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Parity violation in electron-deuteron scattering. II. Break-up channels
ANNALS
OF PHYSICS
136, 435-436
Abstracts
(1981)
of Papers to Appear
in Future
Issues
Symmetry Restoration and !he Background Field Method in Gauge Theories. GRAHAM Lyman Laboratory of Physics, Harvard University, Cambridge. Massachusetts 02 138, LABORATORV OF NUCLEAR STUDIES. Cornell University, Ithaca. New York 14853.
M. AND
SHORE. NEWMAN
Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a firstorder phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants x< = f(B2 ~ E’) and FZ = E . B are discussed. In a simple SU(2) model the critical field strength is found to be gi( /;),,,, =0.057 mt,, m,. being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge. is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter (1. Further applications of these methods are discussed.
Optical Potential Stu& o~C Nuclear States. A. GAL. Brookhaven National York I 1973; G. TOKER. Department of Physics and Astronomy, University Pennsylvania 15260. AND Y. ALEXANDER, Racah Institute of Physics. Jerusalem 9 1904, Israel.
Laboratory. Upton, New of Pittsburgh. Pittsburgh, The Hebrew University.
Single-particle energies and widths of C hypernuclear states are calculated in light systems (A & 40) as energy eigenvalues of the Schrodinger equation for a complex optical potential that fits level shifts and widths of C atoms. The interpretation and significance of Z (normalizable) bound states embedded in the 11 hypernuclear (as well as, sometimes, in the C hypernuclear) continuum are discussed and their properties are studied. primarily in order to identify relatively narrow (I‘< 10 MeV) states. The connection between these calculations and the recently observed Z hypernuclear states suggests that hound states embedded in the Z continuum, rather than (nonnormalizable) Gamow resonant states. arc produced in (K . i[ 1 nuclear reactions.
Parity
Violalion in Electron-Deuferon Scattering. II. Break-up Channels. W-Y. H~NI.EY. AND GERALD A. MILLER. Institute for Nuclear Theory. Department University of Washington, Seattle, Washington 98195
P. HWANG. E. M. of Physics. FM 15.
Parity violation in electron-deuteron inelastic scattering is described. An impulse approximation. modified to incorporate gauge invariance, is employed. Additional meson-exchange currents are included. Normal-parity and abnormal-parity wave function components are generated numerically with a Reid soft-core potential for the former and a general parity-violating weak potential with adjustable coupling constants for the latter. Numerical results for parity-conserving differential cross sections are in good agreement with existing data. For low n-p excitation energies and medium-energy electrons. we find that parity-violating asymmetries are dominated by contributions from neutral weak currents so that the Weinberg-Salam theory can be tested. For low-energy electrons. 5 MeV < E, < 50 MeV, our results Indicate that the asymmetry caused by nuclear parity violation is roughly as important as that due to neutral weak currents. The pion-nucleon parity-violating coupling. ,f,. as well as the rho- and omega
435 Copyright C 1981 by Academic Pres. Inc Ail rights of reproduction in a”) form rr‘w1 -vcd
436
ABSTRACTS
OF PAPERS TO APPEAR IN FUTURE ISSUES
nucleon parity-violating couplings, may be determinable from such experiments. Further, it is possible to check the experiment of Lobashov et al., which detects circular polarization in the thermal-neutron capture reaction.
Fermi-Dirac
of Linear Systems. P. BROADBIUDGE AND C. A, HURST. Department of Physics, The University of Adelaide, G.P.O. Box 498, Adelaide, 5001, South
Quantization
Mathematical Australia.
After discussing the Fermion analogues of classical mechanics, we show that in finite degrees of freedom, the Segal-Weinless construction of the vacuum representation is always possible. This amounts to an explicit construction of a complex structure J which extends real Euclidean space with orthogonal dynamics to a complex Hilbert space with unitary dynamics. Also, we solve the inverse problem, deducing the class of classical Hamiltonians, given the complex structure J. of Two Dimensional Tori in Phase: Projections, Sections and the Wigner Function. A. M. OZORIO DE ALMEIDA AND J. H. HANNAY. H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 lTL, England.
