. If you project them onto a 2d space, aren't they still going to appear to move in straight lines? Or is the projection somehow a nonlinear function that can map straight lines to orbits and other curved paths? Even if so, if the particles in the 3d space move independently of each other, how would any projection create the appearance of dependency?

I don't know if that made any sense; I find this stuff fascinating but am missing a lot of the theoretical background. Trying to get through Penrose's Road to Reality, but I start spacing out on calculus on manifolds...

Try this: take a bunch of massless, interacting particles and let them fly around in three dimensions, crashing into each other and bouncing off as they may. Now take their trajectories, and project them onto a two-dimensional plane. As viewed in the two-dimensional plane, the particles interact as if they had masses, the apparent masses being proportional to their momenta in the direction perpendicular to the plane.

It is possible that the particles we see are all actually massless, their apparent masses corresponding to extra-dimensional momentum components we can't as yet detect.

Sounds like a great simplification, but I'm having difficulty visualizing this. Can you direct me to a website that might have some diagrams of what that 2D projection might look like? All I can think of is a Mercator projection, and I'm sure that's not even in the ballpark.

1. The m is the same in both equations, yes. F is the force of interaction when a particle interacts with another. If particle 'b' perturbs the motion of particle 'a', then the amount of force 'b' exerts on 'a' is measured by F=ma. If you know the mass of 'b', the force exerted upon it by 'a', and the acceleration change from the interaction, you can determine the mass of particle 'a'.
E is the total amount of energy that could be released by converting all the mass of the particle to energy. Not a lot of ways to do this; a matter-antimatter collision of equivalent particles comes to mind. The total energy released by the destruction of the two particles would be their combined masses x c^2. If you know the total amount of energy released in such an instance, and divide by c^2, you get the total mass involved.
2. Yes. F/a is usually a small quantity over another small quantity. E/c^2 is a huge quantity over another huge quantity. It balances out.

And yes, I had to grab scratch paper to be sure :)

I'm not talking about gravity, here, I'm talking about straight Newtonian mechanics.

Ok, I might get it now. So in the hypothetical 3d space all particles have the same resistance to acceleration, but because they have differering velocities in the z direction, their projections on the xy plane will appear to accelerate at different rates given the same force. Is that the idea? If it is, then two further questions:
- Wouldn't observers in the xy plane sometimes see two particles seemingly occupying the same location (because they'd have the same x and y coordinates but different z)?
- How would this model account for gravity? It eliminates inertial mass by making it a function of velocity in the z direction, but what about gravitational mass?

So in the hypothetical 3d space all particles have the same resistance to acceleration, but because they have differering velocities in the z direction, their projections on the xy plane will appear to accelerate at different rates given the same force. Is that the idea?

Just so.

Wouldn't observers in the xy plane sometimes see two particles seemingly occupying the same location (because they'd have the same x and y coordinates but different z)?

[Geek alert: Well, for one thing, the extra dimension would likely be curled up into a very tiny circle, too small to be noticed at our scale. (Call it "periodic boundary conditions", if you prefer.) Alert readers might have wondered why, say, an electron couldn't just have any old momentum in the 5th dimension, and therefore any old mass. It's obviously not that way: electrons all have the same mass. The answer is because, in the tiny extra dimension, the quantum wavefunction of the electron would have to have an integer number of wavefunctions along the circle, to match the boundary conditions. [Super Geek Alert: This means only certain masses would be allowed. Could the higher harmonics represent the muon and the tau? The argument has been made, but we still don't know why only three.] ANYWAY, the wavefunction covers the entire space, so the Pauli Exclusion Principle still applies. [Super Geek Alert: Some theorists postulate that there are LARGE extra dimensions. In such models, it actually is theoretically possible (using polarization) to get electrons to "pass through" each other with a head-on trajectory, because they miss each other in the 5th dimension!]]

How would this model account for gravity? It eliminates inertial mass by making it a function of velocity in the z direction, but what about gravitational mass?

Rather than snow you under with handwaving about Kaluza-Klein towers of gravitons, I'll admit that that's a very technical question, which I'm not qualified to answer. I can tell you that the whole model has not been worked out.

If we were flatlanders, living in a 2D plane, and a 3D rod were perpendicular to our plane, we'd see it only as a 2D circular cross-section of the rod. I see no possibility for simulating mass. If the rod were in motion, passing through our plane, I still see no illusion of mass. I'm obviously missing something. Perhaps if we could, in our 2D way, handle the circle, and test its mass, it would then reveal the effect of the motion, but we wouldn't know about that motion. We'd just think it was massive, and we wouldn't know why. Is that it?

If we were flatlanders, living in a 2D plane, and a 3D rod were perpendicular to our plane, we'd see it only as a 2D circular cross-section of the rod.

The rod isn't perpendicular to the plane (although it could be). The extra dimension is perpendicular to the plane.

If the rod were in motion,

Full stop. The "rod" only represents the trajectory of the particle over time. The particle is pointlike and massless. The particle moves through the space as a massless object. The shadow of the particle on the plane moves on the plane as if it had mass.

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