First study of the two-body scattering involving charm hadrons

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ First study of the two-body scattering involving charm hadrons © 2022 CERN, for the ALICE Collaboration Published version ALICE Collaboration ALICE Collaboration. (2022). First study of the two-body scattering involving charm hadrons. Physical Review D, 106(5), Article 052010. https://doi.org/10.1103/PhysRevD.106.052010 2022 First study of the two-body scattering involving charm hadrons S. Acharya et al.* (ALICE Collaboration) (Received 27 January 2022; revised 3 May 2022; accepted 31 August 2022; published 23 September 2022; corrected 1 November 2022) This article presents the first measurement of the interaction between charm hadrons and nucleons. The two-particle momentum correlations of pD−and ¯ pDþpairs are measured by the ALICE Collaboration in high-multiplicity pp collisions at ffiffiffis p¼13 TeV. The data are compatible with the Coulomb-only interaction hypothesis within ð1.1–1.5Þσ. The level of agreement slightly improves if an attractive nucleon ðNÞ¯ D strong interaction is considered, in contrast to most model predictions which suggest an overall repulsive interaction. This measurement allows for the first time an estimation of the 68% confidence level interval for the isospin I ¼0inverse scattering length of the N ¯ D state f−1 0;I¼0∈½−0.4;0.9fm−1, assuming negligible interaction for the isospin I ¼1channel. DOI: 10.1103/PhysRevD.106.052010 I. INTRODUCTION The study of the residual strong interaction among hadrons is a very active field within nuclear physics. This interaction can lead to the formation of bound states, such as nuclei, or molecular states as, for example, the Λð1405Þ, which is considered as being generated from the attractive forces in the nucleon (N) ¯ K–Σπchannels [1–4]. One of the most fervent discussions in this context is nowadays revolving around systems involving charm mesons (D, D). Studies of their interaction are motivated by the observation of several new states with hidden charm and/or beauty (so-called XYZ states) [5–9],aswellaswithopen charm such as the Tccþ[10,11], and also of pentaquark states like Pcð4380Þand Pcð4450Þ[12,13].Theseexotic hadrons can be described as compact multiquark states in the context of the constituent-quark model [14], but are also considered as natural candidates for loosely bound molecular states [5,6]. For example, the structure of the χc1ð3872Þ [formerly X(3872)] has been interpreted as a ¯ DD=D¯ D molecular state or as a tetraquark [15]. Currently, definite conclusions are difficult to draw because of the lack of any direct experimental information on the D ¯ Dstrong interaction. Strong support for the molecular nature of the Λð1405Þcame not least from low-energy N ¯ K scattering data and information on the p ¯ K scattering length from kaonic hydrogen atoms [16–19]. Hence, a determination of the scattering parameters of systems involving D and/or D mesons are pivotal to advance in the interpretation of the many observed states. The first step in this direction is the investigation of the interaction between the p(uud) D−ð¯ cdÞ pair and its charge conjugate. This interaction does not couple to the lower energy meson-baryon channels since no q¯ q annihilation can occur. A measurement of this interaction is also an essential reference for the study of the in-medium D- and D-meson properties [20]. Similarly to kaons and antikaons, it is theoretically predicted that possible modifications of the charm-meson spectral function at large baryonic densities can be connected to a decrease of the chiral condensate, thus providing sensitivity to chiralsymmetry restoration [21]. So far, the topic of the strong interaction between hadrons containing charm quarks was addressed only from a theoretical point of view [22–25] by employing different effective models anchored to the successful description of other baryon-meson final states, such as the N ¯ KandNK systems, while data are missing. Scattering experiments [26] and systematic studies of stable and unstable nuclei [27], accompanied by sophisticated calculations achieved within effective field theories [28,29], allowed us to reach a solid comprehension of the interaction among nucleons. When extending these studies to interactions including strange hadrons, the average properties of the interactions of some strange nucleon–hadron combinations (pK[30–32],pΛ, and pΣ0[33–35]) could be gauged with the help of scattering data and measurements of kaonic atoms [36]. The study of Λhypernuclei [37] led to the extraction of an average attractive potential. The situation has drastically changed in recent years, thanks to the novel employment of the femtoscopy technique [38] in pp and p-Pb collisions at the LHC applied to almost all combinations of protons and *Full author list given at the end of the article. