Probing in-medium nucleon-nucleon inelastic scattering cross section by using energetic n/p ratio
aa r X i v : . [ nu c l - t h ] J u l Probing in-medium nucleon-nucleon inelastic scattering cross sectionby using energetic n/p ratio
Zong-Zhen Zhang , , Ya-Fei Guo , , , and Gao-Chan Yong , ∗ 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 School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China
The frequently mentioned meson in heavy-ion collisions at intermediate energies is π , which is thelightest meson produced via nucleon-nucleon inelastic scatterings. Nowadays consensus has beenreached on free nucleon-nucleon inelastic scatterings while its in-medium form is still in argumentsand rarely constrained. Based on the Isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU)transport model, the in-medium nucleon-nucleon inelastic scattering is explored. It is found thatthe in-medium modification of nucleon-nucleon inelastic scatterings evidently reduces the neutronto proton ratio n/p at higher kinetic energies. Though the in-medium modification of nucleon-nucleon inelastic scatterings, as expected, affects the value of π − /π + ratio, considering a series ofundetermined properties of delta resonance and π in medium, the energetic neutron to proton ration/p is more suitable to be used to probe the in-medium correction of nucleon-nucleon inelasticscatterings. Dynamical π production in heavy-ion collisions at in-termediate energies was involved into transport modelsmore than three decades ago [1–7]. In the models, π ’s areusually treated via resonance decay while the resonanceexperiences formation and reabsorption in the whole col-lision process [8–12]. And channels relating to resonanceor pion productions are all implemented in free space, thisis because the in-medium corrections of nucleon-nucleoninelastic cross section are still undetermined [13–22]. Un-certainties of in-medium nucleon-nucleon inelastic crosssection in heavy-ion collisions would affect π − /π + ratio[19, 21], an observable which may be used to constrainthe high-density symmetry energy [23–25], the latter hasbeen considered to play crucial role in many aspects in-cluding nuclear physics and astrophysics [26].The reduction of in-medium elastic cross section ofnucleon-nucleon scattering has currently been relativelywell determined, see, e.g., Refs. [18, 27–33]. Althoughgreat efforts have been made to find experimental ob-servables constraining the symmetry energy, little workhas been done so far to probe the in-medium effects ofnucleon or resonance inelastic scattering cross sections,especially in isospin asymmetric nuclear matter. Becausethe in-medium reductions of baryon-baryon scatteringcross section are not only momentum and density depen-dent, but isospin-dependent [18–20, 22], double ratio of π − /π + from neutron-rich and neutron-deficient reactionswith the same isotopes thus cannot cancel out uncertain-ties of the in-medium effects of baryon-baryon scatteringcross section [34]. What is more, for heavy system, suchas Au+Au system, the counterpart with the same iso-topes is not easily to find. Therefore, it is meaningful tofind a way to probe and constrain the in-medium baryon-baryon inelastic scattering cross section. Unfortunately,till now there are rare studies on this subject. π pro- ∗ [email protected] duction in heavy-ion collisions is thought to be not onlyaffected by the in-medium nucleon-nucleon inelastic scat-tering cross section but also the unconstrained symmetryenergy [35]. And π production in heavy-ion collisions isrelated to a series of undetermined properties of delta res-onance in medium, which complicates the question. Sothe straightforward and good way is to use nucleon ob-servable. In heavy-ion collisions at intermediate energies,most energetic nucleons in fact experience