On-shell recursion for massive fermion currents

We analyze the validity of BCFW recursion relations for currents of n - 2 gluons and two massive quarks, where one of the quarks is off shell and the remaining particles are on shell. These currents are gauge-dependent and can be used as ingredients in the unitarity-based approach to computing one-loop amplitudes. The validity of BCFW recursion relations is well known to depend on the so-called boundary behavior of the currents as the momentum shift parameter goes to infinity. With off-shell currents, a new potential problem arises, namely unphysical poles that depend on the choice of gauge. We identify conditions under which boundary terms are absent and unphysical poles are avoided, so that there is a natural recursion relation. In particular, we are able to choose a gauge in which we construct a valid shift for currents with at least n - 3 gluons of the same helicity. We derive an analytic formula in the case where all gluons have the same helicity. As by-products, we prove the vanishing boundary behavior of general off-shell objects in Feynman gauge, and we find a compact generalization of Berends-Giele gluon currents with a generic reference spinor.Comment: 30 pages, 8 figures; v2 minor corrections, journal versio

External leg corrections in the unitarity method

Unitarity cuts diverge in the channel of a single massive external fermion. We propose an off-shell continuation of the momentum that allows a finite evaluation of the unitarity cuts. If the cut is taken with complete amplitudes on each side, our continuation and expansion around the on-shell configuration produces the finite contribution to the bubble coefficient. Finite parts in the expansion of the external leg counterterms must be included explicitly as well.Comment: 28 pages, 9 figures. Published version. Eq. (B.17) corrected, minor clarifications, typos fixe

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

Cuts and coproducts of massive triangle diagrams

Relations between multiple unitarity cuts and coproducts of Feynman integrals are extended to allow for internal masses. These masses introduce new branch cuts, whose discontinuities can be derived by placing single propagators on shell and identified as particular entries of the coproduct. First entries of the coproduct are then seen to include mass invariants alone, as well as threshold corrections for external momentum channels. As in the massless case, the original integral can possibly be recovered from its cuts by starting with the known part of the coproduct and imposing integrability contraints. We formulate precise rules for cuts of diagrams, and we gather evidence for the relations to coproducts through a detailed study of one-loop triangle integrals with various combinations of external and internal masses.Comment: 60 page

Cuts in Feynman Parameter Space

We propose a construction of generalized cuts of Feynman integrals as an operation on the domain of the Feynman parametric integral. A set of on-shell conditions removes the corresponding boundary components of the integration domain, in favor of including a boundary component from the second Symanzik polynomial. Hence integration domains are full-dimensional spaces with finite volumes, rather than being localized around poles. As initial applications, we give new formulations of maximal cuts, and we provide a simple derivation of a certain linear relation among cuts from the inclusion-exclusion principle.Comment: 7 pages, 4 figure

Holography for Coset Spaces

M/string theory on noncompact, negatively curved, cosets which generalize $AdS_{D+1}=SO(D,2)/SO(D,1)$ is considered. Holographic descriptions in terms of a conformal field theory on the boundary of the spacetime are proposed. Examples include $SU(2,1)/U(2)$, which is a Euclidean signature (4,0) space with no supersymmetry, and $SO(2,2)/SO(2)$ and $SO(3,2)/SO(3)$, which are Lorentzian signature (4,1) and (6,1) spaces with eight supersymmetries. Qualitatively new features arise due to the degenerate nature of the conformal boundary metric.Comment: Harvmac, 23 pages. Additions/correction to section 3.

Two-Black-Hole Bound States

The quantum mechanics of $N$ slowly-moving BPS black holes in five dimensions is considered. A divergent continuum of states describing arbitrarily closely bound black holes with arbitrarily small excitation energies is found. A superconformal structure appears at low energies and can be used to define an index counting the weighted number of supersymmetric bound states. It is shown that the index is determined from the dimensions of certain cohomology classes on the symmetric product of $N$ copies of $R^4$. An explicit computation for the case of N=2 with no angular momentum yields a finite nonzero result.Comment: 21 pages, harvmac. Minor corrections to section

Unitarity Cuts with Massive Propagators and Algebraic Expressions for Coefficients

In the first part of this paper, we extend the d-dimensional unitarity cut method of hep-ph/0609191 to cases with massive propagators. We present formulas for integral reduction with which one can obtain coefficients of all pentagon, box, triangle and massive bubble integrals. In the second part of this paper, we present a detailed study of the phase space integration for unitarity cuts. We carry out spinor integration in generality and give algebraic expressions for coefficients, intended for automated evaluation.Comment: 33 pages. v2: notation modified. v3: typos fixe