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.
[1] ChPT is a theory which allows one to study the low-energy dynamics of QCD on the basis of this underlying chiral symmetry.
Due to the running of the strong coupling constant, we can apply perturbation theory in the coupling constant only at high energies.
But in the low-energy regime of QCD, the degrees of freedom are no longer quarks and gluons, but rather hadrons.
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.
Because 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 an alternative method that has proved successful in extracting non-perturbative information.
Using different degrees of freedom, we have to assure that observables calculated in the EFT are related to those of the underlying theory.
This is achieved by using the most general Lagrangian that is consistent with the symmetries of the underlying theory, as this yields the ‘‘most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry.
[2][3] In general there is an infinite number of terms which meet this requirement.
Therefore in order to make any physical predictions, one assigns to the theory a power-ordering scheme which organizes terms by some pre-determined degree of importance.
The ordering allows one to keep some terms and omit all other, higher-order corrections which can safely be temporarily ignored.
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.
In addition to the ordering scheme, most terms in the approximate Lagrangian will be multiplied by coupling constants which represent the relative strengths of the force represented by each term.
The constants can be determined by fitting to experimental data or be derived from underlying theory.
-expansion is constructed by writing down all interactions which are not excluded by symmetry, and then ordering them based on the number of momentum and mass powers.
the pion mass, which breaks the underlying chiral symmetry explicitly (PCAC).
It is also customary to compress the Lagrangian by replacing the single pion fields in each term with an infinite series of all possible combinations of pion fields.
is called the pion decay constant which is 93 MeV.
exist, so that one must choose the value that is consistent with the charged pion decay rate.
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.
, one removes the divergences in the calculation with the renormalization of the low-energy constants (LECs) from the
So if one wishes to remove all the divergences in the computation of a given observable to
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.
For an SU(2) theory the leading order chiral Lagrangian is given by[1] where
is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as
[6] 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.