Order Relativistic Corrections Research Articles

Improved analysis of double J/ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J/\\psi $$\\end{document} production in Z-boson decay

In this paper, we present an improved calculation for the decay rate of the rare Z-boson decay into J/ψ+J/ψ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J/\\psi + J/\\psi .$$\\end{document} This decay is dominated by the photon fragmentation mechanism, i.e., the transition Z→J/ψ+γ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z\\rightarrow J/\\psi + \\gamma ^{*}$$\\end{document} followed by the fragmentation γ∗→J/ψ.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{*}\\rightarrow J/\\psi .$$\\end{document} In our calculation, the amplitude of γ∗→J/ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{*}\\rightarrow J/\\psi $$\\end{document} is extracted from the measured value of Γ(J/ψ→e+e-),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma (J/\\psi \\rightarrow e^+ e^-),$$\\end{document} and the amplitude of Z→J/ψ+γ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z\\rightarrow J/\\psi + \\gamma ^{*}$$\\end{document} is calculate through the light-cone approach. The higher-order QCD and relativistic corrections in the amplitude of γ∗→J/ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{*}\\rightarrow J/\\psi $$\\end{document} and the large logarithms of mZ2/mc2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{_Z}^2/m_c^2$$\\end{document} that appear in the amplitude of Z→J/ψ+γ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Z\\rightarrow J/\\psi + \\gamma ^{*}$$\\end{document} are resummed in our calculation. Besides, the non-fragmentation amplitude is calculated based on the NRQCD factorization, and the next-to-leading order QCD and relativistic corrections are included. The obtained branching fraction for this Z decay channel is 8.66-0.69+1.48×10-11.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$8.66 ^{+1.48} _{-0.69}\ imes 10^{-11}.$$\\end{document}

Relativistic corrections to ψ(nS)→ρπ exclusive decays and their role in the understanding of the ρπ -puzzle

We study relativistic corrections to exclusive $S$-wave charmonium decays into $\rho\pi$ and $\gamma\pi$ final states. The contribution of relative order $v^2$ and the set of associated higher order corrections are calculated using NRQCD and collinear factorisation framework. Numerical estimates show that the dominant effect is provided by the corrections of relative order $v^2$. The numerical values of these contributions are of the same order as the leading-order ones. These results suggest a scenario where the sum of relativistic and radiative QCD corrections could explain the $\rho\pi$-puzzle.

The -order Foldy–Wouthuysen Hamiltonian and relativistic corrections to Coulomb systems

The order nonrelativistic Hamiltonians of scalar, spinor and vector particles in the electromagnetic field are studied by applying the Douglas–Kroll–Hess approach. The result of the spin-1/2 spinor agrees with the Hamiltonian of the nonrelativistic quantum electrodynamics (Zhou et al 2019 Phys. Rev. A 100 012513). Additionally, the -order relativistic correction to few-electron Coulomb systems and the -order relativistic correction to two-electron Coulomb systems are derived. The singularities of Hamiltonians of Coulomb systems are separated out and cancelled. The regularized Hamiltonians up to -order for scalar and electron in the Coulomb field are obtained. Their numerical results agree with the relativistic theory. The regularized Hamiltonian, up to for few-electrons in the Coulomb field, is also derived.

High order relativistic corrections on the electric field gradient within the LRESC formalism.

In this work, we present relativistic corrections to the electric field gradient (EFG) given by the Linear Response Elimination of the Small Component (LRESC) scheme at 1/c2 order and including for the first time spin-dependent (SD) corrections at 1/c4 order. We show that these new terms improve the performance of LRESC as results with this methodology are very close to those calculated at the four-component Dirac-Hartree-Fock (4c-DHF) level. We assess the new corrections in BrY and AtY di-halogen (Y = F, Cl, Br, I, and At) and XZY bi-linear molecules (Z = Zn, Cd, and Hg; X, Y = F, Cl, Br, I, and At). At the 4c-DHF level, we analyze the contributions coming from the large and small components of the relativistic 4c wave function to the electronic part of EFG and compare them with the LRESC corrections to find their electronic origin. For the HgX2 (X = Cl, Br, and I) subset, when the SDcorrecting terms are included, LRESC calculations match very well with 4c-DHF ones and those from the literature, with differences less than 1% for molecules containing up to three heavy atoms. We show that LRESC gives accurate values of EFG, allowing the analysis of the electronic origin of relativistic effects in terms of well-known nonrelativistic operators.

