21 October 1999

Ž .Physics Letters B 465 1999 282–290

Meson mass modification in strange hadronic matter

Subrata Pal, Song Gao, Horst Stocker, Walter Greiner¨Institut fur Theoretische Physik, J.W. Goethe-UniÕersitat, D-60054 Frankfurt am Main, Germany¨ ¨

Received 27 May 1999; received in revised form 1 September 1999; accepted 7 September 1999Editor: P.V. Landshoff

Abstract

Ž . Ž ).We investigate in stable strange hadronic matter SHM the modification of the masses of the scalar s , s and theŽ .vector v, f mesons. The baryon ground state is treated in the relativistic Hartree approximation in the nonlinear s–v and

linear s )–f model. In stable SHM, the masses of all the mesons reveal considerable reduction due to large vacuumpolarization contribution from the hyperons and small density dependent effects caused by larger binding. q 1999 ElsevierScience B.V. All rights reserved.

PACS: 21.65q f; 24.10Jv

The study of the properties of hadrons, in particu-Ž .lar the light vector mesons v, r, f , has recently

attracted wide interest both experimentally and theo-w xretically. Recent experiments from the HELIOS-3 1

w xand the CERES 2 collaborations indicate a signifi-cant amount of strength below the r-meson peak.

w xThis has been interpreted 3 as the decrease of ther-meson mass in the medium. In an effective chiralmodel based on the symmetries of QCD, Brown and

w xRho 4 predicted an approximate scaling law for thein-medium decrease of the masses of the mesons andnucleons. On the other hand, in the quantum hadro-

Ž . w xdynamic QHD model 5,6 based on structurelessbaryons, the meson masses are found to increasewhen particle-hole excitations from the nucleonFermi sea are considered. By also including theparticle-antiparticle excitations from the Dirac sea,the meson masses has been, however, shown to

w xdecrease 7–9 .So far all investigations of in-medium meson

mass modification has been done in the nuclear

matter environment. However, presently there is agrowing interest about the possibility of boundstrange matter. In analogy to the stable strange quark

w x Ž .matter 10 , strange hadronic matter SHM com-posed of nucleons and hyperons has been speculatedw x11 to be absolutely stable or at least metastablewith respect to weak hadronic decays. It is expectedthat such strange matter may possibly be created at

w xRHIC and LHC 12 . It is therefore worth investigat-ing the properties of meson masses in the strangebaryonic matter environment. This study is not onlyof interest in itself but could serve as a signal of theformation of SHM in heavy ion collisions. In thisletter, in a complete self-consistent calculation withinthe framework of the QHD model, we examine themass modifications of both, the scalar and the vectormesons in SHM.

Let us consider the composition of SHM which isstable with respect to particle emission. The analysisof level shifts and widths of Sy atomic level sug-

w xgest 13 a well-depth of S in nuclear matter of

0370-2693r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 01044-8

( )S. Pal et al.rPhysics Letters B 465 1999 282–290 283

U ŽN .f20–30 MeV. While for strong processes,S

SN™LN and SL™J N, the energy released isQf78 and 52 MeV, respectively. Consequently,systems involving S ’s are unstable with respect tothese strong decays. Analysis of binding energies of

y w xL, and emulsion experiments with K beams 13yields well-depths of L and J in nuclear matter ofU ŽN .f28–30 MeV. However, in strong decays,L, J

LL|NJ , the system can be stable since Qf25MeV. Therefore, we consider here only the set of

� 4baryon species B' N, L,J , which can constitutestable SHM.

Up to now all investigations of meson mass modi-fication has been performed in the simple linearWalecka model with nucleons only. Considering herenonlinear self-interactions of the scalar field s inthe QHD model, the total Lagrangian is

m mLLs c ig E ym qg syg g v cŽ .Ý B m B s B v B m BB

1 1m 2 2 mnq E sE sym s qU s y v vŽ .Ž .m s mn2 4

1 2 mq m v v qd LL , 1Ž .v m2

where the summation is over all baryon species� 4 Ž .B' N, L,J . The scalar self-interaction U s s

g s 3r3 qg s 4r4 yields sufficient flexibility to the2 3

effective Lagrangian of the model and they are alsonecessary for a good reproduction of nuclear groundstate properties, in particular the compressibility. Theterm d LL contains renormalization counterterms.This model is not able to reproduce the observedstrongly attractive LL interaction. The situation canbe remedied by introducing two additional meson

