General form of hybrid derivative coupling to study dense nuclear matter and its phase transition to quark matter

PHYSICAL REVIEW C VOLUME 50, NUMBER 2 AUGUST 1994

General form of hybrid derivative coupling to study densenuclear matter and its phase transition to quark matter

Sasabindu Sarkar and Binay MalakarDepartment oj Theoretical Physics, Indian Association for Cultivation of Science,

Jadavpur, Calcutta 7000M, India(Received 4 October 1993; revised manuscript received 30 March 1994)

We propose a general form of hybrid derivative coupling model of scalar mesons to various baryonsin a field theoretical model to study nuclear matter. The strength of Yukawa point coupling andthat of derivative coupling of scalar mesons to various baryons are taken in a suitable proportioncharacterized by a hybridization parameter o.. Analytical relations are set up which express scalarand vector meson coupling constants, compression modulus and the parameter o., in terms of em-pirical values for effective nucleon mass, energy per nucleon, and baryon density of normal nuclearmatter. Density expansions of effective nucleon mass and energy per nucleon, incorporating termsup to eighth power in nuclear matter density, are given. We study equation of states of dense nuclearmatter at zero temperature considering A resonances. We also investigate the phase transition fromhighly compressed nuclear matter to quark matter and its dependence on hybridization parametera and also on coupling constants involving mesons and 4 s. We determine equilibrium pressure,chemical potential, "phase transition density, " baryon densities, and energy densities of hadronicmatter and quark matter which characterize the phase transition region. We also brieQy discusslinear response of the nuclear system in the present model.

PACS number(s): 12.38.Mh, 21.65.+f, 21.60.3z, 21.90.+f

I. INTRODUCTION

The purpose of this paper is to give a general form ofhybrid derivative coupling of scalar mesons (o) to var-ious baryons in the relativistic Geld theoretical modelof symmetric nuclear matter. Zimanyi and Moszkowski

[1] assumed that scalar (cr) mesons couple only to thederivative of the nucleon wave function and to isoscalarvector (u„) mesons, in contrast to Walecka's model [2]for nuclear matter, characterized by too large a compres-sion modulus (K) and too small a nucleon effective mass(M'), in which scalar mesons have Yukawa point cou-pling to nucleons. They [1] obtained a smaller value ofthe compression modulus than that in Ref. [2]. RecentlyGlendening et al. [3) used a hybrid form of derivativecoupling in which a scalar meson couples equally to thebaryon wave function and its derivative and they foundK to be 265 MeV. They [3] further observed that the ef-fective nucleon mass M~ ——0.796M~, appearing in theirmodel is slightly too large in view of recent estimate ofJaminon and Mahaux [4] for M~ ——0.69M~ at satu-ration density. Recently Sharma et al. [5] estimated Kto be 300+25 MeV. Heide and Ellis [6) remarked thatthere is some uncertainty in the above estimates and sothey discussed results calculated for a range of values ofK and M~. In our model we take strength of Yukawapoint coupling and that of derivative coupling in the ratio(1—cs)/cs. Suitable value for the hybridization parametero. is chosen which yields satisfactory results for compres-sion modulus, effective nucleon mass, binding energy pernucleon, saturation nuclear matter density p&. It may benoted that the above-mentioned three models of nuclearmatter correspond to particular values of the parameter

o. appearing in our model. Recently some investigators[7,8] found that the spin-orbit splitting calculated fromthe model of Zimanyi and Moszkowski (o.=l), in con-trast to that obtained from Walecka's model (n=0), istoo small when compared to the experimental result. Inthis case our model for a value of o, =

4 may give a bet-ter result. However, the derivative coupling model is notrenormalizable. In this connection Glendening et al. ob-served that since nuclear Geld theory is an eKective one,this is not a "weighty objection. " Another alternativescheme used by some investigators [6,9,10] is Boguta'smodel [10] of nuclear matter incorporating scalar mesonself-interactions. Boguta's model [10],with nucleon only,is renormalizable but loop expansion may not be conver-gent in view of the fact that the recent result [11] fortwo-loop correction is very large and several times theone-loop correction in the Walecka model. As we includea massive spin-& particle-like delta resonance which islikely to play a significant role in nuclear matter awayfrom equilibrium then even Boguta's model [10] involvingnucleons and delta resonances does not remain renormal-izable. It is found that for M* = 0.7 M at saturation den-sity the coefBcient of o term in Boguta model becomesnegative (implying no lower bound in energy spectrum)[6] when bulk modulus K is less than 320 MeV. In view

of this fact and estimates for K and M* given in [5,4] we

may have some difBculty in working with Boguta's model[1o].

Analytical relations are given which determine thescalar and vector coupling constants and also the hy-bridization parameter o. in terms of empirical values ofbulk properties of nuclear matter like density p, bindingenergy per nucleon (s/p —M~), and effective nucleon

0556-2813/94/50(2)/757(14)/$06. 00 50 757 1994 The American Physical Society

758 SASABINDU SARKAR AND BINAY MALAKAR 50

mass M~ at saturation. Zimanyi and Moszkowski (ZM)[1] gave similar relations which hold for the particularvalue o. = 1. They, however, made the assumption thatthe relativistic energy per nucleon of the Fermi gas givenby ((M* + p ) ~ ) (( ) average over the Fermi sea) canbe replaced by [M' + 2M~To(p/po) ~

] ~, where To isthe average kinetic energy per nucleon at normal densitypo. They [1] remarked that this replacement of p2 by2M~To(p/po) ~ is a good approximation. They furtherobserved that the ratio of the two terms is 15/16 in highlyrelativistic limit. Our relations are free from the aboveassumption of ZM [1]. A relation is also given which ex-presses the compression modulus K as a function of bulkproperties of nuclear matter. An appropriate relation ofa difFerent form of K is given in [1] corresponding to thecase of o;=1. Using the equations derived here we makea list of all the relevant quantities characterizing nuclearmatter for the cases a = 1 (ZM), n = 4, n =

2 (Glen-dening et at. ), o. = 4, and n = 0, for comparison. Theabove relations can also be solved to find appropriate val-ues of hybridization parameter n and coupling constantsfor any set of empirical values of MN, p, and (s/p —MN)at saturation.

Effective nucleon mass MN, energy density r, and pres-sure P for hadronic matter at any arbitrary density pare determined in the usual way from a relativistic fieldequation in the mean-field theory (MFT). We have givendensity expansion of effective nucleon mass M~ and theenergy per nucleon (s/p) showing explicitly leading termscontaining integral powers in nuclear matter density p forany value of hybridization parameter o, and also some im-portant fractional powers of p. ZM [1] did the same foro=1 but retained some terms partly up to p and ex-cluded all fractional powers of p other than p / and p /

in the expression for (s/p), thus neglecting small higher-order relativistic corrections. It is found that the seriesexpansions for M~ and (s/p) derived in this paper arequite accurate even up to saturation density po. In thedomain of very low-density nuclear matter, s/p exhibitsa low maximum and this indicates the existence of liquid-gas phase transition [12].

