Towards the understanding of the genuine three-body interaction for p–p–p and p–p–Λ

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/ Towards the understanding of the genuine three-body interaction for p–p–p and p–p–Λ © CERN for the benefit of the ALICE Collaboration 2023 Published version ALICE Collaboration ALICE Collaboration. (2023). Towards the understanding of the genuine three-body interaction for p–p–p and p–p–Λ. European Physical Journal A, 59, Article 145. https://doi.org/10.1140/epja/s10050-023-00998-6 2023 Eur. Phys. J. A (2023) 59:145 https://doi.org/10.1140/epja/s10050-023-00998-6 Regular Article - Experimental Physics Towards the understanding of the genuine three-body interaction for p–p–p and p–p– ALICE Collaboration CERN, 1211 Geneva 23, Switzerland Received: 17 June 2022 / Accepted: 30 September 2022 / Published online: 3 July 2023 © CERN for the benefit of the ALICE Collaboration 2023 Communicated by C. Munoz Camacho. Abstract Three-bodynuclearforcesplayanimportantrole in the structure of nuclei and hypernuclei and are also incorporatedinmodelsto describe the dynamics of dense baryonic matter, such as in neutron stars. So far, only indirect measurements anchored to the binding energies of nuclei can be used to constrain the three-nucleon force, and if hyperons are considered, the scarce data on hypernuclei impose only weak constraints on the three-body forces. In this work, we present the first direct measurement of the p–p–p and p–p– systems in terms of three-particle correlation functions carried out for pp collisions at √s=13 TeV. Three-particle cumulants are extracted from the correlation functions by applying the Kubo formalism, where the three-particle interaction contribution to these correlations can be isolated after subtracting the known two-body interaction terms. A negative cumulant is found for the p–p–p system, hinting to the presence of a residual three-body effect while for p–p–the cumulant is consistent with zero. This measurement demonstrates the accessibility of three-baryon correlations at the LHC. 1 Introduction One of the open challenges of nuclear physics is the understanding of many-particle dynamics. Studies of the nuclear structure have unambiguously shown that calculations based only on nucleon–nucleon (N–N) interactions fail to accurately describe many experimental observables, such as nuclear binding energies along the periodic table of elements [1],thepositionoftheneutrondriplineforneutronrich nuclei [2] or the properties of the recently observed four-neutrons resonance [3]. A significant improvement in the modelling of nuclear bound objects has been achieved by includingthree-bodyforcesintheoretical calculations.These three-body forces are implemented in chiral effective field theories [4] and in a number of ab initio many-body methods e-mail: [email protected] such as no-core shell model [5], coupled-cluster theory [6,7], self-consistent Green’s function theory [8], similarity renormalisation group [9,10], and quantum Monte Carlo [11]. Studies conducted on intermediate mass neutron-rich nuclei proved that the sensitivity to the three-body forces increases with the number of neutrons in the system [6]. Threebody forces within light and medium-mass nuclei, where the nuclear saturation density corresponds to typical interparticle distances of 2 fm, contribute about 10–15% to the total interaction strength [12,13]. However, at higher densities and shorter inter-particle distances their contribution might increase [2], but no data are available in such a regime and the properties of nuclear matter can be only extrapolated using the available information at saturation densities. The experimental information on the three-body forces involving hyperons is even more scarce since the data available for hypernuclei are much less than the data for nuclei. Recent hypetriton measurements in several colliding systems at RHIC and LHC [14–17] provide important input to the understanding of N–N–forces and future measurements will resolve the current tensions among the different estimations of the binding energy and life-time. Theoretical works assign to the hypertriton a radius of the order of 5 fm [18] and hence a N–distance of 10 fm [13,19] within this state. Heavier hypernuclei are more compact and their size is comparable to that of normal nuclei so that they represent an optimal test bed for the N–N–interaction [20]. However, good fits of the theoretical models to the available hypernuclear data, from 7 Li to 208 Pb [21–24], require a full understanding of the shell-structure of such bound objects as well as accurate experimental constraints on the spin-dependent N–interaction, in particular for the p-wave and higher partial waves. The lack of precise data as well as the limitations in the microscopic description of the structure of hypernuclei cause large ambiguities on the strength of the N–N–three-body force. Further opportunities are provided by the recently observed 3 n bound state [25] and planned experimental programs focused on neutron-rich hypernu- 123 145 Page 2 of 22 Eur. Phys. J. A (2023) 59 :145 clei [26]. Nevertheless, the contribution from three-body forces in bound objects such as nuclei and hypernuclei cannot be separated from the lower-order two-body interactions, hence, complementary experimental methods to investigate three-baryon systems could provide an important contribution to this field. Neutron-rich and dense baryonic matter constitutes an interesting system also because of its connections to the physics of neutron stars (NS) [27]. The structure and composition of the innermost part of NS is not known. Amongst many possible scenarios, some models support the appearance of various hadronic particle species with increasing baryon density inside the star [27,28]. The presence of hadronic degrees of freedom and their relative abundances are sensitive to the two- and three-body interaction models which are used to compute the