Measurement of the Longitudinal Spin Transfer to Lambda and Anti-Lambda Hyperons in Polarised Muon DIS

The longitudinal polarisation transfer from muons to lambda and anti-lambda hyperons, D_LL, has been studied in deep inelastic scattering off an unpolarised isoscalar target at the COMPASS experiment at CERN. The spin transfers to lambda and anti-lambda produced in the current fragmentation region exhibit different behaviours as a function of x and xF . The measured x and xF dependences of D^lambda_LL are compatible with zero, while D^anti-lambda_LL tends to increase with xF, reaching values of 0.4 - 0.5. The resulting average values are D^lambda_LL = -0.012 +- 0.047 +- 0.024 and D^anti-lambda_LL = 0.249 +- 0.056 +- 0.049. These results are discussed in the frame of recent model calculations.


Introduction
The study of the Λ andΛ hyperon polarisation in DIS is important for the understanding of the nucleon structure, the mechanisms of hyperon production and the hyperon spin structure. In particular, it may provide valuable information on the unpolarised strange quark distributions s(x) ands(x) in the nucleon. In this paper measurements of the longitudinal polarisation of Λ andΛ hyperons produced in deep-inelastic scattering (DIS) of polarised muons off an unpolarised isoscalar target are presented.
In a simple parton model a polarised lepton interacts preferentially with a quark carrying a specific spin orientation. After the absorption of the hard virtual photon the remaining system consists of the scattered quark and the target remnant which are both polarised. During the hadronisation, part of the polarisation of the scattered quark or the target remnant is transferred to the produced hyperon. Therefore, the value of hyperon polarisation will depend on the spin dynamics in the hadronisation not only of the scattered quarks, but also of the target remnant. For the kinematic conditions of the COMPASS experiment the role of these effects has been considered in [1]. It occurs that even in the current fragmentation region the spin transfer from the polarised target remnant to the Λ hyperon is substantial. The situation is different for the spin transfer toΛ. A recent analysis [2] shows that at the energy of the COMPASS experiment theΛ polarisation is dominated by the spin transfer from thē s-quarks, and hence is sensitive to thes(x) distribution in the nucleon.
The polarisation of the produced hyperons depends also on their spin structure. For example, in the SU(6) quark model the whole spin of the Λ is carried by the s-quark. Therefore, under the assumption of the quark helicity conservation, even if the struck u-or d-quark were completely polarised this would not lead to a polarisation of the Λ hyperon. As a consequence, if the dominant mechanism of Λ production is the independent u-and d-quark fragmentation, then the polarisation of the directly produced Λ hyperons should be P Λ ∼ 0.
Using SU(3) f symmetry and experimental data for spin-dependent quark distributions in the proton, the authors of [3] predict that the contributions of u-and d-quarks to the Λ spin are negative and substantial, at the level of 20% for each light quark. In this model the spin transfer from u-or d-quarks would lead to a negative spin transfer to Λ.
The Λ andΛ hyperons can also be produced indirectly, via decays of heavy hyperons such as Σ 0 , Σ(1385), Ξ etc. In this case a non-zero spin transfer from u-and d-quarks to Λ is also possible [4], [5].
The spin transfer to Λ andΛ hyperons has been studied extensively in a number of theoretical models [6,7,8,9,10,11,12,13,14] for e + e − collisions and lepton DIS. From a theoretical point of view the simplest case is the polarisation of the hyperons formed in e + e − annihilation at the Zpeak, where hadronisation of the qq-system can be well described using independent fragmentation. The electroweak theory predicts for the s-quarks from Z decay a longitudinal polarisation of -0.94. The corresponding antiquarks have the same degree of polarisation, but with opposite helicity. The fraction of this polarisation that is transferred to the produced Λ(Λ) hyperon was studied using e + e − annihilation by ALEPH [15] and OPAL [16] experiments at LEP. Both experiments find large and negative values for the hyperon longitudinal polarisation, with a strong dependence on the fraction of the primary quark momentum carried by the hyperon. These results clearly indicate that the fragmentation processes preserve a strong correlation between the fragmenting quark helicity and the final hyperon polarisation. However, as was shown in [8], these data are insufficient to distinguish between predictions of the SU(6) or the BJ [3] models.
