Chiral-even axial twist-3 GPDs of the proton from lattice QCD

This work presents the first lattice-QCD calculation of the twist-3 axial quark GPDs for the proton using the large-momentum effective theory approach. We calculate matrix elements with momentum-boosted proton states and a non-local axial operator. The calculation is performed using one ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with a clover term. The ensemble has a volume $32^3\times64$, lattice spacing 0.0934 fm, and corresponds to a pion mass of 260 MeV. The matrix elements are calculated for three values of the proton momentum, namely 0.83, 1.25, and 1.67 GeV. The light-cone GPDs are defined in the symmetric frame, which we implement here with a (negative) 4-momentum transfer squared of 0.69, 1.38, and 2.76 GeV$^2$, all at zero skewness. We also conduct several consistency checks, including assessing the local limit of the twist-3 GPDs and examining the Burkhardt-Cottingham-type as well as Efremov-Teryaev-Leader-type sum rules.