Phenomenological construction of a relativistic nucleon-nucleon interaction for the superfluid gap equation in finite density systems

Transcription

1 Phenomenological construction of a relativistic nucleon-nucleon interaction for the superfluid gap equation in finite density systems Masayuki Matsuzaki a,1 Tomonori Tanigawa b,2 a Department of Physics, Fukuoka University of Education, Munakata, Fukuoka , Japan b Department of Physics, Kyushu University, Fukuoka , Japan We construct phenomenologically a relativistic particle-particle channel interaction which suits the gap equation for nuclear matter. This is done by introducing a density-independent momentumcutoff parameter to the relativistic mean field model so as to reproduce the pairing properties obtained by the Bonn-B potential and not to change the saturation property. The significance of the short-range correlation in the gap equation is also discussed. Key words: Superfluidity; Nuclear matter; Relativistic mean field PACS numbers: f; c; n 1 Electronic address: matsuza@fukuoka-edu.ac.jp 2 Electronic address: tomo2scp@mbox.nc.kyushu-u.ac.jp Preprint submitted to Elsevier Preprint 1 November 1999

2 1 Introduction Pairing correlation between nucleons is a key ingredient to describe the structure of neutron stars and finite nuclei. There are two distinct ways of description of such finite-density nuclear many-body systems; the non-relativistic and the relativistic ones. The latter incorporates the mesons explicitly in addition to the nucleons in terms of a field theory. Both describe the basic properties such as the saturation with a similar quality in different manners. Irrespective of whether non-relativistic or relativistic, however, various theoretical approaches can be classified into two types: One is realistic studies adopting phenomenological interactions constructed for finite-density systems from the beginning (we call this the P-type, indicating phenomenological, hereafter), as often done in the studies of heavy nuclei. And the other is microscopic studies based on bare nucleon-nucleon interactions in free space (the B-type, indicating bare ). In relativistic studies, typical examples of these two types as for the particle-hole (p-h) channel are the relativistic mean field (RMF) model and the Dirac-Brueckner-Hartree-Fock (DBHF) method, respectively. As for the particle-particle (p-p) channel, that is, pairing correlation, a bare interaction was used as the lowest-order contribution in the gap equation [1] in a study of the B-type [2]. The first relativistic study of the P-type of pairing correlation in nuclear matter was done by Kucharek and Ring [3]. They adopted a one-boson exchange (OBE) interaction with the coupling constants of the RMF model, which we call the RMF interaction hereafter, aiming at a fully selfconsistent Hartree-Bogoliubov calculation, which we call the P1-type, in the sense that both the p-h and the p-p channel interactions are derived from a common Lagrangian. But the resulting pairing gaps were about three times larger than those accepted in the non-relativistic studies. The reason can be ascribed to the fact that the coupling constants of the RMF model were determined by physics involving only small momenta (k k F, 2 3π 2 (k F ) 3 = ρ denoting the saturation density of symmetric nuclear matter), and therefore 2

3 the adopted OBE interaction is not reliable at large momenta. After a fiveyear blank, some attempts to improve this were done [4 6]. But their results were insufficient. An alternative way is to adopt another interaction in the p-p channel while the single-particle states are still given by the RMF model. We call this the P2-type. There are some variations of this. The first one, which we call the P2a-type, adopts another phenomenological interaction for the p-p channel. Actually, the non-relativistic Gogny force [7] was used combined with the single-particle states of the RMF model for finite nuclei in Ref.[8] and subsequent works, and gave excellent results. The second variation, which we call the P2b-type, is to adopt a bare interaction that describes the largemomentum part realistically, the Bonn potential, again combined with the single-particle states of the RMF model [9] (see also Ref.[1]). If one assumes that the RMF model simulates roughly the DBHF calculation, this P2b-type can be regarded as simulating the B-type calculation [2] mentioned above. The results of these P2a- and P2b-types are very similar at densities ρ<ρ. This supports a statement that the Gogny force resembles a realistic free interaction in the low-density limit [11]. But a clear difference can be seen at ρ ρ. This difference can also be seen in fully non-relativistic calculations; compare the results in Refs.[12] and [13], for example. The precise origin of this difference has not been understood well. A comparison after taking the polarization effects which have been known to be important at finite densities [14 16] into account may be necessary. The third variation, which we call the P2b -type, is to parameterize the p-p channel interaction in terms of a few scattering parameters. This was actually examined by being combined with the DBHF calculation [17], which we call the B -type (see also Ref.[18]). This method is free from model-dependent ambiguities and meets the viewpoint of modern effective field theories [19,2] but is applicable only to dilute systems. These classifications are summarized in Table 1. 3

