Hadron-hadron interactions measured by ALICE at the LHC
Laura Fabbietti, Valentina Mantovani Sarti, Otón Vázquez Doce
HHadron-hadron interactions measured by ALICE at theLHC
L. Fabbietti, V. Mantovani Sarti, and O.V´azquez Doce Xxxx. Xxx. Xxx. Xxx. YYYY. AA:1–28https://doi.org/10.1146/((please addarticle doi))Copyright © YYYY by Annual Reviews.All rights reserved
Keywords
Strong interaction, correlations, hyperon, kaon, nuclear physics,femtoscopy, lattice QCD, bound state, chiral effective field theory,neutron stars
Abstract
The strong interaction among hadrons has been measured in the pastby scattering experiments. But while this technique was extremely suc-cessful in providing information about the nucleon-nucleon and pion-nucleon interactions, when unstable hadrons are considered the exper-iments become more challenging. In the last few years the analysis ofcorrelations in the momentum space for pairs of stable and unstablehadrons measured in pp and p-Pb collisions by ALICE at the LHCprovided a new method to investigate the strong interaction amonghadrons. In this article we provide a review of the numerous resultsrecently achieved for hyperon–nucleon, hyperon–hyperon and kaon–nucleon pairs showing that the new method opens the possibility ofmeasuring the residual strong interaction of any hadron pair. a r X i v : . [ nu c l - e x ] D ec ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. Determination of the particle emitting source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Probing the strong interaction for strange hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1. Study of the hyperon-nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2. Search for bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3. Coupled channel dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165. Implications for neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1. Introduction
The study of the residual strong interaction between hadrons (colorless bound states ofquarks and anti-quarks) is still an open topic in nuclear physics. While the StandardModel of elementary particle physics provides a satisfactory description at the quark levelin the high-energy regime, the low energy processes, that characterize the interaction amonghadronic degrees of freedom, are not yet described by a fundamental theory and are oftendifficult to access experimentally.The hadron–hadron interactions have been studied in the past by means of scatteringexperiments at low energies (below the nucleon mass) for both stable and unstable beams. Areasonable amount of scattering data (roughly 8000) (1, 2) are available for nucleon–nucleon(NN) reactions but for kaons (mesons with one strange quark) and hyperons (baryons withat least one strange quark) the beam realization is more challenging. Kaon–nucleon inter-actions could be studied because the necessary secondary beams are accessible (3, 4), butfor hyperon–nucleon reactions the data is scarcer (17 data points) (5, 6, 7) because of theunstable nature of the hyperon beams due to the weak and electromagnetic decays. Thelack of statistics in reactions involving unstable hadrons affects the current description ofthe corresponding strong interaction from a theoretical point of view. If we consider onlyhadrons containing u,d and s quarks, most of the predicted interactions are not constrainedexperimentally and this represents not only a limit for nuclear physics but has also implica-tions for astrophysics. Neutron stars (NS), for example, could be constituted from nucleons,hyperons and kaons and their properties strictly depend upon the interactions among thesehadrons (8, 9, 10). Although the dense environment present within neutron stars is noteasy to realize under controlled conditions with terrestrial experiments, the study of two-and three-body interactions among neutrons, protons, hyperons and kaons in vacuum drivethe equation of state (EoS) of NS.In this paper we focus on the novel input provided by the ALICE experiment to thetopic of the interactions among nucleons, hyperons and kaons by means of the femtoscopymethod applied to data from ultra-relativistic pp and p–Pb collisions at the LHC.Historically, the femtoscopy technique can be traced back to the first measurementsof particle interferometry with photons, performed by Hanbury, Brown and Twiss dur-ing the 1950s, used to determine the size of stars (11). The same idea has been sub- equently applied to pairs of identical particles in elementary and heavy-ion collisions(HIC) (12, 13, 14, 15, 16, 17), proving to be an extremely useful tool to determine thespace-time structure of the emitting source. The analysis of pion or kaon pairs, where thequantum statistics together with the Coulomb interaction characterize the shape of thecorrelations, has dominated the femtoscopy studies in the last three decades. Results fromintermediate-energy heavy-ion collisions (HIC) at the Bevalac in the mid 1980s quantita-tively showed that the spatial dynamics of the system was probed and in the following yearsfemtoscopic studies have been performed in several different experiments and energy ranges,from SIS (18, 19), AGS (20, 21), SPS (22, 23), and more recently RHIC (24) to LHC (25) .The plethora of collected data made it possible to study the three-dimensional evolutionof the particle emitting source, helping to characterize the kinematic freeze out of differenthadron species. The typical source sizes measured in HIC ranges from 2 fm for SIS energiesup to 5 − − (31) correlationsmeasured in Au–Au collisions at √ s NN = 200 GeV. These studies showed the limits of themethod applied to heavy-ion collisions where the average inter-particle distances of 7-8 fmreduces the sensitivity to the short-range strong interaction. They also demonstrated thatthe interaction studies require an excellent purity for particle identification and a detailedtreatment of the residual background (32). Following the same approach, the ALICE collab-oration successfully extracted for the first time the Λ–K and baryon–antibaryon scatteringparameters from Pb–Pb collisions at √ s NN = 2 .
76 and 5 .
02 TeV (33, 34). These resultsprovided a first proof of the fact that the correlation function can be exploited to inferinformations on the underlying strong interaction.In the last three years, the ALICE collaboration also applied the femtoscopy technique alsoto pp and p-Pb collisions and showed for the first time the potential to precisely assess thestrong range interaction amongst stable and unstable hadrons. It was demonstrated thatthe average inter-particle distance obtained from such collisions at the LHC is about 1 fmand hence comparable to the range of the strong potentials. This feature, combined withthe excellent particle identification provided by the ALICE apparatus and the large statis-tics collected during the Run 2 data taking, allowed us to precisely measure the followinginteractions: p–p (35), K + –p and K − –p (36), p–Λ (35), p–Σ (37), Λ–Λ(38), p–Ξ − (39),and p–Ω − (40). This review describes the main features of the femtoscopy technique, theadvantages of using it in small colliding systems and the results obtained in the study ofhadron–hadron interactions with strangeness.The paper is structured as follows: the femtoscopy method is presented in Sec. 2. InSec. 3, the modeling of the emitting source in pp collisions is presented and the main featuresof femtoscopy in elementary collisions are discussed. The main results on hadron-hadroninteractions obtained with the ALICE femtoscopy measurements are discussed in Sec. 4with particular emphasis on hyperon-nucleon systems (Sec. 4.1), the possible detection ofbound states (Sec. 4.2) and coupled-channel dynamics (Sec. 4.3). In Sec. 5 the possibleimplications for the presence of hyperons inside neutron stars are considered. Finally inSec. 6 the future prospects for femtoscopy achievements in the next ALICE data-takingperiods (Run 3, Run 4) are discussed and Sec. 7 summarizes the current state of the fieldand points to future theoretical and experimental developments. • Short title 3 . Methodology
The fundamental quantity to be measured in femtoscopy is the correlation function. It isexpressed as a function of the relative distance between two particles (cid:126)r ∗ and its reducedrelative momentum, k ∗ = | (cid:126)p ∗ − (cid:126)p ∗ | /2 in the pair rest frame, with (cid:126)p ∗ = − (cid:126)p ∗ , by theKoonin-Pratt formula (41, 42) : Numerical expressionof k ∗ : k ∗ = (cid:114) a − m m a + m + m , a = ( q inv + m + m )and q inv = | (cid:126)p − (cid:126)p | − | E − E | , with m and m the massof the particles inthe pair and (cid:126)p , theparticle momenta inthe laboratoryreference system. Ifthe two particleshave the same massthen k ∗ = q inv . C ( k ∗ ) = (cid:90) S ( (cid:126)r ∗ ) | ψ ( (cid:126)r ∗ , (cid:126)k ∗ ) | d r. S ( (cid:126)r ), describes the source emitting particles; the second termcontains the interaction part via the two-particle wave function ψ ( (cid:126)r, (cid:126)k ∗ ). The shape of thecorrelation function will be determined from the characteristics of the source function andthe sign and strength of the interaction.An analytical model by Lednick´y and Lyuboshitz (43) exists to compute this correlationfunction. The Lednick´y-Lyuboshitz (LL) model assumes a Gaussian profile that dependsonly on the magnitude of the relative distance for the source function, S ( r ∗ ) = (cid:0) πr (cid:1) − / · exp (cid:18) − r ∗ r (cid:19) , r is the radius parameter that defines the size of the source. The effective rangeapproximation is used to define the complex scattering amplitude as f ( k ∗ ) S = (cid:18) f S + 12 d S k ∗ − ik ∗ (cid:19) − , S the total spin of the particle pair, and f S and d S the scattering length and theeffective range, respectively. The correlation function for uncharged particles becomes then C ( k ∗ ) LL = 1 + (cid:88) S ρ S (cid:34) (cid:12)(cid:12)(cid:12)(cid:12) f ( k ∗ ) S r (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) − d S √ πr (cid:19) ++ 2 (cid:60) f ( k ∗ ) S √ πr F (2 k ∗ r ) − (cid:61) f ( k ∗ ) S r F (2 k ∗ r ) (cid:35) , where F (2 k ∗ r ) and F (2 k ∗ r ) are analytical functions which result from the Gaussiansource approximation, and ρ S is the pair fraction that is emitted into the spin state S.Since the LL approach is based on the effective range expansion it presents limitations forsmall systems, because it does not account for the details of the wave function at smalldistances where the effect of the strong potential is more pronounced.Such limitations motivated the development of the C orrelation A nalysis T ool using the S chr¨odinger equation” (CATS) framework (44), which provides a numerical recipe for thecalculation of the exact solution of the two-body non relativistic S chr¨odinger equationgiven a local interaction potential. The resulting relative wave function combined with aparametrization of the source makes it possible to compute predictions for the differentcorrelation functions by means of Eq. 1. The CATS framework is able to account forboth short range potentials and the Coulomb long range interaction, as well as differentparametrizations of the source function beyond the Gaussian approximation. Moreover theLL model is also implemented in CATS, using the the scattering length f S and the effectiverange d S as inputs for the description of the interaction. he features of the interaction are mapped into the corresponding correlation function.In particular, the effects of the final-state interactions are more evident in the correlationfunction at small k ∗ values. A repulsive interaction, with positive values of the local po-tentials will imply a correlation function with values between 0 and 1. For an attractiveinteraction instead, the resulting correlation function gets values above unity. This intuitivepicture is modified, though, if