Higher-order Perturbation Theory Research Articles

Vanishing of nonlinear tidal Love numbers of Schwarzschild black holes

It is well known that asymptotically flat Schwarzschild black holes in general relativity in four spacetime dimensions have vanishing induced linear tidal response. We extend this result beyond linear order for the polar sector, by solving the static nonlinear Einstein equations for the perturbations of the Schwarzschild metric and computing the quadratic corrections to the electric-type tidal Love numbers. After explicitly performing the matching with the point-particle effective theory at leading order in the derivative expansion, we show that the Love number couplings remain zero at higher order in perturbation theory.

Ab initio dispersion potentials based on physics-based functional forms with machine learning.

In this study, we introduce SAPT10K, a comprehensive dataset comprising 9982 noncovalent interaction energies and their binding energy components (electrostatics, exchange, induction, and dispersion) for diverse intermolecular complexes of 944 unique dimers. These complexes cover significant portions of the intermolecular potential energy surface and were computed using higher-order symmetry-adapted perturbation theory, SAPT2+(3)(CCD), with a large aug-cc-pVTZ basis set. The dispersion energy values in SAPT10K serve as crucial inputs for refining the abinitio dispersion potentials based on Grimme's D3 and many-body dispersion (MBD) models. Additionally, Δ machine learning (ML) models based on newly developed intermolecular features, which are derived from intermolecular histograms of distances for element/substructure pairs to simultaneously account for local environments as well as long-range correlations, are also developed to address deficiencies of the D3/MBD models, including the inflexibility of their functional forms, the absence of MBD contributions in D3, and the standard Hirshfeld partitioning scheme used in MBD. The developed dispersion models can be applied to complexes involving a wide range of elements and charged monomers, surpassing other popular ML models, which are limited to systems with only neutral monomers and specific elements. The efficient D3-ML model, with Cartesian coordinates as the sole input, demonstrates promising results on a testing set comprising 6714 dimers, outperforming another popular ML model, component-based machine-learned intermolecular force field (CLIFF), by 1.5 times. These refined D3/MBD-ML models have the capability to replace the time-consuming dispersion components in symmetry-adapted perturbation theory-based calculations and can promptly illustrate the dispersion contribution in noncovalent complexes for supramolecular assembly and chemical reactions.

Starting Cosmological Simulations from the Big Bang.

The cosmic large-scale structure (LSS) provides a unique testing ground for connecting fundamental physics to astronomical observations. Modeling the LSS requires numerical N-body simulations or perturbative techniques that both come with distinct shortcomings. Here we present the first unified numerical approach, enabled by new time integration and discreteness reduction schemes, and demonstrate its convergence at the field level. In particular, we show that our simulations (i)can be initialized directly at time zero, and (ii)can be made to agree with high-order Lagrangian perturbation theory in the fluid limit. This enables fast, self-consistent, and UV-complete forward modeling of LSS observables.

The development of computational methods for Feynman diagrams

Over the last 70 years, Feynman diagrams have played an essential role in the development of many theoretical predictions derived from the standard model Lagrangian. In fact, today they have become an essential and seemingly irreplaceable tool in quantum field theory calculations. In this article, we propose to explore the development of computational methods for Feynman diagrams with a special focus on their automation, drawing insights from both theoretical physics and the history of science. From the latter perspective, the article particularly investigates the emergence of computer algebraic programs, such as the pioneering SCHOONSCHIP, REDUCE, and ASHMEDAI, designed to handle the intricate calculations associated with Feynman diagrams. This sheds light on the many challenges faced by physicists when working at higher orders in perturbation theory and reveal, as exemplified by the test of the validity of quantum electrodynamics at the turn of the 1960s and 1970s, the indispensable necessity of computer-assisted procedures. In the second part of the article, a comprehensive overview of the current state of the algorithmic evaluation of Feynman diagrams is presented from a theoretical point of view. It emphasizes the key algorithmic concepts employed in modern perturbative quantum field theory computations and discusses the achievements, ongoing challenges, and potential limitations encountered in the application of the Feynman diagrammatic method. Accordingly, we attribute the enduring significance of Feynman diagrams in contemporary physics to two main factors: the highly algorithmic framework developed by physicists to tackle these diagrams and the successful advancement of algebraic programs used to process the involved calculations associated with them.

