In-medium Δ(1232) potential, pion production in heavy-ion collisions and the symmetry energy
EEur. Phys. J. A manuscript No. (will be inserted by the editor)
In-medium ∆ ( ) potential, pion production in heavy-ion collisionsand the symmetry energy M.D. Cozma and M.B. Tsang National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Department of Theoretical Physics, IFIN-HH, Reactorului 30, 077125 Mˇagurele-Bucharest, RomaniaReceived: date / Accepted: date
Abstract
Using the dcQMD transport model, the isoscalarand isovector in-medium potentials of the ∆ (1232) baryonare studied and information regarding their effective strengthis obtained from a comparison to experimental pion produc-tion data in heavy-ion collisions below 800 MeV/nucleonimpact energy. The best description is achieved for an iso-scalar potential moderately more attractive than the nucleonoptical potential and a rather small isoscalar relative effec-tive mass m ∗ ∆ ≈ ∆ (1232) and nucleonpotentials are equal. The density dependence of symmetryenergy can be studied using the high transverse momentumtail of pion multiplicity ratio spectra. Results are howevercorrelated with the value of neutron-proton effective massdifference. This region of spectra is shown to be affectedby uncertain model ingredients such as the pion potential orin-medium correction to inelastic scattering cross-sectionsat levels smaller than 10%. Extraction of precise constraintsfor the density dependence of symmetry energy above satu-ration will require experimental data for pion production inheavy-ion collisions below 800 MeV/nucleon impact energyand experimental values for the high transverse momentumtail of pion multiplicity ratio spectra accurate to better than5%. PACS · · a Email address: [email protected]
The isospin dependent part of the equation of state of nu-clear matter (asy-EoS), commonly known as the symmetryenergy (SE) remains among the most debated topics in nu-clear physics. Its relevance for the structure of rare isotopes,dynamics of heavy-ion collisions and properties of neutronstars and associated phenomena has been long recognizedand has prompted numerous experimental and theoreticalstudies [1,2,3,4]. By combining results for various experi-mental observables with phenomenological models [5,6,7,8,9,10,11] and theoretical many-body simulations of nu-clear matter [12,13,14] a consistent description of SE atsub-saturation densities has been achieved.The recent observation of a binary neutron star mergerby the LIGO-VIRGO collaboration [15,16] has opened upthe possibility of studying the asy-EoS in the vicinity oftwice saturation density (2 ρ ) by means of correlations be-tween tidal polarizability of neutron stars ( Λ ), their radii andultimately symmetry energy [4,17]. However, a unique cor-respondence between Λ and the SE does not exist, due to adegeneracy of the sensitivity to the slope ( L ) and curvature( K sym ) parameters of the asy-EoS around 2 ρ [18]. Nuclearphysics laboratory experiments, astrophysical observationsand theoretical studies are thus needed to provide lackingcomplementary information. More recently, developmentsof theoretical many-body calculations based on chiral effec-tive interactions have made predictions of the asy-EoS up to2 ρ with unprecedented accuracy possible [19], calling forindependent confirmation of these results.Heavy-ion collisions (HIC) provide an unique opportu-nity to study nuclear matter at densities exceeding ρ inthe laboratory. To this end several promising observableshave been identified: the ratio of neutron-to-proton yieldsof squeezed out nucleons [20], charged pion multiplicityratio (PMR) and its spectral ratio [21,22], elliptic flow re- a r X i v : . [ nu c l - t h ] J a n lated observables [23] and others. Using neutron-to-protonand neutron-to-charged particles elliptic flow ratios compat-ible constraints for the value of L have been extracted usingdifferent transport models [24,25, 26,27]. Extrapolations to2 ρ are still uncertain due to limited experimental accuracyand suboptimal average density probed by these observablesin AuAu collisions at 400 MeV/nucleon impact energy.The charged pion multiplicity ratio has attracted consid-erable attention from the community. Reaching at a consis-tent picture for the density dependence of SE has been how-ever elusive up to this moment [28,29,30,22,31,32]. Nu-merous studies have attempted to remedy the problem, buthave only succeeded in unvealing the sensitivity of PMRto additional model ingredients [31,32,33,34,35,36,37,38,39]. In recent years, the Transport Model Evaluation Project(TMEP) has aimed towards understanding differences be-tween existing models and formulating benchmark calcu-lations that every realistic model should reproduce [40,41,42]. The model used in this study is part of that effort.In Refs. [32,33] a Quantum Molecular Dynamics (QMD)model has been employed in an attempt to explain the FOPIexperimental pion production data [43] by inclusion of thresh-old effects [44,45] that arise as a consequence of imposingtotal energy conservation of the system. This requirement isoften not properly treated in semi-classical transport models,in spite of its relevance for the existence of thermodynamicequilibrium [35]. The crucial ingredients for the computa-tion of threshold effects are the in-medium potential energyof nucleons, resonances (only ∆ (1232) close to the vacuumproduction threshold) and pions.The knowledge of the isoscalar ∆ (1232) potential (ISDP)is uncertain, with empirical information contradicting mi-croscopical calculations [46,47,48,49,50,51,52,53]. Discre-pancies among results of microscopical models have alsobeen noted and are often related to details of how pion-nucleon and pion-nucleon-delta couplings have been extrac-ted from few-body experimental data. In particular, includ-ing (or omitting) processes such as N ∆ → N ∆ , N ∆ → ∆ ∆ , NN → ∆ ∆ and ∆ ∆ → ∆ ∆ in models used to describe nucleon-nucleon scattering data was proven to have an impact on thedetermined strength of the ∆ potential [53]. No informationis avaiblable about the isovector component of the ∆ (1232)potential (IVDP). These quantities are also relevant for de-termining the threshold density above which ∆ (1232) oc-curs in neutron stars, with impact on the maximum mass ofsuch objects [54,55,56,57,58] and in the analysis of neu-trino physics experimental data [47].In view of the above, it is customary to set, in transportmodels, the ∆ (1232) potential (DPOT) in terms of that ofnucleons using a simple Ansatz based on the decay channelsof this resonance into nucleon-pion pairs [59]. The signifi-cance of this assumption was recognized and a large sensi-tivity of PMR to the magnitude of these potentials was ev- idenced in Ref. [33]. Subsequently, it was shown that thedensity dependence of the SE can be studied by using PMRsupplemented by the ratio of average transverse momenta ofcharged pions [32]. The latter observable is needed in orderto constrain the strength of IVDP, which was varied usinga scaling parameter. In that study the ISDP was kept fixed,equal to that of the nucleon, in spite of previously provendependence of PMR on its strength [33].Extracting the asy-EoS from low and intermediate en-ergy regime experiments is further complicated by uncer-tainties stemming from the rather poorly constrained mo-mentum/energy dependence of nuclear interactions, usuallyquantified in terms of effective masses [11,60, 61,62,63,64]and the degeneracy of effects induced by the isoscalar mass,the neutron-proton effective mass difference ( δ m ∗ np ) and thedensity dependence of SE on observables [64,65,66].The present study builds on the results of Refs. [32,33].The goal is to describe all pionic observables, not just ratiosof multiplicities or average transverse momenta, in an at-tempt to reduce residual model dependence originating fromthe isoscalar part of the interaction. To achieve this goal theDPOT is treated as an independent quantity. For both iso-scalar and isovector components freedom is built into pa-rametrizations as to allow independent assigning of poten-tial depths at saturation and effective masses. Details of thetransport model, parametrizations used for DPOT and bench-marking calculations for nucleonic observables are presentedin Section 2. The observables relevant for constraining ofDPOT parameters and their extraction from experimentaldata are described in Section 3. In Section 4 the feasibility ofconstraining the density dependence of SE from pionic ob-servables is reassessed, together with a study of the impactof other relevant model parameters, such as δ m ∗ np . A sectiondevoted to summary and conclusions follows. finite spread in phase space, d r i dt = ∂ (cid:104) U i (cid:105) ∂ p i + p i m , d p i dt = − ∂ (cid:104) U i (cid:105) ∂ r i . (1)The average of the potential operator is understood to betaken over the entire phase-space and weighted by the Wignerdistribution of particle i . The potential operator U i is in thiscase the sum of the Coulomb and strong interaction potentialoperators.In the present study a variant developed over the lastcouple of years, dubbed dcQMD, is used [27,32,33]. Ittraces its origin to the T¨ubingen QMD model transport mo-del developed in the 90’s and early 2000’s [69,70,71,72].In the present model, the relativistic relation betweenmass, energy and momentum is used in all kinematic equa-tions. Consequently the kinetic term in Eq. (1) is replacedby its relativistic counterpart. To be complete, the effectiveclassical Hamiltonian reads H = ∑ i (cid:113) p i + m i + ∑ i , j , j > i (cid:20) A u + A l + ˜ τ i ˜ τ j A l − A u (cid:21) u i j (2) + ∑ i , j , j > i (cid:20) ( C l + C u ) + ˜ τ i ˜ τ j ( C l − C u ) (cid:21) u i j + ( p i − p j ) / Λ + ∑ i B σ + [ − x ˜ τ i β i ] u σ i + D [ − y ˜ τ i β i ] u i + ∑ i , j , j > i U Couli j where ˜ τ i =- τ i / T i , u i j = ρ i j / ρ is the partial relative interac-tion density of particles i and j with u i = ∑ j (cid:54) = i u i j and β i isthe isospin asymmetry at the location of particle i . Here T i and τ i denote the isospin and isospin projection of particle i respectively. It is straightforward to show that the momen-tum independent part of the interaction leads to the expres-sion of the energy per particle presented in Eq. (8) up tosymmetry potentials of second and higher order. The mo-mentum dependent term above represents a finite particlenumber approximation to the corresponding expression inEq. (8).The scattering term includes elastic and inelastic two-baryon collisions ( N + N → N + N , N + N → N + R , N + R → N + R (cid:48) , etc.), resonance decays into a pion-nucleon or pion-resonance pairs ( R → N + π and R → R (cid:48) + π ) and singlepion absorption reactions ( π + N → R ). Collision processesthat consist of 3-particle initial or final states (as for examplenon-resonant background pion production N + N → N + N + π ) have not been considered. Non-resonant pion productioncontributions are needed at invariant masses close to the pro-duction threshold to describe experimental data [73, 74,75].Their inclusion in the scattering term is, in the context