equation of state (EoS) of NS matter. The different hypotheses can be tested by deriving the masses and radii of NS for a specific EoS and comparing themwith the corresponding astrophysicalobservations [28]. The suggestion of strange baryons inside NS is motivated by the fact that central densities of NS might become sufficiently large (ρ≈3−4ρ0, where ρ0is the nuclear saturation density) to provide favourable conditions for the onset of strangeness production processes leading to, in particular, the formation of hyperons. The appearance of hyperons in NS matter results in a softening of the EoS which is at variance with astrophysical observations of two solar mass stars [29,30]. However, in Ref. [31], it was shown that by adding a strongly repulsive N–N–interaction, tuned to reproduce the separation energies of hyperons in several hypernuclei, a sufficiently stiff EoS can be obtained and even the massive NS observables can be reproduced. This indicates that three-body forces may have a significant contribution in modelsthat describe the structure of NS.Hence, a direct measurement of the three-body forces involving nucleons and hyperons at small inter-particle distances is required. The femtoscopy technique can be used as a tool to investigate the strong interaction amongst hadrons produced in particle collisions [32–35] and recently has been successfully employed to analyse experimental data. The produced hadrons may undergo final state interactions (FSIs) and the resulting correlation in the momentum space can be studied to test the underlying dynamics using correlation functions [34,35]. The method has been applied by the STAR Collaboration to measure hadron-hadron correlations in Au– Au collisions with a centre-of-mass energy of 200 GeV per nucleon pair [36–38]. In such ultra-relativistic heavy-ion collisions, the average relative distances of emitted particles is about 7–8 fm [35]. In small colliding systems, such as pp and p–Pb collisions at the LHC, particles are produced at distances of the order of 1 fm, hence, the sensitivity of the correlation function to the short-range strong interaction is enhanced. Recently, the method has been employed by ALICE to study FSIs of hadrons produced in such small collidingsystems.Thelargedatasamplesallowedfortheprecise measurement of correlation functions for multiple hadronic pairs (p–p [39], p–K+and p–K−[40], p–[39], p–0[41], –[42], p–−[43], p–−[44], p–φ[45] and baryon– antibaryon [46]). By using these results, several models for thetwo-bodystronginteractioncouldbevalidated(foracomplete review see Ref. [47]). The femtoscopy technique was also employed in the analysis of three- and four-pion correlations measured in pp, p–Pb, Pb–Pb collision systems by ALICE [48,49] to probe coherenthadronproduction. TheKubo’s cumulantexpansion method [50] was used to isolate the genuine three-particle correlation from the two-body contributions where the latter were evaluated by combining two particles from the same event and a third particle taken from another event. Alternatively, the recently developed projector method [51], where either the theoretical or the measured two-body correlation functions are used to obtain the lower-order contributions, can be employed. This method allows a significant reduction of the statistical uncertainties. In this article, the first femtoscopic study of three-baryon correlations is performed for the p–p–p and p–p–systems measured in high-multiplicity (HM) pp collisions at √s=13 TeV. The Kubo’s formalism and the projector method are employed to isolate the genuine three-body correlation and the choice of the reaction system aims to study the interaction at small distances. The article is organised as follows: in Sect. 2.1 the data analysis procedure is presented starting from the event selection; in Sect. 2.2 the definition of the two-particle correlation function is extended to the three-particlecase;inSect.2.3thefemtoscopicthree-particle cumulant is defined; the lower-order two-particle correlation contributions in the measured correlation functions are evaluated in Sect. 2.4 and the decomposition of the cumulant to account for misidentifications and particle feed-down are presented in Section 2.5; the final results are discussed in Sect. 3and the conclusions are given in Sect. 4. 2 Analysis 2.1 Event selection and particle identification The data sample of pp collisions at a centre of mass energy √s=13 TeV was recorded with the ALICE detector [52,53] during the LHC Run 2 (2015–2018). The sample has been collected employing a HM trigger. The triggerisbasedonthemeasuredamplitudeintheV0detectorsys- tem, consisting of two arrays of plastic scintillators located at forward (2.8<η<5.1) and backward (−3.7<η<−1.7) pseudorapidities [54]. The selected HM events correspond to the highest 0.17% multiplicity interval with respect to all 123 Eur. Phys. J. A (2023) 59 :145 Page 3 of 22 145 inelastic collisions with at least one measured charged particle within |η|<1 (INEL>0). This condition results in an average of 30 charged particles in the range |η|<0.5[44]. Charged-particle tracking in the midrapidity region is conducted with the Inner Tracking System (ITS) [52] and the Time Projection Chamber (TPC) [55]. These detectors are immersed in a homogeneous 0.5 T magnetic field parallel to the beam direction. The ITS consists of six cylindrical layers of high position-resolution silicon detectors placed radially between 3.9 and 43 cm around the beam vacuum tube. The TPC consists of a 5 m long, cylindrical gaseous detector with full azimuthal coverage in the pseudorapidity range |η|<0.9. Particle identification (PID) is conducted via the measurement of the specific ionisation energy loss (dE/dx)inthe TPC gas with up to 159 reconstructed space points along the particle trajectory. For high momentum particles, the TPC measurement is combined with information provided by the time-of-flight (TOF) [56] detector system, which is located at a radial distance