ABSTRACT Dyson-Schwinger Equations(DSE) are an infinite set of coupled equations that interrelate... more ABSTRACT Dyson-Schwinger Equations(DSE) are an infinite set of coupled equations that interrelate all of the basic propagators of QCD. In order to truncate these equations, we start with an effective dressed gluon propagator whose asymptotic(ultraviolet) form is borrowed from perturbative QCD calculations. The infrared part of the effective gluon propagator is modelled by a Gaussian form in momentum space. The problem is solved in Euclidean metric and results are analytically continued to the physical region. The solution of the DSE for the quark propagator gives the momentum dependent quark masses and quark charge normalization factors. In formulating the DSE for the quark propagator and in the Bethe-Salpeter Equation(BSE)(for qbarq bound states), the dressed quark-gluon interaction vertex is used. Dressing is done according to the Slavnov-Taylor identity, with the ghost fields neglected throughout the calculation. The quark propagator obtained from the solution of the DSE is tested for the confinement. The BSE for the qbarq bound state is solved numerically using an inverse iteration method. In calculating the pion's electromagnetic form factor, coupling of the photon to the quarks is realized through a dressed Ball-Chiu vertex, which satisfies the Ward-Takahashi identity. It has been determined that the proper normalization of the Bethe-Salpeter wavefunction and using a dressed photon-quark coupling is essential in obtaining the right pion charge. Results for the pion electromagnetic form factor and weak decay constant are obtained. Application to other mesons is underway.
Meson-Nuclear Physics - 1979, 2nd International Topical Conference on Meson-Nuclear Physics, 1979
We calculate both total and differential cross‐sections for the reaction 12C(γ,π−)12Ng.s. for pio... more We calculate both total and differential cross‐sections for the reaction 12C(γ,π−)12Ng.s. for pion laboratory kinetic energies ranging from 0–180 MeV. We pay particular attention to the role of non‐static terms in the photoproduction operator.
Meson-Nuclear Physics - 1979, 2nd International Topical Conference on Meson-Nuclear Physics, 1979
The physical annihilation of pions is included in an optical potential by a nonlocal operator, in... more The physical annihilation of pions is included in an optical potential by a nonlocal operator, instead of the commonly used &rgr;2(r&drarr;) to account for form. The dependence of pion-nucleus elastic differential cross section on the nonlocaLity in the annihilation operator is determined and found to be strongly linked to the annihilation strength.
A. momentum space method for hadronic atoms is developed to incorporate relativistic, nonlocal, c... more A. momentum space method for hadronic atoms is developed to incorporate relativistic, nonlocal, complex hadron-nucleus interactions. The logarithmic singularity due to the Coulomb interaction has been treated by Lande s subtraction technique. Vacuum polarization, and both nuclear and pion finite-size effects have been included in this momentum space method. Precision eigenvalues and eigenfunctions for the Schrodinger, relativistic Schrodinger, Klein-Gordon (of various types), and Dirac equatioris have been calculated using a rapid and convenient inverse iteration method; Reliability of this novel approach is confirmed by comparing with parallel coordinate space methods. Several illustrative applications are made to simple pionic, and kaonic cases to demonstrate possible applications. Fpr example, it is found that: (1) to extract the pion size from pionic atom data, energy shifts must be measured to an accuracy of better than 50 eV; (2) to determine the form of Klein-Gordon equations appropriate for kaonic atoms, one needs a precision of better than 20 eV; (3) the finite-range of the m-N int|:raction plays a non-negligible role and, therefore, should be carefully included in the pion-nucleus interaction. More extensive applications of these methods are suggested. NUCLEAR STRUCTURE Hadronio atoms, momentum space formulation, and its applic ations.
In the photoproduction process the interaction of the meson with the residual nucleus plays a sig... more In the photoproduction process the interaction of the meson with the residual nucleus plays a significant role. The basic process consists of the photon coupling to the electromagnetic currents of the constituent nucleons, the nucleons are thereby moved to act as meson radiators. The distribution of the nucleons in the nucleus and the associated currents are ascertained by using electron scattering and nuclear spectra information to provide nuclear wave functions.
Coulomb interaction, since the full momentum dependence of the basic photopion operator can be in... more Coulomb interaction, since the full momentum dependence of the basic photopion operator can be included.
Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is ... more Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is shown that an infrared finite gluon propagator can lead to quark confinement via generation of complex mass poles in quark propagators. Observables, including electromagnetic form factors, are calculated entirely in Euclidean metric for spacelike values of bound state momentum and final results are extrapolated to the physical region.
