[
Abstract
We perform a coupledchannels study of the lowlying states in C with a covariant energy density functional based microscopic particlecore coupling model. The energy differences of and states in C and C are predicted to be 0.25 MeV and 0.34 MeV, respectively. We find that configuration mixings in the and states of C are the weakest among those of C. It indicates that C provides the best candidate among the carbon hypernuclei to study the spinorbit splitting of hyperon state.
Configuration mixing in lowlying spectra of carbon hypernuclei]Configuration mixing in lowlying spectra of carbon hypernuclei
1]XIA HaoJie 2†]MEI Hua ^{1}^{1}1†Corresponding author (email: meihuayaoyugang@gmail.com) 2]YAO JiangMing
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\hb@xt@1ex^{1}School of Mechanical and Electrical Engineering, Handan University, Handan 056005, Hebei, China 
\hb@xt@1ex^{2}Department of Physics and Astronomy, University of North Carolina, Chape Hill 275993255, North Carolina, USA 
Received April 4, 2017; accepted May 1, 2017; published online May 2, 2017
Hypernuclei, Nuclear Density Functional Theory and Extensions, Energy Spectra
The spectroscopy of hypernuclear lowlying states is very important to understand the structure of hypernuclei and the hyperon impurity effect in atomic nuclei. Several novel phenomena about hyperon have already been discovered by studying energy spectra and electromagnetic transitions of lowlying states in shell hypernuclei. One of them is the shrinkage effect of hyperon in Li, which was indicated from the measurement of ray transition probability from to in Li. The ratio of in Li to in Li has been converted into the reduction of atomic size by about 19% [1]. Another novel finding is about the weak spinorbit splitting () of state in C. The rays from the excited and states to the ground state were measured following the C(,)C reaction. The energy difference between the and states was determined to be (syst) keV [2], which has been interpreted as the spinorbit splitting between 1 and 1 hyperon states in C. Here, this interpretation relies on the assumption that the and states are the pure configuration of and coupled to the ground state () of C [3], respectively. It is worth mentioning that for most hypernuclei, the and states cannot be naively interpreted using such a simple picture due to a large effect of configuration mixings [4, 5, 6, 7]. It has been found in a recent study that the mixing amplitude is negligibly small in spherical and weaklydeformed hypernuclei, but strongly increases as the core nucleus undergoes a transition to a welldeformed shape [8]. This perturbs the interpretation of their energy difference as the spinorbit splitting for the state.
Considering the great success of covariant density functional theory for atomic nuclei [9] and hypernuclei [10], in this letter, we examine the configuration mixing in the lowlying states of C with a novel microscopic particlecore coupling model built on a multireference covariant density functional theory (MRCDFT). The MRCDFT has been successfully adopted to describe the lowlying states of carbon isotopes [11]. It provides a good starting point for studying the hypernuclear lowlying states. Special emphasis will be placed upon the energies and the ingredients of the wave functions for the lowest and states in the carbon isotopes.
In the microscopic particlecore coupling model [5, 6, 7], wave functions of the lowlying states of hypernuclei are constructed as
(1) 
where and are the coordinates of the hyperon and the nucleons, respectively. and are the radial wave function and the spinangular wave function for the particle, respectively. The index distinguishes different core states for a given angular momentum . The nuclear core states are determined from the MRCDFT [14, 15, 13, 12],
(2) 
The symmetryconserving reference states are a set of axially deformed meanfield states projected onto angular momentum () and particle numbers
(3) 
where the , and are projection operators. The meanfield states are obtained by relativistic meanfield plus BCS calculations with a constraint on the quadrupole deformation parameter . The weight function and the energy of the state are the solutions of the HillWheelerGriffin (HWG) equation [16]. The Hamiltonian kernel is calculated with the mixeddensity prescription, i.e., the offdiagonal elements of the energy overlap (sandwiched by two different reference states) take the same energy functional form as that of the diagonal one provided that all the densities and currents are replaced by the mixed ones [14, 15].
The Hamiltonian for the whole hypernucleus is chosen as [7]
(4) 
where is the kinetic energy of the hyperon, and is the manybody Hamiltonian for the core nucleus, satisfying . The third term on the right side of Eq. (4) represents the interaction term between the particle and the nucleons in the core nucleus, where is the mass number of the core nucleus. The nucleon () interaction takes the form derived from a relativistic pointcoupling energy functional [17]
(5)  
(6)  
(7) 
The equation is transformed into coupledchannels equations in relativistic framework. Therefore, the is a fourcomponent wave function that is solved by expanding on a set of radial wave functions of spherical harmonic oscillator. The expansion coefficients are determined by solving the coupledchannel equations, in which all the potentials coupling different channels are related to transition densities between lowlying states of core nuclei from the MRCDFT calculation. More details about the microscopicparticle core coupling model can be found in Refs. [5, 6, 7].
