Calculating $p$-values and their significances with the Energy Test for large datasets

The energy test method is a multi-dimensional test of whether two samples are consistent with arising from the same underlying population, through the calculation of a single test statistic (called the $T$-value). The method has recently been used in particle physics to search for differences between samples that arise from CP violation. The generalised extreme value function has previously been used to describe the distribution of $T$-values under the null hypothesis that the two samples are drawn from the same underlying population. We show that, in a simple test case, the distribution is not sufficiently well described by the generalised extreme value function. We present a new method, where the distribution of $T$-values under the null hypothesis when comparing two large samples can be found by scaling the distribution found when comparing small samples drawn from the same population. This method can then be used to quickly calculate the $p$-values associated with the results of the test.