A free scalar has <scaling dimension> one. Setting the meson dimension to one gives
$$
3\left(1-\frac{N_c+1}{N_f}\right)=1,
\qquad
N_f=\frac32(N_c+1).
$$
The expected interacting <supersymmetric conformal window> is therefore
$$
\boxed{\frac32(N_c+1)<N_f<3(N_c+1)}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 307 / 4 / iv / Solution

The supersymmetric conformal window is the range of flavor numbers for which an asymptotically free supersymmetric gauge theory flows to an interacting infrared fixed point. Its lower edge is often detected when a gauge-invariant chiral operator reaches the scalar unitarity bound.


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