16. Extending the Decay Calculator

The gap, stated honestly

The decays tutorial closes with the Higgs branching-ratio plot — and at the physical \(m_h = 125\) GeV that plot is qualitatively wrong: with one lepton generation and on-shell \(1\to2\) kinematics only, it shows \(\mathrm{BR}(h\to\tau\tau) = 100\%\), because the on-shell \(WW/ZZ\) channels are closed (\(2m_W = 160.8 > 125.25\)). Nature disagrees:

channel	BR at \(m_h=125\) GeV	tier
\(b\bar b\)	58.2%	1 — tree-level \(f\bar f\)
\(WW^*\)	21.4%	2 — off-shell \(VV^*\)
\(gg\)	8.2%	3 — loop-induced
\(\tau^+\tau^-\)	6.3%	1
\(c\bar c\)	2.9%	1
\(ZZ^*\)	2.6%	2
\(\gamma\gamma\)	0.23%	3
\(Z\gamma\)	0.15%	3
\(\mu^+\mu^-\)	0.022%	1

This chapter is the roadmap for closing that gap. The three tiers differ by orders of magnitude in effort, and they differ in kind: Tier 1 is model content plus one API improvement (the \(|\mathcal M|^2\) engine already computes it); Tier 2 is a genuine architectural extension (from “square one vertex” to “assemble a tree diagram”); Tier 3 is the only one that steps outside the tree-level, derive-from-the-Lagrangian ethos — deliberately, and with precedent.

Coverage arithmetic: Tier 1 alone reaches \(\approx 67\%\) of the width and — more importantly — replaces the wrong plot with a right one whose missing pieces are quantified. Adding Tier 2 reaches \(\approx 91\%\); adding Tier 3 completes the canonical picture.

16.1 Tier 1 — all tree-level \(f\bar f\) channels

Channels: \(b\bar b\), \(c\bar c\), \(s\bar s\), \(\tau^+\tau^-\), \(\mu^+\mu^-\). Effort: small. The physics engine needs nothing new: ffs_squared() already produces \(\Gamma(h\to f\bar f) = N_c\,m_h m_f^2\beta^3/8\pi v^2\), and the colour factor \(N_c\) is a solved problem — a colour-triplet channel with a single declared colour component and color_factors={q: 3} on one leg yields exactly \(3\,y_q^2\,m_h\beta^3/16\pi\), verified against the leptonic case.

What is actually missing:

Model content

Declare all three generations of leptons and quarks with realistic Yukawas. examples/sm_scalar_gauge.py already contains the full pattern — SU3c triplets QL/uR/dR, flavour-indexed Yu/Yd, the explicit Python colour sum — so this is a modelling exercise, not new machinery.

The colour over-count trap (verified, and the motivation for the API fix)

There is exactly one wrong way to combine the existing pieces, and it is the natural way. Write the Yukawa as an explicit 3-colour sum (as sm_scalar_gauge.py does — correctly, for vertex extraction), then collapse the three colour components into one particle with particle_map, then apply color_factors. Each step looks right; together they give

\[\Gamma_{\text{wrong}} = 81\,y_b^2\,(\dots) = 27\times\Gamma_{\text{right}}.\]

The mechanism: particle_map merges the three colour couplings coherently — the vertex coupling becomes \(3y_b\), contributing \(9y_b^2\) to \(|\mathcal M|^2\) — and color_factors on both legs multiplies a further \(3\times3\). But colour is an incoherent sum: one final state per colour, \(\Gamma = N_c\,\Gamma_{\text{one colour}}\). The correct recipe today is one colour component in the decay-facing Lagrangian plus color_factor=3 on one leg.

The recommended addition: a post-EWSB DiracParticle

A natural question first — why not declare a DiracFermion instead of two WeylFermions and avoid all pairing bookkeeping? Because in the gauge basis it is impossible: the SM is chiral, \(b_L\) sits in an SU(2) doublet with \(Y=+1/6\) while \(b_R\) is a singlet with \(Y=-1/3\), and a 4-component field carries a single rep assignment — no choice makes the unbroken Lagrangian invariant. This is why DiracFermion raises on construction (fields.py): for input fields it is not a missing feature but a physics impossibility. (Only vector-like fermions, with equal L/R reps as in examples/sm_vll.py, admit one; enabling DiracFermion as sugar that expands to two Weyls internally for that case would be harmless, but does not help the chiral SM fermions this tier needs.)

