Complex Yukawa couplings, the Jarlskog invariant, and the concurrence of baryogenesis and CP violation
Derivation record D3 — extracted from STF from First Principles V7.9, Appendix R. Companion to the framework backbone, First Principles V8.1 (The Two-Clock Theory).
Z. Paz — EXISTS | HAPPENS project — August 2026 · V1.0
ORCID: https://orcid.org/0009-0003-1690-3669
This paper is the derivation record for the framework’s CP-violation mechanism, extracted unchanged from STF from First Principles V7.9 Appendix R. It is published standalone because it is the connective tissue between the Standard Model constants derivation (baryogenesis) and the flavour-sector papers, and belongs to neither.
Untouched by the August 2026 two-clock revision: no dependence on the flyby, the curvature norm, the ghost-freedom proof, or the activation threshold’s normalization. Its status is derived within the compactification sector, conditional on the CICY #7447/ℤ₁₀ choice and on the moduli-mass formula. The resonance-window argument depends on the Weil–Petersson computation published as its companion derivation paper.
Notation note. Throughout Appendices R and S, Θ denotes the Weil-Petersson scalar curvature of the complex-structure moduli space (defined precisely in R.5 and S.2). This is distinct from the expansion scalar Θ = ∇_μu^μ_φ used in Appendix C.6.3, which is a kinematic quantity of the scalar-field congruence (equal to 3H on an FLRW background). The two symbols refer to different objects in different physical contexts; no equation in this appendix or Appendix S involves the expansion scalar.
This appendix derives the origin of CP violation in the STF framework. The mechanism is a direct extension of the baryogenesis result in Appendix K.8: the same oscillating scalar field φ_S that sources the baryon asymmetry also drives a phase lag in the five complex-structure moduli z_α (Appendix Q), freezing a CP-odd component into the Yukawa coupling matrix. The two effects are concurrent — one resonant epoch, two Standard Model outputs.
Appendix K.8 derives the baryon asymmetry via a phase lag mechanism:
\[ \chi_{R} ( \omega ) = \frac{1}{\omega_{0}^{2} - \omega^{2} - i \Gamma_{R} \omega} , \quad \quad \delta_{R} ( \omega ) = a r c t a n \left( \frac{\Gamma_{R} \, \omega}{\omega_{0}^{2} - \omega^{2}} \right) \]
At resonance ω → ω₀, the phase lag δ_R → π/2, giving η_b = (π/2)(α/10)³ = 6.10×10⁻¹⁰ (99.74% of observed 6.12×10⁻¹⁰).
K.11 identifies the quantities that minimal STF does not address: quark mass hierarchies, CKM mixing, and CP violation, all requiring the complex-structure moduli z_α. Appendix Q establishes that exactly 5 such moduli survive the Z₁₀ quotient. Appendix R derives the CP violation mechanism from those 5 moduli using the same phase lag structure as K.8.
The full Kähler potential of the CICY #7447/Z₁₀ compactification is:
\[ K = - 3 \ln \left( T + \bar{T} \right) - \ln \left\lbrack \int_{X} \Omega ( z ) \land \bar{\Omega} \left( \bar{z} \right) \right\rbrack = - 6 \sigma - 3 \ln 2 - K_{\mathrm{cs}} \left( z , \bar{z} \right) , \qquad \mathrm{Re}\,T \equiv e^{2 \sigma} \]
Notation. Here σ is the logarithmic breathing coordinate, defined by Re T = e^{2σ}, so that −3 ln(T+T̄) = −6σ − 3 ln 2; it is distinct from the linear modulus Re T itself. Earlier versions of this appendix wrote −6σ without the constant and without stating this definition, which read as −3 ln(2σ) — the two coordinates share a symbol only through this relation.
where T = σ + iθ is the volume modulus (Re T = σ, with e^{6σ} = Vol(X)) and K_cs is the Weil-Petersson Kähler potential on complex structure moduli space.
The GVW superpotential is:
\[ W = \int_{X} G_{3} \land \Omega ( z ) = n^{a} \Pi_{a} ( z ) \]
where nᵃ are integer flux quanta and Π_a(z) are periods of the holomorphic 3-form Ω(z).
The F-term scalar potential contains the cross-derivative:
\[ \frac{\partial^{2} V}{\partial \left( \text{Re} \, T \right) \, \partial z_{\alpha}} \neq 0 \]
because W depends on both σ = Re T (through e^K) and z_α (through the periods Π_a(z)). This coupling is a direct consequence of the Kähler potential — no additional assumption is made.
