Master Integrals for the 2-loop QCD virtual corrections to the Forward-Backward Asymmetry

TL;DR

The paper delivers a complete analytic determination of the Master Integrals required for the two-loop QCD virtual corrections to the forward-backward asymmetry A_FB^{q\bar{q}} in e^+e^- annihilation, retaining arbitrary space-like Q^2 and finite heavy-quark mass m. By combining IBP/LI reductions with the differential equations method, it identifies 18 new MIs (in addition to 17 previously computed) and expresses their D-4 Laurent coefficients in terms of 1D harmonic polylogarithms up to weight 4. The results include detailed expansions for all 2-loop topologies with 3–6 denominators and provide explicit representations for reducible 6-denominator integrals, alongside high-energy expansions (Q^2 >> m^2) to facilitate cross-checks and phenomenological applications. This analytic, polylogarithmic framework enables precise, scalable evaluation of two-loop form factors relevant to A_FB in current and future collider analyses.

Abstract

We present the Master Integrals needed for the calculation of the two-loop QCD corrections to the forward-backward asymmetry of a quark-antiquark pair produced in electron-positron annihilation events. The abelian diagrams entering in the evaluation of the vector form factors were calculated in a previous paper. We consider here the non-abelian diagrams and the diagrams entering in the computation of the axial form factors, for arbitrary space-like momentum transfer Q^2 and finite heavy quark mass m. Both the UV and IR divergences are regularized in the continuous D-dimensional scheme. The Master Integrals are Laurent-expanded around D=4 and evaluated by the differential equation method; the coefficients of the expansions are expressed as 1-dimensional harmonic polylogarithms of maximum weight 4.

Figures (6)

• Figure 1: The 2-loop vertex diagrams involved in the calculation of $A_{FB}$ at order ${\mathcal{O}}(\alpha_{S}^{2})$. The curly lines are massless gluons; the double straight lines, quarks of mass $m$; the single straight lines, massless quarks. All the external fermion lines are on the mass-shell: $p_1^2 = p_2^2 = -m^{2}$, the double lines; $p_1^2 = p_2^2 = 0$, the single lines. The dashed line on the r.h.s. carries momentum $Q=p_{1}+p_{2}$, with the metrical convention $Q^{2}>0$ when $Q$ is space-like.
• Figure 2: The four independent 6-denominator topologies. Internal straight lines carry mass $m$, internal wavy lines are massless; the mass shell conditions are $p_1^2=p_2^2=-m^2$ for the external straight lines and $p_1^2=p_2^2=0$ for the external wavy lines; the dashed line on the right carries momentum $Q=p_1+p_2$, with $Q^2 > 0$ when $Q$ is space-like.
• Figure 3: The set of the 15 independent 5-denominator topologies. The graphical conventions are the same as in Fig. \ref{['fig1bis']}.
• Figure 4: The set of the 15 independent 4-denominator topologies. The graphical conventions are the same as in Fig. \ref{['fig1bis']}.
• Figure 5: The set of the 2 independent 3-denominator topologies. The graphical conventions are the same as in Fig. \ref{['fig1bis']}.