Chiral perturbation theory
Encyclopedia
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 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" the QCD partition function
, (such that the degrees of freedom in the Lagrangian are replaced by hadrons) then one could extract information about low-energy physics. To date this has not been accomplished. A low-energy effective theory with hadrons as the fundamental degrees of freedom is a possible solution. 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 can be determined by fitting to experimental data or be derived from underlining theory.
There are several power counting schemes in ChPT. The most widely used one is the
-expansion. However, there also exist the 
, 
and 
expansions. All of these expansions are valid in finite volume, (though the 
expansion is the only one valid in infinite volume.) Particular choices of finite volumes require one to use different reorganizations of the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes.
The Lagrangian of the
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
MeV. In general different choices for 
exist and one must specify the value one chooses before beginning any computations.
The theory allows the description of interactions between pion
s, and between pions and nucleon
s (or other matter fields). SU(3) ChPT can also describe interactions of kaon
s 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.
For an SU(2) theory the leading order chiral Lagrangian is given by
where
MeV and 
is the quark mass matrix. In the 
-expansion of ChPT, the small expansion parameters are
In this expansion,
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 one must compute the contact terms that come from the 
Lagrangian (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 
, the propagator
counts as
, while the derivative contributions count as 
. 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 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.
Effective field theory
In physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...
constructed with a Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
consistent with the (approximate) chiral symmetry
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
of 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), as well as the other symmetries of parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
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
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....
is one alternative method that has proved successful in extracting non-perturbative information.
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" the QCD partition function
Partition function (quantum field theory)
In quantum field theory, we have a generating functional, Z[J] of correlation functions and this value, called the partition function is usually expressed by something like the following functional integral:...
, (such that the degrees of freedom in the Lagrangian are replaced by hadrons) then one could extract information about low-energy physics. To date this has not been accomplished. A low-energy effective theory with hadrons as the fundamental degrees of freedom is a possible solution. According to Steven Weinberg
Steven Weinberg
Steven Weinberg is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles....
, 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 can be determined by fitting to experimental data or be derived from underlining theory.
There are several power counting schemes in ChPT. The most widely used one is the
-expansion. However, there also exist the , and expansions. All of these expansions are valid in finite volume, (though the expansion is the only one valid in infinite volume.) Particular choices of finite volumes require one to use different reorganizations of the chiral theory in order to correctly understand the physics. These different reorganizations correspond to the different power counting schemes.The Lagrangian of the
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 iswhere
MeV. In general different choices for exist and one must specify the value one chooses before beginning any computations.The theory allows the description of interactions between 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....
s, and between pions and nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
s (or other matter fields). SU(3) ChPT can also describe interactions of kaon
Kaon
In particle physics, a kaon is any one of a group of four mesons distinguished by the fact that they carry a quantum number called strangeness...
s and eta mesons, while similar theories can be used to describe the vector mesons. Since chiral perturbation theory assumes chiral symmetry
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
, and therefore massless quarks, it cannot be used to model interactions of the heavier quarks.
For an SU(2) theory the leading order chiral Lagrangian is given by
where
MeV and is the quark mass matrix. In the -expansion of ChPT, the small expansion parameters areIn this expansion,
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
Observable
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. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
to
, then one must compute the contact terms that come from the Lagrangian (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 , the propagatorPropagator
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. Propagators are used to represent the contribution of virtual particles on the internal...
counts as
, while the derivative contributions count as . Therefore, since the calculation is valid to , one removes the divergences in the calculation with the renormalization of the low-energy constants (LECLEC
LEC can stand for:* Levelised energy cost, the average whole-lifetime cost of electricity from a power plant.* Lake City , Florida, United States; Amtrak station code LEC* Lake Erie College, in Painesville, Ohio...
'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 some cases, chiral perturbation theory has been successful in describing the interactions between hadrons in the non-perturbative regime of the strong interaction
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
. For instance, it can be applied to few-nucleon systems, and at next-to-next-to-leading order in the perturbative expansion
Perturbation theory
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...
, it can account for three-nucleon forces in a natural way.