The Gaussian CL$_s$ Method for Searches of New Physics

We describe a method based on the CL$_s$ approach to present results in searches of new physics, under the condition that the relevant parameter space is continuous. Our method relies on a class of test statistics developed for non-nested hypotheses testing problems, denoted by $\Delta T$, which has a Gaussian approximation to its parent distribution when the sample size is large. This leads to a simple procedure of forming exclusion sets for the parameters of interest, which we call the Gaussian CL$_s$ method. Our work provides a self-contained mathematical proof for the Gaussian CL$_s$ method, that explicitly outlines the required conditions. These conditions are milder than that required by the Wilks' theorem to set confidence intervals (CIs). We illustrate the Gaussian CL$_s$ method in an example of searching for a sterile neutrino, where the CL$_s$ approach was rarely used before. We also compare data analysis results produced by the Gaussian CL$_s$ method and various CI methods to showcase their differences.