Glauber Series Research Articles

Resummation of Glauber phases in non-global LHC observables for large Nc

The Glauber series for non-global jet observables at hadron colliders simultaneously includes the super-leading logarithms alongside an arbitrary number of Glauber phases. Building on the formalism of [1], it is shown that the leading terms in this series for large Nc can be resummed in closed form in renormalization-group improved perturbation theory. This remarkable observation suggests that large-Nc methods might also be helpful to study other aspects of non-global logarithms at hadron colliders, and to combine our analytic results with amplitude-level parton showers.

Glauber phases in non-global LHC observables: resummation for gluon-initiated processes

The resummation of the “Glauber series” in non-global LHC observables is extended to processes with gluons in the initial state. This series simultaneously incorporates large double-logarithmic corrections, the so-called “super-leading logarithms”, together with higher-order exchanges of pairs of Glauber gluons associated with the large numerical factor (iπ)2. On a technical level, the main part of this work is devoted to the systematic reduction of the appearing color traces and construction of basis structures, which consist of thirteen elements for gg and eleven elements for qg scattering. Numerical estimates for wide-angle gap-between-jet cross sections at the parton level show that, in particular for gg scattering at relatively small vetoes Q0, the contribution involving four Glauber exchanges gives a sizeable correction and should not be neglected.

Quark-Hadron Model for Elastic scattering between two-composite systems at relativistic energies

We have previously developed formulas to calculate the full Glauber series of the elastic scattering between two composite systems their structures are pictured as bag of quarks or as cluster of nucleons. In this work, we have constructed a hybrid model in which the quark-hadron pictures are combined. The elastic cross sections of alpha-proton at center of mass energy √(s )=89 GeV and alpha-alpha at √(s )=126 GeV are evaluated by the hybrid model, the pure quark and conventional nucleon models. The results of different approaches are compared with the experimental data. The comparison shows that the predictions of the three model are identical at forward scattering angles and are significantly different at large momentum transfers. At large angles, the hybrid model calculation fits very well the experimental data than the calculations of quark and nucleon models. The nucleon model prediction provides better agreement with the data than the quark model calculation. The improvement achieved by hybrid model over the quark model at large scattering angles shows that the inclusion of quark–quark interactions is more consistently when the separation between the colliding nucleons is less than the hadron core.

Teaching strategy for introducing beginners to Coherent States

Handling coherent states by undergraduates students may be a hard task, as they have to deal with Glauber's series $e^{-\frac {|\alpha|^2} {2}}\sum\limits_{n=0}^\infty\frac {\alpha^n} {\sqrt{n!}}\phi_n(x)$. We show here that the task can be greatly simplified by introduction of a novel compact formula for Glauber coherent states employed in [ Int. J. Mod. Phys. B {\bf 31} (2017) 175051]. This expression is obtained by solving the basic differential equation associated to coherent states $a|\alpha>=\alpha|\alpha>$.

Integral formula for calculating rigorously the full Glauber series of the elastic scattering between two cluster nuclei

In a previous work, the full Glauber series of the elastic scattering between two cluster nuclei has been evaluated in an approximate way. The present paper introduces a more elaborate technique to calculate such series rigorously. The deuteron-carbon elastic angular distribution is calculated by the two approaches and the results showed that the inaccuracy of the previous approach is significant at large angles. Impressive fit is observed to the experimental data when the phase variation is included in the present analysis.

Full Glauber series analysis for elastic scattering with consistent center-of-mass correlations

Within the formalism of Glauber multiple scattering theory, high-energy elastic collisions between light and medium weight nuclei $(A,Bg~4)$ have been investigated by invoking an exact center-of-mass correlation function in each order of the complete expansion of the phase-shift function. The calculated angular distributions $(\mathrm{n}\mathrm{u}\mathrm{c}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{r}+\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{l}\mathrm{o}\mathrm{m}\mathrm{b}),$ using the developed formulas, show better agreement with the scattering data.

Glauber amplitude for scattering of electrons by hydrogen atoms: 1s-->2s,2p excitations.

The method proposed by Yates [Chem. Phys. Lett. 25, 480 (1974)] has been extended to the case of 1s\ensuremath{\rightarrow}2s,2p excitation of a hydrogen atom by electron impact. An alternative analytical procedure used by Berestetskii, Lifshitz, and Pitaevskii [Quantum Electrodynamics (Pergamon, Oxford, 1982), p. 539] has, however, been adopted for the evaluation of the third term of the Glauber series. In the process Yates's expression for the third Glauber term for electron--hydrogen-atom elastic scattering has been reproduced. The differential scattering cross section for the excitation process has been calculated as suggested by Yates. This cross section is found to be in reasonable agreement in the low-angle region with the exact Glauber results reported by Tai et al. [Phys. Rev. A 1, 1819 (1970)] and by Gien [Phys. Rev. A 20, 1457 (1979)].

Elastic scattering of electrons and positrons by atomic hydrogen and helium at intermediate and high energies

The eikonal Born series (EBS) method is used to analyse the elastic scattering of fast electrons and positrons by atomic hydrogen and helium. The properties of the terms of the Born and Glauber series are discussed, and in particular the dependence of these terms as a function of the (large) incident wavenumber k and the magnitude Delta of the momentum transfer. A more precise EBS treatment of small-angle scattering is presented by improving the calculation of the second Born term in this angular region. The EBS results agree very well with the recent absolute measurements of elastic differential cross sections, both for electron-atomic-hydrogen and electron-helium collisions.

