Three Loop Massive Operator Matrix Elements and Asymptotic Wilson Coefficients with Two Different Masses

TL;DR

The work addresses two-mass effects in three-loop heavy-quark corrections to deep-inelastic scattering and VFNS matching, focusing on charm and bottom with $\eta= m_c^2/m_b^2$. It develops a complete renormalization framework for massive OMEs with two masses, deriving mass and coupling renormalization, operator renormalization, and collinear factorization, and then computes fixed moments $N=2,4,6$ and general-$N$ results for NS, transversity, and $gq$, along with the scalar $A_{gg,Q}$ two-mass diagrams. The paper delivers explicit expressions for the two-mass contributions in various OMEs, analyzes pole structures, and introduces new iterated integrals and GHPLs to express the results, including $z$- and $N$-space forms. The findings enable more accurate VFNS matching and heavy-flavor phenomenology at high-energy colliders by providing essential two-mass input up to $3$-loop order and highlighting the limitations of naive $\eta$-expansions for general $N$.

Abstract

Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of Feynman diagrams carrying quark lines with two different masses. In the case of the charm and bottom quarks, the usual decoupling of one heavy mass at a time no longer holds, since the ratio of the respective masses, $η= m_c^2/m_b^2 \sim 1/10$, is not small enough. Therefore, the usual variable flavor number scheme (VFNS) has to be generalized. The renormalization procedure in the two--mass case is different from the single mass case derived in \cite{Bierenbaum:2009mv}. We present the moments $N=2,4$ and $6$ for all contributing operator matrix elements, expanding in the ratio $η$. We calculate the analytic results for general values of the Mellin variable $N$ in the flavor non-singlet case, as well as for transversity and the matrix element $A_{gq}^{(3)}$. We also calculate the two-mass scalar integrals of all topologies contributing to the gluonic operator matrix element $A_{gg}$. As it turns out, the expansion in $η$ is usually inapplicable for general values of $N$. We therefore derive the result for general values of the mass ratio. From the single pole terms we derive, now in a two-mass calculation, the corresponding contributions to the 3-loop anomalous dimensions. We introduce a new general class of iterated integrals and study their relations and present special values. The corresponding functions are implemented in computer-algebraic form.

Figures (12)

• Figure 1: One of the massive fermion loop insertion is effectively rendered massless via a Mellin--Barnes representation.
• Figure 2: The ratio of the genuine 2-mass contributions to $A_{qq,Q}^{(3), \rm NS}$ to the complete $T_F^2$-part of massive 3-loop OME $A_{qq,Q}^{(3), \rm NS}$ as a function of $x$ and $Q^2$, for $m_c = 1.59~\mathrm{GeV}, m_b = 4.78~\mathrm{GeV}$ in the on-shell scheme. Dash-dotted line: $\mu^2 = 30~\mathrm{GeV}^2$; Dotted line: $\mu^2 = 50~\mathrm{GeV}^2$; Dashed line: $\mu^2 = 100~\mathrm{GeV}^2$; Full line: $\mu^2 = 1000~\mathrm{GeV}^2$. The single mass contributions are given in Ref. Ablinger:2014vwa.
• Figure 3: The ratio of the genuine 2-mass contributions to $A_{qq,Q}^{(3), \rm NS, TR}$ to the complete $T_F^2$-part of the massive 3-loop corrections to $A_{qq,Q}^{(3), \rm NS, TR}$ as a function of $x$ and $Q^2$, for $m_c = 1.59~\mathrm{GeV}, m_b = 4.78~\mathrm{GeV}$ in the on-shell scheme. Dash-dotted line: $\mu^2 = 30~\mathrm{GeV}^2$; Dotted line: $\mu^2 = 50~\mathrm{GeV}^2$; Dashed line: $\mu^2 = 100~\mathrm{GeV}^2$; Full line: $\mu^2 = 1000~\mathrm{GeV}^2$. The single mass contributions are given in Ref. Ablinger:2014vwa.
• Figure 4: The ratio of the genuine 2-mass contributions to $A_{gq,Q}^{(3)}$ to the complete $T_F^2$-part of the massive 3-loop OME $A_{gq,Q}^{(3)}$ as a function of $x$ and $Q^2$, for $m_c = 1.59~\mathrm{GeV}, m_b = 4.78~\mathrm{GeV}$ in the on-shell scheme. Dash-dotted line: $\mu^2 = 30~\mathrm{GeV}^2$; Dotted line: $\mu^2 = 50~\mathrm{GeV}^2$; Dashed line: $\mu^2 = 100~\mathrm{GeV}^2$; Full line: $\mu^2 = 1000~\mathrm{GeV}^2$. The single mass contributions are given in Ref. Ablinger:2014lka.
• Figure 5: Diagram 1. Here both mass assignments $m_a=m_1$, $m_b=m_2$ and $m_a=m_2$, $m_b=m_1$ yield identical results.