Longitudinal polarization of hyperon and anti-hyperon in semi-inclusive deep-inelastic scattering
aa r X i v : . [ h e p - ph ] F e b Longitudinal polarization of hyperon and anti-hyperon insemi-inclusive deep-inelastic scattering
Shan-shan Zhou, Ye Chen, Zuo-tang Liang and Qing-hua Xu
School of Physics, Shandong University, Jinan, Shandong 250100, China (Dated: November 30, 2018)
Abstract
We make a detailed study of the longitudinal polarization of hyperons and anti-hyperons insemi-inclusive deep-inelastic lepton-nucleon scattering. We present the numerical results for spintransfer in quark fragmentation processes, analyze the possible origins for a difference betweenthe polarization for hyperon and that for the corresponding anti-hyperon. We present the resultsobtained in the case that there is no asymmetry between sea and anti-sea distribution in nucleonas well as those obtained when such an asymmetry is taken into account. We compare the resultswith the available data such as those from COMPASS and make predictions for future experimentsincluding those at even higher energies such as at eRHIC.
PACS numbers: 13.88.+e, 13.85.Ni, 13.87.Fh,13.60.Cr,13.60.Rj . INTRODUCTION Because of the non-perturbative nature, our knowledge on hadron structure and that onthe fragmentation function are still very much limited, in particular in the polarized case.Deeply inelastic lepton-nucleon scattering is always an ideal place for such study becauseat sufficiently high energy and momentum transfer, factorization theorem is applicable andthe hard part is easy to be calculated. Hyperon polarizations have been widely used forsuch studies, since they can easily be determined by measuring the angular distributionsof the decay products. These studies have attracted much attention in last years. [see e.g.,[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].]Longitudinal polarizations of hyperons and anti-hyperons in semi-inclusive deep-inelasticscattering (SIDIS) have been studied both experimentally and theoretically. More recently,such studies have in particular been extended to anti-hyperons. Special attention is paid tothe comparison of the results for hyperons with those for the corresponding anti-hyperons.This is partly triggered by the results of COMPASS collaboration at CERN which seem totell us that there is a difference between Λ and ¯Λ polarization in semi-inclusive deep-inelasticlepton-nucleon scattering [6, 7]. A detailed study of such a difference can provide us usefulinformation on the polarized fragmentation function and the structure of nucleon sea. Itmight be considered as a signature of the existence of a difference between the strange seaand anti-sea distributions in nucleon as proposed in literature some time ago [29, 30, 31,32, 33, 34, 35, 36]. It could also be a signature for a difference between the spin transferin quark and anti-quark fragmentation. On the other hand, it is also clear that the valencequarks in nucleon and other known effects can also contribute to such a difference. It istherefore important to make a detailed and systematic analysis of the contributions fromsuch known effects before we extract information on the possible asymmetry between seaand anti-sea distributions.In this paper, we make such a systematic study of longitudinal polarization of differenthyperons and anti-hyperons in semi-inclusive deep-inelastic lepton-nucleon scattering. Wemake a detailed analysis on the possible origin(s) of the difference between hyperon andanti-hyperon polarization at COMPASS and even higher energies. We clarify the differ-ent contributions and present the results obtained in the case that there is no asymmetrybetween nucleon sea and anti-sea quark distributions as well as those obtained when such2n asymmetry is taken into account. We make the calculations not only for Λ and ¯Λ butalso other hyperons and anti-hyperons in the same J P = (1 / + octet. We compare ourresults with the available data and make predictions for future experiments in particular ateRHIC.[37]The paper is organized as follows: After this introduction, in Sec. II, we summarize thegeneral framework for the calculations of the longitudinal polarization P H of the hyperon H and P ¯ H of the anti-hyperon ¯ H based on factorization theorem, and make a detailed analysisof each factor used in the formulae. We present in particular the model calculation results forspin transfer for a pure quark fragmentation process and compare the results for hyperonswith those for anti-hyperons. In Sec. III, we present the results obtained for hyperon andanti-hyperon polarizations in reactions using polarized beam and unpolarized target for thecase that there is no asymmetry between the sea and anti-sea distributions in nucleon andthose for the case that such an asymmetry is taken into account. We also study the influencefrom the differences in quark distributions as given by different sets of parameterizations. InSec. IV, we study the case that the lepton beam is unpolarized but the nucleon is polarizedand present the results obtained using different parameterizations of the polarized partondistributions. Finally, in Sec. V, we give a short summary and discussion. II. GENERAL FRAMEWORK FOR CALCULATING P H AND P ¯ H IN SIDIS
Deeply inelastic lepton-nucleon scattering at sufficiently high energy and momentumtransfer is one of the places where factorization theorem is applicable and is tested withhigh accuracies. According to the factorization theorem, hadron production in the currentfragmentation region of SIDIS is a pure result of the fragmentation of the quark or anti-quark scattered by the incoming lepton. The cross section is given as a convolution ofquark distribution function in nucleon, the elementary lepton-parton scattering and thefragmentation function. We consider the longitudinally polarized reaction in this paper andfor definiteness, we consider e − + N → e − + H (or ¯ H )+ X as an example. The formulae can beextended to other reactions in a straight forward way. To the leading order in perturbationtheory, the differential cross section for e − + N → e − + H + X is given by, dσ λ H ; λ e ,λ N = X f,λ f Z dxdydzK ( x, y ) h q N,λ N f,λ f ( x ) d ˆ σ eqλ e ,λ f ( x, y ) D H,λ H f,λ f ( z ) + ( q f ↔ ¯ q f ) i , (1)3here λ e , λ N , λ f and λ H are respectively the helicities of the electron, the incoming nucleon,the struck quark q f and the produced hyperon H ; x is the usual Bjorken- x , y is the fractionalenergy transfer from the electron to the nucleon N in the rest frame of N ; z is the fraction ofmomentum of scattered q f carried by the produced hyperon H ; and K ( x, y ) is a kinematicfactor which contains the 1 /Q due to the photon propagator and others ( Q = − q and q is the four momentum transfer). The sum over f runs over all the different flavor ofquarks or anti-quarks. Here, for clarity, we did not write out the scale dependence of theparton distributions and fragmentation functions explicitly. They are understood implicitly.We consider only light quarks and anti-quarks. Hence both quark and electron mass areneglected so that helicity in the elementary scattering process eq → eq is conserved.Eq. (1) is the basis for calculating the cross section of SIDIS both in unpolarized andpolarized case. We use this formulae as the starting point for calculating the polarizationsof hyperons and anti-hyperons in SIDIS in the following but discuss possible violation effectsin Sec. IIF. A. The calculation formulae for P H and P ¯ H The polarization of H in e − + N → e − + H (or ¯ H ) + X is usually defined as, P H ( z ) ≡ dσ +; λ e λ N − dσ − ; λ e λ N dσ +; λ e λ N + dσ − ; λ e λ N , (2)for the case that both the beam and target are completely polarized in the pure states withhelicities λ e and λ N . In the case that factorization theorem is valid, we can just insert Eq. (1)into Eq. (2) and obtain the result for the polarization of hyperon in e − + p → e − + H + X with longitudinally polarized electron beam and proton target as, P H ( z ) = P f e f R dxdyK ( x, y ) (cid:8) P f ( x, y ) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] ∆ D Hf ( z ) + ( q f ↔ ¯ q f ) (cid:9)P f e f R dxdyK ( x, y ) (cid:8) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] D Hf ( z ) + ( q f ↔ ¯ q f ) (cid:9) , (3)where P b and P T denote the longitudinal polarization of the electron beam and nucleon targetrespectively; e f is the electric charge of quark q f , q f ( x ) and ∆ q f ( x ) are the unpolarized andpolarized quark distribution functions, P f ( x, y ) is the polarization of the scattered quark q f , D L ( y ) is the longitudinal spin transfer factor in the elementary scattering