Glauber theory for nucleus-nucleus scattering at high incident energies is reformulated so as to become applicable also for the scattering at intermediate energies. We test validity of the eikonal and adiabatic approximations used in the formulation, and discuss the relation between the present theory and the conventional Glauber calculations with either the empirical nucleon-nucleon profile function or the modified one including the in-medium effect.
Calculation of elastic p8Li- and p9Li-scattering differential cross sections, performed at two energies 0.07 and 0.7 GeV/nucleon within Glauber multiple diffraction scattering, are presented and discussed. Three-body wave functions: alpha-t-n (for 8Li) and 7Li-n-n (for 9Li) with realistic potentials of intercluster interactions were used there. Sensitivity of elastic scattering to proton-nucleus interaction and nuclear structure has been studied. In particular, dependence of differential cross section on contribution of higher-order collisions, scattering at core and at periphery nucleons, on …
The series of calculations of differential cross sections and analyzing powers (Ay) of the elastic proton scattering on isotopes 6He and 8He was presented in the framework of the theory of Glauber multiple diffraction scattering. The three-body α-n-n and harmonic oscillator wave functions were used for 6He, for 8He - the function of density distribution in LSSM. For p6He scattering the expansion of the Glauber operator into a series of multiple scattering is written in the form consistent with the picture of weakly bound clusters in halo-nuclei, for p8He scattering the members of single- and d…
Glauber-theory calculations of high-energy nuclear scattering observables using variational Monte Carlo wave functions
Osaka Metropolitan University（大阪公立大学）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南部阳一郎理论与实验物理研究所）；RIKEN Nishina Center（理化学研究所西田中心）；Niigata University（新潟大学）；Argonne National Laboratory（阿贡国家实验室）
A new approach to deal with the scattering amplitudes in Glauber theory is proposed. It relies on the use of generating function, that has been explicitly found. The method is applied to the analytical calculation of the nucleus-nucleus elastic scattering amplitudes in the all interaction orders of the Glauber theory.
We compare two formalism that describe minijet production in pA and AA collisions: pQCD supplemented by Glauber-Gribov multiple semihard parton scatterings (pQCD+Glauber), and the Colour Glass Condensate (CGC). We argue that in a suitable limit they are equivalent to each other, the PQCD+Glauber model being more accurate from a numerical point of view. Finally, we analyze RHIC data on Au-Au integrated charged multiplicities in the pQCD+Glauber framework, and conclude that at least at central rapidity there is no sign of gluon saturation.
Obtaining cross sections for nuclear reactions at intermediate energies based on the Glauber formulation has a long tradition. Only recently the energy regime of a few hundred MeV has become accessible to ab-initio Faddeev calculations of three-body scattering. In order to go to higher energies, the Faddeev equation for three-body scattering is formulated and directly solved without employing a partial wave decomposition. In the simplest form the Faddeev equation for interacting scalar particles is a three-dimensional integral equation in five variables, from which the total cross section, the…
A high-precision test of the Glauber formula for the amplitude of nucleon-deuteron scattering is performed. Nucleon-nucleon amplitudes used in the calculations depend on the spins of interacting particles, phase shifts, and mixing parameters. These amplitudes were derived by using the Nijmegen potentials. The differential cross sections for nucleon-deuteron scattering were calculated for the projectile-nucleon energies of 65, 95, 135, 150, 190, and 250~MeV, and the results of these calculations were compared with experimental data.
The differential elastic cross sections of $^{12}$C -- $^{12}$C and $^{11}$Li -- $^{12}$C nuclei are calculated in the complete Glauber theory. The role of the possible correlations connected to the shell effects in $^{11}$Li nucleus is considered.
A general formalism of the Glauber theory for elementary atom (EA) - target atom (TA) scattering is developed. A second-order approximation of its complete version is considered in the framework of the optical-model perturbative approach. A `potential' approximation of a second-order optical model is formulated neglecting the excitation effects of the TA. Its accuracy is evaluated within the second-order approximation for the complete version of the Glauber EA-TA scattering theory.
