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Generalized Weinberg Sum Rules in Deconstructed QCD

Recently, Son and Stephanov have considered an "open moose" as a possible dual model of a QCD-like theory of chiral symmetry breaking. In this note we demonstrate that although the Weinberg sum rules are satisfied in any such model, the relevant sums converge very slowly and in a manner unlike QCD. Further, we show that such a model satisfies a set of generalized sum rules. These sum rules can be understood by looking at the operator product expansion for the correlation function of chiral currents, and correspond to the absence of low-dimension gauge-invariant chiral symmetry breaking condensates. These results imply that, regardless of the couplings and F-constants chosen, the open moose is not the dual of any QCD-like theory of chiral symmetry breaking. We also show that the generalized sum rules can be "solved", leading to a compact expression for the difference of vector- and axial-current correlation functions. This expression allows for a simple formula for the S parameter (L_10), which implies that S is always positive and of order one in any (unitary) open linear moose model. Therefore the S parameter is positive and order one in any "Higgsless model" based on the continuum limit of a linear moose regardless of the warping or position-dependent gauge-coupling chosen.