Picard–Fuchs Equations for Feynman Integrals

Abstract

We present a systematic method to derive an ordinary differential equation for any Feynman integral, where the differentiation is with respect to an external variable. The resulting differential equation is of Fuchsian type. The method can be used within fixed integer space-time dimensions as well as within dimensional regularisation. We show that finding the differential equation is equivalent to solving a linear system of equations. We observe interesting factorisation properties of the D-dimensional Picard-Fuchs operator when D is specialised to integer dimensions.

The kite integral to all orders in terms of elliptic polylogarithms

“…There is a wide class of Feynman integrals, mainly related to massless theories, which can be expressed in terms of multiple polylogarithms. With the advance of computational techniques in recent years, these may be computed efficiently [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. More challenging are Feynman integrals, which cannot be expressed in terms of multiple polylogarithms.…”

The iterated structure of the all-order result for the two-loop sunrise integral

“…Methods for the numerical evaluation are available [7]. This allows that a wide class of Feynman integrals can be computed systematically to all orders in ε. Algorithms which accomplish this are for example based on nested sums [8][9][10][11][12], linear reducibility [13][14][15] or differential equations [16][17][18][19][20][21][22]. On the mathematical side, multiple polylogarithms are closely related to punctured Riemann surfaces of genus zero [5,23,24].…”

“…The simplest Feynman integral which cannot be expressed in terms of multiple polylogarithms is the two-loop sunrise integral with non-vanishing masses. This Feynman integral has already received considerable attention in the literature [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In this paper we study the two-loop sunrise integral with equal non-zero masses in D = 2 − 2ε space-time dimensions.…”

Differential equations for Feynman integrals beyond multiple polylogarithms

“…The method of differential equations [1][2][3][4][5][6][7][8][9][10] is a powerful tool to tackle Feynman integrals. Let t be an external invariant (e.g.…”

“…The simplest example is given by the two-loop sunrise integral [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] with equal masses. A slightly more complicated integral is the two-loop kite integral [45][46][47][48][49], which contains the sunrise integral as a sub-topology.…”