4 Since we are interested in a wide density range where the 1 S gap exists and would like to respect the selfconsistency in the sense that both the p-h and the p-p channel interactions are derived from a common Lagrangian, here we construct a phenomenological relativistic nucleon-nucleon interaction based on the RMF interaction (the P1-type) by modeling the pairing properties given by the RMF+Bonn calculation (the P2b-type). However, one thing we have to bear in mind is the double counting problem of the short-range correlation [21,22]. We will come back to this issue later. 2 Construction of a relativistic particle-particle channel interaction We start from the ordinary σ-ω model Lagrangian density, L = ψ(iγ µ µ M)ψ ( µσ)( µ σ) 1 2 m2 σσ Ω µνω µν m2 ωω µ ω µ + g σ ψσψ gω ψγµ ω µ ψ, Ω µν = µ ω ν ν ω µ. (1) The antisymmetrized matrix element of the RMF interaction V derived from this Lagrangian is defined by, v(p, k) = ps, ps V ks, ks ps, ps V ks, ks, (2) under an instantaneous approximation, with tildes denoting time reversal. After a spin average and an angle integration are performed to project out the S-wave component, its concrete form is given by v(p, k) = g2 σ 2E p E k { 1+ 4M 2 m 2 σ (E p E k ) 2 4pk ln ( (p + k) 2 + m 2 σ (p k) 2 + m 2 σ )} 4

5 + g2 ( ω M 2 + p 2 + k 2 (E p E k ) 2) ln 2E p E k pk ( (p + k) 2 + m 2 ω (p k) 2 + m 2 ω ). (3) Our policy of constructing a phenomenological interaction proposed above is to introduce a density-independent parameter Λ so as not to change the Hartree part with the momentum transfer q = which determines the single-particle energies, respecting that the original parameters of the RMF are densityindependent. Since the large-momentum part of the RMF interaction does not have a firm experimental basis as mentioned above, we suppose there is room to modify that part. Needless to say, such a modification should be checked by studying independent phenomena, for example, medium-energy heavy-ion collisions. In the preceding letter [23], the upper bounds of the momentum integration in the gap equation, (p) = 1 (k) v(p, k) 8π 2 (E k E kf ) (k) k2 dk, (4) and the nucleon effective mass equation, M = M g2 σ γ m 2 σ 2π 2 M k2 + M 2 v2 k k2 dk, (5) where the spin-isospin factor γ = 4 and 2 indicate symmetric nuclear matter and pure neutron matter, respectively, were cut at a finite value Λ, as usually done in condensed-matter physics, since the gap increases monotonically until reaching Kucharek and Ring s value when the model space is enlarged as shown in Fig.3 of Ref.[6], while v(p, k) is left unchanged. We call this method the sudden cut hereafter. This was done first in Ref.[3] by inspection. We proposed a quantitative method to determine Λ, which is described below, and obtained 3.6 fm 1 for the linear σ-ω parameter set. This value almost coincides with their value, about 3.65 fm 1, for the NL2 parameter set. 5

6 In the present paper, we examine smooth cutoffs that weaken the largemomentum part; a form factor f(q 2 ), q = p k, is applied to each nucleonmeson vertex in v(p, k) while the upper bounds of the integrals are conceptually infinity. They are replaced numerically by a finite number, 2 fm 1 which has been proved to be large enough in Ref.[6]. Since there is no decisive reasoning to choose a specific form, we examine three types, Λ2 monopole: f(q 2 )= Λ 2 +q, 2 ( ) Λ dipole: f(q )=, (6) Λ 2 +q 2 strong: f(q 2 )= Λ2 q 2 Λ 2 +q. 2 Note that the sudden cut above was applied to k, nottoq. The parameter Λ is determined so as to minimize the difference in the pairing properties from the results of the P2b-type RMF+Bonn calculation, which is our model. Here we adopt the Bonn-B potential because this has a moderate property among the available (charge-independent) versions A, B, and C [24]. The pair wave function, φ(k) = 1 (k) 2E qp (k), E qp (k) = (E k E kf ) (k), (7) is related to the gap at the Fermi surface, (k F )= 1 v(k 4π 2 F,k)φ(k)k 2 dk, (8) and its derivative determines the coherence length [25], ( dφ dk ξ = 2 k 2 dk φ 2 k 2 dk ) 1 2, (9) 6

