16
Mass without Mass
The origin of the bulk of the mass of ordinary matter is well accounted for, in a theory based on pure concepts and using no mass parameters—indeed, no mass unit—at all!
Frank Wilczek1
We’re almost there. In this final chapter, I propose to address the principal question concerning the nature of matter in the context of a familiar, everyday substance—water. Well, water in a very specific form. Imagine a cube of ice, measuring a little over one inch (or 2.7 centimetres) in length. Imagine holding this cube of ice in the palm of your hand. It is cold, and a little slippery. It weighs hardly anything at all yet we know it weighs something.
We make our question a little more focused. What is this cube of ice made of? And, an important secondary question: What is responsible for its mass?
To the ancient Greek philosophers, water was one of the four elements along with earth, air, and fire. The Greek atomists argued that material substance cannot suddenly appear from nothing and cannot be divided endlessly into nothing. The atoms of the Greeks possessed specific properties, of size, shape, position, and weight (or mass).
Our cube of ice must therefore be composed of atoms, and atoms cannot exist except within empty space, called the void. Lucretius argued that the very fluid properties of water imply that it contains small atoms with a ‘readiness to roll’, in contrast with honey, which is stickier and must consist of atoms that are not so smooth, or so fine, or so round.2 But as the environment surrounding the water is cooled it expels particles or atoms of heat, thereby cooling and eventually freezing the water.3 We might therefore imagine that as water freezes to ice, its small, spherical atoms are drawn closer and closer together, eventually to lock ranks and form the regular array characteristic of a solid.
The atoms of the mechanical philosophers—of Bacon, Boyle, and Newton, among others—were not much more sophisticated. In Querie 31 of his Opticks, Newton speculated that forces might be at work between the atoms. But he would not be drawn on the precise nature of these forces beyond suggesting that they might be already known to us, in the forms of gravity, electricity, and magnetism.
To understand what a cube of ice is made of, we need to draw on the learning acquired by the chemists. Building on a long tradition established by the alchemists, these scientists distinguished between different chemical elements, such as hydrogen, carbon, and oxygen. Research on the relative weights of these elements and the combining volumes of gases led Dalton and Gay-Lussac to the conclusion that different chemical elements consist of atoms with different weights which combine according to a set of rules involving whole numbers of atoms.
The mystery of the combining volumes of hydrogen and oxygen gas to produce water was resolved when it was realized that hydrogen and oxygen are both diatomic gases, H2 and O2. Water is then a compound consisting of two hydrogen atoms and one oxygen atom, H2O.
This partly answers our first question. Our cube of ice consists of molecules of H2O organized in a regular array. We can also make a start on our second question. Avogadro’s law states that a mole of chemical substance will contain about 6 × 1023 discrete ‘particles’. Now, we can interpret a mole of substance simply as its molecular weight scaled up to gram quantities. Hydrogen (in the form of H2) has a relative molecular weight of 2,* implying that each hydrogen atom has a relative atomic weight of 1. Oxygen (O2) has a relative molecular weight of 32, implying that each oxygen atom has a relative atomic weight of 16. Water (H2O) therefore has a relative molecular weight of 2 × 1 + 16 = 18.
It so happens that our cube of ice weighs about 18 grams, which means that it represents a mole of water, more or less.4 According to Avogadro’s law it must therefore contain about 6 × 1023 molecules of H2O. This would appear to provide a definitive answer to our second question. The mass of the cube of ice derives from the mass of the hydrogen and oxygen atoms present in 6 × 1023 molecules of H2O.
But, of course, we can go further. We learned from Thompson, Rutherford, and Bohr and many other physicists in the early twentieth century that all atoms consist of a heavy, central nucleus surrounded by light, orbiting electrons. We subsequently learned that the central nucleus consists of protons and neutrons. The number of protons in the nucleus determines the chemical identity of the element: a hydrogen atom has one proton, an oxygen atom has eight (this is called the atomic number). But the total mass or weight of the nucleus is determined by the total number of protons and neutrons in the nucleus.
Hydrogen still has only one (its nucleus consists of a single proton—no neutrons). The most common isotope of oxygen has—guess what—sixteen (eight protons and eight neutrons). It’s obviously no coincidence that these proton and neutron counts are the same as the relative atomic weights I quoted above.
If we ignore the electrons, then we would be tempted to claim that the mass of the cube of ice resides in all the protons and neutrons in the nuclei of its hydrogen and oxygen atoms. Each molecule of H2O contributes ten protons and eight neutrons, so if there are 6 × 1023 molecules in the cube and we ignore the small difference in mass between a proton and a neutron, we conclude that the cube contains in total about eighteen times this figure, or 108 × 1023 protons and neutrons.
