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Relativistic Unitary Description of Ππ Scattering Relativistic unitary description of ππ scattering M. A. Pichowsky, A. Szczepaniak, and J. T. Londergan Department of Physics and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47405 ranging from the two-pion threshold up to 1400 MeV. A unitary framework based on the Bakamjian-Thomas This system provides an excellent test for the framework. construction of relativistic quantum mechanics is used to de- A relativistic treatment is quite important when dealing scribe two-pion scattering from threshold to 1400 MeV. The with particles as light as pions and the interplay between framework properly includes unitarity cuts for one-, two- and strong dynamics and chiral symmetry makes this system three-hadron states and provides an excellent description of quite interesting. The ππ system is somewhat simpler the available data for ππ phase shifts and inelasticities. The role and importance of three-hadron cuts are calculated and than others, in that its study requires only a minimal discussed. complication from the proper implementation of rela- tivistic spins, since both of the two-body states (ππ and KK¯ ) involved are comprised of spin-0 particles. Another I. INTRODUCTION attractive aspect of applying the framework to ππ scat- tering is the relative wealth of experimental data for the isoscalar, S-wave channel. A nonperturbative framework capable of describing the One drawback with using ππ scattering as a touchstone relativistic, coupled-channel scattering of hadrons is pre- may be that inelasticities due to open states of three or sented. The approach is based on a relativistic Hamil- more particles, do not appear to be significant for this tonian formulation with model interactions introduced process; that is, the ρππ and ππππ thresholds seem to into the mass operator, and with few-body states imple- have little impact on the S-andP -wave observables. The mented in a way that maintains the unitarity of the the- effects of the opening of these three- and four-body chan- ory. The elementary degrees of freedom in the framework nels seem to be overwhelmed by the opening of the two- are finite-sized hadrons which provide a natural ultravi- body KK¯ channel. Nonetheless, several important as- olet regularization, ensuring that the scattering ampli- pects of the framework can be explored in an application tudes are finite. to ππ scattering. The Hilbert space is truncated to include only one-, The isoscalar-scalar (I =0,J PC =0++) channel of ππ two- and three-body states. A central and novel fea- scattering has been a subject of numerous and extensive ture of this framework is the explicit inclusion of both studies. The study of meson scattering in this low-energy real and imaginary parts of scattering amplitudes arising region may be an ideal test of our understanding of the from the opening of three-body channels. The proper interplay between bound states in QCD and chiral dy- handling of three-body unitarity cuts is crucial to gain- namics. The region near E = 1000 MeV is perhaps most ing a deeper understanding of several well-known scatter- interesting, as it is dominated by the mixing between ππ ing systems; a good example is πN scattering in the P 11 and KK¯ channels and the isoscalar-scalar f (980) meson channel, which exhibits a significant inelasticity arising 0 resonance. The nature of the f (980) resonance, and the from the intermediate three-body ππN state [1]. 0 question of whether it is comprised of valence quarks or It presents a formidable challenge to develop a general, arises purely from meson scattering dynamics, has been relativistic scattering framework to describe the final- addressed by many authors [2–8]. Above this energy re- state interactions between hadrons, that includes the ef- gion, three additional scalar meson resonances have been fects of three-body unitarity cuts. Nonetheless, a practi- well established. These are referred to as the f (1370), cal framework which can treat hadron reactions beyond 0 f (1500) and the f (1710). It is still unclear which of the lowest-order valence quark picture is clearly desirable. 0 0 these should be considered as quark-antiquark bound For example, the systematic analysis of hadron reactions states, glueballs or possibly resonances arising from dy- in the baryon resonance region currently being conducted namical effects of final-state interactions [9–13]. Thus, in Hall B at the Thomas Jefferson National Accelerator the isoscalar-scalar channel remains a source of great in- Facility (TJNAF) requires that such a framework be used terest and mystery for meson phenomenology. to extract information about baryon resonances in this Although there have been previous studies which em- highly complex dynamical region. The framework devel- ploy a framework similar to the one developed here, there oped herein is an attempt to construct a useful, relativis- are some important differences. Most studies of meson tic framework capable of