Stability of the massive graviton around a BTZ black hole in three dimensions

We investigate the massive graviton stability of the BTZ black hole obtained from three dimensional massive gravities which are classified into the parity-even and parity-odd gravity theories. In the parity-even gravity theory, we perform the $s$ -mode stability analysis by using the BTZ black string perturbations, which gives two Schrödinger equations with frequency-dependent potentials. The $s$ -mode stability is consistent with the generalized Breitenlohner-Freedman bound for spin-2 field. It seems that for the parity-odd massive gravity theory, the BTZ black hole is stable when the imaginary part of quasinormal frequencies of massive graviton is negative. However, this condition is not consistent with the $s$ -mode stability based on the second-order equation obtained after squaring the first-order equation. Finally, we explore the black hole stability connection between the parity-odd and parity-even massive gravity theories.     

Stability of Black Holes and Black Branes

We establish a new criterion for the dynamical stability of black holes in D ≥ 4 spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, ${\mathcal{E}}$ , on a subspace, ${\mathcal{T}}$ , of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. This is shown by proving that—apart from pure gauge perturbations and perturbations towards other stationary black holes— ${\mathcal{E}}$ is nondegenerate on ${\mathcal{T}}$ and that, for axisymmetric perturbations, ${\mathcal{E}}$ has positive flux properties at both infinity and the horizon. We further show that ${\mathcal{E}}$ is related to the second order variations of mass, angular momentum, and horizon area by ${\mathcal{E} = \delta^2 M -\sum_A \Omega_A \delta^2 J_A - \frac{\kappa}{8\pi}\delta^2 A}$ , thereby establishing a close connection between dynamical stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamical instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with ${\mathcal{E} < 0}$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of ${\mathcal{E}}$ on ${\mathcal{T}}$ is equivalent to the satisfaction of a “ local Penrose inequality,” thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability. Although we restrict our considerations in this paper to vacuum general relativity, most of the results of this paper are derived using general Lagrangian and Hamiltonian methods and therefore can be straightforwardly generalized to allow for the presence of matter fields and/or to the case of an arbitrary diffeomorphism covariant gravitational action.     

F(T) gravity and k-essence

Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible candidate of dark energy. We perform a thorough reconstruction analysis on the so-called $F(T)$ models, where $F(T)$ is some general function of the torsion term. We derive conditions for the equivalence between of $F(T)$ models with purely kinetic k-essence. We present a new class models of $F(T)$ gravity and k-essence.     

Gauge Theories Labelled by Three-Manifolds

We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional ${\mathcal{N} = 2}$ gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that ${S^{3}_{b}}$ partition functions of two mirror 3d ${\mathcal{N} = 2}$ gauge theories are equal. Three-dimensional ${\mathcal{N} = 2}$ field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional ${\mathcal{N} = 2}$ SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.     

P–V criticality of topological black holes in Lovelock–Born–Infeld gravity

To understand the effect of third order Lovelock gravity, $P$ – $V$ criticality of topological AdS black holes in Lovelock–Born–Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and in some more detail than the previous literature. A detailed analysis of the limit case $\beta \rightarrow \infty $ is performed for the seven-dimensional black holes. It is shown that, for the spherical topology, $P$ – $V$ criticality exists for both the uncharged and the charged cases. Our results demonstrate again that the charge is not the indispensable condition of $P$ – $V$ criticality. It may be attributed to the effect of higher derivative terms of the curvature because similar phenomenon was also found for Gauss–Bonnet black holes. For $k=0$ , there would be no $P$ – $V$ criticality. Interesting findings occur in the case $k=-1$ , in which positive solutions of critical points are found for both the uncharged and the charged cases. However, the $P$ – $v$ diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of the entropy. It is shown that, for any nontrivial value of $\alpha $ , the entropy is always positive for any specific volume $v$ . Since no $P$ – $V$ criticality exists for $k=-1$ in Einstein gravity and Gauss–Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which are absent in the Gauss–Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of the entropy. We also check the Gibbs free energy graph and “swallow tail” behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research.     

