Birth of electroweak theory from an Imperial perspective

Transcription

1 Birth of electroweak theory from an Imperial perspective Tom Kibble King s College London 2 Oct 2012 Electroweak theory Oct

2 Outline Story of spontaneous symmetry breaking in gauge theories and electro-weak unification from viewpoint of Imperial College Physics at Imperial around 1960 Gauge theories the idea of unification Obstacles to unification Goldstone theorem The Higgs mechanism Electroweak unification Later developments Electroweak theory Oct

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 hadronic symmetries, field theory (not just S-matrix theory), especially gauge theories my first contribution in 1961 was to show how gravity could be seen as a gauge theory of the Poincaré group. Electroweak theory Oct

4 Early gauge theories SU(2) gauge theory (of strong interactions) Yang & Mills 1954 also Ronald Shaw, student of Salam s (Cambridge thesis) Calculations impossible in theory with g ~ 1, so interest shifted to weak interactions, especially after V A theory Marshak & Sudarshan (1957), Feynman & Gell-Mann (1958) Schwinger (1956) first suggesed a unified gauge theory of weak and electromagentic interactions mediated by W ± and γ. Glashow (1961) proposed SU(2) x U(1) model with a fourth gauge boson Z 0, to solve parity problem. Salam and Ward (1964), unaware of Glashow s work, proposed a similar model, also based on SU(2) x U(1). But in 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. Electroweak theory Oct

5 Nambu-Goldstone bosons Spontaneous breaking of a continuous symmetry massless spin-0 Nambu-Goldstone bosons. e.g. Goldstone model L = µ φ * µ φ V V = 1 λ(φ * φ η2 ) 2 vacuum breaks symmetry: 0 φ 0 = η 2 eiα choose α = 0 and set φ = 1 2 (η + ϕ 1 + iϕ 2 ) existence of V = 1 2 λη2 ϕ cubic and quartic terms So m 1 2 = λη 2, m 2 2 = 0 (Goldstone boson) This was believed inevitable in a relativistic theory Electroweak theory Oct

6 Goldstone Theorem (Goldstone, Salam & Weinberg 1962): assume µ j µ = 0 and 0 δφ(0) 0 = iε d 3 x 0 [φ(0), j 0 (0,x)] 0 0 Now µ j µ = 0 would seem to imply dq dt = 0, Q = d 3 x j 0 (x) The broken symmetry condition is then i 0 "# φ(0),q$ % 0 = η 0 But if Q is time-independent, the only intermediate states that can contribute are zero-energy states which can only appear if there are massless particles (or other vacua? see later). No observed massless scalars continuous symmetry! no spontaneous breaking of a Weinberg: Nothing will come of nothing; speak again! (King Lear) Electroweak theory Oct

7 Impasse? There were known (not well understood) counter-examples in condensed matter, e.g. superconductivity (Philip Anderson 1963). But this was believed impossible in a relativistic theory because of the Goldstone theorem. There was a lot of debate (at IC and elsewhere) about how to avoid this impasse 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, Richard Hagen. Electroweak theory Oct

8 Higgs mechanism The argument fails in the case of a gauge theory, e.g. scalar electrodynamics. Englert & Brout (1964), Higgs (1964), Guralnik, Hagen & TK (1964) Higgs model (gauged Goldstone model): L = D µ φ * D µ φ 1 4 F µν F µν V D µ φ = µ φ + iea µ φ F µν = µ A ν ν A µ again set φ = 1 2 (η + ϕ 1 + iϕ 2 ) B µ = A µ + 1 eη µ ϕ 2 V = 1 2 λ(φ * φ 1 2 η2 ) 2 F µν = µ B ν ν B µ L = 1 2 µ ϕ 1 µ ϕ F µν F µν 1 2 λη2 ϕ e2 η 2 B µ B µ + cubic terms... Thus the massless gauge and Goldstone bosons have combined to give a massive gauge boson. But: there is more to it. Electroweak theory Oct

9 Field equations are also satisfied for any Gauge modes µ F µν = j ν = e 2 η 2 B ν + so long as With B the Coulomb gauge condition k µ = 0 A k = 0 requires ϕ 2 = 0 (or constant) However the Lorentz gauge condition µ A µ = 0 ϕ satisfy µ 2 µ ϕ 2 = 0 ϕ 2 (gauge invariance of original model) B µ = A µ + 1 eη µ ϕ 2 = 0 To tie down not only B but also A µ and ϕ µ 2, we need to impose a gauge condition: only requires that in this manifestly covariant gauge, the Goldstone theorem does apply, but the Goldstone boson is a pure gauge mode. Electroweak theory Oct

10 How is the Goldstone theorem avoided? This is the particular question asked by Guralnik, Hagen & TK What was wrong with the proof? we assumed µ j µ = 0 dq dt = 0, Q = d 3 x j 0 (x) But this is true only if we can drop a surface integral of j k at infinity justifiable if commutators vanish at spacelike separation, as in manifestly Lorentz-invariant theory, e.g. Lorentz-gauge QED but not in Coulomb gauge QED In fact when there is spontaneous symmetry breaking, Q does not exist Electroweak theory Oct

11 Unitary inequivalence When the symmetry is spontaneously broken, the integral Q = d 3 x j 0 (x) e.g. here does not exist as a self-adjoint operator so does not define a conserved charge operator. We chose 0 φ 0 = η / 2 but could have chosen α φ α = ηe iα / 2 If Q existed, we would expect Q = e 2 η 2 e iαq 0 = α d 3 x B 0 (x) + But in fact 0 α = 0 and indeed 0 φ(x 1 ) φ(x n ) α = 0 for α 0 0 and α belong to orthogonal Hilbert spaces, carrying unitarily inequivalent representations of the canonical commutation relations This is a defining property of spontaneous symmetry breaking Electroweak theory Oct

12 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 nonabelian 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 Salam in lectures at IC in autumn of 1967 and published in a Nobel symposium in 1968 he called it the electroweak theory. Electroweak theory Oct

13 Later developments Both Salam and Weinberg speculated that their theory was renormalizable. This was proved by Gerard t Hooft in 1971 a tour de force using methods developed by his supervisor, Tini Veltman, especially the computer algebra programme Schoonship. In 1973 the key prediction of the theory, the existence of neutral current interactions those mediated by Z 0 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 Hooft and Veltman gained their Nobel Prizes in Finally in 2012 the Higgs boson was (almost surely) found at LHC a fantastic achievement by the two great collaborations! Electroweak theory Oct

14 I owe my involvement to my mentor and inspiration, Abdus Salam Electroweak theory Oct