The wave functions of the meanfield states in Eq.(3) are products of singleparticle wave functions which are obtained by solving the Dirac equation selfconsistently on a harmonic oscillator basis with 12 major shells. The parameter sets PCF1 [18] and PCYS4 [17] are adopted for the effective nucleonnucleon and interaction, respectively. In the particlenumber and angularmomentum projection calculation, the number of mesh points for gauge angle in is chosen to be 9 and that for Euler angle in the interval is chosen to be 16. The radial wave function in Eq.(1) is expanded on the radial part of spherical harmonic oscillator basis with 18 major shells.
Figure 1 shows the lowlying states of C from the MRCDFT calculation, in comparison with data [19]. Since the exact particlenumber projection is adopted in the present calculation, the predicted energy spectra are somewhat different from those presented in Ref. [11], where only an approximate way [13] was adopted to take care of the particlenumber conserving effect. One can see that the main characters of the energy spectra are reproduced rather well. Our calculation gives a vibrationallike spectrum for C and rotationallike spectra for other isotopes.
Table [ lists the binding energies of hyperon for the ground states of C from the microscopic particlecore coupling model calculation. The binding energy is calculated as the energy difference between the state in C and the state in C. It is shown that the calculated is 13.22 MeV for C (compared to the data MeV [20]) and increases globally with neutron number, except for C. For comparison, the intrinsic quadrupole deformation parameters for the hypernuclei C (with the on the lowestenergy state) and core nuclei C from pure meanfield calculations are also provided in Table [. One can see that the quadrupole deformation of the whole hypernuclei is decreased in comparison with that of core nuclei. This finding has already been found in many studies [21, 22, 24, 25, 23, 26].
C  C  C  C  

B  13.22  13.04  13.52  14.20 
()  0.00 (0.00)  0.00 (0.00)  0.27 (0.34)  0.34 (0.41) 
Figure 2 displays the lowlying energy spectra of C. The positiveparity groundstate band with spinparity of in all the four carbon hypernuclei shares a similar structure to that for the corresponding core nucleus and their wave functions are dominated by the configuration of , where denotes the particle in the state. In contrast, the lowlying negativeparity states show an admixture of the and the configurations, as shown in Table [. One can see that the state in C is also an admixture of and with the mixing amplitude of and , while the mixing amplitude for in C is and , respectively. It indicates that the interpretation of the energy difference between the and states as the splitting of single states [3, 1] is questionable. In contrast, the mixing amplitude in C is weaker than that in C and thus C provides a more ideal hypernucleus with which to extract the splitting of state. The stronger configurationmixing amplitude in C than that in C can be traced back to the relatively larger quadrupole collectivity of corresponding core nucleus. Previous study [11] has shown that the energy surface of C as a function of deformation is much softer than that of the closedshell nucleus C and thus C possesses a larger quadrupole collectivity from dynamic shape fluctuation.
Table [ lists the energies of the and states in carbon hypernuclei and their energy differences . We note that the state in all the four hypernuclei is predicted to be higher than the state. For C, the predicted is 0.253 MeV, close to the data MeV [20]. For C, the energy difference of the and states is predicted to be 0.344 MeV, about 0.1 MeV larger than that in C. In contrast to the cases in C, this value is only 67 keV and 33 keV in C, respectively. One can see from Table [ that the energy difference of the and states in C cannot be interpreted as the spinorbit splitting of the state due to the large configuration mixing. As found in the recent study for Sm hypernuclei [8], the energy difference of the and states is monotonically decreasing as the amplitude of configuration mixing increases. A similar phenomenon is also observed in the carbon hypernuclei. In short, our results indicate that C is a more ideal hypernucleus than C to extract the splitting of the state.
C  C  C  C  











C  C  C  C  

12.964  12.224  10.498  10.027  
12.711  11.880  10.431  9.994  
0.253  0.344  0.067  0.033 
We have investigated the lowlying states in C with a microscopic particlecore coupling model based on the MRCDFT. It has been found that the positiveparity groundstate band of hypernuclei with spinparity of is dominated by the configuration of and shares a similar structure to that for the core nucleus with spinparity . In contrast, the lowlying negativeparity states are admixtures of the and the configurations. Among all the four carbon hypernuclei, C stands out as the best candidate to extract the spinorbit splitting of the state because about 97% of the wave functions for the and states are the configurations of and , respectively. Their energy difference turns out to be 0.344 MeV. We have found that the and states in C have a slightly larger configuration mixing than those in C. From this point of view, we conjecture that the previously measured energy difference between the and states in C [1] underestimates the splitting of single states. Therefore, a new measurement on hypernuclear ray spectroscopy for C is suggested to confirm our conclusions. This kind of measurement is feasible with the Japan Proton Accelerator Research Complex (JPARC) facility [20].
