Investigation of the fundamental constants stability based on the
  reactor Oklo burn-up analysis
  New severe constraints on the variation of the fine structure constant have
been obtained from reactor Oklo analysis in our previous work. We investigate
here how these constraints confine the parameter of BSBM model of varying
$\alpha$. Integrating the coupled system of equations from the Big Bang up to
the present time and taking into account the Oklo limits we have obtained the
following margin on the combination of the parameters of BSBM model: $$
|\zeta_m (\frac{l}{l_{pl}})^2|<6\cdot 10^{-7}, $$ where
$l_{pl}=(\frac{G\hbar}{c^3})^{\frac{1}{2}} \approx 1.6 \cdot 10^{-33}$ cm is a
Plank length and $l$ is the characteristic length of the BSBM model. The
natural value of the parameter $\zeta_m$ - the fraction of electromagnetic
energy in matter - is about $10^{-4}$. As a result it is followed from our
analysis that the characteristic length $l$ of BSBM theory should be
considerably smaller than the Plank length to fulfill the Oklo constraints on
$\alpha$ variation.
