Approximate N$$^{3}$$LO parton distribution functions with theoretical uncertainties: MSHT20aN$$^3$$LO PDFs

Abstract

Abstract We present the first global analysis of parton distribution functions (PDFs) at approximate N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO in the strong coupling constant $$\alpha _{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> , extending beyond the current highest NNLO achieved in PDF fits. To achieve this, we present a general formalism for the inclusion of theoretical uncertainties associated with the perturbative expansion in the strong coupling. We demonstrate how using the currently available knowledge surrounding the next highest order (N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO) in $$\alpha _{s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> can provide consistent, justifiable and explainable approximate N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO (aN $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO) PDFs. This includes estimates for uncertainties due the currently unknown N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO ingredients, but also implicitly some missing higher order uncertainties (MHOUs) beyond these. Specifically, we approximate the splitting functions, transition matrix elements, coefficient functions and K -factors for multiple processes to N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO. Crucially, these are constrained to be consistent with the wide range of already available information about N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO to match the complete result at this order as accurately as possible. Using this approach we perform a fully consistent approximate N $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO 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 aN $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO PDFs and the standard NNLO PDF set, and study the impact of using aN $$^{3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow/> <mml:mn>3</mml:mn> </mml:msup> </mml:math> LO PDFs on the LHC production of a Higgs boson at this order. Finally, we provide guidelines on how these PDFs should be used in phenomenological investigations.

Topics & Concepts