#!/usr/bin/env python3 """ Two-point running test for the cube-geometry weak-mixing prediction. The panel-greenlit second-hit test. Two anchors, with different standings: IR anchor : sin^2 theta_W(Q=0) = vertexDegree / (4 pi) = 3/(4 pi) = 0.238732 The one cube-forced weak-mixing number. Lean (Masses/WeakMixingFromCube.lean, weakMixingFromCube_consistent) certifies it lies inside the measured 1-sigma band [0.23851, 0.23883] of the Q=0 Thomson-limit value, ~0.4 sigma from center; identifying the cube ratio with sin^2 theta_W(0) is HYPOTHESIS tier (falsifier: weakMixingFromCube_falsifier). Not a claim about the M_Z value. UV anchor : sin^2 theta_W(M_GUT) = 3/8 = 0.375 NOT cube-forced. 3/8 is the textbook SU(5) tree-level value of sin^2 theta_W at unification (Georgi-Quinn-Weinberg); in Lean, sin2ThetaW_GUT := 3/8 is a definition importing that standard value. It is reached, by construction, at the scale where the GUT-normalized U(1) coupling g1 equals the SU(2) coupling g2. The test (no free parameters): run the SM gauge couplings from M_Z upward with the standard beta functions and find the scale mu* where g1 = g2 (equivalently sin^2 theta_W(mu*) = 3/8). GREEN if mu* lands in the GUT window 1e15 - 1e17 GeV. Inputs: PDG/MS-bar electroweak values at M_Z. Nothing here is fit to the cube. """ import numpy as np from scipy.integrate import solve_ivp # ---- experimental inputs at M_Z (MS-bar, PDG 2024) ---- M_Z = 91.1876 # GeV alpha_em_inv_MZ = 127.951 # alpha_em(M_Z)^-1 (MS-bar) sin2_MZ = 0.23121 # sin^2 theta_W(M_Z) (MS-bar) alpha_s_MZ = 0.1179 # for completeness / g3 cos2_MZ = 1.0 - sin2_MZ # couplings at M_Z. alpha2 = alpha_em / sin2 ; alpha' = alpha_em / cos2 ; alpha1 = (5/3) alpha' alpha_em_MZ = 1.0 / alpha_em_inv_MZ alpha2_MZ = alpha_em_MZ / sin2_MZ alphaP_MZ = alpha_em_MZ / cos2_MZ # U(1)_Y in SM normalization alpha1_MZ = (5.0/3.0) * alphaP_MZ # GUT-normalized U(1) alpha3_MZ = alpha_s_MZ print("== couplings at M_Z ==") print(f" alpha_em^-1 = {alpha_em_inv_MZ}, sin^2 = {sin2_MZ}") print(f" alpha1^-1(GUT-norm) = {1/alpha1_MZ:.4f}") print(f" alpha2^-1 = {1/alpha2_MZ:.4f}") print(f" alpha3^-1 = {1/alpha3_MZ:.4f}") # ---- one-loop SM beta coefficients (GUT-normalized U(1)) ---- # d alpha_i^-1 / dt = - b_i / (2 pi), t = ln(mu) b1 = 41.0/10.0 # +4.1 b2 = -19.0/6.0 # -3.1667 b3 = -7.0 def running_inv(alpha_inv_MZ, b, mu): t = np.log(mu / M_Z) return alpha_inv_MZ - (b / (2*np.pi)) * t # ---- one-loop crossing g1 = g2 (where sin^2 = 3/8) ---- # alpha1^-1(mu) = alpha2^-1(mu) # 1/alpha1_MZ - b1/(2pi) t = 1/alpha2_MZ - b2/(2pi) t # (1/alpha1 - 1/alpha2) = (b1 - b2)/(2pi) t t_cross = (1/alpha1_MZ - 1/alpha2_MZ) / ((b1 - b2) / (2*np.pi)) mu_cross_1loop = M_Z * np.exp(t_cross) print("\n== one-loop g1=g2 crossing (sin^2 = 3/8 by construction) ==") print(f" t* = ln(mu/M_Z) = {t_cross:.3f}") print(f" mu* = {mu_cross_1loop:.3e} GeV") def sin2_at(mu, two_loop=False): """sin^2 theta_W(mu) = alpha' / (alpha2 + alpha') with alpha' = (3/5) alpha1.""" if not two_loop: a1i = running_inv(1.0/alpha1_MZ, b1, mu) a2i = running_inv(1.0/alpha2_MZ, b2, mu) else: a1i, a2i, _ = two_loop_inv(mu) a1 = 1.0/a1i a2 = 1.0/a2i aP = (3.0/5.0) * a1 # back to SM hypercharge return aP / (a2 + aP) print("\n== one-loop sin^2 theta_W at canonical scales ==") for mu in [M_Z, 1e3, 1e6, 1e9, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, mu_cross_1loop]: print(f" mu = {mu:.3e} GeV : sin^2 = {sin2_at(mu):.5f}") # ---- two-loop SM RGEs (GUT-normalized), standard B matrix ---- # d a_i / dt = (1/2pi) b_i a_i^2 + (1/8 pi^2) sum_j B_ij a_i^2 a_j ... use alpha = g^2/4pi # Use the standard two-loop gauge B-matrix (SM, one Higgs doublet): B = np.array([ [199.0/50.0, 27.0/10.0, 44.0/5.0], [9.0/10.0, 35.0/6.0, 12.0], [11.0/10.0, 9.0/2.0, -26.0], ]) b_vec = np.array([b1, b2, b3]) def rge(t, y): # y = alpha_i (not inverse). t = ln mu. a = y dadt = np.zeros(3) for i in range(3): gauge2 = sum(B[i, j] * a[j] for j in range(3)) dadt[i] = (b_vec[i] / (2*np.pi)) * a[i]**2 + (1.0/(8*np.pi**2)) * a[i]**2 * gauge2 return dadt y0 = np.array([alpha1_MZ, alpha2_MZ, alpha3_MZ]) t_span = (0.0, np.log(1e18 / M_Z)) sol = solve_ivp(rge, t_span, y0, dense_output=True, rtol=1e-9, atol=1e-12, max_step=0.05) def two_loop_inv(mu): t = np.log(mu / M_Z) a = sol.sol(t) return 1.0/a[0], 1.0/a[1], 1.0/a[2] # find two-loop g1=g2 crossing from scipy.optimize import brentq def diff12(logmu): mu = np.exp(logmu) a1i, a2i, _ = two_loop_inv(mu) return a1i - a2i lo, hi = np.log(1e10), np.log(1e16) try: logmu_star = brentq(diff12, lo, hi) mu_cross_2loop = np.exp(logmu_star) except Exception as e: mu_cross_2loop = None print("two-loop crossing solve failed:", e) print("\n== two-loop g1=g2 crossing (sin^2 = 3/8) ==") if mu_cross_2loop: print(f" mu* = {mu_cross_2loop:.3e} GeV") print(f" sin^2 at mu* = {sin2_at(mu_cross_2loop, two_loop=True):.5f} (should be 0.375)") print("\n== two-loop sin^2 theta_W at canonical scales ==") for mu in [M_Z, 1e3, 1e9, 1e13, 1e14, 1e15, 1e16, 1e17]: print(f" mu = {mu:.3e} GeV : sin^2 = {sin2_at(mu, two_loop=True):.5f}") # ---- VERDICT ---- print("\n" + "="*64) print("VERDICT (panel criterion: mu* in GUT window 1e15 - 1e17 GeV)") print("="*64) window = (1e15, 1e17) for name, mu in [("1-loop", mu_cross_1loop), ("2-loop", mu_cross_2loop)]: if mu is None: continue inside = window[0] <= mu <= window[1] print(f" {name}: sin^2=3/8 reached at {mu:.3e} GeV -> " f"{'INSIDE' if inside else 'OUTSIDE'} [1e15,1e17]") print("\nIR anchor 3/(4pi) =", 3/(4*np.pi)) print("UV anchor 3/8 =", 3/8) # ---- the decisive caveat: the crossing scale is set by the BSM spectrum, not the cube ---- # Repeat the one-loop g1=g2 crossing with MSSM coefficients to show the verdict flips. b1_mssm, b2_mssm = 33.0/5.0, 1.0 t_mssm = (1/alpha1_MZ - 1/alpha2_MZ) / ((b1_mssm - b2_mssm) / (2*np.pi)) mu_mssm = M_Z * np.exp(t_mssm) print("\n== same test, MSSM beta functions (b1=33/5, b2=1) ==") print(f" g1=g2 (sin^2=3/8) at mu* = {mu_mssm:.3e} GeV -> " f"{'INSIDE' if 1e15 <= mu_mssm <= 1e17 else 'OUTSIDE'} [1e15,1e17]") print(" => the crossing scale is dominated by the (unknown) BSM content, not by cube geometry.")