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**Spontaneous symmetry breaking** is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state.^{ [1] }^{ [2] }^{ [3] } In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry.

In explicit symmetry breaking, if two outcomes are considered, the probability of a pair of outcomes can be different. By definition, spontaneous symmetry breaking requires the existence of a symmetric probability distribution—any pair of outcomes has the same probability. In other words, the underlying laws^{[ clarification needed ]} are invariant under a symmetry transformation.

The system, as a whole^{[ clarification needed ]}, changes under such transformations.

Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions can be described by spontaneous symmetry breaking. Notable exceptions include topological phases of matter like the fractional quantum Hall effect.

Consider a symmetric upward dome with a trough circling the bottom. If a ball is put at the very peak of the dome, the system is symmetric with respect to a rotation around the center axis. But the ball may *spontaneously break* this symmetry by rolling down the dome into the trough, a point of lowest energy. Afterward, the ball has come to a rest at some fixed point on the perimeter. The dome and the ball retain their individual symmetry, but the system does not.^{ [4] }

In the simplest idealized relativistic model, the spontaneously broken symmetry is summarized through an illustrative scalar field theory. The relevant Lagrangian of a scalar field , which essentially dictates how a system behaves, can be split up into kinetic and potential terms,

**(1)**

It is in this potential term that the symmetry breaking is triggered. An example of a potential, due to Jeffrey Goldstone ^{ [5] } is illustrated in the graph at the left.

- .
**(2)**

This potential has an infinite number of possible minima (vacuum states) given by

- .
**(3)**

for any real *θ* between 0 and 2*π*. The system also has an unstable vacuum state corresponding to *Φ* = 0. This state has a U(1) symmetry. However, once the system falls into a specific stable vacuum state (amounting to a choice of *θ*), this symmetry will appear to be lost, or "spontaneously broken".

In fact, any other choice of *θ* would have exactly the same energy, implying the existence of a massless Nambu–Goldstone boson, the mode running around the circle at the minimum of this potential, and indicating there is some memory of the original symmetry in the Lagrangian.

- For ferromagnetic materials, the underlying laws are invariant under spatial rotations. Here, the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking. Below the Curie temperature, however, the magnetization acquires a constant nonvanishing value, which points in a certain direction (in the idealized situation where we have full equilibrium; otherwise, translational symmetry gets broken as well). The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which do not and are thus spontaneously broken.
- The laws describing a solid are invariant under the full Euclidean group, but the solid itself spontaneously breaks this group down to a space group. The displacement and the orientation are the order parameters.
- General relativity has a Lorentz symmetry, but in FRW cosmological models, the mean 4-velocity field defined by averaging over the velocities of the galaxies (the galaxies act like gas particles at cosmological scales) acts as an order parameter breaking this symmetry. Similar comments can be made about the cosmic microwave background.
- For the electroweak model, as explained earlier, a component of the Higgs field provides the order parameter breaking the electroweak gauge symmetry to the electromagnetic gauge symmetry. Like the ferromagnetic example, there is a phase transition at the electroweak temperature. The same comment about us not tending to notice broken symmetries suggests why it took so long for us to discover electroweak unification.
- In superconductors, there is a condensed-matter collective field ψ, which acts as the order parameter breaking the electromagnetic gauge symmetry.
- Take a thin cylindrical plastic rod and push both ends together. Before buckling, the system is symmetric under rotation, and so visibly cylindrically symmetric. But after buckling, it looks different, and asymmetric. Nevertheless, features of the cylindrical symmetry are still there: ignoring friction, it would take no force to freely spin the rod around, displacing the ground state in time, and amounting to an oscillation of vanishing frequency, unlike the radial oscillations in the direction of the buckle. This spinning mode is effectively the requisite Nambu–Goldstone boson.
- Consider a uniform layer of fluid over an infinite horizontal plane. This system has all the symmetries of the Euclidean plane. But now heat the bottom surface uniformly so that it becomes much hotter than the upper surface. When the temperature gradient becomes large enough, convection cells will form, breaking the Euclidean symmetry.
- Consider a bead on a circular hoop that is rotated about a vertical diameter. As the rotational velocity is increased gradually from rest, the bead will initially stay at its initial equilibrium point at the bottom of the hoop (intuitively stable, lowest gravitational potential). At a certain critical rotational velocity, this point will become unstable and the bead will jump to one of two other newly created equilibria, equidistant from the center. Initially, the system is symmetric with respect to the diameter, yet after passing the critical velocity, the bead ends up in one of the two new equilibrium points, thus breaking the symmetry.

