Properties of Strange Hadronic Matter in Bulk and in Finite

1、properties of strange hadronic matter in bulk and in finite the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter comp

2、ared to several a r x i v :n u c l -t h /0005060v 2 19 j u n 2000properties of strange hadronic matter in bulk and in finite systems ju rgen scha?ner-bielich riken bnl research center,brookhaven national laboratory, upton,new york 11973 avraham gal racah institute of physics,the hebrew university,je

3、rusalem 91904,israel the hyperon-hyperon potentials due to a recent su(3)nijmegen soft-core potential model are incorporated within a relativistic mean ?eld calculation of strange hadronic matter.we ?nd considerably higher binding energy in bulk matter compared to several recent calculations which c

4、onstrain the composi-tion of matter.for small strangeness fractions (f s 1),matter is dominated by n composition and the calculated binding energy closely follows that calculated by using the hyperon potentials of our previous calculations.for larger strangeness fractions (f s 1),the calculated bind

5、ing energy increases substantially beyond any previous calculation due to a phase transition into n dominated matter.we also compare bulk matter calculations with ?-nite system calculations,again highlighting the consequences of reducing the coulomb destabilizing e?ects in ?nite strange systems.i.in

6、troduction bodmer and witten independently highlighted the idea that strange quark matter,with roughly equal composition of u ,d and s quarks leading to a strangeness fraction f s =?s/a 1and a charge fraction f q =z/a 0,might provide the absolutely stable form of matter 1,2.metastable strange quark

7、matter has been studied by chin and kerman 3.ja?e and collaborators 4,5subsequently charted the various scenarios possible for the stability of strange quark matter,from absolute stability down to metastability due to weak decays.finite strange quark systems,so called strangelets,have also been cons

8、idered 4,6.for a recent review of theoretical studies and experimental searches for strangelets,see refs. 7,8. less advertised,perhaps,is the observation made in our previous work 9,10that metastable strange systems with similar properties,i.e.f s 1and f q 0,might also exist in the hadronic basis at

9、 moderate values of density,between twice and three times nu-clear matter density.these strange systems are made out of nucleons (n ),lambda ()and cascade ()hyperons.the metastability of these strange hadronic systems was established by extending relativistic mean ?eld (rmf)calculations from ordinar

10、y nuclei (f s =0)to multi-strange nuclei with f s =0.although the detailed pattern of metastability,as well as the actual values of the binding energy,depend speci?cally on the partly unknown hyperon potentials assumed in dense matter,the predicted phenomenon of metastability turned out to be robust

11、 in these calculations 10,11. 1 the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several quite recent

12、ly,stoks and lee12have challenged the generality of the above results for strange hadronic systems.these authors constructed g matrices for coupled baryon-baryon channels,using an su(3)extension13of the nijmegen soft-core nsc97potentials14from the s=0,?1sector(to which data these potentials have bee

13、n?tted)into the unexplored s=?2,?3,?4sector.these g matrices were then employed within a brueckner-hartree-fock(bhf)calculation of strange hadronic matter(shm)in bulk.the results showed that nsystems are only loosely bound,and that charge-neutral strangeness-rich hadronic systems are unlikely to exi

14、st in nature in metastable form,in stark contrast to our earlier ?ndings9,10. this vast di?erence in the predictions for the metastability and binding of shm between following a bhf methodology,which uses an su(3)extrapolated form of the nsc97baryon-baryon potentials,and following a rmf methodology,

15、which is based on mean?elds designed to mimic the consequences of the nijmegen hard-core potential model d15,has prompted us to investigate possible origins of it.in this work we present calculational evidence for the incompleteness of the procedure applied by stoks and lee12.we do so by reproducing

16、 qualitatively their results for the instability and weak binding of nmatter in bulk, within a constrained rmf calculation in which the mean?elds are now designed to mimic the consequences of the nsc97model used by stoks and lee.the constraints imposed by us,as a check,are identical with those impos

17、ed by these authors for the composition of shm (see fig.4of ref.12).we argue that this is not the right way to identify minimum-energy equilibrium con?gurations for shm.indeed,doing the unconstrained rmf calculation with the same nsc97-inspired mean?elds,we?nd qualitatively good agreement,for f s1 w

18、here the bulk matter is ndominated,between these new results and our old results in model210.for f s1,the new unconstrained calculation results in considerably higher binding energies than ever calculated for shm,due to a phase transition into n dominated matter. the paper is organized as follows.in

19、 section ii we describe the methodology of?nding equilibrium con?gurations within the rmf formalism,and the input mean?elds entering the new rmf calculations.section iii includes the results of these new calculations for bulk shm,as well as for?nite multi-strange systems for which bhf calculations h

20、ave not been done to date.the role of the coulomb interaction in stabilizing charge-neutral strange systems is highlighted.our results are summarized and discussed in section iv,where we also comment on the applicability of the su(3)-extended nsc97potential. ii.methodology and input we adopt the rel

