Abstract
The Pentagon Operator Product Expansion represents polygonal Wilson loops in planar $$ \mathcal{N}=4 $$ super Yang-Mills in terms of a series of flux tube excitations for finite coupling. We demonstrate how to re-sum this series at the one loop level for the hexagonal Wilson loop dual to the six-point MHV amplitude. By summing over a series of effective excitations we find expressions which integrate to logarithms and polylogarithms, reproducing the known one-loop result.
Highlights
Sm are symmetry factors, ui are rapidities, and Πdyn and Πmat are referred to as the dynamical part and matrix part respectively
We demonstrate how to re-sum this series at the one loop level for the hexagonal Wilson loop dual to the six-point MHV amplitude
We specialize to the case where r1 = 4 and r2 = 0, corresponding to an NMHV Wilson loop with four units of R-charge
Summary
The resummation of the MHV POPE series can be carried out following a similar strategy to that employed in [7]. Where this integral representation is valid only when Re(c) > Re(b) > 0 After this replacement, the POPE series is converted into the following expression,. All of the integrands are of the form exp [if (t)u] /u2 in the limit of large u, with (1 + e−2τ t)(1 − t) f (t) = 2σ − log t which has a root between 0 and 1 at, t∗ = 1 1 − e2σ+2τ − e2τ + (1 − e2σ+2τ − e2τ )2 + 4e2τ . 2. When t ∈ (t∗, 1), f (t) > 0, the contour closes in the upper half complex plane and picks up a pole at u = i|a|/2 so that the integration at infinity vanishes. We move the summation over helicity inside the integration This summation converges in the collinear limit, and it has a closed form, found from the following simple relation,. A straightforward way to see this simplification is to use symbol methods, as we illustrate in appendix A
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