Effects of the high-momentum tail of nucleon momentum distribution on the isospin-sensitive observables
aa r X i v : . [ nu c l - t h ] N ov Effects of the high-momentum tail of nucleon momentum distribution on theisospin-sensitive observables
Fang Zhang ∗ and Gao-Chan Yong †‡ School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, People’s Republic of China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
Based on the Isospin-dependent transport model Boltzmann-Uehling-Uhlenbeck (IBUU), effectsof the high momentum tail (HMT) of nucleon momentum distribution in colliding nuclei on someisospin-sensitive observables are studied in the semi-central
Au+
Au reactions at incident beamenergy of 400 MeV/nucleon. It is found that the nucleon transverse flow, the difference of neutronand proton transverse flows, the nucleon elliptic flow and free neutron to proton ratio are all lesssensitive to the HMT, while the isospin-sensitive nucleon elliptic flow difference is clearly affectedby the HMT. Except at very high kinetic energies, the kinetic energy distributions of π − , π + andcharged pion ratio π − /π + are all sensitive to the HMT. PACS numbers: 25.70.-z, 21.65.Ef
I. INTRODUCTION
The results of recent high-momentum transfer reac-tions demonstrate that about 20% nucleons form stronglycorrelated nucleon pairs with large relative momen-tum and small center-of-mass momentum (relative tothe Fermi momentum) [1–3]. Such phenomenon is ex-plained by the short range nucleon-nucleon interactions[4–7]. And such short range nucleon-nucleon interactionis strong in neutron-proton pairs but weak in neutron-neutron or proton-proton pairs. Thus in isospin asym-metric nuclei, protons have a larger probability than neu-trons to have momenta higher than the Fermi momentum[8, 9].Nowadays the equation of state (EoS) of isospin sym-metric nuclear matter is relatively well determined butthe EoS of isospin asymmetric nuclear matter, especiallythe density dependence of the nuclear symmetry energyis still uncertain [10–13]. Besides being of great impor-tance in nuclear physics, the symmetry energy also playscrucial roles in many astrophysical processes. Thus thecrucial task is to find experimental observables that aresensitive to the symmetry energy. A number of poten-tial observables have been identified in heavy-ion colli-sions induced by neutron-rich nuclei, such as the free neu-tron/proton ratio [14], the π − /π + ratio [15], the t/ He[16], the nucleon transverse flow [17], the eta produc-tion [18], etc. More related references can be found inRef. [10, 11, 13, 19].Because both isospin-sensitive nucleon transverse andelliptic flows depend on nucleon momentum distribution,they are expected to be sensitive to the HMT. In addi-tion, in heavy-ion collisions the primordial π − /π + ratiois approximately equal to ( N/Z ) at certain momentumspace and the π − ’s are mostly produced from neutron- † Corresponding author ∗ Electronic address: [email protected] ‡ Electronic address: [email protected] neutron collisions while π + ’s are mostly produced fromproton-proton collisions [20, 21]. Thus the kinetic energydistribution of the π − /π + ratio is expected to be alsosensitive to the HMT.Since the HMT of single-nucleon momentum distribu-tion in nucleus was confirmed experimentally at JeffersonLaboratory, it should be considered in the nuclear trans-port simulations. Unfortunately, most isospin-dependenttransport models rarely took the effects of the HMTinto account [22–26]. Recently, the effects of the HMThave been added into the transport model by modifyingthe symmetry potential in nuclear mean-field potential[27, 28]. It is also necessary to see the HMT effects ofnucleon initial momentum distribution of nuclei on somefrequently studied isospin-sensitive observables in heavy-ion collisions, such as nucleon collective flow, free neutronto proton ratio n/p as well as the π − /π + ratio. We makethese studies in Au+
