A study of Feynman integrals with uniform transcendental weights and their symbology

Abstract

Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales may have complicated symbol structures, and we show that twistor geometries of closely related dual conformal integrals shed light on their alphabet and symbol structures. In this paper, first, as a cutting-edge example, we derive the two-loop four-external-mass Feynman integrals with uniform transcendental (UT) weights, based on the latest developments on UT integrals. Then we find that all the symbol letters of these integrals can be explained non-trivially by studying the so-called Schubert problem of certain dual conformal integrals with a point at infinity. Certain properties of the symbol such as first two entries and extended Steinmann relations are also studied from analogous properties of dual conformal integrals.

A nice two-loop next-to-next-to-MHV amplitude in $$ \mathcal{N} $$ = 4 super-Yang-Mills

“…The 8-point penta-box integral I pb can be obtained as a double-collinear limit of the 10-point penta-box integral, which we can compute in principle in Feynman-parameterized form, but the computation turns out to be rather tedious. Fortunately, we find that the 10-point penta-box integral belongs to the family of integrals recently computed in [43] by canonical differential equations: the penta-box integral can be identified as a non-DCI double box integral by sending x 5 to ∞, see figure 2. This non-DCI limit gives us a bijection between the kinematics of these two Feynman integrals,…”

“…We will use the dual conformal invariant (DCI) regularization [42] since these integrands are defined strictly in four-dimensional space: it turns out the n = 8 penta-box and doublebox can be obtained from double-collinear limits of n = 10 finite DCI integrals, and the DCI regularization allows us to extract these divergent integrals from the latter: by sending a dual point to infinity for the n = 10 penta-box integral, it is nicely reduced to some two-loop master integrals with four masses which have been computed very recently using differential equations [43]; the n = 10 double-box evaluates to elliptic multiple polylogarithms [37], but we will see that it suffices to take a double-collinear limit for its one-dimensional integral representation, which gives (MPL) n = 8 double-box. After subtracting the DCIregulated one-loop 8-point MHV amplitude multiplied by the four-mass box, we confirm the cancellation of divergences and end up with a finite ratio function.…”

“…(1) 8,0 . Their results in dual conformal regularization can be obtained as special double-collinear limits of their finite counterparts with n = 10, which are already known from the literature [37,43]. In the rest of this section, we will elaborate the procedure of taking such collinear limits.…”

“…The non-DCI double-box integral I nDCI-db is denoted in [43] as G 1,1,1,1,1,1,1,−1 , which can be expressed as linear combination of the following three master integrals…”

Cutting the traintracks: Cauchy, Schubert and Calabi-Yau

“…Inspired by remarkable mathematical structures such as cluster algebras for a large class of Feynman integrals evaluating to MPL functions (see [68][69][70][71] and references therein), a new method called "Schubert analysis" has been exploited to generate and explain the alphabets of such integrals [72][73][74]. The method has proved to be very powerful even for generating the alphabet of the eMPL symbols for the massive double-box (L = 2 full traintrack) [37].…”