Quark Mixing and the CKM Matrix

The CKM matrix is the mismatch between the left-handed rotations that diagonalize the up- and down-quark mass matrices: $V_{\rm CKM}=U_u^\dagger U_d$. Right-handed rotations disappear from Standard Model charged currents, while unitarity keeps neutral currents flavor diagonal. For three nondegenerate generations, quark rephasings leave three physical mixing angles and one irreducible phase.

Required background. Flavor symmetry and Yukawa spurions supplies weak-basis freedom; charged and neutral weak currents supplies the electroweak interaction. Helpful background. Normal forms, spectra, and projectors supplies singular-value decomposition and degeneracy handling.

Biunitary mass rotations

After electroweak symmetry breaking,

$$M_u=\frac{v}{\sqrt2}Y_u, \qquad M_d=\frac{v}{\sqrt2}Y_d.$$

Choose singular-value decompositions

$$M_u=U_uD_uW_u^\dagger, \qquad M_d=U_dD_dW_d^\dagger,$$

where $D_u,D_d$ are diagonal, nonnegative, and placed in a declared flavor order. Relate weak-basis and mass-basis fields by

$$u_L^{\,w}=U_u u_L,\quad u_R^{\,w}=W_u u_R, \qquad d_L^{\,w}=U_d d_L,\quad d_R^{\,w}=W_d d_R.$$

Then $U_u^\dagger M_uW_u=D_u$ and similarly for $d$. The charged-current interaction becomes

$$\mathcal L_{cc} =-\frac{g}{\sqrt2}\, \bar u_L\gamma^\mu Vd_L\,W_\mu^+ +\text{h.c.}, \qquad V=U_u^\dagger U_d.$$

This derivation fixes the convention used throughout the chapter. A source defining weak-basis fields with inverse rotations may write $U_d^\dagger U_u$; it must also conjugate and relabel every charged-current factor. The invariant amplitude is the translation check.

The right-handed matrices $W_u,W_d$ do not appear because the Standard Model $W^\pm$ current is left handed. Neutral currents contain

$$\bar u_L^{\,w}\gamma^\mu u_L^{\,w} =\bar u_L\gamma^\mu U_u^\dagger U_u u_L =\bar u_L\gamma^\mu u_L,$$

and likewise for $d$. Thus unitary rotations preserve flavor-diagonal tree-level photon and $Z$ currents. Loop-induced flavor-changing neutral currents remain possible and are GIM suppressed; they do not contradict this tree-level result Schwartz 2014, §§ 29.3.2 and 29.4, pp. 595–598 and 603–605.

Counting angles and phases

An $n\times n$ unitary matrix contains

$$\frac{n(n-1)}2$$

rotation angles and $n(n+1)/2$ phases. Rephase the mass eigenfields vectorially,

$$u_i\to e^{i\alpha_i}u_i, \qquad d_j\to e^{i\beta_j}d_j.$$

The masses remain real and positive, while

$$V_{ij}\to e^{-i\alpha_i}V_{ij}e^{i\beta_j}.$$

Of the $2n$ phases, one common baryon-number phase leaves $V$ unchanged, so $2n-1$ phases can be removed. The physical count is therefore

$$N_{\rm angles}=\frac{n(n-1)}2, \qquad N_{\rm phases}=\frac{(n-1)(n-2)}2.$$

For $n=2$, there is one angle and no irreducible phase; for $n=3$, there are three angles and one phase. Exact mass degeneracies enlarge the redefinition freedom and can remove additional coordinates, so these formulas assume nondegenerate quarks of each electric charge. The need for three generations to retain a renormalizable weak-$CP$ phase is the central result of Kobayashi and Maskawa 1973, pp. 652–657.

Standard parameterization and unitarity

Write $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$. The standard convention is

$$V= \begin{pmatrix} c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\\ -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} &c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} &s_{23}c_{13}\\ s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} &c_{23}c_{13} \end{pmatrix}.$$

It is the product $R_{23}U_{13}(\delta)R_{12}$, so each factor is unitary and

$$V^\dagger V=VV^\dagger=\mathbf1.$$

Row and column normalization,

$$\sum_j|V_{ij}|^2=1, \qquad \sum_i|V_{ij}|^2=1,$$

and orthogonality,

$$\sum_kV_{ik}V_{jk}^*=0\quad(i\ne j), \qquad \sum_kV_{ki}V_{kj}^*=0\quad(i\ne j),$$

are exact consequences of the three-generation Standard Model construction, not fitted approximations.

The rephasing-invariant quartet

$$J=\operatorname{Im} \left(V_{us}V_{cb}V_{ub}^*V_{cs}^*\right)$$

has the standard-parameterization value

$$J=c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta.$$

Any quartet with distinct row and column indices has imaginary part $\pm J$. Complex conjugation reverses its sign; rephasing does not change it. A nonzero $\delta$ is not by itself invariant when an angle vanishes or masses become degenerate, whereas the mass-commutator invariant on the preceding page correctly vanishes in every such limit Jarlskog 1985, pp. 1039–1042.

Hierarchical coordinates without losing exactness

The Wolfenstein representation is a useful expansion when a small mixing coordinate $\lambda$ organizes matrix entries. It is not a separate matrix and a finite truncation is not exactly unitary. For example, definitions such as

$$\lambda=s_{12}, \qquad s_{23}=A\lambda^2, \qquad s_{13}e^{-i\delta} =A\lambda^3(\rho-i\eta)$$

become an exact change of coordinates only when inserted into the exact standard parameterization with the corresponding square roots $c_{ij}=\sqrt{1-s_{ij}^2}$. A series truncated at $O(\lambda^k)$ should be checked for closure only through its declared remainder. Triangle apex coordinates are best defined directly by invariant CKM ratios rather than by identifying them with a low-order expansion coefficient.