All electron GW with linearized augmented plane waves for metals and semiconductors

GW approximation is one of the most popular parameter-free many-body methods that go beyond the limitations of the standard density functional theory (DFT) to determine the excitation spectra for moderately correlated materials and in particular the semiconductors. It is also the first step in developing the diagrammatic Monte Carlo method into an electronic structure tool, which would offer a numerically exact solution to the solid-state problem. While most electronic structure packages offer support for GW calculations for band-insulating materials, the level of support for metallic systems is somewhat limited. This limitation can be partly attributed to the relatively minor differences often observed between GW and DFT results in treating metallic systems, which is not expected to persist to higher orders in perturbation theory. Describing metals within the GW framework presents a challenge, as it requires accurate resolution of Fermi surface singularities, which, in turn, calls for a dense momentum mesh. Here we implement the GW algorithm within the all-electron Linear Augmented Plane Wave framework, where we pay special attention to the metallic systems, the convergence with respect to momentum mesh, and proper treatment of the deep laying core states, as needed for the future variational diagrammatic Monte Carlo implementation. Our improved algorithm for resolving Fermi surface singularities allows us a stable and accurate analytic continuation of imaginary axis data, which is carried out for GW excitation spectra throughout the Brillouin zone in both the metallic and insulating materials and is compared to numerically more stable contour deformation integration technique. We compute band structures for elemental metallic systems Li, Na, and Mg as well as for various narrow and wide bandgap insulators such as Si, BN, SiC, MgO, LiF, ZnS, and CdS and compare our results with previous GW calculations and available experiments data. Our results are in good agreement with the available literature. Thus our software allows users to compute full bandstructures for metals and insulators using all-electron potential without downfolding to Wannier orbital basis.

Numerical scattering amplitudes with pySecDec

We present a major update of the program pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The new version enables the evaluation of multi-loop integrals as well as amplitudes in a highly distributed and flexible way, optionally on GPUs. The program has been optimised and runs up to an order of magnitude faster than the previous release. A new integration procedure that utilises construction-free median Quasi-Monte Carlo rules is implemented. The median lattice rules can outperform our previous component-by-component rules by a factor of 5 and remove the limitation on the maximum number of sampling points. The expansion by regions procedures have been extended to support Feynman integrals with numerators, and functions for automatically determining when and how analytic regulators should be introduced are now available. The new features and performance are illustrated with several examples. Program summaryProgram Title: pySecDecCPC Library link to program files:https://doi.org/10.17632/dnrkf5jxzh.4Developer's repository link:https://github.com/gudrunhe/secdec, https://secdec.readthedocs.io (Online documentation)Licensing provisions: GNU Public License v3Programming language: Python, Form, C++, CudaExternal routines/libraries:GSL [1], NumPy [2], SymPy [3], Nauty [4], Cuba [5], Form [6], GiNaC and CLN [7], Normaliz [8], GMP [9].Journal reference of previous version: Comput. Phys. Commun. 273 (2022) 108267 [1].Does the new version supersede the previous version?: Yes.Nature of problem: Scattering amplitudes at higher orders in perturbation theory are typically represented as a linear combination of coefficients — containing the kinematic invariants and the space-time dimension — multiplied with loop integrals which contain singularities and whose analytic representation might be unknown.Solution method: Extraction of singularities in the dimensional regularization parameter as well as in analytic regulators for potential spurious singularities is done using sector decomposition. The combined evaluation of the integrals with their coefficients is performed in an efficient way.Restrictions: Depending on the complexity of the problem, limited by memory and CPU/GPU time.

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.

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

Perturbative Solution of the Einstein Constraints with Spin and Momentum Far Away From a Binary Source in the Bowen‐York Formalism

AbstractThe momentum and Hamiltonian constraints of vacuum Einstein equations, within the Bowen‐York formalism, for two interacting black holes in close separation, with anti‐parallel spins and anti‐parallel linear momenta is studied. An analytical solution using perturbation theory is given. The location and the shape of the apparent horizon which generically depend on all the parameters, angles, and the separation between the black holes are also computed. The solution only works for distances far away from the black holes. To gain more insight close into the black holes, one has to go to the higher orders in perturbation theory, which is a rather cumbersome process. But the solution presented here can be of some use for numerical computations as the latter should match our result for the described problem.

