Bananas of equal mass: any loop, any order in the dimensional regularisation parameter

Abstract

We describe a systematic approach to cast the differential equation for the l-loop equal mass banana integral into an ε-factorised form. With the known boundary value at a specific point we obtain systematically the term of order j in the expansion in the dimensional regularisation parameter ε for any loop l. The approach is based on properties of Calabi-Yau operators, and in particular on self-duality.

A symbol and coaction for higher-loop sunrise integrals

“…[26][27][28][29][30][31][32][33][34][35][36][37][38], one must in general consider integrals that are not themselves Feynman integrals. An example of such a choice that appeared after our initial preprint leverages the notion of Calabi-Yau operators to further decompose the equalmass sunrise and ice cream cone integrals [83][84][85]. It would be interesting to see if these notions can be generalized to the generic-mass case and to other multivariate problems more generally.…”

$\epsilon$-forms for non-planar triangles with elliptic curves at two loops

“…Although 𝐿 (0) 3 (𝑦) is not a Calabi-Yau operator, see [12,13] in the context of Feynman integrals, we can nevertheless construct special normal forms [30] from 𝜓 0 , 𝜓 1 and 𝜓 2 , from which we define a "𝑌 "-invariant:…”

“…The relation between 𝑦 and 𝑞 in (12) only holds in a vicinity of 𝑦 = 0. In the context of elliptic curves, one can analytically continue the above to a bijection relation valid for the whole kinematic regime, i.e., given a kinematic value 𝑦 ∈ R + 𝑖0 (with Feynman 𝑖0 prescription), we can obtain the same value from the following sequence:…”

“…Recently, 𝜀-form differential equations to equal-mass Banana integrals have been derived [12]. In this talk, we will show how the ideas therein can be generalized to general Feynman integrals, like non-planar triangles shown in Fig 1 . Unlike Banana integrals, these two cases have non-trivial subsector dependence.…”

“…A lot of Feynman integrals are beyond MPLs, starting from NNLO; see [3,4] for reviews and references therein. To cut the story short, Feynman integrals can be related to complex manifolds, such as (compact) genus-𝑛 Riemann surfaces [5][6][7] and higher dimensional hypersurfaces like Calabi-Yau manifolds [8][9][10][11][12][13]. As a powerful tool to calculate dimensional-regularized Feynman integrals, the 𝜀-factorized (canonical) differential equations [14] are naturally determined by meromorphic functions on corresponding complex manifolds.…”