Study of a background reconstruction method for the measurement of D-meson azimuthal angular correlations

Study of a background reconstruction method for the measurement of D -meson azimuthal angular correlations 1 corresp first-author malong@fudan.edu.cn malong@fudan.edu.cn malong@fudan.edu.cn https://orcid.org/0000-0001-9167-7209 2 corresp xdong@lbl.gov xdong@lbl.gov xdong@lbl.gov https://orcid.org/0000-0001-9083-5906 1 3 https://orcid.org/0000-0002-6760-2394 1 https://orcid.org/0000-0002-0233-9900 Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA University of California, Los Angeles, California 90095, USA University of California, Los Angeles, California 90095, USA We studied experimental background reconstruction methods for the measurement of the <math id="M1"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> 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 found that the LS method, which describes the combinatorial background of single D 0 meson yields, fails to reproduce the correlated background in the <math id="M2"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation measurement, while the SB background method yields a good description of the background for both single D 0 yields and the correlated background of the <math id="M3"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation measurement. We further examined the validity of the correlation methods under different signal-to-background ratios, providing direct references for experimental measurements. heavy flavor azimuthal correlation PYTHIA 1 Introduction 1 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 high-energy 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 1 - 5 5 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 6 - 10 10 6-10 ]. The nuclear modification factor ( R AA ) for charmed hadrons in heavy-ion collisions is significantly modified compared to the p + p reference [ 11 11 ]. Several models with different energy-loss mechanisms can describe the experimental data [ 12 12 , 13 13 , 18 18 , 19 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 14 - 17 17 14-17 , 20 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., R AA and elliptic flow parameter v 2 ), could be reflected in the azimuthal angle correlations [ 15 15 , 21 21 , 22 22 ]. In p + p collisions, charm-quark pairs are produced through initial back-to-back hard scattering in leading order. In next-to-leading order, the angular correlation between charm-quark pairs widens. In particular, it will show a near-side peak at Δ ϕ ∼0 if the charm pairs are produced through gluon splitting. The measurement of <math id="M4"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> correlations in p + p collisions not only provides a baseline for measurements in heavy-ion collisions, but also offers a good constraint for pQCD calculations. D -mesons inherit most of the initial charm pair correlations, but weak decays smear the correlation significantly. Therefore, the measurement of <math id="M5"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> correlations should be the most sensitive probe for studying charm-quark pair correlations [ 23 23 - 25 25 23-25 , 30 30 ]. The experimental reconstruction of the <math id="M6"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> 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 D 0 mesons through the <math id="M7"><mrow><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup></mrow></math> decay channel, several background methods, such as the like-sign (LS), side-band (SB), and mixed-event (ME) methods, were deployed by experimentalists [ 26 26 , 27 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 D 0 candidates. 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 D 0 peak are used, two symmetric mass regions on both sides of the D 0 peak are usually selected, and the average of these regions is chosen to represent the background underneath the D 0 peak. Both LS and SB techniques can successfully reproduce the background in single D 0 yield measurements with reasonable precision. In this study, we investigated these background reconstruction methods for the experimental measurement of <math id="M8"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> correlations. The ME technique misses the background correlation in the same event and typically needs to be normalized to either LS or SB distributions. In this study, we focused on the comparison of the LS and SB background techniques. 