The Polarised Valence Quark Distribution from semi-inclusive DIS

The semi-inclusive difference asymmetry A^{h^{+}-h^{-}} for hadrons of opposite charge has been measured by the COMPASS experiment at CERN. The data were collected in the years 2002-2004 using a 160 GeV polarised muon beam scattered off a large polarised ^6LiD target and cover the range 0.006<x<0.7 and 1<Q^2<100 (GeV/c)^2. In leading order QCD (LO) the asymmetry A_d^{h^{+}-h^{-}} measures the valence quark polarisation and provides an evaluation of the first moment of Delta u_v + Delta d_v which is found to be equal to 0.40 +- 0.07 (stat.) +- 0.05 (syst.) over the measured range of x at Q^2 = 10 (GeV/c)^2. When combined with the first moment of g_1^d previously measured on the same data, this result favours a non-symmetric polarisation of light quarks Delta u-bar = - Delta d-bar at a confidence level of two standard deviations, in contrast to the often assumed symmetric scenario Delta u-bar = Delta d-bar = Delta s-bar = Delta s.

The COMPASS experiment at CERN has recently published an evaluation of the deuteron spin-dependent structure function g d 1 (x) in the DIS region, based on measurements of the spin asymmetries observed in the scattering of 160 GeV longitudinally polarised muons on a longitudinally polarised 6 LiD target [1]. These measurements provide an accurate evaluation of the first moment of g 1 for the average nucleon N in an isoscalar target g N 1 = (g p 1 + g n 1 )/2, (1) = 0.051 ± 0.003(stat.) ± 0.006(syst.) from which the first moment of the strange quark distribution can be extracted if the value of the octet matrix element (a 8 = 3F − D) is taken from semi-leptonic hyperon decays. 15 At LO in QCD the strange quark polarisation is given by (2) = −0.09 ± 0.01(stat.) ± 0.02(syst.) at Q 2 = 10 (GeV/c) 2 .
Since quarks and antiquarks of the same flavour equally contribute to g 1 , inclusive data do not allow to separate valence and sea contributions to the nucleon spin. We present here additional information on the contribution of the nucleon constituents to its spin based on semi-inclusive spin asymmetries measured on the same data as those used in Ref. [1]. * Corresponding author.
E-mail address: gerhard.mallot@cern.ch (G.K. Mallot). 1 Also at IST, Universidade Técnica de Lisboa, Lisbon, Portugal. 2 On leave of absence from JINR Dubna. 3 Also at Chubu University, Kasugai, Aichi 487-8501, Japan. 4 Also at also at GSI mbH, Planckstr. 1, D-64291 Darmstadt, Germany. 5 Supported by the German Bundesministerium für Bildung und Forschung. 6 Suppported by Czech Republic MEYS grants ME492 and LA242. 7 Supported by DST-FIST II grants, Government of India. 8 Supported by the Shailabala Biswas Education Trust. 9  Deceased. 15 At the precision of the experiment the value of Γ N 1 is unchanged when the evolution of the measured values g 1 (x i , Q 2 i ) to a common Q 2 is done at LO or at NLO in QCD.
The semi-inclusive spin asymmetries for positive and negative hadrons h + and h − are defined by where the arrows indicate the relative beam and target spin orientations.
The data used in the present analysis were collected by the COMPASS Collaboration at CERN during the years 2002-2004. The event selection requires a reconstructed interaction vertex defined by the incoming and scattered muons and located inside one of the two target cells [2]. The energy of the beam muon is required to be in the interval 140 GeV < E μ < 180 GeV and its extrapolated trajectory is required to cross entirely the two cells in order to equalise the fluxes seen by each of them. DIS events are selected by cuts on the photon virtuality (Q 2 > 1 (GeV/c) 2 ) and on the fractional energy of the virtual photon (0.1 < y < 0.9). Final state muons are identified by signals collected behind the hadron absorbers. The hadrons used in the analysis are required to originate from the interaction vertex and to be produced in the current fragmentation region. The latter requirement is satisfied by selecting hadrons with fractional energy z > 0.2. In addition an upper limit z < 0.85 is imposed in order to suppress hadrons from exclusive diffractive processes and to avoid contamination from muons close to the beam axis which escape identification by the muon filters. The hadron identification provided by the RICH detector is not used in the present analysis. The resulting sample contains 30 and 25 million of positive and negative hadrons, respectively.
