High-order Perturbation Research Articles

Observational Detection of Higher-Order Secular Perturbations in Tight Hierarchical Triple Stars

In this work, we search for observational evidence of higher-order secular perturbations in three eclipsing binaries. These are slightly eccentric binaries, and they form the inner pairs of tight, compact, hierarchical triple star systems. Simultaneously, we analyze the high-precision satellite (Kepler and TESS) light curves; eclipse timing variations; combined spectral energy distributions (through catalog passband magnitudes); and, where available, radial velocities of KICs 9714358, 5771589, and TIC 219885468. Besides the determination of the robust astrophysical and dynamical properties of the three systems, we find evidence that the observed unusual eclipse timing variations of KIC 9714358 are a direct consequence of the octupole-order secular eccentricity perturbations forced by unusual, resonant behavior between the lines of the apsides of the inner and outer orbital ellipses. We also show that, despite its evident cyclic eclipse depth variations, KIC 5771589 is an almost perfectly coplanar system (to within 0.3∘), and we explain the rapid eclipse depth variations with the grazing nature of the eclipses. Finally, we find that the inner pair of TIC 219885468 consists of two twin stars; hence, in this triple there are no octupole-order three-body perturbations. Moreover, we show that this triple is also coplanar on the same level as the former one, but due to its deep eclipses, it does not exhibit eclipse depth variations. We intend to follow this work up with further analyses and a quantitative comparison of the theoretical and the observed perturbations.

Symmetries of the squeeze-driven Kerr oscillator

We study the symmetries of the static effective Hamiltonian of a driven superconducting nonlinear oscillator, the so-called squeeze-driven Kerr Hamiltonian, and discover a remarkable quasi-spin symmetry su(2) at integer values of the ratio of the detuning parameter Δ to the Kerr coefficient K. We investigate the stability of this newly discovered symmetry to high-order perturbations arising from the static effective expansion of the driven Hamiltonian. Our finding may find applications in the generation and stabilization of states useful for quantum computing. Finally, we discuss other Hamiltonians with similar properties and within reach of current technologies.

The gravitational force field of proto-pancakes

It is well known that the first structures that form from small fluctuations in a self-gravitating, collisionless, and initially smooth cold dark matter (CDM) fluid are pancakes. We studied the gravitational force generated by such pancakes just after shell crossing and have found a simple analytical formula for the force along the collapse direction, which can be applied to both the single- and multi-stream regimes. We tested the formula on the early growth of CDM proto-haloes seeded by two or three crossed sine waves. Adopting the high-order Lagrangian perturbation theory (LPT) solution as a proxy for the dynamics, we confirm that our analytical prediction agrees well with the exact solution computed via a direct resolution of the Poisson equation, as long as the local caustic structure remains sufficiently one-dimensional. These results are further confirmed by comparisons of the LPT predictions performed this way to measurements in Vlasov simulations performed with the public code ColDICE. We also show that the component of the force orthogonal to the collapse direction preserves its single-stream nature – it does not change qualitatively before or after the collapse – allowing sufficiently high-order LPT acceleration to be used to approximate it accurately as long as the LPT series converges. As expected, solving the Poisson equation on the density field generated with LPT displacement provides a more accurate force than the LPT acceleration itself, as a direct consequence of the faster convergence of the LPT series for the positions than for the accelerations. This may provide a clue as to how we can improve standard LPT predictions. Our investigations represent a very needed first step in the study of gravitational dynamics in the multi-stream regime analytically: we estimate, at the leading order in time and space, the proper backreaction on the gravitational field inside the pancakes.

Efficient algorithm to calculate the optical properties of breast tumors by high-order perturbation theory.

An efficient algorithm to obtain the solutions for n-th order terms of perturbation expansions in absorption, scattering, and cross-coupling for light propagating in human tissue is presented. The proposed solution is free of any approximations and makes possible fast and efficient estimates of mammographic, optical tomographic, and fluorescent images, applying a perturbation order of 30 and more. The presented analysis sets the general limits for the applicability of the perturbation approach as a function of tumor size and optical properties of the human tissue. The convergence tests of the efficient calculations for large absorbing objects show excellent agreement with the reference data from finite element method calculations. The applicability of the theory is demonstrated in experiments on breast-like phantoms with high absorbing and low-scattering lesions.