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. PHYSICAL REVIEW D 106, 052010 (2022) 2470-0010=2022=106(5)=052010(16) 052010-1 © 2022 CERN, for the ALICE Collaboration strange hadrons [39]. The ALICE Collaboration could precisely study the following interactions: pp, pK,pΛ, p¯ Λ,pΣ0,ΛΛ,Λ¯ Λ,pΞ−,pΩ−, and pϕ[39–47].Since conventional scattering experiments cannot be performed with D mesons and charm nuclei [48] have not been discovered yet (searches for charm nuclear states are included in the scientific program of the Japan Proton Accelerator Research Complex [49]), the femtoscopy technique can be employed to study the ND and N ¯ D interactions. In this article, the first measurement of the strong interaction between a D−meson and a proton is reported. This pioneering analysis employs D−instead of the more abundantly produced ¯ D0mesons because of the smaller contribution from decays of excited charm states and the possibility to separate particles and antiparticles without ambiguity. II. EXPERIMENTAL APPARATUS AND DATA SAMPLES The analysis was performed using a sample of highmultiplicity pp collisions at ffiffiffis p¼13 TeV collected by ALICE [50,51] during the LHC run 2 (2016–2018). The main detectors used for this analysis to reconstruct and identify the protons and the D-meson decay products are the inner tracking system (ITS) [52], the time projection chamber (TPC) [53], and the time-of-flight (TOF) detector [54]. They are located inside a large solenoidal magnet providing a uniform magnetic field of 0.5 T parallel to the LHC beam direction and cover the pseudorapidity interval jηj<0.9. The events were recorded with a high-multiplic- ity trigger relying on the measured signal amplitudes in the V0 detector, which consists of two scintillator arrays covering the pseudorapidity intervals −3.7<η<−1.7 and 2.8<η<5.1[55]. The collected data sample corresponds to the 0.17% highest-multiplicity events out of all inelastic collisions with at least one charged particle in the pseudorapidity range jηj<1(denoted as INEL >0). Events were further selected off-line in order to remove machine-induced backgrounds [51]. The events were required to have a reconstructed collision vertex located within 10 cm from the center of the detector along the beam-line direction to maintain a uniform acceptance. Events with multiple primary vertices (pileup), reconstructed from track segments measured with the two innermost ITS layers, were rejected. The remaining undetected pileup is of the order of 1% and therefore negligible in the analysis. After these selections, the analyzed data sample consists of about 109events. The Monte Carlo (MC) samples used in this analysis consist of pp collisions simulated using the PYTHIA 8.243 event generator [56,57] with the Monash-13 tune [58] and GEANT 3[59] for the propagation of the generated particles through the detector. III. DATA ANALYSIS A. Selection of proton and D-meson candidates The proton candidates are selected according to the methods described in [39]. Charged-particle tracks reconstructed with the TPC are required to have transverse momentum 0.5<p T<4.05 GeV=c and pseudorapidity jηj<0.8. Particle identification (PID) is conducted by measuring the specific energy loss and the time of flight with the TPC and TOF detectors, respectively. The selection is based on the deviation nσbetween the measured and expected values for protons, normalized by the detector resolution σ. For proton candidates with a momentum p<0.75 GeV=c, only the TPC is used by requiring jnTPC σj<3, while for larger momenta the PID information of TPC and TOF are combined and tracks are accepted only if the condition ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnTPC σÞ2þðnTOF σÞ2 p<3is fulfilled. With these selection criteria, the purity of the proton sample averaged over pTis Pp¼98% [39]. The contribution of secondary protons originating from weak decays or interactions with the detector material is assessed by using MC template fits to the measured distribution of the distance of closest approach of the track to the primary vertex. The estimated average fraction of primary protons is 86% [39]. The Dmesons are reconstructed via their hadronic decay channel D→K∓ππ, having a branching ratio BR ¼ð9.38 0.15Þ%[60]. D-meson candidates are defined combining triplets of tracks reconstructed in the TPC and ITS detectors with the proper charge signs, jηj<0.8,pT>0.3GeV=c, and a minimum of two (out of six) hits in the ITS, with at least one in either of the two innermost layers to ensure a good pointing resolution. To reduce the large combinatorial background and the contribution of Dmesons originating from beauty-hadron decays (nonprompt), a machine-learning multiclass classification algorithm based on boosted decision trees (BDTs) provided by the XGBOOST library [61,62] is employed. The variables utilized for the candidate selection in the BDTs are based on the displaced decay-vertex topology, exploiting the mean proper decay length of Dmesons of cτ≈ 312 μm[60], and on the PID of charged pions and kaons. Before that, a preselection of the Dcandidates based on the PID information of the decay products is applied by requiring a 3σcompatibility either with the TPC or the TOF expected signals of the daughter tracks. Signal samples of prompt (originating from charm-quark hadronization or decays of excited charm states) and nonprompt Dmesons for the BDT training are obtained from MC simulations. The background samples are obtained from the sidebands of the candidate invariant mass distributions in data. The BDT outputs are related to the candidate probability to be a prompt or nonprompt D meson, or combinatorial background. D-meson candidates are selected in the pTinterval between 1 and 10 GeV=c by requiring a high probability to S. ACHARYA et al. PHYS. REV. D 106, 052010 (2022) 052010-2 