Precision Study of Excited State Effects in Nucleon Matrix Elements

TL;DR

This study investigates excited-state contamination in lattice QCD calculations of nucleon matrix elements, focusing on the axial charge g_A and the first moment of the unpolarized isovector PDF <x>_{u-d}. Using high-statistics data on a Nf=2+1+1 twisted-mass ensemble with non-perturbative renormalization, the authors compare fixed-sink and open-sink methods to quantify contamination as a function of source-sink separation. They find negligible excited-state effects for g_A at m_pi≈380 MeV and a ~10% downward shift for <x>_{u-d}, with an infinite-separation extrapolation giving 0.22(1) versus 0.250(6) at finite separation. The results underscore that excited-state effects are operator-dependent and cannot alone explain the lattice-phenomenology discrepancies, motivating broader systematic studies and methodological advances such as variational approaches for more reliable nucleon matrix elements.

Abstract

We present a dedicated analysis of the influence of excited states on the calculation of nucleon matrix elements. This calculation is performed at a fixed value of the lattice spacing, volume and pion mass that are typical of contemporary lattice computations. We focus on the nucleon axial charge, g_A, for which we use about 7,500 measurements, and on the average momentum of the unpolarized isovector parton distribution, <x>_{u-d}, for which we use about 23,000 measurements. All computations are done employing N_f=2+1+1 maximally-twisted-mass Wilson fermions and using non-perturbatively calculated renormalization factors. Excited state effects are shown to be negligible for g_A whereas they lead to an O(10%) downward shift for <x>_{u-d}.

Figures (4)

• Figure 1: Diagrammatic illustration of the sequential method through the sink (left) and the open sink method (right).
• Figure 2: In the left panel, we show the relative deviation of ETMC lattice results for $g_A$ from the experimental value Nakamura:2010zzi. In the right panel, we show the relative deviation of ETMC lattice results for $\left\langle x \right\rangle_{u-d}$ from a result obtained from a phenomenological analysis Alekhin:2009ni. The lattice values for $N_f=2$ at the various pion masses are from Refs. Alexandrou:2010hfAlexandrou:2011nr. The filled (magenta) diamonds show the results using the $N_f=2+1+1$ ensembles.
• Figure 3: Results for $g_A$ for a range of source-sink separations obtained from the open sink analysis on one $N_f=2+1+1$ ensemble. The light grey band indicates the result obtained from the fixed sink method using a source-sink separation of $12a$ and the dark grey band shows the experimental value.
• Figure 4: Results for $\left\langle x \right\rangle_{u-d}$ for a range of source-sink separations obtained by means of the open sink method. The operator insertion was at a temporal separation from the source of $t'=11a$. The value (including errors) obtained from the fixed sink method using a source-sink separation of $12a$ is indicated by the light grey band. The phenomenologically extracted value is shown with the dark grey band. The blue solid line corresponds to a fit described in the text.