Effective potential of the three-dimensional Ising model: the pseudo-$\epsilon$ expansion study

The ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ that enter the effective potential and small-field equation of state acquire the universal values at criticality. They are calculated for the three-dimensional scalar $\lambda\phi^4$ field theory (3D Ising model) within the pseudo-$\epsilon$ expansion approach. Pseudo-$\epsilon$ expansions for the critical values of $g_6$, $g_8$, $g_{10}$, $R_6 = g_6/g_4^2$, $R_8 = g_8/g_4^3$ and $R_{10} = g_{10}/g_4^4$ originating from the five-loop renormalization group (RG) series are derived. Pseudo-$\epsilon$ expansions for the sextic coupling have rapidly diminishing coefficients, so addressing Pad\'e approximants yields proper numerical results. Use of Pad\'e--Borel--Leroy and conformal mapping resummation techniques further improves the accuracy leading to the values $R_6^* = 1.6488$ and $R_6^* = 1.6490$ which are in a brilliant agreement with the result of advanced lattice calculations. For the octic coupling the numerical structure of the pseudo-$\epsilon$ expansions is less favorable. Nevertheless, the conform-Borel resummation gives $R_8^* = 0.868$, the number being close to the lattice estimate $R_8^* = 0.871$ and compatible with the result of 3D RG analysis $R_8^* = 0.857$. Pseudo-$\epsilon$ expansions for $R_{10}^*$ and $g_{10}^*$ are also found to have much smaller coefficients than those of the original RG series. They remain, however, fast growing and big enough to prevent obtaining fair numerical estimates.