After EWSB, however, the electron is one Dirac particle — and the pheno API currently encodes that fact as three loose dicts: particle_map (which Weyl legs pair up), masses (keyed separately), and color_factors (keyed separately again). Every silent failure mode the tutorial warns about traces to this split:

• omit or misspell a particle_map entry → the L and R currents stay as two half-channels, each missing the \(g_Lg_R\) mass interference (invisible in the massless limit — the worst kind of bug);

• a mass-key mismatch → channels() on DecayCalculator silently skips the leg and the channel vanishes with no error;

• colour multiplicity applied in more than one place → the \(81\times\) over-count above.

The fix is one small dataclass — now delivered (DiracParticle):

DiracParticle(
    "b",
    left=QL.components[1], right=bR.components[0],   # the two Weyl legs
    mass=mb, color=3,                                # bar legs auto-derived
)

with DecayCalculator(model, particles=[...], ...) consuming a list of these instead of the dict triple. The bar legs come from the existing bar_partner registry (fields.py), the mass travels with the particle, and \(N_c\) is applied once, per channel, incoherently — all three traps are now unrepresentable, and any fermion channel claimed by no declared particle is surfaced in calc.unmatched_channels (with a warning) rather than silently dropped. The physics engine was untouched; this was calculator plumbing. See 15. Decay Widths §15.4 for the API and examples/sm_higgs_decays.py for the full \(b\bar b\)-dominated fermionic branching-ratio table.

Running masses

A physical \(\Gamma(h\to b\bar b)\) uses the running mass \(\overline{m}_b(m_h)\approx 2.8\) GeV in the Yukawa, not the pole mass \(\approx 4.8\) GeV — a factor \(\sim 3\) in the rate. This is a numeric input choice (feed the running value), not new machinery; the roadmap notes it so the Tier-1 numbers land near the canonical table rather than mysteriously high.

16.2 Tier 2 — off-shell \(WW^*\) and \(ZZ^*\) ✅ (V–V* delivered)

Channels: \(h\to WW^*\to W f\bar f'\) (21.4%), \(h\to ZZ^*\to Z f\bar f\) (2.6%). Effort: large — this is the architectural jump. Delivered in feynlag.pheno.offshell (see the end of this section); the FFFF and \(t\to bW^*\) topologies remain.

The on-shell VVS width is exactly zero at \(m_h=125\) GeV because the two-body channel is closed; the physical decay proceeds through one on-shell \(V\) and one off-shell \(V^*\) that materialises as a fermion pair. That is a \(1\to3\) process with an internal propagator — three things the current design deliberately does not have:

1. Propagators. An internal line \(\frac{-i\left(g_{\mu\nu} - q_\mu q_\nu/m_V^2\right)}{q^2 - m_V^2 + i m_V\Gamma_V}\) with a Breit–Wigner width (which is itself a decay-calculator output — pleasingly self-referential: \(\Gamma_V\) feeds back from Tier 1).

2. \(1\to3\) phase space. The two-body factor \(\sqrt\lambda/16\pi M^3\) is a closed form; three-body phase space is a 2-dimensional (Dalitz) integral over \((q^2, \cos\theta)\) or \((m_{12}^2, m_{23}^2)\) that has no closed form once the propagator sits inside it — a numerical integration layer (the first in the library). The Dalitz invariants, bounds and the \(1/((2\pi)^3 32M^3)\) constant ThreeBodyKinematics implements follow the standard formulae of the PDG’s Kinematics review [PDG].

3. Diagram assembly. amplitude_squared() squares a single DecayVertex. An off-shell amplitude is vertex × propagator × vertex, and squaring it puts the propagator momentum \(q\) inside the trace algebra — the covariant engine in pheno/lorentz.py handles the traces (that part generalises cleanly), but a new assembly layer must build the chain and route the dot products \(p_i \cdot q\).

The natural staging inside Tier 2: hVV* first with the narrow-\(V\) approximation as a cross-check (\(\Gamma_{h\to V f\bar f} \to \Gamma_{h\to VV}\,\mathrm{BR}\) above threshold), then the full \(q^2\) integration. The same machinery immediately gives three-body decays generally (e.g. \(\mu\to e\nu\bar\nu\) through the FFFF track, top decays \(t\to bW^*\) below threshold in BSM spectra), so it earns its cost beyond the Higgs.

What was built

feynlag.pheno gained four pieces, each general but exercised here on V–V*:

• propagator — the massive-vector Breit–Wigner propagator (numerator \(g_{\alpha\beta}-q_\alpha q_\beta/m^2\) + the \(|q^2-m^2+im\Gamma|^{-2}\) factor).

• ThreeBodyKinematics — the Dalitz invariants \(s_{12},s_{23}\), the \(s_{12}\) bounds, and the on-shell dot map that reduces the covariant \(|M|^2\) (the internal \(q=p_2+p_3\) is just a tensor sum the existing engine contracts — no new trace machinery).