So far, so good. There is nothing in this calculation that the Greek atomists or mechanical philosophers would dispute. Yes, our understanding of the nature and composition of matter is now a lot more sophisticated, but the conclusions are essentially the same. For ‘atoms’ read ‘elementary particles’. Setting aside the small contribution from all the electrons, we trace the mass of our cube of ice to the mass of all the protons and neutrons it contains, as shown in Figure 23.
Figure 23. A cube of ice measuring 2.7 cm in length will weigh about 18 g, (a). It consists of a lattice structure containing a little over 600 billion trillion molecules of water, H2O, (b). Each atom of oxygen contains eight protons and eight neutrons, and each hydrogen atom contains one proton, (c). The cube of ice therefore contains about 10,800 billion trillion (108 × 1023) protons and neutrons.
But we’re not quite done yet. We now know that protons and neutrons are not elementary particles. They consist of quarks. A proton contains two up quarks and a down quark, a neutron two down quarks and an up quark. And the colour force binding the quarks together inside these larger particles is carried by massless gluons.
Okay, so surely we just keep going. If once again we approximate the masses of the up and down quarks as the same we just multiply by three and turn 108 × 1023 protons and neutrons into 324 × 1023 up and down quarks. We conclude that this is where all the mass resides. Yes?
No. This is where our naïve atomic preconceptions unravel. We can look up the masses of the up and down quarks on the Particle Data Group website. The up and down quarks are so light that their masses can’t be measured precisely and only ranges are quoted. The following are all reported in units of MeV/c2. In these units the mass of the up quark is given as 2.3 with a range from 1.8 to 3.0. The down quark is a little heavier, 4.8, with a range from 4.5 to 5.3.* Compare these with the mass of the electron, about 0.51 measured in the same units.
Now comes the shock. In the same units of MeV/c2 the proton mass is 938.3, the neutron 939.6. The combination of two up quarks and a down quark gives us only 9.4, or just one per cent of the mass of the proton. The combination of two down quarks and an up quark gives us only 11.9, or just 1.3 per cent of the mass of the neutron. About ninety-nine per cent of the masses of the proton and neutron seem to be unaccounted for. What’s gone wrong?
To answer this question, we need to recognize what we’re dealing with. Quarks are not self-contained ‘particles’ of the kind that the Greeks or the mechanical philosophers might have imagined. They are quantum wave-particles; fundamental vibrations or fluctuations of elementary quantum fields. The up and down quarks are only a few times heavier than the electron, and we’ve demonstrated the electron’s wave-particle nature in countless laboratory experiments. We need to prepare ourselves for some odd, if not downright bizarre behaviour.
And let’s not forget the massless gluons. Or special relativity, and E = mc2. Or the difference between ‘bare’ and ‘dressed’ mass. And, last but not least, let’s not forget the role of the Higgs field in the ‘origin’ of the mass of all elementary particles. To try to understand what’s going on inside a proton or neutron we need to reach for quantum chromodynamics, the quantum field theory of the colour force between quarks.
Quarks and gluons possess colour ‘charge’. Just what is this, exactly? We have no way of really knowing. We do know that colour is a property of quarks and gluons and there are three types, which physicists have chosen to call red, green, and blue. But, just as nobody has ever ‘seen’ an isolated quark or gluon, so more or less by definition nobody has ever seen a naked colour charge. In fact, quantum chromodynamics (QCD) suggests that if a colour charge could be exposed like this it would have a near-infinite energy. Aristotle’s maxim was that ‘nature abhors a vacuum’. Today we might say: ‘nature abhors a naked colour charge’.
So, what would happen if we could somehow create an isolated quark with a naked colour charge? Its energy would go up through the roof, more than enough to conjure virtual gluons out of ‘empty’ space. Just as the electron moving through its own self-generated electromagnetic field gathers a covering of virtual photons, so the exposed quark gathers a covering of virtual gluons. Unlike photons, the gluons themselves carry colour charge and they are able to reduce the energy by, in part, masking the exposed colour charge. Think of it this way: the naked quark is acutely embarrassed, and it quickly dresses itself with a covering of gluons.
This isn’t enough, however. The energy is high enough to produce not only virtual particles (like a kind of background ‘noise’ or ‘hiss’), but elementary particles, too. In the scramble to cover the exposed colour charge, an anti-quark is produced which pairs with the naked quark to form a meson. A quark is never—but never—seen without a chaperone.
But this still doesn’t do it. To cover the colour charge completely we would need to put the anti-quark in precisely the same place at precisely the same time as the quark. Heisenberg’s uncertainty principle won’t let nature pin down the quark and anti-quark in this way. Remember that a precise position implies an infinite momentum, and a precise rate of change of energy with time implies an infinite energy. Nature has no choice but to settle for a compromise. It can’t cover the colour charge completely but it can mask it with the anti-quark and the virtual gluons. The energy is at least reduced to a manageable level.
