The Magicians

by Marcus Chown

  At first glance, the swimming pool idea chimes perfectly with our experience of mass. A body with a big mass, such as a fridge, is hard to budge – it resists attempts to change its motion. This might plausibly be because there is an invisible medium pushing back. However, the analogy is imperfect – a basic feature of the world, and a foundation stone of Einstein’s special theory of relativity, is that no experiment can reveal whether you are moving at constant speed or stationary. If two people are playing catch on a train and the windows are blacked out and the train is vibration-free, the ball will loop back and forth between them just as it would if they were standing beside the track. They will be unable to tell from the ball’s motion whether they are in motion or not.

  However, if Higgs’ ubiquitous space-filling field was exactly like the water in a swimming pool, it would be possible to distinguish between a body moving through it and one that was stationary, contradicting relativity. Instead, the Higgs field had to appear stationary to every body in the universe, irrespective of its motion.* This property of being, in the jargon, ‘Lorentz invariant’, is true only of ‘scalar fields’ that, like temperature and the height of a billiard table, are characterised by a simple number at each point in space.

  The twist is that the Higgs field resists particles even though they are always stationary with respect to it. A more accurate statement, therefore, is that the field simply interacts with them. It is this interaction that causes intrinsically massless particles to have masses, with their precise mass depending on how strongly each particle interacts with the field.

  A field whose energy is non-zero everywhere in space was something entirely new. In the absence of mass there is no gravitational field, and in the absence of electric charge there is no electromagnetic field. But Higgs imagined a field which existed in otherwise empty space and had no source. The field had the same energy everywhere, and it was this sameness that explained why it had never been noticed before: we are immersed in it, just as we are immersed in the air we breathe.

  Given the opportunity, everything minimises its ‘potential energy’. For instance, a ball will roll to the foot of a hill, where its ‘gravitational potential energy’ is lowest. It had always been assumed that the vacuum was the lowest energy state of the universe; it was where, as surely as a ball rolling to the bottom of a hill, the universe would end up. However, Higgs was suggesting this is wrong and that the lowest-energy state of the universe is actually a vacuum filled with the Higgs field, with a non-zero energy everywhere.

  But arbitrarily introducing into a theory a field which fills all of space in order to give masses to the force carriers is as artificial as inserting their masses by hand. It was also guaranteed to wreck local gauge invariance, which was an essential requirement for a theory without monstrous infinities. What Higgs needed was a way to introduce his field in a natural way, and he had seen how to do it.

  A gauge theory in which all the force carriers are massless is elegant and symmetric; after all, the masses of all the particles are exactly the same. On the other hand, a theory in which the particles have masses that may not be the same is messy and less symmetric – physicists say its symmetry is ‘broken’.

  Examples of a symmetry breaking naturally are easy to find in the everyday world. Imagine a pencil balanced vertically on its sharpened end. It is perfectly symmetric – it looks the same from every direction. However, if it is buffeted by a draught of air, it may fall pointing north, southwest or in any other direction. Its orientation with respect to the vertical is no longer symmetric. This illustrates that although the fundamental laws of physics may be symmetric – in this case, the force of gravity points downwards and favours no direction of the compass – the outcome of those laws may nevertheless be asymmetric.

  The field Higgs had in mind was one that was perfectly symmetric – in effect, ‘switched off’ and incapable of upsetting local gauge invariance – but whose symmetry was spontaneously broken, causing it to switch on and interact with the gauge force carriers to give them masses. The simplest scheme he could imagine involved the energy of the field being given by the height of a marble on a ‘potential’ the shape of a sombrero hat. Initially – and this would have been in the super-high-energy conditions of the Big Bang – the ball would have been in a perfectly symmetric state on the central peak of the hat. However, as the universe cooled down towards its present-day low-energy state, the marble would have rolled into the gutter of the hat, picking a direction and breaking the symmetry.

  An explanation of how the interaction of the massless force carriers with Higgs’ spontaneously broken field generates their masses will have to wait; there is a more immediate and pressing problem with Higgs’ scheme, which other physicists were aware of and which was another reason why quantum field theories were out of favour in the early 1960s.

  Think of the sombrero hat again. The marble characterising the energy of the Higgs field could end up anywhere around the rim, with each location corresponding to a state of the Higgs field. An atom can change from one energy state to another by emitting or absorbing a photon with an energy equal to the energy difference between the states. However, in the case of the Higgs field, each state around the rim of the sombrero is at exactly the same height and so has exactly the same energy. Changing from one state to any other takes no energy, which means it corresponds to a particle with zero mass – such a particle is known as a ‘Goldstone boson’.

  The British physicist Jeffrey Goldstone had discovered that such particles are an unavoidable consequence of spontaneously breaking the symmetry of a scalar field. The problem was that having zero mass, they should be very easy to create and so should have revealed themselves long ago in physicists’ experiments. But not a single Goldstone boson has ever come to light. Physicists had taken the absence of Goldstone bosons as proof that quantum field theories that require spontaneous symmetry breaking to make contact with the real world are a theoretical dead end.