In dense nuclear matter, occurring in neutron star andheavy-ion collision, effects of baryons other than nucle-ons need to be considered. Some investigators [6,9,10]have considered nonstrange baryons like nucleons anddelta (4) resonances at zero temperature, and this is ap-propriate in a heavy-ion collision characterized by rapidcompression. It is found that delta (b, ) particles appearat certain nuclear matter density which depends uponthe parameter o. and the ratio of coupling constant ofscalar meson to delta to that for nucleon. Effective nu-cleon mass and relative delta population in delta excitednuclear rnatter are also studied.

When nuclear matter is greatly compressed we expectdeconfinement of quarks which are the constituents ofhadrons and there is a phase transition from hadronicmat ter to quark mat ter. We determine the phase-transition region where both pressure and chemical equi-librium exist between the two phases for diferent valuesof hybridization parameter o.. It is found that the phasetransition also depends upon the coupling constant of the

scalar meson to nucleon and delta. The MIT bag model[13]of quark matter is assumed here. Knowing the phase-transition pressure we determine the baryon density andenergy density of hadronic matter occurring at the begin-ning of the coexistence region of the two phases and thecorresponding quantities for quark mat ter when hadronicmatter is fully converted into quark matter. It is also ofinterest to find the baryon density in the mixed phaseregion at which energy per baryon for hadronic matteris the same as that for quark rnatter. All these findingsfor various values of n are listed for comparison. We alsostudy equation of state showing pressure versus energydensity in the case of quark matter and delta excited nu-

clear matter for various values of o. , graphically, and alsoindicate there the above-mentioned phase-transition re-gion. The equation of state is useful in the study of thedynamics of compressed nuclear matter.

Finally, we briefly discuss linear response (e.g. , elec-tromagnetic) of the nuclear system. Following the pro-cedures of Matsui [14], Wehrberger [15], and Cohen [16]we give a brief formulation of linear response theory todetermine the response of a nuclear closed core to a per-turbation (due to a valence baryon outside the core) inthe framework of the hybrid derivative coupling model.The above theoretical formulation is done in the nuclearmatter limit.

II. GENERAL FORMOF HYBRID DERIVATIVE COUPLING

IN FIELD THEORY OF HADRONS

As discussed in Sec. I, we take Yukawa point couplingand derivative coupling of scalar meson to baryons ina suitable proportion characterized by the hybridizationparameter o.. Then using notation and convention of [1,3]we consider the following form of Lagrangian density forsymmetric nuclear matter:

L = ). ~

1+ o'~

(0a[~yu~u —gvBgu~u]@B)( gggyl7 4

M~ )

(BuoBuo —mso ) —4~u u)u + 2imv~u~u~ (1)

where g~, o, and w denote the fields of the baryons oftype B, the scalar (o) meson, and vector (w) meson, re-

spectively, mp and m~ are scalar and vector meson mass,respectively, and w& ——0&u —0 w&. As mentioned inSec. I, the nuclear matter field theory of previous in-

vestigators [1—3] can be obtained from (1) by choosing aspecific value of parameter o.. The parameter o. is chosenin such a way as to get satisfactory values for bulk prop-erties of nuclear matter. It is convenient to work withthe following transformed Lagrangian:

L = ) Q~(zpugu —M& —gv~pucuu)Q~B

(~uo.~"o —mso ) —4~u (u"" + ,'mv~uur" (2)—

50 GENERAL FORM OF HYBRID DERIVATIVE COUPLING TO. . . 759

obtained from (1) by rescaling the baryon wave function following manner:

WB~I1+n

M I Ca( gSB&~

Ma )(s) 1 —n(1 —(B)(1 —M* )

The effective mass Ma appearing in (2) is given by

~

1+n ~MB. (4)Ma )

The part of the transformed Lagrangian involving inter-action between scalar mesons and baryons is of the form

( gSao l~int = ) gsa&

~

1+nM ~

PBQBMa )= ) gsa[& n& (gSB/MB) + n tT (gSB/MB)

Bn'o'(—gsa/MB) + ' ' ]OB4'B', (5)

which shows highly nonlinear coupling between scalarmesons and baryons involving higher powers of o..

In the Incan-field approximation, the field equationsfor scalar and vector mesons for uniform static matterare

where MN ——MN/MN and Ma = Ma/MNRelation (4) (for the case when B refers to a nucleon),

(7) and (10) lead to the following transcendental self-consistency relation for effective nucleon mass:

MN = 1 —Cs[1 —n(1 —MN)]

)~ (B B PB f( )(1 —n(1 —(B)(1 —MN))

where

S ( 2 i 2 1 2 (1+aa)' +1)f(aB) = aB~

(1 + aa) aalu2 ( 2 (1+a2B)'~' —1)

(i3)

and

aa = Ma/kFB (i4)&O = ).(gVB/mV)PB

B(6)

For nuclear matter consisting of nucleons only we have

and

mscr = ) gsa[1+ ngsao'/Ma] psa,B

(7)

MN = 1 —Cs MN[1 —n(1 —MN)]4Vr2

N(i5)

where baryon density or vector density pB and scalardensity ppB for baryon B are given by

where

+FN @FN/MN ( FN ™N) (i6)

and

PB —(4BWB) = 7B(kFB/«')

PSB = (Oat/ia)

8In (12) and (15), Cs2 is defined by

Cs = gsN(MN/ms)

Similarly, we define

(i7)

The summation over various baryons in relations (1),(2), and (5)—(7) involves spin-s2delta resonances besidesspin-2 nucleons. However, the nucleon and delta contri-butions to the Fermi-gas energy density, the scalar den-sity, and vector density differ only in their degeneracyfactors so that a simple schematic notation [6] is used inthe above relations. This was done by many investiga-tors like Boguta [10], Waldhauser et al. [9], and Heideand Ellis [6]. The appropriate form of the Lagrangianfor a spin-2 particle involving a spin-2 Rarita Schwingerspinor is used by Wehrberger [15]. In (8) and (9), ( )denotes ground-state expectation value and in (8) pa isthe degeneracy factor and k~B is the Fermi momentum.By writing the ratio of scalar Ineson coupling constantfor baryon B and that for nucleon in the form

Cv gv N (MN/mv ) (18)

4so + 2mv+0 +2 2 ) 'YakFBg( B)

B

and pressure

(i9)

P = —zm&o + zm&ao1 2 2 1 2 2

+ 6, ) pakFB[g(aa) —saa f(aa)]B

(20)

It may be noted that relation (15) reduces to an equiv-alent one given by Barranco [7] when n=l. Followingstandard procedures we obtain from the Lagrangian de-fined by (2) the expressions for energy density s

gSB/gsN = (B(MB/MN) (10)where

and using relation (4), we can express the effective baryonmass MB in terms of effective nucleon mass M~ in the g(aa) = 4(1+aa) + i2aaf(aa) . (2i)

