Long Ma· Xin Dong· Huan-Zhong Huang,3· Yu-Gang Ma

Abstract We studied experimental background reconstruction methods for the measurement of the D-D correlation using PYTHIA simulation. The like-sign (LS) and side-band (SB) background methods, which are widely used in the experimental measurements of single D-meson production yields, were deployed for correlation study. It was foundthatthe LS method,whichdescribesthe combinatorialbackground ofsingleD0 meson yields,failsto reproduce the correlated background in the D0-correlation measurement, while the SB background method yields a good description of the background for both single D0 yields and the correlated background of the D0-correlation measurement.We further examined the validity of the correlation methods under different signal-to-background ratios, providing direct references for experimental measurements.

Keywords Heavy flavor · Azimuthal correlation ·PYTHIA

1 Introduction

Quantum chromodynamics (QCD) is a theory that describes quarks, gluons, and the strong interaction between them. In QCD, heavy flavor quarks (c, b) are mostly produced through initial hard scattering in highenergy collisions of nucleons or nuclei. Because of their large masses, heavy quarks may offer a unique sensitivity for studying the cold and hot QCD medium created in these collisions [1-5]. In proton + proton (p+p) collisions,perturbative QCD (pQCD) calculations reproduce the inclusive heavy-flavor hadron-production cross-section data over a broad range of collision energies and rapidities[6-10]. The nuclear modification factor (RAA) for charmed hadrons in heavy-ion collisions is significantly modified compared to the p+p reference [11]. Several models with different energy-loss mechanisms can describe the experimental data [12, 13, 18, 19].

Recent research suggests that azimuthal correlations Δφ between heavy quark pairs offer a new insight about charm-medium interaction dynamics and therefore can help distinguish different energy-loss mechanisms in a hot QCD medium [14-17, 20]. The theoretical prediction indicates that pure radiative energy loss does not change the initial angular correlation function significantly, while pure collisional energy loss is more efficient at diluting the initial back-to-back charm pair correlation. Furthermore,the momentum broadening in the direction perpendicular to the initial quark momentum, which cannot be probed directly with traditional single-particle measurements(e.g.,RAA and elliptic flow parameter v2), could be reflected in the azimuthal angle correlations [15, 21, 22].

The experimental reconstruction of the D-azimuthal angular correlation is challenging. It requires the reconstruction of two charmed hadrons in a single event.Charmed hadrons must be reconstructed through their hadronic decay channels with small branching ratios.Furthermore, there is often a sizable background in each reconstructed charmed hadron. In single-charmed hadron yield measurements, for instance D0mesons through the K-π+decay channel,several background methods,such as the like-sign (LS), side-band (SB), and mixed-event (ME)methods, were deployed by experimentalists [26, 27]. In the ME technique, background pairs are reconstructed using two daughter tracks from different events.Given that the tracks are produced in different events,the background reconstructed is uncorrelated with the foreground D0candidates. By mixing multiple events, this method has the advantage of reproducing the combinatorial background with good statistics.In the LS technique,the background is generated by pairing the daughter tracks with the same charge sign. It contains the produced background correlated in pairs with opposite charge signs in the same event.In the SB technique, opposite sign pairs with invariant masses away from the D0peak are used, two symmetric mass regions on both sides of the D0peak are usually selected, and the average of these regions is chosen to represent the background underneath the D0peak.Both LS and SB techniques can successfully reproduce the background in single D0yield measurements with reasonable precision.

2 PYTHIA study for D-correlations

The Monte Carlo event generator PYTHIA (version 8.168) was used in this study [28]. We focused on p+p collisions at ■■■s =500 GeV.The parameters were adjusted so that PYTHIA could reproduce the experimental data on the inclusive cˉc production cross section in p+p collisions at 500 GeV,as measured by the STAR experiment at RHIC[29].

Figure 1 shows the cˉc production cross section as a function of the transverse momentum in PYTHIA in comparison with the STAR measurements. The modified PYTHIA parameters in this case were as follows: stronginteraction coupling constant (αs) of the final parton shower (TimeShower:alphaSvalue) set to 0.15; minimum invariant transverse-momentum (pT) threshold for hard QCD process (PhaseSpace:pTHatMin) set to 1.5 GeV/c.With this setting, PYTHIA properly describes both the magnitude and the pT spectrum. It was also found that changing these two parameters has a negligible effect on charm correlations.primary collisions can be eliminated, but considerable background remains, particularly in the low pT region.

Fig. 1 (Coloronline) Charm-paircross section asa functionof transverse momentum in p+p collisions at =500 GeVin PYTHIA (dashed line) compared with STAR measurements (solid circles)

In this study,we did not distinguish the secondary decay vertices in the D0reconstruction.Instead,we combined all kaons and pions at mid-rapidity (｜η｜≤1) in the final stage of the PYTHIA output. This allowed us to study the validity of the background reconstruction methods with different signal-to-background (S/B) ratios of the reconstructed D0candidates.The invariant masses of the unlikesign (US) and (LS) kaon and pion pairs in the same event were calculated. A finite momentum resolution effect was included so that the reconstructed D0signal peak had the width observed in the experiment.

