G-matrix elements with effective masses for mesons and nucleons

00211991)MYAM

July 1991

'OT-Me"I Of

A. HOSAK

et sirs, VnitvrOq qf Pvnn~Ylrania, Philadelphia, PA 19104, USA

rl

I 0j, Pk'sics' Tokyo Memo,

H. T

SSES M

KI

litagt UniversitI', %a~ao>a, T'ok4"® d1 Japan

Received 16 November 1990(Revised 9 January 1991)

:fit-mt: G-matrix elements for finite nuclei are calculated usingeffective masses for mesons and nucleonsat finite density, i .e., those in nuclear medium . We find that the central G-matrix elements are verysensitive to masses of the sigma and omega. By decreasing the mass of the rho, the LS matrixelements are generally enhanced, while the tensor matrix elements are suppressed . A uniformdecrease in various masses yields G-matrix elements whi,,,:h are rather consistent with empiricalmatrix elements obtained by Brown, Richter, Julie aid Wildenthal .

1 . Introducti n

scantly, many sugge "ons have been made for changes of hadron properties innuclear medium --") . &,,~_es the convent :.onal nuclear many-body effects, calcula-tions based on QCD inspired models have been extensively performed at finitetemperature and at finite chemical potential 4) . For example, many calculations inthe Nambu-Jona-Lasinio model -") have been suggesting that the mass of the sigma,in,*,, decreases in medium 6-') . Here the star indicates the effective quantity inmedium. Chiral symmetry essentially determines this behavior . A naive consequenceof this fact is the softening of the equation of state due to the increase of the rangeof attraction . It is also suggested that the masses of the vector mesons decrease inmediaM 3,7) .

He use a the effective masses for mesons and nucleons should have significantinfluence on the nucleon-nucleon interaction in medium and hence on nuclearstructure as well . Compared to a number of microscopic calculations such as thosementioned above, so far, calculations for relevant nuclear phenomena have not beenperformed very much 3'g ) . The purpose of this paper is then to study the G-matrixelements using the effective masses for mesons and nucleons suggested in medium.The G- matrix is the most fundamental two-body interaction which incorporates the

0375-9474/91/$03.50 (K,) 1991 - Elsevier Science Publishers B.V . (North-Holland)

simplest many-body effect, i .e ., the Pauli blocking . We compute the G-matrixelements for finite nuclei, particularly for "0 and compare them with the two-bodymatrix elements in the sd-shell region empirically obtained by Brown, Richter, Julies

hat (

W) ").n fact, such comparisons between the theoretical G-matrix elements and the

empirical matrix elements have been performed by BRJW 9) . There, of course,values for the masses were employed. Compared to the relatively good

arce ent in the central channel, they found some disagreements in the LS andtensor matrix elements. Our interest is then whether a better agreement is achievedby employing the effective masses or not. Furthermore, it is also interesting if wecould find constraints on the way how the effective masses should scale in mediumfrom the phenomenological point of view .

is paper is organized as follows. In sect . 2, a method to solve the Bethe-oldstone equation is briefly explained. The mass number (A) dependence is

properly treated in computing the G-matrix elements to compare them with theempirical matrix elements. As our G-matrix elements are functions only of therelative coordinate, a formula to transform them into the LS coupling scheme ispresented. Then we introduce a few examples for behaviors of mass parameters inmedium. Numerical results are presented in sect . 3, where the G-matrix elementsare calculated using various set of meson and nucleon parameters . The final sectionsummarizes our work.

ka,

Tob / G-mairïr elements

2. Formalism

n this section a method to compute the G-matrix elements for finite nuclei isbriefly outlined . We follow closely our previous work'"'), which is an extensionof the original work known as the M3Y (Michigan three-range Yukawa) inter-action '-') .We start with the Bethe-Goldstone equation 13) :

G(E) = U+'v

Q

G(E) .

