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Elastic scattering based on energy-dependent relativistic Love-Franey model at energies between 20 and 800 MeV

  • In order to investigate the elastic scattering, we fit scattering observables of the weighted fits (WF16) with the relativistic Love-Franey (RLF) model. The masses, cutoff parameters, and initial coupling strengths of RLF are assumed to be independent of energy. Because the energy boundary between low energy and high energy is around 200 MeV, the masses, cutoff parameters, and initial coupling strengths of RLF are obtained by fitting scattering observables of WF16 at an incident energy of 200 MeV. With the masses, cutoff parameters, and initial coupling strengths as the input, the energy-dependent RLF model is constructed over the laboratory energy range of 20 to 800 MeV within a unified fit. To examine the validity of this fit, we investigate p+208Pb elastic scattering for various energies. Although the scattering observables of pp and pn of 200 MeV best fit the values of WF16, the RLF model of 200 MeV without the Pauli blocking (PB) corrections fails to describe the experimental differential cross sections, analyzing powers, and spinrotation functions. When the PB corrections are taken into account for various energies, the RLF model can well describe the experimental data of p+208Pb elastic scattering.
      PCAS:
    • 11.80.Jy(Many-body scattering and Faddeev equation)
  • The interaction between nucleons is very important in understanding the nuclear experiments and the properties of nuclei and neutron stars. However, the interaction between nucleons is still poorly understood, particularly at high densities. The main reason for this is that the interaction is not a directly observable quantity in experiments. The interaction between nucleons as residual interactions of the QCD cannot be solved via perturbation expansion. Therefore, the interaction between nucleons leads to the uncertainty of the symmetry energy. Considerable effort has been directed toward understanding the interaction between nucleons. The global Dirac phenomenological optical potentials, which describe the elastic proton scattering observables of 12C and 208Pb well for incident energies between 20 and 1040 MeV [1, 2], are powerful tools in understanding the interaction between nucleons. They have been widely used to extract the symmetry energy directly or indirectly [3, 4]. The relativistic impulse approximation (RIA) [57] can also be used to describe the elastic proton scattering observables. Theoretically, the symmetry energy can be extracted from the microscopic potentials of the RIA.

    The original RIA with scalar and vector potentials can reproduce the analyzing power and spin-rotation parameter in proton-nucleus scattering for incident energies above 500 MeV [5, 8]. Because the direct and exchange parts of the t-matrix are not separated explicitly in the original RIA, the optical potential of the original RIA diverges at energies below 500 MeV [8]. By separating the direct and exchange parts of the t-matrix, it is found that the pion with pseudoscalar leads to the divergent optical potential. This can be cured by introducing the pion with pseudovector coupling, in a process known as the relativistic Love-Franey (RLF) model [6, 9,10]. The RLF model has been used to describe the elastic proton scattering observables successfully for incident energies of 135, 200, 300, and 400 MeV [6]. The shortcoming of the original RLF model is that the scattering observables were fitted separately at each energy. The separate fits of the scattering observables cannot reflect the energy-dependent trend of the scattering observables. This motivates the proposal of the energy-dependent RLF model [11, 12]. The energy-dependent RLF model has been used to fit empirical amplitudes successfully at incident energies of 50–200 MeV [13], 200–500 MeV [11], and 500–800 MeV [12]. It should be noted that these fits are separated from each other. In this work, we try to construct the energy-dependent RLF model over the laboratory energy range of 20 to 800 MeV within a unified fit.

    The RLF model has been used to study both elastic proton-nucleus scattering [6, 7] and the (p, 2p) reaction [14]. The RLF model without Pauli blocking (PB) corrections cannot well describe the differential cross sections dΩ/dσ, analyzing powers Ay, and spinrotation functions Q of elastic proton-nucleus scattering. With the PB corrections, the RLF model can describe dΩ/dσ, Ay, and Q well [6, 7]. The RLF model is also helpful in understanding the in-medium effect. In-medium nucleon-nucleon (NN) cross sections, which are key inputs of various transport models, can be obtained from RLF. It has been recognized that the theoretical uncertainty in extracting the density dependence of the symmetry energy from data of heavy-ion reactions can be largely ascribed to the poor knowledge and premature treatment of the isospin-dependent in-medium NN cross sections [15, 16]. Although the NN cross section in free space is well-determined, the in-medium NN cross section is still model dependent [1728]. Under the original RIA framework, the in-medium NN cross sections have been studied [24, 28]. However, no such work has been done in the framework of the RLF model. At a low nucleon kinetic energy, it would be interesting to study the in-medium NN cross sections with the RLF model. Additionally, the energy and density dependent symmetry potential of nuclear matter has been investigated with the original RIA and RLF models [24, 28, 29]. The energy and density dependence symmetry potential of the energy-dependent RLF model may have some differences from the results of the original RIA and RLF models.

    The remainder of this paper is organized as follows. In Sec. II, we briefly introduce formulas and approaches used in this work. The formulas include the fits between RIA scattering amplitudes and the empirical amplitudes, as well as the elastic proton scattering observables of 208Pb. Results and discussions are presented in Sec. III. Finally, a summary is given in Sec. IV.

    In order to obtain the on-shell NN matrix elements of RLF scattering amplitudes, one needs to transform the NN matrix elements of RLF scattering amplitudes into those of the non-relativistic NN scattering amplitudes. The relation between the matrix elements of the RLF scattering amplitudes and the matrix elements of the non-relativistic NN scattering amplitudes in the center-of-mass frame can be written as [5, 6, 9, 10, 13]

    (2ikc)1χs1χs2ˆfcmχs1χs2=ˉu(k1,s1)ˉu(k2,s2)ˆFu(k1,s1)u(k2,s2),

    (1)

    where k1(Ec,kc) and k2(Ec,kc) are the incident 4-momenta in the center-of-mass frame. χs represent the usual Pauli spinors. kc represents the momentum in the center of mass frame. The Dirac spinors are normalized (ˉuu=1). ˆfcm and ˆF are the non-relativistic scattering matrix and scattering matrix of RLF respectively. ˆfcm contains five real and imaginary amplitudes [30]:

    (2ikc)1ˆfcm=12[a+b+(ab)σ1nσ2n+(c+d)σ1mσ2m+(cd)σ1lσ2l+e(σ1n+σ2n)],

    (2)

    where a, b, c, d, and e are the non-relativistic real and imaginary amplitudes. The scattering matrix of RLF ˆF also contains five real and imaginary amplitudes:

    ˆF=Ti=SFi(s,t,u)λ(1)iλ(2)i,

    (3)

    where i includes the scalar (S), vector (V), pseudoscalar (P), axial vector (A), and tensor (T). FS, FV, FP, FA, and FT are the relativistic amplitudes of RLF. The Dirac matrices of S, V, P, A, and T are I4×4, γμ, γ5, γ5γμ, and σμν respectively. With these Dirac matrices and Eq. (1), the relation between the non-relativistic amplitudes and the relativistic amplitudes is obtained as follows [5, 9, 10, 13]:

    (abcde)=ikcO5×5(FSFVFPFAFT),

    (4)

    The O5×5 matrix is written as [10, 13]

    O5×5=(cosθcosθcosθcosθ4sinθ111101111011110isinθisinθisinθisinθ4icosθ)×(αhα+pβ0pα+hβ2βhγγpγδ2(p+h)δα(p+h)α0(hp)α2αhγγpγγ2γϵϵ0ϵ2ϵ),

    (5)

    with

    α=cos2θ2,β=1+sin2θ2,γ=sin2θ2,δ=1+cos2θ2,ϵ=EcMsinθ2cosθ2,Ec=(k2c+M2)1/2,h=E2cM2,p=k2cM2,

    (6)

    where θ represents the scattering angle in the two-body center-of-mass, and M represents the free nucleon mass. Now, the relation between the matrix elements of the RLF scattering amplitudes and the matrix elements of the non-relativistic NN scattering amplitudes is clear. For comparison with the empirical amplitudes, the relativistic amplitudes Fi(s,t,u) need to be solved. The Fi(s,t,u) contains direct and exchange contributions:

    Fi(s,t,u)=iM22Eckc[FDi(s,t)+FEi(s,u)],

    (7)

    with

    FDi(s,t)=Nj=1δi,i(j)τ1τ2Tjfj(Ec,|q|),FEi(s,u)=(1)TNNNj=1Ci(j),iτ1τ2Tjfj(Ec,|Q|),

    (8)

    where N represents the number of mesons in this work, Ci(j),i is the Fierz matrix [6, 9, 31], Tj is the isospin of the jth meson, and TNN is the total isospin of the two-nucleon state. When TNN=1, τ1τ2Tj=1 for both Tj=0 and Tj=1. When TNN=0, τ1τ2Tj=1 for Tj=0, and τ1τ2Tj=3 for Tj=1. For instance, when TNN=0, the scalar amplitudes are written as

    FDS(s,t)=fS(Tj=0)(Ec,|q|)3fS(Tj=1)(Ec,|q|),FES(s,u)=j=S,TCj,S[fj(Tj=0)(Ec,|Q|)3fj(Tj=1)(Ec,|Q|)],

    (9)

    where fj(Ec,|q|) and fj(Ec,|Q|) contain the real and imaginary parts. For instance:

    fj(Ec,|q|)=fjR(Ec,|q|)ifjI(Ec,|q|),fjR(Ec,|q|)=g2j(Ec)q2+m2j(1+q2/Λ2j)2,fjI(Ec,|q|)=ˉg2j(Ec)q2+ˉm2j(1+q2/ˉΛ2j)2,

    (10)

    where g2j(Ec), mj, and Λj are the coupling constant, mass, and momentum cutoff of the jth real meson, respectively. ˉg2j(Ec), ˉmj, andˉΛj are the coupling constant, mass, and cutoff parameter of the jth imaginary meson, respectively. The direct momentum transfer |q| and the exchange momentum transfer |Q| are written as [9]

    |q|=|kckc|=2kcsinθ2,|Q|=|kc+kc|=2kcsin(πθ2).

    (11)

    For a given projectile's kinetic energy Tlab, the momentum and energy of the center of mass frame are kc=2MTlab2 and Ec=k2c+M2, respectively. With these fitting formulas, the parameterization of the RLF model can be obtained. Then, the Dirac optical potential is generated with the fitting parameters of the RLF model.

