  We calculate the dependence of the electrostatic persistence length, l_e, of
weakly charged flexible polyelectrolyte chains using a self-consistent
variational theory. The variation of l_e with \kappa, the inverse Debye
screening length, is controlled by the parameter l_0 l_B/A^2, where l_0 is the
bare persistence length, l_B is the Bjerrum length, and A is the mean distance
between charges along the chain. Several distinct regimes for the dependence of
l_e on \kappa emerge depending on the value of l_0 l_B/A^2. We show that when
l_0 l_B /A^2 << 1 we recover the classical result, l_e \propto \kappa^{-2}. For
intermediate values of l_0 l_B /A^2, l_e \propto \kappa^{-1}. In this regime
one can also get l_e \propto \kappa^{-y} with y < 1 depending on the strength
of the Coulomb interaction. Qualitative comparisons between our theory and
simulations as well as other theories are presented.
