Improving constraints on gluon spin-momentum correlations in transversely polarized protons via midrapidity open-heavy-flavor electrons in $p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV

Polarized proton-proton collisions provide a unique opportunity to improve our understanding of gluon contributions to the spin structure of the proton, because they are accessible at leading order, which is not true for lepton-hadron scattering. The complex spin structure of the proton leads to emergent properties such as spin-momentum and spin-spin correlations analogous to the fine and hyperfine structure of atoms. These correlations in protons are experimentally accessible through observables known as transverse single-spin asymmetries (TSSAs). TSSAs quantify azimuthal modulations of particle production in collisions of transversely polarized nucleons with unpolarized particles, and have been measured to reach magnitudes up to 40% in hadron-hadron collisions [1, 2, 3, 4]. Perturbative quantum chromodynamics (pQCD) calculations had predicted TSSAs of $<1$% from purely perturbative contributions [5]; recent calculations suggest small additional perturbative contributions [6].

In $p$$+$$p$ collisions at $\sqrt{s}=200$ GeV, open-heavy-flavor (OHF) production at midrapidity is dominated by gluon-gluon fusion, receiving only a small contribution from quark-antiquark annihilation [23]. In gluon-gluon fusion events, only the first term in Eq. (1) is relevant (as the gluon does not have a transversity distribution in spin $1/2$ nucleons), providing sensitivity to the trigluon correlation functions in polarized protons. The relevant twist-3 correlators for quark-antiquark annihilation and gluon-gluon fusion are the Efremov-Teryaev-Qiu-Sterman ($qgq$) correlator [10, 24], and the trigluon ($ggg$) correlators [25, 26, 27, 28, 29] respectively. Note that the trigluon correlators were introduced in Ref. [25], and were subsequently clarified to be two independent functions [26, 27, 28, 29]. The $qgq$ correlator has been experimentally constrained from global fits, discussed in Ref. [13], while the $ggg$ correlators have received less attention, with few measurements capable of providing indirect constraints [30, 31, 32, 33, 34, 35] or direct constraints [36, 37].

The TSSA for open-charm production in $p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV was calculated in Refs. [38] and [39] within the twist-3 framework. The trigluon correlation functions are defined in Ref. [38] as $T_{G}^{(f)}(x,x)$ (antisymmetric) and $T_{G}^{(d)}(x,x)$ (symmetric), where the $(f)$ and $(d)$ superscripts represent three gluon-field color indices contracting with antisymmetric or symmetric structure tensors. Lack of direct information on the trigluon correlators has led to simple phenomenological models with normalization parameters to the unpolarized gluon PDF. In Ref. [38] (following from Ref. [20]) parameters $\lambda_{f}$ and $\lambda_{d}$ are introduced:

The trigluon correlation functions in Ref. [39] are instead defined as $N(x_{1},x_{2})$ (antisymmetric), and $O(x_{1},x_{2})$ (symmetric), with four independent contributions to TSSAs, $\{N(x,x),N(x,0),O(x,x),O(x,0)\}$. As shown in Ref. [39], at $\sqrt{s}=200$ GeV the asymmetries depend on effective trigluon correlators $N(x,x)-N(x,0)$ and $O(x,x)+O(x,0)$, which are directly related to $T_{G}^{(f)}$ and $T_{G}^{(d)}$ in Ref. [28]. Reference [39] introduces parameters $K_{G}$ and $K_{G}^{\prime}$ with the assumptions:

Note that the assumptions on the trigluon correlators in Eqs. 2, 4, and 5 (e.g., the functional dependence on $x$ and the proportionality to the unpolarized gluon PDF) are oversimplified. For this reason, it is advantageous to compare to different models with various $x$ dependencies. The results presented in this paper place direct constraints on $\lambda_{f},\lambda_{d},K_{G}$ and $K_{G}^{\prime}$.

Open-charm production at the Relativistic Heavy Ion Collider (RHIC) has also been investigated with the TMD factorization approach as a means of constraining the gluon Sivers PDF (see Refs. [40, 41, 42]). The measurements presented here will be useful in providing constraints to the gluon Sivers TMD PDF through constraining the twist-3 trigluon correlators, which are related to $k_{T}$ moments of the gluon Sivers PDF [14].

