Spectral function of the ${\eta }^{\prime }$ meson in nuclear medium based on phenomenological models

The in-medium modification of the spectral function of the ${\eta }^{\prime }$ meson with and without the spatial momentum is studied with the $T\rho$ approximation by employing two phenomenological models for the ${\eta }^{\prime }N$ scattering; one is called coupled channels model and the other the $N\left(1895\right)$-dominance model. In the former model, the ${\eta }^{\prime }N$ scattering amplitude is calculated in the unitarized coupled channel approach involving the ${\eta }^{\prime }N$ channel, while in the latter model the ${\eta }^{\prime }N$ scattering process is dominated by the $N\left(1895\right)$ resonance with the spin and parity $J^P=1/2^{-}$. In the coupled channels model, one single peak of the in-medium ${\eta }^{\prime }$ mode appears in the spectral function and the peak position shifts to higher energies along with the increase of the nuclear density reflecting the repulsive ${\eta }^{\prime }N$ scattering length of the unitarized coupled channel amplitude. On the other hand, two branches related to the ${\eta }^{\prime }$ and $N\left(1895\right)$-hole modes appear in the $N\left(1895\right)$-dominance model. In both models, the shift of the peak position and the width in the spectral function are a few tens of $\mathrm{MeV}$ at the normal nuclear density for the ${\eta }^{\prime }$ meson at rest in the nuclear medium. Once the spatial momentum is turned on, the peak positions in the spectral function approach the energies without the nuclear medium effect. Particularly, in the $N\left(1895\right)$-dominance model, the peak strength of the $N\left(1895\right)$-hole mode gets smaller with the finite momentum and the spectral function comes to have one single peak.

Figure 1

In-medium ${\eta }^{\prime }$ spectral functions ${\mathcal{S}}_{{\eta }^{\prime }}$ calculated with the in-medium ${\eta }^{\prime }$ propagator (15) as functions of the ${\eta }^{\prime }$ energy $\omega$ for the ${\eta }^{\prime }N$ scattering length $a_{{\eta }^{\prime }N}=0.87\mathrm{fm}$ (black-solid), $(0+i0.37)\mathrm{fm}$ (red-dashed), and $(-0.41+i0.04)\mathrm{fm}$ (blue-dotted). The vertical dashed-dotted line denotes the in-vacuum ${\eta }^{\prime }$ mass.

Figure 2

Spectral functions at $\rho ={\rho }_0$ with fixed values of the ${\eta }^{\prime }$ momentum, $p=0$, 0.4 and $0.8\mathrm{GeV}$ as functions of the ${\eta }^{\prime }$ invariant mass ${\left(p_{{\eta }^{\prime }}^2\right)}^{1/2}={({\omega }^2-{\mathit{p}}^2)}^{1/2}$. The scattering length in the $T$-matrix is fixed as $a_{{\eta }^{\prime }N}=(+0.43+i0.37)\mathrm{fm}, a_{{\eta }^{\prime }N}=(0+i0.37)\mathrm{fm}, a_{{\eta }^{\prime }N}=(-0.43+i0.37)\mathrm{fm}$ and $a_{{\eta }^{\prime }N}=+0.87\mathrm{fm}$ from the left plot.

Figure 3

Contour plots of the logarithm of the ${\eta }^{\prime }$ spectral function, $ln{S}_{{\eta }^{\prime }}$, in the $p-\omega$ plane for $\rho ={\rho }_0$. The scattering length in the $T$ matrix is fixed as $a_{{\eta }^{\prime }N}=(+0.43+i0.37)\mathrm{fm}$ (left), $a_{{\eta }^{\prime }N}=(0+i0.37)\mathrm{fm}$ (middle) and $a_{{\eta }^{\prime }N}=(-0.43+i0.37)\mathrm{fm}$ (right). The solid lines are the plots of ${\omega }_R\left(\mathit{p}\right)$ as functions of $p$ and the dotted lines are the ${\eta }^{\prime }$ dispersions in vacuum. In the middle panel, the solid and dotted lines are overlapping with each other.

Figure 4

(a) Scattering amplitude of ${\eta }^{\prime }N\to {\eta }^{\prime }N, f_{{\eta }^{\prime }N,{\eta }^{\prime }N}$, calculated in the coupled channels model as a function of the center-of-mass (c.m.) energy $w$ of the ${\eta }^{\prime }N$ system. (b) Total cross section of ${\pi }^{-}p\to {\eta }^{\prime }n$ calculated in the coupled channels model as a function of the c.m. energy $w$ of the ${\eta }^{\prime }N$ system. The points with error bar are the experimental data taken from Ref. [71].

Figure 5

Diagrams for the $N\left(1895\right)$ contribution in the (a) $s$ channel and the (b) $u$ channel of the ${\eta }^{\prime }N$ scattering. The double line represents the $N\left(1895\right)$ resonance.

