Conformal invariance of the one-loop all-plus helicity scattering amplitudes

Abstract

The massless QCD Lagrangian is conformally invariant and, as a consequence, so are the tree-level scattering amplitudes. However, the implications of this powerful symmetry at loop level are only beginning to be explored systematically. Even for finite loop amplitudes, the way conformal symmetry manifests itself may be subtle, e.g. in the form of anomalous conformal Ward identities. As they are finite and rational, the oneloop all-plus and single-minus amplitudes are a natural first step towards understanding the conformal properties of Yang-Mills theory at loop level. Remarkably, we find that the one-loop all-plus amplitudes are conformally invariant, whereas the single-minus are not. Moreover, we present a formula for the one-loop all-plus amplitudes where the symmetry is manifest term by term. Surprisingly, each term transforms covariantly under directional dual conformal variations. We prove the formula directly using recursive techniques, and check that it has the correct physical factorisations.

Propagators, BCFW recursion and new scattering equations at one loop

“…The recursion relation in (5.11) was deduced by BCFW shifting legs 1 and n of the known one-loop all-plus amplitude, which is a rational function first conjectured in [51] and proven in [52] using off-shell recursion [81]. More recently, the same form of the recursion relation was used to prove a new formula for all-plus amplitudes, based on conformally invariant building blocks [82]. Comparing (5.11) and (4.1), it is easy to check that the tree factorisation terms match.…”

Virtual QCD corrections to gluon-initiated diphoton plus jet production at hadron colliders

“…Remarkably, this amplitude is invariant under conformal transformations, and the expression given here exhibits this property in a manifest way [102]. If all masses are neglected, the SM Lagrangian is conformally invariant.…”

“…Prior to discussing the numerical implementation of all two-loop helicity amplitudes, we would like to comment on the all-plus amplitude, which displays a particularly simple analytic form. We find that the structures appearing are closely related to those appearing in the five-gluon all-plus amplitudes at one [99][100][101][102] and two loops [56,[92][93][94]. We present the finite remainders in the expansion around d s = 2.…”

Full color two-loop six-gluon all-plus helicity amplitude

“…and, in the specific case where K 2 2 = 0, Note that these six-point coefficients are conformally invariant: a feature noticed for the five-point all-plus amplitude in ref. [25].…”