In this work we use the string/gauge duality within the Softwall Model (SW). In this model a dilaton field is introduced in the action for the fields playing the role of a soft infrared (IR) cutoff. The SW model is very useful as it provides linear Regge trajectories for mesons. Here, using a $10-$dimensional SW model, we calculate the corresponding structure functions for deep inelastic scattering (DIS) in which electrons are scattered off hadrons in a kinematical regime where the hadrons are broken apart, with high virtuality $q$, in the exponentially small $x$ (Bjorken parameter) regime. Our results for this regime are consistent with those achieved using other holographic and non-holographic approaches.
We use the AdS/CFT correspondence and the holographic softwall model to investigate the Deep Inelastic Scattering (DIS) in the exponentially small $x$ (Bjorken Parameter) regime. We calculate the corresponding structure functions for scalar fields. Using these results we studied the problem of the saturation line in the strong interactions. Our results are consistent with those achieved using other models.
For an accurate description of the polarized deep inelastic scattering at low $x$ including the logarithmic corrections, $\ln^2(1/x)$, is required. These corrections resummed strongly influence the behaviour of the spin structure functions and their moments. Results of the work of J. Kwiecinski and myself on this problem are reviewed.}
It is shown that the existing four-loop result for the Bjorken polarized sum rule for deep inelastic electron-nucleon scattering obtained within perturbative Quantum Chromodynamics should be supplemented by the calculation of the diagrams of the so called singlet type. We also give an explanation of the interesting coincidence of two different classes of diagrams, one of the non-singlet and one of the singlet type, contributing the $α_s^4$-approximation to the total cross-section of electron-positron annihilation into hadrons.
We examine the uncertainty of perturbative QCD factorization for hadron structure functions in deep inelastic scattering at a large value of the Bjorken variable xB. We analyze the target mass correction to the structure functions by using the collinear factorization approach in the momentum space. We express the long distance physics of structure functions and the leading target mass corrections in terms of parton distribution functions with the standard operator definition. We compare our result with existing work on the target mass correction. We also discuss the impact of a final-state jet…
The possibility of the first measurement of Bjorken unpolarized sum rule for $F_1$ structure function of $νN$ deep-inelastic scattering at Neutrino Factories is commented. The brief summary of various theoretical contributions to this sum rule is given. Using the next-to-leading set of parton distributions functions, we simulate the expected $Q^2$-behavior and emphasize that its measurement can allow to determine the value of the QCD strong coupling constant $α_s$ with reasonable theoretical uncertainty, dominated by the ambiguity in the existing estimates of the twist-4 non-perturbative $1/Q^…
We revisit the role of the chiral "triangle" anomaly in deeply inelastic scattering (DIS) of electrons off polarized protons employing a powerful worldline formalism. We demonstrate how the triangle anomaly appears at high energies in the DIS box diagram for the polarized proton structure function $g_1(x_B,Q^2)$ in both the Bjorken limit of large $Q^2$ and in the Regge limit of small $x_B$. We show that the operator product expansion is not required to extract the anomaly in either asymptotics though it is sufficient in the Bjorken limit. Likewise, the infrared pole in the anomaly arises in bo…
High transverse momentum pi^0 -mesons and charged particles are measured in deep inelastic e-p scattering events at low Bjorken-x taken with the H1 detector at HERA. The production of high-pt particles is strongly correlated to the emission of hard partons in QCD and is therefore sensitive to the dynamics of the strong interaction. For the first time the measurement of single particles has been extended to the region of small angles w.r.t. the proton remnant (forward region). This region is expected to be particularly sensitive to QCD evolution effects in final states. Results are presented as…
Factorization in deep inelastic scattering at Björken limit: Reduction to (1+1)D integrable models
Alikhanyan National Science Laboratory（阿利哈尼扬国家科学实验室）；Beijing Institute of Mathematical Sciences and Applications（北京数学与应用数学研究所）；Freie Universität Berlin（柏林自由大学）
In our recent paper (hep-ph/9501348) we argued that the Bjorken variable $x$ in deep inelastic scattering cannot be interpreted as the light cone momentum fraction $ξ$ even in the Bjorken limit and in zero order of the perturbation theory. The purpose of the present paper is to qualitatively explain this fact using only a few simplest kinematical relations.
