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 V9.7.2 (The Two-Clock Theory and Its Conditional Gravitational Completion).
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. One framing correction carried from the current backbone: the phrase “zero free parameters” in this record reads as no fit within this calculation — the framework as a whole carries structural choices and open normalizations (V8.1 §III.E). 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.