Representation of Integral Dispersion Relations by Local Forms

TL;DR

This paper demonstrates that derivative dispersion relations (DDR) can accurately and mathematically represent integral dispersion relations (IDR) in scattering theory, simplifying their form and correcting misconceptions about their applicability.

Contribution

The authors derive simplified, convergent forms of DDR that exactly match IDR, clarifying their validity across the entire energy range and correcting previous misconceptions.

Findings

DDR can represent IDR with absolute accuracy.

Simplified single-summation forms of DDR are mathematically identical to IDR.

Standard high-energy sDDR forms are incomplete and can lead to errors.

Abstract

The representation of the usual integral dispersion relations (IDR) of scattering theory through series of derivatives of the amplitudes is discussed, extended, simplified, and confirmed as mathematical identities. Forms of derivative dispersion relations (DDR) valid for the whole energy interval, recently obtained and presented as double infinite series, are simplified through the use of new sum rules of the incomplete $\Gamma$ functions, being reduced to single summations, where the usual convergence criteria are easily applied. For the forms of the imaginary amplitude used in phenomenology of hadronic scattering, we show that expressions for the DDR can represent, with absolute accuracy, the IDR of scattering theory, as true mathematical identities. Besides the fact that the algebraic manipulation can be easily understood, numerical examples show the accuracy of these representations up to the maximum available machine precision. As consequence of our work, it is concluded that the standard simplified forms sDDR, originally intended for the high energy limits, are an inconvenient and incomplete separation of terms of the full expression, leading to wrong evaluations. Since the correspondence between IDR and the DDR expansions is linear, our results have wide applicability, covering more general functions, built as combinations of well studied basic forms.