Double-Spin Asymmetry A_{LT} for Polarized Drell-Yan Process in p\bar{p} Collisions: Wandzura-Wilczek Contribution
Yuji Koike, Kazuhiro Tanaka, Shinsuke Yoshida
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4 pages, 2 figures; to appear in the proceedings of the 18th International Symposium on Spin Physics (SPIN20008), October 6 - 11, 2008, Charlottesville, Virginia, USA
The longitudinal-transverse spin asymmetry $A_{LT}$ in the polarized Drell-Yan process depends on twist-3 quark distributions of nucleon. In addition to the contributions associated with the twist-3 operators, these distributions contain the "Wandzura-Wilczek (WW)" part, which is determined by a certain integral of twist-2 distributions. The recently obtained empirical information on the transversity allows a realistic estimate of the WW contribution to $A_{LT}$ for the polarized $p\bar{p}$ collisions. Our results indicate that rather large $A_{LT}$ ($\sim 10$%) can be observed in the proposed spin experiments at GSI, with novel pattern as a function of dilepton mass compared with $A_{TT}$ and $A_{LL}$.
We give the geometric Wandzura-Wilczek-type relations between the meson wave functions of dynamical twist by means of the meson wave functions of geometric twist. We discuss the difference between geometric and dynamical Wandzura-Wilczek-type relations. Additionally, the interrelations between the different twist notations of meson wave functions are discussed.
We derive the higher twist (four and five) nucleon distribution amplitudes in the Wandzura-Wilczek approximation. Our method is based on an analysis of the conformal expansion of nonlocal operators in the spinor formalism.
We study higher twist distribution amplitudes for the SU_F(3) baryon octet. We identify independent functions for all baryons in the isospin symmetry limit and calculate the Wandzura-Wilczek contributions to the twist-4 and 5 distributions amplitudes.
We have calculated the Wandzura-Wilczek contribution to the twist-3 part of $γ^*γ\to 2π$ amplitude. It describes interaction of the longitudinally polarized virtual photon with the real one, and it is suppressed by 1/Q, where $Q^2$ is the virtuality of the $γ^*$, as compared to the twist-2 contribution. We have found that, in the Wandzura-Wilczek approximation, factorization applies to the twist-3 amplitude.
The violation of the so-called Lorentz invariance relations between parton distribution functions is considered in a model independent way. It is shown that these relations are not violated in a generalized Wandzura-Wilczek approximation, indicating that numerically their violation may be small.
We analyze the breaking of the Wandzura-Wilczek relation for the g_2 structure function, emphasizing its connection with transverse momentum dependent parton distribution functions. We find that the relation is broken by two distinct twist-3 terms, and clarify how these can be separated in measurements of double-spin asymmetries in semi-inclusive deep inelastic scattering. Through a quantitative analysis of available g_2 data we also show that the breaking of the Wandzura-Wilczek relation can be as large as 15-30% of the size of g_2.
In order to examine the validity of Wandzura-Wilczek relation for the polarized DIS structure function $g_2(x,Q^2)$, we use the light-front time-ordering pQCD to calculate $g_2(x,Q^2)$ at order $α_s$ on a quark target. We find that the study of the transverse polarized structure function in pQCD is only meaningful if we begin with massive quarks. The result shows that the Wandzura-Wilczek relation for $g_2(x,Q^2)$ is strongly violated by the quark mass in pQCD.
We present the alternative way of derivation of the Wandzura-Wilczek relations between the kinematical twist-3 and twist-2 functions, parameterizing hadronic matrix element in two-photon processes $γ^{\star}π\to γπ$ and $γ^{\star}γ\toππ$. The new equations, providing the independence of the physical cross-sections on the choice of the light-cone direction, are suggested and explored. The amplitude of $γ^{\star}γ\toππ$ up to genuine twist-3 accuracy is found.
If one retains M^2/Q^2 terms in the kinematics, the Nachtmann variable ξseems to be more appropriate to describe deep inelastic lepton-nucleon scattering. Up to the first power of M^2/Q^2, a modified Wandzura-Wilczek relation with respect to ξwas derived. Kinematical correction factors are given as functions of ξand Q^2. A comparison of the modified g_2^WW(ξ) and original g_2^WW(x) with the most recent g_2 data is shown.
