Experimental Tests for Theories of Chiral-Symmetry Breaking

Abstract

Assuming a picture of the strong interactions in which the Hamiltonian is the sum of an $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$-symmetric piece $H_0$ plus a small symmetry-breaking term $\epsilon H_1$, we show how to calculate relations among the corrections of order $\epsilon$ to the symmetry limit. Our techniques are purely group-theoretic and involve no extraneous dynamical assumptions, so that our results provide direct experimental tests for various symmetry-breaking schemes. For example, we show that if $\epsilon H_1$ belongs to the $(3, \overline{3})\otimes (\overline{3}, 3)$ representation of $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$, then there is one sum rule satisfied by the corrections to the generalized Goldberger-Treiman relations for the three decays $n\to p+e+\nu$, $\Sigma \to N+e+\nu$, and $\Lambda \to N+e+\nu$. We also show that the so-called $\Sigma$ terms, which are closely related to $\epsilon H_1$, can be obtained from on-the-mass-shell scattering amplitudes (albeit at an unphysical energy point) if terms of order ${\epsilon }^2$ can be neglected in comparison to terms of order $\epsilon$. The question of whether or not lowest-order calculations of symmetry breaking are meaningful is discussed in some detail.