Logarithmic Terms Research Articles

Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

Threshold resummation of new partonic channels at next-to-leading power

Collider observables involving heavy particles are subject to large logarithmic terms near threshold, which must be summed to all orders in perturbation theory to obtain sensible results. Relatively recently, this resummation has been extended to next-to-leading power in the threshold variable, using a variety of approaches. In this paper, we consider partonic channels that turn on only at next-to-leading power, and show that it is possible to resum leading logarithms using well-established diagrammatic techniques in Quantum Chromodynamics. We first consider deep inelastic scattering, where we reproduce the results of a recent study using an effective theory approach. Next, we consider the quark-gluon channel in both Drell-Yan and Higgs boson production, showing that an explicit all-order form for the leading logarithmic partonic cross section can be obtained. Our results agree with previous conjectures based on fixed-order results.

Accurate prediction of the Drell-Yan distribution in wide dilepton mass and rapidity ranges in pp collisions through NNLO+N3LL

This paper presents high-accuracy predictions for the differential cross ss as a function of the key observable of the neutral-current Drell-Yan (DY) dilepton production in proton-proton (pp) collisions. The differential distributions for the are presented by using the state-of-the-art predictions from the combined calculations of fixed-order perturbative quantum chromodynamics (QCD) corrections at next-to-next-to-leading order (NNLO) accuracy and resummation of large logarithmic terms at next-to-next-to-leading logarithmic (NNLL) and next-to-NNLL (N3LL) accuracies, i.e., NNLO+NNLL and NNLO+N3LL, respectively. The predicted distributions are reported for a thorough set of the DY dilepton invariant mass m ll ranges, spanning a wide kinematic region of 50 < m ll < 1000 GeV both near and away from the Z-boson mass peak, and rapidity y ll ranges in the central detector acceptance region of ∣y ll ∣ < 2.4. The differential distributions in the wide m ll and y ll ranges offer stringent tests to assess the reliability of the predictions, where the m ll and y ll are closely correlated with the parton distribution functions (PDFs) of the incoming partons. The matched predictions through NNLO+N3LL are observed to provide good description of the 13 TeV pp collision data for the (including the dilepton transverse momentum as well) distributions in almost the entire m ll and y ll ranges, apart from the intermediate- to high- region in the lowest mass range 50–76 GeV which is assessed to constitute a challenge for the presented predictions. The predictions at NNLO+N3LL are also reported at 14 TeV for the upcoming high-luminosity running era of the Large Hadron Collider (LHC), in which increasing amount of data is expected to require more accurate and precise theoretical description. The most recent PDF models MSHT20 and CT18, in addition to the NNPDF3.1, are tested for the first time for the matched predictions of the distribution. The differential distributions by the combined predictions through NNLO QCD+NLO EW are finally provided to enable assessment of the impact of the EW corrections for the .

Thermodynamic effects of Bardeen black hole surrounded by perfect fluid dark matter under general uncertainty principle

The thermodynamics of Bardeen black hole surrounded by perfect fluid dark matter is investigated. We calculate the analytical expresses of corresponding thermodynamic variables, e.g., the Hawking temperature, entropy of the black hole. In addition, we derive the heat capacity to analyze the thermal stability of the black hole. We also compute the rate of emission in terms of photons through tunneling. By numerical method, an obvious phase transition behavior is found. Furthermore, according to the general uncertainty principle, we study the quantum corrections to these thermodynamic quantities and obtain the quantum-corrected entropy containing the logarithmic term. Lastly, we investigate the effects of the magnetic charge g, the dark matter parameter k and the generalized uncertainty principle parameter α on the thermodynamics of Bardeen black hole surrounded by perfect fluid dark matter under general uncertainty principle.

Thermodynamic corrections of Gauss–Bonnet black hole under general uncertainty principle

In this paper, the thermodynamics of Gauss–Bonnet black hole is investigated. We calculate the analytical solutions of corresponding thermodynamic variables, e.g. the Hawking temperature, entropy of the black hole at its event horizon. In addition, we derive the heat capacity to analyze the thermal stability of the black hole. We also compute the rate of emission in terms of photons through tunneling. By adopting a numerical method to analyze their properties, an obvious phase transition behavior is found. Furthermore, according to the general uncertainty principle, we study the quantum corrections to these thermodynamic quantities and obtain the quantum-corrected entropy containing the logarithmic term. At last, we investigate the effects of the Gauss–Bonnet coupling parameter [Formula: see text] and generalized uncertainty principle parameter [Formula: see text] on the thermodynamics of Gauss–Bonnet black hole.

