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Azimuthal correlations of prompt D mesons with charged particles in pp and p–Pb collisions at \(\varvec{\sqrt{s_\mathrm{NN}}} = 5.02\ \hbox {TeV}\)

Abstract

The measurement of the azimuthal-correlation function of prompt D mesons with charged particles in pp collisions at \(\sqrt{s} =5.02\ \hbox {TeV}\) and p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) with the ALICE detector at the LHC is reported. The \(\mathrm{D}^{0}\), \(\mathrm{D}^{+} \), and \(\mathrm{D}^{*+} \) mesons, together with their charge conjugates, were reconstructed at midrapidity in the transverse momentum interval \(3< p_\mathrm{T} < 24\ \hbox {GeV}/c\) and correlated with charged particles having \(p_\mathrm{T} > 0.3\ \hbox {GeV}/c\) and pseudorapidity \(|\eta | < 0.8\). The properties of the correlation peaks appearing in the near- and away-side regions (for \(\Delta \varphi \approx 0\) and \(\Delta \varphi \approx \pi \), respectively) were extracted via a fit to the azimuthal correlation functions. The shape of the correlation functions and the near- and away-side peak features are found to be consistent in pp and p–Pb collisions, showing no modifications due to nuclear effects within uncertainties. The results are compared with predictions from Monte Carlo simulations performed with the PYTHIA, POWHEG+PYTHIA, HERWIG, and EPOS 3 event generators.

Introduction

Two-particle angular correlations allow the mechanisms of particle production to be investigated and the event properties of ultra-relativistic hadronic collisions to be studied. In particular, the azimuthal and pseudorapidity distribution of “associated” charged particles with respect to a “trigger” D meson is sensitive to the charm-quark production, fragmentation, and hadronisation processes in proton–proton (pp) collisions and to their possible modifications in larger collision systems, like proton–nucleus (pA) or nucleus–nucleus (AA) [1]. The typical structure of the correlation function, featuring a “near-side” (NS) peak at \((\Delta \varphi ,\Delta \eta ) = (0,0)\) (where \(\Delta \varphi \) is the difference between charged-particle and D-meson azimuthal angles \(\varphi _\mathrm{ch} - \varphi _\mathrm{D}\), and \(\Delta \eta \) the difference between their pseudorapidities \(\eta _\mathrm{ch} - \eta _\mathrm{D}\)) and an “away-side” (AS) peak at \(\Delta \varphi = \pi \) extending over a wide \(\Delta \eta \) range, as well as its sensitivity to the different charm-quark production mechanisms, are described in details in [2].

In this paper, results of azimuthal correlations of prompt D mesons with charged particles at midrapidity in pp and p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) are presented, where “prompt” refers to D mesons produced from charm-quark fragmentation, including the decay of excited charmed resonances and excluding D mesons produced from beauty-hadron weak decays. The study of the near-side correlation peak is strongly connected to the characterisation of charm jets and of their internal structure, in terms of their particle multiplicity and angular profile. Probing the near-side peak features as a function of the charged-particle transverse momentum (\(p_\mathrm{T} \)), possibly up to values of a few GeV/c, gives not only access to the transverse-momentum distribution of the jet constituents, but can also provide insight into how the jet-momentum fraction not carried by the D meson is shared among the other particles produced by the parton fragmentation, as well as on the correlation between the \(p_\mathrm{T} \) of these particles and their radial displacement from the jet axis, which is closely related to the width of the near-side correlation peak. This study provides further and complementary information with respect to the analysis of charm jets reconstructed as a single object through a track-clustering algorithm and tagged by their charm content [3,4,5].

The azimuthal-correlation function of D mesons with charged particles is largely sensitive to the various stages of the D-meson and particle evolution, as hard-parton scattering, parton showering, fragmentation and hadronisation [6]. Its description by the available Monte Carlo event generators like PYTHIA [7, 8], HERWIG [9,10,11], and EPOS 3 [12, 13] or pQCD calculations like POWHEG [14, 15] coupled to event generators handling the parton shower, depends on several features, including the order of the hard-scattering matrix-element calculations (leading order or next-to-leading order), the modelling of the parton shower, the algorithm used for the fragmentation and hadronisation, and the description of the underlying event. The azimuthal-correlation function of D mesons with charged particles in pp collisions at \(\sqrt{s} = 7\ \hbox {TeV}\) measured by ALICE is described within uncertainties by simulations produced using PYTHIA6, PYTHIA8 and POWHEG+PYTHIA6 event generators [2]. However, more precise and differential measurements are needed to set constraints to models and be sensitive to the differences among their expectations.

