Impact Parameter Space Research Articles

Charge and magnetization densities in transverse coordinate and impact parameter space

Electromagnetic form factors obtained from the overlap of light front wave functions (LFWFs) have been used to study the transverse densities of charge and magnetization. The calculations have been carried out to develop a relation between the charge distribution of the quarks inside nucleon in the transverse coordinate space as well as in the impact parameter space. When a comparison is carried out, it is found that the transverse distribution in the impact parameter space, where the longitudinal momentum fraction $x$ can be fixed, falls off faster than the spatial distribution in the transverse coordinate space where there is some contribution from the longitudinal momentum as well. The anomalous magnetization density of the nucleon has also been discussed. Further, we have also presented the results of the QCD transverse AdS charge density inspired from the holographic QCD model.

Lunar-Forming Giant Impact Model Utilizing Modern Graphics Processing Units

Recent giant impact models focus on producing a circumplanetary disk of the proper composition around the Earth and defer to earlier works for the accretion of this disk into the Moon. The discontinuity between creating the circumplanetary disk and accretion of the Moon is unnatural and lacks simplicity. In addition, current giant impact theories are being questioned due to their inability to find conditions that will produce a system with both the proper angular momentum and a resultant Moon that is isotopically similar to the Earth. Here we return to first principles and produce a continuous model that can be used to rapidly search the vast impact parameter space to identify plausible initial conditions. This is accomplished by focusing on the three major components of planetary collisions: constant gravitational attraction, short range repulsion and energy transfer. The structure of this model makes it easily parallelizable and well-suited to harness the power of modern Graphics Processing Units (GPUs). The model makes clear the physically relevant processes, and allows a physical picture to naturally develop. We conclude by demonstrating how the model readily produces stable Earth–Moon systems from a single, continuous simulation. The resultant systems possess many desired characteristics such as an iron-deficient, heterogeneously-mixed Moon and accurate axial tilt of the Earth.

Unified treatment of the QCD evolution of all (un-)polarized transverse momentum dependent functions: Collins function as a study case

By considering semi-inclusive deep-inelastic scattering and the (complementary) ${q}_{T}$-spectrum for Drell-Yan lepton pair production we derive the QCD evolution for all the leading-twist transverse momentum dependent distribution and fragmentation functions. We argue that all of those functions evolve with ${Q}^{2}$ following a single evolution kernel. This kernel is independent of the underlying kinematics and it is also spin independent. Those features hold, in impact parameter space, to all values of ${b}_{T}$. The evolution kernel presented has all of its large logarithms resummed up to next-to-next-to leading logarithmic accuracy, which is the highest possible accuracy given the existing perturbative calculations. As a study case we apply this kernel to investigate the evolution of the Collins function, one of the ingredients that have recently attracted much attention within the phenomenological studies of spin asymmetries. Our analysis can be readily implemented to revisit previously obtained fits that involve data at different scales for other spin-dependent functions. Such improved fits are important to get better predictions---with the correct evolution kernel---for certain upcoming experiments aiming to measure the Sivers function, Collins function, transversity, and other spin-dependent functions as well.

A Study Of Generalized Parton Distributions In Position Space

We present a study of the parton distributions in transverse position or impact parameter space for the chiral odd GPDs of a dressed quark in QED. We also present an analysis of a recent parametrization for the chiral even GPDs inside a proton. We also discuss the region of validity of the parametrizations based on a positivity condition.

Double-scattering mechanism in the exclusiveAA→AAρ0ρ0reaction in ultrarelativistic collisions

We calculate, for the first time, differential distributions for double $\rho^0$ meson production in exclusive ultraperipheral, ultrarelativistic collisions via a double scattering mechanism. The calculations are done in impact parameter space. The cross section for $\gamma A \to \rho^0 A$ is parametrized based on an existing calculation. Smearing of $\rho^0$ masses is taken into account. The results of our calculations are compared at the RHIC energy to the contribution of the two-photon mechanism discussed previously in the literature. The cross section for the double scattering mechanism is found to be an order of magnitude larger at $M_{\rho \rho} <$ 2 GeV and more than two orders of magnitude at $M_{\rho \rho} >$ 3 GeV, than that for the photon-photon mechanism. Compared to the two-photon mechanism the double scattering mechanism populates somewhat larger $\rho^0 \rho^0$ invariant masses and larger rapidity distances between the two $\rho^0$ mesons and gives a significant contribution to the $A A \to A A \pi^+ \pi^- \pi^+ \pi^-$ reaction. Some observables related to charged pions are presented too. We compare the results of our calculation with the STAR collaboration results on four charged pion production. While the shape in invariant mass of the four-pion system is very similar to the measured one, the predicted cross section constitutes only 20% of the measured one. We discuss a possibility of identifying the double scattering mechanism at the LHC.

