Title: Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms Show Chinese title

Title: 计算算符矩阵元的大量3圈图的方法：超对数法

Abstract: We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist $\tau =2$ local operator insertions corresponding to spin $N$. They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version to the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz- and $V$-type graphs, belonging to the genuine 3-loop topologies. In case of the $V$-type graphs with five massive propagators new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of $\sim 30$ square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for $N \in \mathbb{C}$. Integrals with a power-like divergence in $N$--space $\propto a^N, a \in \mathbb{R}, a > 1,$ for large values of $N$ emerge. They still possess a representation in $x$--space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions. Show Chinese abstract

Abstract: 我们计算了包含单个带有扭$\tau =2$局部算符插入的三圈费曼图，这些算符插入对应于自旋$N$。它们对量子色动力学中描述大虚数下深非弹性散射的大质量算子矩阵元做出贡献。 这类图可以使用超对数方法的一个扩展版本来计算，该方法最初是为了处理没有算符的无质量费曼图而设计的。 该方法应用于苯环型和$V$型图，这些图属于真正的三圈拓扑结构。 对于具有五个大质量传播子的$V$型图，出现了新的嵌套求和类型和迭代积分。 这些求和以有限的二项式和逆二项式加权广义循环和的形式给出，而 1 维迭代积分基于一组$\sim 30$平方根值字母。 我们还推导了嵌套求和的渐近表示，并给出了$N \in \mathbb{C}$的解。 在$N$空间中，当$N$取较大值时，会出现幂次发散的积分$\propto a^N, a \in \mathbb{R}, a > 1,$。它们在$x$空间中仍然具有表示形式，这种表示形式在这种情况下由根值迭代积分给出。超对数方法也被用来计算不同算符插入的交叉框图的高阶矩。