Showing papers in "Physical Review D in 1979"
TL;DR: In this article, the existence of fundamental scalar fields constitutes a serious flaw of the Weinberg-Salam theory and a possible scheme without such fields is described, where the symmetry breaking is induced by a new strongly interacting sector whose natural scale is of the order of a few TeV.
Abstract: We argue that the existence of fundamental scalar fields constitutes a serious flaw of the Weinberg-Salam theory. A possible scheme without such fields is described. The symmetry breaking is induced by a new strongly interacting sector whose natural scale is of the order of a few TeV.
1,405 citations
TL;DR: In this article, it was shown that a static, spherically symmetric gravitational field is given by the Schwarzschild metric, and that in the weak-field approximation an antisymmetric field of zero mass and zero spin exists, besides gravitons.
Abstract: A gravitational theory is formulated on the Weitzenb\"ock space-time, characterized by the vanishing curvature tensor (absolute parallelism) and by the torsion tensor formed of four parallel vector fields. This theory is called new general relativity, since Einstein in 1928 first gave its original form. New general relativity has three parameters ${c}_{1}$, ${c}_{2}$, and $\ensuremath{\lambda}$, besides the Einstein constant $\ensuremath{\kappa}$. In this paper we choose ${c}_{1}=0={c}_{2}$, leaving open $\ensuremath{\lambda}$. We prove, among other things, that (i) a static, spherically symmetric gravitational field is given by the Schwarzschild metric, that (ii) in the weak-field approximation an antisymmetric field of zero mass and zero spin exists, besides gravitons, and that (iii) new general relativity agrees with all the experiments so far carried out.
1,075 citations
TL;DR: In this article, the phase diagram of lattice gauge theories coupled to fixed-length scalar (Higgs) fields is studied and it is shown that when the Higgs fields transform like the fundamental representation of the gauge group, the confining phases are smoothly connected, i.e., they are not separated by a phase boundary.
Abstract: We study the phase diagram of lattice gauge theories coupled to fixed-length scalar (Higgs) fields. We consider several gauge groups: ${Z}_{2}$, U(1), and $\mathrm{SU}(N)$. We find that when the Higgs fields transform like the fundamental representation of the gauge group the Higgs and confining phases are smoothly connected, i.e., they are not separated by a phase boundary. When the Higgs fields transform like some representation other than the fundamental, a phase boundary may exist. This is the case for $\mathrm{SU}(N)$ with all the Higgs fields in the adjoint representation and for U(1) with all the Higgs fields in the charge-$N(Ng1)$ representation. We present an argument due to Wegner that indicates the stability of the pure gauge transition. Another phase, free charge or Coulomb, is generally present. In this regime, the spectrum of the theory contains massless gauge bosons (for continuous groups) and finite-energy states that represent free charges.
739 citations
TL;DR: In this paper, it was shown that the dynamical symmetry breakdown of a gauge symmetry can lead to simple relations among the masses of intermediate vector bosons, and that the symmetry breakdown can in some cases lead to a simple relation among the mass of a vector boson.
Abstract: It is shown that the dynamical symmetry breakdown of a gauge symmetry can in some cases lead to simple relations among the masses of intermediate vector bosons.
732 citations
TL;DR: In this paper, a detailed analysis of the double-slit experiment is performed, in which one tries to observe both wave and particle properties of light, and a practically realizable version of this experiment, to which the above analysis applies, is proposed.
Abstract: A detailed analysis of Einstein's version of the double-slit experiment, in which one tries to observe both wave and particle properties of light, is performed. Quantum nonseparability appears in the derivation of the interference pattern, which proves to be surprisingly sharp even when the trajectories of the photons have been determined with fairly high accuracy. An information-theoretic approach to this problem leads to a quantitative formulation of Bohr's complementarity principle for the case of the double-slit experiment. A practically realizable version of this experiment, to which the above analysis applies, is proposed.
523 citations
TL;DR: In this paper, the existence of global maximal and constant mean-curvature time functions in general relativity is studied. But the authors focus on the case where the curvature of the time slices is constant.
