Higgs Yukawa couplings and fermion masses

Insert the lower Higgs component, v plus h divided by square root of two, into a down-type or charged-lepton Yukawa term. The part proportional to v is the mass matrix; the part proportional to h is the same matrix divided by v. For one real positive Yukawa eigenvalue, mass equals y times v divided by square root of two. The coefficient multiplying h times fermion-bar fermion is consequently mass over v, not the original y.

General Yukawa matrices are complex and need not be Hermitian. Two independent unitary rotations, acting on left- and right-chiral fields, produce a diagonal matrix of nonnegative singular values. Treating a generic mass matrix as though one ordinary orthogonal similarity transformation always diagonalizes it misses both phases and the independent chiral spaces. The special symmetric real example below is chosen for hand calculation and is explicitly more restrictive than the general biunitary problem.

Because up- and down-type masses arise from separate Yukawa matrices, their left rotations need not agree. Substituting both rotations into the charged weak current leaves the product of the adjoint up rotation and the down rotation. This is the CKM matrix. The right rotations do not enter the minimal charged current because it couples only left-chiral doublets. Unitarity of the two rotations makes their product unitary when the full three-generation quark space is retained.

Neutral gauge currents act identically on generations with the same charges, so a unitary mass rotation cancels against its adjoint. Tree-level neutral gauge currents remain flavor diagonal. The aligned Higgs matrix is also diagonal in the same mass basis, so this minimal construction has no tree-level flavor-changing neutral Higgs interaction. Extra scalars, nonaligned Yukawa matrices, or higher-dimensional operators can change that conclusion. A small sub-block of an enlarged mixing matrix need not itself be unitary.

At tree level a Higgs decay into a fermion pair carries a squared mass-over-v coupling, along with color multiplicity, phase space, and threshold factors. Comparing two channels solely by their Yukawa eigenvalues is justified only after those other factors are matched or calculated. For quarks, a perturbative running mass at an appropriate scale often organizes the decay prediction, rather than an unrelated mass convention inserted without conversion.

Beyond leading order the relation between an experimentally extracted mass, a renormalized Yukawa coupling, and an inferred Higgs coupling contains matching corrections and scheme dependence. This does not mean the underlying model has lost its predictive relation; it means the compared quantities must be defined consistently. Neutrino masses require additional field content or operators beyond the minimal massless-neutrino inventory, and those choices introduce a separate leptonic mixing problem. CKM unitarity alone does not resolve it or predict the fermion mass hierarchy.