Higgs mechanism
higgs mechanism, higgs mechanism wikipediaIn 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 class of particles, the other being fermions would be considered massless, but measurements show that the W+, W−, and Z bosons actually have relatively large masses of around 80 GeV/c2 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[1] The Large Hadron Collider at CERN announced results consistent with the Higgs particle on March 14, 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature
The mechanism was proposed in 1962 by Philip Warren Anderson,[2] following work in the late 1950s on symmetry breaking in superconductivity and a 1960 paper by Yoichiro Nambu that discussed its application within particle physics A theory able to finally explain mass generation without "breaking" gauge theory was published almost simultaneously by three independent groups in 1964: by Robert Brout and François Englert;[3] by Peter Higgs;[4] and by Gerald Guralnik, C R Hagen, and Tom Kibble[5][6][7] The Higgs mechanism is therefore also called the Brout–Englert–Higgs mechanism or Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism,[8] Anderson–Higgs mechanism,[9] Anderson–HiggsKibble mechanism,[10] Higgs–Kibble mechanism by Abdus Salam[11] and ABEGHHK'tH mechanism by Peter Higgs[11]
On October 8, 2013, following the discovery at CERN's Large Hadron Collider of a new particle that appeared to be the longsought Higgs boson predicted by the theory, it was announced that Peter Higgs and François Englert had been awarded the 2013 Nobel Prize in Physics Englert's coauthor Robert Brout had died in 2011 and the Nobel Prize is not usually awarded posthumously[12]
Contents
 1 Standard model
 11 Structure of the Higgs field
 12 The photon as the part that remains massless
 13 Consequences for fermions
 2 History of research
 21 Background
 22 Discovery
 3 Examples
 31 Landau model
 32 Abelian Higgs mechanism
 33 NonAbelian Higgs mechanism
 34 Affine Higgs mechanism
 4 See also
 5 References
 6 Further reading
 7 External links
Standard model
The Higgs mechanism was incorporated into modern particle physics by Steven Weinberg and Abdus Salam, and is an essential part of the standard model
In the standard model, at temperatures high enough that electroweak symmetry is unbroken, all elementary particles are massless At a critical temperature, the Higgs field becomes tachyonic; the symmetry is spontaneously broken by condensation, and the W and Z bosons acquire masses This is also known as electroweak symmetry breaking; EWSB
Fermions, such as the leptons and quarks in the Standard Model, can also acquire mass as a result of their interaction with the Higgs field, but not in the same way as the gauge bosons
Structure of the Higgs field
In the standard model, the Higgs field is an SU2 doublet ie the standard representation with two complex components called isospin, which is a scalar under Lorentz transformations Its weak hypercharge U1 charge is 1 Under U1 rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other—combining to the standard twocomponent complex representation of the group U2
The Higgs field, through the interactions specified summarized, represented, or even simulated by its potential, induces spontaneous breaking of three out of the four generators "directions" of the gauge group U2 This is often written as SU2 × U1, which is strictly speaking only the same on the level of infinitesimal symmetries because the diagonal phase factor also acts on other fields in particular quarks Three out of its four components would ordinarily amount to Goldstone bosons, if they were not coupled to gauge fields
However, after symmetry breaking, these three of the four degrees of freedom in the Higgs field mix with the three W and Z bosons
W+
,
W−
and
Z
, and are only observable as components of these weak bosons, which are now massive; while the one remaining degree of freedom becomes the Higgs boson—a new scalar particle
The photon as the part that remains massless
The gauge group of the electroweak part of the standard model is SU2 × U1 The group SU2 is the group of all 2by2 unitary matrices with unit determinant; all the orthonormal changes of coordinates in a complex two dimensional vector space