Unveiling the force of weak effects in molecular absorption

This work aims at theoretical consideration of weak effects, some of which are well-known in molecular spectroscopy, and some are not yet fully elaborated. A cohesive element of our consideration consists in the use of the original perturbative approach, which significantly facilitates the derivation of the sought expressions and their employment for applications. We consider several examples of perturbative problems concerning permitted or forbidden electric dipole and permitted magnetic-dipole transitions. The suggested approach can be generalized to other effects involving electronic, spin, and higher-order perturbations in molecular spectra.

Scrutinizing the light scalar quarkonia from LSR at higher orders

We scrutinize, improve some determinations of the masses and couplings of light scalar quarkonia (q¯q and four-quark states) and present new results for the π+π−,K+K−,… molecules using QCD Laplace Sum Rule (LSR) truncated at the D=6 dimension vacuum condensates. We pay a special attention on the higher order perturbative (PT) corrections up to the (estimated) O(αs5) which improve the quality of the analysis. We request that the optimal results obey the rigorous constraint: Resonance ≥ QCD continuum contributions (RP/C≥1) in the LSR which excludes a Breit-Wigner/on-shell (not to be confused with a complex pole) scalar meson mass around (500-600) MeV obtained for values [tc≤(1∼1.5) GeV2] of the QCD continuum threshold. Mass-splittings due to SU(3) breakings are small. We discuss the different assignments of the observed scalar mesons below 2 GeV in the Conclusions where the I=0 states are compared with the scalar gluonia. The results are compiled in Tables 3 to 6. We hope that the systematic analysis done in this paper helps to clarify the complex spectra of the light scalar mesons.

Modelling the next-to-leading order matter three-point correlation function using FFTLog

The study of higher-order statistics, particularly 3-point statistics, of the Large Scale Structure (LSS) of the Universe provides us with unique information on the biasing relation between luminous and dark matter and on deviations from primordial Gaussianity. As a result, much effort has been put into improving measurement techniques as well as theoretical modelling, especially in Fourier space. Comparatively, little progress has been made, instead, in configuration space analyses. This work represents a first step towards filling this gap by proposing a new strategy for modelling 3-point statistics at higher perturbative orders in configuration space. Starting from the next-to-leading order model for the matter bispectrum, we use 2D-FFTLog to generate its counterpart in configuration space. We calibrate the procedure using the leading order predictions for which an analytic model for the 3-point correlation function (3PCF) already exists. Then we assess the goodness of the 3PCF model by comparing its predictions with measurements performed on the matter distribution in collisionless cosmological N-body simulation (DEMNUni). We focus on two redshifts (z = 0.49 and z = 1.05) in the range spanned by current and future galaxy redshift surveys. The χ 2 analysis reveals that the next-to-leading order 3PCF models significantly improve over the leading order one for all triangle configurations in both redshifts, increasing the number of matched configurations at redshift z = 1.05 and z = 0.49, respectively. In particular, a significant improvement is also seen on the Baryonic Acoustic Oscillations (BAO) scale for triangle configurations whose smallest side length is well into the nonlinear regime. The computational cost of the model proposed here is high but not prohibitively large — order of five hours in a 48-core computation — and represents the first step towards a complete 3PC model for the galaxies.

Diagrammatic Monte Carlo for electronic correlation in molecules: High-order many-body perturbation theory with low scaling

Diagrammatic Monte Carlo for electronic correlation in molecules: High-order many-body perturbation theory with low scaling

On the use of an high order perturbation method for numerical time integration in structural dynamics.

On the use of an high order perturbation method for numerical time integration in structural dynamics.

High-Order Perturbation Method for Updating Response Functions for Time-Dependent Transport Calculations

