# Posts Tagged structure function

## Recent Postings from structure function

### High x Structure Function of the Virtually Free Neutron

The pole extrapolation method is applied for the first time to data on semi-inclusive deep-inelastic scattering off the deuteron with tagged spectator protons to extract the high Bjorken x structure function of the neutron. This approach is based on the extrapolation of the measured cross sections at different momenta of the detected spectator proton to the non-physical pole of the bound neutron in the deuteron. The advantage of the method is that it makes it possible to suppress nuclear effects in a maximally model independent way. The neutron structure functions obtained in this way demonstrate surprising x dependence at x> 0.6, indicating the possibility of a rise in the neutron to proton structure function ratio. Such a rise may indicate new dynamics in the generation of high x quarks in the nucleon. One such mechanism we discuss is the possible dominance of short-range isosinglet quark-quark correlations that can enhance the d-quark distribution in the proton resulting in d/u -> 1.

### Mapping the Gas Turbulence in the Coma Cluster: Predictions for Astro-H

Astro-H will be able for the first time to map gas velocities and detect turbulence in galaxy clusters. One of the best targets for turbulence studies is the Coma cluster, due to its proximity, absence of a cool core, and lack of a central active galactic nucleus. To determine what constraints Astro-H will be able to place on the Coma velocity field, we construct simulated maps of the projected gas velocity and compute the second-order structure function, an analog of the velocity power spectrum. We vary the injection scale, dissipation scale, slope, and normalization of the turbulent power spectrum, and apply measurement errors and finite sampling to the velocity field. We find that even with sparse coverage of the cluster, Astro-H will be able to measure the Mach number and the injection scale of the turbulent power spectrum–the quantities determining the energy flux down the turbulent cascade and the diffusion rate for everything that is advected by the gas (metals, cosmic rays, etc). Astro-H will not be sensitive to the dissipation scale or the slope of the power spectrum in its inertial range, unless they are outside physically-motivated intervals. We give the expected confidence intervals for the injection scale and the normalization of the power spectrum for a number of possible pointing configurations, combining the structure function and velocity dispersion data. Importantly, we also determine that measurement errors on the line shift will bias the velocity structure function upward, and show how to correct this bias.

### The 3-Loop Non-Singlet Heavy Flavor Contributions to the Structure Function g_1(x,Q^2) at Large Momentum Transfer

We calculate the massive flavor non-singlet Wilson coefficient for the heavy flavor contributions to the polarized structure function $g_1(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in Quantum Chromodynamics at general values of the Mellin variable $N$ and the momentum fraction $x$, and derive heavy flavor corrections to the Bjorken sum-rule. Numerical results are presented for the charm quark contribution. Results on the structure function $g_2(x,Q^2)$ in the twist-2 approximation are also given.

### The Spin Structure Function $g_1^{\rm p}$ of the Proton and a Test of the Bjorken Sum Rule [Cross-Listing]

New results for the double spin asymmetry $A_1^{\rm p}$ and the proton longitudinal spin structure function $g_1^{\rm p}$ are presented. They were obtained by the COMPASS collaboration using polarised 200 GeV muons scattered off a longitudinally polarised NH$_3$ target. The data were collected in 2011 and complement those recorded in 2007 at 160\,GeV, in particular at lower values of $x$. They improve the statistical precision of $g_1^{\rm p}(x)$ by about a factor of two in the region $x\lesssim 0.02$. A next-to-leading order QCD fit to the $g_1$ world data is performed. It leads to a new determination of the quark spin contribution to the nucleon spin, $\Delta \Sigma$ ranging from 0.26 to 0.36, and to a re-evaluation of the first moment of $g_1^{\rm p}$. The uncertainty of $\Delta \Sigma$ is mostly due to the large uncertainty in the present determinations of the gluon helicity distribution. A new evaluation of the Bjorken sum rule based on the COMPASS results for the non-singlet structure function $g_1^{\rm NS}(x,Q^2)$ yields as ratio of the axial and vector coupling constants $|g_{\rm A}/g_{\rm V}| = 1.22 \pm 0.05~({\rm stat.}) \pm 0.10~({\rm syst.})$, which validates the sum rule to an accuracy of about 9\%.

### Nucleon spin structure II: Spin structure function $g_1^p$ at small $x$

The spin structure function $g_1^p$ of the proton is studied in a two component framework, where the perturbative evolution of parton distributions and nonperturbative vector meson dominance model are used. We predict the $g_1^p$ asymmetric behavior at small $x$ from lower $Q^2$ to higher $Q^2$. We find that the contribution of the large gluon helicity dominates $g_1^p$ at $x>10^{-3}$ but mixed with nonperturbative component which complicates the asymptomatic behavior of $g_1^p$ at $x<10^{-3}$. The results are compatible with the data including the HERA early estimations and COMPASS new results. The predicted strong $Q^2$- and $x$-dependence of $g_1^p$ at $0.01<Q^2<3 GeV^2$ and $x<0.1$ can be checked on the next Electron-Ion Collider.