Rotating the coordinates so that the second basis vector points in the direction of the Higgs boson makes the vacuum expectation value of H the spinor 0, v The generators for rotations about the x, y, and z axes are by half the Pauli matrices σx, σy, and σz, so that a rotation of angle θ about the zaxis takes the vacuum to
0 , v e − 1 2 i θ }i\theta }\right}While the Tx and Ty generators mix up the top and bottom components of the spinor, the Tz rotations only multiply each by opposite phases This phase can be undone by a U1 rotation of angle 1/2θ Consequently, under both an SU2 Tzrotation and a U1 rotation by an amount 1/2θ, the vacuum is invariant
This combination of generators
Q = T z + 1 2 Y +}Y}defines the unbroken part of the gauge group, where Q is the electric charge, Tz is the generator of rotations around the zaxis in the SU2 and Y is the hypercharge generator of the U1 This combination of generators a z rotation in the SU2 and a simultaneous U1 rotation by half the angle preserves the vacuum, and defines the unbroken gauge group in the standard model, namely the electric charge group The part of the gauge field in this direction stays massless, and amounts to the physical photon
Consequences for fermions
In spite of the introduction of spontaneous symmetry breaking, the mass terms preclude chiral gauge invariance For these fields, the mass terms should always be replaced by a gaugeinvariant "Higgs" mechanism One possibility is some kind of Yukawa coupling see below between the fermion field ψ and the Higgs field Φ, with unknown couplings Gψ, which after symmetry breaking more precisely: after expansion of the Lagrange density around a suitable ground state again results in the original mass terms, which are now, however ie, by introduction of the Higgs field written in a gaugeinvariant way The Lagrange density for the Yukawa interaction of a fermion field ψ and the Higgs field Φ is
L F e r m i o n ϕ , A , ψ = ψ ¯ γ μ D μ ψ + G ψ ψ ¯ ϕ ψ , }_ }\phi ,A,\psi =}\gamma ^D_\psi +G_}\phi \psi ,}where again the gauge field A only enters Dμ ie, it is only indirectly visible The quantities γμ are the Dirac matrices, and Gψ is the alreadymentioned Yukawa coupling parameter Now the massgeneration follows the same principle as above, namely from the existence of a finite expectation value  ⟨ ϕ ⟩  Again, this is crucial for the existence of the property mass
History of research
Background
Spontaneous symmetry breaking offered a framework to introduce bosons into relativistic quantum field theories However, according to Goldstone's theorem, these bosons should be massless[13] The only observed particles which could be approximately interpreted as Goldstone bosons were the pions, which Yoichiro Nambu related to chiral symmetry breaking
A similar problem arises with Yang–Mills theory also known as nonAbelian gauge theory, which predicts massless spin1 gauge bosons Massless weaklyinteracting gauge bosons lead to longrange forces, which are only observed for electromagnetism and the corresponding massless photon Gauge theories of the weak force needed a way to describe massive gauge bosons in order to be consistent
Discovery
Philip W Anderson, the first to implement the mechanism in 1962 Five of the six 2010 APS Sakurai Prize Winners – L to R Tom Kibble, Gerald Guralnik, Carl Richard Hagen, François Englert, and Robert Brout Peter Higgs 2009The idea was proposed in 1961 by Julian Schwinger[14] and who was referenced in Philip Warren Anderson's paper[2] on an application of the idea in nonrelativistic field theory, who discussed its consequences for particle physics but did not work out an explicit relativistic model The relativistic model was developed in 1964 by three independent groups – Robert Brout and François Englert;[3] Peter Higgs;[4] and Gerald Guralnik, Carl Richard Hagen, and Tom Kibble[5][6][7] Slightly later, in 1965, but independently from the other publications[15][16][17][18][19][20] the mechanism was also proposed by Alexander Migdal and Alexander Polyakov,[21] at that time Soviet undergraduate students However, the paper was delayed by the Editorial Office of JETP, and was published only in 1966
The mechanism is closely analogous to phenomena previously discovered by Yoichiro Nambu involving the "vacuum structure" of quantum fields in superconductivity[22] A similar but distinct effect involving an affine realization of what is now recognized as the Higgs field, known as the Stueckelberg mechanism, had previously been studied by Ernst Stueckelberg
