Strange nuclear matter within Brueckner-Hartree-Fock Theory

We have developed a formalism for microscopic Brueckner-type calculations of dense nuclear matter that includes all types of baryon-baryon interactions and allows to treat any asymmetry on the fractions of the different species (n, p, $\Lambda$, $\Sigma^0$, $\Sigma^+$, $\Sigma^-$, $\Xi^-$ and $\Xi^0$). We present results for the different single-particle potentials focussing on situations that can be relevant in future microscopic studies of beta-stable neutron star matter with strangeness. We find the both the hyperon-nucleon and hyperon-hyperon interactions play a non-negligible role in determining the chemical potentials of the different species.


I. INTRODUCTION
The properties and composition of dense matter at supranuclear densities determine the static and dynamical behavior of stellar matter [1][2][3][4][5]. The study of matter at extreme densities and temperatures has received a renewed interest due to the possibility of attaining such conditions in relativistic heavy-ion collisions at GSI, and in the near future at CERN and Brookhaven.
It is believed that at extremely high densities, deconfinement will take place resulting in a transition from hadronic to quark matter. The transition point and its characteristics will depend crucially on the equation of state of matter in the hadronic phase. It is well known that the presence of strangeness, in the form of hyperons (Λ,Σ) or mesons (K − ) will soften the equation of state and will delay the transition. Most investigations up to date are made in the framework of the mean field approach, either relativistic [6,7] or non-relativistic with effective Skyrme interactions [8]. Microscopic theories, on the other hand, aim at obtaining the properties of hadrons in dense matter from the bare free space interaction.
In this sense, Brueckner theory was developed long time ago and successfully allowed to understand the properties of (non-strange) nuclear matter starting from interactions that reproduce a huge amount of NN scattering observables. A first attempt to incorporate strangeness in the form of hyperons within Brueckner theory was made in Refs. [9,10], latter extended to investigations of beta-stable nuclear matter [11]. A missing ingredient in these works was the hyperon-hyperon (Y Y ) interaction and the results of single-particle potentials or binding energy per baryon with a finite amount of hyperons were simply orientative.
The recent availability of a baryon-baryon potential [12] covering the complete SU(3)×SU(3) sector has allowed to incorporate the Y Y potential in a microscopic calculation of dense matter with non-zero hyperon fraction [13] . The work of Ref. [13] concentrated mainly on isospin saturated systems, i.e., sys-tems with the same fraction of particles within the same isospin and strangeness multiplet: T = 1/2, S = 0 (neutrons and protons), T = 0, S = −1 (Λ), T = 1, S = −1 (Σ − , Σ 0 , Σ + ) and T = 1/2, S = −2 (Ξ − , Ξ 0 ). In this way, the complications associated to different Fermi seas for each species of the same isospin-strangeness multiplet were avoided and the G-matrix in each sector was independent on the third component of isospin.
It is well known, however, that the presence of electrons makes nuclear star matter to be equilibrated against the weak β-decay reactions for neutron fractions much larger (a factor of 10 or more) than that for protons [14][15][16]. Also, the increase of negatively charged leptons with the baryonic density will turn into a decrease when the appearance of negatively charged baryons becomes energetically more favorable. This is the case of the Σ − hyperon, since neutralizing the proton charge with Σ − instead of e − will remove two energetic neutrons (pΣ − ↔ nn) instead of one (pe − ↔ n). It is clear, therefore, that a microscopic study of β-stable nuclear matter with hyperons requires the treatment of highly asymmetric matter, both in the non-strange sector (protons vs. neutrons) and the hyperonic one (Σ − vs Σ 0 and Σ + ). In the present paper we extend the study of Ref. [13] to allow for different fractions of each species. We will also explore the effect of the recently available Y Y interaction on the single-particle potential of the hyperons, a crucial ingredient to determine the baryonic density at which the different hyperons appear. Our aim is to present a thorough analysis of the properties of the different baryons in dense matter, taking into account their mutual interactions. We will explore different baryonic densities and compositions that are relevant in the study of neutron stars.

