Higgs boson and Yukawa mass matrices

Three of the doublet's four real fields are gauge artefacts once SU(2)_L × U(1)_Y breaks, so choose unitary gauge and write Φ = (0, (v + h)/√2)ᵀ with h real. Then Φ†Φ = (v + h)²/2 and V = μ²Φ†Φ + λ(Φ†Φ)² becomes μ²(v + h)²/2 + λ(v + h)⁴/4. Requiring the term linear in h to vanish gives v² = −μ²/λ; the quadratic term is then λv² h², so ½mₕ² = λv² and mₕ² = 2λv² = −2μ². The cubic and quartic pieces, λv h³ + (λ/4)h⁴, are the self-couplings of h. Measurement runs opposite to the algebra: mₕ = 125.20 GeV and v = 246.22 GeV give λ = mₕ²/(2v²) = 0.129 and |μ| = mₕ/√2 = 88.5 GeV, so both potential parameters are fitted. The trilinear coefficient λv = 31.8 GeV is the one genuine prediction of the potential, and double-Higgs production has so far bounded it only to within a few times that value.

Every mass in the model is a coupling times v, so replace v by v + h in each mass term and the coupling of h falls out. For a boson the mass enters squared: |D_μΦ|² contains M_W² W⁺_μW⁻^μ (1 + h/v)² and ½M_Z² Z_μZ^μ (1 + h/v)², so the terms linear in h are (2M_W²/v) h W⁺W⁻ and (M_Z²/v) h ZZ, which are 52.5 GeV and 33.8 GeV for M_W = 80.37 GeV and M_Z = 91.19 GeV. The Z term lacks the factor 2 only because its two fields are identical; its vertex factor is 2M_Z²/v = 67.5 GeV, the same form as the W's 2M_W²/v. For a fermion the mass enters once: y_f (v + h)/√2 f̄f gives m_f f̄f plus (m_f/v) h f̄f, a dimensionless coupling of 0.70 for the top and 2.1 × 10⁻⁶ for the electron. On log axes of vertex factor against mass the fermion points m_f/v lie on a line of slope one and the boson points 2M_V²/v on a line of slope two, one common v fixing both intercepts. The ATLAS and CMS κ-fits place t, b, τ, W and Z on those lines to about ten per cent, see the muon at the three-σ level, bound the charm to within a few times its predicted value, and say nothing direct about e, u or d.

A Dirac mass m f̄_L f_R is forbidden before breaking: f_L sits in an SU(2) doublet with one hypercharge and f_R in a singlet with another, so the product is not gauge invariant. The doublet Φ is what repairs it. Take Q_L with Y = 1/3, Φ with Y = 1 and d_R with Y = −2/3: Q̄_L Φ d_R is an SU(2) singlet with hypercharge −1/3 + 1 − 2/3 = 0. The up quark needs the conjugate doublet Φ̃ = iσ₂Φ*, which carries Y = −1: −1/3 − 1 + 4/3 = 0 again. Leptons use L̄_L Φ e_R with 1 + 1 − 2 = 0. These three structures, each multiplied by an arbitrary complex 3 × 3 matrix Y^f in generation space, are the only renormalisable gauge-invariant terms that pair left with right, and there is no ν_R in the model to pair with, so no neutrino mass appears. Setting Φ to its vacuum value leaves mass matrices M^f = Y^f v/√2 that are in general neither Hermitian nor symmetric: eighteen real numbers in each sector, with no principle fixing any of them.

Because M^f is a general complex matrix, its eigenvalues are not its masses and one similarity transformation will not do. The right object is the singular value decomposition M = U_L D U_R†, with U_L and U_R unitary and D real, non-negative and diagonal: U_L is the eigenvector matrix of the Hermitian, positive M M†, U_R that of M†M, and both share the eigenvalues mᵢ². The two chiralities rotate independently, f_L′ = U_L† f_L and f_R′ = U_R† f_R, which is legitimate because the kinetic terms f̄_L iγ^μ∂_μ f_L and f̄_R iγ^μ∂_μ f_R are each invariant under any unitary rotation of their own field. In practice you call numpy.linalg.svd(M), which returns U_L, the singular values and U_R†, and the whole verification is checking that U_L D U_R† reproduces M to machine precision. The 2 × 2 toy in the worked examples shows the mechanism by hand: a non-symmetric matrix with entries near 5 and 90 MeV returns m_d = 4.70 MeV, mₛ = 93.0 MeV and a left rotation of 13.0°. Should a negative eigenvalue appear, it is absorbed as a sign in U_R and the mass is still the positive singular value.

Rotating to the mass basis is harmless everywhere except one vertex. A neutral current f̄_L γ^μ f_L becomes f̄_L′ U_L† U_L γ^μ f_L′ = f̄_L′ γ^μ f_L′, diagonal again because U_L is unitary; the same holds for right-handed fields and for the couplings to Z, photon and gluon. That is the GIM mechanism: no tree-level flavour change in neutral currents. The coupling to h stays diagonal for a different reason: it is the mass matrix itself divided by v, so the same two unitaries that diagonalise M diagonalise it too. The charged current ū_L γ^μ d_L mixes two sectors rotated by different matrices, so it becomes ū_L′ (U_L^u† U_L^d) γ^μ d_L′, and V_CKM = U_L^u† U_L^d survives as the misalignment between the up and down left-handed rotations. U_R^u and U_R^d never appear, since the W couples only to left-handed fields, and in the lepton sector the massless neutrino can simply be redefined by U_L^e, so no leptonic mixing exists in the model as written, although neutrino oscillations show that real neutrinos have mass and do mix. A 3 × 3 unitary matrix has nine real parameters; rephasing the six quark fields removes five, one overall phase being invisible, leaving three angles and one phase — Wolfenstein's λ = 0.225, A = 0.83, ρ̄ = 0.16 and η̄ = 0.35 — and that single phase is the only source of CP violation in the quark sector.

Tally what this one Lagrangian term costs. Y^u and Y^d together hold 36 real numbers; the freedom to rotate Q_L, u_R and d_R by U(3)³ removes 27 minus the single unbroken baryon-number phase, leaving 36 − 26 = 10 physical parameters: six quark masses, three CKM angles and one phase. The lepton sector adds three charged masses, so thirteen of the Standard Model's nineteen parameters are Yukawa data, every one fitted and none derived; the Higgs mass itself supplies λ rather than following from it. What the model does predict, and what is actually tested, is that every coupling of h is the corresponding mass over one common v, that V_CKM is unitary, and that neutral currents are flavour-diagonal. Even that test is partial: the couplings to t, b, τ, W and Z are measured, the muon is emerging, the charm is loosely bounded, and for u, d and e the coupling to the Higgs has never been observed. Each first-generation Yukawa in the tables is a number inferred from a mass, not a measurement of the Higgs.