K Nucleus Elastic Scattering and Momentum-Dependent Optical Potentials

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1 Commun. Theor. Phys. (Beijing, China) 41 (2004) pp c International Academic Publishers Vol. 41, No. 4, April 15, 2004 K Nucleus Elastic Scattering and Momentum-Dependent Optical Potentials ZHONG Xian-Hui, LI Lei, CAI Chong-Hai, and NING Ping-Zhi Institute of Physics, Nankai University, Tianjin , China (Received July 17, 2003) Abstract The K nucleus differential elastic scattering cross section for 12 C and 40 Ca at p k = 800 MeV/c is calculated with three momentum-dependent optical potential models, which are density-dependent, relativistic mean field, and hybrid model, respectively. It is found that the forms of momentum-dependent optical potential models proposed by us are reasonable and gain success in the calculations and the momentum-dependent hybrid model is the best model for the K nucleus elastic scattering. PACS numbers: Dr, Xh, Ev Key words: differential elastic scattering cross section, momentum-dependent optical potential, relativistic mean field 1 Introduction The kaon-nuclear physics is the focus of nuclear physicist, K -nucleus elastic and inelastic scattering and kaonic atoms are important objects of experimental and theoretical studies and many investigations have been done in the field. The status on the field kaon theoretically and experimentally has been summarized by Dover and Walker in Ref. [1]. On K -nucleus scattering, K ± elastic and inelastic scattering data at 800 MeV/c on targets of 12 C and 40 Ca have been accumulated at BNL Moby Dick spectrometer facility. [2] A similar but more extensive study of K elastic scattering is due to Rosenthal and Tabakin. [3] In Ref. [4] four forms of antikaon-nucleus optical potentials are used to calculate the K -nucleus elastic scattering at low and intermediate energies, whose theoretical result cannot fit the experimental data well. How to introduce a reasonable and simple form of optical potential to describe the K -nucleus elastic scattering is an important question for discussion and now, some information on the real part of the antikaon potential from all available sources for different antikaon momenta is collected in Ref. [5]. On kaonic atoms, the problem of kaonic atoms has regained interest recently. The phenomenological-densitydependent (DD) potential fitted to the kaon atomic data [6 8] yields a value for its real part in the nuclear [9] interior Re V opt ( 200 ± 20) MeV. Friedman et al. used K atomic data to test several models of the K nucleus interaction and concluded that the best fit antikaon optical potential is found to be strongly attractive with a depth of (180 ± 20) MeV at the nuclear interior, while the studies [10,11 13] of antikaon production from heavy ion collisions suggested an attractive potential of 80 MeV 120 MeV. Obviously there exists inconsistency in the results of the optical potential. Why do the calculations of different models have so many differences? Sibirtsev et al. proposed in Refs. [14] and [15] to attribute this discrepancy to the momentum dependence of the antikaon potential. They find that the K selfenergy at normal nuclear matter density turns out to be 200 MeV at zero momentum in line with kaon atomic data, however, it decreases rapidly in magnitude for higher momenta. Does there exist momentum dependence in antikaon potentials and can we use the optical potentials in line with kaon atomic data to extent to K -nucleus scattering? As the purpose of the present work, starting with the three optical models (DD, relativistic mean field (RMF), hybrid) discussed in Ref. [9], we attempt to investigate their momentum dependence in K nucleus elastic scattering and propose an appropriate form of momentum-dependent optical potential and then calculate the K C, K Ca elastic