The hybrid stochastic deterministic continuous-energy coarse mesh transport method (COMET) has been recently extended for high-fidelity efficient kinetics calculations in highly heterogeneous reactor cores. The method discretizes the time variable as a series of time grids and solves the resulting set of steady-state neutron transport equations. In this work, a high-order perturbation method is developed to update the COMET unperturbed response function library on the fly for changes in the discretized time step size Δ t h . The unperturbed response functions are precomputed with 1 / v Δ t h = 0 . The perturbation expansion coefficients are also generated during the unperturbed response function library precomputation. The adjoint solution needed by the perturbation method is calculated using the reciprocity relation without solving the corresponding adjoint problems. As a result, the method can be easily implemented into any stochastic code to generate the perturbation expansion coefficients together with the unperturbed response functions. The high-order perturbation method is benchmarked by comparing both the response functions and the time-dependent COMET core solution (fission density) with the corresponding reference solutions. It is found that the response functions generated by the perturbation method at second order are in excellent agreement with those directly computed by the Monte Carlo method. When Δ t h changes from 1.0E-4 s to infinity, the average and maximum relative differences in the various response functions were found to be in the range of 0.0000106% to 0.00116% and 0.000011% to 0.00121%, respectively. The fission density as a function of time calculated by COMET using the perturbation method is in excellent agreement with the reference solutions, with an average relative difference of 0.0065% to 0.075%. These comparisons indicate that the perturbation method at second order is highly accurate.

Joint Geometry/Frequency Analyticity of Fields Scattered by Periodic Layered Media

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high-order perturbation of surfaces/asymptotic waveform evaluation (HOPS/AWE) methods for numerically simulating scattering returns from periodic diffraction gratings. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a periodic two-layer structure. Furthermore, we establish joint analyticity of these solutions with respect to both geometry and frequency perturbations. This result provides hypotheses under which a rigorous numerical analysis could be conducted on our recently developed HOPS/AWE algorithm.

Anomalous dimensions from soft Regge constants

Using an effective field theory (EFT) formalism for forward scattering, we reconsider the factorization of 2 → 2 scattering amplitudes in the Regge limit. Expanding the amplitude in gauge invariant operators labelled by the number of Glauber exchanges, allows us to further factorize the standard impact factors into separate collinear and soft functions. The soft functions are universal, and describe radiative corrections to the Reggeized gluon states exchanged by the collinear projectiles. Remarkably, we find that the one-loop soft function for the single Reggeized gluon state is given to mathcal{O} (ϵ) in terms of the two-loop cusp and two-loop rapidity anomalous dimensions. We argue that this iterative structure follows from the simple action of crossing symmetry in the forward scattering limit, which in the EFT allows us to replace the divergent part of a soft loop by a much simpler Glauber loop. We use this correspondence to provide a simple calculation of the two-loop Regge trajectory using the EFT. We then explore its implications at higher perturbative orders, and derive the maximally matter dependent contributions to the Regge trajectory to all loop orders, i.e. the terms sim {alpha}_s^{k+1}{n}_f^k for any k, where nf is the number of massless flavors. These simplifications suggests that the EFT approach to the Regge limit will be helpful to explore and further understand the structure of the Regge limit.

Precision phenomenology with fiducial cross sections in the triple-differential Drell-Yan process

The production of lepton pairs (Drell-Yan process) at the LHC is being measured to high precision, enabling the extraction of distributions that are triply differential in the di-lepton mass and rapidity as well as in the scattering angle described by the leptons. The measurements are performed for a fiducial phase space, defined by cuts on the individual lepton momenta and rapidities. Based on the ATLAS triple-differential Drell-Yan measurement at 8 TeV, we perform a detailed investigation of the phenomenology of this process based on state-of-the-art perturbative predictions in QCD and the electroweak theory. Our results demonstrate the highly non-trivial interplay between measurement variables and fiducial cuts, which leads to forbidden regions at Born level, and induces sensitivity on extra particle emissions from higher perturbative orders. We also investigate the sensitivity of the measurement on parton distributions and electroweak parameters. We derive Standard-Model theory predictions which combine NNLO QCD and NLO EW corrections and include partial N3LO QCD as well as higher-order EW corrections where appropriate. Our results will enable the use of the triple-differential Drell-Yan data in a precise experimental determination of the weak mixing angle.

High-Order Quantum-Mechanical Analysis of Hydrogen Bonding in Hachimoji and Natural DNA Base Pairs.

