Open Loop Amplitudes and Causality to All Orders and Powers from the Loop-Tree Duality

Universal opening of four-loop scattering amplitudes to trees

“…It was recently conjectured in ref. [24] that LTD straightforwardly leads to extremely compact and manifestly causal representations of scattering amplitudes to all orders. This pattern was explicitly proven for a series of multiloop topologies, the maximal loop topology (MLT), next-to-maximal (NMLT) and next-to-next-to-maximal (N 2 MLT) that are characterized by L + 1, L + 2 and L + 3 sets of propagators, respectively, with each set categorized by the dependence on a specific loop momentum or a linear combination of the L independent loop momenta.…”

“…This results in a modification of the infinitesimal complex prescription of the Feynman propagators [18], that needs to be considered carefully to preserve the causal structure of the amplitude. In the context of multiloop scattering amplitudes, the LTD representation is written in terms of nested residues [24]…”

“…The sum over all possible on-shell configurations in A (L) D is understood through the sum of residues. For example, the LTD representation of the multi-banana or MLT topology has the very compact and symmetric form [24]…”

“…The loop-tree duality (LTD) [18][19][20][21][22][23][24] features a manifest distinction between physical and unphysical singularities at integrand level [25,26], opening an alternative framework to perform more efficient calculations. This knowledge was crucial for developing the four dimensional unsubtraction (FDU) [27][28][29][30], which allows to combine real and virtual corrections into a single numerically-stable integral.…”

Two-loop rational terms in Yang-Mills theories

“…In the literature a variety of methods have been proposed that aim at restricting the calculation of loop amplitudes to an integer number of space-time dimensions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. At one loop, the most widely used method is based on the idea of splitting loop amplitudes into two parts according to the dimensionality D n of the numerators of loop integrands.…”