Nonrelativistic Formalism Research Articles

Schrödinger formulation of the nondipole light-matter interaction consistent with relativity

An alternative and powerful Schr\odinger-like equation for describing beyond dipole laser-matter interactions and leading relativistic corrections is derived. It is shown that this particular formulation is numerically very efficient with respect to computational effort and convergence rate of the solutions. Furthermore, its nonrelativistic form turns out to be more compatible with relativity than what seems to be the case with the more common formulations of the nonrelativistic light-matter interaction. Moreover, the extension of this interaction form into the relativistic region preserves, to a large extent, the numerical efficiency. In this work, the formulation is applied to study beyond dipole corrections and relativistic corrections in multiphoton ionization of a hydrogen atom in the x-ray regime.

Preliminary Empirical study of the Rayleigh to Compton scattering ratio for elements at gamma energy 59.5 keV using Si(Li) detector

The characteristic parameters of a material relation to photon interactions such as: mass attenuation coefficient, effective atomic number, effective electron density are required to provide essential data in diverse works such as nuclear diagnostic and cancer radiotherapy, industrial irradiation, radiation dosimetry, radiation protection and shielding, analyzing of the concentration of elements and radioactive isotopes. In this paper, the theoretical models such as non-relativistic form factor (NRFF), relativistic form factor (RFF), and modified form factor (MFF) were used to calculate the ratio Rayleigh-Compton for elements with at gamma energy 59.5 keV. The results showed that there was a discrepancy between the theoretical modes at a high atomic number. The mean value of the Rayleigh-Compton ratio depends on the atomic number, which shows the quadratic function of the correlation coefficient R2 = 0.996 as well. Besides, the experimental system was set-up and measured for some targets such as aluminum, copper, and lead at a scattering angle 150o using 241Am source by Si(Li) detector to confirm the theoretical values. The preliminary result showed that there was a good agreement with experimental and theoretical results is lower than 20%. Further investigation will be measured by the samples for more detailed evaluation.

Bottomonia production and polarization in the NRQCD with k_T-factorization. II: Upsilon (2S) and chi _b(2P) mesons

The Upsilon (2S) production and polarization at high energies is studied in the framework of k_T-factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The direct production mechanism, feed-down contributions from radiative chi _b(3P) and chi _b(2P) decays and contributions from Upsilon (3S) decays are taken into account. The transverse momentum dependent gluon densities in a proton were derived from the Ciafaloni–Catani–Fiorani–Marchesini evolution equation, Kimber-Martin-Ryskin prescription and Parton Branching method. Treating the non-perturbative color octet transitions in terms of the mulitpole radiation theory, we extract the corresponding non-perturbative matrix elements for Upsilon (2S) and chi _b(2P) mesons from a combined fit to Upsilon (2S) transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies sqrt{s} = 7 and 13 TeV and from the relative production rate R^{chi _b(2P)}_{Upsilon (2S)} measured by the LHCb Collaboration at sqrt{s} = 7 and 8 TeV. Then we apply the extracted values to investigate the polarization parameters lambda _theta , lambda _phi and lambda _{theta phi }, which determine the Upsilon (2S) spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.

Elastic Photon Scattering on Hydrogenic Atoms near Resonances

Scattering of light on relativistic heavy ion beams is widely used for characterizing and tuning the properties of both the light and the ion beam. Its elastic component—Rayleigh scattering—is investigated in this work for photon energies close to certain electronic transitions because of its potential usage in the Gamma Factory initiative at CERN. The angle-differential cross-section, as well as the degree of polarization of the scattered light are investigated for the cases of 1 s − 2 p 1 / 2 and 1 s − 2 p 3 / 2 resonance transitions in H-like lead ions. In order to gauge the validity and uncertainty of frequently used approximations, we compare different methods. In particular, rigorous quantum electrodynamics calculations are compared with the resonant electric-dipole approximation evaluated within the relativistic and nonrelativistic formalisms. For better understanding of the origin of the approximation, the commonly used theoretical approach is explained here in detail. We find that in most cases, the nonrelativistic resonant electric-dipole approximation fails to describe the properties of the scattered light. At the same time, its relativistic variant agrees with the rigorous treatment within a level of 10% to 20%. These findings are essential for the design of an experimental setup exploiting the scattering process, as well as for the determination of the scattered light properties.

Are there any challenges in the charmonia production and polarization at the LHC?