the attraction is strong enough to accommodate the presenceof a bound state. In this case a depletion in the values of the correlation function can beseen, depending on the binding energy.The strength of the correlation can also be enhanced by small sizes of the source func-tion, as will be discussed. Other effects, not caused by the final-state interaction, can bevisible at different k ∗ ranges of the correlation function, like quantum-mechanical interfer-ence, resonances, or conservation laws.Figure 1 demonstrates the sensitivity of the femtoscopy method applied to small collidingsystems for the study of the strong interaction. The left upper panel provides the exampleof attractive (orange) and repulsive (azure) potentials and a potential with a shallow boundstate (pink). The left lower panel shows the corresponding squared modulus of the wave- r (fm) - V (r) ( M e V ) RepulsiveAttractivewith a bound state r (fm) (r) | k Y | (r)| k* Y | S(r) r p S (r) ( /f m ) r p C ( k * ) = 1 fm r = 4 fm r Figure 1 (Color online) . Left upper panel: Examples of an attractive (orange), repulsive (azure) and apotential with a shallow bound state (pink). Left lower panel: Modulus squared of the totalwave-functions, as a function of the relative distance r, for each considered interaction. On thesame plot the profile of the emitting source is shown for 1 fm (solid black) and 4 fm (dashedblack). Right panel: resulting correlation function C( k ∗ ) for each interaction, evaluated for thetwo different source sizes r = 1 , function, | ψ ( (cid:126)r, (cid:126)k ∗ ) | , obtained by solving the Schr¨odinger equation employing the CATSframework for the three exemplary potentials. Together with the square of the relativewave functions, the density distribution according to Gaussian profiles with two differentradii ( r = 1 and 4 fm) are provided. The sensitivity of the method to study the stronginteraction depends on the overlap of the square of the wave function with the source den-sity distribution. Typical sizes of the parameter r = 3–6 fm describe the source formedin Pb-Pb collisions at the LHC (45), while pp and p-Pb collisions at the same energieslead to the formation of sources with much smaller radii between 1–1.5 fm (35, 39). Onecan see that for heavy-ion collisions the overlap is minimal and hence the sensitivity to the • Short title 5 hort range interaction is very limited. The typical features of the attractive and repulsiveinteractions and the presence of the bound state are much more pronounced in the case ofthe small source size. The less pronounced correlation function obtained with the largersource is very difficult to measure with sufficient precision.In the case of the bound state, the reason why the correlation function flips around 1 for dif-ferent source sizes is due to the fact that the wave function is very sharply peaked towardsdistances equal to zero, due to the much stronger localization of the bound state. Thistranslates into an increased correlation for small radii, while for large radii only the asymp-totic part of the wave function, which is depleted because of conservation of probability,impacts the correlation function and bring it below one.The method used by ALICE to study the interactions among hadrons consists in thecomparison of the theoretical expectation for the correlation function to a correlation func-tion obtained experimentally. The experimental correlation function is obtained by com-paring the relative momentum distribution of pairs of particles produced in the same event(SE), that constitutes the sample of correlated pairs, with a reference distribution obtainedcombining particles produced in different collisions using the so-called mixed event (ME)technique C ( k ∗ ) = ξ ( k ∗ ) · N SE ( k ∗ ) N ME ( k ∗ ) . ξ ( k ∗ ) in Eq. 4. Such correctionstake into account the finite experimental resolution, and corrections to the ME distributionsin order to assure the same experimental conditions and normalization as for the SE events.In general, they do not account for the contributions from misidentification, weak decaysor residual background induced by mini-jets and event-by-event momentum conservation.These effects are accounted for in the fit of the correlation functions.The experimental correlation function is further distorted by two distinct mechanisms.The sample of particle pairs can include both misidentified particles and feed-down particlesfrom weak decays of resonances. This introduces contributions of different, non-genuine,pairs in the measured correlation function. The treatment of these contributions is describedin detail in (35), and therefore here we only briefly sketch the procedure. The contributionsof the different non-genuine and genuine correlation to the total experimental correlationare indicated by weights called λ parameters. These λ parameters are obtained by pairingsingle particle properties such as the purity (P) and feed-down fractions (f): λ ij = P i P j f i f j .The total correlation function can then be decomposed as: C ( k ∗ ) = 1 + λ genuine · ( C genuine ( k ∗ ) −
1) + (cid:88) ij λ ij ( C ij ( k ∗ ) − , i, j denote all possible impurity and feed-down contributions. Primary andsecondary particles:
The fraction f i ofprimary andsecondary (i.e.produced in a decay)particles enteringthe λ ij parametersare evaluated fromexperimental dataconsidering weak,electromagneticdecays and fakecandidates for eachhadron species. The correlation functions measured by ALICE are compared with theoretical expec-tations obtained according to Eq. 5. For this task the λ parameters are obtained fromexperimental data when possible (e.g.: purity, fractions of secondary particles). Also theexperimental effects denoted by ξ ( k ∗ ) in Eq. 4 are taken into account when modelling thetheoretical correlation function. The only exception is the study of the p–Ξ and p–Ω − cor-relation functions in pp collisions at 13 TeV published in (40), where the experimental datahave been unfolded for all effects and it is directly compared with the genuine theoreticalcorrelation function.In order to account for residual contributions by mini-jets background to the final correla-tions, a baseline with free parameter is multiplied to the correlation function C ( k ∗ ) used o fit the experimental data (35). This baseline assumes different shapes depending on thepair of interest, but contributes with at most few percent to the global correlation strength(39, 37).
3. Determination of the particle emitting source
After the collision and after the hadronization processes are completed, particles mightundergo some inelastic collisions but shortly after their production they propagate freelytowards the detectors. The distribution of the space coordinates at which the differentparticles assume their primary momentum values characterize the particle emitting source.The understanding of this source for the selected colliding system is mandatory in order toextract informations on the underlying strong interaction.Gaussian source profiles in one and three dimensions are typically assumed in fem-toscopic studies performed in heavy-ion collisions (42, 46). However, the presence of acollective expansion can introduce correlations between the position and the momentum ofthe emitted particles. This effect can be seen as a decrease of the extracted gaussian radiiwith increasing pair transverse momentum k T (42, 47). Experimentally, a common scalingof the source size with the transverse mass of protons and kaons pairs has been seen inPb–Pb collisions (48).High-multiplicity pp collisions and heavy-ion systems already showed similar behaviorin several related measured quantities, such as angular correlations and strangeness pro-duction (49, 50, 51, 52). Hence, a similar transverse mass m T scaling of the source size as Transverse mass:
The transverse massof the pair is definedas m T = (cid:112) k T 2 + m ,where m is theaverage mass of theparticle pair and k T = | (cid:126)p T , + (cid:126)p T , | is the relativetransversemomentum. observed in large systems is expected to occur in pp collisions. Measurements of this kind insmall systems are currently available for light meson pairs ( π – π , K–K) (53, 54, 55, 56, 57),accessing only low values of m T but indicating already a dependence of the radius on thetransverse mass.Recently, similar studies of baryon-baryon femtoscopy, for p–p and p–Λ pairs, have beenconducted in high-multiplicity pp collisions and these studies provided for the first time aquantitative measurement of a common scaling in the range m T = 1 . − . / c (58).The explicit inclusion of strongly decaying resonances proved to be a fundamental ingredientfor the description of the data. The presence of feed-down from strong resonances hadalready been suggested as a possible explanation for the description of the different scalingof radii extracted in π – π correlations seen in heavy-ion collisions (59, 60). A similar brokenscaling is observed in pp collisions for p–Λ pairs when a Gaussian source profile is assumedand feed-down effects are not taken into account (58). The extracted p–Λ radii are typically20% larger with respect to the p–p pair results. In the spirit of testing the hypothesisof a common source for small colliding systems, a complete modeling of the resonancecontributions has to be performed.The emitting source S ( r ∗ ) used to fit p–p and p–Λ correlation functions with Eq. 1is composed of a Gaussian core of width r core (see Eq. 2), related to the emission of allprimordial particles S prim ( (cid:126)r ∗ core ) = (cid:0) πr (cid:1) − / · exp (cid:18) − (cid:126)r ∗ r (cid:19) , τ res . • Short title 7 he modification of the relative distance (cid:126)r ∗ of the particle in the pair, entering the finaldescription of the source, linearly depends on both the core distance (cid:126)r ∗ core and on thedistances (cid:126)r ∗ res , i traveled by the resonances i = 1 , (cid:126)p ∗ res , i , mass M res , i andflight time t res , i sampled from the exponential distribution based on the correspondinglifetime τ res , i : (cid:126)r ∗ = (cid:126)r ∗ core + (cid:88) i (cid:126)r ∗ res , i , (cid:126)r ∗ res , i = (cid:126)p ∗ res , i M res , i t res , i . r ∗ = | (cid:126)r ∗ | needs to be evaluated forthe one-dimensional source function S ( r ∗ ) . From the definitions in Eq. 7, the neededingredients are r ∗ core , the momenta, masses and lifetimes of the resonances, as well as theangles formed by (cid:126)r ∗ core , and the resonance distances (cid:126)s ∗ res , and (cid:126)s ∗ res , . c * (MeV/ k * ) k ( C p - p Å p - p (fit) n Coulomb + Argonne p - p Å p - p (fit) n Coulomb + Argonne
100 200 300 ) c * (MeV/ k * ) k ( C = 13 TeV s ALICE pp 0) > % INEL 0.17 - High-mult. (0 c (GeV/ 〉 T m 〈 ( f m ) c o r e r = 13 TeV s ALICE pp 0)>% INEL0.17 − High-mult. (0Gaussian + Resonance Sourcep − p (NLO) Λ− p (LO) Λ− p Figure 2 (Color online) . In both plots: statistical (bars) and systematic uncertainties (boxes) are shownseparately. Left: m T integrated p–p correlation function as a function of k ∗ measured inhigh-multiplicity pp collisions, including the contributions from strong resonances. The width ofthe band (green) represents one standard standard deviation of the systematic uncertainty of thefit. Right: Gaussian core radius r core as a function of (cid:104) m T (cid:105) . Blue crosses correspond to thep–p correlation function fitted with Argonne v (61) as the strong potential. The green squaredcrosses (red diagonal crosses) result from fitting the p–Λ correlation functions with the strong χ EFT LO (62) (NLO (63))
The amount and type of resonances can be estimated from calculations based on thestatistical hadronization model (64, 65, 66) and are found to be similar ( ≈ ∗ for Λ . This differentcomposition of secondary particles translates into a significantly larger average lifetimeand average mass ( M res = 1 .