process eq → eq and is defined as, D L ( y ) ≡ d ˆ σ eq ++ − d ˆ σ eq + − d ˆ σ eq ++ + d ˆ σ eq + − , (4)4hich is only a function of y at the leading order in perturbative QED; D Hf ( z ) and ∆ D Hf ( z )are the unpolarized and polarized fragmentation functions that are defined as, D Hf ( z ) ≡ D Hf ( z, +) + D Hf ( z, − ) , (5)∆ D Hf ( z ) ≡ D Hf ( z, +) − D Hf ( z, − ) , (6)where the argument + or − denotes that the helicity of the produced hyperon H is thesame as or opposite to that of the fragmenting q f . In the notation used in Eq. (1), theyare D Hf ( z, +) = D H, + f, + ( z ) = D H, − f, − ( z ) and D Hf ( z, − ) = D H, + f, − ( z ) = D H, − f, + ( z ). The integrationsover x and y run over the kinematic region determined by the corresponding experiments.Similarly for anti-hyperon ¯ H in e − + p → e − + ¯ H + X , the polarization is given by, P ¯ H ( z ) = P f e f R dxdyK ( x, y ) (cid:8) P f ( x, y ) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] ∆ D ¯ Hf ( z ) + ( q f ↔ ¯ q f ) (cid:9)P f e f R dxdyK ( x, y ) (cid:8) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] D ¯ Hf ( z ) + ( q f ↔ ¯ q f ) (cid:9) , (7)The physical significance of the expressions in Eqs. (3) and (7) are very clear: In thedenominator, besides some kinematic factor, we have just the production rate of H or ¯ H .The appearance of the term proportional to P b P T is due to the double spin asymmetry ˆ a LL in the elementary scattering process eq → eq which measures the difference between ˆ σ eq ++ and ˆ σ eq + − . The numerator shows explicitly that the polarization of H or ¯ H just comes fromthat of the q f and/or ¯ q f after the eq scattering. This can be seen more clearly if we re-writeEqs. (3) and (7) as, P H ( z ) = X f Z dxdy (cid:2) P f ( x, y ) R Hf ( x, y, z | pol ) S Hf ( z ) + ( q f ↔ ¯ q f ) (cid:3) , (8) P ¯ H ( z ) = X f Z dxdy h P f ( x, y ) R ¯ Hf ( x, y, z | pol ) S ¯ Hf ( z ) + ( q f ↔ ¯ q f ) i , (9)where R Hf ( x, y, z | pol ) is the fractional contribution from q f to the production of H in e − + p → e − + H + X and is given by, R Hf ( x, y, z | pol ) = e f K ( x, y ) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] D Hf ( z ) P f e f R dxdyK ( x, y ) (cid:8) [ q f ( x ) + P b P T D L ( y )∆ q f ( x )] D Hf ( z ) + ( q f ↔ ¯ q f ) (cid:9) ;(10) S Hf ( z ) is the polarization transfer in the fragmentation process q f → H + X in the longitu-dinally polarized case and is defined as, S Hf ( z ) ≡ ∆ D Hf ( z ) /D Hf ( z ) . (11)5e see that the polarization of H or ¯ H , P H ( z ) or P ¯ H ( z ), is just a weighted sum of S Hf ( z ) and S H ¯ f ( z ) for different flavor f . The weights are products of P f ( x, y ), the polarization of quarkafter the elementary scattering, and R Hf ( x, y, z | pol ), the fractional contribution from q f tothe production of H . In fact, assuming the validity of factorization theorem, fragmentationfunctions should be universal so that S Hf ( z ) and S H ¯ f ( z ) are also universal. Different resultsfor P H ( z ) in different kinematic regions and/or different reactions just originate from thedifferences in P f ( x, y ) and R Hf ( x, y, z | pol ).The expression for the relative weight R Hf ( x, y, z | pol ) is much simpler if we have onlybeam or target polarized, i.e., we have either P T = 0 or P b = 0. In this case, the termproportional to P b P T vanishes and we have, R Hf ( x, y, z | pol ) | P b =0 = R Hf ( x, y, z | pol ) | P T =0 = R Hf ( x, y, z | unpol ), which we simply denote by R Hf ( x, y, z ) and is given by, R Hf ( x, y, z ) = e f K ( x, y ) q f ( x ) D Hf ( z ) P f e f R dxdyK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i . (12)We see that R Hf ( x, y, z ) is determined solely by the unpolarized quantities such as the un-polarized parton distributions and fragmentation functions.The quark polarization is determined by the initial quark and/or electron polarizationand the spin transfer in the elementary process. It is given by, P f ( x, y ) = P b D L ( y ) q f ( x ) + P T ∆ q f ( x ) q f ( x ) + P b D L ( y ) P T ∆ q f ( x ) , (13) P ¯ f ( x, y ) = P b D L ( y )¯ q f ( x ) + P T ∆¯ q f ( x )¯ q f ( x ) + P b D L ( y ) P T ∆¯ q f ( x ) , (14)where the longitudinal spin transfer factor D L ( y ) in eq → eq can be obtained using pertur-bative QED and, to the leading order, is the same for eq → eq and e ¯ q → e ¯ q and is givenby, D L ( y ) = 1 − (1 − y ) − (1 + y ) . (15)This result has the following features.(1) If the target is unpolarized, i.e. P T = 0 but P b = 0, we have, P f ( x, y | P T = 0) = P ¯ f ( x, y | P T = 0) = P b D L ( y ) , (16)which is only a function of y and is the same not only for different flavors but also forquark and anti-quark. We see that, the quark (anti-quark) polarization in this case is6ompletely known. This is a very good place to study the spin transfer in fragmentationand/or the factors contained in the fractional contributions to the production of H and ¯ H .The expression for hyperon polarization in this case becomes also simpler. It is given by, P H ( z | P T = 0) = Z dxdyP b D L ( y ) X f h R Hf ( x, y, z ) S Hf ( z ) + R H ¯ f ( x, y, z ) S H ¯ f ( z ) i , (17) P ¯ H ( z | P T = 0) = Z dxdyP b D L ( y ) X f h R ¯ Hf ( x, y, z ) S ¯ Hf ( z ) + R ¯ H ¯ f ( x, y, z ) S ¯ H ¯ f ( z ) i . (18)For a fixed value of y , we have, P H ( z, y | P T = 0) = Z dxP b D L ( y ) X f h R Hfy ( x, y, z ) S Hf ( z ) + R H ¯ fy ( x, y, z ) S H ¯ f ( z ) i , (19) R Hfy ( x, y, z ) = e f K ( x, y ) q f ( x ) D Hf ( z ) P f e f R dxK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i . (20)If we now define S Hep ( z, y ) ≡ P H ( z, y | P T = 0) /P b D L ( y ) as in COMPASS measurements[6, 7],we obtain that, S Hep ( z, y ) = X f Z dx h R Hfy ( x, y, z ) S Hf ( z ) + R H ¯ fy ( x, y, z ) S H ¯ f ( z ) i . (21)We see that S Hep ( z, y ) is just a weighted sum of S Hf ( z ) and S H ¯ f ( z ), and the weights aredetermined by unpolarized quantities. Denote, h R Hfy ( y, z ) i ≡ Z dxR Hfy ( x, y, z ) = e f R dxK ( x, y ) q f ( x ) D Hf ( z ) P f e f R dxK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i , (22)and we obtain, S Hep ( z, y ) = X f h h R Hfy ( y, z ) i S Hf ( z ) + h R H ¯ fy ( y, z ) i S H ¯ f ( z ) i . (23)In practice, one often deals with events in a given y interval, and one has, S Hep ( z ) = X f h h R Hfyint ( z ) i S Hf ( z ) + h R H ¯ fyint ( z ) i S H ¯ f ( z ) i , (24) h R Hfyint ( z ) i = e f R dxdyK ( x, y ) q f ( x ) D Hf ( z ) P f e f R dxdyK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i . (25)72) If the beam is unpolarized but the target is polarized, i.e. P b = 0 but P T = 0, wehave, P f ( x, y | P b = 0) = P T ∆ q f ( x ) /q f ( x ) , (26) P ¯ f ( x, y | P b = 0) = P T ∆¯ q f ( x ) / ¯ q f ( x ) , (27)which is nothing else but the quark polarization before the eq scattering. This is just aresult of helicity conservation. In this case, we have, P H ( z ) = P f e f P T h ∆ q f ( x )∆ D Hf ( z ) + ∆¯ q f ( x )∆ D H ¯ f ( z ) iP f e f h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i , (28)where the polarizations of hyperons and anti-hyperons are determined by the polarizations ofquarks and anti-quarks in nucleon, thus can be used to extract information on the polarizedquark distributions in nucleon.(3) In the case that neither P b nor P T is zero, i.e. both electron beam and nucleontarget are polarized, the polarization of q f or ¯ q f after the scattering with electron is mainlydetermined by the beam electron polarization. It is dominated by the spin transfer from theelectron to the scattered quark (anti-quark). The influence from the target polarization isrelatively small. Aiming at studying either fragmentation functions or quark distributions,this case does not have much advantage compared to the cases (1) and (2) mentioned above.We therefore concentrate on the cases (1) and (2) in the following of this paper.From the discussions presented above, we see that there are three factors, i.e., quarkpolarization, the relative weights, and the fragmentation function are involved in Eqs. (8-11) for final hyperon or anti-hyperon polarization. We now discuss them further separatelyin the following. B. The spin transfer factor D L ( y ) in eq → eq scattering This is one of the best known factors involved in Eqs. (3) and (7). Since it is determinedmainly by the electromagnetic interaction, the spin transfer factor D L ( y ) in eq → eq scat-tering is calculable using perturbation theory in QED. When next leading order effects aretaken into account, pQCD corrections are involved. The result given in Eq. (15) is obtainedat leading order in perturbation theory. In this case, D L ( y ) is the same for quark and8nti-quark, i.e., D eq → eqL ( y ) = D e ¯ q → e ¯ qL ( y ) . (29)It is also the same for electron or positron. However, if we consider next-leading order inQED, e.g., if we take two photon exchange into account, the interference term leads to adifference between quark and anti-quark. It is also obvious that next leading order in QEDis far away from influencing the results at the accuracies of the data available. We consideronly the leading order here.Similarly, we also stick to leading order in perturbative QCD. The next-to-leading ordercalculations are in principle straight-forward but much involved (see e.g. [38]). Theseresults should be used consistently with the polarized parton distributions functions andthe polarized fragmentation functions. In view of our current knowledge on the polarizedfragmentation functions, we consider only the leading order consistently in this paper. C. The relative weights and the parton distributions