We analyze the inclusive prompt production of $\psi^\prime$, $\chi_c$, $J/\psi$ and $\eta_c$ mesons at the LHC using the $k_T$-factorization approach. Our consideration is based on the off-shell production amplitudes for hard partonic subprocesses, nonrelativistic QCD formalism for the formation of bound states and transverse momentum dependent (or unintegrated) gluon densities in a proton derived from Catani-Ciafaloni-Fiorani-Marchesini equation. The nonperturbative color octet transitions are treated in terms of the multipole radiation theory. We extract the corresponding long-distance matrix elements from a combined fit to transverse momentum distributions measured at various LHC experiments. We make predictions for the polarization parameters $\lambda_\theta$, $\lambda_\phi$, $\lambda_{\theta \phi}$ and the frame-independent parameter $\lambda^*$ and compare them to the available $\psi^\prime$ and $J/\psi$ data. Finally, we present a universal set of parameters that provides a reasonable simultaneous description for the whole body of the LHC data (on the $p_T$ distributions, relative production rates and polarization observables) for the whole charmonium family.

The coherent to Compton scattering differential cross section ratios of some fifth period elements in the external magnetic field

Some fifth-period elements Y, Zr, Nb, Mo, Ru, Rh, Pd, Ag and Cd have been obtained in powder form and converted into pellets by applying pressure (~ 1 GPa). They have annealed at 1000 °C (except Cd ~ 300 °C). In the external magnetic fields (0.4 and 0.8 T) and without magnetic field, the coherent to Compton differential cross-sectional ratios has been investigated by using energy-dispersive X-ray fluorescence system. The data were analyzed via Origin 9.0 computer program. The results were calculated experimentally and theoretically. The theoretical results were obtained with two different methods, and these results were compared with the experimental results. These methods are relativistic form factor theory and non-relativistic form factor theory at without magnetic field. It was seen that the coherent to Compton differential cross-sectional ratios of the elements used in the present study increases with the increasing of the magnetic field and atomic number.

Bottomonia production and polarization in the NRQCD with k_T-factorization. I: Upsilon (3S) and chi _b(3P) mesons

The Upsilon (3S) production and polarization at high energies is studied in the framework of k_T-factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The transverse momentum dependent (TMD, or unintegrated) gluon densities in a proton were derived from the Ciafaloni-Catani-Fiorani-Marchesini (CCFM) evolution equation as well as from the Kimber–Martin–Ryskin (KMR) prescription. Treating the non-perturbative color octet transitions in terms of the multipole radiation theory and taking into account feed-down contributions from radiative chi _b(3P) decays, we extract the corresponding non-perturbative matrix elements for Upsilon (3S) and chi _b(3P) mesons from a combined fit to Upsilon (3S) transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies sqrt{s} = 7 and 13 TeV and central rapidities. Then we apply the extracted values to describe the CDF and LHCb data on Upsilon (3S) production and to investigate the polarization parameters lambda _theta , lambda _phi and lambda _{theta phi }, which determine the Upsilon (3S) spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.

Measurement of coherent to Compton scattering differential cross-section ratios of cadmium at different momentum transfer factors

The coherent to Compton scattering ratio is a useful technique because it is independent of the density of the material and the material is defined only by the atomic number. Also, this ratio is used where changes in the atomic number are important and changes in the attenuation coefficient are too small to detect. In this study, coherent to Compton scattering ratios of cadmium have been calculated at different scattering angles (90°, 95°, 100°, 105°, 110°, 115°, 120°, 125°, and 130°). A high purity germanium detector was used. The coherent to Compton scattering differential cross-section ratios were plotted as a function of the scattering angles and constituted a best fit curve. It was observed that the coherent to Compton differential cross-section ratios decrease with an increasing scattering angle. The experimental results agree well with the results of the non-relativistic and relativistic form factor method.

ϒ(3S) and χb(3P) production and polarization in the NRQCD with kT–factorization

The ϒ(3S) production and polarization at high energies is studied in the framework of kT–factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The transverse momentum dependent (TMD, or unintegrated) gluon densities in a proton were derived from the CiafaloniCatani-Fiorani-Marchesini (CCFM) evolution equation as well as from the Kimber–Martin–Ryskin (KMR) prescription. Treating the nonperturbative color octet transitions in terms of the mulitpole radiation theory and taking into account feed-down contributions from radiative χb(3P) decays, we extract the corresponding non-perturbative matrix elements for ϒ(3S) and χb(3P) mesons from a combined fit to ϒ(3S) transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies √s = 7 and 13 TeV and central rapidities. Then we apply the extracted values to investigate the polarization parameters λθ, λφ and λθφ, which determine the ϒ(3S) spindensity matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.