The target spins are reversed at regular intervals of 8 hours during the data taking. The spin asymmetries are obtained from the numbers of hadrons collected from each target cell during consecutive periods before and after reversal of the target spins, following the same procedure as for inclusive asymmetries [3]. They are listed in Table 1 and also shown in Fig. 1 as a function of x, in comparison to the SMC [4,5] and HERMES [6] results. The consistency of the results from the three experiments illustrates the weak Q 2 dependence of the semi-inclusive asymmetries. The COMPASS results show a large gain in statistical precision with respect to SMC, especially in the low x region (x < 0.04), while at larger x the COMPASS errors are comparable to those of HERMES. The systematic errors, shown by the bands at the bottom of the figure, result from different sources. The uncertainty on the various factors entering in the asymmetry calculation (beam and target polarisation, depolarisation factor and dilution factor) leads to a relative error of 8% on the asymmetry when combined in quadrature. The uncertainty due to radiative corrections is smaller than in the inclusive case due to the selection of hadronic events and does not exceed 10 −3 in any x bin. The presence of possible false asymmetries due to time-dependent apparatus effects has been studied in the same way as for the inclusive asymmetries: the data sample has been divided into a large number of subsamples, each of them collected in a small time interval. The observed dispersion of the asymmetries obtained for these subsamples has been found compatible with the value expected from their statistical error. This allows to set an upper limit for this type of false asymmetries at about half of the statistical error. Asymmetries, obtained with different settings of the microwave frequency used for dynamic nuclear polarisation of the target, have also been compared and did not reveal any systematic difference.
Under the common assumption that hadrons in the current fragmentation region are produced by independent quark fragmentation, the semi-inclusive asymmetries A h + , A h − can be written in LO approximation as where q(x, Q 2 ) and q(x, Q 2 ) are the polarised and unpolarised parton distribution functions (pdfs) and D h q (z, Q 2 ) the fragmentation function of a parton q into a hadron h. The above formula does not account for the full complexity of the hadronisation process as described, for instance in the Lund string fragmentation model [7], and its validity in low energy fixed target experiments has been questioned [8]. It has nevertheless been shown to hold as a good approximation at the energy of COMPASS [9].
In addition to purely experimental effects such as (x, z) correlations in the spectrometer acceptance, a z dependence of the semi-inclusive asymmetries A h+ , A h− could reveal a break-down of the independent fragmentation formula (Eq. (4)). In order to check the possible presence of this effect, we have reevaluated the asymmetries for each x bin subdivided into three intervals of z. Within their statistical precision the obtained values do not indicate any systematic z dependence.
In the present analysis we use the "difference asymmetry" which is defined as the spin asymmetry for the difference of the cross sections for positive and negative hadrons .
The difference asymmetry approach for the extraction of helicity distributions, introduced in Ref. [10], has been used in the SMC analysis [4] and been further discussed in [11,12]. In LO QCD and under the assumption of isospin and charge conjugation symmetries, the fragmentation functions D h q cancel out from A π + −π − . In addition, in the case of an isoscalar target and assuming s = s, the difference asymmetries for pions and kaons are both equal to the valence quark polarisation where we introduce the valence quark distributions q v = q −q.
Since kaons contribute to the asymmetry in the same way as pi- ons, their identification is not needed, allowing to reduce the statistical errors. The difference asymmetry for (anti)protons A p−p N has also the same value but under more restrictive assumptions and is more likely to be affected by target remnants. Since protons and antiprotons account only for about 10% of the selected hadron sample, the relation is expected to hold as a good approximation in the present analysis. Monte Carlo studies using POLDIS [13] and Lund string hadronisation show that the asymmetries A At higher order in QCD the difference asymmetries still determine the valence quark polarisation without any assumption on the sea and gluon densities [11]. Fragmentation functions no longer cancel out but their effect is expected to be small [12].