Wandzura-Wilczek (WW) relations between matrix-elements of bilocal light-ray operators have recently regained interest in connection with off-forward scattering processes. Originally derived for matrix elements over leading-twist operators, their generalization to off-forward and exclusive processes gets complicated by the presence of higher-twist operators that are total derivatives of leading-twist ones and do not contribute to forward-scattering. We demonstrate that, for exclusive matrix-elements, the inclusion of these operators into WW-relations is essential for fulfilling constraints imp…
We review the study the Wandzura-Wilczek relation for the structure function g_2, with a particular attention on the connection with the framework of Transverse Momentum Dependent factorization. We emphasize that the relation is broken by two distinct twist-3 terms. In the light of these findings, we clarify what can be deduced from the available experimental data on g_2, which indicate a breaking of the order 20-40%, and how to individually measure the twist-3 terms.
We computed the NLO corrections to twist-3, $L \to T$, flavor non-singlet amplitude in DVCS on a nucleon in the Wandzura-Wilczek approximation. Explicit calculation shows that factorization holds for NLO contribution to this amplitude, although the structure of the factorized amplitude at the NLO is more complicated than in the leading-order formula. Next-to-leading order coefficient functions for matrix elements of twist-3 vector and axial-vector quark string operators and their LO evolution equations are presented.
The longitudinal-transverse spin asymmetry A_{LT} in the polarized Drell-Yan process depends on the twist-3 spin-dependent distributions of nucleon. In addition to the contributions expressed as matrix element of the twist-3 operators, these distributions contain the so-called Wandzura-Wilczek part, which is completely determined by a certain integral of the twist-2 spin-dependent parton distributions. We demonstrate that the recently obtained empirical information on the transversity distribution allows a realistic estimate of the Wandzura-Wilczek contribution to A_{LT} for the case of polari…
A complete list of the so-called Lorentz invariance relations between parton distribution functions is given and some of their consequences are discussed, such as the Burkhardt-Cottingham sum rule. The violation of these relations is considered in a model independent way. It is shown that several Lorentz invariance relations are not violated in a generalized Wandzura-Wilczek approximation, indicating that numerically their violation may be small.
We computed twist-4 part of the photon spin-flip amplitude in deeply virtual Compton scattering on a nucleon in the Wandzura-Wilczek approximation. We found a factorizable contribution, which arises from photon scattering on quarks with non-zero angular momentum along the collision axis. As the genuine twist-2 amplitude arises at the NLO, for moderate virtualities of the hard photon, $Q^2 \le 10$ GeV$^2$, kinematical twist-4 correction can give numerically important contribution to the photon helicity-flip amplitude.
We show that quark orbital angular momentum is directly related to off-forward correlation functions which include intrinsic transverse momentum corresponding to a derivative with respect to the transverse coordinates. Its possible contribution to scattering processes is therefore of higher twist and vanishes in the forward limit. The relation of OAM to other twist 2 and 3 distributions known in the literature is derived and formalized by an unintegrated sum rule.
We present a derivation of Wandzura-Wilczek (WW) like relations for skewed parton distributions. It is demonstrated for photon-pion scattering that the skewed twist-3 parton distributions contributing to the DVCS amplitude have discontinuities at the points $x=\pmξ$ in the WW approximation. This may lead to a violation of factorisation for the twist-3 DVCS amplitude with transverse polarization of the virtual photon. We show, however, that the contribution of the divergencies to the scattering of a transversely polarized virtual photon affects DVCS observables only at order $1/Q^2$ and can be …
We compute the transverse single spin asymmetry in forward $p^\uparrow p \to hX$ and $p^\uparrow A \to hX$ collisions from the odderon mechanism originally suggested by Kovchegov and Sievert [1]. Working in the hybrid approach of the Color Glass Condensate effective theory we firstly identify the relevant collinear parton distribution function (PDF) of the transversely polarized proton $p^\uparrow$ as the intrinsic twist-3 $g_T(x)$ distribution. We further argue that the complete polarized cross section also contains contributions from the kinematical and the dynamical twist-3 PDFs, in additio…
