Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

By Z. Bern, L. Dixon, Vladimir A. Smirnov. 2005

896 citations recorded by the source corpus.

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 - 2{epsilon} dimensions, as a Laurent expansion about {epsilon} = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/{epsilon}{sup 2} pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/10c46f4c8fc26c6d80c3a0f4b6b6dd13c2bc5671

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In Nuclear and High Energy Physics

1. S. Caron-Huot
2. L. Alday
3. L. Dixon
4. J. Maldacena
5. J. Drummond
6. R. Roiban
7. P. Vieira
8. J. Henn

Citations in the corpus

• Holography from Conformal Field Theory (2009, background): “However, the only amplitudes for which there is a fairly strong all-orders understanding are the on-shell four and five point functions [15], which are highly constrained by symmetry [16], whereas we need at least the off-shell fourpoint function.”
• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “In the case of maximally helicity-violating (MHV) amplitudes, the relevant piece (obtained by subtracting particular universal infrared-divergent terms [22, 23]) which is not fixed by dual conformal symmetry is called the remainder function.”
• Integrability and transcendentality (2006, background): “For example, it was recently proposed that higher-loop maximally helicity violating (MHV) n-point gluon amplitudes should be iteratively expressible through the (regulated) one-loop amplitudes [1], [2].”
• MHV amplitudes in N=4 super Yang-Mills and Wilson loops (2007, methodology): “8) where F (1) n (0) is the finite remainder of M((1)) n as defined in [1].”
• Feynman integrals and hyperlogarithms (2015, methodology): “It was evaluated first in a very special case in [17] (using a Mellin-Barnes representation) in terms of harmonic polylogarithms and multiple zeta values.”
• Space-time S-matrix and flux tube S-matrix II. Extracting and matching data (2013, methodology): “The remainder function logR is defined as the logarithm of the ratio between the amplitude and the BDS ansatz [16,50]”
• Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions (2014, methodology): “Namely, we can construct the ratio of polygons W using the one loop result — given by the BDS ansatz [28] — and the two-loop correction — first computed in [29] and dramatically simplified in [30].”
• Heptagons from the Steinmann cluster bootstrap (2016, methodology): “The BDS ansatz [27] for the n-particle MHV amplitude (with the Parke-Taylor tree amplitude scaled out) is given by”
• Black hole binary dynamics from the double copy and effective theory (2019, background): “[44, 72] for further details on constructing MB representations and many examples.”