Confinement interaction in nonlinear generalizations of the Wick–Cutkosky model

Abstract

Препринти Iнституту фiзики конденсованих систем НАН України розповсюджуються серед наукових та iнформацiйних установ. Вони також доступнi по електроннiй комп'ютернiй мережi на WWW-сер-верi iнституту за адресою http://www.icmp.lviv.ua/ Анотацiя. Розглядаються узагальнення моделi Вiка-Куткоського з нелiнiйним полем-посередником. Шляхом розкладу за параметром нелiнiйности та вилученням поля-посередника з допомогою кова-рiянтної функцiї Ґрiна отримано лаґранжiян iз часо-нелокальними багато-точковими членами взаємодiї. У нижчих наближеннях теорiї ϕ 3 отримано звичнi дво-струмовi взаємодiї та три-струмову взаємо-дiю типу конфайнменту. Цей же результат отримано точно для версiї дипольної моделi. При гамiльтонiзацiї та канонiчному квантуваннi наближено враховано часову нелокальнiсть. Релятивiстичнi дво-та три-частинковi хвильовi рiвняння отримано варiяцiйним методом iз застосуванням багато-частинкових пробних станiв у просторi Фока. Отримано нерелятивiстичнi границi цих рiвнянь, аналiзуються та коротко обговорюються їх властивостi. On confinement interactions in nonlinear generalizations of the Wick-Cutkosky model Yu.W.Darewych, A.DuviryakAbstract. We consider nonlinear-mediating-field generalizations of the Wick-Cutkosky model. Using an expansion in the nonlinearity parameter and eliminating the mediating field by means of the covariant Green function we derive a Lagrangian with many-point time-nonlocal interaction terms. In low-order approximations of ϕ 3 theory we obtain the usual twocurrent interaction and a three-current confining one. The same result is obtained exactly for a version of the dipole model. The Hamiltonization and canonical quantization is performed with time non-locality taken into account approximately. Relativistic two-and three-particle wave equations are derived variationally by using many-particle Fock space trial states. The non-relativistic limits of these equations are obtained and their properties are analyzed and discussed briefly.

Analysis of inter-quark interactions in classical chromodynamics

“…For the study of inter-quark interactions (and subsequently for the study of the properties of mesons and baryons) it is convenient to use a (formal) solution of the gluon equations of motion (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) to reformulate the Lagrangian, and thus the action, of QCD, so that the gluon propagator appears directly in the interaction terms. Such reformulation has been shown to be useful for the study of inter-particle forces in scalar theory with a nonlinear mediating field [4,5]. The formal solution of (1-19) involves the use of the symmetric Green function of that equation, and this requires a choice of gauge.…”

“…The formal solution of (1-19) involves the use of the symmetric Green function of that equation, and this requires a choice of gauge. We shall use the Lorentz gauge, ∂ µ A µ (a) (x) = 0, whereupon the "glue" equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) can be rewritten as an integral equation,…”

“…We have not included the free-gluon solution of the homogeneous equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) in (2-1), since free gluons do not arise and so free-gluon solutions will play no role in the present considerations.…”

“…The index (1) in (2)(3)(4), as in all that follows, indicates that it is a first-order iterative expression. We note that the gluon fields are expressed here explicitly in terms of the quark fields ψ i q and the propagator D only.…”

“…, it is useful to consider the case of static sources in the non-relativistic limit, as was done for the scalar model [4,5]. For the static case, j…”