The longitudinal spin transfer to Λ(Λ) in DIS was measured in a number of experiments [17,18,19,20,21,22,23,24]. The earlier neutrino DIS experiments [17,18,19] have found an indication for a large negative polarisation of Λ hyperons formed in the target fragmentation region. However, the statistics of each of these experiments does not exceed 500 events. The E665 Collaboration [22] has measured Λ(Λ) production using 470 GeV positive muons scattered off hydrogen, deuterium and other nuclear targets. The total number of events amounts to 750 Λ and 650Λ. The spin transfers of Λ andΛ hyperons were found to have opposite signs: negative for Λ and positive forΛ. The NOMAD Collaboration [20,21] has studied Λ andΛ polarisation in DIS using 43 GeV muon neutrinos. The total number of events amounts to 8087 Λ, mainly in the target fragmentation region, and 649Λ. The results confirm those of earlier experiments [17,18,19] and show a large negative longitudinal polarisation P Λ in the target fragmentation region while no significant spin transfer to Λ was detected in the current fragmentation region. This region was later explored by the HERMES Collaboration [23]. The Λ polarisation was measured using the 27.6 GeV longitudinally polarised positron beam. The total statistics is 7300 Λ. Within the experimental uncertainties, the resulting spin transfer was found to be compatible with zero. The STAR Collaboration at RHIC [24] has measured the longitudinal spin transfer to Λ andΛ in polarised proton-proton collisions at a centre of mass energy √ s = 200 GeV. The data sample comprises 30000 Λ and 24000Λ. The measurement is limited to the mid-rapidity region (|η| < 1) with an average x F = 7.5 · 10 −3 . (The Feynman variable is x F = 2p L /W , where p L is the particle longitudinal momentum in the hadronic centre-of-mass system, whose invariant mass is W .) A small spin transfer, compatible with zero, was found for both Λ andΛ.
The statistical accuracy achieved in the measurements discussed above is quite limited, particularly, in the current fragmentation region. The data on the polarisation of the Λ hyperons tend to be compatible with zero and no clear conclusions about the spin transfer mechanism can be drawn. The dependence of the spin transfer on the Bjorken scaling variable x and x F has been mapped out by the HERMES experiment [23], but for the Λ hyperon only.
The present analysis is based on about 70000 Λ and 42000Λ events. Our data allow to explore the x-dependence of the spin transfer to Λ in a large x-interval and to measure, for the first time, the x-and x F -dependences of the spin transfer toΛ. A substantialΛ polarisation is found, which is important for the investigation of the strange quark distribution in the nucleon.

Data analysis
We have studied Λ andΛ production by scattering 160 GeV polarised µ + off a polarised 6 LiD target in the COMPASS experiment (NA58) at CERN. A detailed description of the COMPASS experimental setup is given elsewhere [25].
The data used in the present analysis were collected during the years 2003-2004. The longitudinally polarised muon beam has an average polarisation of P b = −0.76 ± 0.04 in the 2003 run and of P b = −0.80±0.04 in the 2004 run. The momentum of each beam muon is measured upstream of the experimental area in a beam momentum station consisting of several planes of scintillator strips or scintillating fibres with a dipole magnet in between. The target consists of two 60 cm long, oppositely polarised cells. The data from both longitudinal target spin orientations were recorded simultaneously and averaged in the present analysis. The number of events with Λ(Λ) hyperons for each target spin orientation is the same within 1% accuracy.
The event selection requires a reconstructed interaction vertex defined by the incoming and the scattered muon located inside the target. 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 (0.2 < y < 0.9). The data sample consists of 8.67 · 10 7 DIS events from the 2003 run and 22.5 · 10 7 DIS events from the 2004 run.
The Λ andΛ hyperons are identified by their decays into pπ − andpπ + . To estimate systematic effects, decays of K 0 S → π + π − are also analysed. Events with Λ,Λ and K 0 S decays are selected by demanding that two hadron tracks form a secondary vertex. Particle identification provided by a ring imaging Cherenkov detector and calorimeters is not used for hadrons in the present analysis. The results of the analysis for RICH-identified hadrons will be the subject of a separate paper with larger statistics. In order to suppress background events, the secondary vertex is required to be within a 105 cm long fiducial region starting 5 cm downstream of the target. The angle θ col between the hyperon momentum and the line connecting the primary and the secondary vertex is required to be θ col < 0.01 rad. This cut selects events with the correct direction of the hyperon momentum vector with respect to the primary vertex, which results in a reduction of the combinatorial background. A cut on the transverse momentum p t of the decay products with respect to the hyperon direction of p t > 23 MeV/c is applied to reject e + e − pairs due to γ conversion. Only particles with momenta larger than 1 GeV/c were selected to provide optimal tracking efficiency.
