Higgs Interactions, Production, Decay, and Pole Observables

A defensible Higgs prediction begins with a renormalized electroweak parameter set and a complex-pole definition, builds complete production and decay amplitudes, and only then maps them to inclusive, fiducial, or pseudo-observables with correlated uncertainties. This chain prevents an isolated coupling, diagram, branching fraction, or line-shape parameter from being mistaken for a directly measured quantity.

Required background. Electroweak renormalization and input schemes supplies the parameters, tadpole prescription, pole conventions, and perturbative order. Unstable-particle observables and resonance approximations supplies the analytic pole expansion and the conditions for resonance factorization.

Helpful background. Higgs self-interactions and the scalar potential supplies the distinction between a renormalized self-coupling and a multi-Higgs observable.

The map below separates the evergreen amplitude and scheme relations from the evidence layer. Higgs self-couplings, symmetry breaking, Yukawa couplings, high-energy cancellations, and renormalized pole amplitudes meet at a precisely declared observable before any dated measurement or combination is interpreted.

Higgs predictions require compatible symmetry, renormalization, pole, and observable definitions. Measurements, likelihood versions, combinations, and current interpretations remain in dated evidence records; the diagram is schematic.

Couplings and production mechanisms

For $Q=T_3+Y$, $H\sim(\mathbf1,\mathbf2)_{1/2}$, and

$$D_\mu H= \left(\partial_\mu-igT^aW_\mu^a-ig'\frac12B_\mu\right)H,$$

the tree-level terms linear in the radial field include

$$\mathcal L\supset \frac{2m_W^2}{v}hW_\mu^+W^{-\mu} +\frac{m_Z^2}{v}hZ_\mu Z^\mu -\sum_f\frac{m_f}{v}h\bar ff -\frac{\lambda_3}{3!}h^3 .$$

These vertices seed the standard production classes, but a prediction also requires QCD and electroweak radiation, parton distributions for hadronic beams, unstable-particle treatment, and a measurement definition. The interactions follow from the Higgs kinetic and Yukawa terms as developed in Schwartz 2014, §§29.1 and 29.3, pp. 584–599.

Production class	Amplitude-level origin	Definition and consistency questions
Gluon fusion, $gg\to h+X$	colored-particle loops, with quark Yukawa couplings in the minimal model	quark-mass scheme, exact-mass versus effective description, QCD order, PDFs, jet-bin definition
Vector-boson fusion, $qq\to qqh+X$	two electroweak quark currents joined by the $hVV$ interaction	fiducial VBF definition, interference with other $hjj$ amplitudes, electroweak and nonfactorizable corrections
Associated $Vh$ production	an electroweak current produces an off-shell $V$, or a loop-induced partonic channel, followed by $Vh$	separate partonic channels and $V$-decay treatment; avoid folding a loop-induced component into a different perturbative order
Heavy-quark associated production	$t\bar th$, single-top–Higgs, or bottom-associated amplitudes probe Yukawa interactions	heavy-flavor scheme, mass definition, interference among gauge and Yukawa diagrams, decay and acceptance treatment
Multi-Higgs production	self-interaction diagrams interfere with boxes, Yukawa, gauge, and possible contact terms	parameter basis, complete gauge-invariant amplitude, resonant substructure, and correlated variations

This is a mechanism classification, not a ranking by current rate or sensitivity. The LHC Higgs Cross Section Working Group organizes production, decay, off-shell effects, fiducial quantities, and pseudo-observables in this way in de Florian et al. 2017, Parts I and III.

At fixed order, a label such as “VBF” or “$Vh$” can refer to a calculational signal definition rather than an exactly isolated set of observed events. The page or dataset using the label must state the partonic channels, perturbative orders, phase-space cuts, overlap removal, and interference retained.

Decay amplitudes and branching fractions

The same renormalized couplings generate several structurally different decay classes.