An analysis of elastic scattering reactions with a Fermi–Dirac pomeron opaqueness in impact parameter space

In the Bourrely–Soffer–Wu model we introduce for the pomeron a new opaqueness in impact parameter space in terms of different quark contributions described by a Fermi–Dirac distribution. In order to check the validity of this assumption we consider \(p~p\), \(\bar{p}~p\), and \(\pi ^{\pm }~p\) elastic scattering. We emphasize the role of the gluon above the diffraction peak in the differential cross sections. Once these contributions are determined we extend the model to light nuclei elastic reactions like \(p~d\), \(p~^4\hbox {He}\) and \(\pi ^{\pm }~^4\hbox {He}\). The results obtained show a good description of all these elastic processes over the available experimental energy range and moderate momentum transfer.

Exclusive electromagnetic production of pion pairs in lead-lead collisions at LHC

The cross section for π+ π− and π 0 π 0 meson pairs production in peripheral ultrarelativistic heavy-ion collisions is calculated in the impact parameter space equivalent photon approximation. The cross section is calculated at the energy available at the CERN Large Hadron Collider, √s NN = 3.5 TeV. For the first time the world data for γ γ → ππ are described, both for the total cross section and for the angular distribution. This is obtained simultaneously for γ γ → π+ π− and γ γ → π 0 π 0 at all experimentally available energies.

Diffractive dissociation of gluons into heavy quark–antiquark pairs in proton–proton collisions

We discuss diffractive dissociation of gluons into heavy quark pairs. The particular mechanism is similar to the diffractive dissociation of virtual photons into quarks, which drives diffractive deep inelastic production of charm in the low-mass diffraction, or large β-region. There, it can be understood, with some reservations, in terms of a valence heavy quark content of the Pomeron. The amplitude for the gp→QQ¯p is derived in the impact parameter and momentum space. The cross section for single diffractive pp→QQ¯pX is calculated as a convolution of the elementary cross section and gluon distribution in the proton. Integrated cross section and the differential distributions in e.g. transverse momentum and rapidity of the charm and bottom quark and antiquark, as well as the quark–antiquark invariant mass are calculated for the nominal LHC energy for different unintegrated gluon distributions from the literature. The ratio of the bottom-to-charm cross sections are shown and discussed as a function of several kinematical variables.

Elasticppscattering from the optical point to past the dip: An empirical parametrization from ISR to the LHC

We describe the main features of recent LHC data on elastic pp scattering through a simple parametrization to the amplitude, inspired by a model proposed by Barger and Phillips in 1973, comprising of two exponentials with a relative phase. Despite its simplicity, this parameterization reproduces two essential aspects of the elastic differential cross section, the well known precipitous descent in the forward direction and a sharp `dip' structure. To include a complete description of data sets near -t=0, we correct the original parametrization. We examine two possibilities, the presence of the two-pion threshold singularity or a multiplicative factor reflecting the proton form factor. We find good descriptions of LHC7 and ISR data in either case. The form factor model allows simple predictions for higher energies through asymptotic theorems and asymptotic sum rules in impact parameter space. We present predictions for this model at higher LHC energies, which can be used to test whether asymptotia is reached. The black disk limit in this model is seen to be reached only for energies near 10^6 TeV.

π+π−andπ0π0pair production in photon-photon scattering and ultraperipheral ultrarelativistic heavy-ion collisions