### Thermal conductivity of the neutron star crust: A reappraisal

We use classical and quantum Monte Carlo techniques to study the static structure function $S(q)$ of a one-component ion lattice and use it to calculate the thermal conductivity $\kappa$ of high-density solid matter expected in the neutron star crust. We also calculate the phonon spectrum using the dynamic-matrix method and use it to obtain $\kappa$ in the one-phonon approximation. We compare the results obtained with these methods and assess the validity of some commonly used approximations in the literature. We find that quantum effects became relevant for the calculation of $\kappa$ when the temperature $T\lesssim 0.3~\Omega_\mathrm{P}$, where $\Omega_\mathrm{P}$ is the ion plasma frequency. Dynamical information beyond the static structure becomes relevant when $T\lesssim 0.1~\Omega_\mathrm{P}$. We discuss the implications of these findings for calculations of $\kappa$ in multi-component systems and identify strategies for using Monte Carlo techniques in future work.

### Geometrical scaling behavior of the top structure functions ratio at the LHeC

We consider the ratio of the top structure functions $R^{t}(\tau_{t})$ in top pair production as a probe of the top content of the proton at the LHeC project. We study the top structure functions with the geometrical scaling of gluon distribution at small $x$ and show that top reduced cross section exhibits geometrical scaling in a large range of photon vitualities. This analysis shows that top longitudinal structure function has sizeable impact on the top reduced cross section at $Q^{2}{\approx}~ 4m_{t}^{2}$.

### On the longitudinal structure function in the dipole model

We compare new HERA data for the longitudinal structure function $F_{\rm L}$ with the predictions of different variants of the dipole model. In particular we show that the ratio $F_{\rm L}/F_2$ is well described by the dipole models and is rather insensitive to the details of the fit. Fits to $F_2$ are performed with the help of geometrical scaling (GS). Using the property of GS we derive the bounds for $F_{\rm L}/F_2$ both for the different versions of the dipole model and in the general case. Finally we briefly discuss how the higher Fock components of the photon wave function may affect these bounds.

### On the longitudinal structure function in the dipole model [Replacement]

We compare new HERA data for the longitudinal structure function $F_{\rm L}$ with the predictions of different variants of the dipole model. In particular we show that the ratio $F_{\rm L}/F_2$ is well described by the dipole models and is rather insensitive to the details of the fit. Fits to $F_2$ are performed with the help of geometrical scaling (GS). Using the property of GS we derive the bounds for $F_{\rm L}/F_2$ both for the different versions of the dipole model and in the general case. Finally we briefly discuss how the higher Fock components of the photon wave function may affect these bounds.

### Forward Compton Scattering with weak neutral current: constraints from sum rules [Cross-Listing]

We generalize forward real Compton amplitude to the case of the interference of the electromagnetic and weak neutral current, formulate a low-energy theorem, relate the new amplitudes to the interference structure functions and obtain a new set of sum rules. We address a possible new sum rule that relates the product of the axial charge and magnetic moment of the nucleon to the 0th moment of the structure function $g_5(\nu,0)$. We apply the GDH and the finite energy sum rule for constraining the dispersive $\gamma Z$-box correction to the proton’s weak charge.

### Coronal turbulence and the angular broadening of radio sources - the role of the structure function

The amplitude of density turbulence in the extended solar corona, especially near the dissipation scale, impinges on several problems of current interest. Radio sources observed through the turbulent solar wind are broadened due to refraction by and scattering off density inhomogeneities, and observations of scatter broadening are often employed to constrain the turbulence amplitude. The extent of such scatter broadening is usually computed using the structure function, which gives a measure of the spatial correlation measured by an interferometer. Most such treatments have employed analytical approximations to the structure function that are valid in the asymptotic limits $s \gg l_{i}$ or $s \ll l_{i}$, where $s$ is the interferometer spacing and $l_{i}$ is the inner scale of the density turbulence spectrum. We instead use a general structure function (GSF) that straddles these regimes, and quantify the errors introduced by the use of these approximations. We have included the effects of anisotropic scattering for distant cosmic sources viewed through the solar wind at small elongations. We show that the regimes where the GSF predictions are more accurate than those of the asymptotic expressions are not only of practical relevance, but are where inner scale effects influence estimates of scatter broadening. Taken together, we argue that the GSF should henceforth be used for scatter broadening calculations and estimates of turbulence amplitudes in the solar corona and solar wind.