The asymmetry measurements presented here utilize data recorded in 2015 by the PHENIX experiment at RHIC with collisions of transversely polarized protons on transversely polarized protons at $\sqrt{s}=200$ GeV, and approximately 23 pb^-1 of integrated luminosity. The polarization of each beam at RHIC in 2015 is measured to be $0.58\pm 0.02$ for the clockwise beam and $0.60\pm 0.02$ for the counterclockwise beam, with transverse polarization direction aligned vertically to the accelerator plane [43]. The polarization direction is varied from bunch to bunch (a) to reduce systematics related to detector coverage and performance, and (b) to allow for the polarization of a single beam to be considered at a time by averaging over the polarization directions of the opposing beam. This yields two independent data sets from which the transverse single-spin asymmetries are extracted, validated for consistency, and averaged to obtain the final result.

The PHENIX detector is described in detail in Ref. [44]. Detector subsystems used for midrapidity charged-particle detection comprise two central-arm spectrometers oriented to the left and right of the beam axis, each with acceptance $|\eta|<0.35$ and $\Delta\phi=0.5\pi$, and a silicon vertex detector (VTX) [45, 46] with acceptance of $|\eta|<1$ and $\Delta\phi\approx 0.8\pi$ per arm. The central arms contain drift and pad chambers for tracking [47], electromagnetic calorimeters (EMCal) to measure energy deposition of charged particles and photons [48], and a ring-imaging Čerenkov (RICH) detector for particle identification with $e/\pi$ separation up to $5$ GeV/$c$ [49].

Curating the electron candidate sample follows the same procedure as in Ref. [50]. The electron candidate sample is composed of tracks reconstructed from hits in the drift and pad chambers of the central arm spectrometers coincident with hits in the silicon vertex detector. Tracks within $1.0<p_{T}$ (GeV/c) $<5.0$ that fire at least one photomultiplier (PMT) tube in the RICH detector, and that have a maximum displacement of 5 cm between the track projection and center of the ring of Čerenkov light as measured by the PMTs in the RICH are considered. In order to increase the electron purity, track energy $E$ deposited in the EMCal and track momentum $p$ should have a ratio near unity, as electrons deposit most of their energy in the EMCal while charged hadrons do not. The $E/p$ distribution for electron candidates in Run-15 was fit with an exponential + Gaussian, where the mean $\mu_{E/p}$ and width $\sigma_{E/p}$ of the Gaussian portion were extracted and used to impose the following condition $\left|(E/p-\mu_{E/p})/\sigma_{E/p}\right|<2$. Spatial displacements $\Delta z$ and $\Delta\phi$ of track projections and corresponding electromagnetic showers in the EMCal are required to be separated by no more than 3 standard deviations of the corresponding $\Delta z$ and $\Delta\phi$ distributions, and the probability that an EMCal cluster originates from an electromagnetic shower (as calculated by the shower shape) is required to be above 0.01. Tracks reconstructed in the central arms are projected to the VTX detector and fit to coincidental VTX hits via the iterative algorithm described in Ref. [51] — the fit is required to satisfy $\chi^{2}/ndf<3$. A hit is required in both of the inner two layers of the VTX to veto conversion electrons created by photons interacting with detector material, and an additional hit is required in either of the outer layers of the VTX. The narrow opening angle between $e^{+}e^{-}$ from photonic conversions is exploited to further reduce background from conversions in the beam pipe or inner two layers of the VTX; more details for this and the VTX detector can be found in Ref. [50]. An additional requirement was placed on the number of live trigger counts per bunch crossing because the asymmetry analysis is performed bunch-by-bunch.

TSSAs can be calculated as amplitudes of sinusoidal modulations of azimuthal particle production:

In Eq. (7), $N^{\uparrow,\downarrow}$ are the spin-dependent yields for collisions with $\uparrow,\downarrow$ polarized bunch crossings respectively, and $\mathcal{R}=\mathcal{L}^{\uparrow}/\mathcal{L}^{\downarrow}$ is the relative luminosity, defined as the ratio of luminosities for collisions with oppositely oriented bunch crossing polarization. The azimuthal correction factor $\langle\mid\cos\phi\mid\rangle$ is calculated in each transverse momentum ($p_{T}$) bin for the electron candidate sample to account for detector efficiency effects. To serve as a cross check to Eq. (7), the asymmetries are also calculated with the “square root formula,” as shown in Eq. (8). The difference in asymmetries calculated with the separate methods is taken as a systematic uncertainty $\sigma^{\rm sys}_{\rm diff}$.

The $L,R$ subscripts represent the left and right spectrometer arm with respect to the polarized proton-going direction. The square root formula cannot be used independently on each spectrometer arm, leading to only two independent data sets for cross validation and averaging corresponding to the two beams, rather than four independent data sets as is the case for the relative luminosity formula, corresponding to the two beams and two spectrometer arms. As an additional cross check, $A_{N}$ was calculated as shown in Eq. 6, via sinusoidal fits, with 3 $\phi$ bins per spectrometer arm, yielding consistent results with that of Eqs. 7 and 8.