In particle physics, the force carrier particles are normally specified by field equations with gauge symmetry; their equations predict that certain measurements will be the same at any point in the field. For instance, field equations might predict that the mass of two quarks is constant. Solving the equations to find the mass of each quark might give two solutions. In one solution, quark A is heavier than quark B. In the second solution, quark B is heavier than quark A *by the same amount*. The symmetry of the equations is not reflected by the individual solutions, but it is reflected by the range of solutions.

An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. "Hidden" is a better term than "broken", because the symmetry is always there in these equations. This phenomenon is called *spontaneous* symmetry breaking (SSB) because *nothing* (that we know of) breaks the symmetry in the equations.^{ [6] }^{:194–195}

Chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of the strong interactions in particle physics. It is a property of quantum chromodynamics, the quantum field theory describing these interactions, and is responsible for the bulk of the mass (over 99%) of the nucleons, and thus of all common matter, as it converts very light bound quarks into 100 times heavier constituents of baryons. The approximate Nambu–Goldstone bosons in this spontaneous symmetry breaking process are the pions, whose mass is an order of magnitude lighter than the mass of the nucleons. It served as the prototype and significant ingredient of the Higgs mechanism underlying the electroweak symmetry breaking.

The strong, weak, and electromagnetic forces can all be understood as arising from gauge symmetries. The Higgs mechanism, the spontaneous symmetry breaking of gauge symmetries, is an important component in understanding the superconductivity of metals and the origin of particle masses in the standard model of particle physics. One important consequence of the distinction between true symmetries and *gauge symmetries*, is that the spontaneous breaking of a gauge symmetry does not give rise to characteristic massless Nambu–Goldstone physical modes, but only massive modes, like the plasma mode in a superconductor, or the Higgs mode observed in particle physics.

In the standard model of particle physics, spontaneous symmetry breaking of the SU(2) × U(1) gauge symmetry associated with the electro-weak force generates masses for several particles, and separates the electromagnetic and weak forces. The W and Z bosons are the elementary particles that mediate the weak interaction, while the photon mediates the electromagnetic interaction. At energies much greater than 100 GeV, all these particles behave in a similar manner. The Weinberg–Salam theory predicts that, at lower energies, this symmetry is broken so that the photon and the massive W and Z bosons emerge.^{ [7] } In addition, fermions develop mass consistently.

Without spontaneous symmetry breaking, the Standard Model of elementary particle interactions requires the existence of a number of particles. However, some particles (the W and Z bosons) would then be predicted to be massless, when, in reality, they are observed to have mass. To overcome this, spontaneous symmetry breaking is augmented by the Higgs mechanism to give these particles mass. It also suggests the presence of a new particle, the Higgs boson, detected in 2012.

Superconductivity of metals is a condensed-matter analog of the Higgs phenomena, in which a condensate of Cooper pairs of electrons spontaneously breaks the U(1) gauge symmetry associated with light and electromagnetism.

Most phases of matter can be understood through the lens of spontaneous symmetry breaking. For example, crystals are periodic arrays of atoms that are not invariant under all translations (only under a small subset of translations by a lattice vector). Magnets have north and south poles that are oriented in a specific direction, breaking rotational symmetry. In addition to these examples, there are a whole host of other symmetry-breaking phases of matter — including nematic phases of liquid crystals, charge- and spin-density waves, superfluids, and many others.

There are several known examples of matter that cannot be described by spontaneous symmetry breaking, including: topologically ordered phases of matter, such as fractional quantum Hall liquids, and spin-liquids. These states do not break any symmetry, but are distinct phases of matter. Unlike the case of spontaneous symmetry breaking, there is not a general framework for describing such states.^{ [8] }

The ferromagnet is the canonical system that spontaneously breaks the continuous symmetry of the spins below the Curie temperature and at *h* = 0, where *h* is the external magnetic field. Below the Curie temperature, the energy of the system is invariant under inversion of the magnetization *m*(**x**) such that *m*(**x**) = −*m*(−**x**). The symmetry is spontaneously broken as *h* → 0 when the Hamiltonian becomes invariant under the inversion transformation, but the expectation value is not invariant.

Spontaneously-symmetry-broken phases of matter are characterized by an order parameter that describes the quantity which breaks the symmetry under consideration. For example, in a magnet, the order parameter is the local magnetization.