II. FORMALISM
In this section we present the formalism to obtain, in the Brueckner-Hartree-Fock approximation, the single-particle energies of n, p, Λ, Σ − , Σ 0 , Σ + , Ξ − and Ξ 0 embedded in an infinite system composed of different concentrations of such baryons. We first construct effective baryon-baryon (BB) interactions (G-matrices) starting from new realistic bare BB interactions, which have become recently available for different strangeness channels [12].

A. Effective BB interaction
The effective BB interaction or G-matrix is obtained from the bare BB interaction by solving the corresponding Bethe-Goldstone equation, which in partial wave decomposition and using the quantum numbers of the relative and center-of-mass motion (RCM) reads The starting energy ω corresponds to the sum of non-relativistic single-particle energies of the interacting baryons including their rest masses. Note that we use kinetic energy F . This angle-average is shown in appendix A, together with the expressions that define the Pauli operator in a particular (T, M T ) channel in terms of the basis of physical states. Although we keep the index M T in the bare potential matrix elements they do not really have a dependence on the third component of isospin since we consider charge symmetric and charge independent interactions. Therefore, the dependence of the G-matrix on the third component of isospin comes exclusively from the Pauli operator, since, as can be clearly seen in appendix A, it acquires a dependence on M T when different concentrations of particles belonging to the same isomultiplet (i.e. different values for the corresponding k F 's) are considered.
In comparison with the pure nucleonic calculation, this problem is a little bit more complicated because of its coupled-channel structure. Whereas for the strangeness sectors 0 and −4 there is only one particle channel (NN and ΞΞ respectively) and two possible isospin states (T = 0, 1), in the S = −1(S = −3) sector we are dealing with the ΛN(ΛΞ) and ΣN(ΣΞ) channels, coupled to T = 1/2 In the S = −2 sector we must consider the channels ΛΛ, ΛΣ, ΞN and ΣΣ in isospin states In addition, each box G B 1 B 2 →B 3 B 4 has a 2 × 2 matrix sub-structure to incorporate the couplings between (L, S) states having the same total angular momentum J. This submatrix reads B. The baryon single-particle energy in Brueckner-Hartree-Fock approximation In the Brueckner-Hartree-Fock approximation the single-particle potential of a baryon B 1 which is embedded in the Fermi sea of baryons B 2 is given, using the partial wave decomposition of the G-matrix, by if both types of baryons are identical, or by if they are different. The labels S B 1 , S B 2 (T B 1 , T B 2 ) denote the spin (isospin) of baryons B 1 and B 2 , respectively, and |T M T is the Clebsch-Gordan coefficient coupling to total isospin T . The variable k denotes the relative momentum of the B 1 B 2 pair, which is constrained by with ξ B 1 = M B 2 /M B 1 . Finally, the weight function f (k, k B 1 ), given by results from the analytical angular integration, once the angular dependence of the G-matrix elements is eliminated. This is done by choosing appropriate angular averages for the center- including its own Fermi sea, then its single-particle potential is given by the sum of all the partial contributions where U is the potential of the baryon B i due to the Fermi sea of baryons B j . In this expression k denotes the single-particle momentum of particle B i . The non-relativistic single-particle energy of baryon B is then given by This is precisely the single-particle energy that determines the value of the starting energy ω at which the G B 1 B 2 ↔B 3 B 4 -matrix in Eq. (2) (or (3)) should be evaluated. This implies a self-consistent solution of Eqs. (1), (2) (or (3)) and (7). The Fermi energy of each species is determined by setting k to the corresponding Fermi momentum in the above expression.