    The Dirac optical potential of the RLF model is the same as that of the original RIA model [6, 32, 33]:

    Uopt(q,E)=4πiplabMˉΨ|An=1ˆF(q,E;n)eiqr(n)|Ψ,

    (12)

    where plab represents the laboratory momentum of incident nucleons. q and E represent the momentum transfer and collision energy, respectively. |Ψ represents the A-particle ground state of the target nucleus.

    In proton-nucleus scattering, the scattering process can be approximately treated as incident protons scattered by each of the nucleons in the target nucleus. Assuming the incident proton wave function as U0(r) and the relativistic Hartree wave function of the nucleus ground state as U0(r), the optical potential can act on the incident proton wave function and project into the coordinate space. In this work, we only list the formulas after Fourier transforms [33]:

    r|Uopt(q,E)|U0(r)=4iπplabML[d3rρL(r)tLD(|rr|;E)]λLU0(r)4iπplabML[d3rρL(r,r)tLE(|rr|;E)]λLU0(r),

    (13)

    where

    tLD(|r|;E)]d3q(2π)3tLD(q,E)eiqr,

    (14)

    tLE(|r|;E)]d3q(2π)3tLE(Q,E)eiQr,

    (15)

    with tLD(q,E)(iM2/2Eckc)FLD(q) and tLE(Q,E)(iM2/2Eckc)FLE(Q), respectively. The first term of the above equation represents the direct optical potential in the coordinate space:

    ULD(r,E)=4iπplabMd3rρL(r)tLD(|rr|;E),

    (16)

    and the second term of the above equation represents the exchange optical potential. With the local-density approximation, the exchange optical potential is written as

    ULE(r,E)=4iπplabMd3rρL(r,r)×tLE(|rr|;E)j0(plab|rr|),

    (17)

    j0 is a spherical Bessel function. The nuclear densities of Eqs. (13), (16), and (17) are defined as

    ρL(r,r)occαˉUα(r)λLUα(r),ρL(r)ρL(r,r).

    (18)

    The occupied states indicate that one sums over target protons for the density with pp amplitudes and over target neutrons for the density with pn amplitudes. The densities of target protons and neutrons can be obtained from relativistic mean field theory (RMF) [7, 33]. When the densities of target protons and neutrons predicted by various models are substituted into Eq. (18), the results are almost the same in p+208 Pb elastic scattering. In the following sections, we take the densities predicted by the code of Ref. [33] as the input.

    With above formulas, the Dirac optical potential can be obtained. For the spin-saturated nucleus, the Dirac optical potential only contains scalar, vector, and tensor terms. Because the tensor contribution is small, the tensor contribution has been neglected in the calculation. The Dirac optical potential is simplified as

    Uopt(r;E)=US(r;E)+γ0UV(r;E),UL(r;E)=ULD(r;E)+ULE(r;E).

    (19)

    For the laboratory energy near 200 MeV, the contribution of Pauli blocking (PB) plays an important role in the calculations. The modification of PB with a local-density approximation is [6, 13, 33]

    ULPB(r;E)=[1aL(E)(ρB(r)ρ0)2/3]UL(r;E).

    (20)

    Here, ρB(r) represents the local baryon density of the target nuclus, and ρ0=0.1934 fm3.

    With the Dirac optical potential, the Dirac equation of the projectile is written as

    hU0(r)=EU0(r)={iα+UV(r;E)+β[M+US(r;E)]}U0(r).

    (21)

    By solving the Dirac equation, i.e., Eq. (21), the scattering observables (the differential cross section dσ/dΩ, analyzing power Ay, and spin rotation function Q) are determined. The detailed solution processes are presented in Ref. [33]. In this work, we concentrate on dσ/dΩ, Ay, and Q of p+208Pb elastic scattering.

    The RLF pp and pn scattering amplitudes are written as [9]

    Fi(pp)=Fi(TNN=1),Fi(pn)=12[Fi(TNN=1)+Fi(TNN=0)],

    (22)

    where the relativistic amplitudes FS, FV, FP, and FA of pp and pn and the non-relativistic amplitudes a, b, c, d, and e of pp and pn are generated from Eq. (4). However, the non-relativistic amplitudes are no longer available on the on-line Scattering Analysis Interactive Dial-in (SAID) facility. The scattering observables are still available on SAID. There are several models available for computing the scattering observables on SAID. We choose the scattering observables obtained from the weighted fits (WF16) on SAID [34]. In contrast to previous studies [1113], we fit 10 scattering observables of the weighted fits (WF16) rather than the real and imaginary amplitudes of pp and pn. These 10 scattering observables are written as [30]

    dσdΩσ=12(|a|2+|b|2+|c|2+|d|2+|e|2),D=12(|a|2+|b|2|c|2|d|2+|e|2)/σ,DT=12(|a|2|b|2+|c|2|d|2+|e|2)/σ,Ayy=12(|a|2|b|2|c|2+|d|2+|e|2)/σ,

    P=(ae)/σ,CKP=(de)/σ,CKK=(ad+bc)/σ,CPP=(adbc)/σ,R=[(ab)cos(α+θ2)+(cd)cos(αθ2)(be)sin(α+θ2)],A=[(ab)sin(α+θ2)+(cd)sin(αθ2)(be)cos(α+θ2)],

    (23)

    These 10 scattering observables are the unpolarized differential cross section (dσdΩ), depolarization tensor for polarized beam (D), polarization transfer from beam to recoil particle (DT), polarization correlation for initially unpolarized particles Ayy, CKP, CKK, CPP, polarization of scattered particle (P), and triple scattering parameters A and R. α is a function of the angle:

    α=θ2θlab,

    (24)

    where θlab represents the laboratory scattering angle. Theoretically, the fitting quality with these 10 scattering observables of pp and pn should be as good as that in the previous case with the non-relativistic amplitudes of pp and pn.

    The real isovector pseudoscalar (π), isoscalar vector (ω), and isovector vector (ρ) masses are fixed to their physical values mπ=138 MeV, mω=782 MeV, and mρ= 770 MeV, respectively. The remaining meson masses are set in accordance with Ref. [13]. Because the energy boundary between low energy and high energy is around 200 MeV, we first fit the scattering observables to obtain the initial coupling strengths of RLF. The initial coupling constant g20 (ˉg20) and the cutoff parameter Λ (ˉΛ) are obtained by fitting the scattering observables for Tlab= 200 MeV alone. In order to obtain the best fit to the scattering observables for Tlab= 200 MeV, the Levenberg-Marquadt method has been employed to minimize the value of χ2. χ2 is defined as follows:

    χ2=data(χempiricalχfit)2χ2empirical,

    (25)

    where χ2empirical represents an angle-averaged value and is defined as

    χ2empirical=175θ=5χ2empirical(θ)Nempirical,

    (26)

    where Nempirical=35 represents the number of angles per energy. For a single incident energy, the number of the data points is 700: 35 angles × 10 scattering observables × 2 pp and pn. As a result, the real (imaginary) initial coupling constant g20 (ˉg20), the real (imaginary) meson mass m (ˉm), and the real (imaginary) cutoff parameter Λ (ˉΛ) are presented in Table 1. It should be noted that Λ>m has been employed for a given meson.

    Table 1

    Table 1.  Parameter sets for the energy-dependent RLF over the laboratory energy range of 20 to 800 MeV. The masses m (ˉm) and cutoff parameters Λ (ˉΛ) are in MeV, whereas the remaining parameters are dimensionless.
    Real parameters
    MesonIsospinCoupling typemg20a1a2a3a4Λ
    σ0Scalar (S)600−8.3486.527×1012.773−2.741×102−7.639962.7
    ω0Vector (V)7821.019×101−6.246×101−1.400×1018.492×1012.677×1011155
    t00Tensor (T)5502.794×1013.3031.172×101−6.676×101−3.008×1011945
    a00Axial vector (A)5004.921×1016.0661.953×101−1.089−5.133×1011597
    η0Pseudoscalar (P)9501.180×101−1.694×102−1.350×1025.4175.550×102957.3
    δ1Scalar (S)5006.095×1035.635×1011.287−6.040×102−4.1012945
    ρ1Vector (V)770−1.667×1019.420×1051.644×101−2.905×1022.915×1013001
    t11Tensor (T)600−2.500×101−1.893−7.8484.448×1011.989×1011280
    a11Axial vector (A)650−1.356−4.678−1.529×1018.102×1014.146×101745.1
    π1Pseudoscalar (P)1381.095×101−3.834×109−4.8894.005×1011.082×101677.4
    Imaginary parameters
    MesonIsospinCoupling typeˉmˉg20a1a2a3a4ˉΛ
    σ0Scalar (S)600−2.849−4.637−1.475×1016.301×1014.377×101800.0
    ω0Vector (V)7004.4054.8252.077×101−9.616×101−5.672 ×101716.0
    t00Tensor (T)750−9.457×101−1.474−7.5354.140×1011.914×1011340
    a00Axial vector (A)750−2.198−3.583−1.374×1016.840×1013.762×1011437
    η0Pseudoscalar (P)10001.420×101−1.619×102−2.011×1029.7796.591×1022758
    δ1Scalar (S)6503.0941.5932.9451.614×102−1.328×101751.9
    ρ1Vector (V)600−2.421−9.543×101−3.6449.074×1021.292×101766.6
    t11Tensor (T)7506.140×1011.5135.775−3.011×101−1.557×1011662
    a11Axial vector (A)10002.2655.0711.997×101−1.011−5.450×1011233
    π1Pseudoscalar (P)500−5.8912.029×1012.363×101−9.593×101−7.574×1011342
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    The energy-dependent RLF model has been used to fit the empirical NN data for separate incident energies (50–200 MeV [13], 200–500 MeV [11], and 500–800 MeV [12]). To relate these energy domains, we first fit the scattering observables of various energies (Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV) separately with the same cutoff parameters and meson masses as 200 MeV. As shown in Fig. 1, we find that the coupling strengths, such as g2π, g2η, g2a1, and g2t1, approximately follow a logarithmic or parabolic curve opening to the right. Combining the linear and square terms [11, 12], the energy-dependent coupling strengths can be written as

    Figure 1

    Figure 1.  (color online) The coupling constants (g2π, g2η, g2a1, and g2t1) as a function of the laboratory energies Tlab. The coupling constants are obtained from separate fits with Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV.

    g2(Ec)=g20+a1ln(TlabT0)+a2(TlabT01)+a3[(TlabT0)21]+a4(TlabT01),

    (27)

    with T0=200 MeV. This equation is also valid for the imaginary coupling strengths. When the incident energy is 200 MeV, the energy-dependent coupling strengths should equal the initial coupling strengths. The meson masses and cutoff parameters of RLF are assumed to be independent of the energy in this work.

