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A relativistic quark model for mesons based on numerical solutions of the Bethe-Salpeter equation
ABSTRACTS OF PAPERS TO APPEAR IN FUTURE ISSUES
365
numerically. Coulomb effects are neglected. We derive the relativistic expressions for single scattering and nucleon exchange and discuss the approximations made, including the off-massshell extrapolation of the nucleon-nucleon scattering amplitude. Fits are made to experimental measurements of differential cross sections, nucleon polarizations, and total elastic cross sections. Unitarity is maintained. We tabulate the partial waves for J < 5/2, L < 2. They are consistent with recent Faddeev calculations. We argue that with the additional calculation of double scattering the deuteron D-state percentage can be determined to the same relative uncertainty as the differential cross section. Even without the calculation of double scattering, our results indicate a D-state percentage around 8%. In an effort to provide benchmarks for future work, we have tried to be conscientious in describing our techniques and in tabulating numerical results. Comparisons are also made with earlier analyses. Short-Distance Behaviour of Quantum Electrodynamics and the Callan-Symanzik Equation for the Photon Propagator. EDUARDO DE RAFAEL. Institute for Advanced Study, Princeton, New Jersey and Institut des Hautes etudes Scientifiques, Bures-sur-Yvette, France, and JONATHAN L. ROSNER. Institute for Advanced Study, Princeton, New Jersey and CERN, Geneva, Switzerland. The short-distance behaviour of the photon propagator is discussed within the context of the corresponding Callan-Symanzik equation. The Callan-Symanzik function ,8(a) is calculated in perturbation theory up to sixth order. We find
/3(a) =f(j +f(g)’- &yj8+o[Q4]. The simplicity of this result is to be contrasted with a corresponding perturbation theory calculation of the Gell-Mann-Low function #(z), whose sixth order coefficient contains the transcendental 5(3) (the Riemann zeta function of argument three). A mechanism of cancellations in the calculation of b(a) has been found, and we prove its validity to all orders in perturbation theory. Lower Bounds on Cross Sections Without Unknown Constants and Valid at All Energies. C. LOPEZ AND F. J. YNDURAIN. Departamento de Flsica-C-XI, Universidad Autonoma de Madrid, Canto Blanco, Madrid, Spain. We obtain lower bounds for cross sections (total and differential) which are of the form of integral constraints, and which contain no unknown constants and that are valid at fmite energies (and not only asymptotically). The information that we use to obtain the bounds may be of three different types (giving three different kinds of bounds): a few low energy parameters; a few low energy parameters plus experimental information on a given wave (the D-wave); or one unphysical parameter that may be obtained from other sources (field theoretical calculations with soft pion techniques). The comparison of the bounds with experiment is also discussed. A Relativistic Quark Model for Mesons Based on Numerical Solutions of the Bethe-Salpeter Equation. ALAN H. Guru. Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540. A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin-l/2 quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethtialpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral
366
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vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The h quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for rr, K -+ PY are calculated, and it is found, independent of parameters, thatf, w fK for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for p”, w, 4 + e+e-, r”.+p- are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as w + +‘y. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with JPC = O+-) with masses of about 2 Bev. Invarimt Realization of Functional Integration over a Local Gauge Group. I. A. BATAL~N, E. S. FRADKIN, AND I. E. TAMM. Department of Theoretical Physics, P. N. Lebedev Physical Institute, Moscow, U.S.S.R. An invariant functional formulation of the nonlinear chiral theories as well as of their generalizations is developed. To guarantee the manifest invariance of the generating functional under different choices of the local coordinates in the inner space (parametrization) new variables are utilized which are the left side of the classical equations of motions (with or without the source). In terms of the new variables an invariant regularixation is introduced and an invariant perturbation theory is developed.
