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Particle-hole excitations in 16O in a relativistic field theory of nuclei
Volume 152B, number 5,6
PHYSICS LETTERS
14 March 1985
P A R T I C L E - H O L E EXCITATIONS IN 160 IN A RELATIVISTIC F I E L D THEORY OF NUCLEI ¢~ R.J. FURNSTAHL Institute of Theoretical Physics, Department o f Physics, Stanford University, Stanford, CA 94305, USA Received 11 December 1984
Low-lying negative parity excitations in 160 are studied in an RPA calculation based on a self-consistent, relativistic Hartree ground state. The particle-hole interaction is prescribed by the underlying meson-baryon field theory, with no additional parameters. Retardation in the meson propagators is neglected, but the full Dirac matrix structure is maintained. Reasor.able excitation spectra are obtained.
A relativistic field theory o f nuclear systems and their interactions with external probes has been developed recently at Stanford and elsewhere [1]. The theory is based on baryons and mesons as the explicit dynamical degrees of freedom and is referred to as quanturn hadrodynamics or QHD. QHD offers a complete, consistent, and unified theoretical treatment of nuclei. A thorough discussion of QHD and its consequences for nuclear physics is given in the review article by Serot and Walecka [1]. The most successful results o f QHD have been based on the relativistic Hartree approximation. By fitting a minimal number o f coupling constants and masses to bulk nuclear properties, the relativistic Hartree solutions quantitatively predict charge densities, neutron densities, and rms radii of the ground states of closed-shell nuclei. The observed s p i n - o r b i t splittings between single-particle levels and the existence of the nuclear shell model are also predicted [2]. The same calculations, when combined with the empirical N - N scattering amplitudes and the impulse approximation, yield remarkably accurate results for medium-energy p r o t o n - n u c l e u s cross sections and spin observables, with no further adjustment o f parameters [3]. The success o f the Hartree approximation in these contexts encourages us to extend this approach to the description of the excited state spectrum. * Supported in part by NSF grant PHY-81-07395. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In ref. [4], collective excitations were examined in a T h o m a s - F e r m i approximation using relativistic Hartree ground state densities. The experimental systematics o f giant resonance energies throughout the periodic table were reproduced. However, the present approach offers a more consistent and quantitative description o f excited states. We consider an RPA-like treatment o f the low-lying excited states in closed-shell nuclei based on a relativistic Hartree description o f the ground state. Retardation in the meson propagators is neglected but otherwise the full Lorentz structure is maintained. Given a QHD model, the effective p a r t i c l e - h o l e interaction is completely specified by the approximation, with no additional parameters and no constants fit to excited state properties. We emphasize that a reasonable excitation spectrum is not guaranteed in this approach. The effective interaction involves strong and often sensitive cancellations o f large potentials, unlike the usual approach with nonrelativistic interactions. In addition, the fitted constants in the Hartree approach incorporate ground-state exchange and correlation effects which may not be directly applicable to the description o f excited state properties. By calculating consistently, however, we can identify and examine the limitations o f the approximation. Two models o f quantum hadrodynamics are considered, as discussed by Serot and Walecka [ 1]. QHD-I is based on baryons and neutral vector and scalar mesons (o, w). The model QHD-II includes, in addition, 313
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charged vector (p) and pseudoscalar Or) fields in a renormalizable field theory. These theories are easily extended to include electromagnetic interactions. In QHD calculations of ground state nuclei, only doubly-magic nuclei have been considered. Neglecting the Coulomb interaction, the relativistic Hartree treatment of the ground state of 160 involves only the mesons of QHD-I. The pion and the charged rho meson do not contribute in the Hartree approximation because of the good parity and neutrality of the ground state. However, there are no such constraints in the calculation of the excited state levels. (In addition to the pion and rho meson, the spatial components of the neutral vector meson (~o) will make non-zero contributions.) We expect that any realistic interaction must ultimately include pion and rho meson exchange. By comparing the excitation spectra from the QHD-I and QHD-II models, we can identify those states which are particularly sensitive to these contributions and test our treatment of the charged mesons. Within the framework of the QHD field theory, we consider the particle-hole (polarization) propagator for a closed shell nucleus in the one-meson-exchange approximation. That is, we use a Bethe-Salpeter equation for particle-hole scattering to sum the contributions of ladder and ring diagrams to the propagator, as discussed in ref. [5]. The poles of this propagator occur at the collective excitation energies of the nucleus. Feynman rules for the meson-nucleon vertices and meson propagators in QHD-I and QHD-II are given in ref. [ 1]. For the single-particle nucleon propagator we use the Hartree propagator:
i G n ( x , y ) = ~ Ua(x) Uafy) exp [-ie~(x 0 - YO)] × [0(x0 - Y0) 0(e~ - eF) -
0 ( Y 0 - x 0 ) 0 ( e F -- e a ) ] ,
(1)
where the Ua are the (positive energy) Dirac spinors determined in a Hartree calculation of the ground state. Here a represents the quantum numbers {n, K, m, t} and (following the conventions of ref. [6]),
UnKmt(r)=_ [ i[Gn~(r)/r] ¢bKm(a) ~ ~ _[" ilnKm) ~ . - [FnK(r)/r] cb_rm (~2) ] gt = ~_ [n--ff~)] ~t, (2) 314
14 March 1985