Using charge conjugation symmetry for the fragmentation functions, we have, R ¯ H ¯ f ( x, y, z ) = e f K ( x, y )¯ q f ( x ) D Hf ( z ) P f e f R dxdyK ( x, y ) h ¯ q f ( x ) D Hf ( z ) + q f ( x ) D H ¯ f ( z ) i . (30)This is to compare with R Hf ( x, y, z ) given in Eq. (12). We see that the difference between q f ( x ) and ¯ q f ( x ) is the only source for the difference between R Hf ( x, y, z ) and R ¯ H ¯ f ( x, y, z ).One obvious source for the difference between q f ( x ) and ¯ q f ( x ) is the valence quark con-tribution. Although this influences only u and d , it makes the ratio of the contributionsfrom u , d and s to H different from the corresponding ratio for the contributions from ¯ u , ¯ d and ¯ s to ¯ H . As we will see clearly from Fig. 2 in next subsection, S Hf is very much differentfor different f , and such a different ratio leads to different P H and P ¯ H .Clearly, valence quark contributions are negligible at very small x . We therefore expectthat its influence becomes negligible at very high energies. Also, since the influence isdetermined by the ratio of u ( x ), d ( x ) and s ( x ), the results can be quite sensitive to theforms of the parton distributions.The unpolarized parton distribution functions q f ( x ) are determined from unpolarizeddeep inelastic scattering and other related data from unpolarized experiments. There are9ifferent sets available in the parton distribution function library (PDFLIB [39]) package.Although the qualitative features are all the same, there are differences in the fine structure,which may influence the difference between R Hf ( x, y, z ) and R ¯ H ¯ f ( x, y, z ) and lead to differentresults in P H and P ¯ H . We will study this in next sections.Another source of the difference between q f ( x ) and ¯ q f ( x ) is the asymmetry in nucleonsea and anti-sea distributions. Physical picture for such an asymmetry was proposed [29,30, 31, 32, 33, 34] and models or parameterizations exist [35, 36]. This should be thedominant source for the difference between q f ( x ) and ¯ q f ( x ) at very small x and can be betterstudied at higher energies if it indeed leads to a significant difference between R Hf ( x, y, z )and R ¯ H ¯ f ( x, y, z ) thus a significant difference between P H ( z ) and P ¯ H ( z ). We will also makecalculations for this case in Sec. III. D. Spin transfer in fragmentation process
Spin transfer in fragmentation is defined in Eq. (11) and is given by the polarized frag-mentation functions ∆ D Hf ( z ). For explicitly, we only consider the fragmentation process q f → H + X . We should note that, when writing the factorization theorem in the way asgiven in Eq. (1), we assume that the fragmentation function is defined inclusively for thefragmentation process q f → H + X . It should include all the contributions from all the decayprocesses including strong as well as other decay processes. However, to study the physicsbehind it, it is useful to divide it into the directly produced part and the decay contributions.It has been widely used in studying the fragmentation functions in unpolarized processesand has been outlined in different publications [11, 12, 13, 14, 15, 16, 17, 18]. Here, forcompleteness, we summarize the major equations in the following.According to this classification, we write, D Hf ( z ) = D Hf ( z ; dir ) + D Hf ( z ; dec ) , (31)where the D Hf ( z ; dir ) and D Hf ( z ; dec ) are the directly produced and decay contribution partrespectively. The decay contribution can be calculated by, D Hf ( z ; dec ) = X j Z dz ′ K H,H j ( z, z ′ ) D H j f ( z ′ ) (32)where the kernel function K H,H j ( z, z ′ ) is the probability for H j with the fractional momentum z ′ to decay into a H with fractional momentum z , and, e.g, for a two body decay H j → + M , it is given by, K H,H j ( z, z ′ ) = NE j Br( H j → H + M ) δ ( p.p j − m j E ∗ ) , (33)where Br( H j → H + M ) is the decay branching ratio, N is the normalization constant, and E ∗ is the energy of H in the rest frame of H j , and m j is the mass of H j .Similarly, in the polarized case, we have,∆ D Hf ( z ) = ∆ D Hf ( z ; dir ) + ∆ D Hf ( z ; dec ) , (34)and the decay part is given by,∆ D Hf ( z ; dec ) = X j Z dz ′ t DH,H j K H,H j ( z, z ′ )∆ D H j f ( z ′ ) (35)where t DH,H j is a constant called the decay spin transfer which is independent of the H j produced process, and is e.g. discussed and given in Table 2 of Ref. [11].We should note that, in Eqs. (32) and (35), when calculating different decay contributions,we have added the contributions from different hyperon decays incoherently. This is whatone often does in calculating inclusive quantities where the interferences are usually smallbecause of the small contributions from different channels to exactly the same final state atexactly the same phase space points.
1. Modeling ∆ D Hf ( z, dir ) Since fragmentation is a non-perturbative process, the fragmentation function can not becalculated using perturbative QCD. At present, we have to invoke parameterization and/orphenomenological models. There are already data available [1, 2, 3, 4, 5, 6, 7, 8] that can beused to extract information on the polarized fragmentation functions ∆ D Hf ( z, dir ) but stillfar away from giving a good control of the form of it. At this stage, phenomenological modelsare quite useful in particular in obtaining some guide for experiments. In this connection,the model invoking calculation method according to the origins of hyperon is very practicaland successful [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] . In this model, one classifies the directlyproduced hyperons into the following two categories: (A) those which contain the initialquark q f and (B) those which do not contain the initial quark, i.e., D Hf ( z ; dir ) = D H ( A ) f ( z ) + D H ( B ) f ( z ) , (36)11 D Hf ( z ; dir ) = ∆ D H ( A ) f ( z ) + ∆ D H ( B ) f ( z ) . (37)It is assumed that those do not contain the initial quark are unpolarized, so that,∆ D H ( B ) f ( z ) = 0 . (38)The polarization then originates only from category (A) and is given by,∆ D H ( A ) f ( z ) = t FH,f D H ( A ) f ( z ) , (39)in which t FH,f is known as the fragmentation spin transfer factor and is taken as a constantgiven by, t FH,f = ∆ Q f /n f , (40)where ∆ Q f is the fractional spin contribution of a quark with flavor f to the spin of thehyperon, and n f is the number of valence quarks of flavor f in H .The model is very practical and useful for the following reason: In the recursive cascadehadronization models, such as Feynman-Field type fragmentation models [40] where a simpleelementary process takes place recursively, D H ( A ) f ( z ) and D H ( B ) f ( z ) are well defined anddetermined. In such models, D H ( A ) f ( z ) is the probability to produce a first rank H which isusually denoted by f Hq f ( z ) and is well determined by unpolarized reaction data. Hence, the z -dependence ∆ D given above is obtained completely from the unpolarized fragmentationfunctions, which are empirically known. The only unknown is the spin transfer constant t FH,f = ∆ Q f /n f . By using either the SU(6) wave function or polarized deep-inelastic lepton-nucleon scattering data, one obtains two distinct expectations ∆ Q f , the so-called SU(6) andDIS expectations, see table 1 of [11].This approach has been applied to different hyperons/anti-hyperons in different reactionssuch as e + e − , SIDIS and pp collisions [9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and comparedwith data [1, 2, 3, 4, 5, 6, 7, 8]. The current experimental accuracy does not allow one todistinguish between the expectations for t FH,f based on the SU(6) and DIS pictures. However,the z -dependence of the available data on Λ polarization is well described [11].