Figure 6

In-medium properties of the ${\eta }^{\prime }$ meson at rest in the nuclear medium calculated with the coupled channels model: (a) the spectral functions ${\mathcal{S}}_{{\eta }^{\prime }}$ as functions of $\omega$ for nuclear densities $\rho =0.1{\rho }_0, 0.5{\rho }_0, 1.0{\rho }_0$. The inserted figure is the enlarged view of the plots in the vicinity of $\omega =m_{{\eta }^{\prime }}$. (b) the in-medium mass modification $\mathrm{\Delta }{\omega }_R={\omega }_R-m_{{\eta }^{\prime }}$ and in-medium ${\eta }^{\prime }$ width ${\mathrm{\Gamma }}_{*}$ in units of GeV as a function of density $\rho$. (c) Real and imaginary parts of the wave function renormalization $Z$ as functions of density $\rho$.

Figure 7

Decomposition of the spectral function and the in-medium width to each intermediate channel for the ${\eta }^{\prime }$ meson at rest in the nuclear medium. (a) Partial spectral functions defined in Eq. (29) for $\rho ={\rho }_0$ as functions of the ${\eta }^{\prime }$ energy. (b) Partial widths defined in Eq. (30) as functions of $\rho /{\rho }_0$.

Figure 8

Momentum dependence of the spectral function at $\rho ={\rho }_0$ in the coupled channels model. (a) The spectral functions with fixed ${\eta }^{\prime }$ momenta $p=0,0.4,0.8\mathrm{GeV}$ are plotted as functions of the ${\eta }^{\prime }$ invariant mass $(p_{{\eta }^{\prime }}^2{)}^{1/2}={\left({\omega }^2-{\mathit{p}}^2\right)}^{1/2}$. (b) A contour plot of the logarithm of the ${\eta }^{\prime }$ spectral function, $ln{S}_{{\eta }^{\prime }}$, in the $p-\omega$ plane at $\rho ={\rho }_0$ is shown. The solid and dotted lines are the in-medium dispersion relation ${\omega }_R\left(\mathit{p}\right)$ and the in-vacuum dispersion $\omega ={\left(m_{{\eta }^{\prime }}^2+{\mathit{p}}^2\right)}^{1/2}$.

Figure 9

In-medium properties of the ${\eta }^{\prime }$ meson at rest in the nuclear medium calculated with the $N\left(1895\right)$ resonance model: (a) the spectral functions ${\mathcal{S}}_{{\eta }^{\prime }}$ as functions of $\omega$ for nuclear densities $\rho =0.1{\rho }_0,0.5{\rho }_0,1.0{\rho }_0$. (b) the in-medium mass modifications $\mathrm{\Delta }{\omega }_R={\omega }_R-m_{{\eta }^{\prime }}$ and in-medium ${\eta }^{\prime }$ widths ${\mathrm{\Gamma }}_{*}$ in units of GeV as functions of density $\rho$. (c) Real and imaginary parts of the wave function renormalization $Z$ as functions of density $\rho$.

Figure 10

Pole trajectories $({\omega }_R,-{\mathrm{\Gamma }}_{*}/2)$ of the in-medium ${\eta }^{\prime }$ propagator in the $N\left(1895\right)$-dominance model as the nuclear density $\rho$ varies. The open circle and square are the points given by $\omega =(m_{{\eta }^{\prime }},0)$ and $\omega =(m_{N^{*}}-m_N,-{\mathrm{\Gamma }}_{N^{*}}/2)$ which pole 1 and 2 approach in the limit of $\rho \to 0$, respectively, and the dots are plotted at every $0.1{\rho }_0$.

Figure 11

Real parts of the wave-function renormalizations $\mathrm{Re}\left[Z^{\left(1\right)}\right]$ calculated by the $N\left(1895\right)$-dominance model with the resonance mass $m_{N^{*}}=1.894$, 1.879, and $1.854\mathrm{GeV}$.

Figure 12

Same as Fig. 6 but for $m_{N^{*}}=1.906\mathrm{GeV}$ in the $N\left(1895\right)$-dominance model.

Figure 13

Same as Fig. 8 but for the $N\left(1895\right)$-dominance model. (b) The solid and dotted lines are the in-medium dispersion relation ${\omega }_R^{\left(1\right)}\left(\mathit{p}\right)$ for pole 1 and the in-vacuum ${\eta }^{\prime }$ dispersion $\omega ={\left(m_{{\eta }^{\prime }}^2+{\mathit{p}}^2\right)}^{1/2}$, while the dashed and the dash-dotted lines denote the in-medium dispersion relation ${\omega }_R^{\left(2\right)}\left(\mathit{p}\right)$ for pole 2 and the in-vacuum $N^{*}$ dispersion relation $\omega ={\left(m_{N^{*}}^2+p^2\right)}^{1/2}-m_N$.

Figure 14

Plot of the real and imaginary part of $Z^{\left(1\right)}$ and $Z^{\left(2\right)}$ at $\rho ={\rho }_0$ in the $N\left(1895\right)$-dominance model as functions of the ${\eta }^{\prime }$ spatial momentum $p$.