differential cross sections at the incoming antikaon momentum p k = 800 Mev/c with the momentum-dependent optical potentials and compare the different models (DD, RMF, hybrid) with each other to obtain the best momentum-dependent optical potential model for K nucleus elastic scattering. In Sec. 2 we briefly review the DD, RMF, and hybrid models, and we do some approximations for RMF model to simplify the calculations. The optical potentials of different models are discussed, the momentum-dependent optical potentials for DD, RMF, and hybrid models are introduced and the K nucleus differential elastic scattering cross sections for C and Ca at p k = 800 MeV/c are calculated in Sec. 3. The results of this work are summarized in Sec K -Nucleus Optical Potentials Now let us briefly review the interaction of K with The project supported by National Natural Science Foundation of China under Grant No , and the Research Fund for the Doctoral Program of Higher Education of China under Grant No

2 574 ZHONG Xian-Hui, LI Lei, CAI Chong-Hai, and NING Ping-Zhi Vol. 41 the nucleus and the optical potentials for DD, RMF, and hybrid models. The interaction of K with the nucleus is described by Klein Gordon (KG) equation of the form presented in Ref. [16] ( 2 + k 2 2ε(k)(V opt + V c ) + V 2 c )φ = 0, (1) where k and ε(k) are the K -nucleus wave number and reduced energy in the c.m. system respectively, and V c is the Coulomb interaction of the K with the nucleus. 2.1 DD Optical Potential The phenomenological DD potential of Friedman et al. [6,8] is given at threshold by ( 2µV opt (r) = 4π 1 + µ ) b(ρ)ρ(r), (2) m ( ρ(r) ) α b(ρ) = b 0 + B 0, (3) ρ(0) where µ is the K nucleus reduced mass, b 0 and B 0 are complex parameters determined from fits to the data, in this work b 0 = ( i0.62) fm, B 0 = (1.78 i0.22) fm, α = 0.31, m is the mass of the nucleon and ρ(r) = ρ n (r) + ρ p (r) is the nuclear density distribution normalized to the number of nucleons A and ρ 0 = 0.18 fm 3 is a typical central nuclear density. 2.2 RMF Optical Potential Antikaons are incorporated into the RMF model by using the Lagrangian density of the form [17] L k = µ Ψ µ Ψ m 2 k ΨΨ g σk m k ΨΨσ ig ωk ( Ψ µ Ψω µ Ψ µ Ψω µ ) + (g ωk ω µ ) 2 ΨΨ. (4) L k describes the interaction of the antikaon field ( Ψ) with the scalar (σ) and vector (ω) isoscalar fields. The corresponding equation of motion for K in a Z = N nucleus can be expressed by KG equation (1) with the real part of the optical potential given at threshold by ReV opt = m k µ ( 1 2 S V V 2 2m k ), (5) where S = g σk σ(r) and V = g ωk ω 0 (r) in terms of the mean-isoscalar fields. From Eq. (5) in the mean-field approximation we can obtain ReV opt = m k µ where ( 1 2 α σu (N) s + α ω U (N) v + (α ωu (N) v ) 2 2m k ), (6) U v (N) g 2 ωn = ρ v m 2, U (N) g 2 σn s = ρ s ω m 2, (7) σ ρ v and ρ s are the nuclear vector and scalar densities, respectively, evaluated at nuclear matter density. If we neglect the small components of the Dirac spinor, then ρ v (r) = ρ s (r). The nuclear vector density is equal to nuclear density distribution ρ(r) in fact. To simplify the calculation we neglect the difference of nuclear vector and scalar densities and take the phenomenological nuclear density distribution ρ(r). In the calculation of Re V opt, the nonlinear parametrization (NL1) due to Reinhard et al. [18] and equation (6) are used. The parameters are listed in Table 1. Table 1 The parameters for calculation. m g σn α σ = g σk /g Nσ 0.20 m σ g Nω α ω = g ωk /g ωn m ω m k α Hybrid Optical Potential In the hybrid model, the functional RMF from Eq. (6) is used in the nuclear interior for ReV opt, whereas the purely phenomenological DD from Eq. (2) is used in the surface of the nucleus and beyond. The radius R m where the two forms are matched to each other is chosen as R m R 1/2 fm, where ρ(r 1/2 ) = ρ 0 /2. 