Analytic results on the massive three-loop form factors: gluonic contributions

Abstract

We compute the gluonic contributions to the three-loop heavy-quark form factors for the vector, axial-vector, scalar, and pseudoscalar currents. In the low-energy limit, $q^2/m^2 \rightarrow 0$, we used guessing algorithms to derive closed-form difference and differential equations from the rational sequences associated with the multiple zeta values and other constants appearing in the expansion coefficients, which required up to 26000 coefficients in the most demanding cases. Part of the results are obtained analytically in terms of harmonic polylogarithms and square-root valued iterated integrals by solving the obtained differential equations. For the remaining contributions, arbitrarily deep series expansions around the singularities of the form factors can be obtained by matching local expansions at intermediate points and by exploiting the differential equations obeyed by the functions associated with each transcendental constant in the expansions around $s = q^2/m^2 = 0$. For all these calculations advanced computer algebra methods have been employed. By using analytic continuation methods for differential equations matching the expansions at different values of $s$ and using PSLQ, we derive analytic results in the high-energy limit, $q^2/m^2 \rightarrow \infty$. The expansion coefficients are expressed in terms of multiple zeta values up to weight $w = 6$ together with three additional constants, two related to sixth-root-of-unity letters and one associated with quadratic-form implied iterated integrals. We also derive deep expansions about the threshold and pseudo-threshold. Numerical results are presented in the whole kinematic range and compared to results in the literature.