(1)E-HoHere v is a nucleon-nucleon interaction, Q the Pauli-blocking operator, and Ho asingle-particle hamiltonian . The G-matrix elements are obtained for a finite nucleus,specifically for 160, in order to compare them with the empirical BRJW matrixelements . The harmonic-oscillator hamiltonian is then chosen for Ho

where ju, is the reduced mass of two nucleons and the harmonic-oscillator parameter

w is given by the formula

A. Hosaka, H. Toki / G-matrix elements

431

w =45A_1/3 -25A-2/ 3 =14 MeV

(for '0) .(3)

To solve the Bethe-Goldstone equation (1), the relative coordinate is separated byapplying the method of Eden and Emery for the Q-operator '4) . The gap energy of40 MeV is used as an energy difference between the nucleons below and above theFermi surface's) . Barret-Hewitt-MacCarthy method is then applied to the inversionof eq. (1) see ref. '6) . The resulting G-matrix elements depend on the starting energyE, which has to be determined self-consistently. In practice, we assume that therelevant two nucleons are on the Fermi surface and average them over it.For the nucleon-nucleon interaction we employ a non-relativistic parametrization

of the Bonn potential "). There, the sigma (0+, T= 0), delta (0+, T =1), eta (0-, T=0) pion (0-, T =1), omega (1 -, T= 0) and rho (1 -, T =1) mesons are explicitlyincluded in the one-boson exchange potential. For convenience we summarize theirexplicit form in the appendix .The G-matrix elements obtained from eq. (1) are given in the partial-wave

representation in the relative coordinate. These matrix elements are transformedinto those in the LS coupling scheme to be compared with those by BRJW. Acombination of the spin-tensor transformation, Brody-Moshinsky transformationand recoupling of angular momentum yields a formula 9) :

(abLSJTIGk l cdL'S'JT)

_ (2k+ 1)

L

S

J

E (-).r'+j(2J'+ 1)

L

S

JS L k ,.S L k

x

1

(n1NL1 nalanhlh : L)(n'l'1VL1 n,l~ndld : L')nln'l' Nij

The LS coupling matrix elements on the left-hand side is given for the central(k =0), spin-orbit (k =1) and tensor (k = 2) channels, separately . The two particlesin the states a and b with angular momenta la and l,� respectively, are coupled tothe total angular momentum L: [Ia , Ih

]L. Node quantum numbers are denoted by

na and nh . Similar notations are used for the particle c and d. On the right-handside the sum of quantum numbers of the two particles are decomposed into thecenter-of-mass and relative coordinate representation : n and n' (1 and l') are theinitial and final node (angular momentum) quantum numbers in the relative coordin-ate, and 1V(L) is that in the center-of-mass system . Note that the matrix elementson the right-hand side are diagonal in the center-of-mass coordinate . Standardnotations for the Brody-Moshinsky coefficients (nIÎYLI nalanhlh : L), 6j and square

9j symbols are used . The square 9j symbols are related to the standard 9j symbols

1 S j l' S jx L 0 L L 0 L ([IS]jnlGj[l'S?n') .

L S J' L' S J'

432


by1, Si Ji

lt StA A A A

l' S2 j2 =jJ2LS 1, S2LL S J

L S J

with jI =

2ja + 1 etc . Finally, the matrix elements are properly antisymmetrized by)

(abLSJT I G1,I edL'S'JT)�� ti-syrn

1

pa am;aer a [ref . 20)]

(1+8,,,)(1+S(.,)

{(abLSJTIGkjcdL'S'JT)

(abLSJTIGk.)dcL'S'JT)}.

(6)

Now we briefly state possible changes of meson properties in medium . A simplepicture is drawn by assuming that the hadron dynamics at low energies is approxi-mately governed solely by chiral symmetry. An example of a dynamical model isthe NJL model, where mesons are generated as quark and anti-quark pairs;-7). Abasic quantity in the model is the pion decay constant f*;, in medium. Otherdimensional parameters such as the masses of the sigma and constituent quark thenscale as ,f . If the density is increased, the system tends to recover chiral symmetry,resulting a decrease in f: . Hence, those masses are expected to decrease . The pionis special, since it is a Goldstone boson. Its mass stays essentially constant (zero inthe exact chiral limit) up to the critical densitY .A very simple relation has been proposed by Brown et al. 1 ) . Assuming that the

KSRF relation °x) holds in-medium :

m� = ,\[2 g *f:

as well as in free space, where g* is the (almost density independent) p -> 27 decayconstant, the vector meson mass m* scales as f* . In the constituent quark modelthe nucleon mass M is approximately three times the constituent quark mass.Therefore, one reaches the relation

f*tr

m Cr MI* M*f, m, m,

M .8

Incidentally, it is reasonable to assume that other dimensional quantities also scalein the same way. Therefore, cutoff parameters in the meson-nucleon vertex, A, mayfollow the same relation as (8) .