C. Energy density and binding energy per baryon
The total non-relativistic energy density, ε, and the total binding energy per baryon, B/A, can be evaluated from the baryon single-particle potentials in the following way where ρ is the total barionic density. The density of a given baryon species is given by where

III. RESULTS
We start this section by presenting results for the single-particle potential of each baryon species, as a function of the baryon momentum, for several baryonic densities and various nucleonic and hyperonic fractions. We have focussed on results for the Nijmegen model (e) of the recent parametrization [12], since it gives, together with model (f), the best predictions for hypernuclear observables [17], apart from reproducing the Y N scattering scattering data as well as the other models. We will restrict our calculations to matter composed of neutrons, protons, Λ's and Σ − 's, since these last two hyperons species are the first ones to appear [11].
This is also confirmed on the recent study of β-stable neutron star matter [18] up tp baryonic In Fig. 1 we show our results for non-strange nuclear matter at normal density, ρ 0 = 0.17 fm −3 , and three proton fractions (x p = 0.5x N , 0.25x N and 0), where x N is the fraction of non-strange baryons, which in this case is 1. We also show the hyperon single-particle potentials, denoted with the label "old", obtained with the Nijmegen 1989 version of the Y N interaction [19]. On the right panel, corresponding to symmetric nuclear matter, we see that neutrons and protons have the same single-particle potential, of the order of −79 MeV at zero momentum. Looking at the middle and left panels we see how, as the fraction of protons decreases, the protons gain binding while the neutrons lose attraction. This is a consequence of the different behavior of the NN interaction in the T = 0 and T = 1 channels, the T = 0 channel being substantially more attractive. The potential of the proton is built from more T = 0 than T = 1 pairs and hence becomes more attractive. The Λ single-particle potential in symmetric nuclear matter turns out to be around −38 MeV at k = 0 and has a smooth parabolic behavior as a function of k. This result is larger than the value of −30 MeV obtained when one extrapolates to large A the s-wave Λ single-particle energy of several hypernuclei [20]. It is also much larger in magnitude than the value of Apart from the different size, the new single-particle hyperon potentials also show a totally different behavior with increasing asymmetry than that observed for the potentials obtained with the 1989 Nijmegen Y N interaction. While the old Λ single-particle potential turns to be slightly more attractive with increasing neutron fraction (i.e. going from the right panel to the left one), the new one becomes slightly more repulsive. The changes for the Σ − single-particle potential are more drastic. While the 1989 interaction gives a Σ − potential which shows a little change with increasing neutron fraction, the new Σ − potential becomes strongly attractive. The value at k = 0 for the Σ − potential changes from about −20 MeV in symmetric nuclear matter to −37 MeV in neutron matter. This has important consequences in the composition of dense matter: if hyperons feel substantially more attraction, their appearance in dense matter will happen at lower density. We note that our results with the 1989 Nijmegen interaction are consistent with those shown in [11], where the same Y N interaction is used. Some differences are found in the magnitude of the single-particle potentials which should be adscribed to the use of a continuum spectrum prescription in the case of [11].
Having established how the nucleons affect the single-particle potential of hyperons it is necessary to investigate the influence of a finite fraction of hyperons on the hyperons themselves and on the nucleons. This is visualized in Figs. 2 and 3 that show the singleparticle potentials of the different baryons as functions of the momentum. Figure 2 shows results at ρ = 0.3 fm −3 and a hyperon fraction x Y = 0.1, which is assumed to come from only Σ − (top panels) or split into Σ − and Λ hyperons in a proportion 2 : 1, hence (repulsion) to the neutron (proton) single-particle potential. We also observe that the Σ − feels more attraction, as a consequence of having replaced some repulsive Σ − p pairs by attractive Σ − n ones. The Λ loses binding because the Fermi sea of neutrons is larger and their contribution to the Λ single-particle energy explores higher relative momentum components of the effective Λn interaction, which are less attractive than the small relative momentum ones. Finally, since the Fermi sea of hyperons is small, the differences observed on the potentials by going from the top panels to the corresponding lower ones (which amounts to replacing Σ − hyperons by Λ ones) are also small.
