Short-range tensor interaction and high-density nuclear symmetry energy
aa r X i v : . [ nu c l - t h ] J u l Short-range tensor interaction and high-density nuclear symmetry energy
Ang Li
1, 2 and Bao-An Li ∗ Department of Physics and Astronomy, Texas A & M University-Commerce, Commerce, Texas 75429-3011, USA Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China (Dated: November 21, 2018)Effects of the short-range tensor interaction on the density-dependence of nuclear symmetry energy are ex-amined by applying an approximate expression for the second-order tensor contribution to the symmetry energyderived earlier by G.E. Brown and R. Machleidt. It is found that the uncertainty in the short-range tensor forceleads directly to a divergent high-density behavior of the nuclear symmetry energy.
PACS numbers: 21.65.Cd, 21.65.Ef, 21.30.Fe
The density dependence of nuclear symmetry energy E sym ( r ) encodes the energy related to neutron-proton asym-metry in the Equation of State (EOS) of nuclear matter. Whilethe E sym ( r ) is very important for both nuclear physics andastrophysics [1–14], it is still rather uncertain especially atsupra-saturation densities. Besides promising constraints be-ing extracted from astrophysical observations [16], significantprogress has been made recently in constraining the E sym ( r ) around and below the nuclear matter saturation density r using experiments in terrestrial laboratories [17]. Lookingforward, it is very exciting to note that dedicated experi-ments are currently underway or being planned at severaladvanced radioactive ion beam facilities at CSR/China [15],FRIB/USA [18], GSI/Germany [19], RIKEN/Japan [20] andKoRIA/Korea [21] to pin down the high-density behavior ofthe E sym ( r ) . While essentially all existing many-body theo-ries have been used to predict the E sym ( r ) , the results divergequite widely especially at supra-saturation densities, see, e.g.,ref. [5] for a recent review. Thus, it is necessary to iden-tify fundamental reasons for the uncertain high-density be-havior of the E sym ( r ) . Generally speaking, besides the dif-ferent techniques often used in treating nuclear many-bodyproblems in various theories, our poor knowledge about theisospin dependence of the in-medium nuclear strong interac-tion is at least partially responsible for the uncertain E sym ( r ) .In fact, it has been recognized that the spin-isospin depen-dence of the three-body force, see, e.g., ref. [9, 22], the isospindependence of short-range nucleon-nucleon correlation func-tions, see, e.g., ref. [23], and the short-range tensor force,see, e.g., ref. [24] all play some significant roles in determin-ing the high-density behavior of the E sym ( r ) . In particular,it is easy to understand qualitatively why the nuclear tensorinteraction is important in determining the E sym ( r ) . Withinthe parabolic approximation of the EOS of isospin asymmet-ric nuclear matter, see, e.g., ref. [25], the E sym ( r ) can bewritten as the difference between the nucleon specific energyin pure neutron matter (PNM) and symmetric nuclear mat-ter (SNM), i.e., E sym ( r ) = E PNM ( r ) − E SNM ( r ) . It is wellknown that in the isospin-singlet T = ∗ Corresponding author, Bao-An [email protected] significant tensor component is required to understand proper-ties of deuteron and neutron-proton scattering data, see, e.g.,refs. [26, 27]. Moreover, it has been found consistently inmicroscopic many-body calculations that the T = E sym ( r ) . Ap-plying an approximate expression for the second-order tensorcontribution to the symmetry energy derived earlier by G.E.Brown and R. Machleidt [39], we find that the uncertainty inthe short-range tensor force contributes significantly to the di-vergence of the E sym ( r ) at supra-saturation densities.In the best-studied phenomenology of nuclear forces, i.e.,the one-boson-exchange model, the tensor interaction resultsfrom exchanges of the isovector p and r mesons. For in-stance, the tensor part of the one-pion exchange potential(OPEP) can