Spontaneous breaking of a continuous symmetry is inevitably accompanied by gapless (meaning that these modes do not cost any energy to excite) Nambu–Goldstone modes associated with slow, long-wavelength fluctuations of the order parameter. For example, vibrational modes in a crystal, known as phonons, are associated with slow density fluctuations of the crystal's atoms. The associated Goldstone mode for magnets are oscillating waves of spin known as spin-waves. For symmetry-breaking states, whose order parameter is not a conserved quantity, Nambu–Goldstone modes are typically massless and propagate at a constant velocity.

An important theorem, due to Mermin and Wagner, states that, at finite temperature, thermally activated fluctuations of Nambu–Goldstone modes destroy the long-range order, and prevent spontaneous symmetry breaking in one- and two-dimensional systems. Similarly, quantum fluctuations of the order parameter prevent most types of continuous symmetry breaking in one-dimensional systems even at zero temperature. (An important exception is ferromagnets, whose order parameter, magnetization, is an exactly conserved quantity and does not have any quantum fluctuations.)

Other long-range interacting systems, such as cylindrical curved surfaces interacting via the Coulomb potential or Yukawa potential, have been shown to break translational and rotational symmetries.^{ [9] } It was shown, in the presence of a symmetric Hamiltonian, and in the limit of infinite volume, the system spontaneously adopts a chiral configuration — i.e., breaks mirror plane symmetry.

Dynamical symmetry breaking (DSB) is a special form of spontaneous symmetry breaking in which the ground state of the system has reduced symmetry properties compared to its theoretical description (i.e., Lagrangian).

Dynamical breaking of a global symmetry is a spontaneous symmetry breaking, which happens not at the (classical) tree level (i.e., at the level of the bare action), but due to quantum corrections (i.e., at the level of the effective action).

Dynamical breaking of a gauge symmetry ^{ } is subtler. In the conventional spontaneous gauge symmetry breaking, there exists an unstable Higgs particle in the theory, which drives the vacuum to a symmetry-broken phase. (See, for example, electroweak interaction.) In dynamical gauge symmetry breaking, however, no unstable Higgs particle operates in the theory, but the bound states of the system itself provide the unstable fields that render the phase transition. For example, Bardeen, Hill, and Lindner published a paper that attempts to replace the conventional Higgs mechanism in the standard model by a DSB that is driven by a bound state of top-antitop quarks. (Such models, in which a composite particle plays the role of the Higgs boson, are often referred to as "Composite Higgs models".)^{ [10] } Dynamical breaking of gauge symmetries is often due to creation of a fermionic condensate — e.g., the quark condensate, which is connected to the dynamical breaking of chiral symmetry in quantum chromodynamics. Conventional superconductivity is the paradigmatic example from the condensed matter side, where phonon-mediated attractions lead electrons to become bound in pairs and then condense, thereby breaking the electromagnetic gauge symmetry.

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes. However, if the system is sampled (i.e. if the system is actually used or interacted with in any way), a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed "hidden symmetry", and has crucial formal consequences. (See the article on the Goldstone boson.)

When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate *which* member is distinct, only that *one is*. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be resymmetrized, by taking the average of each of the elements of the group being the distinct one.

The crucial concept in physics theories is the order parameter. If there is a field (often a background field) which acquires an expectation value (not necessarily a *vacuum* expectation value) which is not invariant under the symmetry in question, we say that the system is in the ordered phase, and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter, which specifies a "frame of reference" to be measured against. In that case, the vacuum state does not obey the initial symmetry (which would keep it invariant, in the linearly realized **Wigner mode** in which it would be a singlet), and, instead changes under the (hidden) symmetry, now implemented in the (nonlinear) **Nambu–Goldstone mode**. Normally, in the absence of the Higgs mechanism, massless Goldstone bosons arise.

The symmetry group can be discrete, such as the space group of a crystal, or continuous (e.g., a Lie group), such as the rotational symmetry of space. However, if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry.

On October 7, 2008, the Royal Swedish Academy of Sciences awarded the 2008 Nobel Prize in Physics to three scientists for their work in subatomic physics symmetry breaking. Yoichiro Nambu, of the University of Chicago, won half of the prize for the discovery of the mechanism of spontaneous broken symmetry in the context of the strong interactions, specifically chiral symmetry breaking. Physicists Makoto Kobayashi and Toshihide Maskawa, of Kyoto University, shared the other half of the prize for discovering the origin of the explicit breaking of CP symmetry in the weak interactions.^{ [11] } This origin is ultimately reliant on the Higgs mechanism, but, so far understood as a "just so" feature of Higgs couplings, not a spontaneously broken symmetry phenomenon.

