I have a question about the one-loop computation of the wave-function renormalization factor in SQCD.

According to Seiberg duality, the following electric $\mathrm{SQCD}_{e}$

\begin{gather}

S_{e}(\mu)=\frac{1}{2g_{e}(\mu)^{2}}\left(\int d^{4}x\int d^{2}\theta\mathrm{Tr}(\mathbb{W^{\alpha}\mathbb{W}_{\alpha}})+\int d^{4}x\int d^{2}\bar{\theta}\mathrm{Tr}(\overline{\mathbb{W}}^{\dot{\alpha}}\overline{\mathbb{W}}_{\dot{\alpha}})\right)+ \\

+\frac{Z_{Q}(\Lambda_{e},\mu)}{4}\sum_{f=1}^{F}\int d^{4}x\int d^{2}\theta \int d^{2}\bar{\theta}\left(\widetilde{Q}^{\dagger}_{f}e^{V}\widetilde{Q}_{f}+Q^{\dagger}_{f}e^{-V}Q_{f}\right),

\end{gather}

with gauge group $SU(N)$ and $F$ flavors, is dual to the magnetic $\mathrm{SQCD}_{m}$

\begin{gather}

S_{m}(\mu)=\frac{1}{2g_{m}(\mu)^{2}}\left(\int d^{4}x\int d^{2}\theta\mathrm{Tr}(\mathbb{W^{\alpha}\mathbb{W}_{\alpha}})+\int d^{4}x\int d^{2}\bar{\theta}\mathrm{Tr}(\overline{\mathbb{W}}^{\dot{\alpha}}\overline{\mathbb{W}}_{\dot{\alpha}})\right)+ \\

+\frac{Z_{q}(\Lambda_{m},\mu)}{4}\sum_{f=1}^{F}\int d^{4}x\int d^{2}\theta \int d^{2}\bar{\theta}\left(\tilde{q}^{\dagger}_{f}e^{V}\tilde{q}_{f}+q^{\dagger}_{f}e^{-V}q_{f}\right)+ \\

+\frac{Z_{T}(\Lambda_{m},\mu)}{4}\int d^{4}x\int d^{2}\theta \int d^{2}\bar{\theta}T^{\dagger}T+ \\

+\lambda(\Lambda_{m})\left(\int d^{4}xd^{2}\theta\mathrm{tr}(qT\tilde{q})+\int d^{4}xd^{2}\bar{\theta}\mathrm{tr}(\tilde{q}^{\dagger}T^{\dagger}q^{\dagger})\right),

\end{gather}

with gauge group $SU(F-N)$ and $F$ flavors in the IR.

In the above expressions, $\Lambda_{e}$ and $\Lambda_{m}$ are respectively the UV cutoffs, factors $Z_{Q}$, $Z_{q}$, and $Z_{T}$ are wave-function renormalization constants, $T$ in $\mathrm{SQCD}_{m}$ is an $SU(F-N)$-gauge singlet, and the trace $\mathrm{tr}$ is taken over both flavor and color indices. The Yukawa coupling constant $\lambda(\Lambda_{m})$ is independent of the scale $\mu$ because of the famous Non-Renormalization Theorem.

My questions are about the one-loop computation of the above wave-function renormalization constant.

In QCD, we know that in minimal-subtraction scheme the wave-function renormalization factor for the kinetic term of the fermion is given by

$$Z=1-C_{2}(R)\frac{g^{2}}{8\pi^{2}}\frac{1}{\epsilon}+\mathcal{O}(g^{4}),$$

where $C_{2}(R)$ is the second Casimir of the representation $R$ that the quark field is in. This can be found in many standard QFT textbooks such as Srednicki, equation (73.3).

In this paper, there is a similar formula (equation (7) on page 4) of the wave-function renormalization constant,

$$Z_{Q}(\Lambda,\mu)=1+C_{2}(R)\frac{g^{2}}{4\pi^{2}}\log\frac{\mu}{\Lambda}+\mathcal{O}(g^{4}), \tag{7}$$

given in Wilsonian approach.

**1. Could you enlighten me how to derive equation (7) in Wilsonian approach?**

**2. Please also tell me how to derive equation (13) and (14) on page 6 **

\begin{align}

Z_{q}(\Lambda,\mu)&=1+\left(\frac{g^{2}}{4\pi^{2}}C_{2}(R)-\frac{\lambda^{2}}{8\pi^{2}}F\right)\log\frac{\mu}{\Lambda}+\mathcal{O}(g^{4},g^{2}\lambda^{2},\lambda^{4}), \tag{13} \\

Z_{T}(\Lambda,\mu)&=1-\frac{\lambda^{2}}{8\pi^{2}}(F-N)\log\frac{\mu}{\Lambda}+\mathcal{O}(g^{4},g^{2}\lambda^{2},\lambda^{4}), \tag{14}

\end{align}

**for the magnetic dual theory.**

My final question is a stupid one. As far as I could understand from what I read about Seiberg duality, the conjecture claims that in the conformal window

$$\frac{3}{2}N<F<3N$$

both $\mathrm{SQCD}_{e}$ and $\mathrm{SQCD}_{m}$ are conformal and flow to the same IR fixed point. In $\mathrm{SQCD}_{e}$, the NSVZ $\beta$ function is given by

\begin{align}

\beta(g_{e})&=-\frac{g_{e}^{3}}{16\pi^{2}}\frac{3N-F(1-\gamma(g_{e}))}{1-\frac{Ng^{2}_{e}}{8\pi^{2}}}, \\

\gamma(g_{e})&=-\frac{g_{e}^{2}}{8\pi^{2}}\frac{N^{2}-1}{N}+\mathcal{O}(g_{e}^{4}),

\end{align}

whose zero (Banks-Zaks Fixed Point) is at

$$(g^{\ast}_{e})^{2}=\frac{8\pi^{2}}{3}\frac{N}{N^{2}-1}\epsilon,$$

when $F=3N-\epsilon N$ with small enough $\epsilon$. On the other hand, in the dual theory $\mathrm{SQCD}_{m}$, the paper shows that the dual Banks-Zaks fixed point is at (equation (15) and (16))

\begin{align}

\frac{(g_{m}^{\ast})^{2}}{8\pi^{2}}&=\epsilon\frac{F-N}{(F-N)^{2}-1}\left(1+2\frac{F}{F-N}\right), \tag{15} \\

\frac{(\lambda^{\ast})^{2}}{8\pi^{2}}&=2\epsilon\frac{1}{F-N}. \tag{16}

\end{align}

**3. Since the Yukawa coupling constant $\lambda$ should not run along the RG flow, does the above fixed point $\lambda^{\ast}$ imply that one must tune $\lambda$ to $\lambda^{\ast}$ in the UV so that the theory flows to a fixed point in the IR?**

**4. Is Seiberg duality claiming that $g_{e}^{\ast}$ and $(g_{m}^{\ast},\lambda^{\ast})$ are actually the same point in the theory space?**

Worrying that my question will have no answers, I also posted my question here.