Critical analysis of derivative dispersion relations at high energies

Abstract

We discuss some formal and fundamental aspects related with the replacement of integral dispersion relations by derivative forms, and their practical uses in high energy elastic hadron scattering, in particular $pp$ and $\bar{p}p$ scattering. Starting with integral relations with one subtraction and considering parametrizations for the total cross sections belonging to the class of entire functions in the logarithm of the energy, a series of results is deduced and our main conclusions are the following: (1) except for the subtraction constant, the derivative forms do not depend on any additional free parameter; (2) the only approximation in going from integral to derivative relations (at high energies) concerns to assume as zero the lower limit in the integral form; (3) the previous approximation and the subtraction constant affect the fit results at both low and high energies and therefore, the subtraction constant can not be disregarded; (4) from a practical point of view, for single-pole Pomeron and secondary reggeons parametrizations and center-of-mass energies above 5 GeV, the derivative relations with the subtraction constant as a free fit parameter are completely equivalent to the integral forms with finite (non-zero) lower limit. A detailed review on the conditions of validity and assumptions related with the replacement of integral by derivative relations is also presented and discussed.Comment: Revised version, 30 pages, 16 eps-figures, elsart.cls (included), to appear in Nucl Phys.

Aspects of a dynamical gluon mass approach to elastic hadron scattering at LHC

“…In the first one generated numerical points for the real primitive of the integrand in (16) have been parametrized by polynomials in ln s and then the even prescription has been applied. In the second one we have used first order derivative dispersion relation for even functions [37] applied as…”

Topics on high-energy elastic hadron scattering

“…In that case, as demonstrated in detail in [16], the derivative dispersion relations with one subtraction reads…”

“…In Reference [16] we present a recent review on different results and statements related to this replacement, and a discussion connecting these different aspects with the corresponding assumptions and classes of functions considered in each case.…”

“…Making use of this parametrization, in this section we present the determination of extrema bounds for the Pomeron intercept [15] and a practical analysis on the replacement of IDR by DDR together with a discussion on the role of the subtraction constant [16].…”

“…[16] that for entire functions in the logarithm of the energy the only approximation involved in the replacement of IDR by DDR concerns the lower limit s 0 in the IDR: the high-energy condition is reached by assuming that s 0 = 2m 2 → 0 in Eqs. (13)(14).…”