Higher derivative quartic vertex of $f(R)$ gravity in light-cone gauge

In the recent studies of four-dimensional Einstein-Hilbert action, quite a few interesting results such as the Kawai-Lewellen-Tye (KLT) relations, MHV-Lagrangian and quadratic forms have been reported. These results naturally raise an important question: Whether these results are valid for the modified theories of gravity in 4-dimensions? In this work, we consider $R+\alpha\,R^2$ gravity and derive the complete quartic interaction vertex in light-cone gauge. We then discuss the implications of the results for KLT relations, and computing MHV amplitudes.