SASABINDU SARKAR AND BINAY MALAKAR 50

The mean-field approximation of 0 and u occurring in

(19) and (20) are determined with the help of relations

(12), (4), and (6).It may be noted that in fast heavy ion collision

strangeness changing weak interaction process has littleeffect. In this case at zero temperature we deal withnonstrange baryons like nucleons (N) and deltas (4) asconsidered by Boguta [10]. In this problem the Fermimomentum of delta, k~~, required in the calculation ofM&, e, and I', is determined by the following conditionof chemical equilibrium:

S r (kFr ) = IJ~(kF~),

where in general the chemical potential of baryon B withmomentum k~~ is given by

CV = [IH + 1 —EF] (27)

where

(28)

is the binding energy per nucleon at saturation. Com-bining the relations (4), (15), (13), (6), (19), and (26) wealso find

3

Cv ——[(2 —M* —2o.(1 —M*) }f (a) —EF] . (29)2p

Comparing (27) and (29) we get the following expressionfor hybridization parameter o.:

PB(kFB) —gVB~0 + EFB

Following Boguta [10] we assume

(23) (1™)'6 = —[3+ 1 —'EF —(-1 —-'M*) f(a)] f(a)

gVE = gVN (24) Inserting this value of n in (25) we have

III. DETERMINATION OF COUPLINGCONSTANTS) HYBRIDIZATION PARAMETER

AND COMPRESSIBILITYFROM BULK PROPERTIES OF NUCLEAR

MATTER

1 —M* M3

[1 —o.(1 —M*)]' pf (a)(25)

In the following we evaluate coupling constants de-fined by (17) and (18) and a in terms of bulk propertieslike baryon density pN or equivalently Fermi momentum

kF~, effective nucleon mass M~, and binding energy pernucleon (s/piv —Miv) of normal nuclear matter. Confin-

ing our calculation to nucleons only we omit the subscriptN for nucleon. We derive from (12)

8(l —M')4 f2(a) Mt-"s =

[2IE+2 —EF —M*f(a)]' p(31)

The function f(a) occurring in (30) and (31) and definedby (13) has the following expansion:

f(a) = 1 —io(kF/M*) + ss(kF/M') —5's(kF/M*)

+,",,'s (kF /M') ' + (32)

In view of the presence of the factors (1 —M*) in (30),(1—M*) in (31), and occurrence of the term (1 —EF) inEq. (27) for Cv2 we can conjecture that in the case when

the difference (1 —M ) at saturation is diminished, thequantities 1/n, C&~, and Cv2 are all accordingly decreased.These conjectures are confirmed in Table I which liststhese quantities.

I et us write the compressibility K of nuclear matterfor any value of density p in the form

Using the fact that for normal nuclear matter (P = 0) K = 9(BP/Bp) = 9p(Op/ctp) . (33)s/p = p —P/p = Cv (p/M ) + EF,

we obtain

(26)U~i~g (23) and (13) and differentiating (15) with respectto kF we find

K/(9M) =—1 —M' M*'

f(a) EF ~M+f (a)EF 1 —n(l —M*) )

+ — +kF

3E~

Inserting the value of n given by (30) in (34) and using (26) and (28) we obtain a bulk modulus at saturation:

K M*2 1 —M*

9M EF~~ f (a)(1 —M') f (a)—2+ 3(1 —M') —,

, + 3(M* f (a)EF ) 1 —(1/2) EF —(1/2) M*f (a) + 8

kp2+ +1 —EF+1E.3' (35)


TABLE I. Hybridization parameter a, effective nucleon mass M', Fermi momentum kz, binding

energy 6/p —M, Cs, Cv related to scalar and vector meson coupling constants and compressionalmodulus K.

0.000.250.500.751.00

M*/M0.540.73250.790.820.85

kp(fm ')

1.331.331.331.331.33

6/p —M(MeV)

—15.75—16.00—16.00—16.00—16.00

&s357.40235.00195.20180.00169.50

Q2

273.80136.0092.8773.0059.10

K(MeV)

540307265239255

The right-hand sides of (27), (30), (31), and (35) refer tonormal nuclear matter.

and

P=I3P) (39)

IV. DENSITY EXPANSIONOF EFFECTIVE NUCLEON MASS

AND ENERGY PER NUCLEON

C's po/

Bv = Cvpo/M (37)

It is convenient to express physical quantities in termsof the following dimensionless scalar and vector couplingconstants and normalized baryon density (omitting thesubscript N for nucleon in the following):

K = KFQ/M, (40)

0 P P (41)

where po and Ego are the density and Fermi momentumat saturation and K is the dimensionless Fermi momen-tum for normal nuclear matter. Starting from Eq. (15),using the property of the cubic equation [17], we expressin the following dimensionless effective nucleon mass M*and energy per nucleon 6/p in terms of the above quan-tities and hybridization parameter n:

P=P/Po ) (38) where

3cr+ —+ 2(1+ 6crP) / cosh -cosh 11+9~ + 2r 2c c 3 (1+6aP)3/~

L' = 1Qp(MQ) 2[1 —cr(1 —MQ)]4[1+ 2o(1 —MQ)][1—a(1—Mo)j 3 +a( o) L(2 3 pL' ( & ) p1+2a(1—Mo ) [1—a(1—Mo )] 5 (M')

- 21pB + 1 1—M' + M+ + 3 K2p2/3 — 3 p4/3K4 +.. .2 P 1—a(1—M') 10M. 56M.3

(42)

(43)

(44)

(45)

The above expressions help us to derive density expansions for M* and e'/p. The expansions for the special caseo.=0 corresponding to the Walecka model are given below:

M+ (1—) + 3 "2/3K2 P 9 "4/3K4 + ( / )P1 i3i 25i

lo (1 -)2 56 (1 -)5 (46)

6/(PM) = [1 + p(Bv/B —1)] +——p / K ——p / K + .1— 3 "2323"434(/ 5)p2 1Q 1

— 56 (1 -)4

For the derivative coupling model of ZM [1] corresponding to n=l, we obtain

M' = 1 —p+ 3p —12P + 55P —273P + 1428P — . . + 1Q p / K [p —4P + 21P —120p +.. .]

——p K p ——p+ —p+. ]+.. .9 4/3 4 r — 64 —2 78 —3

(47)

(48)

6/(PM) = 1+ 2 p(Bv/B —1) + p —3p + lip91p5+204ps++sp2/3K2[p4p2+21P2120p4+]9p4/3K4[p64P2+ 76 ps+]+

(49)


The above relations for arbitrary value of o. which includemore terms in the expansion are given in the Appendix.

