  The ``Brownian bees'' model describes an ensemble of $N=$~const independent
branching Brownian particles. The conservation of $N$ is provided by a modified
branching process. When a particle branches into two particles, the particle
which is farthest from the origin is eliminated simultaneously. The spatial
density of the particles is governed by the solution of a free boundary problem
for a reaction-diffusion equation in the limit of $N \gg 1$. At long times, the
particle density approaches a spherically symmetric steady state solution with
a compact support of radius $\bar{\ell}_0$. However, at finite $N$, the radius
of this support, $L$, fluctuates. The variance of these fluctuations appears to
exhibit a logarithmic anomaly [Siboni {\em et al}., Phys. Rev. E. {\bf104},
054131 (2021)]. It is proportional to $N^{-1}\ln N$ at $N\to\infty$. We
investigate here the tails of the probability density function (PDF), $P(L)$,
of the swarm radius, when the absolute value of the radius fluctuation $\Delta
L=L-\bar{\ell}_0$ is sufficiently larger than the typical fluctuations' scale
determined by the variance. For negative deviations the PDF can be obtained in
the framework of the optimal fluctuation method (OFM). This part of the PDF
displays the scaling behavior: $\ln P\propto - N \Delta L^2\, \ln^{-1}(\Delta
L^{-2})$, demonstrating a logarithmic anomaly at small negative $\Delta L$. For
the opposite sign of the fluctuation, $\Delta L > 0$, the PDF can be obtained
with an approximation of a single particle, running away. We find that $\ln P
\propto -N^{1/2}\Delta L$. We consider in this paper only the case, when
$|\Delta L|$ is much less than the typical radius of the swarm at $N\gg 1$.
