In the
standard model of
particle physics, the
Higgs
mechanism is a theoretical framework which explains how
the masses of the
W and Z bosons
arise as a result of electroweak symmetry breaking.
More generally, the
Higgs mechanism is the way that the
gauge bosons in any
gauge theory, like the standard model, get a
nonzero mass. It requires an extra field, a
Higgs
field, which interacts with the gauge fields, and which
has a nonzero value in its lowest energy state, a
vacuum expectation value. This
means that all of space is filled with the background Higgs field,
the so-called
Higgs condensate. Interaction with
this background field changes the low-energy spectrum of the gauge
fields and the gauge bosons become massive.
The Higgs field has a non-trivial self-interaction, like the
Mexican hat potential, which
leads to
spontaneous
symmetry breaking: in the lowest energy state the symmetry of
the potential (which includes the gauge symmetry) is broken by the
condensate. Analysis of small fluctuations of the fields near the
minimum reveals that the gauge bosons and other particles become
massive.
In
particle language, the constant
Higgs field is a
superfluid of charged
particles, and a charged superfluid is a superconductor. Inside a
superconductor, the gauge electric and magnetic fields both become
short-ranged, or massive.
The Higgs field in the standard model is a
SU doublet, a complex spinor with four
real components, which is charged under the standard model U(1).
After symmetry breaking, three of the four degrees of freedom in
the Higgs field mix with the W and Z bosons, 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 find
the Higgs, and allow physicists to determine its
properties.
The Higgs mechanism in the standard model successfully predicts the
mass of the
W^{±}, and Z
weak gauge
bosons, which are naturally massless. If the Higgs mechanism
were not there, these particles would get much smaller masses from
Higgslike
QCD quark condensates instead.
A part of the Higgs field is expected to show up as a new particle,
called the Higgs boson.
In the standard model of particle physics, the same Higgs mechanism
which breaks the standard model gauge group to
electromagnetism is also responsible for
giving all the
leptons and
quarks their masses. The fermions in the standard
model are
chiral and different
chiralities have different charges. The chiral fermions can come
together in pairs to make massive fermions by absorbing Higgs
bosons from the condensate.
History and naming
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, independently
by
Peter Higgs, and by
Gerald Guralnik,
C. R. Hagen, and
Tom
Kibble who worked out the results by the spring of 1963. 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 further anticipated by
earlier work of
Ernst Stueckelberg 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. While each of these
seminal papers took similar approaches, the contributions and
differences among the
1964
PRL Symmetry Breaking papers are noteworthy. Guralnik, Hagen,
Kibble, Higgs, Brout, and Englert were awarded the 2010
J. J. Sakurai Prize for Theoretical Particle
Physics jointly for this work.
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
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 a gauge theory is
combined with a spontaneous symmetry-breaking model the
(unobserved) massless bosons acquire a mass, which are observed,
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
W bosons and
Z boson, acquire masses. The mass can be
interpreted as the result of the interactions of the particles with
the "Higgs ocean".
Fermions, such as the leptons and quarks in the Standard Model,
acquire mass as a result of their interaction with the Higgs field,
but not in the same way as the gauge bosons.
Superconductivity
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. The
wavefunction of the bosons can be described by introducing a
quantum field, \psi, which
obeys the
Schrödinger equation as a
field equation (in units where \hbar, the Planck quantum
divided by 2\pi, is replaced by 1):i{\partial \over \partial t}
\psi = {(\nabla - iqA)^2 \over 2m} \psi\,
The operator \psi(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 \Psi of \psi(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 \psi 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 \rho is real amplitude, which determines the
local density of the condensate. If the condensate were neutral,
the flow would be along the gradients of \theta, the direction in
which the phase of the Schrödinger field changes. If the phase
\theta changes slowly, the flow is slow and has very little energy.
But now \theta 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 \rho 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 |\psi|^2 (=\rho^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.
Abelian Higgs model
In a relativistic
gauge theory, the
vector bosons are natively 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 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
\Phi.
Higgs potential V.
For a fixed value of \lambda the potential is presented
against the real and imaginary parts of \Phi.
The Mexican-hat or champagne-bottle profile
at the ground should be noted.
