Nuclear dependence of the transverse single-spin asymmetry in the production of charged hadrons at forward rapidity in polarized $p+p$, $p+$Al, and $p+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV

We report on the nuclear dependence of transverse single-spin asymmetries (TSSAs) in the production of positively-charged hadrons in polarized $p^{\uparrow}+p$, $p^{\uparrow}+$Al and $p^{\uparrow}+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The measurements have been performed at forward rapidity ($1.4<\eta<2.4$) over the range of $1.8<p_{T}<7.0$ GeV$/c$ and $0.1<x_{F}<0.2$. We observed a positive asymmetry $A_{N}$ for positively-charged hadrons in \polpp collisions, and a significantly reduced asymmetry in $p^{\uparrow}$+$A$ collisions. These results reveal a nuclear dependence of charged hadron $A_N$ in a regime where perturbative techniques are relevant. These results provide new opportunities to use \polpA collisions as a tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSA as a new handle in studying small-system collisions.

We report on the nuclear dependence of transverse single-spin asymmetries (TSSAs) in the production of positively-charged hadrons in polarized p ↑ + p, p ↑ +Al and p ↑ +Au collisions at √ s N N = 200 GeV. The measurements have been performed at forward rapidity (1.4 < η < 2.4) over the range of 1.8 < pT < 7.0 GeV/c and 0.1 < xF < 0.2. We observed a positive asymmetry AN for positivelycharged hadrons in p ↑ +p collisions, and a significantly reduced asymmetry in p ↑ +A collisions. These results reveal a nuclear dependence of charged hadron AN in a regime where perturbative techniques are relevant. These results provide new opportunities to use p ↑ +A collisions as a tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSA as a new handle in studying small-system collisions.
Understanding the transverse-single-spin asymmetries (TSSA), that describe the azimuthal-angular dependence of particle production relative to the transverse-spin direction of the proton in the reaction p ↑ + p → h + X, has been a long-standing puzzle. The first observations in pion production at large Feynman-x (x F ) [1] showed measured TSSAs that were considerably larger than early theoretical predictions [2], which were of order 10 −4 . Surprisingly large measured TSSAs continued to persist in hadronic collisions at high energies up to √ s = 500 GeV [3][4][5][6][7][8][9][10][11][12][13][14]. To explain these large TSSAs, two approaches were proposed within perturbative quantum chromodynamics (pQCD). One approach is called transverse-momentum-dependent (TMD) factorization, in which the TSSA is generated by the Sivers effect [15,16] and the Collins effect [17] coupled with the transversity distribution. The Sivers effect describes correlations between the nucleon's transverse spin direction and the transverse momentum of a parton in the polarized nucleon. The Collins effect arises from the fragmentation of a transversely polarized parton into a finalstate hadron. The TMD factorization requires two scales where only one needs to be hard, such as in semi-inclusive deep-inelastic scattering (SIDIS) with p T ∼ Λ QCD Q, where p T is the hadron transverse momentum and Q is the momentum transferred to the nucleon [18,19]. However, TMD cannot be used directly when only one hard scale is available, such as in high p T inclusive hadron production in p+p collisions. Another approach, the twist-3, collinear-factorization framework [20], is valid for single hadron (with p T Λ QCD ) production in p+p collisions. The full description of TSSAs in p ↑ +p → h + X in the twist-3 collinear factorization includes twist-3 functions from the polarized proton, the unpolarized proton, and the parton fragmentation into final-state hadrons. The twist-3 function which contains quark-gluon-quark correlations and trigluon correlations in the polarized proton has been studied in detail [21][22][23][24][25][26][27][28][29]. The twist-3 effects in the unpolarized proton are shown to be small [30]. Recently, calculations of the twist-3 contribution from parton fragmentation have been carried out and have shown this to be an important mechanism for understanding the TSSA measurements [31][32][33].
The Relativistic Heavy Ion Collider (RHIC) is a unique facility that can accelerate polarized protons and collide them with other (polarized) protons or nuclei. The ex-tension of the transverse-spin asymmetry measurements to p ↑ +A collisions not only gives us a crucial tool for understanding the nature of TSSAs, but also provides a new approach to studying small-system collisions and the parton dynamics inside nuclei. In particular, when measuring hadron production in the proton-going direction, the properties of nuclear gluons in the small-x region can be probed, where x is the fraction of the proton's longitudinal momentum carried by the parton.
