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Precision determination of nucleon iso-vector scalar and tensor charges at the physical point

Ji-Hao Wang, Zhi-Cheng Hu, Xiangdong Ji, Xiangyu Jiang, Yushan Su, Peng Sun, Yi-Bo Yang

Abstract

We report a high precision calculation of the isospin vector charge $g_{S,T}$ of the nucleon using recently proposed ``blending" method which provides a high-precision stochastic estimate of the all-to-all fermion propagator. By combining the current-involved interpolation operator, which can efficiently cancel the major excited state contaminations, we can extract high-precision $g_S$ and $g_T$ directly at the physical pion mass. Using 15 $N_f=2+1$ lattice ensembles which cover 5 lattice spacing, 5 combinations with the same quark masses and lattice spacing but multiple volumes, including three at the physical pion mass, we report so far most precise lattice QCD prediction $g_T^{\rm QCD} = 1.0256[78]_{\rm tot}(58)_{\rm stat} (17)_{a} (44)_{\rm FV} (01)_χ(22)_{\rm ex} (05)_{\rm re}$ and $g_S^{\rm QCD} = 1.107[46]_{\rm tot}(32)_{\rm stat} ( 04)_{a} (29)_{\rm FV} (01)_χ(13)_{\rm ex} (08)_{\rm re}$ at $\overline{\mathrm{MS}}$ 2~GeV, with the systematic uncertainties from continuum, infinite volume, chiral extrapolations, excited state contamination and also renormalization.

Precision determination of nucleon iso-vector scalar and tensor charges at the physical point

Abstract

We report a high precision calculation of the isospin vector charge of the nucleon using recently proposed ``blending" method which provides a high-precision stochastic estimate of the all-to-all fermion propagator. By combining the current-involved interpolation operator, which can efficiently cancel the major excited state contaminations, we can extract high-precision and directly at the physical pion mass. Using 15 lattice ensembles which cover 5 lattice spacing, 5 combinations with the same quark masses and lattice spacing but multiple volumes, including three at the physical pion mass, we report so far most precise lattice QCD prediction and at 2~GeV, with the systematic uncertainties from continuum, infinite volume, chiral extrapolations, excited state contamination and also renormalization.

Paper Structure

This paper contains 10 sections, 26 equations, 16 figures, 4 tables.

Table of Contents

  1. Brief review of the blending method and verification from $g_V$
  2. Excited state contamination
  3. Brief review of GEVP
  4. current-involved excited state contamination
  5. Influence from $N\pi$
  6. Contribution of $\langle \mathcal{N}_X (\vec{x}, t_f) O_X(\vec{y},t) \mathcal{N}^{\dagger}_X(\vec{z},0) \rangle$
  7. Systematic uncertainties
  8. Continuum extrapolation
  9. Chiral extrapolation
  10. Infinite volume extrapolation

Figures (16)

  • Figure 1: The dependence of the source-sink separation $t_f$ for the ratios $\mathcal{R}_X^{\rm mid}(t_f)\equiv \mathcal{R}_X(t_f, t = t_f/2)$ and $\mathcal{R}_X^{\text{FH}}(t_f)$ on the F64P13 ensemble, for the tensor $X=T$ (upper panel) and scalar $X=S$ (lower panel) operator cases. The gray band represents the result from the joint three-state fit. The minimum source-sink separation for fitting is $t_f^{\text{min}}=3a \sim 0.24~\text{fm}$.
  • Figure 2: The finite spatial lattice length $L$ dependence of the $g_S$ and $g_T$ with data points corrected to the continuum limit and physical pion mass.
  • Figure 3: The pion mass dependence of the $g_S$ and $g_T$ with data points corrected to continuum and infinite volume limits.
  • Figure 4: Comparison of $g_T$ (upper panel) and $g_S$ (lower panel) from this work, PNEME 15 Bhattacharya:2015esaBhattacharya:2015wna/16 Bhattacharya:2016zcn/18 Gupta:2018qil/23 Jang:2023zts, ETM22 Alexandrou:2022dtc, Mainz 19 Harris:2019bih/24 Mainz24, RQCD 23 Bali:2023sdi, QCDSF/UUKQCD/CSSM 23 QCDSFUKQCDCSSM:2023qlx, NME 21 Park:2021ypf, $\chi$QCD 21 Liu:2021irg and also the FLAG averages FlavourLatticeAveragingGroupFLAG:2024oxs. There are also lattice calculations Yamanaka:2018uudHasan:2019noyAbramczyk:2019fnfTsuji:2022ricAlexandrou:2019brg which are not included by FLAG due to certain uncontrollable systematics. All the values are renormalized at $\overline{\mathrm{MS}}(2~\mathrm{GeV})$.
  • Figure 5: Left: Source-sink separation $t_f$ dependence of $Z_VR_V(t_f,t_f/2,{\cal N}) _{\overrightarrow{t_f\rightarrow\ \infty}} g_V$ with different $\bar{N}_{\rm e}$, on the H48P32 ensemble, with $Z_V$ from the renormalization condition $Z_V\langle \pi^+|\bar{u}\gamma_t u|\pi^+\rangle=1$ of the pion matrix element of the vector current. Right: $a^2m_\pi^2$ dependence of the hadron-independent normalized $g_V$, and the result in the continuum limit is consistent with 1 within 0.12% statistical uncertainty.
  • ...and 11 more figures