Chiral symmetry: An analytic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> 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 $U=\mathrm{exp}[i\mathbit{\ensuremath{\tau}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{\ensuremath{\pi}}}\ensuremath{\theta}]=\mathrm{cos}\ensuremath{\theta}I+i\mathbit{\ensuremath{\tau}}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{^}{\mathbit{\ensuremath{\pi}}}\mathrm{sin}\ensuremath{\theta}$, where $\mathbit{\ensuremath{\tau}}$ are Pauli matrices $\mathbit{\ensuremath{\pi}}=({\ensuremath{\pi}}_{1},{\ensuremath{\pi}}_{2},{\ensuremath{\pi}}_{3})$ is the pion field. One extends this result to $SU(3)$ deriving an expression which, for Gell-Mann $\mathbit{\ensuremath{\lambda}}$, reads $U=\mathrm{exp}[i\mathbit{v}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}]=\phantom{\rule{0ex}{0ex}}[(F+\frac{2}{3}G)I+(H\stackrel{^}{\mathbit{v}}+\frac{1}{\sqrt{3}}G\stackrel{^}{\mathbit{b}})\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}]+i[(Y+\frac{2}{3}Z)I+(X\stackrel{^}{\mathbit{v}}+\frac{1}{\sqrt{3}}Z\stackrel{^}{\mathbit{b}})\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}]$, with ${v}_{i}=[{v}_{1},\ensuremath{\cdots}{v}_{8}]$, ${b}_{i}={d}_{ijk}{v}_{j}{v}_{k}$, factors $F,\dots{},Z$ written terms elementary functions depending on $v=|\mathbit{v}|$ $\ensuremath{\eta}=2{d}_{ijk}{\stackrel{^}{v}}_{i}{\stackrel{^}{v}}_{j}{\stackrel{^}{v}}_{k}/3$. This does not depend particular meaning attached variable $\mathbit{v}$ used calculate explicitly associated left right forms. When represents pseudoscalar meson fields, classical limit corresponds $⟨0|\ensuremath{\eta}|0⟩\ensuremath{\rightarrow}\ensuremath{\eta}\ensuremath{\rightarrow}0$ yields cyclic structure $U={[\frac{1}{3}(1+2\mathrm{cos}v)I+\frac{1}{\sqrt{3}}(\ensuremath{-}1+\mathrm{cos}v)\stackrel{^}{\mathbit{b}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}]+i(\mathrm{sin}v)\stackrel{^}{\mathbit{v}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}}$, which gives rise a tilted circumference radius $\sqrt{2/3}$ space defined $I$, $\stackrel{^}{\mathbit{b}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}$, $\stackrel{^}{\mathbit{v}}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{\ensuremath{\lambda}}$. For sake completeness, axial transformations also evaluated explicitly.