where K is the angular quantum number, defined by Rose, which uniquely specifies j and l, and ~t is an isospinor. Use of the Hartree propagator instead of the free propagator sums the tadpole diagrams to all orders [2]. The solutions to the Bethe-Salpeter equation give a consistent and fully relativistic approximation to the nuclear structure problem. Since we work within the framework of a relativistic field theory, the level of approximation is manifest; we know what diagrams have been left out. Thus, corrections to this approach are clearly indicated (if not necessarily tractable!). If the propagator s (and consequently the particlehole scattering kernel) were frequency independent, as in the nonrelativistic calculations with static potentials, the Bethe-Salpeter equation for the particlehole propagator could be reduced to the usual RPA equations as in ref. [5]. Instead we have a complicated integral equation because of the q0 factor in the propagator. We neglect this retardation contribution by approximating 1/(q 2 - m 2) by - 1 / ( q 2 + m2). For lowlying excited states such as those considered here (Eexcited ~ m~), this approximation is probably not too important. The inclusion of retardation effects in this type of calculation will be considered in a future investigation. We note that the full four-component Dirac matrix structure is maintained. This Lorentz structure is preserved throughout the calculation, retaining the physics which is inevitably lost in any reduction to twocomponent formalisms [7]. This is particularly useful when incorporating the nuclear structure results into a consistent, relativistic theory of the interaction of nuclei with external probes (e.g. electron scattering [6]). Having neglected retardation, we can use the techniques developed for nonrelativistic shell-model calculations. In particular, the resulting RPA equations are very similar. After coupling the Hartree wave functions to total angular momentum J and isospin T, the RPA equations can be written (we use the notation of ref. [51),
{(ea - eb) -- (En - EO)} qJ}~ab) +
[ JT
t•m"
(n)
ST
(n)
Vab;tm ~kjT(lm) + Uab;lmCS~(Im)] = 0 ,
(3)
Volume 152B, number 5,6
PHYSICS LETTERS
{ - - ( e a -- eb) -- (E n - EO) ) dP(~(ab) _ ~
lm
JT
(n)
JT
(n)
[°ab;trn~afl~(lm) + Uab;tm ~bjT(lm)] = 0 ,
where
0;ca=-
(3 cont'd)
(z/'+ 1)(2T'+ 1) /b h
× [if/b) J ' T ' I u(1,2) I(am) J ' T ' )
½r (4a)
- ( - 1)I+T'(_ 1)/'a÷im+S'((lb) J ' T ' I v(1,2) I(ma) J ' T ' ) ] , JT _l~Jm_Jl_j( I ~ I [ 2 _ I [ 2 _ T , JT Uab;l m = ( . ~-- , t.,ab;m l .
(4b)
In the matrix elements the bras are Dirac adjoint wave functions. In the QHD-II model, v is the sum of contributions from the scalar (s), vector (co), rho (p), and pi (Tr) mesons: v(1,2) = (-gs2/4r0 exp ( - m s r l 2 ) / r I 2 h 2 + ~'13'2x(gto/41r) exp ( - m~orl 2)/rl 2
5 5 + 3'13'2Xl • x2(g2/4~r) e x p ( - m n r l 2 ) / r l 2 .
The description of the pion in nuclear matter in QHD has been studied by Matsui and Serot [8]• As a pseudoscalar pion-nucleon coupling is known to be inadequate in describing 7r-N phenomenology, they sought an alternative approach. They found a chiral transformation which resulted in a renormalizable pseudovector coupling. This remedied many of the drawbacks of the pseudoscalar approach. In this paper, we assume the free-particle mass and coupling constant of the pion (from scattering data) and study the magnitude of its contribution. Both pseudoscalar and pseudovector couplings are considered ,1. They are found to yield significantly different results in the nuclear medium although the free space descriptions are equivalent. In accord with previous investigations, we find that the pseudoscalar pion couples too strongly to certain states• In QHD-II, the pion-nucleon vertex is given by g ~ 7 5 r i for pseudoscalar coupling and by (fTr/mrr) 75~ri for pseudovector coupling• The coupling constants are related by gTr = ( 2M/m•) fTr ,
+ ")'~')'2h¼Xl "~ 2(g2/art) e x p ( - m p r l 2 ) / r l 2 (5)
Here r12 = Irl - r2l. Note that the spatial dependencies are simple Yukawa functions• Only the scalar and neutral vector contributions to v are considered for the QHD-I results• (Pseudoscalar coupling for the pion is given here. Pseudovector results are discussed later•) The nucleus is treated entirely consistently in this model• The nucleon, 60, p, and n masses and the lr-N coupling constant are fixed at the experimental values, leaving only four parameters. The ground state is described in the relativistic Hartree approximation with these parameters (ms, gs, gto, and go) fitted to the properties of bulk nuclear matter and the rms charge radius of 40Ca. The excited state energy levels and wave functions are calculated with the same constants, using the Hartree single-particle energies to form unperturbed levels and the Hartree spinors as the particle-hole basis. The effective interaction is determined by the basic parameters o f the theory: the meson masses and coupling constants, and by the Lorentz structure of the couplings to nucleons. Furthermore, the interaction with external probes can be described within the same minimal framework•
14 March 1985
(6)
where M is the nucleon mass. The pion propagator is the same in either case: oiO(q) = 6i/(q 2 - m , 2 + it/) - I . (7) The matrix elements in the pseudovector case are simplified by using the Dirac equation satisfied by the Hartree spinors to eliminate the ~ term. For N = Z nuclei with no Coulomb interaction included, the only significant difference between the two couplings are factors of [M*(rl)/M] [M*(r2)/M] in the matrix elements in the pseudovector case• (Other terms arei: smaller by a factor of (Ae/2M) ~ 1 where Ae is the difference between the unperturbed p - h an~, ,~,PA energies.) Since M / ) 1 4 vanes from about 1/2 in ~ e nuclear interior to 1 outside the nucleus, this can ~eptesent a substantial damping of the pion-nucleon coupling inside nuclei. In this paper, we present results for the low-lying negative parity states in 160• For simplicity, the Coulomb interaction is neglected so we generate states of good isospin T. The lpl/2 and lp3/2 states were chosen 4:1 Nonlinear couplings of the o and ~rwhich result from the transformation to pseudovector couplings are not included and represent a correction to this approach• 315