### An Unobscured type II quasar candidate: SDSS J012032.19-005501.9

We report the finding of an unobscured type II Active Galactic Nuclei (AGN) candidate, SDSS J012032.19-005501.9 at a relatively high redshift of 0.601,which shows a number of unusual properties. It varies significantly on timescales of years as typical type I AGNs and marginally on timescales of weeks. The color-magnitude relation and the structure function are also consistent with that of type I AGNs, which imply that its variability likely originates from the black hole accretion system .However, no broad emission line is detected in the SDSS spectrum, and the upper limit of the equivalent width of the H$\rm \beta$ broad emission line is much less than that of type I AGNs. These properties suggest that SDSS J012032.19-005501.9 may be an unobscured quasar without broad emission lines intrinsically, namely an unobscured type II AGN or "true" type II AGN. Furthermore, its continuum luminosity is at least one order of magnitude fainter than the average value of thepast century expected from the [OIII] emission line. It indicates that SDSS J012032.19-005501.9 may be switching off. Additional possible scenarios to explain this intriguing source are also discussed. Future deep observations at multi-wavelengths are needed to reveal the nature of this peculiar and intriguing AGN.

### Hidden Color and the $b_1$ structure function of the deuteron [Cross-Listing]

The $b_1$ structure function is an observable feature of a spin-1 system sensitive to non-nucleonic components of the target nuclear wave function. A simple model for hidden-color, six-quark configurations is proposed and found to give substantial contributions for values of $x>0.2$. Good agreement with Hermes data is obtained. Predictions are made for an upcoming JLab experiment.

### Top structure function at the LHeC

The proposed linear and nonlinear behavior for the top structure function at the LHeC is considered. We present the conditions necessary to prediction the top structure function $F_{2}^{t}(x,Q^{2})$ with respect to the different predictions for the bahavior of the gluon at low $x$.

### Top structure function at the LHeC

The proposed linear and nonlinear behavior for the top structure function at the LHeC is considered. We present the conditions necessary to prediction the top structure function $F_{2}^{t}(x,Q^{2})$ with respect to the different predictions for the bahavior of the gluon at low $x$.

### Improved nonsinglet QCD analysis of the fixed-target DIS data

Deep inelastic scattering data on $F_2$ structure function obtained by BCDMS, SLAC and NMC collaborations in fixed-target experiments were analyzed in the non-singlet approximation with next-to-next-to-leading-order accuracy. The strong coupling constant is found to be $\alpha_s(M_Z^2) = 0.1157 \pm 0.0022 {(total exp.error)} + \biggl\{\begin{array}{l} +0.0028 \\ -0.0016 \end{array} {(theor)}$, which is seen to be well compatible with the average world value. Results obtained in the present paper confirm those derived in \cite{KKPS} by carrying out similar fits but with systematic errors in BCDMS data taken into account in a different way.

### Phase Characteristics of the ALMA 3 km Baseline Data

We present the phase characteristics study of the Atacama Large Millimeter/submillimeter Array (ALMA) long (up to 3 km) baseline, which is the longest baseline tested so far using ALMA. The data consist of long time-scale (10 – 20 minutes) measurements on a strong point source (i.e., bright quasar) at various frequency bands (bands 3, 6, and 7, which correspond to the frequencies of about 88 GHz, 232 GHz, and 336 GHz). Water vapor radiometer (WVR) phase correction works well even at long baselines, and the efficiency is better at higher PWV (>1 mm) condition, consistent with the past studies. We calculate the spatial structure function of phase fluctuation, and display that the phase fluctuation (i.e., rms phase) increases as a function of baseline length, and some data sets show turn-over around several hundred meters to 1 km and being almost constant at longer baselines. This is the first millimeter/submillimeter structure function at this long baseline length, and to show the turn-over of the structure function. Furthermore, the observation of the turn-over indicates that even if the ALMA baseline length extends to the planned longest baseline of 15 km, fringes will be detected at a similar rms phase fluctuation as that at a few km baseline lengths. We also calculate the coherence time using the 3 km baseline data, and the results indicate that the coherence time for band 3 is longer than 400 seconds in most of the data (both in the raw and WVR-corrected data). For bands 6 and 7, WVR-corrected data have about twice longer coherence time, but it is better to use fast switching method to avoid the coherence loss.

### Polarizability sum rule across real and virtual Compton scattering processes [Cross-Listing]

We derive a sum rule relating various electromagnetic properties of a spin-1/2 particle and consider its empirical implications for the proton. Given the measured values of the proton anomalous magnetic moment, electric and magnetic charge radii, the slope of the first moment of the spin structure function $g_1$, and the recently determined proton spin polarizability $\gamma_{E1M2}$, the sum rule yields a constraint on the low-momentum behavior of a generalized polarizability appearing in virtual Compton scattering. With the help of the presently ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and HI$\gamma$S facilities, the sum rule will provide a model-independent test of the low-energy spin structure of the nucleon.

### Polarizability sum rule across real and virtual Compton scattering processes [Cross-Listing]

We derive a sum rule relating various electromagnetic properties of a spin-1/2 particle and consider its empirical implications for the proton. Given the measured values of the proton anomalous magnetic moment, electric and magnetic charge radii, the slope of the first moment of the spin structure function $g_1$, and the recently determined proton spin polarizability $\gamma_{E1M2}$, the sum rule yields a constraint on the low-momentum behavior of a generalized polarizability appearing in virtual Compton scattering. With the help of the presently ongoing measurements of different electromagnetic observables at the MAMI, Jefferson Lab, and HI$\gamma$S facilities, the sum rule will provide a model-independent test of the low-energy spin structure of the nucleon.

