A new preprint makes two notable observations about the CKM matrix.

One of them is that only one of the four parameters of the CKM matrix in the Wolfenstein parameterization (see below for background), specifically "A", runs significant with energy scale. The body text notes that:

The running of A has been calculated before, but there is an apparent disagreement in the literature about its running in the SM: Ref. [8] reports an increase in A of about 13% from the weak scale to the GUT scale, while in Fig. 2 of Ref. [9] A increases by about 25%. We resolve this discrepancy. As we explain in § II, we find that recomputing the running of A using the methods of Ref. [9] gives a result which, in fact, agrees with Ref. [8]. Thus, we take the results of both Ref. [8] and Ref. [9] (except for their Fig. 2) to be correct.

Roughly speaking, this parameter quantifies the extent to which the probability of a first or second generation quark becoming a third generation quark, or visa versa, differs in a way that is not a function of the other parameters, from other possible quark generation transitions.

It also notes that below the top quark mass energy scale, the CKM matrix is basically constant and does not meaningfully run with energy scale.

In the low energy scale limit, the measured value of A, whose best fit global average value is 0.814, is consistent with these transitions occurring with two-thirds of the probability derived from consideration of the other terms.

The body text explains why this is the case:

The CKM elements run due to the fact that the Yukawa couplings run. Furthermore, the running of the CKM matrix is related to the fact that the running of the Yukawa couplings is not universal. If all the Yukawa couplings ran in the same way, the matrices that diagonalize them would not run. Thus, it is the nonuniversality of the Yukawa coupling running that results in CKM running.

Since only the Yukawa coupling of the top quark is large, that is, O(1), to a good approximation we can neglect all the other Yukawa couplings. There are three consequences of this approximation:

1. The CKM matrix elements do not run below m(t).

2. The quark mass ratios are constant except for those that involve m(t).

3. The only Wolfenstein parameter that runs is A.

The first two results above are easy to understand, while the third one requires some explanation. A is the parameter that appears in the mixing of the third generation with the first two generations, and thus is sensitive to the running of the top Yukawa coupling. λ mainly encodes 1–2 mixing — that is, between the first and second generations — and is therefore insensitive to the top quark. The last two parameters, η and ρ, separate the 1–3 and 2–3 mixing. Thus they are effectively just a 1–2 mixing on top of the 2–3 mixing that is generated by A. We see that, to a good approximation, it is only A that connects the third generation to the first and second, and thus it is the only one that runs.

The other is that it looks for numerical coincidences that could conjecturally suggest a deeper structure for the CKM matrix, on a brute force basis, comprehensively for at all energy scales up to the Planck energy scale and finds 19 of them, one of which it deems worthy of setting forth in its abstract. With respect to this relation, the body text notes that:

We find one particularly intriguing relation,

|V(td)V(us)| = |V^2(cb)|, (1.3)

that holds in the SM between 10^9 and 10^15 GeV, overlapping the scale where the Higgs quartic vanishes and the GUT scale. In terms of Wolfenstein parameters, this relation can be written as

A^2 = (1 − ρ)^2 + η^2 . (1.4)

Ideally we would like to find a UV model that generates this relation without tuning.

The paper and its abstract are as follows:

We look for relations among CKM matrix elements that are not consequences of the Wolfenstein parametrization. In particular, we search for products of CKM elements raised to integer powers that approximately equal1 . We study the running of the CKM matrix elements and resolve an apparent discrepancy in the literature.To a good approximation onlyA runs, among the Wolfenstein parameters.Using the Standard Model renormalization group we look for CKM relations at energy scales ranging from the electroweak scale to the Planck scale, and we find 19 such relations. These relations could point to structure in the UV, or be numerical accidents.

For example, we find that|VtdVus|=|V2cb| , within2% accuracy, in the109 -1015 GeV range.We discuss the implications of this CKM relation for a Yukawa texture in the UV.

Yuval Grossman, Ameen Ismail, Joshua T. Ruderman, Tien-Hsueh Tsai, "CKM substructure from the weak to the Planck scale" arXiv:2201.10561 (January 25, 2022).

**Background**

In the Standard Model of Particle Physics, W+ bosons can be emitted by up-type quarks to give rise to any of the three down-type quarks (mass-energy conservation in the end states permitting), and W- bosons can be emitted by down-type quarks to give rise to any of the three up-type quarks (mass-energy conservation permitting). The probability of a quark emitting a W boson is quantified with the weak force coupling constant.

The Cabibbo–Kobayashi–Maskawa matrix a.k.a. CKM matrix summarizes the experimentally measured physical constants that quantify the probability that a W boson emission from a particular type of quark will result in a particular type of quark being produced when mass-energy conservation in the end states does not constrain the available options. There are nine components to the CKM matrix (which are complex valued), the absolute value of which when squared equals the probability that an emission of a W boson from a particular kind of quark will produce a particular kind of quark.

*Illustration from Wikipedia. The magnitude of the element combines the real and imaginary components of each element (basically it is the square root of the absolute value of the square of each entry), discarding any purely complex residual after doing so.*

These entries are complex valued because these probabilities are not CP-symmetric. The CP conservation violations quantified by the CKM matrix reflect the fact that the probabilities are not quite the same for matter and antimatter (although CP violation in the CKM matrix is quite small).

The CKM matrix can also be illustrated geometrically as opposed to algebraically as shown below in a Wikipedia illustration:

While the CKM matrix has nine complex valued entries, it can be fully described with four parameters, although there are multiple ways that this parameterization can be done. It is possible to have a parameterization with as few as one complex valued parameter and three real number valued parameters.

Like all of the experimentally measured Standard Model physical constants, the values of the CKM matrix "run" with energy scale according to a "beta function" that can be determined without experimental input from the equations of the Standard Model.

One of the two main parameterization of the CKM matrix is the Wolfenstein parameterization, which is attractive relative to the "standard parameterization" because it is suggestive of a particular underlying structure in the CKM matrix.

*Illustration from Wikipedia.*

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