be written in configuration space as [26] V t p = − f p p m p ( t · t ) S [ ( m p r ) + ( m p r ) + m p r ] exp ( − m p r ) (1)where r is the inter-particle distance and S = ( s · r )( s · r ) r − ( s · s ) is the tensor operator. The r -exchange tensor inter-action V t r has the same functional form as the OPEP, but withthe m p replaced everywhere by m r , and the f p by − f r . Themagnitudes of both the p and r contributions grow quicklywith decreasing r . A proper cancelation of the opposite con-tributions from the p and r exchanges is supposed to give arealistic strength for the nuclear tensor force. However, sincethe tensor coupling is not well determined consistently fromdeuteron properties and/or nucleon-nucleon scattering data,the tensor interaction is by far the most uncertain part of thenucleon-nucleon interaction [27]. Moreover, it is also possiblethat the in-medium r meson mass m r is different from its free-space value [29]. A density-dependent in-medium m r willlead to very different short-range tensor force [30] and affectsthe symmetry energy at high densities [9, 31, 32]. While thereis no community-wide consensus on whether the m r changesor not in the dense medium, it is a possible origin for the un-certain short-range tensor force. In addition, due to both thephysical and mathematical differences in construction [27],various realistic nuclear potentials usually have widely differ-ent tensor components at short range ( r ≤ < V t > is zero. Thus, the first-order ten-sor force does not contribute to the symmetry energy unlessone assumes that all isosinglet neutron-proton pairs behave asbound deuterons with S = < V sym > = e eff < V t ( r ) > (2)where e eff ≈
200 MeV and V t ( r ) is the radial part of the ten-sor force [39]. While this approximate expression may lead tosymmetry energies systematically different from predictionsof advanced microscopic many-body theories using variousinteractions, it is handy to evaluate effects of the differentshort-range tensor forces within the same simple and analyti-cal approach. Of course, it is necessary and also interesting toevaluate the accuracy of Eq. (2) with respect to microscopicmany-body calculations using the same interaction.To apply Eq. (2) we evaluate the expectation value of V sym using the free single-particle wave function ( V − e i k · r ) h l z t ,where h l = ↑ / ↓ and z t = p / n is the spin and isospin wave func-tion, respectively. The direct and exchange matrixes are, re-spectively, h k lt k ′ l ′ t ′ | V sym | k lt k ′ l ′ t ′ i = V Z d r Z d r ′ e − i k · r e − i k ′ · r ′ h † l ( ) h † l ′ ( ) z † t ( ) z † t ′ ( ) × V sym ( , ) e i k · r e i k ′ · r ′ h l ( ) h l ′ ( ) z t ( ) z t ′ ( )= V Z V sym ( r ) d r (3)and h k lt k ′ l ′ t ′ | V sym | k ′ l ′ t ′ k lt i = V Z d r Z d r ′ e − i k · r e − i k ′ · r ′ h † l ( ) h † l ′ ( ) z † t ( ) z † t ′ ( ) × V sym ( , ) e i k ′ · r e i k · r ′ h l ′ ( ) h l ( ) z t ′ ( ) z t ( )= V d ll ′ d tt ′ Z exp [ − i ( k − k ′ ) · r ] V sym ( r ) d r . (4)The expectation value of V sym in the S = , T = < V sym > =
116 12 (cid:229) k lt (cid:229) k ′ l ′ t ′ [ h k lt k ′ l ′ t ′ | V sym | k lt k ′ l ′ t ′ i− h k lt k ′ l ′ t ′ | V sym | k ′ l ′ t ′ k lt i ]= (cid:229) k lt (cid:229) k ′ l ′ t ′ V { Z V sym ( r ) d r − d tt ′ d ll ′ Z exp [ − i ( k − k ′ ) · r ] V sym ( r ) d r } = V ( p ) Z k F d k Z k F d k ′ { Z V sym ( r ) d r − Z exp [ − i ( k − k ′ ) · r ] V sym ( r ) d r } . (5)Noticing that the momentum integral Z k F d ke i k · r = p Z k F k j ( kr ) dk = p k F j ( k F r ) k F r (6)and the particle number density AV = p k F , we can write thetensor contribution to the symmetry energy as < V sym > A = e eff · k F p { Z V t ( r ) d r − Z [ j ( k F r ) k F r ] V t ( r ) d r } . (7)For large k F , the second integral in the above equation ap-proaches zero, the first term is thus expected to dominate athigh densities, leading to an almost linear density dependence.To access quantitatively effects of the short-range tensorforce on the density dependence of nuclear