The three-loop equal-mass banana integral in ε-factorised form with meromorphic modular forms

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

Section: K3-surfaces As Configuration Spaces On the Projective Planementioning

confidence: 99%

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

Giroux

Pokraka

2023

“…In the special case of equal-mass banana integrals in d = 2 dimensions, one can write the result in terms of an integral involving the periods of the Calabi-Yau variety [64]. Specialising to the three-loop case, the underlying geometry is a one-parameter family of K3 surfaces, and it is possible to express the result in terms of iterated integrals of (meromorphic) modular forms [65][66][67]. 2 It was recently shown that at four loops a similar representation exists at higher orders in the dimensional regulator , and this naturally leads one to consider iterated integrals involving the canonical coordinate on the moduli space of a one-parameter family of Calabi-Yau three-folds [72].…”

Section: Introductionmentioning

confidence: 99%

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

Duhr¹,

Klemm

Nega

et al. 2023

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.

“…It has the special property that its Picard-Fuchs operator in two space-time dimensions is a symmetric square [25,26]. It can therefore be treated with methods similar to the elliptic case [27][28][29][30][31]. The ε-factorised form of the differential equation has been given in [31].…”

Section: Introductionmentioning

confidence: 99%

“…It can therefore be treated with methods similar to the elliptic case [27][28][29][30][31]. The ε-factorised form of the differential equation has been given in [31].…”

Section: Introductionmentioning

confidence: 99%

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

Pögel

Wang

Weinzierl

2023

Self Cite

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.