We revisit high energy factorization for gluon mediated inclusive Deep Inelastic Scattering (DIS) for which we propose a new semi-classical approach that accounts systematically for the longitudinal extent of the target in contrast with the shockwave limit. In this framework, based on a partial twist expansion, we derive a factorization formula that involves a new gauge invariant unintegrated gluon distribution which depends explicitly on the Feynman $x$ variable. It is shown that both the Regge and Bjorken limits are recovered in this approach. We reproduce in particular the full one loop inc…
We study deep-inelastic scattering factorization on a nucleon in the end-point regime $x_B \sim 1-{\cal O}(Λ_{\rm QCD}/Q)$ where the traditional operator product expansion is supposed to fail. We argue, nevertheless, that the standard result holds to leading order in $1-x_B$ due to the absence of the scale dependence on $(1-x_B)Q$. Refactorization of the scale $(1-x_B)Q^2$ in the coefficient function can be made in the soft-collinear effective theory and remains valid in the end-point regime. On the other hand, the traditional refactorization approach introduces the spurious scale $(1-x_B)Q$ i…
We calculate the structure functions for unpolarized deep inelastic scattering of baryons using an AdS/QCD soft wall model that considers a dressed mass for the bulk fermionic fields. Considering the regime of large Bjorken parameter x, we compare the results for the proton structure function $F_2$ with experimental results.
We propose a method to assess the quality of geometrical scaling in Deep Inelastic Scattering and apply it to the combined HERA data on $γ^{\ast}p$ cross-section. Using two different approaches based on Bjorken $x$ binning and binning in $γ^{\ast}p$ scattering energy $W$, we show that geometrical scaling in variable $τ\sim Q^{2} x^λ$ works well up to Bjorken $x$'s 0.1. The corresponding value of exponent $λ$ is 0.32 -- 0.34.
In the theory of deep inelastic scattering (DIS) the final state interaction (FSI) between the struck quark and the remnants of the target is usually assumed to be negligible in the Bjorken limit. This assumption, still awaiting a full validation within nonperturbative QCD, is investigated in a model composed by two relativistic particles, interacting via a relativistic harmonic oscillator potential, within light-cone hamiltonian dynamics. An electromagnetic current operator whose matrix elements behave properly under Poincaré transformations is adopted. It is shown that: i) the parton model i…
The phenomenon of scaling in deep inelastic lepton-nucleon scattering is usually explained in terms of the Feynman parton model, and the logarithmic corrections to scaling are explained in the framework of perturbative QCD. For testing the validity of the parton model, we consider the deep inelastic electron scattering in a model in which the system electromagnetic current operator explicitly satisfies relativistic invariance and current conservation. Let the struck particle have the fraction $ξ$ of the total momentum in the infinite momentum frame. Then it is shown that, due to binding of par…
In the Bjorken limit of the present theory of deep inelastic scattering (DIS) the structure functions (up to anomalous dimensions and perturbative QCD corrections) are described by the parton model. However the current operator in the parton model does not properly commute with the representation operators corresponding to the Lorentz group, space reflection and time reversal. To investigate the violation of these symmetries in the parton model we consider a model in which the current operator explicitly satisfies extended Poincare invariance and current conservation. It is shown that due to b…
We study power corrections to the spin-averaged structure functions F_2 and F_L in the semi-inclusive region. The dramatic breakdown of the operator product expansion at large Bjorken x due to the formation of a narrow jet with invariant mass W (W^2=Q^2(1-x)/x) calls for an alternative approach. Our main conjecture is that the dominant contribution at each twist is the one which mixes under renormalization with the leading twist (ultraviolet dominance). At twist two the dominant perturbative corrections at large x due to soft and collinear radiation can be factorized into a jet function. From …
Nuclear deep inelastic scattering is considered in the framework of a model in which the current operator explicitly satisfies Poincare invariance and current conservation. The results considerably differ from the standard ones at small values of the Bjorken variable $x$. In particular, it is impossible to extract the neutron structure functions from the deuteron data at x<0.01 and we predict that the behavior of the deuteron structure functions at low x and large momentum transfer considerably differs from the behavior of the nucleon structure functions at such conditions. We also argue that …
We study deep inelastic scattering at large-'t Hooft coupling and finite $x$ from gauge/string duality beyond single-hadron final states which gives the leading large-$N_c$ contribution. Within the supergravity approximation, we calculate the subleading large-$N_c$ contribution by introducing an extra hadron into the final states. We find the contribution from these double-hadron final states will dominate in the Bjorken limit $q^2\rightarrow \infty$ compared with the single-hadron states. We discuss the implications of our results.
