Generalized unitarity and integral coefficients

An ordinary unitarity cut relates a discontinuity across a physical threshold to a sum over intermediate on-shell states and phase-space integration. Generalized cuts impose additional propagator on-shell conditions, often at complex momenta, to isolate finer information. Those extra conditions need not define a positive real phase-space process. The trees being sewn must use compatible momentum orientation, particle content, helicity sums, and normalization.

At one loop in four dimensions, four generic independent cut equations can leave isolated solutions for the loop momentum. A quadruple cut then selects the box topology carrying those four denominators, while triangle and bubble integrals lack enough denominators to survive that cut. Degenerate external kinematics can change the solution count, so the familiar generic two-solution picture is not an unconditional theorem for every mass assignment or exceptional momentum point.

For a generic two-solution box cut, the standard normalized coefficient is obtained by averaging the appropriate sewn-tree weights over the solutions, including the internal-state sum. Normalization includes the relation to the chosen scalar box cut and its Jacobians and phases. It is not legitimate to average arbitrary raw tree products from different integral conventions and call the result a coefficient.

At the integrand level, cut residues may also contain terms odd between the two solutions or other spurious components that integrate to zero in the chosen reduction. A small algebraic model writes the normalized solution values as a common box coefficient plus or minus one odd term. Averaging removes that odd part, while taking their difference identifies it. It must not be confused with a second physical box coefficient or with a missed species in the state sum.

A rational function with no branch discontinuity can be invisible to the simplest four-dimensional cuts. Extra-dimensional loop-momentum components in dimensional regularization can generate finite rational terms after integration. Their contribution may survive even when it vanishes on a strictly four-dimensional cut. Dimensional unitarity, additional recursion, or theory-specific information can recover these terms; declaring them zero merely because a selected cut is zero is generally wrong.

The integrated amplitude also requires ultraviolet renormalization and consistent infrared regularization. Poles in the regulator and subtraction conventions matter when combining virtual corrections with real radiation. At multiple loops, overlapping cuts, contact terms, integration identities and surface terms complicate the reconstruction. A coefficient match is therefore conditional on basis completeness and cut coverage. Agreement on several cuts is strong evidence, while uniqueness needs an explicit account of the functions or terms those cuts cannot detect.