Chiral symmetry: An analytic SU(3) unitary matrix

The SU(2) unitary matrix U employed in chiral descriptions of hadronic low-energy processes has both exponential and analytic representations, related by U = exp [i tau . (pi) over cap theta] = cos theta I + i tau . (pi) over cap sin theta, where tau are Pauli matrices and pi = (pi(1), pi(2), pi(3)) is the pion field. One extends this result to the SU(3) unitary matrix by deriving an analytic expression which, for Gell-Mann matrices lambda, reads U = exp [iv . lambda] = [(F + 2/3 G)I + (H (v) over cap1/root 3 G (b) over cap) . lambda] + i[(Y + 2/3 Z)I + (X (v) over cap + 1/root 3 z (b) over cap) . lambda], with v(i) = [v(1), ... v(8)], bi = d(ijk)v(j)v(k), and factors F,..., Z written in terms of elementary functions depending on v = vertical bar nu vertical bar and eta 2d(ijk)($) over cap (i)($) over cap (j)($) over cap (k)/3. This result does not depend on the particular meaning attached to the variable nu and the analytic expression is used to calculate explicitly the associated left and right forms. When nu represents pseudoscalar meson fields, the classical limit corresponds to < 0 vertical bar eta vertical bar 0 > -> eta -> 0 and yields the cyclic structure U = {[1/3 (1+2 cos v)I + 1/root 3 (-1 + cos v)(b) over cap . lambda] + i(sin v)(nu) over cap . lambda}, which gives rise to a tilted circumference with radius root 2/3 in the space defined by I, (b) over cap . lambda, and (nu) over cap . lambda. For the sake of completeness, the axial transformations of the analytic matrix are also evaluated explicitly.

Automated summary

This study explores the application of SU(2) and SU(3) unitary matrices in chiral descriptions, providing analytic representations and revealing their intrinsic relationships and properties through mathematical derivations and symbolic calculations.