- Email: [email protected]
Nuclear matter with Δ degrees of freedom
Nuclear Physics A519 (1990) 269c-278c North-Holland
269c
NUCLEAR MATTER WITH A DEGREES OF FREEDOM
Lidia S. FERREIRA Centro de F~sica, da Mat~ria Condensada, Avenida Gama Pinto 2, 1699 Lisboa Portugal Nuclear matter is studied in the framework of the Bethe Brueckner Goldstone theory with the explicit inclusion of the A isobar degrees of freedom. The calculation of the two-body reaction matrix is considered in full coupled channel calculation with potentials contalning all possible 1rNA and 7rAA couplings. 1. INTRODUCTION It is well established by now that the classical many-body perturbation expansion, the Brueckner 1) theory and the variational method2), yield results for the properties of nuclear matter that are in very good agreement with each other 3) but far from the empirical value. The use of different realistic N-N interactions, that fit the available data on the properties of the two-body system leads to nuclear matter saturation values, which lie as a function of density on a band, the Coester band3), being their differences mainly related to the amount of tensor force contained in the corresponding potential. Recently, relativistic effects have been included4). The new results might define a new Coester line approaching the empirical data. Many effects can be discussed that might change significatively the results.
One
possible extension of the theory is to consider excitations of the nucleon like the A(3,3) resonance which has rather high excitation probability. In fact the A isobar is already a very important feature for the NN force s) since the intermediate range attraction, which is conveniently parametrized in OBEP by a a boson~ can be partially replaced by a potential that couples NN states to NA and AA ones. The transition potentials between these states can be obtained from a n o n - relativistic reduction of the OPE Feynman amplitude for such processes with lrNA and r A A couplings. The coupled-channel approach can then be used to construct realistic NN potentials with A degrees of freedom. However one has in these models to compensate phenomenologically for the intermediate range part of the NN interaction, besides the OPE, which could simulate the residual effect of some cross-box diagrams whose structure cannot be reproduced this way. This iterated potential in the coupled channel, will also give the most important part of the T P E 3-nucleon attraction
0375-9474190/$03.50 © 1990 - Elsevicr Science Publishers B.V. (North-Holland)
270c
L.S. Ferreira / Nuclear matter with A degrees of freedom
in 3-body clusters, and repulsive 3-body contributions. They generate then effective many body forces 6) which might possibly be the explanation of the discrepancy between theory and experiment. The effect of the explicit inclusion of the virtual A states on the saturation properties of nuclear matter has been studied by many authors. Estimations were made 7) of the contribution of A excitations on the binding energy of nuclear matter, with an effective potential to take care of the NA transitions. Within the coupled-channel approach, Day and Coester 3) introduced the A couplings in the 1S0 and 3P1 channels and found that, despite the large shifts in the saturation energy and density, the saturation point still lies within the Coester band. However neither the N N - ~ A transitions or the average potential felt by the A in nuclear matter were included which are supposed to be relevant. A complete calculation including all possible NA and A A channels in the two-body interaction was performed by Niephaus, Gari and SommerS), which used a modified Reid potential. Concerning the importance of three-body forces, calculations with phenomenological three-body potentials, derived from these forces through the explicit introduction of non nucleonic degrees of freedom, were performedg). The variational calculation of ref. 10) included a three-body phenomenological potential, adjusted to reproduce the binding energies of the three nucleon systems and nuclear matter. The ring series with A intermediate states was studied by the authors of ref11). Concerning relativistic effects, the Dirac-Brueckner equations were solved with an extended Bonn potential 12), with explicit 27r and 27rp exchange replacing the ~ bosom The results confirmed the ones obtained with the original Bonn potential. The only relativistic full coupled channel calculation with A degrees of freedom was done by terHaar and Malfliet 4). The latter solved the Dirac-Brueckner equation with a two body potential containing NA transitions, and determined from a covariant 3-dimensional reduction of a coupled-channel Bethe-Salpeter equation. All NN scattering data for low and medium energies was determined with few parameters. We will discuss next potentials with explicit isobar transitions, and the modifications in the m a n y - b o d y theory due to those excitations. 2. NUCLEON-NUCLEON POTENTIALS WITH A DEGREES OF F R E E D O M Effective potentials ~) were constructed, within the Feshbach first order method, in order to account for NA and A A transitions in nuclear matter calculations. Within the non-relativistic coupled channel approach, various groups 3,s,13) modified a standard NN
L.S. Ferreira / Nuclear matter with ix degrees of freedom interaction, in order to include A effects.
271c
Recently, a new realistic NN potential was
presented 14) the Argonne v2s, which explicitly includes the A degrees of freedom through all possible ~rNA and 7rAA couplings and the NA and A A scattering channels with a total of 28 operators. The potential gives an excellent fit to the most recent data on deuteron properties and n-p scattering, and has a structure particularly suited for nuclear matter calculations. The authors also presented a conventional NN potential, the v14 model, which can provide a standard of comparison. All allowed OPE couplings represented in Fig. 1 and the major effect of the p in reducing OPE at short distances were included. The crossed-box T P E diagrams d)-h) of Fig. 1 are simulated approximately, and besides the r N A and zrAA coupling constants, taken from the Chew-Low theory and quark model, respectively, there are no more free parameters then the ones contained in the v14 model. As a requirement of the quark model, it also contains a repulsive core in the NA and AA channels comparable to the ones in the NN channels. The v2s can be considered a minimal model with A degrees of freedom and with as few parameters as the data requires.