These physicists discovered that when a gauge theory is combined with an additional field that spontaneously breaks the symmetry group, the gauge bosons can consistently acquire a nonzero mass In spite of the large values involved see below this permits a gauge theory description of the weak force, which was independently developed by Steven Weinberg and Abdus Salam in 1967 Higgs's original article presenting the model was rejected by Physics Letters When revising the article before resubmitting it to Physical Review Letters, he added a sentence at the end,[23] mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete representations of the symmetry group; these are the Higgs bosons
The three papers by Brout and Englert; Higgs; and Guralnik, Hagen, and Kibble were each recognized as "milestone letters" by Physical Review Letters in 2008[24] While each of these seminal papers took similar approaches, the contributions and differences among the 1964 PRL symmetry breaking papers are noteworthy All six physicists were jointly awarded the 2010 J J Sakurai Prize for Theoretical Particle Physics for this work[25]
Benjamin W Lee is often credited with first naming the "Higgslike" mechanism, although there is debate around when this first occurred[26][27][28] One of the first times the Higgs name appeared in print was in 1972 when Gerardus 't Hooft and Martinus J G Veltman referred to it as the "Higgs–Kibble mechanism" in their Nobel winning paper[29][30]
Examples
The Higgs mechanism occurs whenever a charged field has a vacuum expectation value In the nonrelativistic context, this is the Landau model of a charged Bose–Einstein condensate, also known as a superconductor In the relativistic condensate, the condensate is a scalar field, and is relativistically invariant
Landau model
The Higgs mechanism is a type of superconductivity which occurs in the vacuum It occurs when all of space is filled with a sea of particles which are charged, or, in field language, when a charged field has a nonzero vacuum expectation value Interaction with the quantum fluid filling the space prevents certain forces from propagating over long distances as it does in a superconducting medium; eg, in the Ginzburg–Landau theory
A superconductor expels all magnetic fields from its interior, a phenomenon known as the Meissner effect This was mysterious for a long time, because it implies that electromagnetic forces somehow become shortrange inside the superconductor Contrast this with the behavior of an ordinary metal In a metal, the conductivity shields electric fields by rearranging charges on the surface until the total field cancels in the interior But magnetic fields can penetrate to any distance, and if a magnetic monopole an isolated magnetic pole is surrounded by a metal the field can escape without collimating into a string In a superconductor, however, electric charges move with no dissipation, and this allows for permanent surface currents, not just surface charges When magnetic fields are introduced at the boundary of a superconductor, they produce surface currents which exactly neutralize them The Meissner effect is due to currents in a thin surface layer, whose thickness, the London penetration depth, can be calculated from a simple model the Ginzburg–Landau theory
This simple model treats superconductivity as a charged Bose–Einstein condensate Suppose that a superconductor contains bosons with charge q The wavefunction of the bosons can be described by introducing a quantum field, ψ, which obeys the Schrödinger equation as a field equation in units where the reduced Planck constant, ħ, is set to 1:
i ∂ ∂ t ψ = ∇ − i q A 2 2 m ψ \psi = \over 2m}\psi }The operator ψx annihilates a boson at the point x, while its adjoint ψ† creates a new boson at the same point The wavefunction of the Bose–Einstein condensate is then the expectation value ψ of ψx, which is a classical function that obeys the same equation The interpretation of the expectation value is that it is the phase that one should give to a newly created boson so that it will coherently superpose with all the other bosons already in the condensate
When there is a charged condensate, the electromagnetic interactions are screened To see this, consider the effect of a gauge transformation on the field A gauge transformation rotates the phase of the condensate by an amount which changes from point to point, and shifts the vector potential by a gradient:
ψ → e i q ϕ x ψ A → A + ∇ ϕ \psi &\rightarrow e^\psi \\A&\rightarrow A+\nabla \phi \end}}When there is no condensate, this transformation only changes the definition of the phase of ψ at every point But when there is a condensate, the phase of the condensate defines a preferred choice of phase
The condensate wave function can be written as
ψ x = ρ x e i θ x , ,}where ρ is real amplitude, which determines the local density of the condensate If the condensate were neutral, the flow would be along the gradients of θ, the direction in which the phase of the Schrödinger field changes If the phase θ changes slowly, the flow is slow and has very little energy But now θ can be made equal to zero just by making a gauge transformation to rotate the phase of the field
The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy,
H = 1 2 m  q A + ∇ ψ  2 , \leftqA+\nabla \psi \right^,}and taking the density of the condensate ρ to be constant,
H ≈ ρ 2 2 m q A + ∇ θ 2 \over 2m}qA+\nabla \theta ^}Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,
q 2 ρ 2 2 m A 2 \rho ^ \over 2m}A^}When this term is present, electromagnetic interactions become shortranged Every field mode, no matter how long the wavelength, oscillates with a nonzero frequency The lowest frequency can be read off from the energy of a long wavelength A mode,
E ≈ A ˙ 2 2 + q 2 ρ 2 2 m A 2 }^ \over 2}+\rho ^ \over 2m}A^}This is a harmonic oscillator with frequency
1 m q 2 ρ 2 }q^\rho ^}}}The quantity ψ2 = ρ2 is the density of the condensate of superconducting particles
In an actual superconductor, the charged particles are electrons, which are fermions not bosons So in order to have superconductivity, the electrons need to somehow bind into Cooper pairs The charge of the condensate q is therefore twice the electron charge −e The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound The description of a Bose–Einstein condensate of loosely bound pairs is actually more difficult than the description of a condensate of elementary particles, and was only worked out in 1957 by Bardeen, Cooper and Schrieffer in the famous BCS theory
Abelian Higgs mechanism
Gauge invariance means that certain transformations of the gauge field do not change the energy at all If an arbitrary gradient is added to A, the energy of the field is exactly the same This makes it difficult to add a mass term, because a mass term tends to push the field toward the value zero But the zero value of the vector potential is not a gauge invariant idea What is zero in one gauge is nonzero in another
So in order to give mass to a gauge theory, the gauge invariance must be broken by a condensate The condensate will then define a preferred phase, and the phase of the condensate will define the zero value of the field in a gaugeinvariant way The gaugeinvariant definition is that a gauge field is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction
The condensate value is described by a quantum field with an expectation value, just as in the GinzburgLandau model
In order for the phase of the vacuum to define a gauge, the field must have a phase also referred to as 'to be charged' In order for a scalar field Φ to have a phase, it must be complex, or equivalently it should contain two fields with a symmetry which rotates them into each other The vector potential changes the phase of the quanta produced by the field when they move from point to point In terms of fields, it defines how much to rotate the real and imaginary parts of the fields into each other when comparing field values at nearby points
The only renormalizable model where a complex scalar field Φ acquires a nonzero value is the Mexicanhat model, where the field energy has a minimum away from zero The action for this model is
S ϕ = ∫ 1 2  ∂ ϕ  2 − λ  ϕ  2 − Φ 2 2 , }\left\partial \phi \right^\lambda \left\left\phi \right^\Phi ^\right^,}which results in the Hamiltonian
H ϕ = 1 2  ϕ ˙  2 +  ∇ ϕ  2 + V  ϕ  }\left}\right^+\left\nabla \phi \right^+V\left\phi \right}The first term is the kinetic energy of the field The second term is the extra potential energy when the field varies from point to point The third term is the potential energy when the field has any given magnitude
