Probing high-density symmetry energy using heavy-ion collisions at intermediate energies
aa r X i v : . [ nu c l - t h ] J u l Probing high-density symmetry energy using heavy-ion collisions at intermediateenergies
Gao-Chan Yong, Ya-Fei Guo Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China
The nuclear symmetry energy, which describes the energy difference of per proton and neutronin nuclear matter, has been extensively studied within the last two decades. Around saturationdensity, both the value and the slope of the nuclear symmetry energy have been roughly con-strained, its high-density behavior is now still in argument. Probing high-density symmetry energyat terrestrial laboratories is being carried out at facilities that offer radioactive beams worldwide.While relevant experiments are being conducted, we theoretically developed more advanced isospin-dependent transport model including new physics such as nucleon-nucleon short-range correlationsand in-medium isospin-dependence of baryon-baryon scattering cross section. New sensitive probesof high-density symmetry energy are provided, such as squeezed-out neutron to proton ratio, photonand light cluster as well as the production of mesons with strangeness or hidden strangeness. Theblind spots of probing the high-density symmetry energy by sensitive observable are demonstrated.Model dependence of frequently used sensitive probes of the symmetry energy has been studiedthoroughly based on different transport models. A qualitative observable of neutron to proton ratioat high emitting energy is proposed to probe the high-density symmetry energy qualitatively. Theprobed density regions of the symmetry energy are carefully studied. Effects of nucleon-nucleonshort-range correlations on the some sensitive observables of the symmetry energy in heavy-ion col-lisions are explored carefully. Probing the curvature of the symmetry energy by involving the slopeinformation of the symmetry energy at saturation point in the transport model is proposed. Be-sides constraining the high-density symmetry energy using heavy-ion collisions, a lot of neutron-starrelated observations from heaven may also be used to constrain the high-density symmetry energy.
I. INTRODUCTION
To investigate properties of nuclei far from stabil-ity and neutron-star-like objects in heaven, the concept asymmetric nuclear matter holding unequal numbers ofneutron and proton is frequently mentioned. To describethe properties of asymmetric nuclear matter, the equa-tion of state of nuclear matter is frequently used. Theequation of state (EoS) of nuclear matter at density ρ and isospin asymmetry δ ( δ = ( ρ n − ρ p ) / ( ρ n + ρ p )) usu-ally reads as [1–3] E ( ρ, δ ) = E ( ρ,
0) + E sym ( ρ ) δ + O ( δ ) , (1)where E sym ( ρ ) is the nuclear symmetry energy. Appar-ently, the nuclear symmetry energy describes the changeof the single nucleonic energy of nuclei or nuclear mat-ter when replacing protons with neutrons. The EoS ofisospin symmetric nuclear matter E ( ρ,
0) is relatively wellconstrained [4] while the EoS of isospin asymmetric nu-clear matter, especially the high-density symmetry en-ergy, as shown in Fig. 1, is still a subject of debate [5–11].The nuclear symmetry energy not only plays crucialroles in nuclear physics [2, 3] but also, in a density rangeof 0.1 ∼
10 times nuclear saturation density, affects thebirth of neutron stars and supernova neutrinos [12], thecooling rates and the thickness of the crust of neutronstars, the mass-radius relationship and the moment ofinertia of neutron stars [13–17]. The nuclear symme-try energy also plays important roles in the evolution ofcore-collapse supernova [18] and astrophysical r-processnucleosynthesis [19–22], the gravitational-wave frequency [23, 24] and the gamma-ray bursts [25] in neutron starmergers [26, 27].
Neutron Star Observations E sy m ( M e V ) / NL-RMF(18) PC-RMF(3) RHF(2) DD-RMF(2) Gogny-HF(2) SHF(33)
BHF (Vidana) BHF (Z.H. Li) DBHF (Fuchs) DBHF (Sammarruca) Chiral EFT-N LO450 Chiral EFT-N LO600 VMB-APR VMB-FP VMB-WFF1 VMB-WFF2 VMB-WFF3
FIG. 1: Nuclear symmetry energy given by different nuclearmany-body approaches in comparison with the constrainingboundaries extracted from studying properties of neutronstars. Taken from Ref. [6].