The validation of Monte Carlo simulations for angular correlations of heavy-flavour particles in pp collisions is also useful for interpreting the results in nucleus–nucleus collisions, for which the measurements in pp collisions are used as reference. The temperature and energy density reached in nucleus–nucleus collisions at LHC energies are large enough to produce a quark–gluon plasma (QGP), a deconfined state of strongly-interacting matter [16, 17]. The interaction of heavy quarks (charm and beauty) with the QGP should affect the angular-correlation function [1, 18, 19]. First measurements performed at RHIC and the LHC showed modifications of the correlation function in nucleus–nucleus collisions when the trigger was a heavy-flavour particle, where a suppression of the away-side correlation peak and an enhancement of the near-side correlation peak for associated particles with \(p_\mathrm{T} < 2\ \hbox {GeV}/c\) was observed [20, 21]. A comparison of the results in nucleus–nucleus collisions to those in pp collisions, along with a successful description by models, would allow the modifications of the correlation function to be related to the in-medium heavy-quark dynamics [18, 22, 23].

In proton–nucleus collisions, several cold-nuclear-matter effects can influence the production, fragmentation and hadronisation of heavy-flavour quarks. They are induced by the presence of a nucleus in the initial state of the collision and, possibly, by the high density of particles in its final state. The most relevant effect is a modification of the parton distribution functions due to nuclear shadowing [24], which can consequently affect the heavy-flavour production cross section. Measurements of the nuclear modification factor of D mesons and of electrons from heavy-flavour hadron decays in \(\text {p--Pb} \) collisions at \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) [25, 26] point towards a small influence of cold-nuclear-matter effects on the heavy-flavour quark production at midrapidity. Nevertheless, nuclear effects could still affect the fragmentation and hadronisation of heavy quarks. These can be investigated by measuring potential modifications of the shape of the angular correlation between heavy-flavour particles [27] or, more indirectly, between heavy-flavour particles and charged particles.

Additionally, the search and characterisation of collective-like effects in high-multiplicity proton–proton and proton–nucleus collisions are a crucial topic, due to the observation of long-range, ridge-like structures in two-particle angular-correlation functions at RHIC [28, 29] and the LHC [30,31,32,33,34,35], resembling those observed in Pb–Pb collisions. The mechanism leading to these structures in small collision systems is not straightforward to identify. Possible explanations include final-state effects due to a hydrodynamic behaviour of the produced particles [36, 37], colour-charge exchanges [38, 39], initial-state effects, such as gluon saturation as described within the Color-Glass Condensate effective field theory [40, 41], or gluon bremsstrahlung by a quark-antiquark string [42]. In addition, a positive elliptic-flow coefficient was observed also for heavy-flavour particles, from the analysis of their azimuthal correlations with charged particles, by the ALICE [43,44,45], ATLAS [46,47,48], and CMS [49, 50] Collaborations. This approach generally assumes that the jet-induced correlation peaks do not differ in low- and high-multiplicity collisions, i.e. nuclear effects have the same impact on the heavy-quark fragmentation and hadronisation at different event multiplicities. This assumption can be tested by looking for modifications of the azimuthal-correlation function.

The results presented in this paper significantly improve the precision and extend the kinematic reach, with respect to our previous measurements [2] in both pp (at a different energy) and minimum bias p–Pb collisions. Correlations with associated particles at higher \(p_\mathrm{T} \) probe the angular and \(p_\mathrm{T} \) distribution of the hardest jet fragments, which retain more closely the imprint of the hard-scattering topology. The properties of the away-side peak are also studied for the first time. The paper is structured as follows. In Sect. 2, the ALICE apparatus, its main detectors and the data samples used for the analysis are presented. In Sect. 3 the procedure adopted for building the azimuthal-correlation functions, correcting them for experimental effects, and extracting physical quantities is described. Section 4 describes the systematic uncertainties associated to the measurement. The results of the analysis are presented and discussed in Sect. 5. The paper is briefly summarised in Sect. 6.