This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 11575148, and 11305134.
References
 1 K. Tanida, H. Tamura, D. Abe, H. Akikawa, K. Araki, H. Bhang, T. Endo, Y. Fujii, T. Fukuda, O. Hashimoto, K. Imai, H. Hotchi, Y. Kakiguchi, J. H. Kim, Y. D. Kim, T. Miyoshi, T. Murakami, T. Nagae, H. Noumi, H. Outa, K. Ozawa, T. Saito, J. Sasao, Y. Sato, S. Satoh, R. I. Sawafta, M. Sekimoto, T. Takahashi, L. Tang, H. H. Xia, S. H. Zhou, and L. H. Zhu, Phys. Rev. Lett. 86, 1982 (2001).
 2 S. Ajimura, H. Hayakawa, T. Kishimoto, H. Kohri, K. Matsuoka, S. Minami, T. Mori, K. Morikubo, E. Saji, A. Sakaguchi, Y. Shimizu, M. Sumihama, R. E. Chrien, M. May, P. Pile, A. Rusek, R. Sutter, P. Eugenio, G. Franklin, P. Khaustov, K. Paschke, B. P. Quinn, R. A. Schumacher, J. Franz, T. Fukuda, H. Noumi, H. Outa, L. Gan, L. Tang, L. Yuan, H. Tamura, J. Nakano, T. Tamagawa, K. Tanida, and R. Sawafta, Phys. Rev. Lett. 86, 4255 (2001).
 3 E. H. Auerbach, A. J. Baltz, C. B. Dover, A. Gal, S. H. Kahana, L. Ludeking, and D. J. Millener, Phys. Rev. Lett. 47, 1110 (1981).
 4 T. Motoba, H. Bandō, K. Ikeda, and T. Yamada, Prog. Theor. Phys. Suppl. 81, 42 (1985).
 5 H. Mei, K. Hagino, J. M. Yao, and T. Motoba, Phys. Rev. C 90, 064302 (2014).
 6 H. Mei, K. Hagino, J. M. Yao, and T. Motoba, Microscopic study of lowlying spectra of Phys. Rev. C 91, 064305 (2015).
 7 H. Mei, K. Hagino, J. M. Yao, and T. Motoba, Phys. Rev. C 93, 044307 (2016).
 8 H. Mei, K. Hagino, J. M. Yao, and T. Motoba, arXiv preprint arXiv:1704.02258, 2017.
 9 J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, and L.S. Geng, Prog. Part. Nucl. Phys. 57, 470 (2006).
 10 K. Hagino and J. M. Yao, International Review of Nuclear Physics, Vol. 10, edited by J. Meng (World Scientific, Singapore, 2016), pp. 263303.
 11 J. M. Yao, J. Meng, P. Ring, Z. X. Li, Z. P. Li, and K. Hagino, Phys. Rev. C 84, 024306 (2011).
 12 J. M. Yao, K. Hagino, Z. P. Li, J. Meng, and P. Ring, Phys. Rev. C 89, 054306 (2014).
 13 J. M. Yao, H. Mei, H. Chen, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C 83, 014308 (2011).
 14 J. M. Yao, J. Meng, P. Ring, and D. Pena Arteaga, Phys. Rev. C 79, 044312 (2009).
 15 J. M. Yao, J. Meng, P. Ring, and D. Vretenar, Phys. Rev. C 81, 044311 (2010).
 16 P. Ring and P. Schuck, The Nuclear ManyBody Problem, (SpringerVerlag, New York, 1980).
 17 Y. Tanimura and K. Hagino, Phys. Rev. C 85, 014306 (2012).
 18 T. Bürvenich, D. G. Madland, J. A. Maruhn, and P.G. Reinhard, Phys. Rev. C 65, 044308 (2002).
 19 National Nuclear Data Center, [http://www.nndc.bnl.gov/].
 20 O. Hashimoto and H. Tamura, Prog. Part. Nucl. Phys. 57, 564 (2006).
 21 X.R. Zhou, H.J. Schulze, H. Sagawa, C. X. Wu, and E.G. Zhao, Phys. Rev. C 76, 034312 (2007).
 22 Myaing Thi Win and K. Hagino, Phys. Rev. C 78, 054311 (2008).
 23 B.N. Lu, E. Hiyama, H. Sagawa, and S.G. Zhou, Phys. Rev. C 89, 044307 (2014).
 24 Myaing Thi Win, K. Hagino, and T. Koike, Phys. Rev. C 83, 014301 (2011).
 25 B.N. Lu, E.G Zhao, and S.G. Zhou, Phys. Rev. C 84, 014328 (2011).
 26 W. X. Xue, J. M. Yao, K. Hagino, Z. P. Li, H. Mei, and Y. Tanimura, Phys. Rev. C 91, 024327 (2015).