Ž . Ž )fields, the scalar meson f 975 denoted as s0. Ž . w xhereafter and the vector meson f 1020 11 which

Ž .couple only to the hyperons Y . The correspondingŽ .linear Lagrangian is

Y Y ) m)LL s c g s yg g f cŽ .Ý B s B f B m B

BsL , J

1 ) m ) 2 ) 2)q E s E s ym sŽ .m s2

1 1mn 2 my f f q m f f . 2Ž .mn f m4 2

The self-consistent propagator in the medium forbaryon species B can be written as sum of the

w FŽ .x w DŽ .xFeynman G k and density-dependent G kB B

parts:

G k sG F k qG D kŽ . Ž . Ž .B B B

1m ) )s g k qmŽ .m B

) 2 ) 2k ym q ihm B

ip) ) < <q d k yE k u k y k .Ž .Ž . Ž .0 B F

) BE kŽ .B

3Ž .The momentum and energy of baryon B are k ) sm

Ž . ) Ž . w 2k y g v y g f ,k and E k s k q0 v B 0 f B 0 B) 2 x1r2m , and k are the Fermi momenta. The scalarB FB

mean fields shift the mass m of the baryons both inB

the Fermi and Dirac sea to

m) sm yg syg ) s ) . 4Ž .B B s B s B

Ž .In mean field theory MFT , the total energy of thesystem is generated by the presence of all baryons inthe occupied Fermi seas. In contrast, in the relativis-

Ž .tic Hartree approximation RHA , the effect of theinfinite Dirac sea is also included. The renormalizedtotal energy density in RHA is given by

EE sEE qV qV , 5Ž .RHA MFT B s

w xwhere EE is the usual MFT energy density 11 .MFT

The contribution from vacuum fluctuation of all thebaryons is given by

)I mB B) 4 3 )V sy m ln qm m ymŽ .ÝB B B B B2 ž /m8p BB

2 37 132 ) )y m m ym q m m ymŽ . Ž .B B B B B B2 3

425 )y m ym , 6Ž .Ž .B B12

where I s2 Iq1 is the isospin degeneracy of theB

baryon B. The mass shift of the Dirac sea from mB

to m) produces the vacuum fluctuation contribution.B

The renormalized nonlinear s-meson contribution isw x6

4ms 2V s 1ql ql ln 1ql qlŽ . Ž .s 1 2 1 228pŽ .23y l ql y l qlŽ . Ž .1 2 1 22

1 12 4y l l q3l q l , 7Ž . Ž .1 1 2 13 12

where l s2 g srm2 and l s3g s 2rm2 .1 2 s 2 3 s

( )S. Pal et al.rPhysics Letters B 465 1999 282–290284

The meson propagator in the baryonic mediumcan be computed by summing over bubbles whichconsists of repeated insertions of the lowest orderone-loop proper polarization part. This is equivalent

Ž .to relativistic random phase approximation RPA .Ž ) . Ž .Since both, scalar s , s and vector v, f

mesons are present in our model, it is essential toinclude scalar-vector mixing which is a pure densitydependent effect. Therefore it is convenient to definea full scalar-vector meson propagator DD in theab

form of a 5=5 matrix with indices a,b rangingfrom 0 to 4, where 4 corresponds to the scalar mesonand 0 to 3 the components of vector meson. More-

Ž .over, the strangeness violating scalar-vector cou-pling between the strange and nonstrange mesons areprohibited by invoking the OZI rule. We thereforeconsider couplings between s–v and s )–f sepa-rately. We shall present explicitly the calculationsonly for the s–v propagator; the expressions fors )–f propagator would follow similarly. Dyson’sequation for the full s–v propagator DD can bewritten in matrix form as

DDsDD0 qDD0P DD , 8Ž .

where DD0 is the lowest order s–v meson propaga-tor

D0 0mn0DD s , 9Ž .0ž /0 D

expressed in terms of the noninteracting s andv-meson propagators

10D q s , 10Ž . Ž .