In order to get an idea about the convergence of theabove expansion we give below numerical values of pa-rameters that characterize the above expansions. For theWalecka [20] model

B = 0.4937, B~ = 0.3782, K = 0.2734 . (50a)

For the ZM model

B = 0.2520, Bv = 0.0888, K = 0.2804 . (50b)

Zimanyi and Moszkowski [1] "observed" that theWalecka model (n=0) does not saturate if the kineticenergy associated with the K term is neglected but thederivative model (n = 1) saturates even if this term isnot taken into account. This is evident from the aboverelations.

From the structure of relations (41) and (45) we find

that in the expressions for the M' and s/(pM) termsinvolving K associated with fractional powers of p orig-inate &om the nonrelativistic kinetic-energy term of theFermi gas of nucleons. Similar terms involving K andvarious factional powers of p account for the 6rst rel-ativistic correction to the quantity involving K . Theabove terms involving fractional powers of p in generalgive a smaller contribution than those involving integralpowers of p for a g 0. So it is more convenient to workwith density expansion than expansion in terms of Fermimomentum kF for n $0. In the expansion for s/(pM)various powers of Bp = p which occur in the terms asso-

where

f oo+ e ~ ~

kF2 kF4(51)

&' = 4~'/(&N C's)

Similarly for the case o. = 1 (derivative model of ZM

[1]) we have

gi& /4 ] g'&/2

(kF)'/' 4 k~+ 0 ~ ~ (53)

Referring to Eqs. (4) and (5) for B = X and (Al) (givenin the Appendix) we can define an effective scalar mesoncoupling constant g&N by

ciated with K and K signify modification of Fermi-gasenergy due to the density dependence of effective nucleonmass M*. We may note ZM [1] gave a portion of theterms occurring in (48) and (49) involving only integralpowers of p up to p . They further excluded all frac-tional powers of p except p / and p / in the expressionfor (s/pM), thus neglecting all higher-order relativisticcorrections. We may point out that the numerical coef-ficient "12" of the p4 term in the expression for s/(pM)given by ZM [1] is to be replaced by "11" as shown in(49).

Using (15) we can also construct high-density expan-sion of effective nucleon mass M*. The general form ofthis for an arbitrary value of o. is given in the Appendix.

For the special case +=0 (Walecka model) we have,

—1

gs%/gs~ = (i+ "")= 1+n[ jB+ 3a(—pB) —12n (jB) + 55n (pB)

+ 'p~ 'K'(pB—+ 2(1 —»)(~B) + ) —$6(~ K')(pB+ ) + 1. (54)

Relation (54) shows that we can somewhat identify thehybrid model with the well-known Walecka model havingdensity-dependent scalar coupling constant. It is evidentfrom the above relations and (Al) and (A3) that thee8'ective scalar coupling strength is a decreasing function

of density p for 0,=0. This density dependence of g&~may be the reason for the "observation" made by ZM

[1], given after (50b), for a=1. This observation is likelyto hold in general for some nonzero values of 0, . In view

of relations (Al), (A3), (A4), (51), (53), and the fact that

TABLE II. Phase-transition pressure P, chemical potential p, normalized density and energydensity (p/ps)s, (p ps)~; ss, s~ for hadronic matter and quark matter with B i = 178 MeV, fordifferent values of hybridization parameter o and also (p/pp) for which energy per baryon is thesame for both hadronic matter and quark matter.

0.000.250.500.751.00

P(GeV/fm )

0.2450.8151.4452.0753.035

pc(GeV)

1.481.862.122.312.52

(p/po)h.3.38.0

12.71722

(S /~s).6.51319

24.532

(~/~s).4.39.0

13.318.824.8

Eq

(GeV/fm')1.05 1.221.22 1.451.40 1.611.52 1.721.62 1.92


Cs2 decreases with increasing a (see Table II) we can inferthat M* falls more slowly as a function of density withincreasing hybridization parameter a.

where ni denotes the occupation number of nucleons inthe ith state dressed by meson fields (called quasinucleonby Matsui [14]). Different components of the vector fieldcan be expressed as

V. PHASE TRANSITIONFROM HADRONIC MATTER TO QUARK

MATTERgVN +V

Coo——

2 P= g n;mvM

(61)

32pBq =

3 2vq (55)

For sufBciently high baryon density hadronic matter isexpected to undergo a phase transition into quark mat-ter. As we work in the domain of nonstrange hadronicmatter at zero temperature, we consider massless quarksof two Qavors which come in three colors. In this casebaryon density pBq and energy density eq of quark mat-ter are given by

and

&v j;.mvM (62)

In the above j is the macroscopic baryon current whichis assumed to be absent before in deriving Eq. (19).For a closed-shell nucleus at rest j vanishes. The usualvariational methods yield the following relations for theeffective nucleon mass M' and baryon current density:

and

2&q+ B3 42~2 q (56)

Q2M' = 1 — ', [1 —~(1 —M')]' ) n;i

(63)

where pq is the chemical potential of quark matter and Bis the bag constant [13] [note to be confused with dimen-sionless coupling constant B defined by (36)]. PressurePq of quark matter at zero temperature is given by

and

where

~ g~ XiM+2j —g ni (64)

2m2(57)

To determine the phase-transition region at zero temper-ature we use Gibbs criteria of equality of pressures andbaryon chemical potentials of the two phases expressedby

|-"v.X; =k; — j. (65)

Following Landau's prescription Matsui [14] definedthe energy of dressed nucleon ei and nucleon interactionf;~ inside nuclear matter by the following relations:

Ph, =Pq=P, (58)beiv = ) e;bn; (66)

PBh =PBq =3Pq =Pc & (59) and

where suKxes h, q, and c refer to hadronic matter, quarkmatter, and phase-transition region or mixed phase re-gion, respectively.

be; = ) f;,b'n~ .

Using previous relations we have

(67)

VI. LINEAR RESPONSE IN NUCLEAR MATTER Q2e;=p +[x +M'] ~ (68)

We give a brief formulation of the theory of linear re-sponse (e.g. , electromagnetic) of the nuclear system inthe &amework of hybrid derivative coupling model. Re-stricting our analysis to nucleons only, energy density cor-responding to the Lagrangian given by (2) can be writtenin the following form, using the procedure and notationof Matsui [14] in the case of nonvaiiishing baryon currentdensity:

and

Cv M' BM'M' [x'+ M'2]»2 Bn

C'v xM [x'+ M* ]'&' Bn (69)

In the limit when the macroscopic current j —+ 0 we have

ik; —,~l +M'M2 ) (6o)

2 M2 2 M2 2 Cs2 [1 —~(1 —M~)]2- Z/2

and

C2 M*SM2 E*

2

(7o)


Bj k,. ( CV2 p2 1+ V

where

E» (k2 + M»2)1/22 2

Inserting the above relations into (69) we obtain

E'E'i j 4 FN)

C2 M*' ) C2(73)

In the above relations Cg, defined by

Cs = gsNM/m~ I (74)

is related to an equivalent scalar coupling constant gpNwhich is a function of hybridization parameter n and nu-

cleon mass M' at saturation density pp. In general, we

have

gsgyI = go~I [1+2n(1 —M*)] [1 —n(1 —M*)]

x[1+a(1 —M*)((~ —1)] (75)

where the valence baryon associated with the suKx B'may be a nucleon or a hyperon like A. The quantity (~occurring in (75) is already defined by (10).

