The quark model was a beautifully simplifying scheme, but in truth it was not much more than the result of playing with the patterns. There was simply no experimental foundation for it. Gell-Mann didn’t help his cause by being rather cagey about the status of the new particles. Wishing to avoid getting tangled in philosophical debates about the reality of particles that could in principle never be seen, he referred to the quarks as ‘mathematical’. Some interpreted this to mean that Gell-Mann didn’t think that the quarks were made of real ‘stuff’, entities that existed in reality and combined to give real effects.
Zweig was bolder (or, depending on your point of view, more reckless). In his second CERN preprint he had declared: ‘There is also the outside chance that the model is a closer approximation to nature than we may think, and that fractionally charged aces abound within us.’5
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Solid-state physicist Philip Anderson didn’t believe Goldstone’s theorem. It was transparently obvious from many practical examples in solid-state physics that Nambu–Goldstone bosons are not always produced when gauge symmetries are spontaneously broken. Symmetries were being broken all the time, yet solid-state physicists were hardly being overwhelmed by floods of massless, photon-like particles as a result. There were no massless particles generated inside superconductors, for example. Something was not quite right.
In 1963, Anderson suggested that the problems that the quantum field theorists were wrestling with could in some way resolve themselves:6
It is likely, then, considering the superconducting analogue, that the way is now open … without any difficulties involving either zero-mass Yang–Mills gauge bosons or zero-mass [Nambu–]Goldstone bosons. These two types of bosons seem capable of ‘canceling each other out’ and leaving finite mass bosons only.
Could it really be that simple? Was this a case of two wrongs making a right? Anderson’s paper provoked a minor controversy. As arguments and counter-arguments raged in the scientific press, a number of physicists took careful note.
There followed a series of papers detailing mechanisms for spontaneous symmetry-breaking in which the various massless bosons did indeed ‘cancel each other out’, leaving only massive particles. These were published independently by Belgian physicists Robert Brout and François Englert, English physicist Peter Higgs at Edinburgh University, and Gerald Guralnik, Carl Hagen, and Tom Kibble at Imperial College in London.* The mechanism is commonly referred to as the Higgs mechanism (or, in some quarters more concerned with the democracy of discovery, the Brout–Englert–Higgs–Hagen–Guralnik–Kibble – BEHHGK, or ‘beck’ mechanism).
The mechanism works like this. A massless field particle with spin 1 (a boson) moves at the speed of light and has two ‘degrees of freedom’, meaning that its wave amplitude can oscillate in two dimensions that are perpendicular (that is, transverse) to the direction in which it is travelling. If the particle is moving in the z-direction, say, then its wave amplitude can oscillate only in the x- and y-directions (left/right and up/down). For the photon, the two degrees of freedom are associated with left-circular and right-circular polarization. These states can be combined to give the more familiar states of linear polarization: horizontal (x-direction) and vertical (y-direction). For light, there is no polarization in a third dimension.
To change this state it is necessary to introduce a background quantum field, often called the Higgs field, to break the symmetry.* The Higgs field is characterized by the shape of its potential energy curve.
The idea of a potential energy curve is relatively straightforward. Picture a pendulum swinging to and fro. As the pendulum rises in its swing it slows down, stops, and swings back the other way. At this point all the energy of its motion (its kinetic energy) has transferred into potential energy stored in the pendulum. As the pendulum swings back, the potential energy is released into the kinetic energy of motion and it picks up speed. At the bottom of its swing, with the pendulum pointing straight down, the kinetic energy is at a maximum and the potential energy is zero.
If we plot the value of the potential energy against the angle of displacement of the pendulum from the vertical, we get a parabola – see Figure 13(a). The minimum in this potential energy curve is clearly the point at which the displacement of the pendulum is zero.
