The ice cone family and iterated integrals for Calabi-Yau varieties

Abstract

We present for the first time fully analytic results for multi-loop equal-mass ice cone graphs in two dimensions. By analysing the leading singularities of these integrals, we find that the maximal cuts in two dimensions can be organised into two copies of the same periods that describe the Calabi-Yau varieties for the equal-mass banana integrals. We obtain a conjectural basis of master integrals at an arbitrary number of loops, and we solve the system of differential equations satisfied by the master integrals in terms of the same class of iterated integrals that have appeared earlier in the context of equal-mass banana integrals. We then go on and show that, when expressed in terms of the canonical coordinate on the moduli space, our results can naturally be written as iterated integrals involving the geometrical invariants of the Calabi-Yau varieties. Our results indicate how the concept of pure functions and transcendental weight can be extended to the case of Calabi-Yau varieties. Finally, we also obtain a novel representation of the periods of the Calabi-Yau varieties in terms of the same class of iterated integrals, and we show that the well-known quadratic relations among the periods reduce to simple shuffle relations among these iterated integrals.

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.…”

Motivic geometry of two-loop Feynman integrals

“…In [30] it has been found that the Gauss-Manin connection associated with the single scale case, p 2 1 = p 2 3 = 0 and p 2 2 = 0 and all equal internal masses m 1 = • • • = m n+2 = m, for the ice cream cone integrals takes a lower triangle form, and that the associated Gauss-Manin system of differential equations splits as two (inhomogeneous) differential equations for the (n − 1)-loop sunset integrals, in agreement with the result of Proposition 9.1.…”

$\epsilon$-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:…”

“…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.…”