In the heavy-quark limit, the valence Fock-state components in the B mesons are described by a set of two light-cone wavefunctions. We show that these two wavefunctions obey simple coupled differential equations, which are based on the equations of motion in the Heavy Quark Effective Theory (HQET), and the analytic solutions for them are obtained. The results generalize the recently obtained longitudinal-momentum distribution in the Wandzura-Wilczek approximation by including the transverse momenta. We find that the transverse momentum distribution depends on the longitudinal momentum of the c…
The B-meson wavefunction has been studied with the help of the vacuum-to-meson matrix element of the nonlocal operators in the heavy quark effective theory. In order to obtain the Wandzura-Wilczek-type B-meson wavefunction, we solve the equations which are derived from the equation of motion of the light spectator quark in the B meson by using two different assumptions. Under the condition that $ω_0=2\barΛ$, the solutions for the B-meson wavefunction in this paper agree well with the one obtained by directly taking the heavy quark limit $m_b\to\infty$. Our results show that the equation of mot…
New Lorentz Invariance Relations (LIRs) were presented between twist-three Generalized Parton Distributions (GPDs) and transverse momentum, $k_T$, moments of twist-two Generalized Transverse Momentum-Dependent Distributions (GTMDs). By implementing both these LIRs and the QCD Equations of Motion in the quark quark correlation function, we generated a new series of Wandzura Wilczek (WW) relations in the off-forward sector. Two of these WW relations take on a particularly interesting physical meaning in that they provide a clear interpretation of the QCD structure of Orbital Angular Momentum (OA…
We present the complete cross-section for the production of unpolarized hadrons in semi-inclusive deep-inelastic scattering up to power-suppressed ${\cal O}(1/Q^2)$ terms in the Wandzura--Wilczek-type approximation which consists in systematically assuming that $\bar{q}gq$--terms are much smaller than $\bar{q}q$--correlators. We compute all twist-2 and twist-3 structure functions and the corresponding asymmetries, and discuss the applicability of the Wandzura--Wilczek-type approximations on the basis of available data. We make predictions that can be tested by data from Jefferson Lab, COMPASS,…
The off-cone Compton operator of twist-2 is Fourier transformed using a general procedure which is applicable, in principle, to any QCD tensor operator of definite (geometric) twist. That method allows, after taking the non-forward matrix elements, to separate quite effectively their imaginary part and to reveal some hidden structure in terms of appropriately defined variables, including generalized Nachtmann variables. In this way, without using the equations of motion, generalizations of the Wandzura-Wilzcek relation and of the mass-corrected Callan-Gross relation to the non-forward scatteri…
Power corrections for virtual Compton scattering at leading twist are etermined at operator level. From the complete off-cone representation of the twist-2 Compton operator integral representations for the trace, antisymmetric and symmetric part of that operator are derived. The operator valued invariant functions are written in terms of iterated operators and may lead to interrelations. For matrix elements they go over into relations for generalized parton distributions. -- Reducing to the s-channel relevant part one gets operator pre-forms of the Wandzura-Wilczek and the (target mass correct…
We link the recent computation beyond leading twist of the impact factor of the transition gamma*T -> rhoT performed in the light-cone collinear approach, to the dipole picture by expressing the hard part of the process through its Fourier transform in coordinate space. We show that in the Wandzura-Wilczek approximation the impact factor up to twist 3 factorises in the wave function of the photon combined with the distribution amplitudes of the rho-meson and the colour dipole scattering amplitude with the t-channel gluons. We show also that beyond the Wandzura-Wilczek approximation, the hard c…
These proceedings contain the 'physical' derivation of the Wandzura-Wilczek relations for the spin-dependent structure function g2 (amounting to keeping the intrinsic transverse momentum of quarks, and neglecting the physical transverse components of the gluon fields), recently generalized for the case of non-forward processes (hep-ph/0102209). The simple numerical model for the twist-3 contribution to g2 was also suggested.
The distribution of longitudinal gluons in a nucleon, introduced earlier by Gorsky and Ioffe, is estimated from below by making use of the positivity of density matrix and the analogue of the Wandzura-Wilczek relation for the light-cone distributions of polarized gluons in a transversely polarized nucleon.
The nonsinglet spin dependent structure function $g_2(x,Q^2)$ is studied at small x within the double logarithmic approximation of perturbative QCD. Both positive and negative signature contributions are considered, and we find a power-like growth in $1/x$. We discuss how our result fits into the Wandzura-Wilczek relation.
