Parton Structure Functions Research Articles

Quantum information approach to high energy interactions.

High energy hadron interactions are commonly described by using a probabilistic parton model that ignores quantum entanglement present in the light-cone wave functions. Here, we argue that since a high energy interaction samples an instant snapshot of the hadron wave function, the phases of different Fock state wave functions cannot be measured-therefore the light-cone density matrix has to be traced over these unobservable phases. Performing this trace with the corresponding [Formula: see text] Haar integration measure leads to 'Haar scrambling' of the density matrix, and to the emergence of entanglement entropy. This entanglement entropy is determined by the Fock state probability distribution, and is thus directly related to the parton structure functions. As proposed earlier, at large rapidity [Formula: see text] the hadron state becomes maximally entangled, and the entanglement entropy is [Formula: see text] according to QCD evolution equations. When the phases of Fock state components are controlled, for example in spin asymmetry measurements, the Haar average cannot be performed, and the probabilistic parton description breaks down. This article is part of the theme issue 'Quantum technologies in particle physics'.

Frame-independent spatial coordinate z̃ : Implications for light-front wave functions, deep inelastic scattering, light-front holography, and lattice QCD calculations

A general procedure for obtaining frame-independent three-dimensional light-front coordinate-space wave functions is introduced. The third spatial coordinate $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$ is the boost and Lorentz frame-independent coordinate conjugate to the light-front momentum coordinate $x=\frac{{k}^{+}}{{P}^{+}}$ which appears in the momentum-space light-front wave functions underlying generalized parton distributions, structure functions, distribution amplitudes, form factors, and other hadronic observables. These causal light-front coordinate-space wave functions are used to derive a general expression for the quark distribution function of hadrons as an integral over the frame-independent longitudinal distance (the Ioffe time) between virtual-photon absorption and emission appearing in the forward virtual photon-hadron Compton scattering amplitude. Specific examples using models derived from light-front holographic QCD show that the spatial extent of the proton eigenfunction in the longitudinal direction can have a very large extent in $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{z}$.

Polarized generalized parton distributions and structure functions of the ρ meson

The $\rho$ meson polarized generalized parton distribution functions, its structure functions $g_1$ and $g_2$ and its axial form factors ${\tilde G}_{1,2}$ are studied based on a light-front quark model for the first time. Comparing our obtained moments of $g_1$ to lattice QCD calculation, we find that our results are reasonably consistent to the Lattice predictions.

Impact of recent COMPASS data on polarized parton distributions and structure functions

We perform a new extraction of polarized parton distribution functions (PPDFs) from the spin structure function experimental data in the fixed-flavor number scheme (FFNS). In this analysis, we include recent proton and deuteron spin structure functions obtained by the \texttt{COMPASS} collaboration. We examine the impact of the \texttt{new COMPASS} proton and deuteron data on the polarized parton densities and compare with results from our previous study (KATAO PPDFs), which used the Jacobi polynomial approach. We find the extracted PPDFs of the proton, neutron, and deuteron structure functions are in very good agreement with the experimental data. The results for extracted PPDFs are also compared with available theoretical models from the literature.

Study of spin-dependent structure functions of He3 and H3 at NNLO approximation and corresponding nuclear corrections

We determine polarized parton distribution functions (PPDFs) and structure functions from recent experimental data of polarized deep inelastic scattering (DIS) on nuleons at next-to-next-to-leading order (NNLO) approximation in perturbative quantum chromodynamic (pQCD). The nucleon polarized structure functions are computed using the Jacobi polynomial approach while target mass corrections (TMCs) are included in our fitting procedure. Having extracted the polarized spin structure functions, we extend our study to describe $^3{\mathrm He}$ and $^3{\mathrm H}$ polarized structure functions, as well as the Bjorken sum rule. We also explore the importance of the nuclear corrections on the polarized nuclear structure functions at small and large values of $x$. Our results are compared with the recent available and high precision polarized $^3{\mathrm He}$ and $^3{\mathrm H}$ experimental data.

