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. 
M.S. Onegin
astro-ph.CO hep-ph