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The scalar quark condensate $\kappa(\rho)=\langle M|\sum_{i}\bar q_i q_i|M\rangle$ in nuclear matter can be presented as $\kappa(\rho) =\kappa(0) + \kappa_N\rho+S(\rho)$ with $\langle M|$ the ground state of the matter while $q$ are the operators of the light quarks $u$ and $d$. Here $\rho$ is the density of the matters, $\kappa(0)$ is the vacuum value of the condensate. In the second term on the right hand side the matrix element $\kappa_N$ is $\kappa_N=\langle N|\sum_{i}\bar q_i q_i|N\rangle$ with $\langle N|$ standing for the free nucleon at rest. This matrix element can be expressed in terms of the nucleon sigma term $\sigma_N$ related to observables. The various experiments provide the values between 40 MeV and 65 MeV for $\sigma_N$. The first two terms in definition of $\kappa(\rho)$ compose the gas approximation.
The contribution $S(\rho)$ describes the change of $\kappa(\rho)$ caused by the nucleon interactions. It was demonstrated that $S(\rho)$ is due mostly to the pion cloud created by interacting nucleons (see [1] for references). In the latter calculations $S(\rho)$ was obtained employing the nonrelativistic approximation for nucleons of the matter (curve $1$ in Fig.1). The latest results obtained in framework of chiral perturbation theory [2] are shown by the curve $2$.
In the present report the matter is viewed as a relativistic system of nucleons. In the first step we neglect their interactions. We find that the quark condensate can be presented as $\kappa(\rho)=\kappa(0)+\kappa_N \rho F(\rho,m^*(\rho))$ with $F(\rho, m^*(\rho)) =2/(\pi^2\rho)\int^{p_F}_0\,dp\,p^2\,m^*/\sqrt{m^{*2}+p^2}$. Here $p_F$ is the Fermi momentum, $m^*$ is the nucleon Dirac effective mass. Note that the same function $F(\rho,m^*(\rho))$ connects the vector and scalar densities in the Walecka model.
The effective mass $m^*$ can be calculated in a hadron model. In the version of QCD sum rules presented in [3] the right hand side of the scalar channel equation contains the effective mass $m^*(\rho)$ while the left hand side contains the scalar condensate $\kappa(\rho, m^*(\rho))$. Thus we come to self-consistent equation for $m^*(\rho)$ which was solved in [3]. Here we employ these results for calculation of $\kappa(\rho)$ (curve $3$ in Fig.1). One can see that inclusion of the relativistic dynamics of nucleons is as important as that of nucleon interactions.
References:
[1] E.G.Drukarev,M.G.Ryskin,V.A.Sadovnikova, Phys.Atom.Nuclei,v.75,p.334(2012).
[2] S.Goda and D.Jido, Phys. Rev., v.C88,065204(2013).
[3] E.G.Drukarev,M.G.Ryskin,V.A.Sadovnikova,Eur.Phys.J.,v.A55,p.34(2019).
