One-loop same helicity four-point amplitude from shifts
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Springer
Received: March 24, 2020 Accepted: May 23, 2020 Published: June 12, 2020
Pratik Chattopadhyay and Kirill Krasnov School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, U.K.
E-mail: [email protected], [email protected] Abstract: It has been suggested a long time ago by W. Bardeen that non-vanishing of the one-loop same helicity YM amplitudes, in particular such an amplitude at four points, should be interpreted as an anomaly. However, the available derivations of these amplitudes are rather far from supporting this interpretation in that they share no similarity whatsoever with the standard triangle diagram chiral anomaly calculation. We provide a new computation of the same helicity four-point amplitude by a method designed to mimic the chiral anomaly derivation. This is done by using the momentum conservation to rewrite the logarithmically divergent four-point amplitude as a sum of linearly and then quadratically divergent integrals. These integrals are then seen to vanish after appropriate shifts of the loop momentum integration variable. The amplitude thus gets related to shifts, and these are computed in the standard textbook way. We thus reproduce the usual result but by a method which greatly strengthens the case for an anomaly interpretation of these amplitudes. Keywords: Scattering Amplitudes, Anomalies in Field and String Theories ArXiv ePrint: 2002.11390
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)082
JHEP06(2020)082
One-loop same helicity four-point amplitude from shifts
Contents 1 Introduction
1
2 Triangle anomaly 2.1 Computation
3 4 9 18 19
4 Discussion
19
A Feynman rules and shift extraction A.1 Massless QED Feynman rules A.2 Self-dual YM Feynman rules A.3 Extracting the shift
21 21 22 23
1
Introduction
The same helicity Yang-Mills (YM) amplitudes vanish at tree-level, but become non-zero at one-loop.1 The QCD one-loop amplitudes at four (and five) points were computed by the field theory techniques in [1], and via string-inspired technology in [2] (four-points) and [3] (five-points). The result for same helicity five gluon amplitude was then used to conjecture [4] an n-gluon formula. Supersymmetry implies that there is a relation between same helicity one-loop amplitudes in theories with different spin particles (e.g. spin 1 and spin 1/2) propagating in the loop, see [3]. This means that the same helicity one-loop amplitudes in YM are related to those in massless QED. The later were computed in [5] using recursive methods, thus proving the conjecture of [4]. This conjecture received additional support from the consideration of the collinear limits in [6]. At four points, which is the case of main interest for us in this paper, the same helicity amplitude takes the following extremely simple form [2] Aone−loop (1+ , 2+ , 3+ , 4+ ) ∼
[12][34] , h12ih34i
(1.1)
where the spinor helicity notations are used, see below, and the proportionality factor contains a numerical coefficient as well
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