Higgs Precision and Coupling Inference

Higgs precision inference maps production and decay amplitudes to category yields through branching fractions, acceptance, migration, and correlated nuisance parameters. Rate data constrain products of production and decay couplings divided by the total width; absolute couplings require additional assumptions or information. Signal strengths and $\kappa$ modifiers are useful interfaces, while gauge-consistent EFT interpretations require their own basis, truncation, and validity conditions. This page gives that method without presenting a current coupling fit.

Required background. Higgs interactions, production, decay, and pole observables supplies the production modes, partial widths, and pole definitions. Standard Model pseudo-observables and unstable particles supplies the narrow-width hypotheses and the distinction between pole and fiducial quantities.

Helpful background. SMEFT, HEFT, and Standard Model observables supplies gauge-invariant parameterizations when a coupling-modifier interface is insufficient.

Rates are production–decay products

For a sufficiently narrow, isolated Higgs pole and an inclusive-enough category $c$, the expected signal yield can be organized as

$$\nu_c^{\rm sig} =\mathcal L\sum_{i,f} \epsilon_{c,if}\,\sigma_i\, \operatorname{BR}_f, \qquad \operatorname{BR}_f=\frac{\Gamma_f}{\Gamma_H}.$$

Here $i$ labels production mechanisms, $f$ decay channels, $\epsilon_{c,if}$ includes acceptance and efficiency under a stated response model, and $\mathcal L$ is integrated luminosity. The full expectation also contains backgrounds and nuisance dependence. This factorization fails or needs refinement when off-shell contributions, signal–background interference, nearby singularities, or category boundaries resolve the resonance dynamics.

A production–decay signal strength is

$$\mu_{if} =\frac{\sigma_i\operatorname{BR}_f} {\sigma_i^{\rm SM}\operatorname{BR}_f^{\rm SM}}.$$

It is dimensionless, but it is not itself a Lagrangian coupling. It inherits the reference Standard Model prediction, mass and input scheme, perturbative order, and category composition.

In a leading coupling-modifier interface,

$$\sigma_i=\kappa_i^2\sigma_i^{\rm SM}, \qquad \Gamma_f=\kappa_f^2\Gamma_f^{\rm SM}, \qquad \kappa_H^2\equiv\frac{\Gamma_H}{\Gamma_H^{\rm SM}},$$

so that

$$\mu_{if}=\frac{\kappa_i^2\kappa_f^2}{\kappa_H^2}.$$

This expression assumes that the modifier is meaningful for the amplitude and phase-space region involved, that new tensor structures do not alter acceptance unnoticed, and that the narrow-width decomposition is valid. Loop-induced production or decay generally depends on coherent sums of amplitudes; replacing the whole loop by one positive modifier discards sign and phase information unless it is explicitly defined as an effective parameter. The original LHC Higgs Cross Section Working Group recommendations state these hypotheses and distinguish resolved from effective loop modifiers David et al. 2012, §§2.2–3.2, pp. 3–9, PDF.

The total-width flat direction

On-shell rates alone expose a simple degeneracy. Scale every visible production and decay modifier by a common $a$,

$$\kappa_i\to a\kappa_i, \qquad \kappa_f\to a\kappa_f, \qquad \kappa_H^2\to a^4\kappa_H^2.$$

Then $\mu_{if}$ is unchanged. Starting from a Standard Model reference and scaling all visible couplings by $a$, visible partial widths grow as $a^2$, while keeping the on-shell rates fixed requires a total width proportional to $a^4$; an unobserved or otherwise unconstrained contribution can supply the difference where physically allowed. Thus rate-only data do not determine an absolute coupling scale without a width hypothesis David et al. 2012, §2.2, pp. 3–4, PDF.