2 2q ym q ih˜m s

D0 q s g yq q rm2 D0 q , 11Ž . Ž . Ž .Ž .mn mn m n v

y10D q s , 12Ž . Ž .2 2q ym q ihm v

where q2 'q2 yq is the four-momentum carried bym 0

the meson. Note that the effect of nonlinear s-inter-Ž .action the boson loops is to replace the bare meson

2 Ž . 2 2 2 Ž . 2mass m in Eq. 10 by m sm qE U s rEs s˜s s s2 2 Ž w x.m q2 g sq3g s see Ref. 14 .s 2 3

Ž .The polarization insertion of Eq. 8 is also a5=5 matrix

Ž .B M BP q P qŽ . Ž .Ý Ýmn n

B BPs ,Ž .M B p BP q P q q P qŽ . Ž . Ž .Ý Ým s s� 0

B B

13Ž .Ž .where each entries in Eq. 13 is summed over all the

p Ž .baryons, except for the pion loop P q . The effects

of relatively large s-width has been accounted byw xincluding the contribution of pion loop to P 9 ;s

the in-medium modification of the pion loop is ne-glected. The pion propagator D is obtained fromp

Ž . 2 2Eq. 10 with m replaced by m . The pion loop˜ s p

polarization contribution to the s-meson is

d4k3p 2P q s ig m D k D kqq .Ž . Ž . Ž .Hs sp p p p2 42pŽ .

14Ž .

In terms of the baryon propagator the lowest orderŽ .s , v and s–v mixed polarizations for the baryon

loop B are respectively given by

d4kB 2P q syig Tr G k G kqq ,Ž . Ž . Ž .Hs s B B B42pŽ .

15Ž .

P B qŽ .mn

d4k2syig Tr G k g G kqq g ,Ž . Ž .Hv B B m B n42pŽ .

16Ž .

P ŽM .B qŽ .m

d4ks ig g Tr G k g G kqq .Ž . Ž .Hs B v B B m B42pŽ .

17Ž .Ž Ž ..As in the case for the baryon propagator Eq. 3 the

Ž p .above polarization insertions except P can bes

Ž .expressed as the sum of Feynman F part andŽ . B B ŽF .density-dependent D part, i.e. P sP q

P B ŽD.. The finite D-part has the form G D =G D qG D =G F qG F =G D which describes particle-holeexcitations and also includes the Pauli blocking of

( )S. Pal et al.rPhysics Letters B 465 1999 282–290 285

BB excitations. Since the polarizations of the D-partw xare defined in Ref. 15 , we will not refer them here.

The divergent F-part of the polarization insertionscan be rendered finite by adding appropriate coun-

Ž .terterms to the Lagrangian of Eq. 1 . For the s ,each of the baryons loops and the p-loop have to berenormalized separately. For any baryon loop contri-bution to the s the usual counterterm Lagrangianw x5,6 is used

4 a zl sl md LL s s q E sE s . 18Ž .Ýs ml ! 2ls2

The coefficients a and z can be obtained by2 s

imposing the condition that the propagator in vac-Ž ) .uum m sm reproduces the ‘‘physical’’ proper-B B

w xties of s-meson 15 :

P B ŽRF . q2 ;m) smŽ .s m B B

EB ŽRF . 2 )s P q ;m smŽ .s m B B2E qm

s0 at q2 sm2 . 19Ž .m s

The renormalized s-meson self-energy for a baryonloop is

P B ŽRF . qŽ .s

2 2 2g I m yqs B B s m)s q3m m ymŽ .B B B2 42p

29 )q m ymŽ .B B2

13 ) 2 2y dx m yq x 1yxŽ .Ž .H B m20

m) 2 yq2 x 1yxŽ .B m=ln 2 2m ym x 1yxŽ .B s

3 ) 2 2y m ymŽ .B B2

2m1 s= dx ln 1y x 1yx . 20Ž . Ž .H 2ž /m0 B

For the p-loop we employ the renormalization condi-p ŽRF .Ž 2 . 2 2 w xtion in free space, P q s0 at q sm 9 .s m m s

We obtain finally

3g 2 m2 m2 yq2 x 1yxŽ .1sp p p mp ŽRF .P q s dx ln .Ž . Hs 2 2 232p m ym x 1yxŽ .0 p s

21Ž .

