NNLO compatibility between pQCD theory and phenomenology in determination of the $b$-quark pole and \MSbar running masses

Abstract: This contribution attempts to determine the $b$-quark pole mass $M_b$ and \MSbar running mass $\overline{m}_b$ with two different approaches at the next-to-next-to-leading order (NNLO) corrections. At the first approach, we derive a relation between the $b$-quark pole mass $M_b$ and its \MSbar running mass $\overline{m}_b$ at the NNLO corrections based on the perturbative Quantum Chromo Dynamics (pQCD) predictions. At the second approach, we extract numerical values of the $b$-quark pole and \MSbar running masses based on the NNLO phenomenology of H1 and ZEUS Collaborations combined beauty vertex production experimental data. Then we discuss about the compatibility between the pQCD theory results and phenomenology approach in determination of the $b$-quark pole and \MSbar running masses at the NNLO corrections. Also, we investigate the role and influence of the $b$-quark mass as an extra degree of freedom added to the input parameters of the Standard Model Lagrangian, on the improvement of the uncertainty band of the proton parton distribution functions (PDFs) and particularly on the gluon distribution.