Polarized inclusive deep-inelastic scattering is formulated in the light cone expansion. The QCD evolution of the leading twist distribution functions is derived. It is shown that the twist--2 contribution to the structure functions g_2^{D(3)} is obtained via g_1^{D(3)} by a Wandzura--Wilczek relation.
We derive the analog of the Wandzura-Wilczek relation for the light-cone distributions of polarized gluons in a transversely polarized nucleon. The short distance cross-section is entirely due to the intrinsic transverse momenta of gluon in the nucleon, in complete analogy to the quark case. Numerical estimate for the double spin transverse asymmetries at RHIC are presented.
We show that electromagnetic gauge invariance requires a 'spin rotation' of the quarks in the usual twist-2 contribution to the amplitude for deeply virtual Compton scattering. This rotation is equivalent to the inclusion of certain kinematical twist-3 (Wandzura-Wilczek type) terms, which have been derived previously using other methods. The new representation of the twist-3 terms is very compact and allows for a simple physical interpretation.
Certain exact relations among transverse momentum dependent parton distribution functions due to QCD equations of motion turn into approximate ones upon the neglect of pure twist-3 terms. On the basis of available data from HERMES we test the practical usefulness of one such ``Wandzura-Wilczek-type approximation'', namely of that connecting h_{1L}^{\perp(1)a}(x) to h_L^a(x), and discuss how it can be further tested by future CLAS and COMPASS data.
We give a complete re-analysis of the leading twist quark-antiquark light-cone distribution amplitudes of longitudinal and transverse $ρ$ mesons. We derive Wandzura-Wilczek type relations between different distributions and update the coefficients in their conformal expansion using QCD sum rules including next-to-leading order radiative corrections. We find that the distribution amplitudes of quarks inside longitudinally and transversely polarized $ρ$ mesons have a similar shape, which is in contradiction to previous analyses.
The decomposition of nonlocal operators (and of their matrix elements) into an (infinite) series w.r.t. geometric twist is used to introduce (new) parton distributions, generalized parton distributions and hadron wave functions of definite twist. Their comparision with the conventional distributions (of definite dynamical twist) leads to Wandzura-Wilczek like relations, Burkhardt-Cottingham like sum rules and allows for power corrections of the various distribution amplitudes. Some general aspects are reviewed.
We calculate the off-forward structure function of the pion within a simple model where the size of the pion is accounted for using a momentum cut-off. Twist-two and twist-three generalized parton distributions are extracted. Relations between twist-three and twist-two contributions are obtained, the origin of which is not kinematical as they differ from those arising from the Wandzura-Wilczek approximation.
We introduce the chiral-even and chiral-odd quark distributions as forward matrix elements of related bilocal quark operators with well-defined (geometric) twist. Thereby, we achieve a Lorentz invariant classification of these distributions which differ from the conventional ones by explicitly taking into account the necessary trace terms. The relations between both kinds of distribution functions are given and the mismatch between their different definition of twist is discussed. Wandzura-Wilczek--like relations between the conventional distributions (based on dynamical twist) are derived by …
The twist--2 heavy flavor contributions to the polarized structure function $g_2(x,Q^2)$ are calculated. We show that this part of $g_2(x,Q^2)$ is related to the heavy flavor contribution to $g_1(x,Q^2)$ by the Wandzura--Wilczek relation to all orders in the strong coupling constant. Numerical results are presented.