The relation between the difference asymmetries of Eq. (5) and the single hadron asymmetries of Eq. (3) is The ratio of cross sections for negative and positive hadrons r depends on the event kinematics and is obtained as the product of the corresponding ratio of the number of observed hadrons N − /N + by the ratio of the geometrical acceptances a + /a − , The ratio of the number of negative to positive hadrons (Fig. 2, left) decreases with increasing x. This ratio is subject to acceptance corrections because positive and negative hadrons, produced at the same angle, cross different regions of the spectrometer. To this end LEPTO generated Monte Carlo events have been processed through the program simulating the COM-PASS spectrometer performance [2] and reconstructed in the same way as the data. The acceptances a + and a − are indeed found to be different. The ratio a − /a + , which is about 1.0 at low x, increases for x > 0.1 reaching ∼ 1.12 in the highest x bin. Bin migration was found to be negligible. The corrected cross section ratio σ h − /σ h + is also shown in Fig. 2.
The resulting values of the difference asymmetry A h + −h − d as a function of x are shown in Fig. 2 (right) and listed with their statistical and systematic errors in Table 1. The statistical correlation between A h + d and A h − d which is approximately 0.20 over the measured range of x, is taken into account in the evaluation of the error of A h + −h − d . As can be seen from Eq. (8), a singularity appears when the cross section ratio becomes close to one, leading to infinite statistical errors. For this reason, we discard the lowest x bin used in the inclusive g 1 analysis [1] and take x = 0.006 as lower limit for the present analysis. The increase of A h + −h − d for x > 0.1 illustrates the increasing polarisation of valence quarks carrying a larger fraction of the nucleon momentum.
Target remnants may affect current quark fragmentation at low values of the total hadronic energy W . In order to check the possible presence of such effects in our data, we have reevaluated the difference asymmetries A h + −h − d (x) with the cut W 7 GeV. The comparison of the obtained values with those quoted in Table 1 shows that the W cut only affects the two highest intervals of is reduced by about 0.3 σ stat . It will be shown below that these two values can be replaced by more accurate estimations so that the observed W dependence does not affect any further result.
The polarised valence quark distribution u v + d v is obtained by multiplying A h + −h − d by the unpolarised valence distribution of MRST04 at LO [14]. Here two corrections are applied, one accounting for the fact that although R(x, Q 2 ) = 0 at LO, the unpolarised pdfs originate from F 2 's in which R = σ L /σ T was different from zero [15], the other one accounting for deuteron D-state contribution (ω D = 0.05 ± 0.01 [16]) The LO parameterisation of the DNS fit [17] has been used to evolve all values of u v + d v to a common Q 2 fixed at Q 2 0 = 10 (GeV/c) 2 assuming that the difference between u v + d v at the current Q 2 and at Q 2 0 is the same for the data as for the fit [1]. The DNS analysis includes all DIS g 1 data prior to COMPASS, the partial COMPASS data on g 1 from Ref. [3] as well as the SIDIS data from SMC [5] and HER-MES [6]. Two parameterisations of polarised pdfs are provided at LO, corresponding to two different choices of fragmentation functions, KRE [19] and KKP [20]. We have checked that the x dependence of the ratio σ h − /σ h + (Fig. 2) is fairly well reproduced by the LO MRST04 pdfs and the KKP fragmentation functions whereas the KRE parameterisation leads to a much weaker x dependence. For this reason we choose the fit with the KKP parameterisation. The largest corrections to x[ u v (x) + d v (x)] are at large x and Q 2 and do not exceed 0.03. The use of different fits (NLO fit of Ref. [17] or [18]) leads practically to the same results. The resulting values are shown in Fig. 3 (left). The DNS fit, also shown in the figure, is basically defined by the SMC and HERMES semi-inclusive asymmetries. Its good agreement with the COMPASS values (χ 2 = 7.7 for 11 data points) illustrates the consistency between the three experiments.