.....
:2-:]
i / X.~
I--il )q
FIGURE 1 Two pion exchange processes contributing to the nucleon-nucleon interaction. The full, double and dashed lines, represent the nucleon, the delta and the pion, respectively. The v2s is ideal to introduce many-body forces and study the dispersion and Pauli principle effects. Unfortunately the complexity of the potential, due to the increased number of coupled channels allowed by the A, as it can be seen in table 1, makes the nuclear matter calculation numerically quite involved. The calculations can however be quite simplified if a separable potential is used. Since the techniques to obtain separable representations of a given realistic local potentials have reached high standards of perfectionXS), it is possible to reproduce accurately the on and off the energy shell properties of the original potential,
272c
LIS. Ferreira / N u c l e a r m a t t e r with A degrees o f f r e e d o m
and safely replace the original potential by its separable representation. However care has to be taken with the opening of the NA and A A thresholds. The original potential is then represented for each partial wave by the separable form, N
,Ua
CO' ( q , q ' )
=
a
~
v~,(q)A~vl~'j(q)
I
(1)
i,j=l
where a --- { J S T }
specifies the total angular momentum, spin and isospin for a given
channel . The functions v~.~ are the form factors, and the matrix A~ characterizes the strengths and is independent of the relative momenta q, ql. In principle eq. (1) is exact for an infinite rank, however by a convenient choice of the form factors, it is possible to describe some of the most well established nucleon-nucleon interactions with only a few terms in the expansion. 80
3s, 40
NN
NA
AA
1S o
aD o
15 o SD o
1D 2
sS~aD2SD2aG2
5S21D2SD~SG 2
3.00 ap3
ap0 ap~ 5p~ aF:
3p07F0 apl 7F1
ap~-a F 2
ap2 ap~ aF2 s F2
ap2 rp~ aF2 ~F~ TH 2
3Sa-aD 1
3S~3D17D17G1
aD z
aD2~DaTG2
0
(,(3 8 4
200
400
600
Etab (MeV)
TABLE 1 NN, NA and AA partial waves (2S+llj) for the nucleon-nucleon scattering.
FIGURE 2 The sS1 _3 D1 phase shifts calculated with the separable potential (full line), compared with the ones obtained from the original local vzs potential (full dots) as a function energy.
An efficient procedure 16) consists in taking for the functions vZi the deuteron wave function and stationary scattering states of the vzs potential at definite energies, multiplied by the original local potential. This is, in fact, a generalization of the so called EST method is) in order to include thresholds. The on and off the energy shell behaviour of the scattering matrix was accurately reproduced in each partial wave by this separable form. In order to illustrate this point, we show in Fig. 2 a comparison of the scattering phase shifts calculated with the original vzs and its separable representation for the zS1 _3 D1
L.S. Ferreira / Nuclear matter with
a
degrees of freedom
273c
channel. In Fig. 3 a similar comparison is made for the off the energy shell behaviour of the scattering matrix of the 1S0 channel, by plotting the Kowalski-Noyes function below and above the N/X threshold. As it can be seen the behaviour of the local potential is precisely described over a sufficiently large energy range, required for a nuclear matter calculation. Therefore the original potential can be safely replaced by its separable representation without loss of precision.
iI
2! • 0
"~
_
E'lob = 120 MeV i
=~.,~-
5Do ~""~'~_
~f
LL
E'lob = 850 ,
o
i
a .4
,
II] .6
,
, .8
, I.
MeV i
, 1.2
EIob (GeV) FIGURE 3 Kowalski-Noyes function in the NN and N/X sectors of the 1So channel. The full dots and full line correspond to the exact and separable calculations, respectively.
