Eikonal Approximation Research Articles

Non-global logarithms up to four loops at finite-Nc for V/H+jet processes at hadron colliders

We extend our previous work [1] on calculating non-global logarithms in e+e− annihilation to Higgs/vector boson production in association with a single hard jet at hadron colliders. We analytically compute non-global coefficients in the jet mass distribution up to four loops using the anti-kt jet algorithm. Our calculations are performed in the eikonal approximation, assuming strong energy ordering for the emitted gluons, thus capturing only the leading logarithms of the distribution. We compare our analytical results with the all-orders large-Nc numerical solution. In general, the gross features of the non-global logarithm distribution observed in the e+e− case remain valid for the V/H+jet processes.

Amplitude factorization in the electroweak standard model

We provide tools to analyze factorization at the amplitude level for processes involving the entire standard model. We focus in particular on a momentum region, in which the factorization of real and certain virtual corrections appears in a generalized eikonal approximation in which we expand around a quasisoft limit for massive gauge bosons, fermions, and scalars. We use the chirality-flow formalism to express loop exchanges or emissions as operators on chiral structures. This provides key tools for amplitude evolution with parton exchange and branching in the full Standard Model, including the electroweak sector. Published by the American Physical Society 2024

Next-to-eikonal corrections to dijet production in Deep Inelastic Scattering in the dilute limit of the Color Glass Condensate

We analyze the effects of next-to-eikonal corrections on dijet production in Deep Inelastic Scattering off nuclear targets in the framework of the Color Glass Condensate. They require the knowledge of correlators of fields in the target beyond those computed in the standard McLerran-Venugopalan model, specifically those between transverse and boost-enhanced components, and of the recoil of the fields. We neglect the latter, while for the former we develop a linear model valid for large nuclei. We considered the unpolarized cross sections for dijet production in the approximation of a homogenous dilute nucleus, obtaining simple analytic expressions for the cross sections at next-to-eikonal accuracy, valid in the limit of total dijet momentum and dijet momentum imbalance larger than the saturation scale of the nucleus. We perform a numerical study of the results at energies of the Electron Ion Collider, finding O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(10%) effects in the cross sections at large total momentum. We also analyze the azimuthal asymmetries between total momentum and imbalance, finding that non-eikonal corrections induce odd azimuthal harmonics for the situation of jets with equal momentum fractions from the virtual photon, where they are absent in the eikonal approximation. Finally, in the eikonal approximation we have compared the results of our analytic expansion valid in the dilute limit of the target, and the full Color Glass Condensate results in the McLerran-Venugopalan model and their correlation limit. Our analytic expressions match the correlation limit ones in the region where both should be simultaneously valid and reproduce very well the full Color Glass Condensate results in its validity region.

FUNCTIONAL INTEGRAL APPROACH AND THE EIKONAL APPROXIMATION TO THE POLARON ENERGY AND MASS

This article uses the functional integral method and eikonal approximation to calculate the first-order correction for the energy and (effective) mass of the Polaron. The obtained results have demonstrated the effectiveness of the functional integral method compared to conventional perturbation theory when considering the Polaron problem in solid crystal.

On the breakdown of eikonal approximation and survival of Reggeization in presence of dimension-5 Higgs-gluon coupling

We consider the one-loop effective vertex for the interaction of a gluon with a Reggeized gluon and a Higgs boson in the infinite-top-mass limit, which is described by a dimension-5 non-renormalizable operator. This vertex enters the calculation of differential cross sections for the forward inclusive production of a Higgs boson in high-energy proton-proton collisions, possibly in association with a backward jet or identified hadron, in a framework where next-to-leading logarithms of the energy are resummed to all orders. The effective vertex is extracted from the high-energy behavior of two-to-two amplitudes for the Higgs production in parton-parton collisions and relies on the validity of the Regge form for these amplitudes. We find that the usual eikonal approximation (Gribov prescription) for the Regge limit and the known region-expansion technique in this limit lead to an incomplete result for the amplitude. The discrepancy is traced back to the non-renormalizable nature of the involved operator. However, the Regge limit of the exact QCD amplitude agrees with the Regge-pole exchange form at one loop, nontrivially supporting the Reggeization hypothesis.

Інклюзивна реакція фрагментації дейтронів при зіткненні їх з дейтронами мішені

Within the framework of eikonal approximation and the double folding model, a formalism for calculating the inclusive spectra of particles emitted from the deuteron breakup reaction is proposed. The spectra of protons from the reaction 2H(d,p) at an incident particle energy of 59.9 MeV are described. The calculated values of cross-sections are in good agreement with the corresponding experimental data.