2 PYTHIA study for <math id="M9"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> correlations 1 The Monte-Carlo event generator PYTHIA (version 8.168) was used in this study [ 28 28 ]. We focused on p + p collisions at <math id="M10"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> = 500 GeV. The parameters were adjusted so that PYTHIA could reproduce the experimental data on the inclusive <math id="M11"><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo>¯</mo></mover></mrow></math> production cross-section in p + p collisions at 500 GeV, as measured by the STAR experiment at RHIC [ 29 29 ]. Figure 1 Figure 1 shows the <math id="M12"><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo>¯</mo></mover></mrow></math> 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: strong-interaction coupling constant ( α s ) of the final parton shower (TimeShower:alphaSvalue) set to 0.15; minimum invariant transverse-momentum ( p T ) threshold for hard QCD process (PhaseSpace:pTHatMin) set to 1.5 GeV/ c . With this setting, PYTHIA properly describes both the magnitude and the p T spectrum. It was also found that changing these two parameters has a negligible effect on charm correlations. A sample of six-billion PYTHIA minimum bias events with this setting was generated for the <math id="M14"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation study. D 0 mesons can be directly accessed in the PYTHIA simulation based on their particle identification number. To emulate the experimental measurement, D 0 s were reconstructed by pairing the kaon and pion candidate pairs via the typical hadronic decay channel D 0 → K - + π + and its charge conjugate channel for <math id="M15"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> . In a real experiment with a silicon vertex detector, many background tracks from the primary collisions can be eliminated, but considerable background remains, particularly in the low p T region. In this study, we did not distinguish the secondary decay vertices in the D 0 reconstruction. Instead, we combined all kaons and pions at mid-rapidity (| η | le; 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 D 0 candidates. The invariant masses of the unlike-sign (US) and (LS) kaon and pion pairs in the same event were calculated. A finite momentum resolution effect was included so that the reconstructed D 0 signal peak had the width observed in the experiment. Figure 2 Figure 2 shows the D 0 ( <math id="M16"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> ) signal and the combinatorial background from the LS and SB methods. The LS and SB background regions are denoted by the blue and red hatched areas, respectively. The invariant mass distribution of <math id="M17"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> is almost identical to D 0 in both shape and magnitude. The background was found to be flat in PYTHIA within a relative wide invariant mass range. For simplicity, we denote D 0 ( <math id="M18"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> ) candidates from K - π + ( K + π - ) pairs with unlike signs as ‘US’ candidates, and those from K - π - or K + π + pairs with same charge sign as ‘LS’ background. The SB background is denoted as ‘SB’. Fig. 2 Fig. 2 shows that both the LS and SB methods can reasonably reproduce the real background underneath the reconstructed D 0 signals. For single-particle yield measurement, the D 0 and <math id="M19"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> counts were calculated from Eqs. 1 and 2 for the LS and SB background methods, respectively. If the background methods work well for the <math id="M20"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation measurement, the correlation signal between D 0 and <math id="M21"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> can be derived using Eqs. 3 and 4. The asterisks (*) indicate the correlation functions between the pairs. We can also derive the <math id="M22"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation signal from the PYTHIA simulation directly, and compare it to the reconstructed signals using these two background methods. <math display="block" id="M23"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>N</mi><mrow><mtext>LS</mtext></mrow><mrow><mi>D</mi><mo>+</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></msubsup><mo>=</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>+</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>−</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M24"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>N</mi><mrow><mtext>SB</mtext></mrow><mrow><mi>D</mi><mo>+</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></msubsup><mo>=</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>+</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>−</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M25"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>C</mi><mrow><mtext>LS</mtext></mrow><mrow><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></msubsup><mo>=</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>+</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>L</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> <math