of 3.7 m from the nominal interaction point and consists of multigap resistive plate chambers covering the full azimuthal angle in |η|<0.9. The primary vertex (PV) of the event is reconstructed with the combined track information of the ITS and the TPC, and independently with track segments in the two innermost layers of the ITS. The reconstructed PV of the event is required to have a maximal displacement with respect to the nominal interaction point of 10 cm along the beam axis, in order to ensure a uniform acceptance. Pile-up events with multiple primary vertices are removed following the procedure described in Refs. [39,43,57]. This rejects the events with pile-up of collisions occurring in the same or nearby bunch crossings. However, additional clean-up has to be applied on the track selection level to reject particles produced in pile-up collisions in the long TPC readout time. A total of 1.0×109HM events are used for the analysis after event selection. In order to build the three-particle correlation functions of p–p–p and p–p–systems, particle and antiparticle distributions are combined. In the following, p–p–prefersto p–p–p⊕p–p–pandp–p–referstop–p–⊕ p–p–. The proton and candidates as well as their antiparticles need to be selected. As the particle and antiparticle selections are identical, only the particles are explicitly discussed below. Both particle species are reconstructed using the procedure described in Ref. [57], while the related systematic uncertainties are evaluated by varying the kinematic and topological selection criteria used in the reconstruction. In the following text, the systematic variations are enclosed in parentheses. The primary protons are selected in the momentum interval 0.5(0.4,0.6)<pT<4.05GeV/cand|η|<0.8(0.77,0.85). To improve the quality of the tracks a minimum of 80 (70, 90) out of the 159 possible spatial points inside the TPC are required. The PID selections are applied by comparing the measured dE/dxand time-of-flight with the expected values for a proton candidate. The agreement is expressed in multiples (nPID σ) of the detector resolution σ. For protons with pT<0.75 GeV/cthe nPID σis evaluated only based on the specific energy loss in the TPC, while for pT≥0.75 GeV/ca combined TPC and TOF PID selection is applied nPID σ=n2 σ,TPC +n2 σ,TOF.ThenPID σof the acceptedprotoncandidatesis requiredtobe lower than3(2.5, 3.5). To reject particles that are non-primary or come from pile-up collisions, the distance of closest approach (DCA) to the PV of the tracks is required to be less than 0.1 cm in the transverse plane and less than 0.2 cm along the beam axis. The purity of candidates is estimated using Monte Carlo (MC) simulations by taking the ratio of the number of reconstructed true protons produced by the generator and the number of all candidates identified as protons as a function of the reconstructed transverse momentum. The contributions of secondary protons stemming from weak decays of strange baryons and from interactions in the detector material are extracted using MC template fits to the measured distributions of the DCA to the PV [39]. The average purity of the identified protons is 98.3% and 86.6% of them are primaries. The candidates are reconstructed via the weak decay →pπ−(the →pπ+in case of reconstruction). The secondary daughter tracks are selected with similar criteria as for the primary protons regarding |η|and the number of hits in the TPC. However, a less strict PID requirement of nPID σ<5(4)is used. In addition, the daughter tracks are required to have a DCA to the PV of at least 0.05 (0.06) cm and the DCA between the daughter tracks at the secondary vertex must be smaller than 1.5 (1.2) cm. The cosine of the pointing angle (CPA) between the vector connecting the PV to the decay vertex and the 3-momentum of the candidate is required to be larger than 0.99 (0.995). To reject unphysical secondary vertices, reconstructed with tracks stemming from pile-up of pp collisions occurring in different bunch crossings, the decay tracks are required to possess a hit in the two innermost or the two outermost ITS layers or a matched TOF signal [42]. Finally, a selection on the candidate invariant mass (IM) is applied by requiring it to be ina±4MeV/c2interval around the nominal mass [58]. The primary and secondary contributions to the yield of  are extracted employing a similar method as for protons but using the CPA as an observable for the template fits. The hyperons produced in primary interactions contribute to about 58.5% of their total yield. About 19.5% originate from the electromagnetic decays of 0. The number of 0parti- cles is related to their ratio to the hyperons, which is fixed to 1/3 based on predictions from the isospin symmetry and a measurement of the corresponding production ratios [59]. Further, each of the weak decays of −and 0contributes 123 145 Page 4 of 22 Eur. Phys. J. A (2023) 59 :145 about 11 % to the yield of hyperons. The purity of and  has been extracted by fitting the IM spectra of candidates as a function of the three-particle kinematic variable Q3which is defined in Eq. 5. The fits have been performed in the IM range of 1090 to 1150 MeV/c2using a double Gaussian for the signal and a second-order polynomial for the background. The result has been averaged for Q3<1GeV/c, leading to a combined purity of and of 95.6%. The systematic uncertainties are evaluated by performing simultaneous variations of the selection criteria for protons and candidates as well as for the corresponding antiparticles. The variations are randomly combined in 44 sets in which at least one of the selection criteria is varied. Such procedure allows to account for the correlations between the systematic uncertainties. Each random set of variations is accepted for the evaluation of the systematic uncertainties only if the yield of the triplets is varied by less than 10% with respect to the standard selection in the kinematic region Q3<0.4GeV/c. 2.2 Three-particle correlation function The observable of interest in femtoscopy is usually the twoparticle momentum correlation function [35,60], which is defined as the probability to simultaneously find two particles with momenta p1and p2divided by the product of the corresponding single particle probabilities C(p1,p2)≡P(p1,p2) P(p1)P(p2).