symmetry en-ergy, we adopt here several tensor forces used by Otsuka etal. in their recent studies of nuclear structures [43]. Theconsidered tensor forces, including the standard p + r ex-change (labelled as a ), the G-Matrix (GM) [43] (labelled as b ), M3Y [44](labelled as c ) and the Av18 [34] (labelled asAv18), as shown in the left panel of Fig. 1, behave rather dif-ferently at short distance, but merge to the same Av18 tensorforce at longer range. In addition, we add a case ( d ) wherethe tensor force vanishes for r ≤ . p + r exchangeinteraction is fixed by the standard meson-nucleon couplingconstants with a strong r coupling [42], and we use a short-range cut-off at r = . V ( r < . ) = V ( r = . ) .As emphasized by Otsuka et al. [43], the short-range behaviorof the tensor force has no effect on nuclear structures. How-ever, as we shall show in the following, it affects significantlythe E sym ( r ) especially at supra-saturation densities.Shown in the right panel of Fig. 1 are the potential partsof the symmetry energies due to the tensor forces consideredaccording to Eq. (7). As expected, they tend to grow linearlywith increasing density. Since it is the square of the tensorforce that determines its contribution to the symmetry energy, V t e n s o r [ M e V ] r /fm d 0 1 2 3 4 50204060 A v18 < V s y m > / A [ M e V ] / FIG. 1: (Color online) Left panel: radial parts of the tensor interac-tions having different short-range behaviors but the same long-range( r > . E s y m [ M e V ] K i n e ti c p a r t a b c d / B H F Av18
FIG. 2: (Color online) Symmetry energies using various short-rangetensor interactions in Eq. (2) in comparison with the Brueckner-Hartree-Fock prediction using the Av18 potential. tensor forces having larger magnitudes at short distance affectmore significantly the symmetry energy. It is seen that thevariation of the tensor force at short distance affects signifi-cantly the high-density behavior of nuclear symmetry energy.Including also the kinetic part of the symmetry energy k F m ,we show in Fig. 2 the E sym ( r ) . The divergent values of the E sym ( r ) are completely due to the different short-range tensor forces used. To evaluate the accuracy of the results obtainedusing Eq. (2), we compare in Fig. 2 predictions from Eq. (2)and the Brueckner-Hartree-Fock (BHF) [22] both using theAv18 interaction. It is seen that essentially over the wholedensity range considered, the BHF prediction is about 7 MeVhigher. This is qualitatively understandable since the differ-ence in central forces between the isotriplet T = T = E sym ( r ) . There are alsocorrelations among probably several factors that may all affectthe E sym ( r ) individually. For example, the short-range tensorforce also leads to neutron-proton correlations in SNM [46].Consequently, the single-nucleon momentum distribution ob-tains a high momentum tail that will change the average ki-netic energy of nucleons in SNM [47], and thus the kineticpart of the E sym ( r ) [23]. While this effect is not consideredhere, our results based on Eq. (2) are interesting and useful forbetter understanding the role of tensor forces in determiningthe E sym ( r ) .In summary, using an approximate expression for thesecond-order tensor contribution to the symmetry energy de-rived earlier by G.E. Brown and R. Machleidt, we investigatedeffects of the short-range tensor interaction on the density-dependence of nuclear symmetry energy. We found that in-deed the tensor force dominates the potential part of the nu-clear symmetry energy. The uncertain short-range tensor forcecontributes significantly to the divergence of the nuclear sym-metry energy especially at supra-saturation densities.We would like to thank L.W. Chen, W. G. Newton, H.-J.Schulze, C. Xu and W. Zuo for valuable discussions. Thiswork is supported in part by the US National Science Foun-dation under grant PHY-0757839 and PHY-1068022, the USNational Aeronautics and Space Administration under grantNNX11AC41G issued through the Science Mission Direc-torate, the National Basic Research Program of China un-der Grant 2009CB824800, and the National Natural ScienceFoundation of China under Grant 10905048. [1] B. A. Li, C. M. Ko, and W. Bauer, Int. J. Mod. Phys. E , 147(1998).