$\mathcal{R}^2$ Quantum Corrected Scalar Field Inflation
  String theory enjoys an elevated role among quantum gravity theories, since
it seems to be the most consistent UV completion of general relativity and the
Standard Model. However, it is hard to verify the existence of this underlying
theory on terrestrial accelerators. One way to probe string theory is to study
its imprints on the low-energy effective inflationary Lagrangian, which are
quantified in terms of high energy correction terms. It is highly likely, thus,
to find higher order curvature terms combined with string moduli, that is
scalar fields, since both these types of interactions and matter fields appear
in string theory. In this work we aim to stress the probability that the
inflationary dynamics are controlled by the synergy of scalar fields and higher
order curvature terms. Specifically, we shall consider a well motivated quantum
corrected canonical scalar field theory, with the quantum corrections being of
the $\mathcal{R}^2$ type. The reason for choosing minimally coupled scalar
theory is basically because if scalar fields are evaluated in their vacuum
configuration, they will either be minimally coupled or conformally coupled.
Here we choose the former case, and the whole study shall be performed in the
string frame (Jordan frame), in contrast to similar studies in the literature
where the Einstein frame two scalar theory is considered. We derive the field
equations of the quantum-corrected theory at leading order and we present the
form the slow-roll indices obtain for the quantum corrected theory. We
exemplify our theoretical framework by using the quadratic inflation model, and
as we show, the $\mathcal{R}^2$ quantum corrected quadratic inflation model
produces a viable inflationary phenomenology, in contrast with the simple
quadratic inflation model.