Once $A_{N}$ is calculated for the electron candidate sample, background corrections allow for extraction of the asymmetry for OHF decay electrons. The relevant background sources are electrons from other parent particles ($\pi^{0},\eta$, direct photons $\gamma,J/\psi,K^{0}_{S},K^{\pm}$) and charged hadrons misidentified as electrons (primarily $\pi^{\pm}$). To calculate the background-corrected asymmetry, the fraction of each background source present in the data sample needs to be calculated and the background asymmetries need to be measured. Equation (9) shows the formula for extracting the ${\rm OHF}\rightarrow{e}$ asymmetry from the electron-candidate-sample asymmetry,

where $f_{i}$ represent the background fractions, $A_{N}$ is the asymmetry calculated on the electron candidate sample, and $A_{N}^{i}$ are the background asymmetries. The procedure to calculate the background fractions and a more detailed description of background sources can be found in Ref. [50]. This procedure is repeated in this analysis with the relevant $p_{T}$ bins, and uncertainties on calculated background fractions are propagated through Eq. (9) to obtain systematic uncertainties $\sigma^{\rm sys}_{f^{\pm}}$. Figure 1 shows the resulting background fractions for electrons and positrons combined. The Ke3 background source, which consists of Dalitz decays of $K^{\pm}$ and $K^{0}_{S}$, is heavily suppressed over the measured $p_{T}$ range. The transverse single-spin asymmetries for $K^{\pm}$ or $K^{0}_{S}$ have not been measured in $\sqrt{s}=200$ GeV $p^{\uparrow}+p$ collisions. However, given that the Ke3 background fraction is on the order of $10^{-3}$, and is the smallest contributor, it is safely neglected in the background correction procedure. The relevant background fractions are calculated separately for positrons and electrons as shown in Table 1, with resulting background fractions shown in Figures 5 and 5.

TSSAs for each background source have been measured at PHENIX at midrapidity in $p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV. The asymmetries for photonic background sources $\pi^{0},\eta$, and $\gamma$ were all measured by PHENIX to be consistent with zero using the same dataset as this measurement [34, 37]. They are therefore set to zero in Eq. (9), with a systematic uncertainty $\sigma^{\rm sys}_{A_{N}^{B}}$ assigned for setting $A_{N}^{\pi^{0}}=0$ and $A_{N}^{\eta}=0$ based on propagating uncertainties from the measurements, while the uncertainty associated with setting the direct photon TSSA $A_{N}^{\gamma}=0$ is negligible because $f_{\gamma}$ is on the order of $10^{-3}$ (see Table 1). The TSSAs for $J/\psi$ [52] and charged hadrons [53] were measured with previous PHENIX data sets. The TSSA for $J/\psi$ has a large statistical uncertainty [52], and contributes significantly to the statistical uncertainty of this measurement, especially at high $p_{T}$. This is due to the azimuthal angle of the decay lepton becoming more strongly correlated with the azimuthal angle of the $J/\psi$ at higher $p_{T}$. Additionally, the statistical precision of $A_{N}(p^{\uparrow}+p\rightarrow J/\psi+X)$ could not be improved upon in the Run-15 data given the high degree of photonic electron background. The TSSA for midrapidity $J/\psi$ production measured in Ref. [52] was recalculated as a function of decay lepton $p_{T}$ using pythia [54] decay simulations for the $J/\psi\rightarrow e^{+}e^{-}$ channel to apply Eq. (9).

The TSSAs for midrapidity open charm production ($A_{N}^{D}$) predicted in Refs. [38] and [39] were also recalculated as a function of decay lepton $p_{T}$ for all possible semileptonic decay channels, with decay kinematics simulated in pythia [54] to obtain correlations between $p_{T}$ and $\phi$ of the decay lepton and $D$ meson. The $\phi^{e}$ distribution was then weighted in accordance with $w(\phi^{e})=1+A_{N}^{D}(p_{T}^{D})\cos\phi^{D}$ in each $p_{T}$ bin and then fit with $f(\phi)=N_{0}(1+A_{N}^{e}\cos\phi)$ to extract the decay lepton asymmetry. $D^{0}$ and $\bar{D}^{0}$ production was considered for comparisons to results from Refs. [38] and [39], while $D^{+}$ and $D^{-}$ production was additionally considered when comparing to results of Ref. [39]. OHF production is dominated by open charm at the relevant kinematics, for which $D^{0},\bar{D}^{0},D^{+}$, and $D^{-}$ cover a significant fraction. The effect of including $D^{+}$ and $D^{-}$ in comparing to Ref. [39] makes very little difference as supported by our simulations, implying that comparing to $D^{0}$ and $\bar{D}^{0}$ for Ref. [38] is sufficient. A scan over $(\lambda_{f},\lambda_{d})$ parameter space and independent scans over $K_{G}$ and $K_{G}^{\prime}$ were performed to generate a set of theory curves for comparison, allowing for best-fit parameters and confidence intervals to be determined from data.

