The Last of the Finite Loop Amplitudes in QCD Page: 3 of 45
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to yield compact expressions for NMHV tree amplitudes . The same authors and Witten
gave a simple and elegant proof  of the relation using special complex continuations of
the external momenta.
The proof, which we review in section III, is actually quite general, and applies to any
rational function of the external spinors satisfying certain scaling and factorization prop-
erties. Indeed, it has since been applied to amplitudes with massive particles , and in
gravity as well . The generality of the proof suggested that it should be useful for finding
on-shell recursion relations at one loop. We previously wrote down  such relations for
the (infrared and ultraviolet) finite n-gluon amplitudes in QCD, An (1+, 2+, 3+, ... , n+), for
which all gluons (or all but one) have the same helicity. Unlike the situation for massless
tree amplitudes, in an application to general loop amplitudes in a non-supersymmetric the-
ory, factorization in complex momenta is qualitatively different from that in real momenta.
Accordingly, we had to address new issues, in particular the appearance of double poles in
the complex analytic continuation.
In this paper, we examine another application of such on-shell relations, to one-loop
n-point amplitudes with one pair of massless external quarks and (n - 2) positive-helicity
gluons, A (1,2, 3+, ...,n+). This set of helicity amplitudes, together with the above
n-gluon amplitudes and their partners under parity, are zero identically at tree level due to
supersymmetry Ward identities . This is because massless quarks differ from gluinos only
in color manipulations which are essentially trivial at tree level. At one loop, the difference
in color factors between quarks and gluinos permit the amplitudes to be non-vanishing.
However, any infrared and ultraviolet divergences would have to be proportional to the
corresponding tree amplitude, which vanishes. Hence this set of helicity amplitudes is finite.
Because all the "zeroes" have been filled in at one loop, none of the corresponding two-
loop helicity amplitudes can be finite. For example, the first two-loop four-gluon scattering
amplitude to be computed , AV (1+, 2+, 3+, 4+), has infrared divergences similar to a
typical one-loop amplitude. Thus the amplitudes we compute in this paper represent the
last finite loop amplitudes to be computed in massless QCD.
We calculated the five-point amplitude, uAn') (l, 2, 3+, 4+, 5+) (together with all the
other helicity configurations) long ago , but no other results in this class of amplitudes
exist in pure QCD. On the other hand, related QED and mixed QED/QCD amplitudes,
containing a massless external c+c- pair and arbitrarily many positive-helicity photons or
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Bern, Zvi; Dixon, Lance J. & Kosower, David A. The Last of the Finite Loop Amplitudes in QCD, article, May 31, 2005; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc873114/m1/3/: accessed January 21, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.