One of the important features of my PhD project was about understanding the same helicity one-loop gluon amplitudes. The structure of these amplitudes resemble a tree level object, with the result being a rational function of the momenta involved. To our amazement, we found that the same helicity one-loop amplitudes can be constructed from more elementary tree-level blocks, such as the Berends-Giele currents and an effective propagator. An important observation is that to make sense of these one-loop corrected propagators, one needs to introduce a new set of mathematical variables, named as "region momentum variables" or "dual momenta". This then implies that to interpret such a formula, one must resort to momentum twistor space, where the dual momenta are natural objects. Nevertheless, let us state the formula for these amplitudes of any multiplicity:
The large blobs in the pictorial representation of the amplitude stand for the Berends-Giele currents, with the off-shell legs pointing towards the internal line of this diagram. The other quantities are as follows. The object q is the reference (auxiliary) spinor. The sum is taken over partitions of a cyclically ordered set 1, . . . , n with the convention n + 1 = 1 into two subgroups i, . . . , j and j + 1, . . . , i − 1. The bra-ket notation is standard and represents a particular 2-spinor contraction.
There are several previous works that are strongly related to the context of this formula. First, Zvi Bern observed that in the computation of the same helicity one-loop amplitude in Yang-Mills theory, the sum of box, triangle and bubble integrands, with a particular choice of the loop momenta, vanishes. This has later been explored by other authors in the context of world-sheet formulation of YM, and in the related light-cone formalism. There is also a very closely related discussion, again in the light-cone, in the context of chiral higher spin theories. The upshot of these works is that the same helicity one-loop amplitudes in YM can be obtained by performing tree-level calculations, but with insertions of certain helicity violating ”bubble” counterterms. Our new formula is the same observation phrased in the covariant formalism.
Our new interpretation is relevant for the problem of UV divergence of quantum gravity. As we have already discussed in one of our previous articles, this divergence was linked to the non-vanishing of the same helicity one-loop amplitude in gravity. Given that the story with gravity is likely mirroring that in Yang-Mills, the interpretation of this work shows that the non-vanishing of the same helicity one-loop amplitudes can in turn be linked to the non-vanishing of the one-loop bubble. It would thus be very interesting to better understand the significance and interpretation of the non-vanishing of the bubbles.