We present the first global analysis of parton distribution functions (PDFs)
at approximate N3LO in the strong coupling constant αs,
extending beyond the current highest NNLO achieved in PDF fits. To achieve
this, we present a general formalism for the inclusion of theoretical
uncertainties from missing higher orders (MHOs) into a PDF fit. We demonstrate
how using the currently available knowledge surrounding the next highest order
(N3LO) in αs can provide consistent, justifiable and explainable
approximate N3LO (aN3LO) PDFs, including estimates for missing higher
order uncertainties (MHOUs). Specifically, we approximate the splitting
functions, transition matrix elements, coefficient functions and K-factors
for multiple processes to N3LO. Crucially, these are constrained to be
consistent with the wide range of already available information about N3LO
to match the complete result at this order as accurately as possible. Using
this approach we perform a fully consistent approximate N3LO global fit
within the MSHT framework. This relies on an expansion of the Hessian procedure
used in previous MSHT fits to allow for sources of theoretical uncertainties.
These are included as nuisance parameters in a global fit, controlled by
knowledge and intuition based prior distributions. We analyse the differences
between our aN3LO PDFs and the standard NNLO PDF set, and study the impact
of using aN3LO PDFs on the LHC production of a Higgs boson at this order.
Finally, we provide guidelines on how these PDFs should be be used in
phenomenological investigations.Comment: 150 pages, 48 figures, 20 tables. Updated LHAPDF Grids available
which include a correction of a minor bug in the non-singlet splitting
function leading to very small changes in fit quality and PDFs, but with no
significant changes to any results or conclusion