Abstract
We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar mathcal{N} = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program. Based on physical discontinuities and Steinmann relations, we first summarize all possible double discontinuities (or first-two-entries) for (the symbol of) amplitudes and integrals in terms of dilogarithms, generalizing well-known results for n = 6, 7 to all multiplicities. As our main result, we find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic double-pentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops. We find interesting cancellation between contributions from rational and algebraic letters, and for the former we have also tested cluster-adjacency conditions using the so-called Sklyanin brackets. Finally, we propose a list of possible last-two-entries for MHV amplitudes up to 9 points derived from overline{Q} equations, which can be used to reduce the space of functions for higher-point MHV amplitudes.
Highlights
We comment on the status of “Steinmann-like” constraints, i.e. all-loop constraints on consecutive entries of the symbol of scattering amplitudes and Feynman integrals in planar N = 4 super-Yang-Mills, which have been crucial for the recent progress of the bootstrap program
We find that extended-Steinmann relations hold for all finite integrals that we have checked, including various ladder integrals, generic doublepentagon integrals, as well as finite components of two-loop NMHV amplitudes for any n; with suitable normalization such as minimal subtraction, they hold for n = 8 MHV amplitudes at three loops
As already seen for one-loop N2MHV, amplitudes with n ≥ 8 generally involve letters that cannot be expressed as rational functions of Plücker coordinates of the kinematics G(4, n)/T ; more non-trivial algebraic letters appear in computations based on Qequations [31] for two-loop NMHV amplitudes for n = 8 and n = 9 [32, 33], requiring extension of Grassmannian cluster algebras to include algebraic letters
Summary
As a warm-up exercise, let us first present our conjecture for all possible first-two-entries of integrals/amplitudes. It is possible to write the Li2 part of lower-mass boxes in a way that respects Steinmann relations, but the log log part does not, we will discuss these two parts separately for them. Our main conjecture is that for finite integrals and amplitudes to all loops, the first two entries that satisfy Steinmann relations can only be extracted from such box functions where the log log part need to be treated separately. There are n(n−5)(n2 − n + 5)/8 first-two-entries which can be derived from box functions and satisfy Steinmann relations, and we conjecture that they are all we need for finite integrals and Steinmann-respecting amplitudes. We have checked that in all two-loop n = 8 finite Feynman integrals in figure 1 and Idp(1, 3, 5, 7), exactly these 78 first-two-entries for n = 8 can appear.
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