Bethe-Salpeter Approach for Unitarized Chiral Perturbation Theory

PACS: 11.10.St;11.30.Rd; 11.80.Et; 13.75.Lb; 14.40.Cs; 14.40.Aq Keywords: Bethe-Salpeter Equation, Chiral Perturbation Theory, Unitarity, ππ -Scattering, Resonances.
arXiv:hep-ph/9907469v2 10 Mar 2000
The Bethe-Salpeter equation restores exact elastic unitarity in the s− channel by summing up an infinite set of chiral loops. We use this equation to show how a chiral expansion can be undertaken in the two particle irreducible amplitude and the propagators accomplishing exact elastic unitarity at any step. Renormalizability of the amplitudes can be achieved by allowing for an infinite set of counter-terms as it is the case in ordinary Chiral Perturbation Theory. Crossing constraints can be imposed on the parameters to a given order. Within this framework, we calculate the leading and next-to-leading contributions to the elastic ππ scattering amplitudes, for all isospin channels, and to the vector and scalar pion form factors in several renormalization schemes. A satisfactory description of amplitudes and form factors is obtained. In this latter case, Watson’s theorem is automatically satisfied. From such studies we obtain a quite accurate determination of .1 some of the ChPT SU (2)−low energy parameters (¯ l1 − ¯ l2 = −6.1+0 −0.3 and ¯ l6 = 19.14 ± 0.19). We also compare the two loop piece of our amplitudes to recent two–loop calculations.
Bethe-Salpeter Approach for Unitarized Chiral Perturbation Theory.
J. Nieves and E. Ruiz Arriola
Departamento de F´ ısica Moderna Universidad de Granada E-18071 Granada, Spain (February 1, 2008)
I. INTRODUCTION
The dynamical origin of resonances in ππ scattering has been a recurrent subject in low energy particle physics [1]. Analyticity, unitarity, crossing and chiral symmetry have provided main insights into this subject. It is well known that so far no solution is available exactly fulfilling all these requirements. In practical calculations some of the properties mentioned above have to be given up. Standard Chiral Perturbation Theory (ChPT) furnishes exact crossing and restores unitarity order by order in the chiral expansion. In typical calculations the unitarity limit is reached at about a center of mass (CM) energy √ √ s ∼ 4 πf ∼ 670 MeV1 still a low scale compared to the known resonances. Simply because of this reason standard ChPT is unable to describe the physically observed resonances, namely the ρ and the σ . But even if the unitarity limit was much larger, and thus resonances would appear at significantly smaller scales than it, standard ChPT would be unable to generate them since its applicability requires the existence of a gap between the pion states and the hadronic states next in energy, which for ππ scattering are precisely the resonances. Resonances clearly indicate the presence of non perturbative physics, and thus a pole on the second Riemann sheet (signal of the resonance) cannot be obtained in perturbation theory to finite order. Hence, the energy regime for which ChPT holds has to satisfy s << m2 R regardless of the unitarity limit. It so happens that both the resonances and the unitarity limit lead to similar scales, but there is so far no compelling reason to ascribe this coincidence to some underlying dynamical feature or symmetry2 . The desire to describe resonances using standard ChPT as a guide has led some authors to propose several approaches which favor some of the properties, that the exact scattering amplitude should satisfy, respect to others. Thus, Pade Re-summation (PR) [2], Large Nf −Expansion (LNE) [3], Inverse Amplitude Method (IAM) [4] , Current Algebra Unitarization (CAU) [5], Dispersion Relations (DP) [6], Roy Equations [7], Coupled Channel Lippmann-Schwinger Approach (CCLS) [8] and hybrid approaches [9] have been suggested. Besides their advantages and success to describe the data in the low-lying resonance region, any of them has specific drawbacks. In all above approaches except by LNE and CCLS it is not clear which is the ChPT series of diagrams which has been summed up. This is not the case for the CCLS approach, but there a three momentum cut-off is introduced, hence breaking translational Lorentz invariance and therefore the scattering amplitude can be only evaluated in the the CM frame. On the other hand, though the LNE and CAU approaches preserve crossing symmetry, both of them violate unitarity. Likewise, those approaches which preserve exact unitarity violate crossing symmetry. A clear advantage of maintaining elastic unitarity lies in the unambiguous identification of the phase shifts. This is particularly useful to describe resonances since the modulus of