Abstract: We explore the problem of the existence of global maximal ($K=0$) and constant-mean-curvature ($K={K}_{0}$) time functions in general relativity. We attempt a rigorous definition of numerical relativity so as to bridge the gap between the field and mathematical relativity. We point out that numerical relativity can in principle construct any globally hyperbolic solution to Einstein's equations. This involves the construction of Cauchy time functions. Therefore we review what is known about the existence and uniqueness of such functions when their mean curvature is specified to be a constant on each time slice. We note that in strong-field solutions which contain singularities the question of existence is intimately connected to the nature of the singularity. Defining the class of "crushing singularities" we prove new theorems showing that $K=0$ or $K={K}_{0}$ time functions uniformly avoid such singularities (which include both Cauchy horizons and some curvature singularities). We then study the inhomogeneous generalizations of the Oppenheimer-Snyder spherical-dust-collapse spacetimes. These Tolman-Bondi solutions are classified as to their causal structure and found to contain naked singularities of a new type if the collapse is sufficiently inhomogeneous. We calculate the $K=0$ and $K={K}_{0}$ time slices for a variety of these spacetimes. We find that since some extreme dust collapses lead to noncrushing singularities, maximal time slicing can hit the singularity before covering the domain of outer communications of the resulting black hole. Furthermore, the use of $K={K}_{0}$ slices in the presence of a naked singularity is discussed.
438 citations
TL;DR: In this article, it was shown that even if vacuum neutrino oscillations exist, they are effectively inhibited from occurring in collapsing stars because of the high matter density of the stars.
Abstract: It is shown that even if vacuum neutrino oscillations exist, they are effectively inhibited from occurring in collapsing stars because of the high matter density.
402 citations
TL;DR: In this article, an asymptotic series for the renormalized stress-energy tensor is developed as far as the inverse-square term in powers of the distance.
Abstract: Electromagnetic and scalar fields are quantized in the region near an arbitrary smooth boundary, and the renormalized expectation value of the stress-energy tensor is calculated. The energy density is found to diverge as the boundary is approached. For nonconformally invariant fields it varies, to leading order, as the inverse fourth power of the distance from the boundary. For conformally invariant fields the coefficient of this leading term is zero, and the energy density varies as the inverse cube of the distance. An asymptotic series for the renormalized stress-energy tensor is developed as far as the inverse-square term in powers of the distance. Some criticisms are made of the usual approach to this problem, which is via the "renormalized mode sum energy," a quantity which is generically infinite. Green's-function methods are used in explicit calculations, and an iterative scheme is set up to generate asymptotic series for Green's functions near a smooth boundary. Contact is made with the theory of the asymptotic distribution of eigenvalues of the Laplacian operator. The method is extended to nonflat space-times and to an example with a nonsmooth boundary.
365 citations
TL;DR: In this article, an approximate formula for the production of quark pairs in chromoelectric flux tubes by tunneling is derived and various observable implications are studied, and the non-Abelian nature of the SU(3) color group is taken partially into account thus yielding a quantitative estimate for the baryons-to-mesons ratio.
Abstract: Quark confinement is assumed to be implemented by the generation of chromoelectric flux tubes with uniform energy density. Approximate formulas for the production of quark pairs in these tubes by tunneling are derived and various observable implications are studied. The non-Abelian nature of the SU(3) color group is taken partially into account thus yielding a quantitative estimate for the baryons-to-mesons ratio. Reasonable agreement with experiment is obtained.
365 citations
TL;DR: In this article, an intertwining operator is constructed that explicitly expresses the bilocal two-singleton field in terms of one-particle massless fields, and the theory remains self-consistent and unitary to lowest order in the coupling Fields describing states of one or two singletons, with the attendant interesting gauge problems.
Abstract: Gauge-invariant wave equations for massless fields with fixed, but arbitrary, integer spin have been constructed Extended to include interactions with external sources, the theory remains self-consistent and unitary to lowest order in the coupling Fields describing states of one or two singletons, with the attendant interesting gauge problems, are studied An intertwining operator is constructed that explicitly expresses the bilocal two-singleton field in terms of one-particle massless fields
358 citations
TL;DR: In this paper, the general solution to Einstein's equations for a cloud of strings with spherical, plane, and a particular case of cylindrical symmetry are studied. And the solution with spherical symmetry is used to construct a double-layer model of a star.