46 GeV / c , cτ res = 4 . M res = 1 .
36 GeV / c , cτ res = 1 . (cid:126)r ∗ core , are determined fromtransport model simulations within the EPOS framework (67).The total source can be finally decomposed, depending on the origin of each particle inthe pair (either primary or from a resonance), as follows: ( r ∗ ) = P prim P prim × S prim − prim ( r ∗ ) + P prim P res × S prim − res ( r ∗ )+ P res P prim × S res − prim ( r ∗ ) + P res P res × S res − res ( r ∗ ) . P prim and P res are the fractions of primordial and resonance contributions esti-mated from thermal model calculations. Once all the resonance dynamics and compositionfor the considered pair are accounted for, the Gaussian source size r core , related to theprompt emission of particles, remains as unique free parameter to be determined via a fitto the data.A differential m T analysis has been performed on p–p and p–Λ correlations so thatthe core radius has been extracted in each m T bin. In the left plot of Fig. 2, the result-ing m T integrated p–p correlation function is shown, obtained assuming the core-resonancesource model. The genuine p–p term of the correlation is modeled using the CATS frame-work, assuming the Argonne v (61) as the strong potential (including S,P and D waves),and including the Coulomb interaction along with the proper quantum statistics anti-symmetrization of the wave-function. The underlying strong interaction between protons isknown with high precision and it is accurately described by the Argonne v potential (61),allowing for a reliable determination of the r core parameter. The data are nicely reproducedby the modeled correlation and the same fitting procedure has been adopted in the single m T bins, leading to similar results. The p–Λ interaction is less constrained (35, 68, 7, 6, 5),hence both leading order (LO) (62) and next-to-leading order (NLO) (63) chiral effectivefield theory calculations ( χ EFT) have been considered.In the right plot of Fig. 2 the m T dependence of the extracted core radii for the two pairs ispresented. As can be clearly seen, the inclusion of resonances in the modeling of the sourceprovides a common m T scaling for both baryon-baryon pairs, providing the first quanti-tative evidence of a common emitting source in small systems. This results represents afundamental input in the investigation of the strong interaction by means of femtoscopysince it allows to fix the source for any baryon-baryon pair, given the (cid:104) m T (cid:105) of the pair andthe resonance contributions.If the source S ( r ∗ ) is under control from the analysis of particle species for which thefinal state interaction is known, then the relative wave function ψ ( (cid:126)r, (cid:126)k ∗ ) and hence theinteraction for other species can be determined by studying the correlation function (seeEq. 1). Also, the small values found for the core radius r core proved that pp collisions atthe LHC are excellent systems to study to short range strong interaction.
4. Probing the strong interaction for strange hadrons
The interaction between strange hadrons and nucleons is not very well constrained byexperimental data. In particular, the unstable nature of hyperons makes it very complicatedto investigate two- and three-body interactions. The high statistics that have been collectedfor all hyperon species in pp and p-Pb collisions measured by the ALICE collaborationduring Run 1 and Run 2 at the LHC allowed unprecedented precision in the study ofdifferent interactions, including combinations that had been studied already in past bymeans of scattering experiments (p–K + , − and p–Λ) (69, 70, 71, 72, 7, 6, 5, 68) or werenever measured so far (p–Ξ − and p–Ω − ). • Short title 9 he identification, tracking and momentum resolution provided by ALICE for allcharged particles makes it possible to study correlation functions down to relative momentaof 4 −
10 MeV/ c . A precise measurement of the correlation function in this momentumrange is necessary to study the details of the strong interaction. Moreover the large amountof hyperons, including species such as Ξ and Ω, makes it possible to measure hadron pairsnot accessible in the standard scattering experiments. The measurement of these hyperonsalso allow to test predictions from lattice QCD for interactions with nucleon, since for suchheavy hyperons the calculation results are rather solid (73).The femtoscopic measurements performed in small colliding systems such as pp and p-Pbgrant access to the short-range strong interaction, as already shown in Sec. 2. In the fol-lowing sections we will discuss in detail different features such as coupled-channel effectsand formation of bound states, arising from the short-range dynamics of hadron-hadronpotentials. One of the challenging measurements achieved applying the femtoscopy technique to ppand p-Pb collisions and interpreting the observables with the help of the CATS frameworkis the study of the p–Ξ − interaction. The Ξ ± hyperons are reconstructed exploiting theweak decays Ξ ± → Λ + π ± and Λ → p + π − . A total invariant mass resolution below 2MeV/ c (74) is obtained for the reconstructed Ξ ± and the obtained p–Ξ − ⊕ ¯p–Ξ + correlationfunction is shown in the left panel of Fig. 3. The data have been corrected for experimentaleffects, the statistical and systematic errors are displayed by the black lines and grey boxes,respectively, and the green histogram corresponds to the prediction obtained consideringonly the Coulomb interaction. The right panel of Fig. 3 shows the strong potentials pre-dicted by the HAL QCD collaboration for the four allowed spin and isospin states of thep-Ξ − system. One can see that for all cases an attractive interaction and a repulsive corecharacterize the potentials. c * (MeV/ k * ) k ( C data ALICE
Coulomb HAL QCD - Ξ Coulomb + p- - Ω Coulomb + p- - Ω Coulomb + p-data
ALICE
Coulomb HAL QCD - Ξ Coulomb + p- r - - - ( M e V ) V (r) c * (MeV/ k * ) k ( C = 1.4 fm r I = 0, S = 0I = 0, S = 1I = 1, S = 0I = 1, S = 1
Figure 3 (Color online)
Left panel: pΞ − correlation function measured in pp collisions at √ s NN = 13TeV (40) recorded with a high multiplicity trigger. The experimental data are shown by the blacksymbols together with statistical and systematic errors. The green curve represents the predictedcorrelation function assuming only the Coulomb interaction. The pink curve shows the predictionobtained considering the Coulomb and strong interaction provided by the HAL QCD group (73).Right panel: strong potentials for the different spin and isospin configuration of the pΞ − interaction as a function of the inter-particle distance (39). All the potentials are similar at inter-particle distances above 1 .
10 Author et al. hows that the corresponding correlation functions, that have been evaluated considering aGaussian source with a radius equal to r = 1 . − correlation function shown in the left panel of Fig. 3 isobtained including the strong and the Coulomb potentials in the Schr¨odinger equation andcombining the correlation functions for each of the allowed spin and isospin states weightedby the proper Clebsch-Gordon coefficients, following: C p–Ξ − = 18 C ( I =0 , S =0) + 38 C ( I =0 , S =1) +18 C ( I =1 , S =0) + 38 C ( I =1 , S =1) . − pair has been evaluated following the model described inSec. 3, considering the average m T of the pair and the strong resonance contribution forthe protons, leading to a value of r = 1 . ± .