Multistability and period-adding in a logarithmic Lorenz system

In this paper, we design and explore a novel 3D autonomous nonlinear system with logarithmic nonlinearity. It is framed from the Lorenz model by replacing the variable [Formula: see text] in the second equation by the nonlinear logarithmic term [Formula: see text]. The bi-parametric dynamics of the modeled system is presented. Studies reveal the emergence of shrimp-shaped periodic structures in chaotic regime with period-adding phenomenon, and coexisting periodic-chaotic attractors, which do not occur in the Lorenz model. The stability, bifurcation patterns, and chaotic behaviors of the system are explored with the aid of bifurcation diagrams, phase portraits and Lyapunov exponents. The complexity of the basin sets for the coexisting attractors is also studied.

Energy decay for a wave equation of variable coefficients with logarithmic nonlinearity source term

We investigate the variable-coefficient wave equation with nonlinear acoustic boundary conditions and logarithmic nonlinearity source term. In addition, the damping is nonlinear, and plays a role only in part of the boundary. Using the semigroup theory, we state the existence and uniqueness of the local solutions. Accordingly, we prove that under some assumptions on the initial data, the local solution can be extended to be global. Particularly, it has been shown that the decay rates of the system are given implicitly as solutions to a first-order ODE by applying the Riemannian geometry method.

A Study on the logarithm correction of black hole entropy

The logarithm correction of black hole entropy is important in understanding the essence of black hole entropy, providing a more accurate entropy calculation. We reviewed the mainstream method of logarithm correction of black hole entropy, including quantum loop gravity correction, conformal field theory correction, and classical thermal correction. Specifically, the correction of quantum loop gravity presents a stable general expression of logarithm correction, which only depends on the surface area of the black hole and solves the problem of meaningless entropy solution under a large length scale. Besides, the correction of the Cardy formula of conformal field theory is limited for the third term in depends on the mass of the black hole, which will finally lead to the unstable coefficient before the correction term. Finally, the correction deduced by the classical thermal method also gives a general expression of black hole entropy. In contrast, the entropy of BTZ black hole has a different coefficient before the logarithm term comparing to other kinds of the black hole. These results shed light for the research in general logarithm correction of black hole entropy, which is suitable for all kinds of black holes.

Entanglement measures in a nonequilibrium steady state: Exact results in one dimension

Entanglement plays a prominent role in the study of condensed matter many-body systems: Entanglement measures not only quantify the possible use of these systems in quantum information protocols, but also shed light on their physics. However, exact analytical results remain scarce, especially for systems out of equilibrium. In this work we examine a paradigmatic one-dimensional fermionic system that consists of a uniform tight-binding chain with an arbitrary scattering region near its center, which is subject to a DC bias voltage at zero temperature. The system is thus held in a current-carrying nonequilibrium steady state, which can nevertheless be described by a pure quantum state. Using a generalization of the Fisher-Hartwig conjecture, we present an exact calculation of the bipartite entanglement entropy of a subsystem with its complement, and show that the scaling of entanglement with the length of the subsystem is highly unusual, containing both a volume-law linear term and a logarithmic term. The linear term is related to imperfect transmission due to scattering, and provides a generalization of the Levitov-Lesovik full counting statistics formula. The logarithmic term arises from the Fermi discontinuities in the distribution function. Our analysis also produces an exact expression for the particle-number-resolved entanglement. We find that although to leading order entanglement equipartition applies, the first term breaking it grows with the size of the subsystem, a novel behavior not observed in previously studied systems. We apply our general results to a concrete model of a tight-binding chain with a single impurity site, and show that the analytical expressions are in good agreement with numerical calculations. The analytical results are further generalized to accommodate the case of multiple scattering regions.

Critical exponents of block-block mutual information in one-dimensional infinite lattice systems.

