 Solution

Write $t=T-T_c$. Analyticity gives ${\alpha }_2=at+O(t^2)$ with $a>0$ and ${\alpha }_4(T_c)=u>0$. For $t<0$, minimization gives $m^2=-{\alpha }_2/{\alpha }_4\sim -at/u$, so $\beta =1/2$. At the minima,Two temperature derivatives give a finite heat-capacity jump, hence $\alpha =0$. This is the ordinary mean-field transition of Landau theory.

Solved by gpt-5.6-sol high.