Two-point derivative dispersion relations

Abstract

A new derivation is given for the representation, under certain conditions, of the integral dispersion relations of scattering theory through local forms. The resulting expressions have been obtained through an independent procedure to construct the real part, and consist of new mathematical structures of double infinite summations of derivatives. In this new form the derivatives are calculated at the generic value of the energy $E$ and separately at the reference point $E=m$ that is the lower limit of the integration. This new form may be more interesting in certain circumstances and directly shows the origin of the difficulties in convergence that were present in the old truncated forms called standard-DDR. For all cases in which the reductions of the double to single sums were obtained in our previous work, leading to explicit demonstration of convergence, these new expressions are seen to be identical to the previous ones. We present, as a glossary, the most simplified explicit results for the DDR's in the cases of imaginary amplitudes of forms $(E/m{)}^{\lambda }[\ln (E/m){]}^n$, that cover the cases of practical interest in particle physics phenomenology at high energies. We explicitly study the expressions for the cases with $\lambda$ negative odd integers, that require identification of cancelation of singularities, and provide the corresponding final results.

The final publication is available at http://scitation.aip.org/content/aip/journal/jmp

References in corpus (4)

• Measurement of the proton-air cross-section at $\sqrt{s}=57$ TeV with the Pierre Auger Observatory
• Derivative dispersion relations above the physical threshold
• Representation of Integral Dispersion Relations by Local Forms
• Integral and derivative dispersion relations, analysis of the forward scattering data

Cited by in corpus (5)

• A study on analytic parametrizations for proton-proton cross-sections and asymptotia
• New properties of the Lerch's transcendent
• Leading components in forward elastic hadron scattering: Derivative dispersion relations and asymptotic uniqueness
• Structure of Forward pp and pp Elastic Amplitudes at Low Energies
• Exact treatment of dispersion relations in pp and pp elastic scattering