This potential energy, Vz, Φ = λz2 − Φ22,[31] has a graph which looks like a Mexican hat, which gives the model its name In particular, the minimum energy value is not at z = 0, but on the circle of points where the magnitude of z is Φ
Higgs potential V For a fixed value of λ, the potential is presented upwards against the real and imaginary parts of Φ The Mexicanhat or champagnebottle profile at the ground should be notedWhen the field Φx is not coupled to electromagnetism, the Mexicanhat potential has flat directions Starting in any one of the circle of vacua and changing the phase of the field from point to point costs very little energy Mathematically, if
ϕ x = Φ e i θ x }with a constant prefactor, then the action for the field θx, ie, the "phase" of the Higgs field Φx, has only derivative terms This is not a surprise Adding a constant to θx is a symmetry of the original theory, so different values of θx cannot have different energies This is an example of Goldstone's theorem: spontaneously broken continuous symmetries normally produce massless excitations
The Abelian Higgs model is the Mexicanhat model coupled to electromagnetism:
S ϕ , A = ∫ − 1 4 F μ ν F μ ν +  ∂ − i q A ϕ  2 − λ  ϕ  2 − Φ 2 2 }F^F_+\left\left\partial iqA\right\phi \right^\lambda \left\left\phi \right^\Phi ^\right^}The classical vacuum is again at the minimum of the potential, where the magnitude of the complex field φ is equal to Φ But now the phase of the field is arbitrary, because gauge transformations change it This means that the field θx can be set to zero by a gauge transformation, and does not represent any actual degrees of freedom at all
Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero So in the Abelian Higgs model, the gauge field acquires a mass To calculate the magnitude of the mass, consider a constant value of the vector potential A in the xdirection in the gauge where the condensate has constant phase This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy:
E = 1 2  ∂ Φ e i q A x  2 = 1 2 q 2 Φ 2 A 2 }\left\partial \left\Phi e^\right\right^=}q^\Phi ^A^}This energy is the same as a mass term 1/2m2A2 where m = qΦ
NonAbelian Higgs mechanism
The NonAbelian Higgs model has the following action
S ϕ , A = ∫ 1 4 g 2 tr F μ ν F μ ν +  D ϕ  2 + V  ϕ  =\int }\mathop } F^F_+D\phi ^+V\phi }where now the nonAbelian field A is contained in the covariant derivative D and in the tensor components F μ ν } and F μ ν } the relation between A and those components is wellknown from the Yang–Mills theory
It is exactly analogous to the Abelian Higgs model Now the field φ is in a representation of the gauge group, and the gauge covariant derivative is defined by the rate of change of the field minus the rate of change from parallel transport using the gauge field A as a connection
D ϕ = ∂ ϕ − i A k t k ϕ t_\phi }Again, the expectation value of Φ defines a preferred gauge where the vacuum is constant, and fixing this gauge, fluctuations in the gauge field A come with a nonzero energy cost
Depending on the representation of the scalar field, not every gauge field acquires a mass A simple example is in the renormalizable version of an early electroweak model due to Julian Schwinger In this model, the gauge group is SO3 or SU2 − there are no spinor representations in the model, and the gauge invariance is broken down to U1 or SO2 at long distances To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field φa which transforms as a vector a triplet of SO3 If this field has a vacuum expectation value, it points in some direction in field space Without loss of generality, one can choose the zaxis in field space to be the direction that φ is pointing, and then the vacuum expectation value of φ is 0, 0, A, where A is a constant with dimensions of mass c = ℏ = 1
Rotations around the zaxis form a U1 subgroup of SO3 which preserves the vacuum expectation value of φ, and this is the unbroken gauge group Rotations around the x and yaxis do not preserve the vacuum, and the components of the SO3 gauge field which generate these rotations become massive vector mesons There are two massive W mesons in the Schwinger model, with a mass set by the mass scale A, and one massless U1 gauge boson, similar to the photon
The Schwinger model predicts magnetic monopoles at the electroweak unification scale, and does not predict the Z meson It doesn't break electroweak symmetry properly as in nature But historically, a model similar to this but not using the Higgs mechanism was the first in which the weak force and the electromagnetic force were unified
