Rolling tachyon solution in vacuum string field theory

Abstract

We construct a time-dependent solution in vacuum string field theory and investigate whether the solution can be regarded as a rolling tachyon solution. First, compactifying one space direction on a circle of radius $R$, we construct a space-dependent solution given as an infinite number of $*$ products of a string field with center-of-mass momentum dependence of the form ${e}^{\ensuremath{-}b{p}^{2}/4}$. Our time-dependent solution is obtained by an inverse-Wick rotation of the compactified space direction. We focus on one particular component field of the solution, which takes the form of the partition function of a Coulomb system on a circle with temperature ${R}^{2}$. Analyzing this component field both analytically and numerically using Monte Carlo simulation, we find that the parameter $b$ in the solution must be set equal to zero for the solution to approach a finite value in the large time limit ${x}^{0}\ensuremath{\rightarrow}\ensuremath{\infty}$. We also explore the possibility that the self-dual radius $R=\sqrt{{\ensuremath{\alpha}}^{\ensuremath{'}}}$ is a phase transition point of our Coulomb system.