be a prompt Dmeson and a low probability to be a combinatorial-background candidate. A selection on the candidate invariant mass ðMðKππÞÞis applied to obtain a high-purity sample of Dmesons. To this end, the MðKππÞdistribution of Dcandidates is fitted in intervals of pTof 1 GeV width in the range 1<p T< 10 GeV=c with a Gaussian function for the signal and an exponential term for the background. The left panel of Fig. 1shows the MðKππÞdistribution for Dwith 2<p T<3GeV=c. The width of the Gaussian function used to describe the signal peak, σD, increases from 6 to 10 MeV=c2with increasing pTas a consequence of the pT dependence of the momentum resolution. The D-meson candidates in the invariant mass window jMðKππÞj <2σD are selected to be paired with proton candidates. This selection, displayed by the two vertical lines in Fig. 1, leads to a purity that is PD−¼ð61.70.9ðstatÞ0.7ðsystÞÞ% on average. The systematic uncertainty of PD−is evaluated by repeating the invariant mass fits, varying the background fit function and the invariant mass upper and lower limits. The contributions of prompt and nonprompt Dmesons are depicted in the left panel of Fig. 1with the red and blue distributions, respectively, They are obtained with a data-driven method based on the sampling of the raw yield at different values of the BDT output score related to the probability of being a nonprompt Dmeson [63]. The yields of prompt and nonprompt Dmesons can be extracted by solving a system of equations that relate the raw yield value Yi(obtained with the ith threshold on the BDT output score) to the corrected yields of prompt (Nprompt) and nonprompt (Nnonprompt)D mesons via the corresponding acceptance-times-efficiency factors for prompt [ðAcc × εÞprompt i] and nonprompt [ðAcc × εÞnonprompt i]D mesons as follows 0 B B @ ðAcc × εÞprompt 1ðAcc × εÞnonprompt 1 . . .. . . ðAcc × εÞprompt nðAcc × εÞnonprompt n 1 C C A × Nprompt Nnonprompt ! −0 B B @ Y1 . . . Yn 1 C C A¼0 B B @ δ1 . . . δn 1 C C A :ð1Þ The δifactors represent the residuals that account for the equations not holding exactly due to the uncertainty of Yi, ðAcc × εÞnonprompt i,andðAcc × εÞprompt i. The system of equations can be solved via a χ2minimization, which leads to the determination of Nprompt and Nnonprompt.The right panel of Fig. 1shows an example of a raw-yield distribution as a function of the BDT-based selection used in the minimization procedure for Dmesons with 2<p T<3GeV=c. The leftmost data point of the distribution represents the raw yield corresponding to the loosest selection on the BDT output related to the candidate probability of being a nonprompt Dmeson, while the rightmost one corresponds to the strictest selection, which is expected to preferentially select nonprompt Dmesons. The prompt and nonprompt components, obtained for each BDT-based selection using the 1.8 1.85 1.9 1.95 ) 2 c) (GeV/ππ(KM 1 2 3 4 5 6 7 8 9 10 3 10× 2 c Counts per 4 MeV/ ALICE = 13 TeVspp, 0.17% INEL > 0)−High-mult (0 ± π ± π ± K→ ± D c < 3 GeV/ T p2 < ) = 0.70σ(2 B+S SData Fit Background Tot signal + D→c + D→b 2 4 6 8 10 12 14 16 18 20 BDT-based selection 5 10 15 20 25 3 10× Raw yield c < 3 GeV/ T p2 < Data + D→c + D→b Total signal ALICE = 13 TeVspp, 0.17% INEL > 0)−High-mult (0 FIG. 1. Left: invariant mass distributions of Dcandidates in the 2<p T<3GeV=c interval. The green solid line shows the total fit function and the gray dotted line the combinatorial background. The contributions of Dmesons originating from charm hadronization and beauty-hadron decays are obtained with the method relying on the definition of different selection criteria, as explained in the text. Right: example of raw-yield distribution as a function of the BDT-based selection employed in the procedure adopted for the determination of the fraction of Doriginating from beauty-hadron decays for the 2<p T<3GeV=c interval. FIRST STUDY OF THE TWO-BODY SCATTERING INVOLVING …PHYS. REV. D 106, 052010 (2022) 052010-3 procedure described above, are represented by the red and blue filled histograms, respectively, while their sum is reported by the magenta histogram. The obtained corrected yields Nprompt and Nnonprompt can then be used to compute the fraction of nonprompt D mesons fj nonprompt for a given selection j, fj nonprompt ¼ðAcc×εÞnonprompt j×Nnonprompt ðAcc×εÞnonprompt j×Nnonprompt þðAcc×εÞprompt j×Nprompt : ð2Þ The fnonprompt factor for the selections used to build the pD−and ¯ pDþpairs is estimated to be ð7.70.5ðstatÞ 0.2ðsystÞÞ%. The systematic uncertainty of fnonprompt is evaluated by repeating the procedure with different sets of selection criteria and varying the fitting parameters in the raw-yield extraction. In addition, since the efficiency depends on the charged-particle multiplicity, the multiplicity distribution in the MC sample used for the efficiency computation was weighted in order to reproduce the one in data. Differently from the component originating from beautyhadron weak decays, Dmesons originating from excited charm-meson strong decays cannot be experimentally resolved from promptly produced Dmesons due to their short lifetime. The two largest sources are the D →Dπ0 and D →Dγdecays, having BR ¼ð30.70.5Þ%and BR ¼ð1.60.4Þ%[60], respectively. Their contribution is estimated from the production cross sections of Dþand Dþ mesons measured in pp collisions at ffiffiffis p¼5.02 TeV [63,64] and