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. 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. 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. 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. 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 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 be safely, temporarily ignored. There are several power counting schemes in ChPT. The most widely used one is the p {displaystyle p} -expansion. However, there also exist the ϵ {displaystyle epsilon } , δ , {displaystyle delta ,} and ϵ ′ {displaystyle epsilon ^{prime }} expansions. All of these expansions are valid in finite volume, (though the p {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. 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. Values of these constants – also called low-energy constants or LECs – are usually not known. The constants can be determined by fitting to experimental data or be derived from underlying theory. The Lagrangian of the p-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 order is chosen so that ( ∂ π ) 2 + m π 2 π 2 {displaystyle (partial pi )^{2}+m_{pi }^{2}pi ^{2}} is considered in the first-order approximation, where π {displaystyle pi } is the pion field and m π {displaystyle m_{pi }} the pion mass. Terms like m π 4 π 2 + ( ∂ π ) 6 {displaystyle m_{pi }^{4}pi ^{2}+(partial pi )^{6}} are part of other, higher order corrections.