Pion-nucleus scattering in a cluster model

The forward amplitude for pion-nucleus scattering in the ${N}^{*}$ region is studied using a Glauber formalism with a cluster model of the nucleus. It is found that the major features of the data for light nuclei ($^{6}\mathrm{Li}$, $^{7}\mathrm{Li}$, $^{9}\mathrm{Be}$, $^{12}\mathrm{C}$, $^{16}\mathrm{O}$) are reasonably well reproduced by using a simple parametrization of the cluster structure. These results are in sharp contrast to predictions of a nucleon model at a comparable level of simplicity. We argue that a Glauber series representation of the amplitude is viable in a cluster model but not reliable in a nucleon model essentially because the interlocking effects of multiple scatterings and strong particle correlations inside the nucleus are easy to approximate in the former model and are difficult to account for in the latter model. Several modifications of the basic cluster model are considered and estimated to be small. More studies, experimental and theoretical, on angular distributions are urged.NUCLEAR REACTION $^{2}\mathrm{H}$, $^{3}\mathrm{He}$, $^{4}\mathrm{He}$, $^{6}\mathrm{Li}$, $^{7}\mathrm{Li}$, $^{9}\mathrm{Be}$, $^{12}\mathrm{C}$, $^{16}\mathrm{O}$, $^{27}\mathrm{Al}$, $^{32}\mathrm{S}$; forward amplitude in the ${N}^{*}$ region calculated with a cluster model using a Glauber approximation and compared to data.

Elastic scattering of electrons by atomic hydrogen at intermediate energies

It is shown that the eikonal-Born series method agrees well with the recent experimental data of Teubner et al (1973) for elastic scattering of electrons by atomic hydrogen. The method is compared with other theoretical results (Born, Glauber, static potential) and a wavenumber and the momentum transfer dependence of the successive terms in the Born and Glauber series is analysed.

Multiple diffractive analysis of elastic and inelastic scattering by 1 GeV protons on 12C, 58Ni, and 208Pb

Recent elastic and inelastic scattering experiments of 1 GeV protons on 12C, 58Ni, and 208Pb have been analyzed using multiple diffractive scattering theory. Elastic angular distributions have been calculated, including pair correlations in many-particle densities. There is evidence from the Ni and Pb data for the presence of correlations, but some evaluated correlation corrections do not produce detailed agreement. It is shown that the Glauber series for coherent inelastic scattering can be approximately summed to yield an eikonal form of the DWIA. In particular the smaller angle parts of the inelastic distributions are well described in this approximation. The expression for the summed incoherent inelastic cross section is considered as a sum rule. It is found that for 12C the excitation of the three lowest excited states exhaust ≲ 25 % of the sum rule up to − t ≲ 1 fm −2. For larger t quasi-elastic knock-out largely exhausts the sum rule.

Oscillations of the e.m. form factor and p-p elastic cross-section

The Chow-Yang model is used to connect the oscillation observed in a recent measurement of the proton electromagnetic form factor and a similar structure observed in high-precision measuremerts of pp elastic cross sections at ISR energies. In this model, the pp scattering amplitude is approximated with the Glauber series of the scattering amplitudes of elementary particles; the distribution of constiuents inside the proton (hadronicmatter density) is postulated to coincide with the electric-charge dstribution.

Graphs and glauber

Gribov’s derivation of the Glauber series and optical model from Feynman diagrams is generalized to nonzero scattering angles and any one-particle nuclear density.

Comparison of Finitely and Infinitely Composite Models for High-Energy Proton-Proton Scattering

We have extended the Glauber theory of high-energy scattering to the collision of two clusters with $N=2, 3, 4, \mathrm{and} \mathrm{infinitely}$ many constitutents each. For the special case of Gaussian scattering amplitudes and form factors, a convenient matrix formula is derived which reduces the evaluation of the multiple Gaussian integrals to the diagonalization of a symmetric matrix which can be read off from each graph and allows complete summation (by computer) of the multiple-scattering series. The convergence of the Glauber series is investigated and found to be excellent for most of the parameter values of interest. These results have been derived with the intention of comparing the hadronic differential cross sections in different quark-type models. We find that all the models discussed lead to excellent fits to the experimental proton-proton differential cross section. While the $N=3$ quark model or the $N=\ensuremath{\infty}$ droplet model may be preferred on other grounds, we find that the limited freedom one has in deciding to what degree the shape of the scattering amplitude is due to the form factors and to what degree to the elementary scattering amplitudes makes it impossible to differentiate between the two models on the basis of the structure of the differential cross section alone.

On the Glauber series interpretation of scattering data

The Glauber theory of high-energy collisions is used in a discussion of the scattering of composite bodies on composite bodies. The scattering amplitude is developed as a multiple-scattering series, and it is shown that conclusions drawn on the basis of truncated versions (e.g. inclusion of up to triple scattering only) of such series should be viewed with extreme caution. Because of ease of computation a model of Gaussian density distributions and profile functions is considered in detail and applied to the prediction of the α-α elastic cross-section and to a discussion of «quark” model interpretation of the p-p elastic cross-section. Nothing in the results can be interpreted as a special support for the ordinary quark model; mere multiplicity is what counts.