Pseudo-Goldstone boson
Encyclopedia
Pseudo-Goldstone bosons arise in a quantum field theory
with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken
(hidden), and would thus engender massless Goldstone boson
s. The additional explicit symmetry breaking gives these bosons a small mass. The properties of these pseudo-Goldstone bosons can normally be found by an expansion around the (exactly) symmetric theory in terms of the explicit symmetry-breaking parameters.
Quantum chromodynamics
(QCD), the theory of strong particle interactions, provides the best known example in nature; see the article on the QCD vacuum
for details. Experimentally, it is observed that the masses of the octet of pseudoscalar meson
s (such as the pion
) are very much lighter than the next heavier states, ie, the octet of vector meson
s (such as the rho meson
).
In QCD, this is interpreted as a consequence of spontaneous symmetry breaking
of chiral symmetry in a sector of QCD with 3 flavours of light quarks. Such a theory, for idealized massless quarks, has global
chiral flavour
symmetry. Under SSB, this is spontaneously broken to the diagonal SU(3), generating eight Goldstone boson
s, which are the pseudoscalar mesons transforming as an octet representation
of flavour SU(3)
.
In actual full QCD, the small quark
masses further break the chiral symmetry explicitly as well. The masses of the actual pseudoscalar meson octet are found by an expansion in the quark masses, which goes by the name of chiral perturbation theory
. The internal consistency of this argument is further checked by lattice QCD
computations, which allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark masses is as dictated by chiral perturbation theory
, effectively as the square-root of the quark masses.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
with both spontaneous and explicit symmetry breaking. The controlling approximate symmetries, if they were exact, would be spontaneously broken
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
(hidden), and would thus engender massless Goldstone boson
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
s. The additional explicit symmetry breaking gives these bosons a small mass. The properties of these pseudo-Goldstone bosons can normally be found by an expansion around the (exactly) symmetric theory in terms of the explicit symmetry-breaking parameters.
Quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
(QCD), the theory of strong particle interactions, provides the best known example in nature; see the article on the QCD vacuum
QCD vacuum
The QCD vacuum is the vacuum state of quantum chromodynamics . It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate...
for details. Experimentally, it is observed that the masses of the octet of pseudoscalar meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s (such as the pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
) are very much lighter than the next heavier states, ie, the octet of vector meson
Vector meson
In high energy physics, a vector meson is a meson with total spin 1 and odd parity . Compare to a pseudovector meson, which has a total spin 1 and even parity....
s (such as the rho meson
Rho meson
In particle physics, a rho meson is a short-lived hadronic particle that is an isospin triplet whose three states are denoted as , and . After the pions and kaons, the rho mesons are the lightest strongly interacting particle with a mass of roughly for all three states...
).
In QCD, this is interpreted as a consequence of spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
of chiral symmetry in a sector of QCD with 3 flavours of light quarks. Such a theory, for idealized massless quarks, has global
chiral flavourFlavour (particle physics)
In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics, flavour is a global symmetry...
symmetry. Under SSB, this is spontaneously broken to the diagonal SU(3), generating eight Goldstone boson
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
s, which are the pseudoscalar mesons transforming as an octet representation
Adjoint representation
In mathematics, the adjoint representation of a Lie group G is the natural representation of G on its own Lie algebra...
of flavour SU(3)
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
.
In actual full QCD, the small quark
Quark
A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
masses further break the chiral symmetry explicitly as well. The masses of the actual pseudoscalar meson octet are found by an expansion in the quark masses, which goes by the name of chiral perturbation theory
Chiral perturbation theory
Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...
. The internal consistency of this argument is further checked by lattice QCD
Lattice QCD
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics theory of quarks and gluons. It is a lattice gauge theory formulated on a grid or lattice of points in space and time....
computations, which allow one to vary the quark mass and check that the variation of the pseudoscalar masses with the quark masses is as dictated by chiral perturbation theory
Chiral perturbation theory
Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...
, effectively as the square-root of the quark masses.
See also
- Spontaneous symmetry breakingSpontaneous symmetry breakingSpontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
- Goldstone bosonGoldstone bosonIn particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
s - QCD vacuumQCD vacuumThe QCD vacuum is the vacuum state of quantum chromodynamics . It is an example of a non-perturbative vacuum state, characterized by many non-vanishing condensates such as the gluon condensate or the quark condensate...
- Little HiggsLittle HiggsIn 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...