One-loop correlators and BCJ numerators from forward limits

Abstract

We present new formulas for one-loop ambitwistor-string correlators for gauge theories in any even dimension with arbitrary combinations of gauge bosons, fermions and scalars running in the loop. Our results are driven by new all-multiplicity expressions for tree-level two-fermion correlators in the RNS formalism that closely resemble the purely bosonic ones. After taking forward limits of tree-level correlators with an additional pair of fermions/bosons, one-loop correlators become combinations of Lorentz traces in vector and spinor representations. Identities between these two types of traces manifest all supersymmetry cancellations and the power counting of loop momentum. We also obtain parity-odd contributions from forward limits with chiral fermions. One-loop numerators satisfying the Bern-Carrasco-Johansson (BCJ) duality for diagrams with linearized propagators can be extracted from such correlators using the well-established tree-level techniques in Yang-Mills theory coupled to biadjoint scalars. Finally, we obtain streamlined expressions for BCJ numerators up to seven points using multiparticle fields.

Propagators, BCFW recursion and new scattering equations at one loop

“…While we discussed an idea for a worldsheet formula for gluon amplitudes in D-dimensional super Yang-Mills, a direct investigation is beyond the scope of this paper due to the technical difficulties in implementing the BCFW recursion when it involves sums over fermionic degrees of freedom in an RNS-like representation. This would be important to resolve, possibly using the nodal operator constructions of [98] or the more recent numerator constructions in [33,99]. Since we verified that only physical poles contribute, these approaches could be used to identify the residues, and to check whether they agree with the forward-limit and factorisation terms of the loop-level BCFW recursion.…”

“…Contributions from similar poles in the 'linear' 32 We prove these statements in section 8.2. 33 This construction does indeed give the pure Yang-Mills integrand in the 'linear' formalism, i.e., with the 'old' one-loop scattering equations [24]. 34 We give more details on these tadpole-like configurations -known as 'singular solutions' to the scattering equations -in section 8.2.…”

“…This is however more technically involved due to the supersymmetric sum over states in the forward-limit term. It may be possible to resolve this using either the nodal operator construction of [98] or the more recent numerator constructions in [33,99], but both approaches are beyond the scope of this paper.…”

“…Both have been fruitful for studying theories of massless particles, most importantly gauge theory and gravity, and have brought numerous insights. They have also been extended to loop level, in various approaches; see, e.g., [11][12][13][14][15][16][17][18][19][20] for the BCFW recursion relations and [21][22][23][24][25][26][27][28][29][30][31][32][33] for the scattering equations.…”

Efficient calculation of crossing symmetric BCJ tree numerators

“…We have checked that our algorithm can provide complete ten-point results within minutes on a laptop. In a forthcoming paper [78], we will feed these tree-level numerators into the forward-limit machinery to study one-loop numerators with various amount of supersymmetry.…”