    For various incident energies, the initial coupling constant, the meson mass, and the cutoff parameter remain the same. However, the coupling constants g2j(Ec) and ˉg2j(Ec) change with respect to the incident energy. The parameters a1, a2a3, and a4 of Eq. (27) are obtained by fitting the scattering observables for various incident energies (Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV) uniformly. The number of the data points is 7700: 11 energies × 35 angles × 10 scattering observables × 2 pp and pn. The final results for a1, a2a3, and a4 are presented in Table 1.

    To illustrate the fitting quality, we show the unpolarized differential cross section (dσdΩ), depolarization tensor for the polarized beam (D), polarization transfer from beam to recoil particle (DT), polarization correlation for initially unpolarized particles (Ayy), and polarization of scattered particles (P) of pp and pp at Tlab= 20, 100, 200, 400, and 800 MeV in Figs. 26. At Tlab= 200 MeV, as shown in Fig. 4, because the scattering observables of Tlab= 200 MeV are fitted alone, the scattering observables are described well for both np and pp. As the Coulomb corrections dominate the empirical pp scattering observables of forward and backward angles, the Coulomb corrections are included in the pp scattering observables in this work. The Coulomb corrections are obtained by subtracting the TNN=1 amplitudes of SAID from the pp amplitudes of SAID.

    Figure 2

    Figure 2.  (color online) np and pp observables as a function of the center-of-mass scattering angle θ for Tlab = 20 MeV. The dashed curves are generated from Table 1, and the solid curves are the observables of WF16 [34].

    Figure 3

    Figure 3.  (color online) Same as Fig. 2 but for Tlab = 100 MeV.

    Figure 4

    Figure 4.  (color online) Same as Fig. 2 but for Tlab = 200 MeV.

    Figure 5

    Figure 5.  (color online) Same as Fig. 2 but for Tlab = 400 MeV.

    Figure 6

    Figure 6.  (color online) Same as Fig. 2 but for Tlab = 800 MeV.

    For energies above and below 200 MeV, because the fits cover a wide range of energies, it is expected that the scattering observables are not described as well as those at 200 MeV. As shown in Figs. 2 and 3, for energies far from 200 MeV (Tlab= 20 MeV), the unpolarized differential cross section (dσdΩ) and depolarization tensor for the polarized beam (D) of pn are not perfectly described. However, because the Coulomb corrections of all the pp scattering observables are very significant at energies below 200 MeV, the pp scattering observables are well described. For incident energies above 200 MeV, the results are similar to those of previous studies [11, 12]. As shown in Figs. 5 and 6, the polarization transfer from beam to recoil particle (DT) and polarization correlation for initially unpolarized particles (Ayy) of pn are not perfectly described. When the Coulomb corrections are included, the pp scattering observables are better described than those of pn. For energies above and below 200 MeV, although some fits are not perfectly described, they are not far from the experiment observables. In conclusion, the RLF model of the NN interaction has been constructed over the laboratory energy range of 20 to 800 MeV.

    To examine the validity of the above fit, we investigate the p+208Pb elastic scattering for a wide energy range (Tlab=30, 65, 200, 295, 500, and 800 MeV). The differential cross sections dΩ/dσ, analyzing powers Ay, and spinrotation functions Q have been investigated with RLF [6, 7]. However, the RLF model has not been used to describe dΩ/dσ, Ay, and Q for energies above 400 MeV. Moreover, because the parameters of this work are fitted for a wide energy range, the physical quantities may differ somewhat from those in previous studies. For our calculations, we have taken pseudovector coupling for the pion of RLF. Additionally, because the tensor potential is small, it has been neglected in the calculations.

    The PB corrections have been proved to be significant in RLF at low energy [6, 7]. However, the PB correction factors are still model-dependent. The initial PB parameters are obtained via a relativistic Dirac-Brueckner approach [6]. However, the initial PB parameters cannot well describe the dΩ/dσ, Ay, and Q in the energy-dependent RLF model [7]. Therefore, the PB factors of the energy-dependent RLF model are obtained by acquiring the best fits of dΩ/dσ, Ay, and Q. Because the fits of this work cover a wide range of energies, the PB factors of the previous energy-dependent RLF model may not be able to describe the p+208Pb elastic scattering. In order to obtain the PB factors and evaluate our fits, we first compare the optical potentials of this work with the Dirac global optical potentials (GOPs) [1] and the optical potentials of Ref. [7] at 65 MeV. As shown in upper panel of Fig. 7, without considering the PB corrections, the real (Re) values of the optical potentials of this work are similar to those of the previous energy-dependent RLF model, and the imaginary (Im) values of the optical potentials of this work are slightly higher than those of the previous energy-dependent RLF model. These results indicate that the fits of this work are reasonable. The Im values of the optical potentials of both this work and the previous energy-dependent RLF model are higher than those of the GOPs; however, the Re values of the optical potentials of both this work and the previous energy-dependent RLF model are lower than those of the GOPs. The higher Im and lower Re optical potentials imply that the signs of the PB factors of the Im and Re optical potentials in this work should be opposite. At low energy, the extrapolation of Dirac-Brueckner PB factors are too small to properly change the optical potentials. The Im values with PB factors of the phenomenological approach [7] can change the Im values of the optical potentials properly; however, the Re values with PB factors of the phenomenological approach are too low and positive. As shown in the lower panel of Fig. 7, the optical potentials are obtained with the fit of this work and various PB factors. With the extrapolation of Dirac-Brueckner PB factors, the optical potentials do not change much compared with those without the PB corrections. With the PB factors of the phenomenological approach [7], the Im values of the optical potentials are consistent with those of the GOPs, and the Re values of optical potentials are lower than those of GOPs. Therefore, with phenomenological approach, we re-fit the optical potentials by adjusting the PB factors. The PB factors of this work are presented in Table 2. The PB factors of previous studies are also listed, for comparison. The optical potentials with the PB factors of this work are close to the GOPs. When the optical potentials become reasonable, the energy-dependent RLF can well describe the p+208Pb scattering.

    Figure 7

    Figure 7.  (color online) The real (Re) and imaginary (Im) parts of the scalar S and vector V optical potentials for p+208Pb at 65 MeV. The black solid curves of the upper panel are obtained via the fit of this work (TW) without the Pauli blocking (PB) corrections. The violet dot curves of the upper panel were obtained in Li's work (LI) without PB corrections [7]. The blue dashed curves of the lower panel are obtained by considering the PB corrections of this work. The orange short-dash curves of the lower panel are obtained by considering the PB corrections of Li's work. The cyan dash-dot-dot curves of lower panel are obtained by considering the extrapolation of Dirac-Brueckner (DB) PB corrections [6]. The Dirac GOPs (red dash-dot curves) come from Ref. [1].

    Table 2

    Table 2.  Pauli blocking correction factors aL. The phenomenological Pauli blocking factors [7] and the Pauli blocking correction factors of the Dirac-Brueckner approach [6] are shown for comparison.
    The Pauli blocking correction factors of this work
    Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
    30−1.300.95−0.851.05
    65−0.470.85−0.410.88
    200−0.050.650.65
    2950.650.50
    500−0.05
    650−0.1
    800−0.1
    The phenomenological Pauli blocking factors
    Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
    650.0200.680.140.85
    1000.0070.600.110.73
    1350.0050.530.090.60
    1700.0080.420.070.50
    2000.0100.350.6050.42
    The Pauli blocking correction factors of Dirac-Brueckner approach
    Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
    1350.003770.112360.084030.24535
    200−0.00780.0980.06050.207
    300−0.02560.07590.02430.148
    400−0.04340.061−0.01190.089
    DownLoad: CSV
    Show Table

    As shown in Fig. 8, the dΩ/dσ values of Tlab=65 MeV without PB contributions are lower than the experimental data, and Ay and Q fail to describe the experimental data over the entire angular range. With the PB corrections of this work, the experimental dΩ/dσ, Ay, and Q of Tlab=65 MeV can be well described for the entire angular range. Our predictions of 65 MeV are as good as the previous results obtained by changing g2σ and g2ω[7]. The dΩ/dσ, Ay, and Q with the PB corrections of previous studies are also shown, for comparison. In this work, the PB factors of the phenomenological approach [7] and the extrapolation of Dirac-Brueckner fail to describe the dΩ/dσ, Ay, and Q. Generally, different fitting methods need different PB corrections to describe elastic scattering. It should be noted that the PB factors of energies below 200 MeV are far larger than the initial PB parameters generated from the relativistic Dirac-Brueckner approach [6]. In the following, we only consider the PB corrections of this work.

    Figure 8

    Figure 8.  (color online) Differential cross sections (dΩ/dσ), analyzing powers (Ay), and spinrotation functions (Q) as a function of the center-of-mass scattering angle for 65 MeV p+208Pb scattering. The black solid curves do not include the Pauli blocking (PB) corrections. The blue dashed curves include the PB corrections of this work. The orange short-dash curves of the lower panel are obtained by considering the PB corrections of Li's work (LI) [7]. The cyan dash-dot-dot curves are obtained by considering the extrapolation of PB corrections of Dirac-Brueckner (DB) [6]. The open circles indicate the experimental data taken from Ref. [35].

    For energies far below 200 MeV, as shown in Fig. 9, the dΩ/dσ and Ay of Tlab=30 MeV without PB contributions lack the property of fluctuating with respect to the angle in the p+208Pb elastic scattering and cannot describe the experimental data well. When the PB corrections are taken into account, the property of fluctuating with respect to the angle is generated, and the dΩ/dσ and Ay of 30 MeV can be consistent with the experimental data. For scattering angles smaller than 120 degrees, our result of 30 MeV shows some improvement over the previous results based on changing the real σN and ωN coupling constants (g2σ and g2ω) [7]. Because the spin observable Q of 30 MeV lacks experimental data, there may be uncertainties in the predictions of 30 MeV.