describing the nonperturbative, scattering dynamics are based on potential models (as is low-momentum transfer final-state interactions between the framework developed here.) However, most other ap- hadrons in a unitary manner. proaches typically include one- and two-particle channels For the first application of the framework developed only; that is, they include s-channel states and several here, a simple model for ππ scattering is introduced and two-particle channels, such as ππ, KK¯ , σσ,etc.They used to described the S and P partial waves for energies 1 either neglect the possibility of open three-particle chan- Sec. II A a proof of the covariance of this approach is nels altogether or only partially implement them. For ex- provided. Furthermore, Betz and Coester [18] show that ample, in the model developed by the Julich group [14], such a framework can satisfy cluster separability. All of the interaction potentials between two-particle channels, these features are desirable for the study of hadron scat- such as ππ-ππ or ππ-KK¯ interactions are obtained us- tering. ing an instantaneous approximation of a meson-exchange In Sec. II A, the Lorentz covariance of the framework model. Such instantaneous approximations generally do is demonstrated and the fully-interacting mass operator not account for absorptive effects due to the opening of is constructed. It is shown that the framework leads three-body channels. Still, there is no question that the toM Lorentz-invariant on-shell T-matrix elements T (E,P) Julich model quite successfully describes the phase shifts for colliding particles with total momentum P and en- and inelasticities of ππ scattering for S, P and D waves. ergy E; that is, one finds T (E,P)=T (√s)where Alternatively, the Krakow group [15] has developed a √s = √E2 P2 is the invariant mass of the system. separable-potential model for ππ scattering. In their cal- The Poincare− generators act on a Hilbert space which, culation, few-body dynamical effects are incorporated by in general, contains an infinite number of states. The including additional, effective two-body channels, such Hilbert space is truncated to include only those states es- as a σσ channel [16]. Their model also obtains excellent sential to describe the scattering system of interest within results for the ππ phase shifts and inelasticities. a particular energy range. Here, only one-, two- and The outline of this article is as follows. In Sec. II, three-particle states are maintained. Following this trun- the relativistic scattering formalism employed herein is cation, the operator form of the Lippmann-Schwinger briefly discussed, beginning with a short proof of the co- equation can be written as a set of coupled integral equa- variance of observables calculated within this framework. tions. The input that determines the dynamics is given Then, the integral equations that relate the one-, two- in terms of the matrix elements of a model potential V . and three-body scattering T -matrices are provided. Once these are provided, the full scattering problem is In Sec. III, the framework is applied to a study of ππ solved in a straightforward manner. scattering. The particle states that are included in the model are discussed, along with the necessary dynam- ical model parameters. The interactions employed in A. Relativistic covariance this study arise from the meson exchanges which cou- ple states of various numbers of particles to each other. A simple realization of the Poincare algebra for an In our framework, these interactions arise from one-, two- interacting system of a finite number of constituents and three-meson intermediate states which may exhibit is given by the Bakamjian-Thomas construction [17]. production thresholds, resulting in absorptive contribu- This approach has the advantage of providing a Lorentz- tions to the kernels and self-energies appearing in the covariant generalization for a large class of noncovari- Lippmann-Schwinger equations. A simple fitting proce- ant microscopic models, such as the constituent quark dure is shown to provide excellent agreement with data model. In principle, a noncovariant microscopic model for ππ scattering phase shifts and inelasticities. Details could be used to obtain matrix elements of the under- of the relevant model dynamics that produce the various lying elementary hadronic potentials V . In this case, features observed in the resulting phase shifts and inelas- one might consider this framework as a means to extend ticities are discussed. Then, it is shown that the numer- the original noncovariant model dynamics, allowing for a ical methods employed herein are sufficient to maintain Lorentz-covariant treatment of scattering phenomena. the unitarity of the framework to better than one part in The explicit construction of the Poincare algebra pro- a million. Finally, in Sec. IV, the article is summarized ceeds as follows. Starting from a system of noninteracting and plans for future studies are presented.
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