• integrate — the numerical layer: SciPy when the [numeric] extra is installed, else a numpy Gauss–Legendre fallback (SciPy lives in this one module).

• offshell — scalar_vv_squared (the covariant \(|M|^2\)), offshell_scalar_vv_width (analytic inner \(s_{12}\) integral + numeric \(q^2\) integral), and scalar_offshell_vv_width (summed over the \(V^*\) fermion channels, with the factor 2 for distinct \(W^\pm\) vs. no factor for identical \(ZZ\)).

Verified against the Keung–Marciano closed form [KM84, Djouadi08] \(\Gamma(h\to VV^*)=\frac{3g_V^4 m_h}{512\pi^3}\delta_V R(x)\), \(x=m_V^2/m_h^2\): \(\Gamma(h\to WW^*)=0.80\) MeV and \(\Gamma(h\to ZZ^*)=0.089\) MeV, and by narrow-width factorisation (a heavy scalar’s \(1\to3\) width → the Tier-1 \(\Gamma(S\to VV)\times\mathrm{BR}\)). The parity-violating \(\gamma_5\)/ε term drops against the symmetric propagator + polarisation structure (the 1→3 analogue of the \(\gamma_5\) argument in 15. Decay Widths §15) — no ε-tensor algebra needed for this topology. Worked model: examples/sm_higgs_decays.py (the BR table now includes WW*/ZZ*), and the decays tutorial’s §11 walks the off-shell \(1\to3\) with a \(W^*\) line-shape figure and the completed canonical BR chart. Not yet built: the FFFF and \(t\to bW^*\) topologies (same infrastructure, a new assembler each).

16.3 Tier 3 — loop-induced \(gg\), \(\gamma\gamma\), \(Z\gamma\) ✅ (effective vertices delivered)

Channels: \(gg\) (8.2%), \(\gamma\gamma\) (0.23%), \(Z\gamma\) (0.15%). Effort: moderate — if one accepts the recommended route. Delivered in feynlag.pheno.loop (see the end of this section).

These vanish identically at tree level: the Higgs is electrically and colour neutral, so \(h\to gg/\gamma\gamma\) proceed only through a top-quark triangle and (for \(\gamma\gamma\)) a \(W\) loop. A tree-level library cannot reach them from the Lagrangian, period.

Recommended route: effective vertices. The one-loop form factors are standard closed forms — with \(\tau = m_h^2/4m^2\) per loop particle,

\[\Gamma(h\to gg) = \frac{\alpha_s^2 m_h^3}{72\pi^3 v^2} \left|\tfrac{3}{4}\textstyle\sum_q A_{1/2}(\tau_q)\right|^2, \qquad \Gamma(h\to\gamma\gamma) = \frac{\alpha^2 m_h^3}{256\pi^3 v^2} \left|A_1(\tau_W) + \textstyle\sum_f N_c Q_f^2 A_{1/2}(\tau_f)\right|^2,\]

where \(A_{1/2}\) and \(A_1\) are the fermion- and \(W\)-loop functions of the Higgs Hunter’s Guide [GHKD00]. Implemented as effective_hgg(...) / effective_haa(...) helpers that return a coupling, the width is then an ordinary \(1\to2\) the current engine already squares (VVS with massless vectors — the \(-g_{ab}\) polarisation sum is already in polarization_sum()).

Say it plainly: this imports known one-loop results instead of deriving them — a deliberate, documented exception to the derive-from-the-Lagrangian ethos. There is precedent: the CKM sector (14. Quark Flavour Mixing (CKM)) already takes the FeynRules-style shortcut of inserting the standard-parametrisation \(V\) rather than diagonalising symbolic Yukawas, because the honest general computation is intractable and the physical answer is standard. The same judgment applies here.

The alternative — a genuine one-loop engine (Passarino–Veltman tensor reduction, scalar integrals \(B_0/C_0\), UV renormalisation and a scheme choice) — is a project of the same scale as the whole current library, and already sits on the v2-deferred list next to \(R_\xi\) gauges and ghosts. It should stay there until someone wants NLO for its own sake.