Higher-order corrections to spin-orbit and spin-spin tensor interactions in hydrogen molecular ions: Theory and application to H2+

We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spin-spin tensor interactions in hydrogen molecular ions. The $m\alpha^7 \ln(\alpha)$-order radiative correction is derived in the NRQED framework. We present complete numerical calculations, including as well the $m\alpha^6$-order relativistic correction, for the case of H$_2^+$. The theoretical uncertainty is reduced by more than one order of magnitude with respect to the Breit-Pauli level, down to a few ppm. We also compare our results with available rf spectroscopy data.

A novel mechanism for probing the Planck scale

The Planck or the quantum gravity (QG) scale, being 16 orders of magnitude greater than the electroweak scale, is often considered inaccessible by current experimental techniques. However, it was shown recently by one of the current authors that QG effects via the generalized uncertainty principle affects the time required for free wavepackets to double their size, and this difference in time is at or near current experimental accuracies (Villalpando C and Modak S K 2019 Class. Quantum Grav. 36 215016; Villalpando C and Modak S K 2019 Phys. Rev. D 100 024054). In this work, we make an important improvement over the earlier study, by taking into account the leading order relativistic correction, which naturally appears in the systems under consideration, due to the significant mean velocity of the travelling wavepackets. Our analysis shows that although the relativistic correction adds nontrivial modifications to the results of (Villalpando and Modak 2019 Class. Quantum Grav. 36 215016; Villalpando and Modak 2019 Phys. Rev. D 100 024054), the earlier claims remain intact and are in fact strengthened. We explore the potential for these results being tested in the laboratory.

Leading-order relativistic corrections to the g factor of H2+

Relativistic corrections of order ${\ensuremath{\alpha}}^{2}$ to the $g$ factor of ${\mathrm{H}}_{2}^{+}$ are calculated with a high accuracy of nine significant digits for a wide range of rovibrational states. The precision of previous calculations [R. A. Hegstrom, Phys. Rev. A 19, 17 (1979) is improved by about five orders of magnitude by performing nonadiabatic variational calculations and by including recoil corrections. These results allow for nondestructive identification of the internal state through the measurement of spin-flip transition frequencies, which is a crucial requirement for proposed spectroscopy experiments on ${\mathrm{H}}_{2}^{+}$ and its antimatter counterpart ${\overline{\mathrm{H}}}_{2}{}^{\ensuremath{-}}$ in Penning traps [E. G. Myers, Phys. Rev. A 98, 010101(R) (2018)]. Further, they pave the way towards precision calculations of the $g$ factor through the calculation of higher-order QED corrections and hence to an alternative precision route to obtaining the proton-electron mass ratio.

Bc → J/ψ tensor form factors at large momentum recoil

We present a next-to-leading order (NLO) relativistic correction to [Formula: see text] tensor form factors within nonrelativistic QCD (NRQCD). We also consider complete Dirac bilinears [Formula: see text] with [Formula: see text] matrices [Formula: see text] in the [Formula: see text] transition. The relation among different current form factors is given and it shows that symmetries emerge in the heavy bottom quark limit. For a phenomenological extension, we propose to extract the long-distance matrix elements (LDMEs) for [Formula: see text] meson from the recent HPQCD lattice data and the NLO form factors at large momentum recoil.

Lagrange point stability for a rotating host mass binary

In this new era of gravitational wave astrophysics, observations indicate the likely existence of black holes with significant spin. In order to better understand the potential imprint orbital dynamics have on the multimessenger data, we include rotation of the primary mass to leading order in the analysis of the stability boundary pertaining to the triangular equilibrium points, ${L}_{4}$ and ${L}_{5}$, in the relativistic, restricted, circular, three-body problem. For Lagrange point stability, these rotation effects are of the same order as the leading order relativistic corrections ignoring rotation, and they make both ${L}_{4}$ and ${L}_{5}$ more stable for retrograde orbital motion.

Merging of spinning binary black holes in globular clusters

AbstractBased on our current high resolution direct N-body modelling of the Milky Way typical Star Cluster systems dynamical evolution we try to numerically estimate the influence of individual spin values and orientations on gravitational wave (GW) waveforms and observed time-frequency maps during multiple cycles for binary black hole (BBH) mergers. In our up to date N-body dynamical simulations we use the high order relativistic post-Newtonian corrections for the BH binary particles (3.5 post-Newtonian (PN) terms including spin-spin and spin-orbit terms). In the current work, we present the GW waveforms catalogue which covers the large parameter space in mass ratios 0.05 - 0.82 and extreme possible individual spin cases.

