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Nucleon strange electromagnetic form factors from $N_f=2+1+1$ lattice QCD

Constantia Alexandrou, Simone Bacchio, Mathis Bode, Jacob Finkenrath, Andreas Herten, Christos Iona, Giannis Koutsou, Ferenc Pittler, Bhavna Prasad, Gregoris Spanoudes

Abstract

We present the nucleon strange electromagnetic form factors using four lattice QCD ensembles with $N_f=2+1+1$ twisted mass clover-improved fermions and quark masses tuned to approximately their physical values. The four ensembles have similar physical volume and lattice spacings of $a=0.080$ fm, $0.068$ fm, $0.057$ fm and $0.049$ fm allowing us to take the continuum limit directly at the physical pion mass point. We compute nucleon three-point correlation functions with high statistics, where the disconnected fermion loops are evaluated stochastically with spin-color dilution and hierarchical probing. We find non-zero values for both electric and magnetic form factors. We extract the strange electric and magnetic radii, as well as the strange magnetic moment in the continuum limit by studying the momentum dependence of the form factors. We also compute the charm electromagnetic form factors within the same setup, which we find to be consistent with zero within the statistical precision of our data.

Nucleon strange electromagnetic form factors from $N_f=2+1+1$ lattice QCD

Abstract

We present the nucleon strange electromagnetic form factors using four lattice QCD ensembles with twisted mass clover-improved fermions and quark masses tuned to approximately their physical values. The four ensembles have similar physical volume and lattice spacings of fm, fm, fm and fm allowing us to take the continuum limit directly at the physical pion mass point. We compute nucleon three-point correlation functions with high statistics, where the disconnected fermion loops are evaluated stochastically with spin-color dilution and hierarchical probing. We find non-zero values for both electric and magnetic form factors. We extract the strange electric and magnetic radii, as well as the strange magnetic moment in the continuum limit by studying the momentum dependence of the form factors. We also compute the charm electromagnetic form factors within the same setup, which we find to be consistent with zero within the statistical precision of our data.

Paper Structure

This paper contains 22 sections, 42 equations, 15 figures, 14 tables.

Table of Contents

  1. Introduction
  2. Decomposition of the nucleon matrix element
  3. Lattice setup
  4. Lattice correlation functions
  5. Gauge ensembles and statistics
  6. Renormalization
  7. Extraction of nucleon matrix elements
  8. Analysis of extracted form factors
  9. Dipole form
  10. Galster-like parameterization
  11. $z$-expansion
  12. Model Averaging
  13. $Q^2$-dependence of the strange form factors
  14. Galster-like and dipole analysis
  15. $z$-expansion analysis
  16. ...and 7 more sections

Figures (15)

  • Figure 1: Nucleon two-point function (top) and disconnected nucleon three-point function (bottom). The red lines denote all-to-all quark propagators.
  • Figure 2: $Z_{\rm V}$ as a function of $\mu_0^2$ for the four lattice-spacing ensembles, namely B, C, D and E. A joint fit of the form $f (a^2 \mu_0^2) = c_0 (a) + c_1 a^2 \mu_0^2 + c_2 (a^2 \mu_0^2)^2$ is employed in the range $\mu_0^2 \in [20,32]$ GeV$^2$. The extrapolated values $c_0 (a)$ for each lattice spacing are given at $\mu_0^2 = 0$.
  • Figure 3: Extrapolated values of $Z_{\rm V}$ at $a^2 \mu_0^2 = 0$ from momentum fits over multiple fit ranges together with the model-averaged values (bands). The reduced $\chi^2$ of each fit is given by the label above the corresponding data point. The open symbols denotes the fit with the highest weight based on the Akaike Information Criterion.
  • Figure 4: Nucleon effective mass vs the source-sink time separation, $t_s$ for the ensembles analysed in this work. As indicated in the header, the right-pointing green triangles, orange squares, downward pointing blue triangles, and the purple pentagons denote the results for the coarsest to the finest lattice spacing used in this work. The different color bands are the result of the effective mass of the nucleon using a three-state fit, whereas the black dashed line corresponds to the extracted mass of the nucleon $m_{N} = 0.938 \;\rm GeV$.
  • Figure 5: Results on $G_E^s(Q^2=0.305 \rm \;GeV^2)$ (top) and $G_M^s(Q^2=0.045 \rm \;GeV^2)$ (bottom) for the cE211.044.112 ensemble. The left column shows source-sink time separations $t_s = \rm 16a, 18a$ and $20\rm a$ as indicated in the header, for all insertion times $t_\mathrm{ins}$, where the filled points denote those included in the plateau fits. The right column shows the result using plateau fits for $t_s = \rm 14a, 16a, 18a, 20a$ and $22\rm a$. The red band denotes the weighted average of results for all $t_s$ as listed in Table \ref{['tab:disc_range']}, taken as final value.
  • ...and 10 more figures