3 Calculations and Analysis Phenomenological nuclear density distribution ρ(r) is given by the 3-parameter Fermi (3pF) distribution [16] ρ(r) = ρ 0[1 + ω(r/r) 2 ] 1 + exp((r R)/a). (8) In this paper the parameters ω = , R = fm, a = fm for 12 C [19] and ω = 0, R = 1.28 A 1/ A 1/3 fm, a = 3/π fm for 40 Ca. [15] 3.1 DD and RMF Potentials at p k =0 and K Nucleus Elastic Scattering Figure 1 shows the K C and K Ca optical potentials calculated from Eq. (2) (DD model) and Eq. (6) (RMF model) respectively. From Fig. 1 we can see the two models have apparent differences. The DD real potential is larger than that of RMF model in the nuclear interior, while the RMF real potential is a little deeper than that of DD model in nuclear surface and beyond. The DD model yields that the depth of optical potential is about 227 MeV for K C and 220 MeV for K Ca at ρ 0. The RMF model yields that the depth of optical potential is about 196 MeV for K C and 190 MeV for K Ca at ρ 0. The result agrees with the calculations of Ref. [9]. It indicates that the approximations for RMF model in Subsec. 2.2 are reasonable. The imaginary potential yielded by DD model is about 60 MeV for K C and 55 MeV for K Ca at ρ 0. Figure 2 shows our calculations for the K nucleus elastic differential cross sections for 12 C and 40 Ca at lab K momenta 800 MeV/c. The black dotted line is the experimental data obtained from Ref. [2], the solid line shows

3 No. 4 K Nucleus Elastic Scattering and Momentum-Dependent Optical Potentials 575 the results of hybrid model and the dashed line is for DD model at p k = 0. From Fig. 2 we can easily find that neither DD nor hybrid model at p k = 0 can describe the elastic scattering at p k = 800 MeV/c. Fig. 1 The optical potentials of DD and RMF model for K C and K Ca in line with kaon atomic data. The solid line shows the real potentials of DD, the dotted line is for the imaginary potentials of DD model, the short dotted line is for the RMF real potentials. Fig. 2 The elastic differential cross section for K scattering from 12 C and 40 Ca at p k = 800 MeV/c. The experimental data are taken from Ref. [2]. The solid lines show the result from hybrid model, and the dashed lines indicate the result from DD model.

4 576 ZHONG Xian-Hui, LI Lei, CAI Chong-Hai, and NING Ping-Zhi Vol Momentum-Dependent Potentials and K Nucleus Elastic Scattering Why the DD or hybrid potentials at p k = 0 cannot be used to describe the K nucleus elastic scattering at antikaon momentum p k = 800 MeV/c? We note that the potentials are obtained by fitting the kaon atomic data at p k = 0, however, the K momentum is 800 MeV/c in our problem. Do the optical potentials have momentum dependence in K nucleus elastic scattering as suggested by Sibirtsew et al.? [14,15] We propose the optical potentials have momentum dependence with the following forms given by Re U(ρ(r), p k ) = f(p k )Re U(ρ(r)), (9) Im U(ρ(r), p k ) = f (p k )Im U(ρ(r)), (10) where f(p k ) and f (p k ) are the functions of p k, Re U(ρ(r), p k ) is the real part and Im U(ρ(r), p k ) is the imaginary part of the momentum-dependent potentials. Re U(ρ(r)) is the real and Im U(ρ(r)) is the imaginary part of optical potentials at p k =0. It is interesting that another form of real momentum-dependent potential for K nucleus was given by Ref. [14] Re U(ρ B, p k ) ρ B ( exp( 2.5p k )), (11) where ρ B is the baryon density. Comparing Eq. (11) with Eq. (9), we can get f(p k ) exp( 2.5p k) (12) In our present work the K momentum p k = 800 MeV/c, so f(p k=800 ) = c 1 and f (p k=800 ) = c 2 are constants. To define c 1 and c 2, we use the hybrid model to fit the experimental data of K C and K Ca elastic differential cross section at p k = 800 MeV/c. If we let c 1 1/4.1 and c 2 1/1.8 the experimental data for K C elastic scattering at p k = 800 MeV/c can be fitted fairly well. We also get the best