We revisit the Glauber model to study the heavy ion reaction cross sections and elastic scattering angular distributions at low and intermediate energies. The Glauber model takes nucleon-nucleon cross sections and nuclear densities as inputs and has no free parameter and thus can predict the cross sections for unknown systems. The Glauber model works at low energies down to Coulomb barrier with very simple modifications. We present new parametrization of measured total cross sections as well as ratio of real to imaginary parts of the scattering amplitudes for pp and np collisions as a function…
At sufficiently large proton energies, Glauber multiple-scattering theory offers good opportunities for describing the final state interactions in electro-induced proton emission off nuclear targets. A fully unfactorized relativistic formulation of Glauber multiple-scattering theory is presented. The effect of truncating the Glauber multiple-scattering series is discussed. Relativistic effects in the description of the final-state interactions are found not to exceed the few percent level. Also the frequently adopted approximation of replacing the wave functions for the individual scattering n…
We develop the Glauber theory description of final-state interaction (FSI) in quasielastic $A(e,e'p)$ scattering. The important new effect is an interaction between the two trajectories which enter the calculation of FSI-distorted one-body density matrix and are connected with incoherent elastic rescatterings of the struck proton on spectator nucleons. We demonstrate that FSI distortions of the missing momentum distribution are large over the whole range of missing momenta. Important finding is that incoherent elastic rescatterings of the ejected proton lead to a strong quantum mechanical dist…
We develop the Glauber theory description of initial- and final-state interactions (IFSI) in quasielastic A(p,2p) scattering. We study the IFSI-distortion effects both for the inclusive and exclusive conditions. In inclusive reaction the important new effect is an interaction between the two sets of the trajectories which enter the calculation of IFSI-distorted one-body density matrix for inclusive (p,2p) scattering and are connected with incoherent elastic rescatterings of the initial and final protons on spectator nucleons. We demonstrate that IFSI-distortions of the missing momentum distrib…
We perform a parameter-free calculation for the high-energy proton-nucleus scattering based on the Glauber theory. A complete evaluation of the so-called Glauber amplitude is made by using the factorization of the single-particle wave functions. The multiple-scattering or multistep processes are fully taken into account within the Glauber theory. We demonstrate that proton- $^{12}$C, $^{20}$Ne, and $^{28}$Si elastic and inelastic scattering ($J^π=0^+ \to 2^+$ and $0^+ \to 4^+$) processes are very well described in a wide range of the incident energies from $\sim$50 MeV to $\sim$ 1 GeV. We eval…
A systematic analysis of nucleon-nucleon scattering amplitudes is available up to a laboratory energy of $3$~GeV in case of the $pp$ system and up to $1.2$ GeV for $pn$. At higher energies there is only incomplete experimental information on $pp$ elastic scattering, whereas data for the $pn$ system are very scarce. We apply the spin-dependent Glauber theory to calculate spin observables of $pd$ elastic scattering at $3$-$50$ GeV/c using $pp$ amplitudes available in the literature and parametrized within the Regge formalism. The calculated vector $A_y^p$, $A_y^d$ and tensor $A_{xx}$, $A_{yy}$ a…
We systematically calculate the total reaction cross sections of oxygen isotopes, $^{15-24}$O, on a $^{12}$C target at high energies using the Glauber theory. The oxygen isotopes are described with Slater determinants generated from a phenomenological mean-field potential. The agreement between theory and experiment is generally good, but a sharp increase of the reaction cross sections from ^{21}O to ^{23}O remains unresolved. To examine the sensitivity of the diffraction pattern of elastic scattering to the nuclear surface, we study the differential elastic-scattering cross sections of proton…
We study nuclear reaction cross-sections for stable and unstable projectiles and targets within Glauber model, using densities obtained from various relativistic mean field formalisms. The calculated cross-sections are compared with the experimental data in some specific cases. We also evaluate the differential scattering cross-sections at several incident energies, and observe that the results found from various densities are similar at smaller scattering angles, whereas a systematic deviation is noticed at large angles. In general, these results agree fairly well with the experimental data.
