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Simplifying $D$-Dimensional Physical-State Sums in Gauge Theory and Gravity

Dimitrios Kosmopoulos

Abstract

We provide two independent systematic methods of performing $D$-dimensional physical-state sums in gauge theory and gravity in such a way so that spurious light-cone singularities are not introduced. A natural application is to generalized unitarity in the context of dimensional regularization or theories in higher spacetime dimensions. Other applications include squaring matrix elements to obtain cross sections, and decompositions in terms of gauge-invariant tensors.

Simplifying $D$-Dimensional Physical-State Sums in Gauge Theory and Gravity

Abstract

We provide two independent systematic methods of performing -dimensional physical-state sums in gauge theory and gravity in such a way so that spurious light-cone singularities are not introduced. A natural application is to generalized unitarity in the context of dimensional regularization or theories in higher spacetime dimensions. Other applications include squaring matrix elements to obtain cross sections, and decompositions in terms of gauge-invariant tensors.

Paper Structure

This paper contains 19 sections, 60 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background and definitions
  3. Spurious-singularity-free replacement rules
  4. Demonstration in a simple example
  5. The general case
  6. Proof in gauge theory
  7. Proof in gravity
  8. Summary
  9. Gauge theory
  10. Gravity
  11. An example in detail
  12. Generalized Ward identity
  13. Demonstration in a simple example
  14. The general case
  15. Simplifying the physical-state projectors using the GWI
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Examples of calculations where we may apply the techniques of this paper: (a) integrand-level generalized-unitarity cut and (b) squared matrix elements for cross sections. The blobs represent amplitudes. All exposed lines are taken as on shell. The internal exposed lines indicate gauge-particle legs that we intend to sew.
  • Figure 2: The scalar-QED generalized-unitarity cut studied in Sect. \ref{['demoRulesSec']}. The two blobs are Compton amplitudes in this theory. Solid lines correspond to scalar particles and wiggly lines correspond to photons. External momenta are taken outgoing while internal momenta flow to the right. All exposed lines are taken as on shell.
  • Figure 3: The three Feynman diagrams we need to calculate in order to get the scalar-QED amplitudes of Eq. (\ref{['sQEDlrAmpsEq']}). The solid line represents a scalar particle while the wiggly lines represent photons. We take all momenta to be outgoing.
  • Figure 4: A three-loop generalized-unitarity cut relevant in the construction of the conservative two-body Hamiltonian for spinless black holes to order $G^4$. The blobs represent tree-level amplitudes. The solid lines correspond to scalars while the wiggly ones to gravitons. We take the external particles to be outgoing and the internal momenta to go upwards and to the right. All exposed lines are taken as on shell.
  • Figure 5: An example where we need to break the process into two steps. All exposed lines are taken as on-shell. Solid blobs represent tree-level amplitudes. In (b) the hollow blob represents the one-loop quantity we get by performing the physical-state sums for particles 1 and 2 on the two amplitudes on the left in (a).