Do tau lepton branching fractions obey Benford's law?

Abstract

According to Benford's law, the most significant digit in many datasets is not uniformly distributed, but obeys a well defined power law distribution with smaller digits appearing more often. Among one of the myriad particle physics datasets available, we find that the leading decimal digit for the $\tau$ lepton branching fraction shows marginal disagreement with the logarithmic behavior expected from the Benford distribution. We quantify the deviation from Benford's law using a $\chi^2$ function valid for binomial data, and obtain a $\chi^2$ value of 16.9 for nine degrees of freedom, which gives a $p$-value of about 5%, corresponding to a 1.6$\sigma$ disagreement. We also checked that the disagreement persists under scaling the branching fractions, as well as by redoing the analysis in a numerical system with a base different from 10. Among all the digits, `9' shows the largest discrepancy with an excess of $4\sigma$. This discrepancy is because the digit `9' is repeated for three distinct groups of correlated modes, with each group having a frequency of two or three, leading to double-counting. If we count each group of correlated modes only once, the discrepancy for this digit also disappears and we get pristine agreement with Benford distribution.