An on-shell construction using BCFW trees plus vacuum-pair phase-space integrals with inclusion-exclusion signs reproduces the known one- and two-loop all-plus four-gluon amplitudes.
Loops and trees
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abstract
We investigate relations between loop and tree amplitudes in quantum field theory that involve putting on-shell some loop propagators. This generalizes the so-called Feynman tree theorem which is satisfied at 1-loop. Exploiting retarded boundary conditions, we give a generalization to L-loop expressing the loops as integrals over the on-shell phase space of exactly L particles. We argue that the corresponding integrand for L>2 does not involve the forward limit of any physical tree amplitude, except in planar gauge theories. In that case we explicitly construct the relevant physical amplitude. Beyond the planar limit, abandoning direct integral representations, we propose that loops continue to be determined implicitly by the forward limit of physical connected trees, and we formulate a precise conjecture along this line. Finally, we set up technology to compute forward amplitudes in supersymmetric theories, in which specific simplifications occur.
fields
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UNVERDICTED 3representative citing papers
A recursion formula for ℓ-loop planar integrands in colored QFTs is derived from the classical equation of motion via comb components and loop kernels.
Outlines a Schwinger-Keldysh path-integral framework that derives worldline equations of motion and computes weak-field gravitational waveforms independently for unspecified relativistic orbits.
citing papers explorer
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Perturbative construction of amplitudes from on-shell trees with vacuum pairs: the all-plus four-gluon amplitude through order $\boldsymbol{g}^{\boldsymbol{6}}$
An on-shell construction using BCFW trees plus vacuum-pair phase-space integrals with inclusion-exclusion signs reproduces the known one- and two-loop all-plus four-gluon amplitudes.
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Systematic approach to $\ell$-loop planar integrands from the classical equation of motion
A recursion formula for ℓ-loop planar integrands in colored QFTs is derived from the classical equation of motion via comb components and loop kernels.
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Weak-field waveforms for generic relativistic orbits
Outlines a Schwinger-Keldysh path-integral framework that derives worldline equations of motion and computes weak-field gravitational waveforms independently for unspecified relativistic orbits.