We present an explicit OPE analysis for the first moment of $g_1$ up to order $M^2/Q^2$. This result allows to calculate power corrections to the Bjorken and Ellis--Jaffe sum rules.
We consider deep inelastic scattering (DIS) on a large nucleus described as an extremal RN-AdS black hole using the holographic principle. Using the R-current correlators we determine the structure functions as a function Bjorken-x, and map it on a finite but large nucleus with fixed atomic number. The R-ratio of the nuclear structure functions exhibit strong shadowing at low-x.
We analyze geometrical scaling in deep inelastic scattering using experimental data from HERA ep collider. Parallel analyses are performed for energy and Bjorken-x data binnings. In particular, value of parameter λwhich governs x-dependence of saturation scale is found for both binnings: λ_En = 0.352 \pm 0.008 and λ_Bj = 0.302 \pm 0.004. We use the following method: λ_En is found as a value for which ratios σ^{W1}_{γ*p}(τ)/σ^{W2}_{γ*p}(τ) are closest to 1 (σ^{Wi}_{γ*p} is a photon-proton cross section with definite energy W_i and τ= Q^2 x^λ is a scaling variable); λ_Bj is found analogously usi…
Nuclear modification of hadron production in deep inelastic lepton-nucleus scattering can be applied to study the parton propagation mechanism in cold nuclear matter. By means of the analytic parameterization of quenching weight based on BDMPS formalism with the target nuclear geometry effect, the leading-order computations for hadron multiplicity ratios are performed with comparison to the HERMES charged pions production data on the quarks hadronization occurring outside the nucleus. The relation is discovered between quark transport coefficient and the measurable kinematic variables in deep …
The heavy mass limit of quark models based on the Bakamjian-Thomas cons\-truction reveals remarkable features. In addition to previously demonstrated properties of covariance and Isgur-Wise scaling, exact duality, leading to the Bjorken-Isgur-Wise sum rule, is proven, for the first time to our knowledge in relativistic quark models. Inelastic as well as elastic contributions to the sum rule are then discussed in terms of ground state averages of a few number of operators corresponding to the nonrelativistic dipole operator and various relativistic corrections.
So far the analyses of the polarized structure functions of the proton and neutron have been limited to the evaluation of their integrals and comparing them to the prediction of the static quark model of the nucleon. We extended our analysis to the x dependence of the polarized structure functions and observe: the measured structure function excellently agrees with the prediction of the static quark model for Bjorken $x>0.2$ and drops rapidly for $x<0.2$. It is suggested that for Bjorken $x>0.2$ electrons get scattered on the undamaged constituent quarks (alias valence quarks) denoted as quasi…
The HERA data on inelastic lepton proton scattering for values of the Bjorken scaling variable x < 0.05 are confronted with predictions based on Generalized Vector Dominance. Good agreement between theory and experiment is found over the full kinematic range of the squared four-momentum-transfer, Q^2, from Q^2 = 0 (photoproduction) to Q^2 < 350 GeV^2.
An idea of possible anomalous contribution of non-perturbative origin to the nucleon spin was examined by analysing data on spin asymmetries in polarized deep inelastic scattering of leptons on nucleons. The region of high Bjorken x was explored. It was shown that experimental data available at present do not evidence for this effect.
We discuss hadron mass corrections and threshold resummation for deep-inelastic scattering $\ell N\rightarrow\ell' X$ and semi-inclusive annihilation $e^+e^-\rightarrow h X$ processes, and provide a prescription how to consistently combine these two corrections respecting all kinematic thresholds. We find an interesting interplay between threshold resummation and target mass corrections for deep-inelastic scattering at large values of Bjorken $x_B$. In semi-inclusive annihilation, on the contrary, the two considered corrections are relevant in different kinematic regions and do not affect each…
A systematic study is performed of the structure of the twist-2 contributions to the virtual Compton amplitude in deep-inelastic non-forward scattering for general spin states in the non-local light cone expansion. The gauge invariance is proven in this approach for the twist-2 contributions for non-forward scattering. Relations are derived between the contributing non-forward amplitudes.
