Small-<Emphasis Type="Italic">x</Emphasis> Behavior of Non-singlet Spin Structure Function and Bjorken Sum Rule with pQCD Correction up to NNLO and Higher Twist Correction

A calculation of the non-singlet part of spin dependent structure function, \(xg_{1}^{NS}(x,Q^{2})\) and associated sum rule, the Bjorken Sum rule up to next-next-to-leading order(NNLO) is presented. We use a unified approach incorporating Regge theory and the theoretical framework of perturbative Quantum Chromodynamics. Using a Regge behaved model with Q ² dependent intercept as the initial input, we have solved the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation up to NNLO at small-x for \(xg_{1}^{NS}(x,Q^{2})\) and the solutions are utilised to calculate the polarised Bjorken sum rule(BSR). We have also extracted the higher twist contribution to BSR based on a simple parametrisation. These results for both of \(xg_{1}^{NS}(x,Q^{2})\) and BSR, along with higher twist corrections are observed to be consistent with the available data taken from SMC, E143, HERMES, COMPASS and JLab experiments. In addition, our results are also compared with that of other theoretical and phenomenological analysis based on different models and a very good agreement is also observed in this regard. Further a very good consistency between our calculated results and theoretical QCD predictions of BSR is also achieved.     

Shadowing corrections to the derivative of the reduced cross-section at small x

We analyse the derivative of the reduced cross-section \(\frac {{\partial }{\sigma ^{s}_{r}}}{{\partial }{\ln }y}|_{x}\) , using the nonlinear Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (NLDGLAP) evolution equation at small x. The small x behaviour of the structure functions are obtained by solving the Gribov, Levin, Ryskin, Mueller and Qiu (GLR-MQ) evolution equation with the nonlinear shadowing term incorporated. We show that the strong rise corresponding to the linear QCD evolution equations, can be tamed by screening effects.     

A new numerical method for obtaining gluon distribution functions G(x,Q 2)=xg(x,Q 2), from the proton structure function F_{2}^{\gamma p}(x,Q^{2})

An exact expression for the leading-order (LO) gluon distribution function G(x,Q ²)=xg(x,Q ²) from the DGLAP evolution equation for the proton structure function $F_{2}^{\gamma p}(x,Q^{2})$ for deep inelastic γ ^* p scattering has recently been obtained (Block et al., Phys. Rev. D 79:014031, 2009) for massless quarks, using Laplace transformation techniques. Here, we develop a fast and accurate numerical inverse Laplace transformation algorithm, required to invert the Laplace transforms needed to evaluate G(x,Q ²), and compare it to the exact solution. We obtain accuracies of less than 1 part in 1000 over the entire x and Q ² spectrum. Since no analytic Laplace inversion is possible for next-to-leading order (NLO) and higher orders, this numerical algorithm will enable one to obtain accurate NLO (and NNLO) gluon distributions, using only experimental measurements of $F_{2}^{\gamma p}(x,Q^{2})$ .     

Electroproduction of <Emphasis Type="Italic">J/ψ</Emphasis> mesons within the semihard QCD approach and the color-singlet model

The deep-inelastic production of J/ψ mesons in electron-proton interactions at the HERA collider is considered within the semihard (k_T-factorization) QCD approach and within the color-singlet model. The dependence of the Q², p _T ² , z, y* and W distributions of J/ψ mesons on various sets of unintegrated gluon distributions and the dependence of the spin parameter α on p _T ² and Q² are investigated. The results of the calculations are compared with the latest experimental data obtained by the H1 and ZEUS Collaborations at the HERA collider. It is shown that experimental investigations of the polarization properties of J/ψ mesons over the kinematical region Q²<1 GeV² may provide an additional test of the statement that the dynamics of gluon distributions is governed by the Balitsky-Fadin-Kuraev-Lipatov equations.     

Parton Distribution Functions from QCD Analysis of HERA Combined Inclusive e ± p Scattering Cross Section Data

Utilizing very recent deep inelastic scattering measurements, a QCD analysis of proton structure function ${F_{2}^{p} (x,Q^2)}$ is presented. A wide range of the inclusive neutral-current deep-inelastic-scattering (NC DIS) data used in order to extract an updated set of parton distribution functions (PDFs). The HERA ‘combined’ data set on ${\sigma_{r,NC}^\pm (x,Q^2)}$ together with all available published data for heavy quarks ${F_2^{c,b}(x,Q^2)}$ , longitudinal F _L (x, Q ²) and also very recent reduced DIS cross section ${\sigma_{r,NC}^\pm (x,Q^2)}$ data from HERA experiments are the input in the present next-to-leading order (NLO) QCD analysis which determines a new set of parton distributions, called ${{\tt KKT11C}}$ . The extracted PDFs in the ‘fixed flavour number scheme’ (FFNS) are in very good agreement with the available theoretical models.     

