Do tau lepton branching fractions obey Benford's law?

TL;DR

This study investigates whether the leading digits of tau lepton branching fractions follow Benford's law, finding a marginal overall disagreement mainly due to correlated mode counting, which, when corrected, aligns with the law.

Contribution

The paper provides the first analysis of Benford's law in particle physics datasets, specifically tau lepton branching fractions, and identifies the impact of correlated mode counting on digit distribution.

Findings

Marginal disagreement with Benford's law in tau branching fractions.

Discrepancy mainly due to counting correlated modes multiple times.

Correcting for correlations restores agreement with 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.