Proton Distribution Function Research Articles

Impact of inclusive electron ion collider data on collinear parton distributions

A study is presented of the impact of updated simulated inclusive electron ion collider deep inelastic scattering data on the determination of the proton and nuclear parton distribution functions (PDFs) at next-to-next-to-leading and next-to-leading order in QCD, respectively. The influence on the proton PDFs is evaluated relative to the HERAPDF2.0 set, which uses inclusive HERA data only, and also relative to the global fitting approach of the MSHT20 PDFs. The impact on nuclear PDFs is assessed relative to the EPPS21 global fit and is presented in terms of nuclear modification ratios. For all cases studied, significant improvements in the PDF uncertainties are observed for several parton species. The most striking impact occurs for the nuclear PDFs in general and for the region of high Bjorken x in the proton PDFs, particularly for the valence quark distributions. Published by the American Physical Society 2024

Skewness and kurtosis of solar wind proton distribution functions: The normal inverse-Gaussian model and its implications

Context.In the solar wind (SW), the particle distribution functions are generally not Gaussian. They present nonthermal features that are related to underlying acceleration and heating processes. These processes are critical in the overall dynamics of this expanding astrophysical fluid.Aims.The Proton Alpha Sensor (PAS) on board Solar Orbiter commonly observes skewed proton distributions, with a more populated high-energy side in the magnetic field direction than the Gaussian distribution. Our objectives are: (1) to identify a theoretical statistical function that adequately models the observed distributions and (2) to use its statistical interpretation to constrain the acceleration and heating processes.Methods.We analyzed the 3D velocity distribution functions (VDFs) measured by PAS and compared them to model statistical functions.Results.We show that the normal inverse Gaussian (NIG), a type of hyperbolic statistical distribution, provides excellent fits of skewed and leptokurtic proton distributions. NIG can model both the core distribution and the beam, if present. We propose an interpretation that is inspired by the mathematical formulation of the NIG. It assumes that the acceleration or heating mechanism can be modeled as a drifting diffusion process in velocity space, controlled (or subordinated) by the time of interaction of the particles with “accelerating structures”. The probability function of the interaction time is an inverse Gaussian (IG), obtained by considering a random drift across structures of a given size. The control of the diffusion by interaction times that follow an IG probability function formally defines the NIG distribution. Following this model, we show that skewness and kurtosis can be used to estimate the kinetic and thermal energy gains provided by the interaction with structures. For example, in the case studies presented here, the analyzed populations would have gained kinetic energy representing approximately two to four times their thermal energy, with an increase in velocity – due to acceleration – of from one-tenth to one-third of the observed flow velocity. We also show that the model constrains the initial temperature of the populations.Conclusions.Overall, the NIG model offers excellent fits of the observed proton distributions. Combining the skewness and the kurtosis, it also leads to constraints in the part of acceleration and heating due to the interactions with structures in the formation of the proton populations. We suggest that these effects add to the classical thermal evolution of the bulk velocity and temperature resulting from SW expansion.

Lattice calculation of the intrinsic soft function and the Collins-Soper kernel

We calculate the soft function using lattice QCD in the framework of large momentum effective theory incorporating the one-loop perturbative contributions. The soft function is a crucial ingredient in the lattice determination of light cone objects using transverse-momentum-dependent (TMD) factorization. It consists of a rapidity-independent part called intrinsic soft function and a rapidity-dependent part called Collins-Soper kernel. We have adopted appropriate normalization when constructing the pseudoscalar meson form factor that is needed in the determination of the intrinsic part and applied Fierz rearrangement to suppress the higher-twist effects. In the calculation of CS kernel we consider a CLS ensemble other than the MILC ensemble used in a previous study. We have also compared the applicability of determining the CS kernel using quasi TMDWFs and quasi TMDPDFs. As an example, the determined soft function is used to obtain the physical TMD wave functions (WFs) of pion and unpolarized iso-vector TMD parton distribution functions (PDFs) of proton.

Proton and pion distribution functions in counterpoint

Working with proton and pion valence distribution functions (DFs) determined consistently at the same, unique hadron scale and exploiting the possibility that there is an effective charge which defines an evolution scheme for DFs that is all-orders exact, we obtain a unified body of predictions for all proton and pion DFs – valence, glue, and four-flavour-separated sea. Whilst the hadron light-front momentum fractions carried by identifiable parton classes are the same for the proton and pion at any scale, the pointwise behaviour of the DFs is strongly hadron-dependent. All calculated distributions comply with quantum chromodynamics constraints on low- and high-x scaling behaviour and, owing to emergent hadron mass, pion DFs are the most dilated. These results aid in elucidating the sources of similarities and differences between proton and pion structure.