This scenario may be too naive . So far, many results are model dependent . Forexample, it is pointed out that in the sigma-omega model m* does not necessarilyobey the monotonic decreasing tendency at finite density '9) . The vector meson massis even more uncertain . In the extended NJL model, it stays almost constant'),while QCD sum rule calculations predict a decreasing tendency 2 ') . In this paper,

* The KSRF relation (7) is modified to m � = la-gf, at finite density with the density-dependent


3. Numerical results

433

however, we take a point of view as simple as possible. Namely, we consider thescaling rule which is not very far from (8) . A basic assumption is then that môdecreases in medium .

Before showing our numerical results, let us briefly go through the empiricalG- matrix elements by BRJW. They have performed X2 fittings under many differentconditions and compared them. Among them, the following two sets of matrixelements are relevant for our purpose. The first set is the one they call the 66-parameter model-independent procedure, where they have treated all the 66 para-meters in the sd shell as free parameters . The two-body matrix elements include thecentral, LS, tensor and anti-symmetric LS channels . These will be referred to as theWFULL interaction in the following. The last piece of the WFULL interaction, theanti-symmetric LS channel was, however, found to be relatively less important andthey have performed another fitting without it . The resulting set of matrix elementsare called WNOALS in ref. 9 ) . The difference of the WFULL and WNOALSinteractions provides us with some feeling of errors in the itted matrix elements .We mention that the errors are in general about 10% level and they do not modifythe global features of the empirical matrix elements .

In this paper we adopt their second set of parameters, WNOALS, since in ourprocedure matrix elements in the anti-symmetric LS channel vanish. We thenconsider core polarization effects separetely. Slurpin, huo and Strottman havecomputed the effects including higher orders in the folded diagram method -2) . Wethen subtract these contributions from the WNOALS. The subtracted matrix elementswill be referred to as the WNOCP matrix elements . The effects of the core polariz-ations including higher order diagrams are, generally, not very large. For example,in many central matrix elements they have small repulsive corrections to the bareG-matrix elements at most up to 10% (fig . 0.Here we have to mention the limitation in the separation of core polarization

effects . In sect . 2, we have illustrated an example which shows a qualitative changein meson and nucleon masses by using a quark model, e.g . the NJL model. Aninteresting question is then whether such effects expected in a QCD oriented modelis indeed related to those expected in nuclear many-body physics. They may or mayriot be considered as the same thing in different terminologies . Here, we simplyexpect that at least the quark effects are conveniently included by modifying variousmasses .Now we begin with the G-matrix elements derived from the free-space nucleon-

nucleon interaction . We refer to these G-matrix elements as the bare G-matrixelements . In fig . l, the bare G-matrix elements (dots) are compared with the WNOCPmatrix elements (points connected by a solid line), together with the WNOALS(points connected by a dashed line) matrix elements to see the magnitude of the


Matrix Element Number

0

5

10

0 5 10 15Matrix Element Number


Fig . l . The bare G-matrix elements for the central (a), LS (b) and tensor (c) channels derived from thefree-space nucleon-nucleon interaction (dots) compared with the empirical WNOALS (points connectedby a Eiashed line) and WNGCP matrix elements (points connected by a solid line) . In the WNOCPmatrix elements, the core polarizations 22) are subtracted from the WNOALS matrix elements . The same

conventions are used for the numbering of the matrix elements as adopted by BRJW.

core polarizations. The same conventions are used for the numbering of the matrixelements as adopted by B1JW and are summarized in table la-c. The core poiariz-ations contribute generally about 10% corrections for various matrix elements . For


435

TABLE laMatrix element number and the corresponding quantum numbers forthe central channels. Angular momenta of particle a and b (c and d)are coupled to L (L') : [la , I& ([l,, ld] LI ) . S and T are the total spin

and isospin, respectively.