Mathematical Models and Methods in Applied Sciences, Sep 1, 1996
A review of developments in the mathematics and methods for principal value (PV) integrals is pre... more A review of developments in the mathematics and methods for principal value (PV) integrals is presented. These topics include single-pole formulas for simple and higher-order PVs, simple and higher-order poles in double integrals, and products of simple poles in general multiple integrals. Two generalizations of the famous Poincaré-Bertrand (PB) theorem are studied. We then review the following topics: dispersion relations for the advanced, retarded, and causal Green’s functions; Titchmarsh’s theorem; applications of the PB theorem to two- and three-particle loop integrals; and the R and T matrix formalism. Also, various applications of the PV methods to nuclear physics, transport theory, and condensed matter physics are studied. In the appendices several methods for evaluating PV integrals, including the Haftel-Tabakin procedure for calculating the R and T matrices, are reviewed.
It is shown that a single separable potential can give both an attraction and a repulsion. This p... more It is shown that a single separable potential can give both an attraction and a repulsion. This property is used to construct an interaction that fits the two-nucleon, S-wave phase shifts. For this potential, the important e8'ects of the strong short-range repulsion can be included directly in existing three-body progams. I. INTRODUCTION HE three-nucleon problem can be solved exactly for separable potentials. In several papers, separable potentials have been used to calculate the triton binding energy, radius, Coulomb energy, form factor, and the neutron-deuteron scattering cross section. ' lt has also been shown that the hard-core and tensor-force aspects of the internucleon potential play an important role in determining these three-body properties. ' However, inclusion of the hard core in such calculations is dificult. It is generally believed that a single separable potential can represent the attraction, but not the short-range, two-nucleon repulsion. Consequently, additional separable terms are introduced to give the required repulsion. ' These extra terms complicate the three-body calculation considerably. In this paper, we construct a single separable potential that includes both the attraction and repulsion required to 6t the high-energy 5-wave phase shifts. The significance of this result is that one can include a strong repulsion in three-nucleon calculations without major modification of the existing calculational procedures. It also provides an improved starting point for recently suggested methods of solving the three-body problem for local potentials, 4 Research supported in part by the National Science Foundation.
Total and differential cross sections are presented for the reaction /sup 14/N(..gamma..,..pi../s... more Total and differential cross sections are presented for the reaction /sup 14/N(..gamma..,..pi../sup -/)/sup 14/O/sub g.s./ for 0 --200 MeV pions, using the Blomqvist-Laget photoproduction operator and including all requisite pion distorted waves. It is found that phenomenological p-shell wave functions obtained from fits to inelastic electron scattering data from /sup 14/N, when used in the photoproduction calculation, give good agreement with experiment. This agreement is a result of using nuclear wave functions that not only suppress the Gamow-Teller matrix element but also agree with M1 radiative widths. As in the photoproduction of pions off other light nuclei, the total cross section shows a peak at energies far below the (3, 3) resonance energy, which arises from the recoil nuclei being constrained to discrete final states. As a result of the suppression of the Gamow-Teller matrix element for this transition, the reaction is sensitive (even at threshold) to the higher order terms in the photoproduction operator and hence to the curvature of the pion waves and the Fermi motion of the nucleons in the nuclei. This suggests that this reaction provides a sensitive probe of pion distortions and of the Ericson-Ericson effect.
We examine the photoproduction of the light scalar mesons f0-a0(980) with emphasis on isospinviol... more We examine the photoproduction of the light scalar mesons f0-a0(980) with emphasis on isospinviolating f0 and a0 mixing due to the mass difference between neutral and charged kaons. General forms for the invariant amplitude for scalar meson photoproduction are derived, which yield expressions for the invariant mass distribution of the final meson pairs. A final state interaction form factor is obtained which incorporates f0 and a0 mesons, K K threshold effects, and f0 -a0 mixing. This form factor is applied to predict the effective mass distribution of the ππ and K K pairs in the vicinity of the K K threshold. An estimate of the role of isospin mixing near threshold is provided. The potential role of polarized photons and protons is also discussed.
The separable-potential approach of Mitra is extended by introducing an additional interaction wh... more The separable-potential approach of Mitra is extended by introducing an additional interaction which yields two-nucleon S-wave phase shifts to 340 MeV. A calculation of the exact binding energy and wave function of a simplified three-body system is used to determine the role of short-range correlations and the associated off-energy-shell matrix elements of the interaction. We Gnd that the three-body binding energy ranges from 9.33 to 8.40 MeV; the larger value is calculated using a two-body interaction fitting low-energy data only, while the smaller value results from including a hard-shell repulsion. A smooth repulsion, which is equivalent to the hard shell in the two-body problem, yields a three-body binding energy of 8.79 MeV.
An effective potential is presented which matches S, P, and D-wave nucleon-nucleon phase paramete... more An effective potential is presented which matches S, P, and D-wave nucleon-nucleon phase parameters up to 320 MeV without generating the usual strong, short-range correlations. A suitably defined set of separable potentials, including a noncentral force and having small ...
