Threshold Resummation in Momentum Space from Effective Field Theory

Abstract

Methods from soft-collinear effective theory are used to perform the threshold resummation of Sudakov logarithms for the deep-inelastic structure function F2(x,Q2) in the end-point region x-->1 directly in momentum space. An explicit all-order formula is derived, which expresses the short-distance coefficient function C in the convolution F2 = C multiply sign in circle phi q in terms of Wilson coefficients and anomalous dimensions defined in the effective theory. Contributions associated with the physical scales Q2 and Q2(1-x) are separated from nonperturbative hadronic physics in a transparent way. A crucial ingredient to the momentum-space resummation is the exact solution to the integro-differential evolution equation for the jet function, which is derived. The methods developed in this Letter can be applied to many other hard QCD processes.

Precision direct photon andW-boson spectra at highpTand comparison to LHC data

“…The jet function is obtained from the imaginary part of the two-point function of collinear fields (see the middle Feynman diagram on the right in Figure 1). The two loop results for the inclusive quark and gluon jet functions relevant here were obtained in [33] and [34]. The last Feynman diagram in the Figure shows a NLO correction to the soft function, which describes the soft emissions from the energetic partons in both the initial and final state, which are encoded in Wilson lines along the corresponding directions.…”

“…The second ingredient, which we do not include are the non-logarithmic pieces of the jet, soft, and hard functions. The full two-loop jet functions are known [33,34] and also the two-loop soft function has now been computed [35]. The non-logarithmic piece of the two-loop hard function can be extracted from the results of [38,39], and we plan to include the full two-loop matching in the future.…”

“…The evolution factor U H qg can be obtained from the above results using the crossing relationŝ ↔ −û at fixedt , and U H gq follows from U H qg usingt →û. The resummed results for the jet and soft functions can be obtained by solving their RG equations in Laplace space [31]. For the gluon jet function, for example, the result takes the form…”

“…Physically, the threshold logarithms are associated with collinear radiation in the recoiling jet, or soft radiation coming from the jet or the incoming partons. A method to sum these logarithms to all orders in perturbation theory using the renormalization group in SCET was developed in [31,32]. Its application to direct photon production was discussed in detail in [18].…”

New insights into soft gluons and gravitons

“…They are related to the structure of large logarithms in perturbation theory, which must be summed up to all orders in order to obtain sensible results for many collider observables (see [1,2,3,4,5] for a number of different approaches). Furthermore, there are a number of unproven conjectures regarding infrared divergences, such as the so-called dipole formula in QCD [6], which we will see in what follows.…”