For the v, only a wavefunction counterterm LL sv

z v v mnr4 is required to make the F-part P B ŽF .v mn mn

finite. The renormalized v self-energy is P B ŽRF .smn

Ž 2 . B ŽRF .yg q qq q P . Employing the renormal-mn m m nB ŽRF .Ž 2 ) .ization condition in vacuum, P q ;m smm B B

s0 at q2 sm2 , we obtainm v

P B ŽRF . qŽ .mn

g 2 I m) 2 yq2 x 1yxŽ .1v B B B m2s q dx x 1yx ln .Ž .Hm2 2 22p m ym x 1yxŽ .0 B v

22Ž .As mentioned before, the mixed part P ŽM .B of Eq.m

Ž .17 does not contribute to vacuum polarization.Ž .The solution of Dyson’s equation, Eq. 8 , is

0 Ž 0 .DDsDD r 1yDD P . By defining the dielectric func-tion ´ as

´sdet 1yDD0P s´ 2 ´ , 23Ž . Ž .T L

the poles of the s–v propagator which define therespective meson masses in the medium are nowcontained in the dielectric function when ´s0. By

Ž . < <taking qs 0,0,q where qs q , we obtained in Eq.Ž .23 the transverse and longitudinal dielectric func-tions defined as

´ s 1qD0P , 24Ž .Ž .T T

q22m Ž .0 0 0 0 M´ s 1yD P 1yD P q D D P ,Ž . Ž . Ž .L s L 02q

25Ž .Ž .where the polarization insertion of Eq. 16 is now

split into transverse P sP sP and longitudi-T 11 22

nal P sP yP components. Because of baryonL 00 33

current conservation, only the 0-th component of theM Ž .mixed part P survives. Note that in Eqs. 24 and

Ž .25 the polarizations represent those which haveŽ Ž ..been summed over all the baryons see Eq. 13 .

The eigencondition for determining the collectiveŽexcitation spectrum i.e. finding the effective meson

.masses is equivalent to searching for the zeros ofthe dielectric function. In particular, for a given

< <three-momentum transfer q' q , the ‘‘invariant) 2 2Ž . (mass’’ of a meson s or v is m s q yq ,m 0

where q is obtained from the condition ´s0. In0

the present study of meson mass modifications in themedium, we restrict ourselves to the meson branch in

Ž 2 .the time-like region q )0 .m

( )S. Pal et al.rPhysics Letters B 465 1999 282–290286

Since the propagation of the strange s )–f

mesons are decoupled from that of the nonstranges–v mesons, we may follow the same procedure as

Ž . Ž .given in Eqs. 8 – 25 to obtain the effective massesof the strange mesons. In particular, a dielectric

Ž Ž ..function Eq. 23 is obtained but with the masses ofthe mesons and their couplings to the baryons corre-spond to the s ) and f. Moreover, since a linear

) 2 Ž .s interaction is used in this case, m in Eq. 10˜ s

should be replaced by the bare meson mass m2) .s

Ž .The renormalization conditions in RHA in Eq. 62 Ž .is imposed at q s0, while those used in Eq. 19m

for s and s )-mesons are at q2 sm2 and q2 sm2) ,m s m s

respectively. This difference yields an additional termw xin the vacuum fluctuation energy 15 , so the total

energy density for SHM isa qa )s B s B 22 )EEsEE q I m m ym ,Ž .ÝRHA B B B B24pB

26Ž .Ž .where EE is the RHA energy of Eq. 5 andRHA

2 2m 3 m1s sa s q dx ln 1y x 1yx , 27Ž . Ž .Hs B 2 224m m0B B

and a similar expression for a ) for s ) ; fors B

nucleons a ) s0.s B

The field equations are obtained by minimizingthe energy density EE with respect to that field. At agiven baryon density n and strangeness fractionB

Ž .f s n q2n rn , the set of field equations areS L J B

solved self-consistently in conjunction with thechemical equilibrium condition 2m sm qm dueL N J

to the reaction LL|NJ . The chemical potential ofw ) 2 x1r2a baryon species B is m s k q m qB F BB

g v qg f . The four saturation properties ofv B 0 f B 0Ž . y3nuclear matter NM : density n s0.16 fm , bind-0

ing energy ErBsy16 MeV, effective nucleon massm)rm s0.78 and compression modulus Ks300N N

