Higher-order Perturbation Theory Research Articles

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.

Multiphoton Resonance in Attosecond Transient Absorption

We present the theory and simulation of attosecond transient absorption in helium atoms under the single-active-electron approximation. This study investigates the attosecond dynamics of intrinsic atomic states that interact with a field comprising vacuum ultraviolet (VUV) and extreme ultraviolet (XUV) fields. The absorption spectrum of the helium atom is obtained from the response function, which is constructed by numerically solving the three-dimensional time-dependent Schrödinger equation. We observe a fine structure near the intrinsic atomic level, which is modulated with a 0.2 fs period. Based on high-order time-dependent perturbation theory, the frequency-dependent phase of the dipole response induced by the VUV and XUV fields is analytically obtained, and the fine structure is well explained by the phase difference. In addition, the absorption fringes are dependent on the chirp of the VUV field. This study investigates the features of the attosecond transient absorption in the VUV region, which may have valuable applications in the study of ultrafast phenomena in atoms, molecules, and solids.

Graviton particle statistics and coherent states from classical scattering amplitudes

In the two-body scattering problem in general relativity, we study the final graviton particle distribution using a perturbative approach. We compute the mean, the variance and the factorial moments of the distribution from the expectation value of the graviton number operator in the KMOC formalism. For minimally coupled scalar particles, the leading deviation from the Poissonian distribution is given by the unitarity cut involving the six-point tree amplitude with the emission of two gravitons. We compute this amplitude in two independent ways. First, we use an extension of the Cheung-Remmen parametrization that includes minimally coupled scalars. We then repeat the calculation using on-shell BCFW-like techniques, finding complete agreement. In the classical limit, this amplitude gives a purely quantum contribution, proving that we can describe the final semiclassical radiation state as a coherent state at least up to order mathcal{O} (G4) for classical radiative observables. Finally, we give general arguments about why we expect this to hold also at higher orders in perturbation theory.

Iterative solvers for the thermoacoustic nonlinear eigenvalue problem and their convergence properties

The spectrum of the thermoacoustic operator is governed by a nonlinear eigenvalue problem. A few different strategies have been proposed by the thermoacoustic community to tackle it and identify the frequencies and growth rates of thermoacoustic eigenmodes. These strategies typically require the use of iterative algorithms, which need an initial guess and are not necessarily guaranteed to converge to an eigenvalue. A quantitative comparison between the convergence properties of these methods has however never been addressed. By using adjoint-based sensitivity, in this study we derive an explicit formula that can be used to quantify the behaviour of an iterative method in the vicinity of an eigenvalue. In particular, we employ Banach’s fixed-point theorem to demonstrate that there exist thermoacoustic eigenvalues that cannot be identified by some of the iterative methods proposed in the literature, in particular fixed-point iterations, regardless of the accuracy of the initial guess provided. We then introduce a family of iterative methods known as Householder’s methods, of which Newton’s method is a special case. The coefficients needed to use these methods are explicitly derived by means of high-order adjoint-based perturbation theory. We demonstrate how these methods are always guaranteed to converge to the closest eigenvalue, provided that the initial guess is accurate enough. We also show numerically how the basin of attraction of the eigenvalues varies with the order of the employed Householder’s method.

On phase-space integrals with Heaviside functions

We discuss peculiarities that arise in the computation of real-emission contributions to observables that contain Heaviside functions. A prominent example of such a case is the zero-jettiness soft function in SCET, whose calculation at next-to-next-to-next-to-leading order in perturbative QCD is an interesting problem. Since the zero-jettiness soft function distinguishes between emissions into different hemispheres, its definition involves θ-functions of light-cone components of emitted soft partons. This prevents a direct use of multi-loop methods, based on reverse unitarity, for computing the zero-jettiness soft function in high orders of perturbation theory. We propose a way to bypass this problem and illustrate its effectiveness by computing various non-trivial contributions to the zero-jettiness soft function at NNLO and N3LO in perturbative QCD.

Polarized Z bosons from the decay of a Higgs boson produced in association with two jets at the LHC

Investigating the polarization of weak bosons provides an important probe of the scalar and gauge sector of the Standard Model. This can be done in the Higgs decay to four leptons, whose Standard-Model leading-order amplitude enables to generate polarized observables from unpolarized ones via a fully-differential reweighting method. We study the text {Z} -boson polarization from the decay of a Higgs boson produced in association with two jets, both in the gluon-fusion and in the vector-boson fusion channel. We also address the possibility of extending the results of this work to higher orders in perturbation theory.

