We investigate neutrino-driven convection in core collapse supernovae and its ramifications for the explosion mechanism. We begin with an "optimistic" postbounce model in two important respects: (1) we begin with a 15 M ⊙ precollapse model, which is representative of the class of stars with compact iron cores; (2) we implement Newtonian gravity. Our precollapse model is evolved through core collapse and bounce in one dimension using multigroup (neutrino-energy-dependent) flux-limited diffusion (MGFLD) neutrino transport and Newtonian Lagrangian hydrodynamics, providing realistic initial conditions for the postbounce convection and evolution.Our two-dimensional simulation began at 12 ms after bounce and proceeded for 500 ms.We couple two-dimensional (PPM) hydrodynamics to precalculated one-dimensional MGFLD neutrino transport. (The neutrino distributions used for matter heating and deleptonization in our 2D run are obtained from an accompanying 1D simulation. The accuracy of this approximation is assessed.) For the moment we sacrifice dimensionality for realism in other aspects of our neutrino transport. MGFLD is an implementation of neutrino transport that simultaneously (a) is multigroup and (b) simulates with sufficient realism the transport of neutrinos in opaque, semitransparent, and transparent regions. Both are crucial to the accurate determination of postshock neutrino heating, which sensitively depends on the luminosities, spectra, and flux factors of the electron neutrinos and antineutrinos emerging from their respective neutrinospheres.By 137 ms after bounce, we see neutrino-driven convection rapidly developing beneath the shock. By 212 ms after bounce, this convection becomes large-scale, characterized by higher-entropy, expanding upflows and denser, lower-entropy, finger-like downflows. The upflows reach the shock and distort it from sphericity. The radial convection velocities at this time become supersonic just below the shock, reaching magnitudes in excess of 10 9 cm/sec. Eventually, however, the shock recedes to smaller radii, and at ∼500 ms after bounce there is no evidence in our simulation of an explosion or of a developing explosion.Our angle-averaged density, entropy, electron fraction, and radial velocity profiles in our below the electron neutrino and antineutrino gain radii, above which the neutrino luminosities are essentially constant (i.e., the neutrino sources are entirely enclosed), in an effort to assess how spherically symmetric our neutrino sources remain during our 2D evolution, and therefore, to assess our use of precalculated 1D MGFLD neutrino distributions in calculating the matter heating and deleptonization. We find no differences below the neutrinosphere radii, and between the neutrinosphere and gain radii, no differences with obvious ramifications for the supernova outcome.We note that the interplay between neutrino transport and convection below the neutrinospheres is a delicate matter, and is discussed at greater length in another paper (Mezzacappa et al. 1997a).However, ...

We couple two-dimensional hydrodynamics to realistic one-dimensional multigroup Ñux-limited di †u-sion neutrino transport to investigate protoÈneutron star convection in core-collapse supernovae, and more speciÐcally, the interplay between its development and neutrino transport. Our initial conditions, time-dependent boundary conditions, and neutrino distributions for computing neutrino heating, cooling, and deleptonization rates are obtained from one-dimensional simulations that implement multigroup Ñux-limited di †usion and one-dimensional hydrodynamics.The development and evolution of protoÈneutron star convection are investigated for both 15 and 25 models, representative of the two classes of stars with compact and extended iron cores, respectively. M _ For both models, in the absence of neutrino transport, the angle-averaged radial and angular convection velocities in the initial Ledoux unstable region below the shock after bounce achieve their peak values in D20 ms, after which they decrease as the convection in this region dissipates. The dissipation occurs as the gradients are smoothed out by convection. This initial protoÈneutron star convection episode seeds additional convectively unstable regions farther out beneath the shock. The additional protoÈneutron star convection is driven by successive negative entropy gradients that develop as the shock, in propagating out after core bounce, is successively strengthened and weakened by the oscillating inner core. The convection beneath the shock distorts its sphericity, but on the average the shock radius is not boosted signiÐcantly relative to its radius in our corresponding one-dimensional models.In the presence of neutrino transport, protoÈneutron star convection velocities are too small relative to bulk inÑow velocities to result in any signiÐcant convective transport of entropy and leptons. This is evident in our two-dimensional entropy snapshots, which in this case appear spherically symmetric. The peak angle-averaged radial and angular convection velocities are orders of magnitude smaller than they are in the corresponding "" hydrodynamics-only ÏÏ models.A simple analytical model supports our numerical results, indicating that the inclusion of neutrino transport reduces the entropy-driven (lepton-driven) convection growth rates and asymptotic velocities by a factor D3 (50) at the neutrinosphere and a factor D250 (1000) at o \ 1012 g cm~3, for both our 15 and 25 models. Moreover, when transport is included, the initial postbounce entropy gradient is M _ smoothed out by neutrino di †usion, whereas the initial lepton gradient is maintained by electron capture and neutrino escape near the neutrinosphere. Despite the maintenance of the lepton gradient, protoÈ neutron star convection does not develop over the 100 ms duration typical of all our simulations, except in the instance where "" low-test ÏÏ intial conditions are used, which are generated by core-collapse and bounce simulations that neglect neutrinoÈelectron scattering and ionÈion screening...

