We utilize dispersion relations to constrain weakly coupled ultraviolet (UV) completions of gravity when the external particles are scalars with color or flavor indices. We furnish a sum rule which implies that only certain patterns of representations can be ommitted in the UV. For adjoint scalars coupled to gravity, this predicts new particles, or the breaking of the symmetry, in the UV. This is all derived from the boundedness required for the dispersion relation, covariance with the underlying symmetries, and crossing symmetry. Unitarity is not invoked to derive the vanishing constraint.

We present a subtraction scheme for ultraviolet (UV) divergent, infrared (IR) safe scalar Feynman integrals in dimensional regularization with any number of scales. This is done by introducing u-variables, inspired by structures found studying the open-string moduli space, and results in an algorithmic prescription for the computation of a set of convergent integrals, dressed by powers of epsilon, which calculate the Feynman integral in dimensional regularization. This is the first general solution to this problem.

We motivate an analytic bootstrap which defines algebraic varities whose solutions are multi-parameter families of dual resonant tree-level amplitudes. Imposing the strongest form of our assumptions leaves only a single point, the Veneziano amplitude. This defines simple conditions under which the Veneziano amplitude is the unique consistent solution to a bootstrap problem.