Chiral Symmetry in algebraic and analytic approaches

We compare among themselves two different methods for the derivation of results following from the requirement of polynomial boundedness of tree-level chiral amplitudes. It is shown that the results of the algebraic approach are valid also in the framework of the analytical one. This means that the system of Sum Rules and Bootstrap equations previously obtained with the help of the latter approach can be analyzed in terms of reducible representations of the unbroken Chiral group with the known structure of the mass matrix. Novadays it is commonly realised that the Spontaneously Broken Chiral Symmetry (SBChS) plays an essential role in the hadronization mechanism provided by the large-distance forces in OCD. The smallness of the pion mass (compared to the typical hadronic scale of 1 Gev) made it possible to develop the Chiral Perturbation Theory (ChPT) [1, 2] (for recent reviews see also [3, 4, 5]) allowing one to calculate in a systematic manner the corrections to the zero-order (”bare”) amplitude given by a sum of corresponding tree graphs. Thus, if the bare amplitudes are known, we can analyze the low energy processes with the arbitrary high degree of accuracy. Unfortunately, the bare (tree-level) amplitudes themselves contain a lot of free parameters since they depend of infinite number of basically unknown coupling constants describing the off-shell hadron interactions and point-like vertices. Just because of this reason the current situation looks unsatisfactory: we have an excellent tool – ChPT – for the computation of corrections to the quantities which we know not so much about. A remarkable step in the understanding of the bare couplings was done by S.Weinberg [6], first suggested to use the asymptotic boundedness requirements for the chiral amplitudes describing the forward scattering of massless pions on arbitrary targets (the list of the corresponding refs. can be found, e.g., in [7] ; the paper [8] should be added to this list). It was shown that the SBChS – despite of its dynamical origin – manifests itself in a customary algebraic manner. This means that it is possible to classify the states with a given helicity according to linear reducible representations of unbroken Chiral group, the mass matrix being constructed as a sum of chiral scalar and 4-th component of chiral vector. A variety of interesting results derived by many authors in the framework of Weinberg’s scheme clearly demonstrates both the importance and the fruitfulness of the asymptotic requirements. There are known two technically different approaches to the derivation of results from those requirements. Weinberg’s original approach [6] (later on called algebraic) consisting of the group theoretical analysis of mass (squared) and coupling matrices, shows both attractive features and serious limits. One of the most attractive features of this method is that it allows one to work from the very beginning with purely algebraic structures. However, it is applicable only for the case of forward scattering of massless pions; its feasibility is limited on a small number of partial waves taken into account. Moreover (as stressed by Weinberg himself [6]), from the mathematical point of view this method is only valid if the number of resonances is finite. A different approach (first suggested in [9]) has been recently formulated in its final form in [7]. Later on we call this approach analytic. It takes advantage of the analytic structure of a tree-level amplitude, this structure being defined by the requirements of meromorphy and polynomial boundedness. In this method there is no necessity to require that pion is massless. Also, it is not restricted to forward processes only: the nonforward ones can be analyzed on the same ground. The method does not contain