 Solution

Let $X={\Phi }^{†}\Phi$. Under the global transformation of the chiral matrix scalar model,The phase and $U_L^{†}{U}_L$ cancel, and $U_R^{†}{U}_R=1$. The cyclic property of the trace impliesTherefore each of the three terms in the scalar potential is invariant separately. Since $\theta ,U_L,U_R$ are constant, ${\partial }_{\mu }{\Phi }^{\prime }=e^{i\theta }{U}_L({\partial }_{\mu }\Phi )U_R^{†}$, giving invariance of the kinetic matrix trace by the same argument. Henceis a global symmetry group of the Lagrangian density. Its action has a finite central kernel, which does not affect its Lie algebra or the broken-generator counting below. For special parameter choices the symmetry can be larger; the displayed group is a symmetry for the stated potential without such extra assumptions.