ofusing the geometrical Bertsch prescription for collision val-idation [76] and requirement of conservation of total energyof the system [33], technically challenging, leading to a sig-nificant slow down of computations, and has thus not beenattempted.The vacuum Li-Machleidt [77,78] and Cugnon et al . [79]parametrizations of elastic nucleon-nucleon cross-sections are used below and above pion production threshold respec-tively. They are modified in nuclear matter using an empiri-cal factor depending on density and relative momentum, butnot isospin asymmetry. Such a modification has been foundnecessary to describe stopping and flow observables at lowand intermediate energy heavy-ion collision [80,81,82,83].The FU3FP4 parametrization in Ref. [83] has been foundto lead to the best description of stopping and flow, see Sec-tion 2.3. For this choice, elastic cross-sections are multipliedby a factor depending on the local density ρ and relative mo-mentum p of the scattering nucleons F ( ρ , p ) = (cid:40) p > F ρ − +( p / p ) κ + p ≤ F ρ = λ + ( − λ ) Exp [ − ρζ ρ ] . The parameters in the above expression take the followingvalues: p =0.30 GeV/c, κ =8, λ =1/6 and ζ =1/3. Theoreti-cally computed medium-modified cross-sections [77,78] failto lead to a good description of stopping at low impact ener-gies.The Huber et al . parametrizations for vacuum inelasticnucleon-nucleon cross sections [84] are used. They lead tocharged pion production cross-section that underpredict ex-perimental values for nn / pp and np reactions by 20% and40% respectively, at an impact energy of 400 MeV/nucleon.The discrepancy can be alleviated by including non-resonantbackground contributions. Charged pions emitted in HICoriginate predominantly from nn / pp collisions since for thesechannels production cross-sections are an order of magni-tude larger than in np reactions. Consequently, explicit non-resonant terms to pion production multiplicities can be ne-glected at the impact energies of interest for this study, astheir omission can, as a first approximation, be compen-sated by modifying the strength of the ∆ (1232) potentials(see Section 3.1). This approximation becomes better as theinvariant mass of colliding baryons increases. High energypions may thus be a probe of the equation of state less im-pacted by this type of model uncertainties.Inelastic nucleon-nucleon cross-sections are modified in-medium by using a scaling factor that depends on the ef-fective masses of the scattering baryons, in agreement withthe results of the one-pion exchange microscopical modelof Ref. [85]. Within this model in-medium modified inelas-tic NN → N ∆ cross-sections have been determined by in-cluding effects such as in-medium corrections to the pionpropagator, vertex corrections and in-medium effective mas-ses. The dominant effect could be described by a correc-tion factor depending on effective masses of initial and finalstate-baryons and of the medium-modified invariant massobtained by replacing canonical with kinetic momenta.The dynamics of the present model is non-relativisticand consequently modifications of the invariant mass us- ing a relativistic mean field approach is not possible. In-stead we follow the approach in Ref. [33] developed to en-sure total energy conservation of the system, which natu-rally leads to threshold effects and in-medium modificationsof cross-sections. The central assumption of the approach isthat no true two-body scattering processes exist, but ratherthey are modified by interaction with the rest of the sys-tem. Due to energy exchange with the fireball the initial-and final-state invariant masses of the two scattering par-ticles ( s ini and s fin ), determined using vacuum masses andmomenta, differ. Considering the fact that vacuum inelas-tic cross-section for resonance excitation increases with theinvariant mass, the contribution involving two-particles scat-tering with the higher invariant mass dominates the totalscattering amplitude. This approximation is best close tothreshold and was estimated to be valid up to impact en-ergies of about 800 MeV/nucleon. Therefore the mediummodified invariant mass used to determine cross-sectionsreads s ∗ = Max ( s ini , s fin ) . In Ref. [33] it was shown that s fin − s ini > L > σ ( med ) NN → N ∆ ( s ∗ ) = µ ( ini ) ∗ µ ( ini ) µ ( fin ) ∗ µ ( fin ) σ ( vac ) NN → N ∆ ( s ∗ ) (4)with starred and regular variables corresponding to in-mediumand vacuum quantities and µ denoting the reduced mass ofthe system. A similar expression for the modification factorwas obtained in Refs. [86,87,88] on qualitative grounds forelastic nucleon-nucleon cross-sections. For effective massesthe non-relativistic formula is used e.g. m ∗ = m / ( . + mp dUdp ) .The density dependence of effective masses has only a rathersmall impact on pion multiplicities, in spite of modifica-tion factors that amount to values in the range of 0.5-0.7at saturation. Such substantial decreases of cross-sectionsare partially compensated by having also smaller absorption N ∆ → NN rates. The impact of in-medium modifications ofinelastic cross-sections on pion observables due to isospinasymmetry dependence of effective masses were found tobe small during tests and have been therefore neglected inthe present study.The cross-section for the resonance absorption reaction NR → NN is determined using a detailed balance formula[89], d σ ( NR → NN ) d Ω ( s ∗ ) = m R p NN p NR d σ ( NN → NR ) d Ω ( s ∗ ) × (5) (cid:18) π (cid:90) √ s ini − m N m N + m π dMM p (cid:48) NR A R ( M ) (cid:19) − . Due to the difference between s ini and s fin momenta p NN and p NR have to be evaluated using the invariant masses ofthe NN (final) and NR (initial) states respectively. Such a pre-scription can be understood since, in the expression for thecross-section of a 2-body reaction NR → NN , p NR originatesfrom the evaluation of the incoming flux, while p NN arisesfrom the final-state phase space.The pion decay width of resonances is determined usingthe expression [90] Γ R → N π ( √ s ) = Γ R → N π ( √ s ) √ s √ s p p p + Λ p + Λ , (6)depending on the invariant mass √ s and its pole mass value √ s ; p and p are the corresponding pion momenta in therest frame of the resonance. The above formula is a particu-lar case of a more general expression [91] for a value of theorbital angular momentum of the pion-nucleon system equalto 1. The quantity Λ is computed using Λ = (cid:113) ( m R − m N − m π ) + Γ / . , (7)where m R = .
232 GeV and Γ = .
115 GeV (pole massproperties of the resonance, ∆ ( ) in this case); similarlyfor other resonances (N(1440), etc). In the parent TuQMDmodel, as well a in previous publications [32,33], a formulafor the width that is close to the Huber parametrization [84]had been used. It leads to pion absorption cross-sectionsclose to threshold that are too large, by a factor close to 2,as compared to the experimental data. At invariant massesin the vicinity of the resonance’s mass pole realistic valuesare obtained. The above parametrization for the resonancedecay width solves the mentioned problem. It is worth not-ing that a modification of the resonance decay width doesnot require a refit of the Huber OBE model as long as dou-ble ∆ production is negligible, since the difference can beabsorbed in the π N ∆ vertex form-factor.The above expression for the decay width employs ageneric variable s . For the resonance decay R → N π andpion absorption π N → R terms in the transport model theexpression is evaluated using a modified invariant mass s ∗ = Max ( s ini , s fin ) supplemented by the same argumentation asfor baryon-baryon scattering.Contributions of pion optical potentials have been in-cluded by using the Ericson-Ericson parametrization to de-scribe their density, isospin asymmetry and momentum de-pendence, see Ref. [32] for all relevant details. The set ofparameter values for the optical potential commonly knownas Batty-1 [92] has been used extensively in this work, withone exception. In Section 4 the effective S-wave model setof parameters (denoted S’) [32] has been used to study theresidual model dependence on p T spectra of PMR. Meanfield propagation of pions is treated similarly to that of nu-cleons, by associating a Gaussian wave function to them,whose width has been set such that the ratio of pion-to-proton charge radii is close to its experimental value [32]. Threshold effects have been accounted for within theglobal energy conservation (GEC) scenario introduced inRef. [33] and which has been briefly presented above. Ithas been checked that such a scenario is compatible witha system of nucleons, ∆ (1232)s and pions reaching chemi-cal equilibrium. Specifically, this has been achieved by per-forming numerical checks of detailed balance. To this end,nuclear matter in a box at temperature T=60 MeV has beensimulated using the full model. The initial state of the sys-tem consisted of nucleons and pions with relative multiplic-ity abundances of 90% and 10% respectively. Detailed bal-anced for the reactions N + N ↔ N + ∆ and ∆ ↔ π N wasshown to be fulfilled at a few percent level after a time lapseof about 100 fm/c which signals that chemical equilibriumhas been reached. With appropriate settings the model re-produces the benchmark results of the TMEP Collaboration[40,41,42].2.2 Baryon in-medium interactionsThe same parametrization for the equation of state of nuclearmatter as in [27] is used. The potential part reads EN ( ρ , β ) = A u ρ ( − β ) ρ + A l ρ ( + β ) ρ (8) + B σ + ρ σ ρ σ ( − x β ) + D ρ ρ ( − y β )+ ρρ ∑ τ , τ (cid:48) C ττ (cid:48) (cid:90) (cid:90) d p d p (cid:48) f τ ( r , p ) f τ (cid:48) ( r , p (cid:48) ) + ( p − p (cid:48) ) / Λ . Its analytic form is similar to MDI Gogny-inspired parame-trizations [93,94], but differs from these by an extra density-dependent but momentum-independent term, proportionalto the D parameter, that has been introduced in order toallow independent variations of the slope L and curvature K sym parameters of the symmetry energy, while keeping theneutron-proton isovector effective mass difference fixed.The corresponding single-particle nucleon potential isgiven by U τ ( ρ , β , p ) = A u ρ τ (cid:48) ρ + A l ρ τ ρ (9) + B (cid:16) ρρ (cid:17) σ ( − x β ) + τ x B σ + ρ σ − ρ σ β ρ τ (cid:48) + D (cid:16) ρρ (cid:17) ( − y β ) + τ y D ρρ β ρ τ (cid:48) + C ττ ρ (cid:90) d p (cid:48) f τ ( r , p (cid:48) ) + ( p − p (cid:48) ) / Λ + C ττ (cid:48) ρ (cid:90) d p (cid:48) f τ (cid:48) ( r , p (cid:48) ) + ( p − p (cid:48) ) / Λ . In the above expressions ρ , β and p denote the density,isospin asymmetry and momentum variables respectively. The label τ designates the isospin component of the nucleonand takes the value τ =-1/2 (1/2) for neutrons (protons). Forcold nuclear matter it holds f τ ( r , p ) = ( / h ) Θ ( p τ F − p ) ,with p τ F the Fermi momentum of nucleons with isospin τ .It is common practice, within the framework of transportmodels, to set the resonance potentials in terms of the nucle-onic one. This choice is guided by the decay channels of theresonance in question into a final state comprising a nucleonand a pion [59]. This approach is