What was built

loop provides the one-loop form factors — \(f(\tau)\), \(g(\tau)\); the single-argument \(A_{1/2}(\tau)\), \(A_1(\tau)\) for \(gg/\gamma\gamma\); the two-argument \(I_1,I_2\) and \(A^{Z\gamma}_{1/2},A^{Z\gamma}_1\) for \(Z\gamma\) — and three width helpers higgs_gg_width (with an optional NLO-QCD \(K\)-factor), higgs_gammagamma_width, higgs_zgamma_width. DecayCalculator.loop_widths(...) returns all three. Verified against the PDG (a few % each): \(\Gamma(h\to\gamma\gamma)=9.2\) keV, \(\Gamma(h\to gg)=0.20\) MeV at LO and \(0.32\) MeV with the \(K\)-factor (\(\approx8\%\) BR), \(\Gamma(h\to Z\gamma)=6.2\) keV; the form factors hit their decoupling limits \(A_{1/2}\to4/3\), \(A_1\to-7\), and the \(W\)/top interference in \(\gamma\gamma\) is destructive (the physics that makes it a new-physics probe). Worked model: examples/sm_higgs_decays.py now prints the complete Higgs BR table (every visible channel), and the decays tutorial’s §12 walks the loop physics, the full BR-vs-mass plot, and how new physics shows as a deviation from it. The single-argument form factors follow Djouadi [Djouadi08]; the \(Z\gamma\) closed form is verified against Carena et al. [CGHT13]; the NLO-QCD \(gg\) \(K\)-factor follows Spira et al. [SDGZ95].

16.4 Summary

tier	channels	new machinery	BR gained	effort	ethos
1 ✅	\(b\bar b,c\bar c,s\bar s,\mu\mu\)	DiracParticle abstraction (done); model content; running-mass inputs	→ 67%	small	tree-level, in ethos
2 ✅	\(WW^*, ZZ^*\)	propagators + \(1\to3\) phase space + diagram assembly (done); FFFF/\(t\to bW^*\) topologies remain	→ 91%	large	tree-level, in ethos
3 ✅	\(gg,\gamma\gamma,Z\gamma\)	effective one-loop vertices (\(A_{1/2}, A_1\); done)	→ 100%	moderate	documented exception (CKM precedent)

Tier 2 status: the \(V\)–\(V^*\) topology (\(h\to WW^*\), \(h\to ZZ^*\)) is delivered — feynlag.pheno.offshell reproduces the Keung–Marciano widths (\(\Gamma(h\to WW^*)\approx0.80\) MeV, \(\Gamma(h\to ZZ^*)\approx0.089\) MeV) and the Higgs BR table is now \(b\bar b\)-dominated with \(WW^*\) second (~25%), the canonical shape. The general-\(1\to3\) topologies the roadmap also names — the FFFF track (\(\mu\to e\nu\nu\)) and the fermionic \(t\to bW^*\) — reuse the same propagator + Dalitz infrastructure but are not yet assembled; they are the remaining Tier-2 generality.

The recommended order is the table order: Tier 1 fixes the qualitatively wrong plot for the cost of a dataclass; Tier 2 is where the architecture grows; Tier 3 is a bounded, well-understood graft.

See 15. Decay Widths for the implemented engine, the tutorial (Particle_Decays_Tutorial.ipynb) for the traps §16.1 makes unrepresentable, and the v2-deferred list in the repository CLAUDE.md for what stays out of scope.

16.5 References

• [KM84] W.-Y. Keung and W. J. Marciano, “Higgs scalar decays: H → W±X”, Phys. Rev. D 30, 248 (1984). doi:10.1103/PhysRevD.30.248 — the original \(H\to WW^*\) calculation this section’s Tier-2 widths reproduce.

• [Djouadi08] A. Djouadi, “The Anatomy of Electro-Weak Symmetry Breaking I: The Higgs boson in the Standard Model”, Phys. Rept. 457 (2008) 1–216, arXiv:hep-ph/0503172 — the modern compilation this chapter’s \(R(x)\) notation follows, and the source of the Tier-3 \(A_{1/2}\)/\(A_1\) loop functions.

• [GHKD00] J. F. Gunion, H. E. Haber, G. L. Kane, S. Dawson, The Higgs Hunter’s Guide, Front. Phys. 80 (2000) 1–404 (originally Addison-Wesley, 1990) — the one-loop \(A_{1/2}/A_1\) form factors.

• [SDGZ95] M. Spira, A. Djouadi, D. Graudenz, P. M. Zerwas, “Higgs boson production at the LHC”, Nucl. Phys. B 453 (1995) 17, arXiv:hep-ph/9504378 — the NLO-QCD \(K\)-factor for \(h\to gg\).

• [CGHT13] C.-S. Chen, C.-Q. Geng, D. Huang, L.-H. Tsai, “New Scalar Contributions to \(h\to Z\gamma\)”, Phys. Rev. D 87, 075019 (2013), arXiv:1301.4694 — the SM \(h\to Z\gamma\) closed form (loop functions \(I_1,I_2\), prefactor and couplings) this implementation was verified against.

• [PDG] Particle Data Group, Review of Particle Physics — “Kinematics” review (three-body decays and the Dalitz plot); see the current edition at pdg.lbl.gov.