The pπ − andpπ + invariant mass distributions with peaks of Λ andΛ are shown in Fig. 1 for the data of the 2004 run. The main sources of background are events from K 0 S decays and combinatorial background. A Monte Carlo (MC) simulation shows that the percentage of kaon background changes from 1 to 20 %, when cos θ varies from -1 to 1, where θ is the angle between the direction of the decay proton(antiproton) in the Λ(Λ) rest frame and the quantisation axis along the momentum vector of the virtual photon. At small and negative values of cos θ the invariant mass distribution of the kaon background is flat. However, at cos θ ∼ 1, the kaon contribution is concentrated mainly at small values of the invariant mass on the left side of the Λ(Λ) peak in Fig. 1. In this angular region the distribution of the kaon events under the Λ(Λ) peak changes rapidly. In order to minimise the influence of the kaon background, the angular interval was limited to −1 < cos θ < 0.6 (a similar cut was introduced in the analysis of the STAR data [24]). This cut reduces the Λ(Λ) signal by ∼10%.
The total number of events after all selection cuts is N (Λ) = 69500 ± 360, N (Λ) = 41600 ± 310 and N (K 0 S ) = 496000 ± 830. The large amount ofΛ events is a unique feature of the COMPASS experiment.

Experimental results
The distributions of experimental and MC events satisfying the selection cuts are shown in Fig. 2 for Λ andΛ as functions of different kinematic variables for events of the 2004 run. The data from the 2003 run have similar distributions. Both experimental and MC distributions are normalised on the total number of events. The acceptance of the COMPASS spectrometer selects Λ(Λ) only in the current fragmentation region. For this analysis we take Λ(Λ) events in the interval 0.05 < x F < 0.5 with the average valuex F = 0. 22 (0.20). In contrast to other DIS experiments [20,21,22,23], our data cover a large region of Bjorken x (x = 0.03) extending to values of x as low as x = 0.005. The average value of Λ fractional energy isz = 0.27. The average values of y and Q 2 are 0.46 and 3.7 (GeV/c) 2 , respectively.
The distributions of the kinematic variables forΛ produced in DIS differ from the Λ ones, sincē Λ production is suppressed in the target fragmentation region. However, in the current fragmentation region theΛ have practically the same kinematic distributions as the Λ (see Fig. 2).
The shaded histograms in Fig. 2 show the same distributions for MC events. The COMPASS Monte Carlo code is based on the LEPTO 6.5.1 generator [26] providing DIS events which are passed through a GEANT-based apparatus simulation programme and the same chain of reconstruction procedures as the experimental events. To provide better agreement between data and MC, a tuning of several LEPTO (JETSET) parameters has been performed. The values of the modified parameters are compared with the default ones [27] and with those used in previous experiments in Table 1. Here, the PARJ(21) parameter corresponds to the width of the Gaussian transverse momentum distribution for the primary hadrons. The parameters PARJ(23), PARJ(24) are used to add non-Gaussian tails to the transverse momentum distribution, PARJ(41), PARJ(42) are the parameters of the symmetric Lund fragmentation function [27]. Some small but systematic differences were observed between the momentum spectra seen in the data and those given by the MC simulation. To overcome these differences, the reconstructed MC events are weighted in order to provide the same momentum and z distributions as the data. The weighting of the MC events leads to a slight modification of the angular acceptance, which is below 2 % in the whole angular range.

Determination of the angular distributions
The acceptance corrected angular distribution of the decay protons(antiprotons) in the Λ(Λ) rest frame is Here, N tot is the total number of acceptance corrected Λ(Λ), the longitudinal polarisation P L is the projection of the polarisation vector on the momentum vector of the virtual photon, α = +(−)0.642±0.013 is the Λ(Λ) decay parameter, θ is the angle between the direction of the decay proton for Λ (antiproton -forΛ, positive π -for K 0 S ) and the corresponding axis. The acceptance correction was determined using the MC simulation for unpolarised Λ andΛ decays. The angular dependence of the acceptance is quite smooth, it decreases by a factor 1.2-1.3 in the angular interval used.
To determine the Λ(Λ) angular distributions, a sideband subtraction method is used. The events with an invariant mass within a ±1.5 σ interval from the mean value of the Λ(Λ) peak are taken as signal. The background regions are selected from the left and right sides of the invariant mass peak. Each band is 2 σ wide and starts at a distance of 3 σ from the central value of the peak. The bands of the signal, as well as the background regions, are shown in Fig. 1. The Λ(Λ) angular distribution is determined by subtracting the averaged angular distribution of the events in the sidebands from the angular distribution of those in the signal region. Figure 3 shows the acceptance corrected angular distributions for all events of the 2004 run for Λ andΛ. The events of the 2003 run have similar angular distributions.