05 fm. The total p–Ξ − correlation forthe Coulomb and HAL QCD strong interaction shown in the left panel of Fig. 3 lies abovethe Coulomb predictions demonstrating the presence of an additional attractive stronginteraction. These data provide a reference that can now be employed to test any theoreticalcalculation of the p–Ξ − interaction.The p–Ξ − system presents two inelastic channels, n–Ξ and Λ–Λ, just below thresholdand three others, Λ–Σ , Σ –Σ and Σ + –Σ − , well above threshold (75). The latter, sincetheir opening occurs far away from the p–Ξ − mass threshold and theoretical predictionsindicate a shallow interaction for these pairs, will have a negligible effect on the p–Ξ − corre-lation function. In the specific case of the Λ–Λ channel, precise femtoscopic measurementsconfirmed the weak strength of the strong interaction for these hyperons by means of hy-pernuclei data (76). Predictions based on chiral calculations for the n–Ξ channel showa visible effect in the p–Ξ − correlation signal (75) but currently this coupling is not yetpresent in lattice QCD calculations. In the calculation carried out to obtain the theoreticalcorrelation function shown by the pink histogram in the top panel of Fig. 3, the non diago-nal terms of the interactions that contain the contribution of coupled-channels (see section4.3) are neglected. Nevertheless, a direct measurement of these inelastic contributions isnecessary to draw solid conclusions. The Λ–Λ interaction attracted the attention of both theoreticians and experimentalists al-ready many years ago (77) because of the possible existence of the H-dibaryon: a boundstate composed of six quarks (uuddss). From an experimental point of view, the Λ–Λinteraction was addressed studying first the production of double-Λ hypernuclei. The mea-surement of the binding energy (BE) of the hypernucleus
He allowed the estimation ofthe Λ–Λ BE = 6 . ± . → Λp π were carried out (78), but theynever delivered any evidence. A more recent upper limit evaluation of the bound state BEwas obtained from a correlation analysis (38).It was first proposed to study such an interaction by analyzing heavy-ion collisiondata (79) and the first attempt to investigate the Λ–Λ final state via correlations was carriedout by the STAR collaboration in Au–Au collisions at √ s NN = 200 GeV (80). This first • Short title 11 nalysis delivered a scattering length and an effective range of f − = − . ± . +0 . − . fm − and d = 8 . ± . +2 . − . fm, and these values correspond to a repulsive interaction. How-ever, it was shown that the values and the sign of the scattering parameters strongly dependon the treatment of feed-down contributions from weak decays to the measured correlation.A re-analysis of the data outside the STAR collaboration extracted a positive value for f − corresponding to a shallow attractive interaction potential (32). The Λ–Λ correlations - - ) - (fm ( f m ) d - f ALICE = 7 TeV s pp = 13 TeV s pp = 5.02 TeV NN s Pb - p pairs L-L Å L-L < 1 s n < 2 s n s n s n C(k*) Unphys. STARHAL QCDHKMYYFGNDNFNSC89NSC97EhimeESC08fss2 ) c * (MeV/ k * ) k ( C data ALICE
CoulombCoulomb + HAL-QCD elasticCoulomb + HAL-QCD elastic + inelastic - W p- ) c * (MeV/ k
100 150 200 250 * ) k ( C Figure 4 (Color online)
Right panel: Exclusion plot of the scattering parameters for the Λ–Λ interactionevaluated by testing the different values against the Λ–Λ correlation. Left panel: Correlationfunction of p–Ω − pairs measured by ALICE in high multiplicity pp collisions at √ s = 13 TeV (40). The data are shown by the black symbols, the systematic errors are shown bythe grey boxes. The green line represents the expected correlation function by taking into accountonly the Coulomb interaction, its width is determined by the uncertainty in the source radius.The blue and orange bands consider both Coulomb and strong interaction by the HAL QCDcollaboration (81). The orange band considers for the strong interaction only the elasticcontributions, the blue band considers elastic and inelastic contributions, its width represents theuncertainties associated with the lattice QCD calculations, and the grey band represents, inaddition, the uncertainties associated with the determination of the source radius. The sourceradius, determined experimentally, is 0 . ± .
06 fm. The inset shows in detail the correlationfunction around unity. For more details see text. measured in pp and p-Pb collisions by ALICE at √ s NN = 7 ,
13 TeV and 5 .
02 TeV, respec-tively (35, 38) were also employed to study the interaction, and the residual correlationswere treated by means of a novel data-driven method. Since the statistics of the Λ–Λ pairswith small relative momentum was limited, instead of extracting the scattering parametersfrom the fit of the correlation function a different approach was carried out (38). A scanof different combinations of scattering parameters ( f − , d ) in the range f − ∈ [ − , − and d ∈ [0 ,
18] fm was performed. For each combination of values of the scatteringparameters the correlation function is evaluated for several meson-exchange models of theΛ–Λ interaction by using the Lednick´y-Lyuboshitz (LL) method. The agreement with theexperimental correlation function, using all data samples from pp collisions at √ s = 7 , √ s NN = 5 .
02 TeV, is quantified in terms of a confidence level fol-lowing the method in (82). The CATS framework is used to cross check the results from theLL method; the differences in the correlation functions obtained using the two methods arenegligible. Also the gaussian source approximation, inherent to the LL method, is validatedby cross checks using the source profile predicted by the EPOS transport model (67, 44)and considering the effects of short lived resonances. The results, expressed in number ofstandard deviations ( n σ ) are shown in the left panel of Fig. 4. The black hatched area rep-
12 Author et al. trong Strong + CoulombSekihara et al. 0 . . .
54 MeV 2 .
46 MeV
Table 1 Predicted binding energies for the p– Ω − system by references (90) (Seki-hara et al.) and (81) (Iritani et al.) considering the strong and strong + Coulombinteractions. resents the values for which the LL model breaks down for the small source sizes consideredand delivers unphysical correlation functions.This analysis has made it possible to extend the constraint to the scattering parametersand the BE of the Λ–Λ system. The data is compared with models predicting either a strongattractive interaction (83), a Λ–Λ bound state (84, 85), or a shallow attractive interactionpotential (86, 87, 88, 89). Through the comparison shown in Fig. 4 one can see that thedata favors a shallow attractive interaction, being compatible in particular with the (88)and (87) models that are in agreement with hypernuclei data, and with the model (89),that consists of preliminary lattice QCD calculations by the HAL QCD collaboration. Thedata excludes the region corresponding to a strongly attractive or a very weakly bindingshort-range (small f − and small d ) interaction, and also the first results by STAR (80)corresponding to an attractive interaction are excluded. The data does not exclude a Λ–Λbound state with a shallow binding (corresponding to negative f − and small d values).The observables presented in the left panel of Fig. 4 can be related within the effective-range approximation to a corresponding binding energy via:BE ΛΛ = 1 m Λ d (cid:18) − (cid:113) d f − (cid:19) . σ regioncompatible with the existence of a bound state of the results shown in the left panel ofFig. 4 was performed and allows a binding energy, considering statistical and systematicuncertainties, of BE = 3.2 +1 . − . (stat) +1 . − . (syst) MeV (38).An additional final state suited for the search for a baryon-baryon bound state is thep–Ω − channel. Recent studies from phenomenological approaches (90) and first principlecalculations (81) predict an attractive interaction potential at all distances between protonsand Ω − baryons. Both approaches also predict the existence of a p–Ω − bound state withbinding energies of the order of a few MeV, as is summarized in Table 1. As alreadypointed out in Section 2, the presence of a bound state manifests itself in a depletion of thecorrelation function with a strength that depends on the binding energy and the shape ofthe attractive potential. In the presence of shallow bound states the correlation looks verysimilar as in the case of a strongly attractive interaction, but for more deeply bound statethe correlation can also sink below the unity.The p–Ω − correlation function has been recently studied by ALICE (40) using datafrom high-multiplicity pp collisions at 13 TeV and the results are shown in Fig. 4. Thedata, shown by the black points, is corrected for feed-down contributions and experimentaleffects, such as resolution effects at very small k ∗ values, meaning that the data can bedirectly compared to any theoretical prediction given a known emitting source. • Short title 13 he data in Fig. 4 is compared with the predicted correlation function from calculationson the lattice by the HAL QCD collaboration (81) for a Gaussian source with a radius r =0 . ± .
