Variational calculation of the phase shifts in theλφ4model

Abstract

In this paper, we develop Schiffs discretization procedure for calculating the phase shifts in the hq54 model in (D + 1 )-dimensional space-time (D > 0) with the Gaussian wave-functional approach. In 1 + 1 and 2+ 1 dimensions, the phase shifts are negative, which indicates the interaction between two pions is repulsive. In 3+ 1 dimensions, the phase shifts vanish. We also discuss the dependences of the phase shifts upon the scattering energy in detail.PACS number(s): 11.80.Fv, 11.80.EtThe scattering process serves as a major source of information about the properties of various elementary particles, while the phase shift is a crucial quantity characterizing this process. Traditionally, the scattering problem is coped with mainly by the covariant perturbation technique. Recently, however, the development of the Gaussian wave-functional (GWF) approach has hinted that it is possible to analytically calculate the phase shift in quantum field theory. In the last decade, the Gaussian wave-functional approach in the Schrodinger picture field theory which was gradually developed by Schiff [I], Rosen [2], Barnes and Ghandour [3], etc., has become a powerful tool for investigating the vacuum structure [4-101 as well as searching for the bound states in quantum field theory [5,11,12]. Being explicitly computed, the two-particle state energy may be analyzed to extract knowledge on the phase shift. As an attempt, we have investigated the scattering state in the sinh-Gordon and the sine-Gordon models with this approach [13,14]. In this Brief Report, we continue to study the simplest selfcoupling model, the h44 model, with the same approach.The model was originally introduced into quantum field theory to describe the pion-pion interaction [1,1S]. By now it has become an important model in quantum field theory (including gauge theory), finite-temperature field theory, quantum cosmology, and condensed-matter physics. This model is also an ideal theoretical laboratory for various new methods. Since being advocated [16], the Gaussian wave-functional approach has been used for many problems in ~4~ theory, such as spontaneous symConsider a scalar field 4(x)=4, [ x = ( x l , x 2 , . . . ,x,)is the coordinate in D-dimensional space] described by the Hamiltonian where J x = Jdx = Jdx l d~Z . . dxD and V is the gradient operator in D-dimensional space. m and h are the bare mass and coupling parameters, respectively.From [3,4], the Gaussian vacuum state is where y,z = I dy dz = I d y ,dy2 -. dyDdz ,dz2 . . dz,,

Effect of low momentum quantum fluctuations on a coherent field structure

“…We then solve the equations numerically and these numerical solutions are compared with asymptotic solutions. Note also that the approach used in the present work is completely different to that of [12,13] where the wave functionals are constructed using Gaussian approximations to the functional Schrödinger equation for the Sine-Gordon field. However, the particles are considered as higher excited states (in function space) of the linearized field equations.…”

“…However, the particles are considered as higher excited states (in function space) of the linearized field equations. In our treatment the field equations are nonlinear and dynamic and different particles are represented by different nonlinear field configurations, not by higher order Hermite functionals as in [12,13].…”