Fig. 2 (Color online) Invariant mass distribution of all final-stage kaon and pion pairs with opposite signs in PYTHIA data at midrapidity(shown by solid red line,US).The LS method reproduces the combinatorial background shown by the blue solid line. The blue shaded region shows the LS background within a±3σ window of the signal peak. The SB background regions are shaded in red

Fig. 3 (Color online) Upper Panel: cross-correlations of D0-from US candidates and LS backgrounds. The trigger and associated pT cuts were both set to 1 GeV/c with a S/B ratio of approximately 0.3. Lower panel: similar results from the SB method

Fig.4 (color online).D0-correlation as a function of the relative azimuthal angle Δφ in p+p collisions at = 500 GeV calculated using the LS method (upper panel) and SB method (lower panel)based on Eq. 3 in PYTHIA simulation. The transverse-momentum dependence is shown with pT cuts applied to the triggered and associated D mesons. Panels (a)-(d) show correlations of reconstructed D0 mesons under different S/B ratios in comparison with correlations of produced D0- pairs in PYTHIA

To better illustrate the performance of these two background methods in measuring the angular correlations of the D-pairs, we introduced two variables to quantify the goodness of fit for the reconstructed correlation signals with respect to the real D0-correlations from PYTHIA. ΔP and ΔE are defined in Eq. 6 to describe the relative differences between the data points and the statistical errors from this sample. Note that uiand viare the values of the number of i data points of the reconstructed and real correlation signals in each Δφ bin; N is the total number of data points in each correlation signal, assuming the same binning for the histograms. Figure 5 shows the corresponding results from the LS and SB methods,respectively. Note that ΔP in the LS results shows a large deviation from ΔE when the S/B ratio decreases,indicating that the LS method fails to reproduce the real correlation at relatively low S/B ratios. The SB method exhibits good performance throughout the entire S/B ratio region investigated. The increase in both ΔP and ΔE in the low S/B region for the SB method is due to the reduced statistics.We also studied the performance of the two background methods by considering D0from D*decay and non-prompt D0from B-decay. The conclusions concerning the goodness of fit for both methods remain unchanged. Experimentally, as particle misidentification (Mis-PID) may affect the background reconstruction and cause double counting of the signals, we further evaluated such effects on the correlation reconstruction through a toy Monte Carlo simulation based on the PID criteria for p+p collisions in STAR analysis [29]. We found that the mis-PID effect was significantly small (＜1%) in this case.

Fig.5 (color online).Summary plots of the goodness of fit calculated using the LS method (upper panel) and SB method (lower panel) in the PYTHIA simulation. The estimator is shown as a function of the(S/B)ratio.The solid and dashed lines show ΔP and ΔE,respectively

In the LS method,when a K+π+pair is selected,there is a higher probability of finding a K-π-pair than another K+π+pair because of local and global charge conservation. The reconstructed correlation signal after LS background subtraction from Eq. 3 should contain all correlations between K+π-and K-π-pairs, including the D0-correlation of interest, as well as the correlation due to charge conservation.To further demonstrate that the additional correlation observed in the LS method is related to the underlying event instead of the D0-signal, we turned off the D0hadronic decay process in the PYTHIA simulation and ran the same analysis.

Fig. 6 (Color online) Invariant mass distribution of pure Kπ pairs in PYTHIA. The D0 →Kπ hadronic decay process was turned off.Red line:US Kπ pairs.Blue hatched area:LS Kπ pairs within cut window.Red shaded area: SB Kπ pairs within cut window

Fig. 7 (Color online) Cross-correlations of the pure LS and US Kπ pairs in the LS method

Figure 6 shows the invariant mass distribution of pure Kπ pairs without D0decay contribution.Cross-correlations between US/LS and LS/LS pairs are plotted in comparison with the US/US pair correlations in Fig. 7 with different cuts applied to the invariant mass region.Similarly,results from the SB method are shown in Fig. 8. There is a large difference between the LS*US and US*US pair correlations, while there is very a small difference between LS*LS and US*US. This is consistent with our understanding that there is an additional correlation that is not originated from the D0-pairs.

The SB method is not affected by charge conservation.Note that all cross-correlations fall in the same trend, and there is no remaining K+π--K-π+correlation when the D0→K+π-decay is turned off.

3 Conclusion

In summary, we studied background reconstruction methods for azimuthal correlations between D0andpairs using a PYTHIA simulation.

Fig. 8 (Color online) Cross-correlations of SB and US Kπ pairs in the SB method

Boththe LS andSB methodsprovide agood description of the backgroundwhenreconstructing single D0yields.However, when reconstructing the correlation signal, the LS method fails to reproduce the D0-correlation.The residual correlation after the LS background subtraction mainly comes from the underlying event activity, likely due to local or global charge conservation.We demonstrate that the SB method performs well in describing the correlation background and therefore reproduces the originalcorrelation in the S/B rate regions investigated.The upcoming sPHENIX experiment at RHIC will explore the charm correlation in p+p and Au+Au collisions by measuring theazimuthal correlation with full reconstruction of D-mesons through their hadronic decay channels. Our study on correlation methods constitutes an important reference for future experimental measurements.

Author contributionsAll authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Long Ma.The first draft was written by Xin Dong.Revisions of the manuscript were made by Huan-Zhong Huang and Yu-Gang Ma. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.