The Heavy Fermion Contributions to the Massive Three Loop Form Factors

TL;DR

This work delivers the $n_h$ terms of the massive three‑loop form factors for vector, axialvector, scalar, and pseudoscalar currents using an extended large‑moment method that recasts differential equations into recurrences for moments. It combines IBP reduction, a decoupled moment approach, and renormalization group constraints to predict pole terms and obtain substantial analytic control over first‑order factorizing pieces via harmonic polylogarithms, while tackling non‑first‑order contributions with high‑order ε expansions and recurrence solving. The results include detailed analytic forms and high‑precision numerical representations in both the Euclidean region and near threshold, with extensive cross‑checks against known infrared structures and Ward identities. The methods and outputs significantly advance precision predictions for heavy‑flavor effects in three‑loop QCD amplitudes and establish a framework for addressing elliptic and higher structures in multi‑loop calculations.

Abstract

We compute the $n_h$ terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals have to be dealt with only in their moment representation, allowing to also consider quantities which obey differential equations, which are not first order factorizable (elliptic and higher), already at this level. To obtain all the associated recursions, up to 8000 moments had to be calculated. A new technique has been applied to solve the associated differential equation systems. Here the decoupling is performed such, that only minimal depth $ε$--expansions had to be performed for non--first-order factorizing systems, minimizing the calculation of initial values. The pole terms in the dimensional parameter $ε$ can be completely predicted using renormalization group methods, as confirmed by the present results. A series of contributions at $O(ε^0)$ have first order factorizable representations. For a smaller number of color--zeta projections this is not the case. All first order factorizing terms can be represented by harmonic polylogarithms. We also obtain analytic results for the non--first-order factorizing terms by Taylor series in a variable $x$, for which we have calculated at least 2000 expansion coefficients, in an approximation. Based on this representation the form factors can be given in the Euclidean region and in the region $q^2 \approx 0$. Numerical results are presented.

Figures (13)

• Figure 1: Vector form factor $F_{V,1}$: left $\varepsilon^0 n_h^1$, right $\varepsilon^0 n_h^2$, the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.
• Figure 2: Vector form factor $F_{V,2}$: left $\varepsilon^0 n_h^1$, right $\varepsilon^0 n_h^2$, the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.
• Figure 3: Axial vector form factor $F_{A,1}$: left $\varepsilon^0 n_h^1$, right $\varepsilon^0 n_h^2$, the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.
• Figure 4: Axial vector form factor $F_{A,2}$: left $\varepsilon^0 n_h^1$, right $\varepsilon^0 n_h^2$, the approximation with 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.
• Figure 5: Ratios of the approximations with 20, 50, 100, 200, 500 terms and our best approximation using 2000 terms for the vector form factor $F_{V,1}$. Left $\varepsilon^0 n_h^1$, right $\varepsilon^0 n_h^2$. The ratio using 20, 50, 100, 200, 500 terms is shown in brown, red, blue, green and black, respectively.

Theorems & Definitions (2)

• Remark 5.1
• Remark 5.2