An infinite family of elliptic ladder integrals

Abstract

We identify two families of ten-point Feynman diagrams that generalize the elliptic double box, and show that they can be expressed in terms of the same class of elliptic multiple polylogarithms to all loop orders. Interestingly, one of these families can also be written as a dlog form. For both families of diagrams, we provide new 2ℓ-fold integral representations that are linearly reducible in all but one variable and that make the above properties manifest. We illustrate the simplicity of this integral representation by directly integrating the three-loop representative of both families of diagrams. These families also satisfy a pair of second-order differential equations, making them ideal examples on which to develop bootstrap techniques involving elliptic symbol letters at high loop orders.

The soaring kite: a tale of two punctured tori

“…Although a partial analysis for the kite integral was initiated in section 5, investigating whether Z-arguments can, in some way, be derived more directly from Landau analysis is an exciting avenue for future research. This is particularly relevant for the Symbol Prime bootstrap [43,[49][50][51][52]. Since it is possible to extract the symbol of the physical non ε-form kite integral I 1,1,1,1,1 from the boundary condition and differential equation provided in this work (the gauge transformation to ε-form must be undone), it would be interesting to compare our symbol with the result from a bootstrap.…”

Feynman integrals, geometries and differential equations

“…The latter case is already required for rather simple Feynman integrals, as for example the family of banana graphs. In this talk we put an emphasis on Calabi-Yau geometries [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], which are generalisations of elliptic curves (one-dimensional varieties) to higher dimensions.…”