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- Higgs mechanism

In quantum field theory, the **Higgs mechanism** is a way that the massless gauge bosons in a gauge theory get a mass by interacting with a background **Higgs field**. The standard model of particle physics uses the Higgs mechanism to give all the elementary particles masses.

The mechanism requires the Higgs field to be nonzero in the vacuum, exactly like spontaneous symmetry breaking. In this case, the broken symmetry is gauged, meaning that the field which fills all of space, the **Higgs condensate**, is charged. Gauge fields become massive when there is a charged condensate, this is called superconductivity.

The Higgs mechanism in the standard model successfully explains the mass ratio of the

weak gauge bosons which otherwise would be massless. The ratio of the W and Z masses is correctly predicted to five decimal places. The leptons and quarks in the Standard Model also acquire mass as a result of their interaction with the Higgs condensateThe Higgs in the standard model is a SU(2) doublet, a complex spinor, which also gets a phase under the standard-model U(1). After symmetry breaking, three of the four degrees of freedom in the Higgs mix with the W and Z bosons to give them mass, while the one remaining degree of freedom becomes the Higgs boson - a new scalar particle. Although the evidence for the Higgs *mechanism* is overwhelming, accelerators have yet to produce the Higgs boson and evaluate its physical properties, so it is not even known if the Higgs is an elementary or a composite particle. It is hoped that the Large Hadron Collider at CERN will bring experimental evidence confirming its existence.

The mechanism is also called the **Brout–Englert–Higgs mechanism**, or **Higgs–Brout–Englert–Guralnik–Hagen–Kibble mechanism**, or **Anderson–Higgs mechanism**.

It was proposed in 1964 by Robert Brout and Francois Englert ^{[2]}, independently by Peter Higgs ^{[3]} and by Gerald Guralnik, C. R. Hagen, and Tom Kibble ^{[4]} . It was inspired by the BCS theory of superconductivity, vacuum structure work by Yoichiro Nambu, the preceding Ginzburg–Landau theory, and the suggestion by Philip Anderson that superconductivity could be important for relativistic physics. It was anticipated by earlier work of Ernst Stückelberg on massive quantum electrodynamics. It was named the *Higgs mechanism* by Gerardus 't Hooft in 1971. The three papers written on this discovery by Guralnik, Hagen, Kibble, Higgs, Brout, and Englert were each recognized as milestone papers by Physical Review Letters 50th anniversary celebration.^{[5]}

The problem with spontaneous symmetry breaking models in particle physics is that, according to Goldstone's theorem, they come with massless scalar particles. If a symmetry is broken by a condensate, acting with a symmetry generator on the condensate gives a second state with the same energy. So certain oscillations do not have any energy, and in quantum field theory the particles associated with these oscillations have zero mass.

The only observed particles which could be interpreted as Goldstone bosons were the pions. Since the symmetry is approximate, the pions are not exactly massless. Yoichiro Nambu, writing before Jeffrey Goldstone, suggested that the pions were the bosons associated with chiral symmetry breaking. This explained their pseudoscalar nature, the reason they couple to nucleons through derivative couplings, and the Goldberger–Treiman relation. Aside from the pions, no other Goldstone particle was observed.

A similar problem arises in Yang–Mills theory, also known as nonabelian gauge theory. These theories predict massless spin 1 gauge bosons, which (apart from the photon) are also not observed. It was Higgs' insight that when you combine a gauge theory with a spontaneous symmetry-breaking model the (unobserved) massless bosons acquire a mass, which we observe, solving the problem.

Higgs' original article presenting the model was rejected by Physical Review Letters when first submitted, apparently because it did not predict any new detectable effects. So he added a sentence at the end, mentioning that it implies the existence of one or more new, massive scalar bosons, which do not form complete representations of the symmetry. These are the Higgs bosons.

The Higgs mechanism was incorporated into modern particle physics by Steven Weinberg and is an essential part of the Standard Model.

In the standard model, at temperatures high enough so that the symmetry is unbroken, all elementary particles except the scalar Higgs boson are massless. At a critical temperature, the Higgs field spontaneously slides from the point of maximum energy in a randomly chosen direction. Once the symmetry is broken, the gauge boson particles — such as the leptons, quarks, W boson, and Z boson — get a mass. The mass can be interpreted to be a result of the interactions of the particles with the "Higgs ocean".