Similar effects are found in the results reported in Fig. 3, which have been obtained for a baryonic density ρ = 0.6 fm −3 , where it is expected that nuclear matter in β equilibrium already contains hyperons [11]. The single-particle potential of the Λ hyperon is less attractive than that for ρ = 0.3 fm −3 while that of the Σ − is very similar. It just gains somewhat more attraction when the number of neutrons increase relative to that of protons in going from the right panels to the left ones. As for the nucleon single-particle potentials we observe, also on the left panels, that the attractive Σ − n interaction is enhanced at these high densities and makes the neutron spectrum more attractive than the proton one, even in the asymmetric situation when one would expect the protons to be more bound.
To assess the influence of the Y Y interaction we represent the separate contributions building the Λ single-particle potential in Fig. 4 and those for the Σ − one in Fig. 5, for a baryonic density of 0.6 fm −3 . The hyperon fraction of x Y = 0.1 is split into fractions We just have to consider here that the curves in Fig. 6 also contain the kinetic energy of the corresponding Fermi momentum. It is interesting to comment on the high density behavior of the chemical potentials, since this will determine the feasibility of having hyperons in betastable neutron star matter. In symmetric nuclear matter, both the Λ and the Σ − chemical potentials show, from a certain density on, an increase with increasing density which is very mild as compared to that assumed by phenomenological Y N interactions [24]. When the number of neutrons over that of protons is increased (top panels), the Λ chemical potential barely changes because of the similarity between the Λn and Λp interaction. However, the Σ − hyperon acquires more binding due to the dominant Σ − n attractive pairs over the Σ − p repulsive ones. This will favor the appearance of Σ − in dense neutron star matter, through the nn → pΣ − conversion, when the equilibrium between chemical chemical potentials is achieved at both sides. Once a Fermi sea of Σ − hyperons starts to build up, however, the neutrons become more attractive moderating, in turn, the appearance of Σ − hyperons. As we see, the composition of dense neutron star matter in equilibrium will result from a delicate interplay between the mutual influence among the different species. In fact, one needs to find, at each baryonic density, the particle fractions which balance the chemical potentials in the weak and strong reactions that transform the species among themselves. This study, which is beyond the scope of the present work, will be presented in a separate publication [18].
One of the novelties of this work is that we allow for different concentrations for the baryon species. Therefore, we can explicitly treat the dependence of the G-matrix on the third component of isospin which comes from the Pauli operator of species B, B that may have, even when belonging to the same isospin-strangeness multiplet, different Fermi momenta. See appendix A for more details.
In Fig. 7 we report the diagonal ΣN → ΣN G-matrix elements in the 1 S 0 channel, as a functiom of relative momentum for a density ρ = 0. We finish this section by reporting in Fig. 8  implications on the properties of neutron stars are deferred to a future study [18].

IV. CONCLUSIONS
In this work we have developed the formalism for microscopic Brueckner-type calculations of dense nuclear matter with strangeness, allowing for any concentration of the different baryon species.
By relating the Pauli operator to the different pairs of physical particles that contribute to the particular (T, M T ) channel (see appendix A), we have been able to obtain the M T dependence of the effective interaction (G-matrix) between any two species.
We have seen, however, that the dependence of the G-matrix on the third component of isospin is weak enough to allow, in future studies, for a simpler strategy consisting of obtaining the effective interactions in isospin saturated situations (k ). The various single-particle potentials can then be obtained by folding the approximate effective interactions with the Fermi seas of the different species.
