The two-mass contributions to the three-loop massive operator matrix elements $\tilde{A}_{Qg}^{(3)}$ and $Δ\tilde{A}_{Qg}^{(3)}$

TL;DR

This work computes the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $ ilde{A}_{Qg}^{(3)}$ and $Δ ilde{A}_{Qg}^{(3)}$ in QCD, using a semi-analytic approach in $x$-space for a general mass ratio $\eta=m_c^2/m_b^2$. Mellin moments up to $N olinebreak= olinebreak 2000 ext{ (}3000 ext{ for polarized)}$ are obtained independently and cross-checked against the $x$-space results, with the polarized calculation performed in the Larin scheme. The two-mass contributions account for roughly half of the full $O(T_F^2)$ and $O(T_F^3)$ terms, underscoring their phenomenological importance in two-mass VFNS scenarios. A detailed renormalization framework is presented, including mass and coupling renormalization and the treatment of reducible diagrams via the background-field method, and the structure of the two-mass OMEs is analyzed in both Mellin-$N$ and $x$-space representations. While full analytic solutions in Mellin space are not yet possible, the authors develop robust $x$-space semi-analytic techniques, validate them against fixed Mellin moments, and quantify the two-mass impact on the heavy-flavor sector of DIS at three loops.

Abstract

We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $\tilde{A}_{Qg}^{(3)}$ and $Δ\tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 \%$ of the full \textcolor{blue}{$O(T_F^2)$} and \textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.

Figures (4)

• Figure 1: Sample diagrams contributing to the two-mass contributions to the operator matrix element $\tilde{A}_{Qg}^{(3)}$. The thin fermionic lines correspond to heavy quarks of a mass $m_1$ and the thick ones to quarks of a mass $m_2$. The amplitudes are symmetric under the interchange $m_1 \leftrightarrow m_2$.
• Figure 2: The relative difference between different expansions depths of $\tilde{A}_{Qg}^{(3)}$ in $\delta$ to our best approximation which uses 20 expansion terms for $Q^2=50~\rm GeV^2$. The curves show the relative difference to the expansion up to $\delta^{15}$, $k=15$ (blue dotted line), $\delta^{16}$, $k=16$ (red dash-dotted line), $\delta^{17}$, $k=17$ (green dashed line), $\delta^{18}$, $k=18$ (black full line).
• Figure 3: The ratio $R_{Qg}^{(3)}(x,Q^2)$ of the 2-mass contributions to $\tilde{A}_{Qg}^{(3)}$ to the complete contribution of $O(T_F^2)$ and $O(T_F^3)$ to ${A}_{Qg}^{(3)}$ as defined in Eq. \ref{['eq:ratio']} in the unpolarized case. $Q^2=30~\rm GeV^2$ dotted line (blue), $Q^2=50~\rm GeV^2$ dash-dotted line (red), $Q^2=100~\rm GeV^2$ dashed line (green), $Q^2=1000~\rm GeV^2$ full line (black) .
• Figure 4: The ratio $\Delta R_{Qg}^{(3)}(x,Q^2)$ of the 2-mass contributions to $\Delta \tilde{A}_{Qg}^{(3)}$ to the complete contribution of $O(T_F^2)$ and $O(T_F^3)$ to $\Delta {A}_{Qg}^{(3)}$ as defined in Eq. \ref{['eq:ratio']} in the polarized case in the Larin scheme. $Q^2=30~\rm GeV^2$ dotted line (blue), $Q^2=50~\rm GeV^2$ dash-dotted line (red), $Q^2=100~\rm GeV^2$ dashed line (green), $Q^2=1000~\rm GeV^2$ full line (black) .