Instantaneous Bethe-Salpeter equation: improved analytical solution

Studying the Bethe-Salpeter formalism for interactions instantaneous in the rest frame of the bound states described, we show that, for bound-state constituents of arbitrary masses, the mass of the ground state of a given spin may be calculated almost entirely analytically with high accuracy, without the (numerical) diagonalization of the matrix representation obtained by expansion of the solutions over a suitable set of basis states.Comment: 7 page

Instantaneous Bethe-Salpeter equation: utmost analytic approach

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.Comment: LaTeX, 23 pages, 2 figures, version to appear in Phys. Rev.

On the Lorentz structure of the confinement potential

We investigate the Lorentz structure of the confinement potential through a study of the meson spectrum using Salpeter's instantaneous approximation to the Bethe-Salpeter equation. The equivalence between Salpeter's and a random-phase-approximation (RPA) equation enables one to employ the same techniques developed by Thouless, in his study of nuclear collective excitations, to test the stability of the solutions. The stablity analysis reveals the existence of imaginary eigenvalues for a confining potential that transforms as a Lorentz scalar. Moreover, we argue that the instability persists even for very large values of the constituent quark mass. In contrast, we find no evidence of imaginary eigenvalues for a timelike vector potential --- even for very small values of the constituent mass.Comment: 18 pages using RevTeX 3.0, with 8 figures available upon request, FSU-SCRI-94-1

Bound q\bar q Systems in the Framework of the Different Versions of the 3-Dimensional Reductions of the Bethe-Salpeter Equation

Bound q\bar q systems are studied in the framework of different 3-dimensional relativistic equations derived from the Bethe-Salpeter equation with the instantaneous kernel in the momentum space. Except the Salpeter equation, all these equations have a correct one-body limit when one of the constituent quark masses tends to infinity. The spin structure of the confining qq interaction potential is taken in the form $x\gamma_{1}^{0}\gamma_{2}^{0}+(1-x)I_{1}I_{2}$, with $0\leq x \leq 1$. At first stage, the one-gluon-exchange potential is neglected and the confining potential is taken in the oscillator form. For the systems (u\bar s), (c\bar u), (c\bar s) and (u\bar u), (s\bar s) a comparative qualitative analysis of these equations is carried out for different values of the mixing parameter x and the confining potential strength parameter. We investigate: 1)the existence/nonexistence of stable solutions of these equations; 2) the parameter dependence of the general structure of the meson mass spectum and leptonic decay constants of pseudoscalar and vector mesons. It is demonstrated that none of the 3-dimensional equations considered in the present paper does simultaneously describe even general qualitative features of the whole mass spectrum of q\bar q systems. At the same time, these versions give an acceptable description of the meson leptonic decay characteristics.Comment: 22 pages, 5 postscript figures, LaTeX-file (revtex.sty

The stability of the spectator, Dirac, and Salpeter equations for mesons

Mesons are made of quark-antiquark pairs held together by the strong force. The one channel spectator, Dirac, and Salpeter equations can each be used to model this pairing. We look at cases where the relativistic kernel of these equations corresponds to a time-like vector exchange, a scalar exchange, or a linear combination of the two. Since the model used in this paper describes mesons which cannot decay physically, the equations must describe stable states. We find that this requirement is not always satisfied, and give a complete discussion of the conditions under which the various equations give unphysical, unstable solutions

On the validity of the reduced Salpeter equation

We adapt a general method to solve both the full and reduced Salpeter equations and systematically explore the conditions under which these two equations give equivalent results in meson dynamics. The effects of constituent mass, angular momentum state, type of interaction, and the nature of confinement are all considered in an effort to clearly delineate the range of validity of the reduced Salpeter approximations. We find that for $J\not{\hspace*{-1.0mm}=}0$ the solutions are strikingly similar for all constituent masses. For zero angular momentum states the full and reduced Salpeter equations give different results for small quark mass especially with a large additive constant coordinate space potential. We also show that $\frac{1}{m}$ corrections to heavy-light energy levels can be accurately computed with the reduced equation.Comment: Latex (uses epsf macro), 24 pages of text, 12 postscript figures included. Slightly revised version, to appear in Phys. Rev.

Group Treatment Effectiveness for Substance Use Disorders: Abstinence vs. Harm Reduction

The purpose of this systematic review and meta-analysis was to compare group treatment effectiveness for substance use disorders within the U.S. across treatment philosophies as it relates to the primary research question, Is there a significant difference of group treatment effectiveness between Abstinence and Harm Reduction treatment philosophies? It was hypothesized that group treatment will remain an effective intervention to treat substance use disorders between treatment philosophies and that no significant differences exist between-group comparisons. The aim of this study is to provide evidence of treatment effectiveness that will ultimately improve treatment outcomes for substance use disorders, provide guidance for broader implementation of evidence-based treatment approaches within the U.S., and to provide current information for evidence-based decision-making. Targeted studies included randomized and non-randomized controlled trials published in scholarly, peer-reviewed journals within the past 15 years, i.e. 2004-2020. Targeted participants were individuals diagnosed with one or more substance use disorders and/or co-occurring disorders. Treatment outcomes must be measured in quantitative methods with group treatment as the independent variable and substance use disorder outcomes as the dependent variables. The selected studies must indicate treatment philosophy used and provide a direct comparison of Abstinence and Harm Reduction. Random-effects model meta-analysis was used to compute effect sizes for treatment outcomes to compare treatment philosophies. Five studies met eligibility criteria (Miotto et al., 2012; Nyamathi et al., 2011; Rosenblum et al., 2005; Weiss et al., 2007; Weiss et al., 2009). In four of the five included studies, the Harm Reduction condition outperformed the Abstinence condition. However, the meta-analysis indicated that there was not a statistically significant difference between outcomes of substance use by treatment philosophy (Z = 1.29) and (P = 0.20). However, there may be a clinically significant difference due to the aggregate standardized mean difference (-0.15, CI [-0.38, 0.08]) which favors Harm Reduction over Abstinence in the reduction of substance use. Future research focused on clearly identified group treatment philosophy is imperative to provide up-to-date and a more accurate reflection on the effectiveness for treating substance use disorders

On the instantaneous Bethe-Salpeter equation

We present a systematic algebraic and numerical investigation of the instantaneous Bethe-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector types. We explore stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions.Comment: Latex (uses epsf macro), 26 pages of text, 12 postscript figures included

Bethe--Salpeter equation in QCD

We extend to regular QCD the derivation of a confining $q \bar{q}$ Bethe--Salpeter equation previously given for the simplest model of scalar QCD in which quarks are treated as spinless particles. We start from the same assumptions on the Wilson loop integral already adopted in the derivation of a semirelativistic heavy quark potential. We show that, by standard approximations, an effective meson squared mass operator can be obtained from our BS kernel and that, from this, by ${1\over m^2}$ expansion the corresponding Wilson loop potential can be reobtained, spin--dependent and velocity--dependent terms included. We also show that, on the contrary, neglecting spin--dependent terms, relativistic flux tube model is reproduced.Comment: 23 pages, revte