Local CFTs extremise $F$

Many CFTs can be extended to lines of nonlocal CFTs parametrised by the scaling dimension $\Delta$ of the fundamental field appearing in the action. $\Delta=\frac{d}{2}-\zeta$ is set by the exponent of the kinetic term $(-\partial^2)^{\zeta}$, which is nonlocal for noninteger $\zeta$. If $\Delta$ is tuned to $\Delta_\mathrm{local}$, the scaling dimension of the fundamental field in the local CFT, arXiv:1703.05325 showed that we recover the conformal data of that CFT (plus a decoupled sector). One natural question is: how is the local point special on this line of nonlocal CFTs? We prove that these local CFTs lie at the extrema of the (universal part of the) sphere free energy $\tilde{F}(\Delta)=\sin(\frac{\pi d}{2}) \log Z_{S^d}$ of the long-range CFTs: $d\tilde{F}/d\Delta|_{\Delta=\Delta_\mathrm{local}}=0$; and for unitary CFTs they locally maximise it. The simple proof uses the fact that the derivative of $\tilde{F}$ with respect to $\Delta$ receives contributions only from the nonlocal terms in the action. The nonlocal terms must be absent in the limit $\Delta \to \Delta_\mathrm{local}$, and hence the derivative is zero. Demonstrating maximisation then requires a proof of the generalised $F$-theorem in conformal perturbation theory to subleading order. We check our result with the O$(N)$ $\phi^4$ and cubic CFTs in the $\epsilon$ expansion and the large-$N$ limit. This result provides a minimal encoding of the scaling dimension of the fundamental field in many CFTs, and also explains the curious derivative structure of the large-$N$ expansion of $\Delta_\phi$ in the O$(N)$ model. Finally, this nonlocal $F$-extremisation can be viewed as a non-supersymmetric version of the known $c,F,a$-extremisation mechanisms.