Around saturation density both the value of the nu-clear symmetry energy (about 31 . ± . . ± . II. THE ISOSPIN-DEPENDENTBOLTZMANN-UEHLING-UHLENBECK (IBUU)TRANSPORT MODEL
The Boltzmann-Uehling-Uhlenbeck (BUU) equationdenotes time evolution of single particle phase space dis-tribution function f ( ~r, ~p, t ), which reads [50] ∂f∂t + ∇ ~p E · ∇ ~r f − ∇ ~r E · ∇ ~p f = I c . (2)The left-hand side of Eq. (2) describes time evolutionof the particle phase space distribution function due toits transport and mean field, and the right-hand sideaccounts for the modification of the phase space distri-bution function by elastic and inelastic two body colli-sions. E denotes a particle’s total energy, which is equalto kinetic energy E kin plus its average potential energy U . The mean-field potential U of the single particledepends on its position and momentum of the particleas well as its local asymmetry of medium and is givenself-consistently by its phase space distribution function f ( ~r, ~p, t ). The BUU model is one of the most extensivelyused transport models which describes nucleus-nucleuscollisions at different energy regions. At intermediateenergies, its main ingredients are single particle poten-tial and particle-particle scattering cross sections as wellas pauli-blockings of fermions.To study the effects of symmetry energy in heavy-ioncollisions, isospin dependence is extensively involved intodifferent parts of the BUU transport model. The isospin-and momentum-dependent single nucleon mean-field po- tential reads [2, 51, 54] U ( ρ, δ, ~p, τ ) = A u ( x ) ρ τ ′ ρ + A l ( x ) ρ τ ρ + B (cid:16) ρρ (cid:17) σ (1 − xδ ) − xτ Bσ + 1 ρ σ − ρ σ δρ τ ′ + 2 C τ,τ ρ Z d p ′ f τ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ + 2 C τ,τ ′ ρ Z d p ′ f τ ′ ( ~r, ~p ′ )1 + ( ~p − ~p ′ ) / Λ , (3)where ρ denotes the saturation density, τ, τ ′ =1/2(-1/2)is for neutron (proton). ρ n , ρ p denote neutron and protondensities, respectively. The parameter values A u ( x ) =33.037 - 125.34 x MeV, A l ( x ) = -166.963 + 125.34 x MeV,B = 141.96 MeV, C τ,τ = 18.177 MeV, C τ,τ ′ = -178.365MeV, σ = 1 . .
24 MeV/c. With thesesettings, the empirical values of nuclear matter at nor-mal density are reproduced, i.e., the saturation density ρ = 0.16 fm − , the binding energy E = -16 MeV, theincompressibility K = 230 MeV [29, 52], the isoscalareffective mass m ∗ s = 0 . m [53], the single-particle poten-tial U ∞ = 75 MeV at infinitely large nucleon momentumat saturation density in symmetric nuclear matter, thesymmetry energy E sym ( ρ ) = 30 MeV [54]. In Eq. (3), X = -1X = 2 X = 1X = 0 E s y m ( M e V ) / FIG. 2: Density-dependent symmetry energy with different x parameters. Taken from Ref. [54]. different symmetry energy’s stiffness parameters x canbe used in different density regions to mimic differentdensity-dependent symmetry energies. Fig. 2 shows theused symmetry energy derived from the single particlepotential Eq. (3) with different x parameters at low andhigh densities.The isospin-dependent baryon-baryon ( BB ) scatteringcross section in medium σ mediumBB is reduced comparedwith their free-space value σ freeBB by a factor of [54, 55] R BBmedium ( ρ, δ, ~p ) ≡ σ mediumBB elastic,inelastic /σ freeBB elastic,inelastic = ( µ ∗ BB /µ BB ) , (4)where µ BB and µ ∗ BB are the reduced masses of the collid-ing baryon pairs in free space and medium, respectively.The effective mass of baryon in isospin asymmetric nu-clear matter is expressed as [54] m ∗ B m B = 1 / (cid:16) m B p dUdp (cid:17) . (5)More details on the present used model can be found inRef. [54]. III. SENSITIVE PROBES OF NUCLEARSYMMETRY ENERGY IN HEAVY-IONCOLLISIONS
To constrain the density-dependent symmetry energyby heavy-ion collisions, symmetry energy sensitive probesare usually used as counterparts to compare with relevantexperimental data. Since the symmetry potential has op-posite actions for neutrons and protons and the value ofsymmetry potential is always quite small compared to theisoscalar potential, to enlarge the effects of symmetry en-ergy, most observables in fact use differences or ratios ofisospin multiplets of baryons, mirror nuclei and mesons,such as the neutron/proton ratio, neutron-proton differ-ential flow, neutron-proton correlation function, t / He, π − /π + , Σ − / Σ + and K /K + ratios, etc [2, 3].Squeezed-out nucleons emitted in the direction of per-pendicular to the reaction plane in semi-central heavy-ioncollisions carry