Theoretical implications for a new measurement of $K_L\to \pi^0 \ell\ell$

Kaon physics is at an important experimental juncture with respect to the ongoing measurements of several observables. This work will build on the existing status by formulating different phenomenological analyses corresponding to different paths that may lie ahead. Beginning with the golden channels, $K^+\rightarrow\pi^+\nu\bar\nu$ and $K_L\rightarrow\pi^0\nu\bar\nu$, the paper will eventually cast the spotlight on the importance of a precise measurement of BR($K_L\rightarrow\pi^0\ell\bar\ell$). The phenomenological analyses involve sequentially adding kaon physics observables at the projected final precision of their respective measurements to the global fit. More specifically, we consider three different scenarios with different sets of observables assumed at their final precisions. Beginning with BR($K^+\rightarrow\pi^+\nu\bar\nu$) and BR($K_L\rightarrow\pi^0\nu\bar\nu$), we sequentially add BR($K_L\rightarrow\pi^0 e\bar e$) and BR($K_L\rightarrow\pi^0 \mu\bar\mu$) to the global fit. The evolution of the result from one scenario to the next makes a strong case for the consideration of future measurement of BR($K_L\rightarrow\pi^0\ell\bar\ell$).