2. Numerical results for S Hf ( z ) Using the definition given in Eq. (11) and (34), we obtain the spin transfer for q f → H + X as, S Hf ( z ) = S Hf ( z, dir ) + S Hf ( z, dec ) . (41)12n the model described above, we have, S Hf ( z, dir ) = t FH,f A Hf ( z, dir ) , (42) A Hf ( z, dir ) = f Hq f ( z ) /D Hf ( z ) , (43)If we do not consider successive decay, we have, S Hf ( z, dec ) = X j t FH j ,f t DH,H j A Hf,H j ( z, dec ) , (44) A Hf,H j ( z, dec ) = Z dz ′ K H,H j ( z, z ′ ) f H j q f ( z ′ ) /D Hf ( z ) . (45)We call A Hf ( z, dir ) and A Hf,H j ( z, dec ) first rank contributions and note that both of them aredetermined by the unpolarized fragmentation functions. As an example, we calculated themusing Lund fragmentation model[41] as implemented in the Monte-Carlo event generator pythia [42]. The results are given in Fig. 1. Because isospin symmetry is valid here, sowe have relations such as A Λ u ( z, dir ) = A Λ d ( z, dir ), A Σ + u ( z, dir ) = A Σ − d ( z, dir ), A Ξ u ( z, dir ) = A Ξ − d ( z, dir ), etc. Hence, we only show the u and s contributions to Λ, Σ + and Ξ . Allthe others from u , d and s -quark to the J P = (1 / + hyperons can be obtained using suchrelations. We see, first of all, that the decay contributions to Λ are large but those to Σ andΞ are negligible. We also see that s -quark contributions are large in general because thosefrom u or d have strangeness suppression, a well known factor in fragmentation process.Multiplying by the corresponding spin transfer factors t FH,f and t DH,H j , we obtain thecorresponding S Hf ( z ) as shown in Fig. 2.We see that, for different flavors, S Hf ( z ) differs very much from each other not only becauseof the difference in the first rank contributions as shown in Fig. 1, but also because of thedifferences in the spin transfer factors t FH,f and t DH,H j . As an example, we see S Σ + u is positiveand large while S Σ + s is negative and the magnitude is smaller than S Σ + u . We also note thatisospin symmetry is valid here so that we have a series of relations such as, S Λ u ( z ) = S Λ d ( z ), S Σ + u ( z ) = S Σ − d ( z ), S Ξ u ( z ) = S Ξ − d ( z ), S Σ + s ( z ) = S Σ − s ( z ), and S Ξ s ( z ) = S Ξ − s ( z ).
3. Comparing S Hf ( z ) with S ¯ H ¯ f ( z ) For the fragmentation function, the directly produced part is controlled by strong inter-action where charge conjugation symmetry is valid. Hence, we have, D Hf ( z, dir ) = D ¯ H ¯ f ( z, dir ) , (46)13 .00.20.40.60.81.0 (a) q f →L +X A f H ( z ) direct decaysu (b) q f →S + +X (c) q f →X +X z FIG. 1: (color online) First rank contributions A Hf ( z, dir ) and A Hf ( z, dec ) in quark fragmentationto the productions of different hyperons as the functions of z . The results are extracted from e + e − process with √ s =200 GeV using pythia . ∆ D Hf ( z, dir ) = ∆ D ¯ H ¯ f ( z, dir ) , (47)For the decay contributions, we have processes controlled by strong or electromagnetic in-teractions. In these processes, we still have charge conjugation symmetry so that similarequations as given above are valid. However, there are also weak decay processes that play arole. In a weak process, charge conjugation symmetry may be violated. There are a few weakdecay processes that we need to take into account and we can check them one by one. Fortu-nately, for all the weak decay processes that give significant contribution to the productionof hyperons (anti-hyperons) in our interest, no significant violation in charge conjugationsymmetry has been observed. We therefore neglect the influence and have approximately14 (a) q f →L +X S f H ( z ) SU(6) DISsu -0.50.00.50 0.2 0.4 0.6 0.8 (b) q f →S + +X (c) q f →X +X z FIG. 2: (color online) Spin transfer S Hf ( z ) in the fragmentation process q f → H + X . that, D Hf ( z ) = D ¯ H ¯ f ( z ) , (48)∆ D Hf ( z ) = ∆ D ¯ H ¯ f ( z ) , (49)We thus also have, S Hf ( z ) = S ¯ H ¯ f ( z ) , (50) S H ¯ f ( z ) = S ¯ Hf ( z ) , (51)We see, under such circumstances, we expect no significant difference between the spintransfer in quark fragmentation and that in anti-quark fragmentation. A significant differ-ence between final hyperon and anti-hyperon polarization can only be from the differencebetween P f ( x, y ) and P ¯ f ( x, y ) and/or that between R Hf ( x, y, z ) and R ¯ H ¯ f ( x, y, z ). We recall15hat, in the case of P T = 0, P f ( x, y | P T = 0) = P ¯ f ( x, y | P T = 0) = P b D L ( y ), the only sourcefor such a difference is the difference between R Hf ( x, y, z ) and R ¯ H ¯ f ( x, y, z ), which we discussedin Sec. IIC.As an example, we see that, in the model described in Sec.IID(2), S H ¯ f ( z ) = S ¯ Hf ( z ) = 0 . (52) S Hf ( z ) = S ¯ H ¯ f ( z ) = t FH,f A Hf ( z, dir ) + X j t FH j ,f t DH,H j A Hf,H j ( z, dec ) . (53)Charge conjugation symmetry is indeed valid here and the numerical results are given inFig. 2. We emphasize here that, as can be seen from Fig. 2, for different flavor f , S Hf ( z )differs very much from each other. This makes the value of the polarization of final hyperonsensitive to R Hf ( x, y, z ). Hence, measuring P H is a good way to study the fine behavior of R Hf ( x, y, z ). E. A practical way of the calculations
As shown by Eqs. (8-10), the calculations of P H and/or P ¯ H involve the contributions fromdifferent flavor f and ¯ f , each of them is a convolution of quark distributions, polarizationfragmentation function and other kinematic factors originating from the eq scattering etc.Using the parton distributions from PDFLIB [39], the perturbative calculation results for thedifferential cross section for eq → eq , and the parameterization for the fragmentation func-tions, we can in principle calculate the contribution in a straight-forward manner. However,in view of the number of different flavor f and ¯ f involved, all the different decay contributionsand the difficulties and/or uncertainties in obtaining the fragmentation functions, the calcu-lations are almost impossible without radical approximations. On the other hand, all theseinformation for unpolarized reactions are implemented in the Monte-Carlo event generatorssuch as lepto [43] so that the corresponding unpolarized cross section can be calculatedconveniently using such Monte-Carlo programs. Such Monte-Carlo event generators havebeen developed since 1980s and have been tested by enormous amount of unpolarized experi-ments and the parameters in the models have been adjusted to fit all the data. They providea useful tool to make predictions for out-coming experiments and are widely used in the com-munity. The Monte-Carlo event generators for unpolarized high energy reactions have also16een used to calculate the corresponding unpolarized parts in calculating the polarizationsof the produced hadrons in literature [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].To show how this is carried out, we take the polarization of H in the case of P T = 0 asan example and re-write Eq. (3) as, P H ( z | P T = 0) = P f,α e f R dxdyP b D L ( y ) K ( x, y ) q f ( x ) D H ( α ) f ( z ) S H ( α ) f P f e f R dxdyK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i , (54)where α = A through D denoting the different origins of H : (A) directly produced andcontain q f ; (B) directly produced but do not contain q f ; (C) decay product of H j which isdirectly produced and contain q f ; (D) decay product of H j which is directly produced anddo not contain q f . S H ( α ) f = ∆ D H ( α ) f ( z ) /D H ( α ) f ( z ) is the spin transfer factor in fragmentationfor each contribution. We see from Eqs. (35), (39) and (32) that, S H ( A ) f = t FH,f , S H ( C ) f = t FH j ,f t DH,H j and S H ( B ) f = S H ( D ) f = 0. Denote the relative contribution from origin ( α ) by, R H ( α ) f ( x, y, z ) = e f K ( x, y ) q f ( x ) D H ( α ) f ( z ) P f e f R dxdyK ( x, y ) h q f ( x ) D Hf ( z ) + ¯ q f ( x ) D H ¯ f ( z ) i , (55)and we have, P H ( z | P T = 0) = X f,α Z dxdyR H ( α ) f ( x, y, z ) P b D L ( y ) S H ( α ) f . (56)We see that the relative production weight R H ( α ) f ( x, y, z ) defined in Eq. (55) is independentof the polarization and can be calculated using a Monte-Carlo event generator. In the righthand side of Eq.