  The harmonically confined Vicsek model displays qualitative and quantitative
features observed in natural insect swarms. It exhibits a scale free transition
between single and multicluster chaotic phases. Finite size scaling indicates
that this unusual phase transition occurs at zero confinement [Physical Review
E 107, 014209 (2023)]. While the evidence of the scale-free-chaos phase
transition comes from numerical simulations, here we present its mean field
theory. Analytically determined critical exponents are those of the Landau
theory of equilibrium phase transitions plus dynamical critical exponent $z=1$
and a new critical exponent $\varphi=0.5$ for the largest Lyapunov exponent.
The phase transition occurs at zero confinement and noise in the mean field
theory. The noise line of zero largest Lyapunov exponents informs observed
behavior: (i) the qualitative shape of the swarm (on average, the center of
mass rotates slowly at the rate marked by the winding number and its trajectory
fills compactly the space, similarly to the observed condensed nucleus
surrounded by vapor), and (ii) the critical exponents resemble those observed
in natural swarms. Our predictions include power laws for the frequency of the
maximal spectral amplitude and the winding number.