particularly well suited forthe ∆ (1232) baryon which has a branching ratio close to 1for the ∆ → N π decay channel. It is nevertheless applied tothe entire list of resonances included in the given transportmodel. To be specific, U R τ ( ρ , β , p ) = ( − τ / T ) U − ( ρ , β , p ) (10) + ( + τ / T ) U ( ρ , β , p ) , where T and τ are the isospin and its desired projection forthe resonance in question; U − and U represent the neutronand proton potentials respectively, whose expressions can beread from Eq. (9). For an isospin T =3/2 resonance it leadsto U ∆ − = U − = U is + U iv U ∆ = U − + U = U is + U iv U ∆ + = U − + U = U is − U iv U ∆ ++ = U = U is − U iv , (11)which can be split into iso-scalar and iso-vector contribu-tions, denoted above by U is and U iv . Their expression can bereadily found out to be U is ( ρ , β , p ) = A u + A l ρρ + B (cid:16) ρρ (cid:17) σ ( − x β ) (12) + D (cid:16) ρρ (cid:17) ( − y β )+ C l + C u ρ [ I ( p , p nF ) + I ( p , p pF ) ] , U iv ( ρ , β , p ) = A l − A u ρρ β − x B σ + (cid:16) ρρ (cid:17) σ β (13) − y D (cid:16) ρρ (cid:17) β + C l − C u ρ [ I ( p , p nF ) − I ( p , p pF ) ] , with the following notations: C l = C / , / = C − / , − / , C u = C / , − / = C − / , / , p nF and p pF represent the Fermi mo-menta of neutrons and protons respectively; I ( p , p τ F ) standsfor the integrals appearing in Eq. (9), for which an analyticexpression can be derived for the case of zero-temperaturenuclear matter I ( p , p τ F ) = (cid:90) d p (cid:48) f τ ( r , p (cid:48) ) + ( p − p (cid:48) ) / Λ (14) U nucleonU ∆0 standard choice˜U ∆0 =-100.0 MeV˜U ∆0 =0.0 MeVm ∗ ∆ =0.85m ∗ ∆ =0.45 U ∆0 - stiff density dependenceU ∆0 - soft density dependence Fig. 1
Momentum (left panel) and density dependence (right panel) of ISDP for several choices of depth and effective isoscalar mass, as discussedin the text, compared to the nucleon isoscalar potential with the compressibility modulus set to K =245 MeV. Each explanatory key applies to bothplots. In the left panel, the nucleon potential in symmetric matter U and the standard choice U ∆ almost coincide. In the right, panel the ISDPs U ∆ corresponding to different effective masses show similar density dependence. = π h Λ (cid:34) Λ + p F ( τ ) − p Λ p ln Λ + [ p + p F ( τ )] Λ + [ p − p F ( τ )] + p F ( τ ) Λ + (cid:32) arctan p − p F ( τ ) Λ − arctan p + p F ( τ ) Λ (cid:33)(cid:35) . It can be easily seen that the expression above is an odd func-tion of p τ F . As a result the isoscalar and isovector potentialsabove are even and odd functions in the isospin asymmetryvariable β respectively, as required by charge symmetry.It is worth stressing that in the above equations U is and U iv are identical to the corresponding nucleonic potentials,as a direct consequence of the Ansatz in Eq. (11), and theparameters appearing in their expressions are therefore de-termined by reproducing nuclear matter properties.The nucleonic potential in Eq. (9) can be expanded in aTaylor series in terms of the isospin asymmetry parameteraround the point β =0 U τ ( ρ , β , p ) = U ( ρ , p ) + ∑ i = , ∞ U sym , i ( ρ , p ) ( τ β ) i . (15)The first two terms, U ( ρ , p ) and U sym , ( ρ , p ) , represent thenucleon potential in isospin symmetric nuclear matter andthe first-order symmetry potential respectively. Their expres-sions can be derived from those for the isoscalar and isovec-tor nucleon potentials in Eq. (12) and Eq. (13) using the re-lations U ( ρ , p ) = U is ( ρ , β = , p ) , (16) U sym , ( ρ , p ) = lim β → U iv ( ρ , β , p ) β . Naturally, for the case of the Ansatz used in Eq. (11) we have U R ( ρ , p ) = U ( ρ , p ) while U Rsym , ( ρ , p ) = U sym , ( ρ , p ) oncethe replacement 2 τ → τ / T is made in Eq. (15).In this study we depart from the usually made assump-tion in transport models, briefly presented above, that U is and U iv entering Eq. (11) are the corresponding nucleon po-tentials. We do however assume that their expressions interms of density, isospin asymmetry and momentum are thesame but different values for the coupling parameters. In or-der to make this distinction clear we add a superscript “ ∆ ”to relevant quantities, in particular U ∆ is , U ∆ iv , U ∆ and U ∆ sym , .We allow the freedom that the density and momentum de-pendence of resonance potentials be different at intermedi-ate and long ranges as well as at densities below twice sat-uration density. We do however require, for a standard caselabeled accordingly where distinction is relevant, that theirhigh density part is similar to that of nucleons, in view oftheir similar quark structure. This approach is different fromthe one pursued in Refs. [32,33] where both the isoscalarand isovector components of the DPOT were modified by ascaling factor.In the following we present details of how the values ofparameters entering in Eq. (12) and Eq. (13) are fixed in thisstudy. There are six free parameters entering the expressionof the ISDP U ∆ is : ( A l + A u ) / B , σ , D , C u + C l and Λ (asuperscript “ ∆ ” is in order for each of these parameters, butis omitted). For simplicity we set D =0.0 MeV and σ =1.465.The remaining four are determined by requiring that certain U ∆sym , standard choice˜U ∆sym , =90.0 MeV˜U ∆sym , = 0.0 MeV δ m ∗ ∆ = 0.30 δ m ∗ ∆ =-0.10 U ∆sym , - stiff density dependenceU ∆sym , - soft density dependence Fig. 2
Momentum (left panel) and density dependence (right panel) of the leading order symmetry potential U ∆ sym , . The leading order nucleonsymmetry potential corresponding to L =60.5 MeV and K sym =-81.0 MeV is also shown for comparison. Each explanatory key applies to both plots.In the left panel, the leading order nucleon symmetry potential U sym , and the standard choice IVDP U ∆ sym , almost coincide. In the right, panel theIVDPs U ∆ sym , corresponding to different effective mass differences show a very similar density dependence. values for the isoscalar effective mass of the resonance m ∗ ∆ and the potential in symmetric matter at suitable values fordensity and momentum, ˜ U ∆ ≡ U ∆ ( ρ , p = ) , U ∆ ( ρ , p = ) and U ∆ ( ρ , p = ∞ ) , are described. The quoted value for σ ensures that the density dependence of the resulting ISDPis close to that of the nucleon once the values at the threeabove mentioned points fulfill this requirement too.The expression of the IVDP U ∆ iv contains four addition-ally free parameters: ( A l − A u ) / C l − C u , x and y (again,the “ ∆ ” superscript is omitted). The value of the last oneis irrelevant in the context of setting D =0 MeV. The re-maining three are determined by requiring definite valuesfor ˜ U ∆ sym , ≡ U ∆ sym , ( ρ , p = ) , U ∆ sym , ( ρ , p = ) and theisovector mass-splitting δ m ∗ ∆ = ( m ∗ ∆ − − m ∗ ∆ ++ ) / m ∆ , the lastquantity being evaluated at saturation density and β =0.5.The second order symmetry potential U ∆ sym , impacts the valueof the isovector mass-splitting at a few percent level since itscontribution to the symmetry potential is smaller than 10%irrespective of the value of β .The values for the ten model parameters for the casewhen the DPOT is similar to the nucleon’s up to twice satu-ration density and for kinetic energies up to 1.0 GeV are pre-sented in Table (1). For the isovector part, the quoted param-eter values lead to nucleon in-medium interactions that cor-respond to a density dependence of SE with a slope L =60.5MeV and curvature parameter K sym =-81.0 MeV.In Fig. (1) the momentum and density dependence ofISDP in symmetric nuclear matter U ∆ for several cases is Table 1
Input quantities and their values (first and second columns)used to set the DPOT together with the model parameters appearingin Eq. (12) and Eq. (13) and their determined values (third and fourthcolumns). This set of parameters leads to ISDP and IVDP that resemblethe nucleonic potentials closely. Quantities denoted by capital lettersare expressed in units of MeV, while the rest are dimensionless. Theeffective mass m ∗ ∆ is expressed in units relative to the vacuum value ofthe mass of the ∆ (1232) isobar.Input Parameters m ∗ ∆ Λ U ∆ ( ρ , p = ) -67.0 C l + C u -153.82 U ∆ ( ρ , p = ) -55.0 A l + A u -26.15 U ∆ ( ρ , p = ∞ ) +75.0 B D (fixed) 0.0. σ (fixed) 1.465 δ m ∗ ∆ C l − C u U ∆ , sym ( ρ , p = ) +45.0 A l − A u -109.97 U ∆ , sym ( ρ , p = ) +67.5 x y (fixed) 0.0 presented. The corresponding nucleon potential, U , is alsoshown for reference. A standard U ∆ that corresponds to apotential depth at saturation and zero momentum ˜ U ∆ =-67.0MeV and an isoscalar effective mass m ∗ ∆ = 0.65 has beendefined. It mirrors both the momentum and density depen-dence of the nucleon U potential, as can be seen from the left and right panels of Fig. (1) respectively. Sensitivity of pi-onic observables to U ∆ will be studied by varying its depthat saturation ˜ U ∆ in the interval [-100.0,0.0] MeV and theisoscalar effective mass in the range [0.45,0.85]. The po-tentials corresponding to the limits of these intervals areshown in Fig. (1). Modification of the ISDP depth at satu-ration induces also a drastic change of the density depen-dence. Additionally, two potentials denoted as “stiff densitydependence” and “soft density dependence” are also shown.They have been constructed by modifying the value of thepotential at twice saturation density U ∆ ( ρ , p = ) to -5MeV for stiff and -105 MeV for soft and allowing for a non-zero value of the D parameter while keeping parameters σ and y fixed to the values quoted in Table (1). This proce-dure ensures that the IVDP remains unchanged. The modelparameters have been adjusted such as to modify only thedensity dependence above saturation, while keeping the po-tential depth at saturation and half-saturation (both at zeromomentum) and the isoscalar effective mass fixed. Thesetwo potentials will be used to study the impact of stiff andsoft supranormal density dependence of the U ∆ potential onpionic observables.Similarly, in Fig. (2) the momentum and density depen-dence of the leading order symmetry potential of ∆ (1232) U ∆ sym , is shown. A standard choice U ∆ sym , potential is de-fined by requiring that its strength at saturation and zero mo-mentum is ˜ U ∆ sym , =45.0 MeV and the isovector mass splittingamounts to δ m ∗ ∆ =0.175. The corresponding nucleon poten-tial, that leads to a density dependence of symmetry energywith a slope L =60.5 MeV and curvature parameter K sym =-81.0 MeV, is shown for comparison. Sensitivity of pionicobservables to U ∆ sym , will be studied by varying its strengthat saturation ˜ U ∆ sym , in the interval [-15.0,90.0] MeV and theisovector mass splitting δ m ∗ ∆ in the range [-0.10,0.30]. Alsoin this case, two potentials labeled “stiff” and “soft” densitydependence have been constructed by modifying the poten-tial strength at twice saturation density, while keeping thevalues at saturation and half-saturation fixed, all this at p=0.The choices of U ∆ sym , ( ρ , p = ) equal to 117.5 MeV and17.5 MeV have been made for the stiff and soft cases re-spectively. Technically this was achieved by modifying thevalue of quantity D y (redefined as a variable independent of D ) while keeping D equal to zero.2.3 Benchmarking the nucleonic sectorThe time evolution of heavy-ion collisions at impact ener-gies of a few hundred MeV/nucleon is governed by nucle-onic degrees of freedom. In order to realistically describepion production at these energies it is crucial that nucle-onic multiplicity spectra are accurately reproduced in orderto have the correct invariant mass spectra of two-body col- lisions. To this end, before embarking on a study of pionproduction, a theoretical transport model would have to passthe test of comparing predictions for nucleonic observablesto experimental data. In particular a proper description ofstopping and flow observables is mandatory.In a previous publication theoretical predictions for trans-verse and elliptic flows for Au+