3. THE BETHE BRUECKNER GOLDSTONE WITH A DEGREES OF FREEDOM The extension of the Bethe Brueckner Goldstone BBG at the two-hole level to include explicitly A couplings leads to the calculation of a modified two-body reaction matrix G, with respect to the one obtained from a NN potential without A degrees of freedom. One has to solve a set of coupled equations, that in the momentum representation take the form,
< k'lk'2;71'lG(w)]kak2;71 > = < k'ik'2;71'lvlk~k2;z I > + <
k3k,
<
(2)
~ - E¢,(k3, k4)
where the label y indicates the type of particles involved in the channel, namely y = {NN, N/X, AA}, each k labels the momentum, spin and isospin of the corresponding particle, v is the bare potential, the v2s, and Qn is the Pauli operator, which requires the nucleon momenta to be outside the Fermi sea, but allows the/X to have any momentum. The quantity En,, is the sum of the two single particle energies e(k) inside nuclear matter, present in the channel 7/", defined by eN( k ) = k2 / 2 m + UN( k ), eA ( k ) = ( /X - m ) + k 2/2/X + UA (k),
L.S. Ferreira / Nuclearmatter with ix degreesof freedom
274c
being U the self-consistent single particle potential for the nucleon and the A. The nucleon single particle potential can be taken as
UN(k~) = ~
< klkzNNlG(~)lkak2NN >
(3)
with ~a = eN(kl) + eN(kz). In the case of the "continuous choice", the potential will be continuous over the Fermi surface and Eq. (3) is valid for all values of kl. If the "gap choice"is adopted, Eq. (3) holds only only for momenta below kF, and above kF the nucleon single particle potential is assumed to vanish. It can be proved that one is then able to go further than the two-body cluster contribution and include higher order clusters in the BBG expansion, when the "continuous choice" is adopted. Analogously one can define the average potential felt by the A as
U~(kl) =
~
< klk2NAIG(~v)Iklk2NA >
(4)
k2 <_kF
where, according to table 1, only the channels with total isotopic spin T = 1 actually contribute. A partial wave expansion of Eq. (2) has to be m a d e , and in order not to mix different partial waves, the Pauli operator in the energy denominator must be replaced by its angle average. This approximation is expected to be reasonable. From the structure of Eqs. (2), (3) and (4) there is an implicit double self-consistency condition for the nucleon and A single particle potentials, which can be solved by iteration. Having done so, and obtained the reaction matrix, since the total potential energy at the two-hole level is given by
B=~
1
~
(5)
kxk2
where the single particle momenta are inside the Fermi sea, the saturation curve,that is, the binding energy per nucleon as a function of density, is immediately determined. From the structure of Eq. 2, we see that the effect of introducing explicitly isobar degrees of freedom has changed the reaction matrix in two aspects. The first one concerns the presence of the Pauli principle for the nucleons that will exclude certain NA states. The A A states will be unaffected since Q a a = 1. This is named as the Pauli effect. Secondly, the energy denominators for the intermediate NA and A A states are increased, the propagator is modified. This is the dispersion effect. Both effects will reduce the effective coupling to the NA and A A states in comparison with a calculation with an interaction
L.S. Ferreira / Nuclear matter with A degrees of freedom
275c
where these degrees of freedom are not explicit but are integrated phenomenologically in the two-body interaction. It is expected that they will also vary with the density. The solution of Eq. (3) with a separable interaction, Eq. (1), brings significant simplifications. Like in the case of the two-body problem, it becomes reduced to a matrix inversion. In a full coupled channel approach, with A degrees of freedom, this certainly will reduce the computation time.
4. THE EQUATION OF STATE WITH A DEGREES OF FREEDOM The self-consistent Eqs. (2) and (3) were solved with U~ -- 0 by Day and Coester 3) for the couplings from NN to NA states in S and P waves and no AA transitions within the "gap choice". Similar calculation was performed by Niephaus, Gari and Sommer but with a modified Reid potential with extra operators to express the transitions to NA and AA states, refitted to the scattering data. All possible partial waves up to L = 4 were included. The v28 fits more modern data, so there will be an improvement in relation to the modified Reid potential. The use of its separable representation, will also make the calculation in the "continuous choice" simpler. The self-consistent Eqs.
(2-4) were solved iv) by an
iterative procedure in the "continuous choice", with the separable representation 16) of the v2s. The starting point was a parabolic form for the single particle nucleon potential. The A potential was assumed to produce only a constant shift of 40 MeV in the A mass. This is the value obtained near saturation by the authors of reflS). All the channels displayed in table 1, and in addition the single channels 1F3,3F~,l G4 and 3G4 taken from the Argonne v14 were included. The saturation curve obtained from Eq. (5), E = B / A , is reported in Fig. 4 (full line) in comparison with the results obtained from the Argonne v14 potential (dashed line) in the "continuous choice". Since the two potentials have the same structure for the NN part, and fit the same scattering data~ a comparison of the two shows the effect of the isobar. The saturation density is shifted down to a value smaller than the empirical one, while at saturation, nuclear matter is a few MeV less bound. By switching off the A potential the binding energy is increased about 2 MeV. The remaining discrepancy between the two potentials comes from the dispersion effects mainly, and from the Pauli principle. The corresponding saturation point still lies within the Coester band, however pushed towards the lower border of it as it can be seen from Fig. 5. The same calculations in the framework of the "gap choice" lies 1-2 MeV above the
276c
L.S. Ferreira / Nuclear matter with zx degrees o f freedom
one reported in the work of Niephaus et al. s), being this discrepancy essentially due to the use of different potentials.
J
i
I
i
7"~
!
-5
.'i
/ //
*,,..,,×
//"
"", ,,,
-1{
l
".k....
..s
\,,
-,.
, .