Single-spin asymmetry ATsin(2ϕ−ϕS) in π−p Drell-Yan process within TMD factorization

We study the single-spin asymmetry ATsin(2ϕ−ϕS) in the pion-induced Drell-Yan process within the transverse momentum dependent factorization (TMD factorization). The asymmetry can be expressed as the convolution of the Boer-Mulders function and the transversity function. We numerically estimate the asymmetry ATsin(2ϕ−ϕS) at the COMPASS kinematics with the model results for the pion meson distributions from the light-cone wave function approach and the available parametrization for the proton distributions. We also include the TMD evolution formalism both proton and pion parton distribution functions by using two different parametrizations on nonperturbative Sudakov form factor. We find that the asymmetry ATsin(2ϕ−ϕS) as functions of xp, xπ, xF and q⊥ is qualitatively consistent with the recent COMPASS measurement.

Proton Energization by Phase Steepening of Parallel-propagating Alfvénic Fluctuations

Abstract Proton energization at magnetic discontinuities generated by phase-steepened fronts of parallel-propagating, large-amplitude Alfvénic fluctuation is studied using hybrid simulations. We find that dispersive effects lead to the collapse of the wave via phase steepening and the subsequent generation of compressible fluctuations that mediate an efficient local energy transfer from the wave to the protons. Proton scattering at the steepened edges causes nonadiabatic proton perpendicular heating. Furthermore, the parallel electric field at the propagating fronts mediates the acceleration of protons along the mean field. A steady-state is achieved where the proton distribution function displays a field-aligned beam at the Alfvén speed, and compressible fluctuations are largely damped. We discuss the implications of our results in the context of Alfvénic solar wind.

Self-consistent determination of proton and nuclear PDFs at the Electron Ion Collider

We quantify the impact of unpolarized lepton-proton and lepton-nucleus inclusive deep-inelastic scattering (DIS) cross section measurements from the future Electron-Ion Collider (EIC) on the proton and nuclear parton distribution functions (PDFs). To this purpose, we include neutral- and charged-current DIS pseudodata in a self-consistent set of proton and nuclear global PDF determinations based on the NNPDF methodology. We demonstrate that the EIC measurements will reduce the uncertainty of the light quark PDFs of the proton at large values of the momentum fraction $x$ and, more significantly, of the quark and gluon PDFs of heavy nuclei, especially at small and large $x$. We illustrate the implications of the improved precision of nuclear PDFs for the interaction of ultrahigh energy cosmic neutrinos with matter.

The Thermodynamics of Cosmic Gases in Expanding Universes Based on VlasowTheoretical Grounds

Contemporary cosmology is taking for granted that before the phase of matter recombination matter and radiation were in perfect thermodynamic equilibrium. This would mean that protons in this phase of the universe are described by Maxwell distributions and photons are described by a Planckian black body law. When looking, however, a bit deeper into the kinetic theory of the physical processes close to and just after the recombination phase of electrons and protons, it becomes evident that in a homologously expanding universe proton distribution functions will not maintain their Maxwellian profile and connected with it, that their most relevant velocity moments, i.e. their density and their temperature, vary in an unexpected nonclassical, non-adiabatic manner. As consequence of that, in contrast to the classical view, the entropy of free atoms does change with cosmic time contrary to the standard thermodynamically expectation. We shall also demonstrate here that the realistic behaviour of cosmic gases in this phase and later depends on the specific form of the Hubble expansion of the universe, especially an accelerated expansion phase as is discussed nowadays will strongly influence the thermodynamics of the cosmic gas.

Calculation of the Linear Polarization Degree of the Proton-Induced Hα Line Emission in Solar Flares

This paper presents an original analytical calculation of the linear polarization degree of Hα lines caused by anisotropic protons during solar flares, when the temperature of thermal protons in the solar chromosphere is higher than the temperature of thermal electrons. The dependences of the linear polarization degree on the parameters of the distribution function of thermal and nonthermal protons during solar flares are analyzed. Flare plasma diagnostics are discussed.

Polarized parton distribution functions: parametrization and transverse momentum dependence

We extract the polarized parton distribution functions (PPDFs) of proton. To this end we do a global analysis on recent and up-to-dated experimental data of polarized nucleon and deuteron structure functions at the NLO accuracy of perturbative QCD. We use the Jacobi polynomials expansion technique to do the required analyses. To achieve a more realistic estimation of the PPDFs and observable uncertainties we employ the Hessian error propagation. Following that we apply a theoretical approach which is based on imposing Lorentz structure in the parton model to investigate the transverse momentum dependence of these PPDFs. In this approach, the transverse momentum dependent (TMD) polarized parton distributions, which are also called the helicity distributions, are related to the integrated ones. This relation is indicating the priority and importance of data analysis for PPDFs. The results show a convenient general treatment expected for TMD densities which are in fact resulted from data analysis and fitting processes of integrated PPDFs in an adequate and a proper way.

