The long range gravitational potential energy between strings

We calculate the gravitational potential energy between infinitely long parallel strings with tensions τ1 and τ2. Classically, it vanishes, but at one loop, we find that the long range gravitational potential energy per unit length is U/L=24GN2τ1τ2/(5πa2)+⋯, where a is the separation between the strings, GN is Newtonʹs constant, and we set ℏ=c=1. The ellipses represent terms suppressed by more powers of GNτi. Typically, massless bulk fields give rise at one loop to a long range potential between p-branes in space–times of dimension p+2+1. The contribution to this potential from bulk scalars is computed for arbitrary p (strings correspond to p=1) and in the case of three-branes its possible relevance for cosmological quintessence is commented on.