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Wolfgang Lucha

Publications and source records attributed to Wolfgang Lucha.

At least 145 records · Page 8Linked to original sources
arXiv2007

Stability in the instantaneous Bethe-Salpeter formalism: reduced exact-propagator bound-state equation with harmonic interaction

Several numerical investigations of the Salpeter equation with static confining interactions of Lorentz-scalar type revealed that its solutions are plagued by instabilities of presumably Klein-paradox nature. By proving rigorously that the energies of all predicted bound states are part of real, entirely discrete spectra bounded from below, these instabilities are shown, for confining interactions of harmonic-oscillator shape, to be absent for a reduced version of an instantaneous Bethe-Salpeter formalism designed to generalize the Salpeter equation towards an approximate inclusion of the exact propagators of all bound-state constituents.

hep-ph↗
arXiv2007

Systematic errors of bound-state parameters extracted by means of SVZ sum rules

This talk presents the results of our study of systematic errors of the ground-state parameters obtained by Shifman-Vainshtein-Zakharov (SVZ) sum rules. We use the harmonic-oscillator potential model as an example: in this case we know the exact solution for the polarization operator, which allows us to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We extract the parameters of the ground state by making use of the standard procedures of the method of QCD sum rules, and compare the obtained results with their known exact values. We show that if the continuum contribution to the polarization operator is not known and is modelled by some effective continuum threshold, the standard procedures adopted in sum rules do not allow one to gain control over the systematic errors of the extracted ground-state parameters.

hep-ph↗
arXiv2007

Systematic errors of transition form factors extracted by means of light-cone sum rules

This talk presents results of our study of heavy-to-light transition form factors extracted with the help of light-cone sum rules. We employ a model with scalar particles interacting via massless-boson exchange and study the heavy-to-light correlator, relevant for the extraction of the transition form factor. We calculate this correlator in two different ways: by making use of the Bethe-Salpeter wave function of the light bound state and by making use of the light-cone expansion. This allows us to calculate the full correlator and separately the light-cone contribution to it. In this way we show that the off-light cone contributions are not suppressed compared to the light-cone one by any large parameter. Numerically, the difference between the value of the form factor extracted from the full correlator and from the light-cone contribution to this correlator is found to be about 20-30% in a wide range of masses of the particles involved in the decay process.

hep-ph↗
arXiv2007

Stability in the instantaneous Bethe-Salpeter formalism: harmonic-oscillator reduced Salpeter equation

A popular three-dimensional reduction of the Bethe-Salpeter formalism for the description of bound states in quantum field theory is the Salpeter equation, derived by assuming both instantaneous interactions and free propagation of all bound-state constituents. Numerical (variational) studies of the Salpeter equation with confining interaction, however, observed specific instabilities of the solutions, likely related to the Klein paradox and rendering (part of the) bound states unstable. An analytic investigation of this problem by a comprehensive spectral analysis is feasible for the reduced Salpeter equation with only harmonic-oscillator confining interactions. There we are able to prove rigorously that the bound-state solutions correspond to real discrete energy spectra bounded from below and are thus free of any instabilities.

hep-ph↗
arXiv2007

Can one control systematic errors of QCD sum rule predictions for bound states?

We study the possibility to control systematic errors of the ground-state parameters obtained by Shifman-Vainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator, which allows one to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We determine the parameters of the ground state making use of the standard procedures of the method of QCD sum rules, and compare the obtained results with the known exact values. We show that in the situation when the continuum contribution to the polarization operator is not known and is modelled by an effective continuum, the method of sum rules does not allow to control the systematic errors of the extracted ground-state parameters.

hep-ph↗
arXiv2007

Systematic errors of bound-state parameters obtained with SVZ sum rules

We study systematic errors of the ground-state parameters obtained by Shifman-Vainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator, which allows one to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We determine the parameters of the ground state making use of the standard procedures of the method of sum rules, and compare the obtained results with the known exact values. We show that in the situation when the continuum contribution to the polarization operator is not known and is modelled by an effective continuum, the method of sum rules does not allow to control the systematic uncertainties of the extracted ground-state parameters.

