Critical Wess-Zumino models with four supercharges in the functional renormalization group approach

We analyze the $\mathcal{N}=1$ supersymmetric Wess-Zumino model dimensionally reduced to the $\mathcal{N}=2$ supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the $\mathcal{N}=(2,2)$ model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the nontrivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a Kähler metric. The Kähler metric is not protected by supersymmetry and we calculate its explicit form at the fixed point. In addition we determine the exact quantum dimension of the chiral superfield and several critical exponents of interest, including the correction-to-scaling exponent $\omega$, within the functional renormalization group approach. We compare the results obtained at different levels of truncation, exploring also a momentum-dependent wave function renormalization. Finally we briefly describe a tower of multicritical models in continuous dimensions.

Figure 1

Shooting from the origin. Field coordinate ${\rho }_s$ of the singularity in $\zeta (\rho )$ closest to the origin as a function of $g^2$. Here $a=1.7$; the position of the spike provides $g_m^2(a=1.7)=2.53$.

Figure 2

Left panel: Fixed point values $g_m^2$ as provided by shooting from the origin for different $a$. The solid line is a fit; for the interpolating function; see Eq. (62). Right panel: Comparison of shooting from the origin and polynomial truncation. The solid line is located at $g_m^2(a)$ as obtained from the interpolating function. The $g_{*}^2(a,N)$ due to polynomial truncation of order $N$ converge to $g_m^2(a)$.

Figure 3

The critical Kähler metric for $a=1.75$. The solid black line is produced by pseudospectral methods, while the solid gray line shows a ${\rho }^{-1/4}$ decay. Inset: The pseudospectral solution (dotted line) is compared to the one obtained by shooting from the origin (dashed line) with $g_{*}^2=2.576$ corresponding to a maximum of ${\rho }_s$ as in Fig. 1.