Self-consistent flow equations for a general \(O(N)\)-symmetric effective potential without any polynomial truncation are derived, and their numerical solutions are examined. The effective Lagrangian for the \(O(N)\)-model at the scale \(\Lambda\) of the ultraviolet region is \[ {\mathfrak L}_\Lambda= {1\over 2} (\partial_\mu \Phi)^2+ V(\Phi^2); \quad V(\Phi^2)= {\lambda\over 4}(\Phi^2-\Phi_0^2)^2, \] \(\Phi=(\Phi_1, \dots, \Phi_N)\) and the negative sign signals the broken phase. The expansion of the effective action of this model up to order \(O (\partial^4)\) is written \[ \Gamma[\Phi=]\int d^dx\Bigl\{ V(\Phi^2)+\frac 12 Z_1 (\Phi^2) (\partial_\mu \Phi)^2+ \frac 12 Z_2(\Phi^2) (\Phi\partial_\mu \Phi)^2 \Bigr\}. \] By using a heat kernel, the effective action can be written as \[ \Gamma [\Phi]=-{1\over 2}\int d^dx \int^\infty_0 {d\tau\over \tau}f_k \int {d^dp \over (2\pi)^d} \text{tr} e^{-\tau \bigl(p^2-2i p_\mu\partial_\mu-\partial^2 + V''_{ij} (\Phi)\bigr)}, \] \(V_{ij}''= \lambda (\Phi^2- \Phi^2_0) \delta_{ij}+ 2\lambda\Phi_i \Phi_j\). The technical details of the derivation of this expression is given in Appendix A. Introducing differential equations for the blocking functions \(f_k^{(i)}\) the solutions of these equations are given by \[ f_k^{(i)}= {2^i (d-2)!! \over\Gamma (d/2)(d-2+2i)!!} \Gamma\left({d \over 2}+ i,\tau Zk^2 \right). \] Then using the abbreviations \(m^2_\sigma= \lambda(3 \Phi^2- \Phi^2_0)\), \(m^2_\pi= \lambda(\Phi^2- \Phi^2_0)\), the renormalization flow equations are derived as \[ \begin{aligned} \partial_tV & =S_d {k^d\over d}\left[ {1\over 1+2v'+4 \Phi^2 v''} +{N-1\over 1+2v'} \right],\\ -{1\over Z}\partial_tZ & ={2S_d\over \Phi^2 Zk^2} {k^d\over d}\left. \left[1+{1\over 1+4 \Phi^2v'')^2}+ {1\over 2 \Phi^2 v''}\left( {1\over 1+4\Phi^2v''}-1\right) \right] \right |_{ \Phi^2 =\Phi_0^2},\\ \partial_tV & =k{\partial V\over \partial k},\;v^{(i)} = {V^{(i)} \over Zk^2},\;S_d={2\over \Gamma(d/2)(4\pi)^{d/2}}. \end{aligned} \] In the numerical study, these equations are rewritten to the following properly rescaled dimensionless flow equations in \(d\) dimensions: \[ \begin{aligned} \partial_t u(\varphi^2) & =-du +(d-2+\eta) \varphi^2 u'+{S_d\over d}\left[ {1\over 1+2u'+ 4 \varphi^2 u''}+ {N-1\over 1+2u'} \right],\\ -{1\over Z}\partial_tZ & ={2S_d \over \varphi^2 d}\left.\left[ 1+{1\over (1+4\varphi^2 u'')^2}{1\over 2\varphi^2 u''}\left( {1\over 1+4\varphi^2 u''}-1\right) \right] \right|_{\varphi= \varphi_0}. \end{aligned} \] Generalization to finite temperature is given within the Matsubara formalism and it is shown that the zero temperature limits of the finite-temperature threshold functions \[ {\mathcal M}(p,m^2, \alpha)= \sum^\infty_{m= -\infty} {(\omega^2_n)^p \over(1+ \omega^2_n/k^2+ m^2)^\alpha}, \] are \[ \begin{aligned} {\mathcal M}(0,m^2,\alpha) & \to{k\over\pi T}{2\cdot 4 \cdots (2 \alpha-3) \over 1\cdot 3\cdots (2\alpha-2)} {1\over (1+m^2)^{(2\alpha -1)/2}},\\ {\mathcal M}(1,m^2, \alpha) & \to{k^3 \over 2\pi T}{2\cdot 4\cdots (2\alpha-5) \over 1\cdot 3\cdots (2\alpha-4)} {1\over(\alpha-1)} {1\over(1+m^2)^{(2 \alpha-3)/2}}, \end{aligned} \] (Sect. 3. Detailed calculations are not given).
The rest of the paper is devoted to the numerical study of these equations. In Section 4, the numerical results of the solutions of the flow equations are summarized. Then in Section 5, the critical behavior of the system at the transition temperature is studied. The authors state that the results are in perfect agreement with other works and approaches.