  We compute the one-loop partition function for quadratic tachyon background
in open string theory.
  Both closed and open string representations are developed. Using these
representations we study the one-loop divergences in the partition function in
the presence of the tachyon background. The divergences due to the open and
closed string tachyons are treated by analytic continuation in the tachyon mass
squared. We pay particular attention to the imaginary part of the analytically
continued expressions. The last one gives the decay rate of the unstable
vacuum. The dilaton tadpole is also given some partial consideration. The
partition function is further used to study corrections to tachyon condensation
processes describing brane descent relations. Assuming the boundary string
field theory prescription for construction of the string field action via
partition function holds at one loop level we study the one-loop corrections to
the tachyon potential and to the tensions of lower-dimensional branes.