Affine Higgs mechanism
Ernst Stueckelberg discovered[32] a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon Effectively, Stueckelberg's model is a limit of the regular Mexican hat Abelian Higgs model, where the vacuum expectation value H goes to infinity and the charge of the Higgs field goes to zero in such a way that their product stays fixed The mass of the Higgs boson is proportional to H, so the Higgs boson becomes infinitely massive and decouples, so is not present in the discussion The vector meson mass, however, equals to the product eH, and stays finite
The interpretation is that when a U1 gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part The angular part of the Higgs field θ has the following gauge transformation law:
θ → θ + e α A → A + ∂ α \theta &\rightarrow \theta +e\alpha \,\\A&\rightarrow A+\partial \alpha \end}}The gauge covariant derivative for the angle which is actually gauge invariant is:
D θ = ∂ θ − e A HIn order to keep θ fluctuations finite and nonzero in this limit, θ should be rescaled by H, so that its kinetic term in the action stays normalized The action for the theta field is read off from the Mexican hat action by substituting ϕ = H e i θ / H }
S = ∫ 1 4 F 2 + 1 2 D θ 2 = ∫ 1 4 F 2 + 1 2 ∂ θ − H e A 2 = ∫ 1 4 F 2 + 1 2 ∂ θ − m A 2 }F^+}D\theta ^=\int }F^+}\partial \theta HeA^=\int }F^+}\partial \theta mA^}since eH is the gauge boson mass By making a gauge transformation to set θ = 0, the gauge freedom in the action is eliminated, and the action becomes that of a massive vector field:
S = ∫ 1 4 F 2 + 1 2 m 2 A 2 }F^+}m^A^\,}To have arbitrarily small charges requires that the U1 is not the circle of unit complex numbers under multiplication, but the real numbers R under addition, which is only different in the global topology Such a U1 group is noncompact The field θ transforms as an affine representation of the gauge group Among the allowed gauge groups, only noncompact U1 admits affine representations, and the U1 of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy
The Higgs condensate in this model has infinitesimal charge, so interactions with the Higgs boson do not violate charge conservation The theory of quantum electrodynamics with a massive photon is still a renormalizable theory, one in which electric charge is still conserved, but magnetic monopoles are not allowed For nonAbelian gauge theory, there is no affine limit, and the Higgs oscillations cannot be too much more massive than the vectors
See also
 Electroweak interaction
 Electromagnetic mass
 Higgs bundle
 Mass generation
 Quantum triviality
 Yang–Mills–Higgs equations
References
 ^ G Bernardi, M Carena, and T Junk: "Higgs bosons: theory and searches", Reviews of Particle Data Group: Hypothetical particles and Concepts, 2007, http://pdglblgov/2008/reviews/higgs_s055pdf
 ^ a b P W Anderson 1962 "Plasmons, Gauge Invariance, and Mass" Physical Review 130 1: 439–442 Bibcode:1963PhRv130439A doi:101103/PhysRev130439
 ^ a b F Englert; R Brout 1964 "Broken Symmetry and the Mass of Gauge Vector Mesons" Physical Review Letters 13 9: 321–323 Bibcode:1964PhRvL13321E doi:101103/PhysRevLett13321
 ^ a b Peter W Higgs 1964 "Broken Symmetries and the Masses of Gauge Bosons" Physical Review Letters 13 16: 508–509 Bibcode:1964PhRvL13508H doi:101103/PhysRevLett13508
 ^ a b G S Guralnik; C R Hagen; T W B Kibble 1964 "Global Conservation Laws and Massless Particles" Physical Review Letters 13 20: 585–587 Bibcode:1964PhRvL13585G doi:101103/PhysRevLett13585
 ^ a b Gerald S Guralnik 2009 "The History of the Guralnik, Hagen and Kibble development of the Theory of Spontaneous Symmetry Breaking and Gauge Particles" International Journal of Modern Physics A24 14: 2601–2627 arXiv:09073466 Bibcode:2009IJMPA242601G doi:101142/S0217751X09045431
 ^ a b History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism Scholarpedia
 ^ "Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism" Scholarpedia Retrieved 20120616
 ^ Liu, G Z; Cheng, G 2002 "Extension of the AndersonHiggs mechanism" Physical Review B 65 13: 132513 arXiv:condmat/0106070 Bibcode:2002PhRvB65m2513L doi:101103/PhysRevB65132513
 ^ Matsumoto, H; Papastamatiou, N J; Umezawa, H; Vitiello, G 1975 "Dynamical rearrangement in the AndersonHiggsKibble mechanism" Nuclear Physics B 97: 61 Bibcode:1975NuPhB9761M doi:101016/0550321375902151
