Fine Structure Constant Variation from a Late Phase Transition
  Recent experimental data indicates that the fine structure constant alpha may be varying on cosmological time scales. We consider the possibility that such a variation could be induced by a second order phase transition which occurs at late times (z ~ 1 - 3) and involves a change in the vacuum expectation value (vev) of a scalar with milli-eV mass. Such light scalars are natural in supersymmetric theories with low SUSY breaking scale. If the vev of this scalar contributes to masses of electrically charged fields, the low-energy value of alpha changes during the phase transition. The observational predictions of this scenario include isotope-dependent deviations from Newtonian gravity at sub-millimeter distances, and (if the phase transition is a sharp event on cosmological time scales) the presence of a well-defined step-like feature in the alpha(z) plot. The relation between the fractional changes in alpha and the QCD confinement scale is highly model dependent, and even in grand unified theories the change in alpha does not need to be accompanied by a large shift in nucleon masses. 
Z. Chacko, C. Grojean, M. Perelstein