TABLE lbMatrix element numbers and the corresponding quantum numbers forthe spin-orbit channel . The conventions are the same as in table la

Matrixelementnumber

la lb 1, Id L L' S T

1 2 2 2 2 0 0 1 02 2 2 2 2 2 2 1 03 2 2 2 2 4 4 1 04 2 2 2 0 2 2 1 05 2 2 0 0 0 0 1 06 2 0 2 0 2 2 1 07 0 0 0 0 0 0 1 08 2 2 2 2 0 0 0 19 2 2 2 2 2 2 0 110 2 2 2 2 4 4 0 111 2 2 2 0 2 2 0 112 2 2 0 0 0 0 0 113 2 0 2 0 2 2 0 114 0 0 0 0 0 0 0 115 2 2 2 2 1 1 1 116 2 2 2 2 3 3 1 117 2 0 2 0 2 2 1 118 2 2 2 2 1 1 0 019 2 2 2 2 3 3 0 020 2 0 2 0 2 2 0 0

Matrixelementnumber

la l,, h Id L L' S T

1 2 2 2 2 2 2 1 02 2 2 2 2 4 4 1 03 2 2 2 0 2 2 1 04 2 0 2 0 2 2 1 05 2 2 2 2 1 1 1 16 2 2 2 2 3 3 1 17 2 2 2 0 1 2 1 18 2 2 2 0 3 2 1 19 2 0 2 0 2 2 1 1

osaka, H Toki / G-matrix elements

Matrix element numbers and the corresponding quantumnumbers forthe tensor channel. The conventions are the same as in table la

TA13LE lc

the tensor matrix elements, however, the effects are somewhat larger and in somecases they reach up to 40% corrections .

Remarkably, as already noted by BRJW, the bare G-matrix elements agree prettywell with the WNOCP (and to the same extent with the WNOALS) matrix elements,especially for the central channel both in its shape and absolute values*. For theLS channel, the agreement is poor as compared to the central channel . The bareG-matrix elements for T= 0 (matrix element number 1-4) are essentially zero, whilethose for T = 1 (5-9) have non-negligible negative values . However, the empiricalWNOCP matrix elements do not show this tendency . For the tensor channel, theagreement seems even worse. Nevertheless, it is interesting to note that most of thebare G-matrix and the WNOCP matrix elements are very small and lie withint0.2 MeV. There are five (three) relatively large matrix elements in the bare G-matrixelements (in the WNOCP) . They do, however, not always agree each other. Some-times it has been pointed out that the calculated LS matrix elements are too small 8)and the tensor matrix elements are too strong 23 ) . However, in our calculations, itis not so obvious to recognize this tendency .Now it is very interesting to study if modifications of various masses as suggested

by QCD oriented models can improve the situation, expecially, for the LS and

* Although we do not compare our present results with previous calculations using the Paris potential,we just state that the bare G-matrix elements from the Bonn potential are very similar to those derivedfrom the Paris potential .

Matrixelementnumber

1. lb 1, ld L L° S T

1 2 2 2 2 2 0 1 02 2 2 2 2 2 1 03 2 2 2 2 4 2 1 04 2 2 2 2 4 4 1 05 2 2 2 0 0 2 1 06 2 2 2 0 2 2 1 0ï 2 2 2 0 4 2 1 0

2 2 0 0 2 0 1 09 2 0 2 0 2 2 1 0

10 ? 0 0 0 2 0 1 011 _' 2 2 2 1 1 1 112 2 2 2 2 3 1 1 113 2 2 2 2 3 3 1 114 2 2 2 J 1 2 1 115 2 2 2 0 3 2 1 116 2 0 2 0 2 2 1 1

tensor matrix elements, while keeping the agreement in the central channelunchanged . To start with, we investigate the role of each mesons by changing eachmeson mass. The most important in the nucleon-nuclon hiteraction is the sigmaand omega exchanges. In fig. 2 we present the central matrix elements for the massesof the sigma and omega (mô and m*) decreased to 95% of their free-space values,together with the bare G-matrix elements and the WNOCP matrix elements. Thereduction of m*: and mw increases attraction and repulsion, respectively, as it shouldbe. Rather unexpected is the fact that the matrix elements are very sensitive to mQand m!, . Only 5% change of m*: or m* yields as much as 50% change for certainmatrix elements, which is far outside of both errors and corrections due to corepolarizations . This in fact is one of the main conclusions in our present paper: ifwe want to keep the agreement between the calculated and empirical matrix elementsin the central channel, the change in mQ (or ml) must be accompanied by thechange in ml (or m*:) by about the same amount.Although we do not show the result explicitly, it turned out that the LS and

tensor matrix elements are rather stable and°,r the change ofm*! and ml . This mightbe somewhat unexpected for the LS channA, since the LS interaction is mainlydue to the sigma and omega exchanges. The reason is roughly explained from eqs.(A.6)-(A.8) . If the meson mass m* is decreased, the exponential factor exp (-m*r)

