# Posts Tagged structure function

## Recent Postings from structure function

### Amplitude of solar wind density turbulence from 10--45 $R_{\odot}$

We report on the amplitude of the density turbulence spectrum ($C_{N}^{2}$) and the density modulation index ($\delta N/N$) in the solar wind between $10$ and $45 R_{\odot}$. We derive these quantities using a structure function that is observationally constrained by occultation observations of the Crab nebula made in 2011 and 2013 and similar observations published earlier. We use the most general form of the structure function, together with currently used prescriptions for the inner/dissipation scale of the turbulence spectrum. Our work yields a comprehensive picture of a) the manner in which $C_{N}^{2}$ and $\delta N/N$ vary with heliocentric distance in the solar wind and b) of the solar cycle dependence of these quantities.

### Heavy flavour corrections to polarised and unpolarised deep-inelastic scattering at 3-loop order

We report on progress in the calculation of 3-loop corrections to the deep-inelastic structure functions from massive quarks in the asymptotic region of large momentum transfer $Q^2$. Recently completed results allow us to obtain the $O(a_s^3)$ contributions to several heavy flavour Wilson coefficients which enter both polarised and unpolarised structure functions for lepton-nucleon scattering. In particular, we obtain the non-singlet contributions to the unpolarised structure functions $F_2(x,Q^2)$ and $x F_3(x,Q^2)$ and the polarised structure function $g_1(x,Q^2)$. From these results we also obtain the heavy flavour contributions to the Gross-Llewellyn-Smith and the Bjorken sum rules.

### The impact of intrinsic charm on the parton distribution functions

In this work, we present a new investigation about the impact of intrinsic charm (IC) on the physical observables, in particular, on the heavy structure function $F_2^c$. Since IC distribution is dominant at large Bjorken variable $x$, normally, it is expected that it can be explored only at large $x$. But, by studying the correlation of the charm density in the proton with $F_2^c$, we are going to show that the IC component can also be effective at low $x$. To investigate further, we perform three QCD global analyses of parton distribution functions (PDFs), by including the EMC $F_2^c$ data that are recognized as clear evidence for existence of the intrinsic charm in the proton, and also by considering the IC component. Although the fit of the EMC data is extremely poor due to the data points with lower $x$ values, i.e. $x<$0.05, but these analyses can give us new information about the impact of EMC data and IC contribution on the behaviour of PDFs.

### Covariant and infrared-free graviton two-point function in de Sitter spacetime II

The solution to the linearized Einstein equation in de Sitter (dS) spacetime and the corresponding two-point function are explicitly written down in a gauge with two parameters $a$' and $b$'. The quantization procedure, independent of the choice of the coordinate system, is based on a rigorous group theoretical approach. Our result takes the form of a universal spin-two (transverse-traceless) sector and a gauge-dependent spin-zero (pure-trace) sector. Scalar equations are derived for the structure functions of each part. We show that the spin-two sector can be written as the resulting action of a second-order differential operator (the spin-two projector) on a massless minimally coupled scalar field (the spin-two structure function). The operator plays the role of a symmetric rank-$2$ polarization tensor and has a spacetime dependence. The calculated spin-two projector grows logarithmically with distance and also no dS-invariant solution for either structure functions exist. We show that the logarithmically growing part and the dS-breaking contribution to the spin-zero part can be dropped out, respectively, for suitable choices of parameters $a$' and $b$'. Considering the transverse-traceless graviton two-point function, however, shows that dS breaking is universal (cannot be gauged away). More exactly, if one wants to respect the covariance and positiveness conditions, the quantization of the dS graviton field (as for any gauge field) cannot be carried out directly in a Hilbert space and involves unphysical negative norm states. However, a suitable adaptation (Krein spaces) of the Gupta-Bleuler scheme for massless fields, based on the group theoretical approach, enables us to obtain the corresponding two-point function satisfying the conditions of locality, covariance, transversality, index symmetrizer, and tracelessness.

### Covariant and infrared-free graviton two-point function in de Sitter spacetime II [Replacement]

The solution to the linearized Einstein equation in de Sitter (dS) spacetime and the corresponding two-point function are explicitly written down in a gauge with two parameters $a$' and $b$'. The quantization procedure, independent of the choice of the coordinate system, is based on a rigorous group theoretical approach. Our result takes the form of a universal spin-two (transverse-traceless) sector and a gauge-dependent spin-zero (pure-trace) sector. Scalar equations are derived for the structure functions of each part. We show that the spin-two sector can be written as the resulting action of a second-order differential operator (the spin-two projector) on a massless minimally coupled scalar field (the spin-two structure function). The operator plays the role of a symmetric rank-$2$ polarization tensor and has a spacetime dependence. The calculated spin-two projector grows logarithmically with distance and also no dS-invariant solution for either structure functions exist. We show that the logarithmically growing part and the dS-breaking contribution to the spin-zero part can be dropped out, respectively, for suitable choices of parameters $a$' and $b$'. Considering the transverse-traceless graviton two-point function, however, shows that dS breaking is universal (cannot be gauged away). More exactly, if one wants to respect the covariance and positiveness conditions, the quantization of the dS graviton field (as for any gauge field) cannot be carried out directly in a Hilbert space and involves unphysical negative norm states. However, a suitable adaptation (Krein spaces) of the Gupta-Bleuler scheme for massless fields, based on the group theoretical approach, enables us to obtain the corresponding two-point function satisfying the conditions of locality, covariance, transversality, index symmetrizer, and tracelessness.

### Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables [Cross-Listing]

We extend our earlier results for the quark helicity evolution at small $x$ to derive the small-$x$ asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the $g_1$ structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject, performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-$x$ evolution equations by employing the large-$N_c$ limit. All evolution equations resum double-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the single-logarithmic powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin using the infra-red evolution equations. Our numerical solution of the linearized large-$N_c$ evolution equations for the flavor singlet case is presented in the accompanying Letter and is further discussed here.

### Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables

We extend our earlier results for the quark helicity evolution at small $x$ to derive the small-$x$ asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the $g_1$ structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject, performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-$x$ evolution equations by employing the large-$N_c$ limit. All evolution equations resum double-logarithmic powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the single-logarithmic powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin using the infra-red evolution equations. Our numerical solution of the linearized large-$N_c$ evolution equations for the flavor singlet case is presented in the accompanying Letter and is further discussed here.

### Small-$x$ asymptotics of the quark helicity distribution

We construct a numerical solution of the small-$x$ evolution equations recently derived in \cite{Kovchegov:2015pbl} for the (anti)quark helicity TMDs and PDFs as well as the $g_1$ structure function. We focus on the case of large $N_c$ where one finds a closed set of equations. Employing the extracted intercept, we are able to predict directly from theory the behavior of the helicity PDFs at small $x$, which should have important phenomenological consequences. We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called "spin crisis."

### Small-$x$ asymptotics of the quark helicity distribution [Cross-Listing]

We construct a numerical solution of the small-$x$ evolution equations recently derived in \cite{Kovchegov:2015pbl} for the (anti)quark helicity TMDs and PDFs as well as the $g_1$ structure function. We focus on the case of large $N_c$ where one finds a closed set of equations. Employing the extracted intercept, we are able to predict directly from theory the behavior of the helicity PDFs at small $x$, which should have important phenomenological consequences. We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called "spin crisis."

### Tagged spectator DIS on a polarized spin-1 target

We discuss the process of deep-inelastic electron scattering (DIS) on the polarized deuteron with detection of a nucleon in the nuclear fragmentation region ("spectator tagging"). We cover (a) the general structure of the semi-inclusive DIS cross section on a spin-1 target; (b) the tagged structure functions in the impulse approximation, where deuteron structure is described by the $NN$ light-front wave function; (c) the extraction of free neutron structure through on-shell extrapolation in the recoil proton momentum. As an application we consider the extraction of the neutron spin structure function $g_{1n}$ through polarized electron scattering on the longitudinally polarized deuteron with proton tagging and on-shell extrapolation. Such measurements would be possible at an Electron-Ion Collider (EIC) with polarized deuteron beams and forward proton detectors.

### Tagged spectator DIS on a polarized spin-1 target [Cross-Listing]

We discuss the process of deep-inelastic electron scattering (DIS) on the polarized deuteron with detection of a nucleon in the nuclear fragmentation region ("spectator tagging"). We cover (a) the general structure of the semi-inclusive DIS cross section on a spin-1 target; (b) the tagged structure functions in the impulse approximation, where deuteron structure is described by the $NN$ light-front wave function; (c) the extraction of free neutron structure through on-shell extrapolation in the recoil proton momentum. As an application we consider the extraction of the neutron spin structure function $g_{1n}$ through polarized electron scattering on the longitudinally polarized deuteron with proton tagging and on-shell extrapolation. Such measurements would be possible at an Electron-Ion Collider (EIC) with polarized deuteron beams and forward proton detectors.

### Tagged spectator DIS on a polarized spin-1 target [Cross-Listing]

We discuss the process of deep-inelastic electron scattering (DIS) on the polarized deuteron with detection of a nucleon in the nuclear fragmentation region ("spectator tagging"). We cover (a) the general structure of the semi-inclusive DIS cross section on a spin-1 target; (b) the tagged structure functions in the impulse approximation, where deuteron structure is described by the $NN$ light-front wave function; (c) the extraction of free neutron structure through on-shell extrapolation in the recoil proton momentum. As an application we consider the extraction of the neutron spin structure function $g_{1n}$ through polarized electron scattering on the longitudinally polarized deuteron with proton tagging and on-shell extrapolation. Such measurements would be possible at an Electron-Ion Collider (EIC) with polarized deuteron beams and forward proton detectors.

### Ensemble X-ray variability of Active Galactic Nuclei. II. Excess Variance and updated Structure Function

Most investigations of the X-ray variability of active galactic nuclei (AGN) have been concentrated on the detailed analyses of individual, nearby sources. A relatively small number of studies have treated the ensemble behaviour of the more general AGN population in wider regions of the luminosity-redshift plane. We want to determine the ensemble variability properties of a rich AGN sample, called Multi-Epoch XMM Serendipitous AGN Sample (MEXSAS), extracted from the latest release of the XMM-Newton Serendipitous Source Catalogue, with redshift between 0.1 and 5, and X-ray luminosities, in the 0.5-4.5 keV band, between 10^{42} and 10^{47} erg/s. We caution on the use of the normalised excess variance (NXS), noting that it may lead to underestimate variability if used improperly. We use the structure function (SF), updating our previous analysis for a smaller sample. We propose a correction to the NXS variability estimator, taking account of the light-curve duration in the rest-frame, on the basis of the knowledge of the variability behaviour gained by SF studies. We find an ensemble increase of the X-ray variability with the rest-frame time lag tau, given by tau^{0.12}. We confirm an inverse dependence on the X-ray luminosity, approximately as L_X^{-0.19}. We analyse the SF in different X-ray bands, finding a dependence of the variability on the frequency as nu^{-0.15}, corresponding to a softer when brighter trend. In turn, this dependence allows us to parametrically correct the variability estimated in observer-frame bands to that in the rest-frame, resulting in a moderate shift upwards (V-correction). Ensemble X-ray variability of AGNs is best described by the structure function. An improper use of the normalised excess variance may lead to an underestimate of the intrinsic variability, so that appropriate corrections to the data or the models must be applied to prevent these effects.