Estimates of the Bc wave function from the CDF and LHCb production data

In the framework of perturbative QCD and nonrelativistic bound state formalism, we calculate the production of Bc and B*c mesons at the conditions of the CDF andLHCb experiments. We derive first estimations for the Bc wave function from a comparison with the available data.

First estimates of the Bc wave function from the data on the Bc production cross section

In the framework of perturbative QCD and nonrelativistic bound state formalism, we calculate the production of Bc and Bc⁎ mesons at the conditions of the CDF and LHCb experiments. We derive first estimations for the Bc wave function from a comparison with the available data.

An experimental study on the angular dependence of coherent to Compton scattering differential cross-section ratios of calcium

The coherent to Compton scattering cross-section ratio of calcium was measured experimentally for several polar scattering angles (90°, 100°, 110°, 120°, and 130°) and azimuthal scattering angles (30°, 20°, 10°, 0°, -10°, and -20°). A high purity germanium semiconductor detector with a resolution of 199.6 eV at 5.9 keV was used. The theoretical values of the coherent to Compton scattering differential cross-section of calcium were calculated on the basis of nonrelativistic form factors, relativistic form factors, and S-matrix theories. It was observed that the experimental results were most compatible with the nonrelativistic form factor theory.

A mathematical solution for the parameters of three interfering resonances**Supported by National Natural Science Foundation of China (NSFC) (11575017, 11761141009), the Ministry of Science and Technology of China (2015CB856701) and the CAS Center for Excellence in Particle Physics (CCEPP)

The multiple-solution problem in determining the parameters of three interfering resonances from a fit to an experimentally measured distribution is considered from a mathematical viewpoint. It is shown that there are four numerical solutions for a fit with three coherent Breit-Wigner functions. Although explicit analytical formulae cannot be derived in this case, we provide some constraint equations between the four solutions. For the cases of nonrelativistic and relativistic Breit-Wigner forms of amplitude functions, a numerical method is provided to derive the other solutions from that already obtained, based on the obtained constraint equations. In real experimental measurements with more complicated amplitude forms similar to Breit-Wigner functions, the same method can be deduced and performed to get numerical solutions. The good agreement between the solutions found using this mathematical method and those directly from the fit verifies the correctness of the constraint equations and mathematical methodology used.

Light–matter interaction in the long-wavelength limit: no ground-state without dipole self-energy

Most theoretical studies for correlated light–matter systems are performed within the long-wavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light–matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light–matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light–matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called ‘depolarization shift’ is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the standard approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground-states. This is very important for practical calculations and for density-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light–matter ground-state allows one to calculate the Stark shift non-perturbatively.

Decay properties of charm and bottom mesons in a quantum isotonic nonlinear oscillator potential model

Employing generalized quantum isotonic oscillator potential we determine wave function for mesonic system in nonrelativistic formalism. Then we investigate branching ratios of leptonic decays for heavy-light mesons including a charm quark. Next, by applying the Isgur-Wise function we obtain branching ratios of semileptonic decays for mesons including a bottom quark. The weak decay of the $ B_{c}$ meson is also analyzed to study the life time. Comparison with other available theoretical approaches is presented.

Prompt charmonia production and polarization at the LHC in the NRQCD with kT -factorization. III. J/ψ meson

In the framework of $k_T$-factorization approach, the production and polarization of prompt $J/\psi$ mesons at the LHC energies is studied. Our consideration is based on the non-relativistic QCD formalism for bound states and off-shell amplitudes for hard partonic subprocesses. Both the direct production mechanism and feed-down contributions from $\chi_c$ and $\psi(2S)$ decays are taken into account. The transverse momentum dependent (or unintegrated) gluon densities in a proton were derived from Ciafaloni-Catani-Fiorani-Marchesini evolution equation or, alternatively, were chosen in accordance with Kimber-Martin-Ryskin prescription. The non-perturbative color-octet matrix elements were first deduced from the fits to the latest CMS data on $J/\psi$ transverse momentum distributions and then applied to describe the ATLAS and LHCb data on $J/\psi$ production and polarization at $\sqrt s = 7$, $8$ and $13$ TeV. We perform an estimation of polarization parameters $\lambda_\theta$, $\lambda_\phi$ and $\lambda_{\theta \phi}$ which determine $J/\psi$ spin density matrix and demonstrate that treating the soft gluon emission as a series of explicit color-electric dipole transitions within NRQCD leads to unpolarized $J/\psi$ production at high transverse momenta, that is in qualitative agreement with the LHC data.

