Exact results for $$ {Z}_m^{\mathrm{OS}} $$ and $$ {Z}_2^{\mathrm{OS}} $$ with two mass scales and up to three loops
Matteo Fael, Kay Schönwald, Matthias Steinhauser
Abstract
A bstract We consider the on-shell mass and wave function renormalization constants $$ {Z}_m^{\mathrm{OS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mi>m</mml:mi> <mml:mi>OS</mml:mi> </mml:msubsup> </mml:math> and $$ {Z}_2^{\mathrm{OS}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> <mml:mi>OS</mml:mi> </mml:msubsup> </mml:math> up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $$ \sqrt{1-{\tau}^2} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> </mml:math> and $$ \sqrt{1-{\tau}^2}/\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mi>τ</mml:mi> </mml:math> which extends the findings from ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> ( ϵ 2 ) and $$ \mathcal{O} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> ( ϵ ) at two- and three-loop order which are crucial ingredients for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.