Transverse single-spin asymmetries of midrapidity $\pi^0$ and $\eta$ mesons in polarized $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV

We present a measurement of the transverse single-spin asymmetry for $\pi^0$ and $\eta$ mesons in $p^\uparrow$$+$$p$ collisions in the pseudorapidity range $|\eta|<0.35$ and at a center-of-mass energy of 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider. In comparison with previous measurements in this kinematic region, these results have a factor of 3 smaller uncertainties. As hadrons, $\pi^0$ and $\eta$ mesons are sensitive to both initial- and final-state nonperturbative effects for a mix of parton flavors. Comparisons of the differences in their transverse single-spin asymmetries have the potential to disentangle the possible effects of strangeness, isospin, or mass. These results can constrain the twist-3 trigluon collinear correlation function as well as the gluon Sivers function.

We present a measurement of the transverse single-spin asymmetry for π 0 and η mesons in p ↑ +p collisions in the pseudorapidity range |η| < 0.35 and at a center-of-mass energy of 200 GeV with the PHENIX detector at the Relativistic Heavy Ion Collider. In comparison with previous measurements in this kinematic region, these results have a factor of 3 smaller uncertainties. As hadrons, π 0 and η mesons are sensitive to both initial-and final-state nonperturbative effects for a mix of parton flavors. Comparisons of the differences in their transverse single-spin asymmetries have the potential to disentangle the possible effects of strangeness, isospin, or mass. These results can constrain the twist-3 trigluon collinear correlation function as well as the gluon Sivers function.

I. INTRODUCTION
The transverse structure of the proton has attracted increasing experimental and theoretical interest in the past two decades. In particular, transverse single-spin asymmetries (TSSAs) have been one of the primary means to probe transverse partonic dynamics in the nucleon. Large azimuthal asymmetries of up to ≈40% predict small asymmetries on the order of m q /Q [1], where m q is the bare quark mass and Q is the hard scale, indicating that significant nonperturbative effects must dominate the large measured asymmetries. Two different quantum chromodynamics (QCD) approaches have been proposed to describe the large asymmetries observed in hadronic interactions.
In the first approach, nonperturbative parton distribution functions (PDFs) and fragmentation functions (FFs) are explicitly dependent on transverse momentum in the transverse-momentum-dependent (TMD) framework. These functions depend on a soft (k T ) and hard (Q) momentum scale such that Λ QCD < ∼ k T Q. One possible origin of the large TSSAs is the Sivers TMD PDF [2], which correlates the nucleon transverse spin with the parton transverse momentum, k T . Recently, this function has received great theoretical and experimental interest due to the prediction of modified universality between semi-inclusive deep-inelastic lepton-nucleon scattering (SIDIS) and the Drell-Yan (DY) process of quarkantiquark annihilation to dileptons. This prediction is a result of differences in color flow from interactions with remnants in the initial versus final states, related to the parity-time (PT) odd nature of the Sivers TMD PDF and the PT invariance of QCD [3]. Another possible origin of the TSSA is the Collins TMD FF [4], which correlates the transverse polarization of a fragmenting quark to the angular distribution of hadrons.
