Resummation with Wilson lines off the light cone
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esummation with Wilson Lines off the Light Cone1 Hsiangnan Li Institute of Physics, Academia Sinica, Taipei, Taiwan 115, Republic of China Department of Physics, TsingHua University, Hsinchu, Taiwan 300, Republic of China Department of Physics, National ChengKung University, Tainan, Taiwan 701, Republic of China email: [email protected] Abstract—I review the resummation formalism for organizing large logarithms in perturbative expansion of collinear subprocesses through the variation of Wilson lines off the light cone. A master equation is derived, which involves the evolution kernel resulting from this variation. It is then demonstrated that all the known single and doublelogarithm summations for a parton distribution function or a transversemomentum dependent parton distribution can be reproduced from the master equation by applying appropriate soft gluon approximations to the evolution kernel. Moreover, jet substructures, information which is crucial for particle identification at the Large Hadron Collider and usually acquired from event generators, can also be calculated in this formalism. DOI: 10.1134/S106377961404011X 1
1. INTRODUCTION
It is known that radiative corrections in perturba tive QCD (pQCD) produce large logarithms at each order of the coupling constant. Double logarithms appear in processes involving two scales, such as ln2(p+b) with p+ being the large longitudinal momen tum of a parton and b being the impact parameter conjugate to the small parton transverse momentum kT. In the region with a large Bjorken variable x, there exists ln2(1/N) from the Mellin transformation of ln(1 – x)/(1 – x)+, for which the two scales are the large p+ and the small infrared cutoff (1 – x)p+ for gluon emissions from a parton. Single logarithms are generated in processes involving one scale, such as lnp+ and ln(1/x), for which the relevant scales are the large p+ and the small xp+, respectively. To improve perturbative expansion, these logarithmic corrections need to be organized by evolution equations or resum mation techniques. Various methods have been devel oped to organize these logarithmic corrections to a parton distribution function (PDF) or to a transverse momentumdependent distribution function (TMD): the kT resummation for ln2(p+b) [1, 2], the threshold resummation for ln2(1/N) [3–5], the joint resummation [6, 7] that unifies the above two formalisms, the Dok shitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equation for lnp+ [8], the Balitsky–FadinKuraev– Lipatov (BFKL) equation for ln(1/x) [9], and the Cia faloni–Catani–Fiorani–Marchesini (CCFM) equa tion [10] that combines the above two evolution equa tions. The definition of a PDF or a TMD contains Wilson lines along the light cone, which collect gluons colli 1 The article is published in the original.
mated to a beam particle of momentum p and attach ing to other parts of a scattering process. The Wilson lines contain vertical links at infinity, if a TMD is con sidered. To perform resummation, a simple trick is to vary the
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