### Ensemble X-ray variability of Active Galactic Nuclei. II. Excess Variance and updated Structure Function [Replacement]

Most investigations of the X-ray variability of active galactic nuclei (AGN) have been concentrated on the detailed analyses of individual, nearby sources. A relatively small number of studies have treated the ensemble behaviour of the more general AGN population in wider regions of the luminosity-redshift plane. We want to determine the ensemble variability properties of a rich AGN sample, called Multi-Epoch XMM Serendipitous AGN Sample (MEXSAS), extracted from the fifth release of the XMM-Newton Serendipitous Source Catalogue (XMMSSC-DR5), with redshift between 0.1 and 5, and X-ray luminosities in the 0.5-4.5 keV band between 10^42 and 10^47 erg/s. We urge caution on the use of the normalised excess variance (NXS), noting that it may lead to underestimate variability if used improperly. We use the structure function (SF), updating our previous analysis for a smaller sample. We propose a correction to the NXS variability estimator, accounting for the light curve duration in the rest frame on the basis of the knowledge of the variability behaviour gained by SF studies. We find an ensemble increase of the X-ray variability with the rest-frame time lag tau, given by tau^0.12. We confirm an inverse dependence on the X-ray luminosity, approximately as L_X^-0.19. We analyse the SF in different X-ray bands, finding a dependence of the variability on the frequency as nu^-0.15, corresponding to a softer when brighter trend. In turn, this dependence allows us to parametrically correct the variability estimated in observer-frame bands to that in the rest frame, resulting in a moderate shift upwards (V-correction). Ensemble X-ray variability of AGNs is best described by the structure function. An improper use of the normalised excess variance may lead to an underestimate of the intrinsic variability, so that appropriate corrections to the data or the models must be applied to prevent these effects.

### The spin structure of the proton at low $x$ and low $Q^2$ in two-dimensional bins from COMPASS [Cross-Listing]

The longitudinal double spin asymmetries $A_1^p$ and the spin dependent structure function of the proton $g_1^p$ were extracted from COMPASS data in the region of low Bjorken scaling variable $x$ and low photon virtuality $Q^2$. The data were taken in 2007 and 2011 from scattering of polarised muons off polarised protons, resulting in a sample that is 150 times larger than the one from the previous experiment SMC that pioneered studies in this kinematic region. For the first time, $A_1^p$ and $g_1^p$ were evaluated in this region in two-dimensional bins of kinematic variables: $(x,Q^2)$, $(\nu ,Q^2)$, $(x,\nu)$ and $(Q^2,x)$. The following kinematic region was investigated: $4\times 10^{-5}<x<4\times 10^{-2}$, $0.001$~(GeV/$c$)$^2<Q^2<1$~(GeV/$c$)$^2$ and $14$~GeV$<\nu <194$~GeV. The obtained results were confronted with theoretical models.

### Gluonic Transversity from Lattice QCD

We present an exploratory study of the gluonic structure of the $\phi$ meson using lattice QCD (LQCD). This includes the first investigation of gluonic transversity via the leading moment of the twist-two double-helicity-flip gluonic structure function $\Delta(x,Q^2)$. This structure function only exists for targets of spin $J\ge1$ and does not mix with quark distributions at leading twist, thereby providing a particularly clean probe of gluonic degrees of freedom. We also explore the gluonic analogue of the Soffer bound which relates the helicity flip and non-flip gluonic distributions, finding it to be saturated at the level of 80%. This work sets the stage for more complex LQCD studies of gluonic structure in the nucleon and in light nuclei where $\Delta(x,Q^2)$ is an 'exotic glue' observable probing gluons in a nucleus not associated with individual nucleons.

### Gluonic Transversity from Lattice QCD [Cross-Listing]

We present an exploratory study of the gluonic structure of the $\phi$ meson using lattice QCD (LQCD). This includes the first investigation of gluonic transversity via the leading moment of the twist-two double-helicity-flip gluonic structure function $\Delta(x,Q^2)$. This structure function only exists for targets of spin $J\ge1$ and does not mix with quark distributions at leading twist, thereby providing a particularly clean probe of gluonic degrees of freedom. We also explore the gluonic analogue of the Soffer bound which relates the helicity flip and non-flip gluonic distributions, finding it to be saturated at the level of 80%. This work sets the stage for more complex LQCD studies of gluonic structure in the nucleon and in light nuclei where $\Delta(x,Q^2)$ is an 'exotic glue' observable probing gluons in a nucleus not associated with individual nucleons.