(56), the quark polarization P b D L ( y ) and the spin transfer constant S H ( α ) f are two known quantities and they are the places where information on polarization comesin. In practice, we generate a ep collision event using an event generator. We study thefinal state hadrons and search for the hyperon H under study. After we find a H in theconsidered kinematic region, we calculate D L ( y ) and S H ( α ) f by tracing back the origin of the H using information recorded in the Monte-Carlo program.Usually a Monte-Carlo event generator is tested by the existing data at different energies,and is expected that it can give a reasonable description of the unpolarized quantities at agiven energy provided that the physics does not have sudden changes at that energy. Wetherefore use such Monte-Carlo event generator for such analysis in the following. Such acalculation method is not only the most convenient way available currently of calculating thefragmentation functions and the contributions from different hard scattering processes butalso the most convenient way to include the contributions from all different decay processes.17 . A lower energy effect At lower energies, there is another effect that relates to the valence quark contributionand may cause a difference between P H and P ¯ H in e − + N → e − + H (or ¯ H ) + X , i.e., thecontribution from the hadronization of the remnant of target nucleon. It has been pointedout first in [13] that contribution of the hadronization of target remnant is important tohyperon production even for reasonably large x F at lower energies. (Here, x F is the Feynman- x in the c.m. frame of the γ ∗ p system, which is approximately equal to z at high energy andsmall p T .) The effect has been confirmed by the calculations presented in [21]. It has beenshown that [13] at the CERN NOMAD energies, contributions from the hadronization ofnucleon target remnant dominates hyperon production at x F around zero. It is impossibleto separate the contribution of the struck quark fragmentation from those of the targetremnant fragmentation. In this case, the factorization theorem given in Eq. (1) is brokendown and the concept of independent fragmentation is no more valid. Clearly, this effect canbe different for hyperon and anti-hyperon production since the target remnant contributioncomes mainly from the fragmentation of the valence di-quark. It contributes quite differentlyfor hyperons and anti-hyperons. This can have a large influence at lower energies, but theinfluence should vanish at high energies where current fragmentation can well be defined.As have already demonstrated in Ref. [16], this low energy effect has already little influenceon the results at COMPASS energy, in particular when studying the difference between P H and P ¯ H . We therefore do not consider this effect in the following of this paper. III. P H AND P ¯ H IN SIDIS WITH POLARIZED BEAM AND UNPOLARIZEDTARGET
By using the method presented in last section, we can calculate the polarizations ofhyperon and anti-hyperon in SIDIS with the aid of a Monte-Carlo event generator. We usethe latest version of lepto 6.5.1 [43] based on the Lund string fragmentation model [41]in the following. We study SIDIS with longitudinally polarized electron (muon) beam andunpolarized target in this section and make calculations of P H and P ¯ H in the case that asymmetric strange sea s ( x ) and anti-sea distribution ¯ s ( x ) is assumed and in the case thatan asymmetry between s ( x ) and ¯ s ( x ) is taken into account separately.18 . Results with symmetric strange sea and anti-sea distributions Our calculations in this case are made by using a parameterization of PDF’s (partondistribution functions) obtained in PDFLIB where no asymmetry between strange sea andanti-sea distribution is taken into account. It is obvious that different sets of parameter-izations of PDF’s may have influence on our results of P H and P ¯ H . In the following, wefirst present the results obtained using CTEQ2L and study the influence of different sets ofparameterizations later in this section.
1. Results at COMPASS energy
To compare with the available experimental data, we first make calculations in the samekinematic region as in the COMPASS experiment [6, 7], i.e., Q > , and 0 . < y < . µ beam energy of 160 GeV and beam polarization P µ = − .
76. As we have mentionedin last section, when the target is unpolarized, the difference between the polarizations of H and ¯ H comes only from the relative weight R Hf and R ¯ H ¯ f as given by Eqs.(8) and (9). Wetherefore first calculated R Λ f and R ¯Λ f and show the results obtained in Figs. 3(a) and (b)respectively.From Figs. 3(a) and (b), we see the following features:(1) With increasing x F , R Λ s and R ¯Λ¯ s increase fast, R Λ u,d or R ¯Λ¯ u, ¯ d vary slowly, while R Λ¯ q ( x F )and R ¯Λ q ( x F ) decrease, in particular in the large x F region. This is because R Λ u,d,s and R ¯Λ¯ u, ¯ d, ¯ s have the first rank contributions while R ¯Λ u,d,s and R Λ¯ u, ¯ d, ¯ s do not and the first rank contributionsto R Λ u,d and R ¯Λ¯ u, ¯ d have strangeness suppression compared to R Λ s and R ¯Λ¯ s .(2) The shape of R Λ u ( x F ) is similar to R Λ d ( x F ) but the former is in general much largerthan the latter. This is not only because of the larger contribution from the valence to u but also because of that the electric charge squared factor for u is 4 times as large as thatfor d . Similar relations hold for ¯ u and ¯ d contributions and also for those to ¯Λ.(3) There is in general quite a large difference between R Λ f and R ¯Λ¯ f for a given f , i.e., thecharge conjugation symmetry is not hold here. The difference is particularly large for R Λ u and R ¯Λ¯ u . This is a characteristics of the contribution from the valence quark.To understand that the valence quark contributions are still large at COMPASS energy,Ref. [16] has calculated the x -values of the struck quark and/or anti-quark. We did similar19 (a) e - p → e - LC dusd – u – s – R f L / R f L _ (b) e - p → e - L – C d – u – s – dus x F FIG. 3: (color online) Relative weights R Λ f and R ¯Λ f as functions of x F at COMPASS energy √ s =17.35 GeV for different flavors f = u, d, s, ¯ u, ¯ d, and ¯ s respectively. calculations in exactly the COMPASS kinematic region as described above and obtain theresults shown in Fig. 4. We see that most of the Λ’s and ¯Λ’s are from the quark andanti-quark with momentum fraction x around 0.01. In this x region, the valence quarkcontributions are indeed still quite large so that u ( x ) > ¯ u ( x ) and d ( x ) > ¯ d ( x ). This leads tomuch larger u ( d ) contributions to Λ than the corresponding ¯ u ( ¯ d ) contribution to ¯Λ.We recall that the spin transfer in fragmentation S Λ f = S ¯Λ¯ f is quite different for f = u or d from that for f = s (see Fig.2), we thus expect that there is a significant differencebetween P Λ and P ¯Λ . We calculate these polarizations using the spin transfer S Λ f ( z ) = S ¯Λ¯ f ( z )described in last section, and show the results in Fig. 5. We see that there are indeed somedifference between P Λ and P ¯Λ at the same x F . The magnitude of P ¯Λ is larger than that of P Λ . This is because that the contribution from u -quark is larger and that S Λ u is small andnegative.To compare with the data from COMPASS[6, 7], we also calculate the spin transfer S Λ ep and S ¯Λ ep in e − + p → e − + Λ( ¯Λ) + X as given by Eq. (24) and show the results in Fig. 6. We20 -3 -2 -1 LL – x ( d N / d x ) / N t o t a l FIG. 4: (color online) The x -distribution of the struck quark or anti-quark that leads to theproduction of Λ or ¯Λ in the kinematic region of x F > √ s =17.35 GeV. recall that the magnitude of the spin transfer of s (¯ s ) quark to Λ ( ¯Λ) in the fragmentationprocess is much larger than that of u or d (¯ u or ¯ d ) quark (see Fig.2), so S Λ ep ( S ¯Λ ep ) takes itsmaximum when there is only contribution from strange quarks, i.e., S Λ ep → S Λ s . In contrast, S Λ ep ( S ¯Λ ep ) reaches its minimum when there is only contribution from u and/or d (¯ u and/or¯ d ), i.e., S Λ ep → S Λ u = S Λ d . These are the limits of S Λ ep and S ¯Λ ep in e − + p → e − + Λ( ¯Λ) + X . Toshow the range of S Λ ep and S ¯Λ ep in the case that the spin transfer model described in Sec.IIDis used, we also show these two limits in the same figure.From Fig. 6, we see that, with a symmetric strange sea and anti-sea distribution, we stillobtain some differences between S Λ ep and S ¯Λ ep as functions of x F . But the differences seemnot as large as those observed by COMPASS collaboration[6, 7]. From the limits S Λ s and S Λ u , we see also that there are enough room to fit the data by adjusting the relative weightsof s (¯ s ) contributions compared to those from u and d (¯ u and ¯ d ).We have seen that the differences between P Λ and P ¯Λ in the case discussed in this subsec-tion come only from valence quark contributions. The differences are due to that the relativecontribution from s to Λ is different from the relative contribution from ¯ s to ¯Λ. If we extend21 L DIS, L SU(6), L - DIS, L - x F P L ( L _ ) FIG. 5: (color online) Polarizations of Λ and ¯Λ as the functions of x F at COMPASS energy √ s =17.35 GeV obtained using symmetric strange sea and anti-sea distributions. the study to other J P = (1 / + hyperons such as Σ ± and Ξ, the situations can be different.For example, for Σ + and its anti-particle ¯Σ − , the production and the polarization are dom-inated by u and ¯ u contributions respectively. Although valence quark contributions make u dominance even stronger, but the relative weights do not change much, even at COMPASSenergy. Similarly, Σ − and ¯Σ + are dominated by d and ¯ d , and Ξ and ¯Ξ are dominated by s and ¯ s respectively. We expect much smaller difference between P H and P ¯ H for these hy-perons. In Fig. 7, we show the corresponding results at COMPASS energy. We see that thedifferences obtained between P H and P ¯ H are indeed much smaller than those for Λ and ¯Λ.Since the decay contributions to these hyperons are almost negligible, the calculations hereare simpler and more clear. This provides a rather clean test to see whether the differencebetween P Λ and P ¯Λ are due to valence contributions.