.j . .I
/1 "z
i /
;
//
/ 1I
I
1.0
I
1
1.2
I
1.~, Kf {fm -I}
,
I
1.6
1!8
FIGURE 4 Binding energy per nucleon as a function of Fermi momentum in the "continuous choice" for the v2s (full llne) and v]4 (dashed line) models. The dotted line indicates the "empirical" saturation curve. The dash-dotted and dash-double-dotted lines are the relativistic calculation without and with A, ref4), respectively. The dot-double- dashed is refl°). Concerning the effect of three-body forces, the authors of refJ °) included a phenomenological three body force adjusted to reproduce the properties of the three nucleon system. They found that the saturation density is reduced, and the equation of state becomes much stiffer. However they were not able to fit simultaneously the binding energy of the three-body system and nuclear matter. Their estimation of three-body effects for the v14 is also shown in Fig. 4. It should be noted that a Brueckner calculation with explicit isobar degrees of freedom contains the dominant part of the repulsion due to the threebody force. In fact, one of the repulsive diagrams generated by the explicit introduction of NA transitions that contribute to the effective three-body force is naturally included by the self consistency. Therefore in order to compare ref. 17) with refJ °) one should add the attractive part of the three-body force. At kF = 1.3frn -1 the expectation value the attractive part of the force amounts to -4.031°). It is interesting to compare the results of ref.17) with the ones of terHaar and Malfliet*). Their results are also shown in Fig. 4 and 5. A quite close agreement existe with the v2s. Without the A transitions the relativistic effects seem to be strong and shift the saturation point to a lower density. With them, the saturation energy is shifted even to a lower
L.S. Ferreira
/
Nuclear matter with ix degrees of freedom
277c
density, so the agreement can be partially explained by noticing that at low densities the relativistic corrections are expected to be smaller. Another relevant quantity in the many-body problem which is changed with the introduction of A degrees of freedom is the total defect parameter ~. It is now split into three components. The defect parameter gNN expresses the content of two particles-two holes in the ground state, while ~NZX and nz,,x give an indication of the percentage of A particles present in the ground state. When in conventional calculations one finds values for ~ at kF = 1.4 fm -1 of order of .12 with the A effect it becomes .324). The percentage of A in the ground state is then around 12%.
-S
>~ A
txJ
.V2B +HJ
v~c
-10
THMD
.mRm
NRSC.v~.
eOPARIS TH f,,l$ BONN .UGI .BS
-15
OLp
q~J63
-20 -25
,
!
1.2
I
I
1.4 1.6 kF{fm-1)
I
1.8
FIGURE 5 Nuclear matter saturation predicted by various potentials taken from ref3). The values V28C, V14C, V28 and V14 are for the v2s and vz4, in the "continuous" and "gap" choices respectively of refiT), and THMD, THM, the relativistic calculation with and without A of ref4). The calculation of the incompressibility in nuclear matter is a long standing problem. The effect of including the isobar tends to make the saturation curve more shallow then decreasing the value of the incompressibility. A calculation with the v2s gives a number of about 180 MeV.
5. FINAL REMARKS The inclusion of A degrees of freedom in the description of nuclear matter properties, modifies substantially the results. Different calculations agree on the fact that the saturation curve is shifted upwards a few MeV, and the corresponding saturation density lowered. The defect parameter is increased and the incompressibility also affected. However the full
278c
L.S. Ferreira / Nuclear matter with ix degrees of freedom
coupled- channel calculations available do not go beyond the Brueckner two hole-line level, aside from some third order correlations which are included if the "continuous" choice is adopted. The estimations made, trying to sum up the ring series were based on too many approximations to be conclusive. Therefore the story about the A is not yet closed. As far as the description of the "empirical curve" is concerned, one may consider that the attractive part of the three-body forces will play an important role. The coupling to A states is able to generate such forces. A complete calculation at the three-hole line level of approximation with explicit A degrees of freedom is then the natural step to answer this question. The results for the v14 and v2s were obtained in collaboration with M. Baldo, INFN, Sez. of Catania. I thank him for the most valuable collaboration. REFERENCES 1) B.D. Day, Rev. Mod. Phys. 39(1967)719. 2) B. Friedman and V.R. Pandharipande, Nucl. Phys. A361(1981)502. 3) B.D. Day and F. Coester, Phys. Rev. C13(1973)1720. 4) B. terHaar and R. Malfliet, Phys. Rep. 149(1987)207. 5) A. M. Green and P. Haapakoski, Nucl. Phys. A221(1974)429. 6) B. Mckeller and R.Rajaraman, Mesons in Nuclei, vol.I Eds. M. Rho and D. H. Wilkinson, North-Holland Publ. (1979). 7) K. Holinde and R. Machleidt, Nucl. Phys. A280(1977)429. 8) G.H. Niephaus, M. Gari and B. Sommer, Phys. Rev. C20 (1979)1096. 9) R. B. Wiringa , Lecture Notes in Phys. 198(1983). 10) R.B. Wiringa, V. Fiksand and A. Fabrocini, Phys. Rev. C38(1988)1010. 11) H. Muther Prog. Part. Phys. 14(1984)123. 12) R. Machleidt and R. Brockmann, Phys. Left. 160B(1985)364. 13) R. A. Smith and V. R. Pandharipande, Nucl. Phys. A256(1976) 327. 14) R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29(1984)1207. 15) L.S. Ferreira, Lecture Notes in Phys 273(1986)100. 16) M. Baldo and L.S. Ferreira, Phys. Rev C41(1990). 17) M. Baldo and L.S. Ferreira, Nucl. Phys. A(1990), in press. 17) K. Dreissigacker, S. Furui, C. H. Hajduk, P. Sauer and R. Machleidt, Nucl. Phys. A375(1982)334.