hep-ph↗
arXiv2007

Stability of Salpeter Solutions

In the framework of instantaneous approximations to the Bethe-Salpeter formalism for the description of bound states within quantum field theories, depending on the Lorentz structure of the Bethe-Salpeter interaction kernel the solutions of the full Salpeter equation with some confining interaction may exhibit certain instabilities, which are possibly related to the Klein paradox and signal the decay of states assumed to be bound by the confining interactions. They are observed in numerical (variational) studies of the Salpeter equation. The presumably simplest scenario allowing for the fully analytic investigation of this problem is set by the reduced Salpeter equation with harmonic-oscillator interaction. In this case, Salpeter's integral equation simplifies to either an algebraic relation or a second-order homogeneous linear ordinary differential equation, immediately accessible to standard techniques. There one may hope to be able to decide unambiguously whether this setting poses a well-defined eigenvalue problem the solutions of which correspond to stable bound states associated to real energy eigenvalues bounded from below. By analytical spectral analysis the bound-state solutions of this "harmonic-oscillator reduced Salpeter equation" can be shown to be free of such instabilities.

hep-ph↗
arXiv2007

Systematic uncertainties of hadron parameters obtained with QCD sum rules

We study the uncertainties of the determination of the ground-state parameters from Shifman-Vainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator $Π(μ)$, which allows one to obtain both the OPE to any order and the spectrum of states. We start with the OPE for $Π(μ)$ and analyze the extraction of the square of the ground-state wave function, $R\propto|Ψ_0(\vec r=0)|^2$, from an SVZ sum rule, setting the mass of the ground state $E_0$ equal to its known value and treating the effective continuum threshold as a fit parameter. We show that in a limited ``fiducial'' range of the Borel parameter there exists a solution for the effective threshold which precisely reproduces the exact $Π(μ)$ for any value of $R$ within the range $0.7 \le R/R_0 \le 1.15$ ($R_0$ is the known exact value). Thus, the value of $R$ extracted from the sum rule is determined to a great extent by the contribution of the hadron continuum. Our main finding is that in the cases where the hadron continuum is not known and is modeled by an effective continuum threshold, the systematic uncertainties of the sum-rule procedure cannot be controlled.

hep-ph↗
arXiv2007

Ultrarelativistic N-boson systems

General analytic energy bounds are derived for N-boson systems governed by ultrarelativistic Hamiltonians of the form H = sum_{i=1}^N||p_i|| + sum_{1=i<j}^N V(r_{ij}), where V(r) is a static attractive pair potential. It is proved that a translation-invariant model Hamiltonian H_c provides a lower bound to H for all N \ge 2. This result was conjectured in an earlier paper but proved only for N = 2,3,4. As an example, the energy in the case of the linear potential V(r) = r is determined with error less than 0.55% for all N \ge 2.

math-ph↗
arXiv2007

Binding energies of semirelativistic N-boson systems

General analytic energy bounds are derived for N-boson systems governed by semirelativistic Hamiltonians of the form H=\sum_{i=1}^N \sqrt(p_i^2+m^2) + \sum_{1=i \ge generally, and we prove this for N=3, and for N=4 when m=0. The conjecture is also valid generally for the harmonic oscillator and in the nonrelativistic large-m limit. This formulation allows reductions to scaled 3- or 4-body problems, whose spectral bottoms provide energy lower bounds. The example of the ultrarelativistic linear potential is studied in detail and explicit upper- and lower-bound formulas are derived and compared with earlier bounds.

math-ph↗
arXiv2007

Heavy-to-light form factors: sum rules on the light cone and beyond

We report the first systematic analysis of the off-light-cone effects in sum rules for heavy-to-light form factors. These effects are investigated in a model based on scalar constituents, which allows a technically rather simple analysis but has the essential features of the analogous QCD calculation. The correlator relevant for the extraction of the heavy-to-light form factor is calculated in two different ways: first, by adopting the full Bethe-Salpeter amplitude of the light meson and, second, by performing the expansion of this amplitude near the light cone $x^2=0$. We demonstrate that the contributions to the correlator from the light-cone term $x^2=0$ and the off-light-cone terms $x^2\ne 0$ have the same order in the $1/m_Q$ expansion. The light-cone correlator, corresponding to $x^2=0$, is shown to systematically overestimate the full correlator, the difference being $\sim Λ_{\rm QCD}/δ$, with $δ$ the continuum subtraction parameter of order 1 GeV. Numerically, this difference is found to be 10-20%.

