Genesis of electroweak
symmetry breaking - 1
Tom Kibble
Imperial College
13 Sep 2012
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Outline — Part 1
Story of idea of spontaneous symmetry breaking in gauge theories
and electroweak unification
— from my viewpoint at Imperial College
• Physics after WW2: QED, renormalization theory
• Abdus Salam
• Models of strong interactions:
gauge theory, symmetry breaking
• The idea of weak-electromagnetic unification
• Obstacles to unification — the Goldstone theorem
Part 2:
• Overcoming the obstacles — the Higgs mechanism
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Physics after WW2
• During the war, physicists had been working on atomic weapons,
radar, operational research, etc.
• After 1945, they went back to fundamental physics, leading to
very rapid developments, initially in the simplest quantum field
theory, Quantum Electrodynamics (QED) — the theory of
interacting electrons and photons.
• Perturbation theory gave excellent results to lowest order in the
fine structure constant
e2
1


4 0 hc 137
• But — higher order corrections were infinite.
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Renormalization
• Solution was found in 1947, independently by Richard Feynman,
Julian Schwinger and (in 1943) by Sin-Itiro Tomonaga
— all the infinities could be collected into infinite renormalization
constants, relating m and e to m0 and e0. For this work, they shared
the Nobel Prize for Physics in 1965.
• In 1948 Freeman Dyson showed that all three approaches were
equivalent, and gave a proof that renormalization worked to all orders.
• Innovative experiments on the Lamb shift and the magnetic moment
of the electron confirmed the results were correct to unprecedented
accuracy.
• There was a gap in Dyson’s proof, however ...
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Abdus Salam at 14
• Abdus Salam was born in 1926, the son of
a minor civil servant living in Jhang near
Lahore in what
is now Pakistan.
• At 14, he won
a scholarship to
Government
College, Lahore
with the highest marks ever recorded,
making the front page of the local paper.
• He published his first paper at 17 — an
improved solution to an algebraic problem
solved by Srinivas Ramanujan.
• At 20 he won a scholarship to Cambridge.
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Salam in Cambridge
• Salam went to Cambridge in 1946 — an outstanding undergraduate.
• He was excited by recent developments in theoretical physics, and
asked to be taken on as a PhD student by Nicholas Kemmer.
• Kemmer had no specific project for him, but suggested an older
student of Kemmer’s, Paul Matthews, might have ideas for a project.
Matthews suggested he try to fill an outstanding gap in Dyson’s proof
that renormalization works to all orders in perturbation theory; it did not
directly deal with the case of
overlapping divergences, e.g. in
• Matthews returned from a brief holiday to find Salam had
completely solved the problem! This work gained him an instant
international reputation, with an invitation to Princeton.
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Strong interaction models
• The success of QED inspired physicists to look for similar theories of
strong and weak interactions.
— or even better, a unified theory of all of them.
• Initially, strong interactions attracted most interest.
• The best guess as field theory of strong interactions was Hideki
Yukawa’s meson theory: pions as force carriers
 p 
Lint  ig 5  
    ,   (  , 0 ,  )
 n 
— Salam and Matthews (at Princeton) showed that too could be
renormalized.
• But there was a big problem
— no one could make any calculations for a model with g ~ 1.
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Field theory vs S-matrix theory
• Problem with a field theory of strong interactions: perturbation theory
calculations are impossible with a coupling constant ~1.
• During 1950s, many people concluded that field theory had had its day
— the new rage was S-matrix theory, based on analytic properties of
scattering amplitudes, especially Regge poles.
• Many people thought there were no elementary hadrons
— all were bound states of each other – the self-consistent bootstrap.
• But in a few places, the flag of field theory was kept flying
— Imperial College, Harvard, ... .
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Imperial College ca. 1960
Imperial College theoretical physics group was founded in 1956 by
Abdus Salam — in 1959 he became the youngest FRS at age 33
• I arrived in 1959
• 3 permanent faculty:
— Abdus Salam
— Paul Matthews
— John C Taylor
• I joined faculty in 1961
• Numerous visitors:
Murray Gell-Mann,
Stanley Mandelstam,
Steven Weinberg,
Kenneth Johnson, Art Rosenfeld, ...
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Gauge theories
• First example of a gauge theory beyond QED was the Yang-Mills
theory (1954), gauged SU(2)
— intended as a theory of strong interactions, with SU(2)
representing isospin
— same theory also proposed by Salam’s student Ronald Shaw, but
unpublished except as a Cambridge University PhD thesis.
 p 
  
