Negative-parity nucleon excited state in nuclear matter

Spectral functions of the nucleon and its negative-parity excited state in nuclear matter are studied by using QCD sum rules and the maximum entropy method (MEM). It is found that in-medium modifications of the spectral functions are attributed mainly to density dependencies of the $\langle \overline{q}q\rangle$ and $\langle q^{†}q\rangle$ condensates. The MEM reproduces the lowest-energy peaks of both the positive- and negative-parity nucleon states at finite density up to $\rho \sim {\rho }_N$ (normal nuclear matter density). As the density grows, the residue of the nucleon ground state decreases gradually while the residue of the lowest negative-parity excited state increases slightly. On the other hand, the positions of the peaks, which correspond to the total energies of these states, are almost density independent for both parity states. The density dependencies of the effective masses and vector self-energies are also extracted by assuming phenomenological mean-field-type propagators for the peak states. We find that, as the density increases, the nucleon effective mass decreases while the vector self-energy increases. The density dependence of these quantities for the negative-parity state on the other hand turns out to be relatively weak.

Figure 1

The density dependence of $G_{m\mathrm{OPE}}^{\pm }(s,\tau ,\theta )$. The perturbative, chiral condensate $\langle \overline{q}q\rangle$, vector quark condensate $\langle q^{†}q\rangle$ terms and the total are shown at $\tau =0.5{\mathrm{GeV}}^4, \beta =-0.9, \theta =0.108\pi$, and $|\vec{q}|=0$. The $\langle q^{†}q\rangle$ term stands for the sum of the terms proportional to the condensate $\langle q^{†}q\rangle$ in $G_{m1\mathrm{OPE}}(s,\tau ,\theta )$ and $G_{m3\mathrm{OPE}}(s,\tau ,\theta )$.

Figure 2

The positive- (left) and negative-parity (right) spectral functions extracted from $G_{m\mathrm{OPE}}^{\pm }(s,\tau ,\theta )$ of Eq. (14) by MEM. The red, green, blue, magenta, and light blue lines correspond to the spectral functions at the density $0.0{\rho }_N, 0.25{\rho }_N, 0.5{\rho }_N, 0.75{\rho }_N$, and $1.0{\rho }_N$, respectively. Here ${\rho }_N$ denotes the normal nuclear matter density.

Figure 3

The density dependence of the effective masses and vector self-energies of positive (left) and negative-parity (right) states. The red, green, and blue lines correspond to the effective masses, vector self-energies and total energies, respectively. The dashed lines are the results in which the four-quark condensate are assumed to be independent of the density (see Sec. 4a).

Figure 4

The ${\sigma }_N/2m_q$ dependence of the effective masses and vector self-energies of positive- (left) and negative-parity (right) states. The red and green lines correspond to the effective masses and vector self-energies, respectively. The solid lines show the results with full density dependence according to the factorization hypothesis for the four-quark condensates, while the dashed lines correspond to the density-independent ${\langle \overline{q}q\rangle }^2$ case.

Figure 5

The density dependence of the effective masses and vector self-energies of positive- (left) and negative-parity (right) states at the Fermi momentum. The red, green, and blue lines correspond to the effective masses, vector self-energies, and total energies, respectively. For comparison, their values at $|\vec{q}|=0$ are shown as dashed lines.

Figure 6

The red (green) point corresponds to the ratio of the coefficient of the second term to the first term of ${\mathrm{\Pi }}_m^{+}(q_0,|\vec{q}\left|\right) [{\mathrm{\Pi }}_m^{-}(q_0,|\vec{q}\left|\right)]$ in Eq. (25).