Au at an impact energyof 400 MeV/nucleon were compared to experimental FOPIdata in the context of extracting constraints for the densitydependence of the symmetry energy [27]. In this Section, thetheory-experiment comparison is extended by investigatingstopping and system size dependence of observables in the150 to 1000 MeV/nucleon impact energy range.In the left panel of Fig. (3) theoretical predictions for thestopping observable varxz of protons in central
Au+
Auare presented and compared to experimental data [43]. Theimpact of relevant models ingredients is shown in order toassess model uncertainties. The full model predictions (fullcurve) describe low impact energy data very well. At thehigher end of the incident energy interval a slight underpre-diction is however noticeable. For comparison, full modelpredictions employing a different Pauli blocking algorithmthat estimates occupancy fractions making use of the Gaus-sian wave function associated to each nucleon rather thanthe standard TuQMD algorithm [27] are presented (dashedcurve). The difference is small at all incident energies. Theimportance of threshold effects and the multi-nucleon cor-relations they induce is underlined by comparing the pre-dictions of the model with these effects switched off (dash-dotted curve) to the full model (full curve). Their impact islarger at lower incident energies, the difference between thetwo calculations amounting to about 10%. The magnitudeof the effect is surprising in view of the fact that shifts ofthe invariant mass of the colliding nucleons amounts to afew MeV [33]. At a basic level the effect is a consequenceof stronger energy dependence of elastic collision and thenucleon optical potential at lower incident energies.The impact of in-medium modifications of cross-sectionsis demonstrated by switching off these effects to inelasticchannels and then additionally also to the elastic ones. Asexpected, in-medium modifications of inelastic cross-sectionaffect stopping observables only above 500 MeV/nucleonimpact energy. The Ansatz of relating these medium correc-tions to effective masses induces an energy dependence of varxz that deviates visibly from the experimental one eventhough the absolute magnitudes are still reproduced. At lowimpact energies, modifications of elastic nucleon-nucleoncross-sections are crucial to describe experimental data anda momentum dependence of these effects appears to be manda-tory. Similar conclusions have been reached in other studies[80,82,83].The same analysis has also been performed for deute-ron and triton stopping in
Au+
Au collisions for which
Fig. 3 (Left Panel) Model dependence for proton stopping in central AuAu collisions as a function of the impact energy per nucleon. (Right Panel)System size dependence of stopping for protons in central collision for systems of different masses. The FOPI experimental data [43] have beenplotted for comparison. experimental measurement are available [43]. The relevanceof the above discussed model ingredients remains similar,however the overall description of the experimental data ispoorer. Deuteron stopping is under-predicted by approxi-mately 15%, while for tritons the deviation increases to 35%.This is not surprising in the context of triton multiplicitiesbeing under-estimated by a factor of about 2 [27] by the mo-del. Switching off in-medium effects on cross-sections re-duces the discrepancy considerably but the induced energydependence of the observable at the lower limit for the im-pact energy is not realistic.The right panel of Fig. (3) presents predictions of thefull model for proton stopping in central collisions for threedifferent systems:
Au+
Au, Ni+ Ni and Ca+ Ca.Experimental results at impact energies for which data areavailable [43] are also shown. A generally good agreementbetween theory and experiment is observed.Turning to elliptic flow, in the left panel of Fig. (4) pre-dictions for transverse momentum dependent elliptic flowof protons in
Au+
Au collision at an impact energyof 250 MeV/nucleon are presented. Similarly as for stop-ping, the impact of certain model ingredients is shown. Onlyin-medium modifications of elastic cross-sections lead to asignificant departure from the full model predictions, whilethreshold effects and different approaches of computing thenucleon occupancy have a negligible impact. The full modelis in almost in perfect agreement to the corresponding exper-imental data [95]. By comparing the left panels of Fig. (3)and Fig. (4) it is evident that a simultaneous description of both stopping and elliptic flow is not possible by solely in-troducing in-medium modifications of elastic cross-sections.The inclusion of threshold effects appears almost indispens-able. As the incident energy is increased the impact of in-medium modifications of elastic cross-sections on ellipticflow decreases, a good description of the experimental datais still achieved [27]. Investigation of elliptic flow of deuteronsand tritons has lead to the same conclusions.The right panel of Fig. (4) presents predictions for ra-pidity dependent elliptic flow at an impact energy of 400MeV/nucleon for three systems:
Au+
Au, Ru+ Ruand Ca+ Ca. Experimental data are available only forthe first and third systems [95]. An excellent description of
Au+
Au data is observed, the strength of the predictedelliptic flow of protons for Ca+ Ca collisions is slightlyweaker than the experimental one. A similar picture is validfor the elliptic flow of deuterons for the same reactions.A similar study has been performed for transverse flow.None of the model ingredients studied above have a signifi-cant impact for this observable and consequently the qualityof the description of the experimental data is similar to thatof Ref. [27] for all impact energies in the range of interestand for all light cluster species for which experimental datahave been reported in Ref. [95]. ∆ (1232) potential on pionic observables The magnitudes of ISDP and IVDP are poorly known atbest, as already emphasized in previous sections. It is thus Fig. 4 (Left Panel) Model dependence for elliptic flow of protons. Theoretical curves have the same meaning as those in the left panel of Fig. (3).(Right Panel) System size dependence of elliptic flow of protons. Experimental data are taken from Ref. [95]. mandatory to identify a sufficient number of observables toextract both the values for the parameters used to fix thesepotentials and those describing the density dependence ofthe EoS. Originally, the charged pion multiplicity ratio wasproposed as an observable for extracting the value of theslope parameter L of SE [21]. In a previous publication [32]the average transverse momentum of charged pions was usedto constrain the strength of the IVDP relative to the nucleonsymmetry potential. This observable has however been pro-ven very sensitive to the pion optical potential. To avoid ad-ditional model dependence we will restrict the present studyto multiplicity related observables only. An obvious choiceis the total charged pion multiplicity, for which experimen-tal data exist for several systems at various impact ener-gies [43]. Owing to the limited applicability of the approx-imations used in taking into account threshold effects [33]the upper limit of the impact energy will be restricted to800 MeV/nucleon. Consequently the following available ex-perimental data for given systems and impact energies inMeV/nucleon can be used: Ca Ca (400, 600, 800), Ru Ru(400), Zr Zr (400) and Au Au (400, 600, 800). Thesesystems have the following average isospin asymmetry: 0.0(CaCa), 0.08 (RuRu), 0.17 (ZrZr) and 0.20 (AuAu) allowingthe study of both ISDP and IVDP. The rather broad range ofimpact energies will also facilitate the study of their momen-tum dependence. In the near future experimental data withsignificantly better accuracy for Sn Sn, Sn Sn and Sn Sn at an impact energy of 270 MeV/nucleon, slightlybelow threshold, will become available [96] and potentiallyprovide tighter constraints. 3.1 Relevant observablesFor each system two independent observables can be con-structed from charged pion multiplicities: total charged pionmultiplicity (PM) and charged pion multiplicity ratio (PMR).For two systems at the same impact energy, one neutron richand one neutron deficient, two observables, dependent onthe two PMs and two PMRs can be defined: the ratio of totalcharged pion multiplicities and double charged pion multi-plicity ratio. Each of them are useful in studying the impactof DPOT on pionic observables.In the top panels of Fig. (5) the sensitivity of PM toDPOT parameters, compressibility modulus of symmetricnuclear matter and slope L of the symmetry energy is pre-sented. The standard choice for the six mentioned param-eters is (see also Table (1)): ˜ U ∆ =-67.0 MeV, ˜ U ∆ sym , =45.0MeV, m ∗ ∆ =0.65, δ m ∗ ∆ =0.175, K =245 MeV and L =60 MeV.The values of the first four parameters lead to ISDP andIVDP that resemble closely the corresponding ones of nu-cleon. The calculations presented in the figure were per-formed by varying the indicated parameter within a reason-able interval, as represented by the abscissa of the corre-sponding plot, while the values of the other five parametersare kept unchanged to their standard ones. The PM displaysconsiderable sensitivity to the isoscalar potential depth atsaturation ˜ U ∆ and the value of the isoscalar effective mass m ∗ ∆ . There is a comparably much smaller sensitivity, of theorder of 10-20%, to the strength of IVDP at saturation ˜ U ∆ sym , and to the value of the compressibility modulus. The sen-sitivities to the remaining two parameters, isovector mass L [MeV] K [MeV] ˜U ∆0 [MeV] m ∗ ∆ ˜U ∆sym , [MeV] δ m ∗ ∆ Fig. 5