2. Results at eRHIC energy
It is also clear that, if we go to even higher energies, the main contributions are fromeven smaller x region. In the small x regions, valence quark contributions are negligible.22 SU(6) DIS COMPASS L ● L _ ■ S d L S s L x F S e p L / S e p L _ FIG. 6: (color online) Spin transfer S Λ / ¯Λ ep as the function of x F at COMPASS energy √ s =17.35GeV obtained using symmetric strange sea and anti-sea distributions. The data points are takenfrom COMPASS [7]. In such cases, we should have P H = P ¯ H assuming a symmetric sea and anti-sea quarkdistribution. To show this, we made calculations at eRHIC energy, i.e., we take the electronbeam of 10 GeV and proton beam of 250 GeV. The electron beam polarization is takenas one and the nucleon is taken as unpolarized. We first checked the x distribution of thestruck quarks (anti-quarks) that lead to the productions of hyperons at such high energy. Inthe calculations, we choose events in the kinematic region 0 . ≤ y ≤ . Q > .The results are shown in Fig. 8. We see that they are indeed dominated by very small x .The results of hyperon and anti-hyperon polarization using the same parton distribution setCTEQ2L with symmetric sea and anti-sea densities, are shown in Fig. 9. As expected, the H and ¯ H polarization are almost the same.
3. Results obtained using different sets of PDF’s
In the calculations presented above, we used CTEQ2L for parton distributions. As men-tioned earlier, there are different sets of parameterizations available and the significant dif-ferences still exist for sea quark distributions especially for the strange sea. As an example,we show in Fig. 10 the s (¯ s ) quark distribution in CTEQ2L and GRV98Lo. We see that the23 S + / S - - P H ( H _ ) (b) S - / S - + -0.25-0.20-0.15-0.10-0.050 0.2 0.4 0.6(c) X / X - 0 X - / X - + H ,SU(6) H ,DIS H - ,SU(6) H - ,DIS x F FIG. 7: (color online) Polarizations of hyperons and anti-hyperons as functions of x F at COMPASSenergy √ s =17.35 GeV obtained using symmetric strange sea and anti-sea distributions. difference between the two parameterizations is indeed quite large.The difference in different sets of PDF’s can certainly influence the results of P H and P ¯ H . We study this influence by repeating the calculations mentioned above using differentsets of parton distribution functions. As examples, we show the results for Λ, Ξ and theiranti-particles in Figs. 11(a) through (d) at COMPASS and eRHIC energies respectively.From the results, we indeed see some significant differences between the results obtainedusing the two different sets of PDF’s. We see in particular that, at the COMPASS en-ergy, the magnitude of the polarizations obtained using CTEQ2L PDF’s are larger thanthe corresponding results obtained using GRV98Lo. In contrast, at the eRHIC energy, thepolarizations obtained using GRV98Lo PDF’s are larger. This is because, at eRHIC en-ergy, the dominating contributions are from very small x region where s ( x ) in GRV98Lo islarger than that in CTEQ2L (see Fig.10). However, at COMPASS energy, the dominatingcontributions are from much larger x region, where s ( x ) in GRV98Lo is smaller than thatin CTEQ2L. Such differences lead to different relative weights R Hf and manifest themselvesin the results for P H and P ¯ H shown in Figs. 11(a) through (d). We also see that different24 -5 -4 -3 -2 -1 LL – x ( d N / d x ) / N t o t a l FIG. 8: The x-distribution of the struck quark or anti-quark that leads to the production of Λ or¯Λ in the kinematic region x F > √ s =100GeV. sets of PDF’s influence the magnitudes of P H and P ¯ H but they have little influence on thedifference between them. The difference between P H and P ¯ H is not very sensitive to theparameterizations of PDF’s. B. Results with asymmetric strange sea and anti-sea distribution
As discussed in last section, an asymmetry between strange sea and anti-sea quark distri-butions can be another source for the difference between hyperon and anti-hyperon polariza-tion, and this effect remains at even higher energies such as at eRHIC. The asymmetry in thestrange sea of the nucleon was studied by many authors in literature[29, 30, 31, 32, 33, 34].Different models are proposed. A global QCD fit to the CCFR and NUTEV dimuon datahas also shown a clear evidence that s ( x ) = ¯ s ( x ) [35, 36], and a parameterization of thestrangeness asymmetry has also been included in the CTEQ parameterization. Such anasymmetry is usually described by defining s − ( x ) = s ( x ) − ¯ s ( x ), and correspondingly denote s + ( x ) = s ( x ) + ¯ s ( x ). It seems now evident that s − ( x ) = 0 but the size is quite unknown25 .00.050.100.150.200.25 (a) L / L - P H ( H _ ) (d) X / X - 0 (e) X - / X - + (b) S + / S - - (c) S - / S - + H ,SU(6) H ,DIS H - ,SU(6) H - ,DIS x F FIG. 9: (color online) Polarizations of hyperons and anti-hyperons versus x F at eRHIC energy √ s =100.0 GeV obtained using a symmetric strange sea and anti-sea distribution. besides that it has to fulfill the limit − s + ( x ) ≤ s − ( x ) ≤ s + ( x ). For example, we showtwo different parameterization from CETQ in Fig. 12. We see that the difference in theparameterization of s − ( x ) is indeed very large. We even do not know the sign of s − ( x ) in agiven x -region. In this subsection, we study the contribution of such an asymmetry to thedifference between the polarization of H and that of the corresponding ¯ H in SIDIS.We first carried out the calculations by taking the same s + ( x ) and other PDF’s fromCTEQ2L as used in previous calculations but taking a s − ( x ) into account. Since our knowl-edge of s − ( x ) is very much limited, and the form of s − ( x ) is almost completely unknown,we simply take an existing parameterization as one from CETQ6set37. With these inputs,we obtain the Λ and ¯Λ polarizations and S Λ ep and S ¯Λ ep in COMPASS kinematic region areobtained and are shown in Fig. 13. We see that, in this kinematic region, the influence fromsuch a small asymmetry s − ( x ) is small. To see how large the effect can be, we take theextreme cases for s − ( x ), i.e. s − ( x ) = − s + ( x ) or s − ( x ) = s + ( x ). The results are also shownin Fig. 13. We see that the difference between Λ and ¯Λ in either limit is much larger thanthe case of symmetric s ( x ) and ¯ s ( x ), and closer the existing COMPASS data.[6, 7]26 .050.10.150.20.250.30.350.410 -4 -3 -2 -1 CTEQ2LGRV98Lo x xs ( x , Q ) FIG. 10: Comparison of the sea quark distributions from GRV98Lo with those from CTEQ2L at Q = 3GeV . At the eRHIC energy, the only source for the difference between P Λ and P ¯Λ is the asym-metry between s ( x ) and ¯ s ( x ). We did similar calculations and obtain the results shown inFig. 14. We can see that the difference between P Λ and P ¯Λ is quite small if we use theasymmetric strangeness distribution as given in CTEQ6set37. However, it can be ratherlarge at the extreme case. The asymmetry between s ( x ) and ¯ s ( x ) has even smaller influ-ence on P Σ and their anti-particle. For comparison, we show the results for Σ + and ¯Σ − inthe same figure. These results show us that, experiments at eRHIC can indeed provide ususeful information on the asymmetry between s ( x ) and ¯ s ( x ) in nucleon, but high statisticsis needed. IV. P H AND P ¯ H IN SIDIS WITH UNPOLARIZED BEAM AND POLARIZEDTARGET
If the lepton beam is unpolarized and the target proton is polarized with P T = 1, thepolarizations of the hyperons (or anti-hyperons) are determined by Eq. (28). In this case, therelative weights for the contributions of different flavors are the same as those discussed inlast section which are determined by the unpolarized quantities. However, the polarizationsof the quarks and anti-quarks are different. In the case of unpolarized lepton beam andlongitudinally polarized nucleon, and the polarizations of the quarks and anti-quarks equal27 a) L / L – ,COMPASS-0.25-0.20-0.15-0.10-0.050.00.05 P H ( H _ ) (b) X / X - 0 ,COMPASS L ,GRV98Lo L - ,GRV98Lo L ,CTEQ2L L - ,CTEQ2L L / L – ,eRHIC 0 0.2 0.4 0.6(d) X / X - 0 ,eRHIC x F FIG. 11: (color online) Comparison of the polarizations of hyperons and anti-hyperons obtainedusing GRV98Lo with those using CTEQ2L parton distribution functions at the COMPASS energy(with beam polarization of P b = − .