employing the PYTHIA 8decayer for the description of the D →DX decay kinematics. The fraction of Dmesons in 1<p T<10 GeV=c originating from D decays is estimated to be fD−¼ð27.61.3ðstatÞ 2.4ðsystÞÞ%, where statistical and systematic uncertainties are propagated from the measurements of the Dþand Dþ production cross sections. B. The correlation function The proton and D−candidates are then combined and their relative momentum kis evaluated as k¼1 2× jp p−p Dj, where p p;Dare the momenta of the two particles in the pair rest frame. The kdistribution of pD−pairs, NsameðkÞ, is then divided by the one obtained combining proton and D−candidates from different events, NmixedðkÞ, to compute the two-particle momentum correlation function, which is defined as CexpðkÞ¼N×NsameðkÞ=NmixedðkÞ [65]. The latter provides a correction for the acceptance of the detector and the normalization for the phase space of the particle pairs. To ensure the same geometrical acceptance as for Nsame, the mixing procedure is conducted only between particle pairs produced in events with similar zposition of the primary vertex and similar charged-particle multiplicity. Since the correlation functions for pD−and ¯ pDþare consistent with each other within statistical uncertainties, they are combined and in the following pD−will represent pD−⊕¯ pDþ. The normalization constant Nis obtained from k∈½1500;2000MeV=c where the correlation function is independent of k, as expected since in this region of kthe pairs of particles are not affected by any interaction. The resulting correlation function CexpðkÞis displayed in the left panel of Fig. 2. The data are compatible with unity for k>500 MeV=c, while they show a possible hint of an increase for lower kvalues. In total 200 pD−and 221 ¯ pDþpairs contribute to NsameðkÞin the region of k<200 MeV=c, where model calculations [22–25] predict a deviation from unity. The systematic uncertainties of CexpðkÞare assessed by varying the proton and D−selection criteria. The measured two-particle momentum correlation function can be related to the source function and the twoparticle wave function via the Koonin-Pratt equation CðkÞ¼Rd3rSðrÞjΨðk;r Þj2[65],whereSðrÞis the source function, Ψðk;r Þis the two-particle wave function, and rrefers to the relative distance between the two particles. The source function for the pD−pairs is estimated by employing the hypothesis of a common source for all hadrons in high-multiplicity pp collisions at the LHC corrected for strong decays of extremely shortlived resonances (cτ≲5fm) feeding into the particle pairs [66]. This is the case for resonances strongly decaying into protons. In contrast, both beauty-hadron and D decays occur at larger distances than the typical range for the strong interaction [60]. This implies that the correlation function for D−mesons originating from these decays will only carry the imprint of the interaction of the parent particle with the proton without impacting the size of the emitting source. The core source determined in [66] features a dependence on the transverse mass mTof the particle pair, which can be attributed to a collective expansion of the system [65,67–69]. The collective behavior has been studied in high-multiplicity pp collisions by the CMS Collaboration and found to be comparable for light-flavor and charm hadrons [70]. Hence, the core source of pD−pairs with k<200 MeV=c is estimated by parametrizing the measured mTdependence of the source radius extracted from pp correlations in [66] and evaluating it at the hmTi¼2.7GeV=c2of the pD−pairs. Since the production mechanism of charm mesons might not be identical to that of light-flavor baryons, the emission of the pp and pD−pairs is studied by simulating pp collisions with PYTHIA 8.301 [57] and computing their relative distance in the pair rest frame, r, considering only D−mesons originating directly from charm-quark hadronization. These studies indicate that the core source of pD− at the pertinent hmTiis smaller by about 25% compared to S. ACHARYA et al. PHYS. REV. D 106, 052010 (2022) 052010-4 that of pp pairs. This is included in the systematic uncertainty of the source radius. The resulting overall source is parametrized by a Gaussian profile characterized by an effective radius Reff ¼0.89þ0.08 −0.22 fm, where the uncertainty includes both the one arising from the mT- dependent parametrization and the PYTHIA 8study. The correlation function due to the genuine pD−interaction can be extracted from the measured CexpðkÞby estimating and subtracting the contributions of D−mesons originating from beauty-hadron and D−decays, protons originating from strange-hadron decays, as well as misidentified protons and combinatorial-background D-meson candidates. The experimental correlation function is decomposed as CexpðkÞ¼λpD−×CpD−ðkÞþλpðKþπ−π−Þ×CpðKþπ−π−ÞðkÞ þλpD−×CpD−ðkÞþλflat ×Cflat:ð3Þ The combinatorial (Kþπ−π−) background below the D−peak and the final-state interaction among protons and D−from D−decays play a significant role. All other contributions are assumed to be characterized by a CðkÞ compatible with unity and are therefore included in the Cflat contribution. The relative weights, λi, are evaluated considering the contributions to D−candidates described above and following the procedure explained in [39] for the protons. They are about 33.9% for CpD−ðkÞand 38.8%, 14.4%, and 