### Gluonic Transversity from Lattice QCD [Replacement]

We present an exploratory study of the gluonic structure of the $\phi$ meson using lattice QCD (LQCD). This includes the first investigation of gluonic transversity via the leading moment of the twist-two double-helicity-flip gluonic structure function $\Delta(x,Q^2)$. This structure function only exists for targets of spin $J\ge1$ and does not mix with quark distributions at leading twist, thereby providing a particularly clean probe of gluonic degrees of freedom. We also explore the gluonic analogue of the Soffer bound which relates the helicity flip and non-flip gluonic distributions, finding it to be saturated at the level of 80%. This work sets the stage for more complex LQCD studies of gluonic structure in the nucleon and in light nuclei where $\Delta(x,Q^2)$ is an 'exotic glue' observable probing gluons in a nucleus not associated with individual nucleons.

### Gluonic Transversity from Lattice QCD [Replacement]

We present an exploratory study of the gluonic structure of the $\phi$ meson using lattice QCD (LQCD). This includes the first investigation of gluonic transversity via the leading moment of the twist-two double-helicity-flip gluonic structure function $\Delta(x,Q^2)$. This structure function only exists for targets of spin $J\ge1$ and does not mix with quark distributions at leading twist, thereby providing a particularly clean probe of gluonic degrees of freedom. We also explore the gluonic analogue of the Soffer bound which relates the helicity flip and non-flip gluonic distributions, finding it to be saturated at the level of 80%. This work sets the stage for more complex LQCD studies of gluonic structure in the nucleon and in light nuclei where $\Delta(x,Q^2)$ is an 'exotic glue' observable probing gluons in a nucleus not associated with individual nucleons.

### Nuclear longitudinal structure function in eA processes at the LHeC

The nucleon and nuclear longitudinal structure functions are determined by the Kharzeev-Levin-Nardin (KLN) model of the low $x$ gluon distribution. The behavior of the gluon distribution ratio $R_{g}=\frac{G^{A}}{AG^{p}}$ and the ratio $R_{L}^{total}=\frac{F_{L}^{A-total}}{AF_{L}^{p-total}}$ in this processes are found. The heavy longitudinal structure function ratios $R_{L}^{H}=\frac{F_{L}^{H(A)}}{AF_{L}^{H(p)}}$ in eA processes at the LHeC region are discussed. Heavy contributions to the total longitudinal structure function ratio $R^{H}_{L}$ are considerable and should not be neglected especially at smaller $x$ of the LHeC project. In the KLN model the new geometrical scaling for transition from the linear to nonlinear regions in accordance with the LHeC processes is used, whose results intensively depended on the heavy quarks mass effect.

### Nuclear longitudinal structure function in eA processes at the LHeC [Replacement]

The nucleon and nuclear longitudinal structure functions are determined by the Kharzeev-Levin-Nardin (KLN) model of the low $x$ gluon distribution. The behavior of the gluon distribution ratio $R_{g}=\frac{G^{A}}{AG^{p}}$ and the ratio $R_{L}^{total}=\frac{F_{L}^{A-total}}{AF_{L}^{p-total}}$ in this processes are found. The heavy longitudinal structure function ratios $R_{L}^{H}=\frac{F_{L}^{H(A)}}{AF_{L}^{H(p)}}$ in eA processes at the LHeC region are discussed. Heavy contributions to the total longitudinal structure function ratio $R^{H}_{L}$ are considerable and should not be neglected especially at smaller $x$ of the LHeC project. In the KLN model the new geometrical scaling for transition from the linear to nonlinear regions in accordance with the LHeC processes is used, whose results intensively depended on the heavy quarks mass effect.

### Measurements of the Separated Longitudinal Structure Function F_L from Hydrogen and Deuterium Targets at Low Q^2

Structure functions, as measured in lepton-nucleon scattering, have proven to be very useful in studying the quark dynamics within the nucleon. However, it is experimentally difficult to separately determine the longitudinal and transverse structure functions, and consequently there are substantially less data available for the longitudinal structure function in particular. Here we present separated structure functions for hydrogen and deuterium at low four--momentum transfer squared, Q^2< 1 GeV^2, and compare these with parton distribution parameterizations and a k_T factorization approach. While differences are found, the parameterizations generally agree with the data even at the very low Q^2 scale of the data. The deuterium data show a smaller longitudinal structure function, and smaller ratio of longitudinal to transverse cross section R, than the proton. This suggests either an unexpected difference in R for the proton and neutron or a suppression of the gluonic distribution in nuclei.

### Lepton-Induced Reactions on Nuclei [Replacement]

Background: Long-Baseline experiments such as the planned Deep Underground Neutrino Experiment (DUNE) require theoretical descriptions of the complete event in a neutrino-nucleus reaction. Since nuclear targets are used this requires a good understanding of neutrino-nucleus interactions. Purpose: Develop a consistent theory and code framework for the description of lepton-nucleus interactions that can be used to describe not only inclusive cross sections, but also the complete final state of the reaction. Methods: The Giessen-Boltzmann-Uehling-Uhlenbeck (GiBUU) implementation of quantum-kinetic transport theory is used, with improvements in its treatment of the nuclear ground state and of 2p2h interactions. For the latter an empirical structure function from electron scattering data is used as a basis. Results: Results for electron-induced inclusive cross sections are given as a necessary check for the overall quality of this approach. The calculated neutrino-induced inclusive double-differential cross sections show good agreement with data from neutrino- and antineutrino reactions for different neutrino flavors at MiniBooNE and T2K. Inclusive double-differential cross sections for MicroBooNE, NOvA, MINERvA and LBNF/DUNE are given. Conclusions: Based on the GiBUU model of lepton-nucleus descriptions a good theoretical description of inclusive electron-, neutrino- and antineutrino-nucleus data over a wide range of energies, different neutrino flavors and different experiments is now possible. Since no tuning is involved this theory and code should be reliable also for new energy regimes and target masses. \end{description}

