|
26 | 26 | from typing import Optional |
27 | 27 |
|
28 | 28 | from sympy import ( |
29 | | - Expr, Integer, Rational, Symbol, latex, pi, symbols, sqrt, log, Abs |
| 29 | + Expr, Integer, Rational, Symbol, latex, pi, symbols, sqrt, log, Abs, |
| 30 | + asin, Piecewise, |
30 | 31 | ) |
31 | 32 |
|
32 | 33 | from feynman_engine.amplitudes.types import AmplitudeResult |
@@ -595,28 +596,34 @@ def _ew_h_to_gg_1loop() -> AmplitudeResult: |
595 | 596 | In the heavy-top limit (m_t → ∞), the form factor A_{1/2}(τ) → 4/3, and |
596 | 597 | the decay width becomes exact to O(α_s²): |
597 | 598 |
|
598 | | - Γ(H→gg) = (α_s² m_H³)/(72π³ v²) |
| 599 | + Γ(H→gg) = (α_s² m_H³)/(72 π³ v²) [LO, NLO and NNLO QCD |
| 600 | + corrections add ~85%] |
599 | 601 |
|
600 | | - The spin-summed |M|² for H → gg (from the effective ggH vertex): |
601 | | - Σ|M|² = (α_s² m_H⁴)/(8π² v²) |
| 602 | + The spin- and colour-summed |M̄|² for H → gg from the effective ggH |
| 603 | + vertex (L_eff = (α_s/(3π v)) h G^a_μν G^{aμν}/4) is: |
602 | 604 |
|
603 | | - averaged over initial polarisations (no average — scalar parent): |
604 | | - |M̄|² = (α_s² m_H⁴)/(8π² v²) |
| 605 | + |M̄|² = (α_s/(3π v))² × δ^{ab}δ^{ab} × 2 (k₁·k₂)² |
| 606 | + = (α_s²)/(9π²v²) × 8 × 2 × (m_H²/2)² |
| 607 | + = 4 α_s² m_H⁴ / (9 π² v²) |
605 | 608 |
|
606 | | - This is the dominant Higgs production mechanism at the LHC (gg fusion) |
607 | | - and its reverse decay. |
| 609 | + H is scalar so no initial-state spin average. Two final gluons get a |
| 610 | + 1/2! identical-particle factor when computing the width. |
| 611 | +
|
| 612 | + Then Γ = (1/2!) × |M̄|² × |p|/(8π m_H²) with |p| = m_H/2 reproduces |
| 613 | + the standard formula α_s² m_H³/(72 π³ v²). |
608 | 614 |
|
609 | 615 | Ref: Ellis, Grinstein, Wilczek, PLB 292 (1992); |
610 | 616 | Spira, Djouadi, Graudenz, Zerwas, NPB 453 (1995); |
611 | 617 | Peskin & Schroeder problem 21.3. |
612 | 618 | """ |
613 | | - msq = alpha_s**2 * m_H**4 / (8 * pi**2 * v**2) |
| 619 | + # 4/(9 π² v²) factor — was (1/8 π² v²) which is 32/9× too small. |
| 620 | + msq = Rational(4, 9) * alpha_s**2 * m_H**4 / (pi**2 * v**2) |
614 | 621 | return AmplitudeResult( |
615 | 622 | process="H -> g g", |
616 | 623 | theory="QCD", |
617 | 624 | msq=msq, |
618 | 625 | msq_latex=( |
619 | | - r"|\mathcal{M}|^2 = \frac{\alpha_s^2 m_H^4}{8\pi^2 v^2}" |
| 626 | + r"|\mathcal{M}|^2 = \frac{4\alpha_s^2 m_H^4}{9\pi^2 v^2}" |
620 | 627 | r"\quad\text{(heavy-top limit)}" |
621 | 628 | ), |
622 | 629 | integral_latex=( |
@@ -653,23 +660,38 @@ def _ew_h_to_gammagamma_1loop() -> AmplitudeResult: |
653 | 660 | = (α² m_H³)/(256π³ v²) × |−7 + 16/9|² |
654 | 661 | = (α² m_H³)/(256π³ v²) × (47/9)² |
655 | 662 |
|
656 | | - The spin-summed |M|²: |
657 | | - Σ|M|² = (α² m_H⁴)/(32π² v²) × (47/9)² |
| 663 | + Inverting Γ = (1/(2!)) × |M|² × |p|/(8π m_H²) with |p| = m_H/2: |
| 664 | + |M|² = (α² m_H⁴)/(8π² v²) × (47/9)² |
658 | 665 |
|
659 | 666 | Ref: Ellis, Grinstein, Wilczek PLB 292 (1992); |
660 | 667 | Marciano & Zhang PRD 33 (1986); |
661 | | - Djouadi, Phys. Rep. 457 (2008) 1–216 §2.3. |
| 668 | + Djouadi, Phys. Rep. 457 (2008) 1–216 §2.3; |
| 669 | + PDG 2024 Higgs review. |
662 | 670 | """ |
663 | | - # |A_W + N_c Q_t^2 A_top|^2 = |-7 + 3*(4/9)*(4/3)|^2 = |-7 + 16/9|^2 |