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as the hole space. In the relativistic Hartree description of 160 as described in ref. [2], only the lds/2 and 2Sl/2 single particle levels are bound. (There is a d3/2 resonance at about 2.8 MeV.) Thus, to go beyond the smallest configuration space we must be able to deal with continuum states. At present the continuum is discretized by imposing the boundary condition GnK(r=R)= 0 and restricting all integrations to a sphere of radius R (normalizing wave functions to 1 in this volume). With R chosen to be at least several times the ground state radius, the bound state wave functions are negligibly affected and we generate an orthonormal basis which is approximately complete ,2. The advantage of this approach is that bound and continuum states are treated in essentially the same manner. High-lying continuum states should be included to quantitatively describe the region of excitation energy dominated by the giant resonances, as well as to test the convergence properties o f low-lying states with respect to the size of the model space. While the convergence of energies is usually rapid for these states, the calculated transition strength can be dramatically increased by contributions from high-lying configurations, particularly in the case of highly collective states. Here we include configurations with unperturbed energies up to 40 MeV. This is sufficient to ensure stable energies for the levels of interest. In describing the nucleus in a nuclear shell model, the translational invariance of the relativistic theory is broken. This leads to spurious "center-of-mass" components in the nuclear wave functions. In the present case, the problem arises with spurious components in the T = 0, j~r = 1 - states. At present we have no consistent method to insure that the spurious state does not mix with physical states. In self-consistent nonrelativistic RPA calculations, the center-of-mass excitation is orthogonal to the other T = 0, j~r = 1 - states and appears at zero excitation energy. We do not find this result in our calculation. However, the energy of the spurious state is lowered significantly when the model space is enlarged. No convergence of this trend has been observed. In figs. 1 and 2 the energy level diagrams for the low-lying states are shown. In each figure, the first ~:2 For 160 we take R to be 12 fm, so that the first d3/2 state occurs at the resonance energy.
316
14 March 1985
160
T= I --I-
--'1-
20
I 3 "1 2s~lp~
5 3 "1 d~lp~
= ~ ' -I ~_ 2~ :
__1-
__,~: __,-
I"
2-,_ ~:
z
~511':
~: 4"
--2-
,:
° i"
,~2:
~ - --,_--~~,o_- •
,-
4-
d3 -I 21P~
9-"
--I-
o-
__.~3-
3-
- - ~ - r O"
--2-
2"
O-
I I "1 2sglp~ --2-
I0
--I
5 "I -I Ld21p2
~'--2-
---2-
Unper t ur bed
QHD-I
QHD-]I PS Pions
QHD-TI" PV Pions
Experiment
Fig. 1. Negative-parity T= 1 states in 160. Only ld,2s(lp) -1 unperturbed levels are shown.
column shows the unperturbed Hartree p a r t i c l e - h o l e energies which are taken from the ground-state calculation of 160. These levels reflect the nuclear shell structure resulting from the relativistic Hartree ap-
160 20
I 3 "1 2s~tp~
__
T:O
~i 2-
3 I°1 Id~Ip~
F
---0-
15
II-
--2" 2" I I "1 2s~lp}
I0
~
5 r "l Id-~ Ip-~
O-
--4-
--2"
s
~:' ~5-
QHD-I
--.3-
--2--4-
--,2--.0-
--OS I-
--2-
I-
s
Unperturbed
)-
I
--22-
Itl-
0
QHD-'rr PS Pions
,--2'' 3-
QHD-rr PV Pions
I
I~
3-
Experiment
Fig. 2. Negative-parity T= 0 states in 160. Only ld,2s(lp) -1 unperturbed levels are shown.
Volume 152B, number 5,6
PHYSICS LETTERS
proximation. The ordering and approximate location of the unperturbed levels agree (within 2 MeV at worst) with "experimental" particle-hole levels obtained from neighbouring nuclei [9]. The second column shows the QHD-I results (o and 60 only) and the third and fourth columns give the QHD-II levels with pseudoscalar and pseudovector pion-nucleon coupling, respectively ,3. Experimental levels from ref. [9] are given in column five. Since we can only expect to reproduce states which are primarily one-particle one-hole in nature, only these levels are included (as identified by one-nucleon transfer reactions [10]). Both QHD-I and QHD-II reproduce the low-lying clusters of T = 1 levels at energies about 2 MeV too low. The QHD-II effective interaction with the pseudovector coupling provides a superior reproduction of the relative ordering of the lowest four levels. Overall, given the simplicity of the inputs, the agreement with experiment by both models is impressive. We emphasize that even when the levels vary only slightly from their unperturbed energies, the contributions from the individual meson exchanges are not necessarily small. The giant dipole state found experimentally at ~23 MeV is identified with the highly collective state at ~20.5 MeV in the QHD-I calculation. This level contains about 55 percent of the dipole strength up to 40 MeV. The remaining strength is spread among the levels from 25 to 30 MeV and around 35 MeV. The QHD-II interaction pushes the strong dipole state up to ~22.4 MeV and spreads the remaining strength more evenly. A more sophisticated approach to the continuum is necessary to improve the description of the giant resonance region. The difference between the pseudoscalar and pseudovector spectra is greatest in pion-like states (jrr = 0 - , 2-, 4 - ) where the pseudoscalar coupling is strongly attractive. The lowest 0 - state is found about 1.4 MeV below its unperturbed energy in the calculation with pseudoscalar coupling. Experimentally, this state occurs slightly above the unperturbed energy and this result has been cited as an indication that pion condensation does not occur in 160 [11]. The pseudovector coupling damps the strong pseudoscalar contribution and results in a QHD-II prediction of the 0 - state .,3 In fig. 2, the level identified as the spurious state is marked with an s.
