Quark CP Violation and the Unitarity Triangle

Quark $CP$ violation is physical only through combinations unchanged by quark and meson phase conventions. For three nondegenerate generations, the Jarlskog invariant $J$ supplies the unique CKM measure up to sign, and CKM unitarity turns an orthogonality relation into a closed triangle whose area is $|J|/2$ before normalization. Different decay and mixing observables constrain its sides and angles with different short-distance, hadronic, and covariance inputs; no plot alone establishes consistency.

Required background. Neutral-meson mixing and mixing-induced CP violation supplies $q/p$, $\lambda_f$, and time-dependent asymmetries. Helpful background. Discrete and antiunitary symmetries supplies the distinction between basis phases and symmetry violation.

Rephasing-invariant weak CP violation

This page uses the standard CKM convention $V_{ub}=s_{13}e^{-i\delta}$ and defines

$$J=\operatorname{Im} \left(V_{us}V_{cb}V_{ub}^*V_{cs}^*\right) =c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta.$$

Under $u_i\to e^{i\alpha_i}u_i$ and $d_j\to e^{i\beta_j}d_j$, every CKM element changes phase, but the four phases in a closed quartet cancel. All quartets with distinct rows and columns have imaginary part $\pm J$. Complex conjugating the CKM matrix reverses $J$.

For nondegenerate quark masses, $J\ne0$ is equivalent to weak $CP$ violation in the three-generation quark sector. The basis-independent mass condition is stronger than merely observing $\delta\ne0$ in one parameterization:

$$\frac{1}{6i}\operatorname{tr}\!\left( [Y_uY_u^\dagger,Y_dY_d^\dagger]^3 \right) =J\,\Delta_u\Delta_d.$$

If a mixing angle vanishes or two equal-charge Yukawa singular values become degenerate, the right-hand side vanishes and the apparent phase can be removed from physical amplitudes. This is Jarlskog’s convention-independent criterion Jarlskog 1985, pp. 1039–1042.

Weak phases should not be confused with two other uses of “strong”:

• a strong phase in a decay amplitude is a CP-even rescattering or absorptive phase;
• strong $CP$ is the separate QCD invariant $\bar\theta$.

The first is required for a direct rate asymmetry; the second is not part of a unitarity triangle.

Constructing the unitarity triangle

Orthogonality of the $d$ and $b$ columns gives

$$V_{ud}V_{ub}^* +V_{cd}V_{cb}^* +V_{td}V_{tb}^*=0.$$

The three terms are complex vectors that close. Divide by $V_{cd}V_{cb}^*$ and define the rephasing-invariant apex

$$\bar z\equiv\bar\rho+i\bar\eta =-\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*}.$$

Then

$$1-\bar z =-\frac{V_{td}V_{tb}^*}{V_{cd}V_{cb}^*},$$

so the normalized triangle has vertices $0$, $1$, and $\bar z$. Each ratio is invariant because its numerator and denominator acquire the same quark-rephasing phase.

The conventional interior angles are

$$\alpha=\arg\!\left( -\frac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*} \right),$$ $$\beta=\arg\!\left( -\frac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*} \right), \qquad \gamma=\arg\!\left( -\frac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} \right).$$

For a nondegenerate triangle,

$$\alpha+\beta+\gamma=\pi.$$

Let $A=V_{ud}V_{ub}^*$ and $B=V_{cd}V_{cb}^*$. The area of the unnormalized triangle is

$$\mathcal A_\triangle =\frac12|\operatorname{Im}(AB^*)| =\frac{|J|}{2}.$$

After division by $-B$, its area becomes

$$\mathcal A_{\rm norm} =\frac{|J|}{2|V_{cd}V_{cb}^*|^2}.$$

Thus a collapsed triangle and $J=0$ are the same weak-$CP$ null test, provided the side used for normalization is nonzero. The triangle construction and standard phase convention are developed in Schwartz 2014, § 29.3.3, pp. 598–600.

