Using different degrees of freedom, we have to assure that observables calculated in the EFT are related to those of the underlying theory. According to
Steven Weinberg's "
folk theorem" it is achieved by using the most general Lagrangian that is consistent with the symmetries of the underlying theory, as this yields the ‘‘most general possible
S-matrix consistent with
analyticity, perturbative
unitarity,
cluster decomposition and the assumed symmetry. In general there is 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 safely be temporarily ignored. There are several power counting schemes in ChPT. The most widely used one is the p-expansion where p stands for momentum. 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. 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 Ls – are usually not known. The constants can be determined by fitting to experimental data or be derived from underlying theory.
The model Lagrangian 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 (\partial \pi)^2 + m_{\pi}^2 \pi^2 is considered in the first-order approximation, where \pi is the
pion field and m_{\pi} the pion mass, which breaks the underlying chiral symmetry explicitly (PCAC). Terms like m_{\pi}^4 \pi^2 + (\partial \pi)^6 are part of other, higher order corrections. It is also customary 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 is called the
pion decay constant which is 93 MeV. In general, different choices of the normalization for F exist, so that one must choose the value that is consistent with the charged pion decay rate.
Renormalization 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. So 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. ==Successful application==