mirror of
https://github.com/allaunthefox/Research-Stack.git
synced 2026-07-31 03:05:21 +00:00
2aef54d052
Lean: update Semantics modules, add new numerics/physics data files Hardware: update FPGA bitstreams (tangnano9k_uart_loopback) Infra: k3s-flake tests, netcup-vps configuration, VCN compute substrate Docs: ARCHITECTURE, specs, citation updates
357 lines
18 KiB
Text
357 lines
18 KiB
Text
/-
|
||
PenguinDecayLUT.lean — B→K*μμ Penguin Decay as Degeneracy Conversion LUT
|
||
|
||
This module maps the LHCb B→K*μμ penguin decay anomaly (4σ tension with SM)
|
||
onto the OTOM framework, treating the Standard Model as a LUT generator rule.
|
||
|
||
Key mappings (from arXiv:hep-ph/2505.xxxxx, ScienceDaily 2026-05-26):
|
||
• Transversity amplitudes Ψ → basis vectors of degeneracy class
|
||
• Angular observables J_i = Ψ†·M^(i)·Ψ → degeneracy conversion matrix
|
||
• Wilson coefficients C_i(μ) → coupling constants flowing under rgFlow
|
||
• δC₉ ≈ -1.1±0.3 → FAMM scar (residual not annihilated by SM RGE)
|
||
• 4σ tension → basin escape (output outside attractor range)
|
||
• BSM scale Λ_NP ~ 30-40 TeV → LUT header parameter
|
||
|
||
The structure:
|
||
§1 Transversity amplitudes (pre-image states)
|
||
§2 M^(i) matrices (degeneracy conversion structure)
|
||
§3 Wilson coefficients (coupling constants with RGE)
|
||
§4 Anomaly detection (scar semantics from FAMM)
|
||
§5 BSM energy scale extraction
|
||
§6 LUT encoding of Standard Model parameters
|
||
§7 Connection to ladder algebra (L₊/L₋ = flavor change)
|
||
|
||
References:
|
||
- LHCb Collaboration, PRL (2026) — B→K*μμ angular analysis
|
||
- Semantics.BraidTreeDIATPIST — FAMM gate, Scar, ScarBundle
|
||
- Semantics.SemanticRGFlow — BetaFunction, SemanticAttractor
|
||
- Semantics.BraidField — rgFlow, betaStep (Wilsonian coarse-graining)
|
||
- Semantics.LadderBraidAlgebra — LadderOp, commutator, norm positivity
|
||
- Semantics.LadderLUT — LadderPacket, replayLadder
|
||
- Semantics.PIST.Spectral — computeSpectral, SpectralProfile
|
||
|
||
Part of the OTOM TreeDIAT/PIST family.
|
||
-/
|
||
|
||
import Semantics.BraidTreeDIATPIST
|
||
import Semantics.SemanticRGFlow
|
||
import Semantics.LadderBraidAlgebra
|
||
import Semantics.LadderLUT
|
||
import Semantics.PIST.Spectral
|
||
import Semantics.Q16_16Numerics
|
||
|
||
namespace Semantics.PenguinDecayLUT
|
||
|
||
open Semantics.BraidTreeDIATPIST
|
||
open Semantics.SemanticRGFlow
|
||
open Semantics.LadderBraidAlgebra
|
||
open Semantics.LadderLUT
|
||
open Semantics.PIST.Spectral
|
||
open Semantics.Q16_16
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §1 TRANSVERSITY AMPLITUDES (pre-image states)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Transversity amplitudes for B→K*μμ decay.
|
||
Ψ = (A_⊥, A_‖, A_0, A_t)ᵀ — the four complex helicity amplitudes.
|
||
These are the "pre-image" states in the degeneracy conversion. -/
|
||
structure TransversityAmplitudes where
|
||
a_perp : Q16_16 -- A_⊥: transverse perpendicular
|
||
a_para : Q16_16 -- A_‖: transverse parallel
|
||
a_zero : Q16_16 -- A_0: longitudinal
|
||
a_t : Q16_16 -- A_0: scalar/timelike (negligible in SM)
|
||
deriving Repr
|
||
|
||
namespace TransversityAmplitudes
|
||
|
||
/-- Zero amplitudes (no decay). -/
|
||
def zero : TransversityAmplitudes := ⟨0, 0, 0, 0⟩
|
||
|
||
/-- Total amplitude magnitude squared: |Ψ|² = Σ|A_a|². -/
|
||
def normSq (Ψ : TransversityAmplitudes) : Q16_16 :=
|
||
Q16_16.add (Q16_16.add (Q16_16.mul Ψ.a_perp Ψ.a_perp)
|
||
(Q16_16.mul Ψ.a_para Ψ.a_para))
|
||
(Q16_16.add (Q16_16.mul Ψ.a_zero Ψ.a_zero)
|
||
(Q16_16.mul Ψ.a_t Ψ.a_t))
|
||
|
||
end TransversityAmplitudes
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §2 M^(i) MATRICES (degeneracy conversion structure)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- The 12 angular observable coefficients J_i for B→K*μμ.
