Higgs mechanism

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The Higgs mechanism, also called the Brout-Englert-Higgs mechanism, Higgs-Kibble mechanism or Anderson-Higgs mechanism, is a form of superconductivity in the vacuum. It considers all of space filled with a relativistically invariant quantum fluid called the Higgs field, whose motion prevents certain forces from propagating over long distances. Part of the Higgs field mixes with the force-carrying gauge fields to produce massive gauge bosons, while the rest of the Higgs field describes a new particle, called the Higgs boson. The range of the force and the mass of the gauge bosons are inverses in natural units, but the mass of the Higgs boson is different and depends on the details.

The mechanism is the only way elementary vector particles, like the $W^\pm$ or the $\,Z$ , can have a mass. Interactions with the associated Higgs boson gives mass to the quarks and leptons in the standard model. The Higgs mechanism is an example of tachyon condensation where the tachyon is the Higgs field.

It was proposed in 1964 by Robert Brout and Francois Englert ^[1], independently by Peter Higgs ^[2] and by Gerald Guralnik, C. R. Hagen, and Tom Kibble ^[3] following earlier work by Yoichiro Nambu on the structure of the vacuum. It was inspired by the BCS theory of superconductivity, 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.

Although the evidence for the Higgs mechanism is overwhelming, accelerators have yet to produce a Higgs boson, e.g. by determining its mass. Even then it would not be clear if the Higgs is an elementary or a composite particle. For example, one might speculate that, similar to Cooper pairs, which are the carriers of the above-mentioned BCS theory of superconductivity, the Higgs field could finally turn out to consist of two weakly bound W-particles. This would lead to a rough mass estimate of 2x80=160 GeV.

1 General Discussion
2 Superconductivity
3 Abelian Higgs Model
4 Nonabelian Higgs Mechanism
5 References
6 See also
7 External links

[edit] General Discussion

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 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, like a pencil standing on end that falls. 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".

[edit] Superconductivity

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 $q$ . 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 $\hbar$ , the Planck quantum divided by $2π$ , is replaced by 1):

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

The operator $ψ(x)$ annihilates a boson at the point $x$ , while its adjoint $\scriptstyle \psi^\dagger$ creates a new boson at the same point. The wavefunction of the Bose-Einstein condensate is then the expectation value $Ψ$ of $ψ(x)$ , 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.

$\psi \rightarrow e^{iq\phi(x)} \psi \,$

$A \rightarrow A + \nabla \phi \,$

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 wavefunction can be written as

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

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,

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

and taking the density of the condensate $ρ$ to be constant,

$H\approx {\rho^2 \over 2m} (qA+ \nabla \theta)^2. \,$

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

${q^2 \rho^2 \over 2m} A^2. \,$

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,

$E\approx {{\dot A}^2\over 2} + {q^2 \rho^2 \over 2m} A^2. \,$

This is a harmonic oscillator with frequency $\scriptstyle \sqrt{q^2 \rho^2/m}$ . The quantity $| ψ | 2$ (= $ρ 2$ ) 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.

[edit] Abelian Higgs Model

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 $Φ$ to be charged, it must be complex. Equivalently, it should contain two fields with a symmetry which rotates them into each other, the real and imaginary parts. 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 Mexican-hat model, where the field energy has a minimum away from zero.

$S(\phi ) = \int {1\over 2} |\partial \phi|^2 - \lambda\cdot (|\phi|^2 - \Phi^2)^2$

This defines the following Hamiltonian:

$H(\phi ) = {1\over 2} |\dot\phi|^2 + |\nabla \phi|^2 + V(|\phi|)$

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 $\scriptstyle V(z,\Phi)= \lambda\cdot ( |z|^2 - \Phi^2)^2\,$ 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 $Φ$ . An image of the potential is found here:

Higgs potential V. For a fixed value of

λ

the potential is presented against the real and imaginary parts of

Φ

. The Mexican-hat or champagne-bottle profile at the ground should be noted.

When the field $Φ$ (x) is not coupled to electromagnetism, the Mexican-hat 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

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

with a constant prefactor, then the action for the field $θ(x)$ , i.e., the "phase" of the Higgs field Φ(x), has only derivative terms. This is not a surprise. Adding a constant to $θ(x)$ is a symmetry of the original theory, so different values of $θ(x)$ cannot have different energies. This is an example of Goldstone's theorem: spontaneously broken continuous symmetries lead to massless particles.

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

$S(\phi ,A) = \int {1\over 4} F^{\mu\nu} F_{\mu\nu} + |(\partial - i q A)\phi|^2 + \lambda\cdot (|\phi|^2 - \Phi^2)^2.$

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 $θ(x)$ can be set to zero by a gauge transformation, and does not represent any degrees of freedom at all.

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 $m 2 A 2 / 2$ where $m = q Φ$ .

[edit] Nonabelian Higgs Mechanism

The Nonabelian Higgs model has the following action:

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

where now the nonabelian field $\mathbf A$ is contained in D and in the tensor components $F μ,ν$ and $F μ,ν$ (the relation between $\mathbf A$ and those components is well-known 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.

$D\phi = \partial \phi - i A^k t_k \phi \,$

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.

The Higgs mechanism in the standard model is by a field which is a weak SU(2) doublet with weak hypercharge 1/2. The expectation value of this field chooses a preferred direction for both the hypercharge and weak gauge transformations, but not for a particular linear combination of the two. The linear combination which leaves the Higgs field invariant is the unbroken gauge group which is the gauge group of electromagnetism.

In all cases the essential point of the mechanism is the non-vanishing of the vacuum expectation value of the Higgs field, $|\langle\rm{vac}|\phi|\rm{vac} \rangle |\, > \,0$ .

[edit] References

[edit] See also

[edit] External links

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Higgs mechanism

From Wikipedia, the free encyclopedia

Contents

[edit] General Discussion

[edit] Superconductivity

[edit] Abelian Higgs Model

[edit] Nonabelian Higgs Mechanism

[edit] References

[edit] See also

[edit] External links

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