The origin of mass

In this World Year of Physics Essay written for Frontline, the Nobel Prize-winning physicist explains how his own work on subnuclear forces casts new light on the origin of mass from energy.

When only 21 years old and a graduate student at Princeton University, in work with David Gross he defined the properties of colour gluons, which hold atomic nuclei together. He was jointly awarded the Nobel Prize for Physics in 2004, together with Prof. David Gross and Prof. H.D. Politzer, for the discovery of asymptotic freedom.

Professor Wilczek taught at Princeton from 1974 to 1981. Subsequently he held distinguished chairs in physics at the University of California at Santa Barbara and the Institute for Advanced Study, Princeton. In 2000, he moved to the Massachusetts Institute of Technology, where he is currently the Herman Feshbach Professor of Physics.

Professor Wilczek has been a Sloan Foundation Fellow (1975-77) and a MacArthur Foundation Fellow (1982-87). Apart from the Nobel Prize, he has received numerous awards for his contributions to the development of theoretical physics.

He is a member of the National Academy of Sciences in the United States, the Netherlands Academy of Sciences and the American Academy of Arts and Sciences, and is a Trustee of the University of Chicago.

Prof. Wilczek is an award-winning communicator of science and contributes regularly to Physics Today and to Nature, explaining topics at the frontiers of physics to wider scientific audiences.

Two of his pieces have been anthologised in Best American Science Writing (2003, 2005). Together with his wife Betsy Devine, he has written a book, Longing for the Harmonies (W.W. Norton).

EVERYDAY work on the frontiers of modern physics usually involves complex concepts and extreme conditions. We speak of quantum fields, entanglement, or supersymmetry, and analyse the ridiculously small or conceptualise the incomprehensibly large. Just as Willie Sutton famously explained that he robbed banks because "that's where the money is", so we do these things because "that's where the Unknown is". It is an amazing and delightful fact, however, that occasionally this sophisticated work gives answers to childlike questions about familiar things. Here I would like to describe how my own work on subnuclear forces, the world of quarks and gluons, casts brilliant new light on one such childlike question: What is the origin of mass?

This is an especially appropriate topic for the World Year of Physics 2005, because it relates so closely to the circle of ideas around Albert Einstein's most famous equation, E = mc2. That equation, written in that form, immediately suggests the possibility of converting small quantities of mass into large quantities of energy - a suggestion that was realised, of course, with the development of nuclear reactors and nuclear weapons. It is worth noting, however, that this is not the way the equation appears in Einstein's original paper. In that paper you do not find E = mc2, but rather m = E/c2. The difference is trivial algebraically, but profound conceptually, for the second (original) form of the equation suggests something quite different: the possibility to derive mass from energy. For a modern physicist, and even for Einstein in 1905, this sounds a deeper resonance. Energy appears a pervasive, primary concept in modern physics, and there is no real prospect of explaining it in terms of something more basic. For mass the situation is quite different. The title of Einstein's paper is "Does the Inertia of a Body Depend Upon Its Energy Content?". It shows that from the beginning, Einstein was thinking about questioning the foundations of fundamental physics, not making bombs. Modern physics, as I shall now explain, answers his question with a resounding "Yes!"

That a question makes grammatical sense does not guarantee that it is answerable, or even coherent. The concept of mass is one of the first things we discuss in my freshman mechanics class. Classical mechanics is, literally, unthinkable without it. Newton's Second Law of Motion says that the acceleration of a body is given by dividing the force acting upon it by its mass. So a body without mass would not know how to move, because you would be dividing by zero. Also, in Newton's Law of Gravity, the mass of an object governs the strength of the force it exerts. One cannot build up an object that gravitates, out of material that does not, so you cannot get rid of mass without getting rid of gravity.

Finally, the most basic feature of mass in classical mechanics is that it is conserved. For example, when you bring together two bodies, the total mass is just the sum of the individual masses. This assumption is so deeply ingrained that it was not even explicitly formulated as a law. (Though I teach it as Newton's Zeroth Law.) Altogether, in the Newtonian framework it is difficult to imagine what would constitute an "origin of mass", or even what this phrase could possibly mean. In that framework mass just is what it is - a primary concept.