### Gauge Invariance and QCD Twist-3 Factorization for Single Spin Asymmetries [Replacement]

The collinear factorization at twist-3 for Drell-Yan processes is studied with the motivation to solve the discrepancy in literature about the single spin asymmetry in the lepton angular distribution, and to show how QCD gauge invariance is realized in the hadronic tensor. The obtained result here agrees with our early result derived with a totally different approach. In addition to the asymmetry we can construct another two observables to identify the spin effect. We show that the gauge invariance of different contributions in the hadronic tensor is made in different ways by summing the effects of gluon exchanges. More interestingly is that we can show that the virtual correction to one structure function of the hadronic tensor, hence to some weighted SSA observables, is completely determined by the quark form factor. This will simplify the calculation of higher order corrections. The corresponding result in semi-inclusive DIS is also given for the comparison with Drell-Yan processes.

### 3-loop heavy flavor Wilson coefficients in deep-inelastic scattering

We present our most recent results on the calculation of the heavy flavor contributions to deep-inelastic scattering at 3-loop order in the large $Q^2$ limit, where the heavy flavor Wilson coefficients are known to factorize into light flavor Wilson coefficients and massive operator matrix elements. We describe the different techniques employed for the calculation and show the results in the case of the heavy flavor non-singlet and pure singlet contributions to the structure function $F_2(x,Q^2)$.

### The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension

The pure singlet asymptotic heavy flavor corrections to 3-loop order for the deep-inelastic scattering structure function $F_2(x,Q^2)$ and the corresponding transition matrix element $A_{Qq}^{(3), \sf PS}$ in the variable flavor number scheme are computed. In Mellin-$N$ space these inclusive quantities depend on generalized harmonic sums. We also recalculate the complete 3-loop pure singlet anomalous dimension for the first time. Numerical results for the Wilson coefficients, the operator matrix element and the contribution to the structure function $F_2(x,Q^2)$ are presented.

### The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension [Cross-Listing]

The pure singlet asymptotic heavy flavor corrections to 3-loop order for the deep-inelastic scattering structure function $F_2(x,Q^2)$ and the corresponding transition matrix element $A_{Qq}^{(3), \sf PS}$ in the variable flavor number scheme are computed. In Mellin-$N$ space these inclusive quantities depend on generalized harmonic sums. We also recalculate the complete 3-loop pure singlet anomalous dimension for the first time. Numerical results for the Wilson coefficients, the operator matrix element and the contribution to the structure function $F_2(x,Q^2)$ are presented.

### The Nonperturbative Structure of Hadrons

In this thesis we explore a diverse array of issues that strike at the inherently nonperturbative structure of hadrons at momenta below the QCD confinement scale. In so doing, we mainly seek a better control over the partonic substructure of strongly-interacting matter, especially as this relates to the nonperturbative effects that both motivate and complicate experiments — particularly DIS; among others, such considerations entail sub-leading corrections in $Q^2$, dynamical higher twist effects, and hadron mass corrections. We also present novel calculations of several examples of flavor symmetry violation, which also originates in the long-distance properties of QCD at low energy. Moreover, we outline a recently developed model, framed as a hadronic effective theory amenable to QCD global analysis, which provides new insights into the possibility of nonperturbative heavy quarks in the nucleon. This model can be extended to the scale of the lighter mesons, and we assess the accessibility of the structure function of the interacting pion in the resulting framework.

### The Nonperturbative Structure of Hadrons [Cross-Listing]

In this thesis we explore a diverse array of issues that strike at the inherently nonperturbative structure of hadrons at momenta below the QCD confinement scale. In so doing, we mainly seek a better control over the partonic substructure of strongly-interacting matter, especially as this relates to the nonperturbative effects that both motivate and complicate experiments — particularly DIS; among others, such considerations entail sub-leading corrections in $Q^2$, dynamical higher twist effects, and hadron mass corrections. We also present novel calculations of several examples of flavor symmetry violation, which also originates in the long-distance properties of QCD at low energy. Moreover, we outline a recently developed model, framed as a hadronic effective theory amenable to QCD global analysis, which provides new insights into the possibility of nonperturbative heavy quarks in the nucleon. This model can be extended to the scale of the lighter mesons, and we assess the accessibility of the structure function of the interacting pion in the resulting framework.

### Geometrical scaling in charm structure function ratios

By using a Laplace-transform technique, we solve the next-to-leading-order master equation for charm production and derive a compact formula for the ratio $R^{c}=\frac{F^{^{c\overline{c}}}_L}{F^{^{c\overline{c}}}_2}$, which is useful for extracting the charm structure function from the reduced charm cross section, in particular, at DESY HERA, at small x. Our results show that this ratio is independent of xat small x. In this method of determining the ratios, we apply geometrical scaling in charm production in deep inelastic scattering (DIS). Our analysis shows that the renormalization scales have a sizable impact on the ratio Rcat high $Q^{2}$. Our results for the ratio of the charm structure functions are in a goodagreement with some phenomenological models.

