Chiral symmetry: An analytic $SU(3)$ unitary matrix

Abstract

The $SU(2)$ unitary matrix $U$ employed in hadronic low-energy processes has both exponential and analytic representations, related by $U=\exp \left[i\tau \cdot \hat{\pi }\theta \right]=\cos \theta I+i\tau \cdot \hat{\pi }\sin \theta$. One extends this result to the $SU(3)$ unitary matrix by deriving an analytic expression which, for Gell-Mann matrices $\lambda$, reads $U=\exp \left[i\mathbf{v}\cdot \lambda \right]=\left[\left(F+\frac{2}{3}G\right)I+\left(H\hat{\mathbf{v}}+\frac{1}{\sqrt{3}}G\hat{\mathbf{b}}\right)\cdot \lambda \right]+i\left[\left(Y+\frac{2}{3}Z\right)I+\left(X\hat{\mathbf{v}}+\frac{1}{\sqrt{3}}Z\hat{\mathbf{b}}\right)\cdot \lambda \right]$, with $v_i=[v_1,\cdots v_8]$, $b_i=d_{ijk}{v}_j{v}_k$, and factors $F,\cdots Z$ written in terms of elementary functions depending on $v=|\mathbf{v}|$ and $\eta =2d_{ijk}{\hat{v}}_i{\hat{v}}_j{\hat{v}}_k/3$. This result does not depend on the particular meaning attached to the variable $\mathbf{v}$ and the analytic expression is used to calculate explicitly the associated left and right forms. When $\mathbf{v}$ represents pseudoscalar meson fields, the classical limit corresponds to $⟨0|\eta |0⟩\to \eta \to 0$ and yields the cyclic structure $U=\left\{\left[\frac{1}{3}\left(1+2\cos v\right)I+\frac{1}{\sqrt{3}}\left(-1+\cos v\right)\hat{\mathbf{b}}\cdot \lambda \right]+i\left(\sin v\right)\hat{\mathbf{v}}\cdot \lambda \right\}$, which gives rise to a tilted circumference with radius $\sqrt{2/3}$ in the space defined by $I$, $\hat{\mathbf{b}}\cdot \lambda$, and $\hat{\mathbf{v}}\cdot \lambda$. The axial transformations of the analytic matrix are also evaluated explicitly.