Beam intensity effects on the $K^+ \to \pi^+ \nu \bar\nu$ measurement at NA62

The ultra-rare $K^+ \to \pi^+ \nu \bar\nu$ decay, precisely predicted in the Standard Model, is a gateway into exploring new physics at mass scales unattainable at present collider experiments. The NA62 experiment at CERN aims at measuring the branching ratio of the $K^+ \to \pi^+ \nu \bar\nu$ decay to a precision of 15% with any deviation from the Standard Model prediction hinting at New Physics. A Cherenkov detector (CEDAR) is used in NA62 to identify a minority component of particles (kaons) in an unseparated hadron beam. The optical design for a new CEDAR detector optimised to work with hydrogen as radiator gas, Cedar-H, is presented. Cedar-H has been commissioned and installed into NA62 with more than satisfactory performances and a 20% increase in light yield. The single event sensitivity for the $K^+ \to \pi^+ \nu \bar\nu$ decay with data collected in 2021 is computed as $B_{SES} = (5.00 \pm 0.13_{stat}) \times 10^{−11}$. Studies of each component entering into this figure and their variation with the intensity of the NA62 hadron beam are performed. A new optimal beam intensity to maximise the number of $K^+ \to \pi^+ \nu \bar\nu$ events collected is discussed.