Insights on proton structure from lattice QCD: The twist-3 parton distribution function $g_T(x)$

In this work, we present the first-ever calculation of the isovector flavor combination of the twist-3 parton distribution function $g_T(x)$ for the proton from lattice quantum chromodynamics (QCD). We use an ensemble of gauge configurations with two degenerate light, a strange, and a charm quarks ($N_f=2+1+1$) of maximally twisted mass fermions with a clover improvement. The lattice has a spatial extent of 3 fm and lattice spacing of 0.093 fm and reproduces a pion mass of 260 MeV. We use the quasidistribution approach and employ three values of the proton momentum boost, 0.83, 1.25, and 1.67 GeV. We use a source-sink separation of 1.12 fm to suppress excited-states contamination. The lattice data are renormalized nonperturbatively. We calculate the matching equation within Large Momentum Effective Theory, which is applied to the lattice data in order to obtain $g_T$. The final distribution is presented in the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV. We also calculate the helicity distribution $g_1$ to test the Wandzura-Wilczek approximation for $g_T$. We find that the approximation works well for a broad range of $x$. This work demonstrates the feasibility of accessing twist-3 parton distribution functions from novel methods within lattice QCD and can provide essential insights into the structure of hadrons.

Figure 1

Results for $g_T(x)$ as a function of $x$ for three nucleon momenta. $P_3=0.83$, 1.25, and 1.67 GeV are shown with red, green, and blue bands, respectively.

Figure 2

Comparison of $x$-dependence of $g_T$ (blue band) and $g_1$ (orange band) at $P_3=1.67\mathrm{GeV}$.

Figure 3

Comparison of our $g_T(x)$ (blue band) with its WW approximations: lattice-extracted $g_T^{\mathrm{WW}}$ (red band) and calculated from global fits (NNPDF1.1pol [83], orange band, and JAM17 [84], purple band). The proton momentum is $P_3=1.67\mathrm{GeV}$.