### Lepton-Induced Reactions on Nuclei [Replacement]

Background: Long-Baseline experiments such as the planned Deep Underground Neutrino Experiment (DUNE) require theoretical descriptions of the complete event in a neutrino-nucleus reaction. Since nuclear targets are used this requires a good understanding of neutrino-nucleus interactions. Purpose: Develop a consistent theory and code framework for the description of lepton-nucleus interactions that can be used to describe not only inclusive cross sections, but also the complete final state of the reaction. Methods: The Giessen-Boltzmann-Uehling-Uhlenbeck (GiBUU) implementation of quantum-kinetic transport theory is used, with improvements in its treatment of the nuclear ground state and of 2p2h interactions. For the latter an empirical structure function from electron scattering data is used as a basis. Results: Results for electron-induced inclusive cross sections are given as a necessary check for the overall quality of this approach. The calculated neutrino-induced inclusive double-differential cross sections show good agreement with data from neutrino- and antineutrino reactions for different neutrino flavors at MiniBooNE and T2K. Inclusive double-differential cross sections for MicroBooNE, NOvA, MINERvA and LBNF/DUNE are given. Conclusions: Based on the GiBUU model of lepton-nucleus descriptions a good theoretical description of inclusive electron-, neutrino- and antineutrino-nucleus data over a wide range of energies, different neutrino flavors and different experiments is now possible. Since no tuning is involved this theory and code should be reliable also for new energy regimes and target masses. \end{description}

### Lepton-Induced Reactions on Nuclei [Replacement]

Background: Long-Baseline experiments such as the planned Deep Underground Neutrino Experiment (DUNE) require theoretical descriptions of the complete event in a neutrino-nucleus reaction. Since nuclear targets are used this requires a good understanding of neutrino-nucleus interactions. Purpose: Develop a consistent theory and code framework for the description of lepton-nucleus interactions that can be used to describe not only inclusive cross sections, but also the complete final state of the reaction. Methods: The Giessen-Boltzmann-Uehling-Uhlenbeck (GiBUU) implementation of quantum-kinetic transport theory is used, with improvements in its treatment of the nuclear ground state and of 2p2h interactions. For the latter an empirical structure function from electron scattering data is used as a basis. Results: Results for electron-induced inclusive cross sections are given as a necessary check for the overall quality of this approach. The calculated neutrino-induced inclusive double-differential cross sections show good agreement with data from neutrino- and antineutrino reactions for different neutrino flavors at MiniBooNE and T2K. Inclusive double-differential cross sections for MicroBooNE, NOvA, MINERvA and LBNF/DUNE are given. Conclusions: Based on the GiBUU model of lepton-nucleus descriptions a good theoretical description of inclusive electron-, neutrino- and antineutrino-nucleus data over a wide range of energies, different neutrino flavors and different experiments is now possible. Since no tuning is involved this theory and code should be reliable also for new energy regimes and target masses. \end{description}

### Transverse Momentum Dependent Parton Distributions with self-similarity at small $x$ and models of proton structure function

In this paper we make re-analysis of a self-similarity based model of the proton structure function at small $x$ pursued in recent years. The additional assumption is that it should be singularity free in the entire kinematic range $0\leq \textit{x}\leq 1$. Our analysis indicates that the singularity free version of the model is valid in a more restrictive range of $Q^{2}$. We then analyse the defining Transverse Momentum Dependent Parton Distributions (TMD) occurred in the models and show that the proper generalizations and initial conditions on them not only remove the undesired singularity but also results in a QCD compatible structure function with logarithmic growth in $Q^2$. The phenomenological range of validity is then found to be much larger than the earlier versions. We also extrapolate the models to large $x$ in a parameter free way.

### Interpretation of the structure function of rotation measure in the interstellar medium

The observed structure function (SF) of rotation measure (RM) varies as a broken power-law function of angular scales. The systematic shallowness of its spectral slope is inconsistent with the standard Kolmogorov scaling. This motivates us to examine the statistical analysis on RM fluctuations. The correlations of RM constructed by Lazarian & Pogosyan (2016) are demonstrated to be adequate in explaining the observed features of RM SFs through a direct comparison between the theoretically obtained and observationally measured SF results. By segregating the density and magnetic field fluctuations and adopting arbitrary indices for their respective power spectra, we find that when the SFs of RM and emission measure have a similar form over the same range of angular scales, the statistics of the RM fluctuations reflect the properties of density fluctuations. RM SFs can be used to evaluate the mean magnetic field along the line of sight, but cannot serve as an informative source on the properties of turbulent magnetic field in the interstellar medium. We identify the spectral break of RM SFs as the inner scale of a shallow spectrum of electron density fluctuations, which characterizes the typical size of discrete electron density structures in the observed region.

### Interpretation of the structure function of rotation measure in the interstellar medium [Replacement]

The observed structure function (SF) of rotation measure (RM) varies as a broken power-law function of angular scales. The systematic shallowness of its spectral slope is inconsistent with the standard Kolmogorov scaling. This motivates us to examine the statistical analysis on RM fluctuations. The correlations of RM constructed by Lazarian & Pogosyan (2016) are demonstrated to be adequate in explaining the observed features of RM SFs through a direct comparison between the theoretically obtained and observationally measured SF results. By segregating the density and magnetic field fluctuations and adopting arbitrary indices for their respective power spectra, we find that when the SFs of RM and emission measure have a similar form over the same range of angular scales, the statistics of the RM fluctuations reflect the properties of density fluctuations. RM SFs can be used to evaluate the mean magnetic field along the line of sight, but cannot serve as an informative source on the properties of turbulent magnetic field in the interstellar medium. We identify the spectral break of RM SFs as the inner scale of a shallow spectrum of electron density fluctuations, which characterizes the typical size of discrete electron density structures in the observed region.