display="block" id="M26"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msubsup><mi>C</mi><mrow><mtext>SB</mtext></mrow><mrow><mi>D</mi><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></msubsup><mo>=</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>U</mi><mi>S</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>+</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><mo stretchy="false">)</mo><mo>*</mo><mi>S</mi><mi>B</mi><mo stretchy="false">(</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> The di-hadron correlation measurements are usually plotted as a function of the azimuthal angle difference, i.e., <math id="M27"><mrow><mi>Δ</mi><mi>ϕ</mi><mo>=</mo><msub><mi>ϕ</mi><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow></msub><mo>−</mo><msub><mi>ϕ</mi><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></msub></mrow></math> . The upper panel in Fig. 3 Fig. 3 shows the correlations between US candidates and the LS backgrounds as a function of Δ ϕ . The p T > 1.0 GeV/c cut was set for both D 0 and <math id="M28"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> mesons, and the mass-window cuts for US, LS, and SB pairs are shown as colored bands in Fig. 2 Fig. 2 . The plot shows that the correlation between LS and LS background pairs (LS*LS) tends to peak at Δϕ around 0 and that its magnitude is considerably larger than that between the LS background and US candidates (LS*US). The lower panel in Fig. 3 Fig. 3 shows the results of the SB method with the same trigger p T and S/B ratio as the LS method. The correlation between the SB background and US candidates lies between the other two correlation terms. The correlation between two SB background pairs shows a trend similar to that between the SB background and US candidates. Figure 4 Figure 4 shows the reconstructed <math id="M30"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation signals with the LS and SB background methods in comparison with the real correlation signals from PYTHIA directly. Two sets of p T cuts were imposed for the triggered and associated D -mesons, as shown in the upper and lower panels, respectively. Panels in two different columns show the comparisons with two different mass-window cuts, which result in different S/B ratios of the reconstructed D 0 candidates. The red data points represent the correlation signals from the reconstructed D 0 s, whereas the blue data points are the real D 0 correlations directly obtained from PYTHIA with the same kinematic cuts applied. Similar results from the SB method are presented in Fig. 4 Fig. 4 . Note that the reconstructed correlations using the LS method are different from the real <math id="M34"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> signal from PYTHIA. In particular, the reconstruction correlations start to show an enhanced structure in the near-side region when the S/B ratio decreases. Reconstructed correlations using the SB method can reproduce the real <math id="M35"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> signal reasonably well in these kinematic and S/B ratio regions. In addition, the quality of the reproduction does not depend on the transverse-momentum cut. It depends on the S/B ratios of the D 0 candidates. <math display="block" id="M36"><mrow><msub><mi>u</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mrow><mtext>trig</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><mi>Δ</mi><msub><mi>ϕ</mi><mi>i</mi></msub></mrow></mfrac><msub><mo stretchy="false">)</mo><mrow><mtext>reco</mtext></mrow></msub><mn>,</mn><mtext> </mtext><mtext> </mtext><msub><mi>v</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><msub><mi>N</mi><mrow><mtext>trig</mtext></mrow></msub></mrow></mfrac><mfrac><mrow><mtext>d</mtext><mi>N</mi></mrow><mrow><mtext>d</mtext><mi>Δ</mi><msub><mi>ϕ</mi><mi>i</mi></msub></mrow></mfrac><msub><mo stretchy="false">)</mo><mrow><mtext>real</mtext></mrow></msub></mrow></math> <math