269c
NUCLEAR MATTER WITH A DEGREES OF FREEDOM
Lidia S. FERREIRA Centro de F~sica, da Mat~ria Condensada, Avenida Gama Pinto 2, 1699 Lisboa Portugal Nuclear matter is studied in the framework of the Bethe Brueckner Goldstone theory with the explicit inclusion of the A isobar degrees of freedom. The calculation of the two-body reaction matrix is considered in full coupled channel calculation with potentials contalning all possible 1rNA and 7rAA couplings. 1. INTRODUCTION It is well established by now that the classical many-body perturbation expansion, the Brueckner 1) theory and the variational method2), yield results for the properties of nuclear matter that are in very good agreement with each other 3) but far from the empirical value. The use of different realistic N-N interactions, that fit the available data on the properties of the two-body system leads to nuclear matter saturation values, which lie as a function of density on a band, the Coester band3), being their differences mainly related to the amount of tensor force contained in the corresponding potential. Recently, relativistic effects have been included4). The new results might define a new Coester line approaching the empirical data. Many effects can be discussed that might change significatively the results.
One
possible extension of the theory is to consider excitations of the nucleon like the A(3,3) resonance which has rather high excitation probability. In fact the A isobar is already a very important feature for the NN force s) since the intermediate range attraction, which is conveniently parametrized in OBEP by a a boson~ can be partially replaced by a potential that couples NN states to NA and AA ones. The transition potentials between these states can be obtained from a n o n - relativistic reduction of the OPE Feynman amplitude for such processes with lrNA and r A A couplings. The coupled-channel approach can then be used to construct realistic NN potentials with A degrees of freedom. However one has in these models to compensate phenomenologically for the intermediate range part of the NN interaction, besides the OPE, which could simulate the residual effect of some cross-box diagrams whose structure cannot be reproduced this way. This iterated potential in the coupled channel, will also give the most important part of the T P E 3-nucleon attraction
0375-9474190/$03.50 © 1990 - Elsevicr Science Publishers B.V. (North-Holland)
270c
L.S. Ferreira / Nuclear matter with A degrees of freedom
in 3-body clusters, and repulsive 3-body contributions. They generate then effective many body forces 6) which might possibly be the explanation of the discrepancy between theory and experiment. The effect of the explicit inclusion of the virtual A states on the saturation properties of nuclear matter has been studied by many authors. Estimations were made 7) of the contribution of A excitations on the binding energy of nuclear matter, with an effective potential to take care of the NA transitions. Within the coupled-channel approach, Day and Coester 3) introduced the A couplings in the 1S0 and 3P1 channels and found that, despite the large shifts in the saturation energy and density, the saturation point still lies within the Coester band. However neither the N N - ~ A transitions or the average potential felt by the A in nuclear matter were included which are supposed to be relevant. A complete calculation including all possible NA and A A channels in the two-body interaction was performed by Niephaus, Gari and SommerS), which used a modified Reid potential. Concerning the importance of three-body forces, calculations with phenomenological three-body potentials, derived from these forces through the explicit introduction of non nucleonic degrees of freedom, were performedg). The variational calculation of ref. 10) included a three-body phenomenological potential, adjusted to reproduce the binding energies of the three nucleon systems and nuclear matter. The ring series with A intermediate states was studied by the authors of ref11). Concerning relativistic effects, the Dirac-Brueckner equations were solved with an extended Bonn potential 12), with explicit 27r and 27rp exchange replacing the ~ bosom The results confirmed the ones obtained with the original Bonn potential. The only relativistic full coupled channel calculation with A degrees of freedom was done by terHaar and Malfliet 4). The latter solved the Dirac-Brueckner equation with a two body potential containing NA transitions, and determined from a covariant 3-dimensional reduction of a coupled-channel Bethe-Salpeter equation. All NN scattering data for low and medium energies was determined with few parameters. We will discuss next potentials with explicit isobar transitions, and the modifications in the m a n y - b o d y theory due to those excitations. 2. NUCLEON-NUCLEON POTENTIALS WITH A DEGREES OF F R E E D O M Effective potentials ~) were constructed, within the Feshbach first order method, in order to account for NA and A A transitions in nuclear matter calculations. Within the non-relativistic coupled channel approach, various groups 3,s,13) modified a standard NN
L.S. Ferreira / Nuclear matter with ix degrees of freedom interaction, in order to include A effects.
271c
Recently, a new realistic NN potential was
presented 14) the Argonne v2s, which explicitly includes the A degrees of freedom through all possible ~rNA and 7rAA couplings and the NA and A A scattering channels with a total of 28 operators. The potential gives an excellent fit to the most recent data on deuteron properties and n-p scattering, and has a structure particularly suited for nuclear matter calculations. The authors also presented a conventional NN potential, the v14 model, which can provide a standard of comparison. All allowed OPE couplings represented in Fig. 1 and the major effect of the p in reducing OPE at short distances were included. The crossed-box T P E diagrams d)-h) of Fig. 1 are simulated approximately, and besides the r N A and zrAA coupling constants, taken from the Chew-Low theory and quark model, respectively, there are no more free parameters then the ones contained in the v14 model. As a requirement of the quark model, it also contains a repulsive core in the NA and AA channels comparable to the ones in the NN channels. The v2s can be considered a minimal model with A degrees of freedom and with as few parameters as the data requires.