    Figure 9

    Figure 9.  (color online) Differential cross sections (dΩ/dσ), analyzing powers (Ay), and spinrotation functions (Q) as a function of the center-of-mass scattering angle for 30 MeV p+208Pb scattering. The solid curves do not consider the Pauli blocking (PB) corrections, and the dashed curves consider the PB corrections of this work. The open circles indicate the experimental data taken from Refs. [36, 37].

    When Tlab=200 MeV, as shown in Figs. 4 and 10, even though the np and pp scattering observables are the best fit, the experimental dΩ/dσ, Ay, and Q without PB contributions cannot fit the experimental data well. In particular, the spin observable Q is poorly fitted for the entire angular range. When the PB corrections are included, the experimental dΩ/dσ, Ay, and Q are fitted well for scattering angles smaller than 40 degrees. The result of Tlab=200 MeV is similar to that of Ref. [7].

    Figure 10

    Figure 10.  (color online) Same as Fig. 9 but for 200 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Refs. [3840].

    For energies above 200 MeV, the situation of PB corrections is complex. When the energy equals 295 MeV, as shown in Fig. 11, the dΩ/dσ and Ay without PB contributions are not well fitted. When a large imaginary factor of PB contributions is used for Tlab=295 MeV, the dΩ/dσ and Ay are only slightly corrected for scattering angles smaller than 30 degrees. As the energy increases, the factors of PB contributions become small; however, the effect of PB corrections cannot be neglected. For instance, as shown in Figs. 1214, when Tlab=500, 650, and 800 MeV, the dΩ/dσ, Ay, and Q without PB contributions are not described well. When a small factor of the real vector of PB contributions is included, the dΩ/dσ, Ay, and Q are fitted better than those in the case without PB contributions.

    Figure 11

    Figure 11.  (color online) Same as Fig. 9 but for 295 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Ref. [41].

    Figure 12

    Figure 12.  (color online) Same as Fig. 9 but for 497 and 500 MeV p+208Pb scattering. The experimental differential cross sections and analyzing powers of 500 MeV are taken from Ref. [42], and the experimental spinrotation functions of 497 MeV are taken from Ref. [43].

    Figure 13

    Figure 13.  (color online) Same as Fig. 9 but for 650 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Ref. [44].

    Figure 14

    Figure 14.  (color online) Same as Fig. 9 but for 800 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Refs. [45, 46].

    The energy-dependent RLF model has been used to fit empirical amplitudes for various energy domains: 50–200, 200–500, and 500–800 MeV. In this work, we try to relate these energy domains within a unified fit. In contrast to previous studies, we fit the scattering observables of pp and pn of WF16 instead of the non-relativistic amplitudes of pp and pn. Because an energy of 200 MeV is chosen as the energy boundary between low and high energy, we first fit the scattering observables of pp and pn of 200 MeV alone. When the scattering observables of pp and pn of 200 MeV are well fitted, the scattering observables of pp and pn of other energies are fitted by using the energy-dependent RLF model. Even though the scattering observables of other energies are not described as well as those of 200 MeV, their trends are consistent with the experimental value, and their values are not far from the experimental values. In conclusion, we constructed the RLF model of the NN interaction over the laboratory energy range of 20 to 800 MeV.

    To examine the validity of our fit, we use the energy-dependent RLF model in the range of 20 to 800 MeV to investigate p+208 Pb elastic scattering for Tlab=30, 65, 200, 295, 500, and 800 MeV. The differential cross sections dΩ/dσ, analyzing powers Ay, and spin rotation functions Q of p+208 Pb elastic scattering without PB corrections are not well fitted at all the energies considered. Although the scattering observables of pp and pn of 200 MeV best fit the experiment values, the RLF model of 200 MeV without PB corrections fails to describe the experimental dΩ/dσ, Ay, and Q. For a better description, the PB corrections must be taken into account. We vary the PB factors to obtain the best fits of dΩ/dσ, Ay, and Q. In our calculations, the PB factors are larger than those obtained via a relativistic Dirac-Brueckner approach at energies below 295 MeV but similar to those of phenomenological PB effects [7]. For Tlab500 MeV, the RLF model with a small factor of the real vector of PB contributions can better describe the dΩ/dσ, Ay, and Q than those without PB corrections. Our results clearly indicate the importance of PB corrections in the RLF model.

    We thank Ying Kuang and Profs. Zhi-Pan Li and Wei-Zhou Jiang for helpful discussions.

    [1] S. Hama, B. C. Clark, E. D. Cooper et al., Phys. Rev. C 41, 2737 (1990) doi: 10.1103/PhysRevC.41.2737
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    [25] F. Sammarruca, Eur. Phys. J. A 50, 22 (2014) doi: 10.1140/epja/i2014-14022-1
    [26] T. T. Wang, Y. G. Ma, C. J. Zhang et al., Phys. Rev. C 97, 034617 (2018) doi: 10.1103/PhysRevC.97.034617
    [27] R. Wang, Z. Zhang, L. W. Chen et al., Phys. Lett. B 807, 135532 (2020) doi: 10.1016/j.physletb.2020.135532
    [28] S. N. Wei, R. Y. Yang, J. Ye et al., Phys. Rev. C 103, 064604 (2021) doi: 10.1103/PhysRevC.103.064604
    [29] Z. H. Li, L. W. Chen, C. M. Ko et al., Phys. Rev. C 74, 044613 (2006) doi: 10.1103/PhysRevC.74.044613
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    [33] C. J.Horowitz, D. P. Murdock, and B. D. Serot, in Computational Nuclear Physics I, edited by K. Langanke, J. A. Maruhn, and S. E. Koonin (Springer-Verlag, Berlin, 1991), p. 129
    [34] R. L. Workman, W. J. Briscoe, and I. I. Strakovsky, Phys. Rev. C 94, 065203 (2016) doi: 10.1103/PhysRevC.94.065203
    [35] H. Sakaguchi, M. Nakamura, K. Hatanaka et al., Phys. Rev. C 26, 944 (1982) doi: 10.1103/PhysRevC.26.944
    [36] W. T. H. Van Oers et al., Phys. Rev. C 10, 307 (1974) doi: 10.1103/PhysRevC.10.307
    [37] B. W. Ridley and J. F. Turner, Nucl. Phys. 58, 497 (1964) doi: 10.1016/0029-5582(64)90561-9
    [38] D. A. Hutcheon, W. C. Olsen, H. S. Sherif et al., Nucl. Phys. A 483, 429 (1988) doi: 10.1016/0375-9474(88)90078-4
    [39] N. Ottenstein, S. J. Wallace, and J. A. Tjon, Phys. Rev. C 38, 2272 (1988) doi: 10.1103/PhysRevC.38.2272
    [40] L. Lee, T. E. Drake, S. S. M. Wong et al., Phys. Lett. B 205, 219 (1988) doi: 10.1016/0370-2693(88)91653-X
    [41] J. Zenihiro, H. Sakaguchi, T. Murakami et al., Phys. Rev. C 82, 044611 (2010) doi: 10.1103/PhysRevC.82.044611
    [42] G. W. Hoffmann, L. Ray, M. L. Barlett et al., Phys. Rev. Lett. 47, 1436 (1981) doi: 10.1103/PhysRevLett.47.1436
    [43] B. Aas, E. Bleszynski, M. Bleszynski et al., Nucl. Phys. A 460, 675 (1986) doi: 10.1016/0375-9474(86)90531-2
    [44] A. M. Mack, Norton M. Hintz, D. Cook et al., Phys. Rev. C 52, 291 (1995) doi: 10.1103/PhysRevC.52.291
    [45] G. W. Hoffman, G. S. Blanpied, W. R. Coker et al., Phys. Rev. Lett. 40, 1256 (1978) doi: 10.1103/PhysRevLett.40.1256
    [46] R. W. Fergerson, M. L. Barlett, G. W. Hoffmann et al., Phys. Rev. C 33, 239 (1986) doi: 10.1103/PhysRevC.33.239
  • [1] S. Hama, B. C. Clark, E. D. Cooper et al., Phys. Rev. C 41, 2737 (1990) doi: 10.1103/PhysRevC.41.2737
    [2] E. D. Cooper, S. Hama, B. C. Clark et al., Phys. Rev. C 47, 297 (1993) doi: 10.1103/PhysRevC.47.297
    [3] C. Xu, B. A. Li, and L. W. Chen, Eur. Phys. J. A 50, 21 (2014) doi: 10.1140/epja/i2014-14021-2
    [4] X. H. Li, B. J. Cai, L. W. Chen et al., Phys. Lett. B 721, 101 (2013) doi: 10.1016/j.physletb.2013.03.005
    [5] J. A. McNeil, L. Ray, and S. J. Wallace, Phys. Rev. C 27, 2123 (1983) doi: 10.1103/PhysRevC.27.2123
    [6] D. P.Murdock and C. J. Horowitz, Phys. Rev. C 35, 1442 (1987) doi: 10.1103/PhysRevC.35.1442
    [7] Z. P. Li, G. C. Hillhouse, and J. Meng, Phys. Rev. C 78, 014603 (2008) doi: 10.1103/PhysRevC.78.014603
    [8] L. Ray and G. W. Hoffmann, Phys. Rev. C 31, 538 (1985) doi: 10.1103/PhysRevC.31.538
    [9] C. J. Horowitz, Phys. Rev. C 31, 1340 (1985) doi: 10.1103/PhysRevC.31.1340
    [10] C. J. Horowitz and M. J. Iqbal, Phys. Rev. C 33, 2059 (1986) doi: 10.1103/PhysRevC.33.2059
    [11] O.V. Maxwell, Nucl. Phys. A 600, 509 (1996) doi: 10.1016/0375-9474(95)00494-7
    [12] O.V. Maxwell, Nucl. Phys. A 638, 747 (1998) doi: 10.1016/S0375-9474(98)00418-7
    [13] Z. P. Li, G. C. Hillhouse, and J. Meng, Phys. Rev. C 77, 014001 (2008) doi: 10.1103/PhysRevC.77.014001
    [14] E.D. Cooper and O.V. Maxwell, Nucl. Phys. A 493, 468 (1989) doi: 10.1016/0375-9474(89)90098-5
    [15] B. A. Li and L. W. Chen, Phys. Rev. C 72, 064611 (2005) doi: 10.1103/PhysRevC.72.064611
    [16] Z. Q. Feng, Phys. Rev. C 85, 014604 (2012) doi: 10.1103/PhysRevC.85.014604
    [17] D. Klakow, G. Welke, and W. Bauer, Phys. Rev. C 48, 1982 (1993) doi: 10.1103/PhysRevC.48.1982
    [18] G. Q. Li and R. Machleidt, Phys. Rev. C 48, 1702 (1993) doi: 10.1103/PhysRevC.48.1702
    [19] P. Danielewicz, Nucl. Phys. A 673, 375 (2000) doi: 10.1016/S0375-9474(00)00083-X
    [20] C. Fuchs, A. Faessler, and M. El-Shabshiry, Phys. Rev. C 64, 024003 (2001) doi: 10.1103/PhysRevC.64.024003
    [21] J. Y. Liu, W. J. Guo, S. J. Wang et al., Phys. Rev. Lett. 86, 975 (2001) doi: 10.1103/PhysRevLett.86.975
    [22] F. Sammarruca and P. Krastev, Phys. Rev. C 73, 014001 (2006) doi: 10.1103/PhysRevC.73.014001
    [23] H. F. Zhang, Z. H. Li, U. Lombardo et al., Phys. Rev. C 76, 054001 (2007) doi: 10.1103/PhysRevC.76.054001
    [24] W. Z. Jiang, B. A. Li, and L. W. Chen, Phys. Rev. C 76, 044604 (2007) doi: 10.1103/PhysRevC.76.044604
    [25] F. Sammarruca, Eur. Phys. J. A 50, 22 (2014) doi: 10.1140/epja/i2014-14022-1
    [26] T. T. Wang, Y. G. Ma, C. J. Zhang et al., Phys. Rev. C 97, 034617 (2018) doi: 10.1103/PhysRevC.97.034617
    [27] R. Wang, Z. Zhang, L. W. Chen et al., Phys. Lett. B 807, 135532 (2020) doi: 10.1016/j.physletb.2020.135532
    [28] S. N. Wei, R. Y. Yang, J. Ye et al., Phys. Rev. C 103, 064604 (2021) doi: 10.1103/PhysRevC.103.064604
    [29] Z. H. Li, L. W. Chen, C. M. Ko et al., Phys. Rev. C 74, 044613 (2006) doi: 10.1103/PhysRevC.74.044613
    [30] J. Brstricky, F. Lehar, and P. Winternitz, J. Phys. 39, 1 (1978) doi: 10.1051/jphys:019780039010100
    [31] M. Fierz, Z. Phys. 104, 553 (1937) doi: 10.1007/BF01330070
    [32] J. A. McNeil, J. R. Shepard, and S. J. Wallace, Phys. Rev. Lett. 50, 1439 (1983) doi: 10.1103/PhysRevLett.50.1439
    [33] C. J.Horowitz, D. P. Murdock, and B. D. Serot, in Computational Nuclear Physics I, edited by K. Langanke, J. A. Maruhn, and S. E. Koonin (Springer-Verlag, Berlin, 1991), p. 129
    [34] R. L. Workman, W. J. Briscoe, and I. I. Strakovsky, Phys. Rev. C 94, 065203 (2016) doi: 10.1103/PhysRevC.94.065203
    [35] H. Sakaguchi, M. Nakamura, K. Hatanaka et al., Phys. Rev. C 26, 944 (1982) doi: 10.1103/PhysRevC.26.944
    [36] W. T. H. Van Oers et al., Phys. Rev. C 10, 307 (1974) doi: 10.1103/PhysRevC.10.307
    [37] B. W. Ridley and J. F. Turner, Nucl. Phys. 58, 497 (1964) doi: 10.1016/0029-5582(64)90561-9
    [38] D. A. Hutcheon, W. C. Olsen, H. S. Sherif et al., Nucl. Phys. A 483, 429 (1988) doi: 10.1016/0375-9474(88)90078-4
    [39] N. Ottenstein, S. J. Wallace, and J. A. Tjon, Phys. Rev. C 38, 2272 (1988) doi: 10.1103/PhysRevC.38.2272
    [40] L. Lee, T. E. Drake, S. S. M. Wong et al., Phys. Lett. B 205, 219 (1988) doi: 10.1016/0370-2693(88)91653-X
    [41] J. Zenihiro, H. Sakaguchi, T. Murakami et al., Phys. Rev. C 82, 044611 (2010) doi: 10.1103/PhysRevC.82.044611
    [42] G. W. Hoffmann, L. Ray, M. L. Barlett et al., Phys. Rev. Lett. 47, 1436 (1981) doi: 10.1103/PhysRevLett.47.1436
    [43] B. Aas, E. Bleszynski, M. Bleszynski et al., Nucl. Phys. A 460, 675 (1986) doi: 10.1016/0375-9474(86)90531-2
    [44] A. M. Mack, Norton M. Hintz, D. Cook et al., Phys. Rev. C 52, 291 (1995) doi: 10.1103/PhysRevC.52.291
    [45] G. W. Hoffman, G. S. Blanpied, W. R. Coker et al., Phys. Rev. Lett. 40, 1256 (1978) doi: 10.1103/PhysRevLett.40.1256
    [46] R. W. Fergerson, M. L. Barlett, G. W. Hoffmann et al., Phys. Rev. C 33, 239 (1986) doi: 10.1103/PhysRevC.33.239
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Si-Na Wei and Zhao-Qing Feng. Elastic scattering based on energy-dependent relativistic Love-Franey model at energies between 20 and 800 MeV[J]. Chinese Physics C. doi: 10.1088/1674-1137/accb88
Si-Na Wei and Zhao-Qing Feng. Elastic scattering based on energy-dependent relativistic Love-Franey model at energies between 20 and 800 MeV[J]. Chinese Physics C.  doi: 10.1088/1674-1137/accb88 shu
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Elastic scattering based on energy-dependent relativistic Love-Franey model at energies between 20 and 800 MeV