(1) These probabilities are related to the inclusive Lorentz-in- variantspectra P(p1,p2)∝E1E2d6N d3p1d3p2and P(pi)∝Eid3Ni d3pi. In the absence of a correlation signal, the value of C(p1,p2) is constant and normalised to unity. A similar logic can be followed to construct the three-particle correlation functions as C(p1,p2,p3)≡P(p1,p2,p3) P(p1)P(p2)P(p3).(2) Following [61,62], Eq. 1can also be written as C(k∗)=d3r∗S(r∗)|ψ(r∗,k∗)|2,(3) where S(r∗)is the distributionof the relative distances of particle pairs in the pair rest frame (PRF, denoted by the ∗)–the so-called source function. The properties of the source in pp collisions at √s= 13 TeV have been evaluated in Ref. [57], including the effects of short-lived resonance decays which enlarge the effective source size. The wave function of the particle pair relative motion is denoted by ψ(r∗,k∗)where k∗=(p∗ 1−p∗ 2)/2 is the relative momentum. The wave function encapsulates the details of the particle interaction and drives the shape of the correlation function. In case of the three-particle correlation function, the two-particle source function and the wave function of the particle pair relative motion must be replaced by a three-particle source function and wave function. In this analysis, the measured threeparticle correlation functions are not compared to theoretical predictions. The goal here is to extract the three-particle femtoscopic cumulants which provide experimental evidence of the existence, or the absence, of genuine three-particle correlations, as explained in Sect. 2.3. The three-particle correlation function can be written as C(p1,p2,p3)=C(Q3)=NNs(Q3) Nm(Q3),(4) where Ns(Q3)and Nm(Q3)are the same-event and mixedevent distributions of three particle combinations (triplets) as a function of Q3and Nis the normalisation parameter. The Lorentz-invariant variable Q3is defined in [48]as Q3=−q2 12 −q2 23 −q2 31 ,(5) where qij is the norm of the four-vector [35] qμ ij =pi−pjμ−pi−pj·Pij P2 ij Pμ ij,Pij ≡pi+pj, (6) which can be rewritten as qμ ij =2mj mi+mj pμ i−2mi mi+mj pμ j.(7) Here miand mjare the particle iand jmasses, pμ iand pμ j are the particle four momenta, while qμ ij is the relative fourmomentum of the pair ij. In the case of same mass particles, the term (pi−pj)·Pij P2 ij Pμ ij becomes 0. In the non-relativistic case q2 ij =−4k∗ ij 2, where k∗ ij is the relative momentum of the ij pair in the PRF. The mixed-event sample is obtained using event-mixing techniques, in which the particle triplets of interest are generated by combining single particles stemming from three different events. To maintain the same acceptance effects as in the same event sample, the mixing procedure is conducted only for events with similar zposition of the primary vertex andmultiplicity[39].Additionally,inordertocorrectforpossible differences in terms of multiplicity distribution between sameandmixedevents,theyieldofthelatterisre-weightedin each multiplicity interval to have the same statistical weight as the distribution when particles are from the same event. To account for the two-track merging and splitting effects due to the finite two-track resolution in the same-event sample, a minimum value of the distance between two proton tracks (in case of p–pairs, the proton from decay is considered along with the primary proton) on the azimuthal-polar angles plane η–ϕ is applied to both the same- and mixed-event 123 Eur. Phys. J. A (2023) 59 :145 Page 5 of 22 145 samples. The default selection is η2+ϕ2≥0.0172and a systematic variation of +10 % for the value of the minimum distance is applied in the analysis. The normalisation parameter Nis chosen such that the mean value of the correlation function equals unity in a Q3region where the effects of FSIs are negligible. The interval Q3∈(1.0−1.2)GeV/c is chosen for all triplets. 2.3 Three-particle femtoscopic cumulants The measurable three-particle correlation function C(p1,p2,p3)include all interactions at work in the threeparticle system: the two-body interactions among all pairs withintheselectedtripletandthegenuinethree-bodyinterac- tion. To access only the genuine three-body correlations, one can use cumulants. Given random variables Xi, the cumulant for a triplet is defined by Kubo [50]as X1X2X3c=X1X2X3−{X1X2X3 +X2X3X1+X3X1X2} +2X1X2X3, (8) where Xiis the expectation value of the variable Xi and XiXj,XiXjXkare the two- and three-variable joint moments. The three-particle correlation function, defined in Eq. 4, is the three-particle momentum distribution normalised to the mixed-event distribution. The cumulants method can be applied to the numerator which contains the correlated particles, and then the expression is normalised to the mixed-event distribution. The three-particle femtoscopic cumulant c3thus can be defined as c3(p1,p2,p3)=[N3(p1,p2,p3)−N2(p1,p2)N1(p3) −N2(p2,p3)N1(p1)−N2(p3,p1)N1(p2) +2N1(p1)N1(p2)N1(p3)]/ N1(p1)N1(p2)N1(p3), (9) where N3p1,p2,p3and N2pi,pjare the same-event three- and two-particle momentum distributions; N1pi is the single-particle momentum distribution; the product terms N2pi,pjN1pkand N1piN1pjN1pkindicate the mixed event distributions. Thus one can further rewrite the femtoscopic cumulant as c3p1,p2,p3=C(p1,p2,p3)−C([p1,p2],p3) −C([p2,p3],p1)−C([p3,p1],p2)+2. (10) This method has been already successfully applied within the ALICE Collaboration to study the possibility of coherent pion production by measuring three-pion femtoscopic cumulants in Refs. [48,49]. Theorem I from Ref. [50] enunciates that the three-particle cumulant is zero if the variables Xi,Xj,...can be divided into two or more groups that are statistically independent. In case of femtoscopic cumulants, this translates into c3p1,p2,p3=0 in the absence of genuine three-body correlations. Therefore, the measurements of non-vanishing values of c3can be used as an experimental confirmation of the existence of genuine three-body effects. If genuine three-body correlations are not present in the particle triplet, the three-particle correlation function can be expressed using only lower order contributions as follows Ctwo-body(p1,p2,p3)=C([p1,p2],p3)+C([p2,p3],p1) +C([p3,p1],p2)−2.