The (p,2p) reaction in finite range relativistic distorted-wave impulse approximation

The [Formula: see text] reaction on [Formula: see text] at incident proton energy of 300[Formula: see text]MeV is examined in the formalism of finite-range relativistic distorted-wave impulse approximation (FR-RDWIA). In comparison to conventional t-matrix model of Love–Franey, a new form of nucleon–nucleon t-matrix effective interaction is derived at 300[Formula: see text]MeV using Reid soft core potentials for isotopic spin one and taking into account the finite-range effects in the [Formula: see text] interaction at knockout vertex. In comparison to the conventional finite range nonrelativistic and relativistic formalism, the present formalism with a new version of [Formula: see text] t-matrix is effectively reproducing the shape of cross-section energy distributions for [Formula: see text], [Formula: see text] and [Formula: see text] states for asymmetric angle pair of [Formula: see text]–[Formula: see text]. Discrepancies between the experimental cross-section data and finite range theoretical calculations at [Formula: see text][Formula: see text]MeV are reasonably resolved in the present approach. Without any adjustable parameter of bound state, the obtained spectroscopic factors are in reasonably good agreement with the relativistic and nonrelativistic theoretical predictions by [Formula: see text], [Formula: see text] and [Formula: see text] analysis.

Space-time-symmetric extension of nonrelativistic quantum mechanics

In quantum theory we refer to the probability of finding a particle between positions $x$ and $x+dx$ at the instant $t$, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between $x$ and $x+dx$ during [$t$,$t+dt$]. Then, we find a Schr\"odinger-like equation for a "mirror" wave function $\phi(t,x)$ associated with the probability of measuring the system between $t$ and $t+dt$, given that detection occurs at $x$. In this framework, it is shown that energy measurements of a stationary state display a non-zero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to {\it ad hoc} assumptions, is a natural, built-in part of the formalism presented here.

Charmonium Mass Spectrum with Spin-Dependent Interaction in Momentum-Helicity Space

In this paper we have solved the nonrelativistic form of the Lippmann-Schwinger equation in the momentum-helicity space by inserting a spin-dependent quark-antiquark potential model numerically. To this end, we have used the momentum-helicity basis states for describing a nonrelativistic reduction of one-gluon exchange potential. Then we have calculated the mass spectrum of the charmonium ψcc¯, and finally we have compared the results with the other theoretical results and experimental data.

NORSE: A solver for the relativistic non-linear Fokker–Planck equation for electrons in a homogeneous plasma

Energetic electrons are of interest in many types of plasmas, however previous modeling of their properties has been restricted to the use of linear Fokker–Planck collision operators or non-relativistic formulations. Here, we describe a fully non-linear kinetic-equation solver, capable of handling large electric-field strengths (compared to the Dreicer field) and relativistic temperatures. This tool allows modeling of the momentum-space dynamics of the electrons in cases where strong departures from Maxwellian distributions may arise. As an example, we consider electron runaway in magnetic-confinement fusion plasmas and describe a transition to electron slide-away at field strengths significantly lower than previously predicted. Program summaryProgram title: NORSEProgram Files doi:http://dx.doi.org/10.17632/86wmgj758w.1Licensing provisions: GPLv3Programming language: MatlabNature of problem: Solves the Fokker–Planck equation for electrons in 2D momentum space in a homogeneous plasma (allowing for magnetization), using a relativistic non-linear electron–electron collision operator. Electric-field acceleration, synchrotron-radiation-reaction losses, as well as heat and particle sources are included. Scenarios with time-dependent plasma parameters can be studied.Solution method: The kinetic equation is represented on a non-uniform 2D finite-difference grid and is evolved using a linearly implicit time-advancement scheme. A mixed finite-difference-Legendre-mode representation is used to obtain the relativistic potentials (analogous to the non-relativistic Rosenbluth potentials) from the distribution.