    Corresponding author: Zhao-Qing Feng, fengzhq@scut.edu.cn
  • School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510640, China

Abstract: In order to investigate the elastic scattering, we fit scattering observables of the weighted fits (WF16) with the relativistic Love-Franey (RLF) model. The masses, cutoff parameters, and initial coupling strengths of RLF are assumed to be independent of energy. Because the energy boundary between low energy and high energy is around 200 MeV, the masses, cutoff parameters, and initial coupling strengths of RLF are obtained by fitting scattering observables of WF16 at an incident energy of 200 MeV. With the masses, cutoff parameters, and initial coupling strengths as the input, the energy-dependent RLF model is constructed over the laboratory energy range of 20 to 800 MeV within a unified fit. To examine the validity of this fit, we investigate p+208Pb elastic scattering for various energies. Although the scattering observables of pp and pn of 200 MeV best fit the values of WF16, the RLF model of 200 MeV without the Pauli blocking (PB) corrections fails to describe the experimental differential cross sections, analyzing powers, and spinrotation functions. When the PB corrections are taken into account for various energies, the RLF model can well describe the experimental data of p+208Pb elastic scattering.

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    I.   INTRODUCTION
    • The interaction between nucleons is very important in understanding the nuclear experiments and the properties of nuclei and neutron stars. However, the interaction between nucleons is still poorly understood, particularly at high densities. The main reason for this is that the interaction is not a directly observable quantity in experiments. The interaction between nucleons as residual interactions of the QCD cannot be solved via perturbation expansion. Therefore, the interaction between nucleons leads to the uncertainty of the symmetry energy. Considerable effort has been directed toward understanding the interaction between nucleons. The global Dirac phenomenological optical potentials, which describe the elastic proton scattering observables of 12C and 208Pb well for incident energies between 20 and 1040 MeV [1, 2], are powerful tools in understanding the interaction between nucleons. They have been widely used to extract the symmetry energy directly or indirectly [3, 4]. The relativistic impulse approximation (RIA) [57] can also be used to describe the elastic proton scattering observables. Theoretically, the symmetry energy can be extracted from the microscopic potentials of the RIA.

      The original RIA with scalar and vector potentials can reproduce the analyzing power and spin-rotation parameter in proton-nucleus scattering for incident energies above 500 MeV [5, 8]. Because the direct and exchange parts of the t-matrix are not separated explicitly in the original RIA, the optical potential of the original RIA diverges at energies below 500 MeV [8]. By separating the direct and exchange parts of the t-matrix, it is found that the pion with pseudoscalar leads to the divergent optical potential. This can be cured by introducing the pion with pseudovector coupling, in a process known as the relativistic Love-Franey (RLF) model [6, 9,10]. The RLF model has been used to describe the elastic proton scattering observables successfully for incident energies of 135, 200, 300, and 400 MeV [6]. The shortcoming of the original RLF model is that the scattering observables were fitted separately at each energy. The separate fits of the scattering observables cannot reflect the energy-dependent trend of the scattering observables. This motivates the proposal of the energy-dependent RLF model [11, 12]. The energy-dependent RLF model has been used to fit empirical amplitudes successfully at incident energies of 50–200 MeV [13], 200–500 MeV [11], and 500–800 MeV [12]. It should be noted that these fits are separated from each other. In this work, we try to construct the energy-dependent RLF model over the laboratory energy range of 20 to 800 MeV within a unified fit.

      The RLF model has been used to study both elastic proton-nucleus scattering [6, 7] and the (p, 2p) reaction [14]. The RLF model without Pauli blocking (PB) corrections cannot well describe the differential cross sections dΩ/dσ, analyzing powers Ay, and spinrotation functions Q of elastic proton-nucleus scattering. With the PB corrections, the RLF model can describe dΩ/dσ, Ay, and Q well [6, 7]. The RLF model is also helpful in understanding the in-medium effect. In-medium nucleon-nucleon (NN) cross sections, which are key inputs of various transport models, can be obtained from RLF. It has been recognized that the theoretical uncertainty in extracting the density dependence of the symmetry energy from data of heavy-ion reactions can be largely ascribed to the poor knowledge and premature treatment of the isospin-dependent in-medium NN cross sections [15, 16]. Although the NN cross section in free space is well-determined, the in-medium NN cross section is still model dependent [1728]. Under the original RIA framework, the in-medium NN cross sections have been studied [24, 28]. However, no such work has been done in the framework of the RLF model. At a low nucleon kinetic energy, it would be interesting to study the in-medium NN cross sections with the RLF model. Additionally, the energy and density dependent symmetry potential of nuclear matter has been investigated with the original RIA and RLF models [24, 28, 29]. The energy and density dependence symmetry potential of the energy-dependent RLF model may have some differences from the results of the original RIA and RLF models.

