标题： 一阶可分解贡献到三环质量算符矩阵元$A_{Qg}^{(3)}$和$ΔA_{Qg}^{(3)}$ 显示英文标题

标题： The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $ΔA_{Qg}^{(3)}$

摘要： 非极化和极化的质量算子矩阵元$A_{Qg}^{(3)}$和$\Delta A_{Qg}^{(3)}$在确定其主积分的差分或微分方程中包含一阶可分解和非一阶可分解贡献。 我们在单个重质量情况下计算了所有贡献费曼图的一阶可分解贡献。 此外，我们给出了在主积分中出现非一阶可分解贡献的情况下完整的颜色-$\zeta$因子，但这些贡献在最终结果中通过使用任意高阶Mellin矩的方法发现会相互抵消。 个别贡献还依赖于Mellin$N$空间中的广义调和求和以及嵌套的有限二项式和反二项式求和，相应地，在Bjorken-$x$空间中依赖于Kummer-Poincaré和平方根值字母表。 我们对在$N$空间中解析求解当前问题的可能性进行了完整的讨论，并且也讨论了在目前情况下，通过严格方法将给定的$N$空间表达式解析延拓到$N \in \mathbb{C}$的限制。 通过生成函数的表示方式，可以在一个17字母的字母表上实现一阶可分解结果的同步表示。 我们最终得到了在$x$空间中对应字母表上的迭代积分表示，这些表示包含至多权值{\sf w = 5}的特殊常数，这些常数可以有理化为在特定参数下的Kummer-Poincaré迭代积分。 分析 $x$-空间表示需要对区间 $x \in [0,1/4], [1/4,1/2], [1/2,1]$ 和 $x > 1$ 进行单独分析。 我们还推导了第一阶可分解贡献的小和大 $x$ 极限。 显示英文摘要

摘要： The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$ contain first-order factorizable and non-first-order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first-order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color-$\zeta$ factors for the cases in which also non-first-order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$-space, and correspondingly, on Kummer-Poincar\'e and square-root valued alphabets in Bjorken-$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$-space analytically and we also discuss the limitations in the present case to analytically continue the given $N$-space expressions to $N \in \mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first-order factorizable results over a 17-letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$-space, also containing up to weight {\sf w = 5} special constants, which can be rationalized to Kummer-Poincar\'e iterated integrals at special arguments. The analytic $x$-space representation requires separate analyses for the intervals $x \in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first-order factorizable contributions.