For a black hole’s spacetime manifold in the Euclidean signature, its metric is positive definite and therefore a Riemannian manifold. It can be regarded as a gravitational instanton and a topological characteristic which is the Euler number to which it is associated. In this paper we derive a formula for the Euler numbers of four-dimensional rotating black holes by the integral of the Euler density on the spacetime manifolds of black holes. Using this formula, we obtain that the Euler numbers of Kerr and Kerr-Newman black holes are $2.$ We also obtain that the Euler number of the Kerr-Sen metric in the heterotic string theory with one boost angle nonzero is $2,$ which is in accordance with its topology.

A quantitative test for the validity of the semiclassical approximation in gravity is given. The criterion proposed is that solutions to the semiclassical Einstein equations should be stable to linearized perturbations, in the sense that no gauge invariant perturbation should become unbounded in time. A self-consistent linear response analysis of these perturbations, based upon an invariant effective action principle, necessarily involves metric fluctuations about the mean semiclassical geometry, and brings in the two-point correlation function of the quantum energy-momentum tensor in a natural way. This linear response equation contains no state dependent divergences and requires no new renormalization counterterms beyond those required in the leading order semiclassical approximation. The general linear response criterion is applied to the specific example of a scalar field with arbitrary mass and curvature coupling in the vacuum state of Minkowski spacetime. The spectral representation of the vacuum polarization function is computed in n dimensional Minkowski spacetime, and used to show that the flat space solution to the semiclassical Einstein equations for $n=4\mathrm{}$ is stable to all perturbations on distance scales much larger than the Planck length.

We discuss the issue of toy model building for the dark energy component of the universe. Specifically, we consider two generic toy models recently proposed as alternatives to quintessence models, respectively known as Cardassian expansion and the Chaplygin gas. We show that the former is entirely equivalent to a class of quintessence models. We determine the observational constraints on the latter, coming from recent supernovae results and from the shape of the matter power spectrum. As expected, these restrict the model to a behavior that closely matches that of a standard cosmological constant $\Lambda .$

We discuss the theory of dark energy based on maximally extended supergravity and suggest a possible anthropic explanation of the present value of the cosmological constant and of the observed ratio between dark energy and energy of matter.

Carlson, Carone, and Lebed have derived the Feynman rules for a consistent formulation of noncommutative QCD. The results they obtained were used to constrain the noncommutativity parameter in Lorentz violating noncommutative field theories. However, their constraint depended upon an estimate of the matrix element of the quark level operator $\left(p/-m\right)$ in a nucleon. In this paper we calculate the matrix element of $\left(p/-m\right),$ using a variety of confinement potential models. Our results are within an order of magnitude agreement with the estimate made by Carlson et al. The constraints placed on the noncommutativity parameter were very strong, and are still quite severe even if weakened by an order of magnitude.

We present new, nontrivial generalizations of the recent Tomaras-Tsamis-Woodard extension of the original Schwinger formula for charged pair production in a constant electric field. That extension generalized the Schwinger result to electric fields $E_3{(x}_{\pm })$ dependent upon one or the other light-cone coordinates, $x_{+}$ or $x_{-},$ $x_{\pm }{=x}_3\pm x_0;$ the present work generalizes their result to electric fields $E_3{(x}_{+}{,x}_{-})$ dependent upon both coordinates. Displayed in the form of a final, functional integral, or equivalent linkage operation, our result does not appear to be exactly calculable in the general case; and we give a simple, approximate example when $E_3{(x}_{+}{,x}_{-})$ is a slowly varying function of its variables. We extend this result to the more general case where $\overrightarrow{E}$ can point in a varying direction, and where an arbitrary magnetic field $\overrightarrow{B}$ is present; both extensions can be cast into the form of Gaussian-weighted functional integrals over well defined factors, which are amenable to approximations depending on the nature and variations of the fields.

The forward-backward asymmetry in $e^{+}{e}^{-}$ annihilation into a quark-antiquark pair is considered in the double-logarithmic approximation at energies much higher than the masses of the weak bosons. It is shown that after accounting for the exchange of virtual photons and $W,Z$ bosons to all orders one is led to the following effect (asymmetry): quarks with positive electric charge (e.g., u, $\overline{d})$ tend to move in the $e^{+}$ direction whereas quarks with negative charges (e.g., d, $ū)$ tend to move in the $e^{-}$ direction. The value of the asymmetry grows with increasing energy when the produced quarks are within a cone with an opening angle, in the center of mass frame, ${\theta }_0\sim {2M}_Z/\sqrt{s}$ around the $e^{+}{e}^{-}$ beam. Outside this cone, at ${\theta }_0\ll \theta \ll 1,$ the asymmetry is inversely proportional to $\theta .$

We reconstruct the neutrino mass matrix in the flavor basis, in the case of an inverted mass hierarchy (ordering), using all available experimental data on neutrino masses and oscillations. We analyze the dependence of the matrix elements $m_{\alpha \beta }$ on the $\mathrm{CP}$ violating Dirac $\delta$ and Majorana $\rho$ and $\sigma$ phases, for different values of the absolute mass scale. We find that the present data admit various structures of the mass matrix: (i) hierarchical structures with a set of small (zero) elements; (ii) structures with equalities among various groups of elements: e-row and/or μτ-block elements, diagonal and/or off-diagonal elements; (iii) “democratic” structure. We find the values of phases for which these structures are realized. The mass matrix elements can anticorrelate with flavor: inverted partial or complete flavor alignment is possible. For various structures of the mass matrix we identify the possible underlying symmetry. We find that the mass matrix can be reconstructed completely only in particular cases, provided that the absolute scale of the mass is measured. Generally, the freedom related to the Majorana phase $\sigma$ will not be removed, thus admitting various types of mass matrix.