display="block" id="M37"><mrow><mi>Δ</mi><mi>P</mi><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mrow><mo>|</mo><mrow><mfrac><mrow><msub><mi>u</mi><mi>i</mi></msub><mo>−</mo><msub><mi>v</mi><mi>i</mi></msub></mrow><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow></mfrac></mrow><mo>|</mo></mrow><mn>,</mn><mtext> </mtext><mtext> </mtext><mi>Δ</mi><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover></mstyle><mrow><mo>|</mo><mrow><mfrac><mrow><msqrt><mrow><msubsup><mi>σ</mi><mrow><msub><mi>u</mi><mi>i</mi></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow><mn>2</mn></msubsup></mrow></msqrt></mrow><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow></mfrac></mrow><mo>|</mo></mrow></mrow></math> To better illustrate the performance of these two background methods in measuring the angular correlations of the <math id="M38"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> pairs, we introduced two variables to quantify the goodness of fit for the reconstructed correlation signals with respect to the real <math id="M39"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlations from PYTHIA. Δ P and Δ E are defined in Eq. 6 to describe the relative differences between the data points and the statistical errors from this sample. Note that u i and v i are 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. Fig. 5 Fig. 5 shows the corresponding results from the LS and SB methods, respectively. Note that Δ P in the LS results shows a large deviation from Δ E 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 Δ P and Δ E 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 D 0 from D * decay and non-prompt D 0 from 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 29 ]. We found that the mis-PID effect was significantly small ( < 1%) in this case. 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 <math id="M40"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> 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 <math id="M41"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> signal, we turned off the D 0 hadronic decay process in the PYTHIA simulation and ran the same analysis. Figure 6 Figure 6 shows the invariant mass distribution of pure Kπ pairs without D 0 decay contribution. Cross-correlations between US/LS and LS/LS pairs are plotted in comparison with the US/US pair correlations in Fig. 7 Fig. 7 with different cuts applied to the invariant mass region. Similarly, results from the SB method are shown in Fig. 8 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 <math id="M42"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> 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 D 0 → K + π - decay is turned off. 3 Conclusion 1 In summary, we studied background reconstruction methods for azimuthal correlations between D 0 and <math id="M43"><mrow><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> pairs using a PYTHIA simulation. Both the LS and SB methods provide a good description of the background when reconstructing single D 0 yields. However, when reconstructing the correlation signal, the LS method fails to reproduce the <math id="M44"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> 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 original <math id="M45"><mrow><msup><mi>D</mi><mn>0</mn></msup><mo>−</mo><mover accent="true"><mrow><msup><mi>D</mi><mn>0</mn></msup></mrow><mo stretchy="true">¯</mo></mover></mrow></math> correlation 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 the <math id="M46"><mrow><mi>D</mi><mo>−</mo><mover accent="true"><mi>D</mi><mo stretchy="true">¯</mo></mover></mrow></math> azimuthal 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. References R.C. Hwa , Charm correlation as a diagnostic probe of quark matter . Phys. Rev. Lett . 77 , 227 ( 1996 ). doi: 10.1103/PhysRevLett.77.227 http://doi.org/10.1103/PhysRevLett.77.227 Charm correlation as a diagnostic probe of quark matter B. Svetitsky , A. Uziel , Passage of charmed particles through the mixed phase in high-energy heavy-ion collisions . Phys. Rev. D 55 , 2616 ( 1997 ). doi: 10.1103/PhysRevD.55.2616 http://doi.org/10.1103/PhysRevD.55.2616 Passage of charmed particles through the mixed phase in high-energy heavy-ion collisions Z. Lin , M. Gyulassy , Open charm as a probe of preequilibrium dynamics in nuclear collisions . Phys. Rev. C 51 , 2177 ( 1995 ). doi: 10.1103/PhysRevC.51.2177 http://doi.org/10.1103/PhysRevC.51.2177 Open charm as a probe of preequilibrium dynamics in nuclear collisions V. Greco , H. van Hees , R. Rapp , Heavy quark dynamics in the QGP . AIP Conference Proceedings 1422 , 117 ( 2012 ). doi: 10.1063/1.3692206 http://doi.org/10.1063/1.3692206 Heavy quark dynamics in the QGP L.H. Song , L.W. Yan , Y. Liu , Constraining the colored <math id="M47"><mrow><mi>c</mi><mover accent="true"><mi>c</mi><mo>¯</mo></mover></mrow></math> energy loss from J/Ψ production in p-A collisions . Nucl. Sci. Tech . 