Ghost Virasoro zero-mode convention ($L_0^{\mathrm{gh}}|0⟩=0$)

With the ghost oscillator vacuum, one can define $L_0^{\mathrm{gh}}={\sum }_{k>0}k(b_{-k}{c}_k+c_{-k}{b}_k)$ with vacuum eigenvalue zero. The corresponding Virasoro algebra contains a linear shift: $[L_m^{\mathrm{gh}},L_n^{\mathrm{gh}}]=(m-n)(L_{m+n}^{\mathrm{gh}}-{\delta }_{m+n,0})-\frac{26}{12}(m^3-m){\delta }_{m+n,0}$. Shifting the zero mode by minus one removes the linear term but changes the stated vacuum eigenvalue. A BRST charge must use a consistent choice throughout.