The potential energy curve of the Higgs field is subtly different. Instead of the angle of displacement, we plot the displacement or value of the field itself. Towards the bottom of the curve there is a small bump, not unlike the top of a sombrero or a ‘punt’ (the indentation) in the bottom of a champagne bottle. The presence of this bump forces the symmetry to break. As the field cools and loses its potential energy, like the toppling pencil it randomly falls into a valley in the curve (the curve is actually three-dimensional). But this time the lowest point in the curve corresponds to a non-zero value of the field. Physicists refer to this as a non-zero vacuum expectation value. It represents a ‘false’ vacuum, meaning that the vacuum is not completely empty – it contains non-zero values of the Higgs field.
FIGURE 13 (a) In the case of a simple frictionless pendulum, the potential energy curve is shaped like a parabola and zero potential energy corresponds to zero pendulum displacement. However, the potential energy curve for the Higgs field (b) is shaped differently. Now zero potential energy corresponds to a finite displacement (of the field itself), or what physicists call a non-zero vacuum expectation value.
Breaking the symmetry creates a massless Nambu–Goldstone boson. This may now be ‘absorbed’ by the massless spin 1 field boson to create a third degree of freedom (forward/back). The wave amplitude of the field particle can now oscillate in all three dimensions, including the direction in which it is travelling. The particle acquires ‘depth’ (see Figure 14).
In the Higgs mechanism the act of gaining three-dimensionality is like applying a brake. The particle slows down to an extent which depends on the strength of its interaction with the Higgs field.
The photon does not interact with the Higgs field and continues to move unhindered, at the speed of light. It remains massless. Particles which do interact with the field acquire depth, gain energy, and slow down, the field dragging on them like molasses. The particle’s interactions with the field are manifested as a resistance to the particle’s acceleration.*
Does this sound vaguely familiar?
The inertial mass of an object is a measure of its resistance to acceleration. Our instinct is to equate inertial mass with the amount of substance that the object possesses. The more ‘stuff’ it contains, the harder it is to accelerate. The Higgs mechanism turns this logic on its head. We now interpret the extent to which the particle’s acceleration is resisted by the Higgs field as the particle’s (inertial) mass.
FIGURE 14 (a) A massless boson moves at the speed of light and has just two transverse ‘degrees of freedom’, left/right (x) and up/down (y). On interacting with the Higgs field, the particle may absorb a massless Nambu–Goldstone boson, acquiring a third degree of freedom – forward back (z). Consequently, the particle gains ‘depth’ and slows down. This resistance to acceleration is the particle’s mass.
The concept of mass has vanished in a puff of logic. It has been replaced by interactions between otherwise massless particles and the Higgs field.
The Higgs mechanism did not win converts immediately. Higgs himself had some difficulties getting his paper published. He sent it initially to the European journal Physics Letters in July 1964, but it was rejected by the editor as unsuitable. Years later, Higgs wrote:7
I was indignant. I believed that what I had shown could have important consequences in particle physics. Later, my colleague Squires, who spent the month of August 1964 at CERN, told me that the theorists there did not see the point of what I had done. In retrospect, this is not surprising: in 1964…quantum field theory was out of fashion…
Higgs made some amendments to his paper and resubmitted it to the journal Physical Review Letters. It was sent to N
ambu for peer review. Nambu asked Higgs to comment on the relationship between his paper and an article just published in the same journal (31 August 1964) by Brout and Englert. Higgs had not been aware of Brout and Englert’s work on the same problem and acknowledged their paper in an added footnote. He also added a final paragraph to the main text in which he drew attention to the possibility of ‘incomplete multiplets of scalar and vector bosons’,8 a rather obscure reference to the possibility of another, massive zero-spin boson, the quantum particle of the Higgs field.
This would come to be known as the Higgs boson.
Perhaps surprisingly, the Higgs mechanism had little immediate impact on those who might have benefited most from it.