Analysis of DIS structure functions of the nucleon within truncated Mellin moments approach

We present generalized evolution equations and factorization in terms of the truncated Mellin moments (TMM) of the parton distributions and structure functions. We illustrate the x and Q2 dependence of TMM in the polarized case. Using the TMM approach we compare the integrals of g1 with HERMES and COMPASS data from the limited x-ranges.

Parton densities and structure functions of nucleons and light nuclei based on the modified chiral quark-exchange model

We investigate the structure of the light nuclei based on the modified chiral quark model $(\ensuremath{\chi}\mathrm{QM})$, for the first time. To this end, we first calculate the parton distribution functions (PDFs) of the bounded nucleons using the $\ensuremath{\chi}\mathrm{QM}$ and then compute those of the light nuclei. For this purpose, it is first needed to obtain the bare quark densities inside the bounded nucleons of the nucleus. These bare distributions are calculated by applying the quark exchange model (QEM) in this article. Finally, we obtain the quark, antiquark, and gluon distributions and also the structure functions (SFs) of the bound state nucleons and the light nuclei at low ${Q}^{2}$ scale $({Q}^{2}=0.35\phantom{\rule{0.16em}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathrm{GeV}}^{2})$. These distributions can be evolved to the higher scales using DGLAP evolution equations. It is shown that the results of our theoretical framework are in good agreement with the experimental data and they also have appropriate behavior in comparison with the different parametrization models.

Polarized deeply inelastic scattering (DIS) structure functions for nucleons and nuclei

We extract parton distribution functions (PDFs) and structure functions from recent experimental data of polarized lepton--deeply inelastic scattering (DIS) on nucleons at next-to-leading order (NLO) quantum chromodynamics. We apply the Jacobi polynomial method to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution as this is numerically efficient. Having determined the polarized proton and neutron spin structure, we extend this analysis to describe $^{3}\mathrm{He}$ and $^{3}\mathrm{H}$ polarized structure functions, as well as various sum rules. We compare our results with other analyses from the literature.

Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering

We present all-order results for the highest three large-x logarithms of the splitting functions P_qg and P_gq and of the coefficient functions C_phi,q, C_2,g and C_L,g for structure functions in Higgs- and gauge-boson exchange DIS in massless perturbative QCD. The corresponding coefficients have been derived by studying the unfactorized partonic structure functions in dimensional regularization independently in terms of their iterative structure and in terms of the constraints imposed by the functional forms of the real- and virtual-emission contributions together with their Kinoshita--Lee-Nauenberg cancellations required by the mass-factorization theorem. The numerical resummation corrections are small for the splitting functions, but partly very large for the coefficient functions. The highest two (three for C_L,g) logarithms can be resummed in a closed form in terms of new special functions recently introduced in the context of the resummation of the leading logarithms

Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions

We analyze the iterative structure of unfactorized partonic structure functions in the large-x limit, and derive all-order expressions for the leading-logarithmic off-diagonal splitting functions Pgq and Pqg and the corresponding coefficient functions Cϕ,q and C2,g in Higgs- and gauge-boson exchange deep-inelastic scattering. The splitting functions are given in terms of a new function not encountered in perturbative QCD so far, and vanish maximally in the supersymmetric limit CA−CF→0. The coefficient functions do not vanish in this limit, and are given by simple expressions in terms of the above new function and the well-known leading-logarithmic threshold exponential. Our results also apply to the evolution of parton fragmentation functions and semi-inclusive e+e− annihilation.

Spin-dependent parton distributions and structure functions

Nuclear parton distributions and structure functions are determined in an effective chiral quark theory. We also discuss an extension of our model to fragmentation functions.

Diffractive parton distributions from H1 data

We analyse the latest H1 large rapidity gap data to obtain diffractive parton distributions, using a procedure based on perturbative QCD, and compare them with distributions obtained from the simplified Regge factorisation type of analysis. The diffractive parton densities and structure functions are made publically available.