14 March 1985
pushed up slightly. A more dramatic example is the 2 - state which remains close to its unperturbed energy in the pseudovector case (~ 15 MeV), but with pseudoscalar coupling is pushed down below all other states. The general results for the T = 0 levels are less impressive than the T = 1 results. The QHD-I spectrum is generally consistent with experimental 1p - 1h levels except that the lowest 3 - state is pushed down only 1 MeV from its unperturbed energy. Although the addition of the charged mesons pushes this state down to its experimental energy and increases its collectivity, the QHD-II interaction appears to be too attractive in the unnatural parity states, even with pseudovector coupling. The pseudoscalar pion certainly couples too strongly, resulting in a total disruption of the ordering of the lowest levels. The difference between pseudoscalar and pseudovector coupling is particularly striking for the 0 - levels. As the size of the configuration space is increased, the pseudovector energies are relatively stable but the pseudoscalar energies decrease dramatically. Although the use of a pseudovector pion-nucleon coupling improves the spectrum in comparison to experiment, an improved treatment of the charged mesons (possibly incorporating the constraints of chiral symmetry) appears to be needed. By comparing QHD-I and QHD-II results we can identify those states which are particularly sensitive to the effects of the rho and pi mesons. These states will provide the best test of models including charged meson exchange. The most dramatic difference is seen in the 4 - states. The levels in figs. 1 and 2 have also been generated using SchrSdinger-equivalent (two-component) wave functions [12] and a nonrelativistic reduction of the effective interaction. The spectra change on a microscopic level, but the present model is too crude for one to draw conclusions about explicit relativistic effects. However, electron scattering results are more sensitive to these effects [13]. The QHD approach provides a consistent framework in which to describe the interactions of the nucleus with external probes. Using the wave functions generated in the Hartree RPA calculation, we have studied electron scattering in a consistent relativistic theory with a covariant (effective) current. These resuits, including comparisons to nonrelativistic calcula317
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PHYSICS LETTERS
tions, will be discussed in a future publication. In summary, QHD provides a minimal theoretical framework for understanding a wide range of nuclear properties. By calculating within the QHD:I and QHDII models in the relativistic Hartree approximation for the ground state, and RPA (with no retardation) for the excited states, one can generate, with just three coupling constants and one mass (fit to bulk nuclear matter): ground-state charge densities and binding energies, - the single-particle spectrum (including s p i n - o r b i t splittings), - polarized n u c l e o n - n u c l e u s scattering, reasonable excited state spectra. We emphasize again that there was no additional fitting of constants to excited state properties. Since the ground state calculation involves strong cancellations of scalar and vector potenti~ils, it is not obvious that the same values which describe the ground state will adequately describe collective excited states. Since we have a real many-body theory, we can, in principle, calculate corrections to the present results. The effects of retardation in the meson propagators must be examined and the configuration space enlarged to test convergence properties. By including more Feynman diagrams, a more realistic particlehole interaction can be derived, including the effects of relativistic correlations and other higher-order effects. -
-
318
14 March 1985
The author is pleased to thank Professor J.D. Walecka and Professor B.D. Serot for valuable discussions and comments.
References [1] B.D. Serot and J.D. Walecka, in: Advances in nuclear physics, eds. J.W. Negele and E. Vogt, to be published, and references cited in this article. [2] C.J. Horowitz and B.D. Serot, Nuel. Phys. A368 (1981) 503. [3] B.C. Clark, S. Harna, R.L. Mercer, L. Ray and B.D. Serot, Phys. Rev. Lett. 50 (1983) 1644; J.A. McNeil, J.R. Shepard and S.J. Wallace,Phys. Rev. Lett. 50 (1983) 1439. [4] R.J. Furnstahl and B.D. Serot, Stanford University preprint ITP-757 (1983), submitted to Acta Phys. Polonica. [5] A.L. Fetter and J.D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill,New York, 1971). [6] B.D. Serot, Phys. Lett. 107B (1981) 263. [7] E.D. Cooper, A.O. Gattone and M.H. Macfarlane, Phys. Lett. 130B (1983) 359. [8] T. Matsui and B.D. Serot, Ann. Phys. (NY) 144 (1982) 107. [9] F. Ajzenberg-Selove,Nucl. Phys. A375 (1982) 1. [10] M. Waroquier, G. Wenes and K. Heyde, Nucl. Phys. A41M (1983) 298. [11] S.-O. B~'ckmanand W. Weise, in: Mesons in nuclei, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979). [12] R. Brockman andW. Weise, Phys. Rev. C16 (1977) 1282. [13] J.R. Shepard, E. Rost, E.R. Siciliano and J.A. McNeil, Phys. Rev. C29 (1984) 2243.
