The non-Abelian exponentiation theorem for multiple Wilson lines

Abstract

We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions from a Wilson line are described by effective vertices, each having a connected colour factor. The exponent consists of connected graphs made out of these vertices. We show that this readily provides a general colour basis for webs. We further discuss the kinematic combinations that accompany each connected colour factor, and explicitly catalogue all three-loop examples, as necessary for a direct computation of the soft anomalous dimension at this order.

From webs to polylogarithms

“…In particular, all colour factors appearing in the exponent correspond to connected graphs [93]. There are two complementary approaches to exponentiation: one is based on evolution equations [2-15, 18, 19, 21-40], which are ultimately a consequence of multiplicative renormalizability [94][95][96][97], and the second is a diagrammatic approach, the direct computation of the exponent in terms of webs [93,[98][99][100][101][102][103][104]. Following ref.…”

“…Wilson loop [99,100], and was recently generalised to the case of multiple Wilson lines in arbitrary representations of the colour group [93,[101][102][103][104]. In contrast to the Abelian case, where only connected diagrams contribute to the exponent, in a non-Abelian theory certain non-connected diagrams 2 contribute as well.…”

“…In contrast to the Abelian case, where only connected diagrams contribute to the exponent, in a non-Abelian theory certain non-connected diagrams 2 contribute as well. The non-Abelian exponentiation theorem states that such a diagram D contributes to the exponent with a modified, Exponentiated Colour Factor (ECF) C(D), which corresponds to a connected graph [93].…”

“…[103,104], where it was shown that web mixing matrices are idempotent, R 2 = R, namely they act as projection operators, selecting particular linear combinations of colour factors, those corresponding to unit eigenvalues of R, to appear in the exponent. It was then proven [93] that these combinations always correspond to connected graphs. The mixing matrix can also be viewed as acting on the kinematic factors, generating particular linear combinations of {F(D)} in which certain subdivergences cancel, as dictated by the renormalisation properties of the vertex at which the Wilson lines meet [104].…”

“…The contributions to the anomalous dimension need to be summed over all subtracted webs of order n, Γ (n) = −2n i w (n,−1) i (where we discarded running coupling terms). It should be noted that a given web may in general contribute to several different colour factors (these are enumerated by the eigenvectors of the mixing matrix corresponding to unit eigenvalue [93,101]). Also note that different webs may contribute to the same colour factor.…”