# Renormalization of the coupling constant in $\mathcal{N}=1$ SYM

I have been watching the lecture https://youtu.be/lrikIt9MXpQ from the school LACES 2020 by INFN.

The $$\mathcal{N}=1$$ SYM is investigated, with the action: $$\mathcal{L} = \frac{1}{32\pi} \mathrm{Im} \left[\tau \int d^2 \theta \ W^{\alpha} W_\alpha\right]$$ And there is claim, that the holomorphy constraints the dependence of coupling constant on the holomorphic scale $$\bar{\Lambda}$$ to be: $$\tau(\mu) = \underbrace{\frac{b_1}{2 \pi} \log \frac{\bar{\Lambda}}{\mu}}_{\text{1-loop}} + \underbrace{\sum_n c_n \left(\frac{\bar{\Lambda}}{\mu} \right)^{n b_1}}_{\text{instantons}}$$ Where holomorphic scale $$\bar{\Lambda}$$ is defined as: $$\bar{\Lambda} = \Lambda e^{i \theta / b_1} \qquad \Lambda = \mu_0 e^{-8 \pi^2 / b_1 g^2 (\mu)}$$ It claimed (approx 6:10 on the video) that there can be no corrections beyond 1-loop, because instanton dependence on $$g^2 (\mu)$$ has to be via $$\tau$$, and the higher loop corrections would come a powers of $$g^2 (\mu)$$ and they have to be combined with the corresponding power of $$\theta$$, but these terms cannot be generated, because $$\theta$$ enters the Lagrangian only as a term, proportional to the derivative of something ($$F \wedge \tilde{F} = d(A \wedge d A + \frac{2}{3} A \wedge A \wedge A)$$), and does not contribute to the perturbation theory.

I do not understand, how the instanton dependence is connected with the reasoning on the perturbation theory, instanton contributions are in some sense, by definition, non-perturbative phenomena. How does this constrain the perturbative expansion? I guess, that the crucial part in the reasoning should be some holomorphy properties, but I do not see how, they emerge in the present case.

I would be grateful for comments and clarification

The point is that the $$U(1)_{R}$$ current and the conformal current, at the classical level, lives in the same super-multiplet, so an anomaly of one current is related to the other. Indeed at classical level we have

$$\partial_{m}j^{m} = \sigma^{m}_{\alpha\dot\alpha}\bar S_{m}^{\dot\alpha} = \sigma^{m}_{\alpha\dot\alpha}S_{m}^{\alpha}=T^{m}\,_{m}=0$$

where $$j^{m}$$ is the $$U(1)_{R}$$ current, $$(S_{m}^{\alpha},\bar S_{m}^{\dot\alpha})$$ is the supersymmetry current and $$T_{mn}$$ is the energy-momentum tensor. Conformal symmetry means $$T^{m}\,_{m}=0$$.

Since the anomaly of the $$U(1)_{R}$$ current is one-loop exact we know how this symmetry is broken: it is broken by instanton effects. This is so because the anomaly is of the form

$$dj \propto F\wedge F \implies (Q_{2}-Q_{1})=\int_{\partial \Sigma} j = \int_{\Sigma}F\wedge F$$

where $$j$$ is the $$3$$-from current, $$F$$ is the $$2$$-form field strength, $$(Q_{2}-Q_{1})$$ is the charge violation and $$\Sigma$$ is the history of the space-time from time $$1$$ to time $$2$$. Since, by definition, instantons are the only thing that couples with $$\int _{\Sigma}F\wedge F$$, the only thing that couples to the $$U(1)_{R}$$ anomaly are the instantons.

Once we know how the $$U(1)_{R}$$ is broken, by supersymmetry you can learn about how the conformal symmetry is broken and this is why instantons are the only non-perturbative objects that shows up. The relation between these two anomalies is not so straightforward so I will not discuss it here.

You can see more about it in Supersymmetry and the Adler-Bardeen theorem - M. T. Grisaru, P. C. West and also in Supersymmetry and Nonperturbative beta Functions - Nathan Seiberg.

(particularly in the first paragraph bellow equation $$13$$ of the last reference).