Chiral perturbation theory

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 (LECs), are associated with terms in the Lagrangian that can be determined by fitting to experimental data or be derived from underlying theory.

There are several power counting schemes in ChPT. The most widely used one is the ${\displaystyle p}$-expansion. However, there also exist the ${\displaystyle \epsilon }$, ${\displaystyle \delta ,}$ and ${\displaystyle \epsilon ^{\prime }}$ expansions. All of these expansions are valid in finite volume, (though the ${\displaystyle p}$ 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.

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