Spinless Salpeter Equation Research Articles

Spinless Salpeter equation: Laguerre bounds on energy levels

The spinless Salpeter equation may be considered either as a standard approximation to the Bethe-Salpeter formalism, designed for the description of bound states within a relativistic quantum field theory, or as the most simple, to a certain extent relativistic, generalization of the costumary nonrelativistic Schr\"odinger formalism. Because of the presence of the rather difficult-to-handle square-root operator of the relativistic kinetic energy in the corresponding Hamiltonian, very frequently the corresponding (discrete) spectrum of energy eigenvalues cannot be determined analytically. Therefore, we show how to calculate, by some clever choices of basis vectors in the Hilbert space of solutions, for the rather large class of power-law potentials, at least upper bounds on these energy eigenvalues. For the lowest-lying levels, this may be done even analytically.

Relativistic Coulomb problem: Analytic upper bounds on energy levels.

The spinless relativistic Coulomb problem is the bound-state problem for the spinless Salpeter equation (a standard approximation to the Bethe--Salpeter formalism as well as the most simple generalization of the nonrelativistic Schr\odinger formalism towards incorporation of relativistic effects) with the Coulomb interaction potential (the static limit of the exchange of some massless bosons, as present in unbroken gauge theories). The nonlocal nature of the Hamiltonian encountered here, however, renders extremely difficult to obtain rigorous analytic statements on the corresponding solutions. In view of this rather unsatisfactory state of affairs, we derive (sets of) analytic upper bounds on the involved energy eigenvalues.

Erratum: Analytic solution of the relativistic Coulomb problem for a spinless Salpeter equation

Scenarios for electroweak baryogenesis require an understanding of the effective potential at finite temperature near a first-order electroweak phase transition. Working in Landau gauge, we present a calculation of the dominant two-loop corrections to the ring-improved one-loop potential in the formal limit $g^4 \ll \lambda \ll g^2$, where $\lambda$ is the Higgs self-coupling and $g$ is the electroweak coupling. The limit $\lambda \ll g^2$ ensures that the phase transition is significantly first-order, and the limit $g^4 \ll \lambda$ allows us to use high-temperature expansions. We find corrections from 20 to 40\% at Higgs masses relevant to the bound computed for baryogenesis in the Minimal Standard Model. Though our numerical results seem to still rule out Minimal Standard Model baryogenesis, this conclusion is not airtight because the loop expansion is only marginal when corrections are as big as 40\%. We also discuss why super-daisy approximations do not correctly compute these corrections.

Exact numerical solution of the spinless Salpeter equation for the Coulomb potential in momentum space.

The spinless Salpeter equation with the Coulomb potential is solved exactly in momentum space and is shown to agree very well with a coordinate-space calculation. The agreement is so good that it clearly establishes that the procedure discussed can be used to reliably calculate fine-structure effects in any bound-state system without relying on approximations. The procedure works no matter how large the coupling constant. An equation, called the spinless Salpeter equation with retardation, is introduced and this is also able to be solved exactly.

Matrix representation of the nonlocal kinetic energy operator, the spinless Salpeter equation and the Cornell potential.

A new procedure for solving the spinless Salpeter equation is developed. This procedure is implemented with the Cornell potential, where all of the required matrix elements can be calculated from analytic expressions in a convenient basis. Beginning with analytic results for the square of the momentum operator, the matrix elements of the nonlocal kinetic energy operator are obtained from an algorithm that computes the square root of the square of the relativistic kinetic energy operator. Results calculated with the spinless Salpeter equation are compared with those obtained from Schr\"odinger's equation for heavy-quark systems, heavy-light systems, and light-quark systems. In each case the Salpeter energies agree with experiment substantially better than the Schr\"odinger energies.

Energies of quark-antiquark systems, the Cornell potential, and the spinless Salpeter equation.