Dependence of the total charged pion multiplicity in CaCa collisions (top panels), ratio of charged pion multiplicities of AuAu to CaCa(middle panels) and double charged pion multiplicity ratio of AuAu to CaCa (bottom panels) on ∆ (1232) potential depths at saturation ˜ U ∆ is , ˜ U ∆ iv ,effective mass parameters m ∗ ∆ and δ m ∗ ∆ , compressibility modulus of symmetric nuclear matter K and slope of symmetry energy at saturation L forcentral ( b < difference δ m ∗ ∆ and slope parameter L are negligibly small.The sensitivity to ˜ U ∆ sym , is small, but not negligible, for the Ca Ca system. In fact the smallest sensitivity to this pa-rameter was found for Ru Ru. The slope of the depen-dence of PM on ˜ U ∆ sym , for Au Au has an opposite signto that derived from the top panel of Fig. (5) for Ca Ca.This suggests that the net effect is the result of two oppositetrends related to the average isospin asymmetry and fluctu-ations. It is concluded that PM is suitable to fix the param-eters of ISDP. Secondary order corrections due to IVDP arenot negligible and will have to be accounted for.In the middle panels of Fig. (5) a similar analysis is pre-sented for the ratio of total charged pion multiplicities of Au Au to Ca Ca at impact energy of 400 MeV/A.The important feature of this observable is that the hugedependence on ISDP evidenced for total pion multiplici-ties of individual systems almost cancel out, with a remain-ing residual sensitivity of about 10%. The dominant varia-tions are related to the isovector potential depth at saturation˜ U ∆ sym , and isovector effective mass difference δ m ∗ ∆ . Sensi-tivity to the equation of state parameters K and L is also inthis case limited to about 10%. Constraining the isovectorpotential parameters using this observable appears feasiblebut model dependence is not negligible due to relatively im-portant sensitivity to other parameters. Using the standardvalues for model parameters the calculation overestimates the experiment by 30%. Varying IVDP parameters withina conservative interval does not allow a mitigation of thediscrepancy. An investigation of this issue suggests that apossible resolution may involve modifications of in-medium ∆ production cross-sections, a stiff density dependence ofIVDP above saturation (see below) or a larger positive valuefor the neutron-proton effective mass difference (the stan-dard choice being 0.33 β ).The third sensitivity study was performed for the doublecharged pion multiplicity ratio of Au Au to Ca Ca at400 MeV/nucleon impact energy. The results are presentedin bottom panels Fig. (5) for the same model parameters asabove. For this observable the sensitivity to each of the cho-sen model parameters is sizable. This is the only observ-able of the three that shows sizable dependence to asy-EoS.However, the extraction of the value of L is impeded by theunknown values of DPOT parameters. Setting this quantityequal to that of the nucleon has been in the past a choice ofconvenience that generally does not lead to a good descrip-tion of all available experimental data. The alternative ap-proach of modifying in-medium ∆ production cross-sectionis restricted by existing microscopical models [85] that sug-gest that such effects are largely governed by scaling lawsinvolving in-medium effective masses. Such effective mod-ifications of inelastic cross-sections have been included inthe present model. U ∆0 - standardU ∆sym , - standard U ∆0 - soft/stiffU ∆sym , - standard U ∆0 - standardU ∆sym , - soft/stiff Fig. 6
Dependence of the total charged pion multiplicity in CaCa col-lisions (top panel), ratio of charged pion multiplicities of AuAu toCaCa (middle panel) and double charged pion multiplicity ratio ofAuAu to CaCa (bottom panel) on the density dependence of U ∆ and U ∆ sym , potentials above saturation for central ( b < In Section 2.2 soft and stiff density dependent ISDPs andIVDPs have been constructed and displayed in Fig. (1) andFig. (2) respectively. In Fig. (6) their impact on the three ob-servables discussed above is presented and compared withpredictions for the “standard” case potential whose param-eters values are listed in Table (1). Modifying the densitydependence of ISDP above saturation has an impact on to-tal charged pion multiplicities (PM) of at most 10%, a softerdensity dependence of the potential leading to an increase oftotal pion multiplicities. The ratio of total charged pion mul-tiplicities and the double charged pion multiplicities ratio ofAuAu to CaCa at 400 MeV/nucleon are only marginally af-fected by a soft/stiff density dependence of ISDP above sat-uration. Modification of the density dependence of the IVDPhas a visible impact on all three observables. PM is affectedat 5% level, the impact on PM(Au)/PM(Ca) is close to 10%and the effect on the double ratio PMR(Au)/PMR(Ca) is thelargest at 20%. Comparison with Fig. (5) reveals that thesensitivity to the density dependence of IVDP above satu-ration is several times smaller than to the magnitude of thesymmetry potential at saturation ˜ U ∆ sym , . The same observa-tion is true also for ISDP by an even larger margin. Thisprovides an a posteriori justification for the choice of pa-rameters used in this study to fix the density dependence ofDPOT: its strength at saturation and at twice saturation den-sity. Sensitivity of each of the three observables has also beenstudied with respect to the following ingredients of the trans-port model: pion potential, neutron-proton effective massdifference and in-medium modification of both elastic andinelastic cross-sections. The impact is found to lie in the in-terval 5-10% in each case, with the exception of mediummodifications of cross-sections that impact total pion multi-plicities at the level of 20% for light systems such as CaCa.It is concluded that the three observables can be used to ex-tract information on the strength of the ISDP and IVDP atsaturation ˜ U ∆ and ˜ U ∆ sym , and the isoscalar effective massof ∆ (1232) in nuclear matter m ∗ ∆ . Extracting constraints for δ m ∗ ∆ is feasible but dependence of results on other modelparameters (such as the slope parameter of asy-EoS and thedensity dependence of IVDP above saturation) cannot be ne-glected.Constraints for the density dependence of DPOT abovesaturation are necessary for a complete knowledge of thisquantity. The results in this Section do however prove thatthis is not feasible at present given the small impact on thestudied observables, which is comparable or even smallerthan the uncertainties of other not precisely known model in-gredients such as pion potentials or in-medium cross-sections.Consequently, in the following we will only present the im-pact of the high density dependence of DPOT on constraintsfor ˜ U ∆ , ˜ U ∆ sym , , m ∗ ∆ and δ m ∗ ∆ rather than attempting to extractvalues for U ∆ and U ∆ sym , at 2 ρ .A similar study has been performed for momentum re-lated observables. Also in this case the impact of the DPOTis sizable but of comparable relative magnitude to that ofthe pion optical potential. Using such observables would in-duce important model dependence of results, as already ev-idenced in Ref. [32], and will not be pursued here.3.2 Constraining ∆ (1232) potential parametersUsing the insights of the previous Section we proceed to ex-tract constraints for the values of DPOT parameters. Resultsfor the isoscalar component are presented in Fig. (7) as cor-relations between the isoscalar potential depth at saturation˜ U ∆ and isoscalar effective mass m ∗ ∆ . To fix this quantity theavailable experimental data comprise those of the following(nearly) isospin symmetric systems: CaCa at 400, 600 and800 MeV/nucleon and RuRu at 400 MeV/nucleon centralcollisions [43]. In the left panel constraints for ISDP param-eters extracted from collisions of CaCa at 400 MeV/nucleonare presented in the form of 1- σ confidence level contourplots. Besides a calculation employing the full model, cer-tain model ingredients have been modified or switched offto test model dependence. Three additional simulations cor-respond to the full model making use of a Pauli blockingalgorithm based on computation of the occupancy fraction using the Gaussian wave function associated to each nu-cleon [27], full model without the pion potential and fullmodel with vacuum inelastic cross-sections. The first twolead to results compatible with the full model, while for thethird the deviation is more important as a consequence oftotal pion multiplicities being impacted at 20% level by in-medium modifications of inelastic cross-sections. For heav-ier systems (such as AuAu) or light systems at higher im-pact energies the effect of medium modification of inelas-tic cross-sections on multiplicities is smaller, in the 10-15% range. Additionally, constraints extracted using the softand stiff density dependent U ∆ potentials above saturation,introduced in Section 2.2, are also presented. The impactof modifying the strength of U ∆ at 2 ρ is small, similar inmagnitude to the effect due to the pion potential. Calcula-tions using a soft/stiff density dependence of U ∆ sym , are notshown, however results in Fig. (6) allow the inference thattheir impact is of similar small magnitude as for U ∆ . Conse-quently the PM observable can only be used to study the U ∆ potential close or below saturation by determining values for˜ U ∆ and m ∗ ∆ .In the attempt to pin down the momentum dependenceof ISDP, simulations of collisions at different impact en-ergies have been performed and compared to experimentaldata. Constraints for ˜ U ∆ and m ∗ ∆ are presented in the rightpanel of Fig. (7). Simulations for RuRu at 400 MeV/nucleonhave also been performed and since total multiplicity for thissystem has displayed the smallest sensitivity to the isovec-tor component of the potential, as previously mentioned, ithas been added to the comparison. It is evident that the 1- σ CL contour plots for CaCa at different impact energies haveslightly different slopes in the ( ˜ U ∆ , m ∗ ∆ ) plane, converging atsmaller values for m ∗ ∆ . A combined fit of the four reactions issub-optimal with a minimum value of χ /point=2.55. As hasbeen evidenced in the upper panels of Fig. (5) there is stillnon-negligible residual dependence on the IVDP strength. Itwas found that it affects total pion multiplicities of CaCa by15, 10 and 5 % at 400, 600 and 800 MeV/nucleon impactenergies respectively. Once this is taken into account a cor-rected combined fit with a minimum value of χ /point=0.80is obtained. The combined fit of the four reactions has some-what restricted the possible values for ˜ U ∆ and m ∗ ∆ , definitevalues could however not be extracted in part because theexperimental data carry rather large uncertainties but alsobecause at higher impact energies the sensitivity decreases.Near-future availability of experimental data slightly belowpion production threshold for the nearly isospin symmet-ric Sn Sn system by the SPIRIT collaboration may im-prove the present situation significantly. In Fig. (7) the val-ues of parameters leading to an ISDP equal to that of thenucleon are depicted by a star symbol. It departs from the fa-vored parameter values of the combined fit for the four sys-tems at more than 5- σ CL. However, within this approach it is not clear whether this is a model independent conclusionas the favored values for the potential parameters may bethe result of the fit compensating for some drastic approxi-mations.In Ref. [47] the strength of the DPOT was extracted fromthe study of experimental data of quasi-elastic scattering ofelectrons on bound nucleons in nuclei of different masses: Li, C, Al, Ca/Ar and Fe. It was found that the DPOTis more attractive than the empirical nucleon optical poten-tial and the attraction is stronger for heavier nuclei reflectinghigher probed densities. The attraction