.....
:2-:]
i / X.~
I--il )q
FIGURE 1 Two pion exchange processes contributing to the nucleon-nucleon interaction. The full, double and dashed lines, represent the nucleon, the delta and the pion, respectively. The v2s is ideal to introduce many-body forces and study the dispersion and Pauli principle effects. Unfortunately the complexity of the potential, due to the increased number of coupled channels allowed by the A, as it can be seen in table 1, makes the nuclear matter calculation numerically quite involved. The calculations can however be quite simplified if a separable potential is used. Since the techniques to obtain separable representations of a given realistic local potentials have reached high standards of perfectionXS), it is possible to reproduce accurately the on and off the energy shell properties of the original potential,
272c
LIS. Ferreira / N u c l e a r m a t t e r with A degrees o f f r e e d o m
and safely replace the original potential by its separable representation. However care has to be taken with the opening of the NA and A A thresholds. The original potential is then represented for each partial wave by the separable form, N
,Ua
CO' ( q , q ' )
=
a
~
v~,(q)A~vl~'j(q)
I
(1)
i,j=l
where a --- { J S T }
specifies the total angular momentum, spin and isospin for a given
channel . The functions v~.~ are the form factors, and the matrix A~ characterizes the strengths and is independent of the relative momenta q, ql. In principle eq. (1) is exact for an infinite rank, however by a convenient choice of the form factors, it is possible to describe some of the most well established nucleon-nucleon interactions with only a few terms in the expansion. 80
3s, 40
NN
NA
AA
1S o
aD o
15 o SD o
1D 2
sS~aD2SD2aG2
5S21D2SD~SG 2
3.00 ap3
ap0 ap~ 5p~ aF:
3p07F0 apl 7F1
ap~-a F 2
ap2 ap~ aF2 s F2
ap2 rp~ aF2 ~F~ TH 2
3Sa-aD 1
3S~3D17D17G1
aD z
aD2~DaTG2
0
(,(3 8 4
200
400
600
Etab (MeV)
TABLE 1 NN, NA and AA partial waves (2S+llj) for the nucleon-nucleon scattering.
FIGURE 2 The sS1 _3 D1 phase shifts calculated with the separable potential (full line), compared with the ones obtained from the original local vzs potential (full dots) as a function energy.
An efficient procedure 16) consists in taking for the functions vZi the deuteron wave function and stationary scattering states of the vzs potential at definite energies, multiplied by the original local potential. This is, in fact, a generalization of the so called EST method is) in order to include thresholds. The on and off the energy shell behaviour of the scattering matrix was accurately reproduced in each partial wave by this separable form. In order to illustrate this point, we show in Fig. 2 a comparison of the scattering phase shifts calculated with the original vzs and its separable representation for the zS1 _3 D1
L.S. Ferreira / Nuclear matter with
a
degrees of freedom
273c
channel. In Fig. 3 a similar comparison is made for the off the energy shell behaviour of the scattering matrix of the 1S0 channel, by plotting the Kowalski-Noyes function below and above the N/X threshold. As it can be seen the behaviour of the local potential is precisely described over a sufficiently large energy range, required for a nuclear matter calculation. Therefore the original potential can be safely replaced by its separable representation without loss of precision.
iI
2! • 0
"~
_
E'lob = 120 MeV i
=~.,~-
5Do ~""~'~_
~f
LL
E'lob = 850 ,
o
i
a .4
,
II] .6
,
, .8
, I.
MeV i
, 1.2
EIob (GeV) FIGURE 3 Kowalski-Noyes function in the NN and N/X sectors of the 1So channel. The full dots and full line correspond to the exact and separable calculations, respectively.