more information on the property of densematter. Fig. 3 demonstrates transverse momentum de-pendent neutron/proton ratio of mid-rapidity nucleonsemitted in the direction perpendicular to the reactionplane [56]. It is seen in panel (b) that the symmetry en-ergy effect on the n/p ratio increases with the increasingtransverse momentum p t and the effect can be as highas 40% at high transverse momenta. The high p t nucle-ons most likely originate from the high density region inthe early stage in heavy-ion collisions thus is more sensi-tive to the high-density symmetry energy. The n/p ratioof free nucleons at mid-rapidity without azimuthal an-gle cut (shown in panel (a)) is much less sensitive to thesymmetry energy [56].It is practically difficult to measure observables involv-ing neutrons. One question frequently being asked iswhether the triton- He (t - He) pair may carry the infor-mation of nuclear symmetry energy. It is normally diffi-cult to extract reliable information about the symmetryenergy from the individual flows of triton and He clus-ters. To decrease effects of the isoscalar potential whileenhancing effects of the isovector potential are helpful.Fig. 4 shows the triton- He relative and differential flowsas a function of rapidity [57]. From panel (a) of Fig. 4, (b) , ( n / p ) f r ee p t (GeV/c)1.21.51.82.1 (a) Without azimuthal angle cuts x= 0 x= -1 Sn+
Sn, E/A=400 MeV, b=5 fm and |(y/y beam ) c.m. |<0.5 FIG. 3: Effects of symmetry energy on the non-squeezed-out(panel (a)) and squeezed-out (panel (b)) neutron to protonratios in
Sn+
Sn reaction at 400 MeV/nucleon. Takenfrom Ref. [56]. it is shown that the triton- He relative flow is very sen-sitive to the symmetry energy. Effects of the symmetryenergy on the differential flow are shown in panel (b), isrelatively small. Therefore triton- He relative flow is apotential probe used to detect the symmetry energy inheavy-ion collisions.Charged π − /π + ratio is another sensitive probe of nu-clear symmetry energy. Shown in Fig. 5 is ( π − /π + ) like ratio (= π − +∆ − + ∆ π + +∆ ++ + ∆ + ) as a function of time [58]. Dueto more neutron-neutron scatterings than that of proton-proton when two neutron skins start overlapping at thebeginning of the reaction, the value of the ( π − /π + ) like ratio soars in the early stage of the reaction. The( π − /π + ) like ratio saturates at about 25 fm/c indicatingthat a chemical freeze-out stage has been reached. Thesensitivity to the symmetry energy is clearly shown inthe final π − /π + ratio.The hadronic probes usually suffer from distortions dueto the strong interactions in the final state. It is prefer-able to get more clean ways to probe the symmetry en-ergy especially at supranormal densities. Fig. 6 showsthe spectra ratio R / ( γ ) of hard photons with p aγ (= (a)(b) FIG. 4: Effects of symmetry energy on the triton- He relative(panel (a)) and differential (panel (b)) flows as a function ofrapidity in the semi-central reaction of Sn + Sn at 400MeV/nucleon. Taken from Ref. [57]. . × − × ε γ ( β i + β f )) and p bγ (= 2 . × − − y ) α y , y = ε γ /E max , α = 0 . − . β i , β i and β f are theinitial and final velocities, E max is the energy available inthe center of mass) production forms [59]. It is first seenthat calculations with p aγ and p bγ and in-medium NN crosssections all lead to about the same R / ( γ ) within statis-tical errors as expected. The effect of the in-medium NNcross sections is canceled out in the spectra ratio. Freefrom the uncertainties of elementary photon productionand the NN cross sections, the observable R / ( γ ) couldbe considered as a robust probe of the symmetry energy.Comparing the calculations with soft and stiff symme-tries both using the p bγ , it is seen that the R / ( γ ) is verysensitive to the symmetry energy especially for energeticphotons.In the light of sensitivity of the π − /π + ratio to thesymmetry energy at subthreshold energies, the multiplic-ity of η production as a function of incident beam energyis also examined especially below its production thresh-old. Compared to pions, η mesons experience weakerfinal state interactions due to hidden strangeness. Fig. 7shows η multiplicity as a function of beam energy inAu+Au reaction with soft ( x = 0) and stiff ( x = − η declines with decreasing beam energy, especially FIG. 5: Effects of symmetry energy on the ( π − /π + ) like ratioin the central Sn + Sn reaction at 400 MeV/nucleon.Taken from Ref. [58].