The second approach to describe the large asymmetries relies on collinear higher-twist effects with multiparton correlations. In the twist-3 approach, interference arises between scattering amplitudes with one and two collinear partons, which leads to a nonzero TSSA. This approach applies to observables in which only one sufficiently hard momentum scale is measured, such that Q Λ QCD [5]. To keep the multiparton correlation functions process-independent, the initial-and final-state * PHENIX Spokesperson: akiba@rcf.rhic.bnl.gov † Deceased interactions between the struck parton and the proton remnants are included in the hard perturbative part of the twist-3 collinear factorization [6]. Collinear twist-3 correlation functions are split into two types: the quarkgluon-quark functions (qgq) and the trigluon functions (ggg). In the context of initial-state effects, the qgq functions describe the interference from scattering off of one quark versus scattering off of a gluon and a quark of the same flavor, while the ggg functions capture the interference between scattering off of one gluon versus scattering off of two. The twist-3 approach is well suited to describe observed inclusive forward hadron asymmetries because the observed hadron p T can be used as a proxy for the hard scale, and unlike the TMD approach, these correlation functions do not explicitly depend on a soft-scale transverse momentum. However, the twist-3 approach has been related to k T moments of TMD PDFs and TMD FFs and has been shown to be equivalent to the TMD approach in the overlapping kinematic regime [7]. Both SIDIS and DY measurements have shown that certain TMD PDFs are consistent with modified universality provided that effects from TMD evolution with the hard scale Q are small [8,9] and that interactions in the initial or final state can produce nonzero asymmetries [10][11][12][13]. In hadronic interactions where at least one final-state hadron is measured, both initial-and final-state interactions can play competing roles in the measured asymmetries; here TMD-factorization breaking has been predicted due to soft gluon exchanges that are possible in both the initial and final states simultaneously [14]. Additional leading-power spin asymmetries have been predicted in hadronic collisions due to this breakdown, without which these asymmetries would be subleading [15], but further work is needed to connect TMD-factorization breaking to experimentally measured asymmetries. Note that inclusive hadron TSSA measurements in hadronic collisions appear to plateau at p T up to 5 GeV/c [16,17] and have been measured to be nonzero at up to p T ≈ 7 GeV/c [18,19]. Recent studies in the twist-3 framework have successfully described the p T dependence of these forward asymmetries by including twist-3 effects in hadronization [20]. The twist-3 perturbative prediction is that the asymmetry should eventually decrease as the hard scale p T continues increasing [15].
Since the inception of the collinear twist-3 and TMD factorization pictures, there has been theoretical evidence that they could combine to form a unified picture of TSSAs in hard processes. This concept was re-cently tested with the first simultaneous global analysis of TSSAs in SIDIS, DY, e + e − annihilation, and protonproton collisions [5]. This study used quark TMD PDFs and FFs to describe the asymmetries in processes that are sensitive to the soft-scale momentum, i.e. SIDIS, Drell-Yan, and e + e − annihilation. These TMD functions were also used to calculate collinear twist-3 qgq correlation functions which were applied to inclusive forward pion asymmetry measurements from the Relativistic Heavy Ion Collider (RHIC). This simultaneous description of TSSAs across multiple collision species indicates that all TSSAs have a common origin that is related to multiparton correlations.
Additional questions about the origin of the TSSAs in hadronic interactions remain. Forward jet measurements indicate that the TSSA is significantly smaller than neutral pion asymmetries at similar x F and √ s [21]. Nonzero kaon and antiproton asymmetries observed at forward rapidities show that the measured asymmetries cannot be due only to proton valence quark contributions as naively predicted in a valence-like model, where the Sivers effect from sea-quarks and/or gluons is ignored, and that the fragmentation of quarks into hadrons in which they are not valence quarks could play a role in the observed nonzero asymmetries [22,23]. Eta meson measurements, sensitive to potential effects from strange quark contributions, isospin, and/or hadron mass show forward asymmetries similar in magnitude to neutral pions [24]. Even four decades after the initial discovery of large TSSAs in hadronic interactions [25], there remain many unresolved questions about their origin. Therefore it is crucial to continue extending measurements to try to better understand the nonperturbative dynamics which are responsible for the TSSAs in hadronic collisions.
In this paper we report a measurement of the TSSA of π 0 and η mesons in p ↑ +p collisions at √ s = 200 GeV in the midrapidity region |η| < 0.35. The data was taken during the 2015 RHIC run and a total integrated luminosity of approximately 60 pb −1 was collected. This measurement extends previous measurements from RHIC to higher p T and reduces the statistical uncertainties by a factor of three in the overlapping p T region.