06 fm. The source characteristics have been determined following the methodexplained in Sec. 3, for a (cid:104) m T (cid:105) of the p–Ω − pairs of 2 . / c , and taking into accountthe effect produced by short lived resonances. The difference between the blue and orangecolored bands corresponding to the HAL QCD prediction in Fig. 4 reflect the currentuncertainty of the calculations due to the presence of strangeness-rearrangement processesin the p–Ω − channel.For the p–Ω − S-wave interaction, the total angular momentum J can take on valuesof J = 2 or J = 1. Processes such as pΩ − → ΞΛ , ΞΣ can occur (91), affecting thep–Ω − interaction in particular in the J = 1 channel. For the J = 2 channel, the presenceof strangeness-rearrangement processes should be strongly suppressed, since they are onlypossible through D-wave interaction processes. This J = 2 channel is, so far, the onlychannel calculated by the HAL QCD Collaboration (81). In order to compare the latticeQCD calculations with the ALICE data, two extreme assumptions are made for the de-scription of the interaction in the J = 1 channel, following the recipe explained in (91): i)no strangeness-rearrangement processes occur, and the shape of the J = 1 channel showsan attraction analogous to the J = 2 channel as calculated by HAL QCD; ii) the J = 1channel is completely dominated by by strangeness-rearrangement processes, i.e. a com-plete absorption is assumed for this channel. The correlation functions resulting from theassumptions i) and ii) are represented by the orange and blue lines in Fig. 4, respectively.For both predictions the Coulomb interaction is also taken into account, and the coloredwidths of the curves represent the intrinsic uncertainties of the lattice QCD calculations,with the grey curves showing, in addition, the uncertainties related to the experimentaldetermination of the source radius. Clearly, the most attractive solution is preferred bythe data, although the calculations underpredict the ALICE results at all k ∗ values. In theabsence of measurements of the pΩ − → ΞΛ , ΞΣ cross sections, future studies of Λ –Ξ − and Σ –Ξ − correlations will help reducing the uncertainties in the expectations from thetheory by pinning down the contributions of the inelastic channels.An evident depletion is present in the lattice QCD predictions shown in Fig. 4. Bylooking in particular to the region k ∗ ∈ [100,200] MeV/ c , one can see that the correlationfunction reaches values below the Coulomb-only prediction. Such depletion, not confirmedby the present experimental data, is due to the presence of the p–Ω − di-baryon state.The strength of the depletion depends on: i) the characteristics of the interaction; ii)the binding energy of the p–Ω − state; iii) the size of the particle emitting source. Thisdependence has been studied in detail in (91) through the study of the interplay betweenthe scattering length associated to the p–Ω − interaction and the corresponding correlationfunction obtained for different source sizes.The ALICE data shown in Fig. 4 do not follow the depletion predicted by the latticeQCD calculations. In order to obtain firm conclusions on the possible existence of the p–Ω − state, and, if existent, experimentally quantify its binding energy, a differential analysis ofthe p–Ω − correlations in systems with slightly different source sizes is necessary. This canbe done at the LHC by ALICE by studying p–Pb and peripheral Pb–Pb collisions.In order to show more clearly the effect of a possible bound state in the correlationfunction, it is useful to compare several local potentials describing the p–Ω − J = 2 inter-action that are associated either to p–Ω − bound states with different properties or do notpredict any bound state. We make use here of the potentials presented in (92), and labelled
14 Author et al. * (MeV/c) C ( k * ) HAL QCD - W- p I V - W- p II V - W- p III V - W- p r (fm) V (r) ( M e V ) - - - - - HAL QCD - W- p I V - W- p II V - W- p III V - W- p Figure 5
Left panel: Comparison of the strong interaction potentials for p–Ω − from references (92)(dashed lines) and (91) (orange solid line). The potentials , V II (red) and V III (blue) imply ap–Ω − bound state with binding energies due to strong interactions of 0.05 and 24.8 MeV. Nobound state is associated with the V I (green) potential. The orange solid line represents the HALQCD potential with nearly physical quark masses (91) predicting a binding energy of 1.54 MeV.Right panel: Correlation function for p–Ω − pairs corresponding to the potentials shown in the leftpanel for a radius of 0.95 fm. Strong Strong + Coulomb V I – – V II .
05 MeV 0 .
63 MeV V III . . Table 2 Predicted binding energies for the V I , V II and V III potentials for the p– Ω − J = 2 channel from (92). as V I , V II and V III . These potentials are based on dated calculations by the HAL QCDcollaborations with non-physical quark masses (93) ( m π = 875 MeV, m K = 916 MeV). Forthe construction of the V II potential, the lattice QCD data in (93) is fitted by an attrac-tive Gaussian core plus and attractive Yukawa tail, while for the V I and V III potentials therange-parameter at long distance of the fit is varied in order to obtain a weaker and strongerattraction, respectively. The radial shape of such potentials compared with the most recentHAL QCD potential with physical quark masses (91) ( m π = 146 MeV, m K = 525 MeV)can be seen in the left panel of Fig. 5, while the predicted binding energy for each case islisted in table 2.The right panel of Fig. 5 displays the corresponding correlation functions for a sourceof r = 0 . ± .
06 fm. One can compare the limiting cases of a deeply bound p–Ω − boundstate with a binding energy of 24.8 MeV ( V III potential, blue curves) with the case of avery shallow binding energy of 0.05 MeV ( V II potential, red curves) or no bound state ( V I potential, green curves). Although the V I potential is much less attractive than the V III at all distances, the correlation function of the former is higher due to the deep depletioncaused in the latter by the presence of a deeply bound state. Also the comparison betweenthe V II and V III curves shows the same effect, since the shallow binding energy of the V II potential is reflected into a much shallower depletion as well. The correlation functioncorresponding to the most recent HAL QCD potential with physical quark masses (91) isshown by the orange curve. The binding energy of a few MeV is reflected in a correlation • Short title 15 unction that is lower than the one for the V II potential, despite the fact that the latter isless attractive for distances r > . V I , V II and V III potentials for source sizesranging from 2 to 5 fm have been compared with data from ultrarelativistic Au-Au collisionsat a center-of-mass energy of 200 GeV per nucleon pair by the STAR collaboration (31).The combination of a low purity and statistical significance of the data with such a largesystem size reduces the sensitivity of the comparison, as discussed at the end of section 2.The ratio of the correlation function for p–Ω − pairs in peripheral collisions (centralitiesof 40-80%) to the one in central collisions (centralities of 0-40%) at a k ∗ = 20 MeV/ c iscompatible within 1 σ with the V III , and within 3 σ with the V I and V II potentials for anexpanding source. Coupled-channel processes are widely present in hadron-hadron interactions whenever pairsof particles, relatively close in mass, share the same quantum numbers: baryonic chargeB, electric charge Q and strangeness S. The coupling translates into on/off shell processesfrom one system to the other.Whenever present, the multi-channel dynamics deeply affect the hadron-hadron inter-action and is at the origin of several phenomena, such as bound states and resonances,which crucially depend on the coupling between these inelastic channels. A striking exam-ple can be found in the origin of the Λ(1405), a molecular state arising from the coupling ofantikaon-nucleon ( ¯K–N) to Σ– π (94, 95). In the baryon-baryon sector, the coupling betweenN–Λ and N–Σ is of great importance in providing the repulsive behaviour of Λ hyperons indense nuclear matter (63).Since in femtoscopic measurements the final state is fixed (the measured particle pair),the corresponding correlation function represents an inclusive quantity able to show sensi-tivity to all the available initial inelastic channels produced in the collision (75, 96).The effects of coupled-channels on the final measured correlation function depends ontwo main ingredients: the coupling constant strength, stemming from the strong multi-channel dynamics, and the conversion weights, namely the amount of pairs in the corre-sponding channel produced close enough to convert into the final measured state. The cor-relation function in Eq. 1 needs to be modified and, for a system with N coupled-channels,this observable in the i channel that is measured reads (43, 75, 96) C i ( k ∗ ) = (cid:90) d r ∗ S i ( r ∗ ) | ψ i ( k ∗ i , r ∗ ) | + N (cid:88) j (cid:54) = i w j (cid:90) d r ∗ S j ( r ∗ ) | ψ j ( k ∗ j , r ∗ ) | . S i ( r ∗ ) and the one for the incoming inelasticchannels, S j ( r ∗ ), might be different (since the m T distribution of the different pairs candiffer) but the results presented in Sec. 3 and the proximity in mass amongst the differentchannels quantitatively prove that the equality S i ( r ∗ ) = S j ( r ∗ ) can be assumed.The first integral on the right-hand side describes the elastic contribution where initialand final state coincide, while the second integral is responsible for the remaining inelastic
16 Author et al. rocesses j → i . The last integral depends on two main ingredients: the wave function ψ j ( k ∗ j , r ∗ ) for channel j going to the final state i and the conversion weights w j . Thelatter are directly related to the amount of pairs, for each inelastic channel, produced inthe initial collision which are kinematically available to convert into the final measuredstate. Estimates for these weights can be obtained using information on yields from statis-tical hadronization models (64, 65, 66) and on the kinematics of the produced pairs fromtransport models (67).As can be seen from Eq. (11), the correlation function involves contributions fromboth elastic and inelastic components, and in principle, the scattering amplitude of thesingle-channel i cannot be fully isolated from the inelastic contributions. Depending on thecoupling strength, the coupled-channel contributions j modify the C( k ∗ ) in two differentways, whether their opening (the minimum energy at which they can be produced) occursbelow or above the production threshold of the considered pair (the reduced mass of thepair). Inelastic channels opening below threshold do not introduce any shape modificationto the C( k ∗ ), they just act as an effective attraction, increasing the signal strength of thecorrelation function. Channels appearing above threshold instead lead to a modification ofthe k ∗ dependence of the C( k ∗ ) in the vicinity of the opening, which is typically translatedinto a cusp structure, whose height is driven by the coupling strength. coupled-channels p K " (a) (b) 𝜓 𝒌 ,𝒓 K " - p + + 𝜓 𝒌 ,𝒓 K )
0- n 𝜓 𝒌 ,𝒓 π - Σ 𝜓 𝒌 ,𝒓 π - Λ 𝜓 𝒌 ,𝒓 Λ - p + 𝜓 𝒌 ,𝒓 Σ - p 𝜓 𝒌 ,𝒓 Σ + - n p Λ C ( k * ) = 1 fm G r C ( k * ) p - K n K+ Sp n + K+ L p + Sp n + K+ = 5 fm G r (c)(d) (e)(f)
100 150 200 250 300k* (MeV/c)0.90.9511.051.11.151.21.25 C ( k * )
100 150 200 250 300k* (MeV/c)0.90.9511.051.11.151.21.25 C ( k * ) + + Figure 6
