Spontaneous symmetry breaking in gauge theories

1. Spontaneous symmetry breakingin gauge theories Tom Kibble 20 Jan 2014 Electroweak symmetry breaking Jan 2014

2. Symmetry • Symmetry helps to solve problems in physics • But this doesn’t always work — the symmetry may be spontaneously broken This talk is the story of how this idea of spontaneous symmetry breaking came to play a key role in one part of the standard model of particle physics — as I saw it from my viewpoint at Imperial College Electroweak symmetry breaking Jan 2014

3. Imperial College Theory Group • I joined the IC theoretical physics group in 1959, the same year its founder, Abdus Salam, became the youngest FRS at age 33. • It was very lively, with numerous visitors: Murray Gell-Mann, Stanley Mandelstam, Steven Weinberg, ... • Interests in — quantum field theory — symmetries Electroweak symmetry breaking Jan 2014

4. Quantum Field Theory • In classical physics, wave fields (e.g. electric and magnetic fields) and particles are very different — in quantum physics, they are the same — described by quantum fields • Bosons (force carriers) have spin where — e.g. photon has spin — we call it a spin-1 particle • Fermions (constituents of matter)have spin — electrons, protons, neutrons have spin (spin- particles) — fermions obey the Pauli exclusion principle: two fermions cannot occupy the same quantum state, the foundation of atomic structure • Interactions are described by quantum field theory Electroweak symmetry breaking Jan 2014

5. Quantum Electrodynamics • First quantum field theory was Quantum Electrodynamics (QED), the theory of interacting electrons and photons, developed in 1930s. • Method of calculation — perturbation theory — involved a power series expansion in powers of the fine structure constant (e = proton charge = – electron charge) • Lowest-order calculations gave excellent results • But — higher order corrections were all infinite. • Solution — renormalization — was found in 1947, independently by Richard Feynman, Julian Schwinger and (in 1943) by Sin-Itiro Tomonaga • Led to amazingly accurate predictions of electron magnetic moment and Lamb shift (2s1/2 – 2p1/2 energy difference in H), etc. Electroweak symmetry breaking Jan 2014

6. What next? • We distinguish four types of interaction strong nuclear short-range coupling strength ~ 1 electromagnetic long-range strength = ~ 10–2 weak nuclear short-range strength ~ 10–10 gravitational long-range strength ~ 10–40 • After QED’s success, people searched for field theories of other interaction (or even better, a unified theory of all of them). • Most interest in strong interactions — there were candidate field theories, but no one could calculate with them because perturbation theory doesn’t work if the ‘small’parameter is ~ 1. • So perhaps weak interactions were more promising? Electroweak symmetry breaking Jan 2014

7. Symmetries • A huge ‘zoo’of particles were discovered, which could obviously be grouped into families, related by approximate symmetries • Isospin. Heisenberg suggested the very similar proton (p) and neutron (n) could be seen as two states of a nucleon (N: N+ = p, N0 = n), with a symmetry that ‘rotates’ one into the other, likespatial rotations that rotate the spin states of an electron — an SU(2) symmetry — now understood as a symmetry between two lightest quarks (u,d) • Eightfold way. Strongly interacting particles (hadrons) could be grouped into octets and decuplets. Gell-Mann (1961)and Ne’eman(1961)showed that this could be explained by a more approximate SU(3) symmetry — now understood as a symmetry of the three lightest quarks (u,d,s). Electroweak symmetry breaking Jan 2014

8. Gauge theories • Quantum mechanics has a global ‘phase’ symmetry — physics doesn’t change when we make the same ‘phase rotation’ of the wave function everywhere — mathematically, a U(1) symmetry • QED is a gauge theory — it has a special local U(1) symmetry — this means we can make different‘rotations’ at different points of space and time, but it only works if there is an electromagnetic field • This idea of replacing global by local symmetry — the gauge principle — gives a rationale for the existence of the electromagnetic field, and for the zero rest-mass of the photon. • This idea could be applied elsewhere. There was a lot of interest at Imperial College. Salam was convinced from an early stage that a unified theory of all interactions should be a gauge theory. Electroweak symmetry breaking Jan 2014

9. Larger gauge theories • First example of a gauge theory beyond QED was the Yang-Mills theory (1954), a gauge theory of isospin SU(2) symmetry. — same theory also proposed by Salam’s student Ronald Shaw, but unpublished except as a Cambridge University PhD thesis — ultimately not correct theory of strong interactions, but the foundation for all later gauge theories. • Because isospin is an approximate symmetry, this symmetry must be broken in some way — but adding symmetry-breaking terms destroys many of the nice properties of gauge theories. • Because of the difficulty of calculating with a strong-interaction theory, interest began to shift to weak interactions, especially after it was found that they could be explained as proceeding via exchange of bosons W± with spin 1 (units of ), like the photon ( ). — could there be a unified theory of weak and electromagnetic? Electroweak symmetry breaking Jan 2014

10. Similarity and Dissimilarity Weak interaction Electromagnetic interaction exchange of spin-1 W± exchange of spin-1 . But long range short range large parity violating (non-mirror-symmetric) parity conserving (mirror-symmetric) So: Can there be a symmetry relatingandW±? If so it must be broken Electroweak symmetry breaking Jan 2014