An updated analysis of all available neutrino oscillation evidence in solar experiments (SK day and night spectra, global rates from Homestake, SAGE, and GALLEX) including the latest SNO CC and NC data is presented. Assuming that the shape of the SNO CC energy spectrum is undistorted and using the information provided by SNO we obtain, for the fraction of electron neutrinos remaining in the solar beam at energies ≳5 MeV, ${\phi }_{\mathrm{CC}}/{\phi }_{\mathrm{NC}}{=0.34\mathrm{}}_{-0.04}^{+0.05\mathrm{}},$ which is nominally many standard deviations away from the standard value. The fraction of oscillating neutrinos into active ones is computed to be $\left({\Phi }_{\mathrm{NC}}-{\Phi }_{\mathrm{CC}}\right)/({\Phi }_{\mathrm{SSM}}-{\Phi }_{\mathrm{CC}}{)=0.92\mathrm{}}_{-0.20}^{+0.39\mathrm{}},$ nearly $5\sigma$ deviations from the pure sterile oscillation case. The data are still compatible with an important fraction of sterile component in the solar beam (up to 20%–30% of the total). In the framework of two active neutrino oscillations we determine individual neutrino mixing parameters and their errors in the region of no spectrum distortion $(\Delta 〈T_e〉<1\%);$ we obtain $\Delta m^2{=4.5\mathrm{}}_{-1.4}^{+2.7\mathrm{}}\times {10}^{-5}{\mathrm{eV}}^2,{\tan }^2\theta {=0.40\mathrm{}}_{-0.08}^{+0.10\mathrm{}}.$ These parameters are in agreement with those ones of the best ${\chi }^2$ solution in the LMA region.

We introduce the 2-point-particle-irreducible expansion, which sums bubble graphs to all orders. We prove the renormalizibility of this summation. We use it on the Gross-Neveu model to calculate the mass gap and vacuum energy. After an optimization of the expansion, the final results are qualitatively good.

In the TeV energy region and above, the uncertainty in the level of prompt atmospheric neutrinos would limit the search for diffuse astrophysical neutrinos. We suggest that neutrino telescopes may provide an empirical determination of the flux of prompt atmospheric electron and muon neutrinos by measuring the flux of prompt down-going muons. Our suggestion is based on the consideration that prompt neutrino and prompt muon fluxes at sea level are almost identical.

We construct an ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ gauge model based on an $S_{2L}$ permutation symmetry for left-handed $\mu$ and $\tau$ families, which provides the almost maximal atmospheric neutrino mixing and the large solar neutrino mixing of the large mixing angle type. Neutrinos acquire one-loop radiative masses induced by the radiative mechanism of the Zee type as well as tree level masses induced by the type II seesaw mechanism utilizing interactions of lepton triplets with an $\mathrm{SU}\left(3\right)\mathrm{}$-sextet scalar. The atmospheric neutrino mixing controlled by the tree-level and radiative masses turns out to be almost maximal owing to the presence of $S_{2L}$ supplemented by a $Z_4$ discrete symmetry. These symmetries ensure the near equality between the ${\nu }_e-{\nu }_{\mu }$ and ${\nu }_e-{\nu }_{\tau }$ radiative masses dominated by contributions from heavy leptons contained in the third members of lepton triplets, whose Yukawa interactions conserve $S_{2L}$ even after the spontaneous breaking. The solar neutrino mixing controlled by radiative masses, including a ${\nu }_{\mu }-{\nu }_{\tau }$ mass, which are taken to be of similar order, turns out to be described by large solar neutrino mixing angles.

Study of direct photon production in high-energy hadronic collisions provides a clean tool for testing the essential validity of perturbative quantum chromodynamics (PQCD) predictions as well as for constraining the gluon distribution of nucleons. These attractive considerations prompted us to study the characteristics of direct photons at CERN LHC energy $(\sqrt{s}=14\mathrm{}\mathrm{TeV}).\mathrm{}$ In order to validate our simulation results, we first describe the direct photon data at $\sqrt{s}=1.8\mathrm{}\mathrm{TeV}$ in the central pseudorapidity $\left(\eta \right)$ region. We used next-to-leading-order (NLO) QCD calculations and leading-order (LO) PYTHIA estimates with the latest parton distribution function, CTEQ5M1. At $\sqrt{s}=14\mathrm{}\mathrm{TeV},$ the LO and NLO QCD predictions for direct photon cross section are presented as a function of transverse momentum of photon ${(p}_T)$ in the kinematical region $20\mathrm{GeV}<p_T<400\mathrm{}\mathrm{GeV}$ and $|\eta |<3.\mathrm{}$ The sensitivity of the theoretical predictions to the choice of renormalization scales and gluon distributions is also demonstrated. The pseudorapidity $\left(\eta \right)$ and cone size dependence of the direct photon cross section is also discussed.

