Gluon polarization measurements from longitudinally polarized proton-proton collisions at STAR

Jets produced in the pseudo-rapidity range, $-1.0<\eta<1.0$, from $pp$ collisions at RHIC kinematics are dominated by quark-gluon and gluon-gluon scattering processes. Therefore the longitudinal double spin asymmetry $A_{LL}$ for jets is an effective channel to explore the longitudinal gluon polarization in the proton. At STAR, jets are reconstructed in full azimuth, from the charged-particle tracks seen by the Time Projection Chamber and electro-magnetic energy deposited in the Barrel and Endcap electro-magnetic calorimeters at both $\sqrt{s} = $ 200 and 510 GeV. Early STAR inclusive jet $A_{LL}$ results at $\sqrt{s} = $ 200 GeV provided the first evidence of the non-zero gluon polarization at $x>$ 0.05. At $\sqrt{s} = $ 510 GeV, the inclusive jet $A_{LL}$ is sensitive to the gluon polarization as low as $x \sim $ 0.015. In this talk, we will discuss recent STAR inclusive jet and dijet $A_{LL}$ results at $\sqrt{s} = $ 510 GeV and highlight the new techniques designed for this analysis, for example the underlying event correction to the jet transverse energy and its effect on the jet $A_{LL}$. Dijet $A_{LL}$ results are shown for four topologies in regions of pseudo-rapidity, effectively scanning the $x$-dependence of the gluon polarization.


Introduction
Early deep inelastic scattering (DIS) experiments in the 1980s showed that quarks inside the proton make only a small contribution to its total spin [1]. Where the rest of the proton spin comes from has been an outstanding problem awaiting to be explored. Theorists introduced parton distribution functions, PDFs, to describe the probability of a parton with momentum fraction x encountering a probe at energy scale, Q 2 , f (x, Q 2 ). Jaffe and Manohar proposed that not only do quarks contribute to the proton spin, but also gluons and the orbital angular momentum of quarks and gluons [2]. However the kinematics space in x − Q 2 covered by the polarized fixed target experiments through DIS processes only provide limited constraints on the gluon polarization inside the proton [3].
Different from DIS experiments, a polarized hadron-hadron collider at high center-of-mass energy, √ s, such as the Relativistic Heavy Ion Collider (RHIC) [4,5,6] can provide direct access to gluon polarization inside the proton. At RHIC, either transversely or longitudinally polarized proton beams collide at both √ s = 200 and 510 GeV. To explore the gluon polarizations, we measure the longitudinal double-spin asymmetry, A LL , for jets, defined as the fractional difference of the jet cross sections when beams have the same and the opposite helicities. The A LL can be expressed as the sum of convolutions of the polarized PDFs and partonic longitudinal double-spin asymmetryâ LL over all possible partonic processes. The next-to-leading order (NLO) perturbative quantum chromodynamics (pQCD) calculations show that the qg and gg  [7]. Both qg and gg processes have sizableâ LL [8], therefore jet A LL are sensitive to gluon polarizations. The same applies to A LL measurements for hadrons, for example π 0 . Given beam polarizations, P 1 and P 2 , and relative luminosities R = L ++ L −− , A LL is defined experimentally as: 2. Inclusive jet and dijet A LL measurements at STAR Solenoidal Tracker at RHIC (STAR) [9] has published a series of inclusive jet and dijet A LL results at √ s = 200 GeV [10,11,12]. The inclusive jets with transverse momentum, p T , and pseudo-rapidity, η, sample the scattering parton x ≈ x T e ±η , where x T = 2p T √ s . The dijets are able to unfold the initial kinematics x 1 , x 2 , and the scattering angle in the parton scattering rest frame, cosθ * , as in Equations 2, 3 and 4. Since the kinematics of the two scattering partons are simultaneously determined by dijet kinematics, it constrains the shape of polarized gluon distribution function, ∆g(x), as function of x. At √ s = 200 GeV, jets are sensitive to ∆g(x) at x as low as 0.05 when |η| < 1.0. The new prediction from DSSV group who included all the recently published STAR inclusive jet and dijet A LL results at √ s = 200 GeV shows 1 0.01 ∆g(x) = 0.296 ± 0.108 at Q 2 = 10 GeV 2 [13]. However large uncertainties of ∆g(x) still exist at x < 0.01. To explore the low x gluon polarization that is not well constrained by current available experimental data, we need to increase √ s or extend η forward.
In the year 2012, STAR recorded data from 82 pb −1 of longitudinally polarized pp collisions at √ s = 510 GeV, with average beam polarizations for two beams, 54% and 55% respectively [14], and R varing from 0.9 to 1.1. The electro-magnetic calorimeter based jet patch triggers, JP0, JP1 and JP2, are optimized to sample three different ranges over jet p T with thresholds set at 5.4, 7.3 and 14.4 GeV/c. Jets are reconstructed with charged tracks and electro-magnetic towers using the anti-k T algorithm with the parameter R = 0.5 [15].
An off-axis cone method adapted from the ALICE experiment at the LHC [16] is applied to correct the jet transverse energy due to underlying event contributions. It collects particles inside two cones centered at ± π 2 away from the jet in φ and at the same jet η. The correction dp T is taken as dp T =ρ × A, whereρ is the averaged energy density of the two off-axis cones and A is the jet area. This method samples the η dependence of the underlying event activities.
To study its contribution to jet A LL , we measure the longitudinal double-spin dp T asymmetry, A dp T LL as in Equation 5. A constant fit through A dp T LL as a function of jet p T shows that the underlying event correction is consistent with zero, as in Figure 1 [17]. Taking < dp T > ×A dp T LL as a shift of the jet p T due to underlying events, where < dp T > is the average dp T regardless of beam helicities, we estimated the potential contribution is at the level of 10 −4 , which is assigned as a systematic uncertainty.
The systematic uncertainties are studied with an embedding sample where simulated hard QCD scattering events are embedded into zero-bias events that are randomly taken during the collisions. The exponent parameter that controls the √ s dependent cut-off p T,0 in the default Perugia 2012 tune [18] was modified to match the simulated π ± spectra with the previously published STAR measurements [19,20] from pp collisions at √ s = 200 GeV [21,22]. Figure 2 shows the excellent agreement between data and simulation for jet p T spectra for jets satisfying JP0, JP1 and JP2 trigger requirements. Figure 2. Jet p T spectra comparison between data (markers) and embedding (lines) for jets satisfying JP0, JP1 and JP2 triggers separately [17].  [17].