We consider restrictions imposed on the electromagnetic and weak current operators by Poincare invariance and show that some assumptions used in deriving the sum rules in deep inelastic scattering (DIS) have no physical ground. In particular there is no ground to neglect the contribution of nonperturbative effects to these operators, even in the Bjorken limit. Therefore the conclusion that the sum rules in DIS unambiguously follow from QCD is not substantiated.
We study the heavy-quark contributions to the proton structure functions $F_2^i(x,Q^2)$ and $F_L^i(x,Q^2)$, with $i=c,b$, for small values of Bjorken's $x$ variable and provide compact formulas for their ratios $R_i=F_L^i/F_2^i$ that are useful to extract $F_2^i(x,Q^2)$ from measurements of the doubly differential cross section of inclusive deep-inelastic scattering at DESY HERA. Our approach naturally explains why $R_i$ is approximately independent of $x$ and the details of the parton distribution functions in the low-$x$ regime.
We discuss effect of nuclear shadowing in neutrino deep-inelastic scattering in terms of non perturbative parton model. We found that for small Bjorken $x$ and large $Q^2$ the structure function $F_3$ is shadowed in nuclei about two times as stronger as $F_2$. The underlying reason and phenomenological aspects of this observation are discussed.
We fit the reduced cross section for deep-inelastic electron scattering data to a three parameter ln^2 s fit, A + beta ln^2 (s/s_0), where s= [Q^2/x] (1-x) + m^2, and Q^2 is the virtuality of the exchanged photon. Over a wide range in Q^2 (0.11 < Q^2 < 1200 GeV^2) all of the fits satisfy the logarithmic energy dependence of the Froissart bound. We can use these results to extrapolate to very large energies and hence to very small values of Bjorken x -- well beyond the range accessible experimentally. As Q^2 --> infinity, the structure function F_2^p(x, Q^2) exhibits Bjorken scaling, within exp…
We discuss the close relation between shadowing in deep-inelastic lepton-nucleus scattering and diffractive photo- and leptoproduction of hadrons from free nucleons. We show that the magnitude of nuclear shadowing at small Bjorken-x, as measured by the E665 and NMC collaborations, is directly related to HERA data on the amount of diffraction in the scattering from free nucleons.
Multiplicity correlations between the current and target regions of the Breit frame in deep-inelastic scattering processes are studied. It is shown that the correlations are sensitive to the first-order perturbative QCD effects and can be used to extract the behavior of the boson-gluon fusion rates as a function of the Bjorken variable. The behavior of the correlations is derived analytically and analyzed using a Monte Carlo simulation.
Based upon the color-dipole picture, we provide closed analytic expressions for the longitudinal and the transverse photoabsorption cross sections at low values of the Bjorken variable of x<0.1. We compare with the experimental data for the longitudinal-to-transverse ratio of the (virtual) photoabsorption cross section and with our previous fit to the experimental data for the total photoabsorption cross section. Scaling in terms of the low-x scaling variable eta(W^2,Q^2) is analyzed in terms of the reduced cross section of deep inelastic scattering.
Experiments at HERA looking for deep-inelastic electroproduction of neutrons in the proton fragmentation region are in progress. They are aimed to measure the pion structure function at small Bjorken $x$. The important condition for such a study is to establish under what kinematical conditions the dominance of the pion-pole graph in the process is guaranteed. We analyse other sources of theleading neutron, in order to figure out the kinematical region where the one-pion exchange dominates.
In this talk, I will concentrate on $Q^2$-dependence of deep inelastic sum rules. I will first give a modern definition of deep-inelastic sum rules and then discuss physical origins of their scaling violation at finite $Q^2$. Following this, I discuss a few well-known examples, in particular, the Bjorken sum rule, which is at the center of interest of this symposium. Finally, I consider the $Q^2 \to 0$ limit of sum rules using low-energy theorems. I think this can motivate some interesting CEBAF physics.