      The remainder of this paper is organized as follows. In Sec. II, we briefly introduce formulas and approaches used in this work. The formulas include the fits between RIA scattering amplitudes and the empirical amplitudes, as well as the elastic proton scattering observables of 208Pb. Results and discussions are presented in Sec. III. Finally, a summary is given in Sec. IV.

    II.   FORMALISM
    • In order to obtain the on-shell NN matrix elements of RLF scattering amplitudes, one needs to transform the NN matrix elements of RLF scattering amplitudes into those of the non-relativistic NN scattering amplitudes. The relation between the matrix elements of the RLF scattering amplitudes and the matrix elements of the non-relativistic NN scattering amplitudes in the center-of-mass frame can be written as [5, 6, 9, 10, 13]

      (2ikc)1χs1χs2ˆfcmχs1χs2=ˉu(k1,s1)ˉu(k2,s2)ˆFu(k1,s1)u(k2,s2),

      (1)

      where k1(Ec,kc) and k2(Ec,kc) are the incident 4-momenta in the center-of-mass frame. χs represent the usual Pauli spinors. kc represents the momentum in the center of mass frame. The Dirac spinors are normalized (ˉuu=1). ˆfcm and ˆF are the non-relativistic scattering matrix and scattering matrix of RLF respectively. ˆfcm contains five real and imaginary amplitudes [30]:

      (2ikc)1ˆfcm=12[a+b+(ab)σ1nσ2n+(c+d)σ1mσ2m+(cd)σ1lσ2l+e(σ1n+σ2n)],

      (2)

      where a, b, c, d, and e are the non-relativistic real and imaginary amplitudes. The scattering matrix of RLF ˆF also contains five real and imaginary amplitudes:

      ˆF=Ti=SFi(s,t,u)λ(1)iλ(2)i,

      (3)

      where i includes the scalar (S), vector (V), pseudoscalar (P), axial vector (A), and tensor (T). FS, FV, FP, FA, and FT are the relativistic amplitudes of RLF. The Dirac matrices of S, V, P, A, and T are I4×4, γμ, γ5, γ5γμ, and σμν respectively. With these Dirac matrices and Eq. (1), the relation between the non-relativistic amplitudes and the relativistic amplitudes is obtained as follows [5, 9, 10, 13]:

      (abcde)=ikcO5×5(FSFVFPFAFT),

      (4)

      The O5×5 matrix is written as [10, 13]

      O5×5=(cosθcosθcosθcosθ4sinθ111101111011110isinθisinθisinθisinθ4icosθ)×(αhα+pβ0pα+hβ2βhγγpγδ2(p+h)δα(p+h)α0(hp)α2αhγγpγγ2γϵϵ0ϵ2ϵ),

      (5)

      with

      α=cos2θ2,β=1+sin2θ2,γ=sin2θ2,δ=1+cos2θ2,ϵ=EcMsinθ2cosθ2,Ec=(k2c+M2)1/2,h=E2cM2,p=k2cM2,

      (6)

      where θ represents the scattering angle in the two-body center-of-mass, and M represents the free nucleon mass. Now, the relation between the matrix elements of the RLF scattering amplitudes and the matrix elements of the non-relativistic NN scattering amplitudes is clear. For comparison with the empirical amplitudes, the relativistic amplitudes Fi(s,t,u) need to be solved. The Fi(s,t,u) contains direct and exchange contributions:

      Fi(s,t,u)=iM22Eckc[FDi(s,t)+FEi(s,u)],

      (7)

      with

      FDi(s,t)=Nj=1δi,i(j)τ1τ2Tjfj(Ec,|q|),FEi(s,u)=(1)TNNNj=1Ci(j),iτ1τ2Tjfj(Ec,|Q|),

      (8)

      where N represents the number of mesons in this work, Ci(j),i is the Fierz matrix [6, 9, 31], Tj is the isospin of the jth meson, and TNN is the total isospin of the two-nucleon state. When TNN=1, τ1τ2Tj=1 for both Tj=0 and Tj=1. When TNN=0, τ1τ2Tj=1 for Tj=0, and τ1τ2Tj=3 for Tj=1. For instance, when TNN=0, the scalar amplitudes are written as

      FDS(s,t)=fS(Tj=0)(Ec,|q|)3fS(Tj=1)(Ec,|q|),FES(s,u)=j=S,TCj,S[fj(Tj=0)(Ec,|Q|)3fj(Tj=1)(Ec,|Q|)],

      (9)

      where fj(Ec,|q|) and fj(Ec,|Q|) contain the real and imaginary parts. For instance:

      fj(Ec,|q|)=fjR(Ec,|q|)ifjI(Ec,|q|),fjR(Ec,|q|)=g2j(Ec)q2+m2j(1+q2/Λ2j)2,fjI(Ec,|q|)=ˉg2j(Ec)q2+ˉm2j(1+q2/ˉΛ2j)2,

      (10)

      where g2j(Ec), mj, and Λj are the coupling constant, mass, and momentum cutoff of the jth real meson, respectively. ˉg2j(Ec), ˉmj, andˉΛj are the coupling constant, mass, and cutoff parameter of the jth imaginary meson, respectively. The direct momentum transfer |q| and the exchange momentum transfer |Q| are written as [9]

      |q|=|kckc|=2kcsinθ2,|Q|=|kc+kc|=2kcsin(πθ2).

      (11)

      For a given projectile's kinetic energy Tlab, the momentum and energy of the center of mass frame are kc=2MTlab2 and Ec=k2c+M2, respectively. With these fitting formulas, the parameterization of the RLF model can be obtained. Then, the Dirac optical potential is generated with the fitting parameters of the RLF model.

      The Dirac optical potential of the RLF model is the same as that of the original RIA model [6, 32, 33]:

      Uopt(q,E)=4πiplabMˉΨ|An=1ˆF(q,E;n)eiqr(n)|Ψ,

      (12)

      where plab represents the laboratory momentum of incident nucleons. q and E represent the momentum transfer and collision energy, respectively. |Ψ represents the A-particle ground state of the target nucleus.

      In proton-nucleus scattering, the scattering process can be approximately treated as incident protons scattered by each of the nucleons in the target nucleus. Assuming the incident proton wave function as U0(r) and the relativistic Hartree wave function of the nucleus ground state as U0(r), the optical potential can act on the incident proton wave function and project into the coordinate space. In this work, we only list the formulas after Fourier transforms [33]:

      r|Uopt(q,E)|U0(r)=4iπplabML[d3rρL(r)tLD(|rr|;E)]λLU0(r)4iπplabML[d3rρL(r,r)tLE(|rr|;E)]λLU0(r),

      (13)

      where

      tLD(|r|;E)]d3q(2π)3tLD(q,E)eiqr,

      (14)

      tLE(|r|;E)]d3q(2π)3tLE(Q,E)eiQr,

      (15)

      with tLD(q,E)(iM2/2Eckc)FLD(q) and tLE(Q,E)(iM2/2Eckc)FLE(Q), respectively. The first term of the above equation represents the direct optical potential in the coordinate space:

      ULD(r,E)=4iπplabMd3rρL(r)tLD(|rr|;E),

      (16)

      and the second term of the above equation represents the exchange optical potential. With the local-density approximation, the exchange optical potential is written as

      ULE(r,E)=4iπplabMd3rρL(r,r)×tLE(|rr|;E)j0(plab|rr|),

      (17)

      j0 is a spherical Bessel function. The nuclear densities of Eqs. (13), (16), and (17) are defined as

      ρL(r,r)occαˉUα(r)λLUα(r),ρL(r)ρL(r,r).

      (18)

      The occupied states indicate that one sums over target protons for the density with pp amplitudes and over target neutrons for the density with pn amplitudes. The densities of target protons and neutrons can be obtained from relativistic mean field theory (RMF) [7, 33]. When the densities of target protons and neutrons predicted by various models are substituted into Eq. (18), the results are almost the same in p+208 Pb elastic scattering. In the following sections, we take the densities predicted by the code of Ref. [33] as the input.

      With above formulas, the Dirac optical potential can be obtained. For the spin-saturated nucleus, the Dirac optical potential only contains scalar, vector, and tensor terms. Because the tensor contribution is small, the tensor contribution has been neglected in the calculation. The Dirac optical potential is simplified as

      Uopt(r;E)=US(r;E)+γ0UV(r;E),UL(r;E)=ULD(r;E)+ULE(r;E).

      (19)

      For the laboratory energy near 200 MeV, the contribution of Pauli blocking (PB) plays an important role in the calculations. The modification of PB with a local-density approximation is [6, 13, 33]

      ULPB(r;E)=[1aL(E)(ρB(r)ρ0)2/3]UL(r;E).

      (20)

      Here, ρB(r) represents the local baryon density of the target nuclus, and ρ0=0.1934 fm3.

      With the Dirac optical potential, the Dirac equation of the projectile is written as

      hU0(r)=EU0(r)={iα+UV(r;E)+β[M+US(r;E)]}U0(r).

      (21)

      By solving the Dirac equation, i.e., Eq. (21), the scattering observables (the differential cross section dσ/dΩ, analyzing power Ay, and spin rotation function Q) are determined. The detailed solution processes are presented in Ref. [33]. In this work, we concentrate on dσ/dΩ, Ay, and Q of p+208Pb elastic scattering.