### Study of HERA ep Data at Low Q^2 and Low x_Bj and the Need for Higher-Twist Corrections to Standard pQCD Evolution

A detailed comparison of HERA data at low Bjorken-$x$ and low four-momentum-transfer squared, $Q^2$, with predictions based on $\ln{Q^2}$ evolution (DGLAP) in perturbative Quantum Chromo Dynamics suggests inadequacies of this framework. The standard DGLAP evolution was augmented by including an additional higher-twist term in the description of the longitudinal structure function, $F_{\rm L}$. This additional term, $F_{\rm L}~A_{\rm L}^{\rm HT}/Q^2$, improves the description of the reduced cross sections significantly. The resulting predictions for $F_{\rm L}$ suggest that further corrections are required for $Q^2$ less than about 2 GeV$^2$.

### Study of HERA ep Data at Low Q^2 and Low x_Bj and the Need for Higher-Twist Corrections to Standard pQCD Evolution [Cross-Listing]

A detailed comparison of HERA data at low Bjorken-$x$ and low four-momentum-transfer squared, $Q^2$, with predictions based on $\ln{Q^2}$ evolution (DGLAP) in perturbative Quantum Chromo Dynamics suggests inadequacies of this framework. The standard DGLAP evolution was augmented by including an additional higher-twist term in the description of the longitudinal structure function, $F_{\rm L}$. This additional term, $F_{\rm L}~A_{\rm L}^{\rm HT}/Q^2$, improves the description of the reduced cross sections significantly. The resulting predictions for $F_{\rm L}$ suggest that further corrections are required for $Q^2$ less than about 2 GeV$^2$.

### Study of HERA ep Data at Low Q^2 and Low x_Bj and the Need for Higher-Twist Corrections to Standard pQCD Fits [Replacement]

A detailed comparison of HERA data at low Bjorken-$x$ and low four-momentum-transfer squared, $Q^2$, with predictions based on $\ln{Q^2}$ evolution (DGLAP) in perturbative Quantum Chromo Dynamics suggests inadequacies of this framework. The standard DGLAP evolution was augmented by including an additional higher-twist term in the description of the longitudinal structure function, $F_{\rm L}$. This additional term, $F_{\rm L}~A_{\rm L}^{\rm HT}/Q^2$, improves the description of the reduced cross sections significantly. The resulting predictions for $F_{\rm L}$ suggest that further corrections are required for $Q^2$ less than about 2 GeV$^2$.

### Study of HERA ep Data at Low Q^2 and Low x_Bj and the Need for Higher-Twist Corrections to Standard pQCD Fits [Replacement]

A detailed comparison of HERA data at low Bjorken-$x$ and low four-momentum-transfer squared, $Q^2$, with predictions based on $\ln{Q^2}$ evolution (DGLAP) in perturbative Quantum Chromo Dynamics suggests inadequacies of this framework. The standard DGLAP evolution was augmented by including an additional higher-twist term in the description of the longitudinal structure function, $F_{\rm L}$. This additional term, $F_{\rm L}~A_{\rm L}^{\rm HT}/Q^2$, improves the description of the reduced cross sections significantly. The resulting predictions for $F_{\rm L}$ suggest that further corrections are required for $Q^2$ less than about 2 GeV$^2$.

### Exploring the nature of broadband variability in the FSRQ 3C 273

Detailed investigation of broadband flux variability in the blazar 3C 273 allows us to probe the location and size of emission regions and their physical conditions. We report the results on correlation studies of the flaring activity observed between 2008 and 2012. The observed broadband variations were investigated using the structure function and the discrete correlation function, and power spectral density analysis (PSD) methods. The PSD analysis showed that the optical/IR light curve slopes are consistent with the slope of white noise processes, while, the PSD slopes at radio, X-ray and gamma-ray energies are consistent with red-noise processes. The flux variations at gamma-ray and mm-radio bands are found to be significantly correlated. Using the estimated time lag of (110\pm27) days between gamma-ray and radio light curves, we constrained the location of the gamma-ray emission region at a de-projected distance of 1.2\pm0.9 pc from the jet apex. Flux variations at X-ray bands were found to have a significant correlation with variations at both radio and \gamma-rays energies. The correlation between X-rays and gamma-rays light curves suggests presence of two components responsible for the X-ray emission. A negative time lag of -(50\pm20) days, where the X-rays are leading the emission, suggests X-rays are emitted closer to the jet apex from a compact region at a distance of ~(0.5\pm0.4) pc from the jet apex. A positive time lag of (110\pm20) days suggests jet-base origin of the other X-ray component at ~(4--5)~pc from the jet apex. The flux variations at radio frequencies were found to be well correlated with each other such that the variations at higher frequencies are leading the lower frequencies, which could be expected in the standard shock-in-jet model.