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We present an SU (4) model of high-temperature superconductivity having many similarities to dynamical symmetries known to play an important role in microscopic nuclear structure physics and in elementary particle physics. Analytical solutions in three dynamical symmetry limits of this model are found: an SO(4) limit associated with antiferromagnetic order; an SU (2) × SO(3) limit that may be interpreted as a d-wave pairing condensate; and an SO(5) limit that may be interpreted as a doorway state between the antiferromagnetic order and the superconducting order. The model suggests a phase diagram in qualitative agreement with that observed in the cuprate superconductors. The relationship between the present model and the SO(5) unification of superconductivity and antiferromagnetic order proposed by Zhang is discussed.There are compelling arguments that the mechanism leading to high-temperature superconductivity does not correspond to ordinary BCS s-wave pairing. Although experimental evidence implicates singlet (hole) pairs as the carriers of the supercurrent, the interaction leading to the formation of the singlet pairs appears not to be the traditional lattice phonon mechanism underlying the BCS theory, but rather seems to be a collective electronic interaction. Furthermore, the pairing gap is anisotropic, with nodes in the k x -k y plane strongly suggestive of dwave hybridization in the 2-particle wavefunctions, and the mechanism responsible for superconductivity in the cuprates is thought to be closely related to the unusual antiferromagnetic (AF) insulator properties of their normal states.Contrary to the case for BCS superconductors, the formation of Cooper pairs and the formation of a superconducting (SC) condensate of those pairs in high-T c compounds may be distinct, with pair formation corresponding to a higher temperature scale than the condensation of the pairs into the SC state. That is, there appear to be at least two distinct energy scales associated with the formation of the high-temperature SC state. This is reminiscent of grand unified theories in elementary particle physics, where qualitatively different physical phases result from a hierarchy of symmetry breakings occurring on different energy (temperature) scales. This finds its most natural explanation in a Lie group structure that is broken spontaneously (and perhaps explicitly) down to subgroups at different characteristic energy scales. I. DYNAMICS AND SYMMETRIESSuch observations argue strongly for a theory based on continuous symmetries of the dynamical system that is capable of describing more sophisticated pairing than found in the simple BCS picture (which is described by a single complex order parameter), and capable of unifying different collective modes and phases on a equivalent footing. Then such fundamentally different physics as antiferromagnetic order and superconductivity can emerge from the same effective Hamiltonian as concentration variables (e.g., doping parameters) are varied. A. Fermion Dynamical SymmetriesFor approxim...

We compare Newtonian three-flavor multigroup Boltzmann (MGBT) and (Bruenn's) multigroup flux-limited diffusion (MGFLD) neutrino transport in postbounce core collapse supernova environments. We focus our study on quantities central to the postbounce neutrino heating mechanism for reviving the stalled shock. Stationary-state three-flavor neutrino distributions are developed in thermally and hydrodynamically frozen time slices obtained from core collapse and bounce simulations that implement Lagrangian hydrodynamics and MGFLD neutrino transport. We obtain distributions for time slices at 106 ms and 233 ms after core bounce for the core of a 15 M ⊙ progenitor, and at 156 ms after core bounce for a 25 M ⊙ progenitor. For both transport methods, the electron neutrino and antineutrino luminosities, RMS energies, and mean inverse flux factors, all of which enter the neutrino heating rates, are computed as functions of radius and compared. The net neutrino heating rates are also computed as functions of radius and compared.Notably, we find significant differences in neutrino luminosities and mean inverse flux factors between the two transport methods for both precollapse models and for all three time slices. In each case, the luminosities for each transport method begin to diverge above the neutrinospheres, where the MGBT luminosities become larger than their MGFLD counterparts, finally settling to a constant difference maintained to the edge of the core. We find that the mean inverse flux factors, which describe the degree of forward peaking in the neutrino radiation field, also differ significantly between the two transport methods, with MGBT providing more isotropic radiation fields in the gain region.Most important, we find, for a region above the gain radius, net heating rates for MGBT that are as much as ∼ 2 times the corresponding MGFLD rates, and net cooling rates below the gain radius that are typically ∼0.8 times the MGFLD rates. These differences stem from differences in the neutrino luminosities and mean inverse flux factors, which can be as much as 11% and 24%, respectively. They are greatest at earlier postbounce times for a given progenitor mass and, for a given postbounce time, greater for greater progenitor mass. We discuss the ramifications these new results have for the supernova mechanism.

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SU (4) dynamical symmetry is shown to imply a no-double-occupancy constraint on the minimal symmetry description of antiferromagnetism and d-wave superconductivity. This implies a maximum doping fraction of 1 4 for cuprates and provides a microscopic critique of the projected SO(5) model. We propose that SU (4) superconductors are representative of a class of compounds that we term non-abelian superconductors. We further suggest that non-abelian superconductors may exist having SU (4) symmetry and therefore cuprate-like dynamics, but without d-wave hybridization.

An SU (4) model of high-temperature superconductivity and antiferromagnetism has recently been proposed. The SO(5) group employed by Zhang is embedded in this SU (4) as a subgroup, suggesting a connection between our SU (4) model and the Zhang SO(5) model. In order to understand the relationship between the the two models, we have used generalized coherent states to analyze the nature of the SO(5) subgroup. By constructing coherent-state energy surfaces, we demonstrate explicitly that the SU (4) ⊃ SO(5) symmetry can be interpreted as a critical dynamical symmetry interpolating between superconducting and antiferromagnetic phases, and that this critical dynamical symmetry has many similarities to critical dynamical symmetries identified previously in other fields of physics. More generally, we demonstrate with this example that the mathematical techniques associated with generalized coherent states may have powerful applications in condensed matter physics because they provide a clear connection between microscopic many-body theories and their broken-symmetry approximate solutions. In addition, these methods may be interpreted as defining the most general Bogoliubov transformation subject to a Lie group symmetry constraint, thus providing a mathematical connection between algebraic formulations and the language of quasiparticle theory. Finally, we suggest that the identification of the SO(5) symmetry as a critical dynamical symmetry implies deep algebraic connections between high-temperature superconductors and seemingly unrelated phenomena in other field of physics.