PHYSICS LETTERS
14 March 1985
P A R T I C L E - H O L E EXCITATIONS IN 160 IN A RELATIVISTIC F I E L D THEORY OF NUCLEI ¢~ R.J. FURNSTAHL Institute of Theoretical Physics, Department o f Physics, Stanford University, Stanford, CA 94305, USA Received 11 December 1984
Low-lying negative parity excitations in 160 are studied in an RPA calculation based on a self-consistent, relativistic Hartree ground state. The particle-hole interaction is prescribed by the underlying meson-baryon field theory, with no additional parameters. Retardation in the meson propagators is neglected, but the full Dirac matrix structure is maintained. Reasor.able excitation spectra are obtained.
A relativistic field theory o f nuclear systems and their interactions with external probes has been developed recently at Stanford and elsewhere [1]. The theory is based on baryons and mesons as the explicit dynamical degrees of freedom and is referred to as quanturn hadrodynamics or QHD. QHD offers a complete, consistent, and unified theoretical treatment of nuclei. A thorough discussion of QHD and its consequences for nuclear physics is given in the review article by Serot and Walecka [1]. The most successful results o f QHD have been based on the relativistic Hartree approximation. By fitting a minimal number o f coupling constants and masses to bulk nuclear properties, the relativistic Hartree solutions quantitatively predict charge densities, neutron densities, and rms radii of the ground states of closed-shell nuclei. The observed s p i n - o r b i t splittings between single-particle levels and the existence of the nuclear shell model are also predicted [2]. The same calculations, when combined with the empirical N - N scattering amplitudes and the impulse approximation, yield remarkably accurate results for medium-energy p r o t o n - n u c l e u s cross sections and spin observables, with no further adjustment o f parameters [3]. The success o f the Hartree approximation in these contexts encourages us to extend this approach to the description of the excited state spectrum. * Supported in part by NSF grant PHY-81-07395. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
In ref. [4], collective excitations were examined in a T h o m a s - F e r m i approximation using relativistic Hartree ground state densities. The experimental systematics o f giant resonance energies throughout the periodic table were reproduced. However, the present approach offers a more consistent and quantitative description o f excited states. We consider an RPA-like treatment o f the low-lying excited states in closed-shell nuclei based on a relativistic Hartree description o f the ground state. Retardation in the meson propagators is neglected but otherwise the full Lorentz structure is maintained. Given a QHD model, the effective p a r t i c l e - h o l e interaction is completely specified by the approximation, with no additional parameters and no constants fit to excited state properties. We emphasize that a reasonable excitation spectrum is not guaranteed in this approach. The effective interaction involves strong and often sensitive cancellations o f large potentials, unlike the usual approach with nonrelativistic interactions. In addition, the fitted constants in the Hartree approach incorporate ground-state exchange and correlation effects which may not be directly applicable to the description o f excited state properties. By calculating consistently, however, we can identify and examine the limitations o f the approximation. Two models o f quantum hadrodynamics are considered, as discussed by Serot and Walecka [ 1]. QHD-I is based on baryons and neutral vector and scalar mesons (o, w). The model QHD-II includes, in addition, 313
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charged vector (p) and pseudoscalar Or) fields in a renormalizable field theory. These theories are easily extended to include electromagnetic interactions. In QHD calculations of ground state nuclei, only doubly-magic nuclei have been considered. Neglecting the Coulomb interaction, the relativistic Hartree treatment of the ground state of 160 involves only the mesons of QHD-I. The pion and the charged rho meson do not contribute in the Hartree approximation because of the good parity and neutrality of the ground state. However, there are no such constraints in the calculation of the excited state levels. (In addition to the pion and rho meson, the spatial components of the neutral vector meson (~o) will make non-zero contributions.) We expect that any realistic interaction must ultimately include pion and rho meson exchange. By comparing the excitation spectra from the QHD-I and QHD-II models, we can identify those states which are particularly sensitive to these contributions and test our treatment of the charged mesons. Within the framework of the QHD field theory, we consider the particle-hole (polarization) propagator for a closed shell nucleus in the one-meson-exchange approximation. That is, we use a Bethe-Salpeter equation for particle-hole scattering to sum the contributions of ladder and ring diagrams to the propagator, as discussed in ref. [5]. The poles of this propagator occur at the collective excitation energies of the nucleus. Feynman rules for the meson-nucleon vertices and meson propagators in QHD-I and QHD-II are given in ref. [ 1]. For the single-particle nucleon propagator we use the Hartree propagator:
i G n ( x , y ) = ~ Ua(x) Uafy) exp [-ie~(x 0 - YO)] × [0(x0 - Y0) 0(e~ - eF) -
0 ( Y 0 - x 0 ) 0 ( e F -- e a ) ] ,
(1)
where the Ua are the (positive energy) Dirac spinors determined in a Hartree calculation of the ground state. Here a represents the quantum numbers {n, K, m, t} and (following the conventions of ref. [6]),
UnKmt(r)=_ [ i[Gn~(r)/r] ¢bKm(a) ~ ~ _[" ilnKm) ~ . - [FnK(r)/r] cb_rm (~2) ] gt = ~_ [n--ff~)] ~t, (2) 314
14 March 1985