No fitted angle, hierarchy value, or unitarity tension is quoted here. Those are evidence-dependent inferences, not part of the CKM definition.

An exact construction check

A reproducible calculation specifies

$$(s_{12},c_{12})=\left(\frac35,\frac45\right), \quad (s_{23},c_{23})=\left(\frac5{13},\frac{12}{13}\right), \quad (s_{13},c_{13})=\left(\frac7{25},\frac{24}{25}\right), \quad \delta=\frac{\pi}{2}.$$

The exact invariant is

$$J =c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13} =\frac{580608}{13203125}.$$

The companion mass fixture takes

$$D_u=\operatorname{diag}(1,2,3), \qquad D_d=\operatorname{diag}(4,5,6), \qquad M_u=D_u,\quad M_d=VD_d.$$

An independent singular-value decomposition recovers the positive singular values but may permute columns and attach arbitrary phases. Restoring the declared ordering and fixing those phases must reproduce $U_u^\dagger U_d=V$. Comparing raw singular vectors before this repair is not a valid failure test.

From a CKM element to an observable

A CKM entry is a Lagrangian coordinate in a declared convention. A low-energy amplitude has additional layers:

$$\mathcal A(H\to f) =\frac{G_F}{\sqrt2} \sum_{p,i}\lambda_p\, C_i(\mu)\langle f|Q_i(\mu)|H\rangle +\mathcal A_{\rm long},$$

where $\lambda_p$ is a product of CKM entries. Under quark rephasing, $\lambda_p$ and the external-state/operator phases compensate. Under an operator-basis change $Q'=RQ$, $C'=R^{-\mathsf T}C$, so the contraction is unchanged. Renormalization-scale dependence cancels between $C_i(\mu)$ and $\langle Q_i(\mu)\rangle$ to the calculated order.

Claim	Required inputs beyond $V$	Null or consistency check
charged-current rate	current normalization, phase space, radiative corrections, hadronic form factor if applicable	rate is unchanged by quark rephasing
loop-induced FCNC	complete internal-flavor sum and loop functions	constant loop term cancels by CKM unitarity
unitarity relation	all three terms in one row/column orthogonality sum	complex-vector closure
weak-$CP$ quantity	at least one rephasing-invariant quartet and, for direct asymmetry, a strong-phase difference	$J\to0$ when $\delta\to0$ or any $s_{ij}\to0$

The matching, RG, and matrix-element workflow continues on the next route.

Independent checks and limitations

• SVD round trip: verify $U_u^\dagger M_uW_u=D_u$ and $U_d^\dagger M_dW_d=D_d$, with nonnegative ordered diagonals.
• Unitarity: test both $V^\dagger V$ and $VV^\dagger$. Row normalization alone does not establish column orthogonality.
• Neutral currents: insert the same left rotation on both sides of a neutral current; $U^\dagger U$ must remove it.
• Rephasing: change arbitrary $\alpha_i,\beta_j$ and verify $|V_{ij}|$, $J$, and every closed unitarity relation remain fixed.
• Generation limit: for two generations the phase count must be zero. A parameterization that retains a measurable complex phase has not exhausted field rephasings.
• Degeneracy: if two like-charge masses coincide, rotate their degenerate subspace and confirm that purported physical mixing coordinates can change while amplitudes do not.
• Perturbative renormalization: beyond tree level, CKM counterterms and input definitions require a consistent gauge and renormalization prescription. Only complete amplitudes are scheme independent.

Common pitfalls

Using the right-handed rotation in $V$. The charged current couples only $u_L^w$ to $d_L^w$, so the mismatch is $U_u^\dagger U_d$. Right rotations diagonalize masses but do not enter this current.

Treating a truncated Wolfenstein matrix as exactly unitary. Closure errors of the first omitted order are expected. Use the exact standard matrix for invariant or computational checks.

Calling $V_{ij}$ directly observable. Its magnitude can be inferred only through a process with radiative, kinematic, and often hadronic inputs. Its phase alone changes under quark rephasing.

Informal self-check

Apply $u_i\to e^{i\alpha_i}u_i$ and $d_j\to e^{i\beta_j}d_j$ to $V_{us}V_{cb}V_{ub}^*V_{cs}^*$ and show that its imaginary part is invariant.

Answer

The four factors acquire phases $e^{-i\alpha_u+i\beta_s}$, $e^{-i\alpha_c+i\beta_b}$, $e^{+i\alpha_u-i\beta_b}$, and $e^{+i\alpha_c-i\beta_s}$. Their product is one, so the quartet and its imaginary part are unchanged. An individual factor such as $V_{ub}$ does not pass this check.

Handoffs

• Send the exact CKM convention, relevant invariant products, and quark ordering to the weak effective Hamiltonian.
• Send a closed row or column relation and its covariance structure to the unitarity-triangle route, without importing current fit coordinates.
• Send the anomalous phases removed during mass diagonalization to the strong-$CP$ route, where $\theta$ is transformed simultaneously.

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
• Kobayashi, Makoto, and Toshihide Maskawa. “CP-Violation in the Renormalizable Theory of Weak Interaction.” Progress of Theoretical Physics 49 (1973): 652–657. DOI
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014, §§ 29.3.2 and 29.4. DOI