 Orthogonal unitary-gauge slice near a scalar vacuum

For a real orthogonal gauge group representation, the tangent vectors to the vacuum orbit are $v_a=T_a{\varphi }_0$. Choose an independent basis of these vectors. Under an infinitesimal gauge transformation, the derivative of the displayed slice conditions with respect to the broken gauge parameters is the Gram matrix $v_a^T{v}_b$. It is invertible on the independent tangent directions, so the implicit function theorem selects this unitary gauge locally near the vacuum. The remaining scalar fluctuations are orthogonal to the gauge orbit, and their derivative has no quadratic mixing with the massive gauge fields. The statement is local and does not assert a global gauge slice through zeros of the scalar field.