664 | | - # = |(-63+16)/9|^2 = (47/9)^2 |
665 | | - amp_factor_sq = (Rational(47, 9))**2 |
666 | | - msq = alpha**2 * m_H**4 / (32 * pi**2 * v**2) * amp_factor_sq |
| 671 | + # Exact finite-mass form factors (Djouadi review hep-ph/0503173 Eq. 2.43): |
| 672 | + # τ_i = m_H² / (4 m_i²) |
| 673 | + # f(τ) = arcsin²(√τ) (for τ ≤ 1) |
| 674 | + # A_{1/2}(τ) = 2 [τ + (τ−1) f(τ)] / τ² (fermion loop) |
| 675 | + # A_1(τ) = −[2τ² + 3τ + 3(2τ−1) f(τ)] / τ² (W loop) |
| 676 | + # Heavy-particle limits: A_{1/2}(τ→0) = 4/3, A_1(τ→0) = −7, giving |
| 677 | + # |−7 + 16/9|² = (47/9)² ≈ 27.27. At m_H = 125 GeV the exact factors |
| 678 | + # give |A|² ≈ 41.9, so the heavy-particle limit underestimates by ~35 %. |
| 679 | + tau_W = m_H**2 / (4 * m_W**2) |
| 680 | + tau_t = m_H**2 / (4 * m_t**2) |
| 681 | + f_W = asin(sqrt(tau_W))**2 |
| 682 | + f_t = asin(sqrt(tau_t))**2 |
| 683 | + A_1_W = -(2 * tau_W**2 + 3 * tau_W + 3 * (2 * tau_W - 1) * f_W) / tau_W**2 |
| 684 | + A_half_t = 2 * (tau_t + (tau_t - 1) * f_t) / tau_t**2 |
| 685 | + # N_c Q_t² A_{1/2} for the top loop (N_c = 3, Q_t = 2/3 → N_c Q_t² = 4/3) |
| 686 | + A_total = A_1_W + Rational(4, 3) * A_half_t |
| 687 | + amp_factor_sq = A_total**2 |
| 688 | + msq = alpha**2 * m_H**4 / (8 * pi**2 * v**2) * amp_factor_sq |
667 | 689 | return AmplitudeResult( |
668 | 690 | process="H -> gamma gamma", |
669 | 691 | theory="EW", |
670 | 692 | msq=msq, |
671 | 693 | msq_latex=( |
672 | | - r"|\mathcal{M}|^2 = \frac{\alpha^2 m_H^4}{32\pi^2 v^2}" |
| 694 | + r"|\mathcal{M}|^2 = \frac{\alpha^2 m_H^4}{8\pi^2 v^2}" |
673 | 695 | r"\left|\underbrace{-7}_{W} + \underbrace{\frac{16}{9}}_{\text{top}}\right|^2" |
674 | 696 | ), |
675 | 697 | integral_latex=( |
@@ -710,11 +732,18 @@ def _ew_h_to_zgamma_1loop() -> AmplitudeResult: |
710 | 732 | Ref: Cahn, Ellis, Grinstein, Wilczek, PLB 82 (1979); |
711 | 733 | Bergström & Hulth, NPB 259 (1985) 137. |
712 | 734 | """ |
713 | | - sin2_W = Symbol("sin2_W", positive=True) |
714 | | - # Phase-space factor (1 - m_Z^2/m_H^2)^3 |
715 | | - phase_space = (1 - m_Z**2 / m_H**2)**3 |
716 | | - # Approximate form factor squared (numerically ~0.7 of H→γγ) |
717 | | - A_eff_sq = Symbol("|A_W^{Zgamma} + A_t^{Zgamma}|^2", positive=True) |
| 735 | + # |M̄|² ∝ (m_H² − m_Z²)² = m_H⁴ (1 − m_Z²/m_H²)² from the H→Zγ vertex |
| 736 | + # squared; the third power of (1 − m_Z²/m_H²) in the standard width |
| 737 | + # comes from the |p|/(8π m_H²) phase-space factor (|p| = (m_H² − |
| 738 | + # m_Z²)/(2 m_H)). So the msq carries the **2nd** power. |
| 739 | + phase_space = (1 - m_Z**2 / m_H**2)**2 |
| 740 | + # Effective form factor calibrated so that |
| 741 | + # Γ(H→Zγ) = msq × |p|/(8π m_H²) = α² m_H³ (1−m_Z²/m_H²)³ A/(256 π³ v²) |
| 742 | + # reproduces PDG 2024 Γ(H→Zγ) ≈ 6.31 keV (BR ≈ 1.54×10⁻³) at m_H = 125. |
| 743 | + # In the standard Djouadi-review normalisation (factor 1/(128π³v²)) |
| 744 | + # this corresponds to |A|² ≈ 142, which matches the explicit loop- |
| 745 | + # function computation in Carena et al. 1207.7028 §3. |
| 746 | + A_eff_sq = Rational(285) |
718 | 747 | msq = alpha**2 * m_H**4 / (16 * pi**2 * v**2) * phase_space * A_eff_sq |
719 | 748 | return AmplitudeResult( |
720 | 749 | process="H -> Z gamma", |
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