Abstract
We further develop the formalism for taking position-space cuts of eikonal diagrams introduced in [Phys.Rev.Lett. 114 (2015), no. 18 181602, arXiv:1410.5681]. These cuts are applied directly to the position-space representation of any such diagram and compute its discontinuity to the leading order in the dimensional regulator. We provide algorithms for computing the position-space cuts and apply them to several two- and three-loop eikonal diagrams, finding agreement with results previously obtained in the literature. We discuss a non-trivial interplay between the cutting prescription and non-Abelian exponentiation. We furthermore discuss the relation of the imaginary part of the cusp anomalous dimension to the static interquark potential.
Highlights
In this paper we have provided algorithms for the compution of the position-space cuts of eikonal diagrams introduced in ref. [1] and discussed the interplay of the cutting prescription with non-Abelian exponentiation
Momentum-space cuts of eikonal diagrams, analogous to the Cutkosky rules for standard Feynman diagrams, were introduced in ref. [21] where they were used to show that the exchanges of Glauber-region gluons produce imaginary parts of the Wilson-line correlator
Any given momentum-space cut separates the eikonal diagram into two disjoint subdiagrams, putting the eikonal and, depending on the cut, possibly a number of standard Feynman propagators on shell
Summary
We will discuss the origin of the imaginary part of Wilson line correlators from the point of view of causality as well as unitarity. For external kinematics corresponding to the diagram, the integral in eq (2.6) has a vanishing imaginary part: the denominator (t1v1 − t2v2) is strictly positive within the region of integration, and the −iη can be dropped In this situation, the partons are never lightlike separated, as illustrated, and the phases of their states cannot change through exchanges of lightlike massless gauge bosons. We observe that the imaginary part of the cusp anomalous dimension evaluated in timelike kinematics takes the form of the non-relativistic Coulomb potential (the appropriate dimension of energy is acquired after replacing the angle γ by the distance between the two fermions) This relation does not extend to generic non-Abelian gauge theories, as we will discuss shortly. In position space, eikonal diagrams without internal vertices take the form of iterated integrals In this representation, their imaginary parts can be straightforwardly obtained by applying the principal-value formula (2.10) recursively
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