Experimental apparatus and data sample

The ALICE apparatus consists of a central barrel, covering the pseudorapidity region \(|\eta |<\) 0.9, a muon spectrometer with \(-4< \eta < -2.5\) coverage, and forward- and backward-pseudorapidity detectors employed for triggering, background rejection, and event characterisation. A complete description of the detector and an overview of its performance are presented in [51, 52]. The central-barrel detectors used in the analysis presented in this paper, employed for charged-particle reconstruction and identification at midrapidity, are the Inner Tracking System (ITS), the Time Projection Chamber (TPC), and the Time-Of-Flight detector (TOF). They are embedded in a large solenoidal magnet that provides a magnetic field of 0.5 T, parallel to the beams. The ITS consists of six layers of silicon detectors, with the innermost two composed of Silicon Pixel Detectors (SPD). It is used to track charged particles and to reconstruct primary and secondary vertices. The TPC is the main tracking detector of the central barrel. In addition, it performs particle identification via the measurement of the particle specific energy loss (dE/dx) in the detector gas. Additional information for particle identification is provided by the TOF, via the measurement of the charged-particle flight time from the interaction point to the detector. The TOF information is also employed to evaluate the starting time of the event [53], together with the time information provided by the T0 detector, an array of Cherenkov counters located along the beam line, at \(+370\) cm and \(-70\) cm from the nominal interaction point.

The results reported in this paper were obtained on the data samples collected during the 2016 LHC p–Pb run at \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) and the 2017 LHC pp run at \(\sqrt{s} = 5.02\ \hbox {TeV}\), corresponding, after the event selection, to integrated luminosities of \(L_\mathrm{int}\) = (295 ± 11) \(\mu \)b\(^{-1}\) and \(L_\mathrm{int}\) = (19.3 ± 0.4) nb\(^{-1}\), respectively. The events were selected using a minimum bias (MB) trigger provided by the V0 detector [54], a system of two arrays of 32 scintillators each, covering the full azimuthal angle in a pseudorapidity range of \(2.8< \eta < 5.1\) (V0A) and \(-3.7< \eta < -1.7\) (V0C). The trigger condition required at least one hit in both the V0A and the V0C scintillator arrays. This trigger is fully efficient for recording collisions in which a D meson is produced at midrapidity [2]. The V0 time information and the correlation between number of hits and track segments in the SPD were used to reject background events from the interaction of one of the beams with the residual gas in the vacuum tube. Pile-up events, whose probability was below 1% (0.5%) in pp collisions (\(\text {p--Pb} \) collisions), were rejected with almost 100% efficiency by using an algorithm based on track segments, reconstructed with the SPD, to detect multiple primary vertices. The remaining undetected pile-up events are a negligible fraction of the analysed sample. In order to obtain a uniform acceptance of the detectors, only events with a reconstructed primary vertex within ±10 cm from the centre of the detector along the beam line were considered for both pp and \(\text {p--Pb} \) collisions. In \(\text {p--Pb} \) collisions, the \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) energy was obtained by delivering proton and lead beams with energies of 4 TeV and 1.58 TeV per nucleon, respectively. Therefore, the proton–nucleus center-of-mass frame was shifted in rapidity by \(\Delta y_{\mathrm{NN}}\) = 0.465 in the proton direction with respect to the laboratory frame. The azimuthal correlations between D mesons and charged particles in \(\text {p--Pb} \) collisions were studied as a function of the collision centrality. The centrality estimator is based on the energy deposited in the zero-degree neutron calorimeter in the Pb-going direction (ZNA). The procedure used to define the centrality classes and to determine the average number of binary nucleon–nucleon collisions for each class is described in [55].

