Six physically distinct fixed points of the Standard Model's CKM matrix remain fixed through every order of perturbation theory, according to a formal analysis posted as an arXiv preprint. The result concerns the massless one-loop fixed-point picture and says an extra assumption used in an earlier proof is not needed.
The finding concerns the CKM matrix itself, not a complete freeze of the model's Yukawa sector. Yukawa couplings may continue to change under renormalization-group evolution even when the CKM matrix stays at one of these fixed points.
How the all-order proof is built
The work studies the CKM matrix V as a member of the mathematical group of 3-by-3 unitary matrices. It asks whether the special matrices that stop moving in the massless one-loop flow continue to do so at every order in perturbation theory.
The proof follows the way family indices join up inside perturbative diagrams. With the external legs held fixed, internal fermion lines contract over family indices in a pattern consistent with ordinary matrix multiplication. At any order, the resulting beta-functions, the quantities that describe how the matrices change with the renormalization-group variable, are built from products of the relevant matrices, together with traces from internal family-index loops.
The analysis states that this matrix structure carries the one-loop fixed-point pattern into the full perturbative expansion. It is also independent of the Yukawa values, leaving open the possibility that the CKM matrix remains fixed while those couplings continue to run.
Why the fixed points hold
At the fixed points, the transformed matrices remain diagonal because the fixed CKM matrices merely permute the diagonal entries of the two Yukawa-eigenvalue matrices. That action does not generate new non-diagonal structures. In the language of the proof, the matrices controlling the flow lie in the Cartan sub-algebra, the part that commutes with the diagonal matrices in question.
The general CKM flow combines V with two SU(3) Lie-algebra quantities, A and A', in a left-right difference. At the special points, those quantities commute with the relevant diagonal Yukawa matrices, meaning the order of multiplication makes no difference. The off-diagonal commutator terms therefore disappear, leaving diagonal renormalization-group running while the CKM matrix stays fixed.
The analysis lists sixteen fixed-point matrices but identifies only six as physically distinct after quark rephasing. Rephasing changes the phase convention assigned to quark fields, so matrices that differ only in that way do not represent different physical CKM points. The six remaining points are organized as a unitary representation of S3, the symmetric group of three objects.
What remains unresolved
The result does not require every visible feature of every matrix entry to remain unchanged in every phase convention. Entries that are zero stay zero. Nonzero entries may acquire phase changes during renormalization-group evolution, but the analysis treats those changes as removable through quark rephasings. On that basis, the physical CKM point remains fixed even if an individual phase convention makes some entries look different.
There is a remaining qualification at higher loops. The proposed extension that would choose phases so quantities called A and A' vanish at all fixed points is explicitly presented as a conjecture, not a verified result. The all-order fixed-point claim is therefore tied to the phase treatment set out in the proof, while the broader phase-gauge statement remains open.
The scope is perturbative, even though it includes every order in that expansion. It does not establish a nonperturbative fixed-point theorem. Nor does it show that the complete Yukawa sector is itself at a fixed point.
The supplied document is labeled arXiv:2608.28149v1 and dated 28 Aug 2026. Whether the higher-loop phase-gauge conjecture is correct, and whether the result can be extended beyond perturbation theory, remain open questions.
Paper data and sources
Original title: Fixed points of the CKM matrix renormalization group running to all orders in perturbation theory
Authors: Brian P Dolan
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text