inelastic pro-cess, such process may shed interesting light on the ex-ploration of the nucleon-nucleon inelastic scattering crosssection in medium. In this study, the in-medium effect ofnucleon-nucleon inelastic cross section on the neutron toproton ratio n/p at higher kinetic energies is first investi-gated. It is shown that the energetic n/p ratio is indeedsensitive to the in-medium effects of nucleon-nucleon in-elastic cross section.In the present study, the used Isospin-dependentBoltzmann-Uehling-Uhlenbeck (IBUU) transport modelincludes nucleon-proton short-range correlations, isospin-dependent in-medium elastic and free/in-medium in-elastic baryon-baryon cross sections as well as themomentum-dependent isoscalar and isovector pion po-tentials [36–38]. The isospin- and momentum-dependentsingle nucleon potential reads [39–41] U ( ρ, δ, ~p, τ ) = A u ( x ) ρ τ ′ ρ + A l ( x ) ρ τ ρ + B ( ρρ ) 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 , (1)where ρ stands for saturation density, τ, τ ′ = 1 / − / δ is the isospin asymmetry ofmedium. Detailed parameter values can be found in Ref.[40]. The symmetry energy’s stiffness parameter x is op-tional to mimic different forms of the symmetry energypredicted by various many-body theories without chang-ing any property of the symmetric nuclear matter andthe symmetry energy at normal density. In this study,we fix x = 1 [40] since the effects of symmetry energy arenot the question focused here.We assume the potential for ∆ resonances is a weightedaverage of those for neutron and proton. The weightedfactor depending on the charge state of the resonance isthe square of the Clebsch-Gordon coefficients for isospincoupling in the processes ∆ ↔ πN [11], i.e., U ∆ − = U n ,U ∆ = 23 U n + 13 U p ,U ∆ + = 13 U n + 23 U p ,U ∆ ++ = U p . (2)The nucleon-nucleon free inelastic isospin decomposi-tion cross sections σ pp → n ∆ ++ = σ nn → p ∆ − = σ + 12 σ ,σ pp → p ∆ + = σ nn → n ∆ = 32 σ ,σ np → p ∆ = σ np → n ∆ + = 12 σ + 14 σ (3)are parameterized via σ II ′ ( √ s ) = π (¯ hc ) p α ( p r p ) β m Γ ( q/q ) ( s ∗ − m ) + m Γ (4)with I and I ′ being the initial state and final stateisospins of two nucleons. The parameters α, β, m , Γas well as other kinematic quantities can be found inRef. [42]. The cross section for the two-body free inversereaction is calculated by the modified detailed balance,which takes into account the finite width of baryon reso-nance [6, 9] σ N ∆ → NN = m ∆ p f σ NN → N ∆ δ ) p i (cid:30) Z √ s − m N m π + m N dm ∆ π P ( m ∆ ) . (5)We extend the effective mass scaling model for in-medium elastic cross sections to inelastic case in isospinasymmetric matter [18, 40]. Compared with thefree-space baryon-baryon scattering cross section σ free ,the baryon-baryon scattering cross section in medium σ medium is reduced by a factor R medium ( ρ, δ, ~p ) ≡ σ medium /σ free = ( µ ∗ BB /µ BB ) , (6)where µ BB and µ ∗ BB are the reduced masses of the collid-ing baryon pairs in free and medium cases, respectively. R m ed i u m nn np pp (a) (b) p (MeV/c) (c) (d) FIG. 1: (Color online) Reduction factor R medium of nn , np and pp colliding pairs as a function of momentum with differ-ent densities ( ρ/ρ =0.5, 1.0, 1.5 and 2.0). The asymmetry ofnuclear matter is set to be δ =0.2. The effective mass of baryon in medium reads [43] m ∗ B m B = 1 / (1 + m B p dUdp ) . (7)Since it has been argued that the in-medium reductionof baryon-baryon inelastic cross sections are also final-state dependent [20, 44, 45], similar reduction factorsconnecting with the effective masses of final state parti-cles, e.g., as proposed in Refs. [44, 45], are checked in ourmodel. It is found that both forms of the reduction factorof the in-medium inelastic baryon-baryon cross sectionare in fact less sensitive to different final-state outgoingchannels. Compared with our