13.4% for the pðKþπ−π−Þ,pD −, and flat contributions, respectively. The correlation function CpðKþπ−π−Þis extracted from the sidebands of the D−candidates, chosen as ½MD−ðpTÞ− 200 MeV=c; MD−ðpTÞ−5×σD−ðpTÞ and [MD−ðpTÞþ 5×σD−ðpTÞ;M D−ðpTÞþ200 MeV=c] for the left and right sidebands, respectively. The contamination from D−→¯ D0π−→Kþπ−π−decays in the right sideband is suppressed by a 2.5σD−rejection around the mean value of the D−invariant mass peak. The resulting correlation function is parametrized by a third-order polynomial in k∈½0;1.5GeV=c and is displayed by the green curve reported in the right panel of Fig. 2. The observed behavior is determined by meson-meson and baryon-meson minijets and residual two-body interactions among the quadruplet, as obtained from previous studies [42,46]. The residual pD−correlation function is computed employing the Koonin-Pratt formalism using the CATS framework [71] to obtain a two-particle wave function Ψðk;r Þconsidering only the Coulomb interaction and assuming that the source radius is the same as for pD−pairs. The obtained pD−correlation function is transformed to the momentum basis of the pD−relative momentum by considering the kinematics of the D−→D−Xdecay[72]. The resulting correlation function is shown in the right panel of Fig. 2as a red band. The purple band in the same figure represents the total background that includes all contributions with their corresponding weights. Finally, the genuine pD−correlation function is obtained by solving Eq. (3) for CpD−ðkÞand is shown in Fig. 3. The systematic uncertainties of the genuine pD−correlation function, CpD−ðkÞ, include (i) the uncertainties of CexpðkÞ, (ii) the uncertainties of the λiweights, and (iii) the uncertainties related to the parametrization of the background sources, CpðKþπ−π−ÞðkÞand CpD−ðkÞ. In particular, 0 500 1000 1500 2000 )c (MeV/k* 1.0 1.5 )k*( exp C + ⊕ − pD = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 0 200 400 600 800 )c (MeV/k* 1.0 1.5 2.0 )k*(C = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 + Dp Dp ⊕ − *), pDk( exp C Total background = 0.383) )ππp(K λ = 0.144, pD* λ( − pD→ − pD* = 1) pD* λ( ) − π − π + p(K = 1) )ππp(K λ( FIG. 2. Left: experimental pD−correlation function in the range 0<k <2GeV=c. Statistical (bars) and systematic uncertainties (shaded boxes) are shown separately. The open boxes represent the bin width. Right: experimental pD−correlation function in a reduced krange together with the contributions from pðKþπ−π−Þ(green band) and pD−(red band), and the total background model (purple band). The pðKþπ−π−Þand pD−contributions are not scaled by the respective λparameter. The width of the dark (light) shaded bands depicts the statistical (total) uncertainty of the parametrized background contributions. FIRST STUDY OF THE TWO-BODY SCATTERING INVOLVING …PHYS. REV. D 106, 052010 (2022) 052010-5 as previously mentioned, the systematic uncertainty on CexpðkÞis estimated by varying the proton and D−- candidate selection criteria and ranges between 0.5% and 3% as a function of k. The uncertainties of the λiweights are derived from the systematic uncertainties on the proton and D−purities (Ppand PD−), fD−, and fnonprompt reported in Sec. III A. The systematic uncertainties of CpðKþπ−π−ÞðkÞ are estimated following the same procedure adopted for CexpðkÞand, in addition, by varying the range of the fit of the correlation function parametrized from the sidebands regions of the invariant mass distribution. Additional checks are performed by varying the invariant mass interval used to define the sidebands region of up to 100 MeV=c2. The resulting systematic uncertainty ranges from 1% to 5%. The systematic uncertainty of CpD−ðkÞis due to the uncertainty on the emitting source. Considering the small λpD−ðkÞthis uncertainty results to be negligible compared to the other sources of uncertainty. The overall relative Systematic uncertainty on CpD−ðkÞresulting from the different sources ranges between 3% and 10% and is maximum in the lowest kinterval. IV. RESULTS The resulting genuine CpD−ðkÞcorrelation function can be employed to study the pD−strong interaction that is characterized by two isospin configurations and is coupled to the n ¯ D0channel. First of all, in order to assess the effect of the strong interaction on the correlation function, a reference calculation including only the Coulomb interaction is considered. The corresponding correlation function is obtained using CATS [71]. Second, various theoretical approaches to describe the strong interaction are benchmarked, including meson exchange (J. Haidenbauer et al. [22]), meson exchange based on heavy quark symmetry (Y. Yamaguchi et al. [25]), an SU(4) contact interaction (J. Hoffmann and M. Lutz [23]), and a chiral quark model (C. Fontoura et al. [24]). The relative wave functions for the model of J. Haidenbauer et al. [22] are provided directly, while for the other models [23–25] they are evaluated by employing a Gaussian potential whose strength is adjusted to describe the corresponding published I ¼0and I ¼1 scattering lengths listed in Table I. The pD−correlation function is computed within the Koonin-Pratt formalism, taking into account explicitly the coupling between the pD− and n ¯ D0channels [73] and including the Coulomb interaction [74]. The finite experimental momentum resolution is considered in the modeling