    III.   RESULTS AND DISCUSSIONS

      A.   Fitting procedures and results

    • The RLF pp and pn scattering amplitudes are written as [9]

      Fi(pp)=Fi(TNN=1),Fi(pn)=12[Fi(TNN=1)+Fi(TNN=0)],

      (22)

      where the relativistic amplitudes FS, FV, FP, and FA of pp and pn and the non-relativistic amplitudes a, b, c, d, and e of pp and pn are generated from Eq. (4). However, the non-relativistic amplitudes are no longer available on the on-line Scattering Analysis Interactive Dial-in (SAID) facility. The scattering observables are still available on SAID. There are several models available for computing the scattering observables on SAID. We choose the scattering observables obtained from the weighted fits (WF16) on SAID [34]. In contrast to previous studies [1113], we fit 10 scattering observables of the weighted fits (WF16) rather than the real and imaginary amplitudes of pp and pn. These 10 scattering observables are written as [30]

      dσdΩσ=12(|a|2+|b|2+|c|2+|d|2+|e|2),D=12(|a|2+|b|2|c|2|d|2+|e|2)/σ,DT=12(|a|2|b|2+|c|2|d|2+|e|2)/σ,Ayy=12(|a|2|b|2|c|2+|d|2+|e|2)/σ,

      P=(ae)/σ,CKP=(de)/σ,CKK=(ad+bc)/σ,CPP=(adbc)/σ,R=[(ab)cos(α+θ2)+(cd)cos(αθ2)(be)sin(α+θ2)],A=[(ab)sin(α+θ2)+(cd)sin(αθ2)(be)cos(α+θ2)],

      (23)

      These 10 scattering observables are the unpolarized differential cross section (dσdΩ), depolarization tensor for polarized beam (D), polarization transfer from beam to recoil particle (DT), polarization correlation for initially unpolarized particles Ayy, CKP, CKK, CPP, polarization of scattered particle (P), and triple scattering parameters A and R. α is a function of the angle:

      α=θ2θlab,

      (24)

      where θlab represents the laboratory scattering angle. Theoretically, the fitting quality with these 10 scattering observables of pp and pn should be as good as that in the previous case with the non-relativistic amplitudes of pp and pn.

      The real isovector pseudoscalar (π), isoscalar vector (ω), and isovector vector (ρ) masses are fixed to their physical values mπ=138 MeV, mω=782 MeV, and mρ= 770 MeV, respectively. The remaining meson masses are set in accordance with Ref. [13]. Because the energy boundary between low energy and high energy is around 200 MeV, we first fit the scattering observables to obtain the initial coupling strengths of RLF. The initial coupling constant g20 (ˉg20) and the cutoff parameter Λ (ˉΛ) are obtained by fitting the scattering observables for Tlab= 200 MeV alone. In order to obtain the best fit to the scattering observables for Tlab= 200 MeV, the Levenberg-Marquadt method has been employed to minimize the value of χ2. χ2 is defined as follows:

      χ2=data(χempiricalχfit)2χ2empirical,

      (25)

      where χ2empirical represents an angle-averaged value and is defined as

      χ2empirical=175θ=5χ2empirical(θ)Nempirical,

      (26)

      where Nempirical=35 represents the number of angles per energy. For a single incident energy, the number of the data points is 700: 35 angles × 10 scattering observables × 2 pp and pn. As a result, the real (imaginary) initial coupling constant g20 (ˉg20), the real (imaginary) meson mass m (ˉm), and the real (imaginary) cutoff parameter Λ (ˉΛ) are presented in Table 1. It should be noted that Λ>m has been employed for a given meson.

      Real parameters
      MesonIsospinCoupling typemg20a1a2a3a4Λ
      σ0Scalar (S)600−8.3486.527×1012.773−2.741×102−7.639962.7
      ω0Vector (V)7821.019×101−6.246×101−1.400×1018.492×1012.677×1011155
      t00Tensor (T)5502.794×1013.3031.172×101−6.676×101−3.008×1011945
      a00Axial vector (A)5004.921×1016.0661.953×101−1.089−5.133×1011597
      η0Pseudoscalar (P)9501.180×101−1.694×102−1.350×1025.4175.550×102957.3
      δ1Scalar (S)5006.095×1035.635×1011.287−6.040×102−4.1012945
      ρ1Vector (V)770−1.667×1019.420×1051.644×101−2.905×1022.915×1013001
      t11Tensor (T)600−2.500×101−1.893−7.8484.448×1011.989×1011280
      a11Axial vector (A)650−1.356−4.678−1.529×1018.102×1014.146×101745.1
      π1Pseudoscalar (P)1381.095×101−3.834×109−4.8894.005×1011.082×101677.4
      Imaginary parameters
      MesonIsospinCoupling typeˉmˉg20a1a2a3a4ˉΛ
      σ0Scalar (S)600−2.849−4.637−1.475×1016.301×1014.377×101800.0
      ω0Vector (V)7004.4054.8252.077×101−9.616×101−5.672 ×101716.0
      t00Tensor (T)750−9.457×101−1.474−7.5354.140×1011.914×1011340
      a00Axial vector (A)750−2.198−3.583−1.374×1016.840×1013.762×1011437
      η0Pseudoscalar (P)10001.420×101−1.619×102−2.011×1029.7796.591×1022758
      δ1Scalar (S)6503.0941.5932.9451.614×102−1.328×101751.9
      ρ1Vector (V)600−2.421−9.543×101−3.6449.074×1021.292×101766.6
      t11Tensor (T)7506.140×1011.5135.775−3.011×101−1.557×1011662
      a11Axial vector (A)10002.2655.0711.997×101−1.011−5.450×1011233
      π1Pseudoscalar (P)500−5.8912.029×1012.363×101−9.593×101−7.574×1011342

      Table 1.  Parameter sets for the energy-dependent RLF over the laboratory energy range of 20 to 800 MeV. The masses m (ˉm) and cutoff parameters Λ (ˉΛ) are in MeV, whereas the remaining parameters are dimensionless.

      The energy-dependent RLF model has been used to fit the empirical NN data for separate incident energies (50–200 MeV [13], 200–500 MeV [11], and 500–800 MeV [12]). To relate these energy domains, we first fit the scattering observables of various energies (Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV) separately with the same cutoff parameters and meson masses as 200 MeV. As shown in Fig. 1, we find that the coupling strengths, such as g2π, g2η, g2a1, and g2t1, approximately follow a logarithmic or parabolic curve opening to the right. Combining the linear and square terms [11, 12], the energy-dependent coupling strengths can be written as

      Figure 1.  (color online) The coupling constants (g2π, g2η, g2a1, and g2t1) as a function of the laboratory energies Tlab. The coupling constants are obtained from separate fits with Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV.

      g2(Ec)=g20+a1ln(TlabT0)+a2(TlabT01)+a3[(TlabT0)21]+a4(TlabT01),

      (27)

      with T0=200 MeV. This equation is also valid for the imaginary coupling strengths. When the incident energy is 200 MeV, the energy-dependent coupling strengths should equal the initial coupling strengths. The meson masses and cutoff parameters of RLF are assumed to be independent of the energy in this work.

      For various incident energies, the initial coupling constant, the meson mass, and the cutoff parameter remain the same. However, the coupling constants g2j(Ec) and ˉg2j(Ec) change with respect to the incident energy. The parameters a1, a2a3, and a4 of Eq. (27) are obtained by fitting the scattering observables for various incident energies (Tlab= 20, 50, 75, 100, 150, 200, 300, 400, 500, 600, and 800 MeV) uniformly. The number of the data points is 7700: 11 energies × 35 angles × 10 scattering observables × 2 pp and pn. The final results for a1, a2a3, and a4 are presented in Table 1.

      To illustrate the fitting quality, we show the unpolarized differential cross section (dσdΩ), depolarization tensor for the polarized beam (D), polarization transfer from beam to recoil particle (DT), polarization correlation for initially unpolarized particles (Ayy), and polarization of scattered particles (P) of pp and pp at Tlab= 20, 100, 200, 400, and 800 MeV in Figs. 26. At Tlab= 200 MeV, as shown in Fig. 4, because the scattering observables of Tlab= 200 MeV are fitted alone, the scattering observables are described well for both np and pp. As the Coulomb corrections dominate the empirical pp scattering observables of forward and backward angles, the Coulomb corrections are included in the pp scattering observables in this work. The Coulomb corrections are obtained by subtracting the TNN=1 amplitudes of SAID from the pp amplitudes of SAID.

      Figure 2.  (color online) np and pp observables as a function of the center-of-mass scattering angle θ for Tlab = 20 MeV. The dashed curves are generated from Table 1, and the solid curves are the observables of WF16 [34].

      Figure 3.  (color online) Same as Fig. 2 but for Tlab = 100 MeV.

      Figure 4.  (color online) Same as Fig. 2 but for Tlab = 200 MeV.

      Figure 5.  (color online) Same as Fig. 2 but for Tlab = 400 MeV.

      Figure 6.  (color online) Same as Fig. 2 but for Tlab = 800 MeV.

      For energies above and below 200 MeV, because the fits cover a wide range of energies, it is expected that the scattering observables are not described as well as those at 200 MeV. As shown in Figs. 2 and 3, for energies far from 200 MeV (Tlab= 20 MeV), the unpolarized differential cross section (dσdΩ) and depolarization tensor for the polarized beam (D) of pn are not perfectly described. However, because the Coulomb corrections of all the pp scattering observables are very significant at energies below 200 MeV, the pp scattering observables are well described. For incident energies above 200 MeV, the results are similar to those of previous studies [11, 12]. As shown in Figs. 5 and 6, the polarization transfer from beam to recoil particle (DT) and polarization correlation for initially unpolarized particles (Ayy) of pn are not perfectly described. When the Coulomb corrections are included, the pp scattering observables are better described than those of pn. For energies above and below 200 MeV, although some fits are not perfectly described, they are not far from the experiment observables. In conclusion, the RLF model of the NN interaction has been constructed over the laboratory energy range of 20 to 800 MeV.

    • B.   Proton-nucleus scattering results and discussions

    • To examine the validity of the above fit, we investigate the p+208Pb elastic scattering for a wide energy range (Tlab=30, 65, 200, 295, 500, and 800 MeV). The differential cross sections dΩ/dσ, analyzing powers Ay, and spinrotation functions Q have been investigated with RLF [6, 7]. However, the RLF model has not been used to describe dΩ/dσ, Ay, and Q for energies above 400 MeV. Moreover, because the parameters of this work are fitted for a wide energy range, the physical quantities may differ somewhat from those in previous studies. For our calculations, we have taken pseudovector coupling for the pion of RLF. Additionally, because the tensor potential is small, it has been neglected in the calculations.

