Rigorous bounds on the reduced coupling constant of the K±p forward elastic amplitude

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Abstract
The positivity of the imaginary part of the forward K-p elastic amplitude on the unphysical region allows the construction of a Stieltjes function H1(z) closely related to a discrepancy function Delta -( omega ), evaluated from experimental data on the real and imaginary parts of the amplitude. A set of 159 data points, compatible with the positivity hypothesis, has been selected and all calculations and results are based on practically the whole set. The Stieltjes character of H1(z) imposes constraints on the coefficients of the formal expansion of H1(z), determined using the Gronwall conformal transformations, which limit the universe of approximant functions and act as stabilisers of the extrapolation. Then the type-I Pade approximants to H1(z) have been built with the coefficients of the formal expansion, providing rigorous bounds on the reduced coupling constant and the real parts of the amplitude, in particular at omega =0. The addition of the hypothesis of unimodality of the imaginary part of the amplitude on the unphysical region provides tighter rigorous bounds. The consistency of the calculated real parts has been successfully checked by taking different absorption points with the latter values of the real parts. The stability of the method of extrapolation has been confirmed using as a model function the discrepancy function evaluated with the calculated real parts, perturbed randomly according to the experimental errors.