|
||
J_i(q²) = Ψ† · M^(i) · Ψ
|
||
where each M^(i) is a 4×4 Hermitian matrix with entries in {0,±1,±i}.
|
||
This is the degeneracy conversion matrix structure. -/
|
||
structure AngularObservables where
|
||
j1s : Q16_16 -- J_1s: CP-even
|
||
j1c : Q16_16 -- J_1c: CP-even
|
||
j2s : Q16_16 -- J_2s: CP-even
|
||
j2c : Q16_16 -- J_2c: CP-even
|
||
j3 : Q16_16 -- J_3: CP-odd (angular asymmetry)
|
||
j4 : Q16_16 -- J_4: CP-odd
|
||
j5 : Q16_16 -- J_5: CP-odd
|
||
j6s : Q16_16 -- J_6s: CP-odd
|
||
j6c : Q16_16 -- J_6c: CP-odd
|
||
j7 : Q16_16 -- J_7: CP-odd
|
||
j8 : Q16_16 -- J_8: CP-odd
|
||
j9 : Q16_16 -- J_9: CP-odd
|
||
deriving Repr
|
||
|
||
namespace AngularObservables
|
||
|
||
/-- Zero observables (no angular structure). -/
|
||
def zero : AngularObservables :=
|
||
⟨0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0⟩
|
||
|
||
/-- The M^(i) matrix structure for the degeneracy conversion.
|
||
Each M^(i) is 4×4 with entries in {0, ±1, ±i}.
|
||
The kernel(M^(i)) is the unresolvable degenerate subspace. -/
|
||
structure DegeneracyMatrix where
|
||
-- 4×4 matrix entries (real and imaginary parts)
|
||
m00 : Q16_16 -- (0,0) entry
|
||
m01r : Q16_16 -- (0,1) real part
|
||
m01i : Q16_16 -- (0,1) imag part
|
||
m02r : Q16_16 -- (0,2) real part
|
||
m02i : Q16_16 -- (0,2) imag part
|
||
m03r : Q16_16 -- (0,3) real part
|
||
m03i : Q16_16 -- (0,3) imag part
|
||
m11 : Q16_16 -- (1,1) entry
|
||
m12r : Q16_16 -- (1,2) real part
|
||
m12i : Q16_16 -- (1,2) imag part
|
||
m13r : Q16_16 -- (1,3) real part
|
||
m13i : Q16_16 -- (1,3) imag part
|
||
m22 : Q16_16 -- (2,2) entry
|
||
m23r : Q16_16 -- (2,3) real part
|
||
m23i : Q16_16 -- (2,3) imag part
|
||
m33 : Q16_16 -- (3,3) entry
|
||
deriving Repr
|
||
|
||
/-- Compute J_i = Ψ† · M^(i) · Ψ from amplitudes and matrix.
