Chiral perturbation theory (ChPT) is an effective field theory constructed with a lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), 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. As QCD becomes non-perturbative at low-energy, it is impossible to use perturbative methods to extract information from the partition function of QCD. Lattice QCD is one alternative method that has proved successful in extracting non-perturbative information. In physics, an effective field theory is an approximate theory (usually a quantum field theory) that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances (or, equivalently, higher energies). ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... The initialism QCD can mean: Quantum chromodynamics Quintessential Player, formerly known as Quintessential CD Quality, Cost, Delivery, A three-letter acronym used in lean manufacturing This page concerning a three-letter acronym or abbreviation is a disambiguation page — a navigational aid which lists other pages that might otherwise share the... Look up Parity in Wiktionary, the free dictionary Parity is a concept of equality of status or functional equivalence. ... It has been suggested that lattice field theory be merged into this article or section. ...

In the low-energy regime of QCD, the degrees of freedom are no longer quarks and gluons, but rather hadrons. This is a result of confinement. If one could "solve" (in the sense where the degrees of freedom in the Lagrangian are replaced by hadrons) the QCD partition function, then one could extract information regarding low-energy physics. To date this has not been accomplished. Therefore, a low-energy effective theory with hadrons as the fundamental degrees of freedom is a possible solution to extracting the physics. According to Steven Weinberg, an effective theory can be useful if one writes down all terms consistent with the symmetries of the parent theory. In general there are an infinite number of terms which meet this requirement. Therefore in order to make any physical predictions, one assigns the theory a power counting scheme which organizes terms by a pre-specified degree of importance which allows one to keep some terms and reject all others as higher order corrections which can be safely neglected. In addition, unknown coupling constants also called low-energy constants (LEC's) are associated with terms in the Lagrangian that must be determined by fitting to experiment. For other uses of this term, see: Quark (disambiguation) 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, the quarks are subatomic particles thought to be elemental and indivisible. ... In physics, gluons are the bosonic particles which are responsible for the strong nuclear force. ... In particle physics, a hadron is a subatomic particle which experiences the strong nuclear force. ... Confinement is the physics phenomenon that quarks cannot be isolated. ... In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise... Steven Weinberg (born May 3, 1933) is an American physicist. ... LEC can stand for: LucasArts Entertainment Company, George Lucas computer game company Local enterprise company, Scotland Local exchange carrier, a telecommunication service provider Lake Erie College, in Painesville, Ohio Language Evolution and Computation Research Unit, at the University of Edinburgh Light emitting capacitor Category: ...

There are a few different power counting schemes in ChPT. The most widely used one is the p-expansion. However, there also exist the ε, δ, and expansions. All these expansions are also valid if one works in finite volume (though the p expansion is the only one valid in infinite volume) though particular choices of finite volumes requires one to use different reorganizations or the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes.

The Lagrangian of the p expansion is constructed by introducing every interaction of particles which is not excluded by symmetry, and then ordering them based on the number of momentum and mass powers (so that is considered in the first approximation, and terms like are used as higher order corrections). It is also common to compress the Lagrangian by replacing the single pion fields in each term with an infinite series of all possible combinations of pion fields. One of the most common choices is

where f = 132 MeV. In general different choices for f exist and one must specify the value they choose before beginning any computations.

The theory allows the description of interactions between pions, and between pions and nucleons (or other matter fields). SU(3) ChPT can also describe interactions of kaons and eta mesons, while similar theories can be used to describe the vector mesons. Since chiral perturbation theory assumes chiral symmetry, and therefore massless quarks, it cannot be used to model interactions of the heavier quarks. In particle physics, pion (short for pi meson) is the collective name for three subatomic particles: Ï€0, Ï€+ and π−. Pions are the lightest mesons and play an important role in explaining low-energy properties of the strong nuclear force. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... In particle physics, Kaons (also called K-mesons and denoted K) are a group of four mesons distinguished by the fact that they carry a quantum number called strangeness. ... A phenomenon is said to be chiral if it is not identical to its mirror image (see Chirality (mathematics)). The spin of a particle may be used to define a handedness for that particle. ... For other uses of this term, see: Quark (disambiguation) 1974 discovery photograph of a possible charmed baryon, now identified as the Σc++ In particle physics, the quarks are subatomic particles thought to be elemental and indivisible. ...

For an SU(2) theory the leading order chiral Lagrangian is given by

where f = 132 MeV and m_q is the quark mass matrix. In the p-expansion of ChPT, the small expansion parameters are

In this expansion, m_q counts as because to leading order in the chiral expansion.

The effective theory in general is non-renormalizable, however given a particular power counting scheme in ChPT, the effective theory is renormalizable at a given order in the chiral expansion. For example, if one wishes to compute an observable to , then the following will contribute to the calculation. One must compute the contact terms that come from the Lagrangian (Note this is different for an SU(2) vs. SU(3) theory) at tree-level, and the one-loop contributions from the Lagrangian. One can easily see that a one-loop contribution from the Lagrangian counts as by noting that the integration measure counts as p⁴, the propagator counts as p ^{− 2}, while the derivative contributions count as p². Therefore, since the calculation is valid to , one removes the divergences in the calculation with the renormalization of the low-energy constants (LEC's) from the Lagrangian. Therefore, if one wishes to remove all the divergences in the computation of a given observable to , one uses the coupling constants in the expression for the Lagrangian to remove those divergences. In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ... In physics, the adjective renormalizable refers to a theory (usually a quantum field theory) in which all ultraviolet divergences, infinities and other seemingly meaningless results can be cured by the process of renormalization. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In physics, a one-loop Feynman diagram is a connected Feynman diagram with only one cycle (unicyclic). ... In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. ... LEC can stand for: LucasArts Entertainment Company, George Lucas computer game company Local enterprise company, Scotland Local exchange carrier, a telecommunication service provider Lake Erie College, in Painesville, Ohio Language Evolution and Computation Research Unit, at the University of Edinburgh Light emitting capacitor Category: ...

In some cases, chiral perturbation theory has been successful in describing the interactions between hadrons in the non-perturbative regime of the strong interaction. For instance, it can be applied to few-nucleon systems, and at next-to-next-to-leading order in the perturbative expansion, it can account for three-nucleon forces in a natural way. In particle physics, a hadron is a subatomic particle which experiences the strong nuclear force. ... In mathematics and physics, a non-perturbative function or process is one that cannot be described by perturbation theory. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem. ... A three-body force is a force that does not exist in a system of two objects but appears in a three-body system. ...

References and external links

• On the foundations of chiral perturbation theory, H. Leutwyler (Annals of Physics, v 235, 1994, p 165-203)