Quark Flavor and CP

Enter this chapter by asking which quantity survives a change of quark basis. Yukawa eigenvalues and commutator invariants describe flavor breaking; CKM quartets describe charged-current mixing and weak phases; a decay amplitude combines CKM factors, Wilson coefficients, and renormalized matrix elements; neutral-meson observables depend on rephasing-invariant mixing–decay combinations; and strong $CP$ depends on $\bar\theta$, not on the QCD angle or a quark-mass phase separately. The six routes below build that chain without turning basis coordinates, scale-dependent coefficients, or current fit values into observables.

Choose the invariant before the phenomenology

Quark flavor is one subject because each stage removes an unphysical convention before handing a physical quantity to the next:

Question	Convention-dependent ingredients	Comparison layer
How do Yukawas break flavor?	weak-basis matrices $Y_u,Y_d$	singular values and traces or commutators of $Y_uY_u^\dagger$ and $Y_dY_d^\dagger$
How do charged currents mix quarks?	left-handed diagonalization matrices and quark phases	CKM moduli and rephasing-invariant quartets
How is a weak decay calculated?	operator basis, subtraction scheme, scale $\mu$, evanescent operators	the amplitude $\sum_i C_i(\mu)\langle Q_i(\mu)\rangle$
How does a neutral meson oscillate and decay?	phases of $\lvert P^0\rangle$, $\lvert\bar P^0\rangle$, $q/p$, and decay amplitudes	$\lambda_f=(q/p)(\bar A_f/A_f)$, eigenvalues, and rates
How is weak $CP$ tested geometrically?	a chosen CKM parameterization and triangle orientation	$J$, unitarity closure, invariant side ratios, and correlated observables
Why can strong interactions violate $CP$?	$\theta$ and $\arg\det M_q$ separately	$\bar\theta=\theta+\arg\det M_q$ in the convention declared here

The chapter owns the durable construction and inference interfaces. Numerical Yukawa hierarchies, flavor averages, individual decay catalogs, experimental tensions, and axion exclusions require dated evidence records and are not reproduced here.

Check your preparation

This diagnostic is unscored. Each repair link points to the exact capability used by the chapter.

Can you already…	Ready when you can…	Repair
diagonalize a complex mass matrix?	distinguish its left and right singular-vector rotations and track a basis change through an interaction	review Yukawa couplings and fermion masses and normal forms, spectra, and projectors
build an invariant tensor or spurion contraction?	assign transformation laws so a formally invariant operator is easy to check	review multiplets, invariants, and selection rules
follow weak charged and neutral currents?	show why only the charged current contains the mismatch of up- and down-quark rotations	review charged and neutral weak currents
evolve operators and coefficients together?	derive the transpose and sign relation that keeps $C^{\mathsf T}Q$ scale independent	review dual evolution of operators and Wilson coefficients
solve a decaying two-state system?	diagonalize a non-Hermitian $2\times2$ Hamiltonian without renormalizing raw survival probabilities to one	review linear ODEs and evolution operators
track anomalous chiral phases?	explain how a quark rotation moves phase between the mass matrix and the QCD topological term	review the $U(1)_A$ problem and QCD topology

The prerequisite notes on individual leaves are hard dependencies for their derivations. The order below is only a suggested route. A reader interested solely in strong $CP$ may begin at the topology repair and take the final route directly; a weak-decay calculation should preserve the first three dependencies.

Chapter guide

1. Flavor Symmetry and Yukawa Spurions derives the quark kinetic flavor group, spurion transformations, residual symmetries, parameter count, and basis invariants.
2. Quark Mixing and the CKM Matrix performs the two biunitary mass rotations, constructs $V_{\rm CKM}$, counts its physical angles and phase, and checks neutral-current diagonality.
3. Weak Effective Hamiltonians and Flavor-Changing Processes follows matching, operator mixing, threshold running, matrix elements, phase space, and long-distance terms into one physical amplitude.
4. Neutral-Meson Mixing and Mixing-Induced CP Violation solves the decaying two-state system and separates $CP$ violation in mixing, decay, and their interference.
5. Quark CP Violation and the Unitarity Triangle derives $J$, triangle closure and area, direct-$CP$ phase requirements, and the distinct inputs that constrain the triangle.
6. Strong CP and the Axion Interface constructs $\bar\theta$ under anomalous rotations and shows how a QCD axion promotes it to a dynamically relaxed field.

The first five routes form one hard-dependency chain for a full weak-flavor synthesis. The sixth shares Yukawa phases and anomaly bookkeeping but has a different hard prerequisite and a different physical invariant.