11. Early Unified Models • The first suggestion of a gauge theory of weak interactions mediated by W+ and W– was by Schwinger (1956), who suggested there might be an underlying unified theory, incorporating also the photon. • Glashow(1961) proposed a model with symmetry group SU(2) x U(1) and a fourth ‘gauge boson’ Z0, showing that the parity problem could be solved by a mixing between the two neutral gauge bosons. • Salam and Ward(1964), unaware of Glashow’s work, proposed a similar model, also based on SU(2) x U(1). • Butin all these models symmetry breaking, giving the W bosons masses, had to be inserted by hand — and models with spin-1 bosons with explicit masses were known to be non-renormalizable. • Big question: could this be a spontaneously broken symmetry? (first suggested by YoichiroNambu) Electroweak symmetry breaking Jan 2014

12. Nambu-Goldstone bosons • Spontaneous breaking of a continuous symmetry existence of massless spin-0 Nambu-Goldstone bosons. • e.g. Goldstone model: complex scalar field with — vacuum breaks symmetry: — choose and set cubic and quartic terms So (NG boson) • NG bosons correspond to spatial oscillations in Electroweak symmetry breaking Jan 2014

13. Goldstone Theorem • There were known (not well understood) counter-examples in condensed matter, e.g. superconductivity (Nambu 1960, Philip Anderson 1963). • That NG bosons are massless in any relativistic theory was apparently proved by Weinberg & Salam, published with Goldstone (1962) • No observed massless scalars no spontaneous breaking of a continuous symmetry ! • Weinberg: ‘Nothing will come of nothing; speak again!’ (King Lear) • I was very interested when in 1964 Gerald Guralnik (a student of Walter Gilbert, who had been a student of Salam) arrived at Imperial College as a postdoc to find that he had been studying this problem, and already published some ideas about it. We began collaborating, with another US visitor, Carl Richard Hagen. We (and others) solved problem. Electroweak symmetry breaking Jan 2014

14. Mass generation mechanism • Solution was found by three groups — Englert & Brout (1964), Higgs (1964), Guralnik, Hagen & TK (1964) — gauge theories are not like other field theories: masslessness of Nambu–Goldstone bosons and gauge bosons ‘cancels out’, combining to create massive gauge bosons. • All three proposed (from different viewpoints) essentially the same model for spontaneous symmetry breaking in the simplest U(1) gauge theory, i.e. a broken version of electrodynamics — it involves introducing a new scalar (spin-0) field, with ‘sombrero’ potential (as in Goldstone model) — spontaneous symmetry breaking occurs when this field acquires a non-zero average value — this gives mass to other fields it interacts with, in particular the gauge bosons. Electroweak symmetry breaking Jan 2014

15. Electroweak unification • The three papers on the Higgs mechanism attracted very little attention at the time. The boson attracted even less interest. • By 1964 both the mechanism and Glashow’s (and Salam and Ward’s) SU(2) x U(1) model were in place, but it still took three more years to put the two together. • Further work on the detailed application of the mechanism to non-abelian theories (TK, 1967). This work helped, I believe, to renew Salam’s interest. • Unified model of weak and electromagnetic interactions of leptons proposed by Weinberg(1967) — essentially the same model was presented independently by Salamin lectures at IC in autumn of 1967 and published in a Nobel symposium in 1968 — he called it theelectroweak theory. Electroweak symmetry breaking Jan 2014

16. Later developments • Both Salam and Weinberg speculated that their theory was renormalizable. This was proved by Gerard ’t Hooftin 1971 —a tour de force using methods developed by his supervisor, TiniVeltman, especially the computer algebra programmeSchoonship. • In 1973 the key prediction of the theory, the existence of neutral current interactions — those mediated by Z0 — was confirmed at CERN. • This led to the Nobel Prize for Glashow, Salam & Weinberg in 1979 — but Ward was left out (because of the ‘rule of three’?). • In 1983 the W and Z particles were discovered at CERN — then the Higgs boson became important (last missing piece). •’t Hooftand Veltman gained their Nobel Prizes in 1999. Electroweak symmetry breaking Jan 2014

17. The Higgs boson • In 1964, the Higgs boson had been a very minor and uninteresting feature of the mechanism — the key point was the mechanism for giving the gauge bosons masses and escaping the Goldstone theorem. • But after 1983 it started to assume a key importance as the only missing piece of the standard-model jigsaw. The standard model worked so well that the boson (or something else doing the same job) more or less had to be present. • Finding the Higgs was one of the main objectives of the LHC — this succeeded triumphantly in 2012 — led in 2013 to Nobel Prizes for Englert and Higgs Electroweak symmetry breaking Jan 2014

18. Giving mass to particles • How does the Higgs field give mass to the gauge bosons? — the nonzero vacuum value of is equivalent to a sea of Higgs bosons (a ‘condensate’) • The mechanism is similar to refraction: — when a photon enters a plasma, with free electrons, it is slowed — interaction with the electrons gives it an effective mass • The Higgs also gives masses to the other particles it interacts with, e.g. the electron — but most of the mass of the proton or neutron comes from the gluons that bind these particles in the nucleus, by a quite different mechanism Electroweak symmetry breaking Jan 2014

19. The End • I wish to conclude by acknowledging my huge debt to my mentor and inspiration, Abdus Salam • He was a brilliant physicist, an inspiring leader, a skilled diplomat, and a warm and generous man. • It was a very sad loss when he died prematurely in 1996. Electroweak symmetry breaking Jan 2014