value c 1 1/5 and c 2 1/1.8 for K Ca elastic scattering at p k = 800 MeV/c. The calculations are all shown in Fig. 3. From the difference of c 1 for K C and K Ca, it is seen that c 1 is somewhat dependent on the different nucleus. According to Eq. (12), c is very close to c 1 1/4.1 for K C and c 1 1/5 for K Ca. Fig. 3 The elastic differential cross section for K scattering from 12 C and 40 Ca at p k = 800 MeV/c. The experimental data is shown by black dots. The lines show calculations from DD, hybrid, and RMF models, which are denoted in the figure respectively. In Fig. 3, it shows that there is only a little difference from the experimental data for K C. For K Ca except that there exists some obvious difference between the calculations and the experimental data in the region of 14 θ c.m. 20 and θ c.m. 25, in the other region, the calculations fit fairly well with the experimental data. In summary, within the experimental uncertainties our result is in agreement with the K-nucleus elastic scattering experimental data fairy well. From above analysis, it indicates that we have achieved success in describing the K C and K Ca elastic scattering with the momentum-dependent optical potentials at p k = 800 MeV/c. By far, we have developed a new form of momentum-dependent optical potential to describe K- nucleus elastic scattering. The success greatly supports the suggestion of Sibirtsew et al. [14,15] that the K nucleus optical potentials have strong momentum-dependence and confirms that the forms of momentum- dependent optical potentials assumed by us in Eqs. (8) and (9) are reasonable. In the above discussion, the momentum-dependent factors c 1 and c 2 are defined, that is, the momentum dependent optical potentials are defined at p k = 800 MeV/c. Then we calculate the K C and K Ca elastic scattering differential cross sections with the other two (DD,RMF) momentum-dependent optical potential models at p k = 800 MeV/c. Together with the result of hybrid model, the results of DD and RMF model are shown in Fig. 3.

5 No. 4 K Nucleus Elastic Scattering and Momentum-Dependent Optical Potentials 577 Comparing the results of above three models with the experimental data, it is easily seen that the RMF model is worse than the other two models, the hybrid model is better, but the difference is very small with DD model. In summary, the momentum-dependent hybrid model is better than the other two models in describing K nucleus elastic scattering, this agrees with the conclusion of Friedman et al. in Ref. [9]. Why do the momentum-dependent DD and hybrid model describe the K nucleus elastic scattering better than the RMF model? In fact, the three models have the same imaginary optical potential given by DD model, the only difference is in the real part of the optical potential. We note that the DD and hybrid models have the same optical potentials in the surface of nucleus and beyond, while the hybrid model and RMF models have the same optical potentials in the nuclear interior. We can affirm that the most differences of the calculation result for the three models come from the differences of the optical potential in the surface of nucleus and beyond. It indicates that K is sensitive to the optical potentials only in the nuclear surface and beyond. Since the momentumdependent hybrid model is better than the other two models, we conclude that the DD optical potential model in the surface of nucleus is better than the RMF model, while the RMF optical potential model in the nuclear interior is better than the DD model. The K C and K Ca momentum-dependent optical potentials at p k = 800 MeV/c are shown in Fig. 4 respectively. The dotted line presents the real DD potential, the short dotted line is the real hybrid potential, the dashed line shows the RMF potential, and the solid line is the imaginary potential. From