will effectively become negative dominant at certain region and the transition order might turn to first order. Of course, generally, we can consider any power obeying the symmetry. Here we only consider the first powers $F^2(X^+X), F^2(X^+X)^2, F^2(X^+X)^3$ and assume the effect of higher power term would be suppressed by certain mechanism.

According to the above discussion, we will consider the following general form of potential $V(\chi,F^2)$

$$\begin{aligned}
\label{vcf}
V(\chi,F^2) = (\frac{3}{2}  + \lambda_2A_t^{\prime2}\frac{z^4}{e^{4A_s}})\chi^2 \notag+ ( \lambda_4A_t^{\prime2}\frac{z^4}{e^{4A_s}}- v_4)\chi^4 + (\lambda_6A_t^{\prime2}\frac{z^4}{e^{4A_s}} - v_6)\chi^6.\end{aligned}$$

Here, $v_4, v_6$ terms are corresponding to $(X^+X)^2, (X^+X)^3$ term, $\lambda_2, \lambda_4,\lambda_6$ terms are corresponding to $F^2 X^+X, F^2(X^+X)^2, F^2(X^+X)^3$ terms. We have imposed that at finite $\mu$ only $A_t(z)$ is non-vanishing and replaced $F^2$ with it.

=6.5 cm =4.5 cm =6.5 cm =4.5 cm -0.05cm 0.15 cm **( a )** 6.5 cm **( b )**\

Therefore, up to now, from the analysis in chiral limit, we have introduced five parameters $v_4, v_6$ and $\lambda_2, \lambda_4, \lambda_6$ to describe chiral phase transition. Now we can start to describe the phenomenology of real QCD. So in the rest part we will take $m_q=0.005\rm{GeV}$, close to the physical value of $u,d$ quarks. Note that when $\mu=0$, $A_t\equiv0$ and $\lambda_2,\lambda_4,\lambda_6$ will 