Schematic representation of the effects of coupled-channels on the K − –p andp– Λ correlation function . In (a) a picture of the system configuration in femtoscopicmeasurements, where only the final K − –p and p–Λ channels are measured. Upper panels (c,e) :results for radii achieved in pp collisions (1 fm). Lower panels (d,f) : results for radii achieved inheavy-ion collisions (4 fm). In (b) and (c) the correlation function for K − –p, from the pure elasticterm (dotted line) to the full C( k ∗ )(solid line) with all coupled-channels ( ¯K –n, π –Σ, π –Λ)included and the conversion weights are fixed to unity. In (e) and (f) the p–Λ correlation functionobtained assuming LO (LO13 (62)) and two versions of NLO (NLO13 (62),NLO19 (63)) χ EFTcalculations. Contributions from S-,P- and D-waves are included. Dashed lines: results with onlythe elastic term p–Λ → p–Λ in Eq. 11. Solid lines: results with the inclusion of coupled-channelcontributions in Eq. 11 from n–Σ + and p–Σ with conversion weight 1 / • Short title 17 hese two main differences are illustrated in Fig. 6 within for the K − p and p–Λ systems.The K − p system presents couplings to several inelastic channels below threshold such as π Λ, π Σ and, due to the breaking of isospin symmetry, to charge-conjugated ¯K N at roughly4 MeV above threshold corresponding to k ≈
60 MeV / c in the C( k ∗ ). On the left part (a)of Fig. 6, a schematic representation of the collision is shown. From the emitting source,formed after the collision, all the pairs constituting the four coupled-channels are producedand described by the corresponding wave functions ψ j ( k ∗ j , r ∗ ). The correlation of K − p pairscomposing the final-state (channel 1) is measured (b) and its decomposition in the differentchannels contributions is shown in (c) and (d) for two different source sizes. The largestcontributions to the C( k ∗ ) from coupled-channels occur for a small emitting source with R = 1 fm in (c). The C( k ∗ ) signal increases as the inelastic contributions are added andthe cusp structure, visible when the ¯K n channel is explicitly added, indicates the openingof this channel above threshold. For both source radii, this structure already appears whenthe mass difference between K − and ¯K is considered, and it is present also in the elasticK − p → K − p contribution (dotted line). As already mentioned, the explicit inclusion of the¯K –N contribution acts as an ”effective” attraction component, increasing the signal of thecorrelation function and of the cusp accordingly to the strength of the coupling between thetwo channels (short-dashed line). This effect is suppressed when the source size is increasedup to R = 4 fm (d), as in central heavy-ion collisions.A similar trend can be seen when another strongly coupled-channel is introduced, the π –Σ (dashed-dotted line), responsible for the dynamic generation of the molecular stateΛ(1405). The strong coupling to this channel, lying below threshold, is directly translatedinto the correlation function of K − –p pairs, visible as a clear enhancement of the signal atlow momentum with respect to the single-channel contribution.The extreme sensitivity to coupled-channel contributions of the C( k ∗ ) obtained in smallsystems has been recently confirmed by results for the K − –p correlation function, measuredby the ALICE Collaboration in pp collisions at different energies (36). Future measure-ments of this pair, performed in different colliding systems, will also be able to providequantitative constraints on the coupling strength to the ¯K n channel.In Fig. 6 another coupled system, formed by the interaction of respectively a Λ and a Σ withnucleons, is depicted. The strength in the N–Λ ↔ N–Σ conversion is not experimentallywell constrained since scattering measurements cannot currently provide enough precisedata on the p–Λ cross section at momenta close to the opening (5, 6, 7). The onlyexperimental observations of the Σ–p cusp have been extracted in partial-wave analyseson p+p → pK + Λ reactions at low energy but they are strongly affected by Λ–p final-stateinteractions (97, 98).Measurements of the p–Λ correlation function in pp collisions will be able to provide precisedata in this system, thanks to the large production of Λ hyperons at ALICE, and directaccess to the conversion N–Λ ↔ N–Σ . In Fig. 6 the theoretical p–Λ correlation functionobtained within different calculations based on χ EFT, next-to-leading (NLO) (62, 63) andleading order (LO) (62), is shown for a source radius of 1 fm and for a momentum cutoffparameter of 600 MeV/ c . The coupling to the N–Σ (n–Σ + , p–Σ ) occurs already inthe S wave and finds the largest contribution from D waves, hence partial waves up to l = 2 are included. The conversion weights w j for this coupling in Eq. 11 can be fixed to1 / − –p case, leads
18 Author et al. o the appearance of a cusp structure at k ∗ = 289 MeV / c , corresponding to the kinematicopening of the inelastic n–Σ + and p–Σ channels. The low momentum region of the C( k ∗ )is not deeply affected by the explicit inclusion of the inelastic terms since the opening ofthe N–Σ occurs above threshold.The largest differences in the behaviour of the C( k ∗ ), despite the presence or absenceof N–Σ contributions, arise in the LO and NLO descriptions. The LO predictions havealready been ruled out by scattering data in the proximity of the cusp region, since thecalculation deviates significantly from the data, despite the large uncertainties. The twoversions of the NLO calculations (NLO13, NLO19) mainly differ in the description andstrength of the Λ ↔ Σ conversion potential, which leads to significant modifications of theΛ hyperon interaction in dense nuclear matter and to different results for light hypernuclei.As can be seen in Fig. 6, the cusp height predicted from these two approaches is rathersimilar but mild differences are present below and above the cusp. The high precisiondata achieved by the ALICE collaboration in the published analyses on baryon-baryonfemtoscopy during the Run 2 period (58, 74, 40) suggest that a discrimination betweenthese two χ EFT calculations can be achieved. A scenario in which the latest NLO19model, predicting a more attractive Λ single-particle potential in neutron matter, willprovide a better description of the ALICE data will change the current picture on thep–Λ interaction and hence have profound implications for three-body hyperonic forcesand for open problems in astrophysics such as the presence of hyperons in neutronstars (9, 101, 10).In conclusion, femtoscopic measurements in pp collisions are able to probe the inner partof the pair wave-function, in which the coupled-channel dynamics dominate the stronginteraction. Moving to larger source sizes tests the asymptotic part of the wave-functionwhere the inelastic terms are noticeably suppressed and partial access to the pure elasticinteraction can be obtained. This opens the possibility to investigate the dynamics of thecouplings between the elastic and inelastic channels by performing femtoscopic measure-ments of the same pair in different colliding systems, leading to a complete description ofseveral hadron-hadron interactions.
5. Implications for neutron stars
The interaction of hyperons with nucleons is one of the key ingredients needed tounderstand the composition of the most dense objects in our universe: neutron stars(NS) (102, 103). These kinds of stars are the final outcome of supernova explosionsand are typically characterized by large masses ( M ≈ . − . M (cid:12) ) and small radii( R ≈ −
13 km) (104, 105, 106). In the standard scenario, the gravitational pressureis typically counter-balanced by the Fermi pressure of neutrons in the core, which, alongwith electrons, are the only remnants from the mother-star collapse. The high densityenvironment ( ρ ≈ ρ ) supposed to occur in the interior of NS leads to an increasein the Fermi energy of the nucleons, translating into the appearance of new degrees offreedom such as hyperons. This energetically favored production of strange hadronsinduces a softening of the Equation of State (EoS). The behavior of the mass as a function Equation of State:
Thermodynamicalrelation betweenpressure,energy-density andtemperaturedescribing theproperties of nuclearmatter underextreme conditions(high T or highdensity) andstrongly dependenton the constituentsand the interactionsamong them. of the radius has a unique correspondence with the EoS through the solution of theTolman–Oppenheimer–Volkoff equations, hence the mass-radius relation strongly dependson the constituents of the EoS and on their interactions. The inclusion of hyperons leads • Short title 19 o NS configurations unable to reach the current highest mass limit from experimentalobservations of 2 . M (cid:12) (106).For this reason, the presence of hyperons inside the inner cores of NS is still under debate,and this so-called hyperon puzzle is far from being solved (107, 108).A key element in the complete understanding of this puzzle is the interaction of hyperonswith the surrounding medium which strongly affects the properties of the correspondingEoS (8, 109) and can be related to the interaction between hyperons and nucleons (Y–Nand Y–N–N) in vacuum. A repulsive Y–N interaction occurring already at the two-bodylevel can push the appearance of hyperons to larger densities, limiting the possible presenceof these particle species inside NS, stiffening the EoS and leading to larger star masses.The more precisely the hyperon-nucleon two-body and three body interactions are knownin vacuum, the more detailed is the knowledge of the hyperonic content inside NS. A largeinterest in this topic has been triggered also from the recent measurements of gravitationalwaves signals from NS mergers, which opened a new gate to experimentally access theproperties of the matter inside NS.As already shown in the previous sections, femtoscopy is capable of providing more andmore insight into interactions involving nucleons and hyperons which are poorly known ornot accessible with scattering experiments.A key example is given by the femtoscopic measurements of the p–Λ strong interaction.The Λ baryons are typically the first hyperon species that are produced inside NS, due totheir light mass. Their appearance is also theoretically favored by the overall attractivepotential that a Λ feels at the saturation density, U Λ = −
30 MeV (68). The resultsobtained from the ALICE Collaboration on this system, as already mentioned in Sec. 4.3,support recent χ EFT calculations in which an even more attractive interaction of theΛ with the surrounding nucleons, due to the Λ–N ↔ Σ–N dynamics, is predicted. In thiscase the early appearance of Λ hyperons in neutron matter will lead to a too soft EoS andultimately to stable light NS configurations. Such a scenario, in order to co-exist with theastrophysical constraints on NS masses, requires the introduction of repulsive forces thatmight be present in other YN systems and the inclusion of three-body interactions.Repulsive Hyperon-nucleon-nucleon interactions, such as Λ–N–N, have already beenincluded in several approaches to obtain a stiffer EoS (110, 9). However, at the moment,these three-body forces rely on the experimental measurements of hypernuclei bindingenergies ( H, He), in which the determination of the genuine Λ–N–N interaction is notstraightforward and can be affected by many-body effects. For this reason the currenttheoretical understanding of the role played by three-body terms in the strangeness | S | = 1sector inside NS is not yet settled.A major improvement in the understanding of the role played by heavier strange hadronsin the hyperon puzzle has been achieved by the validation of lattice QCD predictions forthe N–Ξ interaction. As shown in Sec. 4.1, ALICE measurements on p–Ξ − pairs (39)confirmed a strong attractive interaction between these two hadrons and provided a directconfirmation of lattice potentials (73). Using this same interaction as a starting pointto extrapolate results in a neutron-rich dense system, a repulsive average interaction ofroughly +6 MeV can be obtained (113). Currently, models for EoS including Ξ hyperonsassume large variations in the values of the single particle potential ( − , +40 MeV) (8) andhence the validated lattice predictions impose a much more stringent constraint. In Fig. 7,the fractions of particles, obtained from mean-field calculations (111, 8, 109, 112, 114),produced in the inner part of NS are shown as a function of the energy density. The
