On Exact Tachyon Potential in Open String Field Theory

In these notes we revisit the tachyon lagrangian in the open string field theory using background independent approach of Witten from 1992. We claim that the tree level lagrangian (up to second order in derivatives and modulo some class of field redefinitions) is given by $L = e^{-T} (\partial T)^2 + (1+T)e^{-T}$. Upon obvious change of variables this leads to the potential energy $-{\varphi }^2\log \frac{{\varphi }^2}{e}$ with canonical kinetic term. This lagrangian may be also obtained from the effective tachyon lagrangian of the p-adic strings in the limit $p\to 1$. Applications to the problem of tachyon condensation are discussed.