We calculate shadowing in the process of deep inelastic interactions of leptons with nuclei in the perturbative regime of QCD. We find appreciable shadowing for heavy nuclei (e.g. Pb) in the region of small Bjorken scaling variable $10^{-5}\leq x \leq 10^{-3}$. This shadowing depends weakly on $Q^2$, but it may be strongly influenced, especially at $x \geq 10^{-3}$, by the existence of real parts of the forward scattering amplitudes.
Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken, the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axial-vector and two non-singlet vector currents and the non-singlet axial-vect…
We study the power corrections (infrared renormalon contributions) to the coefficient functions for non-singlet deep inelastic structure functions due to gluon vacuum polarization insertions in one-loop graphs. Remarkably, for all the structure functions $F_1$, $F_2$, $F_3$ and $g_1$, there are only two such contributions, corresponding to $1/Q^2$ and $1/Q^4$ power corrections. We compute their dependence on Bjorken $x$. The results could be used to model the dominant higher-twist contributions.
The (electromagnetic or weak) current operator responsible for deep inelastic scattering (DIS) should be local and satisfy the well-known commutation relations with the representation operators of the Poincare group. The problem whether these conditions are compatible with the factorization theorem and operator product expansion is investigated in detail. We argue that the current operator contains a nontrivial nonperturbative part which contributes to DIS even in the Bjorken limit. Nevertheless there exists a possibility that many results of the standard theory remain.
We clarify conflicting results in the literature on coefficient functions in front of higher twist operators contributing to the parton sum rules for deep inelastic scattering from polarized targets. The necessary corrections do not affect our calculations of matrix elements, published in Phys.Lett.B242(1990)245, but change final estimates of the $\sim 1/Q^2$ contributions to Bjorken and Ellis--Jaffe sum rules.
This is a review of the program we started in 1968 to understand and generalize Bjorken scaling and Feynman's parton model in a canonical quantum field theory. It is shown that the parton model proposed for deep inelastic electron scatterings can be derived if a transverse momentum cutoff is imposed on all particles in the theory so that the impulse approximation holds. The deep inelastic electron-positron annihilation into a nucleon plus anything else is related by the crossing symmetry of quantum field theory to the deep inelastic electron-nucleon scattering. We have investigated the implica…
Differential cross sections of deep inelastic scattering of charged leptons from hadrons are investigated by using the gauge/string duality. We consider vector mesons derived from different holographic dual models obtaining a general expression. We focus on the strongly coupled regime of dual gauge theories for different values of the Bjorken parameter. We find new predictions which are particularly interesting for differential scattering cross sections of polarized leptons scattered off polarized vector mesons. We also carry out a detailed comparison of the moments of the structure functions …
Deep inelastic structure functions can be calculated from supergravity when the Bjorken parameter $x$ satisfies $ x > 1/\sqrt{gN} $. We consider a gauge theory with very large 't Hooft coupling $gN$ in order to investigate the region $x << 1$. In this case the center of mass energy is large enough to increase the number of hadronic constituents of the final state. We calculate the structure functions in terms of the number of final hadronic constituents. At small $x$ we find a scaling law similar to geometric scaling but with $γ_s = 0.5 $ and $ λ= 1 $.
We present our results for inclusive instanton-induced cross-sections in deep-inelastic scattering, paying in particular attention to the residual renormalization-scale dependencies. A ``fiducial'' kinematical region in the relevant Bjorken variables is extracted from recent lattice simulations of QCD. The integrated instanton-contribution to the cross-section at HERA corresponding to this fiducial region is surprisingly large: It is in the O(100) pb range, and thus remarkably close to the recently published experimental upper bounds.
We calculate twist--4 coefficient functions for the deep inelastic structure function $F_2(x,Q^2)$ associated to 4--gluon operator matrix elements for general values of the Bjorken variable $x$ and study the numerical effect on the slope $\partial F_2(x,Q^2)/\partial \log Q^2$. It is shown that these contributions diminish the strongly rising twist--2 terms towards small values of $x$.
The theory of deep inelastic scattering structure functions is reviewed with an emphasis put on the QCD expectations of their behaviour in the region of small values of the Bjorken parameter $x$.
