Viability of Noether Symmetry of F(R) Theory of Gravity

Recently, we have explored vices and virtues of $R^{\frac{3}{2}}$ term in the action which has in-built Noether symmetry and anticipated that a linear term might improve the situation (Sarkar et al., arXiv:1201.2987 [astro-ph.CO], 2012). In the absence of a conserved current it is extremely difficult to obtain an analytical solution of the said fourth order theory of gravity in the presence of a linear term. Here, we therefore enlarge the configuration space by including a scalar field in addition and also taking some of the anisotropic models (in the absence of a scalar field) into account. We observe that Noether symmetry remains obscure and it does not even reproduce the one that already exists in the literature (Sanyal, Gen. Relativ. Gravit., 37:407, 2005). However, there exists in general, a conserved current for F(R) theory of gravity in the presence of a non-minimally coupled scalar field (Sanyal, Phys. Lett. B, 624:81, 2005; Mod. Phys. Lett. A, 25:2667, 2010), which simplifies the field equations considerably. Here, we briefly expatiate the non-Noether conserved current and show that indeed the situation is modified.     

Condensation of an ideal gas with intermediate statistics on the horizon

We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose?CEinstein condensed state with the Hawking temperature T _c =T _H if the particle number of the system be equal to the number of quantum bits of space-time $N \simeq{A}/{l_{p}^{2}}$ . Entropy of the gas is proportional to the area of the horizon (A) by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics.     

From nonassociativity to solutions of the KP hierarchy

A recently observed relation between ‘weakly nonassociative’ algebras $\mathbb{A}$ (for which the associator ( $\mathbb{A},\mathbb{A}^2 ,\mathbb{A}$ ) vanishes) and the KP hierarchy (with dependent variable in the middle nucleus $\mathbb{A}$ ′ of { $\mathbb{A}$ ) is recalled. For any such algebra there is a nonassociative hierarchy of ODEs, the solutions of which determine solutions of the KP hierarchy. In a special case, and with matrix algebra $\mathbb{A}$ ′, this becomes a matrix Riccati hierarchy which is easily solved. The matrix solution then leads to solutions of the scalar KP hierarchy. We discuss some classes of solutions obtained in this way.     

Stability study of a model for the Klein–Gordon equation in Kerr space-time

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein–Gordon equation, describing the propagation of a scalar field of mass $\mu $ in the background of a rotating black hole. Rigorous results prove the stability of the reduced, by separation in the azimuth angle in Boyer–Lindquist coordinates, field for sufficiently large masses. Some, but not all, numerical investigations find instability of the reduced field for rotational parameters $a$ extremely close to $1$ . Among others, the paper derives a model problem for the equation which supports the instability of the field down to $a/M \approx 0.97$ .     

Dynamical behaviors of FRW universe containing a positive/negative potential scalar field in loop quantum cosmology

The dynamical behaviors of FRW Universe containing a posivive/negative potential scalar field in loop quantum cosmology scenario are discussed. The method of the phase-plane analysis is used to investigate the stability of the Universe. It is found that the stability properties in this situation are quite different from the classical cosmology case. For a positive potential scalar field coupled with a barotropic fluid, the cosmological autonomous system has five fixed points and one of them is stable if the adiabatic index $\gamma $ satisfies $0<\gamma <2$ . This leads to the fact that the universe just have one bounce point instead of the singularity which lies in the quantum dominated area and it is caused by the quantum geometry effect. There are four fixed points if one considers a scalar field with a negative potential, but none of them is stable. Therefore, the universe has two kinds of bounce points, one is caused by the quantum geometry effect and the other is caused by the negative potential, the Universe may enter a classical re-collapse after the quantum bounce. This hints that the spatially flat FRW Universe containing a negative potential scalar field is cyclic.     

Analyticity of \eta \pi isospin-violating form factors and the \tau \rightarrow \eta \pi \nu second-class decay

We consider the evaluation of the \(\eta \pi \) isospin-violating vector and scalar form factors relying on a systematic application of analyticity and unitarity, combined with chiral expansion results. It is argued that the usual analyticity properties do hold (i.e. no anomalous thresholds are present) in spite of the instability of the \(\eta \) meson in QCD. Unitarity relates the vector form factor to the \(\eta \pi \rightarrow \pi \pi \) amplitude: we exploit progress in formulating and solving the Khuri–Treiman equations for \(\eta \rightarrow 3\pi \) and in experimental measurements of the Dalitz plot parameters to evaluate the shape of the \(\rho \) -meson peak. Observing this peak in the energy distribution of the \(\tau \rightarrow \eta \pi \nu \) decay would be a background-free signature of a second-class amplitude. The scalar form factor is also estimated from a phase dispersive representation using a plausible model for the \(\eta \pi \) elastic scattering \(S\) -wave phase shift and a sum rule constraint in the inelastic region. We indicate how a possibly exotic nature of the \(a_0(980)\) scalar meson manifests itself in a dispersive approach. A remark is finally made on a second-class amplitude in the \(\tau \rightarrow \pi \pi \nu \) decay.     

