Improved measurement of the longitudinal spin transfer to $\Lambda$ and $\bar \Lambda$ hyperons in polarized proton-proton collisions at $\sqrt s$ = 200 GeV

The longitudinal spin transfer $D_{LL}$ to $\Lambda$ and $\bar{\Lambda}$ hyperons produced in high-energy polarized proton--proton collisions is expected to be sensitive to the helicity distribution functions of strange quarks and anti-quarks of the proton, and to longitudinally polarized fragmentation functions. We report an improved measurement of $D_{LL}$ from data obtained at a center-of-mass energy of $\sqrt{s}$ = 200 GeV with the STAR detector at RHIC. The data have an approximately twelve times larger figure-of-merit than prior results and cover $|\eta|<$ 1.2 in pseudo-rapidity with transverse momenta $p_T$ up to 6 GeV/c. In the forward scattering hemisphere at largest $p_T$, the longitudinal spin transfer is found to be $D_{LL}$ = -0.036 $\pm$ 0.048 (stat) $\pm$ 0.013(sys) for $\Lambda$ hyperons and $D_{LL}$ = 0.032 $\pm$ 0.043\,(stat) $\pm$ 0.013\,(sys) for $\bar{\Lambda}$ anti-hyperons. The dependences on $\eta$ and $p_T$ are presented and compared with model evaluations.

The longitudinal spin transfer DLL to Λ andΛ hyperons produced in high-energy polarized proton-proton collisions is expected to be sensitive to the helicity distribution functions of strange quarks and anti-quarks of the proton, and to longitudinally polarized fragmentation functions. We report an improved measurement of DLL from data obtained at a center-of-mass energy of √ s = 200 GeV with the STAR detector at RHIC. The data have an approximately twelve times larger figure-of-merit than prior results and cover |η| < 1.2 in pseudo-rapidity with transverse momenta pT up to 6 GeV/c. In the forward scattering hemisphere at largest pT , the longitudinal spin transfer is found to be DLL = -0.036 ± 0.048 (stat) ± 0.013(sys) for Λ hyperons and DLL = 0.032 ± 0.043 (stat) ± 0.013 (sys) forΛ anti-hyperons. The dependences on η and pT are presented and compared with model evaluations. The self-analyzing weak decay of Λ,Λ, and other hyperons has made it possible to study a number of spin phenomena in nature. In high-energy collisions of heavy ion beams, for example, a substantial alignment was recently observed between the angular momentum of the colliding system and the spin of the emitted hyperons [1]. This provides a new way to study the hot and dense matter produced in such collisions. The discovery of substantial transverse polarization in inclusive Λ production at forward rapidities by protons on nuclear targets continues to present a challenge for theoretical models [2]. Sizable longitudinal Λ +Λ polarization effects have been observed in e + + e − annihilation at an energy corresponding to the Z 0 pole [3,4], originating mostly from fragmentation of the strongly polarized strange quarks and anti-quarks from Z 0 decay [5][6][7]. The spin transfer to the struck quarks is expected to play an important role in semi-inclusive deep-inelastic scattering spin-transfer measurements of longitudinally polarized positron and muon beams off unpolarized targets [8][9][10], while neutrino measurements [11,12] are sensitive to fragments of the target remnant.
In longitudinally polarized p + p collisions, the spin transfer D LL to a Λ hyperon is defined as: where σ denotes the (differential) production crosssection and the superscripts + or − denote the helicity of the beam proton or the produced Λ hyperon. The spin transfer forΛ is defined similarly. At hard scales, D LL is sensitive to the internal spin structure of the proton and of the Λ orΛ hyperon. Theory expectations [13][14][15][16][17] describe D LL in factorized frameworks, where it then arises from quark and anti-quark parton distribution functions (PDFs) in the polarized proton, partonic cross-sections that are calculable, and polarized fragmentation functions. Among the hyperons, the Λ andΛ hyperons are attractive probes [16,18] since a substantial fraction of their spin is expected to be carried by strange quarks and anti-quarks, and their hard production rate [19] is comparatively high. Measurements of Λ andΛ D LL can thus contribute insights into longitudinally polarized fragmentation functions and strange quark and anti-quark helicity distributions in ways that are complementary to other constraints [6,[20][21][22][23][24][25].
In this paper we report an improved measurement of the longitudinal spin transfer D LL to Λ andΛ hyperons in longitudinally polarized proton-proton collisions at √ s = 200 GeV. The data were recorded with the STAR experiment [26] in the year 2009 and correspond to an in-tegrated luminosity, L, of about 19 pb −1 with an average longitudinal beam polarization, P beam , of 57%, measured with a relative 4.7% accuracy [27][28][29][30]. This data sample has a figure-of-merit, P 2 beam L, approximately twelve times higher than that of our previous D LL measurement [31].
