Astrid blin an introduction to chiral perturbation theory and its applications
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.
Contents
- Astrid blin an introduction to chiral perturbation theory and its applications
- Chiral perturbation theory
- References
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
The Lagrangian of the p 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
where
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.
For an SU(2) theory the leading order chiral Lagrangian is given by
where
where
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
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.