We discuss ${\gamma}{\gamma}\to {\pi}{\pi}$ reactions, starting from the two-pion threshold up to about $\sqrt{s_{{\gamma}{\gamma}}}$ = 6 GeV. We include the dipion continuum due to pion exchange (for ${\pi}^+{\pi}^-$) and ${\rho}^\pm$ exchange (for ${\pi}^0{\pi}^0$) as well as the dipion ${\sigma}(600)$, $f_0(980)$ and $f_2(1270)$ s-channel resonances. We discuss also a possible contribution of less pronounced scalar resonances $f_0(1500)$ and $f_0(1710)$ as well as the tensor resonances $f'_2(1525)$, $f_2(1565)$ and $f_2(1950)$. We find that the inclusion of the spin-four $f_4(2050)$ resonant state improves the situation at $\sqrt{s_{{\gamma}{\gamma}}} \approx$ 2 GeV. We estimate the relevant diphoton partial decay width to be ${\Gamma}_{f_4 \to {\gamma}{\gamma}} \approx$ 0.7 keV. At higher energies, Brodsky-Lepage and hand-bag mechanisms are included in addition. We nicely describe the world data for ${\gamma}{\gamma}\to {\pi}{\pi}$ for the first time both for the total cross section and angular distributions. The cross section for the production of two-pions in ultraperipheral ultrarelativistic heavy ion collisions is calculated in the impact parameter space Equivalent Photon Approximation (EPA). We obtain the nuclear cross section of 46.7 mb for ${\pi}^+{\pi}^-$ and 8.7 mb for ${\pi}^0{\pi}^0$ at the LHC energy $\sqrt{s_{NN}} =$ 3.5 TeV and minimal cut on pion transverse momentum $p_t>$ 0.2 GeV. Differential distributions in impact parameter, dipion invariant mass, single pion and dipion rapidity, pion transverse momentum as well as pion pseudorapidity are shown. The ${\gamma}{\gamma}\to {\pi}^+ {\pi}^-$ subprocess constitutes a background to $A A \to A {\rho}^0 (\to {\pi}^+ {\pi}^-) A$ process.

Generalized parton distributions of the photon with helicity flip

We present a calculation of the generalized parton distributions (GPDs) of the photon when the helicity of the initial photon is different from the final photon. We calculate the GPDs using overlaps of photon light-front wave functions (LFWFs) at leading order in electromagnetic coupling α and zeroth order in the strong coupling αs, when the momentum transfer is purely in the transverse direction. These involve a contribution of orbital angular momentum of two units in the LFWFs. We express these GPDs in the impact parameter space.

The γγ→J/ψJ/ψ reaction and the J/ψJ/ψ pair production in exclusive ultraperipheral ultrarelativistic heavy ion collisions

We calculate the cross section for the γγ→J/ψJ/ψ process. Two mechanisms are considered: box (two-loop) diagrams of the order of $O(\alpha_{\mathrm{em}}^{2} \alpha_{s}^{2})$ and two-gluon exchange of the order of $O(\alpha_{\mathrm{em}}^{2} \alpha_{s}^{4})$ . The first mechanism is calculated in the heavy-quark non-relativistic approximation while in the second case we also include the effects of quantum motion of quarks in the bound state. The box contribution dominates at energies close to the threshold (W< 15 GeV) while the two-gluon mechanism takes over at W> 15 GeV. Including the bound-state wave function effects for the two-gluon exchange mechanism gives a cross section 0.1–0.4 pb, substantially smaller than that in the non-relativistic limit (0.4–1.6 pb). We also find a strong infrared sensitivity which manifests itself in a rather strong dependence on the mass for the t-channel gluons. The elementary cross section is then used in the Equivalent Photon Approximation (EPA) in the impact parameter space to calculate the cross section for 208Pb+208Pb→208Pb+J/ψJ/ψ+208Pb reaction. Distributions in rapidity of the J/ψJ/ψ pair and invariant mass of the pair are shown. It is shown that in heavy ion collisions the box mechanism significantly dominates over the two-gluon exchange mechanism.

Difference between standard and quasi-conformal BFKL kernels

As it was recently shown, the colour singlet BFKL kernel, taken in Möbius representation in the space of impact parameters, can be written in quasi-conformal shape, which is unbelievably simple compared with the conventional form of the BFKL kernel in momentum space. It was also proved that the total kernel is completely defined by its Möbius representation. In this paper we calculated the difference between standard and quasi-conformal BFKL kernels in momentum space and discovered that it is rather simple. Therefore we come to the conclusion that the simplicity of the quasi-conformal kernel is caused mainly by using the impact parameter space.

Factorization theorem for Drell-Yan at low q T and transverse-momentum distributions on-the-light-cone

We derive a factorization theorem for Drell-Yan process at low q_T using effective field theory methods. In this theorem all the obtained quantities are gauge invariant and the special role of the soft function--and its subtraction thereof--is emphasized. We define transverse-momentum dependent parton distribution functions (TMDPDFs) which are free from light-cone singularities while all the Wilson lines are defined on-the-light-cone. We show explicitly to first order in \alpha_s that the partonic Feynman PDF can be obtained from the newly defined partonic TMDPDF by integrating over the transverse momentum of the parton inside the hadron. We obtain a resummed expression for the TMDPDF, and hence for the cross section, in impact parameter space. The universality of the newly defined matrix elements is established perturbatively to first order in \alpha_s. The factorization theorem is validated to first order in \alpha_s and also the gauge invariance between Feynman and light-cone gauges.

