Calculation of the chiral Lagrangian coefficients from the underlying theory of QCD: A simple approach

Abstract

We calculate the coefficients in the chiral Lagrangian approximately from QCD based on a previous study of deriving the chiral Lagrangian from the first principles of QCD in which the chiral Lagrangian coefficients are defined in terms of certain Green's functions in QCD. We first show that, in the large-Nc limit, the anomaly part contributions to the coefficients are exactly cancelled by certain terms in the normal part contributions, and the final results of the coefficients only concern the remaining normal part contributions depending on QCD interactions. We then do the calculation in a simple approach with the approximations of taking the large-Nc limit, the leading order in dynamical perturbation theory, and the improved ladder approximation, thereby the relevant Green's functions are expressed in terms of the quark self energy Σ(p 2 ). By solving the Schwinger-Dyson equation for Σ(p 2 ), we obtain the approximate QCD predicted coefficients and quark condensate which are consistent with the experimental values.

Derivation of the effective chiral Lagrangian for pseudoscalar mesons from QCD

“…The first one is to introduce a chiral rotation (33) which simplifies the U (x)-dependence in such a way that the U (x)-dependence resides only implicitly in the rotated sources and some ro- (73), our formula reduces to the well-known Pagels-Stokar formula for the pion decay constant F π [11]. The complete calculation of the fifteen coefficients will be presented in a separate paper [12]. The derivation of the effective chiral Lagrangian further including the η ′ meson or the ρ meson, and the application of the present approach to the electroweak chiral Lagrangian are all in preparation.…”

“…So far, we have given the formal QCD definitions of the fourteen coefficients of the chiral Lagrangian up to the p 4order. To get the values of the coefficients, we need to solve the relevant Green's functions which is a hard task, and we shall present the calculations in a separate paper [12].…”

“…In the literature, a further approximation of dropping the last term in (73) is usually taken [18] (It can be shown that to leading order in dynamical perturbation [11], this term can be reasonably ignored [12].). Moreover, to leading order in dynamical perturbation or in the Landau gauge, Z(−p 2 ) = 1.…”

Chiral effective Lagrangian for heavy-light mesons from QCD

“…In order to keep the invariance of the vector part of the chiral transformation, we follow Ref. [6] and take the form of the self-energy of the light quark as Σðx − yÞ ¼ Σð∇ 2 Þδðx − yÞ, where Σ is the self-energy function in the momentum space and ∇ ≡ ∂ − iV Ω is the covariant derivative. This form retains the correct chiral transformation properties in the theory.…”

“…The derivation is based on the standard generating functional of QCD with external bilinear light-quark field sources in the path integral formalism so that, the coefficients in the chiral Lagrangian can be derived from QCD Green functions. Thanks to the developments of Dyson-Schwinger equation (DSE) and lattice QCD methods, although an exact calculation of the nonperturbative QCD Green functions is not feasible so far, this derivation provides a possibility to numerically calculate the low energy constants (LECs) of ChPT from QCD-the fundamental theory of strong interaction [6][7][8][9][10]. The extension of this approach to include other light mesons are also investigated in the literature [11][12][13].…”