Renormalization aspects of \mathcal {N}=1 Super Yang–Mills theory in the Wess–Zumino gauge

The generalized $f(R)$ gravity with curvature–matter coupling in five-dimensional (5D) spacetime can be established by assuming a hypersurface-orthogonal space-like Killing vector field of 5D spacetime, and it can be reduced to the 4D formalism of FRW universe. This theory is quite general and can give the corresponding results for Einstein gravity, and $f(R)$ gravity with both no-coupling and non-minimal coupling in 5D spacetime as special cases, that is, we would give some new results besides previous ones given by Huang et al. in Phys Rev D 81:064003, 2010. Furthermore, in order to get some insight into the effects of this theory on the 4D spacetime, by considering a specific type of models with $f_{1}(R)=f_{2}(R)=\alpha R^{m}$ and $B(L_{m})=L_{m}=-\rho $ , we not only discuss the constraints on the model parameters $m,n$ , but also illustrate the evolutionary trajectories of the scale factor $a(t)$ , the deceleration parameter $q(t)$ , and the scalar field $\epsilon (t),\phi (t)$ in the reduced 4D spacetime. The research results show that this type of $f(R)$ gravity models given by us could explain the current accelerated expansion of our universe without introducing dark energy.     

Toward a Mathematical Holographic Principle

In work started in [17] and continued in this paper our objective is to study selectors of multivalued functions which have interesting dynamical properties, such as possessing absolutely continuous invariant measures. We specify the graph of a multivalued function by means of lower and upper boundary maps \(\tau _{1}\) and \(\tau _{2}.\) On these boundary maps we define a position dependent random map \(R_{p}=\{\tau _{1},\tau _{2};p,1-p\},\) which, at each time step, moves the point \(x\) to \(\tau _{1}(x)\) with probability \(p(x)\) and to \(\tau _{2}(x)\) with probability \(1-p(x)\) . Under general conditions, for each choice of \(p\) , \(R_{p}\) possesses an absolutely continuous invariant measure with invariant density \(f_{p}.\) Let \(\varvec{\tau }\) be a selector which has invariant density function \(f.\) One of our objectives is to study conditions under which \(p(x)\) exists such that \(R_{p}\) has \(f\) as its invariant density function. When this is the case, the long term statistical dynamical behavior of a selector can be represented by the long term statistical behavior of a random map on the boundaries of \(G.\) We refer to such a result as a mathematical holographic principle. We present examples and study the relationship between the invariant densities attainable by classes of selectors and the random maps based on the boundaries and show that, under certain conditions, the extreme points of the invariant densities for selectors are achieved by bang-bang random maps, that is, random maps for which \(p(x)\in \{0,1\}.\)      

Semiclassical strings in supergravity PFT

In this paper, we introduce the bulk viscosity in the formalism of modified gravity theory in which the gravitational action contains a general function \(f(R,T)\) , where \(R\) and \(T\) denote the curvature scalar and the trace of the energy–momentum tensor, respectively, within the framework of a flat Friedmann–Robertson–Walker model. As an equation of state for a prefect fluid, we take \(p=(\gamma -1)\rho \) , where \(0 \le \gamma \le 2\) and a viscous term as a bulk viscosity due to the isotropic model, of the form \(\zeta =\zeta _{0}+\zeta _{1}H\) , where \(\zeta _{0}\) and \(\zeta _{1}\) are constants, and \(H\) is the Hubble parameter. The exact non-singular solutions to the corresponding field equations are obtained with non-viscous and viscous fluids, respectively, by assuming a simplest particular model of the form of \(f(R,T) = R+2f(T)\) , where \(f(T)=\alpha T\) ( \(\alpha \) is a constant). A big-rip singularity is also observed for \(\gamma <0\) at a finite value of cosmic time under certain constraints. We study all possible scenarios with the possible positive and negative ranges of \(\alpha \) to analyze the expansion history of the universe. It is observed that the universe accelerates or exhibits a transition from a decelerated phase to an accelerated phase under certain constraints of \(\zeta _0\) and \(\zeta _1\) . We compare the viscous models with the non-viscous one through the graph plotted between the scale factor and cosmic time and find that the bulk viscosity plays a major role in the expansion of the universe. A similar graph is plotted for the deceleration parameter with non-viscous and viscous fluids and we find a transition from decelerated to accelerated phase with some form of bulk viscosity.     

