  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 $h
\rightarrow 4\ell$ and for the dominant irreducible $q\bar{q} \rightarrow
4\ell$ background, where $4\ell = 2e2\mu, 4e, 4\mu$. 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 $ZZ$, $Z\gamma$ and
$\gamma\gamma$ 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.