 n 
L  i  D  m  41 F  F 
D    21 igA  
F    A   A  gA  A
• Although this ultimately proved not to be the correct theory of strong
interactions, it was the model for all subsequent gauge theories.
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Gauge theories at Imperial
• Salam was convinced from an early stage that a unified theory of all
interactions should be a gauge theory.
• There was a lot of interest in gauge theories at Imperial College
— my own first involvement in 1961 was to show how gravity could be
viewed as a gauge theory of the Poincaré group
— but not a renormalizable one.
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The particle zoo
• Experimental particle physics grew fast
— cosmic ray observation with cloud chambers
— bubble chamber experiments with particle accelerators
• Discovered a huge number of new particles
— could they all be elementary?
• Search for symmetries
— particles arranged in multiplets, related by symmetries
— SU(2) isospin (Heisenberg, Kemmer)
— SU(3) eightfold way (Murray Gell-Mann, Yuval Ne’eman)
— ...
• Now understood in terms of quarks: SU(2) symmetry of (u,d),
SU(3) of (u,d,s)
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Broken symmetries
• These were approximate symmetries, therefore broken in some way
— spontaneously?
• Spontaneous breaking of gauge symmetry, giving mass to the plasmon,
was known (not fully understood) in superconductivity. Nambu (1960)
suggested a similar mechanism could give masses to elementary
particles.
• Nambu and Jona-Lasinio (1961) proposed a specific model
Lint  g[( )2  ( 5 )2 ]
i
— phase symmetry   e  is exact

— chiral symmetry   e 5  is spontaneously broken
  0  m  0
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Unification
• Because of the difficulty of calculating with a theory with large coupling
constant, interest began to shift towards the weak interactions.
• A very important step was the discovery that the weak four-fermion
interactions involved V and A rather than S, T or P.
• V–A theory proposed by Marshak & Sudarshan (1957) and by
Feynman & Gell-Mann (1958)
• This meant that the weak interactions could be seen as due to the
exchange of spin-1 W± bosons. This made them seem very similar to
electromagnetic interactions mediated by photons.
— So the question arose:
could there be a unified theory of weak and electromagnetic
interactions?
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Similarity and Dissimilarity
Electromagnetic
interaction
Weak
interaction
exchange of
spin-1 
exchange of
spin-1 W±
But
long range
 M  0
short range
 MW large
parity conserving
parity violating
So: Can there be a symmetry relating  and W±?
If so it must be broken
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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)
— though neither model used the correct representation of leptons.
• But gauge bosons are naturally massless, and in all these models
symmetry breaking, giving the W bosons masses, had to be inserted
by hand.
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Massive vector bosons
• Gauge theories naturally predicted massless vector bosons.
• If masses were added by an explicit symmetry-breaking term, then the
vector-meson propagator would not be

k  k 
ig 
i
g  2 
but rather
2
2  
2
k m 
m 
k
• Thus we have a much worse divergence, and the theory is clearly not
renormalizable.
• So the question started to be asked: could the symmetry breaking that
gives rise to vector boson masses be spontaneous symmetry breaking?
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Spontaneous Symmetry Breaking
• Particle physics exhibited many approximate symmetries
— it was natural to ask whether they could be spontaneously broken
• Spontaneous breaking of symmetry occurs when the ground state
or vacuum state does not share the symmetry of the underlying
theory. It is ubiquitous in condensed matter physics
• Often there is a high-temperature symmetric phase, and a critical
temperature below which the symmetry is spontaneously broken
— crystallization of a liquid breaks rotational symmetry
— so does Curie-point transition in a ferromagnet
— gauge symmetry is broken in a superconductor
• Could this work in particle physics too?
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Nambu-Goldstone bosons
• But there was a big problem — the Goldstone theorem:
in many cases, spontaneous beaking of a continuous symmetry
implies the existence of massless spin-0 bosons, none of which
had ever been seen.
• This happens because of the existence of degenerate vacuum states,
labelled by a continuous parameter  . We can consider an excitation
in which this parameter varies spatially. Since the different vacuum
states have the same energy, the only
energy cost comes from the
2
gradient terms, giving E   2 
tending to zero in the longwavelength limit k  0. This implies a massless excitation.
 
• For example the Nambu–Jona-Lasinio model has a massless
pseudoscalar, identified with the pion
— N & J-L suggested chiral symmetry was not quite exact even before
spontaneous symmetry breaking, hence pion has a small mass
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Impasse
• Counter-examples to Goldstone were known in condensed matter
• When Steven Weinberg spent a sabbatical at Imperial in 1962, he and
Salam constructed a proof for all relativistic theories (Goldstone, Salam
& Weinberg 1962).
• In a relativistic theory, there seemed to be no escape
— spontaneous symmetry breaking implied the existence of
zero-mass spinless bosons
— since no such bosons had been seen, spontaneous symmetry
breaking was ruled out
— other models with explicit symmetry breaking were clearly
divergent, giving infinite results
• Weinberg commented:
‘Nothing will come of nothing; speak again!’ (King Lear)
Solution tomorrow!
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