Soft Wilson lines in the soft-collinear effective theory

Abstract

The effects of the soft gluon emission in hard scattering processes at the phase boundary are resummed in the soft-collinear effective theory (SCET). In SCET, the soft gluon emission is decoupled from the energetic collinear part, and is obtained by the vacuum expectation value of the soft Wilson-line operator. The form of the soft Wilson lines is universal in deep inelastic scattering, in the Drell-Yan process, in the jet production from e + e − collisions, and in the γ * γ * → π 0 process, but its analytic structure is slightly different in each process. The anomalous dimensions of the soft Wilson-line operators for these processes are computed along the light-like path at leading order in SCET and to first order in α s , and the renormalization group behavior of the soft Wilson lines is discussed.Here the integration path C is determined by the kinematics of the processes, T orders gauge fields A a µ (z) in time, and P orders the generators T a of the SU(N) gauge group along the path C. The difference between W (C) and W T (C) is that the gluon fields in W (C) are ordered along the path C, but not according to time. Therefore on different parts of the path C, the gluon fields are time or anti-time ordered. The minute difference in the definitions of W T (C) and W (C) affects the analytic structure of the soft Wilson lines.The main theme of the paper is to study the analytic structure of the Wilson lines, especially the soft Wilson lines in SCET near the boundary of the phase space. We show that the soft Wilson lines appear universally in deep inelastic scattering, in the Drell-Yan process and in the jet production from e + e − collisions, in which the Wilson lines appear in the matrix elements squared or the discontinuity of the forward scattering amplitude. They

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“…We use the same Wilson line for both the quark and antiquark parts of n . Another possibility is to make different field redefinitions on the particle and antiparticle parts of the fields [52]. In fact, all results are independent of the choice of reference point in the field redefinition; the path is determined entirely by changes the field redefinition induces on the operators and the interpolating fields for the states [53].…”

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“…15 When considering SCET without Glauber gluons these directions are often determined by those of their parent collinear particles and the structure of the product of operators being considered [120,121]. In the theory with Glauber gluons, the choice for the direction of the soft Wilson lines is not relevant, since this dependence is instead captured by contributions from the Glauber region.…”

“…In the calculation of S-matrix elements, the direction of the combined ultrasoft lines is determined by the product of ultrasoft lines generated by J Γ and by the interpolating fields for the incoming and outgoing states (or equivalently those of the external collinear particles) [120,121]. The directions for the ultrasoft Wilson lines in the S-matrix elements are then both (0, ∞) for n-n production, while we have Yn(−∞, 0) and Y † n (0, ∞) for the n-n quark scattering.…”

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“…ultrasoft Wilson lines [94,95], so Y (r) n may also occur with a path from (0, ∞). Indeed for e + e − → dijets all the ultrasoft Wilson lines occur with paths from (0, ∞) (see eg.…”