Abstract: A gauge-invariant version of the Stachel string cloud model is presented. Some formal aspects as well as the energy conditions for the model are studied. The general solution to Einstein's equations for a cloud of strings with spherical, plane, and a particular case of cylindrical symmetry are studied. The solution with spherical symmetry is used to construct a double-layer model of a star.
TL;DR: In this paper, it was shown that an antisymmetric tensor satisfying the Penrose-Floyd equation f/sub munu(p/) = 0 is a Dirac tensor that can be interpreted as the square root of the ordinary total squared angular momentum Casimir operator of the rotation group.
Abstract: It is found that an operator of the form i..gamma../sub 5/..gamma../sup ..mu../(f/sub ..mu..//sup ..nu../del/sub ..nu../ - 1/6..gamma../sup ..nu../..gamma../sup p/f/sub munu(p/) commutes with the Dirac operator ..gamma../sup ..mu../del/sub ..nu../ whenever f/sub munu/ is an antisymmetric tensor satisfying the Penrose-Floyd equation f/sub ..mu..//sub( nu(p/) = 0. Such a tensor exists notably in the Kerr solutions and in the flat-space limit wherein the operator can be interpreted as the square root of the ordinary total squared angular momentum Casimir operator of the rotation group.
TL;DR: In this paper, the behavior of lattice models of quark confinement at high temperature was investigated and it was shown that at high temperatures, free gluons form a plasma which Debye screens the quarks.
Abstract: We consider the behavior of lattice models of quark confinement at high temperature. We find that confinement is strictly a low-temperature phenomenon. At high temperatures a transition to a plasma-like phase occurs. In this phase free gluons form a plasma which Debye screens the quarks.
TL;DR: In this article, the general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give $c$-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases.
Abstract: The general formalism for path integrals expressed in terms of an arbitrary continuous representation (generalized coherent states) is applied to give $c$-number formulations for the canonical algebra and for the spin algebra, and is used to derive meaningful stationary-phase approximations for these two cases. Some clarification of a recent discussion by Jevicki and Papanicolaou for the spin case is given.
TL;DR: In this paper, the instanton-generated CP$-violating perturbation was shown to have a form in the form of a 3-dimensional convex polygonal graph.
Abstract: The instanton-generated $\mathrm{CP}$-violating perturbation is shown to have a form $\frac{3\overline{\ensuremath{\theta}}{m}_{u}{m}_{d}{m}_{s}}{({m}_{u}{m}_{d}+{m}_{u}{m}_{s}+{m}_{d}{m}_{s})}(\overline{\ensuremath{\psi}}i{\ensuremath{\gamma}}_{5}\ensuremath{\psi})$ for $\overline{\ensuremath{\theta}}\ensuremath{\ll}1$. The induced neutron electric dipole moment, evaluated in the framework of the MIT bag model, is ${d}_{n}=8.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}\overline{\ensuremath{\theta}}e$ cm.
TL;DR: In this paper, the methods of quantum detection theory are applied to a resonant-mass gravitational-radiation antenna and the sensitivity limits are found which depend strongly on the quantum state in which the antenna is prepared.
Abstract: The methods of quantum detection theory are applied to a resonant-mass gravitational-radiation antenna. Quantum sensitivity limits are found which depend strongly on the quantum state in which the antenna is prepared. Optimum decision strategies and their corresponding sensitivities are derived for some important initial states. The linear detection limit (${E}_{min}\ensuremath{\sim}\ensuremath{\hbar}\ensuremath{\omega}$) is shown to apply when the antenna is prepared in a coherent state. Preparation of the antenna in an excited energy eigenstate or in a state highly localized in position or momentum space leads to increased sensitivity. A set of minimum-uncertainty states for phase-sensitive detection is introduced.
TL;DR: In this article, the invariant cross-sections of hadrons at large transverse momentum were measured at a laboratory angle of 77 mrad, which corresponds to angles near 90\ifmmode^\circ\else\textdegree\fi{} in the c.m. system of the incident proton and a single nucleon at rest.
