On the Statistical Treatment of the Cabibbo Angle Anomaly

TL;DR

This paper analyzes the statistical methods used to interpret the Cabibbo angle measurements, highlighting differences between direct angle comparisons and CKM unitarity tests, and discusses implications for New Physics signals.

Contribution

It clarifies the distinction between testing Cabibbo angle equality and CKM unitarity, proposing that direct angle comparison is a more comprehensive test of the Standard Model.

Findings

Current data favor New Physics over the SM at over 3 sigma.

CKM unitarity is rejected at around 4.8 sigma.

Different treatments of nuclear beta decays affect the significance levels.

Abstract

We point out that testing the equality of the Cabibbo angle as extracted from $\Gamma(K\rightarrow \pi l\nu)$, the ratio $\Gamma(K\rightarrow l\nu)/\Gamma(\pi\rightarrow l\nu)$ and nuclear $\beta$ decays is not identical to a test of first row unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. The reason is that a CKM unitarity test involves only two parameters, while the degrees of freedom for the assessment of the goodness-of-fit of the universality of the Cabibbo angle entailed by the Standard Model (SM) is equal to the number of measurements minus one. Beyond the SM all different processes could in principle give different Cabibbo angles. Consequently, the difference between the two tests becomes relevant starting from three observables giving results for the Cabibbo angle that are in tension with each other. With current data, depending on the treatment of the nuclear $\beta$ decays, we find that New Physics is favored over the SM at $5.1\,\sigma$ or $3.6\,\sigma$ while CKM unitarity is rejected at $4.8\sigma$ or $3.0\sigma$, respectively. We argue that the best method to test the SM is to test the equality of the Cabibbo angle, because CKM unitarity is only one aspect of the SM.