The dimension-shift conjecture for one-loop amplitudes

Abstract

A conjecture made by Bern, Dixon, Dunbar, and Kosower asserts a simple dimension shifting relationship between the one-loop structure of $$ \mathcal{N} $$ N = 4 MHV amplitudes and all-plus helicity amplitudes in pure Yang-Mills theory. We prove this conjecture to all orders in dimensional regularisation using unitarity cuts, and evaluate the form of these simplest one-loop amplitudes using a generalised D-dimensional unitarity technique which captures the full amplitude to all multiplicities.

One-loop multicollinear limits from 2-point amplitudes on self-dual backgrounds

“…These configurations are interesting for many reasons: they are rational functions when the electron/quark mass is zero, include the leading-colour contribution to 1-loop gluon scattering, and have all-multiplicity formulae (when the electron/quark is massless) [36][37][38][39][40]. Remarkably, the all-plus helicity configuration in massless QCD is related to maximally supersymmetric Yang-Mills amplitudes with two negative helicity gluons [41,42], and this configuration is even linked with ultraviolet divergences in two-loop perturbative gravity [43][44][45].…”

Proving the dimension-shift conjecture

“…Of particular practical interest is the fact that the relation (1) holds to all orders in ε, and thus relates the general-dimensional structure of the integral functions upon which it depends. In [3] the conjecture was verified up to n = 6 but it has now been proven to all-multiplicities, and to all orders in ε [4]. Moreover, the complete all-orders-in-ε all-n amplitudes can be computed in both theories.…”

“…where ε is the usual dimensional-regulator parameter and the 〈i j〉 4 factor is the standard (Weyl) spinor-helicity contraction which compensates for the spinor-weight (or little-group scaling) between the two amplitudes.…”

A Unitarity Approach to Two-Loop All-Plus Rational Terms

“…14: An example of a one-loop-squared cut contributing to the A 12;6,9 partial amplitude. As explained in the text, it contributions to the coefficient of the triple-trace structure ITr(1, 2, 3, 4, 5, 6) ITr (7,8,9) ITr (10,11,12).…”

“…In this respect, they are a good laboratory for exploring aspects of Yang-Mills amplitudes beyond what is probed in N = 4 supersymmetric amplitudes. These amplitudes may also be a portal to exploring hidden connections between different theories: at one loop, there is an intriguing connection [7,8] between the all-plus amplitude and the simplest N = 4 super-Yang-Mills amplitude, scattering all gluons but two of like helicity (MHV). Furthermore, for four and five gluons in the planar limit, the leading transcendental weight parts of all-plus amplitudes at two loops (and three loops for four gluons) have been shown to be dual to those of N = 4 Wilson loops with Lagrangian insertions at one lower loop order [9].…”