Momentum Space Integral Equations for Three Charged Particles: Diagonal Kernels

It has been a long-standing question whether momentum space integral equations of the Faddeev type are applicable to reactions of three charged particles, in particular above the three-body threshold. For, the presence of long-range Coulomb forces has been thought to give rise to such severe singularities in their kernels that the latter may lack the compactness property known to exist in the case of purely short-range interactions. Employing the rigorously equivalent formulation in terms of an effective-two-body theory we have proved in a preceding paper [Phys. Rev. C {\bf 61}, 064006 (2000)] that, for all energies, the nondiagonal kernels occurring in the integral equations which determine the transition amplitudes for all binary collision processes, possess on and off the energy shell only integrable singularities, provided all three particles have charges of the same sign, i.e., all Coulomb interactions are repulsive. In the present paper we prove that, for particles with charges of equal sign, the diagonal kernels, in contrast, possess one, but only one, nonintegrable singularity. The latter can, however, be isolated explicitly and dealt with in a well-defined manner. Taken together these results imply that modified integral equations can be formulated, with kernels that become compact after a few iterations. This concludes the proof that standard solution methods can be used for the calculation of all binary (i.e., (in-)elastic and rearrangement) amplitudes by means of momentum space integral equations of the effective-two-body type.Comment: 36 pages, 2 figures, accepted for publication in Phys. Rev.

Long-range behavior of the optical potential for the elastic scattering of charged composite particles

The asymptotic behavior of the optical potential, describing elastic scattering of a charged particle $\alpha$ off a bound state of two charged, or one charged and one neutral, particles at small momentum transfer $\Delta_{\alpha}$ or equivalently at large intercluster distance $\rho_{\alpha}$, is investigated within the framework of the exact three-body theory. For the three-charged-particle Green function that occurs in the exact expression for the optical potential, a recently derived expression, which is appropriate for the asymptotic region under consideration, is used. We find that for arbitrary values of the energy parameter the non-static part of the optical potential behaves for $\Delta_{\alpha} \rightarrow 0$ as $C_{1}\Delta_{\alpha} + o\,(\Delta_{\alpha})$. From this we derive for the Fourier transform of its on-shell restriction for $\rho_{\alpha} \rightarrow \infty$ the behavior $-a/2\rho_{\alpha}^4 + o\,(1/\rho_{\alpha}^4)$, i.e., dipole or quadrupole terms do not occur in the coordinate-space asymptotics. This result corroborates the standard one, which is obtained by perturbative methods. The general, energy-dependent expression for the dynamic polarisability $C_{1}$ is derived; on the energy shell it reduces to the conventional polarisability $a$ which is independent of the energy. We emphasize that the present derivation is {\em non-perturbative}, i.e., it does not make use of adiabatic or similar approximations, and is valid for energies {\em below as well as above the three-body dissociation threshold}.Comment: 35 pages, no figures, revte

New results from fluctuation analysis in NA49 at the CERN SPS

The exploration of the phase diagram of strongly interacting matter, particularly the study of the phase transition from hadronic to partonic matter and the search for a hypothetical critical endpoint of the first order transition line, is one of the most challenging tasks in present heavy ion physics. In this talk new results on chemical (particle ratio), transverse momentum, multiplicity and azimuthal angle fluctuations will be presented. We also discuss their connection to the onset of deconfinement and to the critical endpoint.Comment: The Proceedings of the International Conference "Critical Point and Onset of Deconfinement - CPOD 2011", Wuhan, November 7-11, 201

Chemical equilibrium study at SPS 158A GeV

A detailed study of chemical freeze-out in nucleus-nucleus collisions at beam energy 158A GeV is presented. By analyzing hadronic multiplicities within the statistical hadronization approach, the chemical equilibration of p-p, C-C, Si-Si and Pb-Pb systems is studied as a function of the number of participating nucleons in the system. Additionally, Two Component statistical hadronization model is applied to the data and is found to be able to explain the observed strangeness hadronic phase space under-saturation.Comment: 4 pages, 3 figures to appear in the proceedings of the ''Strangeness in Quark Matter 2004'' conferenc

