Current Algebras and Approximate Symmetries

Abstract

The components of currents involved in weak and electromagnetic interactions are assumed to generate the algebra of chiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$. For the hadrons, the spurion scheme of broken $U(6,6)\mathrm{}$ symmetry is considered, which implies nonchiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ symmetry for hadrons at rest, and invariance with respect to collinear $U{\mathrm{}\left(6\right)\mathrm{}}_p$ for particles moving in a given direction. The chiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$ commutation relations are evaluated in a single-particle approximation with $U{\mathrm{}\left(6\right)\mathrm{}}_p$ supermultiplets. It is found that one can obtain $U\left(6\right)\mathrm{}$ results, even though currents are involved which are not elements of the algebra of $U{\mathrm{}\left(6\right)\mathrm{}}_p$ or even of nonchiral $U\left(6\right)\mathrm{}\otimes U\left(6\right)\mathrm{}$. There appear, however, factors $\frac{v}{c}$ or $1-\frac{v}{c}$, which either cancel or can be eliminated by taking the appropriate limits. The effect of mass splitting in the single-particle approximation of the commutation relations is evaluated. The relation of these results to the sum rules of Adler and Weisberger is discussed, as well as the relevance of these sum rules for the problem of the damping of semileptonic $\Delta S=1\mathrm{}$ transitions.