Au collisions at a beam energyof 400 MeV/nucleon with an impact parameter b = 7.2fm. The reasons of selecting b = 7.2 fm are that there areFOPI-LAND data for Au+Au at 400 MeV/nucleon withb = 7.2 fm. And also with larger impact parameters, onecan get clear nucleon flow.Our paper is organized as follows. The next sectiongives a brief description of the applied IBUU model. Thefollowing is the detailed studies of the HMT effects onthe collective flow and the free neutron to proton ratio n/p and the π − /π + ratio. The conclusion is given in thelast section. II. THE IBUU TRANSPORT MODEL
To study the effects of the HMT of nucleonmomentum distribution on isospin-sensitive observ-ables, we use the Isospin-dependent Boltzmann-Uehling-Uhlenbeck (IBUU) transport model [20, 29–34]. In thisIBUU model, nucleon coordinates are given in nucleuswith radius R = 1.2 A / , where A is the mass numberof nucleus. Fig. 1 shows nucleon initial momentum distri- -2 -1 AuProton NeutronAV18+TBF n ( k ) k (fm -1 ) FIG. 1: Momentum distributions of neutron and proton innucleus
Au. Taken from Ref. [35]. bution with high-momentum tail in
Au, which is givenby the extended brueckner-hartree-fock (BHF) approachby adopting the AV 18 two-body interaction plus a mi-croscopic three-body-force (TBF) [35]. The instability ofinitial colliding nuclei including the HMT has negligibleeffects on our studies here. As comparison, we also givenucleon momentum distribution n ( k ) = (cid:26) , k ≤ k F ;0 , k > k F . (1)In this model, we use the isospin- and momentum-dependent mean-field single nucleon potential [33, 34, 36,37] 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 , (2)where τ, τ ′ = 1 / − /
2) for neutrons (protons), ρ is nu-clear saturation density. δ = ( ρ n − ρ p ) / ( ρ n + ρ p ) is theisospin asymmetry, and 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 are obtainedby fitting empirical constraints of the saturation density,the binding energy, the incompressibility, the isoscalareffective mass, the single-particle potential, the symme-try energy value and the symmetry potential at infinitely large nucleon momentum at saturation density. f τ ( ~r, ~p )is the phase-space distribution function at coordinate ~r and momentum ~p and solved by using the test-particlemethod numerically. The symmetry energy’s stiffness pa-rameter x in the above single nucleon potential is usedto mimic different forms of the symmetry energy. Sincewe do not intend to study the effect of nuclear symmetryenergy, here we just let x = 0. Note here that, besidesconsidering the HMT in momentum initialization, the ef-fect of the HMT on the single particle potential is alsoconsidered consistently [34]. For ∆ baryon potential, itis divided into nucleon potential by [20, 38] U ∆ − = U ( ρ, δ, ~p, τ = 12 ) , (3) U ∆ = 23 U ( ρ, δ, ~p, τ = 12 ) + 13 U ( ρ, δ, ~p, τ = −
12 ) , (4) U ∆ + = 13 U ( ρ, δ, ~p, τ = 12 ) + 23 U ( ρ, δ, ~p, τ = −
12 ) , (5) U ∆ ++ = U ( ρ, δ, ~p, τ = −
12 ) . (6)We use the reduced baryon-baryon ( BB ) scattering crosssection in medium by a factor of R medium ( ρ, δ, ~p ) ≡ σ medium BB elastic , inelastic /σ free BB elastic , inelastic = ( µ ∗ BB /µ BB ) , (7)where µ BB and µ ∗ BB are the reduced masses of the collid-ing baryon-pair in free space and medium, respectively.Similar with the definition in Ref. [32], the effective massof baryon in isospin asymmetric nuclear matter is ex-pressed by m ∗ B m B = (cid:26) m B p dU B dp (cid:27) , (8)where dU B denotes nucleon or ∆ potentials. III. RESULTS AND DISCUSSION
Nucleon collective transverse flow is the moving par-ticles deflected away from the beam axis in the reactionplane, which reads [31, 39–43] F ( y ) = 1 N ( y ) N ( y ) X i =1 p xi ( y ) . (9)In the above, p x is the momentum of an emitting nucleonin x direction at rapidity y , N ( y ) is the number of freenucleons at rapidity y . Here, free nucleons are identifiedwhen their local densities are less than ρ /
8. Shown inFig. 2 is the effects of the HMT on the nucleon trans-verse flow as a function of reduced rapidity. From Fig.2, it is seen that the effects of the HMT on the trans-verse flow are in fact not evident. This is understandablesince the shape of high-momentum tail roughly exhibits -100-50050100-100-50050 -0.6 -0.3 0.0 0.3 0.6-100-50050 (a) with HMT w/o HMT