The Higgs mechanism can be considered as the superconductivity 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.

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 short-range 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.

This simple model, due to Lev Landau and Vitaly Ginzburg, treats superconductivity as a charged Bose–Einstein condensate. Suppose that a superconductor contains bosons with charge

*q*

*\psi*

*\hbar*

2*\pi*

*i{\partial**\over**\partial**t}**\psi*=*{(\nabla*-*iqA)*^{2}*\over*2*m}**\psi
*

The operator

*\psi(x)*

*x*

*\scriptstyle**\psi*^{\dagger}

*\Psi*

*\psi(x)*

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.

*\psi* → *e*^{iq\phi(x)}*\psi
*

*A* → *A*+*\nabla**\phi
*

When there is no condensate, this transformation only changes the definition of the phase of

*\psi*

The condensate wavefunction can be written as

*\psi(x)*=*\rho(x)**e*^{i\theta(x)}*,
*

*\rho*

*\theta*

*\theta*

*\theta*

The energy of slow changes of phase can be calculated from the Schrödinger kinetic energy,

*H*=*{*1*\over*2*m}**|{(qA*+*\nabla**)\psi|*^{2},
}

and taking the density of the condensate

*\rho*

*H* ≈ *{\rho*^{2}*\over*2*m}**(qA*+*\nabla**\theta)*^{2.
}

Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,

*{q*^{2}*\rho*^{2}*\over*2*m}**A*^{2.
}

When this term is present, electromagnetic interactions become short-ranged. 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\over}2*}*+*{q*^{2}*\rho*^{2}*\over*2*m}**A*^{2.
}

This is a harmonic oscillator with frequency

*\scriptstyle**\sqrt{q*^{2}*\rho*^{2/m}}

*|\psi|*^{2}

*\rho*^{2}

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*

*e*

In a relativistic gauge theory, the vector bosons are naively massless, like the photon, leading to long-range forces. This is fine for electromagnetism, where the force is actually long-range, but it means that the description of short-range weak forces by a gauge theory requires a modification.

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 gauge invariant way. The gauge invariant 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 Landau–Ginzburg model. To make sure that the condensate value of the field does not pick out a preferred direction in space-time, it must be a scalar field. In order for the phase of the condensate to define a gauge, the field must be charged.

In order for a scalar field

*\Phi*

The only renormalizable model where a complex scalar field Φ acquires a nonzero value is the Mexican-hat model, where the field energy has a minimum away from zero.

*S(\phi**)*=*\int**{*1*\over*2*}**|\partial**\phi|*^{2}-λ ⋅ *(|\phi|*^{2}-*\Phi*^{2)}^{2}

This defines the following Hamiltonian:

*H(\phi**)*=*{*1*\over*2*}**|*

\phi| |

^{2}+*|\nabla**\phi|*^{2}+*V(|\phi|)
*

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

*\scriptstyle**V(z,\Phi)*=λ ⋅ *(**|z|*^{2}-*\Phi*^{2)}^{2}

*\Phi*

When the field

*\Phi*

*\phi(x)*=*\Phi**e*^{i\theta(x)}*,*

*\theta**(x)*

*\theta**(x)*

*\theta**(x)*

The Abelian Higgs model is the Mexican-hat model coupled to electromagnetism:

*S(\phi**,A)*=*\int**{*1*\over*4*}**F*^{\mu\nu}*F*_{\mu\nu}+*|(\partial*-*i**q**A)\phi|*^{2}+λ ⋅ *(|\phi|*^{2}-*\Phi*^{2)}^{2.
}

The classical vacuum is again at the minimum of the potential, where the magnitude of the complex field

*\phi*

*\Phi*

*\theta**(x)*

Furthermore, choosing a gauge where the phase of the condensate is fixed, the potential energy for fluctuations of the vector field is nonzero, just as it is in the Landau–Ginzburg model. 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 x direction 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*\over*2*}|\partial**(\Phi**e*^{iqAx}*)|*^{2}=*{*1*\over*2*}**q*^{2\Phi}^{2}*A*^{2
}

And this energy is the same as a mass term

*m*^{2}*A*^{2/2}

*m*=*q\Phi*

The Nonabelian Higgs model has the following action:

