#!/usr/bin/env python3 """ SCOPE NOTE, 2026-10-11 (comp 202). Where this file quotes lambda_SM(M_*) ~ 0.0927 (Buttazzo et al. 2013), the 0.8 percent agreement, the one-sigma Higgs match or comp 176's 0.4 sigma, that is the anchor of its day and is kept as found. comp 202 computes the number at three loops on current inputs: 0.0932 +- 0.0008, 1.6 sigma from e^-1/4, which is what the book states from v32.137. Nothing here is re-run; the substrate side is unaffected. PROVENANCE: PROOF Computation 109 -- Higgs-quartic normalisation + O(4) one-loop self-energy: explicit verification of two referee findings ========================================================================= A referee note on v26.03 flagged two compounding issues in the Higgs-sector derivation that together drive the M_* = 4 pi m_h claim: Finding A: postulate (1) has quartic (1/2) ^2; working potential (23) has (1/4) phi^4. Are they reconcilable? Finding B: paper's one-loop self-energy uses "4 lambda" (N lambda for N=4 real components). Standard O(N) calculation gives (N+2) lambda = 6 lambda. Which is right? This computation performs both verifications symbolically. No code paths are skipped; every contraction is enumerated explicitly. FINDING A: V_7 reduction of postulate (1) ========================================== Postulate (1) in the paper: F[psi, eps] = integral [ -eps + (1/4) ^2 + c |grad psi|^2 ] dmu (1) with psi V_7-valued. V_7 = Im(O) carries the inner product = sum_a v_a w_a (orthonormal basis from the octonion norm; equivalent to the Euclidean inner product on R^7 under the standard identification). Radial reduction: Write psi = phi * n-hat with n-hat in S^6 (unit V_7 vector) and phi a real magnitude. Then: = phi^2 * = phi^2 * 1 = phi^2 ^2 = phi^4 Substituting into (1): F[phi, eps] = int[ -eps phi^2 + (1/4) phi^4 + c|grad phi|^2 + ... ] In the committed doublet convention the postulate quartic is (1/4), so the radial reduction gives (1/4) phi^4, matching the working potential (23). The paper's working potential (Eq 23) reads: F(phi) = -eps phi^2 + (1/4) phi^4 with quadratic coefficient -eps (same as postulate) and quartic (1/4). Reconciliation attempt: a field rescaling phi -> alpha phi gives -eps phi^2 -> -alpha^2 eps phi^2 (1/2) phi^4 -> (alpha^4/2) phi^4 Going from (1/2) -> (1/4) requires alpha^4 = 1/2 i.e. alpha = 2^(-1/4). Then the quadratic becomes -2^(-1/2) eps phi^2 = -eps/sqrt(2) * phi^2. The paper's (23) has quadratic -eps, NOT -eps/sqrt(2). So the rescaling that fixes the quartic spoils the quadratic. There is no field rescaling that takes (1) to (23) simultaneously in both coefficients. VERDICT (A): the referee is correct that the paper, as originally written, stated the quartic in inconsistent normalisations. RESOLUTION (adopted in v26.09): the paper commits to the DOUBLET CONVENTION throughout. In this convention: - is identified with the doublet bilinear Phi^dag Phi (NOT the real-component norm; the two differ by a factor of 2). - lambda_PST = 1/4 is the coefficient of (Phi^dag Phi)^2, the SAME convention as the SM lambda_SM in V_SM = lambda_SM (Phi^dag Phi)^2. - The postulate (1) is corrected to (1/4)^2, so that the radial reduction -> psi^2 gives (1/4)psi^4, matching the working potential (23) with no factor mismatch. - The complex-component sombrero of sec:mstar (F = -eps|psi|^2 + (1/2)|psi|^4) is a different field normalisation used only for the m_h/v/r_0 relations; it is bridged explicitly to the doublet convention in that section. - Z^2 = lambda_SM(M_*)/lambda_PST is then a ratio of two same- convention couplings, = 4 lambda_SM(M_*) ~ 0.37 ~ e^(-1). The choice matters: b enters the bridge premise (B) prediction lambda_SM(M_*) = b * e^(-1). b = 1/4 gives 0.092; b = 1/2 gives 0.184. FINDING B: O(N) one-loop scalar self-energy ============================================ Lagrangian (symmetric phase, all four Higgs components live in a common O(4) multiplet): V = lambda (Phi^dagger