Conventions that keep the chain consistent

The site conventions fix $\gamma_5=i\gamma^0\gamma^1\gamma^2\gamma^3$, $\epsilon^{0123}=+1$, $\epsilon_{0123}=-1$, and the (+---) metric. This chapter adds:

Surface	Convention	Invariant check
CKM matrix	$V=U_u^\dagger U_d$ with the standard three-angle parameterization and $V_{ub}=s_{13}e^{-i\delta}$	$V^\dagger V=\mathbf1$ and quartets are unchanged by quark rephasing
Weak Hamiltonian	$Q^{\rm bare}=ZQ(\mu)$, $\gamma=Z^{-1}\mu\,dZ/d\mu$, so $\mu\,dQ/d\mu=-\gamma Q$ and $\mu\,dC/d\mu=\gamma^{\mathsf T}C$	$\mu\,d[C^{\mathsf T}\langle Q\rangle]/d\mu=0$ to the computed order
Neutral mixing	$\lambda_f=(q/p)(\bar A_f/A_f)$ and, for $\Delta\Gamma=0$, $A_{CP}=[\Gamma(\bar P^0(t)\to f)-\Gamma(P^0(t)\to f)]/\text{sum}$	$A_{CP}=S_f\sin\Delta mt-C_f\cos\Delta mt$ is invariant under flavor-state rephasing
Strong $CP$	$\mathcal L_\theta=(g_s^2\theta/32\pi^2)G^a_{\mu\nu}\widetilde G^{a\mu\nu}$ and $\mathcal L_m=-\bar q_LM_qq_R+\text{h.c.}$	$\bar\theta=\theta+\arg\det M_q$ is unchanged by an anomalous chiral basis transformation

A source using $V=U_d^\dagger U_u$, the opposite $CP$-asymmetry numerator, $\Delta\Gamma=\Gamma_H-\Gamma_L$, or $\bar\theta=\theta-\arg\det Y$ must be translated in full. The invariant checkpoint, not the intermediate sign, decides whether the translation succeeded.

One calculation, four kinds of input

A flavor claim is reproducible only if its layers remain distinct:

Layer	Example object	What must accompany it
Short-distance theory	CKM product and Wilson coefficient	operator normalization, scheme, scale, perturbative order, thresholds
Long-distance QCD	hadronic matrix element or absorptive sum over common states	renormalization match, finite-volume/continuum or model method, covariance
Propagation and kinematics	phase space, line shape, $M_{12}$, $\Gamma_{12}$, time acceptance	mass/width convention, state phases, approximations, detector interface
Inference	fitted invariant, triangle region, or $CP$ asymmetry	dataset identity, likelihood/covariance, nuisance treatment, evidence date

The first three layers can be combined in a stable formal derivation. The last becomes a current-status statement only with a versioned evidence source.

Synthesis check

Choose one weak-flavor amplitude or mixing observable and answer the following without assigning a score:

1. Basis: Which field rephasings or unitary rotations are unphysical? A successful answer identifies an invariant quartet, commutator, or $\lambda_f$. Repair at the invariant map.
2. Matching: Which heavy modes were removed and which operators remain? A successful answer states the basis, matching scale, and power corrections. Repair with the preparation diagnostic.
3. Running: How do coefficient and matrix element scheme/scale dependences cancel? A successful answer writes the paired RG equations and a finite-basis transformation. Repair at the chapter convention table.
4. Phases: Which phases are weak, strong, mixing, or conventional? A successful answer demonstrates rephasing invariance and names the strong phase needed for direct $CP$ violation. Repair through the chapter guide.
5. Evidence ceiling: Which numerical inputs could change? A successful answer separates structural identities from hadronic, experimental, and correlated-fit inputs using the four-layer table.

A complete synthesis can be reconstructed from the declared basis and scales, survives rephasing, reaches a physical rate or invariant, and states at least one limiting or null check.

Continue by purpose

• For electroweak current normalization: continue to Electroweak Theory and the Higgs.
• For general operator matching, running, and evanescent schemes: continue to Renormalization and Effective Field Theory.
• For hadronic form factors entering flavor amplitudes: continue to Hadron Form Factors and Current Structure.
• For exact synthetic checks: reproduce the CKM and neutral-meson fixtures with the conventions stated in this chapter.
• For current combinations, anomalies, or search status: enter Research and require a dated, versioned evidence record.
• For generic axionlike portals rather than the QCD relaxation mechanism: continue to Axionlike and Pseudoscalar Portals.
• For another route through this volume: return to Gauge Theories and the Standard Model.

References

• Buchalla, Gerhard, Andrzej J. Buras, and Markus E. Lautenbacher. “Weak Decays Beyond Leading Logarithms.” Reviews of Modern Physics 68 (1996): 1125–1244. DOI · Open PDF
• D’Ambrosio, Giancarlo, Gian F. Giudice, Gino Isidori, and Alessandro Strumia. “Minimal Flavour Violation: An Effective Field Theory Approach.” Nuclear Physics B 645 (2002): 155–187. DOI · Open PDF
• Grilli di Cortona, Giovanni, Edward Hardy, Javier Pardo Vega, and Giovanni Villadoro. “The QCD Axion, Precisely.” Journal of High Energy Physics 01 (2016): 034. DOI · Open PDF
• Nir, Yosef. “CP Violation in Meson Decays.” Lectures at the CERN–CLAF and Les Houches schools, 2005. arXiv · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, chs. 29 and 31. DOI