The dynamics of gluons in the small-x regime, where the gluon density is predicted to increase drastically, can be described by the color-glass condensate (CGC) formalism [34]. In recent years, substantial attention has been given to an interplay between small-x physics and spin physics by studying TSSAs in transversely-polarized proton and ion collisions (p ↑ +A). Gluon saturation effects in a nucleus are taken into account for various calculations of TSSAs in p ↑ +A collisions [35][36][37][38][39][40][41][42][43]. The hybrid approach for calculating TSSAs [41,42] utilizes the twist-3 framework for the polarized-proton side and the CGC framework for the target-nucleus side. In this approach, an A-dependence of the TSSA arises from the saturation scale Q s , where Q 2 sA ∝ A 1/3 for the target nucleus. A recent calculation using the hybrid approach for the Adependence of the TSSA in p ↑ +A → h + X showed that the contribution from the twist-3 functions in the polarized proton is independent of A for p T Λ QCD [44], while the fragmentation function can generate an A −1/3dependence for Λ QCD p T Q s [45] in the forward (relative to the proton beam direction) region. On the other hand, in [44,45], it was pointed out that phenomenological studies based on TMD factorization may imply an A-dependence for p T < ∼ Q s from a Sivers-type contribution [35]. Furthermore, a Collins-type contribution [36] could result in an A-dependence for p T Q s but no A-dependence for p T Q s . Therefore, measurements of the A-dependence of TSSAs in p ↑ +A → h + X can be a crucial tool for determining the dominant source of TSSAs in p ↑ +p → h + X. We note preliminary results from the STAR collaboration [46] of measured A N for π 0 in p ↑ +p and p ↑ +Au collisions at 2.6 < η < 4.0, 0.2 < x F < 0.7, and p T > 1.5 GeV/c that show small or no A-dependence.
We report here on the observation of a nuclear dependence of the TSSA of positively-charged hadron production at forward rapidity (0.1 < x F < 0.2 and 1.4 < η < 2.4) in collisions between transversely polarized protons and unpolarized protons or nuclei, p ↑ +p, p ↑ +Al, p ↑ +Au at √ s N N = 200 GeV measured with the PHENIX detector. The positively-charged hadron is preferred in the nuclear dependence measurement because the significance for the TSSA of negatively-charged hadrons will be reduced by the partial cancellation of the asymmetry due to opposite signs of TSSA for π − and K − in p ↑ +p collisions [9,10]. In this measurement, we follow the convention to quantify the TSSA as A N , where A N is the azimuthal angular (φ h ) modulation of the hadron production cross section, σ, relative to the azimuthal angle of the transverse spin of the proton φ pol , i.e., The data from transversely polarized p ↑ +p, p ↑ +Al, and p ↑ +Au collisions at √ s N N = 200 GeV were collected with the PHENIX detector during the RHIC 2015 running period. Proton beams were polarized vertically with respect to the beam direction with an average polarization of 58% (clockwise-beam) or 57% (counterclockwisebeam) for p ↑ +p, 58% for p ↑ +Al, and 61% for p ↑ +Au collisions, with a relative uncertainty of 3% due to uncertainty in the polarization normalization. The beams are bunched. To minimize systematic effects due to time dependence of machine and detector performance, the spin configuration of the colliding bunches is alternated every 106 ns. The PHENIX detector comprises two central arms at midrapidity and two muon arms at forward rapidity [47]; only reconstructed tracks from the muon arms are used for this analysis. The two muon spectrometers cover 1.2 < η < 2.4 (polarized p-going direction) and −2.2 < η < −1.2 (A-going direction) in pseudorapidity with full azimuthal angle coverage. Each muon arm has 7.5 nuclear interaction lengths (λ I ) of hadron absorber followed by a muon tracker (MuTr), which is a set of three stations of cathode strip chambers for momentum measurements of charged particles. The MuTr determines the momentum of a charged particle in a radial magnetic field of B · dl = 0.72 T · m with a momentum resolution of δp/p ≈ 0.05 for hadrons in the kinematic range of this analysis. A Muon Identifier (MuID), located behind the MuTr, comprises five layers of stainless-steel absorbers (∼ 5λ I total) and Iarocci tube planes. The MuID helps to identify muons and hadrons based on the penetration depth of the tracks at p z > ∼ 3.5 GeV/c [48]. More details on the muon arms can be found in [49].
