5 w $^{′′}$( ϕ → ±∞ ) ≃ ± ϕ 2 = ⇒ W ′′ ∗ ( φ → ±∞ ) ≃ ± φ $^{2}$. r = lim i →∞ b$_{2}$$_{i}$ b$_{2}$$_{i}$$_{+2}$ , (27) u ′′ = 2 u u 2 − 3 u 4 1 u ′ 2 + 4 π $_{2}$( u 2 + 1) 2 u 2 − 1 (2 u − ϕu $^{′}$) . (29) B$_{i}$ j = ∂ ( ∂$_{t}$b$_{i}$ ) ∂b$_{j}$ , (26) ¯ r = lim i →∞ ¯ b$_{2}$$_{i}$ ¯ b$_{2}$$_{i}$$_{+2}$ = k lim i →∞ b$_{2}$$_{i}$ b$_{2}$$_{i}$$_{+2}$ = k · r . (28) 2 n critical exponents 6 $^{−}$0 . 799 $^{−}$5 . 92 $^{−}$20 . 9 8 $^{−}$0 . 767 $^{−}$4 . 83 $^{−}$14 . 4 $^{−}$38 . 2 10 $^{−}$0 . 757 $^{−}$4 . 35 $^{−}$11 . 5 $^{−}$26 . 9 $^{−}$60 . 8 12 $^{−}$0 . 756 $^{−}$4 . 16 $^{−}$9 . 94 $^{−}$21 . 4 $^{−}$43 . 8 $^{−}$89 . 0 14 $^{−}$0 . 756 $^{−}$4 . 10 $^{−}$9 . 13 $^{−}$18 . 3 $^{−}$35 . 1 $^{−}$65 . 4 $^{−}$123 16 $^{−}$0 . 756 $^{−}$4 . 08 $^{−}$8 . 72 $^{−}$16 . 4 $^{−}$29 . 9 $^{−}$52 . 9 $^{−}$91 . 9 $^{−}$163 18 $^{−}$0 . 756 $^{−}$4 . 08 $^{−}$8 . 54 $^{−}$15 . 2 $^{−}$26 . 4 $^{−}$45 . 0 $^{−}$75 . 0 $^{−}$124 $^{−}$209 2 n ± λ ± b$_{4}$ ± b$_{6}$ ± b$_{8}$ ± b$_{10}$ ± b$_{12}$ 4 1.546 2.305 6 1.590 2.808 6.286 8 1.595 2.873 7.150 13.41 10 1.595 2.873 7.155 13.48 1.212 12 1.595 2.870 7.118 12.90 -8.895 -183.3  0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  0  2  4  6  8  10  12  14  16 | b 2i /b 2i+2 | i FIG. 2. Ratio b 2 i /b 2 i +2 of the Taylor expansion coefficients of the potential as a function of i . TABLE I. Wilson-Fisher fixed point as obtained from the polynomial approximation to w $^{′}$( ϕ ). Note that the discrep- ancy between the fixed-point values of b 10 for the truncation with 2 n = 10 and 2 n = 12 is a truncation effect. Due to our polynomial expansion of w ′ k ( ϕ ), see Eq. (24), we obtain a hi- erachy of flow equations for the couplings b 2 i in which the RG flow of the coupling b 2 n (for given truncation order n ) suffers the most by the finite truncation order n . Therefore we ex- pect that the fixed-point values for the higher-order couplings converge when n is increased, as it is indeed the case for λ , b 2 , b 4 , b 6 . TABLE II. Critical exponents for the Wilson-Fisher fixed point for different truncations. that for the corresponding series with dimensionful field and couplings ¯ b$_{2}$$_{i}$ = b$_{2}$$_{i}$ · k 3 / 2 − i the radius of convergence shrinks with decreasing scale k , Thus, the radius of convergence of the power series ex- pansion of W ′ $_{k}$( φ ) tends to zero for k → 0. in Tabular I. From the stability matrix, Let us now turn to the solution of the partial differ- ential equation (22). We have seen in Eq. (25) that the coupling associated with the IR unstable direction does not feed back into the fixed-point equation. It is therefore sufficient to consider the second derivative of Eq. (22). To simplify the notation we introduce u ( ϕ ) = w ′′ $_{k}$( ϕ ). The fixed-point equation for u reads 2. Partial differential equation we read off that the non-Gaußian fixed points are IR sta- ble. Here, we have set b$_{0}$ = a 2 and b$_{2}$ = λ . These IR stable fixed points are to be considered as supersymmet- ric equivalent of the Wilson-Fisher fixed point. Let us now discuss the critical exponents which are the negative eigenvalues of the stability matrix at the fixed point. The coupling a $^{2}$, which does not feed back into the equations for the higher-orders couplings, defines an IR unstable direction with a critical exponent 1 /ν$_{W}$ = 3 $_{2}$. The critical exponents of the IR stable directions of the Wilson-Fisher fixed point are given in Tabular II. Actu- ally we observe a better convergence of the lowest critical exponents than in two dimensions [13]. From Tabular I we estimate that the radius of conver- gence of the Taylor series (24), given by It is straightforward to see that the above equation has an asymptotic solution u$_{as}$ ( ϕ ) ∼ ϕ $^{2}$. Employing a standard Runge-Kutta solver for ordinary differential equations we find one regular odd solution for the starting condition u (0) = 0 and u $^{′}$(0) ≡ 2 λ ≈ ± 2 · 1 . 59508 = ± 3 . 19016 . For field amplitudes ϕ < 0 . 245 the regular solution is bounded by u < 1 and this inner part of the solution corresponds to the IR stable fixed-point solution found in the polynomial approximation discussed above. Since u = w ′′ we find for large fields In other words the outer part of the regular fixed-point solution connects smoothly to the inner part. Thus we is finite. This can be seen more clearly from Fig. 2, where we plotted the ratios b$_{2}$$_{i}$/b$_{2}$$_{i}$$_{+2}$ as functions of i . Note,