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Higgs was born in Newcastle upon Tyne, England, in 1929. In 1950 he graduated in physics at King’s College, London and secured his PhD four years later. There followed spells at the University of Edinburgh and University and Imperial Colleges in London. He had returned to the University of Edinburgh in 1960 to take up a lectureship in Mathematical Physics. He married Jody Williamson, a fellow activist with the Campaign for Nuclear Disarmament, in 1963.
In August 1965 Higgs took Jody to Chapel Hill for a sabbatical period at the University of North Carolina. Their first son, Christopher, was born a few months later. Shortly afterwards Higgs received an invitation from Freeman Dyson to present a seminar on the Higgs mechanism at the Institute for Advanced Study. Higgs was wary of the likely reception of his theory at what had become commonly known as the Institutes’ ‘shotgun seminars’, but when he delivered the seminar in March 1966 he emerged unscathed. Pauli had died in December 1958, but it is interesting to speculate if Higgs’ arguments would have changed his attitude to Yang’s unfortunate pleadings, a little over twelve years before.
Higgs took this opportunity to fulfil a long-standing request to give a seminar at Harvard University, and made his way there the next day. The audience was equally sceptical, with one Harvard theorist later admitting that they ‘had been looking forward to tearing apart this idiot who thought he could get around the Goldstone theorem.’9
Glashow was in the audience, but it seems that he had by this time quite forgotten his earlier attempts to develop a unified electro-weak theory, a theory which predicted massless W+, W−, and Z0 particles which needed somehow to be massive. ‘His amnesia unfortunately persisted through 1966,’ Higgs wrote.10 In fairness to Glashow, Higgs was preoccupied with the application of his mechanism to the strong force.
But Glashow failed to put two and two together. It would be Glashow’s former high school classmate Steven Weinberg (and, independently, Abdus Salam) who would eventually make the connection.
After receiving a bachelor’s degree from Cornell University in 1954 Weinberg had begun his graduate studies at the Niels Bohr Institute in Copenhagen, returning to complete his PhD at Princeton University in 1957. He completed postdoctoral studies at Columbia University in New York and at the Lawrence Radiation Laboratory in California, before gaining a professorship at the University of California at Berkeley. He took a leave of absence to become a visiting lecturer at Harvard in 1966 and became a visiting professor at MIT the following year.
Weinberg had spent the previous couple of years working on the effects of spontaneous symmetry breaking in strong-force interactions described by an SU(2)×SU(2) field theory. As Nambu and Jona-Lasinio had found a few years before, the result of symmetry-breaking is that protons and neutrons acquire mass. Weinberg believed that the Nambu–Goldstone bosons so created could be approximated as the pions. At the time this all seemed to make sense and, far from trying to evade the Goldstone theorem, he now positively welcomed the predicted extra particles.
But now Weinberg realized that this approach wasn’t going to bear fruit. It was at this point that he was struck by another idea:11
At some point in the fall of 1967, I think while driving to my office at MIT, it occurred to me that I had been applying the right ideas to the wrong problem.
Weinberg had been applying the Higgs mechanism to the strong force. He now realized that the mathematical structures he had been trying to apply to strong-force interactions were precisely what were needed to resolve the problems with weak-force interactions and the massive bosons that these interactions implied. ‘My God,’ he exclaimed to himself, ‘this is the answer to the weak interaction!’12
Weinberg was well aware that if the masses of the W+, W−, and Z0 particles were added by hand, as in Glashow’s SU(2)×U(1) electro-weak field theory, then the result was rendered unrenormalizable. He now wondered if breaking the symmetry using the Higgs mechanism would endow the particles with mass, eliminate the unwanted Nambu–Goldstone bosons, and yield a theory that could in principle be renormalized.
There remained the problem of the weak neutral currents, interactions involving the neutral Z0 particle for which there was still no experimental support. He decided to avoid this problem altogether by restricting his theory to leptons – electrons, muons, and neutrinos. He had by now become wary of the hadrons, particles affected by the strong force, and especially the strange particles, the principal ground for the experimental exploration of weak-force interactions.
