Leading dilaton Weyl anomaly coefficient (${\overline{\beta }}^{\phi }=(D-26)/6+{\alpha }^{\prime }[-\square \phi /2+|\nabla \phi {|}^2]$)

For the conventional positive Euclidean dilaton curvature coupling, with no antisymmetric background field, the leading scalar Weyl coefficient is the displayed expression. The matter/ghost central charge gives $(D-26)/6$. In a geodesic background field expansion of a string sigma model, contracting $\frac{1}{2}{\nabla }_a{\nabla }_b\phi Y^a{Y}^b$ in the curvature coupling produces the counterterm $\delta \phi =-{\alpha }^{\prime }\square \phi /(2\epsilon )$ and hence the Laplacian term. The full embedding equation ${\square }_h{X}^a+{\Gamma }^a{}_{bc}\partial X^b\partial X^c=({\alpha }^{\prime }/2)R^{(2)}{\nabla }^a\phi$ supplies the gradient-square improvement in the dilaton Weyl variation. Setting this coefficient to zero fixes the constant in the dilaton equation from contracted Bianchi identity; the metric equation alone does not. The Lorentzian dilaton coupling sign convention determines the named-field translation.

The noncritical coefficient is also given in Tong's string theory notes, Section 7.4.4.