Abstract
Canonical Feynman integrals are of great interest in the study of scattering amplitudes at the multi-loop level. We propose to construct dlog-form integrals of the hypergeometric type, treat them as a representation of Feynman integrals, and project them into master integrals using intersection theory. This provides a constructive way to build canonical master integrals whose differential equations can be solved easily. We use our method to investigate both the maximally cut integrals and the uncut ones at one and two loops, and demonstrate its applicability in problems with multiple scales.
Highlights
Functions of uniform transcendentality (UT) are of great interest in the studies of scattering amplitudes in quantum field theories
Our method serves as a constructive way to find canonical Feynman integrals without analyzing the differential equations
Ci j=1 where fj(z) are algebraic functions of the Baikov variables. These integrals apparently lead to UT functions, but it is not a priori clear how they are related to Feynman integrals and whether all master integrals in a canonical basis can be represented in this form
Summary
Functions of uniform transcendentality (UT) are of great interest in the studies of scattering amplitudes in quantum field theories. It has been realized that canonical Feynman integrals are closely related to d log-form integrals [26,27,28,29,30,31,32,33,34,35] They lead to beautiful geometric pictures for the scattering amplitudes in planar N = 4 supersymmetric theories. We build all possible d log-form integrals which can be interpreted as Feynman integrals in this topology, and project them back to loop integrals using the intersection theory [37,38,39,40,41,42,43] This amounts to exploiting the geometric picture of the hypergeometric integrals, and computing the “inner-products” of them using concepts from algebraic geometry. Our method serves as a constructive way to find canonical Feynman integrals without analyzing the differential equations
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