Topology Description
Regions of η 3 and η 4 Jets reconstructed from the detector responses in the embedding sample are required to meet the jet patch trigger requirements. Comparing their predicted A LL as a function of jet p T with the unbiased parton level A LL allows to estimate the trigger bias and reconstruction correction and its uncertainty. The 100 equally probable replicas from NNPDFpol1.1 [23], which cover the current uncertainty band of ∆g(x), results in much more precise estimation of the correction and its uncertainty than the previous measurements at √ s = 200 GeV. The STAR 2012 inclusive jet A LL as a function of parton jet x T at √ s = 510 GeV is presented in the right panel of Figure 3 [17], together with STAR 2009 results at √ s = 200 GeV [10]. Both results agree well in the overlapping x T region. The new results are also consistent with recent NLO PDF predictions that imply positive gluon polarization [23,24]. The 510 GeV results extend measurements to lower x T which is sensitive to low x polarized gluons. The sensitivities to ∆g(x) goes to x as low as ∼ 0.015, as in the left panel of Figure 3 [17].  [24] and NNPDFpol1.1 [23] models (solid line with shades) [17].
The dijet events require the opening angle ∆φ > 2π 3 and the asymmetric p T cut, p T,3 > 6 GeV/c and p T,4 > 8 GeV/c, for the two jets. The combinations of the unfolded x 1 and x 2 constrain the shape of ∆g(x). The partonicâ LL depends on cosθ * . Therefore we proposed four η topology binnings as in Table 1. As expected, we see the difference in measured dijet A LL for four η topologies, as in the right panel of Figure 4. The sampled x 1 and x 2 are much narrower than the sampled x g by inclusive jets, as in the left panel of Figure 4 [17].
The preliminary results for the inclusive jet and dijet A LL measurements from STAR 2013   [24] and NNPDFpol1.1 [23] models (solid line with shades) [17].
510 GeV pp collisions were released in 2018 [25]. The integrated luminosity is about four times larger than that of the 2012 data set, however the dijet jet patch triggers were introduced to favorably capture dijet events. The same procedure has been applied in the 2013 inclusive jet A LL measurements. Both results agree with each other. We are finalizing the systematic uncertainties before the future publication.

Other measurements and STAR forward upgrade
The neutral pions, π 0 , are reconstructed from the γ decay in the forward meson spectrometer. The measured π 0 A LL results at √ s = 510 GeV from STAR 2012 and 2013 data, divided into two η ranges, 2.65 < η < 3.15 and 3.15 < η < 3.90, are very small, less than 5 × 10 −3 . The forward η allows to access polarized gluons at x in the order of 10 −3 [26]. The STAR forward upgrade has been fully approved and funded in time for the RHIC 2022 run. It features a forward calorimeter system and a forward tracking system at 2.5 < η < 4.0. The calorimeter includes a hadron calorimeter and an electro-magnetic calorimeter. Silicon disks and small thin gap chambers will be installed for the forward tracking system. The dijet A LL will be one of the highlighted physics programs for this upgrade, with one or both jets inside the forward region. With both jets inside the forward region at √ s = 510 GeV, it allows to sample ∆g(x) at x as low as 10 −3 , where the current model predictions show large uncertainties. The STAR forward upgrade will also lay the ground for the future Electron Ion Collider [27].

Conclusion
In summary, the inclusive jet measurements at STAR probe the magnitude of the ∆g(x) over a wide range of x. The dijet measurements provide additional constraints on the shape of ∆g(x). The first measurements of inclusive jet and dijet A LL at √ s = 510 GeV are sensitive to gluons at x ∼ 0.015. The results are consistent with current model predictions that imply positive gluon polarizations over x > 0.02. The STAR forward upgrade will play an important role in exploring the gluon polarizations at x near 10 −3 , which is loosely constrained by the current world data.