Energies and emission strengths of Kα X-rays from iodine and gold with multiply ionized L shell

The energies and rates of Kα X-ray satellites due to 1s − 2p transitions in iodine and gold with one to multiple vacancies in the L shell are computed using Multi-Configuration Dirac Fock wavefunctions with the inclusion of relativistic and quantum electrodynamics corrections. The calculations have been carried out for I1+ to I6+ with spectator vacancies in the 2p shell and Au1+ to Au8+ with filled and empty 2s shell along with vacancies in the 2p shell. As existing data on the X-ray emission from low ionized iodine and gold are confined to single inner shell vacancy, direct comparison of the present results on the various radiative transitions from iodine and gold with a single vacancy in the K shell, variously ionized L shell and no vacancy in any other shells is possible only for the two well known Kα 1 and Kα 2 diagram lines. Hence to substantiate the reliability of the current results through one to one comparison with existing data on the 1s − 2p transitions, calculations have also been performed on the radiative decay of H-, He-, Be-, and F-like gold with singly ionized 1s , closed 2s , multiply ionized 2p and rest of the shells open for which semi-relativistic data exist. In addition to providing additional check on the reliability of the computed results, this comparison also tries to analyze the impact of higher order relativistic corrections on the K X-ray energies and rates. The intrinsic variations of transition properties with degree of ionization have been analyzed.

Relativistic SZ temperature scaling relations of groups and clusters derived from the BAHAMAS and MACSIS simulations

ABSTRACT The Sunyaev–Zeldovich (SZ) effect has long been recognized as a powerful cosmological probe. Using the BAHAMAS and MACSIS simulations to obtain ${\gt }10\, 000$ simulated galaxy groups and clusters, we compute three temperature measures and quantify the differences between them. The first measure is related to the X-ray emission of the cluster, while the second describes the non-relativistic thermal SZ (tSZ) effect. The third measure determines the lowest order relativistic correction to the tSZ signal, which is seeing increased observational relevance. Our procedure allows us to accurately model the relativistic SZ (rSZ) contribution and we show that a ${\gtrsim}10\!-\!40{{\ \rm per\ cent}}$ underestimation of this rSZ cluster temperature is expected when applying standard X-ray relations. The correction also exhibits significant mass and redshift evolution, as we demonstrate here. We present the mass dependence of each temperature measure alongside their profiles and a short analysis of the temperature dispersion as derived from the aforementioned simulations. We also discuss a new relation connecting the temperature and Compton-y parameter, which can be directly used for rSZ modelling. Simple fits to the obtained scaling relations and profiles are provided. These should be useful for future studies of the rSZ effect and its relevance to cluster cosmology.

Decay properties of singly charmed baryons

The magnetic moments, transition magnetic moments and the radiative decay widths of singly charmed baryons are calculated with $ {J^P = \frac{1}{2}}^+$ and $ {J^P = \frac{3}{2}}^+$ in the constitute quark model. Further, the strong decay rates for $ S $, $ P $ and $ D $ wave transitions are also presented. The singly charmed baryon masses used in the calculations were obtained from the hypercentral Constitute Quark Model (hCQM) without and with first order relativistic correction. Obtained results are compared with experimental observation as well as with the other theoretical predictions.

Relativistic corrections to the form factors of Bc into P-wave orbitally excited charmonium

We investigated the form factors of the Bc meson into P-wave orbitally excited charmonium using the nonrelativistic QCD effective theory. Through the analytic computation, the next-to-leading order relativistic corrections to the form factors were obtained, and the asymptotic expressions were studied in the infinite bottom quark mass limit. Employing the general form factors, we discussed the exclusive decays of the Bc meson into P-wave orbitally excited charmonium and a light meson. We found that the relativistic corrections lead to a large correction for the form factors, which makes the branching ratios of the decay channels B(Bc±→χcJ(hc)+π±(K±)) larger. These results are useful for the phenomenological analysis of the Bc meson decays into P-wave charmonium, which shall be tested in the LHCb experiments.

Relativistic corrections to the form factors of Bc into S -wave charmonium

We investigate the form factors of $B_c$ meson into $S$-wave charmonium within the nonrelativistic QCD effective theory and obtain the next-to-leading order relativistic corrections to the form factors, where both the $B_c$ meson and the charmonium are treated as the nonrelativistic bound states. Treating the charm quark as a light quark in the limit \ $m_c/m_b\to 0$, some form factors are identical at the maximum recoil point, which are consistent with the predictions in the heavy-quark effective theory and the large-energy effective theory. Considering that the branching ratios of $B^+_c \to J/\psi D_s^+$ and $B^+_c \to J/\psi D_s^{*+}$ have been measured by the LHCb and ATLAS Collaborations recently, we employ the form factors of $B_c$ meson into $S$-wave charmonium at the next-to-leading order accuracy to these two decay channels and obtain more precise predictions of their decay rates. Numerical results indicate that the factorizable diagrams dominate the contribution in these two channels, while the colour-suppressed and the annihilation diagrams contribute less than 10 percent. Our results are consistent with the LHCb and ATLAS data.

