Spectral functions for $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons in nuclear matter with partial restoration of chiral symmetry

We investigate the in-medium masses of a $\overline{D}\left(0^{-}\right)$ meson and a ${\overline{D}}_0^{*}\left(0^{+}\right)$ meson, and spectral functions for $\overline{D}$ and ${\overline{D}}_0^{*}$ meson channels in nuclear matter. These mesons are introduced as chiral partners in the chiral-symmetry-broken vacuum, hence they are useful to explore the partial restoration of the broken chiral symmetry in nuclear matter. We consider the linear sigma model to describe the chiral symmetry breaking and to investigate a qualitative tendency of changes of $\overline{D}$ mesons at low density. Our study shows that the loop corrections to $\overline{D}$ and ${\overline{D}}_0^{*}$ meson masses provide a smaller mass splitting at finite density than that in vacuum, a result that indicates a tendency of the restoration of the chiral symmetry. We investigate also the spectral function for ${\overline{D}}_0^{*}$ meson channel, and find three peaks. The first peak, which corresponds to the resonance of the ${\overline{D}}_0^{*}$ meson, is broadened by collisions with nucleons in medium, and the peak position shifts to lower mass due to the partial restoration of chiral symmetry as the density increases. The second peak is identified as a threshold enhancement which shows a remarkable enhancement as the density increases. The third peak is Landau damping. The obtained properties of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons in nuclear matter will provide useful information for experiments.

Figure 1

Density dependence of the mean field ${\sigma }_0^{*}$.

Figure 2

Self-energies of (a) the pion and (b) the $\sigma$ meson. The nucleon loop is calculated by using in-medium propagator ${\tilde{G}}_N(k_0,\vec{k})$ in Eq. (11).

Figure 3

Resummed propagator for pion and $\sigma$ meson.

Figure 4

Spectral functions for (a) pion, ${\rho }_{\pi }^{*}(q_0,\vec{q})$, and (b) $\sigma$ meson, ${\rho }_{\sigma }^{*}(q_0,\vec{q})$, at ${\rho }_B=0.066{\mathrm{fm}}^{-3}$. $q$ is defined by $q=|\vec{q}|$. $q, q_0$, and ${\rho }_{\pi \left(\sigma \right)}^{*}(q_0,\vec{q})$ in these figures are scaled by $m_N^{*}$ so that these are dimensionless quantities. Red curves indicate the dispersion relations of (a) pion and (b) $\sigma$ meson, which are determined by ${\tilde{\mathrm{\Gamma }}}_{\pi }^{*\left(2\right)}(q_0,\vec{q})=0$ or ${\tilde{\mathrm{\Gamma }}}_{\sigma }^{*\left(2\right)}(q_0,\vec{q})=0$, respectively. Then red curves should be understood as the position of the delta function, which shows the one-particle state of the pion and $\sigma$ meson. Bumps in the spacelike region are induced by the Landau damping, which is diagrammatically shown in Fig. 5.

Figure 5

Diagrammatical interpretation of the Landau damping. This is understood as the absorption process of a pion ($\sigma$ meson) by a nucleon in medium, since the four-momentum of the pion ($\sigma$ meson) is in the spacelike domain $q^2<0$.

Figure 6

Hartree-type one-loop corrections ($\delta {\sigma }_0^{*}$) to the self-energies of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons. These corrections provide mass modifications to $\overline{D}$ and $\overline{D}$ mesons, and provide no change in the decay properties. The blobs for a pion and a $\sigma$ meson are defined in Fig. 3.

Figure 7

Density dependence of masses of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons with one-loop corrections in Fig. 6. Blue (red) circles show the mass of the $\overline{D}$ (${\overline{D}}_0^{*}$) meson at each density. Filled circles are the results with mean field and one-loop corrections ${\sigma }_0^{*}+\delta {\sigma }_0^{*}$. Open circles show the masses of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons with only mean field ${\sigma }_0^{*}$ plotted as a reference. The dashed lines refer the mass of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons in the vacuum.

Figure 8

Density dependence of mass difference between ${\overline{D}}_0^{*}$ and $\overline{D}$ mesons with one-loop corrections, ${\mathrm{\Delta }}_m^{*}=m_{{\overline{D}}_0^{*}}^{*}-m_{\overline{D}}^{*}$, for $m_{{\overline{D}}_0^{*}}^{*}$ and $m_{\overline{D}}^{*}$ shown in Fig. 6. Filled purple circles indicate the mass difference when both mean field and one-loop correction, ${\sigma }_0^{*}+\delta {\sigma }_0^{*}$, are considered, and open circles indicate the mass difference when only the mean field ${\sigma }_0^{*}$ is considered. The dashed line refers the mass difference between ${\overline{D}}_0^{*}$ and $\overline{D}$ mesons in the vacuum.

Figure 9

Fock-type one-loop corrections to the self-energy for the $\overline{D}$ meson $\left[{\tilde{\mathrm{\Sigma }}}_{\overline{D}}^{*}(q_0,\vec{q})\right]$.

Figure 10

Fock-type one-loop corrections to the self-energy for the ${\overline{D}}_0^{*}$ meson $\left[{\tilde{\mathrm{\Sigma }}}_{{\overline{D}}_0^{*}}^{*}(q_0,\vec{q})\right]$.

Figure 11

Spectral function for the ${\overline{D}}_0^{*}$ meson channel at several densities. The dashed black curve indicates the spectral function in vacuum, and the vertical dotted line is the threshold of $\overline{D}+\pi$. We can find three peaks, as shown in the figure at ${\rho }_B=0.036 {\mathrm{fm}}^{-3}$. A detailed explanation of these peaks is given in the text.

Figure 12

Diagrammatical interpretation for (a) decay process of ${\overline{D}}_0^{*}\to \overline{D}\pi$ and (b) Landau damping.

Figure 13

Spectral function for the $\overline{D}$ meson channel at ${\rho }_B=0.066{\mathrm{fm}}^{-3}$. The dashed black curve is the spectral function for the $\overline{D}$ meson channel in the vacuum.

Figure 14

Density dependence of mass of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons with all one-loop corrections in Figs. 6, 9, and 10. Blue (red) circles show the mass of the $\overline{D}$ (${\overline{D}}_0^{*}$) meson at each densities. Filled circles are the results with all one-loop corrections. Open circles show the masses of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons with only mean field ${\sigma }_0^{*}$ plotted as a reference. The dashed lines show the massed of $\overline{D}$ and ${\overline{D}}_0^{*}$ mesons in the vacuum.

Figure 15

Density dependence of mass difference between ${\overline{D}}_0^{*}$ and $\overline{D}$ mesons with one-loop corrections shown in Figs. 6, 9, and 10. Filled purple circles indicate the mass difference with all one-loop corrections, and open circles indicate the mass difference with mean field ${\sigma }_0^{*}$. The dashed line refers the mass difference between ${\overline{D}}_0^{*}$ and $\overline{D}$ mesons in the vacuum.