= Chiral matrix scalar model
{title2=$\Phi\mapsto e^{i\theta}U_L\Phi U_R^\dagger$}
A theory of an $N\times N$ matrix of <complex scalar fields> with kinetic term $\operatorname{tr}(\partial\Phi^\dagger\partial\Phi)$ and potential $V=\alpha\operatorname{tr}[(\Phi^\dagger\Phi)^2]/2+\beta[\operatorname{tr}(\Phi^\dagger\Phi)]^2/2+m^2\operatorname{tr}(\Phi^\dagger\Phi)$. It is invariant under global $SU(N)_L\times SU(N)_R\times U(1)$ transformations. For $\alpha,\beta>0$ and $m^2<0$, minimizing the nonnegative eigenvalues of $\Phi^\dagger\Phi$ gives a <vacuum expectation value> $C\mathbf1$ with $C^2=-m^2/(\alpha+N\beta)$, up to a global symmetry transformation. The connected stabilizer is diagonal $SU(N)$, with $N^2$ broken generators. Expanding $\Phi=C\mathbf1+(H+iP)/\sqrt2$ with <Hermitian matrices> gives $N^2$ massless <Goldstone bosons> in $P$ and squared <masses> $2C^2(\alpha+N\beta)$ for the singlet in $H$ and $2\alpha C^2$ for its traceless <Adjoint representation>. The faithful unbroken conjugation action factors through $SU(N)/\mathbb Z_N$.
Back to article page