where K is the angular quantum number, defined by Rose, which uniquely specifies j and l, and ~t is an isospinor. Use of the Hartree propagator instead of the free propagator sums the tadpole diagrams to all orders [2]. The solutions to the Bethe-Salpeter equation give a consistent and fully relativistic approximation to the nuclear structure problem. Since we work within the framework of a relativistic field theory, the level of approximation is manifest; we know what diagrams have been left out. Thus, corrections to this approach are clearly indicated (if not necessarily tractable!). If the propagator s (and consequently the particlehole scattering kernel) were frequency independent, as in the nonrelativistic calculations with static potentials, the Bethe-Salpeter equation for the particlehole propagator could be reduced to the usual RPA equations as in ref. [5]. Instead we have a complicated integral equation because of the q0 factor in the propagator. We neglect this retardation contribution by approximating 1/(q 2 - m 2) by - 1 / ( q 2 + m2). For lowlying excited states such as those considered here (Eexcited ~ m~), this approximation is probably not too important. The inclusion of retardation effects in this type of calculation will be considered in a future investigation. We note that the full four-component Dirac matrix structure is maintained. This Lorentz structure is preserved throughout the calculation, retaining the physics which is inevitably lost in any reduction to twocomponent formalisms [7]. This is particularly useful when incorporating the nuclear structure results into a consistent, relativistic theory of the interaction of nuclei with external probes (e.g. electron scattering [6]). Having neglected retardation, we can use the techniques developed for nonrelativistic shell-model calculations. In particular, the resulting RPA equations are very similar. After coupling the Hartree wave functions to total angular momentum J and isospin T, the RPA equations can be written (we use the notation of ref. [51),
{(ea - eb) -- (En - EO)} qJ}~ab) +
[ JT
t•m"
(n)
ST
(n)
Vab;tm ~kjT(lm) + Uab;lmCS~(Im)] = 0 ,
(3)
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PHYSICS LETTERS
{ - - ( e a -- eb) -- (E n - EO) ) dP(~(ab) _ ~
lm
JT
(n)
JT
(n)
[°ab;trn~afl~(lm) + Uab;tm ~bjT(lm)] = 0 ,
where
0;ca=-
(3 cont'd)
(z/'+ 1)(2T'+ 1) /b h
× [if/b) J ' T ' I u(1,2) I(am) J ' T ' )
½r (4a)
- ( - 1)I+T'(_ 1)/'a÷im+S'((lb) J ' T ' I v(1,2) I(ma) J ' T ' ) ] , JT _l~Jm_Jl_j( I ~ I [ 2 _ I [ 2 _ T , JT Uab;l m = ( . ~-- , t.,ab;m l .
(4b)
In the matrix elements the bras are Dirac adjoint wave functions. In the QHD-II model, v is the sum of contributions from the scalar (s), vector (co), rho (p), and pi (Tr) mesons: v(1,2) = (-gs2/4r0 exp ( - m s r l 2 ) / r I 2 h 2 + ~'13'2x(gto/41r) exp ( - m~orl 2)/rl 2
5 5 + 3'13'2Xl • x2(g2/4~r) e x p ( - m n r l 2 ) / r l 2 .
The description of the pion in nuclear matter in QHD has been studied by Matsui and Serot [8]• As a pseudoscalar pion-nucleon coupling is known to be inadequate in describing 7r-N phenomenology, they sought an alternative approach. They found a chiral transformation which resulted in a renormalizable pseudovector coupling. This remedied many of the drawbacks of the pseudoscalar approach. In this paper, we assume the free-particle mass and coupling constant of the pion (from scattering data) and study the magnitude of its contribution. Both pseudoscalar and pseudovector couplings are considered ,1. They are found to yield significantly different results in the nuclear medium although the free space descriptions are equivalent. In accord with previous investigations, we find that the pseudoscalar pion couples too strongly to certain states• In QHD-II, the pion-nucleon vertex is given by g ~ 7 5 r i for pseudoscalar coupling and by (fTr/mrr) 75~ri for pseudovector coupling• The coupling constants are related by gTr = ( 2M/m•) fTr ,
+ ")'~')'2h¼Xl "~ 2(g2/art) e x p ( - m p r l 2 ) / r l 2 (5)
Here r12 = Irl - r2l. Note that the spatial dependencies are simple Yukawa functions• Only the scalar and neutral vector contributions to v are considered for the QHD-I results• (Pseudoscalar coupling for the pion is given here. Pseudovector results are discussed later•) The nucleus is treated entirely consistently in this model• The nucleon, 60, p, and n masses and the lr-N coupling constant are fixed at the experimental values, leaving only four parameters. The ground state is described in the relativistic Hartree approximation with these parameters (ms, gs, gto, and go) fitted to the properties of bulk nuclear matter and the rms charge radius of 40Ca. The excited state energy levels and wave functions are calculated with the same constants, using the Hartree single-particle energies to form unperturbed levels and the Hartree spinors as the particle-hole basis. The effective interaction is determined by the basic parameters o f the theory: the meson masses and coupling constants, and by the Lorentz structure of the couplings to nucleons. Furthermore, the interaction with external probes can be described within the same minimal framework•
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(6)
where M is the nucleon mass. The pion propagator is the same in either case: oiO(q) = 6i/(q 2 - m , 2 + it/) - I . (7) The matrix elements in the pseudovector case are simplified by using the Dirac equation satisfied by the Hartree spinors to eliminate the ~ term. For N = Z nuclei with no Coulomb interaction included, the only significant difference between the two couplings are factors of [M*(rl)/M] [M*(r2)/M] in the matrix elements in the pseudovector case• (Other terms arei: smaller by a factor of (Ae/2M) ~ 1 where Ae is the difference between the unperturbed p - h an~, ,~,PA energies.) Since M / ) 1 4 vanes from about 1/2 in ~ e nuclear interior to 1 outside the nucleus, this can ~eptesent a substantial damping of the pion-nucleon coupling inside nuclei. In this paper, we present results for the low-lying negative parity states in 160• For simplicity, the Coulomb interaction is neglected so we generate states of good isospin T. The lpl/2 and lp3/2 states were chosen 4:1 Nonlinear couplings of the o and ~rwhich result from the transformation to pseudovector couplings are not included and represent a correction to this approach• 315