(11) InEq.11,C([pi,pj],pk)isbuiltbycombiningparticlesiand jfrom the same event with particle kfrom another event to obtain the numerator N2pi,pjN1pkof the correlation function while the denominator N1p1N1p2N1p3is estimated using three particles from three different events as described in Sect. 2.2. 2.4 Projector method An alternative method to isolate the genuine three-body contribution to the measured three-particle correlation functions is the projector method [51]. This method makes use of the subtraction rule provided by the Kubo’s cumulant decomposition (Eq. 10) but, instead of evaluating them with the datadriven approach based on event mixing described above, it calculates C([pi,pj],pk)using the measured or the calculated two-particle correlation function and the projection of the third non-interacting (spectator) particle. The method is described in Ref. [51]. Given the three-particle correlation function, C(Q3), and the two-body correlation functions, C(k∗ ij), the projector method provides a kinematic transformation from the relative momentum k∗ ij of the interacting pairs ij to the Q3of the three-body system (i−j)−kunder study. The transformation is given by the following integral in the momentum space Cij(Q3)=C(k∗ ij)Wij(k∗ ij,Q3)dk∗ ij,(12) where the indices ij denote the interacting pair and the projector function Wij is equal to [51] Wij(k∗ ij,Q3)=16(αγ −β2)3/2k∗ ij 2 πQ4 3γ2γQ2 3−(αγ −β2)k∗ ij 2. (13) 123 145 Page 6 of 22 Eur. Phys. J. A (2023) 59 :145 0.1 0.2 0.3 0.4 0.5 0.6 0.7 )c (GeV/ 3 Q 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 ) 3 Q( pp)(p C ALICE = 13 TeVs pp 0.17% INEL)High Mult. (0 a) Data-drivenp)pp(pp)(p p Projector methodp)(p 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 )c (GeV/ 3 Q 2 1.5 1 0.5 0 0.5 1 n ALICE = 13 TeVs pp 0.17% INEL>0) High Mult. (0 Fig. 1 The upper panels show the comparison of the two-particle correlationsprojectedonthree-particlephasespaceobtainedusingthedata- driven approach based on event mixing (green points) and the projector method (grey band). The resulting correlation functions are shown for (p–p)–p (a), (p–p)–(b) and p–(p–)(c) cases. The error bars and the boxes represent the statistical and systematic uncertainties, respectively. The grey band includes systematic and statistical uncertainties summed in quadrature. The lower panels show the deviations between the data-driven approach and the projector method, expressed in terms of nσ The constants α,βand γdepend on the particle masses.1 The integral in Eq. 12 can be evaluated using the measured p–p and p–correlation functions from Refs. [41,57,63]. The resulting correlation functions are compared to the ones obtained by employing the data-driven method (Eq. 11) and shown in Fig. 1. Panel (a) shows the (p–p)–p correlation function, the green points are obtained using the data-driven approach and the grey band is obtained with the projector method. The statistical and systematic uncertainties are shown separately for the data driven method, while the width of the grey band represents the sum in quadrature of the statistical and systematic uncertainties for the projector method. The statistical uncertainties of all the measured correlation functions have been estimated using a bootstrap [63] method bysamplingsame-andmixed-eventcountsfromPoissondis- tributions. The statistical uncertainties shown correspond to the central 68% confidence interval and are consistent with theuncertaintiesobtainedemployingthe standard error propagation method. The systematic uncertainties are estimated by varying the selection criteria of the particle candidates as described in Sect. 2.1. Panels (b) and (c) show the same comparison for the (p–p)–and the p–(p–) correlation functions. The number of events used for mixing to obtain 1 α=4m2 k (mi+mk)2+4m2 k (mj+mk)2+4; β=4mk(mi+mj+mk) mi+mjmj (mj+mk)2−mi (mi+mk)2; γ=4(mi+mj+mk)2 (mi+mj)2m2 i (mi+mk)2+ m2 j (mj+mk)2. (p–p)–pand (p–p)–correlation functions is 30. Thenumerator of p–(p–) correlation function requires p–pairs in same event sample, which are less abundant than p–p pairs. Thustoobtaingoodstatisticalprecision,thenumberofevents used for mixing (to account for the third uncorrelated particle) must be increased to 100 in case of p–(p–) triplets. The results from the data-driven and the projector method are in good agreement between each other. The number of deviations nσin each bin are shown in the bottom panels of Fig. 1, where σis the combined statistical and systematic uncertainty for both the experimental data and the projector. The agreement in the region Q3<0.8GeV/chas been evaluated by performing a χ2test. The χ2is calculated combining the nσvalues of each bin. Finally, the p value from the χ2-distribution is computed and the global nσvalues are extracted. The latter amount to 0.167, 0.0006 and 2.75 for (p–p)–p, (p–p)–and p–(p–), respectively. The data-driven method requires the usage of the third particle in the triplet from the mixed-event data sample and consequently the statistical uncertainty depends on the number of events used for mixing, while the projector method does not have this limitation. Thus, the latter significantly reduces the total uncertainty in the evaluation of the two-particle correlation effect on the three-particle correlation functions. For this reason, the projector method is used to calculate the threeparticle cumulants for the p–p–p and p–p–triplets. The total two-particle contribution to the three-particle correlation function is obtained by substituting all terms on the right-hand side of Eq. 11 with the corresponding kinematic transformation, i.e. Ctwo-body(Q3)=C12(Q3)+C23(Q3)+C31(Q3)−2, (14) 123 Eur. Phys. J. A (2023) 59 :145 Page 7 of 22 145 where the indices refer to the label of the correlated pairs. In the case of p–p–p we have Ctwo-body p−p−p(Q3)=3C(p−p)−p(Q3)−2,(15) and in the case of p–p–we have Ctwo-body p−p−(Q3)=C(p−p)−(Q3)+2Cp−(p−)(Q3)−2. (16) The resulting total lower-order contributions to the threeparticle correlation functions (Eqs. 15 and 16) are shown in Fig. 2. The agreement between the data-driven approach and the projector method predictions translate into nσ= 0.167 and nσ= 0.0014 for the p–p–p and p–p–lower-order contributions, respectively. 