We have studied the dependence of the single-particle potentials on the nucleon and hyperon asymmetries, focussing on situations that can be relevant in future studies of betastable neutron star matter with strangeness. This is why, apart from neutrons and protons, we have only considered the Λ and Σ − hyperons, which are the first ones expected to appear. We have compared the symmetric nuclear matter composition (x n = x p = x N ) with the asymmetric case containing a large fraction of neutrons (x n = 3x p = 0.75x N ), for a small, but relevant, hyperon fraction x Y = 0.1. This fraction may be fully composed by Σ − hyperons (x Σ − = x Y ) or contain also a small proportion of Λ's (x Σ − = 2x Λ = 2x Y /3). We find that the presence of hyperons, especially Σ − , modifies substantially the single-particle potentials of the nucleons. The neutrons feel an increased attraction due to the Σ − n effective interaction that only happens through the very attractive T = 3/2 ΣN channel, while the protons feel a repulsion as the Σ − p pairs also receive contributions from the very repulsive By decomposing the Λ and Σ − single-particle potentials in the contributions from the various species, we have seen the relevance of considering the Y Y interaction. For a baryonic density of 0.6 fm −3 , a nuclear asymmetry of x n = 3x p = 0.75x N and a hyperon fraction of , we find that the hyperonic contribution to the Λ single-particle potential at zero momentum is of the order of −10 MeV (1/3 of the total U Λ (0)), and that for the Σ − is of the order of −25 MeV (1/2 of the total U Σ − (0)).
In the absence of hyperonic Fermi seas the Λ and Σ − chemical potentials show a mild increase with increasing baryonic density. The presence of a Fermi sea of Σ − hyperons slows down this increase, especially for the Σ − chemical potential and in the case of asymmetric nuclear matter, due to the very attractive T = 3/2 ΣN interaction acting on Σ − n pairs. This will make the balance between chemical potentials in the strong nn → Σ − p conversion easier and will favor the appearance of Σ − at lower densities.
Finally, we have studied the modifications of the binding energy per baryon in symmetric and asymmetric nuclear matter when some nucleons are replaced either by Λ or Σ − hyperons.
As expected, we observe an increase in the binding energy, which increases with density, mainly as a result of a decrease in kinetic energy because the hyperons can be accommodated in lower momentum states and have a larger mass. This effect will produce a softening in the equation of state that will influence the behavior of dense matter and the structure of neutron stars. and From the above expressions it is easy to see that in isospin saturated matter matter (i.e. ) the dependence on the third component of isospin disappears.

MOMENTA
In this appendix we show how to compute an appropiate angular average of the centerof-mass momentum of the pair B 1 B 2 and the hole momentum k B 2 which enters in the determination of the starting energy in Eqs. (2) and (3) . The center-of-mass momentum K and the relative momentum k of the pair B 1 B 2 are defined in the following way: From the above expressions it is easy to write K and k B 2 in terms of the extrenal momentum k B 1 and the relative momentum k, which is used as integration variable in Eqs.
(2) and (3) The angle average of the center-of-mass momentum is defined as , with θ being the angle between k B 1 and k. The integration runs over all the angles for which | k B 2 | < k . Similarly, for the hole momentum we have . We can distinguish two cases in performing the angular integrals, βk B 1 < αk . In the first case, we have two possibilities: 0 < k < αk all angle values are allowed, giving the result and αk +βk B 1 , which have the following upper limit in the value of cos θ giving the result In the second case, there is only one possibility: βk B 1 −αk the result is the same as in the previous case for the zone αk The result for the values 0 < k < βk B 1 −αk is zero because k B 2 is always larger than its Fermi sea.
This kind of average defines an angle-independent center-of-mass momentum and a hole momentum (and therefore a starting energy) for each pair k B 1 , k so the angular integration in Eqs. (2) and (3) can be performed analytically. Nevertheless, we still require to solve the G-matrix equation for each pair of values k B 1 and k, making the calculation much time consuming. In order to speed up the procedure we introduce another average, which gives equivalent results and saves a lot of time. For each external momentum k B 1 , we will only need to solve the G-matrix equation for two values of the center-of-mas and hole momenta, which are obtained from by limiting the integral over the modulus of k to the two possibilities mentioned above.
As before, we have the same cases βk B 1 < αk . Let's consider the first case. Now, when the integral over k in Eqs. (B.12) and (B.13) is limited to 0 < k < αk whereas in the zone αk When k B 1 = 0 there exists only one zone of integration, 0 < k < αk , and the average is very simple Finally, in the second case, βk B 1 > αk , there is also only one integration zone, +βk B 1 , and the corresponding averages are