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Fixed points of the renormalisation group running of the CKM and PMNS matrices

Brian P. Dolan

TL;DR

This work analyzes the massless 1-loop RG running of CKM (and PMNS) mixing parameters, uncovering six fixed points that correspond to the Weyl group of SU(3) and yield vanishing CP violation. It derives explicit beta functions for the mixing angles and CP phase in the three-generation case and shows these fixed points are tied to a geometric structure on the SU(3) flag manifold, with Jarlskog invariants equal to zero at all fixed points. A key result is a general argument that, under a natural commuting assumption between RG flow and the left Cartan action, the 1-loop fixed points persist to all orders, including non-perturbatively. Although the SM running is phenomenologically tiny in the infrared, the formal framework provides insights into gradient-flow-like metrics on the mixing space and has potential applications to extensions with extra generations or dark sectors.

Abstract

The renormalisation group running of the CKM parameters in the Standard Model is re-examined. For the massless 1-loop running six fixed points are found and their associated operator mixing matrices determined, in terms of the Yukawa couplings of the quarks. CP is an exact symmetry at each of the fixed points. An argument is given that the fixed points found at 1-loop must remain fixed points to all orders and even non-perturbatively, as they are associated with certain differential geometric properties of the space of CKM matrices. The analysis is equally applicable to the CKM matrix in the quark sector and the PMNS matrix in the leptonic sector.

Fixed points of the renormalisation group running of the CKM and PMNS matrices

TL;DR

This work analyzes the massless 1-loop RG running of CKM (and PMNS) mixing parameters, uncovering six fixed points that correspond to the Weyl group of SU(3) and yield vanishing CP violation. It derives explicit beta functions for the mixing angles and CP phase in the three-generation case and shows these fixed points are tied to a geometric structure on the SU(3) flag manifold, with Jarlskog invariants equal to zero at all fixed points. A key result is a general argument that, under a natural commuting assumption between RG flow and the left Cartan action, the 1-loop fixed points persist to all orders, including non-perturbatively. Although the SM running is phenomenologically tiny in the infrared, the formal framework provides insights into gradient-flow-like metrics on the mixing space and has potential applications to extensions with extra generations or dark sectors.

Abstract

The renormalisation group running of the CKM parameters in the Standard Model is re-examined. For the massless 1-loop running six fixed points are found and their associated operator mixing matrices determined, in terms of the Yukawa couplings of the quarks. CP is an exact symmetry at each of the fixed points. An argument is given that the fixed points found at 1-loop must remain fixed points to all orders and even non-perturbatively, as they are associated with certain differential geometric properties of the space of CKM matrices. The analysis is equally applicable to the CKM matrix in the quark sector and the PMNS matrix in the leptonic sector.
Paper Structure (12 sections, 65 equations, 2 figures)

This paper contains 12 sections, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Yukawa couplings
  3. Running couplings
  4. Two generations: the Cabibbo angle
  5. The mixing matrix for three generations: the CKM and the PMNS matrices
  6. Operator mixing
  7. The CKM matrix
  8. Fixed points at all orders
  9. Discussion
  10. Feynman diagrams
  11. The polynomials $\bm P$ and $\bm Q$
  12. Operator mixing

Figures (2)

  • Figure 1: The left action of $U(1)$ on $SU(2)/U(1)$, with the N and S-poles fixed points, reduces the sphere to a line of constant longitude.
  • Figure 2: When $\delta=0$ or $\pi$ the $\theta_{2}=\frac{\pi}{2}$ face of the cube degenerates to a line segment which is topologically equivalent to the line on the right of figure \ref{['fig:SU2']}.