A parameterization of the nucleon-nucleon elastic scattering amplitude is needed for future experiments with nucleon and nuclear beams in the beam momentum range of 2 -- 50 GeV/c/nucleon. There are many parameterizations of the amplitude at $P_{lab} >$ 25--50 GeV/c, and at $P_{lab} \leq$ 5 GeV/c. Our paper is aimed to cover the range between 5 -- 50 GeV/c. The amplitude is used in Glauber calculations of various cross sections and Monte Carlo simulations of nucleon-nucleon scatterings. Usually, the differential nucleon-nucleon elastic scattering cross sections are described by an exponential e…
The importance of the effects of nucleon-nucleon (NN) short-range correlations (SRC) and Gribov inelastic shadowing (IS) on various high energy scattering processes involving nuclear targets is demonstrated within an improved Glauber-Gribov approach.
This is a review of Glauber's asymptotic diffraction theory, in which diffractive scattering is described in terms of interference between semiclassical amplitudes, resulting from a stationary-phase approximation. Typically two such amplitudes are sufficient to accurately describe elastic scattering, but the stationary points are located at complex values of the impact parameter. Their separation controls the interference pattern, and their offsets from the real axis determine the overall fall-off with momentum transfer. Asymptotically, at large momentum transfers, the stationary points move t…
We have measured for the first time the charge-changing cross sections ($σ_{\text{CC}}$) of $^{12-16}$C on a $^{12}$C target at energies below $100A$ MeV. To analyze these low-energy data, we have developed a finite-range Glauber model with a global parameter set within the optical-limit approximation which is applicable to reaction cross section ($σ_{\text{R}}$) and $σ_{\text{CC}}$ measurements at incident energies from 10$A$ to $2100A$ MeV. Adopting the proton-density distribution of $^{12}$C known from the electron-scattering data, as well as the bare total nucleon-nucleon cross sections, a…
Starting with QCD, we derive an effective field theory description for forward scattering and factorization violation as part of the soft-collinear effective field theory (SCET) for high energy scattering. These phenomena are mediated by long distance Glauber gluon exchanges, which are static in time, localized in the longitudinal distance, where $|t| \ll s$. In hard scattering, Glauber gluons can induce corrections which invalidate factorization. With SCET, Glauber exchange graphs can be calculated explicitly, and are distinct from graphs with soft, collinear, or ultrasoft gluons. We derive a…
Monte Carlo evaluation is used to calculate heavy-ion elastic scattering including the center-of-mass correction and the Coulomb interaction.Angular distributions are presented for a number of nuclear pairs over a wide energy range using nucleon-nucleon scattering parameters taken from phase-shift analyses and densities from independent sources. A technique for the efficient expansion of the Glauber amplitude in partial waves is developed.
The Glauber model analysis of the elastic $^{6,8}$He+$p$ scattering data at energies around 700 MeV/nucleon, measured in two separate experiments at GSI-Darmstadt, has been done using several phenomenological parametrizations of the nuclear matter density. By taking into account the new data points measured at the high momentum transfer, the nuclear matter radii of $^{6,8}$He were accurately determined from the Glauber model analysis of the data, with the spin-orbital interaction explicitly taken into account. The well-known geometry for the core and dineutron halo has been used with the new p…
Within the framework of the Glauber diffraction theory, the differential cross sections of the elastic p6He, p8Li, and p9Li scattering were calculated at intermediate energies from 70 to 1000 MeV/nucleon. The use of realistic three-body wave functions alpha-n-n (6He), alpha-t-n (8Li) and alpha-t-2n (9Li), obtained in the framework of modern clear models, and the expansion of the Glauber operator into a series of multiple scatterings in a form that is well adapted to the three-body nuclei configuration, allows the calculation of the matrix elements, by taking into account the rescattering from …
We review the theoretical and experimental progress in the Glauber model of multiple nucleon and/or parton scatterings, after the last 10--15 years of operation with proton and nuclear beams at the CERN Large Hadron Collider (LHC) and with various light and heavy colliding ions at the BNL Relativistic Heavy Ion Collider (RHIC). The main developments and the state-of-the-art of the field are summarized. These encompass measurements of the inclusive inelastic proton and nuclear cross sections, advances in the description of the proton and nuclear density profiles and their fluctuations, inclusio…
We construct a microscopic optical potential including breakup effects for elastic scattering of weakly-binding projectiles within the Glauber model, in which a nucleon-nucleus potential is derived by the $g$-matrix folding model. The derived microscopic optical potential is referred to as the eikonal potential. For $d$ scattering, the calculation with the eikonal potential reasonably reproduces the result with an exact calculation estimated by the continuum-discretized coupled-channels method. As the properties of the eikonal potential, the inaccuracy of the eikonal approximation used in the …
Reaction and elastic differential cross sections are calculated for light nuclei in the framework of the Glauber theory. The optical phase-shift function is evaluated by Monte Carlo integration. This enables us to use the most accurate wave functions and calculate the phase-shift functions without approximation. Examples of proton nucleus (e.g. p-$^6$He, p-$^6$Li) and nucleus-nucleus (e.g. $^6$He$-^{12}$C) scatterings illustrate the effectiveness of the method. This approach gives us a possibility of a more stringent analysis of the high-energy reactions of halo nuclei.
