Upper Limit of Fusion Reactivity in Laser-Driven $p+{^{11}{\rm B}}$ Reaction

We explore the averaged fusion reactivity of the $p+{^{11}{\rm B}}$ reaction in tabletop laser experiments using a plasma expansion model. We investigate the energy distribution of proton beams accelerated by lasers as a function of electron temperature $T_e$ and the dimensionless acceleration time $\omega_{pi} t_{\rm acc}$, where $\omega_{pi}$ is the ion plasma frequency. By combining these distributions with the fusion cross-section, we identify the optimal conditions that maximize the fusion reactivity, with $\left\langle \sigma v \right\rangle = 8.12 \times 10^{-16}\,{\rm cm^3/s}$ at $k_B T_e = 10.0\,{\rm MeV}$ and $\omega_{pi} t_{\rm acc} = 0.503$. These findings suggest that an upper limit exists for the fusion reactivity achievable in laser-driven $p+{^{11}{\rm B}}$ fusion experiments, even under optimized conditions.