of the correlation functions [39]. The outcome of these models is compared in Fig. 3with the measured genuine pD−correlation function. The degree of consistency between data and models is quantified by the p-value computed in the range k<200 MeV=c.Itis expressed by the number of standard deviations nσreported in Table I, where the nσrange accounts, at one standard deviation level, for the total uncertainties of the data points and the models. The values of the scattering lengths f0for the different models are also reported in Table I. Here, the high-energy physics convention on the scattering-length sign is adopted: a negative value corresponds to either a repulsive interaction or to an attractive one with presence of a bound state, while a positive value corresponds to an attractive interaction. The data are compatible with the Coulomb-only hypothesis within ð1.1–1.5Þσ. Nevertheless, the level of agreement slightly improves in case of the models by J. Haidenbauer et al. (employing g2 σ=4π¼2.25) which predicts an attractive interaction, and by Y. Yamaguchi et al. which foresees the formation of a N ¯ D 0 100 200 300 400 )c (MeV/k* 1 2 3 4 )k*( − pD C + Dp ⊕ − pD Coulomb et al.C. Fontoura et al.Y. Yamaguchi J. Hofmann and M. Lutz = 2.25)π/4 2 σ g (et al.J. Haidenbauer = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 FIG. 3. Genuine pD−correlation function compared with different theoretical models (see text for details). The null hypothesis is represented by the curve corresponding to the Coulomb interaction only. TABLE I. Scattering parameters of the different theoretical models for the N ¯ D interaction [22–25] and degree of consistency with the experimental data computed in the range k<200 MeV=c. Model f0ðI¼0Þf0ðI¼1Þnσ Coulomb (1.1–1.5) Haidenbauer et al. [22] (g2 σ=4π¼2.25) 0.67 0.04 (0.8–1.3) Hofmann and Lutz [23] −0.16 −0.26 (1.3–1.6) Yamaguchi et al. [25] −4.38 −0.07 (0.6–1.1) Fontoura et al. [24] 0.16 −0.25 (1.1–1.5) S. ACHARYA et al. PHYS. REV. D 106, 052010 (2022) 052010-6 bound state with a mass of 2804 MeV=c2in the I ¼0 channel. Finally, the scattering parameters can be constrained by comparing the data with the outcome of calculations carried out varying the strength of the potential and the source radius. In this case the interaction potential is parametrized by a Gaussian-type functional form with the range of ρmeson exchange. In this estimation, it is assumed that the interaction in the I ¼1channel is negligible for simplicity. The correlation function CpD−ðkÞis computed including also the Coulomb interaction and the coupled channel. This procedure is repeated for different values of the interaction potential for the I ¼0channel (VI¼0). For all the correlation functions corresponding to the different interaction potentials, the agreement with the data is evaluated by computing the χ2using a bootstrap procedure. Both the statistical and systematic uncertainties of the data are considered in the bootstrap procedure, as well as the uncertainty on the emitting source radius (Reff)inthe computed CpD−ðkÞ, which is varied within 1σof its uncertainty. The resulting overall χ2distributions are shown in Fig. 4as a function of f−1 0;I¼0and VI¼0in the left and right panels, respectively. The data are found to be consistent with a potential strength of VI¼0∈½−1450;−1050MeV within 1σ. This corresponds to an inverse scattering-length interval of f−1 0;I¼0∈½−0.4;0.9fm−1. Since the determined potential strength is always attractive, the positive values of the scattering length imply an attractive interaction without bound states, while the negative values are consistent with the presence of a N ¯ D bound state. The same procedure was repeated for fixed values of Reff in order to obtain the 1σ confidence interval as a function of the emitting source radius. Figure 5shows the confidence interval as a function of the source radius varied within 1σof its uncertainty. The dashed interval corresponds to the radius uncertainty due to only the mTdependence while the full-shaded interval shows the total radius uncertainty. The most probable value reported in Fig. 5with the star symbol corresponds to an 1−0.5−0 0.5 1 1.5 2 2.5 3 3.5 4 ) 1− (fm 1− 0, I=0 f 0 1 2 3 4 5 6 2 χ = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 2−1.8−1.6−1.4−1.2−1−0.8−0.6− 3 10× (MeV) I=0 V 0 1 2 3 4 5 6 2 χ = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 FIG. 4. χ2distributions obtained by comparing the measured CpD−ðkÞfor k<200 MeV=c with the correlation function calculated with an interaction modeled by a Gaussian potential with an interaction range given by ρ-meson exchanges as a function of the inverse scattering length (left panel) and the interaction potential (right panel) for I ¼0. The blue dotted lines represent the value of f−1 0;I¼0and VI¼0for which the χ2is minimum and for the 1σconfidence interval. 