Abstract: Measurements of the invariant cross section $\frac{E{d}^{3}\ensuremath{\sigma}}{{d}^{3}p}$ are presented for the production of hadrons ($\ensuremath{\pi}$, $K$, $p$, and $\overline{p}$) at large transverse momentum (${p}_{\ensuremath{\perp}}$) by 200-, 300-, and 400-GeV protons incident on ${\mathrm{H}}_{2}$, ${\mathrm{D}}_{2}$, Be, Ti, and W targets. The measurements were made at a laboratory angle of 77 mrad, which corresponds to angles near 90\ifmmode^\circ\else\textdegree\fi{} in the c.m. system of the incident proton and a single nucleon at rest. The range in ${p}_{\ensuremath{\perp}}$ for the data is $0.77\ensuremath{\le}{p}_{\ensuremath{\perp}}\ensuremath{\le}6.91$ GeV/c, corresponding to values of the scaling variable ${x}_{\ensuremath{\perp}}=\frac{2{p}_{\ensuremath{\perp}}}{\sqrt{s}}$ from 0.06 to 0.64. For $p\ensuremath{-}p$ collisions, the pion cross sections can be represented in the region ${x}_{\ensuremath{\perp}}g35$ by the form $(\frac{1}{{{p}_{\ensuremath{\perp}}}^{n}}){(1\ensuremath{-}{x}_{\ensuremath{\perp}})}^{b}$, with $n=8$ and $b=9$. The ratio of ${\ensuremath{\pi}}^{+}$ to ${\ensuremath{\pi}}^{\ensuremath{-}}$ production grows as a function of ${x}_{\ensuremath{\perp}}$ to a value larger than 2 at ${x}_{\ensuremath{\perp}}\ensuremath{\gtrsim}0.5$. The ratios of the production of ${K}^{+}$ and protons to ${\ensuremath{\pi}}^{+}$ and of ${K}^{\ensuremath{-}}$ and antiprotons to ${\ensuremath{\pi}}^{\ensuremath{-}}$ also scale with ${x}_{\ensuremath{\perp}}$ for $p\ensuremath{-}p$ collisions. The ${K}^{\ifmmode\pm\else\textpm\fi{}}$, $p$, and $\overline{p}$ fitted values for $n$ and $b$ are given. Particle ratios are also presented for ${\mathrm{D}}_{2}$, Be, Ti, and W targets and the dependences on atomic weight ($A$) are discussed.
TL;DR: In this article, the asymptotic behavior of the form factor in quantum electrodynamics is calculated in case the photon has a mass, and the result is essentially that found by Sudakov.
Abstract: The asymptotic behavior of the form factor in quantum electrodynamics is calculated in case the photon has a mass. The technique used takes into account all logarithms in ${q}^{2}$ but neglects inverse powers of ${q}^{2}$. The result is essentially that found by Sudakov. In order to obtain this result the large-$n$ behavior of ${\ensuremath{\gamma}}_{n}$ is obtained.
TL;DR: In this paper, it was shown that neutrinos are emitted preferentially in the direction opposite to the hole's angular momentum, and that in a rotating thermal radiation there exist equilibrium neutrino and antineutrino currents parallel to the angular velocity vector.
Abstract: Two macroscopic effects of parity nonconservation are considered. (i) Particle emission by rotating black holes is shown to be asymmetric. In particular, neutrinos are emitted preferentially in the direction opposite to the hole's angular momentum. (ii) It is shown that in a rotating thermal radiation there exist equilibrium neutrino and antineutrino currents parallel to the angular velocity vector.
TL;DR: In this paper, the ground-state baryons were discussed in a quark model with flavor-independent confinement and color hyperfine interactions, including the effects of wave-function distortions for unequal quark masses as well as interband mixing via the hyperfine interaction.
Abstract: We discuss the ground-state baryons $N$, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, $\ensuremath{\Xi}$, $\ensuremath{\Delta}$, ${\ensuremath{\Sigma}}^{*}$, ${\ensuremath{\Xi}}^{*}$, and $\ensuremath{\Omega}$ in a quark model with flavor-independent confinement and color hyperfine interactions. We include the effects of wave-function distortions for unequal quark masses as well as interband mixing via the hyperfine interactions and find good agreement with the observed masses.
TL;DR: In this paper, a momentum-space representation of the Feynman propagator for scalar and spin-text-one-half fields propagating in arbitrary curved spacetimes was obtained.