ABSTRACT Dyson-Schwinger Equations(DSE) are an infinite set of coupled equations that interrelate... more ABSTRACT Dyson-Schwinger Equations(DSE) are an infinite set of coupled equations that interrelate all of the basic propagators of QCD. In order to truncate these equations, we start with an effective dressed gluon propagator whose asymptotic(ultraviolet) form is borrowed from perturbative QCD calculations. The infrared part of the effective gluon propagator is modelled by a Gaussian form in momentum space. The problem is solved in Euclidean metric and results are analytically continued to the physical region. The solution of the DSE for the quark propagator gives the momentum dependent quark masses and quark charge normalization factors. In formulating the DSE for the quark propagator and in the Bethe-Salpeter Equation(BSE)(for qbarq bound states), the dressed quark-gluon interaction vertex is used. Dressing is done according to the Slavnov-Taylor identity, with the ghost fields neglected throughout the calculation. The quark propagator obtained from the solution of the DSE is tested for the confinement. The BSE for the qbarq bound state is solved numerically using an inverse iteration method. In calculating the pion's electromagnetic form factor, coupling of the photon to the quarks is realized through a dressed Ball-Chiu vertex, which satisfies the Ward-Takahashi identity. It has been determined that the proper normalization of the Bethe-Salpeter wavefunction and using a dressed photon-quark coupling is essential in obtaining the right pion charge. Results for the pion electromagnetic form factor and weak decay constant are obtained. Application to other mesons is underway.
Meson-Nuclear Physics - 1979, 2nd International Topical Conference on Meson-Nuclear Physics, 1979
We calculate both total and differential cross‐sections for the reaction 12C(γ,π−)12Ng.s. for pio... more We calculate both total and differential cross‐sections for the reaction 12C(γ,π−)12Ng.s. for pion laboratory kinetic energies ranging from 0–180 MeV. We pay particular attention to the role of non‐static terms in the photoproduction operator.
Meson-Nuclear Physics - 1979, 2nd International Topical Conference on Meson-Nuclear Physics, 1979
The physical annihilation of pions is included in an optical potential by a nonlocal operator, in... more The physical annihilation of pions is included in an optical potential by a nonlocal operator, instead of the commonly used &rgr;2(r&drarr;) to account for form. The dependence of pion-nucleus elastic differential cross section on the nonlocaLity in the annihilation operator is determined and found to be strongly linked to the annihilation strength.
A. momentum space method for hadronic atoms is developed to incorporate relativistic, nonlocal, c... more A. momentum space method for hadronic atoms is developed to incorporate relativistic, nonlocal, complex hadron-nucleus interactions. The logarithmic singularity due to the Coulomb interaction has been treated by Lande s subtraction technique. Vacuum polarization, and both nuclear and pion finite-size effects have been included in this momentum space method. Precision eigenvalues and eigenfunctions for the Schrodinger, relativistic Schrodinger, Klein-Gordon (of various types), and Dirac equatioris have been calculated using a rapid and convenient inverse iteration method; Reliability of this novel approach is confirmed by comparing with parallel coordinate space methods. Several illustrative applications are made to simple pionic, and kaonic cases to demonstrate possible applications. Fpr example, it is found that: (1) to extract the pion size from pionic atom data, energy shifts must be measured to an accuracy of better than 50 eV; (2) to determine the form of Klein-Gordon equations appropriate for kaonic atoms, one needs a precision of better than 20 eV; (3) the finite-range of the m-N int|:raction plays a non-negligible role and, therefore, should be carefully included in the pion-nucleus interaction. More extensive applications of these methods are suggested. NUCLEAR STRUCTURE Hadronio atoms, momentum space formulation, and its applic ations.
In the photoproduction process the interaction of the meson with the residual nucleus plays a sig... more In the photoproduction process the interaction of the meson with the residual nucleus plays a significant role. The basic process consists of the photon coupling to the electromagnetic currents of the constituent nucleons, the nucleons are thereby moved to act as meson radiators. The distribution of the nucleons in the nucleus and the associated currents are ascertained by using electron scattering and nuclear spectra information to provide nuclear wave functions.
Coulomb interaction, since the full momentum dependence of the basic photopion operator can be in... more Coulomb interaction, since the full momentum dependence of the basic photopion operator can be included.
Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is ... more Pion and kaon observables are calculated using a Dyson-Schwinger Bethe-Salpeter formalism. It is shown that an infrared finite gluon propagator can lead to quark confinement via generation of complex mass poles in quark propagators. Observables, including electromagnetic form factors, are calculated entirely in Euclidean metric for spacelike values of bound state momentum and final results are extrapolated to the physical region.