Some of the corrections for the azimuthal-correlation functions described in Sect. 3 were evaluated by exploiting Monte Carlo simulations, which included a detailed description of the apparatus geometry and of the detector response, using the GEANT3 package [56], as well as the luminous region distribution during the pp and \(\text {p--Pb} \) collision runs. For the evaluation of the charged-particle reconstruction efficiency, pp collisions were simulated with the PYTHIA8 event generator [8] with Monash-2013 tune [57], while \(\text {p--Pb} \) collisions were simulated using the HIJING 1.36 event generator [58] in order to describe the charged-particle multiplicity and detector occupancy observed in data [59]. For the corrections requiring the presence of a D meson in the event, enriched Monte Carlo samples were used, obtained by generating pp collisions containing a \(\mathrm{c}\overline{\mathrm{c}}\) or \(\mathrm{b}\overline{\mathrm{b}}\) pair in the rapidity range \([-1.5,1.5]\), employing PYTHIA 6.4.21 with Perugia-2011 tune. For p–Pb collisions, an underlying event generated with HIJING 1.36, was superimposed to each heavy-quark enhanced PYTHIA event.

Fig. 1
figure 1

Invariant mass (mass-difference) distributions of \(\mathrm{D}^{0}\), \(\mathrm{D}^{+} \) (\(\mathrm{D}^{*+} \)), and charge conjugates, candidates in three \(p_{\mathrm{T}}^{\mathrm{D}} \) intervals for pp collisions at \(\sqrt{s} = 5.02\ \hbox {TeV}\) (top row) and p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 5.02\ \hbox {TeV}\) (bottom row). The curves show the fit functions applied to the distributions. For the \(\mathrm{D}^{0}\), the dashed line represents the combinatorial background including the contribution of reflection candidates (see [62])

Data analysis

The analysis largely follows the procedure described in detail in [2]. It consists of three main parts: (1) reconstruction and selection of D mesons and primary charged particles (see [60] for the definition of primary particle); (2) construction of the azimuthal-correlation function and corrections for detector-related effects, secondary particle contamination, and beauty feed-down contribution; (3) extraction of correlation properties via fits to the average D-meson azimuthal-correlation functions with charged particles.

Selection of D mesons and primary charged particles

The analysis procedure begins with the reconstruction of D mesons (\(\mathrm{D}^{0}\), \(\mathrm{D}^{*}(2010)^{+} \), and \(\mathrm{D}^{+} \) and their charge conjugates), defined as “trigger” particles, and primary charged particles, considered as “associated” particles. The D mesons are reconstructed from the following hadronic decay channels: \({\mathrm{D}^{0} \rightarrow \mathrm{K}^{-}\pi ^{+}}\) (BR = 3.89 ± 0.04%), \({\mathrm{D}^{+} \rightarrow \mathrm{K}^{-}\pi ^{+}\pi ^{+}}\) (BR = 8.98 ± 0.28%), and \({\mathrm{D}^{*+} \rightarrow \mathrm{D}^{0}\pi ^{+}} \rightarrow \mathrm{K}^{-}\pi ^{+}\pi ^{+}\) (BR = 2.63 ± 0.03%) [61] in the transverse-momentum interval \(3< p_\mathrm{T} < 24\ \hbox {GeV}/c\). The D-meson selection strategy, described in detail in [25, 62], exploits the displaced topology of the decay and utilises the particle identification capabilities of the TPC and TOF to select on the D-meson decay particles. A dedicated optimisation on the selection variables was done, where the selections were tightened to increase the signal-to-background ratio of the D-meson invariant mass peaks. A gain up to a factor 5 at low \(p_{\mathrm{T}}^{\mathrm{D}} \) was obtained with respect to the selection defined in [25, 62], at the expenses of a reduction of the raw yield. This allowed reducing the impact of the D-meson combinatorial background, whose subtraction induces the largest source of statistical uncertainty on the correlation functions. With the adopted candidate selection, the D-meson reconstruction efficiency is of the order of few percent for \(p_{\mathrm{T}}^{\mathrm{D}} = 3\ \hbox {GeV}/c\) and increases up to 35% (50%) for \(p_{\mathrm{T}}^{\mathrm{D}} =24\ \hbox {GeV}/c\) in case of \(\mathrm{D}^{0}\) and \(\mathrm{D}^{+} \) (\(\mathrm{D}^{*+} \)) both in pp and in p–Pb collisions.

