= Orthogonal unitary-gauge slice near a scalar vacuum
{title2=$(\phi-\phi_0)^TT_a\phi_0=0$}
For a real orthogonal <gauge group representation>, the tangent vectors to the vacuum orbit are $v_a=T_a\phi_0$. 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> $v_a^Tv_b$. 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.
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