Gfitter - A Generic Fitter Project for HEP Model Testing

Introduction
Constraints on Higgs couplings
Total width of the Higgs boson
Invisible and undetected branching fraction
Assumptions and caveats

Introduction

Modified Higgs couplings are studied in the κ framework, where modifications of the Higgs-boson couplings are described by coupling strength modifiers at tree level. The coupling to vector bosons is modified by κ_V (assuming custodial symmetry, κ_Z = κ_W), the couplings to fermions by κ_F or by individual modifiers κ_t, κ_b, κ_c, κ_τ, κ_μ, and the loop-induced processes by κ_g, κ_γ and κ_Zγ. In the Standard Model all κ_i = 1. More details are given in the report by the Higgs boson working group.

Electroweak precision observables (EWPO) constrain the Higgs sector through the virtual contributions of the Higgs boson to the gauge-boson self-energies. At leading-logarithmic accuracy, a universal rescaling of the HVV coupling modifies the oblique parameters S and T, with U = 0. This provides a constraint on κ_V that is independent of the Higgs signal-strength measurements and does not require any assumption on the total width. Combining the two gives access to the total Higgs-boson width and to decays into final states that are not experimentally accessible.

The effective description is not renormalisable by itself, so a cut-off scale is required. We use Λ = 4πv ≈ 3 TeV as default, the naive-dimensional-analysis estimate of the strong-coupling scale of the electroweak symmetry-breaking sector, and also quote results for Λ = 1 TeV. (This scale is written as Λ with a tilde in the figures; the cut-off of the effective description is Λ_* = Λ / |1 - κ_V²|^1/2.)

The EWPO input is described in the SM section. The Higgs signal strengths are taken from the most recent ATLAS and CMS combinations. Full details are given in our latest publication.

Constraints on Higgs couplings

Using the general parametrisation, in which all fermion couplings and loop-induced contributions have independent modifiers, the ATLAS and CMS signal-strength measurements alone give results consistent with the Standard Model. For the HVV modifier the combined fit gives κ_V = 1.035^+0.051_-0.053. The largest deviation is found for κ_Zγ = 1.47^+0.25_-0.28, corresponding to about 1.7σ.

From the EWPO alone, without any Higgs signal-strength input, we obtain

κ_V = 1.009^+0.012_-0.010 for Λ = 3 TeV

κ_V = 1.011^+0.017_-0.013 for Λ = 1 TeV

This agrees with the determination from the Higgs signal strengths in the resolved parametrisation, κ_V = 1.005 ± 0.017, but is more precise, owing to the high experimental and theoretical precision of the EWPO, and it does not require the assumption of no invisible or undetected decays.

Constraints on the κ_i modifiers in the general parametrisation, using Higgs signal-strength measurements from ATLAS (blue), CMS (red) and their combination (green). Since all κ_i enter quadratically, only their absolute values are constrained. Arrows indicate central values outside the plotting range.	.pdf
Constraints in the κ_V-κ_F plane from Higgs signal-strength measurements in the resolved parametrisation (orange), from the EWPO (green), and from their combination (blue). Fits including EWPO use Λ = 3 TeV. The 68% and 95% contours are computed for one degree of freedom in the EWPO-only fit, and for two degrees of freedom otherwise.	.pdf

Total width of the Higgs boson

By providing an independent constraint on κ_V, the EWPO reduce the freedom in the Higgs-coupling fit and allow a determination of the total Higgs-boson width without specifying a particular extension of the Standard Model. The total-width modifier κ_H² = Γ_H,tot / Γ_H,SM is treated as a free parameter of the fit. The results below are for Λ = 3 TeV.

bosonic parametrisation (11 signal strengths + 24 EWPO)	Γ_H,tot = 4.36^+0.63_-0.50 MeV
effective parametrisation (31 signal strengths + EWPO)	Γ_H,tot = 4.08^+0.43_-0.37 MeV
resolved parametrisation (59 signal strengths + EWPO)	Γ_H,tot = 3.78^+0.30_-0.27 MeV

These values are to be compared with the SM prediction Γ_H,SM = 4.10 ± 0.06 MeV. Including more signal-strength information tightens the constraint, at the cost of stronger assumptions on the Higgs-boson couplings. The effective and resolved parametrisations determine the total width with a relative precision of about 10% or better.

Reducing Λ from 3 TeV to 1 TeV shifts the central values by -0.04 MeV, independently of the parametrisation, and increases the uncertainties by about 5%, 10% and 15% for the bosonic, effective and resolved parametrisations, respectively.

Scans of Δχ² versus the total-width modifier κ_H² for the different parametrisations. Solid and dashed lines correspond to Λ = 3 TeV and 1 TeV, respectively.

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Invisible and undetected branching fraction

Higgs-boson decays into final states that are not experimentally accessible at the LHC enter the total width through Γ_H,tot = Γ_H(κ) / (1 - B_inv - B_undet), where B_inv and B_undet are the invisible and undetected branching fractions. The total-width constraint from the combined EWPO and signal-strength fit therefore translates directly into a constraint on B_i+u = B_inv + B_undet. In this formulation B_i+u is determined from the data and does not rely on the SM prediction for the total width, nor on direct searches for invisible Higgs decays.

The limits below are at 95% CL, obtained by integrating the likelihood over the physical region B_i+u > 0, for Λ = 3 TeV. Expected limits are given in parentheses.

bosonic parametrisation	B_i+u < 0.27 (0.23 expected)
general parametrisation	B_i+u < 0.17 (0.16 expected)
resolved parametrisation	B_i+u < 0.09 (0.08 expected)

In the bosonic parametrisation only the HVV coupling is modified, and all other coupling modifiers, most notably the fermionic ones, are set to their SM values. In the general parametrisation B_i+u is fitted together with all κ_i modifiers, so that no assumption on any coupling modifier is needed. The resolved parametrisation gives the tightest bound, owing to its stronger assumptions on the fermion couplings and the loop-induced processes.

None of these bounds rely on the assumption |κ_V| ≤ 1, which is commonly used when only signal-strength measurements are available. Imposing that assumption with signal strengths alone gives B_i+u < 0.07 at 95% CL.

Scans of Δχ² versus B_i+u = B_inv + B_undet from the combined fit to EWPO and Higgs signal strengths, for the different parametrisations. Solid and dashed lines correspond to Λ = 3 TeV and 1 TeV, respectively. The signal-strength-only result with |κ_V| ≤ 1 is shown for comparison.

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Assumptions and caveats

The constraints on κ_V, Γ_H,tot and B_i+u given above rely on a restricted effective interpretation, and the following assumptions should be kept in mind:

The EWPO enter only through the leading-logarithmic contribution of a universal HVV rescaling to the oblique parameters S and T, with U = 0.
The SM loop structure is otherwise unchanged, custodial symmetry is unbroken, and there are no BSM effects beyond oblique corrections. Modified fermion couplings, vertex corrections and direct loop contributions from new states to the EWPO are not included.
A rescaled HVV coupling alone is not a complete renormalisable theory, so the cut-off dependence should be regarded as a leading-logarithmic estimate. A systematic effective-field-theory treatment would require the relevant operators, counterterms and short-distance assumptions.
Beyond the one-loop approximation, additional cut-off-sensitive contributions may arise unless they are suppressed or cancelled by the UV completion.
The quoted results assume Λ = 3 TeV unless stated otherwise; the dependence on this choice is quantified above.

The bounds therefore apply to this specific effective description rather than to a general BSM scenario.