Closed Bosonic String Field Theory at Quintic Order: Five-Tachyon Contact Term and Dilaton Theorem

We solve the geometry of the closed string field theory five-point vertex. Our solution is calculated in terms of quadratic Strebel differentials which are found numerically all over the relevant subspace of the moduli space of spheres with five punctures. Part of the boundary of the reduced moduli space is described in terms of an algebraic curve, while the remaining part has to be evaluated numerically. We use this data to compute the contact term of five tachyons and estimate its uncertainty to be of about 0.1%. To put to a test the theory and the computations done, we calculate the contact term of five dilatons. In agreement with the dilaton theorem, it is found to cancel the term obtained from the tree level Feynman diagrams built with three- and four-vertices. This cancellation, achieved with a precision of about 0.1%, is within the estimated margin error on the contact term and is therefore a very good evidence that our computations are reliable. The techniques and numerical algorithm developed in this paper make it possible to compute the contact amplitude of any five off-shell closed bosonic string states.