present used form, the re-duction factor from Ref. [44] is too small and the formfrom Ref. [45] is quite close to our present used form.So in the present study, for different scattering outgoingchannels, we do not consider the splitting of the reductionfactor.Fig. 1 shows reduction factors of nucleon-nucleon scat-tering cross section in medium as a function of nucleonmomentum with different densities. Compared with free-space values, it is seen that the in-medium cross sectionsare not only reduced, but the reduction factors R medium of nn and pp split. The reduction factor of neutron in-volved is overall larger than that of proton involved. Tosee the in-medium effects of inelastic cross section onsome observables, we in the following make comparativestudies of relevant observables with free or in-mediumbaryon-baryon inelastic cross sections.With increase of baryon-baryon colliding energy, moreand more inelastic scatterings occur, thus energetic nu-cleon spectrum should be affected by the in-mediumbaryon-baryon inelastic cross sections more or less. Fig. 2shows the effects of the in-medium inelastic scatteringcross section on the n/p ratio in the central collision
150 300 4501.21.41.6 150 300 450 x=0x=1 (a) 400 MeV n / p E Kin (MeV) x=1 free medium (b) 270 MeV x=0
FIG. 2: (Color online) Effects of the in-medium baryon-baryon inelastic scattering cross section on the n/p ratio inthe central collision
Sn+
Sn at beam energies of 400 (a)and 270 (b) MeV/nucleon. For the in-medium cases, to seethe effects of symmetry energy, the parameter x in Eq.(1) isswitched from x=1 (soft) to x=0 (stiff). Sn+
Sn at beam energies of 400 MeV/nucleon and270 MeV/nucleon, respectively. In left and right panelsof Fig. 2, since elastic baryon-baryon scatterings dom-inate at lower colliding energies, one can see that atlower kinetic energies, effects of the in-medium baryon-baryon inelastic scattering cross section on the neutronto proton ratio n/p are less pronounced. While then/p ratio of emitted energetic nucleons is clearly affectedby the in-medium inelastic baryon-baryon cross section.With increase of nucleon-nucleon colliding energy, inelas-tic scatterings gradually become dominant, it is thus notsurprising to see a larger effect of in-medium inelasticcross section on the energetic neutron to proton ration/p. As the reduction of scattering cross section, lessnucleons accumulate more energies through many-timescatterings, thus numbers of energetic nucleons decrease.And positively charged particles like proton suffer fromthe Coulomb interactions which, to some extend, cancelout the reduction of proton involved scattering processes.Therefore relatively the decrease of the number of ener-getic neutron is more than that of proton in the finalstage of reaction. So the reduction of the in-medium in-elastic cross section lowers the value of energetic neutronto proton ratio. From both panels of Fig. 2, it is seenthat the effects of the in-medium baryon-baryon inelasticscattering cross section on the energetic n/p ratio reachabout measurable 15%.It is known that the squeezed-out energetic neutron toproton ratio n/p is very sensitive to the symmetry energy[46, 47]. To see whether the present analyzed energeticn/p ratio is sensitive to the symmetry energy or not,for the in-medium cases in Fig. 2, the symmetry energyparameter from x= 1 (soft) to x= 0 (stiff) is switched.
300 400 50010 -3 -2 -1 med.free med.freep Sn+ Sn d N / d E k i n ( M e V - ) (a)n
100 20010 -5 -4 -3 (b)med.freemed.free E kin (MeV) 100 200 56789 (c)med.free / t=20 fm/c FIG. 3: (Color online) Effects of the in-medium baryon-baryon inelastic scattering cross section on energetic nucle-ons (a), resonances (b) and the ratio of charged resonancesat maximum compression stage (c) in the central collision
Sn+
Sn at the beam energy of 400 MeV/nucleon.
