  The $F$-theorem states that in three dimensions the sphere free energy of a
field theory must decrease between ultraviolet and infrared fixed points of the
renormalization group flow, and it has been proven for unitary conformal field
theories (CFTs). We consider here the long-range bosonic $O(N)^3$ model on a
spherical background, at next-to-next-to-leading order of the $1/N$ expansion.
The model displays four large-$N$ fixed points and we test and confirm the
$F$-theorem holds in this case. This is non-trivial as one of the couplings is
imaginary, and therefore the model is non-unitary at finite $N$. Despite this,
several tests indicating that the large-$N$ CFTs are in fact unitary have been
performed: for instance all the OPE coefficients computed so far in the
large-$N$ limit are real, and the spectrum of bilinear operators is real and
above unitarity bounds. Our result, namely that the F theorem holds at large
$N$, can be viewed as further indication that such theories are unitary. As an
added bonus, we show how conformal partial waves expansions in conformal field
theory can be used to resum infinite classes of vacuum diagrams.
Non-perturbatively, the jump in the value of the free energy has the
interpretation of the inclusion at the ultraviolet fixed point of an extra
non-normalizable contribution in the conformal partial wave expansion. This can
be seen in perturbation theory as the reversal of the sign of an infinite class
of diagrams due to the flow of a coupling constant.
