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.

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.

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.

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.

**Background**

*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.*

*Illustration from Wikipedia.*