INDUCING INTRINSIC γ-RAY EMISSION OF THE INTERSTELLAR MEDIUM BY INTENSE FLUXES OF PROTONS AND α-PARTICLES IN ACTIVE GALACTIC NUCLEI

Cosmic-ray induced γ-ray emission inactive galactic nuclei (AGN) has been examined in thisstudy for the first time. Cross-sections for the formation ofγ-quanta in such cosmic-ray collisions were selected in the1-150 MeV energy range. Synthetic γ-ray spectra werecomputed for both interstellar gas and dust. At the sameenergies of particle collisions and induced emission of γ-quanta, energy intervals of diagnostic interest weredetermined. Specific characteristics of emission weredetected in the energy ranges of 5-15 MeV and 23-30 MeVfor most investigated elements. Diffuse continuous γ-rayspectra for the other energy ranges were less informativewith regard to the determination of the chemicalcomposition of interstellar gas. It has been shown thatexploring cosmic-ray fluxes in the vicinity of galacticcentres by employing the examined γ-ray spectra yields apattern of their energy distribution. Diagrams of cross-sections for γ-quanta formation were computed for the α-process elements. Synthetic γ-ray spectra of interstellar gasand dust were calculated individually and collectively. Ithas been indicated that, under certain conditions in AGN,cross-sections for ionisation of atoms due to energy lossesby cosmic-ray induced ionisation and γ-quanta emissionresulting from collisions with cosmic rays are similar. It hasbeen found that when the maximum of proton and α-particle energy distribution function falls within theinvestigated range of energies, it leads to the formation ofthe peak flux of γ-rays. This is particularly important forthe interpretation of observations in the energy ranges of 5-15 MeV and 23-30 MeV. Synthetic induced γ-ray spectraof interstellar dust were computed, and methods of theirobservations for silicate and carbon-containing dust, whichaccount for 80% and 20% of the interstellar-dust total mass,respectively, were determined. It has been deduced that thecontribution of Compton processes to γ-quanta emissioncan be neglected at the investigated energies.

Kinetic Alfvén wave generation by velocity shear in collisionless plasmas

The evolution of a linearly polarized, long-wavelength Alfvén wave propagating in a collisionless magnetized plasma with a sheared parallel-directed velocity flow is here studied by means of two-dimensional hybrid Vlasov–Maxwell (HVM) simulations. The unperturbed sheared flow has been represented by an exact solution of the HVM set of equations of (Malara et al., Phys. Rev. E, vol. 97, 2018, 053212), thus avoiding spurious oscillations that would arise from the non-stationary initial state and inevitably affect the dynamics of the system. We have considered the evolution of both a small and a moderate amplitude Alfvén wave, in order to separate linear wave–shear flow couplings from kinetic effects, the latter being more relevant for larger wave amplitudes. The phase mixing generated by the shear flow modifies the initial perturbation, leading to the formation of small-scale transverse fluctuations at scales comparable with the proton inertial length/Larmor radius. By analysing both the polarization and group velocity of perturbations in the shear regions, we identify them as kinetic Alfvén waves (KAWs). In the moderate amplitude run, kinetic effects distort the proton distribution function in the shear region. This leads to the formation of a proton beam, at the Alfvén speed and parallel to the magnetic field. Such a feature, due to the parallel electric field associated with KAWs, positively compares with solar wind observations of suprathermal ion populations, suggesting that it may be related to the presence of ion-scale KAW-like fluctuations.

Fine Harmonic Structure of Equatorial Noise with a Quasiperiodic Modulation

AbstractEquatorial noise emissions (fast magnetosonic waves) are electromagnetic waves observed routinely in the equatorial region of the inner magnetosphere. They propagate with wave vectors nearly perpendicular to the ambient magnetic field; that is, they are limited to frequencies below the lower hybrid frequency. The waves are generated by instabilities of ring‐like proton distribution functions, which result in their fine harmonic structure with intensity maxima close to harmonics of the proton cyclotron frequency in the source region. Although most equatorial noise emissions are continuous in time, some events exhibit a clear quasiperiodic time modulation of the wave intensity, with typical modulation periods on the order of minutes. We analyze 72 such events (17 observed by the Cluster spacecraft, 55 observed by the Van Allen Probes spacecraft) for which high‐resolution data were available. The analysis of the observed harmonic structure allows us to determine source radial distances of the events. It is found that the calculated source radial distances are generally close to the radial distances where the events were observed. The harmonic numbers where the events are generated range between about 12 and 30. Two events for which the spacecraft passed through the generation region were identified and analyzed. No simultaneous ultra‐low‐frequency magnetic field pulsations and no periodic plasma number density variations were observed. Although the in situ measured proton distribution functions were shown to be responsible for the wave growth, an insufficient resolution of the particle instruments prevented us from detecting a quasiperiodic modulation possibly present in the particle data.

