Three-loop relation of quark $$\overline {MS} $$ and pole masses

We calculate, exactly, the next-to-leading correction to the relation between the $$\overline {MS} $$ quark mass, $$\bar m$$ , and the scheme-independent pole mass,M, and obtain $$\begin{gathered} \frac{M}{{\bar m(M)}} \approx 1 + \frac{4}{3}\frac{{\bar \alpha _s (M)}}{\pi } + \left[ {16.11 - 1.04\sum\limits_{i = 1}^{N_F - 1} {(1 - M_i /M)} } \right] \hfill \\ \cdot \left( {\frac{{\bar \alpha _s (M)}}{\pi }} \right)^2 + 0(\bar \alpha _s^3 (M)), \hfill \\ \end{gathered} $$ as an accurate approximation forN F−1 light quarks of massesM i <M. Combining this new result with known three-loop results for $$\overline {MS} $$ coupling constant and mass renormalization, we relate the pole mass to the…