[2] Isospin Physics in Heavy-Ion Collisions at Intermediate Ener-gies, Eds. B. A. Li and W. Uuo Schr¨oder (Nova Science Pub-lishers, Inc, New York, 2001).[3] P. Danielewicz, R. Lacey and W.G. Lynch, Science , 1592 (2000).[4] V. Baran, M. Colonna, V. Greco and M. Di Toro, Phys. Rep. , 335 (2005).[5] B. A. Li, L. W. Chen and C. M. Ko, Phys. Rep. , 113 (2008).[6] K. Sumiyoshi and H. Toki, ApJ, , 700 (1994).[7] J. M. Lattimer and M. Prakash, ApJ, , 426 (2001); Science , 536 (2004).[8] A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys.Rep. , 325 (2005).[9] C. Xu and B. A. Li, Phys. Rev. C 81, 044603 (2010).[10] L. W. Chen, C. M. Ko and B. A. Li, Phys. Rev. Lett. , 032701(2005); B. A. Li and L. W. Chen, Phys. Rev. C , 064611(2005).[11] M. B. Tsang, Y. X. Zhang, P. Danielewicz, M. Famiano, Z.X. Li, W. G. Lynch, and A. W. Steiner, Phys. Rev. Lett. ,122701 (2009).[12] M. Centelles, X. Roca-Maza, X. Vinas and M. Warda, Phys.Rev. Lett. , 122502 (2009).[13] Z. G. Xiao, B. A. Li, L. W. Chen, G. C. Yong and M. Zhang,Phys. Rev. Lett. , 062502 (2009).[14] C. Xu, B. A. Li, and L. W. Chen, Phys. Rev. C , 054607(2010).[15] Z. G. Xiao et al., J. Phys. G , 064040 (2009).[16] A. W. Steiner, J.M. Lattimer and E.F. Brown, APJ , 33(2010).[17] W. G. Lynch, talk given at the Second InternationalSymposium on Nuclear Symmetry Energy (NuSYM11),June 17-20, 2011, Northampton, Massachusetts, .[18] Symmetry Energy Project (SEP) http://groups.nscl.msu.edu/hira/sep.htm .[19] Roy Lemmon for the ASY-EOS collaboration, talk given at theSecond International Symposium on Nuclear Symmetry Energy(NuSYM11), June 17-20, 2011, Northampton, Massachusetts, .[20] RIKEN Samurai collaboration http://rarfaxp.riken.go.jp/RIBF-TAC05/10SAMURAI.pdf .[21] Byungsik Hong, talk given at the Second InternationalSymposium on Nuclear Symmetry Energy (NuSYM11),June 17-20, 2011, Northampton, Massachusetts, .[22] W. Zuo, A. Lejeune, U. Lombardo and J.F. Mathiot, Eur. Phys.J. A14 , 469 (2002).[23] C. Xu and B. A. Li, arXiv:1104.2075v1.[24] V. R. Pandharipande and V. K. Garde, Phys. Lett. B, , 608 (1972).[25] I. Bombaci and U. Lombardo Phys. Rev. C , 1892 (1991).[26] R. Machleidt, Adv. Nucl. Phys. , 189 (1989).[27] R. Machleidt and I. Slaus, J. Phys. G: Nucl. Part. Phys. , R69(2001).[28] A. E. L. Dieperink, Y. Dewulf, D. Van Neck, M. Waroquier, andV. Rodin, Phys. Rev. C , 064307 (2003).[29] G. E. Brown and M. Rho, Phys. Rev. Lett. , 2720 (1991);Phys. Rep. , 1 (2004).[30] G. E. Brown and M. Rho, Phys. Lett. B237 , 3 (1990).[31] H. Dong, T.T.S. Kuo and R. Machleidt, Phys. Rev. C , 065803(2009); ibid , 054002 (2011).[32] Hyun Kyu Lee, Byung-Yoon Park and Mannque Rho, Phys.Rev. C , 025206 (2011).[33] M. Lacombe et al., Phys. Rev. C , 861 (1980).[34] R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, Phys. Rev. C , 38 (1995).[35] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. deSwart, Phys. Rev. C , 2950 (1994).[36] I. Tanihata, Mod. Phys. Lett. A 25 , 1886 (2010).[37] R Subedi et al., Science , 1475 (2008).[38] H. Baghdasaryan et al. (CLAS ollaboration), Phys. Rev. Lett. , 222501 (2010).[39] G. E. Brown and R. Machleidt, Phys. Rev. C , 1731 (1994).[40] T. T. S. Kuo and G. E. Brown, Phys. Lett. , 54 (1965).[41] G. E. Brown, J. Speth, and J. Wambach, Phys. Rev. Lett. ,1057 (1981).[42] S.-O. B ¨ a ckman, G. E. Brown, and J. A. Niskanen, Phys. Rep. , 1 (1985).[43] T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe, and Y. Akaishi,Phys. Rev. Lett. , 232502 (2005); T. Otsuka, T. Matsuo, andD. Abe, Phys. Rev. Lett. , 162501 (2006).[44] G. Bertsch, J. Borysowicz, H. McManus, and W. G. Love, Nucl.Phys. A284 , 399 (1977).[45] J. Decharge and D. Gogny, Phys. Rev. C , 1568 (1980).[46] R. Schiavilla, R. B. Wiringa, Steven C. Pieper, and J. Carlson,Phys. Rev. Lett. , 132501 (2007).[47] H.A. Bethe, Ann. Rev. Nucl. Part. Sci.,21