Scalar mass matrix ($M_{ij}^2={\partial }_i{\partial }_{j}V$)

For real scalar fields with canonically normalized quadratic kinetic terms, the tree-level scalar mass matrix is the Hessian matrix of the scalar potential at a stationary vacuum:Its eigenvalues are squared masses in units $c=\hslash =1$. A negative eigenvalue is a local instability of that candidate vacuum.

The kinetic normalization is essential. More generally, expand about ${\varphi }_0$ withwhere $H$ is the potential Hessian matrix and $K$ is a real symmetric positive-definite matrix. A mode $\eta =v{e}^{-ip\cdot x}$ obeys the generalized eigenvalue problemEquivalently, canonical field normalization gives the symmetric squared-mass matrix $K^{-1/2}H{K}^{-1/2}$. For one field with kinetic coefficient $Z>0$ and quadratic potential $m_0^2{\varphi }^2/2$, the physical squared mass is $m_0^2/Z$.

Broken continuous global-symmetry directions are zero vectors of the potential Hessian matrix, producing Goldstone bosons when the kinetic form is nonsingular. A Higgs mechanism changes which scalar modes remain independent physical particles; explicit symmetry breaking can lift a zero mass. Thus both the potential and kinetic structure are needed to read a physical spectrum.

An accidental massless scalar has zero quadratic mass without being required by the stated broken continuous symmetry. A singlet with potential $\kappa z^4$ around $z=0$ is an example: its Hessian matrix vanishes although no continuous symmetry of that single real field protects a mass term. Such a zero mass can require tuning once quantum corrections are included.