40 60 80 100 1201.11.21.31.4 p a , x= 1, mediumnp p b , x= 1, mediumnp p b , x= 1, freenp p b , x= -1, mediumnp (MeV) R / () E/A=50 MeVand b=0 fm
FIG. 6: Effects of symmetry energy on the spectra ratio ofhard photons in reactions of Sn + Sn and Sn + Sn at 50 MeV/A, see text for details. Taken from Ref. [59]. for the soft symmetry energy. And it saturates at inci-dent energy of about 10 GeV/nucleon. The effect of nu-clear symmetry energy on the η production is much moreevident than the π − /π + ratio especially in the deepersub-threshold region. IV. BLIND SPOTS OF PROBING THEHIGH-DENSITY SYMMETRY ENERGY INHEAVY-ION COLLISIONS
Based on the chemical equilibrium condition of nuclearmatter [61], if the symmetry energy does not change withdensity, the liquid-gas phase transition cannot occur.Therefore, in case the symmetry energy is less density- -2 -1 M u l t i p li c i t y E beam /A (GeV) x= 0x= -2 Au+Au FIG. 7: Effects of symmetry energy on the multiplicity of η production as a function of incident beam energy in Au+Aureactions. Taken from Ref. [60]. dependent, the effects of symmetry energy on the final-state observable may disappear.To show the blind spots of probing the high-densitysymmetry energy, the neutron to proton ratio ratio inthe central Au+Au reaction at 300 MeV/nucleon is usedas an example [62]. In the used transport model, the ef-fects of neutron-proton short-range-correlations are takeninto account [54, 63]. Also the transition momentumof the proton is set to be the same as that of neu-tron [64]. The in-medium inelastic baryon-baryon col-lisions and the pion in-medium transport are included aswell [65, 66]. For initialization, nucleon density distribu-tion in colliding nuclei is calculated using the Skyrme-Hartree-Fock with Skyrme M ∗ force parameters [67].And the proton and neutron momentum distributionswith high-momentum tails in initial nucleus are repro-duced [54, 63, 68–70].Since below saturation density, the symmetry energy isroughly constrained [71, 72], we simulate the low-densitysymmetry energy with parameter x = 1. From Fig. 8, itis seen that with parameter x = 1 the simulated symme-try energy is well consistent with the current constraints.In the study, we vary the high-density symmetry energywith parameter x in the range of x = -1, 1, 2, whichcover the current uncertainties of the high-density sym-metry energy [5, 72].Fig. 9 shows the effects of high-density symmetry en-ergy on the neutron to proton ratio n/p of dense matterformed in central Au+Au reaction at 300 MeV/nucleon.In the study, the same low-density symmetry energy asshown in Fig. 8 is used. From Fig. 9 (a) and (c), onesees that the stiffer/soft high-density symmetry energycauses a smaller/large asymmetry of dense matter. While E sy m ( / ) ( M e V ) / x= 1 HIC’s blind spots
Esym at high densities c on s t r a i n t s FIG. 8: The density-dependent symmetry energy at low andhigh densities. The symmetry energy below saturation den-sity is fixed as experimental constraints while the high-densitysymmetry energy is alterable. Taken from Ref. [62]. from Fig. 9 (b), it is seen that the less density-dependenthigh-density symmetry energy ( x = 1, shown in Fig. 8)almost does not affect the asymmetry of dense matter.For isospin-fractionation, there is a chemical equilibriumcondition [61, 73–76] E sym ( ρ ) δ = E sym ( ρ ) δ , (6)where E sym ( ρ ) , E sym ( ρ ) denote symmetry energies atdifferent density regions and δ , δ are, respectively, theirasymmetries. In nuclear matter there is a dynamicalprocess of nucleon movement determined by Eq. (6)according to the density dependent symmetry energy.Since the high-density symmetry energy with parame-ter x = 1, E sym ( ρ ) ≈ E sym ( ρ ), the asymmetry of densematter δ ≈ δ . One thus sees in Fig. 9 (b), the lessdensity-dependent high-density symmetry energy almostdoes not affect the asymmetry of dense matter formed inheavy-ion collisions.Since the symmetry potential acts directly on nucleons,the neutron/proton ratio n/p of nucleon emissions maybe one of the best observables to probe the symmetryenergy [77, 78]. Fig. 10 shows the effects of high-densitysymmetry energy on the kinetic energy distribution ofthe free neutron to proton ratio n/p in central Au+Aureaction at 300 MeV/nucleon. From Fig. 10 (a), (c), itis seen that the stiffer/soft high-density symmetry en-ergy causes a larger/small n/p ratio. While from Fig. 10(b) one sees that the less density-dependent high-densitysymmetry energy almost does not affect the free n/p ra-tio. Therefore, the less density-dependent high-densitysymmetry energy cannot be probed effectively in heavy-ion collisions. w/o Esymat x= -1Au+Au300 MeV ( n / p ) / (a)
10 20 30 (b) x= 1 t (fm/c) 10 20 30 (c) x= 2
FIG. 9: Effects of the high-density symmetry energy on theneutron to proton ratio n/p in dense matter ( ρ/ρ ≥ V. MODEL DEPENDENCE OFSYMMETRY-ENERGY-SENSITIVE PROBESAND QUALITATIVE PROBE
