Semi - inclusive polarized DIS in terms of Melin moments . I . Light sea quark polarized

In connection with the semi-inclusive polarized DIS, it is proposed to consider the first Melin moments ∆q of the polarized quark and antiquark densities, instead of the respective variables δq(x), local in Bjorken x themselves. This gives rise to a very essential simplification of the next to leading order (NLO) QCD and, besides, allows one to use the respective QCD sum rules. An expression for ∆ū−∆d̄ in NLO is obtained which is just a simple combination of the directly measured asymmetries and of the quantities taken from the unpolarized data. Delivered on October 10 at Physics Workshop ”Compass week in Dubna” (JINR, Dubna, 2000) under the title ”Polarized sea-quark flavor asymmetries and COMPASS.” E-mail address: [email protected] 1 Investigation of the quark structure of the nucleon is one of most important tasks of modern high energy physics. In this respect deep inelastic scattering (DIS) is of special importance. Thus, the very impressive result of the New Muon Collaboration (NMC) experiment was obtained in 1991, when the unpolarized structure functions of the proton and neutron, F p 2 (x) and F n 2 (x), were precisely measured within a wide range of Bjorken’s x, and, it was established that the integral ∫ 1 0 dx x [F p 2 (x)−F n 2 (x)] does not equal 1/3 (Gottfried sum rule) but has a much smaller value 0.235 ± 0.0026. This means that the densities of u and d sea quarks, ū(x) and d̄(x), in the proton have different values, and ∫ 1 0 dx [d̄(x)− ū(x)] = 0.147± 0.039 6= 0. In polarized DIS, instead of the unpolarized total q = q+ q↓, sea q̄ and valence qV = q− q̄ quark densities, the set of the respective polarized quantities δq(x,Q) = q(x,Q)− q↓(x,Q ), δq̄(x,Q) = q̄(x,Q) − q̄↓(x,Q ) and δqV (x,Q ) = δq(x,Q) − δq̄(x,Q) is the subject of the investigation. So, the question arises: does the difference between the polarized u and d sea quark densities δū(x,Q) − δd̄(x,Q) also differ from zero? Recently, a series of theoretical papers appeared ([1-4]) where it was predicted that the quantity δū(x,Q)−δd̄(x,Q) does not equal zero. However, the model-dependent results for δū(x,Q) − δd̄(x,Q) essentially differ each from other in these papers. So, it is very desirable to find a reliable way to extract this quantity directly from experiment data. For this purpose it is not sufficient to use just the inclusive polarized DIS data, and one has to investigate semi-inclusive polarized DIS processes like ~μ+ ~ p(~ d) → μ+ h +X. Such processes provide direct access to the individual polarized quark and antiquark distributions via measurements of the respective spin asymmetries. 3 Unfortunately, the description of semi-inclusive DIS processes turns out to be much more complicated in comparison with the traditional inclusive polarized DIS. First, the fragmentation functions are involved, for which no quite reliable information is available. Second (and this is the most serious problem), the consideration even of the next to leading (NLO) QCD order turns out to be extremely difficult, since it involves double convolution products. So, to achieve a reliable description it is very desirable, on the one hand, to exclude from consideration the fragmentation functions, whenever possible, and, on the other hand (and this is the main task), to try to simplify the NLO consideration as much as possible, without which one can say nothing about the reliability and stability of results obtained within the quark-parton model (QPM). It is well known that within QPM one can completely exclude the fragmentation functions from the expressions for the valence quark polarized distributions δqV through experimentally measured asymmetries. To this end, instead of the usual virtual photon asymmetry AγN ≡ A h 1N (which is expressed in terms of the directly measured asymmetry Aexp = (n h ↑↓−n h ↑↑)/(n h ↑↓+n h ↑↑) as A1N = (PBPTfD) Aexp), one has to measure so called ”difference asymmetry” A h−h N [6] (see also [5,7]) which is expressed in terms of the respective counting rates as A N (x,Q ; z) = 1 PBPTfD (n↑↓ − n h̄ ↑↓)− (n h ↑↑ − n h̄ ↑↑) (n↑↓ − n h̄ ↑↓) + (n h ↑↑ − n h̄ ↑↑) , (1) Such a kind of measurements were performed by SMC and HERMES experiments and are also planned by the COMPASS collaboration. For discussion of this subject see, for example [5] and references therein. 2 where the event densities n↑↓(↑↑) = dN h ↑↓(↑↑)/dz, i.e. n h ↑↓(↑↑)dz are the numbers of events for antiparallel (parallel) orientations of here muon and target nuclear (proton or deutron here) spins for the hadrons of type h registered in the interval dz. Coefficients PB and PT , f and D are the beam and target polarizations, dilution and depolarization factors, respectively,(for details on these coefficients see, for example, [8,9] and references therein). Then, the QPM expressions for the difference asymmetries look like (see, for example, COMPASS project [10], appendix A) A −π p = 4δuV − δdV 4uV − dV ; A −π n = 4δdV − δuV 4dV − uV ; A −π d = δuV + δdV uV + dV ; A −K p = δuV uV ; A −K d = A π−π d , (2) i.e., on the one hand, they contain only valence quark polarized densities, and, on the other hand, have the remarkable property to be free of any fragmentation functions. All this is very good, but we are interested here in the sea quark polarized distributions, and, besides, the main question arises what will happen with all this beauty in the next to leading order QCD? We propose to investigate the integral quantities, namely, the first Melin momentsM(δq) ≡ ∫ 1 0 dx [δq(x)] ≡ ∆q (q = u, d, s, ...) instead of the local polarized quark densities δq(x) themselves. This provides very essential advantages: First. Even if the local quantity has a very small 5 value at each point x, the integral of this quantity over the whole range of x-variables may already have quite a considerable value, and, one can hope that QPM turns out to be a good approximation for integral quantities like ∆ū−∆d̄ ≡ ∫ 1 0 dx [δū(x)− δd̄(x)]. (3) An argument in favor of such a hope (for (3)) is the circumstance that all the model predictions [1-4] have one common feature: the local quantity δū(x)− δd̄(x) does not change sign when x varies over its entire range 0 ≤ x ≤ 1. Second. To investigate integral quantities like (3) one can use QCD sum rules. In particular, one can apply such a well established sum rule as the Bjorken sum rule ∫ 1 0 dx[g 1 − g n 1 ] = 1 6 gA gV (1− αs(Q ) π +O(α s)), (4) gA/gV = 1.2537± 0.0028 to express the quantity ∆ū − ∆d̄ of interest via the quantity ∆uV − ∆dV which, in turn, is expressed via the measured difference asymmetries A −π p and A π−π d . 5 Notice, however, that the latest theoretical paper [4] on this subject predicts that the difference between the polarized densities δū and δd̄ should be even more significant than the difference between the unpolarized sea quark densities: |δū− δd̄| ≥ |ū− d̄|. Throughout the paper, all the quantities considered in NLO are given in the MS scheme. 3 Third (and we consider this the most important advantage of the proposed procedure) Application of the Melin moments, instead of the local quantities themselves, results in a remarkable simplification of the NLO QCD consideration of the semi-inclusive polarized DIS, that is extremely complicated in terms of the local quantities. Thus, let us consider the NLO [11] expression for the structure function g 1