Decay class	Leading structure	Required qualifications
$h\to f\bar f$	Yukawa coupling $m_f/v$	fermion-mass and Yukawa scheme, QCD/QED radiation, infrared-safe final-state definition
$h\to WW$ or $ZZ$, followed by decays	$hVV$ coupling and electroweak currents	off-shell vector propagators when required, spin correlations, identical-particle interference, fiducial final state
$h\to gg$	colored-particle loop	QCD order, mass dependence, effective-theory validity, inclusive radiation convention
$h\to\gamma\gamma$ or $Z\gamma$	coherent sum of charged-particle loops	relative phases and interference, electroweak inputs, $Z$-decay and photon isolation definitions
multi-scalar, invisible, or unobserved channels	only when allowed by the declared spectrum and parameterization	thresholds, new-state assumptions, total-width closure, and model dependence

For a mutually exclusive and complete set of decay definitions,

$$\Gamma_H=\sum_i\Gamma_i, \qquad \mathcal B_i=\frac{\Gamma_i}{\Gamma_H}.$$

Completeness is an assumption to check, not a notational fact. If channels are omitted or an unobserved component is allowed, the total width needs an explicit additional parameter. Likewise, partial widths are theory-defined pseudo-observables only after soft radiation, off-shell daughter states, and phase-space conventions are fixed.

Pole expansion and pseudo-observables

Let $s$ be the invariant carried by a scalar resonance. Near an isolated simple pole, an amplitude can be organized as

$$\mathcal A_{i\to f}(s) =\frac{R_{if}}{s-s_H}+N_{if}(s), \qquad s_H=\mu_H^2-i\mu_H\gamma_H .$$

The pole $s_H$, its consistently defined residue, and the full amplitude are gauge-independent; selected resonant diagrams away from the pole need not be. Analytic pole expansion provides the clean separation between resonance parameters and regular background Stuart 1991, pp. 113–119.

For a single nondegenerate state, the residue factorizes into production and decay factors after external-state conventions are fixed. This motivates pole pseudo-observables such as effective residues and partial widths. They remain derived quantities: an experiment observes distributions of stable final states, while theory and a likelihood map those distributions to the pole parameterization.

Near a narrow, isolated resonance one often uses

$$\sigma(i\to H\to f) \simeq \sigma(i\to H)\, \frac{\Gamma_f}{\Gamma_H}.$$

This narrow-width approximation requires all of the following:

• $\gamma_H/\mu_H$ is small over the relevant resolution scale;
• production, decay, luminosity, and phase-space factors vary slowly across the pole;
• cuts and bin boundaries do not strongly distort the resonance region;
• interference with $N_{if}$ is negligible or corrected;
• nearby thresholds or resonances do not spoil the single-pole expansion.

If any condition fails, compute the stable-final-state amplitude including the pole and continuum together. An off-shell distribution can constrain the same couplings that appear on shell, but a conversion into a total-width statement is model-dependent unless the relation between on-shell and off-shell amplitudes is specified.

Inclusive, fiducial, and template observables

An inclusive cross section integrates a defined final state over a broad phase space. A fiducial cross section instead includes a particle-level measurement function $F_{\rm fid}$:

$$\sigma_{\rm fid} =\sum_{a,b}\int dx_1dx_2\, f_a(x_1,\mu_F)f_b(x_2,\mu_F) \int d\Phi\, \left|\mathcal M_{ab\to f}\right|^2 F_{\rm fid}(\Phi).$$

This schematic equation records the ingredients that must agree between theory and measurement: incoming-state convention, PDFs, scales, stable final state, radiation and recombination rules, and fiducial cuts. Detector unfolding and migration introduce another response model and covariance.

Template cross sections partition a fiducial region into bins designed to retain production or kinematic information. Their virtue is a shorter extrapolation than a total inclusive rate; their limitation is dependence on bin definitions, migration modeling, and the theory used to connect bins to couplings. The working-group treatment of fiducial and simplified-template observables is given in de Florian et al. 2017, Part III, chs. 2–3.

Do not combine an inclusive theory prediction with a fiducial measured number by applying an acceptance from a different model without propagating the resulting uncertainty. Also do not call a fitted signal-strength parameter a coupling until the production, decay, width, and unseen-channel assumptions have been written down.

Correlated prediction uncertainties

For small variations of the partial widths,

$$\frac{\delta\mathcal B_i}{\mathcal B_i} =\frac{\delta\Gamma_i}{\Gamma_i} -\sum_j\mathcal B_j \frac{\delta\Gamma_j}{\Gamma_j}.$$

Thus a change in one partial width moves every branching fraction through the shared denominator. If $C_\Gamma$ is the partial-width covariance, define

$$J_{ij}=\frac{\partial\mathcal B_i}{\partial\Gamma_j} =\frac{\delta_{ij}-\mathcal B_i}{\Gamma_H}, \qquad C_{\mathcal B}=J\,C_\Gamma J^{\mathsf T}.$$

Independent quadrature of branching-fraction errors discards this anticorrelation. The same issue appears across production bins when a common scale, PDF eigenvector, coupling, shower model, or luminosity nuisance moves several bins coherently.

