Transverse spin-dependent azimuthal correlations of charged pion pairs measured in p$^\uparrow$+p collisions at $\sqrt{s}$ = 500 GeV

The transversity distribution, which describes transversely polarized quarks in transversely polarized nucleons, is a fundamental component of the spin structure of the nucleon, and is only loosely constrained by global fits to existing semi-inclusive deep inelastic scattering (SIDIS) data. In transversely polarized $p^\uparrow+p$ collisions it can be accessed using transverse polarization dependent fragmentation functions which give rise to azimuthal correlations between the polarization of the struck parton and the final state scalar mesons. This letter reports on spin dependent di-hadron correlations measured by the STAR experiment. The new dataset corresponds to 25 pb$^{-1}$ integrated luminosity of $p^\uparrow+p$ collisions at $\sqrt{s}=500$ GeV, an increase of more than a factor of ten compared to our previous measurement at $\sqrt{s}=200$ GeV. Non-zero asymmetries sensitive to transversity are observed at a $Q^2$ of several hundred GeV and are found to be consistent with the former measurement and a model calculation. %we observe consistent with the former measurement are observed.} We expect that these data will enable an extraction of transversity with comparable precision to current SIDIS datasets but at much higher momentum transfers where subleading effects are suppressed.


Introduction
The proton is the fundamental bound state of quantum chromodynamics (QCD). In spite of its importance for our understanding of this theory, our knowledge of the proton structure remains incomplete [1]. In particular, the proton wave function cannot be computed ab-initio in perturbative QCD (pQCD), but has to be constrained by measurements. In deep inelastic scattering (DIS) experiments of electrons or muons off nuclei at high energies, the wavefunction of the proton is accessed on the lightcone. In this frame, the wavefunction can be expanded in the squared 4-momentum transfer Q 2 of the interaction. The leading coefficients in this expansion can be identified with three parton distribution functions (PDFs). In the parton model, PDFs have a probabilistic interpretation as the probability of finding a parton that carries a momentum fraction x of the parent proton. The moderate Q 2 dependence, which arises from the parton splitting functions [2,3,4], is computed using evolution equations. We assume a Q 2 dependence in the following discussion even when not explicitly written. Two of the PDFs, the parton helicity averaged PDF f 1 (x), and the helicity PDF g 1 (x) appear at leading twist respectively in the spin averaged and longitudinally polarized inclusive DIS cross-section [5]. They are therefore fairly well determined experimentally [6]. The third one, the transversity distribution h 1 (x), does not appear at leading twist in the inclusive DIS cross-section since it is connected to a chiral-odd helicity-flip amplitude. Instead, it is accessed in processes where it couples to the chiral-odd transverse spin dependent fragmentation function (FF) [7]. The transversity PDF can be interpreted as the probability of finding a transversely polarized quark in a transversely polarized proton, and the FF serves as a quark polarimeter.
