Mass spectrum of spin-one hadrons in dense two-color QCD: Novel predictions by extended linear sigma model

We construct an extended version of the linear sigma model in such a way as to describe spin-1 hadrons as well as spin-0 hadrons in two-color QCD (${\mathrm{QC}}_2\mathrm{D}$) by respecting the Pauli-Gürsey $SU(4)$ symmetry. Within a mean-field approximation, we therefrom examine a mass spectrum of the spin-1 hadrons at finite quark chemical potential (${\mu }_q$) and zero temperature. Not only mean fields of scalar mesons and scalar-diquark baryons but also of vector mesons and vector-diquark baryons are incorporated. As a result, we find that, unless all of those four types of mean fields are taken into account, neither lattice result for the critical ${\mu }_q$ that corresponds to the onset of baryon superfluidity nor for ${\mu }_q$ dependence of the pion mass can be reproduced. We also find that a slight suppression of the $\rho$ meson mass in the superfluid phase, which was suggested by the lattice simulation, is reproduced by subtle mixing effects between spin-0 and spin-1 hadrons. Moreover, we demonstrate the emergence of an axial-vector condensed phase and possibly of a vector condensed phase by identifying the values of ${\mu }_q$ at which the corresponding hadron masses vanish. The possible presence of isotriplet $1^{-}$ diquarks that may be denoted by a tensor-type quark bilinear field is also discussed.

Figure 1

${\mu }_q$ dependence of the mean fields: ${\sigma }_0$, $\mathrm{\Delta }$, $\overline{\omega }$, and $\overline{V}$, with $C=12$. The value indicated by the arrow represents the asymptotic constant value of $\mathrm{\Delta }$; $\mathrm{\Delta }\sim 0.34m_1=1.74{\sigma }_0^{(\mathrm{H})}$.

Figure 2

${\mu }_q$ dependence of ${\sigma }_0$ and $\mathrm{\Delta }$ (top) and of the quark-number density ${\tilde{n}}_q$ (bottom) with and without spin-0 and spin-1 mixing, $C=12$ and $C=0$.

Figure 3

Schematic picture of corrections to the hadron masses from various mixings.

Figure 4

${\mu }_q$ dependence of the masses of negative-parity (left) and positive-parity (right) spin-1 hadrons for $(g_{\mathrm{\Phi }},C)=(10,12)$. In this figure the masses are normalized by $m_{\pi }^{(\mathrm{H})}$.

Figure 5

${\mu }_q$ dependence of the masses of positive-parity (left) and negative-parity (right) spin-0 hadrons for $(g_{\mathrm{\Phi }},C)=(10,12)$. In this figure the masses are normalized by $m_{\pi }^{(\mathrm{H})}$.

Figure 6

Same as Fig. 4 but for $(g_{\mathrm{\Phi }},C)=(10,16)$.

Figure 7

Same as Fig. 4 but for $(g_{\mathrm{\Phi }},C)=(10,8)$.

Figure 8

Critical chemical potential for the axial-vector condensation as a function of mixing strength $C$. We take $g_{\mathrm{\Phi }}=10$.

Figure 9

Same as Fig. 4 but for $(g_{\mathrm{\Phi }},C)=(5,12)$.

Figure 10

Critical chemical potential for the axial-vector condensation as a function of the coupling $g_{\mathrm{\Phi }}$. We take $C=12$.

Figure 11

${\mu }_q$ dependence of all the $1^{+}$ and $1^{-}$ hadron masses for $(g_{\mathrm{\Phi }},C)=(10,16)$. The chiral partner structures are clearly shown by the degeneracy.