We report the first-ever calculation of the isovector flavor combination of the chiral-odd twist-3 parton distribution $h_L(x)$ for the proton from lattice QCD. We employ gauge configurations with two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3 fm and lattice spacing of 0.093 fm. The values of the quark masses lead to a pion mass of $260$ MeV. We use a source-sink time separation of 1.12 fm to control contamination from excited states. Our calculation is based on the quasi-distri…
The effects, originating from the longitudinal gluon polarization, are considered. We derive the analogue of the Wandzura-Wilczek relation for the light-cone distributions of polarized gluons in a transversely polarized nucleon. The short distance cross-section is entirely due to the intrinsic transverse momenta of gluon in the nucleon, in complete analogy to the quark case. Numerical estimate for the double spin transverse asymmetries are presented. The distribution of longitudinal gluons in a nucleon, introduced earlier by Gorsky and Ioffe, is estimated from below by making use of the positi…
Two- and three-particle distribution amplitudes of heavy pseudoscalar mesons of well-defined geometric twist are introduced. They are obtained from appropriately parametrized vacuum-to-meson matrix elements by applying those twist projectors which determine the enclosed light-cone operators of definite geometric twist and, in addition, observing the heavy quark constraint. Comparing these distribution amplitudes with the conventional ones of dynamical twist we derive relations between them, partially being of Wandzura-Wilczek type; also sum rules of Burkhardt-Cottingham type are derived.The de…
We show that respecting the underlying Lorentz structure in the parton model has very strong consequences. Failure to insist on the correct Lorentz covariance is responsible for the existence of contradictory results in the literature for the polarized structure function g_2(x), whereas with the correct imposition we are able to derive the Wandzura-Wilczek relation for g_2(x) and the target-mass corrections for polarized deep inelastic scattering without recourse to the operator product expansion. We comment briefly on the problem of threshold behaviour in the presence of target-mass correctio…
In the framework of nonlocal light-cone expansion of two current operators we construct bilocal as well as trilocal QCD light-cone operators with definite geometric twist. We are able to decompose uniquely the appearing QCD light-cone operators into all their twist parts on the light-cone. The quark distribution functions and the vector meson distribution amplitudes which are defined as forward and skewed nucleonic matrix elements of nonlocal light-cone operators are classified with respect to geometric twist. Hereby we found well-kown and new Wandzura-Wilczek type relations between the quark …
We extend the study of quark spin-orbit correlations in the nucleon to the case of transverse polarization. At the leading-twist level, this completes the spin structure of the quark kinetic energy-momentum tensor. In particular, we revisit the transversity decomposition of angular momentum proposed a decade ago by Burkardt and introduce a new transverse correlation, namely between quark transversity and orbital angular momentum. We also provide for the first time the Wandzura-Wilczek expression for the second Mellin moment of twist-3 transversity generalized parton distributions, along with a…
We investigate the helicity dependent twist-two and twist-three generalized parton distributions in light-front Hamiltonian QCD for a massive dressed quark target. Working in the kinematical region $ξ< x <1$, we obtain the splitting functions for the evolutions of twist-two quark and gluon distributions in a straightforward way. For the twist-three distribution, we find that all contributions are proportional to the quark mass and thus the twist-three distribution is directly related to the chiral symmetry breaking dynamics in light-front QCD. We also show that the off-forward Wandzura-Wilczek…
Twist-3 partonic distributions contain important information that characterizes nucleon's structure. In this work, we show our lattice exploration of the twist-3 PDFs $g_T(x)$, and $h_L(x)$. We also present our preliminary results on the twist-3 GPD $\tilde{G}_2(x)$. We use the quasi-distribution approach to connect the lattice-extracted matrix elements, renormalized in the RI/MOM scheme, to light-cone distributions, applying the matching procedure that we developed in parallel. We also calculate the twist-2 counterparts of $g_T(x)$ and $h_L(x)$, i.e. $g_1(x)$, and $h_1(x)$, and test the Wandz…
We present the numerical solution of the one-loop QCD evolution equation for the genuine twist-three part of the orbital angular momentum (OAM) distributions of quarks and gluons inside a longitudinally polarized nucleon. This is based on the observation that the evolution is identical to that of the Efremov-Teryaev-Qiu-Sterman function for transverse single spin asymmetry. Together with the known evolution of the Wandzura-Wilczek part, the one-loop evolution of OAM distributions is now practically under control. We also study, for the first time, the scale dependence of the potential angular …