Fig. 4 we can see the momentum-dependent optical potential of DD model is about 56 MeV for K C and 45 MeV for K Ca and the momentum-dependent optical potential of RMF model is about 50 MeV for K C and 38 MeV for K Ca at the normal nuclear density ρ 0. The imaginary optical potential is about 32 MeV for K C and 35 MeV for K Ca at the normal nuclear density ρ 0. From hybrid model in Fig. 3, we can easily see the DD real potentials region in the surface of nucleus, as we know from the previous analysis, K is sensitive in this region, thus, we can estimate K C interaction region is about 2.2 fm < r < 5 fm and K Ca interaction region is about 3.8 fm < r < 8 fm at p k = 800 MeV/c. Fig. 4 The momentum-dependent potentials for K scattering on 12 C and 40 Ca at p k = 800 MeV/c. The lines show calculations from DD, hybrid, and RMF models. 4 Summary In this paper we use the DD and hybrid model at p k = 0 calculate the K nucleus differential elastic scattering cross section for C and Ca at p k = 800 MeV/c and find neither DD nor hybrid model can describe the elastic scattering at all. Then, we consider the momentum-dependence of optical potentials and propose a form of momentum dependent optical potentials starting with the DD, RMF, and hybrid models at p k = 0 to calculate the K nucleus differential elastic scattering cross sections for C and Ca at p k = 800 MeV/c and achieve success. The success indicates that we have developed a new momentum-dependent optical potential model to describe K nucleus scattering and confirms that there exists strong momentum-dependence for K nucleus optical potentials in elastic scattering. We compare the result of different models with each other and find the momentum-dependent hybrid model is still the best model in describing the elastic scattering at p k = 800 MeV/c and agrees with the conclusion in Ref. [9]. We also obtain the momentum-dependent optical potentials for K C and K Ca and find that the real potential is about MeV, which is in agreement with the result of Ref. [14] and the inmedium potentials evaluated from the measurement of kaonic atoms. [6,7] The imaginary potential is about 35 MeV at ρ 0 at p k = 800 MeV/c. From the analysis, we estimate

6 578 ZHONG Xian-Hui, LI Lei, CAI Chong-Hai, and NING Ping-Zhi Vol. 41 K C interaction region is about 2.2 fm < r < 5 fm and K Ca interaction region is about 3.8 fm < r < 8 fm at p k = 800 MeV/c. It must be noted that, because K is not sensitive to the optical potential in nuclear interior, the depth of the above optical potentials is not very trusty. References [1] Carl B. Dover and George E. Walker, Phys. Rep. 89 (1982) 1. [2] D. Marlow, et al., Phys. Rev. C25 (1982) [3] A.S. Rosenthal and F. Tabakin, Phys. Rev. C22 (1980) 711. [4] C. Garcia-Recio and A.J. Melgarejo, nucl-th/ [5] A. Sibirtsev and W. Cassing, nucl-th/ [6] E. Friedman, A. Gal, and C.J. Batty, Phys. Lett. B308 (1993) 6. [7] C.J. Batty, E. Friedman and A. Gal, Phys. Rep. 287 (1997) 385. [8] E. Friedman, A. Gal, and C.J. Batty, Nucl. Phys. A579 (1994) 518. [9] E. Friedman, A. Gal, and J. Mares, Phys. Rev. C60 (1999) [10] G.Q. Li, C.M. Ko, and X.S. Fang, Phys. Lett. B ) 149. [11] W. Cassing, et al., Nucl. Phys. A614 (1997) 415. [12] E.L. Bratkovskaya, W. Cassing, and U. Mosel, Nucl. Phys. A622 (1997) 593; Phys. Lett. B424 (1998) 224. [13] W. Cassing and E.L. Bratkovskaya, Phys. Rep. 308 (1999) 65. [14] A. Sibirtsev and W. Cassing, Nucl. Phys. A641 (1998) 476. [15] A. Sibirtsev and W. Cassing, nucl-th/ [16] C.J. Batty, E. Friedman, and A. Gal, Phys. Rep. 287 (1997) 385. [17] J. Schaffner, A. Gal, and I.N. Mishustin, et al., Phys. Lett. B334 (1994) 268; J. Schaffner and I.N. Mishustin, Phys. Rev. C53 (1996) [18] P.G. Reinhard, M. Rufa, and J. Friedrich, Z. Phys. A625 (1997) 272. [19] H. de Vries and C.W. de Jager, et al., At. Data Nucl. Data Tables 36 (1987) 495.