The D-meson raw yields were extracted from fits applied to the invariant mass (M) distributions of \(\mathrm{D}^{0}\) and \(\mathrm{D}^{+} \) candidates, and to the distribution of the mass difference \(\Delta M = M(\mathrm{K}\pi \pi ) - M(\mathrm{K}\pi )\) for \(\mathrm{D}^{*+} \) candidates, for several sub-ranges in the interval \(3< p_\mathrm{T} < 24\ \hbox {GeV}/c\). The fit function was composed of two terms, one for the signal and one for the background. The signal was described by a Gaussian, while the background was modelled by an exponential term for \(\mathrm{D}^{0}\) and \(\mathrm{D}^{+} \) mesons, and by a threshold function multiplied by an exponential for the \(\mathrm{D}^{*+} \) meson, as detailed in [2]. Examples of the invariant mass distributions in pp and in p–Pb collision systems are shown in Fig. 1 for \(\mathrm{D}^{0}\), \(\mathrm{D}^{+} \), and \(\mathrm{D}^{*+} \) mesons in different \(p_\mathrm{T} \) intervals.

Associated particles are defined as charged primary particles with \(p_{\mathrm{T}}^{\mathrm{assoc}} > 0.3\ \hbox {GeV}/c\) and with pseudorapidity \(|\eta | < 0.8\). As additional requirement, for this study only pions, kaons, protons, electrons and muons are considered as associated particles. The associated-particle sample does not include the decay products of the trigger D meson. Reconstructed charged-particle tracks with at least 70 space points out of 159 in the TPC, 2 out of 6 in the ITS, and a \(\chi ^2\)/ndf of the momentum fit in the TPC smaller than 2 were considered. The contamination of non-primary particles was largely suppressed by requiring the distance of closest approach (DCA) of the track to the primary vertex to be less than 1 cm in the transverse (xy) plane and along the beam line (z-direction). This selection identifies primary particles with a purity varying from 95% to 99% (increasing with \(p_{\mathrm{T}}^{\mathrm{assoc}} \)) and rejects a negligible amount of primary particles. In particular, less than 1% of the primary particles originating from decays of heavy-flavour hadrons are discarded. For the \(\mathrm{D}^{0}\) mesons produced in \(\mathrm{D}^{*+} \rightarrow \mathrm{D}^{0}\pi ^{+}\) decays, the low-\(p_\mathrm{T} \) pion accompanying the \(\mathrm{D}^{0}\) was removed from the sample of associated particles by rejecting tracks that, combined with the \(\mathrm{D}^{0}\), yielded a \(\Delta M\) consistent within 3\(\sigma \) with the \(\mathrm{D}^{*+} \) mass peak. It was verified with Monte Carlo simulations that this selection rejects more than 99% of the pions from \(\mathrm{D}^{*+} \) decays in all D-meson \(p_\mathrm{T} \) intervals considered and has an efficiency larger than 99% for primary particles with \(p_{\mathrm{T}}^{\mathrm{assoc}} > 0.3\ \hbox {GeV}/c\). The selection criteria described above provided an average track reconstruction efficiency for charged particles with \(p_{\mathrm{T}}^{\mathrm{assoc}} > 0.3\ \hbox {GeV}/c\) of about 83% (82%) in pp (p–Pb) collisions in the pseudorapidity interval \(|\eta | < 0.8\), with an increasing trend as a function of \(p_{\mathrm{T}}^{\mathrm{assoc}} \) up to \(\approx 1\ \hbox {GeV}/c\), followed by saturation at about 90%. As the track reconstruction efficiency has a sudden drop below \(\approx 0.3\ \hbox {GeV}/c\), caused by the TPC requirements in the track selection, this transverse momentum value was chosen as the minimum \(p_{\mathrm{T}}^{\mathrm{assoc}} \) for the analysis.