Computing Mellin Representations and Asymptotics of Nested Binomial Sums in a Symbolic Way: The RICA Package

Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums.

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.

Stability of φ^{4}-vector model: Four-loop ε expansion study.

In most papers, φ^{4}-field theory with the vector (d-component) field φ_{α} is considered as a particular case of the n-component field model for n=d and O(n) symmetry. However, in such a model the symmetry O(d) admits an addition to the action of a term proportional to the squared divergence of the field ∼h(∂_{α}φ_{α})^{2}. From the point of view of renormalization group analysis, it requires a separate consideration, because it may well change the nature of the critical behavior of the system. Therefore, this frequently neglected term in the action requires a detailed and accurate study on the issue of the existence of new fixed points and their stability. It is known that within the lower order of perturbation theory the only infrared stable fixed point with h=0 exists but the corresponding positive value of stability exponent ω_{h} is tiny. This led us to analyze this constant in higher orders of perturbation theory by calculating the four-loop renormalization group contributions for ω_{h} in d=4-2ɛ within the minimal subtraction scheme, which should be enough to infer positivity or negativity of this exponent. The value turned out to be undoubtedly positive, although still small even in higher loops: 0.0156(3). These results cause the corresponding term to be neglected in the action when analyzing the critical behavior of the O(n)-symmetric model. At the same time, the small value of ω_{h} shows that the corresponding corrections to the critical scaling are significant in a wide range.

Multi-time scale control and optimization via averaging and singular perturbation theory: From ODEs to hybrid dynamical systems

Multi-time scale techniques based on singular perturbations and averaging theory are among the most powerful tools developed for the synthesis and analysis of feedback control algorithms. This paper introduces some of the recent advances in singular perturbation theory and averaging theory for continuous-time dynamical systems modeled as ordinary differential equations (ODEs), as well as for hybrid dynamical systems that combine continuous-time dynamics and discrete-time dynamics. Novel multi-time scale analytical tools based on higher-order averaging and singular perturbation theory are also discussed and illustrated via different examples. In the context of hybrid dynamical systems, a class of sufficient Lyapunov-based conditions for global stability results are also presented. The analytical tools are illustrated through various new architectures and algorithms within the context of adaptive and extremum-seeking systems. These tools are suitable for the study of model-free optimization and stabilization problems that require the synergistic use of continuous-time and discrete-time feedback. The paper aims to acquaint the reader with a range of modern tools for studying multi-time scale phenomena in optimization and control systems, providing some guidelines for future research in this field.

Celestial amplitude for 2d theory

We explore celestial amplitude corresponding to 2d bulk mathcal{S} -matrix. We consider scalar particles with identical mass and show that the celestial amplitude becomes the fourier transform of the 2d mathcal{S} -matrix written in the rapidity variable. We translate the crossing and unitarity conditions into the conditions on the celestial amplitude. For the 2d Sinh-Gordon model, we calculate the celestial amplitude perturbatively in coupling constant and check that the crossing and unitarity conditions are satisfied for the celestial amplitude. Imposing the crossing and unitarity conditions to the celestial amplitude, we want to find amplitudes to the higher order in perturbation theory from the lower order i.e., to provide a “proof of principle” to show we can apply the bootstrap idea to the celestial amplitude. We find that imposing the crossing and unitarity conditions is not enough for bootstrapping celestial amplitude, there is an extra term which can’t be fixed by the crossing and unitarity conditions. We also study the gravitational dressing condition in 2d QFT for massless particles in celestial space and see that for the gravitationally dressed celestial amplitude, the poles on the right half-plane get erased for several ansatzes.

Beyond Hard-Core Bosons in Transmon Arrays

Arrays of transmons have proven to be a viable medium for quantum information science and quantum simulations. Despite their widespread popularity as qubit arrays, there remains yet untapped potential beyond the two-level approximation or, equivalently, the hard-core boson model. With the higher excited levels included, coupled transmons naturally realize the attractive Bose-Hubbard model. The dynamics of the model has been difficult to study due to unfavorable scaling of the dimensionality of the Hilbert space with the system size. In this work, we present a framework for describing the effective unitary dynamics of highly-excited states of coupled transmons based on high-order degenerate perturbation theory. This allows us to describe various collective phenomena -- such as bosons stacked onto a single site behaving as a single particle, edge-localization, and effective longer-range interactions -- in a unified, compact and accurate manner. While our examples deal with one-dimensional chains of transmons for the sake of clarity, the theory can be readily applied to more general geometries.

