Rolling tachyon solution in vacuum string field theory

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-bp2/4. 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 R2. 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 x0→∞. We also explore the possibility that the self-dual radius R=√(α') is a phase transition point of our Coulomb system