Using the method of derivation of (71) and the nota-tion of Cohen [16],the total baryon current for closed coreplus a baryon (which may be difFerent &om a nucleon),with momentum p~ and associated vector coupling con-stant gv~ can be written in the following manner:

t ggI3igsI3 M~hh= ——~

Pp0 q ms Ep

(1 —VpV. pa/E* ) Ips 'I (78)

with

&T(qp = o, ~q~= q) = 1 —gvD(q)IIT'(q)

1D(q) = (, ,

).

(79)

(80)

where the subscript B' refers to nucleon or A. Thischange in the Hamiltonian perturbs the wave function ofthe core nucleons which can now be expanded in a com-plete set of positive- and negative-energy states. Usingthe standard procedure of linear response theory [15,16]one can evaluate the additional baryonic current con-tributed by the nucleons of the core perturbed by thebaryon outside the Fermi sphere and derive relation (76).In electromagnetic response a photon can interact withthe valence baryon directly (single-particle magnetic mo-

ment) or indirectly by polarizing the closed-shell core sothat the perturbed core is associated with nonvanishingcurrent [15].

It is of interest to note that II& defined by (77) is ob-tained from the polarization insertion IIT (qp, q) depend-ing upon four-momentum transfer q„by putting qp = 0and q = 0+. For elastic magnetic scattering of electronby nucleus we need to know IIT (qp = 0, q) to determinethe core response of the target nucleus. We may notethat the transverse dielectric function eT (q~), which isdefined as the ratio between the total transverse currentdensity j(q~) in the medium and the applied external cur-rent jp(qp), is related to the polarization insertion IIT (qp)in the following manner for the case qp

——0 and q g 0:

I gV&gVNII~

I +v III ~~ (76)An explicit form of IIT (q) is given by Furnstahl [18]. Forstatic uniform perturbation and p~ or k~ [occurring in

(76) and (71)] equal to kF, the above dielectric functionreduces to the following form:

where

T EgFN

The above relation, derived by using the self-consistency method of Matsui [14], is cast in the formwhich was obtained in the linear response theory byCohen [16]. In (76) IIT is the polarization insertion[random-phase approximation (RPA) -type ring at zero-energy momentum transfer] evaluated with the nucleonHartree-MFT propagator in the case of uniform staticperturbation caused by the valence baryon outside theFermi sphere. Here IIT is determined in the nuclear mat-ter limit. In (76) and subsequent relations 0 = A/p,where A is the atomic number of the nucleus and p is thedensity of nuclear matter.

For calculation in linear response theory [15,16] weneed to determine the perturbation in the appropriateDirac Hamiltonian due to some modification in the meanfields caused by a valence baryon of momentum p~ addedto the closed nuclear core. This is given by

eT (qp——0+, q = 0) = 1 + gv

m~ EFN EPN(81)

Then the total convection current density defined by (76)for a valence nucleon just outside the Fermi sphere be-comes

1 ky0 p

(s2)

It is seen &om (82) that there is practically no en-

hancement of the convection current j and the result forisoscalar magnetic moment agrees with the Schmidt value

[15]. The above result for energy momentum transfer

qp ——0 is independent of any reasonable model of nuclearmatter which should yield empirically known values ofbinding energy per nucleon and saturation density. Inthe case of elastic magnetic scattering of electron by anucleus, transverse dielectric function eT (qp

——0, ~q~ g 0)and j explicitly depend upon effective nucleon mass M'at saturation density so that different models of nuclearmatter characterized by different M* at saturation can

GENERAL FORM OF HYBRID DERIVATIVE COUPLING TO. . . 765

where IIs, the scalar polarization insertion, is given by

k211, = lli')(g, = 0) = —) n,

( (o)

EF~ )2 /1, M'2

kgb~ +, kp~' E2

3M,2l kF +E2 M* )

(84)

(85)

We should note the similarity in the structure of theresponse corrections (involving II& and Iis) determinedin the hybrid derivative coupling model characterized bysome equivalent coupling constant gs~ [see (75)] and ap-propriate efFective nucleon mass M' at saturation withthe corresponding one in the Walecka model. Cohen [16]observed that meson (especially K+) nucleus scatteringprovides a potentially useful way of determining ps.

When the valence baryon in relation (76) is A (i.e.,B = A) characterized by difFerent coupling constant (i.e.,

gv p g gv ~), total isoscalar convection current becomessensitive to the actual value of effective nucleon mass atsaturation in contrast to that when the valence baryonis nucleon. Consequently, this current depends upon theparticular model of nuclear matter used in the calcula-tion. Knowledge of isoscalar convection current of thenuclear core is necessary in the evaluation of isoscalarmagnetic moment of the system of A plus a closed-shellnucleus. For a particle-hole state in nuclear matter rel-evant isoscalar convention current j and correction toscalar density, ps, are obtained by subtracting &om theexpression given by (76) and (83) appropriate terms in amanner prescribed by Cohen [16].

VII. RESULTS AND DISCUSSION

In any calculation for properties of hadronic matter weneed to specify the coupling constants of various baryonsrelative to those of nucleons. A number of investigators[6,9,10,19) considered the role of delta resonances (4)in nuclear matter away &om equilibrium. These inves-tigators in general assumed equality of vector coupling

be compared in the case q g 0.In a similar manner total scalar density ps for the core

plus a valence baryon of momentum p~ can be deter-mined by evaluating Ops /Bnz with the help of the expres-sion for BM'/Bnz given by (70). Expressing the resultfor pg in the notation of Cohen [16], we have

ps= ps +ps(o) (i)

kg*

(2') ss E~

1 + 1 gsB9sNIIi

1 — Nll0 E~ m2s ( m2s )

constants, i.e., gy~ ——g~~. In the case of scalar me-son coupling they observed that the choice gs&/gs~ =(~(M~/M~) for (~ = 1, unlike the value gs~/gs~ = 1,leads to some efFective nucleon mass which is never nega-tive in the case of hadronic matter at high temperature.The consequence of the choice of other values of (~ (g 1)is explored in some cases as done by previous investiga-tors [9,10,19]. Recently Wehrberger et al. [21] observedthat electronuclear physics provides an important con-straint on different possible couplings of scalar meson todelta resonances. They showed that in the theoreticalcalculation the scalar coupling constant ratio gs~/gs~determined the position of the delta peak that occurs inthe measurement of transverse response in quasielasticelectron scattering. They also found &om a comparisonof their "calculation" based on Walecka's model with theexperimental result that the ratio gs~/gs~ cannot begreater than 1.2 when g~~/gv ~=1. Still there are someuncertainties which can affect the ratio gs~/g~~. Thiswas also pointed out by Heide and Ellis [6]. If the samecalculation is repeated using Boguta's model [10] or hy-brid derivative coupling model it is perhaps likely thatdifferent possible values of the ratio gs~/gs~ is to bechosen to get satisfactory results.