We study the virtual Compton amplitude in the generalized Bjorken region (q^2 -> Infinity, t small) in QCD by means of a light-cone expansion of the product of e.m. currents in string operators in coordinate space. Electromagnetic gauge invariance (transversality) is maintained by including in addition to the twist-2 operators 'kinematical' twist-3 operators which appear as total derivatives of twist-2 operators. The non-forward matrix elements of the elementary twist-2 operators are parametrized in terms of two-variable spectral functions (double distributions), from which twist-2 and 3 skewe…
The leading twist transverse momentum dependent parton distribution functions h_{1L}^{\perp} and h_{1T}^\perp, which is sometimes called "pretzelosity," are studied. For h_{1L}^{\perp} we consider a "Wandzura-Wilczek-type" approximation, which follows from QCD equations of motion upon the neglect of pure twist-3 terms and allows to express it in terms of transversity. On the basis of available data from HERMES we test the practical usefulness of this approximation and discuss how it can be further tested by future CLAS and COMPASS data. We review the theoretical properties of retzelosity and o…
The leading- and higher-twist distribution amplitudes and light-cone wave functions of real and virtual photons are analyzed in chiral quark models. The calculations are performed in the nonlocal quark model based on the instanton picture of QCD vacuum, as well as in the spectral quark model and the Nambu--Jona-Lasinio model with the Pauli-Villars regulator, which both treat interaction of quarks with external fields locally. We find that in all considered models the leading-twist distribution amplitudes of the real photon defined at the quark-model momentum scale are constant or remarkably cl…
We study twist-three effects in spin, charge, and azimuthal asymmetries in deeply virtual Compton scattering on a spin-zero target. Contributions which are power suppressed in 1/Q generate a new azimuthal angle dependence of the cross section which is not present in the leading twist results. On the other hand the leading twist terms are not modified by the twist three contributions. They may get corrected at twist four level. In the Wandzura-Wilczek approximation these new terms in the Fourier expansion with respect to the azimuthal angle are entirely determined by the twist-two skewed parton…
The worm-gear transverse momentum dependent (TMD) function $g_{1T}$ is one of the eight leading-twist TMDs and has the probabilistic interpretation of finding a longitudinally polarized quark inside a transversely polarized hadron. In this work, we present the first simultaneous extraction of $g_{1T}$ from all the available experimental measurements. The study involves the analysis of COMPASS, HERMES, and Jefferson Lab data on semi-inclusive deep-inelastic scattering using Monte Carlo techniques. We also compare $g_{1T}$ obtained from this experimental data with different theoretical approache…
We define in a systematic way, based on the light-cone collinear factorization method, the chiral-odd generalized parton distributions (GPDs) of a pseudoscalar hadron (such as the pi0) up to twist 5. For that, we introduce the relevant matrix elements for 2-parton non-local operators, as well as matrix elements for 3-parton non-local correlators. Their detailed parametrization is fixed based on parity, charge conjugation and time reversal invariance. This leads to the introduction of 20 real GPDs, whose symmetry properties are explicitely given. The reduction of these GPDs to a minimal set is …
Motivated by a novel origin of transverse single spin asymmetry (SSA) in semi-inclusive Deep Inelastic Scattering (SIDIS) uncovered by some of us, we quantitatively investigate its impact on the theoretical understanding of the mechanism responsible for SSA. This new contribution from the quark-initiated channel first appears in two-loop perturbation theory and involves the $g_T(x)$ distribution. We point out another entirely analogous piece from the gluon-initiated channel proportional to the gluon helicity distribution $ΔG(x)$. Both contributions are solely expressed in terms of twist-two po…
We develop a general theoretical framework for the description of higher-twist baryon operators which makes maximal use of the conformal symmetry of the QCD Lagrangian. The conformal operator basis is constructed for all twists. The complete analysis of the one-loop renormalization of twist-4 operators is given. The evolution equation for three-quark operators of the same chirality turns out to be completely integrable. The spectrum of anomalous dimensions coincides in this case with the energy spectrum of the twist-4 subsector of the SU(2,2) Heisenberg spin chain. The results are applied to g…
Matching relations for gluon TMDs up to one-loop accuracy
Università degli Studi di Pavia（帕维亚大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Università degli Studi di Cagliari（卡利亚里大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari（意大利国家核物理研究所卡利亚里分部）
We derive a unitarity relationship between the spin structure function $g_{LT}(x,Q^{2})=g_{1}(x,Q^{2})+g_{2}(x,Q^{2})$, the LT interference diffractive structure function and the spin-flip coupling of the pomeron to nucleons. Our diffractive mechanism gives rise to a dramatic small-$x$ rise $g_{LT}(x,Q^{2}) \sim g^{2}(x,\bar{Q}^{2}) \sim (1/x)^{2(1+δ_{g})}$, where $δ_{g}$ is an exponent of small-$x$ rise of the unpolarized gluon density in the proton $g(x,\bar{Q}^{2})$ at a moderate hard scale $\bar{Q}^{2}$ for light flavour contribution and large hard scale $\bar{Q}^{2} \sim m_{f}^{2}$ for he…