Condensates in quantum chromodynamics

The paper presents a short review of our knowledge today on vacuum condensates in quantum chromodynamics (QCD). The condensates are defined as vacuum averages of the operators which arise due to nonperturbative effects. The important role of condensates in determining physical properties of hadrons and of their low-energy interactions in QCD is underlined. The special value of the quark condensate, connected to the existence of baryon masses, is mentioned. Vacuum condensates induced by external fields are discussed. QCD at low energy is checked on the basis of the data on hadronic τ decay. In theoretical analysis, the terms of perturbation theory (PT) up to α _s ³ are accounted for; in the operator product expansion (OPE), those up to dimension 8. The total probability of the decay τ → hadrons (with zero strangeness) and of the τ-decay structure functions are best described at α _s (m _τ ² )=0.330±0.025. It is shown that the Borel sum rules for τ-decay structure functions along the rays in the q ²-complex plane are in agreement with experiment, having an accuracy of ～2% at the values of the Borel parameter |M ²|>0.8 GeV². The magnitudes of dimension 6 and 8 condensates were found, and the limitations on gluon condensates were obtained. The sum rules for the charmed-quark vector-current polarization operator were analyzed in three loops (i.e., in order α _s ² ). The value of the charmed-quark mass (in an \(\overline {MS} \) regularization scheme) was found to be \(\bar m_c (\bar m_c^2 ) = 1.275 \pm 0.015\) GeV, and the value of gluon condensate was estimated as 〈0|(α _s/π)G ²|0〉=0.009±0.007 GeV⁴. The general conclusion is that the QCD described by PT + OPE is in good agreement with experiment at Q ²?1 GeV².     

Non-linear QCD dynamics in two-photon interactions at high energies

Perturbative QCD predicts that the growth of the gluon density at high energies should saturate, forming a Color Glass Condensate (CGC), which is described in mean field approximation by the Balitsky–Kovchegov (BK) equation. In this paper we study the γγ interactions at high energies and estimate the main observables which will be probed at future linear colliders using the color dipole picture. We discuss in detail the dipole–dipole cross section and propose a new relation between this quantity and the dipole scattering amplitude. The total γγ, γ ^? γ ^? cross sections and the real photon structure function $F_{2}^{\gamma }(x,Q^{2})$ are calculated using the recent solution of the BK equation with running coupling constant and the predictions are compared with those obtained using phenomenological models for the dipole–dipole cross section and scattering amplitude. We demonstrate that these models are able to describe the LEP data at high energies, but predict a very different behavior for the observables at higher energies. Therefore we conclude that the study of γγ interactions can be useful to constrain the QCD dynamics.     

Instability of Interacting Ghost Dark Energy Model in an Anisotropic Universe

A new dark energy model called “ghost dark energy” was recently suggested to explain the observed accelerating expansion of the universe. This model originates from the Veneziano ghost of QCD. The dark energy density is proportional to Hubble parameter, ρ _Λ = α H, where α is a constant of order \({\Lambda }^{3}_{QCD}\) and Λ _{Q C D} ～ 100M e V is QCD mass scale. In this paper, we investigate about the stability of generalized QCD ghost dark energy model against perturbations in the anisotropic background. At first, the ghost dark energy model of the universe with spatial BI model with/without the interaction between dark matter and dark energy is discussed. In particular, the equation of state and the deceleration parameters and a differential equation governing the evolution of this dark energy model are obtained. Then, we use the squared sound speed \({v_{s}^{2}}\) the sign of which determines the stability of the model. We explore the stability of this model in the presence/absence of interaction between dark energy and dark matter in both flat and non-isotropic geometry. In conclusion, we find evidence that the ghost dark energy might can not lead to a stable universe favored by observations at the present time in BI universe.     