Different models of nuclear matter considered by Serotand Walecka [20], ZM [1], and Glendening et al. [3] cor-respond to values of a=0, 1, and 2, respectively, of ourgeneral form of hybrid derivative coupling model. In Ta-ble I we list the coupling constants Cs, C&, bulk prop-erties like Fermi momentum kF, binding energy per nu-cleon IE = (s/p) —M, effective nucleon mass M', andcompression modulus K at saturation which are evalu-ated with the help of analytical relations derived here,for the above models and also for the cases o. =

4and 4. It can be seen from Table I that K=307 MeVfor the case n =

4 is quite close to the correspondingvalue of 300+25 MeV as estimated by Sharma et al.[5]. Recently Jaminon and Mahaux [4] obtained effec-tive mass M'=0.74 M but observed that Dirac mass issmaller than this nonrelativistic-type mass. In view ofthis M*=0.7325 M for the case n =

4 given in Ta-ble I is quite satisfactory. Given (c/p —M) = —16MeV, k~=1.33 fm, and M'=0.70 M we find from (30)that the appropriate value of hybridization parameter isn=0.177 and from (35)K=330 MeV.

In connection with the appropriate choice of hybridiza-tion parameter o. it may be mentioned that recently someinvestigators [7,8] observed that purely derivative cou-pling model (ca=1) of ZM [1]unlike the Yukawa point cou-pling (n=0) of Walecka [2] leads to rather small spin-orbit"splitting" in comparison with experimental findings. Wemay note that the above "splitting" is caused by the spin-orbit interaction (5/2M)(1/r)d/dr(Uv —Ug)L. S. Thescalar and vector potentials Us and U~, involving cou-pling constants, are of the form

Us = —Cs(ps/M )(1+aggro/M)

and U~ = C&2(p/M ). The main reason for the above-mentioned findings [7,8] lies in the fact that coupling con-stants Cs and C& for the case o. = 1 are much smaller

766 SASABINDU SARKAR AND BINAY MALAKAR

than those for the case o. = as can be seen from Table I.Taking o. to about 4 and using corresponding Cs and C&from Table I, we may obtain a better result for spin-orbitsplitting.

Table I shows that as o. increases &om zero, M* ac-cordingly increases but the compressional modulus pro-gressively decreases, leading to a softer equation of state.This softening is evident &om Fig. 1 where binding en-ergy (s/p —M) is plotted against (p/po) /s for the caseso.=0, 4, and 1. In the very low-density region as shown1

in a different scale in Fig. 1, there is a low maximum of(z/p —M) at some value of (p/po) / beyond which thepressure is negative until (p/pp) = 1. This unstable re-gion implies the existence of liquid-gas phase transition.

Density expansion relation (46), (48), and (Al) for di-

mensionless effective nucleon mass M' = M*/M givesquite accurate results even for saturation density. Termsup to (p/po) in the above expansion lead to results whichagree within 1% of exact results for the case o. = 0 and

4 and within 2.7% of exact value for the case a =z in

normal nuclear matter. When terms up to (p/po) areconsidered the above "agreement" becomes almost com-plete for n = 0 and 4 and falls short of 1% of exact valuefor o = z~. The series expansions (47), (49), (A2) givingbinding energy per nucleon (s/p —M) for different valuesof n have similar accuracy. It may be noted that in thedensity expansion of (s/p —M) according to the Skyrmemodel [22], generally terms up to (p/po)2 are taken intoaccount.

Using analytical relation (12) for effective nucleon massM', we plot (M'/M) versus (p/po) / in Fig. 2, both forpurely nuclear matter and delta (b, ) excited nuclear mat-ter, for different models characterized by o. = 0, 4, 2, 4,1 and gs~/gs~ = M~/M~. Figure 2 shows that there

641.0

0.548——

-0.5

1.0l

0.5J—

/4I

&,='I/2

0!

5=0

(p p )"'3

FIG. 2. Ratio of e8'ective nucleon mass and bare nucleonmass as a function of (p/po)'/ for pure nuclear matter shown

by a solid line and delta excited nuclear matter by a dashedline. Results are given for a = 0, -', -', —,1.

is a rapid drop in effective nucleon mass M~ as soon asdelta resonances are excited. One of the possible reasonsfor this may be understood from the structure of rela-tion (7) involving scalar densities ps~ and ps~ whichdetermine o. and consequently MN. The associated fac-tor gg~(1 + ogs~o/M~) of pea is greater than theassociated factor gg~(1+ ogs~cr/M/v) of pg/v for thechoice (g~~/gs/v = M~/M~) ) 1 in (7). When deltasare excited the above choice leads to a considerable con-tribution of delta particles in the determination of cr andthis is likely to cause a more rapid drop in M* with den-sity than what occurs when only nucleons are considered.In this connection we should keep in mind that the rela-tive population of deltas goes to 80% in the large-densitylimit. We further found that M* decreases more andmore slowly when n is gradually increased. The possi-ble reason for this is that effective scalar coupling con-stant g&~ determined by (54) is a decreasing function ofdensity and further g&N decreases more rapidly with in-

creasing o.. In this connection it is to be noted that thevariation of effective nucleon mass affects the calculationof production of new particles in heavy-ion collision [8].

Relative delta particle population as a function of(p/po) ~/s is shown in Fig. 3 for different cases correspond-

-1,01.0

g =Q g, =l//'

0— C:o Q. Q— 3/4 Q 1

—160 0.5

CLr

a

FIG. 1. Binding energy per nucleon (c/p —M) as a func-tion of (p/po) /, p/po being the normalized nuclear matterdensity, for three values of hybridization parameter: o. = 0(Walecka), o, = — (present), and o = 1 (ZM).

FIG. 3. Relative delta population, pa/(par+pa), as a func-tion of (p/po) / for n = 0, 4, ~, —,aud 1.


ing to o=0, 4, —,', 4, 1, and gs~/gsiv = M~/MN. Itappears from Fig. 3 that any particular value of relativedelta population occurs at increasing value of normal-ized density p/po when a is gradually enhanced. Thismay be due to the fact that M~ and also the difference(M& —MN ) decrease with density more slowly when o. isincreased. We may note in this connection that recentlyEhehalt [23] mentioned that in Au-Au and Ne-Ne colli-sions about 30% of baryons are deltas (b) when densityof compressed matter is about three times the ground-state nuclear matter density. Figure 3 shows that relativepopulation versus compression in the case a = 4, unlikethat for other values of n, is fairly compatible with thefindings of Ehehalt [23].