One hundred years of Moseley's law: An undergraduate experiment with relativistic effects

As the centenary of H. G. H. Moseley's untimely passing approached, the authors undertook an evaluation of Moseley's work as the basis for an undergraduate laboratory experiment. Using a modern solid-state detector and low activity sources, characteristic K X-rays were excited and detected without the use of a vacuum chamber, from a selection of elemental metal foil samples from Ti to Er. The authors introduce first order relativistic corrections into the experimental analysis to obtain better fits for K X-ray energies that deviate substantially from Moseley's original law for elements beyond the first row transition metals. The experimental set up is described in detail, and the data and conclusions are readily reproducible by undergraduate students in the span of a couple of typical lab sessions.

Revisiting radiative decays of 1^{+-} heavy quarkonia in the covariant light-front approach

We revisit the calculation of the width for the radiative decay of a 1^{+-} heavy Q bar{Q} meson via the channel 1^{+-} rightarrow 0^{-+} +gamma in the covariant light-front quark model. We carry out the reduction of the light-front amplitude in the non-relativistic limit, explicitly computing the leading and next-to-leading order relativistic corrections. This shows the consistency of the light-front approach with the non-relativistic formula for this electric dipole transition. Furthermore, the theoretical uncertainty in the predicted width is studied as a function of the inputs for the heavy-quark mass and wave function structure parameter. We analyze the specific decays h_{mathrm{c}}(1P) rightarrow eta _{mathrm{c}}(1S) + gamma and h_{mathrm{b}}(1P) rightarrow eta _{mathrm{b}}(1S) + gamma . We compare our results with experimental data and with other theoretical predictions from calculations based on non-relativistic models and their extensions to include relativistic effects, finding reasonable agreement.

6Pe—6Po transitions in boron-like ions with Z = 8–13

The energies, fine structure splittings, transition rates, and lifetimes of inner-shell excited sextet states 1s2s2p2nl, (n = 2–7; l = s, p, d) and 1s2p33p of the boron isoelectronic sequence (Z = 8–13) are investigated using the multi-configuration Rayleigh–Ritz variation method. The mass polarization effect and relativistic corrections are included by first-order perturbation theory. Configuration structures of the high-n inner-shell excited sextet series 6Se,o(m) and 6Pe,o(m) (m = 1–5) of boron-like Na6+ ion are assigned. The wavelengths and transition rates of electric-dipole transitions between 6Pe(m) and 6Po(m) (m = 1–5) states are calculated. The quantum electrodynamics (QED) effects and higher order relativistic corrections are also considered to obtain more accurate transition wavelengths. The predicted transition wavelengths agree well with the available theoretical and experimental data. The lifetimes for the inner-shell excited sextet states 6Pe(m) (m = 1–5) are also reported and discussed with the increase of nuclear charge number, Z. These theoretical data are useful for the identification of spectral lines in experiments and the design of XUV and soft X-ray lasers.

Precision studies of the hydrogen molecular ion and its isotopologues

We present systematic calculations of the leading order relativistic corrections including the hyperfine splitting (HFS) for a wide range of rotational and vibrational states of the HT+ molecular ion. We also calculate the DC and AC Stark effects for the molecular hydrogen ions \(H_{2}^{+}\) and HD+ in the non-relativistic approximation. The influence of the DC Stark polarizability effect on the hyperfine substates of a ro-vibrational state is carefully analyzed. Our results enable the detailed evaluation of certain systematic shifts of the transition frequencies for the purpose of ultra-high-precision optical, microwave, or radio-frequency spectroscopy of the hydrogen molecular ions in a trap.

Bivelocity Picture in the Nonrelativistic Limit of Relativistic Hydrodynamics

We discuss the nonrelativistic limit of the relativistic Navier-Fourier-Stokes (NFS) theory. The next-to-leading order relativistic corrections to the NFS theory for the Landau-Lifshitz fluid are obtained. While the lowest order truncation of the velocity expansion leads to the usual NFS equations of nonrelativistic fluids, we show that when the next-to-leading order relativistic corrections are included, the equations can be expressed concurrently with two different fluid velocities. One of the fluid velocities is parallel to the conserved charge current (which follows the Eckart definition) and the other one is parallel to the energy current (which follows the Landau-Lifshitz definition). We compare this next-to-leading order relativistic hydrodynamics with bivelocity hydrodynamics, which is one of the generalizations of the NFS theory and is formulated in such a way to include the usual mass velocity and also a new velocity, called the volume velocity. We find that the volume velocity can be identified with the velocity obtained in the Landau-Lifshitz definition. Then, the structure of bivelocity hydrodynamics, which is derived using various nontrivial assumptions, is reproduced in the NFS theory including the next-to-leading order relativistic corrections.