The STAR subsystems used in the measurement include the Time Projection Chamber (TPC) [32], which is able to track charged particles in the pseudo-rapidity range |η| < 1.3 with full coverage in the azimuthal angle φ. Particle identification was provided via measurements of specific energy loss, dE/dx, due to ionization from charged particles passing through the TPC gas. The Barrel and Endcap Electromagnetic Calorimeters (BEMC and EEMC) [33,34] are lead-sampling calorimeters covering |η| < 1 and 1.1 < η < 2, respectively, with full coverage in φ. The BEMC and EEMC were used as trigger detectors to initiate the recording of data. Collision events were recorded if they satisfied a jet-patch trigger condition in the BEMC or EEMC, which required transverse energy deposits that exceeded thresholds of ≃ 5.4 GeV (JP1, prescaled) or ≃ 7.3 GeV (L2JetHigh) in a patch of calorimeter towers spanning a range of ∆η×∆φ = 1 × 1 in pseudo-rapidity and azimuthal angle. For our previous D LL measurements [31] the BEMC covered only η > 0 and lower trigger thresholds were used. The EEMC was not used.
A longitudinal Λ orΛ polarization component, P Λ(Λ) , manifests itself through a dependence of the number of observed hyperons on the angle θ * of the decay proton or anti-proton in the hyperon rest frame in the weak decay channel Λ → pπ − orΛ →pπ + : where A is the detector acceptance, α Λ(Λ) is the weak decay parameter, and θ * is the angle between the Λ(Λ) momentum direction (i.e. longitudinal polarization) and the (anti-)proton momentum in the Λ(Λ) rest frame. The dependence of A on θ * and other observables is omitted in this notation. The analysis methods are very similar to those of our prior D LL measurement [31]. The Λ andΛ candidates were identified from the topology of their dominant weak decay channels, Λ → pπ − andΛ →pπ + , each having a branching ratio of 63.9% [35]. TPC tracks were required to be formed by a minimum of 15 hits on the pads of the 45 TPC pad-rows. The beam-collision vertex was reconstructed event-by-event from charged particle tracks reconstructed with the TPC. This vertex, the primary event vertex, was required to be along the beam axis and within 60 cm of the TPC center to ensure uniform tracking efficiency. The data for each beamcollision event were then searched for (anti-)proton and pion tracks with curvatures of opposite sign. The (anti-)protons and charged pions were identified by requiring that dE/dx was within three standard deviations of the respective nominal values. The tracks were then paired to form a Λ(Λ) candidate and topological selections were applied to reduce combinatorial and K 0 S backgrounds. The selections included criteria for the distance of closest approach (DCA) between the paired tracks, the DCA of the reconstructed candidate track to the primary event vertex, the DCAs of the (anti-)proton and pion tracks to the primary event vertex, the decay length of the Λ(Λ) candidate, and the cosine of the angle between the Λ(Λ) candidate momentum and its trajectory from the primary event vertex, cos( r · p). These criteria were tuned in p T intervals so as to keep as much signal as possible while keeping the residual background at an acceptable level of about 10% [36]. Table I summarizes the track and candidate selection criteria, the number of Λ(Λ) candidates used in the analysis, and the estimated residual background. The larger number ofΛ than Λ in the analysis has its origins primarily in the trigger conditions and thresholds for the recorded event sample and the energy deposit in the calorimeters associated with the annihilation of anti-protons fromΛ decay. In minimum-bias proton-proton collisions, theΛ yield is below the one for Λ [19]. Figure 1a) shows the invariant mass distribution for the reconstructed Λ (filled circles) andΛ (open circles) candidates with |η Λ(Λ) | < 1.2 and 3 < p T < 4 GeV/c from the data sample obtained with the JP1 trigger condition. Figure 1b) shows cos θ * versus invariant mass m Λ(Λ) for these events. Besides residual combinatorial background, the distribution shows a band originating from K 0 S particle decays when a fraction of the decay pions with p T > 1.2 GeV/c is misidentified as a (anti-)proton. The Λ andΛ candidates within the invariant mass range 1.110 < m Λ(Λ) < 1.122 GeV/c 2 were kept for further analysis. In addition, the Λ andΛ baryons were required to be associated with a reconstructed jet that satisfied the trigger conditions. For this purpose, a jet sample was reconstructed using the mid-point cone algorithm [37] as in several previous STAR jet analyses [38][39][40]. The association required that the reconstructed η and φ of the Λ orΛ candidate were within the jet cone of radius ∆R = (∆η) 2 + (∆φ) 2 = 0.7. Reconstructed jets were required to have p T > 5 GeV/c. The fraction of Λ (Λ) hyperons associated with a nearside trigger jet increases with increasing hyperon p T from about 43% (55%) for 2 < p T < 3 GeV/c to 62% (72%) for 5 < p T < 8 GeV/c. The larger fraction forΛ than for Λ is due to the aforementioned energy deposit from annihilation of decay anti-protons in the calorimeters.