Evaluation and correction of the azimuthal-correlation functions

Selected D-meson candidates with an invariant mass in the range \(|M - \mu |~<~2\sigma \) (peak region), where \(\mu \) and \(\sigma \) denote the mean and width of the Gaussian term of the invariant mass fit function, were correlated to the primary charged particles selected in the same event. A two-dimensional angular-correlation function \(C(\Delta \varphi , \Delta \eta )_{\mathrm{peak}}\) was evaluated by computing the difference of the azimuthal angle and the pseudorapidity of each pair. The azimuthal-correlation functions were studied in four D-meson \(p_\mathrm{T} \) intervals: \(3< p_{\mathrm{T}}^{\mathrm{D}} < 5\ \hbox {GeV}/c\), \(5< p_{\mathrm{T}}^{\mathrm{D}} < 8\ \hbox {GeV}/c\), \(8<p_{\mathrm{T}}^{\mathrm{D}} < 16\ \hbox {GeV}/c\), and \(16< p_{\mathrm{T}}^{\mathrm{D}} < 24\ \hbox {GeV}/c\) and in the following \(p_\mathrm{T} \) ranges of the associated tracks: \(p_{\mathrm{T}}^{\mathrm{assoc}} > 0.3\ \hbox {GeV}/c\), \(0.3< p_{\mathrm{T}}^{\mathrm{assoc}} < 1\ \hbox {GeV}/c\), \(1<p_{\mathrm{T}}^{\mathrm{assoc}} < 2\ \hbox {GeV}/c\), and \(2<p_{\mathrm{T}}^{\mathrm{assoc}} < 3\ \hbox {GeV}/c\), significantly extending both transverse momentum coverages with respect to the previous measurements reported in [2].

The two-dimensional correlation functions are affected by the limited detector acceptance and reconstruction efficiency of the associated tracks (\(\hbox {A}^\mathrm{assoc}\times \epsilon ^\mathrm{assoc}\)), as well as the variation of those values for prompt D mesons (\(\hbox {A}^\mathrm{trig}\times \epsilon ^\mathrm{trig}\)) inside a given \(p_{\mathrm{T}}^{\mathrm{D}} \) interval. In order to correct for these effects, a weight equal to 1/(\(\hbox {A}^\mathrm{assoc}\times \epsilon ^\mathrm{assoc}) \times 1\)/(\(\hbox {A}^\mathrm{trig}\times \epsilon ^\mathrm{trig}\)) was assigned to each correlation pair, as described in detail in [2]. A weight of 1/(\(\hbox {A}^\mathrm{trig}\times \epsilon ^\mathrm{trig}\)) was applied also to the entries in the D-meson invariant mass distributions, used for the evaluation of the amount of signal \(S_{\mathrm{peak}}\) and background \(B_{\mathrm{peak}}\) triggers in the peak region.

The two-dimensional correlation function \(C(\Delta \varphi , \Delta \eta )_{\mathrm{peak}}\) also includes correlation pairs obtained by considering D-meson candidates from combinatorial background as trigger particles. This contribution was subtracted by evaluating the per-trigger correlation function obtained selecting D mesons with an invariant mass in the sidebands, \(1/B_{\mathrm{sidebands}}\times C(\Delta \varphi ,\Delta \eta )_{\mathrm{sidebands}}\), and multiplying it by \(B_{\mathrm{peak}}\). The term \(B_{\mathrm{sidebands}}\) is the amount of background candidates in the sideband region, i.e. \(4\sigma< |M -\mu | < 8\sigma \) (\(5\sigma< M - \mu < 10\sigma \), for \(\mathrm{D}^{*+} \) mesons) of the invariant mass distributions weighted by the inverse of the prompt D-meson reconstruction efficiency.

The event-mixing technique was used to correct the correlation functions \(C(\Delta \varphi ,\Delta \eta )_{\mathrm{peak}}\) and \(C(\Delta \varphi ,\Delta \eta )_{\mathrm{sidebands}}\) for the limited detector acceptance and its spatial inhomogeneities. The peak and sideband region event-mixing functions \(\mathrm {ME}(\Delta \varphi , \Delta \eta )_{\mathrm{peak}}\) and \(\mathrm {ME}(\Delta \varphi , \Delta \eta )_{\mathrm{sidebands}}\) were evaluated as explained in [2]. The inverse of these functions was used to weight the functions \(C(\Delta \varphi ,\Delta \eta )_{\mathrm{peak}}\) and \(C(\Delta \varphi ,\Delta \eta )_{\mathrm{sidebands}}\), respectively.