NUCLEAR PARTON DENSITIES AND STRUCTURE FUNCTIONS

We calculate nuclear parton distribution functions (PDFs), using the constituent quark model. We find the bounded valon distributions in a nuclear to be related to free valon distributions in a nucleon. By using improved bounded valon distributions for a nuclear with atomic number A and the partonic structure functions inside the valon, we can calculate the nuclear structure function in x space. The results for nuclear structure-function ratio [Formula: see text] at some values of A, are in good agreement with the experimental data.

Leading twist nuclear shadowing: Uncertainties, comparison to experiments, and higher twist effects

Using the leading twist approach to nuclear shadowing, which is based on the relationship between nuclear shadowing and diffraction on a nucleon, we calculate next-to-leading order nuclear parton distribution functions (nPDFs) and structure functions in the region $0.2>x>{10}^{\ensuremath{-}5}$ and ${Q}^{2}\ensuremath{\ge}4\text{ }\text{ }{\mathrm{G}\mathrm{e}\mathrm{V}}^{2}$. The uncertainties of our predictions due to the uncertainties of the experimental input and the theory are quantified. We determine the relative role of the small ($\ensuremath{\sim}{Q}^{2}$) and large ($\ensuremath{\gg}{Q}^{2}$) diffractive masses in nuclear shadowing as a function of $x$ and find that the large mass contribution, which is an analog of the triple Pomeron exchange, becomes significant only for $x\ensuremath{\le}{10}^{\ensuremath{-}4}$. Comparing our predictions to the available fixed-target nuclear deep inelastic scattering data, we argue, based on the current experimental studies of the leading twist diffraction, that the data at moderately small $x\ensuremath{\sim}0.01$ and ${Q}^{2}\ensuremath{\sim}2\text{ }\text{ }{\mathrm{G}\mathrm{e}\mathrm{V}}^{2}$ could contain significant higher twist effects hindering the extraction of nPDFs from that data. Also, we find that the next-to-leading order effects in nuclear shadowing in the ratio of the nucleus to nucleon structure functions ${F}_{2}$ are quite sizable. Within the same formalism, we also present results for the impact parameter dependence of nPDFs. We also address the problem of extracting the neutron ${F}_{2n}(x,{Q}^{2})$ from the deuteron and proton data. We suggest a simple and nearly model-independent procedure of correcting for nuclear shadowing effects using ${F}_{2}^{A}/{F}_{2}^{D}$ ratios.

Generalized parton distributions and structure functions from full lattice QCD

We present here the latest results from the QCDSF collaboration for (moments of) structure functions and generalized form factors in full QCD with N f = 2 O ( a ) -improved Wilson fermions based on simulations closer to the chiral and continuum limit.

Next-to-Leading order approximation of polarized valon and parton distributions

Polarized parton distributions and structure functions of the nucleon are analyzed in the improved valon model. The valon representation provides a model to represent hadrons in terms of quarks, providing a unified description of bound state and scattering properties of hadrons. Polarized valon distributions are seen to play an important role in describing the spin dependence of parton distributions in the leading order (LO) and next-to-leading order (NLO) approximations. In the polarized case, a convolution integral is derived in the framework of the valon model. The Polarized valon distribution in a proton and the polarized parton distributions inside the valon are necessary to obtain the polarized parton distributions in a proton. Bernstein polynomial averages are used to extract the unknown parameters of the polarized valon distributions by fitting to the available experimental data. The predictions for the NLO calculations of the polarized parton distributions and proton structure functions are compared with the LO approximation. It is shown that the results of the calculations for the proton structure function, $xg_1^p$, and its first moment, $\Gamma_{1}^p$, are in good agreement with the experimental data for a range of values of $Q^{2}$. Finally the spin contribution of the valons to the proton is calculated.