2.5 Decomposition of the three-particle cumulants The experimental determination of the correlation function is mainly distorted by two distinct impurities in the candidate sample: misidentified particles and feed-down particles originating from weakly decaying particles. This introduces additional contributions to the correlation function of interest. These contributions are either assumed to be flat or, when the interaction is known, they are explicitly considered as discussed in Ref. [39]. The contributions to the correlation function stemming from decaying particles or impurities of the sample are weighted with the so-called λparameters. By adopting this technique the residual correlations can be included in the final description of the experimental correlation function of two particles as C(k∗)=1+λ00(C00(k∗)−1)+ ij=00 λij(Cij(k∗)−1), (17) where the ij = 00 denote all possible impurity and feeddown contributions and the ij =00 is the correctly identified primary particle contribution. These λparameters are obtained employing single particle properties such as the purity and feed-down probability. The underlying mathematical formalism is outlined in Ref. [39]. This mechanism has been extended to the three-particle case and the genuine three-particle cumulants can be obtained by subtracting the impurity and feed-down contributions from the measured cumulants. The full mathematical derivation is presented in Appendix C. The final expression of the genuine three-particle cumulants is c(X0Y0Z0)=1 λX0Y0Z0(XYZ)⎛ ⎝c(XYZ) − i,j,k=(X0Y0Z0) λi,j,k(XYZ)c(XiYjZk)⎞ ⎠, (18) where X,Yand Zrepresent three generic particle species, the index 0 refers to correctly identified primary particle and the indexes i,j,krefer to misidentified or to secondary particles of a generic particle species. As shown in Appendix C, the specific weights λdepend on the purity and feed-down fraction of the single particles and are found to be equal to λX0Y0Z0(ppp)=0.618 and λX0Y0Z0(pp) =0.405 for the p–p–p and p–p–cumulants, respectively. Only 60% (40%) of the (p–p–) triplets correspond to correctly identified primary particles. In the following, the results for the p–p–p cumulants will be corrected according to the evaluated λparameters assuming that all the three-particle contributions stemming from feed-down and impurities are flat in the momentum space. This assumption is supported by the observation that the measured p–p–cumulants are consistent with zero within uncertainties (see Fig. 4and the discussion in Sect. 3). The correction is not applied to the p–p–cumulants because the shape of the feed-down contribution is not known and also because the statistical uncertainties are too large to provide any sensitivity to the three particle correlations. 3 Results The measured three-particle correlation functions for p–p–p and p–p–triplets are shown in Fig. 3on the left and right panels, respectively. The number of events used for mixing for both cases is 30. The total number of same event triplets that are present at the range Q3<0.8 GeV/care 17840 for p–p–p, 10980 for p–p–p, 9191 for p–p–and 5886 for p–p–. The green symbols represent the data points with their statistical and systematic uncertainties, while the grey bands correspond to the lower-order two-body interaction contributions obtained using the projector method already shown in Fig. 2. The non-femtoscopic contributions to the measuredcorrelationfunctions, evaluatedusing Monte Carlo simulations,arefoundtobenegligiblysmall(seeAppendixA for a detailed discussion). In the low Q3region, the measured correlation functions deviatefromthe projectedlowerordercontributionsobtained using only two-particle correlations. The genuine three-body effects are then isolated by evaluating the cumulants c3(Q3)=C(Q3)−Ctwo-body(Q3). (19) The lower-order contribution Ctwo-body(Q3)obtained with the projector method is used. The results for p–p–p and p–p–tripletsare shown in Fig.4onthe left and right panels, respectively. The p–p–p cumulant, already corrected for the feed-down contributions, is negative for 0.16 <Q3<0.22 GeV/c, whilethelargestatisticaluncertaintyinthelowest Q3interval prevents a conclusion on the sign for Q3<0.16 GeV/c.The 123 145 Page 8 of 22 Eur. Phys. J. A (2023) 59 :145 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 )c (GeV/ 3 Q 0 1 2 3 4 5 6 7 ) 3 Q( two-body C ALICE = 13 TeVs pp 0.17% INEL)High Mult. (0 p Data-drivenpp p Projector methodpp ALICE = 13 TeVs pp 0.17% INEL>0) High Mult. (0 Fig. 2 Comparison of the total two-particle contribution to the threeparticle correlation functions obtained using the data-driven approach (green points) and the projector method (grey band). The resulting correlation functions are shown for p–p–p (left panel) and p–p–(right panel). The error bars and the boxes represent the statistical and systematic uncertainties, respectively. The grey band includes systematic and statistical uncertainties summed in quadrature 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 )c (GeV/ 3 Q 0.5 1 1.5 2 2.5 3 3.5 4 4.5 ) 3 Q(C ALICE = 13 TeVs pp 0.17% INEL)High Mult. (0 Datapppppp p Two-particle correlations,pp projector method ALICE = 13 TeVs pp 0.17% INEL>0) High Mult. (0 Fig. 3 Measured