Impacts of the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ reaction rate on $^{56}{\rm Ni}$ nucleosynthesis in pair-instability supernovae

Nuclear reactions are key to our understanding of stellar evolution, particularly the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate, which is known to significantly influence the lower and upper ends of the black hole (BH) mass distribution due to pair-instability supernovae (PISNe). However, these reaction rates have not been sufficiently determined. We use the $\texttt{MESA}$ stellar evolution code to explore the impact of uncertainty in the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate on PISN explosions, focusing on nucleosynthesis and explosion energy by considering the high resolution of the initial mass. Our findings show that the mass of synthesized radioactive nickel ($^{56}{\rm Ni}$) and the explosion energy increase with $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ rate for the same initial mass, except in the high-mass edge region. With a high (about twice the $\texttt{STARLIB}$ standard value) rate, the maximum amount of nickel produced falls below 70 $M_\odot$, while with a low rate (about half of the standard value) it increases up to 83.7 $M_\odot$. These results highlight that carbon burning plays a crucial role in PISNe by determining when a star initiates expansion. The initiation of expansion competes with collapse caused by helium photodisintegration, and the maximum mass that can lead to an explosion depends on the $^{12}\rm{C}\left({\alpha},{\gamma}\right)^{16}\!\rm{O}$ reaction rate.