= Leading dilaton Weyl anomaly coefficient
{title2=$\overline\beta^\varphi=(D-26)/6+\alpha'[-\Box\varphi/2+|\nabla\varphi|^2]$}
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 $\tfrac12\nabla_a\nabla_b\varphi\,Y^aY^b$ in the curvature coupling produces the <counterterm> $\delta\varphi=-\alpha'\Box\varphi/(2\varepsilon)$ and hence the Laplacian term. The full embedding equation $\Box_hX^a+\Gamma^a{}_{bc}\partial X^b\partial X^c=(\alpha'/2)R^{(2)}\nabla^a\varphi$ 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 https://davidtong.org/pdfs/teaching/string-theory/string7.pdf[Tong's string theory notes, Section 7.4.4].
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