transition could be well described by such a simple model. However, when one analyze the effect of baryon chemical potential $\mu$ one finds that the phase transition is always second order for any value of $\mu$. There is no first order transition region for any value of $\mu$. It is inconsistent with the expected phase diagram in $T-\mu$ plane for real QCD. Therefore, additional term in $V_X$ might be necessary. A direct extension is including the higher power $(X^+X)^3$ which satisfies the symmetry and considering effective potential $$\begin{aligned}
V(\chi, F^2)=-\frac{3}{2}\chi^2+v_4 \chi^4+v_6\chi^6.\end{aligned}$$

Similarly, we take chiral limit to study the phase structure first. We take $v_4=0$ and different values of $v_6=20,\mu=0.1\rm{GeV}$(black solid line), $v_6=20,\mu=0.5\rm{GeV}$(black dashed line), $v_6=100, \mu=0.1\rm{GeV}$(blue solid line), $v_6=100, \mu=0.5\rm{GeV}$(blue dashed line) in Fig.\[simplev4v6\](b). From the figure, we find that the effect of $v_6$ is very similar to $v_4$. It will not affect the location of $T_c$, while it suppresses chiral condensate for a given value of $T$ and $\mu$. The transition order of chiral transition is still kept as second order one for any value of $\mu$.

Then we turn on the effect of $v_4, v_6$ simultaneously, and we find that when both $v_4$ and $v_6$ are positive, 