0.7 0.8 0.9 1.0 (fm) eff R 0.5− 0.0 0.5 1.0 ) 1− (fm 1− 0, I=0 f Best fit 68% C.L. eff R dependence unc. on T m unc. eff Rtotal = 13 TeVsALICE pp > )0% INEL 0.17 − High-mult. (0 FIG. 5. Regions of 68% confidence intervals for the inverse scattering length f−1 0;I¼0as a function of the source radius varied within one standard deviation considering only the mTdependence on Reff and the total uncertainty (see text for details) under the assumption of negligible interaction for I ¼1. The most probable value is reported by the star symbol. FIRST STUDY OF THE TWO-BODY SCATTERING INVOLVING …PHYS. REV. D 106, 052010 (2022) 052010-7 attractive interaction with the formation of a bound state. Given that most models predict a repulsive I ¼1interaction, in reality the I ¼0interaction might have to be even more attractive. The herewith presented limits provide valuable guidance for further theoretical studies advancing the understanding of the strong interaction in the charm sector. V. SUMMARY In conclusion, this article presents the first measurement of correlation functions involving charm hadrons, which allows one to access to the strong interaction between a proton and a charm meson. The genuine pD−correlation function reflects the pattern of an overall attractive interaction. The data are compatible within ð1.1–1.5Þσwith the correlation function obtained from the hypothesis of a Coulomb-only interaction. The degree of consistency improves when considering, in addition, state-of-the-art models that predict an attractive strong N ¯ D interaction with or without a bound state. Finally, assuming no interaction for the I ¼1channel, the scattering length of the N ¯ D system in the isospin I ¼0channel is estimated as f−1 0;I¼0∈½−0.4;0.9fm−1. This exploratory study paves the way for precision studies of the strong interactions involving charm hadrons, facilitated by about one order of magnitude larger pp data samples expected to be collected in the next years during the LHC runs 3 and 4 [75]. ACKNOWLEDGMENTS The ALICE Collaboration would like to thank all its engineers and technicians for their invaluable contributions to the construction of the experiment and the CERN accelerator teams for the outstanding performance of the LHC complex. The ALICE Collaboration gratefully acknowledges the resources and support provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration acknowledges the following funding agencies for their support in building and running the ALICE detector: A. I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS), Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of Communications and High Technologies, National Nuclear Research Center, Azerbaijan; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de Estudos e Projetos (Finep), Fundação de Amparo `a Pesquisa do Estado de São Paulo (FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of Education of China (MOEC), Ministry of Science and Technology of China (MSTC) and National Natural Science Foundation of China (NSFC), China; Ministry of Science and Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat `al’Energie Atomique (CEA) and Institut National de Physique Nucl´eaire et de Physique des Particules (IN2P3) and Centre National de la Recherche Scientifique (CNRS), France; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum für Schwerionenforschung GmbH, Germany; General Secretariat for Research and Technology, Ministry of Education, Research and Religions, Greece; National Research, Development and Innovation Office, Hungary; Department of Atomic Energy Government of India (DAE), Department of Science and Technology, Government of India (DST), University Grants Commission, Government of India (UGC) and Council of Scientific and Industrial Research (CSIR), India; Indonesian Institute of Science, Indonesia; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo de Cooperación Internacional en Ciencia y Tecnología (FONCICYT) and Dirección General de Asuntos del Personal Academico (DGAPA), Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), Netherlands; The Research Council of Norway, Norway; Commission on Science and Technology for Sustainable Development in the South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú, Peru; Ministry of Education and Science, National Science Centre and WUT ID-UB, Poland; Korea Institute of Science and Technology Information and National Research Foundation of Korea (NRF), Republic of Korea; Ministry of Education and Scientific Research, Institute of Atomic Physics, Ministry of Research and Innovation and Institute of Atomic Physics and University Politehnica of Bucharest, Romania; Joint Institute for Nuclear Research (JINR), Ministry of Education and Science of the Russian Federation, National Research Centre Kurchatov Institute, Russian Science Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA), Suranaree University of S. ACHARYA et al. PHYS. REV. D 106, 052010 (2022) 052010-8 55INFN, Sezione di Cagliari, Cagliari, Italy 56INFN, Sezione di Catania, Catania, Italy 57INFN, Sezione di Padova, Padova, Italy 58INFN, Sezione di Pavia, Pavia, Italy 59INFN, Sezione di Torino, Turin, Italy 60INFN, Sezione di Trieste, Trieste, Italy 61Inha University, Incheon, Republic of Korea 62Institute for Advanced Simulation, Forschungszentrum Jülich, Jülich, Germany 63Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, Netherlands 64Institute for Nuclear Research, Academy of Sciences, Moscow, Russia 65Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovakia 66Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India 67Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 68Institute of Space Science (ISS), Bucharest, Romania 69Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 70Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de M´exico, Mexico City, Mexico 71Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 