Imaginary Parts and Discontinuities of Wilson Line Correlators

Abstract

We introduce a notion of position-space cuts of eikonal diagrams, the set of diagrams appearing in the perturbative expansion of the correlator of a set of straight semi-infinite Wilson lines. The cuts are applied directly to the position-space representation of any such diagram and compute its imaginary part to the leading order in the dimensional regulator. Our cutting prescription thus defines a position-space analog of the standard momentum-space Cutkosky rules. Unlike momentum-space cuts which put internal lines on shell, positionspace cuts constrain a number of the gauge bosons exchanged between the energetic partons to be lightlike, leading to a vanishing and a non-vanishing imaginary part for space-and timelike kinematics, respectively. Introduction.-The infrared singularities of gauge theory scattering amplitudes play a fundamental role in particle physics for both phenomenological and theoretical reasons. Knowing the structure of long-distance singularities is necessary for combining the real and virtual contributions to the cross section, as the divergences of the separate contributions only cancel once they are added. In addition, infrared singularities dictate the structure of large logarithmic contributions to the cross section, allowing such terms to be resummed. Long-distance singularities, moreover, have several highly interesting properties. They have a universal structure among different gauge theories; their exponentiation properties [1][2][3][4][5][6] and their relation to the renormalization of Wilson line correlators [7,8] allow the exploration of the all-order structure of their perturbative expansion, a feat currently unattainable for complete scattering amplitudes.The key tool for computing the infrared singularities of scattering amplitudes is provided by the eikonal approximation in which each parton i emerging from the hard scattering acts as a source of soft gluon radiation and is replaced by a semi-infinite path ordered Wilson line

Lagrangian insertion in the light-like limit and the super-correlators/super-amplitudes duality

“…It is arbitrary which of the two Wilson loops we choose to consider. 9 The arguments are presented in Feynman gauge but it is simple to extend the arguments to more general gauges. 10 There will also be vectors (xi+1 − y1) κ but they will come with a factor of 1 − t. As in the argument leading to (4.24), we will discard these terms.…”

Path description of coordinate-space amplitudes

“…We will then be ready to apply our analysis to expectations of Wilson lines, and derive some of the interesting results found in Refs. [3,4] from our general viewpoint. We go on to discuss how to treat factors associated with spin, and finally how to use dimensional regularization and incorporate masses.…”

“…It is natural ask whether there is a useful "dual" formalism for coordinate-space Green functions, which are a fundamental element in quantum field theory, and which also occur naturally in the evaluation of expectation values of Wilson lines [3,4]. We will see that a program of perturbative light-cone integrals can indeed be carried out in coordinate space, following a pattern similar to that in momentum space.…”

“…To compare with the results of Ref. [3], we can identify the real part of the product of denominators in (76) by expanding each of them into a real principal value and imaginary delta function. The imaginary part can be found from replacing either of the two remaining denominators with a delta function, which would correspond to vanishing light-cone deficits between the light-cone distances computed along a (two-gluon) path between two of the "external" vertices along the Wilson lines.…”

“…V we study discontinuities in coordinate-space integrals, and rederive and reinterpret the results of Refs. [3,4]. Finally, a brief discussion of the extension to dimensional regularization and massive fields is given as Sec.…”