Later developments in physics make the concept of mass seem less irreducible. Einstein's famous equation for the interconvertibility of mass and energy, already mentioned, was the watershed. In modern particle accelerators, this possibility comes to life. For example, in the Large Electron Positron collider (LEP), at the CERN laboratory near Geneva, beams of electrons and antielectrons (positrons) were accelerated to enormous energies. Powerful, specially designed magnets controlled the paths of the particles, and caused them to circulate in opposite directions around a big storage ring. The paths of these beams intersected at a few interaction regions, where collisions could occur. When a collision between a high-energy electron and a high-energy positron occurs, we often observe that many particles emerge from the event. The total mass of these particles can be thousands of times the mass of the original electron and positron. Thus mass has been created, physically, from energy.

Having convinced ourselves that the question of the origin of mass might make sense, let us now come to grips with it, in the concrete form that it takes for ordinary matter. Ordinary matter is made from atoms. The mass of atoms is overwhelmingly concentrated in their nuclei. The surrounding electrons are of course crucial for discussing how atoms interact with each other - and thus for chemistry, biology, and electronics. But they provide less than a part in a thousand of the mass! Nuclei, which provide the lion's share of mass, are assembled from protons and neutrons. All this is a familiar, well-established story, dating back to 70 years or more. Newer and perhaps less familiar, but by now no less well-established, is the next step: protons and neutrons are made from quarks and gluons. So most of the mass of matter can be traced, ultimately, back to quarks and gluons.

The theory of quarks and gluons is called quantum chromodynamics, or QCD. QCD is a generalisation of quantum electrodynamics (QED). For a nice description of quantum electrodynamics, I highly recommend "QED: The Strange Theory of Electrons and Light", written by a Massachusetts Institute of Technology (MIT) graduate who made good, Richard Feynman. The basic concept of QED is the response of photons to electric charge. The elementary act in QED is emission of a photon by a charged particle. From this elementary act, the whole theory can be built up deductively, using the powerful rules of special relativity and quantum mechanics. The rules for electric and magnetic forces, from atomic to cosmic scales, and for radiation and absorption of light and radio waves - what the great physicist Paul Dirac called "all of chemistry and most of physics" - all emerge by deduction from the elementary act.

It is like making constructions with TinkerToys. The particles are different kinds of sticks you can use, and the elementary act provides the hubs that join them. Given these elements, the rules for construction are completely determined. In this way all the content of Maxwell's equations for radio waves and light, Schrodinger's equation for atoms and chemistry, and Dirac's more refined version including spin - all this, and more, are faithfully encoded in QED.

At this most primitive level QCD is a lot like QED, but bigger. The diagrams look similar, and the rules for evaluating them are similar, but there are more kinds of sticks and hubs. More precisely, while there is just one kind of charge in QED - namely, electric charge - QCD has three different kinds of charge. They are called colours, for no good reason. We could label them red, green and blue. Every quark has one unit of one of the colour charges. In addition, quarks come in different species, or "flavours". The only two that play a role in ordinary matter are two flavours called u and d, for up and down. (Of course, quark "flavours" have nothing to do with how anything tastes. And, these names for u and d do not imply that there is any real connection between flavours and directions. Don't blame me; when I get the chance, I give particles dignified scientific-sounding names like axion and anyon.)

There are u quarks with a unit of red charge, d quarks with a unit of green charge, and so forth, for six different possibilities altogether. And instead of one photon that responds to electric charge, QCD has eight colour gluons that can either respond to different colour charges or change one into another. So there is quite a large variety of sticks, and there are also many different kinds of hubs that connect them. With all these possibilities, it seems like things could get terribly complicated and messy. And so they would, were it not for the overwhelming symmetry of the theory. If you interchange red with blue everywhere, for example, you must still get the same rules. The more complete symmetry allows you to mix the colours continuously, forming blends, and the rules must come out the same for blends as for pure colours.

I shall not be able to do justice to the mathematics here, of course. But the final result is noteworthy and easy to convey: there is one and only one way to assign rules to all the possible hubs so that the theory comes out fully symmetric. Intricate it may be, but messy it is not! With these understandings, QCD is faithfully encoded in a single elementary act and its symmetric cousins. We thereby arrive at definite rules, realised as precise equations, which predict how quarks and gluons behave and interact. Solving the equations can be very difficult, but if they are solved, there is no ambiguity about the outcome. The theory is either right or wrong - there is nowhere to hide.