A Complete Proof of the Confinement Limit of One-Dimensional Dirac Particles

The validity of the confinement limit obtain by Unanyan et al. (Phys Rev A 79:044101, 2009) is extended by including non-symmetric vector and scalar potentials. It shows that the confinement limit of one-dimensional Dirac particles in vector and scalar potentials is \(\lambda _C/\sqrt{2}\) , with \(\lambda _C\) being the Compton wavelength.     

ScannerS: constraining the phase diagram of a complex scalar singlet at the LHC

We present the first version of a new tool to scan the parameter space of generic scalar potentials, ScannerS (Coimbra et al., ScannerS project., 2013). The main goal of ScannerS is to help distinguish between different patterns of symmetry breaking for each scalar potential. In this work we use it to investigate the possibility of excluding regions of the phase diagram of several versions of a complex singlet extension of the Standard Model, with future LHC results. We find that if another scalar is found, one can exclude a phase with a dark matter candidate in definite regions of the parameter space, while predicting whether a third scalar to be found must be lighter or heavier. The first version of the code is publicly available and contains various generic core routines for tree level vacuum stability analysis, as well as implementations of collider bounds, dark matter constraints, electroweak precision constraints and tree level unitarity.     

Self-Dual Conformal Gravity

We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M, g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci-flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel sections of a certain connection on a complex rank-four vector bundle over M. They provide a natural generalisation of the Bach tensor which vanishes identically for anti-self-dual conformal structures. We use the obstructions to demonstrate that LeBrun’s anti-self-dual metrics on connected sums of \({\mathbb{CP}^2}\) s are not conformally Ricci-flat on any open set. We analyze both Riemannian and neutral signature metrics. In the latter case we find all anti-self-dual metrics with a parallel real spinor which are locally conformal to Einstein metrics with non-zero cosmological constant. These metrics admit a hyper-surface orthogonal null Killing vector and thus give rise to projective structures on the space of β-surfaces.     

A special exact spherically symmetric solution in f(T) gravity theories

A non-diagonal spherically symmetric tetrad field, involving four unknown functions of radial coordinate $r$ r , is applied to the equations of motion of f(T) gravity theory. A special exact vacuum solution with one constant of integration is obtained. The scalar torsion related to this special solution vanishes. To understand the physical meaning of the constant of integration we calculate the energy associated with this solution and show how it is related to the gravitational mass of the system.     

Conformally related metrics and Lagrangians and their physical interpretation in cosmology

Conformally related metrics and Lagrangians are considered in the context of scalar–tensor gravity cosmology. After the discussion of the problem, we pose a lemma in which we show that the field equations of two conformally related Lagrangians are also conformally related if and only if the corresponding Hamiltonian vanishes. Then we prove that to every non-minimally coupled scalar field, we may associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential. The latter result implies that the field equations of a non-minimally coupled scalar field are the same at the conformal level with the field equations of the minimally coupled scalar field. This fact is relevant in order to select physical variables among conformally equivalent systems. Finally, we find that the above propositions can be extended to a general Riemannian space of $n$ n -dimensions.     

Stationary scalar configurations around extremal charged black holes

We consider the minimally coupled Klein-Gordon equation for a charged, massive scalar field in the non-extremal Reissner-Nordström background. Performing a frequency domain analysis, using a continued fraction method, we compute the frequencies $\omega $ for quasi-bound states. We observe that, as the extremal limit for both the background and the field is approached, the real part of the quasi-bound states frequencies $\mathcal{R }(\omega )$ tends to the mass of the field and the imaginary part $\mathcal{I }(\omega )$ tends to zero, for any angular momentum quantum number $\ell $ . The limiting frequencies in this double extremal limit are shown to correspond to a distribution of extremal scalar particles, at stationary positions, in no-force equilibrium configurations with the background. Thus, generically, these stationary scalar configurations are regular at the event horizon. If, on the other hand, the distribution contains scalar particles at the horizon, the configuration becomes irregular therein, in agreement with no hair theorems for the corresponding Einstein-Maxwell-scalar field system.     