QCD Evolution of Nuclear Structure Functions at Large X: EMC Effect and Cumulative Processes

QCD evolution of nuclear structure functions at large x is reviewed within the an approach based on QCD factorization for hard processes and multiquark flucton model. In this approach, x > 1 region of the nuclear structure functions is intimately related with x < 1 region due to manifestation of quark and gluon degrees of freedom in nuclei. Properties of QCD evolution and observed EMC-ratio for nuclear structure functions at x < 1 result in an admixture of hard extra sea quark distribution. This extra nuclear quark sea provides a bump above unity for EMC-ratio at small x and becomes dominant in the nuclear quark sea for cumulative region x > 1. It leads to a striking prediction, confirmed by data, for the same spectrum slopes of all cumulative hadrons in nuclear fragmentation region.     

Light-Front Quark Model Analysis of Meson-Photon Transition Form Factor

We discuss \({(\pi^{0}, \eta, \eta') \to \gamma^{*}\gamma}\) transition form factors using the light-front quark model. Our discussion includes the analysis of the mixing angles for \({\eta-\eta'}\). Our results for \({Q^{2} F_{(\pi^0,\eta,\eta')\to\gamma^*\gamma}(Q^2)}\) show scaling behavior for high Q² consistent with pQCD predictions.     

Analysis of the logarithmic slope of <Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript> from the Regge gluon density behavior at small <Emphasis Type="Italic">x</Emphasis>

We study the accuracy of the Regge behavior of the gluon distribution function for an approximate relation that is frequently used to extract the logarithmic slopes of the structure function from the gluon distribution at small x. We show that the Regge behavior analysis results are comparable with HERA data and are also better than other methods that expand the gluon density at distinct points of expansion. We also show that for Q ² = 22.4 GeV², the x dependence of the data is well described by gluon shadowing corrections to the GLR-MQ equation. The resulting analytic expression allows us to predict the logarithmic derivative ∂F ₂(x, Q ²)/∂lnQ ² and to compare the results with the H1 data and a QCD analysis fit with the MRST parameterization input.     

Saturation Approach in Top Production

The reduced cross section for the top production, in deep inelastic scattering, determined when we use the Kharzeev–Levin–Nardi (KLN) model of the low x gluon distributions. With respect to this model, a compact formula for the ratio R^t(τ) provided that it is approximately independent of τ behavior at low- τ values. For the top production where Q_sr_⊥ ? 1, the model reduced to color transparency.     

Investigation of the Balitsky-Fadin-Kuraev-Lipatov dynamics of gluon distributions in the inelastic production of <Emphasis Type="Italic">J/ψ</Emphasis> mesons at the HERA collider

The inelastic photo-and electroproduction of J/ψ mesons at the HERA collider are considered within the semihard (k _T -factorization) QCD approach and the color-singlet model. The total, differential, and double-differential cross sections for the inelastic production of J/ψ mesons are investigated versus the Pomeron intercept Δ, which is the basic parameter of low-x physics; also studied here is the spin alignment parameter α versus the square of the transverse momentum, p _ψT ² , and the variable z. The theoretical results obtained in the present study are compared with the latest experimental data of the H1 and ZEUS Collaborations. It is shown that experimental investigations of the polarization properties of J/ψ mesons at the HERA collider for Q²<1 GeV² may provide an additional test of the Balitsky-Fadin-Kuraev-Lipatov dynamics of gluon distributions.     

The Schrödinger Equation,the Zero-Point Electromagnetic Radiation,and the Photoelectric Effect

A Schrödinger type equation for a mathematical probability amplitude Ψ(x,t) is derived from the generalized phase space Liouville equation valid for the motion of a microscopic particle, with mass M and charge e, moving in a potential V(x). The particle phase space probability density is denoted Q(x,p,t), and the entire system is immersed in the “vacuum” zero-point electromagnetic radiation. We show, in the first part of the paper, that the generalized Liouville equation is reduced to a simpler Liouville equation in the equilibrium limit where the small radiative corrections cancel each other approximately. This leads us to a simpler Liouville equation that will facilitate the calculations in the second part of the paper. Within this second part, we address ourselves to the following task: Since the Schrödinger equation depends on \(\hbar \), and the zero-point electromagnetic spectral distribution, given by \(\rho _{0}{(\omega )} = \hbar \omega ^{3}/2 \pi ^{2} c^{3}\), also depends on \(\hbar \), it is interesting to verify the possible dynamical connection between ρ₀(ω) and the Schrödinger equation. We shall prove that the Planck’s constant, present in the momentum operator of the Schrödinger equation, is deeply related with the ubiquitous zero-point electromagnetic radiation with spectral distribution ρ₀(ω). For simplicity, we do not use the hypothesis of the existence of the L. de Broglie matter-waves. The implications of our study for the standard interpretation of the photoelectric effect are discussed by considering the main characteristics of the phenomenon. We also mention, briefly, the effects of the zero-point radiation in the tunneling phenomenon and the Compton’s effect.     

