The properties of two massless wave equations relevant to quark bound states are examined. We establish general conditions on the Lorentz-scalar and Lorentz-vector potentials which yield arbitrary leading Regge trajectories for the case of circular classical motion. A semiclassical approximation which includes radial motion reproduces remarkably well the exact solutions. The conditions for a mesonic tower structure are discussed.For over 25 years it has been empirically known that all light-quark hadrons lie upon linear Regge trajectories.' While it may have been pointed out in the past2-4 we would like to reemphasize here the relation between linear trajectories, linear confinement, and relativistic dynamics. It seems inescapable that massless quarks bound by a linear confinement potential generate a family of parallel linear Regge trajectories. We find, however, that the Regge slopes and daughter spacings depend on the Lorentz nature and other properties of the interaction. We initially observed this connection from numerically exact solutions to the massless spinless Salpeter wave equation and the quantized straight string.5 In each of the above cases the Regge slope was a different constant. We will demonstrate here that the Regge slopes can be accurately calculated within a classical framework and that the intercepts are all accurately estimated semiclassically. Precociously straight trajectories at low angular momentum are also found in all cases investigated. First-order wave equations of the Schrodinger type, H$=ia$/at, form our main interest here. A generalized spinless Salpeter (GSS) equation with Lorentz-scalar and -vector potentials is given by6where for simplicity we consider here equal-mass particles. This equation can also be generalized to the case of unequal masses.'A closely related second-order equation is the generalized Klein-Gordon ( G K G ) equation6For a pure scalar potential ( V =O) the G K G equation is exactly the square of the GSS Eq. (1). When V f 0 the two wave equations differ but they have in common the same classical and semiclassical solutions.In Sec. I we motivate the GSS equation and discuss its relation to string and string-potential models. In Sec. I1 we discuss pure rotational solutions to the above equation in the massless limit using a classical analysis. In this section we establish the class of potentials which yield a specified leading Regge trajectory. Section 111 contains semiclassical solutions to the GSS or G K G equation with massless quarks. In Sec. IV our solutions are found to accurately compare to "exact" numerical solutions and to exact analytic results where available. Our conclusions are given in Sec. V. I. THE GENERALIZED SPINLESS SALPETER WAVE EQUATIONThe spinless Salpeter (SS), or square-root equation, has long held interest8 as the simplest generalization of the nonrelativistic Schrodinger equation. The SS equation is obtained from Eq. (1) by setting the potential S ( r ) to zero. Our approach in motivating Eq. (1) begins with a classical Lagrangian for ...
We explore the dynamics of mesons composed of spinless quarks connected by a straight flux tube. The mesons are quantized and the constituent motion is relativistic. The methods developed are applied to mesons containing equal mass quarks and heavy-light quarks. For massless quarks a nearly straight leading Regge trajectory with Nambu slope is accompanied by nearly parallel equally spaced daughter trajectories. For heavy-light mesons an analogous structure is found but with double the usual Nambu slope. Comparison with six observed spin-averaged heavy-light meson states yields good agreement. PACS number(s): 12.38.Aw, 12.38.Lg
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.