We derive the splitting kernels for partons produced in large $Q^2$ scattering processes that subsequently traverse a region of strongly-interacting matter using a recently-developed effective theory \SCETG. We include all corrections beyond the small-$x$ approximation, consistent with the power counting of \SCETG. We demonstrate how medium recoil, geometry and expansion scenarios, and phase space cuts can be implemented numerically for phenomenological applications. For the simplified case of infinite transverse momentum kinematics and a uniform medium, we provide closed-form analytic results…
We present quasi-elastic neutrino-nucleus cross sections in the energy range from 150 MeV up to 5 GeV for the target nuclei 12C and 56Fe. A relativistic description of the nuclear dynamics and the neutrino-nucleus coupling is adopted. For the treatment of final-state interactions (FSI) we rely on two frameworks succesfully applied to exclusive electron-nucleus scattering: a relativistic optical potential and a relativistic multiple-scattering Glauber approximation. At lower energies, the optical-potential approach is considered to be the optimum choice, whereas at high energies a Glauber appro…
Glauber-theory analysis of nuclear reactions on 12C target with variational Monte Carlo wave functions
Osaka Metropolitan University（大阪公立大学）；Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University（大阪公立大学南部阳一郎理论与实验物理研究所）；RIKEN Nishina Center（理化学研究所西村中心）；Niigata University（新潟大学）；Argonne National Laboratory（阿贡国家实验室）
It is shown that the ratio of the deuteron and proton analysing powers in proton-deuteron elastic scattering at small angles is sensitive to subtle effects in a theoretical description. These include the transverse spin-spin term in the elementary nucleon-nucleon amplitudes and double-scattering corrections. On the other hand there is far less sensitivity to the spin-orbit amplitude and to binding or other kinematic effects associated with the use of the deuteron, as either target or projectile. The available data are in agreement with the results of a refined Glauber theory model.
Antiproton scattering off $^3\He$ and $^4\He$ targets is considered at beam energies below 300 MeV within the Glauber-Sitenko approach, utilizing the $\bar N N$ amplitudes of the Jülich model as input. A good agreement with available data on differential $\bar p ^4\He$ cross sections and on $\bar p ^3\He$ and $\pbar ^4\He$ reaction cross sections is obtained. Predictions for polarized total $\bar p ^3$He cross sections are presented, calculated within the single-scattering approximation and including Coulomb-nuclear interference effects. The kinetics of the polarization buildup is discussed.