Phi)^2 = (lambda / 4) (phi . phi)^2 = (lambda / 4) sum_{a,b} phi_a^2 phi_b^2 4-point vertex tensor: V_{ijkl} = partial^4 V / partial phi_i partial phi_j ... evaluated at phi = 0 Direct computation (see Part 2 below): V_{ijkl} = 2 lambda (delta_{ij} delta_{kl} + delta_{ik} delta_{jl} + delta_{il} delta_{jk}) Feynman rule = -i V_{ijkl}. One-loop seagull self-energy: close legs k, l into the loop with propagator i/p^2. Loop integral = i * Lambda^2 / (16 pi^2) (Euclidean cutoff Lambda). Symmetry factor 1/2 (two internal legs are interchangeable). Sigma_{ij} = (1/2) * (-i V_{ijkl}) * (i / loop) * delta_{kl} = (1/2) * V_{ijkl} * delta_{kl} * Lambda^2 / (16 pi^2) Contracting V_{ijkl} with delta_{kl}: V_{ijkl} delta_{kl} = 2 lambda [ delta_{ij} delta_{kl} delta_{kl} + delta_{ik} delta_{jl} delta_{kl} + delta_{il} delta_{jk} delta_{kl} ] = 2 lambda [ delta_{ij} * N + delta_{ij} + delta_{ij} ] = 2 lambda (N + 2) delta_{ij} Sigma_{ij} = (1/2) * 2 lambda (N + 2) delta_{ij} * Lambda^2 / (16 pi^2) = (N + 2) lambda * delta_{ij} * Lambda^2 / (16 pi^2) For N = 4: Sigma = 6 lambda * Lambda^2 / (16 pi^2). The paper writes "4 lambda * M_*^2 / (16 pi^2)" (Eq 107). That is N lambda, keeping only the delta_{ij} delta_{kl} contraction and dropping the +2 from the symmetric tensor structure of V_{ijkl}. Cross-check via broken-phase contributions (SM textbook): - physical Higgs (h) self-loop: vertex 3! lambda v^2 -> 3 lambda - 3 Goldstone modes, each contributing 1 lambda - total: 3 lambda + 3 * 1 lambda = 6 lambda (agrees) VERDICT (B): the referee is correct. The one-loop self-energy coefficient is (N+2) lambda = 6 lambda for N=4, not 4 lambda. COMPOUND IMPACT ON M_* PREDICTION ================================== With C_H = (loop coefficient) * lambda_PST and m_h^2 = C_H * M_*^2 / (16 pi^2): M_* = 4 pi m_h / sqrt(C_H) For the four combinations of (count, lambda_PST): count=4, lambda=1/4 (original reading, superseded) -> C_H = 1 -> M_* = 4 pi m_h ~= 1573 GeV count=6, lambda=1/4 (paper b, fix B) -> C_H = 3/2 -> M_* ~= 1285 GeV count=4, lambda=1/2 (fix A, keep B) -> C_H = 2 -> M_* ~= 1113 GeV count=6, lambda=1/2 (both fixes) -> C_H = 3 -> M_* ~= 909 GeV The headline 1573 GeV is the unique corner where both errors hold. Either correction alone shifts M_*. PYTHON VERIFICATION ==================== Symbolic verification of the V_{ijkl} tensor and the loop contraction for arbitrary N, then evaluation at N = 4. """ import math from itertools import product def vijkl(i, j, k, l, lam): """ Vertex tensor V_{ijkl} = 2*lambda * (delta_{ij} delta_{kl} + delta_{ik} delta_{jl} + delta_{il} delta_{jk}). Derived from V = (lambda/4) (sum_a phi_a^2)^2 via four derivatives. """ def kron(a, b): return 1 if a == b else 0 return 2 * lam * (kron(i, j) * kron(k, l) + kron(i, k) * kron(j, l) + kron(i, l) * kron(j, k)) def loop_contraction_coefficient(N, lam): """ Compute Sigma_{ii} after closing the seagull loop: Sigma_{ij} = (1/2) * V_{ijkl} * delta_{kl} * (loop integral) Return the coefficient in front of Lambda^2 / (16 pi^2). Verify that Sigma_{ij} is isotropic (proportional to delta_{ij}). """ sigma_diag = 0 sigma_off = 0 for k in range(N): sigma_diag += (1 / 2) * vijkl(0, 0, k, k, lam) sigma_off += (1 / 2) * vijkl(0, 1, k, k, lam) return sigma_diag, sigma_off def main(): print("=" * 72) print("Computation 109: Higgs-quartic normalisation + O(4) self-energy") print("=" * 72) print() print("FINDING A: V_7 radial reduction of postulate (1)") print("-" * 72) print(" psi V_7-valued, = sum_a psi_a^2 = phi^2 (radial)") print(" ^2 = phi^4") print() print(" Postulate (1) coefficient: (1/2) ^2 -> (1/2) phi^4") print(" Working potential (23): (1/4) phi^4") print() print(" Rescaling test: phi -> alpha*phi to send (1/2) -> (1/4)") print(" requires alpha^4 = 1/2, so alpha = 2^(-1/4) ~= 0.8409") print(f" alpha = {2 ** (-1/4):.6f}") print(f" new quadratic coeff = -alpha^2 * eps = -{2 ** (-1/2):.6f} eps") print(" paper (23) quadratic = -eps (no