is stronger by about20 MeV at momenta close to p =0.5 GeV/c for the heaviestnuclei for which the analyses was performed. Additionally,an arguably stronger energy dependence was evidenced formomenta significantly above the Fermi sea, which may sug-gest a lower isoscalar effective mass of ∆ (1232). A previoussimilar study [46] has reached similar conclusions. Thesequalitative results are in full agreement with findings of thepresent study for the ISDP, as shown in the right panel ofFig. (7) due to a similar similar approach.Comparison with microscopic calculations reveals im-portant differences. Many-body calculations of pion-nucleusscattering or absorption performed in the framework of theDelta-hole model arrived at the conclusion of an ISDP lessattractive than that of the nucleon at saturation density: ˜ U ∆ ≈ -30 MeV [48,49,50,51]. Contributions such as non-resonantbackground pion production, the spin-orbit component ofthe ∆ potential and short-range corrections to interactionvertices are crucial for the quoted result. From the upperpanels of Fig. (5) it is evident that by inclusion of non-resonant background contributions to pion production in thescattering term of the transport model a less attractive ISDPwill be favored. Ab-initio calculations, that have used wellestablished microscopical nucleon-nucleon potentials (suchas Argonne v ) as input, performed within the framework ofthe Bethe-Brueckner-Goldstone method [53] or one-bosonexchange nucleon-nucleon potentials in the relativistic Dirac-Brueckner model allowing a good reproduction of the elasticpion-nucleon P phase-shift [52], have arrived at a mildlyrepulsive ISDP. This is in part due to dominant repulsivecontributions of total isospin I=2, a channel which cannotbe sufficiently constrained by elastic nucleon-nucleon scat-tering data.To proceed to extraction of constraints for IVDP param-eters ˜ U ∆ sym , and δ m ∗ ∆ specific choices need to be made forISDP parameters. The choice ˜ U ∆ =-78 MeV and m ∗ ∆ =0.45allows, as evidenced in the right panel of Fig. (7), a gooddescription of pion multiplicity for isospin symmetric (ornearly so) systems. In Fig. (8) the favored values for ˜ U ∆ sym , and δ m ∗ ∆ , resulting from comparing theoretical and exper-imental values of PMR for central AuAu collisions at 400MeV/nucleon incident energy, are shown as 1- σ CL contourplots. Results for three different values of the slope parame- m ∗ ∆ ˜U ∆0 [MeV] Density Dependence U ∆0 :DefaultSoftStiff CaCa @ 400 MeV/A m ∗ ∆ ˜U ∆0 [MeV] Case: Med Inel CS + SP Pion Pot
CaCa @ 400CaCa @ 600CaCa @ 800RuRu @ 400combined fitcorr comb fit
Fig. 7 (Left Panel) Model dependence of constraining ISDP depth at saturation density and zero momentum ˜ U ∆ and isoscalar effective mass m ∗ ∆ for central CaCa collisions at 400 MeV/nucleon. Results for six different cases are shown: full model, different Pauli blocking algorithm, no pionpotential, no in-medium effects on inelastic channels cross-sections and soft/stiff density dependence of U ∆ above saturation. The star representsthe choice of potential parameters that would render the ISDP equal to nucleon’s. (Right Panel) Constraints for the same parameters by making useof the FOPI experimental data [43] for central collisions of RuRu at 400 MeV/nucleon and CaCa at 400, 600 and 800 MeV/nucleon. The combinedresult for the four reactions is represented by the contour plot labeled “combined fit”. For the contour plot labeled “corr comb fit” corrections dueto sensitivity to IVDP have been applied, as described in the text. All contour plots correspond to 1- σ confidence level of fitting total charged pionmultiplicity. ter L of SE are shown, evidencing an important dependenceon this parameter. Additionally, the soft and stiff density de-pendent U ∆ sym , introduced in Section 2.2 lead to differencesin the extracted constraints of similar magnitude as thoseinduced by L . It has been verified that the extracted con-straints are sensitive also to the value of the neutron-protoneffective mass difference. Results obtained by fitting the ex-perimental value of PMR for ZrZr central collisions at 400MeV/nucleon are also shown for L =60 MeV and the stan-dard density dependence above saturation. They are compat-ible with the corresponding ones for AuAu. Adding the to-tal multiplicity of charged pions for these isospin asymmet-ric systems in the fit, slightly restricts the allowed ranges,by disfavoring regions of higher values for ˜ U ∆ sym , . The starsymbol represents the choice for the potential parametersthat would lead to an identical isovector potential for nucle-ons and ∆ (1232) isobars. Constraints extracted for a widelyused value of the slope parameter, L =60 MeV, depart fromthis commonly made choice, but by a smaller margin com-pared to the isoscalar case.Fitting available experimental data of PMR for AuAuat higher impact energy does not bring additional informa-tion mainly due to the larger experimental error for this ob-servable. The second observable of interest for constrainingthe isovector ∆ (1232) potential, the ratio of total charged- pion multiplicities, has proven ineffective, nearly half of theprobed parameter space in Fig. (8) leading to theoretical pre-dictions in accord to experiment.It becomes clear that a unique extraction of DPOT usingmultiplicity observables alone is not possible at present. Inprinciple, the analysis can be extended to include existinginformation related to momenta of pions. Published resultsfor the ratio of average p T of charged pions exists in theliterature [43] and have been used for this purpose in thepast [32]. The additional induced model dependence fromthe isoscalar channel, as shown in the left panel of Fig. (7),is not negligible and the extraction of the stiffness of SEwill carry an even larger model dependence. Determiningthe slope of the SE is in principle possible by performinga five parameter fit of multiplicity and momentum relatedobservables. This avenue has been explored. The resultingvalue for L does however carry large uncertainties.Close to the pion production threshold it is possible topartially avoid the issue of the not uniquely extracted DPOT.By fitting experimental multiplicities the 4-dimensional pa-rameter space fixing DPOT at saturation is projected ontoa 2-dimensional subspace. For any choice of the remainingtwo unconstrained parameters pion spectra are almost iden-tical. This is a consequence of the fact that close to threshold ∆ degrees of freedom have no impact on the time evolution δ m ∗ ∆ ˜U ∆sym , [MeV] AuAu@400
Dens Dep U ∆sym , :DefaultSoftStiff ZrZr@400
Fig. 8
Constraints for the IVDP parameters, potential depth at satura-tion density and zero momentum ˜ U ∆ sym , and isovector mass difference δ m ∗ ∆ , extracted from FOPI experimental data [43] for PMR in centralZrZr and AuAu collisions at 400 MeV/nucleon. The star represents thechoice of potential parameters that would render IVDP equal to nu-cleon’s. All contour plots correspond to 1- σ confidence level. ˜U ∆0 m ∗ ∆ ˜U ∆sym , δ m ∗ ∆ Fig. 9
Numerical proof that any choice for DPOT parameters, cor-responding to both the isoscalar and isovector potentials, that lie onthe 2-dimensional subspace of the 4-dimensional parameter space thatresults from fitting experimental total charged pion multiplicities andpion multiplicity ratios lead to simulated spectra are the same up to un-certainties induced by experimental data uncertainties. Parameter val-ues for five such choices together with the theoretical multiplicity andPMR p T spectra are shown for mid-central (0.25 ≤ b ≤ K =245 MeV and L =60 MeV). of the reaction in view of their scarcity. Consequently, nu-cleon spectra, which determine the distribution of invariantmasses at which inelastic collisions take place, remain forall practical purposes unaffected by the depth or momentumdependence of DPOT. At these energies the DPOT plays therole of normalization constants (zeroth order moments) forthe spectra, allowing for a reduction of model dependence ofhigher order moments. Fitting pion multiplicities will thuspreserve any asy-EoS dependence of these quantities. Thesituation at higher impact energies, close to 1 GeV/nucleonand above, where the fraction of nucleons excited to reso-nances in the high density fireball is non-negligible [97], isdifferent.Results of numerical calculations, that prove invarianceof spectra to arbitrary choices of model parameters in the2-dimensional subspace left unconstrained after experimen-tal multiplicities have been fitted, are presented in Fig. (9).A four dimensional fit for PM and PMR for mid-centralAuAu collisions at 400 MeV/nucleon has been performed.Five combinations of parameter values for ˜ U ∆ , m ∗ ∆ , ˜ U ∆ sym , and δ m ∗ ∆ for which the fit is perfect ( χ /point=0.0) havebeen chosen such that the sets of values are diverse. Thesechoices are listed in the legend of Fig. (9). The resulting totalmultiplicity and pion multiplicity ratio spectra as a functionof the transverse momentum p T are shown. They are defi-nitely close to each other though not identical. Differencesare due to experimental accuracy of these observables thatwere used to compute the value of χ /point and to the in-terpolation in a 4-dimensional parameter space using a verylimited number of points (3 for each dimension) spanninga rather large parameter space. Nevertheless, the spectra arefor the majority of cases within a few percent of each other.Means of improving this numerical proof are obvious andwith predictable results. The standard density dependenceof U ∆ and U ∆ sym , above saturation has been used. Extendingthe fit to a 6-dimensional one, thus including two additionalparameters that can be used to change U ∆ ( ρ , p = ) and U ∆ sym , ( ρ , p = ) will lead quantitatively to the same mul-tiplicity and single ratio spectra.In practice the following approach will be used. Twoparameters of the DPOT, m ∗ ∆ and δ m ∗ ∆ , will be set to wellchosen values. The remaining two, ˜ U ∆ and ˜ U ∆ sym , will beuniquely determined from a fit to experimental data for PMand PMR. It should be stressed that such a procedure de-stroys the predictive power of the model. The determinedset of parameters can only be used for the particular com-bination of systems and impact energies used in the fit. Nofirm conclusions can be drawn from a possible description(or failure to do so) of a different system. In the next Sectionthe choice m ∗ ∆ =0.45 and δ m ∗ ∆ =0.0 will be used. Fig. 10 (Left Panel) Pion multiplicity and (Right Panel) pion multiplicity ratio p T dependent spectra obtained after fitting DPOT parameters tomultiplicity observables for three values of the SE slope parameter L and δ m ∗ np =0.33 β . For each value of L the impact on spectra of addingto the fitted observables also transverse momentum related ones (average combined transverse momentum and average p T ratio) is also shown.The simulations correspond to mid-central AuAu collisions at 400 MeV/nucleon impact energy. Unpublished FOPI experimental data [98] arerepresented by full circle symbols (left panel) and band (right panel), the shown uncertainties being statistical. . < b < .