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as the hole space. In the relativistic Hartree description of 160 as described in ref. [2], only the lds/2 and 2Sl/2 single particle levels are bound. (There is a d3/2 resonance at about 2.8 MeV.) Thus, to go beyond the smallest configuration space we must be able to deal with continuum states. At present the continuum is discretized by imposing the boundary condition GnK(r=R)= 0 and restricting all integrations to a sphere of radius R (normalizing wave functions to 1 in this volume). With R chosen to be at least several times the ground state radius, the bound state wave functions are negligibly affected and we generate an orthonormal basis which is approximately complete ,2. The advantage of this approach is that bound and continuum states are treated in essentially the same manner. High-lying continuum states should be included to quantitatively describe the region of excitation energy dominated by the giant resonances, as well as to test the convergence properties o f low-lying states with respect to the size of the model space. While the convergence of energies is usually rapid for these states, the calculated transition strength can be dramatically increased by contributions from high-lying configurations, particularly in the case of highly collective states. Here we include configurations with unperturbed energies up to 40 MeV. This is sufficient to ensure stable energies for the levels of interest. In describing the nucleus in a nuclear shell model, the translational invariance of the relativistic theory is broken. This leads to spurious "center-of-mass" components in the nuclear wave functions. In the present case, the problem arises with spurious components in the T = 0, j~r = 1 - states. At present we have no consistent method to insure that the spurious state does not mix with physical states. In self-consistent nonrelativistic RPA calculations, the center-of-mass excitation is orthogonal to the other T = 0, j~r = 1 - states and appears at zero excitation energy. We do not find this result in our calculation. However, the energy of the spurious state is lowered significantly when the model space is enlarged. No convergence of this trend has been observed. In figs. 1 and 2 the energy level diagrams for the low-lying states are shown. In each figure, the first ~:2 For 160 we take R to be 12 fm, so that the first d3/2 state occurs at the resonance energy.
316
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160
T= I --I-
--'1-
20
I 3 "1 2s~lp~
5 3 "1 d~lp~
= ~ ' -I ~_ 2~ :
__1-
__,~: __,-
I"
2-,_ ~:
z
~511':
~: 4"
--2-
,:
° i"
,~2:
~ - --,_--~~,o_- •
,-
4-
d3 -I 21P~
9-"
--I-
o-
__.~3-
3-
- - ~ - r O"
--2-
2"
O-
I I "1 2sglp~ --2-
I0
--I
5 "I -I Ld21p2
~'--2-
---2-
Unper t ur bed
QHD-I
QHD-]I PS Pions
QHD-TI" PV Pions
Experiment
Fig. 1. Negative-parity T= 1 states in 160. Only ld,2s(lp) -1 unperturbed levels are shown.
column shows the unperturbed Hartree p a r t i c l e - h o l e energies which are taken from the ground-state calculation of 160. These levels reflect the nuclear shell structure resulting from the relativistic Hartree ap-
160 20
I 3 "1 2s~tp~
__
T:O
~i 2-
3 I°1 Id~Ip~
F
---0-
15
II-
--2" 2" I I "1 2s~lp}
I0
~
5 r "l Id-~ Ip-~
O-
--4-
--2"
s
~:' ~5-
QHD-I
--.3-
--2--4-
--,2--.0-
--OS I-
--2-
I-
s
Unperturbed
)-
I
--22-
Itl-
0
QHD-'rr PS Pions
,--2'' 3-
QHD-rr PV Pions
I
I~
3-
Experiment
Fig. 2. Negative-parity T= 0 states in 160. Only ld,2s(lp) -1 unperturbed levels are shown.
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proximation. The ordering and approximate location of the unperturbed levels agree (within 2 MeV at worst) with "experimental" particle-hole levels obtained from neighbouring nuclei [9]. The second column shows the QHD-I results (o and 60 only) and the third and fourth columns give the QHD-II levels with pseudoscalar and pseudovector pion-nucleon coupling, respectively ,3. Experimental levels from ref. [9] are given in column five. Since we can only expect to reproduce states which are primarily one-particle one-hole in nature, only these levels are included (as identified by one-nucleon transfer reactions [10]). Both QHD-I and QHD-II reproduce the low-lying clusters of T = 1 levels at energies about 2 MeV too low. The QHD-II effective interaction with the pseudovector coupling provides a superior reproduction of the relative ordering of the lowest four levels. Overall, given the simplicity of the inputs, the agreement with experiment by both models is impressive. We emphasize that even when the levels vary only slightly from their unperturbed energies, the contributions from the individual meson exchanges are not necessarily small. The giant dipole state found experimentally at ~23 MeV is identified with the highly collective state at ~20.5 MeV in the QHD-I calculation. This level contains about 55 percent of the dipole strength up to 40 MeV. The remaining strength is spread among the levels from 25 to 30 MeV and around 35 MeV. The QHD-II interaction pushes the strong dipole state up to ~22.4 MeV and spreads the remaining strength more evenly. A more sophisticated approach to the continuum is necessary to improve the description of the giant resonance region. The difference between the pseudoscalar and pseudovector spectra is greatest in pion-like states (jrr = 0 - , 2-, 4 - ) where the pseudoscalar coupling is strongly attractive. The lowest 0 - state is found about 1.4 MeV below its unperturbed energy in the calculation with pseudoscalar coupling. Experimentally, this state occurs slightly above the unperturbed energy and this result has been cited as an indication that pion condensation does not occur in 160 [11]. The pseudovector coupling damps the strong pseudoscalar contribution and results in a QHD-II prediction of the 0 - state .,3 In fig. 2, the level identified as the spurious state is marked with an s.
