Fermion mass diagonalization and the CKM matrix

Write the weak-basis fields as $u'_L=L_u u_L$ and $u'_R=R_u u_R$, with analogous equations for down-type quarks. The unprimed columns contain mass eigenfields. A general complex mass matrix connects left and right chiralities, so two different unitary matrices are required. Choose $L_u^\dagger M_uR_u=D_u$, where D has real nonnegative masses on its diagonal. This is a singular-value decomposition, not an assumption that the original mass matrix is Hermitian. Degenerate masses permit further basis freedom inside their common eigenspace.

Insert these definitions into the weak charged current before simplifying. The up-quark conjugate supplies the dagger on its left rotation; the down field supplies its undaggered rotation. Thus $V=L_u^\dagger L_d$ multiplies the down-type mass column. Other authors may define their rotation matrices in the opposite direction, producing apparently reversed daggers with identical physics. A useful diagnostic is to reconstruct the current from the stated field definitions, rather than copying the matrix formula without its convention.

Because both left rotations are unitary, V is unitary. Each row and column therefore has squared magnitudes summing to one, and distinct rows or columns have vanishing inner products. A weak transition amplitude contains a particular CKM entry; an isolated rate contribution contains its squared magnitude. Measured hadron decay rates additionally include phase space, strong-interaction matrix elements, electroweak factors, and sometimes competing amplitudes. Their raw branching fractions do not form a CKM probability table without accounting for these ingredients.

A two-family illustration isolates the geometry. Define a real rotation R whose first row is cosine, sine and whose second row is minus sine, cosine. If the up and down left rotations have angles alpha and beta, the current contains the rotation through beta minus alpha. A common rotation of both sectors disappears. Mixing consequently measures a relative orientation, not the size of either diagonalizing rotation considered separately. This cancellation is also why choosing an up-diagonal weak basis does not remove physical down-sector mixing.

A general three-dimensional unitary matrix has nine real parameters: three rotation angles and six phases. Rephasing each of the six quark mass fields changes row or column phases of V. One common phase transformation changes no matrix entry, leaving five useful phase removals. The remaining physical parameters are three angles and one CP phase, provided the relevant masses are nondegenerate. For two families the analogous phase count leaves no charged-current CP phase. Complex-looking entries alone therefore do not demonstrate CP violation.

The right rotations remain essential for constructing positive masses, even though they cancel from the displayed charged current. Likewise, mass hierarchies and mixing angles are separate empirical properties; small masses do not algebraically require small CKM angles. Precision flavor analysis combines several processes to determine the matrix and test unitarity under declared theoretical assumptions. A fitted discrepancy can reflect an incorrect hadronic input, experimental uncertainty, or additional interactions. Identifying its origin requires more information than a single matrix element extracted from one decay channel.