Nonfactorizable corrections to Higgs boson production in weak boson fusion

We discuss the nonfactorizable corrections to Higgs boson production in weak boson fusion at the Large Hadron Collider. Such corrections depend on the finite part of the two-loop virtual amplitude qQ→q′Q′+H which, up to now, has only been computed in the eikonal approximation. We combine this contribution with real-virtual and double-real nonfactorizable QCD corrections and study their impact on the various observables in weak boson fusion. We find that the nonfactorizable corrections are strongly dominated by the two-loop virtual contributions, while all other contributions play a very minor role. This striking imbalance between real and virtual contributions is caused by a process-specific kinematic suppression of the former and a particular enhancement of the virtual corrections related to a Glauber phase. Published by the American Physical Society 2024

Circular geodesics in the field of double-charged dilatonic black holes

A non-extreme dilatonic charged (by two “color electric” charges) black hole solution is examined within a four-dimensional gravity model that incorporates two scalar (dilaton) fields and two Abelian vector fields. The scalar and vector fields interact through exponential terms containing two dilatonic coupling vectors. The solution is characterized by a dimensionless parameter a(0<a<2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0< a < 2)$$\\end{document}, which is a specific function of dilatonic coupling vectors. The paper presents solutions for timelike and null circular geodesics that may play a crucial role in different astrophysical scenarios, including quasinormal modes of various test fields in the eikonal approximation. For a=1/2,1,3/2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a = 1/2,1, 3/2,2 $$\\end{document}, the radii of the innermost stable circular orbit are presented and analyzed.

Methodological derivation of the eikonal equation

Usually, when working with the eikonal equation, reference is made to its derivation in the monograph by Born and Wolf. The derivation of this equation was done rather carelessly. Understanding this derivation requires a certain number of implicit assumptions. For a better understanding of the eikonal approximation and for methodological purposes, the authors decided to repeat the derivation of the eikonal equation, explicating all possible assumptions. Methodically, the following algorithm for deriving the eikonal equation is proposed. The wave equation is derived from Maxwell’s equation. In this case, all conditions are explicitly introduced under which it is possible to do this. Further, from the wave equation, the transition to the Helmholtz equation is carried out. From the Helmholtz equation, with the application of certain assumptions, a transition is made to the eikonal equation. After analyzing all the assumptions and steps, the transition from the Maxwell’s equations to the eikonal equation is actually implemented. When deriving the eikonal equation, several formalisms are used. The standard formalism of vector analysis is used as the first formalism. Maxwell’s equations and the eikonal equation are written as three-dimensional vectors. After that, both the Maxwell’s equations and the eikonal equation use the covariant 4-dimensional formalism. The result of the work is a methodically consistent description of the eikonal equation.

Effective field theory analysis of the Coulomb breakup of the one-neutron halo nucleus 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{19}$$\\end{document}C

We analyse the Coulomb breakup of 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{19}$$\\end{document}C measured at 67A MeV at RIKEN. We use the Coulomb-Corrected Eikonal (CCE) approximation to model the reaction and describe the one-neutron halo nucleus 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{19}$$\\end{document}C within Halo Effective Field Theory (Halo EFT). At leading order we obtain a fair reproduction of the measured cross section as a function of energy and angle. The description is insensitive to the choice of optical potential, as long as it accurately represents the size of 18\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}^{18}$$\\end{document}C. It is also insensitive to the interior of the 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}^{19}$$\\end{document}C wave function. Comparison between theory and experiment thus enables us to infer asymptotic properties of the ground state of 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{19}$$\\end{document}C: these data put constraints on the one-neutron separation energy of this nucleus and, for a given binding energy, can be used to extract an asymptotic normalisation coefficient (ANC). These results are confirmed by CCE calculations employing next-to-leading order Halo EFT descriptions of 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}^{19}$$\\end{document}C: at this order the results for the Coulomb breakup cross section are completely insensitive to the choice of the regulator. Accordingly, this reaction can be used to constrain the one-neutron separation energy and ANC of 19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}^{19}$$\\end{document}C.

Scalar propagator in a background gluon field beyond the eikonal approximation

We investigate the path integral representation of the scalar propagator in a background gluon field, extending beyond the eikonal approximation by considering all gauge field components and incorporating its x− dependence. Utilizing the worldline formalism, we integrate the Schwinger proper time to express the scalar propagator in light-cone coordinates, facilitating a direct comparison with known results in the literature. The derived propagator captures the change of longitudinal momentum of the projectile within the medium. In the high-energy limit, our result simplifies to the effective gluon propagator employed in the BDMPS-Z formalism. Hence, we propose that our outcome serves as a foundational point for investigating corrections to the BDMPS-Z spectrum arising from the longitudinal momentum transfer of the radiated gluon with the medium, as well as for studying collisional energy loss phenomena. Lastly, by employing an expansion around the classical saddle point solution, we systematically derive an eikonal expansion in inverse powers of the boost parameter, encompassing corrections related to longitudinal momentum transfer and interactions of the projectile with the transverse component of the field.

