Flavor, CP Violation and Precision Fits

The cross-sector synthesis calculation for the Standard Model: how the CKM matrix organizes quark flavor, how its single phase produces the observed CP violation, and how the global electroweak fit turns dozens of precision measurements into a stringent, over-constrained test of the whole theory — one that predicted the top and Higgs masses before their discovery.

Conventions: $\hslash =c=1$; $V=V_{\text{CKM}}$.

The CKM matrix and the unitarity triangle

The charged current mixes quark generations through the CKM matrix $V$, a $3\times 3$ unitary matrix with, after removing unphysical phases, three mixing angles and one CP-violating phase. Unitarity, ${\sum }_k{V}_{ik}{V}_{jk}^{\ast }=0$, gives relations that can be drawn as triangles in the complex plane; the most-studied is

$$V_{ud}{V}_{ub}^{\ast }+V_{cd}{V}_{cb}^{\ast }+V_{td}{V}_{tb}^{\ast }=0.$$

This is a closed triangle with unit base; its area is a measure of CP violation and its angles $(\alpha ,\beta ,\gamma )$ are measured independently in $B$-meson decays. The triumph of flavor physics is that many independent measurements — sides from FCNC mixing and semileptonic decays, angles from CP asymmetries — all intersect at a single apex, over-determining the triangle and confirming that a single CKM phase describes all quark-sector CP violation.

CP violation from one phase

CP violation requires a physical complex phase that cannot be rotated away. In the SM this exists only with three or more generations: with two generations the CKM matrix is real (the Cabibbo angle only), so Kobayashi and Maskawa (1973) predicted a third generation purely to accommodate the observed CP violation in kaons — before charm, bottom, or top were known. The measured CP violation in the $K$ and $B$ systems (direct and mixing-induced asymmetries) is quantitatively consistent with the single KM phase, a remarkable success. A Jarlskog invariant $J\approx 3\times 10^{-5}$ quantifies it, phase-convention-independently.

Two independent CP puzzles remain, unrelated by the SM (see the two CP problems): the large CKM phase (measured, understood) and the mysteriously tiny strong-CP $\theta$-angle (unexplained). Notably, the SM's CP violation is far too small to explain the cosmic matter–antimatter asymmetry, one of the clear pointers to BSM physics.

The global electroweak fit

The deepest test of the SM is consistency across many observables. The electroweak sector has only a handful of free parameters (traditionally $\alpha ,G_F,m_Z$, plus $m_t,m_h,{\alpha }_s$), but predicts dozens of measured quantities — $m_W$, all $Z$-pole asymmetries, ${\sin }^2{\theta }_W^{\text{eff}}$, partial widths — through calculable radiative corrections. Because there are far more observables than parameters, the fit is massively over-constrained:

• Historic predictions. The oblique corrections' sensitivity to $\Delta \rho \propto m_t^2$ let the LEP/SLC data predict $m_t\approx 170\mathrm{GeV}$ before the Tevatron discovery (1995, $m_t=173\mathrm{GeV}$), and the logarithmic $m_h$ sensitivity predicted a light Higgs ($m_h\lesssim 200\mathrm{GeV}$) before the LHC found $125\mathrm{GeV}$ (2012).
• Present status. With $m_t$ and $m_h$ now measured, the fit is fully closed and every observable is predicted with no free parameters — and they agree at the per-mille level. This global consistency is the strongest quantitative validation of the Standard Model as a renormalizable quantum theory, and simultaneously a tight filter on new physics (which would show up as a bad fit).

What the fit tests, together

The fit weaves together every sector: QED running $\alpha (m_Z)$, QCD ${\alpha }_s$ corrections to the hadronic widths, electroweak tree + loop structure, and Higgs and top loops — a genuinely cross-sector confrontation that no single theory piece could deliver.

Summary

• The CKM matrix (3 angles + 1 phase) organizes quark flavor; its unitarity triangle is over-determined by independent measurements that all agree.
• A single CP phase describes all quark-sector CP violation and required a third generation (KM, 1973) — but is too small for cosmic baryogenesis.
• The global electroweak fit predicts dozens of observables from a few parameters; it predicted $m_t$ and $m_h$ and now closes with per-mille consistency — the SM's strongest test and a stringent BSM filter.

Where this fits

• The theory: Standard Model — the cross-sector content.
• The observables: Standard Model observables, electroweak observables.
• Companions: Gauge-Boson Decays and Precision Electroweak, GIM and FCNC, Neutrino Masses and the Seesaw.

References

• Kobayashi & Maskawa, Prog. Theor. Phys. 49, 652 (1973).
• Particle Data Group, Review of Particle Physics, "CKM matrix", "Electroweak model".
• Branco, Lavoura & Silva, CP Violation.