p–p–p (left panel) and p–p–(right panel) threeparticle correlation functions. The green points show the experimental results, the error bars and the boxes represent the statistical and systematic uncertainties, respectively. The grey bands represent the expectations for the lower-order two-particle correlations obtained using the projector method and the band width is obtained including systematic and statistical uncertainties summed in quadrature 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 )c (GeV/ 3 Q 6 4 2 0 2 4 ) 3 Q( 3 c p genuine cumulant, flat feed-downpp genuine cumulant, flat feed-downppp ALICE = 13 TeVs pp 0.17% INEL) High Mult. (0 ALICE = 13 TeVs pp 0.17% INEL>0) High Mult. (0 Fig. 4 Three-particle cumulants for p–p–p (left panel, blue square symbols) and p–p–(right panel) triplets obtained by subtracting the lower-order contributions from the measured three-particle correlation functionsshowninFig.3.The p–p–pcumulant inthe leftpanel isfurther corrected for the feed-down contributions from decaying particles and representsthus,thecumulantforthecorrectlyidentifiedprimaryprotons (see Sect. 2.5 for details). The dashed lines correspond to the assumption that there are no genuine three-body correlations c3(Q3)=0. The red open circles in left panel represent the cumulant for p–p–ptriplets (for more details see the main text) agreement between the measured cumulant and the assumption that there are no genuine three-body effects is evaluated using the χ2test in the region Q3<0.4GeV/c, where the two-body interactions are prominent. There is no theoretical or experimental knowledge on the exact Q3range where three-body effects become relevant, however they are expected to contribute at lower or same Q3values as the twobody interactions. For this reason, the region of two-body forces was chosen. The obtained p-value corresponds to 6.7 standard deviations. If the cumulant is obtained using datadrivenmethodto estimatelowerordercontributions,itresults in 6.0 standard deviations. This result hints to the presence of 123 Eur. Phys. J. A (2023) 59 :145 Page 15 of 22 145 References 1. P. Navrátil, S. Quaglioni, G. Hupin, C. Romero-Redondo, A. Calci, Unified ab initio approaches to nuclear structure and reactions. Phys. Scr. 91(5), 053002 (2016) 2. K. Hebeler, J. Holt, J. Menéndez, A. Schwenk, Nuclear forces and their impact on neutron-rich nuclei and neutron-rich matter. Annu. Rev. Nucl. Part. Sci. 65(1), 457–484 (2015). https://doi.org/ 10.1146/annurev-nucl-102313-025446 3. M. 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Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia 123 145 Page 20 of 22 Eur. Phys. J. A (2023) 59 :145 2AGH University of Science and Technology, Cracow, Poland 3Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine 4Department of Physics, Bose Institute, Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India 5California Polytechnic State University, San Luis Obispo, CA, USA 6Central China Normal University, Wuhan, China 7Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana, Cuba 8Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and Mérida, Mexico 9Chicago State University, Chicago, IL, USA 10 China Institute of Atomic Energy, Beijing, China 11 Chungbuk National University, Cheongju, Republic of Korea 12 Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Bratislava, Slovak Republic 13 COMSATS University Islamabad, Islamabad, Pakistan 14 Creighton University, Omaha, NE, USA 15 Department of Physics, Aligarh Muslim University, Aligarh, India 16 Department of Physics, Pusan National University, Pusan, Republic of Korea 17 Department of Physics, Sejong University, Seoul, Republic of Korea 18 Department of Physics, University of California, Berkeley, CA, USA 19 Department of Physics, University of Oslo, Oslo, Norway 20 Department of Physics and Technology, University of Bergen, Bergen, Norway 21 Dipartimento di Fisica, Università di Pavia, Pavia, Italy 22 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy 23 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy 24 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy 25 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Bologna, Italy 26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Catania, Italy 27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN, Padua, Italy 28 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo Collegato INFN, Salerno, Italy 29 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy 30 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy 31 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy 32 European Organization for Nuclear Research (CERN), Geneva, Switzerland 33 Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia 34 Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway 35 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic 36 Faculty of Physics, Sofia University, Sofia, Bulgaria 37 Faculty of Science, P.J. Šafárik University, Kosice, Slovak Republic 38 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 39 Fudan University, Shanghai, China 40 Gangneung-Wonju National University, Gangneung, Republic of Korea 41 Department of Physics, Gauhati University, Guwahati, India 42 Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany 43 Helsinki Institute of Physics (HIP), Helsinki, Finland 44 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico 45 Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania 46 Indian Institute of Technology Bombay (IIT), Mumbai, India 47 Indian Institute of Technology Indore, Indore, India 48 INFN, Laboratori Nazionali di Frascati, Frascati, Italy 49 INFN, Sezione di Bari, Bari, Italy 50 INFN, Sezione di Bologna, Bologna, Italy 51 INFN, Sezione di Cagliari, Cagliari, Italy 52 INFN, Sezione di Catania, Catania, Italy 53 INFN, Sezione di Padova, Padua, Italy 54 INFN, Sezione di Pavia, Pavia, Italy 123 Eur. Phys. J. A (2023) 59 :145 Page 21 of 22 145 55 INFN, Sezione di Torino, Turin, Italy 56 INFN, Sezione