### Nuclear medium effects in $F_{2A}^{EM}(x,Q^2)$ and $F_{2A}^{Weak}(x,Q^2)$ structure functions

Recent phenomenological analysis of experimental data on DIS processes induced by charged leptons and neutrinos/antineutrinos beams on nuclear targets by CTEQ collaboration has confirmed the observation of CCFR and NuTeV collaborations, that weak structure function $F_{2A}^{Weak} (x,Q^2)$ is different from electromagnetic structure function $F_{2A}^{EM} (x,Q^2)$ in a nucleus like iron, specially in the region of low $x$ and $Q^2$. In view of this observation we have made a study of nuclear medium effects on $F_{2A}^{Weak} (x,Q^2)$ and $F_{2A}^{EM} (x,Q^2)$ for a wide range of $x$ and $Q^2$ using a microscopic nuclear model. We have considered Fermi motion, binding energy, nucleon correlations, mesonic contributions from pion and rho mesons and shadowing effects to incorporate nuclear medium effects. The calculations are performed in a local density approximation using a relativistic nucleon spectral function which includes nucleon correlations. The numerical results in the case of iron nucleus are compared with the experimental data.

### Nuclear medium effects in $F_{2A}^{EM}(x,Q^2)$ and $F_{2A}^{Weak}(x,Q^2)$ structure functions [Replacement]

Recent phenomenological analysis of experimental data on DIS processes induced by charged leptons and neutrinos/antineutrinos beams on nuclear targets by CTEQ collaboration has confirmed the observation of CCFR and NuTeV collaborations, that weak structure function $F_{2A}^{Weak} (x,Q^2)$ is different from electromagnetic structure function $F_{2A}^{EM} (x,Q^2)$ in a nucleus like iron, specially in the region of low $x$ and $Q^2$. In view of this observation we have made a study of nuclear medium effects on $F_{2A}^{Weak} (x,Q^2)$ and $F_{2A}^{EM} (x,Q^2)$ for a wide range of $x$ and $Q^2$ using a microscopic nuclear model. We have considered Fermi motion, binding energy, nucleon correlations, mesonic contributions from pion and rho mesons and shadowing effects to incorporate nuclear medium effects. The calculations are performed in a local density approximation using a relativistic nucleon spectral function which includes nucleon correlations. The numerical results in the case of iron nucleus are compared with the experimental data.

### Investigation of pionic contribution in the lepton and anti-lepton production cross section in p-Cu and p-Pt collision

For detailed explanation of the experimental results of lepton production cross section in hadronic collisions such as nucleon-nucleon or nucleon-nuclei, it is of great importance to use quarks and sea quarks distribution function inside free and bound nucleons. In this paper the role of pion cloud inside the nucleus in the structure function of Cu and Pt nuclei and the EMC ratio of these nuclei were investigated by using harmonic oscillator model.

### Investigation of pionic contribution in the lepton and anti-lepton production cross section in p-Cu and p-Pt collision [Cross-Listing]

For detailed explanation of the experimental results of lepton production cross section in hadronic collisions such as nucleon-nucleon or nucleon-nuclei, it is of great importance to use quarks and sea quarks distribution function inside free and bound nucleons. In this paper the role of pion cloud inside the nucleus in the structure function of Cu and Pt nuclei and the EMC ratio of these nuclei were investigated by using harmonic oscillator model.

### Progress on nuclear modifications of structure functions [Cross-Listing]

We report progress on nuclear structure functions, especially on their nuclear modifications and a new tensor structure function for the deuteron. To understand nuclear structure functions is an important step toward describing nuclei and QCD matters from low to high densities and from low to high energies in terms of fundamental quark and gluon degrees of freedom beyond conventional hadron and nuclear physics. It is also practically important for understanding new phenomena in high-energy heavy-ion collisions at RHIC and LHC. Furthermore, since systematic errors of current neutrino-oscillation experiments are dominated by uncertainties of neutrino-nucleus interactions, such studies are valuable for finding new physics beyond current framework. Next, a new tensor-polarized structure function $b_1$ is discussed for the deuteron. There was a measurement by HERMES; however, its data are inconsistent with the conventional convolution estimate based on the standard deuteron model with D-state admixture. This fact suggests that a new hadronic phenomenon should exist in the tensor-polarized deuteron at high energies, and it will be experimentally investigated at JLab from the end of 2010's.

### Progress on nuclear modifications of structure functions

We report progress on nuclear structure functions, especially on their nuclear modifications and a new tensor structure function for the deuteron. To understand nuclear structure functions is an important step toward describing nuclei and QCD matters from low to high densities and from low to high energies in terms of fundamental quark and gluon degrees of freedom beyond conventional hadron and nuclear physics. It is also practically important for understanding new phenomena in high-energy heavy-ion collisions at RHIC and LHC. Furthermore, since systematic errors of current neutrino-oscillation experiments are dominated by uncertainties of neutrino-nucleus interactions, such studies are valuable for finding new physics beyond current framework. Next, a new tensor-polarized structure function $b_1$ is discussed for the deuteron. There was a measurement by HERMES; however, its data are inconsistent with the conventional convolution estimate based on the standard deuteron model with D-state admixture. This fact suggests that a new hadronic phenomenon should exist in the tensor-polarized deuteron at high energies, and it will be experimentally investigated at JLab from the end of 2010's.