### Clustering structure of nuclei in deep inelastic processes

A clustering aspect is explained for the $^9$Be nucleus in charged-lepton deep inelastic scattering. Nuclear modifications of the structure function $F_2$ are studied by the ratio $R_{\rm EMC} = F_2^A /F_2^D$, where $A$ and $D$ are a nucleus and the deuteron, respectively. In a JLab experiment, an unexpectedly large nuclear modification slope $|dR_{\rm EMC}/dx|$ was found for $^9$Be, which could be related to its clustering structure. We investigated a mean conventional part of a nuclear structure function $F_2^A$ by a convolution description with nucleon momentum distributions calculated by antisymmetrized (or fermionic) molecular dynamics (AMD) and also by a simple shell model. We found that clustering effects are small in the conventional part, so that the JLab result could be associated with an internal nucleon modification or a short-range nuclear correlation which is caused by high densities due to cluster formation.

### The QCD analysis of the combined set for the F_3 structure function data based on the analytic approach [Replacement]

We apply analytic perturbation theory to the QCD analysis of the xF_3(x,Q^2) structure function considering a combined set of deep inelastic scattering data presented by several collaborations, and extract values of the scale parameter Lambda_{QCD}, the parameters of the form of the xF_3 structure function, and the x-shape of the higher twist contribution. We study the difference between the results obtained within the standard perturbative and analytic approaches in comparison with the experimental errors and state that the greatest difference occurs for large x.

### The QCD analysis of the combined set for the F_3 structure function data based on the analytic approach

We apply analytic perturbation theory to the QCD analysis of the xF_3(x,Q^2) structure function considering a combined set of deep inelastic scattering data presented by several collaborations, and extract values of the scale parameter Lambda_{QCD}, the parameters of the form of the xF_3 structure function, and the x-shape of the higher twist contribution. We study the difference between the results obtained within the standard perturbative and analytic approaches in comparison with the experimental errors and state that the greatest difference occurs for large x.

### The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity

We calculate the massive flavor non-singlet Wilson coefficient for the heavy flavor contributions to the structure function $F_2(x,Q^2)$ in the asymptotic region $Q^2 \gg m^2$ and the associated operator matrix element $A_{qq,Q}^{(3), \rm NS}(N)$ to 3-loop order in Quantum Chromodynamics at general values of the Mellin variable $N$. This matrix element is associated to the vector current and axial vector current for the even and the odd moments $N$, respectively. We also calculate the corresponding operator matrix elements for transversity, compute the contributions to the 3-loop anomalous dimensions to $O(N_F)$ and compare to results in the literature. The 3-loop matching of the flavor non-singlet distribution in the variable flavor number scheme is derived. All results can be expressed in terms of nested harmonic sums in $N$ space and harmonic polylogarithms in $x$-space. Numerical results are presented for the non-singlet charm quark contribution to $F_2(x,Q^2)$.

### The ratio of the beauty structure functions $R^b = F^b_L/F^b_2$ at low x

We study the structure functions $F_{k}^{b}(x,Q^{2})$ ($k=2, L$) and the reduced cross section $\sigma_{r}^{b}(x,Q^{2})$ for small values of Bjorken$^{,}$s $x$ variable with respect to the hard (Lipatov) pomeron for the gluon distribution and provide a compact formula for the ratio $R^{b}$ that is useful to extract the beauty structure function from the beauty reduced cross section, in particular at DESY HERA. Also we show that the effects of the nonlinear corrections to the gluon distribution tame the behavior of the beauty structure function and the beauty reduced cross section at low $x$.

### Long-term optical and radio variability of BL Lacertae

Well-sampled optical and radio light curves of BL Lacertae in B, V, R, I bands and 4.8, 8.0, 14.5 GHz from 1968 to 2014 were presented in this paper. A possible $1.26 \pm 0.05$ yr period in optical bands and a $7.50 \pm 0.15$ yr period in radio bands were detected based on discrete correlation function, structure function as well as Jurkevich method. Correlations among different bands were also analyzed and no reliable time delay was found between optical bands. Very weak correlations were detected between V band and radio bands. However, in radio bands the variation at low frequency lagged that at high frequency obviously. The spectrum of BL Lacertae turned mildly bluer when the object turned brighter, and stronger bluer-when-brighter trends were found for short flares. A scenario including a precessing helical jet and periodic shocks was put forward to interpret the variation characteristics of BL Lacertae.