Experiment is the ultimate arbiter of scientific truth. There are many experiments that test the basic principles of QCD. Most of them require rather sophisticated analysis, basically because we do not get to see the underlying simple stuff, the individual quarks and gluons, directly. But there is one kind of experiment that comes very close to doing this, and that is what I would like to explain now.

I shall be discussing what was observed at LEP. Before entering into details, I would like to highlight a fundamental point about quantum mechanics, which is necessary background for making any sense at all of what happens. According to the principles of quantum mechanics, the result of an individual collision is unpredictable. We can, and do, control the energies and spins of the electrons and positrons precisely, so that precisely the same kind of collision occurs repeatedly.

Nevertheless, different results emerge. By making many repetitions, we can determine the probabilities for different outcomes. These probabilities encode basic information about the underlying fundamental interactions; according to quantum mechanics, they contain all the meaningful information.

When we examine the results of collisions at LEP, we find there are two broad classes of outcomes. Each happens about half the time.

In one class, the final state consists of a particle and its antiparticle moving rapidly in opposite directions. These could be an electron and an antielectron, a muon and an antimuon, or a tau and an antitau. The electron, muon and tau have one unit of negative electric charge, while their anti-particles have one unit of positive electric charge. These particles, collectively called leptons, are all closely similar in their properties.

Leptons do not carry colour charges, so their main interactions are with photons, and thus their behaviour should be governed by the rules of QED.

This is reflected, first of all, in the simplicity of their final states. Once produced, any of these particles could - in the language of elementary acts - attach a photon using a QED hub, or alternatively, in physical terms, radiate a photon. The basic coupling of photons to a unit charge is fairly weak, however. Therefore each additional attachment is predicted to decrease the probability of the process being described, and so the most usual case is no attachment. In fact the final state that includes a photon does occur, with about 1 per cent of the rate of the particles simply scattering off each other (and similarly for the other leptons). By studying the details of these 3-particle events, such as the probability for the photon to be emitted in different directions (the "antenna pattern") and with different energy, we can check all aspects of our hypothesis for the elementary act. This provides a wonderfully direct and incisive way to check the soundness of the basic conceptual building block from which we construct QED. We can then go on to address the extremely rare cases (.01 per cent) where two photons get radiated, and so forth. For future reference, let us call this first class of outcomes "QED events".

The other broad class of outcomes contains an entirely different class of particles, and is in many ways far more complicated. In these events the final state typically contains ten or more particles, selected from a menu of pions, rho mesons, protons and antiprotons, and many more. These are all particles that in other circumstances interact strongly with one another, and they are all constructed from quarks and gluons. Here, they make a smorgasbord of the Greek and Latin alphabet. It is such a mess that physicists have pretty much given up on trying to describe all the possibilities and their probabilities in detail.

Fortunately, however, some simple patterns emerge if we change our focus from the individual particles to the overall flow of energy and momentum.

Most of the time - in about 90 per cent of the cases - the particles emerge all moving in either one of two possible directions, opposite to one another. We say there are back-to-back jets. (Here, for once, the scientific jargon is both vivid and appropriate.) About 9 per cent of the time, we find flows in three directions; about .9 per cent of the time, four directions; and by then we are left with a very small remainder of complicated events that are hard to analyse this way. I shall call the second broad class of outcomes "QCD events".

Now if you squint a little, you will find that the QED events and the QCD events begin to look quite similar. Indeed, the pattern of energy flow is qualitatively the same in both cases, that is, heavily concentrated in a few narrow jets. There are two main differences. One, relatively trivial, is that multiple jets are more common in QCD than in QED. The other is much more profound. It is that, of course, in the QED events the jets are just single particles, while in the QCD events the jets are sprays of several particles.

In 1973, while I was working as a graduate student with David Gross at Princeton University, I discovered the explanation of these phenomena. The key was a theoretical discovery I shall describe momentarily, which we christened asymptotic freedom. Actually, our discovery of asymptotic freedom preceded these specific experiments. We were inspired by much less direct evidence. As things actually happened, therefore, we were able to predict the properties of these jets, which exhibit the fundamentals with ideal simplicity, before they were observed.

