线性可约椭圆型Feynman积分的全阶结构和有效计算
All orders structure and efficient computation of linearly reducible elliptic Feynman integrals
摘要：
我们定义线性可约椭圆型费曼积分，我们表明，他们可以算法解决的任意顺序的维调节器在1维积分的polylogarithmic被积函数，我们称之为内部polylogarithmic部分（IPP）。通过直接积分的费曼参数表示的解决方案。当IPP依赖于一条椭圆曲线（并且没有其他代数函数）时，这类费曼积分可以通过使用分部积分以椭圆多重对数（eMPLs）的形式进行算法求解。然后我们详细介绍了微分方程方法。具体来说，我们证明了IPP可以映射到一个广义积分拓扑满足一组微分方程的$ε$ -形式。在例子中，我们认为正则微分方程可以直接求解的eMPLs到任意阶的尺寸调节器。可以执行剩余的1维积分以完全根据eMPLS来表达这样的积分。我们应用这些方法来解决两点和三点积分的eMPLs。我们解析地继续这些积分的物理区域，通过使用它们的1维积分表示。
英文摘要：
We define linearly reducible elliptic Feynman integrals, and we show that they can be algorithmically solved up to arbitrary order of the dimensional regulator in terms of a 1-dimensional integral over a polylogarithmic integrand, which we call the inner polylogarithmic part (IPP). The solution is obtained by direct integration of the Feynman parametric representation. When the IPP depends on one elliptic curve (and no other algebraic functions), this class of Feynman integrals can be algorithmically solved in terms of elliptic multiple polylogarithms (eMPLs) by using integration by parts identities. We then elaborate on the differential equations method. Specifically, we show that the IPP can be mapped to a generalized integral topology satisfying a set of differential equations in $ε$-form. In the examples we consider the canonical differential equations can be directly solved in terms of eMPLs up to arbitrary order of the dimensional regulator. The remaining 1-dimensional integral may be performed to express such integrals completely in terms of eMPLs. We apply these methods to solve two- and three-points integrals in terms of eMPLs. We analytically continue these integrals to the physical region by using their 1-dimensional integral representation.
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