Chiral symmetry: An analytic $SU(3) $ unitary matrix
The $SU(2)$ unitary matrix $U$ employed in hadronic low-energy processes has both exponential and analytic representations, related by $ U = \exp\left[ i \mathbfτ \cdot \hat{\mathbfπ} θ\,\right] = \cosθI + i \mathbfτ \cdot \hat{\mathbfπ} \sinθ$. One extends this result to the $SU(3)$ unitary matrix by deriving an analytic expression which, for Gell-Mann matrices $\mathbfλ$, reads $ U= \exp\left[ i \mathbf{v} \cdot \mathbfλ \right] = \left[ \left( F + \tfrac{2}{3} G \right) I + \left( H \hat{\mathbf{v}} + \tfrac{1}{\sqrt{3}} G \hat{\mathbf{b}} \right) \cdot \mathbfλ \, \right] + i \left[ \left( Y + \tfrac{2}{3} Z \right) I + \left( X \hat{\mathbf{v}} + \tfrac{1}{\sqrt{3}} Z \hat{\mathbf{b}} \right) \cdot \mathbfλ \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 $η= 2\, d_{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 $\langle 0|η|0\rangle \rightarrow η\rightarrow 0$ and yields the cyclic structure $ U = \left\{ \left[ \tfrac{1}{3} \left( 1 + 2 \cos v \right) I + \tfrac{1}{\sqrt{3}} \left( -1 + \cos v \right) \hat{\mathbf{b}}\cdot \mathbfλ \right] + i \left( \sin v \right) \hat{\mathbf{v}}\cdot \mathbfλ \right\} $, which gives rise to a tilted circumference with radius $\sqrt{2/3}$ in the space defined by $I$, $\hat{\mathbf{b}}\cdot \mathbfλ $, and $\hat{\mathbf{v}}\cdot \mathbfλ $. The axial transformations of the analytic matrix are also evaluated explicitly.