Target mass corrections to electroweak structure functions and perturbative neutrino cross sections

We provide a complete and consistent framework to include subasymptotic perturbative as well as mass corrections to the leading twist (tau=2) evaluation of charged and neutral current weak structure functions and the perturbative neutrino cross sections. We revisit previous calculations in a modern language and fill in the gaps that we find missing for a complete and ready-to-use "NLO xi-scaling" formulary. In particular, as a new result we formulate the mixing of the partonic and hadronic structure function tensor basis in the operator approach to deep inelastic scattering. As an underlying framework we follow the operator product expansion a la Georgi & Politzer that allows the inclusion of target mass corrections at arbitrary order in QCD and we provide explicit analytical and numerical results at NLO. We compare this approach with a simpler collinear parton model approach to xi-scaling. Along with target mass corrections we include heavy quark mass effects as a calculable leading twist power suppressed correction. The complete corrections have been implemented into a Monte Carlo integration program to evaluate structure functions and/or integrated cross sections. As applications, we compare the operator approach with the collinear approximation numerically and we investigate the NLO and mass corrections to observables that are related to the extraction of the weak mixing angle from a Paschos-Wolfenstein-like relation in neutrino-iron scattering. We expect that the interpretation of neutrino scattering events in terms of oscillation physics and electroweak precision physics will benefit from our results.

Parton densities and structure functions at next-to-next-to-leading order and beyond

We summarize recent results on the evolution of unpolarized parton densities and deep-inelastic structure functions in massless perturbative QCD. Due to the last year's extension of the integer-moment calculations of the three-loop splitting functions, the next-to-next-to-leading order (NNLO) evolution of the parton distributions can now be performed reliably at momentum fractions x ≳ 10−4, facilitating a considerably improved theoretical accuracy of their extraction from data on deep-inelastic scattering. The NNLO corrections are not dominated, at relevant values of x, by their leading small-x terms. At large x the splitting-function series converges very rapidly; hence, employing results on the three-loop coefficient functions, the structure functions can be analysed at next-to-next-to-next-to-leading order (N3LO) for x > 10−2. The resulting values for αs do not significantly change beyond the NNLO, their renormalization scale dependence reaches about ±1% at N3LO.

Simulation of dimensionally reduced super Yang-Mills-Chern-Simons theory

A supersymmetric formulation of a three-dimensional SYM-Chern-Simons theory using light-cone quantization is presented, and the supercharges are calculated in light-cone gauge. The theory is dimensionally reduced by requiring all fields to be independent of the transverse dimension. The result is a non-trivial two-dimensional supersymmetric theory with an adjoint scalar and an adjoint fermion. We perform a numerical simulation of this SYM-Chern-Simons theory in 1+1 dimensions using SDLCQ (Supersymmetric Discrete Light-Cone Quantization). We find that the character of the bound states of this theory is very different from previously considered two-dimensional supersymmetric gauge theories. The low-energy bound states of this theory are very ``QCD-like.'' The wave functions of some of the low mass states have a striking valence structure. We present the valence and sea parton structure functions of these states. In addition, we identify BPS-like states which are almost independent of the coupling. Their masses are proportional to their parton number in the large-coupling limit.

Running coupling Balitskiĭ-Fadin-Kuraev-Lipatov anomalous dimensions and splitting functions

I explicitly calculate the anomalous dimensions and splitting functions governing the Q^2 evolution of the parton densities and structure functions which result from the running coupling BFKL equation at LO, i.e. I perform a resummation in powers of ln(1/x) and in powers of beta_0 simultaneously. This is extended as far as possible to NLO. These are expressed in an exact, perturbatively calculable analytic form, up to small power-suppressed contributions which may also be modelled to very good accuracy by analytic expressions. Infrared renormalons, while in principle present in a solution in terms of powers in alpha_s(Q^2), are ultimately avoided. The few higher twist contributions which are directly calculable are extremely small. The splitting functions are very different from those obtained from the fixed coupling equation, with weaker power-like growth ~x^(-0.25), which does not set in until extremely small x indeed. The NLO BFKL corrections to the splitting functions are moderate, both for the form of the asymptotic power-like behaviour and more importantly for the range of x relevant for collider physics. Hence, a stable perturbative expansion and predictive power at small x are obtained.