There are many factors affecting nuclear reactiontransport simulation, e.g., the initialization, the nucleon-nucleon interaction or single nucleon potential, thenucleon-nucleon scattering cross sections, and the frame-work of transport models. It is thus necessary to make astudy among different models, to see how large the dif-ferences are on the values of isospin sensitive observables.Within the frameworks of isospin-dependent trans-port models Boltzmann-Uehling-Uhlenbeck (IBUU04)[58] and Ultrarelativistic Quantum Molecular Dynam-ics (UrQMD) [79, 80], the model dependences of fre-quently used isospin-sensitive observables π − /π + ra-tio and n/p ratio of free nucleons have been demon-strated [81]. Fig. 11 shows the n/p ratio of free nu-cleons and π − /π + ratio as a function of kinetic energyin the Au + Au reaction at a beam energy of 400MeV/nucleon simulated by the IBUU and the UrQMDmodels. From panel (a) one can see that both modelsgive the same trend of n/p ratio as a function of nucle-onic kinetic energy. The result of the UrQMD model isoverall larger than that of the IBUU model due to theirdifferent forms of symmetry potential [81]. From panel(b) of Fig. 11, one can see that there is a cross betweenthe π − /π + ratios from the UrQMD model and that fromthe IBUU model. At lower kinetic energies, the value of π − /π + ratio from the UrQMD is much larger than that
100 150 2001.21.31.41.51.61.7 w/o Esymat x= -1 ( n / p ) f r ee (a) 100 150 200(b)x= 1 E kin (MeV) 100 150 200(c)x= 2 Au+Au300 MeV
FIG. 10: Effects of high-density symmetry energy on the freeneutron to proton ratio n/p as a function of kinetic energyin central Au+Au reaction at 300 MeV/nucleon. The solidline denotes the case without high-density symmetry energy,panels (a), (b), (c) indicate cases with stiff, soft and super-softsymmetry energies, respectively. Taken from Ref. [62]. (a) (b)
FIG. 11: Model dependence of n/p ratio of free nucleons and π − /π + ratio as a function of kinetic energy in Au + Au at 400 MeV/nucleon. Panels (a) and (b) show n/p and π − /π + ratio cases, respectively. Taken from Ref. [81]. from the IBUU model. And from panel (a) of Fig. 12,one can see that nucleonic transverse flow given by theIBUU model shows large isospin effect than that withthe UrQMD model. From panel (b) of Fig. 12, it is seenthat the slope of neutron-proton differential flow is ev-idently larger for the UrQMD model than that for theIBUU model.Since most observables of the symmetry energy aremodel dependent, on the first step, what is cruciallyneeded currently is a qualitative observable to probewhether the symmetry energy at high densities is stiffor soft.Figure 13 shows nucleon kinetic energy distributionof n/p ratios of free (gas) and bound (liquid) nucleons (a) (b) FIG. 12: Same as Fig. 11, but for the rapidity distributions ofthe transverse flow < p x ( y ) > for neutrons and protons (panel(a)) and the neutron-proton differential transverse flow F xn − p (panel (b)). Taken from Ref. [81]. (a) (b) FIG. 13: kinetic energy distribution of neutron to proton ratio n/p of free (panel (a)) and bound nucleons (panel (b)) in thecentral Au + Au reaction at 400 MeV/A with positive(left, x= -1.2) and negative (right, x= 1) symmetry potentialsat supra-saturation densities. Given by the IBUU transportmodel. Taken from Ref. [5]. in the central Au + Au reaction at 400 MeV/Awith positive and negative symmetry potentials at supra-saturation densities given by the IBUU transport model[5]. From panel (a), it is seen that the value of n/p of nu-clear gas phase is larger than that of liquid phase in thewhole kinetic energy distribution with the positive sym-metry potential. However, it is interesting to see thatthe value of n/p of gas phase is smaller than that of liq-uid phase at higher kinetic energies with the negativesymmetry potential at high densities. The same situa-tion is demonstrated in Fig. 14 with the UrQMD model.The only discrepancy is that the kinetic energy of tran-sition point given by the UrQMD model is lower thanthat given by the IBUU model. With positive/negativesymmetry potential at high densities, for energetic nucle-ons, the value of neutron to proton ratio of free nucleonsis larger/smaller than that of bound nucleon fragments.Compared with extensively studied quantitative observ-ables of the nuclear symmetry energy, the normal or ab-normal isospin-fractionation of energetic nucleons can bea qualitative probe of nuclear symmetry energy at highdensities. (a) (b) FIG. 14: Same as Figure 13 but given by the UrQMD trans-port model with positive (left, γ = 1.5) and negative (right,a= 1.6) symmetry potentials at supra-saturation densities.Taken from Ref. [5]. Pb+
Pb, 600 MeV w/o Esym 0-0.5 , = 2 0-1 , = 2 0-1.5 , = 2 0-3 , = 2 ( - / + ) li k e t (fm/c) (a) (b) Sn+
Sn, 300 MeV w/o Esym 0-0.5 , = 2 0-1 , = 2 0-1.5 , = 2 0-2 , = 2 FIG. 15: Evolution of the relative sensitivity of the sym-metry energy observable ( π − /π + ) like ratio in Sn+
Snat 300 MeV/nucleon (panel (a)) and
Pb+
Pb at 600MeV/nucleon (panel (b)). Taken from Ref. [86].
