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Polynomial Structures in One-Loop Amplitudes
A general one-loop scattering amplitude may be expanded in terms of master
integrals. The coefficients of the master integrals can be obtained from
tree-level input in a two-step process. First, use known formulas to write the
coefficients of (4-2epsilon)-dimensional master integrals; these formulas
depend on an additional variable, u, which encodes the dimensional shift.
Second, convert the u-dependent coefficients of (4-2epsilon)-dimensional master
integrals to explicit coefficients of dimensionally shifted master integrals.
This procedure requires the initial formulas for coefficients to have
polynomial dependence on u. Here, we give a proof of this property in the case
of massless propagators. The proof is constructive. Thus, as a byproduct, we
produce different algebraic expressions for the scalar integral coefficients,
in which the polynomial property is apparent. In these formulas, the box and
pentagon contributions are separated explicitly.Comment: 44 pages, title changed to be closer to content, section 2.1 extended
to section 2.1 and 2.2 to be more self-contained, references added, typos
corrected, the final version to appear in JHE
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