The per-trigger angular-correlation function was obtained by subtracting the sideband-region correlation function from the peak-region one, as follows:

$$\begin{aligned}&\tilde{C}_{\mathrm{inclusive}}(\Delta \varphi , \Delta \eta ) \\&\quad =\frac{p_{\mathrm{prim}}(\Delta \varphi )}{S_{\mathrm{peak}}} \left( \frac{C(\Delta \varphi , \Delta \eta )}{\mathrm {ME}(\Delta \varphi , \Delta \eta )} \bigg |_{\mathrm{peak}}\right. \\&\qquad \left. \phantom {\left( \frac{C(\Delta \varphi , \Delta \eta )}{\mathrm {ME}(\Delta \varphi , \Delta \eta )} \bigg |_{\mathrm{peak}}\right. }- \frac{B_{\mathrm{peak}}}{B_{\mathrm{sidebands}}}\frac{C(\Delta \varphi , \Delta \eta )}{\mathrm {ME}(\Delta \varphi , \Delta \eta )} \bigg |_{\mathrm{sidebands}}\right) . \end{aligned}$$
(1)

The division by \(S_{\mathrm{peak}}\) provides the normalisation to the number of D mesons. In our notation per-trigger quantities are specified by the \(\tilde{C}\) symbol. In Eq. 1, \(p_{\mathrm{prim}}(\Delta \varphi )\) is a correction for the residual contamination of non-primary associated particles not rejected by the track selection (purity correction). This was evaluated with Monte Carlo simulations based on PYTHIA6 (Perugia-2011 tune) by quantifying the fraction of primary particles, among all the tracks satisfying the selection criteria. The correction was applied differentially in \(\Delta \varphi \), since from Monte Carlo studies it was verified that this contamination shows a \(\Delta \varphi \) modulation, typically of about 1–2%. The largest value of the contamination was found in the near-side region, for the lowest \(p_\mathrm{T} \) range of the associated tracks, where \(p_{\mathrm{prim}}(\Delta \varphi )\) approaches 95%.

Statistical fluctuations prevented a (\(\Delta \varphi \), \(\Delta \eta \))-double-differential study of the correlation peak properties. Therefore, the per-trigger azimuthal-correlation function \(\tilde{C}_{\mathrm{inclusive}}(\Delta \varphi )\) was obtained by integrating \(\tilde{C}_{\mathrm{inclusive}}(\Delta \varphi ,\Delta \eta )\) in the range \(|\Delta \eta | < 1\).

A fraction of reconstructed D mesons originates from the decay of beauty hadrons (feed-down D mesons). It was verified with Monte Carlo simulations that azimuthal correlations of prompt and feed-down D mesons with charged particles show different functions. This is a result of the different fragmentation of beauty and charm quarks, as well as of the additional presence of beauty-hadron decay particles in the correlation function of feed-down D-meson triggers. The contribution of feed-down D-meson triggers to the measured angular-correlation function was subtracted using templates of the azimuthal-correlation function of feed-down D mesons with charged particles, obtained with Monte Carlo simulations at generator level (i.e. without detector effects and particle selection), as detailed in [2].

Before performing this subtraction, \(\tilde{C}_{\mathrm{inclusive}}(\Delta \varphi )\) has to be corrected for a bias which distorts the shape of the near-side region of the feed-down contribution, induced by the D-meson topological selection. For feed-down D-meson triggers, indeed, the selection criteria are more likely to be satisfied by decay topologies with small angular opening between the trigger D meson and the other products of the beauty-hadron decay. This induces an enhancement of correlation pairs from feed-down D-meson triggers at \(\Delta \varphi \approx 0\) and a depletion at larger \(\Delta \varphi \) values. This bias was accounted for as a systematic uncertainty in [2]. In this paper, instead, a \(\Delta \varphi \) dependent correction factor (\(c_{\mathrm{FD-bias}}(\Delta \varphi )\)) was determined by comparing Monte-Carlo templates of feed-down D mesons and associated particles at generator level and after performing the event reconstruction and particle selection as on data. This correction factor ranges between 0.6 at \(\Delta \varphi \approx 0\) and 1.3 at \(\Delta \varphi \approx \pi /4\), decreasing then to 1, and was applied to the feed-down contribution to \(\tilde{C}_{\mathrm{inclusive}}(