Highlights of the Beam Energy Scan from STAR

The first part of the beam energy scan (BES) program at RHIC was successfully completed in the years 2010 and 2011. First STAR results from particle yield measurements are in good agreement with previously published data from SPS and AGS experiments whereas other results like azimuthal HBT and $K/\pi$ event-by-event fluctuations differ at some energies. In addition, new observations like the centrality dependence of chemical freeze-out parameters ($T_{\rm{ch}}$ and $\mu_{B}$) or the smoothly increasing difference with decreasing energy in the elliptic flow $v_{2}$ between particles and corresponding anti-particles, are discussed.Comment: CPOD 2011 proceedings, 5 pages, 4 figure

Three charged particles in the continuum. Astrophysical examples

We suggest a new adiabatic approach for description of three charged particles in the continuum. This approach is based on the Coulomb-Fourier transformation (CFT) of three body Hamiltonian, which allows to develop a scheme, alternative to Born-Oppenheimer one. The approach appears as an expansion of the kernels of corresponding integral transformations in terms of small mass-ratio parameter. To be specific, the results are presented for the system $ppe$ in the continuum. The wave function of a such system is compared with that one which is used for estimation of the rate for triple reaction $p+p+e\to d+\nu,$ which take place as a step of $pp$-cycle in the center of the Sun. The problem of microscopic screening for this particular reaction is discussed

Bone graft substitutes and bone morphogenetic proteins for osteoporotic fractures: What is the evidence?

Despite improvements in implants and surgical techniques, osteoporotic fractures remain challenging to treat. Among other major risk factors, decreased expression of morphogenetic proteins has been identified for impaired fracture healing in osteoporosis. Bone grafts or bone graft substitutes are often used for stabilizing the implant and for providing a scaffold for ingrowth of new bone. Both synthetic and naturally occurring biomaterials are available. Products generally contain hydroxyapatite, tricalcium phosphate, dicalcium phosphate, calcium phosphate cement, calcium sulfate (plaster of Paris), or combinations of the above. Products have been used for the treatment of osteoporotic fractures of the proximal humerus, distal radius, vertebra, hip, and tibia plateau. Although there is generally consensus that screw augmentation increased the biomechanical properties and implant stability, the results of using these products for void filling are not unequivocal. In osteoporotic patients, Bone Morphogenetic Proteins (BMPs) have the potential impact to improve fracture healing by augmenting the impaired molecular and cellular mechanisms. However, the clinical evidence on the use of BMPs in patients with osteoporotic fractures is poor as there are no published clinical trials, case series or case studies. Even pre-clinical literature on in vitro and in vivo data is weak as most articles focus on the beneficial role for BMPs for restoration of the underlying pathophysiological factors of osteoporosis but do not look at the specific effects on osteoporotic fracture healing. Limited data on animal experiments suggest stimulation of fracture healing in ovariectomized rats by the use of BMPs. In conclusion, there is only limited data on the clinical relevance and optimal indications for the use of bone graft substitute materials and BMPs on the treatment of osteoporotic fractures despite the clinical benefits of these materials in other clinical indications. Given the general compromised outcome in osteoporotic fractures and limited alternatives for enhancement of fracture healing, clinicians and researchers should focus on this important topic and provide more data in this field in order to enable a sound clinical use of these materials in osteoporotic fractures

Methods to study event-by-event fluctuations in the NA61/SHINE experiment at the CERN SPS

Theoretical calculations locate the critical point of strongly interacting matter (CP) at energies accessible at the CERN SPS. Event-by-event transverse momentum and multiplicity fluctuations are considered as one of the most important tools to search for the CP. Pilot studies of the energy dependence and the system size dependence of both $p_T$ and multiplicity fluctuations were performed by the NA49 experiment. The NA61/SHINE ion program is a continuation of these efforts. After briefly recalling the essential NA49 results on fluctuations we will discuss the technical methods (removing Non-Target interactions) which we plan to apply for future transverse momentum and multiplicity fluctuation analyses.Comment: Proceedings of CPOD 2010, 23-29 August, JINR, Dubn