The sea contribution to the unpolarised structure function F 2 decreases rapidly with increasing x and becomes smaller than 0.1 for x > 0.3. Due to the positivity conditions | q| q and | q| q, the polarised sea contribution to the nucleon spin also becomes negligible in this region. In view of this, the evaluation of the valence spin distribution of Eq. (10) can be replaced by a more accurate one obtained from inclusive interactions. Indeed at LO one obtains The values obtained by taking only the first term on the r.h.s. for x > 0.3 are also shown in Fig. 3. They agree very well with the DNS curve, which is based on previous experiments where the same procedure had been applied [5,6]. The upper limit of the neglected sea quark contribution, derived from the saturation of the positivity constraint | q| q is included in the systematic error.
The first moment of the polarised valence distribution, truncated to the measured range of x, derived from the difference asymmetry for x < 0.3 and from g d 1 for 0.3 < x < 0.7, is shown in Fig. 3 (right). Practically no dependence on the lower limit is observed for x min < 0.03. We obtain for the full measured range of x (13) Γ v (0.006 < x < 0.7) = 0.40 ± 0.07(stat.) ± 0.06(syst.) at Q 2 = 10 (GeV/c) 2 , with contributions of 0.26 ± 0.07 and 0.14 ± 0.01 for x < 0.3 and x > 0.3, respectively. The uncertainty due to the unpolarised valence quark distributions (≈ 0.04) has been estimated by comparing different LO parameterisations and been included in the systematic error. It should be noted that removing the factor (1 + R) in Eq. (10) would increase the value of Γ v to 0.42 ± 0.08 ± 0.06. Our value of Γ v confirms the HERMES result obtained at Q 2 = 2.5 (GeV/c) 2 over a smaller range of x and is also consistent with the SMC result which has three times larger errors (Table 2). The factor (1 + R) was also used in the analyses of the previous experiments.
The difference between our measured value of Γ v (0.006 < x < 0.3) and the integral of g N 1 over the same range of x gives a global measurement of the polarised sea. Indeed, re-ordering Eq. (11) we obtain 0.30 0.006 Table 2 Estimates of the first moments u v + d v and ū + d from the SMC [5], HERMES [6], COMPASS data and also from the DNS fit at LO [17]  The unmeasured low x contribution to Γ v is expected to be negligible since the integral shows no significant variation when its lower limit is varied between 0.006 and 0.02. We thus estimate the first moment as (15) Γ v (0 < x < 1) = 0.41 ± 0.07(stat.) ± 0.06(syst.).
The assumption of a fully flavour symmetric sea ū = d = s = s obviously leads to Γ v (0 < x < 1) = a 8 . As shown in Fig. 3 (right), our experimental value is two standard deviations below the value of a 8 = 0.58 ± 0.03 derived from hyperon β decays [21]. It has been suggested that a value of the valence contribution Γ v smaller than a 8 (as expected from the constituent quark models) could be a hint that a so far unmeasured part of the nucleon's spin resides at x = 0 [22].
An estimate of the light sea quark contribution to the nucleon spin can be obtained by combining the values of Γ v (Eq. (13)), Γ N 1 (Eq. (1)) and a 8 , and the result is found to be zero (Table 2). Possible deviations from the nominal value of a 8 due to SU(3) f symmetry violation in hyperon decays are generally assumed to be of the order of 10% [23] and are included in the systematic error. The zero value of ū + d is in contrast with the non-zero value obtained for s + s (Eq. (2)) and suggests that ū and d , if different from zero, must be of opposite sign. Previous estimates by SMC and HERMES, also given in Table 2, are compatible with this hypothesis. The DNS parameterisation finds a positive ū and a negative d , about equal in absolute value. Opposite signs of ū and d are predicted in several models, e.g. in Ref. [24] (see also [25] and references therein).
Forthcoming COMPASS data on a proton target will provide separate determinations of ū and d .
In conclusion, we have determined at LO QCD the polarised valence quark distribution from the difference asymmetry for oppositely charged hadrons in DIS of muons on a polarised isoscalar target. Its first moment at Q 2 = 10 (GeV/c) 2 over the measured range of x (0.006-0.7) is found to be 0.40 ± 0.07(stat.) ± 0.06(syst.). This value disfavours the assumption of a flavour symmetric polarised sea at a confidence level of two standard deviations and suggests that ū and d are of opposite sign.