The Baker–Akhiezer Function and Factorization of the Chebotarev–Khrapkov Matrix

A new technique is proposed for the solution of the Riemann–Hilbert problem with the Chebotarev–Khrapkov matrix coefficient \({G(t) = \alpha_{1}(t)I + \alpha_{2}(t)Q(t)}\) , \({\alpha_{1}(t), \alpha_{2}(t) \in H(L)}\) , I = diag{1, 1}, Q(t) is a \({2\times2}\) zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann–Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker–Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann–Hilbert problem requires the finding of the \({\rho}\) zeros of the Baker–Akhiezer function ( \({\rho}\) is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree- \({\rho}\) polynomial and solution of a certain linear algebraic system of \({\rho}\) equations.     

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

We study the ground state of the Pauli Hamiltonian with a magnetic field in ${\mathbb{R}^{2d}}$ , d > 1. We consider the case where a scalar potential W is present and the magnetic field B is given by ${B=2i\partial\bar{\partial} W}$ . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.     

On the total mass of closed universes

The total mass, the Witten type gauge conditions and the spectral properties of the Sen–Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression $\mathtt{M}$ M for the total mass density of closed universes is vanishing if and only if the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen–Witten operator; and it is vanishing if and only if Witten’s gauge condition admits a non-trivial solution. Here we generalize slightly the result above on the zero-mass configurations: $\mathtt{M}=0$ M = 0 if and only if the spacetime is holonomically trivial with toroidal spatial topology. Also, we show that the multiplicity of the eigenvalues of the (square of the) Sen–Witten operator is even, and a potentially viable gauge condition is suggested. The monotonicity properties of $\mathtt{M}$ M through the examples of closed Bianchi I and IX cosmological spacetimes are also discussed. A potential spectral characterization of these cosmological spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the Sen–Witten and the 3-surface twistor operators, is also indicated.     

Scalar electron and zino production on and beyond theZ resonance ine + e − annihilation

We study the production of scalar electrons ine ⁺ e ^? collisions on and above theZ resonance. By calculating the cross-section for \(e^ + e^ - \to e^ + e^ - \tilde \gamma \tilde \gamma \) we show that scalar electrons with mass above the beam energies \((\sqrt s /2)\) can be identified. In particular if a zino with mass \(m_{\tilde z}< \sqrt s - m_{\tilde \gamma } \) exists then zino production and decay can give a contribution which dominates the γ exchange contributions. We present final state distributions.     

Nonlinear multidimensional gravity and the Australian dipole

The existing observational data on possible variations of fundamental physical constants (FPC) confirm more or less confidently only a variability of the fine structure constant $\alpha $ in space and time. A model construction method is described, where variations of $\alpha $ and other FPCs (including the gravitational constant $G$ ) follow from the dynamics of extra space-time dimensions in the framework of curvature-nonlinear multidimensional theories of gravity. An advantage of this method is a unified approach to variations of different FPCs. A particular model explaining the observable variations of $\alpha $ in space and time has been constructed. It comprises a FRW cosmology with accelerated expansion, perturbed due to slightly inhomogeneous initial data.     

A Renormalizable 4-Dimensional Tensor Field Theory

We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)⁴ is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the ${\phi^6}$ rather than of the ${\phi^4}$ type, since two different ${\phi^6}$ -type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent ${(\int \phi^2)^2}$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity.     

Gravitational collapse with tachyon field and barotropic fluid

A particular class of space-time, with a tachyon field, $\phi $ , and a barotropic fluid constituting the matter content, is considered herein as a model for gravitational collapse. For simplicity, the tachyon potential is assumed to be of inverse square form i.e., $V(\phi )\sim \phi ^{-2}$ . Our purpose, by making use of the specific kinematical features of the tachyon, which are rather different from a standard scalar field, is to establish the several types of asymptotic behavior that our matter content induces. Employing a dynamical system analysis, complemented by a thorough numerical study, we find classical solutions corresponding to a naked singularity or a black hole formation. In particular, there is a subset where the fluid and tachyon participate in an interesting tracking behaviour, depending sensitively on the initial conditions for the energy densities of the tachyon field and barotropic fluid. Two other classes of solutions are present, corresponding respectively, to either a tachyon or a barotropic fluid regime. Which of these emerges as dominant, will depend on the choice of the barotropic parameter, $\gamma $ . Furthermore, these collapsing scenarios both have as final state the formation of a black hole.