"""v202 -- The rare kaon sector K -> pi nu nubar from the TFPT CKM point. The TFPT kernel fixes the FULL CKM matrix (magnitudes s12,s23,s13 + CP phase delta_CKM) without a flavor fit (v18/v88); feeding that exact point into the STANDARD short-distance formulas (Brod-Gorbahn-Stamou) gives two sharp, just-measured branching ratios. This is a downstream FLAVOR readout, NOT a compiler power: the short-distance functions X_t, P_c, kappa_pm are external electroweak/QCD input, so the branching ratios are typed [C] (a physical bridge), while the CKM point feeding them is [I]/[N] (already in the suite). [I] 1. TFPT CKM POINT (imported, v88): s12 = lambda_C = sqrt(phi0(1-phi0)), s23 = phi0/(1+lambda_C), s13 = lambda_C^3/3, delta = pi/3 + 3 lambda_C^2 = 1.19823 rad. The exact apex (rhobar,etabar) = (0.1374, 0.3509) and lambda_t = V_ts* V_td, lambda_c = V_cs* V_cd follow from the PDG matrix. [C] 2. BRANCHING RATIOS (external SD input): with X_t=1.462, P_c, dP_cu, Delta_EM, kappa_pm (Brod-Gorbahn-Stamou) the TFPT CKM point gives BR(K+ -> pi+ nu nubar) = 9.45e-11 and BR(K_L -> pi0 nu nubar) = 3.33e-11. NOT parameter-free (the SD functions are external), hence [C]. (Recomputed from the CURRENT CKM point; an earlier scaffold quoted BR(K_L)=3.47e-11 from a slip in Im(lambda_t)/lambda^5 -- corrected here.) [N] 3. NA62 CONSISTENCY: BR(K+) = 9.45e-11 lands essentially ON the NA62 full 2016-2024 combination (9.6 +1.9/-1.8)e-11 (La Thuile 2026), +0.08 sigma; the earlier 2016-2022 observation was (13.0 +3.3/-3.0)e-11 (~1.2 sigma above). A strong consistency hit for the CKM point -- but NOT a unique TFPT-vs-SM discriminator (the SM value (8.6 +- 0.4)e-11 also sits inside the NA62 error, so the channel confirms the CKM point rather than separating TFPT from the SM). [I] 4. GROSSMAN-NIR: BR(K_L)/BR(K+) = 0.352, safely inside the model- independent isospin corridor (<= 4.3). A measured ratio above 4.3 would signal new physics outside the SM operator basis (not a TFPT-specific kill, but it would void the matching ansatz). [X] 5. KILL TEST (dated, operational): a stable NA62 BR(K+) OUTSIDE [7,12]e-11, or a KOTO-II BR(K_L) incompatible with the GN-plane point, breaks the TFPT flavor bridge for this sector. The delta_CKM tension (~2 sigma, v88) makes the corridor genuinely discriminating. Python-only (numerical SD evaluation + external electroweak input; the CKM point it rests on is the already-mirrored v88 reading). """ import mpmath as mp from tfpt_constants import check, summary, reset, phi0, PI # standard short-distance input (Brod-Gorbahn-Stamou 2021; PDG-convention) X_T = mp.mpf('1.462') # Inami-Lim top function (NLO QCD + EW) DP_CU = mp.mpf('0.04') # long-distance charm-up interference DELTA_EM = mp.mpf('-0.003') # EM correction to K+ EPS_K = mp.mpf('2.228e-3') # indirect CP violation NA62_C, NA62_LO = mp.mpf('9.6e-11'), mp.mpf('1.8e-11') # NA62 full 2016-2024 combination # (9.6 +1.9/-1.8)e-11 (La Thuile 2026); historical 2016-2022 obs was (13.0 +3.3/-3.0)e-11 NA62_2022_C, NA62_2022_LO = mp.mpf('13.0e-11'), mp.mpf('3.0e-11') BR_SM = mp.mpf('8.6e-11') # SM (meson mixing); also inside NA62 error def ckm_point(): lam = mp.sqrt(phi0 * (1 - phi0)) s12, s23, s13 = lam, phi0 / (1 + lam), lam**3 / 3 c12, c23, c13 = mp.sqrt(1 - s12**2), mp.sqrt(1 - s23**2), mp.sqrt(1 - s13**2) delta = PI / 3 + 3 * lam**2 ed = mp.e**(1j * delta) V = { 'ud': c12 * c13, 'us': s12 * c13, 'ub': s13 * mp.e**(-1j * delta), 'cd': -s12 * c23 - c12 * s23 * s13 * ed, 'cs': c12 * c23 - s12 * s23 * s13 * ed, 'cb': s23 * c13, 'td': s12 * s23 - c12 * c23 * s13 * ed, 'ts': -c12 * s23 - s12 * c23 * s13 * ed, 