# A Question about Wave-Function Renormalization Factor in SQCD

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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.

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