### Progress on nuclear modifications of structure functions [Cross-Listing]

We report progress on nuclear structure functions, especially on their nuclear modifications and a new tensor structure function for the deuteron. To understand nuclear structure functions is an important step toward describing nuclei and QCD matters from low to high densities and from low to high energies in terms of fundamental quark and gluon degrees of freedom beyond conventional hadron and nuclear physics. It is also practically important for understanding new phenomena in high-energy heavy-ion collisions at RHIC and LHC. Furthermore, since systematic errors of current neutrino-oscillation experiments are dominated by uncertainties of neutrino-nucleus interactions, such studies are valuable for finding new physics beyond current framework. Next, a new tensor-polarized structure function $b_1$ is discussed for the deuteron. There was a measurement by HERMES; however, its data are inconsistent with the conventional convolution estimate based on the standard deuteron model with D-state admixture. This fact suggests that a new hadronic phenomenon should exist in the tensor-polarized deuteron at high energies, and it will be experimentally investigated at JLab from the end of 2010's.

### Progress on nuclear modifications of structure functions [Cross-Listing]

We report progress on nuclear structure functions, especially on their nuclear modifications and a new tensor structure function for the deuteron. To understand nuclear structure functions is an important step toward describing nuclei and QCD matters from low to high densities and from low to high energies in terms of fundamental quark and gluon degrees of freedom beyond conventional hadron and nuclear physics. It is also practically important for understanding new phenomena in high-energy heavy-ion collisions at RHIC and LHC. Furthermore, since systematic errors of current neutrino-oscillation experiments are dominated by uncertainties of neutrino-nucleus interactions, such studies are valuable for finding new physics beyond current framework. Next, a new tensor-polarized structure function $b_1$ is discussed for the deuteron. There was a measurement by HERMES; however, its data are inconsistent with the conventional convolution estimate based on the standard deuteron model with D-state admixture. This fact suggests that a new hadronic phenomenon should exist in the tensor-polarized deuteron at high energies, and it will be experimentally investigated at JLab from the end of 2010's.

### A new approach to the variability characterization of active galactic nuclei

The normalized excess variance is a popular method used by many authors to estimate the variability of active galactic nuclei (AGNs), especially in the X-ray band. We show that this estimator is affected by the cosmological time dilation, so that it should be appropriately corrected when applied to AGN samples distributed in wide redshift intervals. We propose a formula to modify this estimator, based on the use of the structure function. To verify the presence of the cosmological effect and the reliability of the proposed correction, we use data extracted from the XMM-Newton Serendipitous Source Catalogue, data release 5 (XMMSSC-DR5), and cross-matched with the Sloan Digital Sky Survey quasar catalogue, of data release 7 and 12.

### Pion structure function from leading neutron electroproduction and SU(2) flavor asymmetry [Cross-Listing]

We examine the efficacy of pion exchange models to simultaneously describe leading neutron electroproduction at HERA and the $\bar{d}-\bar{u}$ flavor asymmetry in the proton. A detailed $\chi^2$ analysis of the ZEUS and H1 cross sections, when combined with constraints on the pion flux from Drell-Yan data, allows regions of applicability of one-pion exchange to be delineated. The analysis disfavors several models of the pion flux used in the literature, and yields an improved extraction of the pion structure function and its uncertainties at parton momentum fractions in the pion of $4 \times 10^{-4} \lesssim x_\pi \lesssim 0.05$ at a scale of $Q^2$=10 GeV$^2$. Based on the fit results, we provide estimates for leading proton structure functions in upcoming tagged deep-inelastic scattering experiments at Jefferson Lab on the deuteron with forward protons.

### Pion structure function from leading neutron electroproduction and SU(2) flavor asymmetry

We examine the efficacy of pion exchange models to simultaneously describe leading neutron electroproduction at HERA and the $\bar{d}-\bar{u}$ flavor asymmetry in the proton. A detailed $\chi^2$ analysis of the ZEUS and H1 cross sections, when combined with constraints on the pion flux from Drell-Yan data, allows regions of applicability of one-pion exchange to be delineated. The analysis disfavors several models of the pion flux used in the literature, and yields an improved extraction of the pion structure function and its uncertainties at parton momentum fractions in the pion of $4 \times 10^{-4} \lesssim x_\pi \lesssim 0.05$ at a scale of $Q^2$=10 GeV$^2$. Based on the fit results, we provide estimates for leading proton structure functions in upcoming tagged deep-inelastic scattering experiments at Jefferson Lab on the deuteron with forward protons.

### Recent progress in some exclusive and semi-exclusive processes in proton-proton collisions

We present the main results of our recent analyses of exclusive production of vector charmonia ($J/\psi$ and $\psi'$) in $k_t$-factorization approach and for $\gamma \gamma$ production of charged dilepton pairs in exclusive and semiinclusive processes in a new approach, similar in spirit to $k_t$-factorization. The results for charmonia are compared with recent results of the LHCb collaboration. We include some helicity flip contributions and quantify the effect of absorption correction. The effect of $c \bar c$ wave function is illustrated. We present uncertainties related to $F_2$ structure function which are the main ingredient of the approach. Our results are compared with recent CMS data for dilepton production with lepton isolation cuts imposed.

### Helicity Evolution at Small-x [Replacement]

We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.

### Helicity Evolution at Small-x [Replacement]

We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.

### Helicity Evolution at Small-x [Replacement]

We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.

### Helicity Evolution at Small-x [Replacement]

We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.

### Helicity Evolution at Small-x

We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\alpha_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\alpha_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. As a cross-check, in the ladder approximation, our equations map onto the same ladder limit of the infrared evolution equations for $g_1$ structure function derived previously by Bartels, Ermolaev and Ryskin.