One can clearly see that the symmetry energy really playsnegligible role in energetic non-squeezed-out neutron toproton ratio n/p.To further demonstrate the in-medium baryon-baryoninelastic scattering cross section on particle produc-tion, we plot Fig. 3, energetic nucleons, resonances andthe ratio of charged resonances at maximum compres-sion stage in the central
Sn+
Sn reactions at 400MeV/nucleon. As discussed in Fig. 2, it is seen fromthe left panel of Fig. 3 that the in-medium baryon-baryon inelastic scattering cross section decreases ener-getic nucleon, especially energetic neutron productions.This is the reason why one sees the in-medium baryon-baryon inelastic scattering cross section reduces the ratioof energetic neutron to proton ratio as shown in Fig. 2.Since energetic nucleon-nucleon collision produce deltaresonances and proton-proton (neutron-neutron) colli-sion produces ∆ ++ (∆ − ), one sees similar in-mediumbaryon-baryon inelastic scattering cross section on ener-getic resonance production as shown in the middle panelof Fig. 3. In the right panel of Fig. 3, effect of in-mediumbaryon-baryon inelastic scattering cross section on the ra-tio of ∆ − / ∆ ++ is clearly shown. Because the productionsof energetic ∆ − / ∆ ++ directly connect with the energeticneutron-neutron/proton-proton collisions and the num-ber of energetic nucleons reduces more than that of ener-getic proton, the reduction of in-medium baryon-baryoninelastic scattering cross section thus decrease the ratio of∆ − / ∆ ++ . Note here that the energetic nucleons studiedhere is not the result of iso-fractionation, but the resultof nucleon-nucleon scatterings. So the energetic neutronto proton ratio is a direct reflection of energetic nucleonin dense matter, rather than the other way round.Since ∆ ++ (∆ − ) decays into π + ( π − ), it is expectedthat the reduction of inelastic baryon-baryon scattering
100 200 300 4001234 x=1x=0 free medium
Sn+
Sn, 400 MeV E Kin (MeV)
FIG. 4: (Color online) Effects of in-medium baryon-baryoninelastic scattering cross section on the kinetic energy distri-bution of the π − /π + ratio in the central collision Sn+
Snat the beam energy of 400 MeV/nucleon. For the in-mediumcase, to see the effects of symmetry energy, the parameter x in Eq.(1) is switched from x=1 (soft) to x=0 (stiff). cross section in medium also decreases the ratio of en-ergetic π − /π + . Fig. 4 shows the kinetic energy dis-tribution of energetic π − /π + ratio in the central reac-tion Sn+
Sn at 400 MeV/nucleon incident beamenergy with free or in-medium baryon-baryon inelasticscattering cross sections. It is seen that the ratio of en-ergetic π − /π + can reduce about 15% when employingin-medium inelastic baryon-baryon scattering cross sec-tion. Although the π − /π + ratio is frequently mentionedin probing the high-density symmetry energy, its kineticdistribution at high energy shown here is not very sensi- tive to the symmetry energy. This result in fact does notconflict with that shown in Ref. [48]. The latter is forsqueezed-out pion emission whereas our present analysisis for general pion emission. As it is mentioned beforethat pion production in heavy-ion collision relates to aseries of undetermined properties of delta resonance andpion in medium, thus it is not suitable to probe the in-medium baryon-baryon inelastic scattering cross section.In summary, based on the transport model, effects ofthe in-medium baryon-baryon inelastic scattering crosssection on energetic nucleons, delta resonances and pionmesons are studied. It is found that in Sn+