29 , 159 ( 2018 ). doi: 10.1007/s41365-018-0502-1 http://doi.org/10.1007/s41365-018-0502-1 Constraining the colored cc¯ energy loss from J/Ψ production in p-A collisions C. Albajar , M.G. Albrow , O.C. Allkofer et al . , A study of the D* content of jets at the CERN pp collider . Phys. Lett. B 244 , 566 - 572 ( 1990 ). doi: 10.1016/0370-2693(90)90363-B http://doi.org/10.1016/0370-2693(90)90363-B A study of the D* content of jets at the CERN pp collider F. Abe , D. Amidei , G. Apollinari et al . , Measurement of the B-meson and b-quark cross sections at <math id="M48"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> =1.8 TeV using the exclusive decay B±→J/ψK± . Phys. Rev. Lett . 68 , 3403 ( 1992 ). doi: 10.1103/PhysRevLett.68.3403 http://doi.org/10.1103/PhysRevLett.68.3403 Measurement of the B-meson and b-quark cross sections at s =1.8 TeV using the exclusive decay B±→J/ψK± D. Acosta , T. Affolder , M.H. Ahn et al . Measurement of Prompt Charm Meson Production Cross Sections in <math id="M49"><mrow><mi>p</mi><mover accent="true"><mi>p</mi><mo stretchy="true">¯</mo></mover></mrow></math> Collisions at <math id="M50"><mrow><msqrt><mi>s</mi></msqrt><mo>=</mo><mn>1.96</mn><mtext>TeV</mtext></mrow></math> . Phys. Rev. Lett . 91 , 241804 ( 2003 ). doi: 10.1103/PhysRevLett.91.241804 http://doi.org/10.1103/PhysRevLett.91.241804 Measurement of Prompt Charm Meson Production Cross Sections in pp¯ Collisions at s=1.96TeV B. Abelev , A. Abrahantes Quintana , D. Adamová et al . Measurement of charm production at central rapidity in proton-proton collisions at <math id="M51"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> = 7 TeV . J. High Energy Phy . 01 , 128 ( 2012 ). doi: 10.1007/JHEP01(2012)128 http://doi.org/10.1007/JHEP01(2012)128 Measurement of charm production at central rapidity in proton-proton collisions at s = 7 TeV B. Abelev , J. Adam , D. Adamová et al . , Measurement of charm production at central rapidity in proton-proton collisions at <math id="M52"><mrow><msqrt><mi>s</mi></msqrt></mrow></math> = 2.76 TeV . J. High Energy Phy . 1207 , 191 ( 2012 ). doi: 10.1007/JHEP07(2012)191 http://doi.org/10.1007/JHEP07(2012)191 Measurement of charm production at central rapidity in proton-proton collisions at s= 2.76 TeV Z.B. Tang , W.M. Zha , Y.F. Zhang , An experimental review of open heavy flavor and quarkonium production at RHIC . Nucl. Sci. Tech . 31 , 81 ( 2020 ). doi: 10.1007/s41365-020-00785-8 http://doi.org/10.1007/s41365-020-00785-8 An experimental review of open heavy flavor and quarkonium production at RHIC M. Cacciari , M. Greco , P. Nason , The pT spectrum in heavy-flavour hadroproduction . J. High Energy Phy . 9805 , 007 ( 1998 ). doi: 10.1088/1126-6708/1998/05/007 http://doi.org/10.1088/1126-6708/1998/05/007 The pT spectrum in heavy-flavour hadroproduction M. Cacciari , S. Frixione , P. Nason , The pT spectrum in heavy-flavour photoproduction . J. High Energy Phy . 0103 , 006 ( 2001 ). doi: 10.1088/1126-6708/2001/03/006 http://doi.org/10.1088/1126-6708/2001/03/006 The pT spectrum in heavy-flavour photoproduction S. Cao , G.Y. Qin , S.A. Bass , Energy loss, hadronization, and hadronic interactions of heavy flavors in relativistic heavy-ion collisions . Phys. Rev. C 92 , 024907 ( 2015 ). doi: 10.1103/PhysRevC.92.024907 http://doi.org/10.1103/PhysRevC.92.024907 Energy loss, hadronization, and hadronic interactions of heavy flavors in relativistic heavy-ion collisions S. Cao , T. Luo , G.Y. Qin , et al . , Heavy and light flavor jet quenching at RHIC and LHC energies . Phys. Lett. B 777 , 255 ( 2018 ). doi: 10.1016/j.physletb.2017.12.023 http://doi.org/10.1016/j.physletb.2017.12.023 Heavy and light flavor jet quenching at RHIC and LHC energies M. Nahrgang , J. Aichelin , P. Gossiaux et al . , Heavy-flavor azimuthal correlations of D mesons . J. Phys. Conf. Ser . 509 , 012047 ( 2013 ). doi: 10.1088/1742-6596/509/1/012047 http://doi.org/10.1088/1742-6596/509/1/012047 Heavy-flavor azimuthal correlations of D mesons M. Rohrmoser , P. Gossiaux , T. Gousset et al . , Constraining in-medium heavy-quark energy-loss mechanisms via angular correlations between heavy and light mesons . J. Phys. Conf. Ser . 779 , 012032 ( 2017 ). doi: 10.1088/1742-6596/779/1/012032 http://doi.org/10.1088/1742-6596/779/1/012032 Constraining in-medium heavy-quark energy-loss mechanisms via angular correlations between heavy and light mesons T. Sjostrand , S. Mrenna , P. Skands , PYTHIA 6.4 physics and manual. 0605, 026 ( 2006 ). doi: 10.1088/1126-6708/2006/05/026 http://doi.org/10.1088/1126-6708/2006/05/026 PYTHIA 6.4 physics and manual. 0605, 026 J. Chen , Z.Q. Feng , J.S. Wang , Nuclear in-medium effects on η dynamics in proton–nucleus collisions . Nucl. Sci. Tech . 