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Exact results for ZmOS and Z2OS with two mass scales and up to three loops

2020/08/03 by Matteo Fael, Kay Schönwald, Matthias Steinhauser
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Function (biology) #Harmonic #Harmonic oscillator #Iterated function #Mathematical functions and polynomials #Numerical analysis #Order (exchange) #Quantum Chromodynamics and Particle Interactions #Quark #Renormalization #hep-ph

paper · pdf · doi:10.1007/jhep10(2020)087

14 pages, 3 figures

arxiv created 2020/08/03 · openalex created_date 2020/08/10 · openalex publication_date 2020/10/01 · arxiv updated 2020/10/28 · openalex updated_date 2026/08/05

Abstract

A bstract We consider the on-shell mass and wave function renormalization constants ZmOS <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 Z2OS <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 √1-τ2 <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 √1-τ2/τ <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 O <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> </mml:math> ( ϵ 2 ) and 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.

Citations