Polarized and unpolarized gluon PDFs: Generative machine learning applications for lattice QCD matrix elements at short distance and large momentum

Lattice quantum chromodynamics (QCD) calculations share a defining challenge by requiring a small finite range of spatial separation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>z</a:mi></a:math> between quark/gluon bilinears for controllable power corrections in the perturbative QCD factorization, and a large hadron boost <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>p</c:mi><c:mi>z</c:mi></c:msub></c:math> for a successful determination of collinear parton distribution functions (PDFs). However, these two requirements make the determination of PDFs from lattice data very challenging. We present the application of generative machine learning algorithms to estimate the polarized and unpolarized gluon correlation functions utilizing short-distance data and extending the correlation up to <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>z</e:mi><e:msub><e:mi>p</e:mi><e:mi>z</e:mi></e:msub><e:mo>≲</e:mo><e:mn>14</e:mn></e:math>, surpassing the current capabilities of lattice QCD calculations. We train physics-informed machine learning algorithms to learn from the short-distance correlation at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>z</g:mi><g:mo>≲</g:mo><g:mn>0.36</g:mn><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>fm</g:mi></g:math> and take the limit, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>p</i:mi><i:mi>z</i:mi></i:msub><i:mo stretchy="false">→</i:mo><i:mi>∞</i:mi></i:math>, thereby minimizing possible contamination from the higher-twist effects for a successful reconstruction of the polarized gluon PDF. We also expose the bias and problems with underestimating uncertainties associated with the use of model-dependent and overly constrained functional forms, such as <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msup><l:mi>x</l:mi><l:mi>α</l:mi></l:msup><l:mo stretchy="false">(</l:mo><l:mn>1</l:mn><l:mo>−</l:mo><l:mi>x</l:mi><l:msup><l:mo stretchy="false">)</l:mo><l:mi>β</l:mi></l:msup></l:math> and its variants to extract PDFs from the lattice data. We propose the use of generative machine learning algorithms to mitigate these issues and present our determination of the polarized and unpolarized gluon PDFs in the nucleon.