/- SMEFTExtension.lean — Standard Model Effective Field Theory This module formalizes the SMEFT framework for extending the Standard Model to account for observed anomalies (muon g-2, B→K*μμ, W mass, etc.). The key equation: H_SMEFT = H_SM + Σ_i (C_i^(6)/Λ²) O_i^(6) + Σ_j (C_j^(8)/Λ⁴) O_j^(8) Where: H_SM = Standard Model Hamiltonian Λ = energy scale of new physics C_i = Wilson coefficients (tuning knobs) O_i = higher-dimensional operators (new interactions) References: - arXiv:2505.xxxxx — LHCb B→K*μμ penguin decay anomaly - Grzadkowski et al. (2010) — SMEFT operator basis - Brivio & Trott (2017) — SMEFT pedagogical review Part of the OTOM TreeDIAT/PIST family. -/ import Semantics.FixedPoint import Semantics.ForceModifiedArrhenius import Semantics.Q16_16Numerics namespace Semantics.SMEFTExtension open Semantics.Q16_16 open Semantics.ForceModifiedArrhenius -- ═══════════════════════════════════════════════════════════════════════════ -- §1 WILSON COEFFICIENTS -- ═══════════════════════════════════════════════════════════════════════════ /-- Wilson coefficients for dimension-6 operators. These are the "tuning knobs" that encode BSM physics. -/ structure WilsonCoefficients6 where C_1 : Q16_16 -- O_1: (L̄γ_μL)(L̄γ^μL) - same-flavor lepton current C_2 : Q16_16 -- O_2: (q̄γ_μq)(L̄γ^μL) - quark-lepton current C_3 : Q16_16 -- O_3: (q̄γ_μT^Aq)(L̄γ^μT^AL) - color-octet C_4 : Q16_16 -- O_4: (q̄γ_μq)(q̄γ^μq) - four-quark C_5 : Q16_16 -- O_5: (q̄q̄)(ll) - scalar four-fermion C_6 : Q16_16 -- O_6: (q̄q̄)(q̄q) - scalar four-quark C_7 : Q16_16 -- O_7: (ēγ_μe)(H†D^μH) - dipole-like C_8 : Q16_16 -- O_8: (q̄σ_μνq)(Hσ^μνH) - chromomagnetic C_9 : Q16_16 -- O_9: (q̄γ_μq)(ēγ^μe) - vector lepton current ← THE ANOMALOUS ONE C_10 : Q16_16 -- O_10: (q̄γ_μq)(ēγ^μγ₅e) - axial lepton current deriving Repr /-- SM values at μ = m_b (approximate). -/ def smWilson : WilsonCoefficients6 := { C_1 := Q16_16.ofRawInt (-196608) -- ~-3.0 , C_2 := Q16_16.ofRawInt 13107 -- ~0.2 , C_3 := Q16_16.zero , C_4 := Q16_16.ofRawInt (-65536) -- ~-1.0 , C_5 := Q16_16.zero , C_6 := Q16_16.zero , C_7 := Q16_16.ofRawInt (-69478) -- ~-0.3 , C_8 := Q16_16.ofRawInt (-45875) -- ~-0.7 , C_9 := Q16_16.ofRawInt 279835 -- ~+4.27 ← SM prediction , C_10 := Q16_16.ofRawInt (-262144) } -- ~-4.0 -- ═══════════════════════════════════════════════════════════════════════════ -- §2 BSM DEVIATIONS (what anomalies tell us) -- ═══════════════════════════════════════════════════════════════════════════ /-- The deviation in Wilson coefficients from BSM physics. -/ structure BSMDelta where delta_C7 : Q16_16 -- electromagnetic penguin shift delta_C9 : Q16_16 -- vector lepton current shift ← THE 4σ ANOMALY delta_C10 : Q16_16 -- axial lepton current shift deriving Repr /-- LHCb fit result: δC_9 ≈ -1.1 ± 0.3. -/ def lhcbAnomaly : BSMDelta := { delta_C7 := Q16_16.ofRawInt 0 -- consistent with 0 , delta_C9 := Q16_16.ofRawInt (-72089) -- ≈ -1.1 , delta_C10 := Q16_16.ofRawInt 0 } -- consistent with 0 /-- Effective Wilson coefficients with BSM contribution. -/ def effectiveWilson (sm : WilsonCoefficients6) (bsm : BSMDelta) : WilsonCoefficients6 := { sm with C_7 := Q16_16.add sm.C_7 bsm.delta_C7 C_9 := Q16_16.add sm.C_9 bsm.delta_C9 C_10 := Q16_16.add sm.C_10 bsm.delta_C10 } -- ═══════════════════════════════════════════════════════════════════════════ -- §3 ENERGY SCALE EXTRACTION -- ═══════════════════════════════════════════════════════════════════════════ /-- Extract BSM energy scale from Wilson coefficient deviation. Λ² ~ g²/(4G_F |V_tb V_ts*| α |δC_9|) For δC_9 ≈ -1.1: Λ ~ 30-40 TeV -/ def extractEnergyScale (bsm : BSMDelta) : Q16_16 := -- Λ ≈ 35 TeV / √(|δC_9|/1.1) (simplified) let abs_DC9 := Q16_16.abs bsm.delta_C9 let scale_35TeV := Q16_16.ofRawInt 2293760 -- 35 TeV in Q16_16 units Q16_16.div scale_35TeV (Semantics.Q16_16Numerics.sqrt abs_DC9) -- ═══════════════════════════════════════════════════════════════════════════ -- §4 OPERATOR STRUCTURE -- ═══════════════════════════════════════════════════════════════════════════ /-- The dimension-6 operators as field bilinears. O_i = (ψ̄₁Γψ₂)(ψ̄₃Γψ₄) where Γ are Dirac matrices. -/ structure Dimension6Operator where name : String fermion1 : String -- first fermion line fermion2 : String -- second fermion line dirac : String -- Dirac structure (V, A, S, P, T) color : String -- color structure (singlet, octet) deriving Repr /-- The key operators for B→K*μμ anomalies. -/ def relevantOps : Array Dimension6Operator := #[ ⟨"O_7", "s_bar", "b", "T", "singlet"⟩ , ⟨"O_9", "s_bar", "b", "V", "singlet"⟩ , ⟨"O_10", "s_bar", "b", "A", "singlet"⟩ ] -- ═══════════════════════════════════════════════════════════════════════════ -- §5 DECAY RATE COMPUTATION -- ═══════════════════════════════════════════════════════════════════════════ /-- Differential decay rate for B→K*μμ: dΓ/dq² ∝ |C_9^eff F_⊥ + (2m_b/q²) C_7 F_T|² + |C_10|² |F_⊥|² -/ def differentialRate (C9eff C7 C10 F_perp F_T mb q2 : Q16_16) : Q16_16 := let term1 := Q16_16.add C9eff (Q16_16.div (Q16_16.mul (Q16_16.ofRawInt 131072) mb) q2) let term2 := Q16_16.mul term1 F_perp let term3 := Q16_16.mul (Q16_16.div (Q16_16.mul (Q16_16.ofRawInt 131072) mb) q2) (Q16_16.mul C7 F_T) let amp_sq := Q16_16.add (Q16_16.mul (Q16_16.add term2 term3) (Q16_16.add term2 term3)) (Q16_16.mul (Q16_16.mul C10 C10) (Q16_16.mul F_perp F_perp)) amp_sq -- ═══════════════════════════════════════════════════════════════════════════ -- §6 LEPTOQUARK MODEL (BSM candidate) -- ═══════════════════════════════════════════════════════════════════════════ /-- Leptoquark parameters. A scalar leptoquark S₁ mediating b→sμμ. -/ structure Leptoquark where mass : Q16_16 -- M_LQ in TeV lambda_b : Q16_16 -- coupling to b quark lambda_s : Q16_16 -- coupling to s quark deriving Repr /-- Extract leptoquark mass from δC_9. M_LQ = √(π |λ_b λ_s*| / (α |V_tb V_ts*| |δC_9|)) -/ def extractLeptoquarkMass (lq : Leptoquark) (delta_C9 : Q16_16) : Q16_16 := let alpha := Q16_16.ofRawInt 7733 -- α ≈ 1/137 let vtb_vts := Q16_16.ofRawInt 2687 -- |V_tb V_ts*| ≈ 0.041 let pi := Q16_16.ofRawInt 205887 -- π in Q16_16 let numerator := Q16_16.mul pi (Q16_16.abs (Q16_16.mul lq.lambda_b lq.lambda_s)) let denominator := Q16_16.mul (Q16_16.mul alpha vtb_vts) (Q16_16.abs delta_C9) Semantics.Q16_16Numerics.sqrt (Q16_16.div numerator denominator) -- ═══════════════════════════════════════════════════════════════════════════ -- §7 EXECUTABLE WITNESSES -- ═══════════════════════════════════════════════════════════════════════════ -- SM Wilson coefficient C_9 #eval smWilson.C_9 -- expect: 279835 (≈ +4.27) -- Effective C_9 with BSM def testEff := effectiveWilson smWilson lhcbAnomaly #eval testEff.C_9 -- expect: 279835 + (-72089) = 207746 (≈ +3.17) -- Energy scale from anomaly #eval extractEnergyScale lhcbAnomaly -- expect: ~35 TeV / √1.0 = ~35 TeV -- Leptoquark mass (λ = 1, δC_9 = -1.1) def testLQ : Leptoquark := ⟨Q16_16.ofRawInt 65536, Q16_16.one, Q16_16.one⟩ #eval extractLeptoquarkMass testLQ (Q16_16.ofRawInt (-72089)) -- expect: ~1-10 TeV end Semantics.SMEFTExtension