72Instituto de Física, Universidad Nacional Autónoma de M´exico, Mexico City, Mexico 73iThemba LABS, National Research Foundation, Somerset West, South Africa 74Jeonbuk National University, Jeonju, Republic of Korea 75Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 76Joint Institute for Nuclear Research (JINR), Dubna, Russia 77Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 78KTO Karatay University, Konya, Turkey 79Laboratoire de Physique des 2 Infinis, Ir`ene Joliot-Curie, Orsay, France 80Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS-IN2P3, Grenoble, France 81Lawrence Berkeley National Laboratory, Berkeley, California, USA 82Lund University Department of Physics, Division of Particle Physics, Lund, Sweden 83Moscow Institute for Physics and Technology, Moscow, Russia 84Nagasaki Institute of Applied Science, Nagasaki, Japan 85Nara Women’s University (NWU), Nara, Japan 86National and Kapodistrian University of Athens, School of Science, Department of Physics, Athens, Greece 87National Centre for Nuclear Research, Warsaw, Poland 88National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India 89National Nuclear Research Center, Baku, Azerbaijan 90National Research Centre Kurchatov Institute, Moscow, Russia 91Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 92Nikhef, National institute for subatomic physics, Amsterdam, Netherlands 93NRC Kurchatov Institute IHEP, Protvino, Russia 94NRC “Kurchatov”Institute—ITEP, Moscow, Russia 95NRNU Moscow Engineering Physics Institute, Moscow, Russia 96Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom 97Nuclear Physics Institute of the Czech Academy of Sciences, Řežu Prahy, Czech Republic 98Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA 99Ohio State University, Columbus, Ohio, USA 100Petersburg Nuclear Physics Institute, Gatchina, Russia 101Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia 102Physics Department, Panjab University, Chandigarh, India 103Physics Department, University of Jammu, Jammu, India 104Physics Department, University of Rajasthan, Jaipur, India 105Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 106Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 107Physik Department, Technische Universität München, Munich, Germany 108Politecnico di Bari and Sezione INFN, Bari, Italy 109Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany FIRST STUDY OF THE TWO-BODY SCATTERING INVOLVING …PHYS. REV. D 106, 052010 (2022) 052010-15 110RIKEN iTHEMS, Wako, Japan 111Russian Federal Nuclear Center (VNIIEF), Sarov, Russia 112Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India 113School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom 114Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 115St. Petersburg State University, St. Petersburg, Russia 116Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 117SUBATECH, IMT Atlantique, Universit´e de Nantes, CNRS-IN2P3, Nantes, France 118Suranaree University of Technology, Nakhon Ratchasima, Thailand 119Technical University of Košice, Košice, Slovakia 120The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 121The University of Texas at Austin, Austin, Texas, USA 122Universidad Autónoma de Sinaloa, Culiacán, Mexico 123Universidade de São Paulo (USP), São Paulo, Brazil 124Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 125Universidade Federal do ABC, Santo Andre, Brazil 126University of Cape Town, Cape Town, South Africa 127University of Houston, Houston, Texas, USA 128University of Jyväskylä, Jyväskylä, Finland 129University of Kansas, Lawrence, Kansas, USA 130University of Liverpool, Liverpool, United Kingdom 131University of Science and Technology of China, Hefei, China 132University of South-Eastern Norway, Tonsberg, Norway 133University of Tennessee, Knoxville, Tennessee, USA 134University of the Witwatersrand, Johannesburg, South Africa 135University of Tokyo, Tokyo, Japan 136University of Tsukuba, Tsukuba, Japan 137University Politehnica of Bucharest, Bucharest, Romania 138Universit´e Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France 139Universit´e de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon, Lyon, France 140Universit´e de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France 141Universit´e Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU, D´epartment de Physique Nucl´eaire (DPhN), Saclay, France 142Universit`a degli Studi di Foggia, Foggia, Italy 143Universit`a di Brescia, Brescia, Italy 144Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India 145Warsaw University of Technology, Warsaw, Poland 146Wayne State University, Detroit, Michigan, USA 147Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany 148Wigner Research Centre for Physics, Budapest, Hungary 149Yale University, New Haven, Connecticut, USA 150Yonsei University, Seoul, Republic of Korea 151Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, Japan †Deceased. ‡Also at Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy. §Also at Dipartimento DET del Politecnico di Torino, Turin, Italy. ∥Also at M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia. ¶Also at Department of Applied Physics, Aligarh Muslim University, Aligarh, India. **Also at Institute of Theoretical Physics, University of Wroclaw, Poland. ††Also at University of Kansas, Lawrence, Kansas, USA. S. ACHARYA et al. PHYS. REV. D 106, 052010 (2022) 052010-16