E Lo

21

0


Central Component

"

:bare Go

: m(sigma) = 95 %o

: m(omega) = 95

0 5 10 15 20Matrix Element Number

437

Fig . 2 . The G-matrix elements in the central channels with mô reduced to 95% from its free-spa,.e value(squares) and with ml, reduced to 95% from its free-space value (circles) compared with the WNOC1'

and the bare G-matrix elements . The same conventions are used as in fig . 1 .

the volume integral, which is however cancelled by the factor m*2/m*:!

in front (M* is kept unchanged).Now let us discuss the role of other mesons : the delta, eta, pion and rho mesons.

,e performed calculations with the mass of each meson decreased . The deltand eta are not important in the nucleon-nucleon interaction and hence the change

in their masses does essentially nothing . The decrease in the pion mass is not veryimportant again as in the

se of m 411, (or m*l) for the LS channel. The decrease inthe mass of the rho ( ), however, has significant effects on the tensor and LSmatrix elements, i.e., it increases the p meson component in relevant interactions .For example, in the, ensor force, the cancellation ofthe attractive one-pion exchangecontribution due to p-exchange is increased with decreas ; n the mass of the p,resulting in the suppression of the tensor matrix elements . As shown in fig. 3, thistendency, is clearly seen, which is consistent with the observation made by Brownand Rho ' ). However, this does not always improve the agreement between thecalculated

-matrix elements and the WNOCP matrix elements . Another importantrole of the decrease in m

is seen in the LS matrix elements . These matrix elementsare generally increased and the agreement is improved particularly for the largematrix elements (number 5 and 8).Now we investigate the uniform scaling of various masses as implied in eq. (8) .

Specifically we consider the scaling ratio of 8®°/®, which is a typical number expected

osak ,

iob / G-matriv elements

N

Tensor Component

® :bare Gm(rho) = 80 %


Fie . 3 . The G-matrix elements in the tensor channels with mP reduced to 80% of its free-space value(squares) compared with the corresponding WNOCP and the G-matrix elements . The same conventions

are used as in fig . 1 .

439

for the nuclear matter. This ratio may be too large for the purpose of shell modelcalculations, since the relevant interaction happens between valence nucleons wherethe medium density is effectively smaller than for the nuclear matter. Here, however,we are only interested in qualitative behavior under the scaling and the specificchoice of this ratio is not very important.

Explicitly, using the notation Ra --- ml/ma, we consider the scaling

with


Rp = Rs =R,1 =0.8, RQ^-R,,-0.8

(9)

M* A*M

=A=0.8 .

(1Q)

Among these scalings, RS and R.. are not very important as discussed before . Thereason that RQ and R,,, are not exactly equal will be explained later. We do notconsider the change in the mass of the pion, since it is protected by chiral symmetry.In changing the nucleon mass M*, the oscillator parameter w is scaled inverselyas M*, since the length scale of the oscillator wave functions is 1/ ,%IM*W. By doingthis the nuclear wave function and hence the size of the nucleus are unchanged.Another manipulation is necessary when M* is changed. There are several placeswhere M* appears in the nucleon-nucleon interaction after the non-relativisticreduction ofthe one-boson exchange potential . The change in M* affects the strengthof the potential . Thus the one-pion exchange part is also modified . We, however,keep the pion strength unchanged in medium, since it is the part which is relativelywell understood . Brown and Rho argue that the decrease in the 7rN: :̀ ~;ouplingconstant in medium partly compensates the decrease in the nucleon effective massM* in the denominator of eq. (A.9) [ref. 23)] .

