Conformality loss and short-range crossover in long-range conformal field theories

We study the conformality loss of theories with long-range interactions. We consider the $O(2)\times O(N)$ multiscalar model with coupling $r^{-d-\delta}$ in $d=4-\epsilon$ dimension. We compute the critical exponents of the long-range fixed points (LRFPs) to three loops. The phase diagram of the model is dominated by two processes: the short-range crossover and merger-annihilation of LRFPs. The two processes intersect at the lower edge of the conformal window, below which the LRFPs disappear into the complex plane. We propose a novel scenario for the short-range crossover of complex LRFPs, in which the short-range crossover occurs on a vertical line in the complex plane of $\delta$. The complex marginal operator generates renormalization group flow on the transition line and significantly enriches the short-range crossover of complex LRFPs.