The STF field φ_S is the canonically normalised volume modulus: φ_S = √24 M_Pl · σ = √24 M_Pl · Re T (Appendix L.3.2). Therefore ∂/∂φ_S = (√24 M_Pl)⁻¹ · ∂/∂(Re T), and the cross-derivative in terms of φ_S is:
\[ \frac{\partial^{2} V}{\partial \phi_{S} \, \partial z_{\alpha}} = \frac{1}{\sqrt{24} \, M_{\mathrm{Pl}}} \cdot \frac{\partial^{2} V}{\partial \left( \text{Re} \, T \right) \, \partial z_{\alpha}} \neq 0 \]
The non-vanishing is preserved under this rescaling, and the prefactor (√24 M_Pl)⁻¹ is absorbed into the effective coupling coefficient on the right-hand side of the driven EOM (V6.0) in R.3. The 5 moduli z_α are sourced whenever φ_S oscillates.
The volume modulus φ_S oscillates at the reheating epoch with amplitude A and frequency ω = m_s:
\[ \delta \phi_{S} ( t ) = A \cos \left( m_{s} t \right) \]
The driven equation of motion for the complex structure modulus z_α, linearised around the vacuum z₀, is:
\[ \boxed{\delta \overset{¨}{z}_{\alpha} + 3 H \, \delta \dot{z}_{\alpha} + m_{z}^{2} \, \delta z_{\alpha} = \left. \frac{\partial^{2} V}{\partial \phi_{S} \, \partial z_{\alpha}} \right|_{z_{0}} \cdot A \cos \left( m_{s} t \right)} \]
The steady-state solution is:
\[ \delta z_{\alpha} ( t ) = \left| \chi_{z} \left( m_{s} \right) \right| \cdot \left. \frac{\partial^{2} V}{\partial \phi_{S} \, \partial z_{\alpha}} \right|_{z_{0}} \cdot A \cos \left( m_{s} t - \delta_{z} \right) \]
with susceptibility χ_z(ω) = 1/(m_z² − ω² − i·3H·ω) and phase lag:
\[ \boxed{\delta_{z} \left( m_{s} \right) = a r c t a n \left( \frac{3 H \, m_{s}}{m_{z}^{2} - m_{s}^{2}} \right)} \]
This is structurally identical to the K.8 formula with the replacement Γ_R → 3H. The same physics — driven harmonic oscillator with damping — produces the same phase lag. At exact resonance m_z = m_s:
\[ \delta_{z} \rightarrow \frac{\pi}{2} \]
Phase lag verification (computed):
| H/m_z | δ_z near resonance (ω = 0.9999 m_z) |
|---|---|
| 0.1 | 89.96° |
| 0.01 | 89.62° |
| 0.001 | 86.19° |
| 0.0001 | 56.31° |
At exact resonance (ω = m_z): δ_z → 90° analytically for all H > 0.
The holomorphic Yukawa coupling of quarks in the heterotic compactification is:
\[ Y_{\mathrm{ij}} \left( z_{\alpha} \right) = \int_{X} \Omega ( z ) \land A_{i} \land A_{j} \]
where A_i are bundle-valued (0,1)-forms representing the quark wavefunctions. Expanding around the vacuum z_0:
\[ Y_{\mathrm{ij}} ( t ) = Y_{\mathrm{ij}}^{( 0 )} + \sum_{\alpha = 1}^{5} \left. \frac{\partial Y_{\mathrm{ij}}}{\partial z_{\alpha}} \right|_{z_{0}} \cdot \left| \delta z_{\alpha} \right| \cdot \cos \left( m_{s} t - \delta_{z} \right) \]
The sum runs over exactly 5 terms — a direct consequence of the character decomposition in Appendix Q.3. The 40 non-invariant moduli do not appear because they are projected out by the Z₁₀ orbifold.
Note on Z₅ decoupling at the symmetric locus. At the Z₁₀-symmetric locus z₂=…=z₅=0, and with the stabilising flux n*=(−247,−266,0,−3) lying in H³(X̃,ℤ)^{Z₁₀} (so the full background preserves Z₅), the cross-derivatives ∂²V/∂φ_S∂z_α vanish for α=2,…,5 by symmetry: φ_S is Z₅-invariant while z_α (α>1) transforms with phase ω^(α−1), making a linear coupling term φ_S z_α non-invariant and therefore forbidden. This decoupling is symmetry-exact — not a leading-order approximation — since the flux by construction lies in the Z₁₀-invariant sublattice and loop corrections cannot generate Z₅-violating terms when the symmetry is exact. At the physical vacuum φ_res≈0.420, only z₁ contributes to f; the contributions of z₂,…,z₅ vanish identically and would only reappear if the background broke Z₅, which it does not.
