Non-Static Effects in a Form Factor of a Relativistic Three-Body Bound State in 1+1 Dimensions

T. Ino, Y. Munakata, J. Sakamoto, F. Yamamoto, T. Nakamae
1990 Progress of theoretical physics  
835 We argue the form factor of a relativistic three·body bound state presented recently, whose wave function includes a new dynamical phase factor. The new factor is found not to influence the asymptotic power behaviour with 'respect to Q2. We propose a new-type of formul'a to relativize a form factor given by a non-relativistic model. Recently Glockle, Nogami and FukuF) (GNF) found an interesting model of two~body composite system in one space dimension, which satisfies all the requirements
more » ... the requirements of quantum mechanics and special relativity, including' the Lorentz contraction of the composite system. The model is analytically solvable and clarifies the structure of a relativistic composite system and its relation to the overall translational motion. Therefore the model is able to reveal an intriguing relation between the form factor of the composite system (simulating the form factor of the deuteron determined by electron scattering) and the static density distribution of the system. As a natural extension of the GNF model (and the work in Ref. 2)), the authors have, in a more recent work,3) presented a model for three Dirac particles interacting through pairwise direct instantaneous interactions in 1 + 1 dimensions and obtained an exact solution of relativistic three-body bound state (which is normal, that is, has a correspondence with the solution in the non-relativistic problem 4 ))_ (The work of Ref. 3) is referred to as I hereafter.) In this work, we shall calculate and discuss the form factor of the relativistic three-body bound state in I, which simulates the form factor of the triton (or that of the proton). In conclusion, we shall propose a formula to relativize a form factor given by a non-relativistic model in the real world. First we state the Breit-type equation and its exact solution in 1. The Hamiltonian, the boost operator and the Breit-type equation in I are H=Hl + H2+ H3+ 1123+ ViI + Vi2, K=Kl + K2+ K3+ X 23 1123+ X 31 ViI + Xi2 Vi2, (1) wh<;re Hi~aiPi+/3im, Vij=-g(l-aiaJo(xi-Xj) , K;=(XiHi+ Hixi)/2, Xij=(xi+xJ/2. (2) Here i and j refer to the three particles 1, 2 and 3, and a and /3 are the Dirac matrices taken to be a=G ~) and /3=(~ _~), and the momentum and position of the particle i are denoted by Pi and Xi ([Xi, Pj] = io;;). The three particles are assumed to have equal
doi:10.1143/ptp.83.835 fatcat:6lzejhgk25b23poeues7oztxqq