di Trieste, Trieste, Italy 57 Inha University, Incheon, Republic of Korea 58 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht University/Nikhef, Utrecht, The Netherlands 59 Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republic 60 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India 61 Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 62 Institute of Space Science (ISS), Bucharest, Romania 63 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt, Frankfurt, Germany 64 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Mexico City, Mexico 65 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil 66 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City, Mexico 67 iThemba LABS, National Research Foundation, Somerset West, South Africa 68 Jeonbuk National University, Jeonju, Republic of Korea 69 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany 70 Korea Institute of Science and Technology Information, Daejeon, Republic of Korea 71 KTO Karatay University, Konya, Turkey 72 Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France 73 Laboratoire de Physique Subatomique et de Cosmologie, CNRS-IN2P3, Université Grenoble-Alpes, Grenoble, France 74 Lawrence Berkeley National Laboratory, Berkeley, CA, USA 75 Lund University Department of Physics, Division of Particle Physics, Lund, Sweden 76 Nagasaki Institute of Applied Science, Nagasaki, Japan 77 Nara Women’s University (NWU), Nara, Japan 78 National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece 79 National Centre for Nuclear Research, Warsaw, Poland 80 National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India 81 National Nuclear Research Center, Baku, Azerbaijan 82 National Research and Innovation Agency - BRIN, Jakarta, Indonesia 83 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark 84 Nikhef, National institute for subatomic physics, Amsterdam, The Netherlands 85 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, UK 86 Nuclear Physics Institute of the Czech Academy of Sciences, Husinec- ˇ Rež, Czech Republic 87 Oak Ridge National Laboratory, Oak Ridge, TN, USA 88 Ohio State University, Columbus, OH, USA 89 Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia 90 Physics Department, Panjab University, Chandigarh, India 91 Physics Department, University of Jammu, Jammu, India 92 Physics Department, University of Rajasthan, Jaipur, India 93 Physics Program and International Institute for Sustainability with Knotted Chiral Meta Matter (SKCM2), Hiroshima University, Hiroshima, Japan 94 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen, Germany 95 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 96 Physik Department, Technische Universität München, Munich, Germany 97 Politecnico di Bari and Sezione INFN, Bari, Italy 98 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany 99 Saga University, Saga, Japan 100 Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India 101 School of Physics and Astronomy, University of Birmingham, Birmingham, UK 102 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del Perú, Lima, Peru 103 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria 104 SUBATECH, IMT Atlantique, CNRS-IN2P3, Nantes Université, Nantes, France 123 145 Page 22 of 22 Eur. Phys. J. A (2023) 59 :145 105 Suranaree University of Technology, Nakhon Ratchasima, Thailand 106 Technical University of Košice, Kosice, Slovak Republic 107 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland 108 The University of Texas at Austin, Austin, TX, US 109 Universidad Autónoma de Sinaloa, Culiacán, Mexico 110 Universidade de São Paulo (USP), São Paulo, Brazil 111 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil 112 Universidade Federal do ABC, Santo Andre, Brazil 113 University of Cape Town, Cape Town, South Africa 114 University of Houston, Houston, TX, USA 115 University of Jyväskylä, Jyvaskyla, Finland 116 University of Kansas, Lawrence, KS, USA 117 University of Liverpool, Liverpool, UK 118 University of Science and Technology of China, Hefei, China 119 University of South-Eastern Norway, Kongsberg, Norway 120 University of Tennessee, Knoxville, TN, USA 121 University of the Witwatersrand, Johannesburg, South Africa 122 University of Tokyo, Tokyo, Japan 123 University of Tsukuba, Tsukuba, Japan 124 University Politehnica of Bucharest, Bucharest, Romania 125 CNRS/IN2P3, LPC, Université Clermont Auvergne, Clermont-Ferrand, France 126 Institut de Physique des 2 Infinis de Lyon, CNRS/IN2P3, Université de Lyon, Lyon, France 127 CNRS, IPHC UMR 7178, Université de Strasbourg, 67000 Strasbourg, France 128 Départment de Physique Nucléaire (DPhN), IRFU, Université Paris-Saclay Centre d’Etudes de Saclay (CEA), Saclay, France 129 Università degli Studi di Foggia, Foggia, Italy 130 Università del Piemonte Orientale, Vercelli, Italy 131 Università di Brescia, Brescia, Italy 132 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India 133 Warsaw University of Technology, Warsaw, Poland 134 Wayne State University, Detroit, MI, USA 135 Westfälische Wilhelms-Universität Münster, Institut für Kernphysik, Münster, Germany 136 Wigner Research Centre for Physics, Budapest, Hungary 137 Yale University, New Haven, CT, USA 138 Yonsei University, Seoul, Republic of Korea 139 Zentrum für Technologie und Transfer (ZTT), Worms, Germany 140 Affiliated with an institute covered by a cooperation agreement with CERN, Geneva, Switzerland 141 Affiliated with an international laboratory covered by a cooperation agreement with CERN, Geneva, Switzerland aAlso at Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA), Bologna, Italy bAlso at Dipartimento DET del Politecnico di Torino, Turin, Italy cAlso at Department of Applied Physics, Aligarh Muslim University, Aligarh, India dAlso at Institute of Theoretical Physics, University of Wroclaw, Wrocław, Poland eAlso at University of Kansas, Lawrence, KS, USA fAlso at An institution covered by a cooperation agreement with CERN, Geneva, Switzerland †Deceased 123