Extension of Hayward black hole in $f(R)$ gravity coupled with a scalar field

Abstract: This study looks into regular solutions in a theory of gravity called $f(R)$ gravity, which also involves a scalar field. The $f(R)$ theory changes Einstein's ideas by adding a new function related to something called the Ricci scalar. This lets us tweak the equations that describe how gravity works. Adding a scalar field makes the theory more interesting, giving us more ways to investigate and understand it. { The main goal of this research is to create regular black holes using a combination of $f(R)$ gravitational theory and a scalar field.} Regular solutions don't have any singularities, which are points where certain physical quantities, like invariants, become really big or undefined. { In this context, we find two regular black hole solutions by using a spherical space with either an equal or unequal approach.} For the solutions where we use the equal approach, we figure out the shape of $f(R)$ and how it changes, along with its first and second derivatives. We demonstrate that Hayward's solution in this theory stays steady because all the shapes of $f(R)$ and their first and second derivatives are positive. Next, we focus on the case where the metric isn't equal and figure out the black hole solution. We also find out what $f(R)$ and the scalar field look like in this situation. We demonstrate that the solution in this case is a broader version of the Hayward solution. When certain conditions are met, we end up back at the scenario where the metrics are equal. We also prove that this model is stable because $f(R)$, along with its first and second derivatives, are all positive. { We analyze the trajectories of these black hole solutions and determine the forms of their conserved quantities that remain same along those trajectories.