Adelic Amplitudes and Intricacies of Infinite Products

TL;DR

This paper explores the properties of adelic amplitudes in string theory, revealing explicit evaluations in special cases, the non-analytic nature of the 5-point amplitude, and proposing new regularization methods that ensure unitarity.

Contribution

It introduces new regularization procedures for adelic amplitudes, explicitly evaluates 5-point amplitudes in certain regimes, and analyzes their positivity for unitarity.

Findings

Explicit evaluation of 5-point adelic amplitude ratios involving Riemann zeta functions.

Demonstration that the 5-point adelic amplitude is not a single analytic function.

Proposal of a new formalism yielding piecewise analytic amplitudes consistent with unitarity.

Abstract

For every prime number $p$ it is possible to define a $p$-adic version of the Veneziano amplitude and its higher-point generalizations. Multiplying together the real amplitude with all its $p$-adic counterparts yields the adelic amplitude. At four points it has been argued that the adelic amplitude, after regulating the product that defines it, equals one. For the adelic 5-point amplitude, there exist kinematic regimes where no regularization is needed. This paper demonstrates that in special cases within this regime, the adelic product can be explicitly evaluated in terms of ratios of the Riemann zeta function, and observes that the 5-point adelic amplitude is not given by a single analytic function. Motivated by this fact to study new regularization procedures for the 4-point amplitude, an alternative formalism is presented, resulting in non-constant amplitudes that are piecewise analytic in the three scattering channels, including one non-constant adelic amplitude previously suggested in the literature. Decomposing the residues of these amplitudes into weighted sums of Gegenbauer polynomials, numerical evidence indicates that in special ranges of spacetime dimensions all the coefficients are positive, as required by unitarity.