  Yet the idea that local gauge symmetry might spawn the fundamental forces was hugely appealing to Higgs. It was elegant, beautiful and it felt right – he just could not let it go. For years, he had mulled it over in his mind, and then he had his breakthrough. Three weeks earlier, he had sent his paper to Physics Letters, in which he set out a striking result. Higgs’ breakthrough was to find that Goldstone bosons go away if a quantum field theory is locally gauge invariant – that is, if there are also gauge force carriers. To appreciate why, it is necessary to know about a key distinction between massless and massive particles.

  A subatomic particle, as pointed out, is simply a wave propagating through a quantum field, much like a wind rippling through a field of wheat. The world in which we live has three dimensions of space, so it seems obvious that a wave representing a particle can oscillate in three mutually perpendicular directions. This intuition is correct for a particle which has a mass, but it is not true for a massless particle such as a photon, which travels at the speed of light.

  A photon is associated with an electromagnetic wave, whose electric and magnetic fields oscillate in a plane perpendicular to the direction of the wave’s travel. In addition to these two ‘transverse’ oscillations, it might be expected that the wave would also oscillate in its direction of motion, but such a ‘longitudinal’ wave would necessarily alternate between moving slower than the speed of light and faster than the speed of light. This is impossible because, according to Einstein, the speed of light is the ultimate cosmic speed limit. The upshot is that while a massive particle has three independent ways of oscillating, a massless one has only two.

  Higgs’ breakthrough was to realise that the Goldstone bosons go away miraculously in a theory with massless gauge particles. They are, in effect, ‘swallowed’ by the force carriers. Not only does the ‘Higgs mechanism’ get rid of the troublesome Goldstone bosons but, in being swallowed, they endow the massless gauge particles with a third way of oscillating, giving them masses.9

  It should be
pointed out that this mechanism for acquiring mass is different from the one previously described in which particles interact with the ubiquitous Higgs field, encountering resistance like a swimmer ploughing through a swimming pool. Indeed, nature has seen fit to provide two separate mechanisms for endowing particles with mass. One, which gives mass to nature’s force carriers, involves spontaneous symmetry breaking; the other, which gives mass to nature’s building blocks – the quarks and leptons – involves a more straightforward interaction with the Higgs field.

  The Higgs mechanism is a miracle that kills two birds with one stone: it simultaneously gets rid of the Goldstone bosons and endows the gauge force carriers with masses. The role of Goldstone bosons, according to Steven Weinberg, is changed ‘from that of unwanted intruders to that of welcome friends’.10 Beneath the mass-inducing cloak of the Goldstone bosons, the force carriers remain massless and, consequently, they continue to be described by a renormalisable, infinity-free local gauge theory.

  *

  Higgs had spelled all this out in two short papers in the summer of 1964. In the first, he showed how it was possible to get rid of Goldstone bosons in a quantum field theory, as long as gauge bosons are also present. In his second paper, he outlined how the gauge bosons acquire masses, by cannibalising the Goldstone bosons.11 But as he sat in his office, with his rejected second paper sitting on the desk before him, he still faced the problem of what to add to ensure it would be accepted for publication. Maybe a physical consequence of his idea?

  Every quantum field has a particle or particles associated with it because every field can be rippled, and a ripple propagating through the field is all a particle actually is. Higgs knew his symmetry-breaking field would be no exception.

  He thought again about the sombrero hat potential that governed his field. The Goldstone bosons arose because the marble could oscillate around the rim of the sombrero, but that was not the only type of oscillation that was possible. The marble could also oscillate in a radial direction, up and down the valley formed by the rim of the sombrero. Such an oscillation would require a minimum amount of energy to excite it, and energy, according to Einstein, has an equivalent mass. If sufficient energy were pumped into a small area of space, it would be possible to create such a particle: a ripple in the Higgs field.

  Higgs’ annoyance at the rejection of his second paper by Physics Letters had not simply been because its editor failed to appreciate the importance of his work. Jacques Prentki had made things even worse by suggesting not that he resubmit his revised paper to Physics Letters, which ran all the papers it received past independent scientific ‘referees’, but rather to Il Nuovo Cimento, the Italian journal which did not bother with referees at all. Prentki seemed to think the paper was irrelevant and worthless, and that stung.12

  Consequently, Higgs was damned if he was going to follow Prentki’s advice. He resolved instead to send the revised paper not to Physics Letters but to its American rival, Physical Review Letters. However, first he needed to add something, and at last he knew what. He picked up a pen and supplemented his paper with a final, two-sentence paragraph. In the first sentence, he wrote, ‘It is worth noting that an essential feature of the type of theory which has been described in this note is the prediction of incomplete multiplets of scalar and vector bosons.’ What Higgs meant in this highly technical statement was that there would be a particle left over from the symmetry-breaking process. A Goldstone boson with mass, a hitherto unexpected fundamental particle.

  What Higgs did not know was that five other physicists had come to exactly the same conclusion at pretty much the same time. In London there was a ‘gang of three’ consisting of Tom Kibble, Gerry Guralnik and Dick Hagen, and in Brussels there was a ‘gang of two’ consisting of Robert Brout and François Englert. Higgs, as he would later refer to himself, was the ‘gang of one’.