The analysis presented here investigates a channel in which transversity couples to the spin dependent dihadron FF H 1 (z, M ) [8,9,10], which, for historical reasons, is also known as the interference fragmentation function (IFF). Here, z is the fraction of the parent parton energy carried by the hadron pair, and M is the invariant mass of the pair. Presently, transversity is only loosely constrained by fits [11,12] to available SIDIS [13,14,15,16,17] and e + e − [18,19] data. The e + e − data are necessary to constrain the polarization dependent fragmentation functions. Data on proton targets are currently limited in the valence region to x < 0.2, restricting the knowledge of u quark transversity at high x. Probing transversity in p + p collisions provides better access to the d-quark transversity than is possible in SIDIS, due to the fact that there is no charge weighting in the hard scattering QCD 2 → 2 processes in p+p collisions. A precision determination of both u and d-quark transversity are important in particular for the determination of the zeroth moment of transversity, the tensor charge Recently, g T has attracted increased interest. One reason is that it can be calculated precisely using lattice QCD [20,21,22,23,24], which makes it one of the few observables involving transverse polarization where experiments can be compared with first order pQCD calculations. In fact, g T is the first nucleon matrix element that could be extrapolated to the physical limit. Furthermore, g T determines the effective tensor coupling constant for beyond the standard model contributions to low energy scattering [25]. This determination is particularly important for planned electric dipole moment experiments where a precise knowledge of g T is needed to determine the contributions of possible new CP violating phases [26]. Due to its chiral-odd property, gluon polarization contributions to transversity in a spin-1 2 target vanish [27]. This characteristic is one reason g T is dominated by medium to high x region. Precision data from transversely polarized p + p collisions at high √ s and p T are crucial to access transversity at high Q 2 , where theoretical uncertainties are well under control. The kinematic region covered by the STAR experiment at these energies overlaps the reach of current SIDIS experimental data on transversity in the upper part of the covered x range (see Fig. 1). The STAR kinematics is obtained from the transverse momentum of the mid-rapidity jet containing the hadron pair since this is the relevant scale in p + p collision and approximately equal to The results presented in this letter at √ s = 500 GeV use more than 10 times the integrated luminosity than our previously reported result at √ s = 200 GeV [28], where a significant signal of transversity was observed in an exploratory measurement of di-pion correlations [28]. The calculations reported in [12] found hints of universality where the phase space of the √ s = 200 GeV p + p and the SIDIS data overlap.
Since the calculations are performed in a collinear framework, this was already postulated. However, since factorization is not proven in this process and has been explicitly shown to be broken in other transverse polarization dependent processes in p + p [29], this was a crucial finding to support the inclusion of the data in global analyses. In the future, a comparison between di-hadron asymmetries, with measurements of azimuthal asymmetries of pions in jets by STAR [30], will provide further tests of universality and factorization. The former asymmetries can be described in a collinear framework, while the latter include an explicit dependency on intrinsic transverse momenta (for more details see [31,32]). The collinear framework is well understood and describes the unpolarized p + p cross-section well [33], but the transverse momentum dependent (TMD) framework is still being developed, and questions remain about universality, factorization and evolution.