II. ANALYSIS
The asymmetries are measured with transversely polarized proton beams where the average polarization of the clockwise beam was 0.58 ± 0.02 and that of the counter-clockwise beam was 0.60 ± 0.02 [26]. The direction of the beam polarization was found to be consistent with the vertical within statistical uncertainties. The polarization direction of each beam independently changes bunch to bunch which reduces systematic uncertainties associated with variations in detector performance with time. The changes in polarization direction also allow for polarization-averaged measurements and, for a singlespin asymmetry analysis, provides two ways to measure the TSSA with the same data set. This is done by sorting the particle yields for the polarization directions of one beam at a time, effectively averaging over the polarization of the other beam. These statistically independent asymmetries are used to verify the analysis and are averaged together for the final result.
The data analysis procedure is similar to our previous measurements [16]. Neutral pion and eta mesons are reconstructed via their two-photon decays by using the midrapidity electromagnetic calorimeter (EMCal). The EMCal is located in two central arms, each covering ∆φ = π/2 in azimuth and |η| < 0.35 in pseudorapidity, centered at φ = π/16 and 15π/16. The EMCal comprises two different types of calorimeters: six sectors of sampling lead-scintillator calorimeters and two sectors of Cherenkov lead-glass calorimeters [27]. The two calorimeter systems have different granularity (δφ × δη = 0.011 × 0.011 in the lead scintillator and 0.008×0.008 in the lead glass) and also different responses to charged hadrons, which provides important systematic cross checks for these measurements. A tracking system includes a drift chamber to measure track momentum and pad chamber stations to measure the charged particle hit position [28]. The measurement of the track positions in front of the calorimeter is used to veto charged particles from the photon sample. The beam-beam counters (BBC) are arrays of quartz Cherenkov radiators that surround the beam pipe and are placed ±144 cm away from the nominal collision point. The BBC covers full azimuth and 3.0 < |η| < 3.9 in pseudorapidity. They measure the z-vertex position; a vertex cut of ±30 cm around the nominal collision point is used for this analysis. The minimum bias trigger requires at least one charged particle to be measured in both the north and south sides of the BBC. This analysis is based on the data sample selected with the EMCal-based high-energy-photon trigger with energy threshold of 1.5 GeV, which is taken in coincidence with the minimum bias trigger.
Photons are identified as clusters in the EMCal and are required to pass a shower profile cut which suppresses clusters from hadrons. High-p T trigger photons are paired with another photon in the same event that is also on the same side of the detector. A charged track veto cut eliminates clusters that geometrically match with a measured charged track, reducing background from electrons. The contribution of EMCal detector noise is reduced by a minimum energy cut of 0.5 GeV and a time-of-flight cut of |T OF | < 5 ns. The timing of the cluster is measured by the EMCal and the time zero reference of the event is provided by the BBC. Each photon pair is required to pass an energy asymmetry cut: α = |E 1 − E 2 |/(E 1 + E 2 ) < 0.8. The π 0 yields comprise photon pairs with invariant mass in the signal region ±25 MeV/c 2 from the π 0 mass peak and η meson yields are measured in the range ±70 MeV/c 2 around the η mass peak.
The transverse single-spin asymmetries are determined with the "relative luminosity" formula which compares the yield of particles for when the beam was polarized up versus down. Here P is the beam polarization, N refers to the meson yield, and the arrows refer to the up (↑) or down (↓) directions of beam polarization. The relative luminosity, R = L ↑ L ↓ , is the ratio of the integrated luminosity with bunches that were polarized in opposite directions. This is determined by the number of times each crossing fires the minimum bias trigger and is measured to better than 10 −4 . The acceptance factor, cos(φ) , accounts for the detector azimuthal coverage, where φ = 0 points 90 • from the (vertical) spin axis. This correction is calculated as a function of photon pair p T because the diphoton azimuthal acceptance depends heavily on the decay angle and ranges from 0.95 at low p T to 0.89 at high p T . The asymmetry is calculated separately for the two detector arms and then the average weighed by the statistical error is taken for the final result. As written, Eq. (1) is for the arm to the left of the beam to be taken as polarized, while an overall minus sign is needed for the arm to the right of the beam to be taken as polarized.
