Effective interactions of the open bosonic string via field theory

We describe a method to extract an effective Lagrangian description for open bosonic strings, at zero transcendentality. The method relies on a particular formulation of its scattering amplitudes derived from color-kinematics duality. More precisely, starting from a $(DF{)}^2+\text{YM}$ quantum field theory, we integrate out all the massive degrees of freedom to generate an expansion in the inverse string tension ${\alpha }^{\mathrm{\prime }}$. We explicitly compute the Lagrangian terms through $\mathcal{O}({\alpha }^{\mathrm{\prime }4})$, and target the sector of operators proportional to $F^4$ to all orders in ${\alpha }^{\mathrm{\prime }}$.