21、ativistic mean field(rmf)model to describe strange hadronic matter (shm)in bulk and for?nite systems of nucleons and hyperons.the model is an e?ective model where the parameters are adjusted to the known properties of nuclei and hypernuclei. we include in our extended rmf model all the1/2+baryons of

22、 the lowest su(3)?avor octet, as well as hidden-strangeness meson exchange to allow for possibly strong hyperon-hyperon (y y)interactions.here we use model1and model2of ref.9.the basic ingredients of these models are the octet baryons matrix b,the matrices v(8) and v(1) of the vector meson octet and

23、 singlet,respectively,and the two scalar mesonsand?.in addition,a coulomb term is included in?nite system calculations.the lagrangian is given as 2 the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of

24、 strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several l=trb(i?gb?g?b?m b)b ?133?c2 ?m2?2 ?g(8)v trb v(8),b +(1?)trb v(8),b ?g(1)v trbbtr v(1)?12tr m2v v?v+1 3gn= 1 2 g=g gn= 1 2 the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core

25、potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several for thenuclear interaction,measurements of the?nal-state interaction ofhyperons produced in the(k?,k+)reaction on12c

26、 in experiments e224at kek24and e885at the ags25indicate a nonrelativistic potential u(n) ,nr of about?16and?14mev or less, respectively.below we will actually vary the value for u(n) to check its e?ect on the binding energy of shm. the hyperon(y)potentials u(y) y in hyperon(y)matter,in the absence

27、of direct ex- perimental data,depend to a large extent on the assumptions made on the underlying y y interactions.in model1,which does not use?andexchanges,the potentials u(y) y are rather weak,less than10mev deep.the exchange of these hidden-strangeness mesons is included in model2,where the?coupli

28、ng to hyperons is adjusted so that the potential of a single hyperon,embedded in a bath ofmatter at nuclear saturation density0,becomes u()(0)=u() (0)=?40mev,(4) in accordance with the attractive y y interactions of the nijmegen potential model d10. the resulting u() is about?20mev,considerably more

29、 attractive than in model1.indeed the few doublehypernuclear events observed so far in emulsion require a relatively strong attractive interaction26,which lends support to model2over model1,but the actual situation for the other,unknown,y ychannels could prove more complex than allowed for by either

30、 model.all that may be said at present is that,as far as theinteraction strength is concerned,model2is a more realistic one than model1. since there appears some confusion in the recent literature12,27regarding how to calculate self consistently the properties of shm in bulk,we will ponder on the th

31、ermody-namically consistent methodology in more detail.as we will demonstrate in the following, a major property of shm within the su(3)-extended nsc97model might have been over-looked in these works.here we focus on the thermodynamically correct treatment in the rmf approximation.the extension to b

32、hf calculations is then straightforward.very recently,thermodynamically consistent bhf calculations of-stable strange matter in neu-tron stars have been performed by baldo et al.28,using the nsc89model29for the y n interactions,and by vida?n a et al.30,using the su(3)-extended nsc97model13for the y

33、n and y y interactions. in general,we can describe the system by the grand-canonical thermodynamic potential ?,which depends on the temperature t,the volume v,and the independently conserved chemical potentials.at t=0,the pressure is given by: p()=?(,t=0)/v.(5) for shm in bulk,since the isospin depe

34、ndence is usually suppressed,there are just two conserved charges in bulk which are the baryon number b and the strangeness number s. the chemical potentials of the individual baryons can be related to the corresponding baryon chemical potentialb and strangeness chemical potentials by i=b ib+s is.(6

35、) this ensures that the system is in chemical equilibrium or,in other words,that the strangeness and baryon numbers are conserved in all possible strong-interaction reactions in the medium,such as 4 the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core potential model are incorpora

36、ted within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several +n ?+n +?+n +?+ (7) the hugenholtz van-hove theorem relates the fermi energy of each baryon to its chemical potential in equilibrated matter i =e

37、 f,i = 62,(9) where i is the spin-isospin degeneracy factor.if the solution results in an imaginary fermi momentum,the particle is not present in the system and the corresponding density is set to zero.in brueckner theory,one has to solve for an equation of the form i =e i (k f,i )=m i +k 2f,i 2 m 2

38、2?b 4 4?12m 220+12m 220 + i =b,l i 2m 2 2+b 44+12m 220+32m 220+ i =b,l i k 2+m ?i 2,(11) respectively.the binding energy per baryon is then obtained by subtracting the properly weighted combination of the rest masses from the energy density of the system 5 the hyperon-hyperon potentials due to a rec

39、ent su(3) nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several e/a= 1 b = +2 the hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core p

40、otential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several u,in models1and29,10.for f s=0,there are only nucleons in the system and one gets the standard equation of state of n

