Using the supplied <Dynkin indices> and $p=N+4$, the <one-loop beta function of a supersymmetric gauge theory> is
$$
b_0=\frac32(2N)-\frac12\left[(N+2)+p\right]
=3N-\frac12(2N+6)=2N-3.
$$
Thus $b_0>0$ for every nontrivial $SU(N)$ in this family, and the electric theory is <asymptotic freedom>[asymptotically free].

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 307 / 4 / iii / Solution

{title2=$b_0$}

With the index convention $I(\mathbf N)=1$ for an $SU(N)$ fundamental, an N=1 supersymmetric gauge theory has $b_0=\tfrac32I(\mathrm{adj})-\tfrac12\sum_i I(R_i)$. Positive $b_0$ gives asymptotic freedom, while negative $b_0$ makes the gauge interaction infrared free near the Gaussian fixed point.


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