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Spontaneous symmetry
breaking
in gauge theories
Tom Kibble
20 Jan 2014
Electroweak symmetry breaking
Jan 2014
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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
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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
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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
h
• Bosons (force carriers) have spin 0, ,2 , where
=
2p
— e.g. photon has spin
— we call it a spin-1 particle
• Fermions (constituents of matter) have spin 21 , 32 ,
— electrons, protons, neutrons have spin 21 (spin- 21 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
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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
e2
1
a=
»
= – electron charge)
2e 0hc 137
• 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.
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What next?
• We distinguish four types of interaction
strong nuclear
short-range
coupling strength ~ 1
electromagnetic
long-range
strength = a ~ 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?
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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, like spatial 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).
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Gauge theories
• Quantum mechanics has a global ‘phase’ symmetry
— physics doesn’t change when we make the same ‘phase rotation’
ia
of the wave function everywhere y (x) ® y (x)e
— 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
ia (x)
space and time, y (x) ® y (x)e
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.
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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 ( g ).
— could there be a unified theory of weak and electromagnetic?
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Similarity and Dissimilarity
Electromagnetic
interaction
Weak
interaction
exchange of
spin-1 .g
exchange of
spin-1 W±
But
long range
Þ Mg = 0
short range
Þ MW large
parity violating
parity conserving
(non-mirror-symmetric)
(mirror-symmetric)
So: Can there be a symmetry relating g 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).
• 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.
• Big question: could this be a spontaneously broken symmetry?
(first suggested by Yoichiro Nambu)
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Nambu-Goldstone bosons
• Spontaneous breaking of a continuous symmetry Þ existence of
massless spin-0 Nambu-Goldstone bosons.
• e.g. Goldstone model: complex scalar field
f with
— vacuum breaks symmetry:
0f 0 =
h
e ia — choose a = 0
2
1
f
=
(h + j1 + ij 2 )
and set
2
cubic and quartic terms
So m12 = lh 2 , m22 = 0 (NG boson)
• NG bosons correspond to spatial
oscillations in a
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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 Þ 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.
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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.
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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.
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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 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 Hooft and Veltman gained their Nobel Prizes in 1999.
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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
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Giving mass to particles
• How does the Higgs field give mass to the gauge bosons?
— the nonzero vacuum value of f 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
2
w pl
e
ne
2
w
=
mg = 2 ,
pl
e 0me
c
• 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
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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.
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