This kind of thing also goes on inside the proton and neutron. Within the confines of their host particles, the three quarks rattle around relatively freely. But, once again, their colour charges must be covered, or at least the energy of the exposed charges must be reduced. Each quark produces a blizzard of virtual gluons that pass back and forth between them, together with quark–anti-quark pairs. Physicists sometimes call the three quarks that make up a proton or a neutron ‘valence’ quarks, as there’s enough energy inside these particles for a further ‘sea’ of quark–anti-quark pairs to form. The valence quarks are not the only quarks inside these particles.
What this means is that the mass of the proton and neutron can be traced largely to the energy of the gluons and the sea of quark–anti-quark pairs that are conjured from the colour field.
How do we know? Well, it must be admitted that it is actually really rather difficult to perform calculations using QCD. The colour force is extremely strong, and the corresponding energies of colour-force interactions are therefore very high. Remember that the gluons also carry colour charge, so everything interacts with everything else. Virtually anything can happen, and keeping track of all the possible virtual and elementary-particle permutations is very demanding.
This means that although the equations of QCD can be written down in a relatively straightforward manner, they cannot be solved analytically, ‘on paper’. Also, the mathematical sleight-of-hand used so successfully in QED no longer applies—because the energies of the interactions are so high we can’t apply the techniques of renormalization. Physicists have had no choice but to solve the equations on a computer instead.
Considerable progress was made with a version of QCD called ‘QCD-lite’. This version considered only massless gluons and up and down quarks, and further assumed that the quarks themselves are also massless (so, literally, ‘lite’). Calculations based on these approximations yielded a proton mass that was found to be just ten per cent lighter than the measured value.
Let’s stop to think about that for a bit. A simplified version of QCD in which we assume that no particles have mass to start with nevertheless predicts a mass for the proton that is ninety per cent right. The conclusion is quite startling. Most of the mass of the proton comes from the energy of the interactions of its constituent quarks and gluons.
Since these calculations were performed further progress has been made with a version of the theory called lattice QCD. The magnitudes of the quantum fields representing the quarks are defined only at specific points on a three-dimensional grid or lattice (rather than continuously through spacetime, as would be required for a continuous field). The magnitudes of the gluon fields are then defined on the links connecting neighbouring points on the lattice. The spacing between lattice points is of the order of a few tenths to a few hundredths of a femtometre (10−15 metres), and the more this is reduced the closer we get to a ‘continuum’ version of QCD. To reduce the number of computer calculations required, physicists perform calculations at smaller and smaller spacings and then extrapolate their results to zero spacing.
In lattice QCD it becomes possible to relax the approximations that were required in QCD-lite. We can involve more generations of quarks and include their masses. Great accuracy is possible but this comes at a cost; the most rigorous lattice QCD calculations require the world’s largest supercomputers.
If all this isn’t bad enough, we must remember that quarks also carry electrical charge and, just like the electron, there will be contributions to the mass arising from the quarks’ interactions with their own self-generated electromagnetic fields. This can be treated using QED and the renormalization methods pioneered in the late 1940s.
Lattice QCD calculations reported in November 2008 yielded a proton mass that is within a few per cent of the measured value.5 Lattice QCD and combined QED calculations of the small mass difference between the proton and the neutron were reported in March 2015.6 The latter results are particularly interesting. If we assume that the electrical charge carried by the proton is uniformly distributed, then we might be tempted to conclude that, just like the electron, interactions with its own electromagnetic field would add an ‘electromagnetic mass’ to the proton that the neutron simply can’t possess. In the absence of any other effects, we would conclude that the proton must therefore be a little heavier than the neutron. But the opposite is true—the neutron is in fact a little heavier than the proton.
Let’s dig a little deeper. We concluded above that the mass of two down quarks plus one up quark (neutron) is about 11.9 MeV/c2. The mass of two up quarks plus one down quark (proton) is about 9.4, a difference of 2.5 MeV/c2, which is obviously just the difference in the masses of a single down quark versus an up quark. Now this is larger than the difference between the masses of the neutron and proton (which is about 1.3 MeV/c2).
We can guess what happens. If we assume that the strengths (and hence energies) of the quark–gluon and quark–quark interactions are similar for both up and down quarks, then these interactions add equally to the masses of the proton and neutron. The difference in their masses is then just the difference in the masses of the down versus up valence quarks, offset by a small net contribution from the electromagnetic mass of the proton.