Longitudinal spin transfer
The spin transfer coefficient D Λ LL ′ describes the probability that the polarisation of the struck quark along the primary quantisation axis L is transferred to the Λ hyperon along the secondary quantisation axis L ′ . In our case the primary and the secondary axes are the same L = L ′ and coincide with the virtual photon momentum. The longitudinal spin transfer relates the longitudinal polarisation of the hyperon P L to the polarisation of the incoming lepton beam P b : The longitudinal polarisation P L is determined from a fit of the angular distribution (1). The virtual photon depolarisation factor D(y) is given by: To evaluate the spin transfer, the product P b · D(y) is calculated for each event passing the selection criteria. The beam polarisation P b is parametrised as a function of the incident muon momentum [25]. The P b ·D(y) distribution is determined by subtraction of the averaged distribution of the sideband events from the distribution of the events in the signal region, marked by solid lines in Fig. 1. The average value of the P b · D(y) distribution is used to calculate the D Λ LL according to Eq. (2). The weighted averages of the spin transfers for the 2003 and the 2004 data are: DΛ LL = 0.249 ± 0.056(stat) ± 0.049(sys),x F = 0.20.
The systematic errors are mainly due to the uncertainty of the acceptance correction determined by the Monte Carlo simulation. The presence of possible systematic effects was checked by looking at the result of a physical process with no polarisation effects, i.e. the longitudinal spin transfers to the K 0 S and by checking the stability of the result by varying the selection cuts. The longitudinal spin transfer to the K 0 S turns out to be D K 0 S LL = 0.016 ± 0.010. The value of the longitudinal spin transfer for kaons is taken as an estimate for the corresponding systematic error δ(K 0 S ). Some systematic effect δ(θ) appears due to variation of the cut on cos θ. The uncertainty of the sidebands subtraction method, δ(ss) is estimated by varying the width of the central band of Fig. 1 from ±1.5 σ to ±1, ±1.25, ±1.75 and ±2 σ. Another source of systematic errors is the uncertainty in the beam polarisation, δ(P b ). The relative error in the value of beam polarisation is 0.05.
The values of the systematic errors are given in Table 2. The total systematic error was obtained by summing the various contributions in quadrature.  A similar difference between Λ andΛ spin transfers is observed in the x F dependence (Fig. 5). The spin transfer toΛ tends to increase with x F , while the Λ one does not show any significant x F dependence.

4
Discussion of the results A comparison of the x F dependence of the longitudinal spin transfer to Λ andΛ for COMPASS and other experiments [20,21,22,23] is shown in Fig. 6. For Λ there is general agreement between the present results and existing data. ForΛ the measurement of the E665 Collaboration [22] indicated a positive spin transfer (see Fig. 6b). The present result confirms this observation with a much better statistical precision. The NOMAD Collaboration has found [21] that the spin transfer toΛ is DΛ LL = 0.23 ± 0.15 ± 0.08 at x F = 0.18 in a good agreement with the present result (5). The measured DΛ LL increases with x F , the same trend was found for the Λ polarisation in the experiments at LEP [15,16].
The main conclusion from the our results is that the longitudinal spin transfers to Λ andΛ hyperons in DIS are not equal. To understand this phenomenon let us consider the leading order (LO) parton model, where the spin transfer to Λ(Λ) produced on an unpolarised target by polarised leptons is given by (see for example [9]): Here e q is the quark charge, q(x) is the unpolarised quark distribution function, D Λ(Λ) q (z) and ∆D Λ(Λ) q (z) are the unpolarised and the polarised quark fragmentation functions.
Practically all models of the Λ spin structure predict that the contribution from the s-quark to the Λ spin is dominant. This contribution varies from 100% for the SU(6) model to 60-70% in the BJmodel [3] or the lattice-QCD calculation [28]. It means that scattering off u-or d-quarks is important for the Λ(Λ) production but not for the spin transfer to Λ(Λ). Accordingly, the polarised fragmentation functions ∆D Λ s entering in the numerator of Eq. (6) is expected to be much larger than its light quarks counterparts. Therefore, one may assume that ∆D Λ q (z) = ∆DΛ q (z) ∼ 0 for q = u, d,ū,d. Then the sum in the numerator of (6) is reduced to the strange quark contribution: From (7)- (8) it is seen that the spin transfer from the polarised lepton to Λ andΛ must be different even if s(x) =s(x). The reason is that the denominators of (7) and (8) are proportional to the Λ(Λ) production cross section. Due to the combined effect of the u-quark dominance and the favoured fragmentation of u-quark to Λ, as opposed toΛ, the cross section for Λ is expected to be larger. This expectation is confirmed by the measured yields of Λ andΛ, reported in Sect. 2. Therefore, one may expect a smaller spin transfer to Λ than toΛ.
