+ dpi / dp2 (PiP2\Gb'E',B'B' (eB' (Pi) + &B' (P2 ))\Pi P2)a
+ 1 V I dpi i dp2 (PiP2\G B'' B' ,B'' B' (eB'' (pi)+ eB' (P2))\PiP2) •
A self-consistent solution of the BBG equations is a formidable numerical task, because of non-linear coupling of integral equations for G-matrices. No wonder that such calculations became feasible only in the late 1990s (Schulze et al. 1998; Vidana et al. 2000a,b; Nishizaki et al. 2002; also see the review paper by Baldo & Burgio 2001 and references therein). Because of a tremendous complexity, an evaluation of the three-body correlation contribution a la Bethe-Faddeev has not been carried out up to now (till 2006). However, in view of uncertainties of the NH and HH interactions in dense matter, even the calculation of the EOS at the BHF level is so model-dependent, that the unsolved problem of higher-order correlations seems of second priority. Additional large uncertainties result from our ignorance of three-body interaction involving hyperons. As visualized by the BBG calculations of Nishizaki et al. (2002), three body interactions can play a decisive role in the EOS of hyperonic matter (see § 6.5.5). The EOS based on the two-body interactions only is usually too soft to support 1.44 M© of the Hulse-Taylor pulsar. Inclusion of realistic three-body interactions involving hyperons is therefore a major task for the many-body calculations of the EOS.
The mean-field method is particularly suitable for calculating the EOS of a uniform multi-component dense matter. Strong interactions of hadrons are invariant with respect to rotation in the isospin space, and therefore L is an iso-scalar. In particular, the coupling constants g$B do not depend on I3B. The generalization of the RMF model equations to the full baryon octet is then straightforward: the total source of a meson field is a sum of contributions from all baryon fields. For the sake of generality, we will write all the equations in the form which enables one to include baryons beyond the lowest-mass baryon octet. In this section we will use units in which h = c = 1.
The equations for the non-vanishing components of the w and p fields are:
In the equation for the a field, the nucleon term has to replaced by a sum over all baryon terms, taking due account of the spin degeneracy,
B 2n2 Jo Vk2 + (mB - JaBO)2 -JaN [bmN (Ja o)2 + C (ja o)3] , (5.84)
where JB is the baryon spin. The Fermi momenta of baryons B are related to their number density nB by nB = kFb . (5.85)
In order to calculate the EOS of the ground-state matter, one needs the energy spectrum of baryons. It is obtained from the Dirac equations for bispinors ^B:
eB(k) = y/k2 + (mB - JaBo)2 + JfBUo + JpBI3BP0 . (5.86)
The only modification in the expressions for the energy density and pressure consists in replacing nucleon contributions by sums over all baryons,
+1 mp(p0)2 + 2 mu+ 1 bmN(giNa)3 + 4 c (giNa)4 , (5.87)
22 ml a
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