You might be tempted to conclude that this subtle horse-trading between quantum fields and forces inside protons and neutrons is all very interesting, but ultimately irrelevant. Well, you’d be quite wrong. The simple fact that the neutron is slightly heavier than the proton underpins much of the structure of the physical world that we tend to take for granted. Put it this way, if the difference in mass was much smaller, then the proton would lose its stability and become radioactive. It would become susceptible to inverse beta-decay, transforming into a neutron with the emission of a W+ particle and an electron-neutrino. If the difference in mass was much larger, then the fusion of protons to form helium nuclei in the centres of stars would become difficult to impossible, and no heavier elements could be formed.
Either way, the universe would be a very different place and we would certainly not be here to witness it.
This is all well and good, but it raises one final question. If the relative stability of the proton and neutron—and hence our very existence—depends on the difference in the masses of the up and down quarks, then what is the origin of this difference? Why is the down quark heavier than the up quark?
Recall from Chapter 14 that all elementary particles, including matter particles such as quarks and electrons, derive their mass through interactions with the Higgs field. In the case of matter particles (which are all fermions, remember) the interaction is referred to as a Yukawa interaction, named for Japanese theorist Hideki Yukawa. But the principle is the same. Particles that would otherwise be massless, ‘two-dimensional’, and moving at the speed of light interact with the Higgs field, ‘absorb’ a Higgs boson, gain a third dimension, and slow down. The interaction results in the appearance of terms related to m2ϕ2 in the equations, and the particles ‘gain mass’.
Our final question then becomes: why does the down quark interact (or ‘couple’) more strongly with the Higgs field when compared with the up quark? It is here that we reach the limit of our present understanding. The Higgs mechanism is a fundamental component of the current standard model of particle physics and the recent discovery of the Higgs boson tells us that it is likely to be correct. But on the question of the relative strengths of the interactions of different matter and force particles with the Higgs field the model is stubbornly silent.
It may well be that as we learn more about the Higgs boson (now that we know we can produce it at the LHC) we will discover more of its secrets. But any revelations must await us in the future.
John Wheeler used the phrase ‘mass without mass’ to describe the effects of superpositions of gravitational waves which could concentrate and localize energy such that a black hole is created. If this were to happen, it would mean that a black hole—the ultimate manifestation of super-high-density matter—had been created not from the matter in a collapsing star but from fluctuations in spacetime. What Wheeler really meant was that this would be a case of creating a black hole (mass) from gravitational energy.
But Wheeler’s phrase is more than appropriate here. Frank Wilczek, one of the architects of QCD, used it in connection with his discussion of the results of the QCD-lite calculations in a couple of papers published in 20027 and the MIT Physics Annual published in 2003.8 If much of the mass of a proton and neutron comes from the energy of interactions taking place inside these particles, then this is indeed ‘mass without mass’, meaning that we get the behaviour we tend to ascribe to mass without the need for mass as a property.
Does this sound familiar? Recall that in Einstein’s seminal addendum to his 1905 paper on special relativity the equation he derived is actually m = E/c2. This is the great insight (not E = mc2). And Einstein was surely prescient when he wrote: ‘the mass of a body is a measure of its energy content’.9 Indeed, it is. In his book The Lightness of Being, Wilczek wrote:10
If the body is a human body, whose mass overwhelmingly arises from the protons and neutrons it contains, the answer is now clear and decisive. The inertia of that body, with 95% accuracy, is its energy content.
In the fission of a U-235 nucleus, some of the energy of the colour fields inside its protons and neutrons is released, with potentially explosive consequences. In the proton–proton chain involving the fusion of four protons, the conversion of two up quarks into two down quarks, forming two neutrons in the process, results in the release of a little excess energy from its colour fields. Mass does not convert to energy. Energy is instead passed from one kind of quantum field to another.
Where does this leave us? We’ve certainly come a long way since the ancient Greek atomists speculated about the nature of material substance, 2,500 years ago. But for much of this time we’ve held to the conviction that matter is a fundamental part of our physical universe. We’ve been convinced that it is matter that has energy. And, although matter may be reducible to microscopic constituents, for a long time we believed that these would still be recognizable as matter—they would still possess the primary quality of mass.
Modern physics teaches us something rather different, and deeply counter-intuitive. As we worked our way ever inwards—matter into atoms, atoms into sub-atomic particles, sub-atomic particles into quantum fields and forces—we lost sight of matter completely. Matter lost its tangibility. It lost its primacy as mass became a secondary quality, the result of interactions between intangible quantum fields. What we recognize as mass is a behaviour of these quantum fields; it is not a property that belongs or is necessarily intrinsic to them.
Despite the fact that our physical world is filled with hard and heavy things, it is instead the energy of quantum fields that reigns supreme. Mass becomes simply a physical manifestation of that energy, rather than the other way around.
This is conceptually quite shocking, but at the same time extraordinarily appealing. The great unifying feature of the universe is the energy of quantum fields, not hard, impenetrable atoms. Perhaps this is not quite the dream that philosophers might have held fast to, but a dream nevertheless.