We perform Glauber model calculations of the antiproton-nucleus elastic and quasielastic scattering and absorption in the beam momentum range $\sim 0.5\div10$ GeV/c. A good agreement of our calculations with available LEAR data and with earlier Glauber model studies of the $\bar p A$ elastic scattering allows us to make predictions at the beam momenta of $\sim 10$ GeV/c, i.e. at the regime of the PANDA experiment at FAIR. The comparison with the proton-nucleus elastic scattering cross sections shows that the diffractive minima are much deeper in the $\bar p A$ case due to smaller absolute valu…
The differential cross sections of the forward coherent $ω$ and $ϕ$ mesons photoproduction from nuclei have been calculated using the Glauber model for the nuclear reaction. The measured cross section of the elementary reaction, i.e., $γN \to ω(ϕ) N$ reaction, are used as input in the Glauber model. The experimentally determined free-space scattering parameters of the quoted mesons have been used to evaluate the meson nucleus interactions. The sensitivity of the cross section to the Fermi-motion of the nucleon and the nucleon-nucleon short-range correlation in the nucleus are studied. The calc…
We calculate differential cross sections for proton-nucleus elastic scattering by using a Glauber theory in the optical limit approximation and nucleon distributions that can be obtained in the framework of macroscopic nuclear models in a way dependent on the equation of state of uniform nuclear matter near the saturation density. We find that the peak angle calculated for unstable neutron-rich nuclei in the small momentum transfer regime increases as the parameter L characterizing the density dependence of the symmetry energy decreases. This is a feature associated with the L dependence of th…
Various spin observables (analyzing powers and spin-correlation parameters) in $pd$ elastic scattering at $T_p = 800-1000$ MeV are analyzed within the framework of the refined Glauber model. The theoretical model uses as input spin-dependent $NN$ amplitudes obtained from the most recent partial-wave analysis and also takes into account the deuteron $D$ wave and charge-exchange effects. Predictions of the refined Glauber model are compared with the existing experimental data. Reasonable agreement between the theoretical calculations and experimental data at low momentum transfers $|t| \lesssim …
Spin-dependent total pbar-d cross sections are considered using the optical theorem. For this aim the full spin dependence of the forward pbar-d elastic scattering amplitude is considered in a model independent way. The single-scattering approximation is used to relate this amplitude to the elementary amplitudes of pbar-p and pbar-n scattering and the deuteron form factor. A formalism allowing to take into account Coulomb-nuclear interference effects in polarized pbar-d cross sections is developed. Numerical calculations for the polarized total pbar-d cross sections are performed at beam energ…
The scattering amplitude of polarized nucleons has been found within the framework of the Klein Gordon with the phenomenological spin - orbit potential. It has the Glauber type representation. The differential cross sections of polarized nucleon are considered and discussed.The Yukawa potential is applied for this problem to determine the polarization of high energy scattering nucleons.
We apply the GLV Reaction Operator formalism to compute the effects of multiple elastic scatterings of jets propagating through dense matter. We derive the elastic Reaction Operator and demonstrate that the recursion relations have a closed form solution that reduces to the familiar Glauber form. We also investigate the accuracy of the Gaussian dipole approximation for jet transverse momentum broadening.
Basing on the thickness (profile) function, previously obtained for the realistic Fermi type distribution of nucleons in nuclei, calculations are made of the microscopic eikonal phases of the nucleus-nucleus scattering and the total reaction cross- sections. In so doing, the phase is deduced to the one-dimensional integral provided that the Gaussian density distribution for the projectile nucleus and an arbitrary shape of the thickness density for the target nucleus are used. The problems of obtaining parameters of the "point" nucleon density are considered. A possibility of approximating the …
The Glauber method is extensively used to describe the motion of a hadronic projectile in interaction with the surrounding nuclear medium. One of the main approximations consists in the linearization of the wave equation for the interacting particle. We have studied the consequences of such an assumption in the case of the $^{12}\mathrm{C(e,e}'\mathrm{p)}^{11}\hbox{\rm B}^*$ reaction at high proton momenta by comparing the results with the predictions obtained when all the ingredients of the calculation are unchanged but the second-order differential equation for the scattered wave, which is s…
The differential cross sections of the p^{10}B scattering at energies of 197, 600 and 1000 MeV have been calculated within the framework of the Glauber theory. The operator of multiple scattering takes into account the contributions of single and double collisions. The contributions from protons scattering on the nucleons of different shells have been estimated in the single-scattering cross-section. The comparison with the experiment at 197 MeV and with DWBA calculation showed the adequate description of cross-sections in the region of the front angles.
