# Michael Engelhardt

## Biography

Dr. Engelhardt’s research in theoretical physics is focused on the low-energy sector of Quantum Chromodynamics (QCD), the gauge field theory which describes the strong interaction. At low energies, the strong interaction is characterized by a number of phenomena which can only be analyzed using non-perturbative methods. Most prominent among these are the confinement of quarks and gluons into hadrons, the spontaneous breaking of chiral symmetry, and the axial U(1) anomaly which arises due to topological properties of the gauge field. Currently, Dr. Engelhardt is predominantly active in two areas, namely, Lattice QCD studies of hadron structure (including Transverse Momentum-Dependent Parton Distributions (TMDs), Generalized Parton Distributions (GPDs), and hadron polarizabilities), as well as the center vortex picture of the strongly interacting vacuum.

#### Recent Publications

Quark orbital angular momentum in the proton evaluated using a direct derivative method – Jan 2019

**Abstract: **Quark orbital angular momentum (OAM) in the proton can be calculated directly given a Wigner function encoding the simultaneous distribution of quark transverse positions and momenta. This distribution can be accessed via proton matrix elements of a quark bilocal operator (the separation in which is Fourier conjugate to the quark momentum) featuring a momentum transfer (which is Fourier conjugate to the quark position). To generate the weighting by quark transverse position needed to calculate OAM, a derivative with respect to momentum transfer is consequently required. This derivative is evaluated using a direct derivative method, i.e., a method in which the momentum derivative of a correlator is directly sampled in the lattice calculation, as opposed to extracting it a posteriori from the numerical correlator data. The method removes the bias stemming from estimating the derivative a posteriori that was seen to afflict a previous exploratory calculation. Data for Ji OAM generated on a clover ensemble at pion mass $m_{\pi } = 317\, \mbox{MeV} $ are seen to agree with the result obtained via the traditional Ji sum rule method. By varying the gauge connection in the quark bilocal operator, also Jaffe-Manohar OAM is extracted, and seen to be enhanced significantly compared to Ji OAM.

Generalized Wandzura Wilczek Relations and Orbital Angular Momentum – Feb 2018

**Abstract: **New Lorentz Invariance Relations (LIRs) were presented between twist-three Generalized Parton Distributions (GPDs) and transverse momentum, $k_T$, moments of twist-two Generalized Transverse Momentum-Dependent Distributions (GTMDs). By implementing both these LIRs and the QCD Equations of Motion in the quark quark correlation function, we generated a new series of Wandzura Wilczek (WW) relations in the off-forward sector. Two of these WW relations take on a particularly interesting physical meaning in that they provide a clear interpretation of the QCD structure of Orbital Angular Momentum (OAM) in the nucleon. In particular, they provide a solution to the outstanding puzzle of how OAM could be simultaneously described by twist-two GTMDs and twist-three GPDs. Additional relations were discussed, in particular, for the helicity configurations that can be detected analyzing specific spin asymmetries: one corresponding to a longitudinally polarized quark in an unpolarized proton, associated with spin-orbit correlations, and one for transverse proton polarization, as a generalization of the relation obeyed by the $g_2$ structure function; finally, we defined a relation connecting the off-forward extension of the Sivers function to an off-forward Qiu-Sterman term.

More articles and publications can be found here.