A complete theory-uncertainty statement separates and correlates, as applicable:

• missing higher orders in QCD and electroweak expansions, including the scale-variation prescription;
• PDFs, $\alpha_s$, running masses, and other parametric inputs;
• heavy-mass expansions, matching, resummation, and threshold choices;
• pole, narrow-width, interference, and off-shell approximations;
• parton shower, hadronization, underlying-event, and acceptance modeling;
• numerical integration, interpolation, and finite simulation samples.

Scale variation probes sensitivity to uncalculated terms; it is not a probability distribution by itself. Scheme and matching variations can supply complementary diagnostics, but overlapping variations should not be counted twice.

Reproducible evidence interface

Before a numerical prediction or inference is compared, record

Layer	Minimum information
Physics state	model and parameter basis; Higgs pole, residue, and total-width conventions
Calculation	process and decay definition; perturbative orders; scales; PDFs; masses; electroweak input and tadpole schemes
Approximation	effective operators, resummation, narrow-width or off-shell treatment, interference, validity tests
Measurement	collision system and energy; dataset period; stable final state; fiducial bins; unfolding or response model
Uncertainty	named sources, correlations, covariance or nuisance implementation, missing-order prescription
Provenance	collaboration or author, release identifier, version, publication date, files or tables used, corrections and supersession
Inference	likelihood or test statistic, priors if any, parameter ranges, fixed assumptions, and goodness-of-fit diagnostics

Current masses, widths, rates, limits, combinations, production rankings, and compatibility statements are time-dependent evidence claims. They should be stated only with a named release or calculation, its dataset period and version, a dated evidence cutoff, and the likelihood or covariance needed to reproduce the inference. No such current numerical or status claim is made on this durable methods page.

Checks and failure modes

Pole check. Vary gauge-fixing parameters in the complete calculation. The complex pole and physical stable-state observable must remain unchanged through the computed order.

Factorization check. Compare the full line shape with the narrow-width result under the actual cuts. A small $\gamma_H/\mu_H$ alone does not control interference or acceptance variation.

Closure check. Verify whether the fitted branching fractions sum to one by assumption. If an unseen width is allowed, include it explicitly in $\Gamma_H$ and the covariance.

Coupling check. Change one renormalized coupling only within a gauge-consistent parameterization. Rescaling a selected diagram can violate Ward identities and misstate interference.

Provenance check. A table copied from a later theory release may use different PDFs, masses, or uncertainty correlations from the likelihood. Match versions before combining them.

Common pitfalls

Equating a production label with an observed event category. Categories have migrations and contributions from several mechanisms. Use a response matrix or a documented purity model.

Turning an off-shell tail directly into a width. The inference requires assumptions relating on-shell residues, off-shell amplitudes, and any new continuum contributions.

Adding theory uncertainties as independent percentages. Common inputs and shared denominators create correlations. Propagate a covariance or explicit nuisance model.

Handoff

A reusable Higgs prediction or inference passes

$$\left\{s_H,R_{if},N_{if};\ \text{production and decay definitions};\ \text{fiducial map};\ \mathcal S,\mu_R,\mu_F,\text{order};\ C_{\rm theory};\ \text{likelihood and provenance}\right\}.$$

Use this object in Precision Standard Model. Time-dependent experimental combinations and reinterpretations belong to Research: EFT and Standard Model Tests, with their dated evidence and likelihood artifacts attached.

References

• de Florian, D., C. Grojean, F. Maltoni, C. Mariotti, A. Nikitenko, M. Pieri, P. Savard, M. Schumacher, R. Tanaka, et al., eds. Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector. CERN Yellow Reports: Monographs 2/2017, CERN-2017-002-M, 2017, pp. 1–869. DOI. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §§29.1 and 29.3, pp. 584–599. DOI.
• Stuart, Robin G. “Gauge Invariance, Analyticity and Physical Observables at the $Z^0$ Resonance.” Physics Letters B 262, no. 1 (1991): 113–119. DOI.