Geometry
The invariant manifolds (or “classical eigenstates”) in the phase space of bound integrable dynamical systems are known to be tori. Sections and projections of general, and special, two dimensional tori in four dimensional phase space are considered. Particular attention is paid to the families of projections accessed by linear canonical transformation since these can (in a certain sense) be considered to be different views of the same torus. The Wigner phase space representation of the corresponding semiclassical quantum eigenstate for a torus of any dimensionality is examined following the analysis of M. V. Berry (Phil. Trans. Roy. Sot. 287 (1977), 237) for one dimensional tori. In this, the value of the semiclassical Wigner function at any phase space point depends on the behaviour of the chords of the torus centred on that point. It is found that for a two dimensional torus the number of such chords is always even. The three dimensional surfaces across which the number of chords changes constitute a (double) fold catastrophe on which the function oscillates with large amplitude. On the torus manifold itself this “Wigner caustic” generally exhibits a hyperbolic umbilic singularity (possibly interspersed with elliptic regions). At special lines and points on the torus, however, higher catstrophes up to E, are generic. Legendre
Transforms
and r-Parricle
Irreducibility
in Quantum
Field
Theory:
The Formalism
for
r = 1,2.
ALAN COOPER, JOEL FELDMAN, AND LON ROSEN. Mathematics Department, University of British Columbia, Vancouver, British Columbia, V6T lY4, Canada. We analyze the first and second Legendre transforms f(‘) (r = 1, 2) of the generating functional G for connected Green’s functions in Euclidean boson field theories. By using Spencer’s idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that r”’ generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, f(*) generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.
OF PHYSICS
136, 435-436
Abstracts
(1981)
of Papers to Appear
in Future
Issues
Symmetry Restoration and !he Background Field Method in Gauge Theories. GRAHAM Lyman Laboratory of Physics, Harvard University, Cambridge. Massachusetts 02 138, LABORATORV OF NUCLEAR STUDIES. Cornell University, Ithaca. New York 14853.
M. AND
SHORE. NEWMAN
Spontaneously broken gauge theories in a constant external electromagnetic field are shown to exhibit a firstorder phase transition to a restored symmetry phase when the external field exceeds a certain critical value. The effects of fields characterized by various values of the two Lorentz invariants x< = f(B2 ~ E’) and FZ = E . B are discussed. In a simple SU(2) model the critical field strength is found to be gi( /;),,,, =0.057 mt,, m,. being the vector boson mass. A number of theoretical developments in the background field formalism are presented. A new gauge-fixing term, the background field R gauge. is introduced. The configuration space heat kernel method for evaluating functional determinants, extended to allow the use of dimensional regularization, is employed, and it is shown how to perform background field calculations in a gauge specified by an arbitrary parameter (1. Further applications of these methods are discussed.
Optical Potential Stu& o~C Nuclear States. A. GAL. Brookhaven National York I 1973; G. TOKER. Department of Physics and Astronomy, University Pennsylvania 15260. AND Y. ALEXANDER, Racah Institute of Physics. Jerusalem 9 1904, Israel.
Laboratory. Upton, New of Pittsburgh. Pittsburgh, The Hebrew University.
Single-particle energies and widths of C hypernuclear states are calculated in light systems (A & 40) as energy eigenvalues of the Schrodinger equation for a complex optical potential that fits level shifts and widths of C atoms. The interpretation and significance of Z (normalizable) bound states embedded in the 11 hypernuclear (as well as, sometimes, in the C hypernuclear) continuum are discussed and their properties are studied. primarily in order to identify relatively narrow (I‘< 10 MeV) states. The connection between these calculations and the recently observed Z hypernuclear states suggests that hound states embedded in the Z continuum, rather than (nonnormalizable) Gamow resonant states. arc produced in (K . i[ 1 nuclear reactions.