      The PB corrections have been proved to be significant in RLF at low energy [6, 7]. However, the PB correction factors are still model-dependent. The initial PB parameters are obtained via a relativistic Dirac-Brueckner approach [6]. However, the initial PB parameters cannot well describe the dΩ/dσ, Ay, and Q in the energy-dependent RLF model [7]. Therefore, the PB factors of the energy-dependent RLF model are obtained by acquiring the best fits of dΩ/dσ, Ay, and Q. Because the fits of this work cover a wide range of energies, the PB factors of the previous energy-dependent RLF model may not be able to describe the p+208Pb elastic scattering. In order to obtain the PB factors and evaluate our fits, we first compare the optical potentials of this work with the Dirac global optical potentials (GOPs) [1] and the optical potentials of Ref. [7] at 65 MeV. As shown in upper panel of Fig. 7, without considering the PB corrections, the real (Re) values of the optical potentials of this work are similar to those of the previous energy-dependent RLF model, and the imaginary (Im) values of the optical potentials of this work are slightly higher than those of the previous energy-dependent RLF model. These results indicate that the fits of this work are reasonable. The Im values of the optical potentials of both this work and the previous energy-dependent RLF model are higher than those of the GOPs; however, the Re values of the optical potentials of both this work and the previous energy-dependent RLF model are lower than those of the GOPs. The higher Im and lower Re optical potentials imply that the signs of the PB factors of the Im and Re optical potentials in this work should be opposite. At low energy, the extrapolation of Dirac-Brueckner PB factors are too small to properly change the optical potentials. The Im values with PB factors of the phenomenological approach [7] can change the Im values of the optical potentials properly; however, the Re values with PB factors of the phenomenological approach are too low and positive. As shown in the lower panel of Fig. 7, the optical potentials are obtained with the fit of this work and various PB factors. With the extrapolation of Dirac-Brueckner PB factors, the optical potentials do not change much compared with those without the PB corrections. With the PB factors of the phenomenological approach [7], the Im values of the optical potentials are consistent with those of the GOPs, and the Re values of optical potentials are lower than those of GOPs. Therefore, with phenomenological approach, we re-fit the optical potentials by adjusting the PB factors. The PB factors of this work are presented in Table 2. The PB factors of previous studies are also listed, for comparison. The optical potentials with the PB factors of this work are close to the GOPs. When the optical potentials become reasonable, the energy-dependent RLF can well describe the p+208Pb scattering.

      Figure 7.  (color online) The real (Re) and imaginary (Im) parts of the scalar S and vector V optical potentials for p+208Pb at 65 MeV. The black solid curves of the upper panel are obtained via the fit of this work (TW) without the Pauli blocking (PB) corrections. The violet dot curves of the upper panel were obtained in Li's work (LI) without PB corrections [7]. The blue dashed curves of the lower panel are obtained by considering the PB corrections of this work. The orange short-dash curves of the lower panel are obtained by considering the PB corrections of Li's work. The cyan dash-dot-dot curves of lower panel are obtained by considering the extrapolation of Dirac-Brueckner (DB) PB corrections [6]. The Dirac GOPs (red dash-dot curves) come from Ref. [1].

      The Pauli blocking correction factors of this work
      Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
      30−1.300.95−0.851.05
      65−0.470.85−0.410.88
      200−0.050.650.65
      2950.650.50
      500−0.05
      650−0.1
      800−0.1
      The phenomenological Pauli blocking factors
      Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
      650.0200.680.140.85
      1000.0070.600.110.73
      1350.0050.530.090.60
      1700.0080.420.070.50
      2000.0100.350.6050.42
      The Pauli blocking correction factors of Dirac-Brueckner approach
      Energy/MeVReal scalarImaginary scalarReal vectorImaginary vector
      1350.003770.112360.084030.24535
      200−0.00780.0980.06050.207
      300−0.02560.07590.02430.148
      400−0.04340.061−0.01190.089

      Table 2.  Pauli blocking correction factors aL. The phenomenological Pauli blocking factors [7] and the Pauli blocking correction factors of the Dirac-Brueckner approach [6] are shown for comparison.

      As shown in Fig. 8, the dΩ/dσ values of Tlab=65 MeV without PB contributions are lower than the experimental data, and Ay and Q fail to describe the experimental data over the entire angular range. With the PB corrections of this work, the experimental dΩ/dσ, Ay, and Q of Tlab=65 MeV can be well described for the entire angular range. Our predictions of 65 MeV are as good as the previous results obtained by changing g2σ and g2ω[7]. The dΩ/dσ, Ay, and Q with the PB corrections of previous studies are also shown, for comparison. In this work, the PB factors of the phenomenological approach [7] and the extrapolation of Dirac-Brueckner fail to describe the dΩ/dσ, Ay, and Q. Generally, different fitting methods need different PB corrections to describe elastic scattering. It should be noted that the PB factors of energies below 200 MeV are far larger than the initial PB parameters generated from the relativistic Dirac-Brueckner approach [6]. In the following, we only consider the PB corrections of this work.

      Figure 8.  (color online) Differential cross sections (dΩ/dσ), analyzing powers (Ay), and spinrotation functions (Q) as a function of the center-of-mass scattering angle for 65 MeV p+208Pb scattering. The black solid curves do not include the Pauli blocking (PB) corrections. The blue dashed curves include the PB corrections of this work. The orange short-dash curves of the lower panel are obtained by considering the PB corrections of Li's work (LI) [7]. The cyan dash-dot-dot curves are obtained by considering the extrapolation of PB corrections of Dirac-Brueckner (DB) [6]. The open circles indicate the experimental data taken from Ref. [35].

      For energies far below 200 MeV, as shown in Fig. 9, the dΩ/dσ and Ay of Tlab=30 MeV without PB contributions lack the property of fluctuating with respect to the angle in the p+208Pb elastic scattering and cannot describe the experimental data well. When the PB corrections are taken into account, the property of fluctuating with respect to the angle is generated, and the dΩ/dσ and Ay of 30 MeV can be consistent with the experimental data. For scattering angles smaller than 120 degrees, our result of 30 MeV shows some improvement over the previous results based on changing the real σN and ωN coupling constants (g2σ and g2ω) [7]. Because the spin observable Q of 30 MeV lacks experimental data, there may be uncertainties in the predictions of 30 MeV.

      Figure 9.  (color online) Differential cross sections (dΩ/dσ), analyzing powers (Ay), and spinrotation functions (Q) as a function of the center-of-mass scattering angle for 30 MeV p+208Pb scattering. The solid curves do not consider the Pauli blocking (PB) corrections, and the dashed curves consider the PB corrections of this work. The open circles indicate the experimental data taken from Refs. [36, 37].

      When Tlab=200 MeV, as shown in Figs. 4 and 10, even though the np and pp scattering observables are the best fit, the experimental dΩ/dσ, Ay, and Q without PB contributions cannot fit the experimental data well. In particular, the spin observable Q is poorly fitted for the entire angular range. When the PB corrections are included, the experimental dΩ/dσ, Ay, and Q are fitted well for scattering angles smaller than 40 degrees. The result of Tlab=200 MeV is similar to that of Ref. [7].

      Figure 10.  (color online) Same as Fig. 9 but for 200 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Refs. [3840].

      For energies above 200 MeV, the situation of PB corrections is complex. When the energy equals 295 MeV, as shown in Fig. 11, the dΩ/dσ and Ay without PB contributions are not well fitted. When a large imaginary factor of PB contributions is used for Tlab=295 MeV, the dΩ/dσ and Ay are only slightly corrected for scattering angles smaller than 30 degrees. As the energy increases, the factors of PB contributions become small; however, the effect of PB corrections cannot be neglected. For instance, as shown in Figs. 1214, when Tlab=500, 650, and 800 MeV, the dΩ/dσ, Ay, and Q without PB contributions are not described well. When a small factor of the real vector of PB contributions is included, the dΩ/dσ, Ay, and Q are fitted better than those in the case without PB contributions.

      Figure 11.  (color online) Same as Fig. 9 but for 295 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Ref. [41].

      Figure 12.  (color online) Same as Fig. 9 but for 497 and 500 MeV p+208Pb scattering. The experimental differential cross sections and analyzing powers of 500 MeV are taken from Ref. [42], and the experimental spinrotation functions of 497 MeV are taken from Ref. [43].

      Figure 13.  (color online) Same as Fig. 9 but for 650 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Ref. [44].

      Figure 14.  (color online) Same as Fig. 9 but for 800 MeV p+208Pb scattering. The open circles indicate the experimental data taken from Refs. [45, 46].

    IV.   SUMMARY
    • The energy-dependent RLF model has been used to fit empirical amplitudes for various energy domains: 50–200, 200–500, and 500–800 MeV. In this work, we try to relate these energy domains within a unified fit. In contrast to previous studies, we fit the scattering observables of pp and pn of WF16 instead of the non-relativistic amplitudes of pp and pn. Because an energy of 200 MeV is chosen as the energy boundary between low and high energy, we first fit the scattering observables of pp and pn of 200 MeV alone. When the scattering observables of pp and pn of 200 MeV are well fitted, the scattering observables of pp and pn of other energies are fitted by using the energy-dependent RLF model. Even though the scattering observables of other energies are not described as well as those of 200 MeV, their trends are consistent with the experimental value, and their values are not far from the experimental values. In conclusion, we constructed the RLF model of the NN interaction over the laboratory energy range of 20 to 800 MeV.

      To examine the validity of our fit, we use the energy-dependent RLF model in the range of 20 to 800 MeV to investigate p+208 Pb elastic scattering for Tlab=30, 65, 200, 295, 500, and 800 MeV. The differential cross sections dΩ/dσ, analyzing powers Ay, and spin rotation functions Q of p+208 Pb elastic scattering without PB corrections are not well fitted at all the energies considered. Although the scattering observables of pp and pn of 200 MeV best fit the experiment values, the RLF model of 200 MeV without PB corrections fails to describe the experimental dΩ/dσ, Ay, and Q. For a better description, the PB corrections must be taken into account. We vary the PB factors to obtain the best fits of dΩ/dσ, Ay, and Q. In our calculations, the PB factors are larger than those obtained via a relativistic Dirac-Brueckner approach at energies below 295 MeV but similar to those of phenomenological PB effects [7]. For Tlab500 MeV, the RLF model with a small factor of the real vector of PB contributions can better describe the dΩ/dσ, Ay, and Q than those without PB corrections. Our results clearly indicate the importance of PB corrections in the RLF model.

    ACKNOWLEDGMENT
    • We thank Ying Kuang and Profs. Zhi-Pan Li and Wei-Zhou Jiang for helpful discussions.

Reference (46)

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