VI. DETERMINATION OF THE DENSITYREGION OF THE SYMMETRY ENERGYPROBED BY THE π − /π + RATIO AND NUCLEONOBSERVABLES
To investigate the symmetry energy at high densities,the π − /π + ratio has been frequently proposed in the lit-erature [58, 82, 83]. Aiming at probing the high-densitysymmetry energy by the π − /π + ratio, Sn+Sn reactionsat about 270 MeV/nucleon experiments are being carriedout at RIKEN in Japan [47, 84, 85]. Does the π − /π + ra-tio in heavy-ion collisions at intermediate energies alwaysprobe the symmetry energy above saturation density?Alternatively, in what conditions does the π − /π + ratioprobe the symmetry energy at high densities?To answer the above questions, we studied the decom-position of the sensitivity of the symmetry energy ob-servables. From Figure 15(a), it is demonstrated that,in the medium-mass nuclei Sn+
Sn central reactionat 300 MeV/nucleon, the effects of the symmetry energyin the densities below 0.5 ρ are larger than that in thedensities above 1.5 ρ . Below saturation density the ef-fects of the symmetry energy are roughly equal to thatabove saturation density. However, from Figure 15(b),one can see that, the effects of the symmetry energy inthe densities below 0.5 ρ are obviously smaller than thatin the densities above 1.5 ρ . The effects of the symme-try energy below saturation density are obviously smallerthan that above saturation density. Therefore, to probethe high-density symmetry energy by the π − /π + ratio,heavy system and at relatively higher incident beam en-ergies is a preferable.Similar studies were also carried out for some otherfrequently discussed observables [87]. The symmetry en-ergy sensitive observable n/p ratio in the Sn+
Snreaction at 0.3 GeV/nucleon is shown to be sensitive tothe symmetry energy below 1 . ρ . Nucleon elliptic flowcan probe the symmetry energy from low to high den-sities when changing the beam energies from 0.3 to 0.6GeV/nucleon in the semi-central Sn+
Sn reaction.And nucleon transverse and elliptic flows in the semi-central
Au+
Au reaction at 0.6 GeV/nucleon aremore sensitive to the high-density symmetry energy.
VII. EFFECTS OF SHORT-RANGECORRELATIONS IN TRANSPORT MODEL (a) 0.4 GeV/nucleon
Au+Au
FOPI data
FOPI data (b) 1 GeV/nucleon (=p max /p F ) N + FIG. 16: Effects of short-range correlations on the numberof produced π + meson production in Au + Au collisions at0.4 (panel (a)) and 1 GeV/nucleon (panel (b)) beam energies.Taken from Ref. [70]. Recent proton-removal experiments showed that onlyabout 80% nucleons participate in the independent par-ticle motion [68, 88, 89]. Also it is shown that nucleonsin nuclei can form pairs with larger relative momentaand smaller center-of-mass momenta [90, 91]. This is in-terpreted by the nucleon-nucleon short-range tensor in-teraction [92, 93]. The nucleon-nucleon short-range cor- relations (SRC) in nuclei cause a high-momentum tail(HMT) in single-nucleon momentum distribution aboveFermi momentum [94–98]. And the high-momentumtail’s shape is almost identical for all nuclei [99–102], i.e.,roughly exhibits a
C/k distribution [69, 103–105].The short-range correlations in nuclei surely affect π , K , η and nucleon emission in heavy-ion collisions. Fig. 16shows π + production as a function of high-momentumtail cutoff parameter λ in Au + Au collisions at 0.4 and1 GeV/nucleon incident beam energies [70]. Larger high-momentum cutoff parameter causes larger nucleon aver-age kinetic energy, especially proton average kinetic en-ergy, thus more π + ’s are produced. As incident beamenergy increases, the initial movement of nucleons in nu-clei becomes less important in nucleus-nucleus collisions. (a) Au+
Au400MeV/nucleon with HMT w/o HMT - + - / + E c.m.Kin (MeV) (b) (c) FIG. 17: Effects of short-range correlations on the kineticenergy distributions of π − (panel (a)), π + (panel (b)) and π − /π + ratio (panel (c)) in the reaction of Au +
Au at400 MeV/nucleon. Taken from Ref. [106].