When the field \Phi(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 \theta (x), i.e., the "phase" of the Higgs field Φ(x), has
only derivative terms. This is not a surprise. Adding a constant to
\theta (x) is a symmetry of the original theory, so different
values of \theta (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 \phi is equal to \Phi. But
now the phase of the field is arbitrary, because gauge
transformations change it. This means that the field \theta (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\Phi.
Nonabelian Higgs mechanism
The Nonabelian Higgs model has the following action:S(\phi ,\mathbf
A) = \int {1\over 4g^2} \mathop{\textrm{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^{\mu \nu} and F_{\mu \nu} (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
\phi 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.
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 which transforms as a vector (a triplet) of
SO(3). If this field has a vacuum expectation value, it points in
some direction in field space. Without loss of generality, one can
choose the z-axis in field space to be the direction that \phi is
pointing, and then the vacuum expectation value of \phi is (0,0,A),
where A is a constant with dimensions of mass (\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, and this is the
unbroken gauge group. Rotations around the x and y axis do not
preserve the vacuum, and the components of the SO(3) 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 U(1) 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.
Standard model Higgs mechanism
The gauge group of the electroweak part of the standard model is
\mathrm{SU}(2)\times \mathrm{U}(1). The Higgs mechanism is by a
scalar field which is a weak SU(2) doublet with weak hypercharge
−1, it has four real components or two complex components, and it
transforms as a spinor under SU(2) and gets multiplied by a phase
under U(1) rotations. Note that this is not the same as two complex
spinors which mix under U(1), which would have eight real
components, rather this is the spinor representation of the group
U(2)--- multiplying by a phase mixes the real and imaginary part of
the complex spinor into each other.
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 makes the vacuum expection value of H the spinor
(A,0). The generators for rotations about the x,y,z axes are by
half the Pauli matrices \sigma_x,\sigma_y,\sigma_z, so that a
rotation of angle \theta about the z axis takes the vacuum
to:
- : (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, which multiplies by the
opposite phase, since the Higgs has charge −1. Under both an SU(2)
z-rotation and a U(1) rotation by an amount \theta/2, the vacuum is
invariant. This combination of generators:
- : Q = W_z + {Y/2} \,
defines the unbroken gauge group, where W_z is the generator of
rotations around the z-axis in the SU(2) and Y is the generator of
the U(1). This combination of generators--- perform a z rotation in
the SU(2) and simultaneously perform a U(1) rotation by half the
angle--- preserves the vacuum, and defines the unbroken gauge group
in the standard model. The part of the gauge field in this
direction stays massless, and this gauge field is the actual
photon.
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. To make all
the Y-charges in the standard model integers, you can rescale the Y
part of the formula by tripling all the Y-charges if you like, and
rewrite the charge formula as Q = W_z + Y/6, but the normalization
with Y/2 is the universal standard.
Affine Higgs mechanism
Ernst Stueckelberg discovered a
version of the Higgs mechanism by analyzing the theory of quantum
electrodynamics with a massive photon.
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 disappears. The vector
meson mass is equal to the product eH, and stays finite.
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 has the following gauge
transformation law:
- : \theta + e\alpha\,
- : A \rightarrow 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 fluctuations finite and nonzero in this
limit, \theta 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 \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 is the gauge boson mass. By making a gauge
transformation to set \scriptstyle \theta=0, the gauge freedom in
the action is eliminated, and the action becomes that of a massive
vector field:
- : 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 transforms as an affine representation of the gauge group.
Among the allowed gauge groups, only non-compact U(1) admits affine
representations, and the U(1) 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.
References
- G. Bernardi, M. Carena, and T. Junk: "Higgs bosons: theory and
searches", Reviews of Particle Data Group: Hypothetical particles
and Concepts, 2007,
http://pdg.lbl.gov/2008/reviews/higgs_s055.pdf
- Higgs–Brout–Englert–Guralnik–Hagen–Kibble Mechanism
on Scholarpedia
- Physical Review Letters – 50th Anniversary Milestone
Papers
- American Physical Society – J. J. Sakurai Prize
Winners
See also
Further reading
General readers:
- Schumm, Bruce A. (2004) Deep Down Things. Johns
Hopkins Univ. Press. Chpt. 9.
External links