 ^ a b Close, Frank 2011 The Infinity Puzzle: Quantum Field Theory and the Hunt for an Orderly Universe Oxford: Oxford University Press ISBN 9780199593507
 ^ "Press release from Royal Swedish Academy of Sciences" PDF 8 October 2013 Retrieved 8 October 2013
 ^ "Guralnik, G S; Hagen, C R and Kibble, T W B 1967 Broken Symmetries and the Goldstone Theorem Advances in Physics, vol 2" PDF
 ^ Schwinger, Julian 1961 "Gauge Invariance and Mass" Phys Rev 125 1: 397–8 Bibcode:1962PhRv125397S doi:101103/PhysRev125397
 ^ AM Polyakov, A View From The Island, 1992
 ^ Farhi, E, & Jackiw, R W 1982 Dynamical Gauge Symmetry Breaking: A Collection Of Reprints Singapore: World Scientific Pub Co
 ^ Frank Close "The Infinity Puzzle" 2011, p158
 ^ Norman Dombey, "Higgs Boson: Credit Where It's Due" The Guardian, July 6, 2012
 ^ Cern Courier, Mar 1, 2006
 ^ Sean Carrol, "The Particle At The End Of The Universe: The Hunt For The Higgs And The Discovery Of A New World", 2012, p228
 ^ A A Migdal and A M Polyakov, "Spontaneous Breakdown of Strong Interaction Symmetry and Absence of Massless Particles", JETP 51, 135, July 1966 English translation: Soviet Physics JETP, 24, 1, January 1967
 ^ Nambu, Y 1960 "Quasiparticles and Gauge Invariance in the Theory of Superconductivity" Physical Review 117 3: 648–663 Bibcode:1960PhRv117648N doi:101103/PhysRev117648
 ^ Higgs, Peter 2007 "Prehistory of the Higgs boson" Comptes Rendus Physique 8 9: 970–972 Bibcode:2007CRPhy8970H doi:101016/jcrhy200612006
 ^ "Physical Review Letters – 50th Anniversary Milestone Papers" Prlapsorg Retrieved 20120616
 ^ "American Physical Society – J J Sakurai Prize Winners" Apsorg Retrieved 20120616
 ^ Department of Physics and Astronomy "Rochester's Hagen Sakurai Prize Announcement" Pasrochesteredu Archived from the original on 20080416 Retrieved 20120616
 ^ FermiFred 20100215 "CR Hagen discusses naming of Higgs Boson in 2010 Sakurai Prize Talk" Youtubecom Retrieved 20120616
 ^ Sample, Ian 20090529 "Anything but the God particle by Ian Sample" Guardian Retrieved 20120616
 ^ G 't Hooft; M Veltman 1972 "Regularization and Renormalization of Gauge Fields" Nuclear Physics B 44 1: 189–219 Bibcode:1972NuPhB44189T doi:101016/0550321372902799
 ^ "Regularization and Renormalization of Gauge Fields by t'Hooft and Veltman PDF" PDF Archived from the original PDF on 20120707 Retrieved 20120616
 ^ Goldstone, J 1961 "Field theories with " Superconductor " solutions" Il Nuovo Cimento 19: 154–164 Bibcode:1961NCim19154G doi:101007/BF02812722
 ^ Stueckelberg, E C G 1938, "Die Wechselwirkungskräfte in der Elektrodynamik und in der Feldtheorie der Kräfte", Helv Phys Acta 11: 225
Further reading
 Schumm, Bruce A 2004 Deep Down Things Johns Hopkins Univ Press Chpt 9
 EnglertBroutHiggsGuralnikHagenKibble mechanism Tom W B Kibble Scholarpedia, 41:6441 doi:104249/scholarpedia6441
External links
 For a pedagogic introduction to electroweak symmetry breaking with step by step derivations, not found in texts, of many key relations, see http://wwwquantumfieldtheoryinfo/Electroweak_Sym_breakingpdf
 Guralnik, GS; Hagen, CR; Kibble, TWB 1964 "Global Conservation Laws and Massless Particles" Physical Review Letters 13 20: 585–87 Bibcode:1964PhRvL13585G doi:101103/PhysRevLett13585
 Mark D Roberts 1999 "A Generalized Higgs Model"
 2010 Sakurai Prize  All Events  YouTube
 From BCS to the LHC  CERN Courier Jan 21, 2008, Steven Weinberg, University of Texas at Austin
 Higgs, dark matter and supersymmetry: What the Large Hadron Collider will tell us Steven Weinberg  YouTube on YouTube 06112009
 Gerry Guralnik speaks at Brown University about the 1964 PRL papers
 Guralnik, Gerald 2013 "Heretical Ideas that Provided the Cornerstone for the Standard Model of Particle Physics" SPG MITTEILUNGEN March 2013, No 39, p 14
 Steven Weinberg Praises Teams for Higgs Boson Theory
 Physical Review Letters – 50th Anniversary Milestone Papers
 Imperial College London on PRL 50th Anniversary Milestone Papers
 Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia
 History of Englert–Brout–Higgs–Guralnik–Hagen–Kibble Mechanism on Scholarpedia
 The Hunt for the Higgs at Tevatron
 The Mystery of Empty Space on YouTube A lecture with UCSD physicist Kim Griest 43 minutes
 

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B♭ (musical note)
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