Mathematical Models and Methods in Applied Sciences, Sep 1, 1996
A review of developments in the mathematics and methods for principal value (PV) integrals is pre... more A review of developments in the mathematics and methods for principal value (PV) integrals is presented. These topics include single-pole formulas for simple and higher-order PVs, simple and higher-order poles in double integrals, and products of simple poles in general multiple integrals. Two generalizations of the famous Poincaré-Bertrand (PB) theorem are studied. We then review the following topics: dispersion relations for the advanced, retarded, and causal Green’s functions; Titchmarsh’s theorem; applications of the PB theorem to two- and three-particle loop integrals; and the R and T matrix formalism. Also, various applications of the PV methods to nuclear physics, transport theory, and condensed matter physics are studied. In the appendices several methods for evaluating PV integrals, including the Haftel-Tabakin procedure for calculating the R and T matrices, are reviewed.
It is shown that a single separable potential can give both an attraction and a repulsion. This p... more It is shown that a single separable potential can give both an attraction and a repulsion. This property is used to construct an interaction that fits the two-nucleon, S-wave phase shifts. For this potential, the important e8'ects of the strong short-range repulsion can be included directly in existing three-body progams. I. INTRODUCTION HE three-nucleon problem can be solved exactly for separable potentials. In several papers, separable potentials have been used to calculate the triton binding energy, radius, Coulomb energy, form factor, and the neutron-deuteron scattering cross section. ' lt has also been shown that the hard-core and tensor-force aspects of the internucleon potential play an important role in determining these three-body properties. ' However, inclusion of the hard core in such calculations is dificult. It is generally believed that a single separable potential can represent the attraction, but not the short-range, two-nucleon repulsion. Consequently, additional separable terms are introduced to give the required repulsion. ' These extra terms complicate the three-body calculation considerably. In this paper, we construct a single separable potential that includes both the attraction and repulsion required to 6t the high-energy 5-wave phase shifts. The significance of this result is that one can include a strong repulsion in three-nucleon calculations without major modification of the existing calculational procedures. It also provides an improved starting point for recently suggested methods of solving the three-body problem for local potentials, 4 Research supported in part by the National Science Foundation.
Total and differential cross sections are presented for the reaction /sup 14/N(..gamma..,..pi../s... more Total and differential cross sections are presented for the reaction /sup 14/N(..gamma..,..pi../sup -/)/sup 14/O/sub g.s./ for 0 --200 MeV pions, using the Blomqvist-Laget photoproduction operator and including all requisite pion distorted waves. It is found that phenomenological p-shell wave functions obtained from fits to inelastic electron scattering data from /sup 14/N, when used in the photoproduction calculation, give good agreement with experiment. This agreement is a result of using nuclear wave functions that not only suppress the Gamow-Teller matrix element but also agree with M1 radiative widths. As in the photoproduction of pions off other light nuclei, the total cross section shows a peak at energies far below the (3, 3) resonance energy, which arises from the recoil nuclei being constrained to discrete final states. As a result of the suppression of the Gamow-Teller matrix element for this transition, the reaction is sensitive (even at threshold) to the higher order terms in the photoproduction operator and hence to the curvature of the pion waves and the Fermi motion of the nucleons in the nuclei. This suggests that this reaction provides a sensitive probe of pion distortions and of the Ericson-Ericson effect.
We examine the photoproduction of the light scalar mesons f0-a0(980) with emphasis on isospinviol... more We examine the photoproduction of the light scalar mesons f0-a0(980) with emphasis on isospinviolating f0 and a0 mixing due to the mass difference between neutral and charged kaons. General forms for the invariant amplitude for scalar meson photoproduction are derived, which yield expressions for the invariant mass distribution of the final meson pairs. A final state interaction form factor is obtained which incorporates f0 and a0 mesons, K K threshold effects, and f0 -a0 mixing. This form factor is applied to predict the effective mass distribution of the ππ and K K pairs in the vicinity of the K K threshold. An estimate of the role of isospin mixing near threshold is provided. The potential role of polarized photons and protons is also discussed.
The separable-potential approach of Mitra is extended by introducing an additional interaction wh... more The separable-potential approach of Mitra is extended by introducing an additional interaction which yields two-nucleon S-wave phase shifts to 340 MeV. A calculation of the exact binding energy and wave function of a simplified three-body system is used to determine the role of short-range correlations and the associated off-energy-shell matrix elements of the interaction. We Gnd that the three-body binding energy ranges from 9.33 to 8.40 MeV; the larger value is calculated using a two-body interaction fitting low-energy data only, while the smaller value results from including a hard-shell repulsion. A smooth repulsion, which is equivalent to the hard shell in the two-body problem, yields a three-body binding energy of 8.79 MeV.
An effective potential is presented which matches S, P, and D-wave nucleon-nucleon phase paramete... more An effective potential is presented which matches S, P, and D-wave nucleon-nucleon phase parameters up to 320 MeV without generating the usual strong, short-range correlations. A suitably defined set of separable potentials, including a noncentral force and having small ...
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Papers by Frank Tabakin