Analytic expressions for quasinormal modes and grey-body factors in the eikonal limit and beyond

Abstract Although the WKB series converges only asymptotically and guarantees the exact result solely in the eikonal regime, we have managed to derive concise analytical expressions for the quasinormal modes and grey-body factors of black holes, expanding beyond the eikonal approximation. Remarkably, these expressions demonstrate unexpectedly strong accuracy. We suggest a comprehensive approach for deriving analytical expressions for grey-body factors and quasinormal modes at various orders beyond the eikonal approximation. Two cases are examined as examples: the Schwarzschild-de Sitter black hole and hairy black holes within the framework of Effective Field Theory. We have publicly shared a generic code that calculates analytical expressions for grey-body factors and quasinormal modes of spherical black holes.

Electromagnetic interaction models for Monte Carlo simulation of protons and alpha particles

Electromagnetic interactions of protons and alpha particles are modeled in a form that is suitable for Monte Carlo simulation of the transport of charged particles. The differential cross section (DCS) for elastic collisions with neutral atoms is expressed as the product of the DCS for collisions with the bare nucleus and a correction factor that accounts for the screening of the nuclear charge by the atomic electrons. The screening factor is obtained as the ratio of the DCS for scattering of the projectile by an atom with a point nucleus and the parameterized Dirac–Hartree–Fock–Slater (DHFS) electron density, calculated from the eikonal approximation, and the Rutherford DCS for collisions with the bare point nucleus. Inelastic collisions, which cause electronic excitations of the material, are described by means of the plane-wave Born approximation, with an empirical simple model of the generalized oscillator strength (GOS) that combines several extended oscillators with resonance energies and strengths determined from the atomic configurations and from the empirical mean excitation energy of the material. The contributions from inner subshells are renormalized to agree with realistic ionization cross sections calculated numerically from the DHFS self-consistent model of atoms by means of the plane-wave Born approximation. The resulting DCS allows analytical random sampling of individual hard inelastic interactions.

Dynamic diffraction of planar X-rays in a crystal lattice with a weak deformation field

In this paper, we consider the problem of the formation of a wave field during dynamic X-ray diffraction under conditions of weak deformation of the crystal lattice induced by a temperature gradient directed along the diffraction vector. Within the framework of the Takagi equations, which describe the propagation of the amplitudes of transmitted and reflected waves in the grating, analytical solutions dependence on the deformation parameters of these amplitudes are given. The angular width of the reflected X-ray radiation, as well as its dependence on the thickness of the single crystal, has been studied. It is shown that with increasing deformation, the rocking curves of the crystal slowly broaden, and their maximum intensity increases, reaching unity (the case of complete transfer). With a further increase in the deformation of the reflecting planes, the rocking curves continues to widen, and their maximum slowly decreases. It is also shown that the angular width of the complete transfer x-rays is directly proportional to the thickness of the sample under study. All observed phenomena are given a theoretical interpretation based on the general theory of dynamic scattering based on the Takagi equations and the eikonal approximation for weakly deformed crystal lattice fields.

Quantum gravity phenomenology from the perspective of quantum general relativity and quadratic gravity

Multi-messenger astronomy provides us with the possibility of discovering phenomenological signatures of quantum-gravity effects. This should be of paramount importance in the pursuit of an elusive quantum theory for the gravitational interactions. Here we discuss feasible explorations within the effective field theory (EFT) treatment of general relativity. By exploring current techniques borrowed from modern amplitude methods, we calculate leading quantum corrections to the classical radiated momentum and spectral waveforms. The lessons drawn from these low-energy results are that phenomenological applications in gravitational-wave physics can be discussed in line with the EFT approach. In turn, we also examine possible phenomenological surveys from the perspective of a UV completion for quantum gravity which employs the metric as the fundamental dynamical variable, namely quadratic gravity. Being more specific, by resorting to the eikonal approximation, we compute the leading-order time delay/advance in the scattering of light by a heavy object and find a possible significant deviation from the standard general-relativity prediction. This allows us to probe causal uncertainty due to quantum fluctuations of the gravitational field as a genuine prediction from Planck-scale physics.

