p-Adic open string amplitudes with Chan-Paton factors coupled to a constant B-field

TL;DR

This paper rigorously regularizes p-adic open string amplitudes with Chan-Paton factors and a B-field using multivariate local zeta functions, ensuring meromorphic continuation and divergence-free results, with special prime number conditions.

Contribution

It introduces new mathematical objects—multivariate local zeta functions depending on physical parameters—and applies them to regularize string amplitudes.

Findings

Amplitudes admit meromorphic continuation in kinematic parameters.

Regularized amplitudes are free of ultraviolet divergences.

Limit p→1 connects to noncommutative field theory amplitudes.

Abstract

We establish rigorously the regularization of the p-adic open string amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan-Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters, this result allows us to regularize the Goshal-Kawano amplitudes, the regularized amplitudes do not have ultraviolet divergences. Due to the need of a certain symmetry, the theory works only for prime numbers which are congruent to 3 modulo 4. We also discuss the limit p tends to 1 in the noncommutative effective field theory and in the Ghoshal-Kawano amplitudes. We show that in the case of four points, the limit p tends to 1 of the regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes attached to the limit p tends to 1 of the noncommutative Gerasimov-Shatashvili Lagrangian.