Au+
Au400MeV/nucleon F x n ( M e V / c ) (b) F x p ( M e V / c ) (c) (y/y beam ) c.m. F x n + p ( M e V / c ) FIG. 2: (Color online) Rapidity distribution of nucleon trans-verse flow (upper: neutron, middle: proton, bottom: nucleon)in the semi-central reaction
Au +
Au at incident beamenergy of 400 MeV/nucleon, with and without the HMT, re-spectively. a C/k form [8, 27], the number of nucleons with mo-menta larger than the Fermi momentum is just a smallproportion of the total number of nucleons in a nucleus.Nevertheless, the strength of the nucleon transverse flowbecomes somewhat larger with the HMT than that with-out the HMT.In the high-momentum tail of nucleon momentum dis-tribution, since nucleonic component is strongly isospin-dependent, i.e., the number of n-p pairs is about 18 timesthat of the p-p or n-n pairs [3], one expects larger effectsof the HMT on the isospin-sensitive collective flow.Fig. 3 shows the rapidity distribution of the free neu-tron to proton ratio n/p and the difference of neutronand proton transverse flows. Since n-p pairs dominate inthe HMT, from Fig. 3, one sees lower value of the n/p ratio at larger rapidities with the HMT. And also withthe HMT, relatively small absolute value of the differenceof neutron and proton flows ( F nx ( y ) − F px ( y )) at larger ra-pidities is seen in the lower panel. From Fig. 2 - 3, onecan see that the effects of the HMT on the free neutronto proton ratio n/p and the nucleon transverse flow areclear but not large. This is really good news for some (a) Au+
Au, 400MeV/nucleon with HMT w/o HMT (y/y beam ) c.m. F x n - F x p ( M e V / c )( n / p ) f r ee (b) FIG. 3: (Color online) Rapidity distribution of free neu-tron to proton ratio (upper panel), the difference of neu-tron and proton transverse flows (lower panel) in the semi-central
Au +
Au reaction at incident beam energy of400 MeV/nucleon with and without the HMT. physical goals such as studying nuclear symmetry energyby these observables in heavy-ion collisions. Neverthe-less, one has to see if other isospin-sensitive observablesare sensitive to the HMT.We now turn to study the effects of the HMT on thedifferential elliptic flow of unbound nucleons. The ellipticflow corresponds to the second Fourier coefficient in thetransverse-momentum distribution and can be expressedas [31, 43–45] v = h p x − p y p x + p y i . (10)Where p x is the nucleon transverse momentum along x axis in the reaction plane, p y is the nucleon transversemomentum along y axis perpendicular to the reactionplane. A negative value of v describes the dominantout-of-plane particle emission while a positive value v indicates the leading in-plane particle emission. Shownin Fig. 4 is the transverse momentum distribution of nu-cleon elliptic flow at mid-rapidity with or without theHMT. It is seen that the values of both neutron and pro-ton elliptic flows are negative, which indicating the lead- -10-8-6-4-2 100 200 300 400 500-12-10-8-6-4-2 (a) Au+
Au400MeV/nucleon v ( % ) v ( % ) (b) with HMT w/o HMT p tc.m. (MeV) |(y/y beam ) c.m. |<=0.5 FIG. 4: (Color online) Transverse momentum distribution ofneutron (upper panel) and proton (lower panel) elliptic flowsin the semi-central
Au +
Au collisions at a beam energyof 400 MeV/nucleon with and without the HMT, respectively. ing out-of-plane nucleon emission at such incident beamenergy. The strength of proton elliptic flow is larger thanthat of neutron, which indicating a strong Coulomb re-pulsion to protons. And one can also see that the effectof the HMT on the proton elliptic flow is larger than thatof neutron. This is because in neutron-rich reaction sys-tem, protons have great probabilities than neutrons to beaffected by the short-range correlations [8]. Anyway, theeffects of the HMT on nucleon elliptic flow are not largein the low transverse momentum region but somewhatlarger in the high transverse momentum region.Since the effects of the HMT on nucleon elliptic flow aregenerally small, we expect the effects of the HMT on thedifference of neutron and proton elliptic flows ( v n − v p )to be larger. For comparison, the total nucleon ellipticflow v n + p is also presented in the upper panel of Fig.5. Shown in the lower panel of Fig. 5, the effects ofthe HMT on the difference of neutron and proton ellipticflows ( v n − v p ) is larger than that of the total nucleonelliptic flow. Because the proton elliptic flow is negativeand stronger than the neutron elliptic flow, the differenceof neutron and proton elliptic flows ( v n − v p ) is positive -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. 