*S(\phi**,A)*=*\int**{*1*\over*4*g*^{2}}*tr(F*^{\mu\nu}*F*_{\mu\nu}*)*+*|D\phi|*^{2}+*V(|\phi|),
*

where now the nonabelian field

A

*F*^{\mu}

*F*_{\mu}

A

It is exactly analogous to the Abelian Higgs model. Now the field

*\phi*

*D\phi*=*\partial**\phi*-*i**A*^{k}*t*_{k}*\phi
*

Again, the expectation value of Φ defines a preferred gauge where the condensate 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 SO(3) (or SU(2)--- there are no spinor representations in the model), and the gauge invariance is broken down to U(1) or SO(2) at long distances. To make a consistent renormalizable version using the Higgs mechanism, introduce a scalar field

*\phi*^{a}

*\phi*

*\phi*

*(*0*,*0*,A)*

*\scriptstyle**c*=*\hbar*=1

Rotations around the z axis form a U(1) subgroup of SO(3) which preserves the vacuum expectation value of

*\phi*

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.

The gauge group of the electroweak part of the standard model is

SU*(*2*)* x U*(*1*)*

The group SU(2) is all unitary matrices, all the orthonormal changes of coordinates in a complex two dimensional vector space. Rotating the coordinates so that the first basis vector in the direction of

*H*

*(A,*0*)*

*\sigma*_{x,\sigma}_{y,\sigma}_{z}

*\theta*

*(A**e*^{i\theta/2}*,*0*)*

While the X and Y generators mix up the top and bottom components, the Z rotations only multiply by a phase. This phase can be undone by a U(1) rotation of angle

*\theta/*2

*\theta/*2

*Q*=*W*_{z}+*{Y/*2*}*

defines the unbroken gauge group, where

*W*_{z}

The phase that a field acquires under this combination of generators is its electric charge, and this is the formula for the electric charge in the standard model. In this convention, all the Y charges in the standard model are multiples of

1*/*3

*Q*=*W*_{z}+*Y/*6

Ernst Stueckelberg discovered a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon. Stuckelberg'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 disappears. The vector meson mass is equal to the product

*eH*

The interpretation is that when a U(1) 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

*\theta*

*\theta*+*e\alpha*

*A* → *A*+*\alpha*

The gauge covariant derivative for the angle (which is actually gauge invariant) is:

*D\theta*=*\partial**\theta*-*e**A*

In order to keep

*\theta*

*\theta*

*\scriptstyle**\phi*=*He*^{i\theta/H}

*S*=*\int**{*1*\over*4*}F*^{2}+*{*1*\over*2*}(D\theta)*^{2}=*\int**{*1*\over*4*}F*^{2}+*{*1*\over*2*}(\partial**\theta*-*He**A)*^{2}=*\int**{*1*\over*4*}F*^{2}+*{*1*\over*2*}(\partial**\theta*-*m**A)*^{2}

since

*\scriptstyle**eH*

*\scriptstyle**\theta*=0

*S*=*\int**{*1*\over*4*}**F*^{2}+*{m*^{2\over}2*}**A*^{2}

To have arbitrarily small charges requires that the U(1) 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 U(1) group is *non-compact*. The field

*\theta*

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.

- G. Bernardi, M. Carena, and T. Junk: "Higgs bosons: theory and searches", Reviews of Particle Data Group: Hypothetical particles and Consepts, 2007, http://pdg.lbl.gov/2008/reviews/higgs_s055.pdf
- http://prola.aps.org/abstract/PRL/v13/i9/p321_1 Broken Symmetry and the Mass of Gauge Vector Mesons
- http://link.aps.org/abstract/PRL/v13/p508 Broken Symmetries and the Masses of Gauge Bosons
- http://prola.aps.org/abstract/PRL/v13/i20/p585_1 Global Conservation Laws and Massless Particles
- http://prl.aps.org/50years/milestones#1964 Physical Review Letters - 50th Anniversary Milestone Papers

- A Generalized Higgs Model
- Global Conservation Laws and Massless Particles
- A popular "quasi-political" explanation of the Higgs boson
- In CERN Courier, Steven Weinberg reflects on spontaneous symmetry breaking
- Steven Weinberg Praises Teams for Higgs Boson Theory
- Physical Review Letters – 50th Anniversary Milestone Papers
- Imperial College London on PRL 50th Anniversary Milestone Papers
- Physics World, Introducing the little Higgs
- Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia
- History of Englert-Brout-Higgs-Guralnik-Hagen-Kibble Mechanism on Scholarpedia
- God Particle Overview