Abstract: We obtain a momentum-space representation of the Feynman propagator $G(x,{x}^{\ensuremath{'}})$ for scalar and spin-\textonehalf{} fields propagating in arbitrary curved spacetimes. The construction uses Riemann normal coordinates with origin at the point ${x}^{\ensuremath{'}}$ and is therefore only valid for points $x$ lying in a normal neighborhood of ${x}^{\ensuremath{'}}$. We show that the resulting momentum-space representation is equivalent to the DeWitt-Schwinger propertime representation. Our momentum-space representation permits one to apply momentum-space techniques used in Minkowski space to arbitrary curved spacetimes. The usefulness of this representation in discussing the renormalizability of interacting field theories in curved spacetime is illustrated by an explicit renormalization, to second order in the coupling constant, of a quartically self-interacting scalar field theory in an arbitrary spacetime.
TL;DR: In this article, the scale dependence of a scale-invariant mass is parametrized by a scale invariant mass, which is a pure number which is exactly deducible from one-loop results.
Abstract: Massless quantum chromodynamics cannot be renormalized on-shell; various possible off-shell renormalization prescriptions yield different definitions of a scale-dependent coupling constant $g$. We show how to relate physical predictions computed in different renormalization schemes. In particular, we compute the dimensionally regularized two- and three-point functions at the symmetric point in momentum space through one-loop order, and deduce the relation between ${g}_{min}$ defined by minimal subtraction and ${g}_{\mathrm{mom}}$ defined by momentum-space subtraction. We find that ${g}_{\mathrm{mom}}$ is fairly insensitive to which vertex one chooses to define it, and only weakly gauge dependent. ${g}_{min}$ is shown to depend strongly on the dimensional-regularization procedure, and can therefore differ quite dramatically from ${g}_{\mathrm{mom}}$. The scale dependence of $g$ is conventionally parametrized by a scale-invariant mass $\ensuremath{\Lambda}$; the ratio of $\ensuremath{\Lambda}$'s defined by any two renormalization schemes is a pure number which we show is exactly deducible from our one-loop results.
TL;DR: In this article, strong-interaction corrections to the non-leptonic weak interaction Hamiltonian were calculated in the leading-logarithmic approximation using quantum chromodynamics.
Abstract: Strong-interaction corrections to the nonleptonic weak-interaction Hamiltonian are calculated in the leading-logarithmic approximation using quantum chromodynamics. Starting with a six-quark theory, the W boson, t quark, b quark, and c quark are successively considered as "heavy" and the effective Hamiltonian calculated. The resulting effective Hamiltonian for strangeness-changing nonleptonic decays involves u, d, and s quarks and has possible CP-violating pieces both in the usual (V-A)×(V-A) terms and in induced, "penguin"-type terms. Numerically, the CP-violating compared to CP-conserving parts of the latter terms are close to results calculated on the basis of the lowest-order "penguin" diagram.
TL;DR: In this paper, the theory and phenomenology of such a situation were examined as a guide for future experimental searches in theories with more scalar multiplets than the minimal Weinberg-Salam model, where there will exist charged physical Higgs particles in addition to the neutral Higgs particle.
Abstract: In theories with more scalar multiplets than the minimal Weinberg-Salam model, there will exist charged physical Higgs particles, in addition to the neutral Higgs particle. We examine the theory and phenomenology of such a situation as a guide for future experimental searches.
TL;DR: In this paper, it was shown that there exists a massless phase in the middle between the ordered and disordered Ising-type phases in a two-dimensional spin system.
Abstract: It is shown that in a two-dimensional ${Z}_{p}$ spin system for $p$ not too small there exists a massless phase in the middle between the ordered and disordered Ising-type phases. A similar thing happens in a four-dimensional ${Z}_{p}$ gauge theory, where a massless QED-like phase appears between the screened and the confined phases. The existence of the middle phase is deduced logically from the existence of such a phase in the continuous O(2)-invariant models using self-duality and correlation inequalities. For the spin case the transition towards this phase is analyzed using a Kosterlitz type of renormalization group suggesting an essential singularity of the correlation length at both transition points. A Hamiltonian strong-coupling expansion up to ninth order is applied to the ${Z}_{p}$ spin system. The results of the Pad\'e analysis of this expansion are consistent with the phase structure described above. For $p\ensuremath{\le}4$ the analysis suggests two phases with a conventional singularity behavior at the transition. In the nontrivial case of $p=3$, critical exponents are calculated and found to give good agreement with experiment. For $p\ensuremath{\ge}5$ the analysis favors three phases with an essential singularity at the transition.