Energy eigenvalues for heavy-quarkonium and heavy-light systems are determined from the spinless Salpeter equation for the Cornell potential. These are calculated by diagonalizing the matrix representation of the Hamiltonian operator in a basis set constructed from the products of centrifugal barrier factors, Laguerre polynomials, and a common exponential. The Salpeter eigenvalues are compared with eigenvalues obtained from Schroedinger's equation and with spin-averaged experimental results. We present analytic expressions for the matrix elements of both the Coulomb and linear parts of the Cornell potential. We also present analytic results for the matrix elements of the Schroedinger kinetic energy operator. Thus, the Schroedinger problem can also be treated as a matrix diagonalization problem. The relativistic kinetic energy operator is evaluated in momentum space. New expressions are derived for the Fourier transforms of the [ital S]- and [ital P]-state radial functions. We find that the measured energies of the heavy-quark systems are better fit by Salpeter's equation than by Schroedinger's, in agreement with an earlier calculation of Jacobs, Olsson, and Suchyta. We also find this to be true for [ital B]-flavor and charmed mesons.

Significance of relativistic wave equations for bound states.

We scrutinize the relevance of relativistic wave equations for the description of quark-antiquark bound states. By comparing the predictions of the nonrelativistic Schroedinger equation (with only the lowest-order relativistic corrections of the famous Breit-Fermi Hamiltonian), the spinless Salpeter equation, and a new semirelativistic wave equation (which incorporates relativistic kinematics and the complete relativistic corrections to the static interaction potential) for light and heavy quarkonia within three different potential models, we discuss the extent to which the use of relativistic wave equations is reasonable or necessary in order to reproduce the experimentally observed meson mass spectra. We are forced to conclude that---contrary to one's physical intuition---a relativistic treatment of bound states in a potential model provides no improvement at all compared to the corresponding nonrelativistic description.

Spinless Salpeter equation as a simple matrix eigenvalue problem.

We propose a new method for solving the spinless Salpeter equation. Choosing a special set of orthonormal basis functions we are able to calculate analytically the integral over the corresponding kernel as well as the matrix elements of a class of potentials. In this way the problem can be reduced to the solution of a simple matrix eigenvalue problem with explicitly known matrices, which can be solved very fast numerically. The method is demonstrated in detail for $S$ waves using a typical interquark potential as an example.

Comparing the Schrödinger and spinless Salpeter equations for heavy-quark bound states

We develop a unified approach to the solution of the Schr\odinger and the spinless Salpeter equations. By adjusting the potential and quark masses to best account for the spin-averaged energy levels, we can compare in a realistic way the effect of using a relativistic kinetic energy. Emphasis is given to consideration of relativistic wave-function corrections. We find that the relativistic equation better accounts for the observed energy levels and gives improved results for s- and p-wave annihilation rates and E1 transitions.

Analytic solution of the relativistic Coulomb problem for a spinless Salpeter equation

We construct an analytic solution to the spinless $S$-wave Salpeter equation for two quarks interacting via a Coulomb potential, $[2{(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{m}^{2})}^{\frac{1}{2}}\ensuremath{-}M\ensuremath{-}\frac{\ensuremath{\alpha}}{r}] \ensuremath{\psi}(r)=0$, by transforming the momentum-space form of the equation into a mapping or boundary-value problem for analytic functions. The principal part of the three-dimensional wave function is identical to the solution of a one-dimensional Salpeter equation found by one of us and discussed here. The remainder of the wave function can be constructed by the iterative solution of an inhomogeneous singular integral equation. We show that the exact bound-state eigenvalues for the Coulomb problem are ${M}_{n}=\frac{2m}{{(1+\frac{{\ensuremath{\alpha}}^{2}}{4{n}^{2}})}^{\frac{1}{2}}}$, $n=1, 2, \dots{}$, and that the wave function for the static interaction diverges for $r\ensuremath{\rightarrow}0$ as $C{(\mathrm{mr})}^{\ensuremath{-}\ensuremath{\nu}}$, where $\ensuremath{\nu}=(\frac{\ensuremath{\alpha}}{\ensuremath{\pi}})(1+\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}+\ensuremath{\cdots})$ is known exactly.