'tb': c23 * c13, } return lam, s12, s23, s13, delta, V def run(): reset() mp.mp.dps = 30 print("v202 rare kaon K -> pi nu nubar from the TFPT CKM point ([C] downstream readout)") lam, s12, s23, s13, delta, V = ckm_point() # 1. the imported CKM point check("TFPT CKM POINT [I]: s12=lambda_C=%.5f, s23=phi0/(1+lambda_C)=%.5f " "(|V_cb|), s13=lambda_C^3/3=%.6f, delta=pi/3+3 lambda_C^2=%.5f rad " "(v88) -- fixed by the kernel, no flavor fit" % (float(s12), float(s23), float(s13), float(delta)), abs(s23 - mp.mpf('0.043428')) < 1e-5 and abs(delta - mp.mpf('1.19823')) < 1e-4) lam_t = mp.conj(V['ts']) * V['td'] lam_c = mp.conj(V['cs']) * V['cd'] apex = -(V['ud'] * mp.conj(V['ub'])) / (V['cd'] * mp.conj(V['cb'])) check("EXACT APEX [I]: (rhobar,etabar) = (%.4f, %.4f) from the full matrix " "(not the leading Wolfenstein approx); lambda_t = %.3e + i %.3e" % (float(apex.real), float(apex.imag), float(lam_t.real), float(lam_t.imag)), abs(apex.real - mp.mpf('0.1374')) < 2e-3 and abs(apex.imag - mp.mpf('0.3509')) < 2e-3) # 2. branching ratios from the standard SD formulas A = phi0 / (lam**2 * (1 + lam)) Pc = (mp.mpf('0.2255') / lam)**4 * mp.mpf('0.3604') kp = mp.mpf('5.173e-11') * (lam / mp.mpf('0.225'))**8 kL = mp.mpf('2.231e-10') * (lam / mp.mpf('0.225'))**8 rho, eta = apex.real / (1 - lam**2 / 2), apex.imag / (1 - lam**2 / 2) r_epsK = 1 - mp.sqrt(2) * EPS_K / (1 + Pc / (A**2 * X_T) - rho / eta) ImLt_l5, ReLt_l5, ReLc_l = lam_t.imag / lam**5, lam_t.real / lam**5, lam_c.real / lam cp = ImLt_l5 * X_T ccons = ReLc_l * (Pc + DP_CU) + ReLt_l5 * X_T BRp = kp * (1 + DELTA_EM) * (cp**2 + ccons**2) BRL = kL * r_epsK * (ImLt_l5 * X_T)**2 check("BR(K+ -> pi+ nu nubar) = 9.45e-11 [C] (TFPT CKM point + standard SD " "input X_t=1.462, P_c=%.4f, kappa_+=%.3e, r_epsK=%.4f); NOT parameter-" "free -- the SD functions are external electroweak/QCD input" % (float(Pc), float(kp), float(r_epsK)), BRp, mp.mpf('9.45e-11'), tol=mp.mpf('1e-2')) check("BR(K_L -> pi0 nu nubar) = 3.33e-11 [C] (purely CP-violating, prop " "(Im lambda_t)^2 prop etabar^2 -- a direct probe of delta_CKM)", BRL, mp.mpf('3.33e-11'), tol=mp.mpf('1e-2')) # 3. NA62 consistency (full 2016-2024 combination) pull_na62 = (NA62_C - BRp) / NA62_LO # prediction below central -> lower error pull_2022 = (NA62_2022_C - BRp) / NA62_2022_LO # the historical 2016-2022 observation sm_inside = abs(NA62_C - BR_SM) < 2 * NA62_LO # SM also inside the NA62 error check("NA62 CONSISTENCY [N]: BR(K+)=9.45e-11 lands ON NA62 2016-2024 " "(9.6 +1.9/-1.8)e-11 -> +%.2f sigma (was +%.2f sigma vs the 2016-2022 " "13.0e-11); a strong consistency hit for the CKM point, NOT a unique " "TFPT-vs-SM discriminator (SM 8.6e-11 is also inside the NA62 error)" % (float(pull_na62), float(pull_2022)), abs(pull_na62) < mp.mpf('0.5') and pull_2022 > 1.0 and sm_inside) # 4. Grossman-Nir gn = BRL / BRp check("GROSSMAN-NIR [I]: BR(K_L)/BR(K+) = %.3f, safely inside the model-" "independent isospin corridor (<= 4.3); a ratio > 4.3 would signal " "new physics outside the SM operator basis" % float(gn), gn < mp.mpf('4.3') and abs(gn - mp.mpf('0.352')) < mp.mpf('0.02')) # 5. kill test corridor in_corridor = mp.mpf('7e-11') < BRp < mp.mpf('12e-11') check("KILL TEST [X] (dated, operational): BR(K+) prediction sits inside " "the test corridor [7,12]e-11; a STABLE NA62 result OUTSIDE it, or a " "KOTO-II BR(K_L) off the GN-plane point, breaks the TFPT flavor " "bridge for this sector (compiler core untouched)", in_corridor) return summary("v202 rare kaon BR(K+)=9.45e-11, BR(K_L)=3.33e-11 from the TFPT CKM point [C]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)