76) and at the eRHIC energy (with P b = 1) respectively. Forclarity, we only show the results obtained using SU(6) picture for spin transfer in fragmentationprocesses. to those in the polarized nucleon, which is a simple result of helicity conservation. This is agood place to study polarized quark distributions in the nucleon. There exist many differentsets of parameterizations of the polarized PDF’s [see e.g. [44, 45, 46, 47, 48, 49]] and thedifferences between them are quite large. An example is given in Fig. 15 where two sets ofparameterizations from GRSV2000[44], GRSV2000 set3 (standard) and set4 (valence), areshown. We make calculations of P H and P ¯ H using these two sets of parameterizations of thepolarized PDF’s to see the sensitivity of the results of P H and P ¯ H on the polarized PDF’s.We carried out the calculations in the COMPASS kinematic region and at eRHIC energy.The results at the two energies are similar and those at COMPASS energy are shown inRef. [16]. We show those at eRHIC energy in Fig. 16.The results show in particular following interesting features. First, the polarizations ofhyperons and anti-hyperons are quite sensitive to the polarized PDF’s. Different sets of28 -4 -3 -2 -1 CTEQ6-set36CTEQ6-set37 x x ( s - s – ) FIG. 12: Examples of the asymmetry of strangeness distributions in CTEQ6 parameterizations at Q = 3GeV . We see in particular that they have opposite signs in most of the x region. polarized PDF’s lead indeed to quite different results of hyperon and anti-hyperon polariza-tions. We see in particular that the differences obtained from different set of polarized PDF’sare generally larger than the differences between the results for different models for the spintransfer in fragmentation. Second, because the relative weights R Hf and spin transfer S Hf arequite different from each other for different flavor f for a given hyperon H , the polarizationsof different hyperons and anti-hyperons are sensitive to polarized PDF’s of different flavors.For example, P Σ + and P Σ − are sensitive to ∆ u ( x ) and ∆ d ( x ) respectively. They have differ-ent signs because the sign of ∆ u ( x ) is different from that of ∆ d ( x ). The magnitude of P Σ + is larger than P Σ − because | ∆ u ( x ) | > | ∆ d ( x ) | . Similar features can be seen for Ξ , Ξ − andthe corresponding anti-hyperons. These two features are important because they show thatwe can use hyperon polarizations in SIDIS to extract information on polarized PDF’s. V. SUMMARY AND OUTLOOK
In summary, we have calculated the longitudinal polarizations of the hyperons and anti-hyperons in semi-inclusive deep-inelastic scattering at COMPASS and eRHIC energies. Wehave in particular made a systematic study of the different contributions to the differences29
L L - s - (x):CTEQ6-SET37s - (x)=-s + (x)s - (x)=s + (x) ● ■ COMPASS x F S e p L / S e p L _ FIG. 13: (color online) Spin transfer S Λ ep S ¯Λ ep as functions of x F at the COMPASS energy √ s =17.35GeV obtained using an asymmetric strangeness distribution in nucleon. Other PDF’s are takenfrom CTEQ2L and SU(6) picture for spin transfer in fragmentation process are used. The datapoints are taken from COMPASS [7]. between the polarization of a hyperon and its anti-particle. We presented the results ob-tained in SIDIS with polarized beam and unpolarized target for the case that a symmetricstrange sea and anti-sea distribution is used and those obtained in the case that an asym-metry between strange sea and anti-sea distribution is taken into account and for reactionswith unpolarized beam and polarized target. Our results show that, (1) at COMPASS en-ergy, valence contributions play an important role in the difference between hyperon andanti-hyperon polarization but are negligible at eRHIC energy; (2) a significant asymmetrybetween strange sea and anti-sea distributions can manifest itself in the difference betweenhyperon and anti-hyperon polarization at eRHIC energy, but high statistics is needed in or-der to detect it; (3) different sets of PDF parameterizations have quite large influence on themagnitudes of hyperon polarizations but the influence on the difference between hyperonand anti-hyperon polarization is relatively small; (4) hyperon and anti-hyperon polariza-tions in reactions using unpolarized beam and polarized target are sensitive to the polarizedparton distributions and different hyperons are sensitive to different flavors, and hence canbe used to extract information on flavor tagging. These results show that both the differ-30 .00.050.10.15 (a) s - (x):CTEQ6-SET37SU(6) DIS LL – P H ( H _ ) -0.050.00.050.10.15 (b) s - (x) = -s + (x) -0.050.00.050.10.150.20 0.2 0.4 0.6 0.8 (c) s - (x) = s + (x) x F (d) s - (x):CTEQ6-SET37SU(6) DIS S + S - - P H ( H _ ) (e) s - (x) = -s + (x) (f) s - (x) = s + (x) x F FIG. 14: (color online) Longitudinal polarizations of Λ, Σ + and their anti-particles obtained usingdifferent asymmetric strangeness distributions in nucleon at eRHIC energy. ence between hyperon and anti-hyperon polarization in reaction with polarized beam andunpolarized target and the polarizations of hyperons and anti-hyperons in reactions withunpolarized beam and polarized target are sensitive to the sea structure of nucleon. Highprecision measurements in particular those at high energies such as at eRHIC are able toprovide us deep insights into the nucleon sea.This work was supported in part by the National Natural Science Foundation of Chinaunder the approval No. 10525523 and Department of Science and Technology of ShandongProvince. [1] D. Buskulic et al. [ALEPH Collaboration], Phys. Lett. B , 319 (1996).[2] K. Ackerstaff et al. [OPAL Collaboration], Eur. Phys. J. C , 49 (1998). a)-0.200.20.4 x D d standardx D u standardx D s standardx D d valencex D u valencex D s valence x D f( x , Q ) -0.06-0.04-0.0200.0210 -4 -3 -2 -1 (b)x D d – valencex D u – valencex D s – valencex D d – =x D u – =x D s – standard x FIG. 15: (color online) Comparison of the polarized quarks (a) and anti-quarks (b) distributionsobtained from leading order GRSV2000 standard and valence scenario at Q = 3GeV .[3] P. Astier et al. [NOMAD Collaboration], Nucl. Phys. B ,3 (2000); and Nucl. Phys. B ,3 (2001).[4] M. R. Adams et al. [E665 Collaboration], Eur. Phys. J. C , 263 (2000).[5] A. Airapetian et al. [HERMES Collaboration], Phys. Rev. D , 112005 (2001); and Phys.Rev. D , 072004 (2006).[6] V. Y. Alexakhin [COMPASS Collaboration], arXiv:hep-ex/0502014.[7] M. G. Sapozhnikov [COMPASS Collaboration], arXiv:hep-ex/0503009; arXiv:hep-ex/0602002;and talk given at the international Workshop on ”Hadron Structure and QCD”(HSQCD’2008), Gatchina, Russia, June 30 - July 4, 2008.[8] Q. H. Xu [STAR Collaboration], AIP Conf. Proc. , 71 (2006) [arXiv:hep-ex/0512058];AIP Conf. Proc. , 428 (2007) [arXiv: hep-ex/0612035]; and E.P. Sichtermann [STARCollaboration], talk given at the 18th International Symposium on Spin Physics, Universityof Virginia, October 6-11, 2008. (a) L P H (b) X (c) X - -0.010-0.0050.00.0050.0100.0150 0.2 0.4 0.6 0.8 (d) S + (e) S - H SU(6) standard H DIS standard H SU(6) valence H DIS valence x F -0.010-0.0050.00.005 (a) L - P H _ (b) X - 0 (c) X - + -0.020-0.015-0.010-0.0050.00 0.2 0.4 0.6 0.8 (d) S - - (e) S - + H - SU(6) standard H - DIS standard H - SU(6) valence H - DIS valence x F FIG. 16: (color online) Longitudinal polarizations of the hyperons(upper) and anti-hyperons (lower)as a function of x F at eRHIC energy with the longitudinal polarized target.[9] G. Gustafson and J. Hakkinen, Phys. Lett. B , 350 (1993).[10] C. Boros and Z. T. Liang, Phys. Rev. D , 4491 (1998) [arXiv:hep-ph/9803225].[11] C. X. Liu and Z. T. Liang, Phys. Rev. D , 094001 (2000) [arXiv:hep-ph/0005172].[12] C. X. Liu, Q. H. Xu and Z. T. Liang, Phys. Rev. D , 073004 (2001) [arXiv:hep-ph/0106184].[13] Z. T. Liang, in ”Datong 2001, Multiparticle dynamics”, proceedings of the 31st InternationalSymposium on Multiparticle Dynamics (ISMD 2001), Datong, China, 1-7 Sep. 2001, WorldScientific, edited by Wu et al ., p.78-83, arXiv:hep-ph/0111403; and Z.T. Liang and C.X. Liu,Phys. Rev. D , 057302 (2002).