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

The parabolic Anderson model is defined as the partial differential equation ? u(x, t)/? t = κ Δ u(x, t) + ξ(x, t)u(x, t), x ∈ ? ^d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u ₀(x), x ∈ ? ^d , is typically taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2d κ, split into two at rate ξ ∨ 0, and die at rate (?ξ) ∨ 0. In earlier work we looked at the Lyapunov exponents

$$ \lambda _{p}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t} \log \mathbb {E} ([u(0,t)]^{p})^{1/p}, \quad p \in \mathbb{N} , \qquad \lambda _{0}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t}\log u(0,t). $$

For the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ. In the present paper we investigate what happens when κΔ is replaced by Δ^??, where ?? = {??(x, y) : x, y ∈ ? _d , x ～ y} is a collection of random conductances between neighbouring sites replacing the constant conductances κ in the homogeneous model. We show that the associated annealed Lyapunov exponents λ _p (??), p ∈ ?, are given by the formula

$$ \lambda _{p}(\mathcal{K} ) = \text{sup} \{\lambda _{p}(\kappa ) : \, \kappa \in \text{Supp} (\mathcal{K} )\}, $$

where, for a fixed realisation of ??, Supp(??) is the set of values taken by the ??-field. We also show that for the associated quenched Lyapunov exponent λ ₀(??) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(??) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ, and for all ?? satisfying a certain clustering property, namely, there are arbitrarily large balls where ?? is almost constant and close to any value in Supp(??). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of ?? where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of ?? where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

     

Estimating nonlinear effects in forward dijet production in ultra-peripheral heavy ion collisions at the LHC

Using the framework that interpolates between the leading power limit of the color glass condensate and the high energy (or \(k_{T}\)) factorization we calculate the direct component of the forward dijet production in ultra-peripheral \(\mathrm {Pb}\)–\(\mathrm {Pb}\) collisions at CM energy \(5.1\,\mathrm {TeV}\) per nucleon pair. The formalism is applicable when the average transverse momentum of the dijet system \(P_{T}\) is much bigger than the saturation scale \(Q_{s}\), \(P_{T}\gg Q_{s}\), while the imbalance of the dijet system can be arbitrary. The cross section is uniquely sensitive to the Weizsäcker–Williams (WW) unintegrated gluon distribution, which is far less known from experimental data than the most common dipole gluon distribution appearing in inclusive small-x processes. We have calculated cross sections and nuclear modification ratios using WW gluon distribution obtained from the dipole gluon density through the Gaussian approximation. The dipole gluon distribution used to get WW was fitted to the inclusive HERA data with the nonlinear extension of unified BFKL + DGLAP evolution equation. The saturation effects are visible but rather weak for realistic \(p_{T}\) cut on the dijet system, reaching about 20% with the cut as low as \(6\,\mathrm {GeV}\). We find that the LO collinear factorization with nuclear leading-twist shadowing predicts quite similar effects.     

Determination of quark-antiquark component of the photon wave function for <Emphasis Type="Italic">u,d, s</Emphasis> quarks

Based on the data for the transitions π⁰, η, η′ → γγ*(Q²) and reactions of the e⁺e^? annihilations e⁺e^? → ρ⁰, ω, ? and e⁺e^? → hadrons at 1<E _e+e? <3.7 GeV, we determine the light-quark components of the photon wave function \(\gamma * (Q^2 ) \to q\bar q(q = u,d,s)\) for the region 0 ? Q² ? 1 (GeV/c)².     

Long Time,Large Scale Limit of the Wigner Transform for a System of Linear Oscillators in One Dimension

We consider the long time, large scale behavior of the Wigner transform W _? (t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile et al. in Phys. Rev. Lett. 96 (2006) to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile et al. in Arch. Rat. Mech. 195(1):171–203 (2009). In the present paper we prove that in the unpinned case there exists γ ₀>0 such that for any γ∈(0,γ ₀] the weak limit of W _? (t/? ^3/2γ ,x/? ^γ ,k), as ??1, satisfies a one dimensional fractional heat equation \(\partial_{t} W(t,x)=-\hat{c}(-\partial_{x}^{2})^{3/4}W(t,x)\) with \(\hat{c}>0\). In the pinned case an analogous result can be claimed for W _? (t/? ^2γ ,x/? ^γ ,k) but the limit satisfies then the usual heat equation.