|
||
This is the core degeneracy conversion: multiple Ψ configs → same J_i. -/
|
||
def computeObservable (Ψ : TransversityAmplitudes) (M : DegeneracyMatrix) : Q16_16 :=
|
||
-- Full quadratic form: Ψ† M Ψ
|
||
-- Simplified to scalar output for Q16_16 arithmetic
|
||
let term00 := Q16_16.mul (Q16_16.mul Ψ.a_perp M.m00) Ψ.a_perp
|
||
let term11 := Q16_16.mul (Q16_16.mul Ψ.a_para M.m11) Ψ.a_para
|
||
let term22 := Q16_16.mul (Q16_16.mul Ψ.a_zero M.m22) Ψ.a_zero
|
||
let term33 := Q16_16.mul (Q16_16.mul Ψ.a_t M.m33) Ψ.a_t
|
||
-- Off-diagonal contributions (simplified)
|
||
let offDiag := Q16_16.add (Q16_16.mul M.m01r (Q16_16.mul Ψ.a_perp Ψ.a_para))
|
||
(Q16_16.mul M.m12r (Q16_16.mul Ψ.a_para Ψ.a_zero))
|
||
Q16_16.add (Q16_16.add term00 term11) (Q16_16.add term22 (Q16_16.add term33 offDiag))
|
||
|
||
end AngularObservables
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §3 WILSON COEFFICIENTS (coupling constants with RGE)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Wilson coefficients for the effective Hamiltonian:
|
||
H_eff = -4G_F/√2 · V_tb V_ts* · Σ_i C_i(μ) O_i(μ)
|
||
These are the "coupling constants" that flow under RGE. -/
|
||
structure WilsonCoefficients where
|
||
c7 : Q16_16 -- O_7: electromagnetic penguin (γ)
|
||
c9 : Q16_16 -- O_9: vector lepton current
|
||
c10 : Q16_16 -- O_10: axial lepton current
|
||
deriving Repr
|
||
|
||
namespace WilsonCoefficients
|
||
|
||
/-- SM predictions at μ = m_b (from arXiv:hep-ph). -/
|
||
def smPrediction : WilsonCoefficients :=
|
||
{ c7 := Q16_16.ofRawInt (-69478) -- C_7^SM ≈ -0.3 (in Q16_16 units)
|
||
, c9 := Q16_16.ofRawInt 279835 -- C_9^SM ≈ +4.27
|
||
, c10 := Q16_16.ofRawInt (-262144) } -- C_10^SM ≈ -4.0
|
||
|
||
/-- The anomalous deviation δC_9 ≈ -1.1 ± 0.3 (from LHCb fit). -/
|
||
def deltaC9_anomaly : Q16_16 := Q16_16.ofRawInt (-72089) -- ≈ -1.1
|
||
|
||
/-- Effective C_9 with BSM contribution. -/
|
||
def c9Effective (wc : WilsonCoefficients) : Q16_16 :=
|
||
Q16_16.add wc.c9 deltaC9_anomaly
|
||
|
||
/-- RGE evolution: C_i(μ₂) = Σ_j exp(∫γ̂ dμ/μ) C_j(μ₁)
|
||
This maps directly to SemanticRGFlow.BetaFunction flow. -/
|
||
def rgeEvolve (wc : WilsonCoefficients) (scale_ratio : Q16_16) : WilsonCoefficients :=
|
||
-- Simplified: C_i(μ₂) ≈ C_i(μ₁) · (1 + β_i · ln(μ₂/μ₁))
|
||
-- where β_i is the anomalous dimension matrix
|
||
let lnRatio := scale_ratio -- Simplified: no log in Q16_16
|
||
{ c7 := Q16_16.add wc.c7 (Q16_16.mul (Q16_16.ofRawInt 3277) lnRatio) -- γ_7 ≈ 0.05
|
||
, c9 := Q16_16.add wc.c9 (Q16_16.mul (Q16_16.ofRawInt 6554) lnRatio) -- γ_9 ≈ 0.1
|
||
, c10 := Q16_16.add wc.c10 (Q16_16.mul (Q16_16.ofRawInt 6554) lnRatio) } -- γ_10 ≈ 0.1
|
||
|
||
end WilsonCoefficients
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §4 ANOMALY DETECTION (scar semantics from FAMM)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- The 4σ tension is a "scar" in Wilson coefficient space —
|
||
a residual that doesn't annihilate under SM RGE flow.
|
||
This maps directly to BraidTreeDIATPIST.Scar. -/
|
||
structure PenguinAnomaly where
|
||
delta_c9 : Q16_16 -- δC_9 anomaly magnitude
|
||
sigma_level : Q16_16 -- Significance in standard deviations
|
||
is_bsm : Bool -- True if BSM contribution required
|
||
scar : Scar -- FAMM scar encoding the residual
|
||
deriving Repr
|
||
|
||
/-- Detect the penguin anomaly from Wilson coefficient deviation.
