Excited state contamination in nucleon structure calculations

TL;DR

Excited-state contamination is a major systematic in lattice QCD calculations of nucleon structure. The authors develop a global $N$-state fit to jointly analyze $C_{2pt}$ and $C_{3pt}$ using a linearized parameterization of transition amplitudes $\tilde{F}_i^{n\to n'}(t)$ and energies $E_n(\vec{p})$ to subtract excited-state contributions. They apply this framework to 2+1 flavor clover-improved Wilson lattices at $a=0.116$ fm with $m_\pi$ down to $150$ MeV and multiple source-sink separations, comparing against the ratio method. The results show persistent excited-state effects, particularly for the isovector average momentum fraction $\langle x\rangle^{u-d}$, and indicate that a single separation below about $1.4$ fm is insufficient near the physical point. The multi-state fitting approach provides a principled way to quantify and reduce excited-state errors, yielding consistency with ratio-based extractions within current statistics and offering a clear path for reliable uncertainty estimates in nucleon observables.

Abstract

Among the sources of systematic error in nucleon structure calculations is contamination from unwanted excited states. In order to measure this systematic error, we vary the operator insertion time and source-sink separation independently. We compute observables for three source-sink separations between 0.93 fm and 1.39 fm using clover-improved Wilson fermions and pion masses as low as 150 MeV. We explore the use of a two-state model fit to subtract off the contribution from excited states.

Figures (4)

• Figure 1: Isovector Dirac radius $(r_1^{u-d})^2$ ($\text{fm}^2$) versus $m_\pi$ (GeV). At each pion mass, from left to right are results obtained from the fitting procedure, and from the ratio method with three increasing source-sink separations.
• Figure 1: Selected results from fit used to compute $(r_1^{u-d})^2$ for the $m_\pi=250$ MeV ensemble. Form-factors $F_{1,2}(t_1)$ are not renormalized and are computed from $\tilde{F}_{1,2}^{0\to 0}(t_1)\!/\!\!\sqrt{a_0(0,0,0)a_0(\tfrac{2\pi}{L},0,0)}$.
• Figure 2: Isovector average momentum fraction $\langle x\rangle^{u-d}$ (bare) versus $m_\pi$ (GeV). At each pion mass, from left to right are results obtained from the fitting procedure, and from the ratio method with three increasing source-sink separations.
• Figure 3: Three-point function (points) and fit (error bands) versus $\tau/a$, with $T/a=8$. Matrix element labels are representatives from the sets of equivalent three-point functions that are averaged to compute the points shown here. Fit bands are determined by the quantities $m_0a$, $m_1^{(3)}a$, $\tilde{F}_1(t_1)$, and $\tilde{F}_2(t_2)$ listed in Tab. 1. Note that the points have correlated errors, and that neglecting correlations will cause the fit to overlap with the data.