Large N approach to Kaon decays and mixing 28 years later: \Delta I=1/2 rule, \hat{B}_K,and \Delta M_K

We review and update our results for $K\rightarrow \pi \pi $ decays and $K^0$ – $\bar{K}^0$ mixing obtained by us in the 1980s within an analytic approximate approach based on the dual representation of QCD as a theory of weakly interacting mesons for large $N$ , where $N$ is the number of colors. In our analytic approach the Standard Model dynamics behind the enhancement of $\hbox {Re}A_0$ and suppression of $\hbox {Re}A_2$ , the so-called $\Delta I=1/2$ rule for $K\rightarrow \pi \pi $ decays, has a simple structure: the usual octet enhancement through the long but slow quark–gluon renormalization group evolution down to the scales $\mathcal{O}(1\, {\hbox { GeV}})$ is continued as a short but fast meson evolution down to zero momentum scales at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark–gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark–gluon evolution. While this physical picture did not yet emerge from lattice simulations, the recent results on $\hbox {Re}A_2$ and $\hbox {Re}A_0$ from the RBC-UKQCD collaboration give support for its correctness. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. Though the current–current operators dominate the $\Delta I=1/2$ rule, working with matching scales $\mathcal{O}(1 \, {\hbox { GeV}})$ we find that the presence of QCD-penguin operator $Q_6$ is required to obtain satisfactory result for $\hbox {Re}A_0$ . At NLO in $1/N$ we obtain $R=\hbox {Re}A_0/\hbox {Re}A_2= 16.0\pm 1.5$ which amounts to an order of magnitude enhancement over the strict large $N$ limit value $\sqrt{2}$ . We also update our results for the parameter $\hat{B}_K$ , finding $\hat{B}_K=0.73\pm 0.02$ . The smallness of $1/N$ corrections to the large $N$ value $\hat{B}_K=3/4$ results within our approach from an approximate cancelation between pseudoscalar and vector meson one-loop contributions. We also summarize the status of $\Delta M_K$ in this approach.     

Maximal Distance Travelled by N Vicious Walkers Till Their Survival

We consider N Brownian particles moving on a line starting from initial positions \(\mathbf{{u}}\equiv \{u_1,u_2,\ldots u_N\}\) such that \(0 . Their motion gets stopped at time \(t_s\) when either two of them collide or when the particle closest to the origin hits the origin for the first time. For \(N=2\) , we study the probability distribution function \(p_1(m|\mathbf{{u}})\) and \(p_2(m|\mathbf{{u}})\) of the maximal distance travelled by the \(1^{\text {st}}\) and \(2^{\text {nd}}\) walker till \(t_s\) . For general N particles with identical diffusion constants \(D\) , we show that the probability distribution \(p_N(m|\mathbf{u})\) of the global maximum \(m_N\) , has a power law tail \(p_i(m|\mathbf{{u}}) \sim {N^2B_N\mathcal {F}_{N}(\mathbf{u})}/{m^{\nu _N}}\) with exponent \(\nu _N =N^2+1\) . We obtain explicit expressions of the function \(\mathcal {F}_{N}(\mathbf{u})\) and of the N dependent amplitude \(B_N\) which we also analyze for large N using techniques from random matrix theory. We verify our analytical results through direct numerical simulations.     

Reunion Probabilities of N One-Dimensional Random Walkers with Mixed Boundary Conditions

In this work we extend the results of the reunion probability of \(N\) one-dimensional random walkers to include mixed boundary conditions between their trajectories. The level of the mixture is controlled by a parameter \(c\) , which can be varied from \(c=0\) (independent walkers) to \(c\rightarrow \infty \) (vicious walkers). The expressions are derived by using Quantum Mechanics formalism (QMf) which allows us to map this problem into a Lieb-Liniger gas (LLg) of \(N\) one-dimensional particles. We use Bethe ansatz and Gaudin’s conjecture to obtain the normalized wave-functions and use this information to construct the propagator. As it is well-known, depending on the boundary conditions imposed at the endpoints of a line segment, the statistics of the maximum heights of the reunited trajectories have some connections with different ensembles in Random Matrix Theory. Here we seek to extend those results and consider four models: absorbing, periodic, reflecting, and mixed. In all four cases, the probability that the maximum height is less or equal than \(L\) takes the form \(F_N(L)=A_N\sum _{\varvec{k}\in \Omega _{\text {B}}} \mathrm{e}^{-\sum _{j=1}^Nk_j^2}\mathcal {V}_N(\varvec{k})\) , where \(A_N\) is a normalization constant, \(\mathcal {V}_N(\varvec{k})\) contains a deformed and weighted Vandermonde determinant, and \(\Omega _{\text {B}}\) is the solution set of quasi-momenta \(\varvec{k}\) obeying the Bethe equations for that particular boundary condition.     