The basic concept of asymptotic freedom is that the probability for a fast-moving quark or gluon to radiate away some of its energy in the form of other quarks and gluons depends on whether this radiation is "hard" or "soft". Hard radiation is radiation that involves a substantial deflection of the particle doing the radiating, while soft radiation is radiation that does not cause such a deflection. Thus hard radiation changes the flow of energy and momentum, while soft radiation merely distributes it among additional particles, all moving together. Asymptotic freedom says that hard radiation is rare, but soft radiation is common.

This distinction explains why on the one hand there are jets, and on the other hand why the jets are not single particles. A QCD event begins as the materialisation of quark and antiquark, similar to how a QED event begins as the materialisation of lepton-antilepton. They usually give us two jets, aligned along the original directions of the quark and the antiquark, because only hard radiation can change the overall flow of energy and momentum significantly, and asymptotic freedom tells us hard radiation is rare. When a hard radiation does occur, roughly 10 per cent of the time at

LEP, we have an extra jet! But we do not see the original quarks, antiquarks, or gluons individually because they are always accompanied by their soft radiation, which is common.

By studying the antenna patterns of the multi-jet QCD events we can check all aspects of our hypotheses for the underlying hubs. Just as for QED, such antenna patterns provide a wonderfully direct and incisive way to check the soundness of the elementary acts from which we construct QCD.

Through the analyses of this and many other applications, physicists have acquired complete confidence in the fundamental correctness of QCD. By now experimenters use it routinely to design experiments searching for new phenomena. I have lived to see the same activity which used to be called "testing QCD" become described as "calculating backgrounds"!

Following the flow of energy and momentum in violent collisions allows us to check the fundamental ideas of the theory, but using that theory to calculate the masses of proton and other strongly interacting particles presents additional challenges. The difficulty is with the soft radiation, which we cannot ignore in this context. Since such radiation is emitted very easily, it is difficult to keep track of it all. To meet that challenge, a radically different strategy is required. Instead of calculating the paths of individual quarks and gluons through space and time, we let each segment of space-time keep track of how many quarks and gluons it contains. We then treat these segments as an assembly of interacting subsystems.

Actually in this context "we" means a collection of hard-working CPUs of a large number of powerful computers. Skilfully orchestrated, and working at teraflop speeds for months at a time, they manage to calculate the properties of the protons and other strongly interacting particles that emerge as the possible stable arrangements of quarks and gluons - including, of course, their masses. The calculated masses agree quite accurately with the observed ones. In my opinion, this accurate calculation of the origin of masses, starting with a tight fundamental theory embodying profound physical concepts and mathematical symmetry, is one of the greatest scientific achievements ever. So that is the origin of (most) the mass of the proton and other strongly interacting particles.

With the answer in hand, let us interpret what we have got. For our purposes it is instructive to compare two versions of QCD, an idealised version I call QCD Lite, and the realistic Full-Bodied version. QCD Lite is cooked up from massless gluons, massless u and d quarks, and nothing else. (Now you can fully appreciate the wit of the name.) If we use this idealisation as the basis for our calculation, we get the proton mass low by about 5 per cent. Realistic, Full-Bodied QCD differs from QCD Lite in two ways. First, it contains four additional flavours of quarks. These do not appear directly in the proton, but they do have some effect on the calculation. Second, it allows for non-zero masses of the u and d quarks. The realistic value of these masses, though, turns out to be small, just a few per cent of the proton mass. Together these corrections change the predicted mass of the proton by about 5 per cent, as we pass from QCD Lite to Full-Bodied QCD. So we find that 95 per cent of the proton (and neutron) mass, and therefore 95 per cent of the mass of ordinary matter, emerges from an idealised theory whose ingredients are entirely massless.

Now I have shown you the theory that describes quarks and gluons and therefore has to account for most of the mass of matter. I have described some of the experiments that validate theory. The calculations of particle masses employ cutting-edge computer technology with massive parallelism, and even then some approximations must be introduced to make the computations feasible. These results are a remarkable embodiment of the vision that elements of reality can be reproduced by purely conceptual constructions - "Its from Bits" - because the underlying theory, based on profoundly symmetrical equations, contains very few adjustable parameters.

But simply having a computer spit out the answer, after gigantic and totally opaque calculations, does not satisfy our hunger for understanding. It is particularly unsatisfactory in the present case, because the answer appears to be miraculous. The computers construct for us massive particles using building blocks - quarks and gluons - that are themselves massless. The equations of QCD Lite output Mass without Mass. It sounds suspicious, like Something for Nothing. How did it happen?