The beam-beam counters (BBCs) [50], at z = ±144 cm from the nominal interaction point, comprise two arrays of 64 quartzČerenkov detectors and cover the full azimuth and 3.1 < |η| < 3.9. The BBCs are used to determine the collision vertex z-position as well as to provide a minimum-bias (MB) trigger with efficiencies of 55% for p+p, 72% for p+Al, and 84% for p+Au collisions. The A-going side of the BBC is also used to determine the event centrality based on the distribution of the charge sum [51]. The recorded events are sampled by the MB trigger combined with muon triggers to enrich good muon and hadron tracks. The MuID provides a trigger for events containing one or more hadron or muon candidates. Momentum-sensitive triggering is provided by hit information from the MuTr to enrich tracks with p T > 3 GeV/c [52].
This analysis uses only charged tracks that stop in the middle of the MuID planes (third or fourth plane out of five planes) due to a hadronic interaction with the absorber material. In the kinematic region of 0.1 < x F < 0.2, where the longitudinal momentum of particles is larger than 10 GeV/c, positively-charged hadron candidates mostly comprise π + and K + .
The particle composition in the measured chargedhadron sample was estimated with a method developed in [48,53], based on RHIC data [54][55][56], pythia [57] and hijing [58] event generators, and geant4-based [59] detector simulation. The K + /π + ratio in midrapidity at p T ∼ 2 GeV/c (typical for our data) was measured to be ∼ 0.35 in both p+p and d+Au collisions [54][55][56], which according to pythia and hijing event generators remains approximately unchanged in p+Al and p+Au collisions, and from the midrapidity region to the muon arm rapidity 1.2 < η < 2.2. The p/π + ratio at midrapidity was found to increase from ∼ 0.25 in p+p collisions to ∼ 0.35 in d+Au collisions [54][55][56], which according to pythia for p+p (hijing for d+Au) increases to ∼ 0.3 (∼ 0.5) at the muon arm rapidity, with the ratio in p+Au consistent with that in d+Au, and the ratio in p+Al slightly smaller than that in d+Au. The initial charged hadron composition is significantly modified due to particle interaction in the detector material, which according to geant4 detector simulation modifies the initial K + /π + (p/π + ) ratio by a factor of 2.7 (0.4), which varies by ≈5% for different hadron interaction models [59]. As a result, the π + /K + /p particle composition in our measured positively charged hadron sample is evaluated to be 45%/47%/5% in p+p collisions, with increased proton fraction to 7% (9%) in p+Al (p+Au) collisions.
The unbinned maximum-likelihood method for extracting A N was established in a previous study [60] which used the same detectors. Compared to binned approaches, this method is robust even for low-statistics data. The log-likelihood is defined to be where φ i is the azimuthal angle of each track with respect to the direction of the polarized proton beam, φ pol is the azimuthal angle for the beam polarization direction, which in the 2015 PHENIX run takes the values +/ − π 2 for ↑ / ↓ spin-signed beam bunches, respectively, and P is the beam polarization. The asymmetry A N is determined by maximizing log L. For p ↑ +p collisions, both beams are polarized, therefore the values of A N were measured separately for each beam and then averaged. For p ↑ +A collisions, only the clockwise proton beam was polarized. The statistical uncertainty was calculated from the second derivative of the log-likelihood estimator, The A N calculated from the likelihood method is compared with the following azimuthal-fitting method based on the polarization formula [61]: where A N (φ) is the simple count-based transverse single spin asymmetry in each of the 16 azimuthal φ-bins, σ ↑ , σ ↓ are cross sections for each polarization of spin up or down, N ↑ , N ↓ are yields, and R = L ↑ /L ↓ is the luminosity ratio (relative luminosity) between bunches with spin up and down, determined from the number of sampled MB triggers corresponding to different spin orientations. From this, A N is extracted from the fit of A N (φ) with a function A N · sin(φ pol↑ − φ i ). The relative variation between this method and the log likelihood method is included in the systematic uncertainty. Figure 1 shows the sine modulation of positivelycharged hadrons for 0.1 < x F < 0.2 and 1.8 < p T < 7.0 GeV/c in p ↑ +p, p ↑ +Al, and p ↑ +Au collisions at √ s N N = 200 GeV, as calculated using Eq. 3. The relatively larger statistical uncertainty in the bin at φ ∼ 0.6 rad is caused by a known detector inefficiency. The χ 2 /N DF of the fits are 9.8/15 for p ↑ +p, 16.3/15 for p ↑ +Al, and 12.9/15 for p ↑ +Au. The p ↑ +p results show a clear nonzero modulation, while the p ↑ +Al results show a weaker modulation. In p ↑ +Au collisions, the modulation is consistent with zero within the statistical uncertainty.
