Towards quasi-transverse momentum dependent PDFs computable on the lattice

Markus A. Ebert, Iain W. Stewart, Yong Zhao
2019 Journal of High Energy Physics  
Transverse momentum dependent parton distributions (TMDPDFs) which appear in factorized cross sections involve infinite Wilson lines with edges on or close to the light-cone. Since these TMDPDFs are not directly calculable with a Euclidean path integral in lattice QCD, we study the construction of quasi-TMDPDFs with finite-length spacelike Wilson lines that are amenable to such calculations. We define an infrared consistency test to determine which quasi-TMDPDF definitions are related to the
more » ... e related to the TMDPDF, by carrying out a one-loop study of infrared logarithms of transverse position b T ∼ Λ −1 QCD , which must agree between them. This agreement is a necessary condition for the two quantities to be related by perturbative matching. TMDPDFs necessarily involve combining a hadron matrix element, which nominally depends on a single light-cone direction, with soft matrix elements that necessarily depend on two light-cone directions. We show at one loop that the simplest definitions of the quasi hadron matrix element, the quasi soft matrix element, and the resulting quasi-TMDPDF all fail the infrared consistency test. Ratios of impact parameter quasi-TMDPDFs still provide nontrivial information about the TMD-PDFs, and are more robust since the soft matrix elements cancel. We show at one loop that such quasi ratios can be matched to ratios of the corresponding TMDPDFs. We also introduce a modified "bent" quasi soft matrix element which yields a quasi-TMDPDF that passes the consistency test with the TMDPDF at one loop, and discuss potential issues at higher orders. ( 1.5) where O(Λ 2 QCD /P 2 z , M 2 /P 2 z ) terms are power corrections. Here f j (y, µ) for −1 < y < 0 corresponds to the anti-quark PDF. The C ij are perturbative matching coefficients which come from a hard region of momentum space, see refs. [43, 45] for further details. Recently, significant progress has been made on various aspects of the LaMET procedure, including the renormalization and matching [43, of the quasi-PDF, the power corrections [73] [74] [75] , as well as the lattice calculation of the x-dependence of PDFs and distribution amplitudes [62, 73, [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] . Notably, the most recent lattice results at physical pion mass [84, 85, 87, 89, 91] and large nucleon momenta [85, 89, 91] have shown encouraging signs that the LaMET approach can lead to a precise determination of the PDFs. Due to the interest in TMDPDFs it is natural to consider the extension of the LaMET approach to transverse momentum observables. Due to the required focus on spatial matrix elements for TMDPDFs, studies based on LaMET are actually related to the lattice methods developed in refs. [37] [38] [39] . While applying LaMET to TMDPDFs might seem straightforward, the richer structure of TMD factorization, which we review in section 2, actually makes this quite non-trivial. In contrast to the case for collinear factorization, TMD physics is plagued by so-called rapidity divergences and the need for combining collinear 1 proton matrix elements with soft vacuum matrix elements. Such soft matrix elements retain a minimal amount of information about both incoming protons (their direction and the color charge of the probing parton). The importance of the soft matrix 1 Note that the second use of the word "collinear" here is in the context of factorized collinear and soft fields as defined for example in SCET, not to distinguish between collinear and TMD factorization.
doi:10.1007/jhep09(2019)037 fatcat:wl4atf5r4vhu7nvmbzlrcoqymy