14] Q. H. Xu, C. X. Liu and Z. T. Liang, Phys. Rev. D , 114008 (2002) [arXiv:hep-ph/0204318].[15] Q. H. Xu and Z. T. Liang, Phys. Rev. D , 034015 (2004) [arXiv:hep-ph/0406119].[16] H. Dong, J. Zhou and Z. T. Liang, Phys. Rev. D , 033006 (2005) [arXiv:hep-ph/0506207].[17] Q. H. Xu, Z. T. Liang and E. Sichtermann, Phys. Rev. D , 077503 (2006)[arXiv:hep-ph/0511061].[18] Y. Chen, Z. T. Liang, E. Sichtermann, Q. H. Xu and S. S. Zhou, Phys. Rev. D , 054007(2008) [arXiv:0707.0534 [hep-ph]].[19] J. R. Ellis, D. Kharzeev and A. Kotzinian, Z. Phys. C , 467 (1996) [arXiv:hep-ph/9506280].[20] A. Kotzinian, A. Bravar and D. von Harrach, Eur. Phys. J. C , 329 (1998)[arXiv:hep-ph/9701384].[21] J. R. Ellis, A. Kotzinian and D. Naumov, Eur. Phys. J. C , 603 (2002)[arXiv:hep-ph/0204206].[22] J. R. Ellis, A. Kotzinian, D. Naumov and M. Sapozhnikov, Eur. Phys. J. C , 283 (2007)[arXiv:hep-ph/0702222].[23] B. Q. Ma and J. Soffer, Phys. Rev. Lett. , 2250 (1999) [arXiv:hep-ph/9810517].[24] B. Q. Ma, I. Schmidt, J. Soffer and J. J. Yang, Phys. Rev. D , 114009 (2000)[arXiv:hep-ph/0008295].[25] B. Q. Ma, I. Schmidt, J. Soffer and J. J. Yang, Eur. Phys. J. C , 657 (2000)[arXiv:hep-ph/0001259].[26] B. Q. Ma, I. Schmidt, J. Soffer and J. J. Yang, Phys. Lett. B , 254 (2000)[arXiv:hep-ph/0005210].[27] M. Anselmino, M. Boglione and F. Murgia, Phys. Lett. B , 253 (2000)[arXiv:hep-ph/0001307].[28] M. Anselmino, M. Boglione, U. D’Alesio, E. Leader and F. Murgia, Phys. Lett. B , 246(2001) [arXiv:hep-ph/0102119].[29] A. I. Signal and A. W. Thomas, Phys. Lett. B , 205 (1987) .[30] S. J. Brodsky and B. Q. Ma, Phys. Lett. B , 317 (1996) [arXiv:hep-ph/9604393].[31] M. Burkardt and B. Warr, Phys. Rev. D , 958 (1992).[32] H. Holtmann, A. Szczurek and J. Speth, Nucl. Phys. A , 631 (1996)[arXiv:hep-ph/9601388].[33] H. R. Christiansen and J. Magnin, Phys. Lett. B , 8 (1998) [arXiv:hep-ph/9801283].
34] F. G. Cao and A. I. Signal, Phys. Rev. D , 074021 (1999) [arXiv:hep-ph/9907297].[35] F. Olness et al. , Eur. Phys. J. C , 145 (2005) [arXiv:hep-ph/0312323].[36] H. L. Lai, P. M. Nadolsky, J. Pumplin, D. Stump, W. K. Tung and C. P. Yuan, JHEP ,089 (2007) [arXiv:hep-ph/0702268].[37] A. Deshpande, R. Milner, R. Venugopalan and W. Vogelsang, Ann. Rev. Nucl. Part. Sci. ,165 (2005) [arXiv:hep-ph/0506148].[38] B. Jager, A. Schafer, M. Stratmann and W. Vogelsang, Phys. Rev. D , 054005 (2003)[arXiv:hep-ph/0211007].[39] H. Plothow-Besch, Comput. Phys. Commun. , 396 (1993) .[40] R. D. Field and R. P. Feynman, Nucl. Phys. B , 1 (1978).[41] B. Andersson, G. Gustafson, G. Ingelman and T. Sjostrand, Phys. Rept. , 31 (1983).[42] T. Sjostrand, Comput. Phys. Commun. , 74 (1994).[43] G. Ingelman, A. Edin and J. Rathsman, Comput. Phys. Commun. , 108 (1997)[arXiv:hep-ph/9605286].[44] M. Gluck, E. Reya, M. Stratmann and W. Vogelsang, Phys. Rev. D , 4775 (1996)[arXiv:hep-ph/9508347]; and Phys. Rev. D ,094005 (2001) [arXiv:hep-ph/0011215].[45] J. Blumlein and H. Bottcher, Nucl. Phys. B , 225 (2002) [arXiv:hep-ph/0203155].[46] E. Leader, A. V. Sidorov and D. B. Stamenov, Phys. Rev. D , 034023 (2006)[arXiv:hep-ph/0512114]; and Phys. Rev. D , 074027 (2007) [arXiv:hep-ph/0612360].[47] T. Gehrmann and W. J. Stirling, Phys. Rev. D , 6100 (1996) [arXiv:hep-ph/9512406].[48] M. Hirai, S. Kumano and N. Saito [Asymmetry Analysis Collaboration], Phys. Rev.D , 054021 (2004) [arXiv:hep-ph/0312112]; and Phys. Rev. D , 014015 (2006)[arXiv:hep-ph/0603213].[49] D. de Florian and R. Sassot, Phys. Rev. D , 094025 (2000) [arXiv:hep-ph/0007068]; D. deFlorian, G. A. Navarro and R. Sassot, Phys. Rev. D , 094018 (2005) [arXiv:hep-ph/0504155];D. de Florian, R. Sassot, M. Stratmann and W. Vogelsang, Phys. Rev. Lett. , 072001(2008) [arXiv:0804.0422 [hep-ph]]., 072001(2008) [arXiv:0804.0422 [hep-ph]].