An alternative method of calculating the asymmetry is the "square root" formula which is used as a cross check. This formula combines data from the two arms (left and right) and both beam polarization directions (up and down). The subscripts in Eq.
(2) refer to the yields to the left (L) and right (R) side of the polarized beam momentum direction. The measured asymmetries are also corrected for background where r is the fractional contribution of photon pairs from combinatorial background within the invariant mass peak. The background fraction is calculated from fits to the invariant mass spectra where a Gaussian is used to describe the invariant mass peak, and a third order polynomial is used to describe the combinatorial background, as shown in the green curves in Fig. 1. Using this method, the contribution of combinatorial background under the π 0 peak is determined to vary from 10% in the lowest p T bin to 6% in the highest. Under the η meson invariant mass peak, the background fraction varies from 71% to 47% in the lowest to highest p T bins. In Eq. (3), the background asymmetry, A BG N , is evaluated with photon pairs in side band regions located on either side of the signal peak, as represented in the red regions in Fig. 1. For the π 0 analysis these side band regions are 47 < M γγ < 97 MeV/c 2 and 177 < M γγ < 227 MeV/c 2 , and for the η meson analysis these regions are 300 < M γγ < 400 MeV/c 2 and 700 < M γγ < 800 MeV/c 2 . The background asymmetries are consistent with zero across all p T bins. Tables I and II show the asymmetries with statistical and systematic uncertainties. The total systematic uncertainty is the sum of the three sources of systematic uncertainty added in quadrature. The systematic uncertainty on the asymmetry due to the background fraction in Eq. (3) is determined by varying the fit ranges when computing r and calculating how much the backgroundcorrected asymmetry changes. While the asymmetries calculated with the "relative luminosity" (Eq. (1)) and the "square root" (Eq. (2)) formulas were found to be statistically consistent, their difference was assigned as a systematic uncertainty due to possible variations in detector performance and beam conditions. This dominates the total systematic uncertainty for most p T bins.
Bunch shuffling is a technique used to investigate po-  tential sources of systematic uncertainty that could cause the measured asymmetry results to vary from their true values beyond statistical fluctuations. Bunch shuffling involves randomizing the assigned bunch-by-bunch polarization directions of the beam such that the physical asymmetry disappears, thereby isolating the statistical variations present in the data. All asymmetry values have bunch shuffling results consistent with statistical variations except for the lowest p T bin where there is 7% and 6% more variation beyond what is expected from statistical fluctuations in the π 0 and η meson analyses, respectively. These values are used to assign additional systematic uncertainties to the lowest p T bin of the π 0 and η meson asymmetries and dominate the total systematic uncertainty for those bins.
Additional cross checks included examining the asymmetries in the two arms separately using Eq. (1) and measuring the asymmetry as an explicit function of φ. All checks were statistically consistent with the main asymmetry results. Figure 2 shows the A N of neutral pions at midrapidity in p ↑ +p collisions at √ s =200 GeV, where the bands represent the systematic uncertainty and the bars represent the statistical uncertainty. The comparison to previous results [16] demonstrates the improvement in statistical precision. The inset in Fig. 2 shows a zoomed-in comparison at small p T . The new measurement is consistent with our previous measurement and improves the precision on average by a factor of 3. The new measurement of A N of neutral pions is consistent with zero in the entire p T range.

III. RESULTS AND DISCUSSION
The measurement of A N of η mesons in p ↑ +p collisions at √ s = 200 GeV is shown in Fig. 3. This measurement is also compared to the previous result, similarly to Fig. 2. The new measurement is consistent with the previous result and with zero across the entire p T range. Comparisons of π 0 and η meson TSSAs may indicate additional effects from strange quarks, isospin differences, or hadron mass. At forward rapidity, existing measurements [24,29] do not yet clearly resolve whether the η meson asymmetry is larger than the π 0 asymmetry as predicted in some models [30]. At midrapidity, there is a larger contribution from gluon dynamics and, as shown in Fig. 4, both asymmetries are consistent with zero and therefore show no evidence for differences due to strangeness, isospin, or mass. Figure 5 shows this π 0 TSSA result plotted with theoretical predictions. The qgq curve shows the predicted contribution from collinear twist-3 qgq functions from both the polarized proton and the process of hadronization. This curve was calculated with fits that were published in Ref. [5] and has been reevaluated in the rapidity range of PHENIX. Midrapidity π 0 production includes a large fractional contribution from gluons in the proton, so a complete collinear twist-3 description of the midrapidity π 0 TSSA would also need to include the contribution from the trigluon correlation function. Given the small expected contribution from the qgq correlation function, this measurement can constrain future calculations of the ggg correlation function, such as those in Ref. [32].