Towards Asymptotic Completeness of Two-Particle Scattering in Local Relativistic QFT

We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let ${\tilde{p}_1}$ and ${\tilde{p}_2}$ be two energy-momentum vectors of a massive particle and let ${\Delta}$ be a small neighbourhood of ${\tilde{p}_1 + \tilde{p}_2}$ . We construct asymptotic observables (two-particle Araki–Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of ${\tilde{p}_1}$ and ${\tilde{p}_2}$ . We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of ${\Delta}$ . Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag–Ruelle scattering states with total energy-momenta in ${\Delta}$ . The result holds under very general conditions which are satisfied, for example, in ${\lambda{\phi}_{2}^{4}}$ . The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.     

Asymptotic Behavior for a Version of Directed Percolation on the Triangular Lattice

We consider a version of directed bond percolation on the triangular lattice such that vertical edges are directed upward with probability $y$ , diagonal edges are directed from lower-left to upper-right or lower-right to upper-left with probability $d$ , and horizontal edges are directed rightward with probabilities $x$ and one in alternate rows. Let $\tau (M,N)$ be the probability that there is at least one connected-directed path of occupied edges from $(0,0)$ to $(M,N)$ . For each $x \in [0,1]$ , $y \in [0,1)$ , $d \in [0,1)$ but $(1-y)(1-d) \ne 1$ and aspect ratio $\alpha =M/N$ fixed for the triangular lattice with diagonal edges from lower-left to upper-right, we show that there is an $\alpha _c = (d-y-dy)/[2(d+y-dy)] + [1-(1-d)^2(1-y)^2x]/[2(d+y-dy)^2]$ such that as $N \rightarrow \infty $ , $\tau (M,N)$ is $1$ , $0$ and $1/2$ for $\alpha > \alpha _c$ , $\alpha < \alpha _c$ and $\alpha =\alpha _c$ , respectively. A corresponding result is obtained for the triangular lattice with diagonal edges from lower-right to upper-left. We also investigate the rate of convergence of $\tau (M,N)$ and the asymptotic behavior of $\tau (M_N^-,N)$ and $\tau (M_N^+ ,N)$ where $M_N^-/N\uparrow \alpha _c$ and $M_N^+/N\downarrow \alpha _c$ as $N\uparrow \infty $ .     

On the {{\mathcal{N}}=2} Superconformal Index and Eigenfunctions of the Elliptic RS Model

We define an infinite sequence of superconformal indices, ${{\mathcal{I}}_n}$ , generalizing the Schur index for ${{\mathcal{N}}=2}$ theories. For theories of class ${{\mathcal{S}}}$ we then suggest a recursive technique to completely determine ${{\mathcal{I}}_n}$ . The information encoded in the sequence of indices is equivalent to the ${{\mathcal{N}}=2}$ superconformal index depending on a maximal set of fugacities. Mathematically, the procedure suggested in this note provides a perturbative algorithm for computing a set of eigenfunctions of the elliptic Ruijsenaars–Schneider model.     

Lovelock gravity from entropic force

In this paper, we first generalize the formulation of entropic gravity to ( $n+1$ )-dimensional spacetime and derive Newton’s law of gravity and Friedmann equation in arbitrary dimensions. Then, we extend the discussion to higher order gravity theories and propose an entropic origin for Gauss–Bonnet gravity and more general Lovelock gravity in arbitrary dimensions. As a result, we are able to derive Newton’s law of gravitation as well as the corresponding Friedmann equations in these gravity theories. This procedure naturally leads to a derivation of the higher dimensional gravitational coupling constant of Friedmann/Einstein equation which is in complete agreement with the results obtained by comparing the weak field limit of Einstein equation with Poisson equation in higher dimensions. Our strategy is to start from first principles and assuming the entropy associated with the apparent horizon given by the expression previously known via black hole thermodynamics, but replacing the horizon radius $r_+$ with the apparent horizon radius $R$ . Our study shows that the approach presented here is powerful enough to derive the gravitational field equations in any gravity theory and further supports the viability of Verlinde’s proposal.     