Experiment
The   [13,14,15,16,17]. The kinematics of the STAR data points correspond to the lower panel of Fig. 3.
patch must have E T > 6.4, 9.0 or 13.9 GeV, respectively. Particles are identified by measuring their average specific ionization energy loss, dE/dx , as they traverse the TPC and comparing this measured value with the associated parameterized expectation for each particle species as a function of η and momentum. Cuts on the number of standard deviations from the pion dE/dx peak (-1σ to 2σ) and the number of hits used to determine dE/dx (> 20) are applied to achieve an 85±2.5% pion pair purity across the entire kinematic range. The pion pair purity is the probability that both particles in a pair are pions. The momentum, p, of each particle is required to be greater than 2 GeV/c. Each proton beam in the RHIC ring consists of bunches that alternate between being transversely polarized up or down with respect to the accelerator plane. However, when the single spin asymmetry measurement is carried out with respect to a given beam, the polarization of the other beam is integrated over to effectively be unpolarized. Polarimeters, which measure the elastic scattering of protons on ultra thin carbon ribbon targets several times during a fill, were used to measure the polarization of each beam.
These polarimeters were calibrated with a polarized hydrogen gas jet target [35].

Analysis
The azimuthal angles in the scattering system used to calculate the π + π − azimuthal correlation follow the definition in ref. [37] and are shown in    by the momentum vectors from each pion ( p h,1 and p h,2 ) in the pair. The difference vector R = p h,1 − p h,2 lies in the di-hadron plane. The pions are chosen to be in close proximity to each other in η − φ space with (∆η) 2 + (∆φ) 2 ≤ 0.7 and the sum of the transverse momenta, p T , for each pair is required to be greater than 3.75 GeV/c. Throughout the rest of this paper, p T is the transverse momentum of the pion pair and p h,1 corresponds to the positive pion and p h,2 to the negative pion. We define the unit vectorsp = p/| p|.
The angle between the scattering plane and the polarization of the incident beam, s a , is φ S . The angle between the scattering plane and the di-hadron plane is φ R , which is used to define φ RS = φ R − φ S , where φ R and φ S are calculated using Eqs. (2)-(5). The angle φ RS modulates the asymmetry due to the product of transversity and the IFF by sin(φ RS ).    Figure 7: The azimuthal asymmetry as a function of invariant mass in the highest p T bin compared with predictions from fits to existing SIDIS and e + e − data provided by the same authors as [12]. Details on the calculation can be found in [36].
The π + π − azimuthal correlation observable, A U T , is defined in Eq. (6), where P is the beam polarization and N ↑(↓) is the number of pion pairs when the polarization of the beam is pointing up (down). The combination of different polarization directions and detector hemispheres removes luminosity and efficiency dependencies from the asymmetry calculation to leading order [38].
A U T is calculated for eight φ RS bins of equal width in the range [0, π], which are then fitted with a single-parameter function, A U T · sin(φ RS ), to extract the amplitude. This procedure is carried out as a function of the pseudorapidity of the pion pair, which is denoted as η for the remainder of this report. η > 0 is forward with respect to the polarized beam direction. A U T is also measured as a function of invariant mass, M inv , and p T .
The scale uncertainty due to the beam polarization in this analysis is 4.5%. We investigated a potential bias of the triggered events towards pions that come from quark jets, which could result in an enhancement of the measured asymmetries, since gluons are not expected to contribute to transversity. To investigate this bias, particles produced in p+p simulated events from PYTHIA 6.426 [39] with the Perugia-0 tune [40], were processed through a detector simulator (GSTAR package based upon GEANT 3.21/08T [41]), and then used to estimate the quark/parton ratio of a biased sample over the quark/parton ratio in an unbiased sample.
In STAR the trigger decision is based on the energy deposit in a defined segment in one of the calorimeters.
We expect therefore that a potential trigger bias effect will be strongest for low p T parent jets, since at high jet p T the impact of a shape difference between quark or gluon initiated jets will be negligible for the trigger decision. For this reason we investigated the trigger bias as a function of the transverse momentum of the hadron pair. Within our statistical uncertainties, we do not observe a significant trigger bias and thus decided not to correct for this effect. Instead, the statistical uncertainty with which one can determine the ratio of the fractions of quark initiated jets in the triggered over the non-triggered sample was assigned as a systematic uncertainty, being ∼20% at low p T and ∼5% at high p T . Note that the trigger bias does not affect the statistical significance of the measurement because the scaling applies to the asymmetry and its uncertainty equally. Since the trigger efficiency is higher for larger jet energies, the selection of pion pairs might be biased towards lower z pairs. Using the same simulation as for the trigger bias, we estimate this effect to be ∼8% at low p T and ∼4% at high p T .
Finally, the pion pair purity previously mentioned was used to estimate the asymmetric asymmetry dilution due to π − K and π − p pairs and found to be about 15% and is represented as rectangles above (below) positive (negative) data points. This estimate assumes the π − K and π − p asymmetries are no larger than the π + − π − asymmetries and have the same sign.