In actual computations, we find a parameter set such that the central G-matrixelement of number 1 agrees approximately with the corresponding WNOCP matrixelement by adjusting either m* or m* .. This is always possible from the previousarguments in fig . 2 . Reduction ratios of m* and rnI are then RQ = 0.80 and R4, = 0.81,respectively .

In fig. 4a-c, we show the results for the central, LS and tensor matrix elementsfor the uniformly scaled parameters . It is interesting that the central G-matrixelements are very similar to those ofthe bare G-matrix elements (except for elements10 and 13) . This is not accidental : if all the dimensional parameters are scaled inexactly the same way, the resulting matrix elements should also scale in the sameway. In our calculations, however, the oscillator parameter is scaled as 1/(massparameters) and also the one-pion exchange potential is fixed. Therefore, the exactscaling law is violated and nontrivial behavior of the matrix elements can happen.This is particularly the case for the LS and the tensor channels . Although theagreement between the calculated G-matrix elements and the WNOCP matrixelements is not improved apparently, we see systematic changes in the matrix

440

0> ~ô ~

c(DOa}

LS Component

® : bare Go : uniform scaling


Central Component

" - bare Ge : uniform scaling

(a)

0 5 10 15 20


Tensor Component

0

5

10



Fig . 4. The G-matrix elements for the central (a), LS (b) and tensor (c) channels for the uniformlyscaled masses : RQ = 0.80, R,,, = 0.81 and RN = RS = R,0 = M*/M = A */A = 0.8, compared with theWNOCP and the bare G-matrix elements . The pion part is fixed to that in free space. The same conventionsare used as in fig . 1 . For some matrix elements, the bare G-matrix results and the uniform scaling results

are almost identical . They are overlapping and appear as closed squares .

A. Hosaka, H. Toki / G-matrix elements 441

elements . Here the use of the effective nucleon mass plays an important role . Asseen in (A.6)-(A.11) there are factors proportional to (m*/M*)2 in the LS andtensor interactions . If the meson mass is decreased while the nucleon mass is keptunchanged, the factors become less than unity, which cancels the increase in thevolume integral due to the increase in the interaction range. However, if m* andM* are simultaneously decreased the factors remain unity hence the increase in thevolume integral dominates, thus increasing the strength of the relevant matrixelements . In the LS channel, the sigma, omega and rho contribute to increase thematrix elements . In the tensor channel, the pion strength is fixed (m,,, is not varied)and the increase in the rho exchange suppresses the total matrix elements.From these schematic calculations, the scaling law (8) with the pion strength fixed

seems to work pretty well . In fact we have tried other combinations of parametersbut many of them did not work as well as the case we have shown here . Especially,the agreement in the central channel seems to have significant meaning: the scalingof m**, must be accompanied by the scaling of ml to compensate the increasedattraction.

4. Conclusion

In this paper we have investigated the G-matrix elements using meson and nucleonparameters modified in medium . These property changes have been advocated forsome time in various context, but have not been tested in many cases in nuclearphysics . Here we have investigated the G-matrix elements using various sets ofmeson and nucleon masses and compared them with the empirical matrix elementsderived from the shell model calculations in the sd-shell region . From a phenomeno-logical point of view, we could have searched the "best fit parameters" whichoptimally reproduce the empirical matrix elements as has been done by BRJW 9) .On the other hand, however, we have to know the limitation of our procedure tocompute the G-matrix elements . For example, our G-matrix elements certainly donot have density dependence, which is averaged over a nucleus . Probably, suchrefinements would be necessary to discuss in detail the calculated G- matrix elementsin comparison with the empirical ones . In this paper, therefore, we have tried toinvestigate rather qualitative behavior of the G-matrix elements under the change

of meson-nucleon parameters predicted by QCD inspired models.Among various findings in the present study, one of the most important results

is the fact that the central G-matrix elements are already well reproduced by using

the free-space parameters and that they are extremely sensitive to the masses of the

sigma and omega mesons . This indeed provides a strong constraint from the

phenomenological side on the way the meson masses, particularly mQ and ml,

should scale in medium . In fact, it turned out that they have to be correlated such

that m**/ m? - mw/ m,,, as long as coupling constants are kept unchanged . This scaling

rule is actually consistent with the one suggested by Brown et al. ' ) .