3. THE BETHE BRUECKNER GOLDSTONE WITH A DEGREES OF FREEDOM The extension of the Bethe Brueckner Goldstone BBG at the two-hole level to include explicitly A couplings leads to the calculation of a modified two-body reaction matrix G, with respect to the one obtained from a NN potential without A degrees of freedom. One has to solve a set of coupled equations, that in the momentum representation take the form,
< k'lk'2;71'lG(w)]kak2;71 > = < k'ik'2;71'lvlk~k2;z I > + <
k3k,
<
(2)
~ - E¢,(k3, k4)
where the label y indicates the type of particles involved in the channel, namely y = {NN, N/X, AA}, each k labels the momentum, spin and isospin of the corresponding particle, v is the bare potential, the v2s, and Qn is the Pauli operator, which requires the nucleon momenta to be outside the Fermi sea, but allows the/X to have any momentum. The quantity En,, is the sum of the two single particle energies e(k) inside nuclear matter, present in the channel 7/", defined by eN( k ) = k2 / 2 m + UN( k ), eA ( k ) = ( /X - m ) + k 2/2/X + UA (k),
L.S. Ferreira / Nuclearmatter with ix degreesof freedom
274c
being U the self-consistent single particle potential for the nucleon and the A. The nucleon single particle potential can be taken as
UN(k~) = ~
< klkzNNlG(~)lkak2NN >
(3)
with ~a = eN(kl) + eN(kz). In the case of the "continuous choice", the potential will be continuous over the Fermi surface and Eq. (3) is valid for all values of kl. If the "gap choice"is adopted, Eq. (3) holds only only for momenta below kF, and above kF the nucleon single particle potential is assumed to vanish. It can be proved that one is then able to go further than the two-body cluster contribution and include higher order clusters in the BBG expansion, when the "continuous choice" is adopted. Analogously one can define the average potential felt by the A as
U~(kl) =
~
< klk2NAIG(~v)Iklk2NA >
(4)
k2 <_kF
where, according to table 1, only the channels with total isotopic spin T = 1 actually contribute. A partial wave expansion of Eq. (2) has to be m a d e , and in order not to mix different partial waves, the Pauli operator in the energy denominator must be replaced by its angle average. This approximation is expected to be reasonable. From the structure of Eqs. (2), (3) and (4) there is an implicit double self-consistency condition for the nucleon and A single particle potentials, which can be solved by iteration. Having done so, and obtained the reaction matrix, since the total potential energy at the two-hole level is given by
B=~
1
~
(5)
kxk2
where the single particle momenta are inside the Fermi sea, the saturation curve,that is, the binding energy per nucleon as a function of density, is immediately determined. From the structure of Eq. 2, we see that the effect of introducing explicitly isobar degrees of freedom has changed the reaction matrix in two aspects. The first one concerns the presence of the Pauli principle for the nucleons that will exclude certain NA states. The A A states will be unaffected since Q a a = 1. This is named as the Pauli effect. Secondly, the energy denominators for the intermediate NA and A A states are increased, the propagator is modified. This is the dispersion effect. Both effects will reduce the effective coupling to the NA and A A states in comparison with a calculation with an interaction
L.S. Ferreira / Nuclear matter with A degrees of freedom
275c
where these degrees of freedom are not explicit but are integrated phenomenologically in the two-body interaction. It is expected that they will also vary with the density. The solution of Eq. (3) with a separable interaction, Eq. (1), brings significant simplifications. Like in the case of the two-body problem, it becomes reduced to a matrix inversion. In a full coupled channel approach, with A degrees of freedom, this certainly will reduce the computation time.
4. THE EQUATION OF STATE WITH A DEGREES OF FREEDOM The self-consistent Eqs. (2) and (3) were solved with U~ -- 0 by Day and Coester 3) for the couplings from NN to NA states in S and P waves and no AA transitions within the "gap choice". Similar calculation was performed by Niephaus, Gari and Sommer but with a modified Reid potential with extra operators to express the transitions to NA and AA states, refitted to the scattering data. All possible partial waves up to L = 4 were included. The v28 fits more modern data, so there will be an improvement in relation to the modified Reid potential. The use of its separable representation, will also make the calculation in the "continuous choice" simpler. The self-consistent Eqs.
(2-4) were solved iv) by an
iterative procedure in the "continuous choice", with the separable representation 16) of the v2s. The starting point was a parabolic form for the single particle nucleon potential. The A potential was assumed to produce only a constant shift of 40 MeV in the A mass. This is the value obtained near saturation by the authors of reflS). All the channels displayed in table 1, and in addition the single channels 1F3,3F~,l G4 and 3G4 taken from the Argonne v14 were included. The saturation curve obtained from Eq. (5), E = B / A , is reported in Fig. 4 (full line) in comparison with the results obtained from the Argonne v14 potential (dashed line) in the "continuous choice". Since the two potentials have the same structure for the NN part, and fit the same scattering data~ a comparison of the two shows the effect of the isobar. The saturation density is shifted down to a value smaller than the empirical one, while at saturation, nuclear matter is a few MeV less bound. By switching off the A potential the binding energy is increased about 2 MeV. The remaining discrepancy between the two potentials comes from the dispersion effects mainly, and from the Pauli principle. The corresponding saturation point still lies within the Coester band, however pushed towards the lower border of it as it can be seen from Fig. 5. The same calculations in the framework of the "gap choice" lies 1-2 MeV above the
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L.S. Ferreira / Nuclear matter with zx degrees o f freedom
one reported in the work of Niephaus et al. s), being this discrepancy essentially due to the use of different potentials.
J
i
I
i
7"~
!
-5
.'i
/ //
*,,..,,×
//"
"", ,,,
-1{
l
".k....
..s
\,,
-,.
, .