Parity
Violalion in Electron-Deuferon Scattering. II. Break-up Channels. W-Y. H~NI.EY. AND GERALD A. MILLER. Institute for Nuclear Theory. Department University of Washington, Seattle, Washington 98195
P. HWANG. E. M. of Physics. FM 15.
Parity violation in electron-deuteron inelastic scattering is described. An impulse approximation. modified to incorporate gauge invariance, is employed. Additional meson-exchange currents are included. Normal-parity and abnormal-parity wave function components are generated numerically with a Reid soft-core potential for the former and a general parity-violating weak potential with adjustable coupling constants for the latter. Numerical results for parity-conserving differential cross sections are in good agreement with existing data. For low n-p excitation energies and medium-energy electrons. we find that parity-violating asymmetries are dominated by contributions from neutral weak currents so that the Weinberg-Salam theory can be tested. For low-energy electrons. 5 MeV < E, < 50 MeV, our results Indicate that the asymmetry caused by nuclear parity violation is roughly as important as that due to neutral weak currents. The pion-nucleon parity-violating coupling. ,f,. as well as the rho- and omega
435 Copyright C 1981 by Academic Pres. Inc Ail rights of reproduction in a”) form rr‘w1 -vcd
436
ABSTRACTS
OF PAPERS TO APPEAR IN FUTURE ISSUES
nucleon parity-violating couplings, may be determinable from such experiments. Further, it is possible to check the experiment of Lobashov et al., which detects circular polarization in the thermal-neutron capture reaction.
Fermi-Dirac
of Linear Systems. P. BROADBIUDGE AND C. A, HURST. Department of Physics, The University of Adelaide, G.P.O. Box 498, Adelaide, 5001, South
Quantization
Mathematical Australia.
After discussing the Fermion analogues of classical mechanics, we show that in finite degrees of freedom, the Segal-Weinless construction of the vacuum representation is always possible. This amounts to an explicit construction of a complex structure J which extends real Euclidean space with orthogonal dynamics to a complex Hilbert space with unitary dynamics. Also, we solve the inverse problem, deducing the class of classical Hamiltonians, given the complex structure J. of Two Dimensional Tori in Phase: Projections, Sections and the Wigner Function. A. M. OZORIO DE ALMEIDA AND J. H. HANNAY. H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 lTL, England.
Geometry
The invariant manifolds (or “classical eigenstates”) in the phase space of bound integrable dynamical systems are known to be tori. Sections and projections of general, and special, two dimensional tori in four dimensional phase space are considered. Particular attention is paid to the families of projections accessed by linear canonical transformation since these can (in a certain sense) be considered to be different views of the same torus. The Wigner phase space representation of the corresponding semiclassical quantum eigenstate for a torus of any dimensionality is examined following the analysis of M. V. Berry (Phil. Trans. Roy. Sot. 287 (1977), 237) for one dimensional tori. In this, the value of the semiclassical Wigner function at any phase space point depends on the behaviour of the chords of the torus centred on that point. It is found that for a two dimensional torus the number of such chords is always even. The three dimensional surfaces across which the number of chords changes constitute a (double) fold catastrophe on which the function oscillates with large amplitude. On the torus manifold itself this “Wigner caustic” generally exhibits a hyperbolic umbilic singularity (possibly interspersed with elliptic regions). At special lines and points on the torus, however, higher catstrophes up to E, are generic. Legendre
Transforms
and r-Parricle
Irreducibility
in Quantum
Field
Theory:
The Formalism
for
r = 1,2.
ALAN COOPER, JOEL FELDMAN, AND LON ROSEN. Mathematics Department, University of British Columbia, Vancouver, British Columbia, V6T lY4, Canada. We analyze the first and second Legendre transforms f(‘) (r = 1, 2) of the generating functional G for connected Green’s functions in Euclidean boson field theories. By using Spencer’s idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that r”’ generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, f(*) generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.