Frozen CP-odd component. At the epoch when φ_S oscillation ceases (H drops below m_z), the phase-lagged displacement δz_α freezes at its current value. The imaginary part of the Yukawa matrix is:
\[ \boxed{I m \left( Y_{\mathrm{ij}} \right) = - \sum_{\alpha = 1}^{5} \left. \frac{\partial Y_{\mathrm{ij}}}{\partial z_{\alpha}} \right|_{z_{0}} \cdot \left| \delta z_{\alpha} \right|_{\mathrm{frozen}} \cdot \sin \left( \delta_{z} \right)} \]
At resonance (δ_z = π/2, sin(δ_z) = 1):
\[ I m \left( Y_{\mathrm{ij}} \right) |_{\mathrm{res}} = - \sum_{\alpha = 1}^{5} \left. \frac{\partial Y_{\mathrm{ij}}}{\partial z_{\alpha}} \right|_{z_{0}} \cdot \left| \delta z_{\alpha} \right|_{\mathrm{max}} \]
Jarlskog invariant. From the Jarlskog construction J = Im(det[Y_u Y_u†, Y_d Y_d†]):
\[ \boxed{J \propto \sin^{2} \left( \delta_{z} \right) \times f \left( \left. \frac{\partial Y}{\partial z} \right|_{z_{0}} , \, | \delta z | \right)} \]
where f encodes the geometric factor from the Yukawa derivatives and moduli displacements. At resonance:
\[ J |_{\mathrm{res}} = f \]
Phase lag table — sin²(δ_z) vs ρ = m_z/m_s, evaluated at the freeze-out epoch H = m_z (computed):
(Each row gives the phase lag that freezes when the Hubble rate drops to H = m_z for that row’s ρ. Substituting H = m_z and ω = m_s into (V6.1): δ_z = arctan(3m_z · m_s/(m_z² − m_s²)) = arctan(3ρ/(ρ² − 1)).)
| ρ = m_z/m_s | δ_z | sin²(δ_z) | J/f |
|---|---|---|---|
| 1.0 (resonance) | 90.0° | 1.0000 | 1.0000 |
| 1.5 | 74.5° | 0.9284 | 0.9284 |
| 2.0 | 63.4° | 0.8000 | 0.8000 |
| 3.0 | 48.4° | 0.5586 | 0.5586 |
| 3.303 (boundary) | 45.0° | 0.5000 | 0.5000 |
| 5.0 | 32.0° | 0.2809 | 0.2809 |
| 10.0 | 16.9° | 0.0841 | 0.0841 |
| 540.6 | 0.318° | 3.08×10⁻⁵ | 3.08×10⁻⁵ |
| 1.59×10¹⁰ (LVS) | ~0° | 3.56×10⁻²⁰ | 3.56×10⁻²⁰ |
Mass formula. The GVW F-term mass matrix for the z_α moduli at the critical point D_αW = 0 is:
\[ M^{2_{\alpha \bar{\beta}}} = m_{3 / 2}^{2} \times \Theta_{\alpha \bar{\beta}} \]
where Θ_{αβ̄} = g^{γδ̄}∂γ∂{δ̄}K_cs is the Weil-Petersson curvature tensor. At the Z₁₀-invariant locus z_α = z_0, the symmetry forces g_{αβ̄} = g·δ_{αβ̄} (all moduli are equivalent under the residual symmetry), giving:
\[ \boxed{m_{z} = m_{3 / 2} \times \sqrt{\Theta} , \quad \quad \Theta \equiv g^{\alpha \bar{\alpha}} \partial_{\alpha} \partial_{\bar{\alpha}} K_{\mathrm{cs}} |_{z_{0}}} \]
In the KKLT-type stabilization of Appendix O.4, the gravitino mass satisfies m_{3/2} ~ m_s: both are set by the same three-term flux superpotential, with m_{3/2} = e^{K/2}|W₀| and m_s² ∝ V’’(σ₀)/M²_Pl, related via the same W₀ and volume at the KKLT minimum [33]. Therefore:
\[ m_{z} = m_{s} \times \sqrt{\Theta} \]
Θ is pure geometry: it depends only on the period matrix of CICY #7447/Z₁₀ at the Z₁₀-invariant locus, and is computable via the Picard-Fuchs system of Appendix S. The flux integers nᵃ cancel in the ratio m_z/m_s.