Study of Alpha Decay Chains of Superheavy Nuclei and Magic Number Beyond Z = 82 and N = 126

We study various $\alpha $ -decay chains on the basis of the preformed cluster decay model. Our work targets the superheavy elements, which are expected to show extra stability at shell closure. Our computations identify the following combinations of proton and neutron numbers as the most stable nuclei: $Z=112$ , $N=161, 163$ ; $Z=114$ , $N=171, 178, 179$ ; and $Z=124$ , $N=194$ . We also investigate the alternative of heavy cluster emissions in the decay chain of ³⁰¹120, instead of $\alpha $ decay. Our study of cluster radioactivity shows that the half-life for ¹⁰Be decay in ²⁸⁹114 is larger, indicating enhanced stability at $Z=114$ , $N=175$ . Similar calculations concerning the emission of $\ ^{14}{\rm C}$ and $\ ^{34}{\rm Si}$ from ³⁰¹120 find the more stable combinations $Z=114$ , $N=173$ , and $Z=106$ , $N=161$ , respectively. From the same parent, ³⁰¹120, the emission of a $\ ^{49-51}{\rm Ca}$ cluster yielding a $Z=100$ , $N=152$ daughter is the most probable.     

Gauge Theories on ALE Space and Super Liouville Correlation Functions

We present a relation between ${\mathcal{N}=2}$ quiver gauge theories on the ALE space ${\mathcal{O}_{\mathbb{P}^1}(-2)}$ and correlators of ${\mathcal{N}=1}$ super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the ${\mathcal{N}=2^*}$ instanton partition function is given by the product of the character of ${\widehat{SU}(2)_2}$ times the super Virasoro conformal block on the torus with one puncture. Moreover, we match the perturbative gauge theory contribution with super Liouville three-point functions.     

Charmonium rescattering effects in \psi(2S) \rightarrow \gamma \eta_{c}^{} decay

Charmonium rescattering effects in the M1 transition of $ \psi$ (2S) $ \rightarrow$ $ \gamma$ $ \eta_{c}^{}$ are investigated by modeling a $ \chi_{{cJ}}^{}$ or J/ $ \psi$ rescattering into a $ \eta_{c}^{}$ final state. The absorptive and dispersive part of the transition amplitudes for the rescattering loops of $ \eta$ $ \psi$ ( $ \gamma^{{\ast}}_{}$ ) and $ \gamma$ $ \chi$ ( $ \psi$ ) are separately evaluated. The numerical results show that the contribution from the $ \gamma$ $ \chi$ ( $ \psi$ ) rescattering process is negligible. Compared with the virtual D $ \bar{{D}}$ (D ^*) rescattering processes, the $ \eta$ $ \psi$ ( $ \gamma^{{\ast}}_{}$ ) process may be regarded as the next-leading order of the hadronic loop mechanism, which only offers the partial decay width of ～ 0.045 keV to the $ \psi$ (2S) $ \rightarrow$ $ \gamma$ $ \eta_{c}^{}$ .     

Quantum billiards in multidimensional models with branes

A gravitational $D$ -dimensional model with $l$ scalar fields and several forms is considered. When a cosmological-type diagonal metric is chosen, an electromagnetic composite brane ansatz is adopted and certain restrictions on the branes are imposed; the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions asymptotic solutions to WDW equation are found in the limit of the formation of the billiard walls which reduce the problem to the so-called quantum billiard on the $(D+ l -2)$ -dimensional Lobachevsky space. Two examples of quantum billiards are considered. The first one deals with $9$ -dimensional quantum billiard for $D = 11$ model with $330$ four-forms which mimic space-like $M2$ - and $M5$ -branes of $D=11$ supergravity. The second one deals with the $9$ -dimensional quantum billiard for $D =10$ gravitational model with one scalar field, $210$ four-forms and $120$ three-forms which mimic space-like $D2$ -, $D4$ -, $FS1$ - and $NS5$ -branes in $D = 10$ $II A$ supergravity. It is shown that in both examples wave functions vanish in the limit of the formation of the billiard walls (i.e. we get a quantum resolution of the singularity for $11D$ model) but magnetic branes could not be neglected in calculations of quantum asymptotic solutions while they are irrelevant for classical oscillating behavior when all $120$ electric branes are present.     

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.