45) AuAu collisions at anincident energy of 400 MeV/nucleon have been performed.The sole motivation for the choice of this system has beenthe availability of experimental data. Nevertheless, this dataset is preliminary and does not account for systematical un-certainties, only statistical uncertainties being depicted forexperimental data in figures of this Section. They have beenemployed in previous similar studies [31,32] and can be use-ful in estimating the feasibility of studying the symmetry en-ergy using this observable and the accuracy of the transportmodel.In Fig. (10) a comparison of model prediction with ex-perimental p T dependent individual pion multiplicity (leftpanel) and PMR (right panel) spectra is presented. One setof calculations (full curves) correspond to DPOT parame-ters extracted from a fit to PM and PMR observables, as de-scribed in Section 3.2. Simulations for which DPOT param-eters have been determined from a fit of both multiplicityand average transverse momentum observables are also dis-played (dashed curves). The fitted momentum observablesare: average transverse momentum of charged pions (cid:104) p cT (cid:105) = M π − (cid:104) p π − T (cid:105) + M π + (cid:104) p π + T (cid:105) M π − + M π + and average transverse momentum ra- tio (cid:104) p π + T (cid:105)(cid:104) p π − T (cid:105) . Each observable contributes to the total χ withthe same weight. For each set, calculations for three valuesof the slope parameter of SE are provided: L =15, 60 and106 MeV. The value for the neutron-proton effective massdifference has been set to its default value δ m ∗ np =0.33 β .The left panel of Fig. (10) presents calculations for π − (top plot) and π + (bottom plot) multiplicity spectra. Differ-ences between the two sets of calculations are largest for π + spectra at low and intermediate p T . Theoretical predic-tions for π + spectra are seen to deviate by important marginsfrom experimental data at large values of p T . Uncertaintiesin other model parameters, such as the neutron-proton ef-fective mass difference δ m ∗ np , may explain this discrepancy(see below). Finer tuning of the symmetric part of EoS, inparticular the compressibility modulus K and nucleon iso-scalar effective mass, preserving a consistent description ofnucleonic observables, may also improve the description athigh p T spectra of both π − and π + mesons.In the right panel of Fig. (10) the PMR p T dependentspectrum is presented. Theoretical predictions display sen-sitivity to L for all values of p T , in relative terms they arelargest at higher transverse momenta where predictions forthe stiffest and softest choices of asy-EoS differ by a factorof almost 2. At higher p T values the two sets of predictionsare nearly identical suggesting that this range of transversemomenta is free of model dependence originating in left-over uncertainties of the DPOT. The two sets of calculationsbecome similar to each other for values of p T for whichthe strength in the multiplicity spectrum is below 10% ofits peak value. The inclusion of momentum observables in the fit doesnot allow for a perfect fit,
Min ( χ ) =
0, to be obtained any-more, but the the minimum of the merit function dependson other model parameters such as L or the strength of theoptical pion potential. The quality of the fit when momen-tum observables are included can in principle be improvedby also varying m ∗ ∆ and δ m ∗ ∆ , rather than using the valuesmentioned at the end of Section 3.2. In practice the discrep-ancy between model and experiment at low/moderate p T isreduced only modestly at the expense of performing a 4-dimensional fit (explicit calculations have been performedfor the L =60 MeV case). The reason lies in the fact thatto describe the spectra, moments of p T multiplicity distri-butions of order larger or equal to 1 need to be describedby the model. The performed 4-dimensional fit only ensuresthat the 0 th order moments are reproduced. In principle thisapproach can be used to constrain the asy-EoS parameters,but given the strong dependence of momentum observableson pion optical potentials, constraints extracted in this man-ner are rather imprecise [32], as already argued in the previ-ous Section. The high p T region appears thus better suitedfor studies of the SE. This will become clearer after othersources of model dependence of predictions in this regionwill be addressed below.The sensitivity of PMR spectra to other two parametersof the EoS, δ m ∗ np and value of SE at ρ =0.10 fm − , is pre-sented in Fig. (11). The former quantity is varied within arange that includes most constraints for its value availablein the literature: -0.33 β < δ m ∗ np < . β [64]. Theoreticalcalculations reveal that the sensitivity to this parameter isalmost as large as to the slope parameter of SE. This is ahardly surprising result since a different momentum depen-dence of the interaction results in different magnitudes ofthreshold shifts which in turn have been previously shownto have a large impact on PMR [31,33]. To our best knowl-edge the impact of δ m ∗ np on PMR has not been previouslyaddressed, which may have contributed to a certain extentto the conflicting results for the density dependence of SEobtained using this observable.The latter parameter represents a substitute to fixing thesymmetry energy at saturation, a point where it is not ac-curately known at present. It has been however possible toextract precise values at sub-saturation densities from ex-perimental data of static properties of nuclei [6,8,9]. Suchempirical findings are in good agreement with many bodycalculations of the neutron matter EoS that use as input mi-croscopical N LO chiral perturbation theory effective poten-tials [12,13,14]. The empirical value S( ˜ ρ )=25.5 MeV, with˜ ρ =0.1 fm − , extracted in Ref. [8] has been used as part of thestandard input to the model. The sensitivity to this parameterhas been studied by varying it in the extremely conservativeinterval 22.5 MeV < S( ˜ ρ ) < Fig. 11
Sensitivity of the p T dependent PMR spectra to the magnitudeof neutron-proton effective mass difference δ m ∗ np and value of the sym-metry energy at ρ =0.10 fm − . The same details regarding the reactionas for Fig. (10) are in order. Fig. 12
Model dependence PMR spectra to the pion potential and in-medium effects on inelastic cross-sections. The same details regardingthe reaction as for Fig. (10) are in order.