Fig. 17 shows effects of short-range correlations on thekinetic energy distributions of π − , π + as well as π − /π + ratio in the semi-central reaction Au +
Au at 400MeV/nucleon [106]. It is clearly demonstrated that thekinetic energy distributions of both π − and π + are sensi-tive to the HMT. The ratio of π − /π + is sensitive to theHMT except in the high kinetic energy region. The short-range corrections increase kinetic energies of neutrons -8-6-4-20 100 200 300 400 5000.00.51.01.5 (a) Au+
Au400MeV/nucleon with HMT w/o HMT |(y/y beam ) c.m. |<=0.5 (b) p tc.m. (MeV) ( v - v )( % ) v + p ( % ) FIG. 18: Effects of short-range correlations on the nucleonelliptic flow (panel (a)) and the difference of neutron and pro-ton elliptic flow (panel (b)) in the semi-central reaction of
Au +
Au at 400 MeV/nucleon. Taken from Ref. [106]. and protons, thus more pion’s are produced. With theshort-range correlations, protons have a larger probabil-ity than neutrons to have larger momenta, one thus seesa lower value of the π − /π + ratio with the HMT. FromFig. 18, one can see that the effects of the HMT on thedifference of neutron and proton elliptic flows ( v n − v p )is also crucial [106].In neutron-rich matter, neutron and proton may havevery different Fermi momenta. If each correlated neutronand proton have 1 /k distributions starting from their re-spective Fermi momenta, based on the n-p dominancemodel [69], the correlated neutron and proton cannothave the same momentum magnitude. It is thus reason-able to think that in neutron-rich matter proton has thesame transition momentum as that of neutron. Fig. 19shows nucleon momentum distribution in Ca. One cansee that a HMT above the nuclear Fermi momentum isproduced. Proton has greater probability than neutronto have momenta greater than the nuclear Fermi momen-tum. Compared case A with case B, it is seen that withthe starting point of neutron Fermi momentum, protonhas even more greater probability to have high momenta. -2 -1 p n n ( k ) ( f m ) k (fm -1 ) Ca (a): n_k F n , p_k F p (all) (b): n_k F n , p_k F p (low) + n_k F n , p_k F n (high) gap FIG. 19: Momentum distribution of nucleon with differentstarting points of protons in Ca. Panels (a), (b) show caseswithout and with proton momentum gap in nucleon momen-tum distributions, respectively. Taken from Ref. [107]. + BA N - cm BA - / + Sn+
Sn, 300 MeV, b=3 fm
FIG. 20: The ratio of π − /π + in Sn+
Sn reactions at300 MeV/nucleon with different proton starting momenta inthe HMT. Taken from Ref. [107].
The dynamics of heavy-ion collisions could be affected byproton momentum gap. Fig. 20 demonstrates the ratioof π − /π + in Sn+
Sn reactions with different protonstarting momenta in the HMT. As expected, there is anevident decline of the value of π − /π + ratio when varyingthe starting momentum of proton in the HMT. From theinserted figure, it is seen that mainly the π + productionis affected. To interpret related experiments at Radioac-tive Isotope Beam Facility (RIBF) at RIKEN in Japan[47, 84], it is necessary to study how the π − /π + ratio isaffected by the proton momentum gap.0 VIII. CROSS-CHECKING THE SYMMETRYENERGY AT HIGH DENSITIES with SRC, with medium σ inelasticBB w/o SRC, with medium σ inelasticBB w/o SRC, w/o medium σ inelasticBB FOPI dataAu+Au, 400 MeV/nucleon π − / π + x FIG. 21: Effects of symmetry energy on the π − /π + ratio inAu+Au reaction at 400 MeV/nucleon with different short-range correlations and in-medium cross section. Taken fromRef. [65]. with SRC w/o SRC FOPI-LAND data
Au+Au400 MeV/nucleon V / V x FIG. 22: Effects of symmetry energy on the ratio of neutronand proton elliptic flows V n /V p in Au+Au reaction at 400MeV/nucleon. Taken from Ref. [65]. Since the constraints on the symmetry energy are al-ways model dependent, it is instructive to make cross-check of the symmetry energy using different probes.Fig. 21 shows the π − /π + ratio given by the IBUU model.It is seen that larger π − /π + ratio corresponds softer sym-metry energy. One can also see that both the short-rangecorrelations and in-medium cross section affect the valueof π − /π + ratio evidently. From Fig. 21, it is concluded that the FOPI pion data supports a softer symmetry en-ergy ( x = 1, 2, even x = 3).Fig. 22 shows comparison of simulated nucleon ellip-tic flow and experimental data. Since stiffer symmetryenergy generally causes more neutrons to emit in the di-rection perpendicular to the reaction plane, larger valuesof elliptic flow ratios of neutron and proton V n /V p areseen with stiffer symmetry energies. With the SRC ofnucleon-nucleon, values of the V n /V p ratio are largerthan that without the SRC. Fig. 22 indicates the FOPI-LAND elliptic flow experimental data does not favor verysoft symmetry energy ( x = 2, 3). Combining the studiesof nucleon elliptic flow and π − /π + ratio, one can roughlyobtain the symmetry energy stiffness parameter x = 1,i.e., a mildly soft density-dependent symmetry energy atsupra-saturation densities is obtained as shown in Fig. 2. IX. PROBING THE CURVATURE OFNUCLEAR SYMMETRY ENERGY K sym AROUND SATURATION DENSITY