hep-ph↗
arXiv2006

Dispersion representations and anomalous singularities of the triangle diagram

We discuss dispersion representations for the triangle diagram $F(p_1^2,p_2^2,q^2)$, the single dispersion representation in $q^2$ and the double dispersion representation in $p_1^2$ and $p_2^2$, with special emphasis on the appearance of the anomalous singularities and the anomalous cuts in these representations. For the double dispersion representation in $p_1^2$ and $p_2^2$, the appearance of the anomalous cut in the region $q^2>0$ is demonstrated, and a new derivation of the anomalous double spectral density is given. We point out that the double spectral representation is particularly suitable for applications in the region of $p_1^2$ and/or $p_2^2$ above the two-particle thresholds. The dispersion representations for the triangle diagram in the nonrelativistic limit are studied and compared with the triangle diagram of the nonrelativistic field theory.

hep-ph↗
arXiv2006

Chiral properties of the constituent quark model

We show that, in a model based exclusively on constituent-quark degrees of freedom interacting via a potential, the full axial current is conserved if the spectrum of \bar QQ states contains a massless pseudoscalar. The current conservation emerges nonperturbatively if the model satisfies certain constraints on (i) the axial coupling g_A of the constituent quark and (ii) the \bar QQ potential at large distances. We define the chiral point of the constituent quark model as that set of values of the parameters (such as the masses of the constituent quarks and the couplings in the \bar QQ potential) for which the mass of the lowest pseudoscalar \bar QQ bound state vanishes. At the chiral point the main signatures of the spontaneously broken chiral symmetry are shown to be present, namely: the axial current is conserved, the decay constants of the excited pseudoscalar bound states vanish, and the pion decay constant has a nonzero value.

hep-ph↗
arXiv2006

Generalized Instantaneous Bethe-Salpeter Equation and Exact Quark Propagators

A three-dimensional reduction of the homogeneous Bethe-Salpeter equation retaining, in contrast to the Salpeter equation, the exact propagators (crucial for, e.g., a proper incorporation of dynamical chiral symmetry breakdown) is proposed. This formalism may be easily extended to Bethe-Salpeter equations for bound states composed of particles that are not, or not all, identical to spin-1/2 fermions as well as to three-dimensional reductions of the Bethe-Salpeter equation different from Salpeter's equation.

hep-ph↗
arXiv2006

Constituent quarks, chiral symmetry, and chiral point of the constituent quark model

We construct the full axial current of the constituent quarks by a summation of the infinite number of diagrams describing constituent-quark soft interactions. By requiring that the conservation of this current is violated only by terms of order $O(M_π^2)$, where $M_π$ is the mass of the lowest pseudoscalar $\bar QQ$ bound state, we derive important constraints on (i) the axial coupling $g_A$ of the constituent quark and (ii) the $\bar QQ$ potential at large distances. We define the chiral point of the constituent quark model as those values of the parameters, such as the masses of the constituent quarks and the couplings in the $\bar QQ$ potential, for which $M_π$ vanishes. At the chiral point the main signatures of the spontaneously broken chiral symmetry are shown to be present, namely: the axial current of the constituent quarks is conserved, the leptonic decay constants of the excited pseudoscalar bound states vanish, and the pion decay constant has a nonzero value.

hep-ph↗
arXiv2006

Gravitating semirelativistic N-boson systems

Analytic energy bounds for N-boson systems governed by semirelativistic Hamiltonians of the form H=\sum_{i=1}^N(p_i^2 + m^2)^{1/2} - sum_{1=i 0, are derived by use of Jacobi relative coordinates. For gravity v=c/N, these bounds are substantially tighter than earlier bounds and they are shown to coincide with known results in the nonrelativistic limit.

math-ph↗