High-order quantum chemistry is applied to hydrogen-bonded natural DNA nucleobase pairs [adenine:thymine (A:T) and guanine:cytosine (G:C)] and non-natural Hachimoji nucleobase pairs [isoguanine:1-methylcytosine (B:S) and 2-aminoimidazo[1,2a][1,3,5]triazin-4(1H)-one:6-amino-5-nitropyridin-2-one (P:Z)] to see how the intermolecular interaction energies and their energetic components (electrostatics, exchange-repulsion, induction/polarization, and London dispersion interactions) vary among the base pairs. We examined the Hoogsteen (HG) geometries in addition to the traditional Watson-Crick (WC) geometries. Coupled-cluster theory through perturbative triples [CCSD(T)] extrapolated to the complete basis set (CBS) limit and high-order symmetry-adapted perturbation theory (SAPT) at the SAPT2+(3)(CCD)δMP2/aug-cc-pVTZ level are used to estimate highly accurate noncovalent interaction energies. Electrostatic interactions are the most attractive component of the interaction energies, but the sum of induction/polarization and London dispersion is nearly as large, for all base pairs and geometries considered. Interestingly, the non-natural Hachimoji base pairs interact more strongly than the corresponding natural base pairs, by -21.8 (B:S) and -0.3 (P:Z) kcal mol-1 in the WC geometries, according to CCSD(T)/CBS. This is consistent with the H-bond distances being generally shorter in the non-natural base pairs. The natural base pairs are energetically more stabilized in their Hoogsteen geometries than in their WC geometries. The Hoogsteen geometry makes the A:T base pair slightly more stable, by -0.8 kcal mol-1, and it greatly stabilizes the G:C+ base pair, by -15.3 kcal mol-1. The G:C+ stabilization is mainly due to the fact that C has typically added a proton when found in Hoogsteen geometries. By contrast, Hoogsteen geometries are substantially less favorable than WC geometries for non-natural Hachimoji base pairs, by 17.3 (B:S) and 13.8 (P:Z) kcal mol-1.

An adaptive divided-difference perturbation method for solving stochastic problems

Most stochastic problems are based on the independent and symmetrically distributed random variables. Using this principle, in this study, we propose an efficient calculation scheme to obtain the statistical moments based on high-order stochastic perturbation (SP) expansion. This solves the long-existing problem of the SP method, namely, the low calculation efficiency of statistical moments using high-order perturbation expansion, and the complexity of the calculation format. The scheme involves adopting the divided-difference method to approximate the partial derivative items of the SP method. The proposed method, which is similar to the collocation point method, only requires the deterministic calculation result for each node in the divided-difference method. In addition, as the adoption of high-order perturbation expansion is limited in multidimensional random problems due to the curse of dimensionality, an adaptive partial derivative items selection method is proposed in this paper to efficiently and reasonably ignore the expansion items that have an insignificant influence on statistical moments. This considerably improves the calculation efficiency of the statistical moments. Finally, the adaptive divided-difference perturbation (ADDP) method proposed in this paper is compared with the quasi-Monte Carlo method and the adaptive sparse grid methods using three numerical examples. The results prove that the ADDP method not only has high efficiency but also leads to significant improvements in the accuracy of statistical moments.

Numerical convergence and error analysis for the truncated iterative generalized stochastic perturbation-based finite element method

The main aim of this work is to present a numerical analysis of convergence and accuracy of the selected generalized perturbation-based schemes in linear and nonlinear problems of solid mechanics. An algorithm for determining the basic probabilistic characteristics has been developed using the iterative generalized stochastic higher-order perturbation method adjacent to symmetrically truncated Gaussian random variables. It has been confirmed that usage of a sufficiently high order of the Truncated Iterative Stochastic Perturbation Technique (TISPT) allows for achieving any desired accuracy in determining up to the fourth-order probabilistic characteristics of static structural response. The semi-analytical probabilistic approach is the reference solution in this study, which is based upon the same composite response functions determined with the use of the Least Squares Method created using specific series of FEM experiments. The entire methodology has been provided for the given extreme value of the coefficient of variation αmax of the input uncertainty source. On the other hand, a selection procedure of the stochastic perturbation method order to achieve 1% numerical accuracy in all up to the fourth-order probabilistic moments has been proposed. Computational experiments include simply supported elastic Euler–Bernoulli beam, a set of steel diagrid structures, nonlinear tension of steel round bar as well as homogenization procedure of some particulate composite.