27 , 73 ( 2016 ). doi: 10.1007/s41365-016-0069-7 http://doi.org/10.1007/s41365-016-0069-7 Nuclear in-medium effects on η dynamics in proton–nucleus collisions A.G. Yang , Y. Zhang , L. Cheng et al . , Squeezed back-to-back correlation of <math id="M53"><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mover accent="true"><mi>D</mi><mo>¯</mo></mover><mn>0</mn></msup></mrow></math> in relativistic heavy-ion collisions . Chin. Phys. Lett . 35 , 052501 ( 2018 ). doi: 10.1088/0256-307X/35/5/052501 http://doi.org/10.1088/0256-307X/35/5/052501 Squeezed back-to-back correlation of D0D¯0 in relativistic heavy-ion collisions S. Cao , G.Y. Qin , S.A. Bass , Modeling of heavy-flavor pair correlations in Au-Au collisions at 200AGeV at the BNL Relativistic Heavy Ion Collider . Phys. Rev. C 92 , 054909 ( 2015 ). doi: 10.1103/PhysRevC.92.054909 http://doi.org/10.1103/PhysRevC.92.054909 Modeling of heavy-flavor pair correlations in Au-Au collisions at 200AGeV at the BNL Relativistic Heavy Ion Collider C. Shen , L. Yan , Recent development of hydrodynamic modeling in heavy-ion collisions . Nucl. Sci. Tech . 31 , 122 ( 2020 ). doi: 10.1007/s41365-020-00829-z http://doi.org/10.1007/s41365-020-00829-z Recent development of hydrodynamic modeling in heavy-ion collisions H. Wang , J.H. Chen , Y.G. Ma et al . , Charm hadron azimuthal angular correlations in Au + Au collisions at <math id="M54"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt></mrow></math> = 200 GeV from parton scatterings . Nucl. Sci. Tech . 30 , 185 ( 2019 ). doi: 10.1007/s41365-019-0706-z http://doi.org/10.1007/s41365-019-0706-z Charm hadron azimuthal angular correlations in Au + Au collisions at sNN = 200 GeV from parton scatterings X. Zhu , M. Bleicher , S.L. Huang , et al . , DD- correlations as a sensitive probe for thermalization in high energy nuclear collisions . Phys. Lett. B 647 , 366 ( 2007 ). doi: 10.1016/j.physletb.2007.01.072 http://doi.org/10.1016/j.physletb.2007.01.072 DD- correlations as a sensitive probe for thermalization in high energy nuclear collisions X. Zhu , N. Xu , P. Zhuang , Effect of partonic “wind” on charm quark correlations in high-energy nuclear collisions . Phys. Rev. Lett . 100 , 152301 ( 2008 ). doi: 10.1103/PhysRevLett.100.152301 http://doi.org/10.1103/PhysRevLett.100.152301 Effect of partonic “wind” on charm quark correlations in high-energy nuclear collisions L. Adamczyk , G. Agakishiev , M.M. Aggarwal et al . Measurements of D0 and D* production in p+p collisions at <math id="M55"><mrow><msqrt><mi>s</mi></msqrt><mstyle mathvariant="bold" mathsize="normal"><mo>=</mo></mstyle><mn>200</mn><mtext>GeV</mtext></mrow></math> . Phys. Rev. D 86 , 072013 ( 2012 ). doi: 10.1103/PhysRevD.86.072013 http://doi.org/10.1103/PhysRevD.86.072013 Measurements of D0 and D* production in p+p collisions at s=200GeV L. Adamczyk , J.K. Adkins , G. Agakishiev et al . , Observation of D0 meson nuclear modifications in Au+Au collisions at <math id="M56"><mrow><msqrt><mrow><msub><mi>s</mi><mrow><mtext>NN</mtext></mrow></msub></mrow></msqrt><mo>=</mo><mn>200</mn><mtext>GeV</mtext></mrow></math> . Phys. Rev. Lett . 113 , 142301 ( 2014 ). doi: 10.1103/PhysRevLett.113.142301 http://doi.org/10.1103/PhysRevLett.113.142301 Observation of D0 meson nuclear modifications in Au+Au collisions at sNN=200GeV T. Sjöstrand , S. Mrenna , P. Skands , A brief introduction to PYTHIA 8.1. Comput . Phys. Commun . 178 : 852 - 867 ( 2008 ). doi: 10.1016/j.cpc.2008.01.036 http://doi.org/10.1016/j.cpc.2008.01.036 A brief introduction to PYTHIA 8.1. Comput D. Tlusty (for STAR collaboration ) , Open charm measurements in p + p collisions at STAR . J. Phys. Conf. Ser . 509 , 012078 ( 2014 ). doi: 10.1088/1742-6596/509/1/012078 http://doi.org/10.1088/1742-6596/509/1/012078 Open charm measurements in p + p collisions at STAR F. Prino , R. Rapp , Open heavy flavor in QCD matter and in nuclear collisions . J. Phys. G Nucl. Part. Phys . 43 , 093002 ( 2016 ). doi: 10.1088/0954-3899/43/9/093002 http://doi.org/10.1088/0954-3899/43/9/093002 Open heavy flavor in QCD matter and in nuclear collisions All 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. 10.1007/s41365-021-00896-w This work was supported in part by the National Natural Science Foundation of China (Nos. 11890710, 11890714, 11905034) and the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDB34000000). 2021-01-20 2021-04-18 2021-04-19 2021-06-14 2021-06-01 2021-07-01 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2021 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-021-00896-w http://dx.doi.org/10.1007/s41365-021-00896-w . 2021 Nuclear Science and Techniques 06 32