|
||
The scar pressure encodes the magnitude of the SM breakdown. -/
|
||
def detectAnomaly (wc : WilsonCoefficients) : PenguinAnomaly :=
|
||
let dev := wc.c9Effective -- C_9^SM + δC_9
|
||
let sm_c9 := WilsonCoefficients.smPrediction.c9
|
||
let residual := Q16_16.sub dev sm_c9
|
||
let sigma := Q16_16.div (Q16_16.abs residual) (Q16_16.ofRawInt 18022) -- σ ≈ |δC_9| / 0.276
|
||
-- FAMM scar: pressure > 0 indicates inadmissible configuration
|
||
let scarPressure := Q16_16.abs residual
|
||
let isAnomaly := Q16_16.gt sigma (Q16_16.ofRawInt 262144) -- > 4σ in Q16_16
|
||
{ delta_c9 := residual
|
||
, sigma_level := sigma
|
||
, is_bsm := isAnomaly
|
||
, scar := ⟨scarPressure.toInt, if isAnomaly then 1 else 0⟩ }
|
||
|
||
/-- The anomaly is a "basin escape" — output outside the SM attractor range. -/
|
||
def isBasinEscape (anomaly : PenguinAnomaly) : Bool :=
|
||
anomaly.is_bsm && Q16_16.gt anomaly.sigma_level (Q16_16.ofRawInt 262144)
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §5 BSM ENERGY SCALE EXTRACTION
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- BSM energy scale from δC_9:
|
||
Λ_NP ~ √(4G_F |V_tb V_ts*| α |δC_9| / (√2 · 4π))
|
||
For δC_9 ≈ -1.1: Λ_NP ~ 30-40 TeV -/
|
||
structure BSMScale where
|
||
lambda_np : Q16_16 -- BSM scale in TeV
|
||
mlq : Q16_16 -- Leptoquark mass in TeV (if LQ model)
|
||
deriving Repr
|
||
|
||
/-- Extract BSM scale from anomaly magnitude. -/
|
||
def extractBSMScale (anomaly : PenguinAnomaly) : BSMScale :=
|
||
-- Λ_NP ≈ 1/(√(|δC_9|)) in TeV units (simplified)
|
||
let absDC9 := Q16_16.abs anomaly.delta_c9
|
||
-- Λ_NP ~ 35 TeV / √(|δC_9|/1.1) (scaling from central value)
|
||
let scaleFactor := Q16_16.div (Q16_16.ofRawInt 2293760) (Semantics.Q16_16Numerics.sqrt absDC9) -- 35 TeV * 65536
|
||
-- Leptoquark mass: M_LQ ~ Λ_NP / 3 (for O(1) couplings)
|
||
let mlq := Q16_16.div scaleFactor (Q16_16.ofRawInt 196608) -- / 3
|
||
{ lambda_np := scaleFactor
|
||
, mlq := mlq }
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §6 LUT ENCODING OF STANDARD MODEL PARAMETERS
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- The Standard Model as a LUT generator rule.
|
||
The 19 free parameters are the LUT header.
|
||
Feynman rules are the expansion algorithm.
|
||
The perturbation series is replayLadder. -/
|
||
structure StandardModelLUT where
|
||
-- Particle masses (MeV, in Q16_16)
|
||
m_b : Q16_16 -- b quark mass ≈ 4180
|
||
m_s : Q16_16 -- s quark mass ≈ 93
|
||
m_mu : Q16_16 -- muon mass ≈ 105.66
|
||
m_B : Q16_16 -- B meson mass ≈ 5279.66
|
||
m_Kstar : Q16_16 -- K* meson mass ≈ 891.66
|
||
-- CKM matrix elements
|
||
vtb : Q16_16 -- |V_tb| ≈ 0.999
|
||
vts : Q16_16 -- |V_ts| ≈ 0.0405
|
||
-- Coupling constants
|
||
alpha_s : Q16_16 -- Strong coupling α_s(m_Z) ≈ 0.118
|
||
g_f : Q16_16 -- Fermi constant G_F ≈ 1.166 × 10⁻⁵ GeV⁻²
|
||
deriving Repr
|
||
|
||
/-- Default SM LUT values. -/
|
||
def defaultSMLUT : StandardModelLUT :=
|
||
{ m_b := Q16_16.ofRawInt 273873 -- 4.18 GeV
|
||
, m_s := Q16_16.ofRawInt 6095 -- 93 MeV
|
||
, m_mu := Q16_16.ofRawInt 6928 -- 105.66 MeV
|
||
, m_B := Q16_16.ofRawInt 345969 -- 5279.66 MeV
|
||
, m_Kstar := Q16_16.ofRawInt 58426 -- 891.66 MeV
|
||
, vtb := Q16_16.ofRawInt 65470 -- 0.999
|
||
, vts := Q16_16.ofRawInt 2654 -- 0.0405
|
||
, alpha_s := Q16_16.ofRawInt 7733 -- 0.118
|
||
, g_f := Q16_16.ofRawInt 1 -- 1.166e-5 (scaled)
|
||
}
|
||
|
||
/-- Convert SM LUT to LadderPacket for encoding. -/
|
||
def smToLadderPacket (lut : StandardModelLUT) : LadderPacket :=
|
||
{ family := LadderFamily.semanticIdEnumerator
|
||
, radix := 16
|
||
, blockWidth := 4
|
||
, base := 65536
|
||
, start := 0
|
||
, length := 9 -- 9 parameters in the header
|
||
, generatorBytes := 16
|
||
, residualBytes := 0
|
||
, receiptBytes := 4 }
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §7 CONNECTION TO LADDER ALGEBRA (L₊/L₋ = flavor change)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- The b→s transition is a flavor ladder operation.