Neutral currents would still be predicted, but in a model consisting only of leptons these currents would involve the neutrino. The neutrino had proved difficult enough to find experimentally in the first place, and Weinberg may have figured that finding weak-force neutral currents involving these particles would present such insurmountable experimental challenges that he could predict them with little fear of contradiction.
Weinberg published a paper detailing an electro-weak unified theory for leptons in November 1967. This was an SU(2)×U(1) field theory reduced to the U(1) symmetry of ordinary electromagnetism by spontaneous symmetry-breaking, giving mass to the W+, W−, and Z0 particles whilst leaving the photon massless. He estimated the mass-scales of the weak-force bosons, about 85 times the proton mass for the W particles and 96 times the proton mass for the Z0. He was not able to prove that the theory was renormalizable but felt confident that it was.
In 1964, Higgs had referred to the possibility of the existence of a Higgs boson, but this was not in relation to any specific force or theory. In his electro-weak theory, Weinberg had found it necessary to introduce a Higgs field with four components. Three of these would give mass to the W+, W−, and Z0 particles. The fourth would appear as a physical particle – the Higgs boson. What had earlier been a mathematical possibility had now become a prediction. Weinberg even estimated the strength of the coupling between the Higgs boson and the electron. The Higgs took a critically important step towards becoming a ‘real’ particle.
In Britain, Abdus Salam had been introduced to the Higgs mechanism by Tom Kibble. He had worked earlier on an SU(2)×U(1) electro-weak field theory and immediately saw the possibilities afforded by spontaneous symmetry-breaking. When he saw a preprint of Weinberg’s paper applying the theory to leptons he discovered that both he and Weinberg had independently arrived at precisely the same model. He decided against publishing his own work until he had had an opportunity properly to incorporate hadrons. But, try as he might, he could not get around the problem of the weak neutral currents.
Both Weinberg and Salam believed that the theory was renormalizable, but neither was able to prove this. They were also unable to predict the mass of the Higgs boson.
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Nobody took much notice. Those few who did pay attention tended to be critical. The mass problem had been fixed through some ‘smoke-and-mirrors’ trick involving a hypothetical field which implied another hypothetical boson. It seemed as though the quantum field theorists were continuing to play games with fields and particles, according to obscure rules that few understood.
Particle physicists simply ignored them and got on with their science.
5
I Can Do That
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In which Gerard ’t Hooft
proves that Yang–Mills field theories can be renormalized and Murray Gell-Mann and Harald Fritzsch develop a theory of the strong force based on quark colour
Aside from the absurd fractional electric charges, there was another big problem with the quark model. As constituents of ‘matter particles’ such as protons and neutrons, the quarks were required to be fermions, with half-integral spins. This meant that, according to Pauli’s exclusion principle, the hadrons could not accommodate more than one quark in each of the possible quantum states.
But the quark model insisted that the proton should consist of two up-quarks and a down-quark. This was a bit like saying that an atomic orbital should contain two spin-up electrons and one spin-down electron. It was just not possible. The symmetry properties of the electron wavefunctions forbade it. There could be only two electrons, one spin-up and one spin-down. There was no room for a third. Likewise, if the quarks were fermions, then there could be no room for two up-quarks in the proton.
This problem had been identified shortly after publication of Gell-Mann’s first quark paper. Physicist Oscar Greenberg suggested in 1964 that quarks might actually be parafermions, which was tantamount to saying that the quarks could be distinguished by other ‘degrees of freedom’ apart from the one for which the quantum numbers were up, down, and strange. As a result there would be different kinds of upquarks, for example. So long as they were of different kinds, two up-quarks could sit happily alongside each other in a proton without occupying the same quantum state.
But there were problems with this model, too. Greenberg’s solution opened the door for baryons to behave like bosons, condensing into a single macroscopic quantum state like a beam of laser light. This was just not acceptable.
Higgs:The invention and discovery of the 'God Particle' Page 8