### The structure function of Galactic \HI opacity fluctuations on AU scales based on MERLIN, VLA and VLBA data

We use MERLIN, VLA and VLBA observations of Galactic \HI absorption towards 3C~138 to estimate the structure function of the \HI opacity fluctuations at AU scales. Using Monte Carlo simulations, we show that there is likely to be a significant bias in the estimated structure function at signal-to-noise ratios characteristic of our observations, if the structure function is constructed in the manner most commonly used in the literature. We develop a new estimator that is free from this bias and use it to estimate the true underlying structure function slope on length scales ranging $5$ to $40$~AU. From a power law fit to the structure function, we derive a slope of $0.81^{+0.14}_{-0.13}$, i.e. similar to the value observed at parsec scales. The estimated upper limit for the amplitude of the structure function is also consistent with the measurements carried out at parsec scales. Our measurements are hence consistent with the \HI opacity fluctuation in the Galaxy being characterized by a power law structure function over length scales that span six orders of magnitude. This result implies that the dissipation scale has to be smaller than a few AU if the fluctuations are produced by turbulence. This inferred smaller dissipation scale implies that the dissipation occurs either in (i) regions with densities $\gtrsim 10^3$cm$^-3$ (i.e. similar to that inferred for "tiny scale" atomic clouds or (ii) regions with a mix of ionized and atomic gas (i.e. the observed structure in the atomic gas has a magneto-hydrodynamic origin).

### The structure function of Galactic \HI opacity fluctuations on AU scales based on MERLIN, VLA and VLBA data [Replacement]

We use MERLIN, VLA and VLBA observations of Galactic \HI absorption towards 3C~138 to estimate the structure function of the \HI opacity fluctuations at AU scales. Using Monte Carlo simulations, we show that there is likely to be a significant bias in the estimated structure function at signal-to-noise ratios characteristic of our observations, if the structure function is constructed in the manner most commonly used in the literature. We develop a new estimator that is free from this bias and use it to estimate the true underlying structure function slope on length scales ranging $5$ to $40$~AU. From a power law fit to the structure function, we derive a slope of $0.81^{+0.14}_{-0.13}$, i.e. similar to the value observed at parsec scales. The estimated upper limit for the amplitude of the structure function is also consistent with the measurements carried out at parsec scales. Our measurements are hence consistent with the \HI opacity fluctuation in the Galaxy being characterized by a power law structure function over length scales that span six orders of magnitude. This result implies that the dissipation scale has to be smaller than a few AU if the fluctuations are produced by turbulence. This inferred smaller dissipation scale implies that the dissipation occurs either in (i) regions with densities $\gtrsim 10^3$cm$^-3$ (i.e. similar to that inferred for "tiny scale" atomic clouds or (ii) regions with a mix of ionized and atomic gas (i.e. the observed structure in the atomic gas has a magneto-hydrodynamic origin).

### Dynamical behavior connection of the gluon distribution and the proton structure function at small $x$

We make a critical study of the relationship between the singlet structure function $F_{2}^{S}$ and the gluon distribution $G(x,Q^{2})$ proposed in the past two decades, which is frequently used to extract the gluon distribution from the proton structure function. We show that a simple relation is not generally valid in the simplest state. We complete this relation by using a Laplace transform method and hard-pomeron behavior at LO and NLO at small $x$. Our study shows that this relation is dependent on the splitting functions and initial conditions at $Q^{2} = Q^{2}_{0}$ and running coupling constant at NLO. The resulting analytic expression allows us to predict the proton structure function with respect to the gluon distributions and to compare the results with H1 data and a QCD analysis fit. Comparisons with other results are made and predictions for the proposed best approach are also provided.

### Photon structure function revisited [Replacement]

The flux of papers from electron positron colliders containing data on the photon structure function ended naturally around 2005. It is thus timely to review the theoretical basis and confront the predictions with a summary of the experimental results. The discussion will focus on the increase of the structure function with x (for x away from the boundaries) and its rise with log Q**2, both characteristics beeing dramatically different from hadronic structure functions. The agreement of the experimental observations with the theoretical calculations of the real and virtual photon structure is a striking success of QCD. It also allows a new determination of the QCD coupling constant which very well agrees with the value quoted in the literature.

### Photon structure function revisited [Replacement]

The flux of papers from electron positron colliders containing data on the photon structure function ended naturally around 2005. It is thus timely to review the theoretical basis and confront the predictions with a summary of the experimental results. The discussion will focus on the increase of the structure function with x (for x away from the boundaries) and its rise with log Q**2, both characteristics beeing dramatically different from hadronic structure functions. The agreement of the experimental observations with the theoretical calculations of the real and virtual photon structure is a striking success of QCD. It also allows a new determination of the QCD coupling constant which very well agrees with the value quoted in the literature.

### Photon structure function revisited

The flux of papers from electron positron colliders containing data on the photon structure function ended naturally around 2005. It is thus timely to review the theoretical basis and confront the predictions with a summary of the experimental results. The discussion will focus on the increase of the structure function with x (for x away from the boundaries) and its rise with log Q**2, both characteristics beeing dramatically different from hadronic structure functions. Comparing the data with a specific QCD prediction a new determination of the QCD coupling coupling constant is presented. The agreement of the experimental observations with the theoretical calculations of the real and virtual photon structure is a striking success of QCD.