MeV are used to fix the nucleon coupling constantsg , g and the parameters g and g of the ss N v N 2 3

self-interaction. The coupling constants for pure NMwithout hyperons are shown in Table 1. When hyper-ons are included, i.e. for SHM, they will contributeto EE from their vacuum fluctuations V even if theirB

Fermi states are empty. This entails a redetermina-tion of the coupling constants for the nucleons. TheV depends on the effective baryon mass, which inB

Table 1The nucleon-meson coupling constants in RHA obtained by repro-

Ž .ducing the nuclear matter saturation properties see text . TheŽ .results are for nuclear matter NM with strangeness fraction

Ž .f s0 and for strange hadronic matter SHM which includes alsoS

the vacuum polarization of hyperons. All the couplings are dimen-sionless except g which has the dimension of fmy1. The bare2

hadronic masses are m s939 MeV, m s1116 MeV, m sN L J

1313 MeV, m s550 MeV, m s783 MeV, m ) s975 MeV,s v s

and m s1020 MeVf

g g g gs N v N 2 3

NM 8.060 8.498 13.962 1.273SHM 7.917 8.498 9.471 15.017

turn depends on the scalar-baryon coupling constantsŽ Ž . Ž .. )see Eqs. 4 and 6 . Therefore, the s and s

Ž .couplings to the hyperons Y should be predeter-Ž .mined. For this purpose, we adopt the SU 6 model,

i.e. g rg s2r3, g rg s1r3 for the s–YsL s N sJ s N') )couplings, and g rg s 2 r3, g rg ss L s N s J s N

)'2 2 r3 for the s –Y couplings. Note that the nu-cleons do not couple to strange mesons, i.e. g ) ss N

g s0. The coupling constants of nucleons for thefN

SHM are given in Table 1. The couplings of v tothe hyperons can be obtained from the well-depth ofY in saturated NM: U ŽN .sg s eq yg v eq withY s Y v Y

U ŽN .s30 MeV and U ŽN .s28 MeV. The fyYL J

couplings are obtained by fitting them to a well-depth U ŽJ .f40 MeV, for a L or J in a JL, J

w x‘‘bath’’ with n ,n 11 . To determine the cou-J 0

plings of the pion to scalar mesons, g and g ) ,sp s p

we adjust them to reproduce the widths of s ands ) in free space, i.e. G 0 syI P p ŽRF .rm ats s s

q2 sm2. For a conservative estimate we considerm s

G 0 s300 MeV and G )

0 s70 MeV.s s

The stability of strange hadronic matter may beexplored by considering its binding energy definedas ErBsEErn yÝ Y m , where the abundance YB i i i i

sn rn . In Fig. 1, we present the binding energyi B

ErB as a function of baryon density n at variousB

strangeness fractions f . With increasing f , theS S

binding energy of SHM is found to increase and thesaturation point is shifted to higher density. This is a

Ž .consequence of the opening of new strangenessdegrees of freedom in the form of hyperons. At highdensities for large f values when the Fermi energyS

of nucleons exceeds the effective masses of the

( )S. Pal et al.rPhysics Letters B 465 1999 282–290 287

Fig. 1. Binding energy ErB versus baryon density n for strangeB

hadronic matter with various strangeness fraction f .S

hyperons minus their associated interaction, the con-version of nucleons at its Fermi surface to hyperonsof increasingly larger mass is favored. This conver-sion lowers the Fermi energy of the nucleons leadingto enhanced binding. A stable SHM with a maximumbinding of ErBsy38.46 MeV is obtained for alarge f f1.35 at a relatively high density n f3n .S B 0

A further increase of f enforces the Fermi energy ofS

the hyperons to increase resulting in the decrease ofbinding. This finding is quite similar to that obtained

w xin the quark-meson coupling model 16 .The variation of baryon effective masses m) isB

shown in Fig. 2 as a function of density n forB

varying strangeness f . It is observed that m) de-S B

creases with increasing n , with m) having theB N

largest decrease rate and m) the smallest at each fJ S

value. Furthermore, with increasing f the effectiveS

nucleon mass increases while the effective masses ofhyperons decrease at any density. This effect stemsfrom decrease of the nonstrange meson fields s andv and increase of the strange meson fields s ) andf with increasing f . Since the nucleons couple onlyS

to s and v, m) is increased. The hyperons, how-N

ever, do couple to all the meson fields resulting in adecrease of their effective masses with increasing f ,S

especially for J which has the strongest coupling tothe strange mesons.