7 which measures the spatial size of the Cooper pairs. These expressions indicate that (k F )andξcarry independent information, φ and dφ, respectively, in dk strongly-coupled systems, whereas they are intimately related to each other in weakly-coupled ones. Therefore we search for Λ which minimizes χ 2 = 1 2N ( ) 2 ( ) 2 (kf ) RMF (k F ) Bonn ξrmf ξ Bonn + (k F ) Bonn ξ Bonn, (1) k F where the subscripts RMF and Bonn denote the RMF interaction including Λ and the Bonn-B potential, respectively, while the single-particle states are determined by the original RMF model in both cases. The actual numerical task is to solve the gap equation (4) and the effective mass equation for the nucleon (5). They couple to each other through E k = k 2 + M 2 + g ω ω, v 2 (k) = 1 E k E kf 1 (Ek. (11) 2 E kf ) (k) The parameters of the standard σ-ω model that we adopt are gσ 2 = 91.64, gω 2 = 136.2, m σ = 55 MeV, m ω = 783 MeV, and M = 939 MeV [26]. N in χ 2 is taken to be 11; k F =.2,.3,...,1.2fm 1. In the following, including those of the sudden cut case [23], the results for symmetric nuclear matter are presented. Those for pure neutron matter are very similar except that (k F ) is a little larger due to a larger effective mass M as shown in Fig. 1 (b) of Ref.[23]. Minimizations of χ 2 gave 7.26, 1.66, and 1.98 fm 1 for the three types of form factor, respectively, as shown in Fig.1(a). Figure 1(b) shows that the Λ-dependence of these smooth cutoff cases is very mild in comparison with the sudden cut case, as expected. Form factors with similar Λ are also suggested in a study of medium-energy heavy-ion collisions [27]. This indicates that the present results have a physical meaning. 7

8 3 Pairing properties obtained by using the constructed interaction Figure 2 presents the results for (k F )andξas functions of the Fermi momentum k F, obtained by using the cutoff parameters so determined. All the four cases reproduce the results from the Bonn-B potential very well in a wide and physically relevant density range, in the sense that pairing in finite nuclei occurs near the nuclear surface [28] where density is lower than the saturation point and that the calculated range of k F almost corresponds to that of the inner crust of neutron stars [29]. This is our first conclusion. The overall slight peak shift to higher k F in (k F ) and the deviation in ξ at the highest k F are brought about by the systematic deviation in the critical density where the gap closes, between the calculations adopting bare interactions and those adopting phenomenological ones as mentioned above. The deviation at the lowest k F is due to the feature that the present model is based on the mean-field picture for the finite-density systems. Actually, in such an extremely dilute system, the effective-range approximation for free scattering holds well [17]. Next we look into the momentum dependence at k F =.9fm 1,where (k F ) becomes almost maximum. Figure 3 (a) shows φ(k). It is evident that all the four cases give the result identical to the Bonn-potential case. This demonstrates clearly the effectiveness of the interaction constructed here. This quantity peaks at k = k F as seen from Eq.(7). The width of the peak represents the reciprocal of the coherence length. Equation (7) shows that φ(k) iscomposed of (k) and the quasiparticle energy E qp (k). Figure 3 (b) graphs the former. The gaps of all the five cases are almost identical up to k 2k F, and deviations are seen only at larger momenta where E qp (k) are large and accordingly pairing is not important. This is not a trivial result since the bare interaction is more repulsive than the phenomenological ones constructed here even at the momentum region where (k) are almost identical as shown in Fig. 3 (c). Finally we turn to the dependence on r, the distance between the two nucleons 8

