On the Statistical Treatment of the Cabibbo Angle Anomaly

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 beyond the Standard Model (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 the SM is rejected 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.

For brevity, we use V ij to denote the magnitude of a CKM matrix element. Because of the smallness of V 2 ub 1.6 · 10 −5 [7] we can neglect V 2 ub in the first row CKM unitarity relation, resulting in Eq. (4) has been extensively employed in order to probe for, or constrain, new physics (NP) [5,6,[22][23][24][25][26][27][28][29][30][31][32]. Equivalently to Eq. (4), we can parametrize V ud and V us in the SM up to corrections of order O(λ 6 ) 0.0001 by using the Cabibbo angle describing the mixing of the first two generations V ud = cos θ C , V us = sin θ C , i.e. we practically have a two-generational model. A high-order Wolfenstein expansion in the Wolfenstein parameter λ can be found e.g. in Ref. [33].
In order to denote the origin of an extraction of the respective CKM matrix element (or their ratio) from experimental data, we use the notation V K l3 us , (V us /V ud ) K l2 and V β ud , respectively. As of now, there are two anomalies: Firstly, there is a significant tension of us , (V us /V ud ) K l2 and V β ud with CKM unitarity [34]. Second, there is an even higher tension between V K l3 us and V β ud [8,34,35]. Motivated by these developments, in this paper we discuss the statistical methodology and the differences between testing the SM versus testing CKM unitarity, Eq. (4). In a general model beyond the SM (BSM) n different processes that we use as measurements of the Cabibbo angle could result in n different values. Thus, a test of the SM is a comparison of a one-parameter fit to a trivial fit containing as many parameters as different extractions for the Cabibbo angle that are available. However, tests of CKM unitarity involve only two parameters, namely V us and the violation of unitarity ∆ (see Eq. (11) below). In that case we compare a one-parameter fit to a two-parameter fit only. No matter how many measurements are available, the degree of freedom of the CKM unitarity test is always fixed. In the past, when only two out of three measurements in Eqs. (1)-(3) showed a tension between each other, this difference was not significant. However, when tensions between all three measurements are present, as is the current situation, one gets sensitive to the fact that in general the significances for the rejection of the SM and CKM unitarity are different.
The point that there is more to test in the measurements of V us and V ud than CKM unitarity was made in specific cases before [5,24,36]. Our aim here is to generalize this observation and give a universal methodology for SM tests with an arbitrary number of measurements of θ C .
We emphasize that the point in this paper is only about the methodology of testing the SM with data on V us and V ud . We do not advocate any of the extractions of V ud , which we use as examples, and are agnostic about the validity of the used models. Especially, we do not claim that the SM is excluded at or beyond 5 σ.
In Sec. II we analyze the difference between testing the SM and CKM unitarity. Subsequently, in Sec. III we present our likelihood ratio tests of the SM and CKM unitarity with current data. In Sec. IV we discuss a specific NP model. In Sec. V we conclude.

A. SM test
Beyond the SM, the analysis of n different observables could in principle result in n different mixing angles of the first two generations θ 1 , θ 2 , . . . , θ n .
We assume here for simplicity that measurements of the same observable by different experiments are already averaged.
In the SM, all of the angles of Eq. (6) are equal to the Cabibbo angle Eq. (7) is the one-parameter SM null hypothesis. It has a constrained nested parameter space relative to the general BSM parameter space, Eq. (6). By "nested" we mean as usual that one parameter space can be obtained by fixing some of the parameters of the other.
The difference in dimensionality between the two theory spaces, i.e. the number of relatively fixed parameters ν SM test of the two hypotheses, is always one less than the total number of observables. The trivial χ 2 fit of the most general BSM parameter space gives by design always the result We perform a likelihood ratio test of SM vs. BSM by calculating the two-sided p-value and the significance z of the rejection of the SM as (see e.g. Refs. [7,37,38]) Here, P ν/2 (∆χ 2 /2) is the regularized lower incomplete gamma function and we use ν = ν SM test and ∆χ 2 = ∆χ 2 SM test with

B. CKM unitarity test
In order to test CKM unitarity with n observables one uses two parameters V us and ∆, the latter of which is used as a measure for the deviation from unitarity. We choose to employ ∆ for the parametrization of V ud in the form We test the null hypothesis ∆ = 0 against the general case including ∆ = 0, which is effectively the same as varying V us and V ud freely. We use the ∆ notation in order to make completely clear that the two models that we compare are nested. We denote the corresponding minimal χ 2 values as χ 2 unitary and χ 2 non-unitary , respectively, and define for the CKM unitarity test