In Fig. 3, we display the ‘‘invariant mass’’ m)

m2 2 ) Ž(s q yq of nonstrange s-meson, m , left pan-0 s

. ) Ž .els and v -meson, m , right panels as a functionv

of baryon density n for different strangeness frac-B

tions f . We consider first the features observed forS) < <m for small three-momentum transfer qs q s1s

Ž . )MeV top-left panel . For f s0, m is found toS s

decrease with increasing n for small values ofB

n Fn . This reduction is caused due to two compet-B 0

ing effects. The vacuum polarization which leads toreduction of m) dominates over the density depen-s

dent dressing of the meson propagator which causesan increase in m). In fact, this reduction can bes

Ž . ŽFig. 2. Effective masses of nucleons top panel , lambda central. Ž .panel and cascade bottom panel in strange hadronic matter with

strangeness fraction f from 0 to 1.5 in steps of 0.5.S

( )S. Pal et al.rPhysics Letters B 465 1999 282–290288

Ž .Fig. 3. The in-medium masses of s-meson left panels andŽ .v-meson right panels as a function of baryon density for various

< <f with qs q s1, 500, and 1000 MeV. The s-meson massesS

are for f from 0 to 1.5 in steps of 0.5. In the right panels, theS

solid and dashed lines correspond to longitudinal and transversev-meson masses, respectively.

traced back to the corresponding reduction of m) -NŽ Ž ..m in the medium see Eq. 20 . However, theN

decrease in the scalar polarization depends on m) 2,B

m2 , and m2 in a complicated fashion. For densitiesB s

Ž .above n , the density-dependent D part becomes0

increasingly dominant resulting in increase of m). Ins

SHM with f /0, the vacuum polarization contribu-S

tion from the hyperons and the nucleons causes aconsiderable suppression of m) at low densities. Ats

higher densities m) increases, however, the rate ofs

increase above ;n becomes smaller with increas-0

ing f . An explanation to this effect is as follows.S

The D-part of s propagator is primarily determinedby nucleons to which it has the strongest coupling.The increase of f causes an increase in the FermiS

momenta of the hyperons while that of the nucleonsdecrease. Consequently, with increasing f the de-S

creasing D-part of the nucleons results in a slowerrate of increase of m) with n .s B

ŽAt small value of qs1 MeV, the density depen-. Ždent effect of scalar-vector mixing is negligible seeŽ ..Eq. 25 . On the other hand, for large values of

Ž .qs500 MeV central-left panel and qs1 GeVŽ .bottom-left panel , the mixing is more effective.ŽFor the latter value, the present model may have

.been stretched to the extremes of applicability. Itgives rise to a repulsion which again leads to de-crease of m) at high density. With increasing f ,s S

the onset of this decrease or the peak positions in thefigure are found to shift to higher densities. This is amanifestation of the shift of saturation value ErB to

Ž .higher n as f increases see Fig. 1 .B S

For small values of qs1 MeV, the transverseand longitudinal invariant v-meson mass, m) andvT

m) are practically identical as evident in Fig. 3vL

Ž .top-right panel . In contrast to s mass, the vacuumŽ Ž ..polarization effect is much stronger see Eq. 22

and the density dependent effect is much weaker forthe heavier v mass. This causes for f s0 a substan-S

tial decrease of m) up to n f2n , and a subse-v B 0

quent small increase with density. The reduction ismore pronounced for SHM where the total vacuum

Žpolarization effect is stronger and the D-part mainly. )controlled by N is weaker. In fact, for f s1.5, mS v

decreases considerably even up to large densities. Asa consequence, the crossing between m) and m) ats v

qs1 MeV is shifted to lower n for the SHM.B

At large values of qs500 MeV and 1 GeV, the) Ž .longitudinal mass m solid line and transversevL

) Ž .mass m dashed line get well separated. It isvL

found that m) is reduced near nuclear matter den-vL

sity and finally increases with density attaining val-ues higher than m) . As in the qs1 MeV case, thevT

reduction in mass is stronger for SHM. Because ofstrong repulsion from the mixing at these highthree-momentum values, m) and m) never crosss v L

each other.In Fig. 4 the ‘‘invariant mass’’ of the strange

) ) Ž . )