Ion-scale Electromagnetic Waves in the Inner Heliosphere

Abstract Understanding the physical processes in the solar wind and corona that actively contribute to heating, acceleration, and dissipation is a primary objective of NASA’s Parker Solar Probe (PSP) mission. Observations of circularly polarized electromagnetic waves at ion scales suggest that cyclotron resonance and wave–particle interactions are dynamically relevant in the inner heliosphere. A wavelet-based statistical study of circularly polarized events in the first perihelion encounter of PSP demonstrates that transverse electromagnetic waves at ion resonant scales are observed in 30–50% of radial field intervals. Average wave amplitudes of approximately 4 nT are measured, while the mean duration of wave events is on the order of 20 s; however, long-duration wave events can exist without interruption on hour-long timescales. Determination of wave vectors suggests propagation parallel/antiparallel to the mean magnetic field. Though ion-scale waves are preferentially observed during intervals with a radial mean magnetic field, we show that measurement constraints, associated with single spacecraft sampling of quasi-parallel waves superposed with anisotropic turbulence, render the measured coherent ion-wave spectrum unobservable when the mean magnetic field is oblique to the solar wind flow; these results imply that the occurrence of coherent ion-scale waves is not limited to a radial field configuration. The lack of radial scaling of characteristic wave amplitudes and duration suggests that the waves are generated in situ through plasma instabilities. Additionally, observations of proton distribution functions indicate that temperature anisotropy may drive the observed ion-scale waves.

Proton–Proton Collisions in the Turbulent Solar Wind: Hybrid Boltzmann–Maxwell Simulations

Abstract The mechanism of heating for hot, dilute, and turbulent plasmas represents a long-standing problem in space physics, whose implications concern both near-Earth environments and astrophysical systems. In order to explore the possible role of interparticle collisions, simulations of plasma turbulence—in both collisionless and weakly collisional regimes—have been compared by adopting Eulerian Hybrid Boltzmann–Maxwell simulations, being proton–proton collisions explicitly introduced through the nonlinear Dougherty operator. Although collisions do not significantly influence the statistical characteristics of the turbulence, they dissipate nonthermal features in the proton distribution function and suppress the enstrophy/entropy cascade in the velocity space, damping the spectral transfer toward large Hermite modes. This enstrophy dissipation is particularly effective in regions where the plasma distribution function is strongly distorted, suggesting that collisional effects are enhanced by fine velocity–space structures. A qualitative connection between the turbulent energy cascade in fluids and the enstrophy cascade in plasmas has been established, opening a new path to the understanding of astrophysical plasma turbulence.

Constraining the Evolution of the Proton Distribution Function in the Heliotail

Abstract We use Interstellar Boundary Explorer (IBEX) measurements of energetic neutral atoms (ENAs) to constrain the proton (mostly pickup ion, PUI) distribution in the heliotail. In our previous study, we solved the Parker transport equation and found that the velocity diffusion coefficient D(v) for PUIs is approximately D(v) ∼ 1.1 × 10−8 km2 s−3 (v/v 0)1.3, assuming the initial proton distribution processed by the termination shock (TS), f p,0, is a kappa distribution with kappa index κ p,0 = 1.63. In this study, we test different forms for f p,0. We find that if f p,0 is kappa-distributed and D(v) = D 0(v/v 0)1.3, any kappa index in the range 1.5 < κ p,0 < 10 is consistent with IBEX data if D 0 ∼ 0.8–1.3 × 10−8 km2 s−3. While the case where D(v) ∝ v 1.3 yields ENA fluxes that appear to best reproduce IBEX data for any κ p,0, it is possible for D(v) to scale close to ∼v 2/3 or ∼v 2 within our uncertainties by changing D 0. We also show that an upstream PUI filled-shell distribution that is heated by a quasi-stationary TS, generating a downstream filled-shell with large cutoff speed, yields an excess of ENAs >2 keV compared to IBEX. However, using a fully kinetic particle-in-cell simulation to process a PUI filled-shell across the TS yields ENA spectra consistent with IBEX, reinforcing the significance of self-consistent, preferential PUI heating and diffusion at the TS. Interestingly, an upstream PUI distribution inferred from the particle-in-cell simulation to reproduce Voyager 2 observations of the nose-ward TS is inconsistent with IBEX observations from the heliotail, suggesting differences in the upstream PUI distribution or TS properties.