In Fig. 4, we plot pressure P against p4 for pure nu-clear matter and delta excited nuclear matter for differ-ent coupling constant ratios gs~/gsiv=1. 313, 1.4, and 1.5in the case when the hybridization parameter is n =

4and also for the quark matter with B ~ = 178 MeV[24]. The "crossing" of the curve for quark matter withthe other curves in Fig. 4 determines the pressure P,chemical potential p, for phase transition from hadronicmatter to quark matter. We also plot in Fig. 5 curvesfor the above cases showing (s/p —M) as a function ofcompression (p/po). This enables us to determine, fromthe crossing, the value of the compression (p/po), (alsocalled phase-transition normalized density) for which en-ergy per baryon (s/p), or energy density s, is the samefor both quark matter and hadronic matter (a = 4).With onset of delta excitation in nuclear matter en-

ergy per baryon is decreased and we have softer equa-tion of state. Consequently, the above-mentioned phase-transition density increases from pure nuclear matter todelta excited matter and also when gs~/gsN is increased.Figure 5 further shows that the density, at which thedelta channel opens, is decreased when coupling con-stant ratio gsa/gsN is enhanced from 1.313 to 1.4. Itis found that when gs~/gsN is further increased to 1.5delta excited hadronic matter has a second minimum in(s/p —M) at —10.51 MeV, implying a possible existence

464

304—

~ 224—I

CL,

144—

-16~ I

Q 1 3 4 5 6 7 8

Compression I /(Q

I

10

FIG. 5. (s/p —M) as a function of compression (p/po) for—- —-, quark matter with B = 178 MeV; —,pure nu-clear matter; and delta excited nuclear matter with couplingconstant ratios ——,gsa/gsN = 1.313;——,gs&/gsN = 1 4;—"—,gsg/gsN = 1.5. The parameter a is taken to 4.

of metastable state [10]. In Fig. 5, we have progres-sively increased the ratio gsa/gsiv to explore possibleexistence of metastable state which is found to occurwhen gsn/gsN=1. 5 for ot = 4. This metastable state isruled out if comparison of experimental result, concern-ing quasielastic electron scattering, with appropriate the-oretical calculation unambiguously shows that gsa/gsNshould be less than 1.5 when calculations are performedin the present model.

For a comparative study of different hybrid derivativecoupling models we plot curves showing P versus p inFig. 6, similar to those in Fig. 4, for hybridization pa-rameter a = 0 [20], a = 4i, a = 2i [3], a = 4, andcr = 1 [1] and obtain the phase-transition pressure P andchemical potential p in a similar way. In this case it isassumed that for hadronic matter gsa/gsN. = Ma/MNand for quark matter B /' = 178 MeV, considered by

890—3575

E

) 590—

1

ui 290—CL

—100 10

(GeV )

15

FIG. 4. Pressure P as a function ofp, p being the chemicalpotential, for ————,quark matter with B = 178 MeV;—pure nuclear matter; and delta excited nuclear matter withcoupling constant ratios ——,gsa/gsN = Ma/MN = 1.313;

gsa/gsN —1.4; and —"—,gsa/gsN = 1.5 withCI = 4.1

m

—2375)—1775L

~~1175Q

575

-250 4 8 12 16 20 24 28 32 36 40 44

(GeV )

FIG. 6. P as a function of p, for several values of hy-bridization parameter o. = 0, —, —, —, and 1 takinggs&/gsN = M&/MN Similar curves a. re shown for quarkmatter with B = 178 and 145 MeV.


Serot and Walecka [24], and 145 MeV, used to fit themass of hadrons [25]. It may be noted that lattice gaugecalculation shows that B / —200 MeV [26]. For a sim-ilar comparison we display in Fig. 7 curves, like thosein Fig. 5, showing (s/p —M) against (p/pp) for valuesof n, gs~ jgsiv and B / as considered in Fig. 6. It isfound from Figs. 6 and 7 that for increasing value ofhybridization parameter o., which corresponds to softerequation of state, the above-mentioned phase transitionoccurs at a higher density of compressed nuclear matter.This occurs also when the bag constant B is increased.From the crossing of the curves for nuclear matter andquark matter as in Fig. 7 and also from Table II we getthe ratio (p/po), in the phase-transition region for whichthe energy per baryon is the same for both nuclear mat-ter and quark matter. This occurs for (p/po), =4.3, 9.0,13.3, 18.8, and 24.9 when the hybridization parametero. = 0, 4, 2, 4, and 1, respectively. Phase-transitionpressure P and chemical potential p, for these values ofo. can be obtained from Fig. 6 and Table II.

Equation of states of delta excited nuclear matter (tak-ing gg~/gsN = 1.313) corresponding to cr = 0, 4, 2, 4, 1

and also for quark matter with B ~ =178 MeV are shownin Fig. 8 by plotting pressure P against energy density E.Phase equilibrium exists along the "horizontal line seg-ments" which connect curves of nuclear matter and quarkmatter at appropriate points. To determine baryon den-sity for any value of chemical potential p, we have plottedp against p/po in Fig. 9 for the above-mentioned difFer-ent models of hadronic matter and quark matter.

Knowing some phase-transition pressure P, (also re-lated to horizontal line segments in Fig. 8) and chemicalpotential p, [defined by (58) and (59)] from Fig. 6, weare able to determine compression (p/po)~ for hadronicmatter with the help of Fig. 9. From Figs. 6, 8, and 9we can determine normalized density (p/po)h and energydensity Ep, in the case of delta excited hadronic matterfor o. = 0, —, 2, —, 1 which exist at the beginning ofphase equilibrium region and similar quantities (p/pp)&,Eq for quark matter when hadronic matter is completelyconverted into quark matter. Barz [27] observed that thelatent heat of the Brst-order phase transition, E'q E'p be-tween hadronic matter and quark matter can be consid-

C3

D

1.0

0.5I—

0'—

//8X'

/

—0.5I—

—1.0—1.0

!—0. 5 0 0.5

log ( (GeV/ frn )10

1.0 1.5

FIG. 8. Pressure P as a function of energy density c fordelta excited matter for difFerent values of hybridization pa-rameter o. —- —- o; = 0 ——a. = l. 1

)—.. .—,a = 4; ——,a = 1 taking gsa/gsN = Ma/M~, andalso for quark matter —,8 = 178 MeV.

80

ered as one of the observable signatures of quark matter.Table II lists the above-mentioned quantities characteriz-ing phase transition for 8 ~ =178 MeV. %e also show inTable II normalized density (p/po), for which the energyper baryon is the same for both delta excited nuclearmatter and quark matter which can be obtained fromFlg. 7.