  As Higgs took his amended paper to the departmental secretary to be retyped, he felt a warm glow of satisfaction. He had no idea how it might fit into the framework of particle physics, but he was sure it was important. He also did not have the slightest inkling that the sentence he had added would earn him immortality; in fact, it would win him the Nobel Prize.

  *

  Higgs’ work created no sensation and grabbed no headlines. As Tom Kibble, one of the ‘gang of six’, would later say, ‘Our work was greeted with deafening silence.’ In devising his mechanism for supplying mass to the gauge force carriers, Higgs had been hoping to make sense of the strong nuclear force that binds together the components of the atomic nucleus, but that problem was not ripe for the picking. The strong force was believed to act principally between protons and neutrons, but these were composite particles and the strong force glued together their constituent quarks, something that was only recognised by the physicists Murray Gell-Mann and George Zweig in the mid-1960s.

  If this was not enough of an issue in trying to understand the strong force, there was another fundamental difficulty: it turned out that the short range of the force was not, as might be expected, a result of its force carriers being massive and therefore being conjured only fleetingly from the vacuum. The ‘gluons’ are massless, so the Higgs mechanism, as far as the strong force is concerned, is irrelevant. Nature has thrown us a curve ball and has chosen to use an entirely different mechanism for giving the strong force its short range.

  Just as the restoring force in a piece of elastic becomes stronger the more it is stretched, so the force between two quarks gets stronger the further they are pulled apart. In fact, so much energy must be put into separating a pair of quarks that it conjures into existence the mass-energy of a quark–antiquark pair, which in turn have to be pulled apart, conjuring another quark–antiquark pair into existence … Since it is never possible to completely separate two quarks, the strong force never gets the opportunity to operate over anything but an ultra-short range.

  Local gauge symmetry is maintained in the case of the strong force for the simple reason that the gauge force carriers – the gluons – retain their masslessness. Along with the quarks, they are imprisoned inside protons and neutrons, and permanently concealed from view. Whereas the massless symmetry of the weak force is hidden by symmetry breaking, the massless symmetry of the strong force is hidden by ‘quarks confinement’.

  The upshot of all this is that Higgs, in inventing a mechanism for giving mass to force-carrying particles, had been thinking about the wrong force. The right force was in fact the weak nuclear force, something recognised in the late 1960s by Steven Weinberg in the US and Abdus Salam in the UK. The pair were engaged in a struggle to show that the weak and electromagnetic forces have a common origin.

  Recall that the great triumph of nineteenth-century physics was the discovery by James Clerk Maxwell that the electric and magnetic forces have a common origin.13 But in that case, there had been strong hints that they were related. Hans Christian Ørsted had shown that a changing electric field creates a magnetic field, and Michael Faraday had shown that a changing magnetic field generates an electric field. But with the electromagnetic force and the weak force, it was not at all obvious that there was a connection.

  The electromagnetic force has an infinite reach, whereas the weak force has a range barely of 1 per cent of the diameter of a proton. And whereas the electromagnetic force merely moves charged particles around in space, the weak force can shift electric charge between particles, magically transforming one kind of particle into another – for instance, a neutron into a proton in the process of radioactive beta decay (though what actually happens is the weak force turns one ‘flavour’ of quark – a down-quark – inside a neutron into another flavour of quark, an up-quark).

  The weak force is critically important in the Sun because nuclear reactions which require the weak force are rare. (In the quantum world, weak is synonymous with infrequent.) The rareness of the first step in the chain of sunlight-generating nuclear reactions is the reason the Sun will use up its fuel gradually over ten billio
n years or so and not squander it explosively in one go. This has enabled the Sun to shine steadily for the billions of years necessary for the evolution of complex life. And if this is not enough to be thankful for, the weak force is also of key importance in the nuclear processes inside massive stars that built up elements such as carbon, oxygen and iron that have been crucial to life on Earth.†

  The weak force appears so different from the electromagnetic force that the claim that they have a common origin is a brave one. How in the world can rainbows and radioactive decay be aspects of the same fundamental phenomenon? But this is exactly what Schwinger suggested in 1956.14 In the 1960s, Weinberg, Salam and others set out to demonstrate that he was right. It was in the ‘unification’ of the electromagnetic and weak force into the ‘electroweak’ force that Higgs’ idea proved its worth.

  The weak force’s short range is indeed explained by its gauge force carriers having large masses – almost one hundred times the mass of a proton. Because weak-induced beta decay adds positive electric charge to a neutron to make a proton, there must be a force carrier with a positive charge: this is the W+. And because beta decay can work in reverse, adding a negative charge to a proton to make a neutron, a W– must also exist. The existence of the W+ and W– was, in fact, predicted by Schwinger. For technical reasons, there must also exist a weak force carrier with no electric charge: this is the Z0, predicted by American physicist Sheldon Glashow in 1960. Together, the W+, W– and Z0 are known as the ‘weak vector bosons’.