The longitudinal spin transfer D LL was extracted in small intervals of cos θ * from the ratio: where α Λ = 0.642 ± 0.013 [35], αΛ = −α Λ , N + (N − ) is the number of Λ orΛ hyperons in the cos θ * interval when the beam is positively (negatively) polarized, cos θ * is the average value of cos θ * in this interval, and R = L + /L − denotes the corresponding luminosity ratio for the two beam polarization states. The detector acceptance cancels in this ratio [31]. Eq. 3 follows from Eqs. 1 and 2 and parity conservation in the hyperon production processes. The observed (raw) yields contain the produced hyperons as well as residual background. In addition, both beams at RHIC are polarized. In the analysis, the (raw) candidate yields n ++ , n +− , n −+ , and n −− by helicity configuration of the RHIC beams were weighted with the corresponding relative luminosities to determine the single spin yields used in Eq. 3. The relative luminosities were measured with Beam-Beam Counters (BBC) [28]. The (raw) spin transfer values were then averaged over the entire cos θ * range and a correction was applied for the effects from the residual backgrounds: where the fraction of residual background, r, within the accepted mass interval 1.110 < m Λ(Λ) < 1.122 GeV/c 2 , and the spin transfer for the residual background, D bg LL , were estimated using side-bands 1.094 < m Λ(Λ) < 1.103 GeV/c 2 and 1.129 < m Λ(Λ) < 1.138 GeV/c 2 on either side of the Λ orΛ mass peak. Simulation shows that less accurate results are obtained if this correction procedure is applied in each cos θ * interval and the resulting D LL (cos θ * ) values are then combined. D bg LL was found to be consistent with zero to within its statistical uncertainties. The statistical uncertainties in the background- corrected D LL values were calculated according to: which thus contains r and the statistical uncertainty in background D bg LL . The uncertainty in r is accounted for in a contribution to the systematic uncertainty. Figure 2a) shows D raw LL obtained from Eq. 3 versus cos θ * for Λ baryons with 3 < p T < 4 GeV/c for intervals of positive and negative pseudorapidity with respect to the momentum of the incident polarized proton. Only statistical uncertainties are shown and the data satisfied the JP1 trigger condition. Figure 2b) shows the corre-spondingΛ results. The Λ andΛ results are constant with cos θ * , as expected and as confirmed by the fit quality of the averages. A null-measurement was performed by analyzing the spin transfer to the spinless K 0 S meson, δ LL , through the K 0 S → π + π − decay channel. This decay channel has a topology similar to the Λ → pπ − and Λ →pπ + channels. The values for δ LL were determined with an artificial weak decay parameter α = 1 using otherwise identical methods as for the hyperon spin transfer measurements. The results are shown in Fig. 2c) and are consistent with zero as expected. No significant asymmetries A L , defined as the cross-section asymmetry for positive and negative beam helicity in single polarized proton-proton scattering, were observed either, as expected at √ s = 200 GeV. The asymmetries A LL , defined as the cross-section asymmetry for aligned and opposed beam helicity configurations in double polarized protonproton scattering, do not necessarily vanish. While no statistically significant values were observed for the Λ andΛ hyperons, an average value of A LL = 0.006 ± 0.002 was observed for K 0 S mesons associated with jets for p T > 1 GeV/c. Figure 3 shows a comparison of the results of D LL to the Λ (top) andΛ (bottom) for negative (left) and positive (right) hyperon pseudorapidities obtained from the JP1 and L2JetHigh triggered data samples in comparison with previously published data [31] in the region of kinematic overlap. The error bars show the size of the  statistical uncertainties, while the boxes indicate the size of the total systematic uncertainty. The central values along the x-axis have been shifted slightly to higher p T values for the L2JetHigh data for visual clarity, while the previously published results have been offset to slightly smaller values. The present data are seen to surpass the prior results in precision and kinematic range. The size of the total systematic uncertainties ranges from 0.006 to 0.017, varying with p T . The improvement in overall size compared to our previous previous D LL measurement [31] is due mostly to the refined treatment of background (c.f. Eqs. 4 and 5) made practicable with the larger data sample. The size of the systematic uncertainty was estimated by considering contributions from uncertainties in the decay parameter, the beam polarization, residual transverse beam polarization components, the relative luminosities, as well as contributions from uncertainties in the fraction of residual background, uncertainties caused by event overlap (pileup) in the detector, and uncertainties introduced by the trigger conditions [36]. Among these, the dominant