Recently several investigators [28—30] studied theabove phase-transition problem. Ellis [30] found that thenumerical values of above-mentioned normalized densi-ties and energy densities (p/po) a, (p/po) ~; sa, s~ char-acterizing the erst-order phase transition between nu-clear matter and quark matter are 8, 15.5; 1.53 and 3.42GeV/fm, respectively. The corresponding quantities forthe case o. =

4 characterizing the hybrid derivative cou-pling model are quite close to these findings of Ellis [30]in comparison with other values of o. as can be seen &om

984~

60

-84I-

— 584.

d, c0!

45M eV, )

78 f8eV )

~ 40

3 84i- 20

184'-

0 12 16 20

( I'/f'0)24

I

28

00

FIG. 7. (s/p —M) as a function of p/pct for the cases men-tioned in Fig. 6.

FIG. 9. Fourth power of chemical potential (p, ) as a func-tion of compression (p/po) for the cases considered in Fig. 8.


the Table II. The model characterized by o. =4 also

yields bulk properties of nuclear matter shown in TableI, which are in close agreement with plausible empiricalresults.

VIII. SUMMARY

In our study of the general form of the hybrid deriva-tive coupling model we consider a suitable combination(characterized by a parameter a) of Yukawa point cou-pling and derivative coupling in order to obtain satis-factory values for the bulk properties of nuclear matter.The hybrid model with a smaller value of the parametern is likely to yield better results for the spin-orbit split-ting than the ZM model (characterized by a = 1). Wealso brie6y discuss linear response of the nuclear systemin the &amework of the hybrid model. Analytic formulasare set up which determine meson coupling constants andhybridization parameter a in terms of the bulk propertiesof nuclear matter.

Expressions for the density expansions of both the ef-fective nucleon mass M* and energy per nucleon aregiven. We study the equation of state of dense nuclear

matter consisting of nucleons and delta resonances whichcan occur in heavy-ion collision for difFerent values of o.and scalar meson coupling constant. We also study vari-ation of eHective nucleon mass and relative population ofdeltas, in delta excited nuclear matter, with density fordiferent values of o.. This has some important implica-tions on the experimental findings of nuclear physics.

In greatly compressed nuclear matter we determinethe usual characteristics of the possible phase transition&om nuclear matter to quark matter for increasing val-ues of the parameter o. corresponding to softer equationof states. We also discuss possible liquid-gas phase tran-sition in the low-density region. For the value of thehybridization parameter n =

4 and the correspondingcoupling constants given in Table I we find pp = 0.16fm 3, binding energy= —16 Mev, M"/M = 0.725, bulkmodulus K = 307 MeV, and phase-transition normalizeddensity (p/p6), = 9.0. The above results for the param-eter o, =

4 are in quite good agreement with the recentexperimental findings or other theoretical results. How-ever, the most appropriate value of the parameter o. isto be determined from the more accurate empirical re-sults for the properties of nuclear matter at saturationand possible phase transition at high density.

APPENDIX

General expressions for density expansions of effective nucleon xnass M' and energy per nucleon 6/(pM), obtainedfrom (41)—(45), for arbitrary value of hybridization parameter n are given below:

M = 1 —p+3o.p —12m p +55o. p —273o. p +1428m p —7752m p +43263n p

+16p / K [p+ 2(1 —3n)p +3(1—6n+ 12n )p +4(1 —9a+ 33o. —55o. )p~

+5(1 —12n + 63m —182n + 273n )p + ]9 "4/3K4[ —+ 128(1 3~) —2 + 78(1 18~ + 18 2) -3

+184(1 6 + 66 2 11 3) —4+ ]+ (Al)

and

1 (B6/(pM) = 1+ -p

~

—1~+ np —3cx P + lln p ——n~p~ —204a p —969n p + 9807o.

+16p / K [1+p+ (1 —3n)p + (1 —6o. + 12a )p + (1 —9n+ 33n —55n )p

+(1 —12a + 63cx —182n + 300n )p + ]3 -4/3K4[1+ 96 —+ 234(1 3 )p2 + 92(1 18~+ 18~2)p3

+ 6 (1 —6ot+ 8 n —lln )p + .]+.159 66 2 3 -4(A2)

Similarly effective mass M for any value of a has the following form in the high-density limit:

2

QI—2 g2 I

—4+F3 k—6

k~~m) + 1 o, + (1 o)2

3 8 1 + 15a + 45n2 + 30n3 4 t A. 1124+ ln —+ — A3 k~8+(1 —a)3 g 2 2)

where

X = A'/(1 —~)' . (A4)

A' in (A4) has already been def1ned by (52).


[2]

[3]

[4]

[6][7][8]

[10][11]

[12]

[13]

[14]

J. Zimanyi and S. A. Moszkowski, Phys. Rev. C 42, 1416(1990).J. D. Walecka, Ann. Phys. (N.Y.) 83, 491 (1974).N. K. Glendening, F. Weber, and S. A. Moszkowski,Phys. Rev. C 45, 844 (1992).M. Jaminon and C. Mahaux, Phys. Rev. C 40, 354(1989).M. M. Sharma, W. T. A. Borghols, S. Brandenburg, S.Crona, A. van der Woude, and M. N. Harakeh, Phys.Rev. C $8, 2562 (1988).E. Heide and P. J. Ellis, Nucl. Phys. A52B, 703 (1991).M. Barranco, Phys. Rev. C 44, 178 (1991).Jian Kang Zhang and D. S. Onley, Phys. Rev. C 44, 2230(1991).B. M. Waldhauser, J. Theis, J. A. Maruhn, H. Stocker,and W. Greiner, Phys. Rev. C $B, 1019 (1987).J. Boguta, Phys. Lett. 109B, 2516 (1982).J. Furnstahl, R. J. Perry, and B. D. Serot, Phys. Rev. C40, 321 (1989).H. P. Boal, Phys. Rev. C 30, 119 (1984) and referencestherein; 30, 749 (1984).A. Chodas, R. L. JaKe, K. Johnson, C. B.Thorn, and U.Weisskopf, Phys. Rev. D 9, 3471 (1974).T. Matsui, Nucl. Phys. A370, 365 (1981).

[15] K. Wehrberger, Phys. Rep. 225, 273 (1993).[16] J. Cohen, Phys. Rev. C 48, 1346 (1993).[17] S. Neumark, Solution of Cubic and Quadratic Equations

(Pergamon, New York, 1965).[18] R. J. Furnstahl, Phys. Rev. C $8, 370 (1988).[19] P. Levai, B. Lukacs, B. Waldhauser, and J. Zimanyi,

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[25] T. De Grand, R. L. Jaffe, K. Johnson, and J. Kiskis,Phys. Rev. D 12, 2060 (1975).

[26] H. Satz, Phys. Rep. 88, 349 (1982).[27] H. W. Barz, Phys. Rev. D 32, 115 (1985).[28] Alv. K. Holme, Phys. Rev. D 40, 3735 (1989).[29] M. Kutschera, W. Broniowski, and A. Kotlorz, Nucl.

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Documents

General form of hybrid derivative coupling to study dense nuclear matter and its phase transition to quark matter