The ${\rm OHF}\rightarrow{e^{\pm}}$ TSSAs are plotted in Fig. 5 alongside theoretical predictions of $A_{N}(p^{\uparrow}+p\rightarrow(D^{0}/\bar{D^{0}}\rightarrow e^{\pm})+X)$ from Ref. [38] in (red/blue) solid lines, and $A_{N}(p^{\uparrow}+p\rightarrow(D/\bar{D}\rightarrow e^{\pm})+X)$ from Ref. [39] in (red/blue) dashed and dotted lines, with $\lambda_{f}$, $\lambda_{d}$, $K_{G}$ and $K_{G}^{\prime}$ chosen to best fit the data for the separate charges simultaneously. The measurements are consistent with zero, and are statistically more precise than previous heavy-flavor measurements. The total systematic uncertainties come from combining those associated with the background fractions, background asymmetries, and the difference in calculating $A_{N}$ with Eqs. (7) and (8); there is no dominant source of systematic uncertainty across charges and $p_{T}$ bins. The systematic uncertainty reaches at most 37% of the corresponding statistical uncertainty (see Table 3), while it is typically suppressed by an order of magnitude or more. The placement of the theoretical curves in Fig. 5 differs for $e^{+}$ vs $e^{-}$ due to the contribution of the symmetric trigluon correlator having opposing signs in charm vs anticharm production, leading to constructive vs destructive interference with the antisymmetric trigluon correlator contribution for the separate charges. This allows for constraining power on all parameters. Summaries for final asymmetries with statistical and systematic uncertainties are given in Table 3 for OHF positrons $A_{N}^{{\rm OHF}\rightarrow{e^{+}}}$ and electrons $A_{N}^{{\rm OHF}\rightarrow{e^{-}}}$ and in Table 3 for nonphotonic (NP) positrons $A_{N}^{{\rm NP}{e^{+}}}$ and electrons $A_{N}^{{\rm NP}{e^{-}}}$.

To determine theoretical parameters that fit the data best, $\chi^{2}(\lambda_{f},\lambda_{d})$, $\chi^{2}(K_{G})$, and $\chi^{2}(K_{G}^{\prime})$ were calculated for the separate charges and summed to extract minimum values. The results along with $1\sigma$ confidence intervals are $\lambda_{f}=-0.01{\pm}0.03$ GeV and $\lambda_{d}=0.11{\pm}0.09$ GeV for parameters introduced Ref. [38], and $K_{G}=0.0006^{+0.0014}_{-0.0017}$, and $K_{G}^{\prime}=0.00025{\pm}0.00022$ for parameters introduced in Ref. [39]. This corresponds to the first constraints on $(\lambda_{f},\lambda_{d})$, and is in agreement with previous constraints on $K_{G}$ and $K_{G}^{\prime}$ derived in Ref. [39]. Figure 5 summarizes the results of the statistical analysis performed to extract best-fit parameters $\lambda_{f}$ and $\lambda_{d}$, where the theoretical asymmetries depend on both parameters. Nicely illustrated are the constraining power of the individual charges and the necessity of combining the charges in the statistical analysis. Both charges predict that contributions from trigluon correlations are small, indicating that $\lambda_{f}$ and $\lambda_{d}$ values that result in cancellation of their contributions to the asymmetry calculation are preferred.

In summary, the PHENIX experiment has measured the transverse single-spin asymmetry of midrapidity open-heavy-flavor decay electrons and positrons as a function of $p_{T}$ in $p^{\uparrow}+p$ collisions at $\sqrt{s}=200$ GeV. Open-heavy-flavor production at RHIC is an ideal channel for probing trigluon correlations in polarized protons because initial-state $qgq$ correlations in the proton and final-state twist-3 correlations in hadronization contribute negligibly. This measurement provides constraints for the antisymmetric and symmetric trigluon correlation functions in transversely polarized protons, including the first constraints on $\lambda_{f}$ and $\lambda_{d}$ as $\lambda_{f}=-0.01{\pm}0.03$ GeV and $\lambda_{d}=0.11{\pm}0.09$ GeV —  a necessary step forward in our understanding of proton structure through correlations between proton spin and gluon momentum.