Proper definition and evolution of generalized transverse momentum dependent distributions

Abstract

We consider one of the most fundamental sets of hadronic matrix elements, namely the generalized transverse momentum dependent distributions (GTMDs), and argue that their existing definitions lack proper evolution properties. By exploiting the similarity of GTMDs with the much better understood transverse momentum distributions, we argue that the existing definitions of GTMDs have to include an additional dependence on soft gluon radiation in order to render them properly defined. With this, we manage to obtain the evolution kernel of all (un)polarized quark and gluon GTMDs, which turns out to be spin independent. As a byproduct, all large logarithms can be resummed up to next-to-next-to-leading-logarithmic accuracy with the currently known perturbative ingredients. I. MOTIVATIONIt has been known for quite some time that hadronic matrix elements such as generalized parton distributions [1-3] (GPDs) and transverse momentum dependent parton distribution/fragmentation functions [4-6] (TMDs in general) are indispensable objects for studying fundamental properties of hadrons. With them one can probe nucleon tomography by investigating their spin and three-dimensional momentum distributions.More recently a new class of hadronic matrix elements was introduced, which generalizes both GPDs and TMDs. In [7], correlation functions which describe off-forward scattering amplitudes were introduced, where the bi-local partonic fields are separated in all three light-front coordinates: (z + , z − and z ⊥ ). Given some kinematics imposed by an underlying scattering process, one can consider a restricted class of matrix elements for which either z + = 0 or z − = 0. In this case the matrix elements reduce to what is known as generalized transverse momentum dependent parton distribution functions (GTMDPDFs or simply GTMDs) [7-9] (see also [10,11], and a recent review in [12]). These hadronic quantities are off-forward matrix elements with explicit dependence on the longitudinal and the transverse momentum components of partons inside hadrons. As such, they are hybrid constructs of both classes of functions: GPDs and TMD bi-local correlators. Notice that here we make a distinction between TMDs and TMD correlators, where the latter, also called unsubtracted TMDs, are ill-defined due to uncanceled spurious rapidity divergences (see discussions in, e.g., [4][5][6]).In this work we argue that the hadronic matrix elements called in the literature GTMDs, as currently formulated and analyzed, are improperly defined. The last assertion results from the observation that these matrix elements lack proper evolution properties with respect to the renormalization scale µ and the rapidity scale Q. Also their operator product expansion into generalized parton distributions breaks down even for large enough transverse momentum, contrary to what the case should be. Definitely these facts severely limit the predictive power of the underlying theory: QCD. The upshot is that such matrix elements, although formulated from the basic QCD fields and being g...

Quark Wigner distributions and spin-spin correlations

“…The Wigner distribution of quarks with longitudinal polarization λ in a longitudinally polarised proton Λ is defined for Γ = γ + 1+λγ 5 …”

“…It is well known that GPDs and TMDs encode informations about the three dimensional partonic structure of hadrons. Recently, generalized transverse momentum dependent PDFs or GTMDs are introduced [3][4][5]. Gluon GTMDs have been discussed in [6].…”

Observables for Generalized TMDs of Quarks

“…[26], which is widely used in the TMD case. More information on this point can be found, for instance, in [27].…”

Transverse phase space and its multipole decomposition

“…A more consistent definition of the GTMDs should in principle also include a soft factor contribution [19], but the latter has no real impact on the following discussions and can therefore be omitted. We choose to work in the symmetric frame defined by…”