- Autocatalytic reactions and order creation
- Catastrophe theory
- Chiral symmetry breaking
- CP-violation
- Fermi ball
- Gauge gravitation theory
- Goldstone boson
- Grand unified theory
- Higgs mechanism
- Higgs boson
- Higgs field (classical)
- Irreversibility
- Magnetic catalysis of chiral symmetry breaking
- Mermin–Wagner theorem
- Norton's dome
- Second-order phase transition
- Spontaneous absolute asymmetric synthesis in chemistry
- Symmetry breaking
- Tachyon condensation
- 1964 PRL symmetry breaking papers

**^**Note that (as in fundamental Higgs driven spontaneous gauge symmetry breaking) the term "symmetry breaking" is a misnomer when applied to gauge symmetries.

In theoretical physics, **quantum chromodynamics** (**QCD**) is the theory of the strong interaction between quarks and gluons, the fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of quantum field theory called a non-abelian gauge theory, with symmetry group SU(3). The QCD analog of electric charge is a property called *color*. Gluons are the force carrier of the theory, just as photons are for the electromagnetic force in quantum electrodynamics. The theory is an important part of the Standard Model of particle physics. A large body of experimental evidence for QCD has been gathered over the years.

The **Standard Model** of particle physics is the theory describing three of the four known fundamental forces in the universe, as well as classifying all known elementary particles. It was developed in stages throughout the latter half of the 20th century, through the work of many scientists around the world, with the current formulation being finalized in the mid-1970s upon experimental confirmation of the existence of quarks. Since then, confirmation of the top quark (1995), the tau neutrino (2000), and the Higgs boson (2012) have added further credence to the Standard Model. In addition, the Standard Model has predicted various properties of weak neutral currents and the W and Z bosons with great accuracy.

**Technicolor** theories are models of physics beyond the Standard Model that address electroweak gauge symmetry breaking, the mechanism through which W and Z bosons acquire masses. Early technicolor theories were modelled on quantum chromodynamics (QCD), the "color" theory of the strong nuclear force, which inspired their name.

In particle physics, a **gauge boson** is a force carrier, a bosonic particle that carries any of the fundamental interactions of nature, commonly called forces. Elementary particles, whose interactions are described by a gauge theory, interact with each other by the exchange of gauge bosons—usually as virtual particles.

In particle and condensed matter physics, **Goldstone bosons** or **Nambu–Goldstone bosons** (**NGBs**) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in particle physics within the context of the BCS superconductivity mechanism, and subsequently elucidated by Jeffrey Goldstone, and systematically generalized in the context of quantum field theory. In condensed matter physics such bosons are quasiparticles and are known as Anderson-Bogoliubov modes.

In the Standard Model of particle physics, the **Higgs mechanism** is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W^{+}, W^{−}, and Z^{0} bosons actually have relatively large masses of around 80 GeV/c^{2}. The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W^{±}, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

In quantum mechanics, **superselection** extends the concept of selection rules.

A **chiral** phenomenon is one that is not identical to its mirror image. The spin of a particle may be used to define a **handedness**, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called **chiral symmetry**.

In particle physics, **little Higgs** models are based on the idea that the Higgs boson is a pseudo-Goldstone boson arising from some global symmetry breaking at a TeV energy scale. The goal of little Higgs models is to use the spontaneous breaking of such approximate global symmetries to stabilize the mass of the Higgs boson(s) responsible for electroweak symmetry breaking.

The **QCD vacuum** is the vacuum state of quantum chromodynamics (QCD). It is an example of a *non-perturbative* vacuum state, characterized by non-vanishing condensates such as the gluon condensate and the quark condensate in the complete theory which includes quarks. The presence of these condensates characterizes the **confined phase** of quark matter.

The **J. J. Sakurai Prize for Theoretical Particle Physics**, is presented by the American Physical Society at its annual April Meeting, and honors outstanding achievement in particle physics theory. The prize consists of a monetary award, a certificate citing the contributions recognized by the award, and a travel allowance for the recipient to attend the presentation. The award is endowed by the family and friends of particle physicist J. J. Sakurai. The prize has been awarded annually since 1985.

In particle physics, **chiral symmetry breaking** is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was awarded the 2008 Nobel prize in physics for describing this phenomenon.

**François, Baron Englert** is a Belgian theoretical physicist and 2013 Nobel prize laureate.

**Christopher T. Hill** is an American theoretical physicist at the Fermi National Accelerator Laboratory who did undergraduate work in physics at M.I.T., and graduate work at Caltech. Hill's Ph.D. thesis, "Higgs Scalars and the Nonleptonic Weak Interactions" (1977) contains one of the first discussions of the two-Higgs-doublet model.