Results
The single spin asymmetry, A U T , was measured as a function of η for five p T bins. It is shown as a function of η in Fig. 3 for the largest p T bin with p T = 13 GeV/c. The other four p T bins have smaller asymmetries compared to the p T bin in Fig. 3. Using the particles produced in PYTHIA and processed through GEANT as mentioned previously, the kinematic variables x and z were estimated. The bottom panel of Fig. 3 shows x and z as a function of pion pair pseudorapidity. As shown in Fig. 3, a strong rise of the measured signal is observed toward higher η where we reach the highest values of x. This is consistent with the expectation that the transversity distribution is largest at high-x.
A U T as a function of M inv for η > 0 is shown in Fig. 4 for the five p T bins. Because A U T is consistent across all p T bins for η < 0 it is shown only for the two largest p T bins. A significant signal is seen in the highest p T bin, with an enhancement near the ρ mass at mid-M inv . In models of the IFF, this enhancement is expected due to the interference of vector meson decays in a relative p-wave with the non-resonant background in a relative s-wave [9]. To test this hypothesis, the same-charge, momentumordered (| p h,1 | > | p h,2 |) asymmetry was calculated and is shown in Fig. 5. This plot shows a significantly smaller asymmetry around the ρ mass compared to the charge-ordered calculation. We note that this suppressed asymmetry can also be explained in single hadron emission models like the Nambu and Jona-Lasinio jet model [42] where the parton producing the lower ranked same-charge pion will carry less of the spin information and is more likely to have a transverse momentum direction correlated (instead of anti-correlated) with the higher ranked pion.
A U T as a function of p T for η > 0 is shown in Fig. 6 for five M inv bins. A significant asymmetry is observed at high p T for M inv > 0.4 GeV/c 2 . Though not shown here, the asymmetry as a function of p T for η < 0 is small compared to the results for η > 0. Supplemental tables containing the numerical results shown in the figures discussed above are available online. Figure 7 shows a comparison of a theoretical calculation with the azimuthal asymmetry as a function of the invariant mass measured in p ↑ + p collisions at √ s = 500 GeV for the highest p T bin. The gray band represents the range of the 68% confidence interval of the fit to SIDIS and e + e − data. The fit has been provided by the authors of reference [12], which was first compared to the STAR results at √ s = 200 GeV [28]. The smaller M inv range for the theory band is due to the fact that this specific model calculation has only been performed up to M inv ≈ 1.2 GeV/c. The asymmetry comparison shows close agreement within statistical uncertainty between the data and the theory band, which further hints at the universality of the mechanism producing azimuthal correlations in SIDIS, e + e − , and p+p data. These high-precision √ s = 500 GeV results can further constrain global fits of transversity parton distribution functions to SIDIS, e + e − , and p + p data, and in particular, improve the statistical significance for x > 0.1.

Conclusions
STAR has measured the first π + π − transverse spin-dependent azimuthal asymmetries in p ↑ + p collisions at √ s = 500 GeV for several pseudorapidity, invariant mass, and transverse momentum bins. These data show significant signals at high p T and M inv for η > 0. IFF models predict an enhancement around the ρ mass due to the interference of vector meson decays in a relative p-wave with the non-resonant background in a relative s-wave. This prediction is consistent with the data reported in the paper. These data probe transversity at much higher Q 2 ≈400 GeV 2 and sample a different mixture of quark flavors compared to the charge weighted coupling in SIDIS. These results can be used to test the universality of the mechanism producing azimuthal correlations in SIDIS, e + e − , and p + p. In the future, a comparison between di-hadron asymmetries with measurements of azimuthal asymmetries of pions in jets will provide further tests of universality and factorization. Additionally, the high-precision of these results can further constrain global fits to world data, especially in the region x > 0.1.

Acknowledgements
We thank Marco Radici and Alessandro Bacchetta from the Department of Physics at the University of Pavia for helpful discussions and for providing theory curves for this work. We thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL, and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Office of Nuclear Physics within the U.S.   Fig. 4, where σstat is the statistical uncertainty, σ pid is the particle identification systematic uncertainy, and σ tb is the trigger bias systematic uncertainty.  Table A.3: A U T as a function of M inv for η < 0, where σstat is the statistical uncertainty, σ pid is the particle identification systematic uncertainy, and σ tb is the trigger bias systematic uncertainty.  Table A.5: A U T for same-charge, momentum-ordered pion pairs as a function of M inv for η < 0 corresponding to Fig. 5, where σstat is the statistical uncertainty, σ pid is the particle identification systematic uncertainy, and σ tb is the trigger bias systematic uncertainty.