Direct CP violation needs two phase differences

Write a bounded two-amplitude decay as

$$A_f =a_1e^{i(\delta_1+\phi_1)} +a_2e^{i(\delta_2+\phi_2)},$$ $$\bar A_{\bar f} =a_1e^{i(\delta_1-\phi_1)} +a_2e^{i(\delta_2-\phi_2)}.$$

The $a_i$ are nonnegative magnitudes, $\delta_i$ are CP-even strong phases, and $\phi_i$ are CP-odd weak phases. Direct subtraction gives

$$|\bar A_{\bar f}|^2-|A_f|^2 =4a_1a_2 \sin(\delta_1-\delta_2) \sin(\phi_1-\phi_2).$$

A direct rate asymmetry therefore requires both a weak-phase difference and a strong-phase difference. A complex CKM product alone is insufficient, and a strong phase alone is CP even. With more amplitudes the result is a sum over pairs with the same structure.

Each $a_i e^{i\delta_i}$ generally contains

$$\sum_k C_k(\mu)\langle f|Q_k(\mu)|H\rangle.$$

The separation between coefficient and matrix element is scheme and scale dependent, while the full amplitude and its CP-conjugate are not. Under $Q'=RQ$ and $C'=R^{-\mathsf T}C$, every physical asymmetry is unchanged. A claimed weak phase extracted without the operator convention, hadronic strong phases, and their covariance is incomplete.

Mixing and decay interference

For a neutral meson decaying to a common final state,

$$\lambda_f=\frac{q}{p}\frac{\bar A_f}{A_f}$$

combines the mixing phase and decay phase invariantly. In the $\Delta\Gamma=0$ convention used in this chapter,

$$C_f=\frac{1-|\lambda_f|^2}{1+|\lambda_f|^2}, \qquad S_f=\frac{2\operatorname{Im}\lambda_f}{1+|\lambda_f|^2},$$ $$A_{CP}(t) =S_f\sin(\Delta m\,t)-C_f\cos(\Delta m\,t),$$

where the numerator is $\Gamma(\bar P^0(t)\to f)-\Gamma(P^0(t)\to f)$. The phase of $q/p$ or $\bar A_f/A_f$ separately changes when the flavor states are rephased; $\lambda_f$ does not.

CP mechanism	Minimal invariant statement	Phase requirement
decay	$\lvert A_f\rvert\ne\lvert\bar A_{\bar f}\rvert$	different weak and strong phases
mixing	$\lvert q/p\rvert\ne1$	dispersive and absorptive mixing not CP aligned
interference	CP-odd $\lambda_f$; often $S_f\ne0$ under stated assumptions	mixing and decay paths reach the same final state

These mechanisms and their phase conventions are classified in Nir 2005, §§ III.C–III.D, pp. 25–27.

What constrains a triangle

Different constraints do not measure the same object:

Constraint class	CKM information	Additional theory input	Essential provenance
tree-level semileptonic rate	side magnitude	current normalization, form factors, radiative and phase-space corrections	matrix-element method and full covariance
tree-level interference in decays	angle such as $\gamma$	amplitude topology, strong phases, suppressed contributions	decay model or symmetry assumptions and experimental likelihood
$\Delta F=2$ oscillation	side/angle combination through $M_{12}$	loop matching, RG, hadronic matrix element	operator scheme/scale and nonperturbative covariance
mixing-induced asymmetry	phase through $\lambda_f$	mixing convention, subleading amplitudes, final-state CP	time acceptance, tagging, resolution, numerator convention
direct CP asymmetry	weak-phase combination	at least two hadronic amplitudes and strong phases	correlated rates and long-distance treatment
global closure test	common apex and consistency	all of the above, including shared nuisances	exact dataset versions, likelihood, correlations, and fit assumptions

A “tree” label reduces sensitivity to new heavy particles in the short-distance amplitude but does not remove hadronic or experimental inputs. A “loop” constraint may be sensitive to physics beyond the Standard Model and cannot be combined as if it directly measured a CKM coordinate without the stated theory hypothesis.