C. Comparison of SM test and CKM unitarity test
The null hypothesis fits of the SM test and the CKM unitarity test are equivalent. They are both one-parameter fits and lead to the same χ 2 : For the two tests however, we compare these fits to different opposite hypotheses. For the CKM unitarity test the difference of dimensionality of the two theory spaces that we compare is always fixed to For the SM test it is Furthermore, the non-unitary fit allowing for ∆ = 0 is nontrivial, resulting in general in Whether or not the SM test and CKM unitarity test give the same results depends on the number of observables n that are taken into account, as we show in Table I.
• n = 1 is the trivial case where no SM test is needed or possible at all, because SM and BSM hypothesis are degenerate. Also, no violation of unitarity can possibly be detected, so everything is in agreement equally with the SM and unitarity.
• For n = 2, the unconstrained fits give the same result (namely a vanishing χ 2 ), because we can always explain two measurements with two free parameters. The tests have also the same number of degrees of freedom and z SM test = z unitarity test .
• n ≥ 3: In this case in general the unconstrained two-parameter CKM unitarity fit cannot explain the data perfectly anymore, i.e. χ 2 min, non-unitary > 0 and therefore we have in general ∆χ 2 SM test > ∆χ 2 unitarity test . The point is that some patterns in the data can not be accounted for by just employing a two parameter fit without unitarity, i.e. this procedure does not account for the generality of possible BSM models.
Note that for n ≥ 3 one cannot a priori say if z SM test > z unitarity test or vice versa, because this does not only depend on ∆χ 2 SM test and ∆χ 2 unitarity test but also on the specific value of ν SM test ≥ 2 vs. ν unitarity test = 1, that is the number of observables n. For example, for given values of ∆χ 2 SM test = 20 and ∆χ 2 unitarity test = 10, we have z SM test > z unitarity test if n = 5 or z SM test < z unitarity test if n = 8, see Fig. 1.
The above discussion makes clear what are the differences between unitarity tests and SM tests in a completely general perspective. In Sec. III we apply the above formalism to the current status of the data.

III. APPLICATION OF FORMALISM TO CURRENT DATA
Current data provides n = 3 precision determinations of the Cabibbo angle where θ K l3 , θ K l2 and θ β could all be different in BSM models. On the right hand side of CKM unitarity on the other hand implies Eqs. (19) and (20) are the SM and CKM unitarity null hypotheses, respectively.
We summarize the latest determinations of V us and V ud in Table II. The obtained value for V ud depends on the details of the treatment of nuclear β decays. There are extractions available from Seng, Gorchtein, Patel, Ramsey-Musolf (SGPRM) [18][19][20] and Czarnecki, Marciano, Sirlin (CMS) [21] using different estimates for the radiative corrections.
Our fit results are shown in Table III. Therein, also subsets of observables are considered for illustration purposes. As discussed in Sec. II, for n = 2 fits the SM test and CKM unitarity test give the same results, and for the current full data set with n = 3 they differ.

IV. NEW PHYSICS MODELS
In this section we demonstrate the ability of a concrete BSM model to describe the data with χ 2 min, BSM = 0, while pointing out that it is not even clear how to formulate the corresponding fit in terms of a CKM unitarity test. We emphasize that this serves as a toy example for illustration only, that is, we did not apply all the available constraints. We employ the model and notation of Ref. [23] and show that BSM couplings of righthanded (RH) quarks [16,32,[40][41][42][43][44][45][46][47][48] to the W boson, i.e. RH currents, could remove the tensions presented in Table III.
Following the notation of Ref. [23], we denote the respective coupling of RH strange quarks by ε s and the one of down quarks by ε ns . Furthermore, the measured values of the CKM matrix elements given in Table II are interpreted as effective ones and are related to the mixing angle and the RH couplings as [23] V β ud = |cos θ C + ε ns | .
Note that ε s and ε ns are in general complex. However, to keep things simple for our purposes it is enough to study the real case here. The SM is obtained in the limit ε s = ε ns = 0 .
Considering Eqs.  In a CKM unitarity test one compares a constrained fit with a fit of free floating V us and V ud . The latter can not necessarily describe the data as well as a BSM model, in case the patterns go beyond just violating unitarity. This matters starting from three independent observables being taken into account. In a SM test one compares a fit of the Cabibbo angle which gives the same absolute χ 2 as the CKM unitarity fit with a completely general BSM scenario where all measured effective angles could be different, as we demonstrated explicitly for a concrete model.
That means the significance of SM tests can in general be different from the one of CKM unitarity fits once more than two observables are considered. In the foreseeable future, τ decays may provide a further precision determination of the Cabibbo angle via the ratio Γ(τ → Kν τ )/Γ(τ → πν τ ), see Refs. [49,50], and the number of precision observables for the determination of the Cabibbo angle rises to four. Further input is also coming up from pion beta decays [51]. With more measurements in the future the differences between SM tests and CKM unitarity tests could become even more significant.
Consequently, we encourage to test the SM by testing for the universality of the Cabibbo angle, rather than testing for CKM unitarity only, with the general methodology laid out above.

ACKNOWLEDGMENTS
The work of YG is supported in part by the NSF grant PHY1316222. SS is supported by a DFG Forschungsstipendium under contract no. SCHA 2125/1-1. This work is supported in part by the U.S. Department of Energy (contract DE-AC05-06OR23177) and National Science Foundation (PHY-1714253).