发表机构
Deutsches Elektronen–Synchrotron DESY; TU Dortmund; Johannes Kepler University Linz（德国电子同步加速器; 多特蒙德工业大学; 林茨约翰内斯开普勒大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解析计算了三圈重夸克形状因子的胶子贡献，在低能和高能极限下通过猜测算法与微分方程求解获得闭式表达式，并给出全运动学范围的数值结果。
AI 中文摘要
我们计算了矢量、轴矢量、标量和赝标量流的三圈重夸克形状因子的胶子贡献。在低能极限 $q^2/m^2 \rightarrow 0$ 下，我们使用猜测算法从与展开系数中出现的多重zeta值和其它常数相关的有理序列推导出闭式差分方程和微分方程，在最苛刻的情况下需要多达26000个系数。部分结果通过求解所获得的微分方程，以调和多重对数函数和平方根值迭代积分的形式解析获得。对于其余贡献，可以通过在中间点匹配局部展开并利用与围绕 $s = q^2/m^2 = 0$ 展开中每个超越常数相关的函数所满足的微分方程，获得围绕形状因子奇点的任意深度的级数展开。所有这些计算都采用了先进的计算机代数方法。通过使用微分方程的解析延拓方法，匹配不同 $s$ 值处的展开，并利用PSLQ算法，我们推导出高能极限 $q^2/m^2 \rightarrow \infty$ 下的解析结果。展开系数以权重最高为 $w = 6$ 的多重zeta值以及三个额外常数表示，其中两个与六次单位根字母相关，一个与二次型隐含的迭代积分相关。我们还推导了关于阈值和伪阈值的深度展开。在整个运动学范围内给出了数值结果，并与文献中的结果进行了比较。
英文摘要
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.
Comments48 pages, 2 figures