)s -meson, m , left panels and f -meson, m ,s f

Ž .right panels are shown as a function of baryondensity n for different strangeness fractions. In thisB

case the masses are determined by the hyperons asnucleons do not couple to the strange mesons. As forthe nonstrange mesons, the masses m)

) and m) ins f

general decreases with increasing n . However, theB

( )S. Pal et al.rPhysics Letters B 465 1999 282–290 289

) Ž .Fig. 4. Same as Fig. 3 but for the s -meson left panels andŽ .f -meson right panels .

decrease is more enhanced over the large densityrange explored here. This arises due to large vacuumfluctuation contribution from the hyperons, and, inparticular, a small density-dependent part of the hy-perons stemming from the large binding in SHM. Athigh densities n G4n , in contrast to nonstrangeB 0

meson masses, the strange meson masses has highervalues for larger f . This reversal in behavior for theS

m)

) and m) results from the increase of the Fermis f

momenta and hence the D-part of the hyperons forlarge f . The scalar mass m)

) however becomesS s

sensitive to high values of q and drops at f s1.5S

below that for f s0.5.S

We have investigated the meson mass modifica-tion in strange hadronic matter as a function ofbaryon density for a fixed strangeness fraction. Theresults correspond to a metastable SHM. For a givenn , an absolute stable SHM could be obtained byB

determining the absolute minimum of the bindingenergy ErB as a function of f . In this situation, weS

have found that the masses of the mesons in SHMundergo a drastic reduction over a wide range of

density. Besides the large vacuum part, the minimumin the energy per baryon ErB at each density en-forces a smallest possible Fermi momentum andhence the density-dependent for all the baryons.

It is worth mentioning here that in the linearw xWalecka model, it has been demonstrated 5,6,17Ž .that a self-consistent inclusion of exchange Fock

term has an insignificant contribution to bindingenergy when the two parameters of this model, gs N

and g , are renormalized to reproduce the samev N

NM saturation properties of baryon density and bind-ing energy. Even the predicted values of the effectivenucleon mass, m) , and incompressibility, K , in thisN

Hartree-Fock calculation are almost identical to thew xHartree results 5 . However, the applicability of the

linear Walecka model to moderate and high densityphenomena can be misleading if the two parametersm) and K at n are not under control. In fact, it wasN 0

w xshown 18 that in the nonlinear Walecka modelwhen the four parameters, g , g , g , and gs N v N 2 3Ž .the latter two are from the s self-interaction arefitted to the same NM saturation properties of n ,0

) Ž .ErB, m , and K as in our calculation , nearly allN

the properties obtained in the relativistic Hartreecalculation differ from the mean field results only by

Žf3% even at ns10n . This is in contrast to the0.linear Walecka model results. It is therefore ex-

pected that by further inclusion of exchange termsŽ .in all the baryonic sectors and pseudoscalar mesonsin the present nonlinear Walecka model, and per-

Žforming a self-consistent calculation with the pa-rameters renormalized to the same NM saturation

. )properties , the results for ErB and m will beB

practically unaltered from the Hartree calculation.Consequently, exchange corrections from the nucle-onic and strangeness sectors should also have aninsignificant contribution and therefore neglected inthe present study of the modification of meson massesin the medium.

In summary, we have investigated the masses ofŽ .baryons N, L,J and, in particular, the nonstrange

Ž . Ž ) .s , v and strange s , f mesons in stablestrange hadronic matter. The ground state propertiesof the SHM in the relativistic Hartree approximationis obtained by using a nonlinear s–v and linears )–f Lagrangians. With increasing strangenessfraction, the effective mass of the nucleons increaseswhile that of the hyperons decreases. The masses of

( )S. Pal et al.rPhysics Letters B 465 1999 282–290290

all the mesons reveal a considerable reduction over awide density range with increasing strangeness. Thismay be attributed to a large contribution from thevacuum polarization of the hyperons which causesthe decrease in the meson masses at small densities.The larger binding for the SHM and therefore asmaller density-dependent part helps to reduce me-son masses at high densities.

Acknowledgements

S.P. and S.G. acknowledge support from theAlexander von Humboldt Foundation.

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