Checks of asymptotia in pp elastic scattering at LHC

We parametrize TOTEM data for the elastic differential pp cross section at s=7 TeV in terms of two exponentials with a relative phase. We employ two previously derived sum rules for pp elastic scattering amplitude in impact parameter space to check whether asymptotia has been reached at the LHC. A detailed study of the TOTEM data for the elastic differential cross section at s=7 TeV is made and it is shown that, within errors, the asymptotic sum rules are satisfied at LHC. We propose to use this parametrization to study forthcoming higher energy data.

Holographic Pomeron and the Schwinger mechanism

We revisit the problem of dipole-dipole scattering via exchanges of soft Pomerons in the context of holographic QCD. We show that a single closed string exchange contribution to the eikonalized dipole-dipole scattering amplitude yields a Regge behavior of the elastic amplitude; the corresponding slope and intercept are different from previous results obtained by a variational analysis of semiclassical surfaces. We provide a physical interpretation of the semiclassical worldsheets driving the Regge behavior for $(\ensuremath{-}t)&gt;0$ in terms of worldsheet instantons. The latter describe the Schwinger mechanism for string pair creation by an electric field, where the longitudinal electric field ${E}_{L}={\ensuremath{\sigma}}_{T}\mathrm{tanh}(\ensuremath{\chi}/2)$ at the origin of this nonperturbative mechanism is induced by the relative rapidity $\ensuremath{\chi}$ of the scattering dipoles. Our analysis naturally explains the diffusion in the impact parameter space encoded in the Pomeron exchange; in our picture, it is due to the Unruh temperature of accelerated strings under the electric field. We also argue for the existence of a ``micro-fireball'' in the middle of the transverse space due to the soft Pomeron exchange, which may be at the origin of the thermal character of multiparticle production in ep/pp collisions. After summing over uncorrelated multi-Pomeron exchanges, we find that the total dipole-dipole cross section obeys the Froissart unitarity bound.

Imaging the Transverse (Spin) Structure of Hadrons

Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, exhibit a significant deviation from axial symmetry when the target and/or quark are transversely polarized. Connections between this deformation and transverse single-spin asymmetries as well as with quark-gluon correlations are discussed. The sign of transverse deformation of impact parameter dependent parton distribu- tions in a transversely polarized target can be related to the sign of the contribution from that quarkflavor to the nucleon anomalous magnetic moment. Therefore, the signs of the Sivers function for u and d quarks, as well as the signs of quark-gluon correlations embodied in the polarized structure function g2 can be understood in terms of the proton and neutron anomalous magnetic moments.

Exclusive production ofπ+π−andπ0π0pairs in photon-photon and in ultrarelativistic heavy ion collisions

The $\gamma\gamma\to\pi\pi$ reactions are discussed. To describe those processes, we include dipion continuum, resonances, high-energy pion-pion rescatterings, $\rho$ meson exchange and pQCD Brodsky-Lepage mechanisms. The cross section for the production of pion pairs in photon-photon collisions in peripheral heavy ion collisions is calculated with the help of Equivalent Photon Approximaption (EPA) in the impact parameter space. We show predictions at $\sqrt{s_{NN}}=3.5$ TeV which could be measured e.g. by the ALICE collaboration at the LHC.

Transverse (spin) structure of hadrons

Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs, exhibit a significant deviation from axial symmetry when the target and/or quark are transversely polarized. Connections between this deformation and transverse single-spin asymmetries as well as with quark–gluon correlations are discussed. The sign of transverse deformation of impact parameter dependent parton distributions in a transversely polarized target can be related to the sign of the contribution from that quark flavor to the nucleon anomalous magnetic moment. Therefore, the signs of the Sivers function for u and d quarks, as well as the signs of quark–gluon correlations embodied in the polarized structure function g2 can be understood in terms of the proton and neutron anomalous magnetic moments.

Pion Tensor Generalized Parton Distributions in a Covariant Constituent Quark Model

The pion tensor generalized parton distributions are evaluated within a covariant, analytic constituent quark model. The generalized form factors for the first two Mellin moments and the probability densities of polarized quarks in the impact parameter space are derived and compared with lattice QCD and quark model results.