The other theory curves in Fig. 5   This π 0 asymmetry result plotted with theory calculations for the asymmetry in both the collinear twist-3 [5] and TMD [31] theoretical frameworks. See text for details. the midrapidity π 0 TSSA generated by the Sivers TMD PDF. These curves include contributions from both the quark and gluon Sivers functions and have been evaluated for x F = 0, which approximates the measured kinematics. These calculations use the generalized parton model (GPM) which takes the first k T moment of the Sivers function (e.g. k T · q(k T )) and does not include next-to-leading-order interactions with the proton fragments. The "GPM" curve uses the parameters stated in Eq. (32) of Ref. [31]. The color-gauge-invariant generalized parton model (CGI-GPM) expands on the GPM by including initial-and final-state interactions through the one-gluon exchange approximation. This model has been shown to reproduce the predicted sign change for the quark Sivers function in SIDIS and Drell-Yan. The CGI-GPM curves plotted in Fig. 5 show two different scenarios for this model, the specifics of which can be found in Eq. (34) of Ref. [31]. The values that are used for the Scenario 1 curve are chosen to maximize the open heavy flavor TSSA generated by the gluon Sivers function while still keeping this asymmetry within the statistical error bars of the published result in Ref. [33] and simultaneously describing the previously published midrapidity π 0 TSSA from Ref. [16]. The values used in the Scenario 2 curve are similarly calculated, except that they minimize the open heavy flavor TSSA within the range of the published statistical error bars. As shown in the zoomed-in inset of Fig. 5, this π 0 TSSA result has the statistical precision at low p T needed to distinguish between the GPM and CGI-GPM frameworks, preferring CGI-GPM Scenario 2.
Measurements of TSSAs in p+p collisions are essential to understanding the underlying nonperturbative processes which generate them. In particular, further measurements are necessary to clarify certain questions in the interpretations of the TSSAs. For example, the small forward jet asymmetries measured in Ref. [21] have been interpreted as a cancellation of up and down quark asymmetries, implying that the comparatively forward large neutral pion asymmetries include significant contributions from spin-momentum correlations in hadronization [20]. Additionally, the p T dependence of these forward rapidity measurements remains to be clearly understood; measurements of nonzero asymmetries out to even higher p T would help confirm that these twist-3 observables eventually fall off with increasing hard scale. While the midrapidity measurements here are all consistent with zero, they still provide the highest available statistical precision and p T reach available at the PHENIX experiment. While forward rapidity light hadron TSSAs are dominated by valence quark spin-momentum correlations in the polarized proton, these midrapidity TSSA measurements are sensitive to both quark and gluon dynamics at leading order. Thus these data also provide further constraints to gluon spin-momentum correlations in transversely polarized protons [32,34].

IV. SUMMARY
The measurements presented here were motivated by the outstanding questions regarding the physical origin of TSSAs. The TSSAs of π 0 and η mesons were measured at midrapidity in p+p collisions at √ s = 200 GeVby the PHENIX experiment. The measured π 0 (η) meson asymmetry is consistent with zero in the presented p T range, up to precision of 3 × 10 −4 (2 × 10 −3 ) in the lowest p T bins. Both measurements have a significant reduction in uncertainty from previous measurements at midrapidity at RHIC. These data extend previous constraints to any presence of gluon spin-momentum correlations in transversely polarized protons.