This conclusion of the LO parton model has been confirmed by the calculation of [2], shown in Figs. 4,5. The model is based on the LEPTO [26] MC event generator, in which the independent fragmentation is replaced by the hadronisation of the string formed by the struck quark and the target remnant. All contributions, including those from the target remnant and from decays of heavy hyperons are taken into account. Figs. 4,5 show that indeed calculations of the model [2] lead to a larger spin transfer toΛ than to Λ. The same trend was found in the recent calculation of [12].
Another indication from the parton model Eqs. (7)- (8) is that the contribution from the strange quarks (antiquarks) is essential for the spin transfer to Λ(Λ). This observation is also confirmed by the results of [2]. In Fig. 7 the degree of the sensitivity to the strange parton distributions is illustrated by the comparison of the results obtained with the CTEQ5L [29](solid line) and GRV98LO [30](dashed line) parton distributions. The GRV98 set is chosen because of its assumption that there is no intrinsic nucleon strangeness at a low scale and the strange sea is of pure perturbative origin. The CTEQ Collaboration allows non-perturbative strangeness in the nucleon. The amount of this intrinsic strangeness is fixed from the dimuon data of the CCFR and NuTeV experiments [31]. As a result, the s(x) distribution of CTEQ is larger than the GRV98 one by a factor of about two in the region x = 0.001 − 0.01. The results in Fig. 7 show that the data on Λ can not discriminate between the predictions since the spin transfer to Λ is small. For theΛ hyperon the use of CTEQ5L set leads to a prediction which is nearly twice larger than the one with the GRV98LO and much closer to the data. This behaviour reflects the difference in the correspondings-quark distributions. If one completely switches off the spin transfer from the s(s) quarks, the spin transfer to Λ(Λ) practically vanishes (dash-dotted line). This feature is independent of the model of Λ spin structure. Calculations in the BJ-model [3], where the spin transfer from the u-and d-quarks(antiquarks) is possible, demonstrate the same absence of the spin transfer to hyperon without contribution from the s(s)-quarks (dotted line).
At present the strange and antistrange quark distributions s(x) ands(x) are directly accessible only through the measurement and study of dimuon events in neutrino and antineutrino DIS [32]. The spin transfer toΛ could provide an additional experimental information for determination of strange quark distributions in the nucleon. To match this goal the present experimental precision must be increased and the theoretical uncertainties should be clarified. For instance, the parameters of the model [2] have been fixed by fitting the NOMAD data [20] at comparatively large x and its predictions can be considered as an illustration of the possible effects.

Conclusions
The longitudinal polarisation transfer from polarised muons to semi-inclusively produced Λ and Λ hyperons has been studied in deep-inelastic scattering at the COMPASS experiment. The present data are the most precise measurements to date of the longitudinal spin transfer to Λ andΛ in DIS. The results show that the spin transfer to Λ is small with D Λ LL = −0.012 ± 0.047 ± 0.024 atx F = 0.22. The spin transfer toΛ is larger with DΛ LL = 0.249 ± 0.056 ± 0.049 atx F = 0.20. These values are in agreement with results of previous measurements [20,21,22,23].
We have also measured the x and x F dependences of the longitudinal spin transfer which are different for Λ andΛ hyperons. The spin transfer to Λ is small, compatible with zero, in the entire domain of the measured kinematic variables. In contrast, the longitudinal spin transfer toΛ increases with x F reaching values of D Λ LL = 0.4−0.5. Comparison with theory shows that the spin transfer to theΛ hyperon strongly depends on the antistrange quark distributions(x) and therefore the precise measurements of theΛ spin transfer will provide useful information about the antistrange quark distributions(x).      for the CTEQ5L without spin transfer from the s-quark (dash-dotted lines). The SU(6) model for the Λ spin structure is assumed. The dotted lines corresponds to the calculations for the CTEQ5L without spin transfer from the s-quark in the BJ-model of Λ spin [3].