Sn re-actions at intermediate energies the in-medium baryon-baryon inelastic scattering cross section reduces emittingnumbers of energetic nucleon, especially energetic neu-tron. Therefore the in-medium baryon-baryon inelasticscattering cross section decreases the energetic neutronto proton ratio n/p. As energetic nucleon in the reactiondirectly relates to energetic delta resonance production,resonance yields and ratio are also sensitive to the in-medium baryon-baryon inelastic scattering cross section.Because pion production is closely connected with deltaresonance, the ratio of π − /π + is also sensitive to the in-medium baryon-baryon inelastic scattering cross section.In the study, it is also seen that, due to non-squeezed-outemissions, both the energetic neutron to proton ratio n/pand the energetic charged pion ratio π − /π + are not verysensitive to the symmetry energy. Since pion productionin heavy-ion collision relates to a series of undeterminedproperties of delta and pion in medium, energetic neu-tron to proton ratio n/p is more suitable than energetic π − /π + ratio to be used to probe the in-medium baryon-baryon inelastic scattering cross section.This work is supported in part by the National NaturalScience Foundation of China under Grant Nos. 11775275,11435014. [1] J. Cugnon, Phys. Rev. C , 1885 (1980).[2] Y. Kitazoe, M. Sano, H. Toki, S. Nagamiya, Phys. Lett.B , 35 (1986).[3] G. F. Bertsch and S. Das Gupta, Phys. Rep. , 189(1988).[4] W. Bauer, Phys. Rev. C , 715 (1989).[5] B. A. Li and W. Bauer, Phys. Rev. C , 450 (1991).[6] P. Danielewicz and G. F. Bertsch, Nucl. Phys. A ,712 (1991).[7] Taesoo Song and Che Ming Ko, Phys. Rev. C , 014901(2015).[8] U. Mosel, Phys. Rev. C , 065501 (2015).[9] B. A. Li, Nucl. Phys. A , 605 (1993).[10] B. A. Li and C. M. Ko, Phys. Rev. C , 2037 (1995).[11] B. A. Li, Nucl. Phys. A , 365 (2002).[12] J. Xu, L. W. Chen, C. M. Ko, B. A. Li and Y. G. Ma,Phys. Rev. C , 067601 (2013).[13] V. Prassa, G. Ferini, T. Gaitanos, H. H. Wolter, G. A. Lalazissis, M. Di Toro, Nucl. Phys. A , 311 (2007).[14] A. B. Larionov, W. Cassing, S. Leupold, U. Mosel, Nucl.Phys. A , 747 (2001).[15] A. B. Larionov, U. Mosel, Nucl. Phys. A , 135 (2003).[16] G. F. Bertsch, G. E. Brown, V. Koch, B. A. Li, Nucl.Phys. A , 745 (1988).[17] G. J. Mao, Z. X. Li, Y. Z. Zhuo and E. G. Zhao, Phys.Rev. C , 792 (1997).[18] B. A. Li and L. W. Chen, Phys. Rev. C , 064611 (2005).[19] G. C. Yong, Eur. Phys. J. A , 399 (2010).[20] Q. Li, Z. Li, Phys. Lett. B , 557 (2017).[21] Z. Zhang, C. M. Ko, Phys. Rev. C , 064604 (2017).[22] Y. Cui, Y. X. Zhang, Z. X. Li, Phys. Rev. C , 054605(2018).[23] B. A. Li, Phys. Rev. Lett. , 192701 (2002).[24] Z. G. Xiao, B. A. Li, L. W. Chen, G. C. Yong, M. Zhang,Phys. Rev. Lett. , 062502 (2009).[25] G. C. Yong, Y. Gao, G. F. Wei, Y. F. Guo, W. Zuo, J. Phys. G: Nucl. Part. Phys. , 105105 (2019).[26] B. A. Li, `A. Ramos, G. Verde and I. Vida˜na (eds.), Eur.Phys. J. A (2014).[27] G. Q. Li and R. Machleidt, Phys. Rev. C , 1702 (1993).[28] G. Q. Li and R. Machleidt, Phys. Rev. C , 566 (1994).[29] T. Alm, G. Riipke, W. Bauer, F. Daffin, M. Schmidt,Nucl. Phys. A , 815 ( 1995).[30] T. Gaitanos, C. Fuchs, H.H. Wolter, Phys. Lett. B ,241 (2005).[31] Qingfeng Li and Zhuxia Li and Sven Soff and MarcusBleicher and Horst St¨ocker, J. Phys. G: Nucl. Part. Phys. , 407 (2006).[32] G. C. Yong, W. Zuo, X. C. Zhang, Phys. Lett. B ,240 (2011).[33] Yingxun Zhang, D. D. S. Coupland, P. Danielewicz,Zhuxia Li, Hang Liu, Fei Lu, W. G. Lynch, M. B. Tsang,Phys. Rev. C , 024602 (2012).[34] W. M. Guo, G. C. Yong, Wei Zuo, Phys. Rev. C ,044605 (2014).[35] B. A. Li, G. C. Yong and W. Zuo, Phys. Rev. C ,014608 (2005). [36] G. C. Yong, Phys. Lett. B
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