A renormalizable effective theory for leading logarithms in ChPT

The theory of the strong interaction, Quantum Chromo Dynamics, does not allow the application of the powerful perturbation techniques in the low–energy region. It is however possible to use the symmetries of QCD to construct an effective theory in the low–energy region, chiral perturbation theory (ChPT) [1–3], which allows a systematic perturbative expansion of Green functions in powers of external momenta and quark masses. In actual calculations in ChPT one expects the dominant contribution to stem from the leading effective Lagrangian, which generates the leading chiral logarithm. Even if the latter do not always dominate, it would be very interesting to know the leading chiral logarithms to every order in the perturbative expansion, and to sum them up. In a recent publication [4], we presented a procedure which allows the calculation of leading logarithms of certain Green functions in the chiral limit rather easily. In the present article, we address the question whether it is possible to sum up these leading logarithms to all orders. In a given renormalizable quantum field theory, resummation of logarithms is based on the renormalization group equations (RGE). However, chiral perturbation theory is not renormalizable, and the structure of the RGE is therefore more involved [5]. In order to avoid the problems introduced by the nonrenormalizable nature of chiral perturbation theory, we consider a theory which is renormalizable and reproduces the leading logarithms of chiral perturbation theory. It is then natural to expect that the summation of logarithms in this renormalizable theory can be performed by use of the RGE. It is known since long that the tree–level graphs of the linear sigma model reproduce, at small momenta, the results of current algebra. In the modern language, this means that they agree with the tree–level graphs of ChPT. It