The **Higgs boson** is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson with zero spin, no electric charge, and no colour charge. It is also very unstable, decaying into other particles almost immediately.

The **1964 PRL symmetry breaking papers** were written by three teams who proposed related but different approaches to explain how mass could arise in local gauge theories. These three papers were written by

In theoretical physics, a **spurion** is a fictitious, auxiliary field in a quantum field theory that can be used to parameterize any symmetry breaking and to determine all operators invariant under the symmetry.

The **goldstino** is the Nambu−Goldstone fermion emerging in the spontaneous breaking of supersymmetry. It is the close fermionic analog of the Nambu−Goldstone bosons controlling the spontaneous breakdown of ordinary bosonic symmetries.

In theoretical physics, a **mass generation** mechanism is a theory that describes the origin of mass from the most fundamental laws of physics. Physicists have proposed a number of models that advocate different views of the origin of mass. The problem is complicated because the primary role of mass is to mediate gravitational interaction between bodies, and no theory of gravitational interaction reconciles with the currently popular Standard Model of particle physics.

In particle physics, **composite Higgs models** (CHM) are speculative extensions of the Standard Model (SM) where the Higgs boson is a bound state of new strong interactions. These scenarios are models for physics beyond the SM presently tested at the Large Hadron Collider (LHC) in Geneva.

- ↑ Miransky, Vladimir A. (1993).
*Dynamical Symmetry Breaking in Quantum Field Theories*. p. 15. ISBN 9810215584. - ↑ Arodz, Henryk; Dziarmaga, Jacek; Zurek, Wojciech Hubert, eds. (30 November 2003).
*Patterns of Symmetry Breaking*. p. 141. ISBN 9781402017452. - ↑ Cornell, James, ed. (21 November 1991).
*Bubbles, Voids and Bumps in Time: The New Cosmology*. p. 125. ISBN 9780521426732. - ↑ Edelman, Gerald M. (1992).
*Bright Air, Brilliant Fire: On the Matter of the Mind*. New York: BasicBooks. p. 203. - ↑ Goldstone, J. (1961). "Field theories with " Superconductor " solutions".
*Il Nuovo Cimento*.**19**(1): 154–164. Bibcode:1961NCim...19..154G. doi:10.1007/BF02812722. S2CID 120409034. - ↑ Steven Weinberg (20 April 2011).
*Dreams of a Final Theory: The Scientist's Search for the Ultimate Laws of Nature*. Knopf Doubleday Publishing Group. ISBN 978-0-307-78786-6. - ↑ A Brief History of Time, Stephen Hawking, Bantam; 10th anniversary edition (1998). pp. 73–74.
^{[ ISBN missing ]} - ↑ Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010). "Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order".
*Phys. Rev. B*.**82**(15): 155138. arXiv: 1004.3835 . Bibcode:2010PhRvB..82o5138C. doi:10.1103/physrevb.82.155138. S2CID 14593420. - ↑ Kohlstedt, K.L.; Vernizzi, G.; Solis, F.J.; Olvera de la Cruz, M. (2007). "Spontaneous Chirality via Long-range Electrostatic Forces".
*Physical Review Letters*.**99**(3): 030602. arXiv: 0704.3435 . Bibcode:2007PhRvL..99c0602K. doi:10.1103/PhysRevLett.99.030602. PMID 17678276. S2CID 37983980. - ↑ William A. Bardeen; Christopher T. Hill; Manfred Lindner (1990). "Minimal dynamical symmetry breaking of the standard model".
*Physical Review D*.**41**(5): 1647–1660. Bibcode:1990PhRvD..41.1647B. doi:10.1103/PhysRevD.41.1647. PMID 10012522. - ↑ The Nobel Foundation. "The Nobel Prize in Physics 2008".
*nobelprize.org*. Retrieved January 15, 2008.

Wikiquote has quotations related to: Spontaneous symmetry breaking |

- For a pedagogic introduction to electroweak symmetry breaking with step by step derivations, not found in texts, of many key relations, see http://www.quantumfieldtheory.info/Electroweak_Sym_breaking.pdf
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- Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia
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- The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
- International Journal of Modern Physics A: The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles
- Guralnik, G S; Hagen, C R and Kibble, T W B (1967). Broken Symmetries and the Goldstone Theorem. Advances in Physics, vol. 2 Interscience Publishers, New York. pp. 567–708 ISBN 0-470-17057-3
- Spontaneous Symmetry Breaking in Gauge Theories: a Historical Survey

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