Single-valued harmonic polylogarithms and the multi-Regge limit

By L. Dixon, Claude Duhr, Jeffrey Pennington. 2012

154 citations recorded by the source corpus.

A bstractWe argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar $ \mathcal{N}=4 $ super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w, w∗). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (ν, n) which are related to (w, w∗) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (ν, n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to N3LLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/e853fa85baa04269aa423ebf7e126821be8dddec

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In Nuclear and High Energy Physics

1. L. Dixon
2. S. Caron-Huot
3. J. Drummond
4. M. Hippel
5. P. Vieira
6. M. Spradlin
7. A. Sever
8. B. Basso

Citations in the corpus

• Quantum gravity from conformal field theory (2017, background): “They are constructed in general in [50] and appear in many contexts to discuss the perturbative contributions to the correlation functions 〈p1p2p3p4〉 [40, 42] as well as in multi-Regge kinematics of scattering amplitudes [51, 52] and Feynman integral calculations [53, 54].”
• Space-time S-matrix and flux tube S-matrix II. Extracting and matching data (2013, methodology): “It turns out to be a powerful constraint on the four loops hexagon and, when combined with the techniques of [13] (see also [54]), it allowed the authors of [14] to successfully determine the symbol of the four loops remainder function.”
• Hexagon functions and the three-loop remainder function (2013, methodology): “The coefficient functions h (L) r (w,w∗) can be determined simply from the g (L) r , by using a crossing relation from the 3 → 3 channel [50, 55].”
• The four-loop six-gluon NMHV ratio function (2015, background): “Here the p (L) r (w,w∗) and q (L) r (w,w∗) are composed of functions known as single-valued harmonic polylogarithms (SVHPLs) [69, 97].”
• Bootstrapping an NMHV amplitude through three loops (2014, methodology): “[56]:”
• When does the gluon reggeize? (2013, background): “On the other hand, NNLL calculations are certainly within the reach of present-day scattering amplitude technology in planar N = 4 super Yang-Mills, see for example [87, 112, 113].”
• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, background): “More generally, it is expected that the integrability of the theory [40] will play an instrumental role in determining its S-matrix, and apart from the collinear limit it has also led to all-loop expressions in the multi-Regge limit [41], another kinematical regime that has provided significant info”
• Motivic amplitudes and cluster coordinates (2013, background): “A fair amount of recent work has considered the behavior of amplitudes in various restricted domains, such as two-dimensional or multi-Regge kinematics (see for example [64–68] or [69–74], respectively), where in either case considerable simplification occurs.”