Deconstruction and elastic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:math>scattering in Higgsless models

Abstract

We study elastic pion-pion scattering in global linear moose models and apply the results to a variety of Higgsless models in flat and anti-de Sitter (AdS) space using the equivalence theorem. In order to connect the global moose to Higgsless models, we first introduce a block-spin transformation which corresponds, in the continuum, to the freedom to perform coordinate transformations in the Higgsless model. We show that it is possible to make an ``$f$-flat'' deconstruction in which all of the $f$-constants ${f}_{j}$ of the linear moose model are identical; the phenomenologically relevant $f$-flat models are those in which the coupling constants of the groups at either end of the moose are small---corresponding to the global linear moose. In studying pion-pion scattering, we derive various sum rules, including one analogous to the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSRF) relation, and use them in evaluating the low-energy and high-energy forms of the leading elastic partial-wave scattering amplitudes. We obtain elastic unitarity bounds as a function of the mass of the lightest $KK$ mode and discuss their physical significance.