20 Author et al.
Radius (km) ) M a ss ( M J0740+6620J0348+0432J1614-2230
Energy density ε / ε P a r t i c l e nu m be r pe r ba r y on M=1.4M O . M max = 2.13M O . ΛΣ − Ξ − pe - µ − n m * /m = 0.65 U Λ = -28MeVU Ξ = -4MeVU Σ = +15MeV Figure 7
Left: fraction of particles produced in the inner core of a NS as a function of the energy density, inunits of energy density (cid:15) at the nuclear saturation point. The single-particle potential depths insymmetric nuclear matter (SNM) for Λ, Σ and Ξ hyperons are displayed. The vertical dotted linesindicate the central energy densities reached for a standard NS of 1 . M (cid:12) and for the maximummass, 2 . M (cid:12) , reached within this specific EoS. The mean-field calculations (111, 8, 109, 112)have been tuned in order to reproduce the lattice predicted value of U Ξ in pure neutron matter(PNM) obtained in (113), using the in-vacuum results validated by ALICE data in (39). The EoSobtained with these constraints provides a stable NS with a maximum mass of M max = 2 . M (cid:12) ,as seen on the mass-radius plot on the right and is compatible with recent astrophysicalmeasurements of heavy NS, indicated by the blue (106), green (105) and orange (104) bands. single-particle potentials for Λ and Σ hyperons have been fixed to the current valuesconstrained from scattering data and hypernuclei, and confirmed by ALICE femtoscopicmeasurements. The isovector couplings to the Ξ have been adjusted to reproduce thepredicted results in pure neutron matter (PNM) obtained from HAL QCD calculationsat finite density (113), stemming from the predictions in vacuum validated by ALICEfemtoscopic measurements on p–Ξ − shown in Sec. 4.1. The slight repulsion acquired by aΞ − in pure neutron matter directly translates into larger energy densities, and hence largernuclear densities, for the appearance of this hyperon species. In the right plot of Fig. 7, theresulting mass-radius relation obtained by assuming the predicted HAL QCD Ξ interactionin medium is shown. The production of cascade hyperons occurring at higher densitiesleads to a maximum NS mass of 2 . M (cid:12) , fully compatible with the recent measurements,indicated by the colored bands, of NS close to and above two solar masses (104, 105, 106).The results obtained in recent years from femtoscopic measurements in small collidingsystems have proven that femtoscopy can play a central role in understanding the dynamicsamongst hyperons and nucleons in vacuum. A comparison between hadronic models andthese data are necessary in order to constrain calculations at finite density and to pin downthe hyperons behaviour in a dense matter environment. The great possibility to investigate,within the femtoscopy technique, different YN interactions and to extend the measurementsto three-body forces, can finally provide quantitative input to the long-standing hyperonpuzzle. • Short title 21 . Outlook
A complete program of new measurements in pp collisions at 14 TeV has been approved forthe upcoming Runs 3 and 4 of the LHC with ALICE (115).In order to address further questions on two and three-body forces involving hyperons,correlation studies constitute one of the main foci of such program. Studies will benefitfrom data-taking with increased instantaneous luminosity and readout speed, plus bettertracking and vertexing performances of the new ALICE apparatus. Moreover, the newdata acquisition system will make it possible to select events with very high multiplicity,up to 16 times the average multiplicity of minimum bias pp collisions. Accessing such aregime of multiplicities in pp events is particularly beneficial for measurements includingstrange hadrons due to the enhanced production of strangeness in collisions with high-multiplicity (51). Assuming an acquired luminosity of 200 pb − and a selection of eventswith a number of produced charged particles ( N ch ) seven times higher than the meannumber of charged particles in minimum-bias collisions N ch > (cid:104) N ch (cid:105) , an overall increaseup to a factor 50 for particle pairs per event is expected for the Run 3 high-multiplicitydata with respect to the sample collected in Run 2 (115).Several new analyses can be performed with the Run 3 and Run 4 ALICE data thatwere not possible with the Run 1 and Run 2 statistics, and the question of three-body forcesincluding hyperons can finally be experimentally addressed. • K − –d correlations: Following the measurement of K − –p correlations in pp byALICE (36), the study of the correlation function of K − –deuteron pairs will be realized.Together with the planned measurements at threshold using kaonic atoms by SIDDHARTA-2 (116), the K − –d femtoscopy will allow us to determine for the first time the full isospindependence of the K–N interaction, a fundamental problem in the strangeness sector in thelow-energy regime of QCD. • p– Σ correlations: The investigation on the p–Σ correlation will provide precisedata on an interaction that, in contrast to the N–Λ interaction, is currently very poorlyknown experimentally. A first measurement (37) of this correlation was performed byALICE with Run 2 data, using high multiplicity pp collisions, demonstrating the feasibilityof the approach, although with large statistical uncertainties and relatively low signal purity.The minimum-bias pp √ s = 14 TeV Run 3 data will allow to have a yield of p–Σ pairs tentimes higher than in the Run 2 data, that will deliver the first precise data in the field. Atthe same time lattice QCD calculations are expected to reach precision also in the | S | = 1sector within the next few years, and hence the ALICE measurement will contribute to thevalidation of the state-of-the-art theoretical calculations. • Λ – Ξ correlations: The enhancement of the yield of strange particles in the high-multiplicity data from Run 3 will allow us to study as well the Λ–Ξ interaction with highprecision. Such a study will complement the measurements of p–Ω − correlations, and thecomparison to the lattice QCD calculations. For the p–Ω − interaction, the J = 1 channellacks of any prediction so far, since it is dominated by absorption in the Λ–Ξ and Σ–Ξchannels. The study of the Λ–Ξ will hence provide the first experimental constraints to thecontribution of the coupled channels for the p–Ω − system. • Ω – Ω correlations: The HAL QCD collaboration has provided Lattice QCD calcula-tions at the physical point for the Ω–Ω system, predicting the existence of ”the most strangedibaryon” with a binding energy of around 1.6 MeV (117), implied by the strong attractivecharacter of the Ω–Ω strong interaction and the fact that the Pauli principle does not applyfor this system.