The finite momentum and azimuthal angle φ resolution in the MuTr and the interactions of particles with the materials prior to entering the MuTr lead to a kinematic smearing for the A N measurement. This smearing effect was studied and corrected with a full detector geant4 simulation. The effect due to the φ smearing was found to be negligible. The momentum smearing effect was evaluated by resolving a set of linear equations connecting A N for the true x F bins (A truth N ) and A N for the reconstructed x F bins (A reco N ): where A reco,m N is A N for the m-th reconstructed x F bin from this measurement and A truth,i N is that for the i-th   Table I. The difference between the obtained A truth N and the measured A reco N is small compared to the statistical uncertainty and is accounted for in the systematic uncertainty. Table I also summarizes the systematic uncertainties for the A N measurements. The difference of A N extracted with two methods, Eqs. 1 and 3, is shown as δA method N . The difference between the obtained A truth N and measured A reco N is assigned as a conservative systematic uncertainty due to the smearing effect, δA smear N . The total systematic uncertainty δA syst N is calculated as a quadratic sum of these two uncertainties. Figure 2 shows A N of positively-charged hadrons in p ↑ +p, p ↑ +Al, and p ↑ +Au collisions vs A 1/3 and the average number of nucleon-nucleon collisions N avg coll . Both quantities are related to the effective thickness of the target. The N avg coll is calculated using the Glauber model [62] for each centrality class in p ↑ +A collisions [51]. The recent efforts to calculate A N in p ↑ +p [33] and p ↑ +A collisions, accounting for gluon saturation effects [44,45] suggested that A N could be A-independent or A −1/3 -dependent for the different contributions to A N in the hybrid approach. The results in this paper strongly disfavor the A-independent scenario. This may suggest that a contribution from the A-dependent terms can be the dominant source of A N in p ↑ +p → h + X in the kinematic range of this measurement. However, we note the p T ∼ 2.9 GeV/c in our results is larger than the saturation scale in the Au nucleus [63,64].
The N avg coll -dependence of A N also suggests the decrease of A N is related to the density of nuclear matter inside the target nucleus which the projectile proton traverses. This N avg coll -dependence of A N could be related to novel effects in p+A collisions, such as multiple scattering of partons in the initial and/or final stages of the hard scattering, which is also indicated in the recent results of the nuclear modification of single hadron production and transverse momentum broadening in dihadron correlations in p+A collisions [56,65,66]. Another possibility is interaction of the parton with hot QCD matter produced in p+A collisions, as suggested by recent results in small systems [67][68][69].
To summarize, we have reported A N of positivelycharged hadrons for 1.4 < η < 2.4, 0.1 < x F < 0.2, and 1.8 < p T < 7.0 GeV/c in p ↑ +p, p ↑ +Al, and p ↑ +Au collisions at √ s N N = 200 GeV. For the first time, we observed an A-dependent A N in light hadron production in p+A collisions, with the asymmetry values dropping from ∼3% in p+p collisions to a value consistent with zero in p+Au collisions. These results may provide new insights into the origin of A N and a unique tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSA as a new approach to studying the small-system collisions.
We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions. We also