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numerically. Coulomb effects are neglected. We derive the relativistic expressions for single scattering and nucleon exchange and discuss the approximations made, including the off-massshell extrapolation of the nucleon-nucleon scattering amplitude. Fits are made to experimental measurements of differential cross sections, nucleon polarizations, and total elastic cross sections. Unitarity is maintained. We tabulate the partial waves for J < 5/2, L < 2. They are consistent with recent Faddeev calculations. We argue that with the additional calculation of double scattering the deuteron D-state percentage can be determined to the same relative uncertainty as the differential cross section. Even without the calculation of double scattering, our results indicate a D-state percentage around 8%. In an effort to provide benchmarks for future work, we have tried to be conscientious in describing our techniques and in tabulating numerical results. Comparisons are also made with earlier analyses. Short-Distance Behaviour of Quantum Electrodynamics and the Callan-Symanzik Equation for the Photon Propagator. EDUARDO DE RAFAEL. Institute for Advanced Study, Princeton, New Jersey and Institut des Hautes etudes Scientifiques, Bures-sur-Yvette, France, and JONATHAN L. ROSNER. Institute for Advanced Study, Princeton, New Jersey and CERN, Geneva, Switzerland. The short-distance behaviour of the photon propagator is discussed within the context of the corresponding Callan-Symanzik equation. The Callan-Symanzik function ,8(a) is calculated in perturbation theory up to sixth order. We find
/3(a) =f(j +f(g)’- &yj8+o[Q4]. The simplicity of this result is to be contrasted with a corresponding perturbation theory calculation of the Gell-Mann-Low function #(z), whose sixth order coefficient contains the transcendental 5(3) (the Riemann zeta function of argument three). A mechanism of cancellations in the calculation of b(a) has been found, and we prove its validity to all orders in perturbation theory. Lower Bounds on Cross Sections Without Unknown Constants and Valid at All Energies. C. LOPEZ AND F. J. YNDURAIN. Departamento de Flsica-C-XI, Universidad Autonoma de Madrid, Canto Blanco, Madrid, Spain. We obtain lower bounds for cross sections (total and differential) which are of the form of integral constraints, and which contain no unknown constants and that are valid at fmite energies (and not only asymptotically). The information that we use to obtain the bounds may be of three different types (giving three different kinds of bounds): a few low energy parameters; a few low energy parameters plus experimental information on a given wave (the D-wave); or one unphysical parameter that may be obtained from other sources (field theoretical calculations with soft pion techniques). The comparison of the bounds with experiment is also discussed. A Relativistic Quark Model for Mesons Based on Numerical Solutions of the Bethe-Salpeter Equation. ALAN H. Guru. Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08540. A study is made to determine if the results of the nonrelativistic quark model can be reproduced by a fully relativistic model of deeply bound spin-l/2 quarks. It is found that the relativistic model does not reproduce the nonrelativistic results, even when the quarks have nonrelativistic momenta. However, the model is rather successful in accounting for the known properties of mesons. Numerical solutions to the Bethtialpeter equation are obtained for pseudoscalar and vector bound states of equal mass quark-antiquark pairs, with either a scalar, pseudoscalar, or neutral
366
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OF
PAPERS
TO
APPEAR
IN
FUTURE
ISSUES
vector exchange interaction. The interaction function corresponds to single particle exchange, with the addition of either one or two regulating terms. It is found that the second regulator allows the internal quark momentum to be nonrelativistic, but that the spinor structure of the wave function remains highly relativistic. Only the scalar interaction can account for the observed spectrum of states. The pseudoscalar interaction yields a vector state of lower mass than the pseudoscalar state, and the vector interaction leads to a vector state which lies approximately one quark mass above the pseudoscalar state. The h quark is taken as slightly heavier than the p and n, and the perturbation treatment of the mass difference leads to a quadratic mass formula. The decay amplitudes for rr, K -+ PY are calculated, and it is found, independent of parameters, thatf, w fK for either a scalar or vector interaction, in agreement with experiment and in contrast with the nonrelativistic model. The amplitudes for p”, w, 4 + e+e-, r”.+p- are also calculated, but in this case the ratios (again parameter independent) are in minor discrepancy with experiment. The question of the additivity of quark amplitudes is examined by calculating (with significant restrictions) the magnetic moments of the vector mesons and the amplitudes for magnetic transitions such as w + +‘y. The magnetic moments of the vector mesons have the same (trivial) ratios to each other as in the nonrelativistic model, but they are strongly enhanced over the sum of the quark magnetic moments. The amplitude for magnetic transitions, however, is related to the quark magnetic moments in approximately the same ratio as in the nonrelativistic model. The model is also used to obtain parameter dependent predictions for the masses and decay amplitudes. These predictions are not experimentally correct, but are generally well within an order of magnitude for a wide range of the parameters. The most significant defect discovered of the model is the presence of ghost states (the daughters of the vector mesons, with JPC = O+-) with masses of about 2 Bev. Invarimt Realization of Functional Integration over a Local Gauge Group. I. A. BATAL~N, E. S. FRADKIN, AND I. E. TAMM. Department of Theoretical Physics, P. N. Lebedev Physical Institute, Moscow, U.S.S.R. An invariant functional formulation of the nonlinear chiral theories as well as of their generalizations is developed. To guarantee the manifest invariance of the generating functional under different choices of the local coordinates in the inner space (parametrization) new variables are utilized which are the left side of the classical equations of motions (with or without the source). In terms of the new variables an invariant regularixation is introduced and an invariant perturbation theory is developed.
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