Hierarchy of quasisymmetries and degeneracies in the CoSi family of chiral crystal materials

In materials, certain approximated symmetry operations can exist in a lower-order approximation of the effective model but are good enough to influence the physical responses of the system, and these approximated symmetries were recently dubbed ``quasisymmetries'' [Nat. Phys. 18, 813 (2022)]. In this paper, we reveal a hierarchy structure of the quasisymmetries and the corresponding nodal structures that they enforce via two different approaches of the perturbation expansions for the effective model in the chiral crystal material CoSi. In the first approach, we treat the spin-independent linear momentum ($k$) term as the zero-order Hamiltonian. Its energy bands are fourfold degenerate due to an $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ quasisymmetry. We next consider both the k-independent spin-orbit coupling (SOC) and full quadratic $k$ terms as the perturbation terms and find that the first-order perturbation leads to a model described by a self-commuting ``stabilizer code'' Hamiltonian with a U(1) quasisymmetry that can protect nodal planes. In the second approach, we treat the SOC-free linear $k$ term and $k$-independent SOC term as the zero order. They exhibit an SU(2) quasisymmetry, which can be reduced to U(1) quasisymmetry by a choice of quadratic terms. Correspondingly, a twofold degeneracy for all the bands due to the SU(2) quasisymmetry is reduced to twofold nodal planes that are protected by the U(1) quasisymmetry. For both approaches, including higher-order perturbation will break the U(1) quasisymmetry and induce a small gap $\ensuremath{\sim}1\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$ for the nodal planes. These quasisymmetry protected near degeneracies play an essential role in understanding recent quantum oscillation experiments in CoSi.

Investigations of the Spin-Hamiltonian Parameters and Defect Structure for the (TiO4)5− Tetrahedral Clusters in ZrSiO4: Ti3+ Crystal

The spin-Hamiltonian parameters (g factors g//, g⊥ and hyperfine structure constants 47Ai, 49Ai, where i = // or ⊥, 47Ai and 49Ai denote the values of isotopes 47Ti3+ and 49Ti3+) of Ti3+ ions at the tetragonal tetrahedral Si4+ sites in ZrSiO4 crystal are computed from the high-order perturbation formulas based on the cluster approach where the contributions due to both the d orbitals of central dn ion and p orbitals of ligand ion are considered. The calculated results are reasonably coincident with the experimental values. The defect structure of tetragonal (TiO4)5− impurity cluster which differs from the replaced host (SiO4)4− cluster in ZrSiO4 is evaluated from the calculation. The results are discussed.

Localization of higher order exceptional points from finite element model and their applications to duct acoustics.

This work reviews the state of the art for high order perturbation method for parametric eigenvalue problems and propose some extensions for the multiparameters case. This approach allows to locate high order exceptional points (EP) arising in eigenvalue problems. EP correspond to a particular tuning of some complex-valued parameters which render the problem degenerate. These non-Hermitian degeneracies have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems (PT-symmetry, thermo-acoustic or fluid-structure instability, etc.) and their numerical solution. For applications dealing with dissipative acoustic waveguides, strong modal attenuation can be achieved close to EP and a maximum of attenuation occurs at EP of high order corresponding to the coalescence of more than two modes. The method is based on the automatic computation of the successive derivatives of some selected eigenpairs with respect to the parameters so that, after recombination, regular functions can be constructed. This algebraic manipulations permit to build a reduced order model allowing i) to quickly solve the eigenvalue problem for other parameters values, ii) to follow modal branches, iii) to locate higher order EPs. The method is applied to the particular case of a circular duct with a locally reacting liner at its wall which admittance varies with azimuthal position.

Dynamical Behaviour Changes in Response to Various Functional Responses: Temporal and Spatial Plankton System

In this paper, we focus on the comparison between ratio-dependent and Holling type-II functional response in the phytoplankton dynamics with the susceptible and infected populations with or without diffusion. The dynamics are investigated in terms of local stability using both the response function for the systems and the definition of the Hopf-bifurcation, which has been demonstrated to occur in the Holling type-II functional response using the half-saturation time (i.e. $$\alpha$$ ) as the bifurcation parameter. We observed that the system’s dynamical behaviour is affected by the response functions and the system is more stable in the case of the ratio-dependent functional response. Furthermore, the study of higher-order perturbation terms reveals the heterogeneity of in-homogeneous spatiotemporal perturbation around temporal steady state outside the linear regime. The system behaviours to both types of response functions are studied numerically in the temporal and spatiotemporal domain (i.e. pattern formation in the diffusive population). Numerical simulation is used to examine the results.