Common ways to break or restrict the direction introduce assumptions that belong in the result:

Interface	Additional information or hypothesis	Limitation to state
No unobserved width	$\Gamma_H$ is the sum of the modeled visible partial widths	Excludes invisible, undetected, or exotic contributions by construction
Coupling bound	One or more modifiers are bounded by a model assumption	Not a model-independent consequence of rate data
Direct line shape	The width is resolved from the resonance profile	Requires instrumental resolution and a complete pole/background model
Off-shell comparison	On- and off-shell amplitudes share the assumed coupling relations	Sensitive to new continuum amplitudes, phases, energy dependence, and EFT validity
Gauge-invariant EFT	Multiple rates and shapes constrain Wilson coefficients	Depends on basis, input scheme, truncation, running, and kinematic domain

An off-shell analysis is therefore not simply “a width measurement.” Its interpretation requires the same on/off-shell coupling structure, interference model, absence or parameterization of new continuum effects, and validity over the probed invariant masses.

Differential categories and response

When shapes matter, replace one efficiency by a response matrix. For truth or fiducial bins $k$ and reconstructed categories $c$,

$$\nu_c(\mathbf p,\boldsymbol\eta) =\mathcal L\sum_k R_{ck}(\boldsymbol\eta)\, \sigma_k(\mathbf p,\boldsymbol\eta) +b_c(\boldsymbol\eta).$$

$R_{ck}$ includes migration and selection under a stated object definition; $b_c$ is the background expectation. New Lorentz structures can change both $\sigma_k$ and $R_{ck}$, so reweighting only an inclusive rate is not generally sufficient. A forward-folded likelihood retains the observed categories and response nuisances. An unfolded spectrum is a derived estimate whose covariance also depends on the response, regularization, and prior choices.

The reusable hierarchy is:

Parameterization	Object fitted	What it preserves	What it does not supply automatically
Category signal strengths	Independent or grouped yield multipliers	Close connection to released categories	A unique coupling or field-theory interpretation
Production–decay signal strengths	$\mu_{if}$	Factorized rate information	Absolute width, phases, or altered acceptance
$\kappa$ modifiers	Ratios of selected amplitudes, rates, or widths	Compact departures around a reference model	Gauge completion, loop renormalization, or EFT validity
Higgs pseudo-observables	Pole residues or form-factor coefficients	Amplitude structures under explicit projections	A unique ultraviolet model
SMEFT or HEFT	Wilson coefficients in a basis and expansion	Symmetry relations and systematic operator content	Validity beyond the truncation or measured phase space

For an EFT prediction in bin $k$,

$$\sigma_k =\sigma_k^{\rm SM} +\sum_a\frac{C_a}{\Lambda^2}\sigma_{k,a}^{(1)} +\sum_{a,b}\frac{C_aC_b}{\Lambda^4}\sigma_{k,ab}^{(2)} +\cdots.$$

Keeping the quadratic dimension-six term while dropping all dimension-eight interference is a truncation choice, not a theorem. The fit must state which orders are retained, how missing terms are assessed, and which bins pass a predeclared validity condition. A complete SMEFT review of input schemes, basis dependence, and Higgs interpretations is given in Brivio and Trott 2019, §§4–6.

Likelihood and theory correlations

Let $n_c$ be observed category counts and $a$ auxiliary measurements constraining nuisances. A schematic statistical model is

$$L(\mathbf p,\boldsymbol\eta) =\prod_c \operatorname{Pois} \!\left(n_c\mid\nu_c(\mathbf p,\boldsymbol\eta)\right) \prod_r p_r(a_r\mid\boldsymbol\eta).$$

The same luminosity, calibration, branching-ratio, scale, PDF, or missing-order nuisance can affect many categories and channels. Naming it independently in each channel destroys the correlation; multiplying two likelihoods that each already constrain the same auxiliary nuisance counts that constraint twice. A reusable result should publish the statistical model or, when a Gaussian approximation is justified, an ordered covariance with enough information to identify its sources.

Theory variations need a causal interpretation. A common scale variation across production bins may induce a coherent normalization and migration pattern; a branching-ratio input can correlate distinct decay channels; an uncertainty shared between production and decay must be propagated through the ratio $\Gamma_f/\Gamma_H$. Treating every quoted percentage as an independent log-normal nuisance usually overstates the number of directions and can distort the flat direction.