41、uclear matter as function of baryon density.the equilibrium density of nuclear matter is determined by minimizing the binding energy with respect to the baryon density.the resulting minimum value of binding energy per nucleon is shown then at f s=0in the plots of fig.1.next,we increase the strangene

42、ss fraction from zero on,and the system of equations adjusts itself at each?xed value of f s to?nd the corresponding baryon densities ensuring chemical equilibrium(eq.(6).the minimum value of the binding energy per baryon as function of baryon density at each?xed strangeness fraction is then plotted

43、 in fig.1for the corresponding value of f s.in this way,one gets the binding energy of shm as function of the strangeness fraction.it turns out thathyperons do not appear at any value of f s in both models1and2.to display the dependence on the nuclear potential we chose three di?erent values,u=?10,?

44、18,?28mev.the variation in the plots of model1is quite pronounced.for u=?28mev,the minimum is at a ?nite value f s=0.6,with a binding energy per baryon of?17.4mev.for shallower potentials,this minimum disappears and slightly strange matter with f s0.1is the most strongly bound con?guration.on the ot

45、her hand,in model2,varying udoes not lead to drastic changes.the minimum in the binding energy per baryon for u=?28mev,at f s=1.3with e/a=?24.6mev,is shifted to e/a=?21.5mev for u=?18mev and to e/a=?19.6mev at a slightly higher value f s=1.4for u=?10mev.the reason is that in model2the minimum is gen

46、erated by the y y interactions which have been adjusted according to eq.(4),so that the binding energy curves in model2are not as much a?ected by changing uas compared to the e?ect of this change in model1.note that the constraint (4)ensures that purematter(f s=2)has the same binding energy,e/a=?8.9

47、mev,in all three cases.purematter is always unbound in model1due to the missing attraction in the y y channels. substantial departures from the universality(eq.(4)assumed in refs.9,10for the y y interactions occur in the most recent su(3)-extension of the nijmegen soft-core potential model nsc9713.i

48、n particular,theandinteractions are predicted to be highly attractive in some channels,leading to bound states.we wish to examine the consequences of this model in our rmf calculation of shm.the y y interactions of ref.13are implemented in our calculation by adjusting the coupling constants of the?m

49、eson?eld to reproduce qualitatively the hyperon binding energy curves shown in fig.2of ref.12for set nsc97f. all the other coupling constants are held?xed,so that we still get the hyperon potentials of eq.(3)in nuclear matter.the resulting binding energy curves,of each baryon species j in its own ma

50、tter b(j)j,are depicted in fig.2as function of density.for nucleons,we again use the parameterization tm1so as to get the correct binding energy at the correct saturation density0.note that nsc97f does not reproduce the correct nuclear matter saturation point,but gives a too shallow minimum at a too

51、 high density(see fig.2of ref. 12).no binding occurs forhyperons,and b()reaches+20mev already at rather low density,=0.1fm?3.this strong repulsive“potential”is due to the very weak underlying interaction in the extended nsc97f model which is incompatible with the fairly strong attraction necessary t

52、o explain the observed doublehypernuclear events(see26,39 and references therein).on the other hand,matter is deeply bound,by?33mev per baryon at=0.58fm?3which is twice as deep as ordinary nuclear matter,andmatter has a binding energy of?23mev per baryon at=0.39fm?3.it is clear from fig.2that a 7 th

53、e hyperon-hyperon potentials due to a recent su(3) nijmegen soft-core potential model are incorporated within a relativistic mean field calculation of strange hadronic matter. we find considerably higher binding energy in bulk matter compared to several 0.00.2 0.40.60.8 1.0 density (fm ?3) ?40?30?20

54、?1001020 b i n d i n g e n e r g y p e r a (m e v )tm1n fig.2.binding energy per nucleon (n )in nucleon matter,compared to the binding energy per hyperon (,)in its own hyperonic matter.the hyperonic parameters were chosen to reproduce the binding energy minima of fig.2in ref.12.mixture of and matter

55、 must be very deeply bound too,unless there is an overwhelmingly repulsive interaction between and hyperons.actually,the interaction between and hyperons is the most attractive one in the extended nsc97f model,giving rise to the deepest bound dibaryon state 13.independently,increasing the number of

56、degrees of freedom will also result in a more deeply bound state.this is the case,for example,when going from unbound neutron matter (=2)to bound nuclear matter (=4).therefore,one expects that matter is in fact more deeply bound than or matter alone.in the following,we will denote the parameterizati

57、on responsible for the curves of fig.2as model n. fig.3shows the binding energy of shm per baryon in model n as function of strangeness fraction.for comparison,the curves for model 1and 2from fig.1are also plotted.we performed two di?erent calculations for model n:one where the hyperon fractions i =i /b are held equal by hand (=)as done in ref.12,and the self-consistent one where the hyperon fractions are determined so as to