Exploring Expansions of the Potential and Dipole Surfaces Used for Vibrational Perturbation Theory.

A scheme for evaluating expansions of the potential and dipole moment surfaces for vibrational perturbation theory is described. The approach is based on numerical differentiation of the Hessian in the coordinates of interest. It is shown that performing these calculations in internal coordinates generates expansions that are transferable among isotopologues of the molecule of interest. Additionally, re-expressing the expansion of the potential in terms of functions of the internal coordinates, for example, cosines of angles or exponential functions of the bond length displacements, provides expansions that can be used for higher-order perturbation theory calculations. The approach is explored and the results are discussed for water, HOD, ammonia, isomers of HNO3, and halogenated methane.

Approximated Risk Probability of having Atherosclerosis, Cardiovascular Disease, and Stroke during the COVID-19 Period Using the Pre-COVID Period’S Dataset as Baseline of Calculations and Higher-Order Perturbation Equations from Quantum Mechanics Based on GH-Method: Math-Physical Medicine (No. 465)

This paper describes the author’s assessment on his risk probability percentage of having atherosclerosis, cardiovascular disease (CVD), and stroke. During the 7-quarters of the COVID-19 period from 2/5/2020 to 11/4/2021, he utilized his earlier 7-quarters data from 5/5/2018 to 2/4/2020 or the “pre-COVID period” and higher-order perturbation theory. The purpose of this study is to predict the present and future period’s CVD/stroke risks based on the previous period’s CVD/stroke risks via an effective approximation method of perturbation theory. In summary, this approximation method of perturbation equation from quantum mechanics offers high prediction accuracies on the present and future period’s (the “COVID period”) risks using the past pre-COVID period’s dataset and waveform as baseline calculations. The three orders of perturbation equations have provided the following high prediction accuracies in comparison against the COVID period’s calculated risks which includes the past 5-quarters of calculated risks and the future 2-quarters speculated risks of the COVID period.

Localization in a one-dimensional alloy with an arbitrary distribution of spacing between impurities: Application to Lévy glass.

We have studied the localization of waves in a one-dimensional lattice consisting of impurities where the spacing between consecutive impurities can take certain values with given probabilities. In general, such a distribution of impurities induces correlations in the disorder. In particular, with a power-law distribution of spacing, this system is used as a model for light propagation in Lévy glasses. We introduce a method of calculating the Lyapunov exponent which overcomes limitations in the previous studies and can be easily extended to higher orders of perturbation theory. We obtain the Lyapunov exponent up to fourth order of perturbation and discuss the range of validity of perturbation theory, transparent states, and anomalous energies which are characterized by divergences in different orders of the expansion. We also carry out numerical simulations which are in agreement with our analytical results.

Next-to-soft-virtual resummed rapidity distribution for the Drell-Yan process to NNLO+NNLL¯

We present the differential predictions for the rapidity distribution of a pair of leptons through the Drell-Yan (DY) process at the LHC taking into account the soft-virtual (SV) as well as next-to-soft-virtual (NSV) resummation effects in QCD perturbation theory to next-to-next-to-leading-order plus next-to-next-to-leading-logarithmic ($\mathrm{NNLO}+\overline{\mathrm{NNLL}}$) accuracy. We perform the resummation in two-dimensional Mellin space using our recent formalism [A. H. Ajjath et al., Phys. Rev. D 103, L111502 (2021)] by limiting ourselves to contributions only from quark-antiquark ($q\overline{q}$) initiated channels. The resummed corrections to the fixed-order results are computed through a matched formula using the minimal prescription procedure. We find that the resummation at next-to-leading-logarithmic (next-to-next-to-leading-logarithmic) level brings about 3.98% (1.24%) corrections. We also observe that the sensitivity to the renormalization scale gets improved substantially by the inclusion of NSV resummed predictions at $\overline{\mathrm{NNLL}}$ accuracy. Further, the lack of quark gluon ($qg$) initiated contributions to the NSV part in the $\overline{\mathrm{NNLL}}$ resummed predictions leaves large factorization scale dependence, indicating their importance at NSV level as we go to higher orders in perturbation theory.