5: (Color online) Total nucleon elliptic flow (upperpanel) and the difference of neutron and proton elliptic flow(lower panel) in the semi-central reaction of
Au +
Auat a beam energy of 400 MeV/nucleon with and without theHMT, respectively. and becomes smaller due to neutron-proton correlationsin the HMT. Comparing Fig. 5 with Fig. 3, it is foundthat the isospin-sensitive elliptic flow observable ( v n − v p )is sensitive to the HMT.Finally, we examine in Fig. 6 the kinetic energy dis-tributions of π − , π + as well as π − /π + ratio with andwithout the HMT in the same semi-central reaction Au +
Au at a beam energy of 400 MeV/nucleon(while simulating central collisions of
Au +
Au atthe same beam energy, we obtained the same physicalresults). It is seen that kinetic energy distributions ofboth π − and π + are very sensitive to the HMT. The ra-tio of π − /π + is also very sensitive to the HMT exceptin the high kinetic energy region. The neutron-protonshort-range corrections increase kinetic energies of a cer-tain proportion of neutrons and protons, thus more pionmesons are produced. This is the reason why the valueof the kinetic energy distribution of pion meson is higherwith the HMT than that without the HMT. Because inisospin asymmetric reaction system, protons have a largerprobability than neutrons to have larger momenta [8, 9] (a) Au+
Au400MeV/nucleon with HMT w/o HMT - + - / + E c.m.Kin (MeV) (b) (c) FIG. 6: (Color online) Kinetic energy distributions of π − (upper panel), π + (middle panel) as well as π − /π + (bottompanel) ratio with and without the HMT in the reaction of Au +
Au at incident beam energy of 400 MeV/nucleon. and proton-proton collision mainly produces π + , one seeslower value of the π − /π + ratio with the HMT than thatwithout the HMT. The effects of the HMT on the π − /π + ratio become insensitive in the high kinetic region. How-ever, in the high-energy tail of π spectra, the π − /π + ratiocould be even more sensitive to the symmetry energy at high densities [46, 47]. This is surely interesting to thosewho use the observable π − /π + ratio to probe the sym-metry energy [20, 47–52] without including the HMT intheir transport models. IV. SUMMARY
In the framework of the isospin-dependent transportmodel, we studied the effects of the high momentum tailof nucleon momentum distribution in colliding nuclei onsome isospin-sensitive observables. We find that in thesame semi-central reaction
Au +
Au at a beam en-ergy of 400 MeV/nucleon, the isoscalar nucleon flows,including transverse and elliptic flows are less affectedby the high momentum tail of nucleon momentum distri-bution in colliding nuclei. The isovector nucleon ellipticflow is sensitive to the high-momentum tail of nucleonmomentum. Except the energetic π − /π + ratio, the ki-netic energy distributions of π − , π + and charged pionratio π − /π + are all sensitive to the high momentum tailof nucleon momentum distribution.Because the nucleon-nucleon correlation is hard toembed into the transport model, to probe the symme-try energy using isospin-sensitive observables, the HMT-insensitive observable such as the difference of neutronand proton transverse flows [17, 43–45] and π − /π + ra-tio at higher energies [46, 47] are recommendable. Andalso because the HMT itself is isospin-dependent, it isnecessary to further study the competitive relation ofthe HMT and the symmetry energy on frequently usedisospin-sensitive observables. Acknowledgments
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