QCD sum rules for the neutron, $\mathrm{\Sigma }$, and $\mathrm{\Lambda }$ in neutron matter

The nuclear density dependencies of the neutron and $\mathrm{\Sigma }$ and $\mathrm{\Lambda }$ hyperons are important inputs in the determination of the neutron star mass as the appearance of hyperons coming from strong attractions significantly changes the stiffness of the equation of state (EOS) at iso-spin asymmetric dense nuclear matter. In-medium spectral sum rules have been analyzed for the nucleon, $\mathrm{\Sigma }$, and $\mathrm{\Lambda }$ hyperon to investigate their properties up to slightly above the saturation nuclear matter density by using the linear density approximation for the condensates. The construction scheme of the interpolating fields without derivatives has been reviewed and used to construct a general interpolating field for each baryon with parameters specifying the strength of independent interpolating fields. Optimal choices for the interpolating fields were obtained by requiring the sum rules to be stable against variations of the parameters and the result to be consistent with known phenomenology. The optimized result shows that Ioffe's choice is not suitable for the $\mathrm{\Lambda }$ hyperon sum rules. It is found that, for the $\mathrm{\Lambda }$ hyperon interpolating field, the up and down quark combined into the scalar diquark structure $u^{T}C{\gamma }_{5}d$ should be emphasized to ensure stable sum rules. The quasi-$\mathrm{\Sigma }$ and -$\mathrm{\Lambda }$ hyperon energies are always found to be higher than the quasineutron energy in the region $0.5<\rho /{\rho }_0<1.5$ where the linear density approximation in the sum-rule analysis is expected to be reliable.

Figure 1

Diagrammatic description for the ${\mathrm{\Sigma }}^0$ correlation function. Set (a) shows the self-correlation function of basis (16). Set (b) shows the self-correlation function of basis (17). In each set, the left diagram shows the partonic propagation over short distance and the right diagram shows the possible lowest dimensional quark condensate contribution. Only strange quarks can propagate to the different helicity state from the initial helicity state.

Figure 2

Sum-rule result in the $\{\tilde{a},\tilde{b}\}$ plane for (a) vacuum mass $M_{\mathrm{\Lambda }}$, and for (b) the ratio of the in-medium quasi-$\mathrm{\Lambda }$ state energy $E_{\mathrm{\Lambda }}$ to $M_{\mathrm{\Lambda }}$. The point denoted by the red-filled circle corresponds to the result obtained with the Ioffe's choice for the interpolating fields (31). Borel mass is set at $M^2=1.1{\mathrm{GeV}}^2$.

Figure 3

Ratios of the in-medium quasi-$\mathrm{\Lambda }$ state energy, scalar, and vector self-energies to its vacuum mass for $\tilde{a}=-2.0$ (left) and −4.0 (right) as a function of $\tilde{b}$. Black dotted line represents 0.99.

Figure 4

Ratios of the Borel-transformed-continuum contribution and highest-dimensional condensate terms to the Borel transformed subtracted total OPE of the $\mathrm{\Lambda }$ for the three different invariants (a) ${\mathrm{\Pi }}_{\mathrm{\Lambda },s}$, (b) ${\mathrm{\Pi }}_{\mathrm{\Lambda },u}$ and (c) ${\mathrm{\Pi }}_{\mathrm{\Lambda },q}$. The black dotted line represents 50%.

Figure 5

(a) Ratios and (b), (c) sum rules for the quasi-$\mathrm{\Lambda }$ self-energies and the quasi-$\mathrm{\Lambda }$ pole. $\rho ={\rho }_0$ in graph (a). In the linear density approximation, in-medium $\mathrm{\Lambda }$ sum rules do not depend on iso-spin asymmetry of the surrounding matter. Graph (b) has been plotted with constant quasi-anti $\mathrm{\Lambda }$ pole and graph (c) has been plotted with density dependent quasi-anti $\mathrm{\Lambda }$ pole. Units for the graphs in (b) and (c) are GeV.

Figure 6

(a) Ratios and (b), (c) sum rules for neutron self-energies and the quasineutron pole in the neutron matter $[\rho ={\rho }_0$ in graph (a)]. Graph (b) has been plotted with constant quasi-antineutron pole and graph (c) has been plotted with density dependent quasi-antineutron pole. Units for the graphs in (b) and (c) are GeV.

Figure 7

(a), (b) Ratios and (c) sum rules for ${\mathrm{\Sigma }}^{+}$ self-energies and the quasi-${\mathrm{\Sigma }}^{+}$ pole. Graph (a) is plotted in iso-spin -symmetric condition ($I=0, \rho ={\rho }_0$) and graph (b) is plotted in neutron matter condition ($I=1, \rho ={\rho }_0$). For the in-medium ${\mathrm{\Sigma }}^{+}$ sum rules, it is negligible for the difference between the constant and density-dependent quasi-anti ${\mathrm{\Sigma }}^{+}$ pole case. Units for the graph in panel (c) are GeV.

Figure 8

Comparison of the density behavior between $\mathrm{\Lambda }, {\mathrm{\Sigma }}^{+}$ hyperon and neutron sum rules in the neutron matter. Only graph (a) where constant quasi-antipoles are assigned shows the cross of quasibaryon energy. Units are GeV.

Figure 9

Factorization-parameter dependence of the sum rule for the quasi-$\mathrm{\Lambda }$ state ($\rho ={\rho }_0, M^2=1.1{\mathrm{GeV}}^2$). (a) $k_1$ dependence from the factorization ${\langle \overline{q}q\overline{q}q\rangle }_{\mathrm{vac}}\Rightarrow k_1{\langle \overline{q}q\rangle }_{\mathrm{vac}}^2$. (b) $k_2$ dependence from the factorization ${\langle \overline{q}q\overline{s}s\rangle }_{\mathrm{vac}}\Rightarrow k_2{\langle \overline{q}q\rangle }_{\mathrm{vac}}{\langle \overline{s}s\rangle }_{\mathrm{vac}}$. (c) $f_2$ dependence from the factorization ${\langle \overline{q}q\overline{s}s\rangle }_{\mathrm{med}.}\Rightarrow f_2({\langle \overline{s}s\rangle }_p{\langle \overline{q}q\rangle }_{\mathrm{vac}}+{\langle {\left[\overline{q}q\right]}_0\rangle }_p{\langle \overline{s}s\rangle }_{\mathrm{vac}})\rho$.

Figure 10

Sum rules with various $T_{uu}^6$ ($\rho ={\rho }_0$). (a) The quasineutron sum rules in the neutron matter. (b) The quasi-$\mathrm{\Lambda }$ sum rules in the neutron matter. (c) Nuclear symmetry energy. The OPE terms for the symmetry energy can be found in Ref. [30]. Units for graph (c) are GeV.