The density-dependent symmetry energy at saturationcan be Taylor expanded as [108], E sym ( ρ ) = E sym ( ρ ) + L (cid:18) ρ − ρ ρ (cid:19) + K sym (cid:18) ρ − ρ ρ (cid:19) , (7)where E sym ( ρ ) is the value of the symmetry energy atsaturation and the quantities L , K sym are, respectively,its slope and curvature at saturation, L = 3 ρ ∂E sym ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12) ρ = ρ , K sym = 9 ρ ∂ E sym ( ρ ) ∂ρ (cid:12)(cid:12)(cid:12) ρ = ρ . (8)The most probable magnitude E sym ( ρ ) = 31 . ± . L = 58 . ± . K sym is still unknown, probably to be inthe range of − ≤ K sym ≤
100 MeV [109, 110]. Fig. 23shows density-dependent symmetry energies derived fromsingle nucleon potential (Eq. (3)) in the transport modeland that of Taylor expansion (Eq. (7)). It is seen that thesymmetry energies obtained from different combinationsof slope values with L = 40 , ,
80 MeV and curvatures K sym = − , L=80, Ksym=0, expansion L=80, Ksym=0, buu used L=60, Ksym=0, expansion L=60, Ksym=0, buu used L=40, Ksym=0, expansion L=40, Ksym=0, buu used L=80, Ksym=-400, expansion L=80, Ksym=-400, buu used L=60, Ksym=-400, expansion L=60, Ksym=-400, buu used L=40, Ksym=-400, expansion L=40, Ksym=-400, buu used E sy m ( M e V ) / FIG. 23: Density-dependent symmetry energies for six com-binations of curvatures K sym = − , L = 40 , ,
80 MeV obtained from single nucleon potentialEq. (3) (symbols) and Taylor expansion Eq. (7) (lines). - M e V K sy m = M e V |(y/y beam ) c.m. | 0.5 L MeV b= 7 fm75 n / p p c.m.t (GeV/c) - 400 MeV K sym = 0 MeV Au+ Au (b) 600 MeV (a) 400 MeV FIG. 24: Effects of the curvature of nuclear symmetry en-ergy on the squeezed-out neutron to proton ratio in the semi-central reaction of Au+Au at 400 (panel (a)) and 600 (panel(b)) MeV/nucleon. The bandwidths denote uncertainties ofthe slope L . Taken from Ref. [116]. symmetry energy on the squeezed-out neutron to pro-ton ratio n/p are quite evident, especially at high trans-verse momenta. The values of the squeezed-out n/p with K sym = 0 are higher than that with K sym = −
400 MeV.It is also seen that, the effects of the curvature of the sym-metry energy on the squeezed-out n/p are much larger
FIG. 25: Effects of the curvature of nuclear symmetry en-ergy on the difference of neutron and proton elliptic flowin the semi-central reaction
Sn + sn reaction at 270MeV/nucleon. than the ratio of integrating neutron and proton ellipticflows [117]. Fig. 25 shows the effects of the curvatureof nuclear symmetry energy on the difference of neutronand proton elliptic flow in the semi-central reaction of
Sn + sn at 270 MeV/nucleon. It is clearly seen thatthe difference of neutron and proton elliptic flow is alsoquite sensitive to the curvature of nuclear symmetry en-ergy. The isospin-dependent squeezed-out nucleon emis-sion thus may be one potential probe of the high-densitysymmetry energy.
X. SUMMARY AND PERSPECTIVE
Nuclear symmetry energy is an old but fundamentaland critical physical quantity in isospin nuclear physics,especially in astrophysics relating to neutron stars’s pro-duction, evolution and merger etc . Due to small asym-metry of compressed nuclear matter formed in generalheavy-ion collisions in terrestrial laboratory, the effectsof nuclear symmetry energy on many observables aregenerally less than 10 or 20%. Therefore uncertaintiesfrom transport models or some other unclear physical in-puts inevitably affect interpretation of nuclear symme-try energy from experimental data. Topics on modeldependence, qualitative observables, sensitive probes,many-observable cross-constraints, probed density-regionof sensitive observables and the effects of short-range cor-relations should be demonstrated before constraining thedensity-dependent symmetry energy from experimentaldata comparisons.2Besides constraining the nuclear symmetry energyfrom heavy-ion collisions [43, 65], the symmetry energycould be also determined via the studies of neutron starmerger [118, 119]. It is fascinating to see the match ofconstraints of the high-density symmetry energy fromheaven [120] and earth [121, 122] in the near future.
XI. ACKNOWLEDGMENTS
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