Operator Growth Bounds from Graph Theory

Let $A$ and $B$ be local operators in Hamiltonian quantum systems with $N $ degrees of freedom and finite-dimensional Hilbert space. We prove that the commutator norm $\lVert [A(t),B]\rVert$ is upper bounded by a topological combinatorial problem: counting irreducible weighted paths between two points on the Hamiltonian's factor graph. Our bounds sharpen existing Lieb-Robinson bounds by removing extraneous growth. In quantum systems drawn from zero-mean random ensembles with few-body interactions, we prove stronger bounds on the ensemble-averaged out-of-time-ordered correlator $\mathbb{E}\left[ \lVert [A(t),B]\rVert_F^2\right]$. In such quantum systems on Erd\"os-R\'enyi factor graphs, we prove that the scrambling time $t_{\mathrm{s}}$, at which $\lvert [A(t),B]\rVert_F=\mathrm{\Theta}(1)$, is almost surely $t_{\mathrm{s}}=\mathrm{\Omega}(\sqrt{\log N})$; we further prove $t_{\mathrm{s}}=\mathrm{\Omega}(\log N) $ to high order in perturbation theory in $1/N$. We constrain infinite temperature quantum chaos in the $q$-local Sachdev-Ye-Kitaev model at any order in $1/N$; at leading order, our upper bound on the Lyapunov exponent is within a factor of 2 of the known result at any $q>2$. We also speculate on the implications of our theorems for conjectured holographic descriptions of quantum gravity.

Three-loop off-forward evolution kernel for axial-vector operators in Larin’s scheme

Evolution equations for leading-twist operators in high orders of perturbation theory can be restored from the spectrum of anomalous dimensions and the calculation of the special conformal anomaly at one order less using conformal symmetry of QCD at the Wilson-Fisher critical point at noninteger $d=4\ensuremath{-}2ϵ$ space-time dimensions. In this work, we generalize this technique to axial-vector operators. We calculate the corresponding three-loop evolution kernels in Larin's scheme and derive explicit expressions for the finite renormalization kernel that describes the difference to the vector case to restore the conventional modified minimal subtraction scheme. The results are directly applicable to deeply virtual Compton scattering and the transition form factor ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\pi}$.

Chiral correlators in mathcal{N} = 2 superconformal quivers

We consider a family of mathcal{N} = 2 superconformal field theories in four dimensions, defined as ℤq orbifolds of mathcal{N} = 4 Super Yang-Mills theory. We compute the chiral/anti-chiral correlation functions at a perturbative level, using both the matrix model approach arising from supersymmetric localisation on the four-sphere and explicit field theory calculations on the flat space using the mathcal{N} = 1 superspace formalism. We implement a highly efficient algorithm to produce a large number of results for finite values of N , exploiting the symmetries of the quiver to reduce the complexity of the mixing between the operators. Finally the interplay with the field theory calculations allows to isolate special observables which deviate from mathcal{N} = 4 only at high orders in perturbation theory.

Measurements of the differential cross sections of the production of Z + jets and \u03b3 + jets and of Z boson emission collinear with a jet in pp collisions at sqrt{s} = 13 TeV

Measurements of the differential cross sections of Z + jets and γ + jets production, and their ratio, are presented as a function of the boson transverse momentum. Measurements are also presented of the angular distribution between the Z boson and the closest jet. The analysis is based on pp collisions at a center-of-mass energy of 13 TeV corresponding to an integrated luminosity of 35.9 fb−1 recorded by the CMS experiment at the LHC. The results, corrected for detector effects, are compared with various theoretical predictions. In general, the predictions at higher orders in perturbation theory show better agreement with the measurements. This work provides the first measurement of the ratio of the differential cross sections of Z + jets and γ + jets production at 13 TeV, as well as the first direct measurement of Z bosons emitted collinearly with a jet.

The cosmic neutrino background as a collection of fluids in large-scale structure simulations

A significant challenge for modelling the massive neutrino as a hot dark matter is its large velocity dispersion. In this work, we investigate and implement a multi-fluid perturbation theory that treats the cosmic neutrino population as a collection of fluids with a broad range of bulk velocities. These fluids respond linearly to the clustering of cold matter, which may be linear and described by standard linear perturbation theory, or non-linear, described using either higher-order perturbation theory or N-body simulations. We verify that such an alternative treatment of neutrino perturbations agrees closely with state-of-the-art neutrino linear response calculations in terms of power spectrum and bispectrum predictions. Combining multi-fluid neutrino linear response with a non-linear calculation for the cold matter clustering, we find for a reference νΛCDM cosmology with neutrino mass sum ∑ m ν = 0.93 eV an enhancement of the small-scale neutrino power by an order of magnitude relative to a purely linear calculation. The corresponding clustering enhancement in the cold matter, however, is a modest ∼ 0.05%. Importantly, our multi-fluid approach uniquely enables us to identify that the slowest-moving 25% of the neutrino population clusters strongly enough to warrant a non-linear treatment. Such a precise calculation of neutrino clustering on small scales accompanied by fine-grained velocity information would be invaluable for experiments such as PTOLEMY that probe the local neutrino density and velocity in the solar neighbourhood.

