Adelic Formulas for Four-Particle String and Superstring Tree Amplitudes in One-Class Quadratic Fields

Regularized adelic formulas for gamma and beta functions for arbitrary fields of algebraic numbers and arbitrary quasicharacters (ramified or not) are constructed under the condition that the corresponding quasicharacter of the idèle group of the field is trivial on the subgroup of principal idèles of this field. The problem of regularizing divergent infinite products for gamma and beta functions of local fields is solved. For the field of rational numbers and for the one-class quadratic fields, specific expressions for the adelic formulas are given. Applications to four-tachyon tree string amplitudes, generalized Veneziano amplitudes, Virasoro amplitudes with arbitrary perturbations, massless four-particle tree amplitudes (for open and closed superstrings), Neveu–Schwarz–Ramond superstring amplitudes, and amplitudes for four charged particles of a heterotic superstring are discussed.