TL;DR: In this paper, a bootstrap program for determining Green's functions from an exact $S$ matrix is carried out for the simplest soliton field theory of a scalar field with $S-matrix operator.
Abstract: The bootstrap program for determining Green's functions from an exact $S$ matrix is carried out for the simplest soliton field theory of a scalar field with $S$-matrix operator $S={(\ensuremath{-}1)}^{\frac{N(N\ensuremath{-}1)}{2}}$, where $N$ is the total number operator. Despite the formal simplicity of the $S$ matrix, the Green's functions derived have a rich structure. The results can be checked since this field theory is none other than that of the order variable of the Ising model in the scaling limit above the critical temperature.
TL;DR: In this article, a one-boson-exchangepotential model is presented, which consists of local potentials due to exchanges of members of the pseudoscalar, vector, and scalar-meson nonets.
Abstract: The $\mathrm{NN}$ and $\mathrm{YN}$ results are presented from a one-boson-exchange-potential model. It consists of local potentials due to exchanges of members of the pseudoscalar, vector, and scalar-meson nonets. SU(3) relations are assumed for the axial-vector couplings of the pseudoscalar mesons, for the electric and magnetic couplings of the vector mesons, for the direct couplings of the scalar mesons, and for the hard-core radii. In the fit to $\mathrm{NN}$ the nonstrange-meson-nucleon couplings are determined. The simultaneous $\mathrm{YN}$ analysis determines the $\frac{F}{D}$ ratios and the SU(3) parameters of the scalar-meson nonet. The description of the $\mathrm{NN}$ data is good ($\frac{{\ensuremath{\chi}}^{2}}{\mathrm{data}}=2.17$), and also of the $\ensuremath{\Lambda}p$, ${\ensuremath{\Sigma}}^{+}p$, and ${\ensuremath{\Sigma}}^{\ensuremath{-}}p$ data up to the pion production threshold. Very close to the $\ensuremath{\Sigma}N$ threshold we find a $\ensuremath{\Lambda}N$ resonance, which is dominantly in the $^{3}D_{1}$ wave. The $\ensuremath{\Lambda}p$ cross section is maximal just below the ${\ensuremath{\Sigma}}^{+}n$ threshold at $E=2128.918$ MeV, in agreement with the experimental results of $\ensuremath{\Lambda}p$ final state interactions. The poles belonging to the $\ensuremath{\Lambda}p$ resonance are located on the second Riemann sheet at $E=2131.77\ifmmode\pm\else\textpm\fi{}i2.39$ MeV.
TL;DR: In this article, the authors examined dynamical chiral symmetry breaking in quantum chromodynamics (QCD) and obtained exact representations for the pion decay constant and the electromagnetic form factor.
Abstract: We examine dynamical chiral symmetry breaking in quantum chromodynamics (QCD). To lowest order in a double perturbation expansion in $g$, the gauge coupling, and $\ensuremath{\lambda}=\mathrm{exp}(\ensuremath{-}\frac{1}{b{g}^{2}})$ we obtain exact representations for the pion decay constant ${f}_{\ensuremath{\pi}}$ and electromagnetic form factor ${F}_{\ensuremath{\pi}}({q}^{2})$. Using the experimental values for ${f}_{\ensuremath{\pi}}$ and ${F}_{\ensuremath{\pi}}({q}^{2})$, we estimate the dynamically generated quark mass to be 270-300 MeV, in good agreement with a value obtained from ${e}^{+}{e}^{\ensuremath{-}}$ annihilation data. We also calculate the finite electromagnetic self-energy of a quark in QCD and discuss the implications.
TL;DR: In this paper, the effect of one-loop contributions of conformally invariant matter fields on the evolution of homogeneous, isotropic, spatially flat classical geometries containing classical radiation was investigated.
Abstract: The use of the effective-action method to calculate quantum effects in the early universe is described. An application is made to the calculation of the effect of one-loop contributions of conformally invariant matter fields on the evolution of homogeneous, isotropic, spatially flat classical geometries containing classical radiation.