 Vanishing disordered-side heat-capacity amplitude in Landau theory

For $F=F_{\mathrm{reg}}+a{m}^2/2+c{m}^6/6$, $c>0$, $a=a_0(T-T_c)$, the ordered global minimum gives $F-F_{\mathrm{reg}}=-(-a{)}^{3/2}/(3\sqrt{c})$ and $C_{\mathrm{sing},-}=T{a}_0^2/(4\sqrt{c}\sqrt{-a})$. On the disordered side the minimum is exactly $m=0$, so the polynomial's heat capacity is zero. The tricritical scaling label $\alpha =1/2$ can be used with zero upper-side amplitude, but no nonzero upper-side power law follows from this saddle alone. A smooth background such as $F_{\mathrm{reg}}=-C_{0}T\log (T/T_{\ast })$ gives a finite upper-side heat capacity $C_0$ without changing the phase diagram.