.j . .I
/1 "z
i /
;
//
/ 1I
I
1.0
I
1
1.2
I
1.~, Kf {fm -I}
,
I
1.6
1!8
FIGURE 4 Binding energy per nucleon as a function of Fermi momentum in the "continuous choice" for the v2s (full llne) and v]4 (dashed line) models. The dotted line indicates the "empirical" saturation curve. The dash-dotted and dash-double-dotted lines are the relativistic calculation without and with A, ref4), respectively. The dot-double- dashed is refl°). Concerning the effect of three-body forces, the authors of refJ °) included a phenomenological three body force adjusted to reproduce the properties of the three nucleon system. They found that the saturation density is reduced, and the equation of state becomes much stiffer. However they were not able to fit simultaneously the binding energy of the three-body system and nuclear matter. Their estimation of three-body effects for the v14 is also shown in Fig. 4. It should be noted that a Brueckner calculation with explicit isobar degrees of freedom contains the dominant part of the repulsion due to the threebody force. In fact, one of the repulsive diagrams generated by the explicit introduction of NA transitions that contribute to the effective three-body force is naturally included by the self consistency. Therefore in order to compare ref. 17) with refJ °) one should add the attractive part of the three-body force. At kF = 1.3frn -1 the expectation value the attractive part of the force amounts to -4.031°). It is interesting to compare the results of ref.17) with the ones of terHaar and Malfliet*). Their results are also shown in Fig. 4 and 5. A quite close agreement existe with the v2s. Without the A transitions the relativistic effects seem to be strong and shift the saturation point to a lower density. With them, the saturation energy is shifted even to a lower
L.S. Ferreira
/
Nuclear matter with ix degrees of freedom
277c
density, so the agreement can be partially explained by noticing that at low densities the relativistic corrections are expected to be smaller. Another relevant quantity in the many-body problem which is changed with the introduction of A degrees of freedom is the total defect parameter ~. It is now split into three components. The defect parameter gNN expresses the content of two particles-two holes in the ground state, while ~NZX and nz,,x give an indication of the percentage of A particles present in the ground state. When in conventional calculations one finds values for ~ at kF = 1.4 fm -1 of order of .12 with the A effect it becomes .324). The percentage of A in the ground state is then around 12%.
-S
>~ A
txJ
.V2B +HJ
v~c
-10
THMD
.mRm
NRSC.v~.
eOPARIS TH f,,l$ BONN .UGI .BS
-15
OLp
q~J63
-20 -25
,
!
1.2
I
I
1.4 1.6 kF{fm-1)
I
1.8
FIGURE 5 Nuclear matter saturation predicted by various potentials taken from ref3). The values V28C, V14C, V28 and V14 are for the v2s and vz4, in the "continuous" and "gap" choices respectively of refiT), and THMD, THM, the relativistic calculation with and without A of ref4). The calculation of the incompressibility in nuclear matter is a long standing problem. The effect of including the isobar tends to make the saturation curve more shallow then decreasing the value of the incompressibility. A calculation with the v2s gives a number of about 180 MeV.
5. FINAL REMARKS The inclusion of A degrees of freedom in the description of nuclear matter properties, modifies substantially the results. Different calculations agree on the fact that the saturation curve is shifted upwards a few MeV, and the corresponding saturation density lowered. The defect parameter is increased and the incompressibility also affected. However the full
278c
L.S. Ferreira / Nuclear matter with ix degrees of freedom
coupled- channel calculations available do not go beyond the Brueckner two hole-line level, aside from some third order correlations which are included if the "continuous" choice is adopted. The estimations made, trying to sum up the ring series were based on too many approximations to be conclusive. Therefore the story about the A is not yet closed. As far as the description of the "empirical curve" is concerned, one may consider that the attractive part of the three-body forces will play an important role. The coupling to A states is able to generate such forces. A complete calculation at the three-hole line level of approximation with explicit A degrees of freedom is then the natural step to answer this question. The results for the v14 and v2s were obtained in collaboration with M. Baldo, INFN, Sez. of Catania. I thank him for the most valuable collaboration. REFERENCES 1) B.D. Day, Rev. Mod. Phys. 39(1967)719. 2) B. Friedman and V.R. Pandharipande, Nucl. Phys. A361(1981)502. 3) B.D. Day and F. Coester, Phys. Rev. C13(1973)1720. 4) B. terHaar and R. Malfliet, Phys. Rep. 149(1987)207. 5) A. M. Green and P. Haapakoski, Nucl. Phys. A221(1974)429. 6) B. Mckeller and R.Rajaraman, Mesons in Nuclei, vol.I Eds. M. Rho and D. H. Wilkinson, North-Holland Publ. (1979). 7) K. Holinde and R. Machleidt, Nucl. Phys. A280(1977)429. 8) G.H. Niephaus, M. Gari and B. Sommer, Phys. Rev. C20 (1979)1096. 9) R. B. Wiringa , Lecture Notes in Phys. 198(1983). 10) R.B. Wiringa, V. Fiksand and A. Fabrocini, Phys. Rev. C38(1988)1010. 11) H. Muther Prog. Part. Phys. 14(1984)123. 12) R. Machleidt and R. Brockmann, Phys. Left. 160B(1985)364. 13) R. A. Smith and V. R. Pandharipande, Nucl. Phys. A256(1976) 327. 14) R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29(1984)1207. 15) L.S. Ferreira, Lecture Notes in Phys 273(1986)100. 16) M. Baldo and L.S. Ferreira, Phys. Rev C41(1990). 17) M. Baldo and L.S. Ferreira, Nucl. Phys. A(1990), in press. 17) K. Dreissigacker, S. Furui, C. H. Hajduk, P. Sauer and R. Machleidt, Nucl. Phys. A375(1982)334.