442


ncouraged by this fact, we have calculated the G-matrix elements using thevarious meson-nucleon masses scaled (almost) in the same way while keeping thepion strength unchanged . This time our interest is concerned with the LS and tensormatrix elements, since they are generally in poor agreement if the free space massesare employed. By using the scaled masses, however, we could not see significantimprovements in the comparison of the calculated and empirical matrix elements .However, systematic changes of the LS and tensor matrix elements may be worthpointing out. Generally, the LS matrix elements are enhanced by the increase ofthe sigma, omega and rho contributions, and the tensor matrix elements are sup-pressed by the increase in the repulsive components due to the rho exchange .

One of the authors (A.H.) thanks Niels Walet for helpful discussions and formany helps for numerical computations . He also thanks Alex Brown for valuablediscussions and providing him with various matrix elements . His thanks are alsodue to Gerry Brown and Andy Jackson for their interest in this subject and for theiruseful comments. This work is supported in part by the National Science Foundation .

with

Appendix

In this appendix we summarize the one-boson exchange potential used in theBonn potential . In the non-relativistic reduction the potential is classified into threeparts: the central, LS and tensor components. The central interaction is induced bythe exchanges of scalar, pseudo-scalar and vector mesons . In coordinate space theyare

c

gs

1 ms2

V0 (s) - -4,7r ms 1- 4 M

Y(msr)

gs ms-

47r 4M2[®2 Y(msr)+ Y(msr)v2] , (A.1)

2

i~ô(v) gV

m~= m� 1 + 1 +fV]Y(mvr)

47r 2 gV M2

47r 4M2

/

2

VCT(ps) l~=gpS MPS

Ps

47r 12 ( M ) Y(mpsr), (A.3)

2V( (v) 9'"= m" m"

47r+f~

6(M

1)Y(mvr) (A.4)

gV

Y(x) = exp ( --x)/x . (A.5)

Here the masses and coupling constants for the scalar, pseudo-scalar and vectormesons are distinguished by s, ps and v, respectively, and M is the nucleon m:as.In momentum space, the coupling constant g includes the monopole form factor .A convenient way to incorporate it in coordinate space is found in ref. ") . Thevector-to-tensor coupling ratio of the vector meson nucleon coupling is denotedby f. These parameters fitted to scattering data are found in ref."). In the o,interaction (A.3) and (A.4), the relevant ~ matrices are suppressed . The T and o--,rinteractions are derived from the corresponding isovector mesons and their interac-tion forms have the same expressions .The LS interaction is due to the scalar and vector exchanges:

with

with


443

2 m m 2VLS(s) = - gs

S

5

Zl(msr) ,41r 2 (M

derived from the isovector mesons with the same functional form.

References

(A.6)

VLs (v) _ _ g~

2mV 3+2 .Î"

( mV Z,(m,,r) ,(A .7)41r 2 gV M

where the l - s operators are suppressed . The T components are derived from thecorresponding isovector mesons with the same functional forms .

Finally the tensor force is induced by the pseudo-scalar and vector mesons:

3

3 ) exp (-x)

(A.11)x x

x

where the tensor operators S12 are suppressed . Similarly, the T components are

1) G.E. Brown, H . Wither and M . Prakash, Nucl . Phys. A506 (1990) 5652) T. Ainsworth, G.E . Brown, M. Prakash and W. Weise, Phys . Lett . B200 (1987) 4133) G.E. Brown, C. Dover, P. Siegel and W. Weise, Phys . Rev. Lett . 60 (1988) 27234) T. Hatsuda, `Hadrons in the hot and dense medium', presented at QCD `90 at Montpellier, France

(1990), preprint INT-4-1990 (Sept. 1990)5) Y . Nambu and G. Jona-Lasinio, Phys . Rev . 122 (1961) 345 ; 124 (1961) 2466) T . Hatsuda and T. Kunihiro, Phys. Rev. Lett . 55 (1985) 158 ; Prog . Theor . Phys . 74 (1985) 765 ; Phys.

Lett . B185 (1987) 304

2VTrr

(ps)m

= mps MSZ(mpsr) , (A.9)

4Tr

VTN(v)

gV my mV

_ _ V(1+fv 2Z(m,r) , (A.10)4Tr 12 g� M

444


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G-matrix elements with effective masses for mesons and nucleons