Chiral perturbation theory

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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 underlining theory.

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

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 (\partial \pi)^2 + m_{\pi}^2 \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\left\{\frac{i}{F}  \begin{pmatrix} \pi^0  &  \sqrt{2}\pi^+ \\ \sqrt{2}\pi^- & - \pi^0 \end{pmatrix}\right\}
where F = 93 MeV. In general different choices of the normalization for F exist and one must choose the value that is consistent with the charged pion decay rate.

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


\mathcal{L}_{2}=\frac{F^2}{4}{\rm tr}(\partial_{\mu}U \partial^{\mu}U^{\dagger})+\frac{\lambda F^3}{4}{\rm tr}(m_q U+m_q^{\dagger}U^{\dagger})

where F = 93 MeV and m_q is the quark mass matrix. In the p-expansion of ChPT, the small expansion parameters are


\frac{p}{\Lambda_{\chi}}, \frac{m_{\pi}}{\Lambda_{\chi}}.

where \Lambda_{\chi} is the chiral symmetry breaking scale, of order 1 GeV (sometimes estimated as \Lambda_{\chi} = 4\pi F). In this expansion, m_q counts as \mathcal{O}(p^2) because m_{\pi}^2=\lambda m_q F to leading order in the chiral expansion (Gell-Mann, M.; Renner, B. (1968). "Behavior of Current Divergences under SU_{3}×SU_{3}". Physical Review 175 (5): 2195. Bibcode:1968PhRv..175.2195G. doi:10.1103/PhysRev.175.2195. ).

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 \mathcal{O}(p^4), then one must compute the contact terms that come from the \mathcal{O}(p^4) Lagrangian (this is different for an SU(2) vs. SU(3) theory) at tree-level and the one-loop contributions from the \mathcal{O}(p^2) Lagrangian.) One can easily see that a one-loop contribution from the \mathcal{O}(p^2) Lagrangian counts as \mathcal{O}(p^4) by noting that the integration measure counts as p^4, the propagator counts as p^{-2}, while the derivative contributions count as p^2. Therefore, since the calculation is valid to \mathcal{O}(p^4), one removes the divergences in the calculation with the renormalization of the low-energy constants (LECs) from the \mathcal{O}(p^4) Lagrangian. Therefore, if one wishes to remove all the divergences in the computation of a given observable to \mathcal{O}(p^n), one uses the coupling constants in the expression for the \mathcal{O}(p^n) Lagrangian to remove those divergences.

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.

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