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in Tabular I. From the stability matrix, 

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 

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, 

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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 , 

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Thus, the radius of convergence of the power series ex- pansion of W ′ $_{k}$( φ ) tends to zero for k → 0. 

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 

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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 

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In other words the outer part of the regular fixed-point solution connects smoothly to the inner part. Thus we 

## 2. Partial differential equation 

