Chiral phase transition with $2+1$ quark flavors in an improved soft-wall AdS/QCD model

We study the chiral phase transition with $2+1$ quark flavors in an improved soft-wall AdS/QCD model, which can produce the light meson spectrum and many other low-energy quantities consistent with experiments in the two-flavor case. The chiral transition behaviors of the quark condensates at different quark masses are analyzed in detail, and the ($m_{u,d}$, $m_s$) phase diagram for the quark sector has been obtained from the improved soft-wall model. We find that the features of the calculated phase diagram are completely consistent with the standard scenario, which is supported by lattice simulations and theoretical arguments. The evidence of a tricritical point on the $m_{u,d}=0$ boundary of the ($m_{u,d}$, $m_s$) phase diagram is first clearly presented in the bottom-up AdS/QCD.

Figure 1

The expected phase diagram in the quark mass plane ($m_{u,d}$, $m_s$) [6]. The lateral axis denotes the up/down quark mass $m_{u,d}$, and the vertical axis denotes the strange quark mass $m_s$. The blue curves represent the second-order lines of the phase transition, which divide the phase diagram into three parts. The light green regions denote the first-order phase transition, while the grey part denotes the crossover transition. The diagonal line corresponds to the flavor-symmetric case with $m_u=m_d=m_s$.

Figure 2

The calculated ($m_{u,d}$, $m_s$) phase diagram in the improved soft-wall model, where the parameters are taken to be ${\mu }_c=1180\mathrm{MeV}$, ${\mu }_g=440\mathrm{MeV}$, $\gamma =-22.6$, and $\lambda =16.8$. The blue curve denotes the second-order transition line.

Figure 3

Upper panel: The chiral transition behavior of ${\sigma }_u$ and ${\sigma }_s$ with temperature $T$ at the physical point $m_u=3.336\mathrm{MeV}$, $m_s=95\mathrm{MeV}$. Lower panel: The solutions of VEVs (${\chi }_u$, ${\chi }_s$) at temperatures $T=100,150,200\mathrm{MeV}$.

Figure 4

Upper panel: The chiral transition behavior of ${\sigma }_u$ and ${\sigma }_s$ with temperature $T$ at the point $(m_u,m_s)=(1\mathrm{MeV},20\mathrm{MeV})$. Lower panel: The three solutions of the VEVs (${\chi }_u$, ${\chi }_s$) at $T=135\mathrm{MeV}$, where the corresponding condensates (${\sigma }_u$, ${\sigma }_s$) in the upper panel (see the insert) have been dotted with the same color as that of the solutions in the lower panel.

Figure 5

The chiral transition behavior of the condensate (${\sigma }_q\equiv {\sigma }_u={\sigma }_s$) with temperature $T$ at $m_q=10,15,20.1,25\mathrm{MeV}$ in the flavor-symmetric case ($m_q\equiv m_u=m_d=m_s$).

Figure 6

The chiral transition behavior of the condensates (${\sigma }_u$, ${\sigma }_s$) with temperature $T$ at $m_u=20,30,32.7,40\mathrm{MeV}$ in the $m_s=0$ case.

Figure 7

The chiral transition behavior of the condensates (${\sigma }_u$, ${\sigma }_s$) with the temperature $T$ at $m_s=100,150,212.7,250\mathrm{MeV}$ in the $m_{u,d}=0$ case.

Figure 8

Upper panel: The calculated ($m_{u,d}$, $m_s$) phase diagrams in the improved soft-wall model with different values of $\gamma$, where the red dot denotes the tricritical point and the black dot denotes the physical point. Lower panel: The chiral transition behavior of the condensate with temperature $T$ at $m_{u,d}=m_s=0$ for different values of $\gamma$.