Can we ignore the scalar field (dilaton) term in the Polyakov sigma-model action when deriving the classical equations of motion?

Question

I have the full Polyakov sigma model action:

\[\begin{equation} \begin{split} &S=S_P + S_B + S_\Phi = \\ &- {1 \over 4 \pi \alpha'} \Big[ \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu \partial_b X^\nu G_{\mu\nu}(X)\, + \\ &+\epsilon^{ab} B_{\mu\nu}(X) \partial_a X^\mu \partial_b X^\nu \, +\alpha'\Phi(X) R^{(2)}(\sigma) \Big] \,. \end{split} \end{equation}\]

and I want to derive the classical equations of motion by varying \(X = X + \delta X\). I am confused as to what to do with the last term. It is of a higher power of \(\alpha '\), so I am thinking it can just be ignored, as it's variation will be of a higher order. Is this thinking correct?

Does this question even make sense, as I'm trying to derive classical equations from a sigma-model, which as far as I have seen, is used when quantizing the string?

Answer 1

"Strings in strong gravitational fields" - Gary T. Horowitz and Alan R. Steif, Phys. Rev. D 42, 1950 confirms the argument in the question:

"For the remainder of this section we consider a purely classical string. Since the dilaton term is multiplied by $\alpha'$, it is a quantum correction and does not directly affect the motion of a classical string."