Scale dimension of chiral symmetry breaking and rigorous constraints on K ℓ3 decay

Abstract

Using new techniques for deriving bounds satisfying the requirements of unitarity and analiticity, optimal constraints on the scalar K ℓ3 form factor are obtained, given the s-wave, I = 1 2 , Kπ elastic phase shift. It is assumed that the propagators of current divergences satisfy unsubtracted dispersion relations, that axial divergences are dominated by pion and kaon poles and that the chiral symmetry breaking Hamiltonian transforms as (3, 3) + ( 3,3) . Recent SLAC-Santa Cruz data satisfy these constraints, as does the Callan-Treiman relation. The constraint on the slope of the form factor, given either the Callan-Treiman relation or the data, is badly violated by the current algebra prediction of Dashen and Weinstein. This violation suggests the necessity for subtraction, corresponding to a chiral symmetry breaking Hamiltonian density with scale dimension greater than or equal to two. The results depend only weakly on the elastic phase shift.