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A background-charge scalar field is a free scalar coupled linearly to the worldsheet Ricci scalar. This curvature coupling produces the improved stress tensor of a linear dilaton conformal field theory.

Differentiating $X(z)X(w)\sim -({\alpha }^{\prime }/2)\log (z-w)$ gives the contractions needed for Wick theorem. The free-boson part of $T$ supplies $1/[2(z-w{)}^4]$, corresponding to central charge one. The cross-contractions between $:\partial X\partial X:$ and ${\partial }^{2}X$ produce the required lower poles but no fourth-order scalar term. Finally,so the product of the two improvement terms contributes $3{\alpha }^{\prime }{q}^2/(z-w{)}^4$. Matching this with $c/[2(z-w{)}^4]$ in the stress-tensor operator-product expansion gives the linear dilaton conformal field theory

The first term of the action is the free boson conformal field theory. For the curvature coupling, insert the supplied first variation of $\sqrt{g}{R}^{(2)}$, integrate by parts twice and discard the boundary term. Its metric variation is the stress-tensor improvementin the conventions of the question. In a locally flat complex coordinate, the holomorphic component of the complete stress-energy tensor is thereforeas required. Thus the coupling $q\int \sqrt{g}X{R}^{(2)}$ makes $X$ a background-charge scalar field, equivalently a worldsheet coordinate in a linear dilaton conformal field theory.