Symmetry properties of nonlocal quark bilinear operators on a Lattice (LP3 Collaboration) JWC is partly supported by the Ministry of Science and Technology (105-2112-M-002-017-MY3) and the Kenda Foundation. TI is supported by Science and Technology Commission of Shanghai Municipality (16DZ2260200). TI and LCJ are supported by the Department of Energy, Laboratory Directed Research and Development (LDRD) funding of BNL (DE-EC0012704). The work of HL is supported by US National Science Foundation (PHY 1653405). JHZ is supported by the SFB/TRR-55 grant 'Hadron Physics from Lattice QCD', and a grant from National Science Foundation of China (11405104). YZ is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DE-SC0011090 and within the framework of the TMD Topical Collaboration

Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimension-three operators will always mix with dimension-four operators, even if chiral symmetry is preserved. Also, the number of dimension-four operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken-x dependence of the parton distribution functions using the quasi-distribution method on a Euclidean lattice. The requirement of using large hadron momentum in this approach makes the control of errors from dimension-four operators even more important.