The release record should bind the following fields:

Field	Minimum content
Observable/category	Stable-particle or reconstructed definition, bins, units, production/decay composition
Reference prediction	Parameter inputs, mass and width convention, perturbative order, calculation version
Response	Acceptance and migration model, nuisance dependence, validity domain
Inference object	Observed and auxiliary data, likelihood or covariance version, nuisance definitions
Correlations and overlap	Sources shared across categories/releases; common events or constraints
Identity and lifecycle	Dataset/table identifiers, DOI or stable URL, exact version, corrections and supersession

Independent checks

Standard Model limit. Setting every modifier to one and any BSM width to zero must reproduce the reference prediction category by category, not merely in the inclusive sum.

Flat-direction test. Apply the common rescaling above. An on-shell rate-only likelihood must remain invariant when the total-width degree of freedom is allowed to transform accordingly. Apparent sensitivity signals an implicit width prior or acceptance dependence that must be exposed.

Response normalization. For each truth bin, the sum over reconstructed destinations equals the modeled acceptance and cannot exceed one for an ordinary probability response. Check migration orientation before multiplying matrices.

Dimensions and positivity. $\sigma$ has mass dimension minus two, $\Gamma$ mass dimension one, and $\mu$ and $\kappa$ are dimensionless. Expected event yields and physical widths must remain nonnegative; amplitude-level interference terms need not be.

Scheme and group factors. Coupling normalizations, loop color factors, electroweak inputs, and branching fractions must use the same reference convention. Reproducing a benchmark partial width or cross section tests this interface independently.

EFT stability. Repeat the inference with the declared high-energy bins removed and with plausible higher-order terms. Large motion relative to the stated precision indicates a truncation or validity problem, not automatically evidence for a parameter shift.

Common pitfalls

Calling $\mu$ a coupling. A signal strength is a ratio of production times branching fraction. Translate it through the width and acceptance model before assigning a coupling meaning.

Fixing the width invisibly. If $\Gamma_H$ is computed only from modeled channels, the fit has imposed a no-unobserved-decay assumption. State it and test an alternative when the question requires model independence.

Applying constant modifiers to loop amplitudes. Interfering Standard Model and new amplitudes carry signs, phases, and kinematic dependence. Use an amplitude-level parameterization when those features matter.

Choosing EFT bins after seeing the fit. A post-fit validity mask changes the sampling procedure and can bias inference. Freeze and version the mask before fitting.

Informal self-check

Assume all visible Higgs couplings are scaled by $a>1$ relative to the reference model. What additional unobserved width preserves every on-shell production–decay rate in the leading factorized approximation?

Solution

Production cross sections and visible partial widths scale as $a^2$, so rate numerators scale as $a^4$. The total width must therefore be $\Gamma_H=a^4\Gamma_H^{\rm SM}$. The visible contribution is $a^2\Gamma_H^{\rm SM}$ if the reference visible channels exhaust the reference width, so the required extra contribution is

$$\Gamma_{\rm extra} =(a^4-a^2)\Gamma_H^{\rm SM}.$$

This construction is a demonstration of the rate degeneracy, not a claim that such an extra width is realized.

Handoffs

• Return to pseudo-observables and unstable particles for pole expansions and narrow-width limitations.
• Use collider measurements and likelihood provenance to connect categories to response matrices and released statistical models.
• Use correlated Standard Model fits to combine channels with shared nuisances and goodness-of-fit tests.
• Use SMEFT, HEFT, and Standard Model observables for a symmetry-complete interpretation with declared power counting.

References

• Brivio, Ilaria, and Michael Trott. “The Standard Model as an Effective Field Theory.” Physics Reports 793 (2019) 1–98. DOI · Open PDF
• David, André, et al. “LHC HXSWG Interim Recommendations to Explore the Coupling Structure of a Higgs-Like Particle.” CERN-PH-TH-2012-284 (2012). Open PDF
• de Florian, Daniel, et al., eds. Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector. CERN Yellow Reports: Monographs 2 (2017). DOI · Open PDF