Two-loop Integrand Decomposition into Master Integrals and Surface Terms

Loop amplitudes are conveniently expressed in terms of master integrals whose coefficients carry the process dependent information. Similarly before integration, the loop integrands may be expressed as a linear combination of propagator products with universal numerator-tensors. Such a decomposition is an important input for the numerical unitarity approach, which constructs integrand coefficients from on-shell tree amplitudes. We present a new method to organise multi-loop integrands into a direct sum of terms that integrate to zero (surface terms) and remaining master integrands. This decomposition facilitates a general, numerical unitarity approach for multi-loop amplitudes circumventing analytic integral reduction. Vanishing integrals are well known as integration-by-parts identities. Our construction can be viewed as an explicit construction of a complete set of integration-by-parts identities excluding doubled propagators. Interestingly, a class of 'horizontal' identities is singled out which hold as well for altered propagator powers.