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Abstract It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper we show that such sectors may have additional symmetries. First of all, self-duality, which was first observed in Feynman integrals related to Calabi-Yau geometries, often carries over to non-Calabi-Yau Feynman integrals. Secondly, we show that in addition there can exist Galois symmetries relating integrals. In the simplest case of two master integrals within a sector, whose definition involves a square root r, we may choose a basis (I 1, I 2) such that I 2 is obtained from I 1 by the substitution r → −r. This pattern also persists in sectors, which a priori are not related to any square root with dependence on the kinematic variables. We show in several examples that in such cases a suitable redefinition of the integrals introduces constant square roots like 3 $$ \sqrt{3} $$ . The new master integrals are then again related by a Galois symmetry, for example the substitution 3 $$ \sqrt{3} $$ → − 3 $$ -\sqrt{3} $$ . To handle the case where the argument of a square root would be a perfect square we introduce a limit Galois symmetry. Both self-duality and Galois symmetries constrain the differential equation.

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.

Abstract We show that the differential equation for the three-loop equal-mass banana integral can be cast into an ε-factorised form with entries constructed from (meromorphic) modular forms and one special function, which can be given as an iterated integral of meromorphic modular forms. The ε-factorised form of the differential equation allows for a systematic solution to any order in the dimensional regularisation parameter ε. The alphabet of the iterated integrals contains six letters.

Abstract We study a two-loop four-point function with one internal mass. This Feynman integral is one of the simplest Feynman integrals depending on two elliptic curves. We transform the associated differential equation into an ε-form. We study the entries of the differential equation, and in particular the entries which depend on both elliptic curves.

Abstract We show that the half-integrands in the CHY representation of tree amplitudes give rise to the definition of differential forms — the scattering forms — on the moduli space of a Riemann sphere with n marked points. These differential forms have some remarkable properties. We show that all singularities are on the divisor ℳ ¯ 0 , n \ ℳ 0 , n $$ {\overline{\mathrm{\mathcal{M}}}}_{0,n}\backslash {\mathrm{\mathcal{M}}}_{0,n} $$ . Each singularity is logarithmic and the residue factorises into two differential forms of lower points. In order for this to work, we provide a threefold generalisation of the CHY polarisation factor (also known as reduced Pfaffian) towards off-shell momenta, unphysical polarisations and away from the solutions of the scattering equations. We discuss explicitly the cases of bi-adjoint scalar amplitudes, Yang-Mills amplitudes and gravity amplitudes.