Exact geometric and covariance checks

For the exact CKM fixture used by a reproducible calculation,

$$J=\frac{580608}{13203125}.$$

The checks are:

1. evaluate the three complex terms in the $db$ column relation and verify exact closure;
2. calculate $\bar z=-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*)$ and verify the normalized sides end at $0$, $1$, and $\bar z$;
3. rephase every quark field arbitrarily and recover the same $J$, $\bar z$, angles, and area.

The calculation’s illustrative two-coordinate covariance is

$$\mathsf C_\triangle =\frac1{400} \begin{pmatrix} 4&1\\ 1&9 \end{pmatrix}, \qquad \det\mathsf C_\triangle=\frac{35}{160000}.$$

A synthetic constraint centered on the planted apex has $\chi^2=0$ there; the exact displacement $(1/20,-1/10)$ gives correlated $\chi^2=29/35$. These values test matrix inversion and correlation handling. They are not current flavor evidence and must never be drawn as a world-fit contour.

Independent checks and limitations

• Rephasing: verify every side ratio, angle ratio, $J$, and $\lambda_f$ is unchanged under arbitrary quark and meson state phases.
• Closure: sum the three complex terms before plotting. A triangle drawn from inconsistent magnitudes can look closed by construction.
• Area: compute it both from complex vectors and from $|J|/2$ in the unnormalized plane.
• CP limit: set $\delta=0$ or any $s_{ij}=0$; $J$, the invariant area, and weak CP-odd observables must vanish.
• Degenerate-mass limit: the full commutator invariant vanishes even if a chosen CKM parameterization retains an apparent phase.
• Direct-asymmetry limit: set either $\Delta\phi=0$ or $\Delta\delta=0$ in the two-amplitude derivation; the rate difference must vanish.
• Scheme cancellation: transform coefficients and matrix elements together. A stable apex cannot be inferred from scheme-mismatched amplitudes.
• Correlations: a global region requires the joint likelihood or covariance and shared nuisance model. Overlaying one-dimensional intervals is not an equivalent fit.
• Domain: this page supplies invariant geometry and the theory-to-observable map, not current apex coordinates, baryogenesis, or a complete neutral-meson phenomenology.

Common pitfalls

Calling $\delta$ itself basis invariant. It is a coordinate in the standard parameterization. $J$ and invariant CKM ratios determine whether its effects are physical.

Confusing a hadronic strong phase with strong $CP$. Rescattering phases conserve CP and enable direct weak-CP rate asymmetries. $\bar\theta$ is a separate flavor-diagonal QCD parameter.

Treating all bands as independent. Shared CKM, lattice, theory, and experimental nuisance inputs correlate constraints. A visual overlap without covariance is not a closure test.

Informal self-check

Show that $\bar z=-V_{ud}V_{ub}^*/(V_{cd}V_{cb}^*)$ is rephasing invariant and that the normalized triangle closes.

Answer

The product $V_{ud}V_{ub}^*$ acquires $e^{i(\beta_d-\beta_b)}$, and $V_{cd}V_{cb}^*$ acquires the same factor, so their ratio is fixed. Dividing $V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0$ by $V_{cd}V_{cb}^*$ gives $-\bar z+1+V_{td}V_{tb}^*/(V_{cd}V_{cb}^*)=0$, or $1-\bar z=-V_{td}V_{tb}^*/(V_{cd}V_{cb}^*)$.

Handoffs

• Send each decay or mixing constraint with its Wilson coefficients, matrix-element scheme, strong phases, and covariance back to the weak-Hamiltonian method.
• Send $q/p$, $A_f$, $\bar A_f$, $\Delta m$, $\Delta\Gamma$, and the asymmetry numerator to the neutral-meson formalism.
• Reproduce the exact rephasing, closure, area, and covariance checks before fitting data.
• Send current apex regions, tensions, and combination claims to a dated Research record with the complete likelihood and version identity.

References

• Jarlskog, Cecilia. “Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation.” Physical Review Letters 55 (1985): 1039–1042. DOI
• Nir, Yosef. “CP Violation in Meson Decays.” Lectures at the CERN–CLAF and Les Houches schools, 2005, §§ II.E and III.C–III.D. arXiv · Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, §§ 29.3.3 and 29.5.1–29.5.2. DOI