Boer-Mulders effect in the unpolarized pion induced Drell\u2013Yan process at COMPASS within TMD factorization

We investigate the theoretical framework of the cos 2phi azimuthal asymmetry contributed by the coupling of two Boer-Mulders functions in the dilepton production unpolarized pi p Drell–Yan process by applying the transverse momentum dependent factorization at leading order. We adopt the model calculation results of the unpolarized distribution function f_1 and Boer-Mulders function h_1^perp of pion meson from the light-cone wave functions. We take into account the transverse momentum dependent evolution effects for both the distribution functions of pion and proton by adopting the existed extraction of the nonperturbative Sudakov form factor for the pion and proton distribution functions. An approximate kernel is included to deal with the energy dependence of the Boer-Mulders function related twist-3 correlation function T_{q,F}^{(sigma )}(x,x) needed in the calculation. We numerically estimate the Boer-Mulders asymmetry nu _{BM} as the functions of x_p, x_pi , x_F and q_T considering the kinematics at COMPASS Collaboration.

Nonlinear Generation Mechanism of EMIC Falling Tone Emissions

AbstractWe have conducted a self‐consistent hybrid simulation, successfully reproducing electromagnetic ion cyclotron (EMIC) emissions with falling tone frequencies. The hybrid simulation is implemented with a parabolic ambient magnetic field. In the simulation, strong oxygen band EMIC emissions are generated through nonlinear wave growth. The cold ion density is modulated by electrostatic structures which are induced by the forward and backward propagating oxygen band EMIC waves. Along with the growth of the oxygen band, the helium band waves also grow because of the initial linear growth followed by the nonlinear growth. The nonlinear growth of the helium band waves is affected by the cold plasma density modulation, and there appear short wave packets of helium band emissions. The short wave packets entrap energetic protons efficiently, resulting in electromagnetic proton hills in the velocity phase space. The proton hill forms a nonlinear resonant current causing the falling frequency of the EMIC waves. We find strong deformation of the velocity distribution function of the energetic protons due to the proton hill being guided by the increasing resonance velocity.

Transition to kinetic turbulence at proton scales driven by large-amplitude kinetic Alfvén fluctuations

Space plasmas are dominated by the presence of large-amplitude waves, large-scale inhomogeneities, kinetic effects and turbulence. Beside the homogeneous turbulence, generation of small scale fluctuations can take place also in other realistic configurations, namely, when perturbations superpose to an inhomogeneous background magnetic field. When an Alfv\'en wave propagates in a medium where the Alfv\'en speed varies in a direction transverse to the mean field, it undergoes phase-mixing, which progressively bends wavefronts, generating small scales in the transverse direction. As soon as transverse scales get of the order of the proton inertial length $d_p$, kinetic Alfv\'en waves (KAWs) are naturally generated. KAWs belong to the branch of Alfv\'en waves and propagate nearly perpendicular to the ambient magnetic field, at scales close to $d_p$. Many numerical, observational and theoretical works have suggested that these fluctuations may play a determinant role in the development of the solar-wind turbulent cascade. In the present paper, the generation of large amplitude KAW fluctuations in inhomogeneous background and their effect on the protons have been investigated by means of hybrid Vlasov-Maxwell direct numerical simulations. Imposing a pressure balanced magnetic shear, the kinetic dynamics of protons has been investigated by varying both the magnetic configuration and the amplitude of the initial perturbations. Of interest here is the transition from quasi-linear to turbulent regimes, focusing, in particular, on the development of important non-Maxwellian features in the proton distribution function driven by KAW fluctuations.

Colliding Alfvénic wave packets in magnetohydrodynamics, Hall and kinetic simulations

The analysis of the Parker–Moffatt problem, recently revisited in Pezzi et al. (Astrophys. J., vol. 834, 2017, p. 166), is here extended by including Hall magnetohydrodynamics and two hybrid kinetic Vlasov–Maxwell numerical models. The presence of dispersive and kinetic features is studied in detail and a comparison between the two kinetic codes is also reported. Focus on the presence of non-Maxwellian signatures shows that – during the collision – regions characterized by strong temperature anisotropy are recovered and the proton distribution function displays a beam along the direction of the magnetic field, similar to some recent observations of the solar wind.