The key, again, is asymptotic freedom. Previously, I discussed asymptotic freedom in terms of hard and soft radiation. Hard radiation is rare, soft radiation is common. There is another way of looking at it, mathematically equivalent, that is useful here. From the classical equations of QCD, one would expect a force field between quarks that falls off as the square of the distance, as in ordinary electromagnetism (Coulomb's Law). Its enhanced coupling to soft radiation, however, means that when quantum mechanics is taken into account, a ``bare'' colour charge, inserted into empty space, will start to surround itself with a cloud of virtual colour gluons. These colour gluon fields themselves carry colour charge, so they are sources of additional soft radiation. The result is a self-catalysing enhancement that leads to runaway growth. A small colour charge, in isolation, builds up a big colour thundercloud.

All this structure costs energy, and theoretically the energy for a quark in isolation is infinite. That is why we never see individual quarks. Having only a finite amount of energy to work with, nature must find a way to short-circuit the colour thundercloud catastrophe. One way is to bring in an antiquark. If the antiquark could be placed right on top of the quark, their colour charges would exactly cancel each other, and the thundercloud would never get triggered. There is also another more subtle way to cancel the colour charge by bringing together three quarks, one of each colour.

In practice neither of these cancellations can be exact, however. Quarks obey the rules of quantum mechanics. It is wrong to think of them simply as tiny particles; rather, they are quantum mechanical wavicles. Quarks are subject, in particular, to Heisenberg's Uncertainty Principle, which implies that if you try to pin down their position too precisely, their momentum will be wildly uncertain. To support the possibility of large momentum, they must acquire large energy. In other words, it takes work to pin quarks down. Wavicles want to spread out. So there is a competition between two effects. To cancel the colour charge completely, we would like to put the quark and the antiquark at precisely the same place; but those wavicles resist localisation, so the cancellation comes at a price.

A number of stable compromise solutions can be found, where the quark and the antiquark (or three quarks) are brought close together but are not perfectly coincident. Their distribution is described by quantum mechanical wave functions. Each possible stable wave-patterns corresponds to - indeed, in a profound sense it is - a different kind of particle that you can observe. There are patterns for protons and neutrons, and for each entry in our whole Greek and Latin smorgasbord. Each pattern has some characteristic energy, because the colour fields are not entirely cancelled and because the wavicles are somewhat localised. And that energy, through Einstein's m = E/c2, is the origin of mass.

A similar mechanism, though much simpler, works in atoms. Negatively charged electrons feel an attractive electric force from the positively charged nucleus, and from that point of view they would like to snuggle right on top of it. Electrons are wavicles, though, and that inhibits them.

In its absence, I shall substitute a classic metaphor. The wave patterns that describe protons, neutrons and their relatives resemble the vibration patterns of musical instruments. In fact, the mathematical equations that govern these superficially very different realms are quite similar. Musical analogies go back to the prehistory of science. Pythagoras, partly inspired by his discovery that harmonious notes are sounded by strings whose lengths are in simple numerical ratios, proposed that "All things are Numbers". Kepler spoke of the music of the spheres, and his longing to find their hidden harmonies sustained him through years of tedious calculations and failed guesses before he identified the true patterns of planetary motions. Einstein, when he learned of Bohr's atomic model, called it "the highest form of musicality in the sphere of thought". Yet Bohr's model, wonderful as it is, appears to us now as a very watered down version of the true wave-mechanical atom; and the wave-mechanical proton is more intricate and symmetric by far!

Mass, a seemingly irreducible property of matter, and a byword for its sluggishness and resistance to change, turns out to emerge from a harmonious interplay of symmetry, uncertainty, and energy. Using these concepts, and the algorithms they suggest, pure computation outputs the numerical values of the masses of particles that we observe.

In conclusion, let me emphasise that our understanding of the origin of mass is by no means complete. We have achieved a beautiful and profound understanding of the origin of most of the mass of ordinary matter, but not of all of it. The value of the electron mass, in particular, remains deeply mysterious even in our most advanced speculations about the grand unification of fundamental forces and string theory. And ordinary matter, we have recently learned, supplies only a small fraction of the mass in the universe as a whole. More beautiful and profound revelations surely await discovery. We continue to search for concepts and theories that will allow us to understand the origin of mass in all its forms, by unveiling more of nature's hidden beauty.