### A lower bound on the Longitudinal Structure Function at small x from a self-similarity based model of Proton

Self-similarity based model of proton structure function at small \textit{x} was reported in the literature sometime back. The phenomenological validity of the model is in the kinematical region $6.2\, \times \, 10^{-7} \leq x \leq 10^{-2}$ and $0.045 \leq Q^{2} \leq 120 \, \mathrm{GeV^{2}}$. We use momentum sum rule to pin down the corresponding self-similarity based gluon distribution function valid in the same kinematical region. The model is then used to compute bound on the longitudinal structure function $F_{L}\left(x,Q^{2} \right)$ for Altarelli-Martinelli equation in QCD and is compared with the recent HERA data.

### Color dipole cross section and inelastic structure function

Instead of starting from a theoretically motivated form of the color dipole cross section in the dipole picture of deep inelastic scattering, we start with a parametrization of the deep inelastic structure function for electromagnetic scattering with protons, and then extract the color dipole cross section. Using the Donnachie-Landshoff parametrization of $F_2(x,Q^2)$, we find the dipole cross section from an approximate form of the presumed dipole cross section convoluted with the perturbative photon wave function for virtual photon splitting into a color dipole with massless quarks. The color dipole cross section determined this way works quite well in the massive case, reproducing the original Donnachie-Landshoff structure function for $0.1$ GeV$^2\leq Q^2\leq 10$ GeV$^2$. We discuss the large and small form of the dipole cross section and compare with other parameterizations.

### Color dipole cross section and inelastic structure function [Replacement]

Instead of starting from a theoretically motivated form of the color dipole cross section in the dipole picture of deep inelastic scattering, we start with a parametrization of the deep inelastic structure function for electromagnetic scattering with protons, and then extract the color dipole cross section. Using the parametrizations of $F_2(\xi=x \ {\rm or}\ W^2,Q^2)$ by Donnachie-Landshoff and Block et al., we find the dipole cross section from an approximate form of the presumed dipole cross section convoluted with the perturbative photon wave function for virtual photon splitting into a color dipole with massless quarks. The color dipole cross section determined this way reproduces the original structure function within about 10\% for $0.1$ GeV$^2\leq Q^2\leq 10$ GeV$^2$. We discuss the large and small form of the dipole cross section and compare with other parameterizations.

### Q2-evolution of parton densities at small x values. Effective scale for combined H1 and ZEUS F2 data

We use the Bessel-inspired behavior of the structure function F2 at small x, obtained for a flat initial condition in the DGLAP evolution equations. We fix the scale of the coupling constant, which eliminates the singular part of anomalous dimesnions at the next-to-leading order of approximation. The approach together with the "frozen" and analytic modifications of the strong coupling constant is used to study the precise combined H1 and ZEUS data for the structure function F2 published recently.

### The effect on PDFs and $\alpha_S(M_Z^2)$ due to changes in flavour scheme and higher twist contributions

I consider the effect on MSTW partons distribution functions (PDFs) due to changes in the choices of theoretical procedure used in the fit. I first consider using the 3-flavour fixed flavour number scheme instead of the standard general mass variable flavour number scheme used in the MSTW analysis. This results in the light quarks increasing at all relatively small $x$ values, the gluon distribution becoming smaller at high values of $x$ and larger at small $x$, the preferred value of the coupling constant $\alpha_S(M_Z^2)$ falling, particularly at NNLO, and the fit quality deteriorates. I also consider lowering the kinematic cut on $W^2$ for DIS data and simultaneously introducing higher twist terms which are fit to data. This results in much smaller effects on both PDFs and $\alpha_S(M_Z^2)$ than the scheme change, except for quarks at very high $x$. I show that the structure function one obtains from a fixed input set of PDFs using the fixed flavour scheme and variable flavour scheme differ significantly for $x \sim 0.01$ at high $Q^2$, and that this is due to the fact that in the fixed flavour scheme there is a slow convergence of large logarithmic terms of the form $(\alpha_S\ln(Q^2/m_c^2))^n$ relevant for this regime. I conclude that some of the most significant differences in PDF sets are largely due to the choice of flavour scheme used.

### The effect on PDFs and $\alpha_S(M_Z^2)$ due to changes in flavour scheme and higher twist contributions [Replacement]

I consider the effect on MSTW partons distribution functions (PDFs) due to changes in the choices of theoretical procedure used in the fit. I first consider using the 3-flavour fixed flavour number scheme instead of the standard general mass variable flavour number scheme used in the MSTW analysis. This results in the light quarks increasing at all relatively small $x$ values, the gluon distribution becoming smaller at high values of $x$ and larger at small $x$, the preferred value of the coupling constant $\alpha_S(M_Z^2)$ falling, particularly at NNLO, and the fit quality deteriorates. I also consider lowering the kinematic cut on $W^2$ for DIS data and simultaneously introducing higher twist terms which are fit to data. This results in much smaller effects on both PDFs and $\alpha_S(M_Z^2)$ than the scheme change, except for quarks at very high $x$. I show that the structure function one obtains from a fixed input set of PDFs using the fixed flavour scheme and variable flavour scheme differ significantly for $x \sim 0.01$ at high $Q^2$, and that this is due to the fact that in the fixed flavour scheme there is a slow convergence of large logarithmic terms of the form $(\alpha_S\ln(Q^2/m_c^2))^n$ relevant for this regime. I conclude that some of the most significant differences in PDF sets are largely due to the choice of flavour scheme used.