rescaling factor)") print(" => no single rescaling reconciles BOTH coefficients.") print() print(" VERDICT: postulate (1) and working (23) are in different") print(" normalisations. Cannot be unified by field rescaling.") print() print("FINDING B: O(N) one-loop self-energy") print("-" * 72) print(" Vertex V_{ijkl} = 2*lambda * (d_ij d_kl + d_ik d_jl + d_il d_jk)") print() print(" Sample V_{ijkl} values (lambda = 1):") for (i, j, k, l) in [(0, 0, 0, 0), (0, 0, 1, 1), (0, 1, 0, 1), (0, 1, 1, 0)]: v = vijkl(i, j, k, l, 1.0) print(f" V_{{{i}{j}{k}{l}}} = {v}") print() print(" One-loop contraction (1/2) * V_{ijkl} * delta_{kl} for") print(" N in {1, 2, 3, 4, 5} with lambda = 1:") print() print(f" {'N':>3} | {'diag (Sigma_00)':>15} | {'off (Sigma_01)':>15} | expected (N+2)*lambda") print(f" {'-'*3}-+-{'-'*15}-+-{'-'*15}-+-{'-'*21}") for N in [1, 2, 3, 4, 5]: diag, off = loop_contraction_coefficient(N, 1.0) expected = N + 2 match = "OK" if abs(diag - expected) < 1e-12 else "MISMATCH" print(f" {N:>3} | {diag:>15.6f} | {off:>15.6f} | " f"{expected:>21} ({match})") print() print(" Off-diagonal (Sigma_01) is zero -> Sigma is isotropic, as expected.") print(" Diagonal coefficient matches (N+2) * lambda exactly for all N.") print() print(" VERDICT: one-loop self-energy = (N+2) lambda * Lambda^2/(16 pi^2)") print(" For N = 4: 6 lambda, NOT 4 lambda.") print() print(" Cross-check: broken-phase total") print(" physical Higgs h-loop (vertex 6 lambda v^2): contributes 3 lambda") print(" 3 Goldstones, each 1 lambda: contributes 3 lambda") print(" SUM: 3 + 3 = 6 lambda. Agrees with O(4) symmetric-phase result.") print() print("COMPOUND IMPACT ON M_* PREDICTION") print("-" * 72) print(" M_* = 4 pi m_h / sqrt(C_H), with C_H = (count) * lambda_PST") print(f" m_h = 125.25 GeV => 4 pi m_h = {4 * math.pi * 125.25:.2f} GeV") print() print(f" {'count':>5} | {'lambda_PST':>10} | {'C_H':>6} | " f"{'M_* (GeV)':>12} | note") print(f" {'-'*5}-+-{'-'*10}-+-{'-'*6}-+-{'-'*12}-+-{'-'*30}") for count, lam, note in [(4, 0.25, "original reading, superseded"), (6, 0.25, "fix B only (paper's b)"), (4, 0.50, "fix A only (paper's count)"), (6, 0.50, "both fixes")]: c_h = count * lam m_star = 4 * math.pi * 125.25 / math.sqrt(c_h) print(f" {count:>5} | {lam:>10.3f} | {c_h:>6.2f} | " f"{m_star:>12.2f} | {note}") print() print(" The 1573 GeV headline is the unique corner where both undercounts") print(" simultaneously hold.") print() print("IMPACT ON BRIDGE PREMISE (B) NUMERICAL CLAIM") print("-" * 72) print(" lambda_SM(M_*) = b * Z_H(beta_KO) = b * e^(-1)") print(f" e^(-1) = {math.exp(-1):.6f}") print() print(f" {'b':>6} | {'b * e^(-1)':>11} | {'vs lambda_SM ~= 0.0927':>22} | match?") print(f" {'-'*6}-+-{'-'*11}-+-{'-'*22}-+-{'-'*10}") for b, label in [(0.25, "paper's b"), (0.50, "postulate's b (Finding A)")]: pred = b * math.exp(-1) ratio_pct = 100 * abs(pred - 0.0927) / 0.0927 print(f" {b:>6.3f} | {pred:>11.6f} | " f"{ratio_pct:>21.1f}% | {label}") print() print(" If Finding A holds (b = 1/2), the 0.8% match becomes ~100% off.") print() print("=" * 72) print("Summary") print("=" * 72) print() print(" Finding A (1/2 vs 1/4 in postulate vs working potential):") print(" confirmed by direct V_7 radial reduction; no rescaling") print(" reconciles both coefficients.") print() print(" Finding B (4 lambda vs 6 lambda in one-loop self-energy):") print(" confirmed by explicit O(N) vertex contraction for all N tested;") print(" agrees with the broken-phase textbook calculation 3*lambda") print(" (physical Higgs) + 3*lambda (Goldstones) = 6*lambda for N=4.") print() print(" Combined: M_* = 4 pi m_h ~= 1573 GeV is the unique corner where") print(" both undercounts hold. Any correction shifts M_* into") print(" the 900-1285 GeV range and breaks the clean 4 pi factor.") print() print(" This computation does not propose a fix. It confirms the referee's") print(" technical claims at proof-detail level.") if __name__ == "__main__": main()