Chiral perturbation theory

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Chiral perturbation theory is an effective field theory constructed on a lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics, as well as the other symmetries of parity and charge conjugation. The Lagrangian is constructed by introducing every interactions of particles which is not excluded by symmetry, and then ordering them based on the number of momentum and mass powers (so that $(\partial \pi)^2 + m_{\pi} \pi^2$ is considered in the first approximation, and terms like $m_{\pi}^4 \pi^2 + (\partial \pi)^6$ 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

$U = exp(\frac{i}{f_{\pi}} \begin{pmatrix} \pi^0 & \sqrt{2} \pi^+ \\ \sqrt{2} \pi^- & - \pi^0 \end{pmatrix})$

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 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.