9 that form a Cooper pair, in order to look into the physical contents further. The gap equation, before the angle integration that results in Eq. (4), can be Fourier-transformed to the local form, (r) = v(r)φ(r)inr-space in the non-relativistic limit [3]. One can see from this expression that, assuming (r) is finite, φ(r) is pushed outwards when v(r) has a repulsive core, as the Brueckner wave functions [21,22]. This is the reason why the use of the G-matrices in the gap equation is said to cause the double counting of the short-range correlation. The r-space pair wave functions, φ(r) = 1 φ(k)j 2π 2 (kr)k 2 dk, (12) where j (kr) is a spherical Bessel function, at k F =.9fm 1 are shown in Fig. 4 (a). Appreciable differences are seen only in the core region. The coherence length, that is a typical spatial scale of pairing correlation, is about 6 fm at this k F as shown in Fig. 2 (b); this is almost one order of magnitude larger than the size of the core region. Therefore, practically we can safely use the p-p channel interaction, including the sudden cut one, constructed here for the gap equation. Figure 4 (b) shows the corresponding (r). The gaps are positive at the outside of the core and negative inside in all cases. Note here that the gap equation is invariant with respect to the overall sign inversion; we defined as (k F ) >. Their depths at the inside region reflect the heights of the repulsive core. The sudden cut case behaves somewhat differently from others due to the lack of large-momentum components. The additional staggering in the strong form factor case stems from v(p, k) =atλ= q = p k ;this gives an additional oscillatory structure in r-space with a period π/λ. In this sense, the monopole and the dipole ones are the best whereas two others are also practically usable. 9

10 4 Concluding remarks These analyses prove that some p-p interactions with strongly different shortrange behavior can give almost identical pairing properties. In other words, very details of the large-momentum interaction are irrelevant to pairing phenomena of a typical spatial scale ξ 6 fm, although some of such contributions are necessary for their quantitative description (see also Ref.[9]). This is another aspect of the short-range correlation in the gap equation. Note that the difference at short distance is reflected in a wide region in k-space as shown in Fig. 3 (c). Therefore the character of the interactions constructed here based on the RMF is not only to simulate crudely The coupling constants must be density-dependent in order to simulate the G-matrices quantitatively. the G-matrices in the DBHF calculations in the sense that the q = Hartree part reproduces the saturation but to give physical pairing properties by improving the q part. To summarize, we have phenomenologically constructed a relativistic particleparticle channel interaction which suits the gap equation for infinite nuclear matter based on the RMF. This has been accomplished by introducing a density-independent momentum-cutoff parameter to the standard RMF so as to reproduce the pairing properties obtained by adopting the Bonn-B potential and not to change the saturation property. Three types of form factor were examined. All of them as well as the sudden cut case studied in the preceding letter [23] give practically identical results. Among them the monopole and the dipole ones have the best properties. This investigation has also clarified that some interactions with strongly different short-range behavior can give practically identical pairing properties. This is another aspect of the short-range correlation in the gap equation. From a theoretical interest, now we are ready to study the polarization effects 1

11 which are known to be important in the non-relativistic calculations [14 16]. In particular, the behavior of the gap near the saturation density is to be studied. From a viewpoint of applications, on the other hand, the present method opens up a way to incorporate pairing-correlation energies consistently into the relativistic equation of state of nuclear matter, in principle. In order to extend the present study to asymmetric matter, it is mandatory to take into account isovector mesons and the non-linear self-coupling terms which are known to be necessary for quantitative calculations. They are also crucial for description of finite nuclei. These will be studied in forthcoming papers. References [1] A.B. Migdal, Sov. Phys. JETP, 13 (1961) 478. [2] Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen and E. Osnes, Phys. Rev. Lett. 77 (1996) [3] H. Kucharek and P. Ring, Z. Phys. A 339 (1991) 23. [4] F.B. Guimarães, B.V. Carlson and T. Frederico, Phys. Rev. C 54 (1996) [5] F. Matera, G. Fabbri and A. Dellafiore, Phys. Rev. C 56 (1997) 228. [6] M. Matsuzaki, Phys. Rev. C 58 (1998) 347. [7] J. Dechargé and D. Gogny, Phys. Rev. C 21 (198) [8] T. Gonzalez-Llarena, J. L. Egido, G. A. Lalazissis and P. Ring, Phys. Lett. B 379 (1996) 13. [9] A. Rummel and P. Ring, report (1996); P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193. [1] M. Matsuzaki and T. Tanigawa, Phys. Lett. B 445 (1999) 254. [11] G.F. Bertsch and H. Esbensen, Ann. Phys. 29 (1991)