If the uncertainty of 1.0 MeV quoted in Ref. [8] for S( ˜ ρ ) istaken into account as a more realistic interval of variation,the sensitivity drops to a few percent.The sensitivity of results to a few extra model ingredi-ents has additionally been studied. In Fig. (12) the impactof modifying the pion optical potential, either by choosinga different S-wave optical potential or discarding it com-pletely, and switching off in-medium effects on inelastic cross-sections in PMR spectra is shown. In relative terms the im-pact of these model ingredients is largest at low p T values.Nevertheless, in the high p T region, of interest for SE stud-ies, a 10% effect is still observed. Additionally it has been < b < < y/y P < Fig. 13 (Left Panel) Density dependence of the symmetry energy for the three values of L for which results were shown in Fig. (10). For the L =60MeV case two additional EoS’es that have the same density dependence below saturation as the standard one but have different values for the slopeparameter L above that point are shown. (Right Panel) Transverse momentum dependent PMR spectra for the three EoS’es with L =60 MeV butwith different values of the slope above saturation for mid-central AuAu collisions at various impact energies. As the impact energy is increasedthe transverse momentum above which spectra are insensitive to residual DPOT dependence (see Fig. (10)) also increases, requiring computationof spectra up to higher values of this variable. For AuAu collisions at 300 MeV/nucleon impact energy, the total charged pion multiplicity andratio are determined by extrapolating existing experimental data for central AuAu reactions [43] which leads to the approximate values of 1.0 and4.25 for the two observables respectively. To obtain the corresponding result for mid-central collisions the experimentally observed fact that pionmultiplicity divided by the number of participants is constant as a function of impact parameter is used. investigated what the impact on PMR spectra of setting theDPOT equal to nucleon’s or to a rather arbitrary strength( ˜ U ∆ =-25 MeV, m ∗ ∆ =0.85, ˜ U ∆ sym , = 0 MeV and δ m ∗ ∆ =-0.15)would be (not shown). In either case the deviation from thestandard full model calculation in Fig. (12) amounts to 20%in the high p T region. Fitting multiplicity observables to ex-tract DPOT parameters is thus a minimum requirement tokeep model dependence at reasonable levels.The results presented above lead to the conclusion thatstudies of PMR cannot provide a constraint for the densitydependence of SE but rather a correlation of the parame-ter used to adjust its stiffness (here the slope L ) with thevalue of δ m ∗ np . To lift this degeneracy an independent con-straint or information for the latter quantity needs to be pro-vided from other sources. Elliptic flow ratios of neutrons-to-charged particles, double ratio of n / p multiplicity spectraand dipole polarizability of nuclei have been identified aspromising such sources [27,11,66]. To minimize model de-pendence, a third observable providing a constraint for thenucleon isoscalar effective mass may be required. 4.2 Probed density and impact energy dependencePMR ratio has been proposed as a probe of the density de-pendence of SE above saturation. A few studies that addressthis question are available [99,100], but neither of these mod-els include threshold effects. A proof that pion productionprobes densities significantly above saturation is providedin the following. In the left panel of Fig. (13) the densitydependence of SE for the three choices of L employed inthis Section is presented. Two additional EoS’es that lead todifferent density dependence above saturation for the L =60MeV case have been constructed by modifying the slope pa-rameter above saturation to L =100 MeV and L =20 MeV toreproduce a stiff and a soft density dependence above thispoint respectively. The three L =60 MeV EoS’es have identi-cal density dependence below saturation enforced by usingin each case also a common value for the curvature parame-ter K sym .To avoid numerical problems generated by discontinu-ous derivatives of the SE with respect to density, model pa-rameters that govern the density dependence of SE become C functions of this variable in a narrow interval around ρ ,its width being set to 0.05 ρ . As a consequence, additionalcontributions to forces proportional to the derivatives with respect to density of coupling constants will need to be in-cluded to obey energy conservation. For the two-body termin Eq. (2) these corrections lead to computational require-ments that scale with the third power of the number of nu-cleons, instead of the second power for the ordinary case. Toavoid this issue, the coupling constant of the two-body termhas been kept the same below and above saturation. Only thethree-body contributions, proportional to the coupling con-stants x and y in Eq. (2), have been modified with the conse-quence that above saturation the values of L and K sym cannotbe chosen independently anymore. The advantage of this ap-proach resides in the fact that energy conservation violationis small, of the order of a few hundred KeV per event, evenwithout including contributions to forces due to the densitydependence of the two coupling constants in the vicinity ofsaturation.In the right panel of Fig. (13) theoretical values for PMRspectra are presented for the three asyEoS’es that are identi-cal below saturation but differ above this point. Results formid-central AuAu collisions at four impact energies in the300-800 MeV/nucleon range are shown. Noteworthy differ-ences between the stiffest and softest choices for the SE thatamount to a factor close to 2 in the region p T > ∆ (1232) are first excited is preserved to a largeextent in spite of the fact that pions that survive up to thefinal state of the reaction undergo, on average, a few ab-sorption/decay processes. To determine the density at whichPMR is most sensitive to, calculations with different combi-nations of values for the slope parameter below and abovesaturation have to be performed. The average probed den-sity can then be extracted using the approach described inRef. [11]. The sensitivity to the asy-EoS above saturationis approximately independent on impact energy advocatingexperimental measurements at higher impact energy in viewof less required beam-time for similar statistical accuracy. The dcQMD model, an offspring of the T¨ubingen QMDtransport model, has been further developed by implement-ing in-medium nucleonic resonance potentials that can beset independently of the nucleon optical potential and aredescribed in terms of intuitive quantities such as potentialdepths, at saturation and zero momentum, and effective mas-ses. This effort has been prompted by results of phenomeno-logical studies and ab-initio calculations that suggest a ∆ (1232) potential that is different from that of the nucleon.The two approaches have led however to different resultswhich has contributed in the past to adopting the Ansatz of equal resonance and nucleon potentials in semi-classicaltransport models for heavy-ion collisions at intermediate im-pact energies. This model extension has been deemed neces-sary as the accurate description of observables carrying in-formation about the isospin dependent part of the equationof state of nuclear matter requires a proper understanding ofresidual effects induced by uncertainties of our knowledgeof the equation of state of symmetric nuclear matter or otherquantities leading to isoscalar contributions to observables.The upgraded model has been employed in the study ofpion production from slightly above threshold to impact en-ergies of 800 MeV/nucleon. One of the objectives has beenthe extraction of effective isoscalar and isovector ∆ (1232)potential strengths and masses from a comparison to avail-able experimental data for Ca Ca, Ru Ru, Zr Zrand Au Au provided by the FOPI Collaboration. Theanalysis has been performed separately for the isoscalar andisovector components of the ∆ (1232) potential following theidentification of observables that are predominantly sensi-tive to one of the two potentials: total charged pion multi-plicity for the former and ratio of total charged multiplicityfor systems with different isospin asymmetry for the latter.The charged pion multiplicity, an observable proposed in thepast for the study of the density dependence of symmetry en-ergy, has been shown to be equally sensitive to both the iso-scalar and isovector ∆ (1232) potentials. It has been shownthat available experimental data for nucleonic observablessuch as stopping, transverse and elliptic flow for systems ofdifferent masses and at different impact energies can be ac-curately described by the model, a pre-requisite for studyingpion emission close to threshold.The extraction of the isoscalar ∆ (1232) potential (ISDP)parameters has been attempted using the experimental datafor total charged pion multiplicities for Ca Ca and also Ru Ru systems at impact energies of 400, 600 and 800MeV/nucleon (only the first impact energy for the latter sys-tem). A precise extraction of the potential depth and iso-scalar effective mass was not possible due to sub-optimalaccuracy of experimental data and a decrease of sensitiv-ity at higher impact energies. However, an effective isosca-lar potential that is more attractive and a smaller isosca-lar effective mass are favored for ∆ (1232) as compared tothose corresponding to the nucleon. The result is in agree-ment with similar information extracted from quasi-elasticelectron-nucleus scattering but is incompatible with micro-scopical model calculation. A possible reason for the latter isthe omission of non-resonant pion production contributions,which would lead to a less attractive potential and may alsoimpact its required momentum dependence.For the isovector ∆ (1232) potential (IVDP) the study hasproven more challenging. Comparing model predictions forthe ratio of total charged multiplicity for systems with dif-ferent isospin asymmetry to experiment has only led to ex- tremely loose constraints for the IVDP parameters, spanninghalf of the probed parameter space. Using the pion multi-plicity ratio for isospin asymmetric systems instead, moreprecise constraints, in the form of correlations between po-tential depth and isovector effective mass difference, couldbe obtained. These have however proven to be rather sen-sitive to values of the slope parameter of symmetry energyat saturation, the value of the neutron-proton effective massdifference and the assumed stiffness for the density depen-dence of IVDP above saturation. Adding the total chargedpion multiplicity to the fit was shown to exclude the morerepulsive IVDP scenarios.Without an accurate knowledge of the ∆ (1232) potentiala study of the symmetry energy using multiplicity observ-ables alone is not possible. An alternative, previously stud-ied in Ref. [32], is to include average transverse momen-tum observables among the fitted quantities. The additionaluncertainties induced by the ISDP results however in evenmore uncertain constraints than before.Studying pion multiplicity ratio spectra has proven morefruitful. It has been shown that by including average trans-verse momenta in the fit of DPOT parameters, a value of p T above which spectra are insensitive to uncertainties in the ∆ (1232) potential can be determined. Residual model de-pendence due to pion optical potential and in-medium mod-ifications of cross-sections uncertainties are below 10% inthis high p T region. Extraction of information regarding thesymmetry energy and related quantities is thus feasible froma comparison theory-experiment of the high p T tail of pionmultiplicity ratio spectra. It has been shown that due to in-clusion of threshold effects the sensitivity to the value ofthe neutron-proton effective mass difference has to be takeninto consideration. Without input from other sources onlya correlation between the values of the slope of symmetryenergy and neutron-proton effective mass can be extractedfrom pion production close to threshold. The sensitivity tothe magnitude of symmetry energy at ρ =0.10 fm − , the den-sity for which it is kept fixed in the present model, was foundto be small, of the order of a few percent. The sensitivity onthe density dependence of the symmetry energy above satu-ration was however found appreciable in spite of the fact thatsurviving pions undergo, on average, a few absorption/decaycycles and was proved to be approximately independent ofimpact energy.It is concluded that more accurate experimental data forpion production in heavy-ion collisions from threshold to800 MeV/nucleon incident energy, that provide sufficientstatistical accuracy but are below the point where the frac-tion of excited nucleons into resonances becomes non neg-ligible, will be one of the pre-requisites for the extraction ofconstraints for the symmetry energy at supranormal densi-ties from terrestrial laboratory experiments. However, pre-cise information from other sources regarding the momen- tum dependence of the isovector component of the nucleonpotential will be needed for providing a precise answer re-garding the value of the symmetry energy around 2 ρ . The authors acknowledge financial support from the U.S.Department of Energy, USA under Grant Nos. DE-SC0014530, US National Science Foundation, United States GrantNo. PHY- 1565546. The research of M.D.C. has been fi-nancially supported in part by the Romanian Ministry ofEducation and Research through Contract No. PN 19 0601 01/2019-2022. M.D.C. acknowledges the hospitality ofNSCL / MSU where part of this study was performed. Theauthors express their gratitude to Maria Colonna, Pawel Da-nielewicz, Justin Estee, Che-Ming Ko, William Lynch, Her-mann Wolter and TMEP Collaboration for stimulating dis-cussions. The computing resources have been partly pro-vided by the Institute for Cyber-Enabled Research (ICER)at Michigan State University.
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