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In Modern Supersymmetry: Dynamics and Duality, on page 134 and 135 in section 8.2, the authors studied the wavefunction renormalization of the Wess-Zumino model.

The kinetic terms are given by

$$\mathcal{L}_{kin}=Z(m,m^{\dagger},\lambda,\lambda^{\dagger},\mu,\Lambda)\partial_{\mu}\phi^{\ast}\partial^{\mu}\phi+iZ(m,m^{\dagger},\lambda,\lambda^{\dagger},\mu,\Lambda)\bar{\psi}\bar{\sigma}^{\mu}\partial_{\mu}\psi,$$

where $m$ and $\lambda$ are spurion fields (i.e. coupling coustants treated as background chiral superfield) in the tree level superpotential

$$W_{eff}=\frac{m}{2}\Phi^{2}+\frac{\lambda}{3}\Phi^{3}\equiv W_{tree},$$

where $\Phi(x,\theta,\bar{\theta})$ is the chiral superfield, and $\Lambda$ is the UV-cutoff.

Next, the authors claimed two things:

  1. If we integrate out modes $\mu>m$, we obtain the wavefunction renormalization factor

$$Z=1+c|\lambda|^{2}\ln\left(\frac{\Lambda^{2}}{\mu^{2}}\right).$$

  1. If we integrate out modes $\mu$ below $m$, we obtain the wavefunction renormalization factor

$$Z=1+c|\lambda|^{2}\ln\left(\frac{\Lambda^{2}}{|m|^{2}}\right).$$

In the above expressions, $c$ is a constant determined by the perturbative calculations.

Can anybody give me the perturbative calculation (at one-loop) in detail? I don't know how to get the above expressions for $Z$.

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