22 Author et al. o far, no experimental data is available for this interaction. The measurement of theΩ–Ω correlation function is extremely challenging and constantly requested by theoreticians.During Run 3, the data acquisition of ALICE will be implemented with a dedicated triggerfor Ω decays that can sample the whole 200 pb − of the pp data taking, resulting in a total of2 · reconstructed and recorded Ω − ⊕ Ω + . For the correlation studies, a total of about 500Ω–Ω pairs are expected to be reconstructed with low relative momentum, k ∗ <
200 MeV / c .On the theoretical side, the predictions from HAL QCD Collaboration (91) provide the S channel of the Ω–Ω interaction alone. This is the channel with the smallest contribution(weight 1/16), and an attractive interaction is also expected for the S channel (weight5/16), but the calculations are not available yet.A projection of the measurement of the Ω–Ω correlation function with the ALICERun 3 data is represented by the black points in Fig. 8 (left panel), and it is comparedwith the Coulomb-only scenario (red curve), and three different curves with the additionalstrong interaction predicted by the HAL QCD Collaboration. One can see that for thelattice calculations ( S channel) there are substantial differences in the expected correlationfunction considering different integration times (t/a parameter). This is true in particularfor a very small source like the one expected for Ω–Ω pairs in pp collisions, with a radiusof around r = 0 . k ∗ = 25 MeV / c is expected in thecorrelation function using a bin width of 50 MeV / c . This would constitute a groundbreakingmeasurement of the Ω–Ω interaction, delivering the first constraint on the lattice QCDcalculations and with the potential for experimentally determining for the first time everthe sign of the strong interaction between Ω–Ω pairs. ) c * (MeV/ k * ) k ( C W = 14 TeV, s pp -1 = 200 pb int L - W - - W Coulomb + HAL QCD t/a=16Coulomb + HAL QCD t/a=17Coulomb + HAL QCD t/a=18Coulomb
Figure 8
Left panel: Expected precision of the Ω-Ω correlation function with ALICE Run 3 data (blackpoints). The red curve represents the Coulomb-only scenario, and the green, black and blue linescontain, in addition, the strong interaction potential by the HAL QCD collaboration withdifferent integration times, t/a = 16, t/a = 17, and t/a = 18, respectively. The gaussian sourceused for the calculations has a radius of r = 0 . • Λ –d correlations: Complementing the studies on Λ–p correlations by ALICE in Run • Short title 23 and Run 2, the study of the Λ–d correlation function provides additional information onthe Λ–N interaction. Experimental access to information on direct Λ–d scattering is evenharder to obtain than that from Λ–p scattering, and correlation studies will constitute anadditional and independent source of information for this channel. Moreover the measure-ment of the Λ–d correlation function in small systems complements the measurement of thehypertriton binding energy and delivers information on many-body forces (118).There are two different spin configurations in the S-wave Λ–d interaction: the doublet S / and the quartet S / states. With no scattering data available for the Λ–d channel,the scattering parameters in the doublet state are constrained by measurements of thelifetime of the bound state found in this partial wave, the hypertriton, H Λ . The hypertritonbinding energy is related to the scattering parameters in the effective range approximationvia the Bethe formula (119). The higher spin configurations are not binding, and they arecurrently not tested by any experimental data. For the constraint of the quartet state,chiral SU(3) calculations (120) are used so far.The expected precision of the measurement of the Λ–d correlation function with thepp √ s = 14 TeV high multiplicity data sample with N ch > (cid:104) N ch (cid:105) during Run 3 at k ∗ = 50 MeV / c is on the order of 5% (with a bin width of 20 MeV / c ). Such a precisemeasurement will complement the hypertriton binding energy measurements, scanning thefull isospin dependence of the interaction and, as has been suggested, possibly providingas well an insight into the coalescence process. These studies in pp can be complementedby studies in larger systems (Pb–Pb) leading to a better knowledge and understanding ofmany-body forces acting in light hypernuclei (118). • Three-body forces:
In addition to the Λ–deuteron measurement, the study of three-body interactions involving hyperons, that are extremely important for the understandingof the structure of neutron stars, will be experimentally accessible with high precision forthe first time. Exclusive measurements of p–p–Λ and correlations with a newly developedmathematical formalism using cumulants that allow the study of the correlation functionof three particles with non-identical masses will enable accessing the final state interaction.For this purpose, the data acquisition will be implemented to sample the whole data takingrecording events where at least two proton candidates and one Λ candidate are reconstructedusing an on-line trigger selection.
7. Summary
We have explained the methodology of femtoscopy and discussed the extraction of the sourcein small systems. The strong resonances have been carefully modelled and the hypothesisof an universal source for all hadron-hadron pairs is demonstrated and also exploited inall the presented analysis. The method has been tested by analysing the p–p and p–Λcorrelations, where the interaction is rather well known, especially for p–p pairs. Followingthis scheme the results achieved for several hyperon–nucleon, hyperon–hyperon and kaon–nucleon combinations have been presented. We have shown the first measurement of theattractive p–Ξ − interaction, confirmed by lattice calculations by the HAL QCD group, themost precise measurement of the Λ–Λ interaction and the upper limit for a possible Hdibaryon state, the first measurement of the attractive strong p–Ω − interaction. The datain the latter case do not show yet any clear evidence for the existence of a bound state. Wehave also discussed the evidence of the coupled channel dynamics with the help of the K − –pand p–Λ channels that have been precisely measured and show the presence of the so far not
24 Author et al. easured coupling K –N ↔ K − –p and N–Λ ↔ N–Σ . We have discussed the consequencesof the new measurements for the physics of neutron stars with one example related to theΞ − strong interaction to show the impact of the new measurements on astrophysics. In thefuture, we want to include even other measurements and possibly results about three-bodyinteractions among hyperons and nucleons to achieve even more stringent constraints forneutron stars. Since we are currently waiting for the start of the Run 3 data taking atthe LHC, we have also discussed the physics opportunities of the imminent future. Weplan to measure all interactions involving strange hadrons, including the most rare of thecombinations such as Ω − Ω pairs.In general, we have shown that the femtoscopy technique applied to small colliding systemsat the LHC is a very promising tool to investigate the strong interactions. The ALICEcollaboration demonstrated that a new laboratory to study hadron-hadron interaction hasbeen established at the LHC, with the capability to unveil the strong interaction amongany hadron–hadron pair
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdingsthat might be perceived as affecting the objectivity of this review.
ACKNOWLEDGMENTS
The authors would like to thank Dimitar Mihaylov, Alice Ohlson and Thomas Humanicfor their inputs and comments to this manuscript. A special thanks also to JuergenSchaffner-Bielich, Debarati Chatterjee, Suprovo Gosh and Benjamin Doenigus for the in-teresting discussions on neutron stars physics. This work has been supported by DFG EXC2094 390783311 ORIGINS, GSI TMLRG1316F, BmBF 05P15WOFCA, SFB 1258, DFGFAB898/2-2.
LITERATURE CITED
1. Arndt R, Briscoe W, Strakovsky I, Workman R.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. D
J. Phys. G
Phys. Lett. B
Phys. Rev.
Phys. Rev.
Nucl. Phys.
A881:62 (2012)9. Lonardoni D, Lovato A, Gandolfi S, Pederiva F.
Phys. Rev. Lett.
Eur. Phys. J. A
Nature
Phys. Rev.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rept.
Fiz. Elem. Chast. Atom. Yadra • Short title 25
8. Adamczewski-Musch J, et al.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Lett. B
Z. Phys. C
Phys. Lett. B
25. Khachatryan V, et al.
Phys. Rev. Lett.
Comput. Phys. Commun.
Phys. Rev. C
Phys. Atom. Nucl.
Phys. Lett. B
Phys. Rev. C
Phys. Lett. B
Phys. Rev. C
Phys. Rev. Lett.
Phys. Lett. B
Phys. Lett.
B797:134822 (2019)39. Acharya S, et al.
Phys. Rev. Lett.
Nature
Phys. Rev. D
Ann. Rev. Nucl. Part. Sci.
Sov. J. Nucl. Phys.
Eur. Phys. J. C
Phys. Rev. C
Eur. Phys. J. C
Phys. Rev. C
Phys. Lett. B
JHEP
Nature Phys.
Phys. Rev. C
Phys. Rev. C
Eur. Phys. J. C
Phys. Rev. D
Phys. Lett. B
JHEP
P hys.Lett.B
59. Sinyukov Y, Shapoval V, Naboka V.
Nucl. Phys. A
Phys. Rev. C
Phys. Rev. C
Nucl. Phys. A
Eur. Phys. J. A
Comput. Phys. Commun.
Eur. Phys. J. C
Comput. Phys. Commun.
Phys. Rev. C
26 Author et al.
8. Hashimoto O, Tamura H.
Prog. Part. Nucl. Phys.
Phys. Rev.
Phys. Rev.
Nucl. Phys.
B139:61 (1978)72. Hadjimichef D, Haidenbauer J, Krein G.
Phys. Rev.
C66:055214 (2002)73. Sasaki K, et al.
Nucl. Phys. A
Phys. Rev. Lett.
Nucl. Phys.
A981:1 (2019)76. Takahashi H, et al.
Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Lett. B
Nucl. Phys. A
Phys. Rev. Lett.
Phys. Lett. B
Progress of Theoretical Physics
Phys. Rev. D
Phys. Rev. D
Progress of Theoretical Physics Supplement
Nuclear Physics A
Phys. Rev. C
Front. Phys. (Beijing)
Phys. Rev.
C98:015205 (2018)91. Morita K, et al.
Phys. Rev. C
Phys. Rev. C
Nucl. Phys. A
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Eur. Phys. J. A
Phys. Lett. B
EPJ Web Conf.
Rom. Rep. Phys.
Eur. Phys. J. A
Ann. Rev. Astron. Astrophys.
Astrophys. J. Lett.
Nature
Science
Nature Astron.
Phys. Rev. C
Prog. Part. Nucl. Phys.
Phys. Rev.
C85:065802 (2012), [Erratum:Phys. Rev.C90,no.1,019904(2014)]110. Haidenbauer J, Meißner UG, Kaiser N, Weise W.
Eur. Phys. J. A
Phys. Rev. C
Phys. Rev. C
PoS
INPC2016:277 (2016) • Short title 27
14. Chatterjee D, Gosh S, Schaffner-Bielich J.
To be published
Nucl. Phys. A
Phys. Rev. Lett.
Phys. Rev.
Nucl. Phys. A915:24 (2013)