The full spectrum of AdS5/CFT4 II: Weak coupling expansion via the quantum spectral curve

We continue the effort to optimise and generalise the solution of the spectral problem of AdS5/CFT4 in the planar limit via integrability. We present a simple strategy to solve the quantum spectral curve (QSC) perturbatively for general states by focussing on the P μ-system. A Mathematica notebook with an implementation of this algorithm is provided, as well as an extensive database with a user-friendly interface containing more than 8000 solutions of the QSC. When investigating the solution space, we observe a curious phenomenon: existence of solutions for which the Q-system degenerates in the limit g → 0. These degeneracies are lifted at higher orders in perturbation theory. The degenerating solutions have auxiliary Bethe roots merging with branch points at weak coupling.

On the Method of Correct Determination of Eigenvalues of a Truncated Hamiltonian Matrix on the Example of a Morse Oscillator

A method of precise determination of the eigenvalues of energies of the molecular Hamiltonian based on high-order perturbation theory has been developed and implemented for a diatomic molecule. The proposed method makes it possible not only to obtain the energy values, but also to estimate the accuracy of their prediction and the applicability limits for the employed model. Numerical calculations have been performed for the extended Morse oscillator with corrections for the potential functions up to the sixth power of the Morse coordinate. The results obtained are compared with the results of calculation for the model of the truncated Hamiltonian matrix. The possibilities of application of the method are analyzed compared to other approaches to the determination of the potential functions for polyatomic molecules.

Running Coupling Constant from Position-Space Current-Current Correlation Functions in Three-Flavor Lattice QCD.

In this Letter, we provide a determination of the coupling constant in three-flavor quantum chromodynamics (QCD), α_{s}^{MS[over ¯]}(μ), for MS[over ¯] renormalization scales μ∈(1,2) GeV. The computation uses gauge field configuration ensembles with O(a)-improved Wilson-clover fermions generated by the Coordinated Lattice Simulations (CLS) consortium. Our approach is based on current-current correlation functions and has never been applied before in this context. We convert the results perturbatively to the QCD Λ parameter and obtain Λ_{MS[over ¯]}^{N_{f}=3}=342±17 MeV, which agrees with the world average published by the Particle Data Group and has competing precision. The latter was made possible by a unique combination of state-of-the-art CLS ensembles with very fine lattice spacings, further reduction of discretization effects from a dedicated numerical stochastic perturbation theory simulation, combining data from vector and axial-vector channels, and matching to high-order perturbation theory.

An effective field theory approach for electroweak interactions in the high energy limit

We present an effective action for the electroweak sector of the standard model valid for the calculation of scattering amplitudes in the high energy (Regge) limit. Gauge invariant Wilson lines are introduced to describe reggeized degrees of freedom whose interactions are generated by effective emission vertices. From this approach previous results at leading logarithmic accuracy for electroweak boson Regge trajectories are reproduced together with the corresponding interaction kernels. The proposed framework lays the path for calculations at higher orders in perturbation theory.

Hamiltonian formulations for perturbed dissipationless plasma equations

The Hamiltonian formulations for the perturbed Vlasov–Maxwell equations and the perturbed ideal magnetohydrodynamics (MHD) equations are expressed in terms of the perturbation derivative ∂F/∂ϵ≡[F,S] of an arbitrary functional F[ψ] of the Vlasov–Maxwell fields ψ=(f,E,B) or the ideal MHD fields ψ=(ρ,u,s,B), which are assumed to depend continuously on the (dimensionless) perturbation parameter ϵ. Here, [ , ] denotes the functional Poisson bracket for each set of plasma equations and the perturbation action functional S is said to generate dynamically accessible perturbations of the plasma fields. The new Hamiltonian perturbation formulation introduces a framework for functional perturbation methods in plasma physics and highlights the crucial roles played by polarization and magnetization in Vlasov–Maxwell and ideal MHD perturbation theories. One application considered in this paper is a formulation of plasma stability that guarantees dynamical accessibility and leads to a natural generalization to higher-order perturbation theory.

Application of Higher-Order Interpolation Perturbation Theory on Postprandial Plasma Glucose Waveform of a Single Lunch Based on GH-Method: Math-Physical Medicine (No. 426)

In this particular research note, the author arbitrarily selects the postprandial plasma glucose (PPG) data and waveforms from his lunch on 3/31/21 as the reference base. He then applies the higher-order, including first-order, second-order, and third-order interpolation perturbation theory to generate three approximated PPG waveforms to compare them against the measured nonlinear PPG waveform. The two major purposes of this study is to check the waveform shape similarity via correlation coefficients and the prediction accuracy via comparison of the averaged PPG values using the “higher order” interpolation perturbation theory. One obvious conclusion can be drawn from this article is that the higher-order perturbation equation provides a better approximate solution. However, in reality, there are very few patients who understand the perturbation theory of quantum mechanics, let alone to be able to conduct the complex mathematical calculations in order to achieve their desired glucose projection.