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pushed up slightly. A more dramatic example is the 2 - state which remains close to its unperturbed energy in the pseudovector case (~ 15 MeV), but with pseudoscalar coupling is pushed down below all other states. The general results for the T = 0 levels are less impressive than the T = 1 results. The QHD-I spectrum is generally consistent with experimental 1p - 1h levels except that the lowest 3 - state is pushed down only 1 MeV from its unperturbed energy. Although the addition of the charged mesons pushes this state down to its experimental energy and increases its collectivity, the QHD-II interaction appears to be too attractive in the unnatural parity states, even with pseudovector coupling. The pseudoscalar pion certainly couples too strongly, resulting in a total disruption of the ordering of the lowest levels. The difference between pseudoscalar and pseudovector coupling is particularly striking for the 0 - levels. As the size of the configuration space is increased, the pseudovector energies are relatively stable but the pseudoscalar energies decrease dramatically. Although the use of a pseudovector pion-nucleon coupling improves the spectrum in comparison to experiment, an improved treatment of the charged mesons (possibly incorporating the constraints of chiral symmetry) appears to be needed. By comparing QHD-I and QHD-II results we can identify those states which are particularly sensitive to the effects of the rho and pi mesons. These states will provide the best test of models including charged meson exchange. The most dramatic difference is seen in the 4 - states. The levels in figs. 1 and 2 have also been generated using SchrSdinger-equivalent (two-component) wave functions [12] and a nonrelativistic reduction of the effective interaction. The spectra change on a microscopic level, but the present model is too crude for one to draw conclusions about explicit relativistic effects. However, electron scattering results are more sensitive to these effects [13]. The QHD approach provides a consistent framework in which to describe the interactions of the nucleus with external probes. Using the wave functions generated in the Hartree RPA calculation, we have studied electron scattering in a consistent relativistic theory with a covariant (effective) current. These resuits, including comparisons to nonrelativistic calcula317
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tions, will be discussed in a future publication. In summary, QHD provides a minimal theoretical framework for understanding a wide range of nuclear properties. By calculating within the QHD:I and QHDII models in the relativistic Hartree approximation for the ground state, and RPA (with no retardation) for the excited states, one can generate, with just three coupling constants and one mass (fit to bulk nuclear matter): ground-state charge densities and binding energies, - the single-particle spectrum (including s p i n - o r b i t splittings), - polarized n u c l e o n - n u c l e u s scattering, reasonable excited state spectra. We emphasize again that there was no additional fitting of constants to excited state properties. Since the ground state calculation involves strong cancellations of scalar and vector potenti~ils, it is not obvious that the same values which describe the ground state will adequately describe collective excited states. Since we have a real many-body theory, we can, in principle, calculate corrections to the present results. The effects of retardation in the meson propagators must be examined and the configuration space enlarged to test convergence properties. By including more Feynman diagrams, a more realistic particlehole interaction can be derived, including the effects of relativistic correlations and other higher-order effects. -
-
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The author is pleased to thank Professor J.D. Walecka and Professor B.D. Serot for valuable discussions and comments.
References [1] B.D. Serot and J.D. Walecka, in: Advances in nuclear physics, eds. J.W. Negele and E. Vogt, to be published, and references cited in this article. [2] C.J. Horowitz and B.D. Serot, Nuel. Phys. A368 (1981) 503. [3] B.C. Clark, S. Harna, R.L. Mercer, L. Ray and B.D. Serot, Phys. Rev. Lett. 50 (1983) 1644; J.A. McNeil, J.R. Shepard and S.J. Wallace,Phys. Rev. Lett. 50 (1983) 1439. [4] R.J. Furnstahl and B.D. Serot, Stanford University preprint ITP-757 (1983), submitted to Acta Phys. Polonica. [5] A.L. Fetter and J.D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill,New York, 1971). [6] B.D. Serot, Phys. Lett. 107B (1981) 263. [7] E.D. Cooper, A.O. Gattone and M.H. Macfarlane, Phys. Lett. 130B (1983) 359. [8] T. Matsui and B.D. Serot, Ann. Phys. (NY) 144 (1982) 107. [9] F. Ajzenberg-Selove,Nucl. Phys. A375 (1982) 1. [10] M. Waroquier, G. Wenes and K. Heyde, Nucl. Phys. A41M (1983) 298. [11] S.-O. B~'ckmanand W. Weise, in: Mesons in nuclei, eds. M. Rho and D. Wilkinson (North-Holland, Amsterdam, 1979). [12] R. Brockman andW. Weise, Phys. Rev. C16 (1977) 1282. [13] J.R. Shepard, E. Rost, E.R. Siciliano and J.A. McNeil, Phys. Rev. C29 (1984) 2243.