LVS is excluded. The Large Volume Scenario would give m_z/m_s ~ √Vol = e^{3σ₀} = 1.59×10¹⁰ (computed from σ₀ = 7.83 of Appendix O.4). At this mass ratio:
\[ \sin^{2} \left( \delta_{z} \right)_{\mathrm{LVS}} \approx \left( \frac{3 m_{s}}{m_{z}} \right)^{2} = \frac{9}{\mathrm{Vol}} = 3.56 \times 10^{-20} \]
To achieve J = J_obs = 3.18×10⁻⁵ (PDG 2024), the geometric factor f would need to be f ~ 8.93×10¹⁴ — unphysically large. The factor f encodes holomorphic Yukawa derivatives ∂Y_ij/∂z_α and moduli displacements |δz_α|. The Yukawa couplings Y_ij = ∫_X Ω ∧ A_i ∧ A_j are integrals of a normalised holomorphic 3-form against bundle-valued (0,1)-forms over a compact CY3; in string units (where the manifold has volume O(1)), this gives Y_ij = O(1) and therefore ∂Y_ij/∂z_α = O(1) ([31]; [32]). The moduli displacement |δz_α| is bounded by the moduli space metric and is O(1) at resonance. Therefore f = O(1) is the natural expectation, and f ~ 10¹⁴ would require anomalously large period integrals with no geometric mechanism to produce them. LVS is excluded as a mechanism for CP violation. This is not an assumption: the Appendix O.4 potential is a three-term KKLT-type flux potential [33], and LVS requires additional α’ corrections to K not present in that minimal action. The framework has already committed to KKLT.
Resonance window derivation. Requiring sin²(δ_z) ≥ 1/2 (>50% CP transfer efficiency, sufficient for J_obs with O(1) geometric factor f):
\[ \arctan \left( \frac{3 \rho}{\rho^{2} - 1} \right) \geq 45 {^\circ} , \quad \rho \equiv m_{z} / m_{s} \]
\[ \Rightarrow \frac{3 \rho}{\rho^{2} - 1} \geq 1 \Rightarrow \rho^{2} - 3 \rho - 1 \leq 0 \Rightarrow \rho \leq \frac{3 + \sqrt{13}}{2} = 3.303 \]
\[ \boxed{\Theta \in \left\lbrack 1 , \ 10.9 \right\rbrack} \quad \Leftrightarrow \quad \rho = m_{z} / m_{s} \in \left\lbrack 1 , \ 3.30 \right\rbrack \quad \Leftrightarrow \quad \sin^{2} \left( \delta_{z} \right) \geq 0.50 \]
This is the resonance window: the range of WP curvature values for which the CP violation mechanism operates at ≥ 50% efficiency. Whether Θ(φ*) lands in this window is determined by the geometry of CICY #7447/Z₁₀ — see Appendix S.
V5.0 K.8 establishes that baryogenesis occurs at the epoch H_reh ~ m_s, when φ_S first enters resonance with the spacetime curvature oscillation. Appendix S shows that if Θ ~ O(1), then m_z ~ m_s, and the complex-structure moduli z_α also enter resonance at the same epoch.
The single epoch H ~ m_s therefore produces both Standard Model outputs simultaneously:
| Epoch H ~ m_s | Mechanism | Output |
|---|---|---|
| φ_S resonates with curvature R | K.8 baryogenesis phase lag δ_R | η_b = (π/2)(α/10)³ = 6.10×10⁻¹⁰ |
| φ_S drives z_α oscillation | Appendix R phase lag δ_z | Im(Y_ij) ≠ 0 → J ∝ sin²(δ_z) × f |
η_b / J ratio (computed):
\[ \frac{\eta_{b}}{J_{\mathrm{obs}}} = \frac{6.10 \times 10^{-10}}{3.18 \times 10^{-5}} = 1.92 \times 10^{-5} \]
(Using the PDG 2024 value J_obs = 3.18×10⁻⁵. The ratio is O(10⁻⁵) regardless of the ±5% uncertainty in J_obs between PDG editions.)
When f is computed from the Yukawa derivatives ∂Y_ij/∂z_α via the period matrix (Route B, Appendix S), this ratio becomes a parameter-free prediction of the framework.
The CP violation mechanism makes the following predictions, each independently testable:
| Θ value | ρ = m_z/m_s | sin²(δ_z) | Prediction |
|---|---|---|---|
| Θ ∈ [1, 10.9] | 1–3.30 | ≥ 0.50 | J = sin²(δ_z(Θ)) × f → full parameter-free prediction |
| Θ < 1 | < 1 | overdamped | No frozen CP phase; J ~ 0 (mechanism fails) |
| Θ > 10.9, Θ ≪ Vol | 3.30–√Vol | < 0.50 | J ~ (3/√Θ)² × f, sub-resonant suppression |
| Θ = Vol (LVS) | 1.59×10¹⁰ | 3.56×10⁻²⁰ | J ~ 10⁻²⁴ × f, excluded by analysis above |
Critical test: When Θ(φ*) is computed via Route B (Appendix S), it immediately determines which row applies. If Θ ∈ [1, 10.9] is confirmed, the framework predicts sin²(δ_z) to four significant figures, and J = sin²(δ_z) × f with f computed from the same period matrix. This is a fully predictive, parameter-free result.