We study the mutual information between two lattice blocks in terms of von Neumann entropies for one-dimensional infinite lattice systems. Quantum q-state Potts model and transverse-field spin-1/2XY model are considered numerically by using the infinite matrix product state approach. As a system parameter varies, block-block mutual information exhibit singular behaviors that enable us to identify the critical points for the quantum phase transitions. As happens with von Neumann entanglement entropy of single block, at critical points, block-block mutual information for two adjacent blocks show a logarithmic leading behavior with increasing the size of the blocks, which yields the central charge c of the underlying conformal field theory, as it should be. It seems that two disjoint blocks show a similar logarithmic growth of the mutual information as a characteristic property of critical systems but the proportional coefficients of the logarithmic term are very different from the central charges. As the separation between the two lattice blocks increases, the mutual information reveals a consistent power-law decaying behavior for various truncation dimensions and lattice-block sizes. The critical exponent of block-block mutual information in the thermodynamic limit is estimated by extrapolating the exponents of power-law decaying regions for finite truncation dimensions. For a given lattice-block size ℓ, the critical exponents for the same universality classes seem to have very close values each other. Whereas the critical exponents have different values to a degree of distinction for the different universality classes. Asthelattice-block size becomes bigger, the critical exponent becomes smaller. We discuss a relation between the exponents of block-block mutual information and correlation with the Shatten one-norm of block-block correlation.

Volume complexity for the nonsupersymmetric Janus AdS5 geometry

We compute holographic complexity for the non-supersymmetric Janus deformation of AdS$_5$ according to the volume conjecture. The result is characterized by a power-law ultraviolet divergence. When a ball-shaped region located around the interface is considered, a sub-leading logarithmic divergent term and a finite part appear in the corresponding subregion volume complexity. Using two different prescriptions to regularize the divergences, we find that the coefficient of the logarithmic term is universal.

Gravitational multipole renormalization

We compute the effect of scattering gravitational radiation off the static background curvature, up to second order in Newton constant, known in literature as tail and tail-of-tail processes, for generic electric and magnetic multipoles. Starting from the multipole expansion of composite compact objects, and as expected due to the known electric quadrupole case, both long- and short-distance (UV) divergences are encountered. The former disappears from properly defined observables, the latter are renormalized and their associated logarithms give rise to a classical renormalization group flow. UV divergences alert for incompleteness of the multipolar description of the composite source, and are expected not to be present in a UV-complete theory, as explicitly derived in literature for the case of conservative dynamics. Logarithmic terms from tail-of-tail processes associated to generic magnetic multipoles are computed in this work for the first time.

Massless positivity in graviton exchange

We formulate Positivity Bounds for scattering amplitudes including exchange of massless particles. We generalize the standard construction through dispersion relations to include the presence of a branch cut along the real axis in the complex plane for the Maldestam variable $s$. In general, validity of these bounds require the cancellation of divergences in the forward limit of the amplitude, proportional to $t^{-1}$ and $\log(t)$. We show that this is possible in the case of gravitons if one assumes a Regge behavior of the amplitude at high energies below the Planck scale, as previously suggested in the literature, and that the concrete UV behaviour of the amplitude is uniquely determined by the structure of IR divergences. We thus extend previous results by including a sub-leading logarithmic term, which we show to be universal. The bounds that we present here have the potential of constraining very general models of modified gravity and EFTs of matter coupled to gravitation.

On discretizing integral norms of exponential sums

In this paper we study Lp Marcinkiewicz-Zygmund type inequalitiesc1∑1≤j≤Nwj|g(xj)|p≤‖g‖Lp(K)p≤c2∑1≤j≤Nwj|g(xj)|p for general exponential sums of the form g(x)=∑1≤j≤naje〈λj,x〉,x,λj∈Rd,aj∈R. One of the main results of the paper asserts that when 1≤p<∞,K=[a,b] and the exponents satisfy relations λj+1−λj≥ϵ,1≤j≤n−1,max1≤j≤n⁡|λj|≤Λ then Marcinkiewicz-Zygmund type inequalities hold for certain point sets of cardinalityN≤cnln1p+1⁡Λϵ. Since the dependence of cardinality on the parameters ϵ,Λ appears only in the logarithmic term, this bound is “almost” degree and separation independent. Moreover, the discrete meshes xj and weights wj=xj+1−xj are given explicitly and they are universal in the sense that they work for any exponential sum as above. This result will rely on some new degree independent Bernstein-Markov type inequality for exponential sums. Moreover we will extend our considerations to multivariate exponential sums. In addition, it will be shown that much stronger results hold for exponential sums with nonnegative coefficients.

