Probing the hadron mass spectrum in dense two-color QCD with the linear sigma model

We investigate modifications of hadron masses at finite quark chemical potential in two-flavor and two-color QCD, data of which are available from lattice simulations, within a linear sigma model based on approximate Pauli-Gursey $SU(4)$ symmetry. The model describes not only ground-state scalar diquarks and pseudoscalar mesons but also the excited pseudoscalar diquarks and scalar mesons; each ground-state diquark (meson) has the corresponding excited diquark (hadron) with opposite parity as a chiral partner. Effects of chiral symmetry breaking and diquark condensates are incorporated by a mean-field treatment. We show that various mixings among the hadrons, which are triggered by the breakdown of baryon number conservation in the superfluid phase, lead to a rich hadron mass spectrum. We discuss the influence of $U(1{)}_A$ anomaly on the density dependence of the mass spectrum and also manifestations of the chiral partner structures as the density increases in the superfluid phase. The predicted hadron masses are expected to provide future lattice simulations with useful information on such symmetry properties in dense two-color QCD.

Figure 1

The ${\mu }_q$ dependence of ${\sigma }_0$ and $\mathrm{\Delta }$ (left panel) and that of the scaled baryonic density $\tilde{\rho }$ (right panel). The vertical line corresponds to ${\mu }_q^{*}=m_{\pi }^{\mathrm{vac}}/2$, which distinguishes the normal and baryon superfluid phases.

Figure 2

Left and right panels represent the ${\mu }_q$ dependence of $0^{+}$ hadron masses ($a_0$, $\sigma$, $B$, and $\overline{B}$) and that of $0^{-}$ hadron masses ($\pi$, $\eta$, $B^{\prime }$, and ${\overline{B}}^{\prime }$), respectively, calculated with the parameter set (I). These hadron masses are normalized by the vacuum pion mass $m_{\pi }^{\mathrm{vac}}$. The vertical dotted line corresponds to the critical chemical potential ${\mu }_q^{*}$.

Figure 3

Same as Fig. 2 but with the parameter set (II).

Figure 4

Same as Fig. 2 but with the parameter set (III).

Figure 5

The ${\mu }_q$ dependence of the mass of the lowest branch of the $\eta \text{−}{B}^{\prime }\text{−}{\overline{B}}^{\prime }$ mixed state in the baryon superfluid phase calculated for several $c$’s. The mass is normalized by $m_{\pi }^{\mathrm{vac}}$.

Figure 6

The ${\mu }_q$ dependence of all the normalized hadron masses calculated with the parameter set (I). The solid and dashed curves denote the positive and negative parity hadrons, respectively, and the vertical dotted line corresponds to ${\mu }_q^{*}$. The two states indicated in each parenthesis are the respective chiral partners realized at sufficiently large ${\mu }_q$.