The quasielastic scattering kaon-nucleus experiment performed at BNL is analyzed in a finite nucleus continuum random phase approximation framework, treating the reaction mechanism in Glauber theory up to two-step inelastic processes. A good description of the data is achieved, also providing a useful constraint on the strength of the effective particle-hole interaction in the scalar-isoscalar channel at intermediate momentum transfers.
The proton structure function is re-deduced from the data of deep inelastic electron-proton scattering after enhanced correction that is made due to the multiple scattering effect. The Glauber approach is used to account for the multiple scattering of electron with the quark constituents of the proton. The results are compared with the prediction of the parton model. Appreciable deviation is observed only at low values of the Bjorken scaling, which correspond to the extremely deep inelastic scattering.
The interpretation of recent Jlab experimental data on the exclusive process A(e,e'p)B off few-nucleon systems are analyzed in terms of realistic nuclear wave functions and Glauber multiple scattering theory, both in its original form and within a generalized eikonal approximation. The relevance of the exclusive process 4He(e,e'p)^3H for possible investigations of QCD effects is illustrated.
Nuclear surface provides useful information on nuclear radius, nuclear structure as well as properties of nuclear matter. We discuss the relationship between the nuclear surface diffuseness and elastic scattering differential cross section at the first diffraction peak of high-energy nucleon-nucleus scattering as an efficient tool in order to extract the nuclear surface information from limited experimental data involving short-lived unstable nuclei. The high-energy reaction is described by a reliable microscopic reaction theory, the Glauber model. Extending the idea of the black sphere model,…
This paper shows a brief review on CDCC and the microscopic reaction theory as a fundamental theory of CDCC. The Kerman-McManus-Thaler theory for nucleon-nucleus scattering is extended to nucleus-nucleus scattering. New development of four-body CDCC is presented. An accurate method of treating inclusive reactions is presented as an extension of CDCC and the Glauber model.
The model of multiple Dirac eikonal scattering of incident proton by target-nucleus nucleons is developed, in which new expressions for the elastic $pA$-scattering amplitudes are obtained from the multiple scattering Watson series with employing the eikonal approximation for the Dirac propagators of the free proton motion between successive scattering acts on nucleons. Basing on this model, calculations for the complete set of observables of the elastic $p+^{40}$Ca and $p+^{208}$Pb at 800 MeV have been performed with using proton-nucleon amplitudes determined from the phase analysis and the nu…
We present updated calculations for observables in the processes $^3$He($e,e^\prime p$)$^2$H, $^4$He($e,e^\prime p$)$^3$H, and $^4$He($\vec{e},e^\prime \vec{p}\,$)$^3$H. This update entails the implementation of improved nucleon-nucleon ($NN$) amplitudes to describe final state interactions (FSI) within a Glauber approximation and includes full spin-isospin dependence in the profile operator. In addition, an optical potential, which has also been updated since previous work, is utilized to treat FSI for the $^4$He($e,e^\prime p$)$^3$H and $^4$He($\vec{e},e^\prime \vec{p}\,$)$^3$H reactions. Th…
We present a study of coherent pion production in neutrino-nucleus interactions using the formalism based on partially conserved axial current theorem which connects the neutrino-nucleus cross section to the pion-nucleus elastic scattering cross section. Pion-nucleus elastic scattering cross section is calculated using Glauber model which takes three inputs, nuclear densities, pion-nucleon cross section and $α_{πN}$ which is the ratio of real to imaginary part of $πN$ forward scattering amplitude, for which the parametrizations are obtained from measured data. We calculate the differential and…
In this article, we present the charged and neutral current coherent pion production in the neutrino-nucleus interaction in the resonance region using the formalism based on the partially conserved axial current (PCAC) theorem which relates the neutrino-nucleus cross section to the pion-nucleus elastic cross section. The pion nucleus elastic cross section is calculated using the Glauber model approach. We calculate the integrated cross sections for neutrino-carbon, neutrino-iron and neutrino-oxygen scattering. The results of integrated cross-section calculations are compared with the measured …