Numerical study of time delay singularly perturbed parabolic differential equations involving both small positive and negative space shifts

Abstract A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

On Asymptotic Behavior of Galactic Rotation Curves in Superfluid Vacuum Theory

The logarithmic superfluid theory of physical vacuum predicts that gravity is an induced phenomenon, which has a multiple-scale structure. At astronomical scales, as the distance from a gravitating center increases, gravitational potential and corresponding spacetime metric are dominated by a Newtonian (Schwarzschild) term, followed by a logarithmic term, finally by linear and quadratic (de Sitter) terms. Correspondingly, rotation curves are predicted to be Keplerian in the inner regions of galaxies, mostly flat in the outer regions, and non-flat in the utmost outer regions. We compare theory's predictions with the furthest rotation curves data points available for a number of galaxies: using a two-parameter fit, we perform a preliminary estimate which disregards the combined effect of gas and stellar disc, but is relatively simple and uses minimal assumptions for galactic luminous matter. The data strongly points out at the existence of a crossover transition from flat to non-flat regimes at galactic outskirts.

Asymptotic behavior for nonlinear Schrödinger equations with critical time-decaying harmonic potential

Time-decaying harmonic oscillators yield dispersive estimates with weak decay, and change the threshold power of the nonlinearity between the short and the long range. In the non-critical case for the time-decaying harmonic oscillator, this threshold can be characterized by polynomial nonlinearities. However, in the critical case, it is difficult to characterize the threshold using only polynomial terms, and thus we use logarithmic nonlinear terms.

Investigating strong gravitational lensing with black hole metrics modified with an additional term

Gravitational lensing is one of the most impressive celestial phenomena, which has interesting behaviors in its strong field limit. Near such limit, Bozza finds that the deflection angle of light is well-approximated by a logarithmic term and a constant term. In this way he explicitly derived the analytic expressions of deflection angles for a few types of black holes. In this paper, we study the explicit calculation to two new types of metrics in the strong field limit: (i) the Schwarzschild metric extended with an additional r−n(n≥3) term in the metric function; (ii) the Reissner-Nordstrom metric extended with an additional r−6 term in the metric function. With such types of metrics, Bozza's original way of choosing integration variables may lead to technical difficulties in explicitly expressing the deflection angles, and we use a slightly modified version of Bozza's method to circumvent the problem.

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi3 QFT in 6 Dimensions

We analyze the asymptotically free massless scalar $\phi^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each repeated in an infinite series. These expansions are in terms of the renormalized coupling. The resonant structure leads to powers of logarithmic terms at higher levels of the trans-series, analogous to logarithmic terms arising from interactions between instantons and anti-instantons, but arising from a purely perturbative formalism rather than from a semi-classical analysis.

Droplet condensation in the lattice gas with density functional theory.

A density functional for the lattice gas with next-neighbor attractions (Ising model) from fundamental measure theory is applied to the problem of droplet states in three-dimensional, finite systems. The density functional is constructed via an auxiliary model with hard lattice gas particles and lattice polymers to incorporate the attractions. Similar to previous simulation studies, the sequence of droplets changing to cylinders and to planar slabs is found upon increasing the average density ρ[over ¯] in the system. Owing to the discreteness of the lattice, additional effects in the state curve for the chemical potential μ(ρ[over ¯]) are seen upon lowering the temperature away from the critical temperature [oscillations in μ(ρ[over ¯]) in the slab portion and spiky undulations in μ(ρ[over ¯]) in the cylinder portion as well as an undulatory behavior of the radius of the surface of tension R_{s} in the droplet region]. This behavior in the cylinder and droplet region is related to washed-out layering transitions at the surface of liquid cylinders and droplets. The analysis of the large-radius behavior of the surface tension γ(R_{s}) gave a dominant contribution ∝1/R_{s}^{2}, although the consistency of γ(R_{s}) with the asymptotic behavior of the radius-dependent Tolman length seems to suggest a weak logarithmic contribution ∝lnR_{s}/R_{s}^{2} in γ(R_{s}). The coefficient of this logarithmic term is smaller than a universal value derived with field-theoretic methods.