Precise description of medium-induced emissions

We study jet fragmentation via final-state parton splittings in the medium. These processes are usually calculated theoretically by invoking the large-Nc limit. In this paper we perform the first computation of a 1 → 2 parton splitting in a thermal medium at finite numbers of colors Nc, for arbitrary momentum-sharing fraction z and with full transverse dynamics. We show how the problem can be transformed into a system of coupled Schrödinger equations, that we solve numerically. The novel numerical results are used to estimate the accuracy of several widely used approximations. We check the error introduced while going from finite Nc (i.e. Nc = 3) to the large-Nc limit, which we find to be small. For unbalanced splittings, e.g. when z → 0, only one of the partons is affected by transverse momentum exchanges with the medium. The emission process then separates into a term responsible for the 1 → 2 splitting and the subsequent independent broadening of the daughter partons. This is also referred to as the factorizable term. For finite z, further contributions arise that are responsible for the coherent color dynamics of the two-parton system, and these are referred to as non-factorizable terms. These were argued to be small for soft (unbalanced) splittings and for large media. In this work we therefore determine the accuracy of keeping only the factorizable term of the large-Nc solution. We find that the error is insignificant at a small splitting fraction z ~ 0, but can be sizable in a more balanced splitting with z ~ 0.5. Finally, we also examine the eikonal approximation, which amounts to approximating the partons’ paths through the medium as straight lines. We find that it is associated with a substantial error for the parameter values we explored in this work.

Chaos Detection by Fast Dynamic Indicators in Reflecting Billiards.

The propagation of electromagnetic waves in a closed domain with a reflecting boundary amounts, in the eikonal approximation, to the propagation of rays in a billiard. If the inner medium is uniform, then the symplectic reflection map provides the polygonal rays' paths. The linear response theory is used to analyze the stability of any trajectory. The Lyapunov and reversibility error invariant indicators provide an estimate of the sensitivity to a small initial random deviation and to a small random deviation at any reflection, respectively. A family of chaotic billiards is considered to test the chaos detection effectiveness of the above indicators.

Small x physics beyond eikonal approximation: an effective Hamiltonian approach

Understanding the spin structure of hadrons in the small x regime is an important direction to unravel the spin puzzle in hadronic physics. To include spin degrees of freedom in the small x regime requires going beyond the usual eikonal approximation in high energy QCD. We developed an effective Hamiltonian approach to study spin related observables in the small x regime using the shockwave formalism. The small-x effective Hamiltonian incorporates both quark and gluon propagators in the background fields and the background field induced interaction vertices up to next-to-eikonal order. A novel feature of sub-eikonal interactions is the background gluon field induced gluon radiation inside the shockwave. Its relation to chromo-electrically polarized Wilson line correlator is established both in small x helicity evolution and in longitudinal double-spin asymmetry for gluon production.

Diverse mechanisms in proton knockout reactions from the Borromean nucleus ^{17}text {Ne}

Nucleon knockout experiments using beryllium or carbon targets reveal a strong dependence of the quenching factors, i.e., the ratio (hbox {R}_{textrm{s}}) of theoretical to the experimental spectroscopic factors (hbox {C}^2{textrm{S}}), on the proton-neutron asymmetry in the nucleus under study. However, this dependence is greatly reduced when a hydrogen target is used. To understand this phenomenon, exclusive ^{1}text {H}(^{17}text {Ne},2hbox {p}~^{16}text {F}) and inclusive ^{12}text {C}(^{17}text {Ne},2hbox {p}~^{16}text {F})hbox {X}, ^{12}text {C}(^{17}text {Ne},^{16}text {F})hbox {X} as well as ^{1}text {H}(^{17}text {Ne},^{16}text {F})hbox {X} (X-denotes undetected reaction products) reactions with ^{16}text {F} in the ground and excited states were analysed. The longitudinal momentum distribution of ^{16}{textrm{F}} and the correlations between the detached protons were studied. In the case of the carbon target, there is a significant deviation from the predictions of the eikonal model. The eikonal approximation was used to extract spectroscopic factor values hbox {C}^2{textrm{S}}. The experimental hbox {C}^2hbox {S} value obtained with C target is markedly lower than that for H target. This is interpreted as rescattering due to simultaneous nucleon knockout from both reaction partners, ^{17}text {Ne} and ^{12}text {C}.

Non-factorizable virtual corrections to Higgs boson production in weak boson fusion beyond the eikonal approximation

Non-factorizable virtual corrections to Higgs boson production in weak boson fusion at next-to-next-to-leading order in QCD were estimated in the eikonal approximation [1]. This approximation corresponds to the expansion of relevant amplitudes around the forward limit. In this paper we compute the leading power correction to the eikonal limit and show that it is proportional to first power of the Higgs boson transverse momentum or the Higgs boson mass over partonic center-of-mass energy. Moreover, this correction can be significantly enhanced by the rapidity of the Higgs boson. For realistic weak boson fusion cuts, the next-to-eikonal correction reduces the estimate of non-factorizable contributions to fiducial cross section by (20) percent.