sources of the systematic uncertainty are from pileup and from trigger bias. These causes of systematic uncertainty are uncorrelated and their effects act primarily as offsets to the data. The effects of pileup were studied with the data by considering variations of the reconstructed spin-sorted hyperon yields per beam-collision event for different collision rates. The reconstructed hyperon yield per collision event is expected to be constant in the absence of pileup. Constant and linear extrapolations to small collision rates, where pileup vanishes, were then used to estimate the contribution from any existing pileup effects in the data to the systematic uncertainty. The resulting uncertainty contribution is found to be largest for small values of p T . The trigger conditions can affect the composition of the recorded data sample in several ways. For example, it could change the relative fractions of the underlying hard scattering processes. The trigger conditions can also distort the sampling for different momentum fractions in the fragmentation. In addition, the trigger conditions could affect the contributions to the Λ orΛ yields from the decays of heavier hyperons. Each of these effects was studied with Monte Carlo simulated events that were generated with PYTHIA 6.4.28 [41], using the Perugia 2012 tune [42] further adapted to the conditions at RHIC [43] and an increased K-factor [19]. These events were then passed through the STAR detector response package based on GEANT 3 [44]. The effects from differences caused by the trigger conditions were then evaluated using the model expectations for D LL from Ref. [17]. Their size was found to increase with p T and forms the dominant contribution to the systematic uncertainty at large p T .
The results for the different trigger conditions in Fig. 3 are point-by-point consistent, as demonstrated by χ 2 = 17 for 16 degrees of freedom. The results from the JP1 and L2JetHigh trigger conditions were thus combined, using statistical weights. Figure 4 shows the combined  data on the spin transfer D LL to the Λ and to theΛ as functions of p T for positive η. The data provide no evidence for a difference between Λ andΛ D LL , as indicated by χ 2 = 1.5 for 4 degrees of freedom. The curves show the theory expectations from Ref. [13,45], which considers D LL for Λ andΛ combined, and from Ref. [17], which considers D LL separately for Λ and forΛ. The theory expectations are seen to increase in size with increasing p T . They increase also with increasing η. The differences between the curves from Ref. [13,45] arise primarily from assumptions for the polarized fragmentation functions, which are thus far only poorly constrained. The data do not provide evidence for a non-vanishing spin transfer signal; their comparison with zero yields χ 2 = 3 for 8 degrees of freedom. However, the data tend to be below a theory expectation based on the extreme assumption that the quark polarized fragmentation functions are flavorindependent, commonly referred to as the DSV scenario 3 expectation [13,45]. The overall probability for DSV scenario 3 to yield a dataset with all central values anywhere below the expectation is less than 1% if eight data points are considered and about 6% for four data points. This corresponds to what is seen from the data if the Λ andΛ points are considered separately and if the Λ and Λ data are combined for each p T value. Table II contains the numerical values of the data points in Fig. 4 as well as the corresponding data for negative η. STAR has recently made the first measurement of the transverse spin transfer, D T T , for Λ andΛ hyperons produced in transversely polarized proton-proton collisions [46]. D T T is sensitive to the quark transversity distributions. In addition, STAR is expanding its acceptance by means of an ongoing upgrade to the inner sectors of the TPC and has proposed an instrument upgrade that would enable a program of measurements, including D LL and other Λ andΛ measurements, at very forward rapidities [47].
In summary, we report an improved measurement of the longitudinal spin transfer, D LL , to Λ hyperons andΛ anti-hyperons in longitudinally polarized protonproton collisions at √ s = 200 GeV. The data correspond to an integrated luminosity of 19 pb −1 with an average beam polarization of 57% and were obtained with the STAR experiment in the year 2009. The Λ andΛ data cover |η| < 1.2 and p T up to 6 GeV/c. The longitudinal spin transfer is found to be D LL = -0.036 ± 0.048 (stat) ± 0.013 (sys) for Λ hyperons and D LL = 0.032 ± 0.043 (stat) ± 0.013 (sys) forΛ anti-hyperons produced with η = 0.5 and p T = 5.9 GeV/c, where the corresponding theory expectations reach their largest sizes. While the data do not provide conclusive evidence for a spin transfer signal, the data tend to be below a theory expectation, DSV scenario 3 [13,45], based on the extreme assumption that the quark polarized fragmentation functions are flavor-independent.