Charmed baryon ${\mathrm{\Lambda }}_c$ in nuclear matter

Density dependences of the mass and self-energies of ${\mathrm{\Lambda }}_c$ in nuclear matter are studied in the parity projected QCD sum rule. Effects of nuclear matter are taken into account through the quark and gluon condensates. It is found that the four-quark condensates give dominant contributions. As the density dependences of the four-quark condensates are not known well, we examine two hypotheses. One is based on the factorization hypothesis (F-type) and the other is derived from the perturbative chiral quark model (QM-type). The F-type strongly depends on density, while the QM-type gives a weaker dependence. It is found that, for the F-type dependence, the energy of ${\mathrm{\Lambda }}_c$ increases as the density of nuclear matter grows, that is, ${\mathrm{\Lambda }}_c$ feels repulsion. On the other hand, the QM-type predicts a weak attraction ($\sim 20$ MeV at the normal nuclear density) for ${\mathrm{\Lambda }}_c$ in nuclear matter. We carry out a similar analysis of the $\mathrm{\Lambda }$ hyperon and find that the F-type density dependence is too strong to explain the observed binding energy of $\mathrm{\Lambda }$ in nuclei. Thus we conclude that the weak density dependence of the four-quark condensate is more realistic. The scalar and vector self-energies of ${\mathrm{\Lambda }}_c$ for the QM-type dependence are found to be much smaller than those of the light baryons.

Figure 1

The positive and negative parity OPE, $G_{m\mathrm{OPE}}^{\pm }\left(\tau \right)$, in vacuum.

Figure 2

The density dependences of the positive parity OPE $G_{m\mathrm{OPE}}^{+}\left(\tau \right)$. The LO-perturbative, the NLO-perturbative, the four-quark, and the vector quark condensate $\langle q^{†}q\rangle$ terms are shown. Here, ${\rho }_N$ means normal nuclear matter density.

Figure 3

The density dependences of (a) the energy $E_{{\mathrm{\Lambda }}_c}$ (MeV), (b) the effective mass $M_{{\mathrm{\Lambda }}_c}^{*}$ (MeV) and (c) the vector self-energy ${\mathrm{\Sigma }}_{{\mathrm{\Lambda }}_c}^v\left(\mathrm{MeV}\right)$. The solid lines show the results with the density dependence of the four-quark condensate according to the factorization hypothesis while dashed lines are the case where its density dependence are estimated from the perturbative chiral quark model.

Figure 4

The positive and the negative parity OPE $G_{m\mathrm{OPE}}^{\pm }\left(\tau \right)$ of $\mathrm{\Lambda }$ in vacuum.

Figure 5

The density dependences of the energy $E_{\mathrm{\Lambda }}$ (MeV). The solid line shows the result with the density dependence of the four-quark condensate according to the factorization hypothesis while the dashed line is the case where its density dependence is estimated from the perturbative chiral quark model.

Figure 6

The density dependences of (a) the energy $E_{{\mathrm{\Lambda }}_b}$ (MeV), (b) the effective mass $M_{{\mathrm{\Lambda }}_b}^{*}$ (MeV), and (c) the vector self-energy ${\mathrm{\Sigma }}_{{\mathrm{\Lambda }}_b}^v\left(\mathrm{MeV}\right)$. The solid lines show the results with the density dependence of the four-quark condensate according to the factorization hypothesis while dashed lines are the case where its density dependence are estimated from the chiral quark model.