We examine the prospects for using lead as a supernova-neutrino detector by considering the spectrum of electrons produced, and the number of one- and two-neutron events. We show that the electron energy spectrum from charged-current reactions could be used to extract information about the high-temperature component of the neutrino spectrum. Some degree of electron neutrino oscillation is expected in the supernova envelope. We examine the prospects for untangling the signatures of various oscillation scenarios, including, e.g., normal or inverted hierarchies, and different values for the small mixing angle ${\theta }_{13}.$

We discuss the worldvolume description of intersecting D-branes, including the metric on the moduli space of deformations. We impose a choice of static gauge that treats all the branes on an equal footing and describes the intersection of D-branes as an embedded special Lagrangian three-surface. Some explicit solutions to these equations are given and their interpretation in terms of a superpotential on moduli space is discussed. These surfaces are interpreted in terms of the flat direction of a non-Abelian superpotential and imply the existence of noncompact $G_2$ manifolds.

It is shown that the vacuum state of weakly interacting quantum field theories can be described, in the Heisenberg picture, as a linear combination of randomly distributed incoherent paths that obey classical equations of motion with constrained initial conditions. We call such paths “pseudoclassical” paths and use them to define the dynamics of quantum fluctuations. Every physical observable is assigned a time-dependent value on each path in a way that respects the uncertainty principle, but in consequence, some of the standard algebraic relations between quantum observables are not necessarily satisfied by their time-dependent values on paths. We define “collective observables” which depend on a large number of independent degrees of freedom and show that the dynamics of their quantum fluctuations can be described in terms of unconstrained classical stochastic processes without reference to any additional external system or to an environment. Our analysis can be generalized to states other than the vacuum. Finally, we compare our formalism to the formalism of coherent states and highlight their differences.

The orbital motion of the Laser Interferometer Space Antenna (LISA) introduces modulations into the observed gravitational wave signal. These modulations can be used to determine the location and orientation of a gravitational wave source. The complete LISA response to an arbitrary gravitational wave is derived using a coordinate free approach in the transverse-traceless gauge. The general response function reduces to that found by Cutler for low frequency, monochromatic plane waves. Estimates of the noise in the detector are found to be complicated by the time variation of the interferometer arm lengths.

We develop a straightforward method to compute charge radii and quadrupole moments for baryons both with and without strangeness, when the number of QCD color charges is $N_c.$ The minimal assumption of the single-photon exchange ansatz implies that only two operators are required to describe these baryon observables. Our results are presented so that SU(3) flavor and isospin symmetry breaking can be introduced according to any desired specification, although we also present results obtained from two patterns suggested by the quark model with gluon exchange interactions. The method also permits to extract a number of model-independent relations; a sample is $r_{\Lambda }^2{/r}_n^2{=3/(N}_c+3),$ independent of SU(3) symmetry breaking.

We present a topologically nontrivial generalization of gauged $N=16\mathrm{}$ supergravity on the coset $E_{8(+8\mathrm{})\mathrm{}}/SO\left(16\right)\mathrm{}$ in three dimensions. This formulation is based on a combination of a BF term and a Chern-Simons term for an $\mathrm{SO}\left(16\right)\mathrm{}$ gauge field $A_{\mu }^{\mathrm{IJ}}.$ The fact that an additional vector field $B_{\mu }^{\mathrm{IJ}}$ is physical and propagating with couplings to σ-model fields makes our new gauging nontrivial and different from the conventional one. Even though the field strength of the $A_{\mu }^{\mathrm{IJ}}$ field vanishes on shell, the action is topologically nontrivial due to a nonvanishing ${\pi }_3$ homotopy. We also present additional modifications by an extra Chern-Simons term. As by-products, we give also an application to $N=9\mathrm{}$ supergravity coupled to a σ model on the coset $F_{4\left(-20\right)}/SO\left(9\right),$ and a new BF-Chern-Simons theory coupled to ${}^{\forall }N$ extended supergravity.

The gravitational field in a neighborhood of a particle of small mass $\mu$ moving through curved spacetime is naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through $O\left(\mu \right).\mathrm{}$ One part is an inhomogeneous field which looks like the $\mu /r$ field tidally distorted by the local Riemann tensor. The other part is a homogeneous field that completely determines the self-force of the particle interacting with its own gravitational field, which changes the worldline at $O\left(\mu \right)$ and includes the effects of radiation reaction. Surprisingly, a local observer measuring the gravitational field in a neighborhood of a freely moving particle sees geodesic motion of the particle in a perturbed vacuum geometry and would be unaware of the existence of radiation at $O\left(\mu \right).\mathrm{}$ In the light of all previous work this is quite an unexpected result.