12 [12] R.A. Broglia, F. De Blasio, G. Lazzari, M. Lazzari and P.M. Pizzochero, Phys. Rev. D 5 (1994) [13] Ø. Elgarøy, L. Engvik, E. Osnes, F.V. De Blasio, M. Hjorth-Jensen and G. Lazzari, Phys. Rev. D 54 (1996) [14]J.M.C.Chen,J.W.Clark,E.KrotscheckandR.A.Smith,Nucl.Phys.A451 (1986) 59. [15] J. Wambach, T.L. Ainsworth and D. Pines, Nucl. Phys. A 555 (1993) 128. [16] H.-J. Schulze, J. Cugnon, A. Lejeune, M. Baldo and U. Lombardo, Phys. Lett. B 375 (1996) 1. [17] Ø. Elgarøy and M. Hjorth-Jensen, Phys. Rev. C 57 (1998) [18] T. Papenbrock and G.F. Bertsch, Phys. Rev. C 59 (1999) 252. [19] S. Weinberg, Phys. Lett. B 251 (199) 288. [2] B.D. Serot and J.D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515. [21] M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A 515 (199) 49. [22] Ø. Elgarøy, L. Engvik, M. Hjorth-Jensen and E. Osnes, Nucl. Phys. A 64 (1996) 466. [23] T. Tanigawa and M. Matsuzaki, Prog. Theor. Phys. 12, (1999) 897. [24] R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189. [25] F.V. De Blasio, M. Hjorth-Jensen, Ø. Elgarøy, L. Engvik, G. Lazzari, M. Baldo and H.-J. Schulze, Phys. Rev. C 56 (1997) [26] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [27] P.K. Sahu, A. Hombach, W. Cassing, M. Effenberger and U. Mosel, Nucl. Phys. A 64 (1998) 493. [28] for example, M. Baldo, U. Lombardo, E. Saperstein and M. Zverev, Nucl. Phys. A 628 (1998)

13 [29] T. Takatsuka and R. Tamagaki, Prog. Theor. Phys. Suppl. 112 (1993) 27. [3] L.N. Cooper, R.L. Mills and A.M. Sessler, Phys. Rev. 114 (1959) Table 1 Classification of various relativistic approaches to nucleon-nucleon pairing Type p-h channel p-p channel(lowest order) References B DBHF bare [2] B DBHF effective range [17] P1 RMF RMF [3 6,23] P2a RMF another phenomenological [3,9] P2b RMF bare [9,1] P2b RMF effective range 13

14 Determination of Λ χ sudden cut monopole.2 dipole (a) strong Λ (fm -1 ) (kf=.9fm -1 ) (MeV) (b) sudden cut monopole dipole strong Cutoff dependence Λ (fm -1 ) Fig. 1. (a): Curvature of χ 2 in Eq.(1) with respect to the cutoff parameter Λ. (b): Λ-dependence of the pairing gap at the Fermi surface, k F =.9fm 1. 14

15 (kf) (MeV) (a) Gap at Fermi surface Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) k F (fm -1 ) Coherence length ξ (fm) (b) Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) k F (fm -1 ) Fig. 2. (a) Pairing gap at the Fermi surface, and (b) coherence length, as functions of the Fermi momentum k F, obtained by adopting the Bonn-B potential, the sudden cut in Ref.[23] and the three types of effective interaction constructed in this study. 15

16 k-space pair wave function Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) φ(k).2.1 (a) k (fm -1 ) (k) (MeV) k-space gap Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) -2 (b) k (fm -1 ) v pp (kf=.9fm -1,k) (fm 2 ) k-space p-p int. Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) -4 (c) k (fm -1 ) Fig. 3. (a) Pair wave function, (b) pairing gap, and (c) matrix element v(k F,k), as functions of the momentum k, calculated at a Fermi momentum k F =.9 fm 1, by adopting the Bonn-B potential, the sudden cut in Ref.[23] and the three types of effective interaction constructed in this study. These figures are drawn up to very large momenta in order to confirm that actually they do not contribute to the pair wave function, and consequently to the pairing gap at the Fermi surface. 16

17 φ(r) (fm -3 ) r-space pair wave function Bonn-B sudden cut(3.6) monopole(7.26) dipole(1.66) strong(1.98) (a) r (fm) 1 r-space gap (r) (MeV/fm 3 ) Bonn-B sudden cut(3.6) -4 monopole(7.26) dipole(1.66) -5 strong(1.98) (b) r (fm) Fig. 4. (a) Pair wave function, and (b) pairing gap, as functions of the distance r, calculatedatafermimomentumk F =.9 fm 1, by adopting the Bonn-B potential, the sudden cut in Ref.[23] and the three types of effective interaction constructed in this study. 17