### Nonlinear correction to the longitudinal structure function at small x

We computed the longitudinal proton structure function $F_{L}$, using the nonlinear Dokshitzer-Gribov-Lipatov-Altarelli-parisi (NLDGLAP) evolution equation approach at small $x$. For the gluon distribution, the nonlinear effects are related to the longitudinal structure function. As, the very small $x$ behavior of the gluon distribution is obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. We show, the strong rise that is corresponding to the linear QCD evolution equations, can be tamed by screening effects. Consequently, the obtained longitudinal structure function shows a tamed growth at small $x$. We computed the predictions for all detail of the nonlinear longitudinal structure function in the kinematic range where it has been measured by $H1$ collaboration and compared with computation Moch, Vermaseren and Vogt at the second order with input data from MRST QCD fit.

### The approximation method for calculation of the exponent of the gluon distribution-$\lambda_{g}$ and the structure function-$\lambda_{S}$ at low $x$ [Replacement]

We present a set of formulae using the solution of the QCD Dokshitzer-Gribov-Lipatov-Altarelli-parisi (DGLAP) evolution equation to the extract of the exponent $\lambda_g$ gluon distribution and $\lambda_S$ structure function from the Regge- like behavior at low $x$. The exponents are found to be independent of $x$ and to increase linearly with ln$Q^{2}$ and compared with the most data from H1 Collaboration. We also calculated the structure function $F_{2}(x,Q^{2})$ and the gluon distribution $G(x,Q^{2})$ at low $x$ assuming the Regge- like behavior of the gluon distribution function at this limit and compared with NLO QCD fit to the H$1$ data, two Pomeron fit, multipole Pomeron exchange fit and MRST (A.D.Martin, R.G.Roberts, W.J.Stirling and R.S.Thorne), DL(A.Donnachie and P.V.Landshoff), NLO-GRV(M.Gluk, E.Reya and A.Vogt) fit results, respectively.

### The approximation method for calculation of the exponent of the gluon distribution-$\lambda_{g}$ and the structure function-$\lambda_{S}$ at low $x$

We present a set of formulae using the solution of the QCD Dokshitzer-Gribov-Lipatov-Altarelli-parisi (DGLAP) evolution equation to the extract of the exponent $\lambda_g$ gluon distribution and $\lambda_S$ structure function from the Regge- like behavior at low $x$. The exponents are found to be independent of $x$ and to increase linearly with ln$Q^{2}$ and compared with the most data from H1 Collaboration. We also calculated the structure function $F_{2}(x,Q^{2})$ and the gluon distribution $G(x,Q^{2})$ at low $x$ assuming the Regge- like behavior of the gluon distribution function at this limit and compared with NLO QCD fit to the H$1$ data, two Pomeron fit, multipole Pomeron exchange fit and MRST (A.D.Martin, R.G.Roberts, W.J.Stirling and R.S.Thorne), DL(A.Donnachie and P.V.Landshoff), NLO-GRV(M.Gluk, E.Reya and A.Vogt) fit results, respectively.

### Hard- Pomeron behavior of the Longitudinal Structure Function $F_{L}$ in the Next- to- Leading- Order at low $x$

We present an analytic formula to extract the longitudinal structure function in the next- to -leading order of the perturbation theory at low $x$, from the Regge- like behavior of the gluon distribution and the structure function at this limit. In this approach, the longitudinal structure function has the hard- Pomeron behavior. The determined values are compared with the $H1$ data and MRST model. All results can consistently be described within the framework of perturbative QCD which essentially show increases as $x$ decreases.

### Analytical approach for the approximate solution of the longitudinal structure function with respect to the GLR-MQ equation at small x

We show that the nonlinear corrections to the longitudinal structure function can be tamed the singularity behavior at low x values, with respect to GLR-MQ equations. This approach can determined the shadowing longitudinal structure function based on the shadowing corrections to the gluon and singlet quark structure functions. Comparing our results with HERA data show that at very low x this behavior completely tamed by these corrections.

### Analytical approach for the approximate solution of the longitudinal structure function with respect to the GLR-MQ equation at small x [Replacement]

We show that the nonlinear corrections to the longitudinal structure function can be tamed the singularity behavior at low x values, with respect to GLR-MQ equations. This approach can determined the shadowing longitudinal structure function based on the shadowing corrections to the gluon and singlet quark structure functions. Comparing our results with HERA data show that at very low x this behavior completely tamed by these corrections.