Quark orbital dynamics in the proton from lattice QCD: From Ji to Jaffe-Manohar orbital angular momentum

Abstract

Given a Wigner distribution simultaneously characterizing quark transverse positions and momenta in a proton, one can directly evaluate their cross-product, i.e., quark orbital angular momentum. The aforementioned distribution can be obtained by generalizing the proton matrix elements of quark bilocal operators which define transverse momentum-dependent parton distributions (TMDs); the transverse momentum information is supplemented with transverse position information by introducing an additional nonzero momentum transfer. A gauge connection between the quarks must be specified in the quark bilocal operators; the staple-shaped gauge link path used in TMD calculations yields the Jaffe-Manohar definition of orbital angular momentum, whereas a straight path yields the Ji definition. An exploratory lattice calculation, performed at the pion mass mπ = 518 MeV, is presented which quasi-continuously interpolates between the two definitions and demonstrates that their difference can be clearly resolved. The resulting Ji orbital angular momentum is confronted with traditional evaluations based on Ji's sum rule. Jaffe-Manohar orbital angular momentum is enhanced in magnitude compared to its Ji counterpart.

One-loop renormalization of staple-shaped operators in continuum and lattice regularizations

“…(50) coincide with those in Eq. (46). The only dependence on the shape regards the Dirac structure of the operator which mixes with O Γ .…”

“…To recover the desired infinite length one checks for convergence as the length increases, and an extrapolation to η→∞ is applied. More recently, the connection between nonlocal operators with staple-shaped Wilson line and orbital angular momentum [46,47] has been discussed. This relies on a comparison between straight and staple-shaped Wilson lines, with the staple-shaped path yielding the Jaffe-Manohar [48,49] definition of quark orbital angular momentum, and the straight path yielding Ji's definition [49][50][51].…”

Nucleon generalized form factors from two-flavor lattice QCD

“…For recent discussions of this topic see, e.g., Refs. [25][26][27] and the articles cited therein. In this article we will not review the many fascinating aspects of GPDs but concentrate on our lattice calculation of the nucleon GFFs using well-established techniques for the calculation of Mellin moments of GPDs; see, e.g., Ref.…”

Generalized TMDs in the exclusive double Drell-Yan process

“…It is remarkable that the same relation between F 1,4 and the quark OAM holds for both commonly used OAM definitions -the (canonical) one by Jaffe and Manohar (L JM ) [8], and the one by Ji (L Ji ) [9]. Moreover, it gives access to the so far elusive L JM in quantum chromodynamics (QCD) on the lattice [6,10].…”