By using the AdS/CFT correspondence we study the deep inelastic scattering of an R-current off a N=4 supersymmetric Yang-Mills (SYM) plasma at finite temperature and strong coupling. Within the supergravity approximation valid when the number of colors is large, we compute the structure functions by solving Maxwell equations in the space-time geometry of the AdS_5 black three-brane. We find a rather sharp transition between a low energy regime where the scattering is weak and quasi-elastic, and a high-energy regime where the current is completely absorbed. The critical energy for this transiti…
Nuclear shadowing corrections to the structure functions of deep inelastic scattering of intermediate-mass nuclei are calculated at very low values of Bjorken x and small values of Q^2 (Q^2<5 GeV^2). The two-component approach (generalized VMD plus hard pomeron) is used for a description of the underlying virtual photon - nucleon interaction. The hard component of the nucleon structure function is calculated in a framework of the colour dipole model with dipole cross section having Regge-type energy dependence. Numerical results for nuclear shadowing corrections are compared with available dat…
Deep inelastic scattering of leptons off nuclear targets is analized within the convolution model taking into account nucleon-nucleon correlations. We show that in the nuclear medium nucleons are distributed according to a function that exhibits a sizeable Q**2-dependence and reduces to the ordinary light-cone distribution in the Bjorken limit. At Q**2 < 50 (GeV/c)**2 and x > 1 this Q**2-dependence turns out to be stronger than the one associated with the nucleon structure function, predicted by pertubative quantum chromodynamics.
The proton structure function is re-deduced from the data of deep inelastic electron-proton scattering after enhanced correction that is made due to the multiple scattering effect. The Glauber approach is used to account for the multiple scattering of electron with the quark constituents of the proton. The results are compared with the prediction of the parton model. Appreciable deviation is observed only at low values of the Bjorken scaling, which correspond to the extremely deep inelastic scattering.
We analyze deep inelastic scattering at small Bjorken x, using the approximate conformal invariance of QCD at high energies. Hard pomeron exchanges are resummed eikonally, restoring unitarity at large values of the phase shift in the dual AdS geometry. At weak coupling this phase is imaginary, corresponding to a black disk in AdS. In this saturated regime, cross sections exhibit geometric scaling and have a simple universal form, which we test against available experimental data for the proton structure function F_2(x,Q^2). We predict, in particular, the dependence of the cross section on the …
We consider a new form of analytical perturbation theory expansion in the massless $SU(N_c)$ theory, for the non-singlet part of the $e^+e^-$-annihilation to hadrons Adler function $D^{ns}$ and of the Bjorken sum rule of the polarized lepton-hadron deep-inelastic scattering $C_{ns}^{Bjp}$, and demonstrate its validity at the $O(α_s^4)$-level at least. It is a two-fold series in terms of powers of the conformal anomaly and of $SU(N_c)$ coupling $α_s$. Explicit expressions are obtained for the $\{β\}$-expanded perturbation coefficients at $O(α_s^4)$ level in $\bar{\rm MS}$ scheme, for both consi…
We study the transverse momentum ($p_T$) spectrum of charged particles produced in deep inelastic scattering (DIS) at small Bjorken $x$ in the central region between the current jet and the proton remnants. We calculate the spectrum at large $p_T$ with the BFKL $\ln (1/x)$ resummation included and then repeat the calculation with it omitted. We find that data favour the former. We normalize our BFKL predictions by comparing with HERA data for DIS containing a forward jet. The shape of the $x$ distribution of DIS + jet data are also well described by BFKL dynamics.
Infrared renormalons and $1/Q^2$ power corrections in deep-inelastic sum rules are studied. The renormalization of operators with power divergence are discussed. The higher-twist terms in the operator product expansion are shown to account for the residual soft contributions survived from the Kinoshita-Lee-Nauenberg type of cancellation in Feynman diagrams. The presence of some degree of arbitrariness in the twist separation allows one to define the most convenient higher-twist operators suitable for a particular non-perturbative method. The discussion is focused on the Bjorken sum rule, for w…
We study polarized deep inelastic scattering of charged leptons from spin-$1/2$ hadrons at low values of the Bjorken parameter and large 't Hooft coupling in terms of the gauge/string theory duality. We calculate the structure functions from type IIB superstring theory scattering amplitudes. We discuss the role of the non-Abelian Chern-Simons term and the Pauli term from the five-dimensional $SU(4)$ gauged supergravity. Furthermore, the exponentially small-$x$ regime where Regge physics becomes important is analyzed in detail for the antisymmetric structure functions. In this case the holograp…