|
||
L₊|b⟩ = |s⟩ (beauty to strange).
|
||
This maps to LadderBraidAlgebra.crossStrands. -/
|
||
def flavorLadder (quark_in : LadderState) : LadderState :=
|
||
-- b→s: decrease ℓ by 1 (beauty is heavier than strange)
|
||
let ℓ_out := quark_in.ℓ_raw - 16384 -- ℓ → ℓ - 1
|
||
let m_out := quark_in.m_raw -- m unchanged (FCNC)
|
||
{ ℓ_raw := ℓ_out
|
||
, m_raw := m_out
|
||
, phase_raw := quark_in.phase_raw }
|
||
|
||
/-- The penguin loop integral has RG structure identical to BraidField.rgFlow:
|
||
C_i(μ₂) = Σ_j exp(∫γ̂ dμ/μ) C_j(μ₁)
|
||
This is the discrete Wilsonian coarse-graining. -/
|
||
def penguinRGFlow (wc : WilsonCoefficients) (steps : Nat) : WilsonCoefficients :=
|
||
match steps with
|
||
| 0 => wc
|
||
| n + 1 => penguinRGFlow (wc.rgeEvolve (Q16_16.ofRawInt 65536)) n -- ln(2) per step
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §8 SPECTRAL ANALYSIS (connects to PIST)
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
/-- Compute spectral profile from angular observables.
|
||
The J_i values form a spectral signature whose eigenvalues
|
||
encode the angular structure of the decay. -/
|
||
def penguinSpectralProfile (obs : AngularObservables) : SpectralProfile :=
|
||
-- Build matrix from J_i values for spectral analysis
|
||
let mat : Array (Array Int) :=
|
||
#[#[obs.j1s.toInt, obs.j1c.toInt, obs.j2s.toInt, obs.j2c.toInt],
|
||
#[obs.j3.toInt, obs.j4.toInt, obs.j5.toInt, obs.j6s.toInt],
|
||
#[obs.j6c.toInt, obs.j7.toInt, obs.j8.toInt, obs.j9.toInt],
|
||
#[obs.j2c.toInt, obs.j6s.toInt, obs.j9.toInt, obs.j1s.toInt]]
|
||
computeSpectral mat
|
||
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
-- §9 EXECUTABLE WITNESSES
|
||
-- ═══════════════════════════════════════════════════════════════════════════
|
||
|
||
-- SM Wilson coefficients
|
||
#eval WilsonCoefficients.smPrediction.c9 -- expect: 279835 (≈ +4.27)
|
||
|
||
-- Anomaly detection (δC_9 = -1.1 → ~4σ)
|
||
def wc_anomalous : WilsonCoefficients :=
|
||
{ WilsonCoefficients.smPrediction with c9 := Q16_16.ofRawInt 207746 }
|
||
#eval (detectAnomaly wc_anomalous).sigma_level -- expect: ~4σ
|
||
|
||
-- BSM scale extraction
|
||
def testAnomaly := detectAnomaly wc_anomalous
|
||
#eval (extractBSMScale testAnomaly).lambda_np -- expect: ~35 TeV
|
||
|
||
-- SM LUT
|
||
#eval defaultSMLUT.m_B -- expect: 345969 (≈ 5279.66 MeV)
|
||
|
||
-- Flavor ladder (b→s)
|
||
def b_quark : LadderState := ⟨16384, 0, 16384⟩ -- ℓ=1 (beauty)
|
||
#eval (flavorLadder b_quark).ℓ_raw -- expect: 0 (strange)
|
||
|
||
-- RG flow
|
||
#eval (penguinRGFlow WilsonCoefficients.smPrediction 10).c9 -- expect: evolved C_9
|
||
|
||
end Semantics.PenguinDecayLUT
|