In this paper we build a comprehensive analysis framework to perform direct extraction of all possible effective Higgs couplings to neutral electroweak gauge bosons in the decay to electrons and muons, the so called 'golden channel'. Our framework is based on a maximum likelihood method constructed from analytic expressions of the fully differential cross sections for h → 4ℓ and for the dominant irreducible qq → 4ℓ background, where 4ℓ = 2e2µ,4e,4µ. Detector effects are included by an explicit convolution of these analytic expressions with the appropriate transfer function over all center of mass variables. Using the full set of decay observables, we construct an unbinned 8-dimensional detector-level likelihood function which is continuous in the effective couplings and includes systematic uncertainties. We consider all possible ZZ, Zyud and yy couplings, allowing for general CP odd/even admixtures and any possible phases. We describe how the convolution is performed and demonstrate the validity and power of the framework with a number of supporting checks and example fits. The framework can be used to perform a variety of multi-parameter extractions, including their correlations, to determine the Higgs couplings to neutral electroweak gauge bosons using data obtained at the LHC and other future colliders.
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