Strangeness of nucleons from $N_f=2+1+1$ lattice QCD

We present the strange electromagnetic form factors of the nucleon using lattice QCD simulations with degenerate light, a strange, and a charm quark in the sea with masses tuned to their physical values. For the first time, the strange electromagnetic form factors are computed at the continuum limit using only ensembles simulated with physical quark masses, eliminating the need for chiral extrapolations and their associated systematic uncertainty. We obtain the momentum transfer dependence of the form factors using the $z$-expansion and provide the strange electric and magnetic radii, as well as the strange magnetic moment. When combining our statistical errors and systematic uncertainties stemming from the momentum transfer dependence fit, our errors are an order of magnitude smaller than those associated with experimental determinations of the strange electromagnetic form factor.

💡 Research Summary

This paper presents a high-precision lattice Quantum Chromodynamics (QCD) calculation of the nucleon’s strange electromagnetic form factors, which describe how the strange quark sea contributes to the proton and neutron’s charge and magnetization distributions. The key achievement is performing this calculation directly at the physical pion mass and taking the continuum limit without relying on chiral extrapolations, thereby eliminating a major source of systematic uncertainty prevalent in earlier studies.

The work utilizes four ensembles of gauge field configurations generated by the Extended Twisted Mass Collaboration (ETMC). These ensembles feature N_f=2+1+1 flavors of clover-improved twisted mass fermions, with the light (up/down), strange, and charm quark masses all tuned to their physical values. The ensembles have different lattice spacings (ranging from ~0.08 fm to ~0.049 fm), enabling a controlled extrapolation to the continuum limit (where the lattice spacing goes to zero). The strange quark contribution is isolated by calculating the disconnected diagram, a computationally expensive task that involves evaluating a closed quark loop. The authors employ advanced techniques like hierarchical probing and deflation to compute these loops efficiently and gather high statistics across hundreds of configurations and source positions.

The nucleon matrix elements of the strange vector current are extracted from ratios of three-point to two-point correlation functions via plateau fits. The momentum transfer (Q²) dependence of the Sachs electric (G_s_E) and magnetic (G_s_M) form factors is then modeled using three different parameterizations: a dipole form, a Galster-like form, and the model-independent z-expansion. A novel single-step fitting procedure is employed, where the fit parameters for the Q²-dependence are themselves modeled as linear functions of the squared lattice spacing (a²). This allows for a simultaneous extraction of the continuum-limit form factor shape. The final results are obtained by performing a model-average over fits with different Q² cutoffs using the Akaike Information Criterion (AIC), providing both statistical and systematic uncertainties.

The primary results are precise values for the slope and intercept of the form factors at Q²=0:

• Strange electric charge radius squared: ⟨r²_E⟩^s = -0.00545(49)(26) fm²
• Strange magnetic radius squared: ⟨r²_M⟩^s = -0.01212(280)(72) fm²
• Strange magnetic moment: μ_s = -0.01792(195)(18)

These small negative values indicate that the net effect of the strange quark-antiquark sea is a slight negative charge and diamagnetic contribution. The uncertainties are an order of magnitude smaller than those from experimental determinations via parity-violating electron scattering (e.g., SAMPLE, HAPPEX, G0 experiments). When compared graphically at a benchmark momentum transfer of Q²=0.1 GeV², the 95% confidence region from this lattice calculation is drastically smaller than the ellipses from experimental global fits.

The study concludes that these state-of-the-art lattice QCD results, free from chiral extrapolation systematics, provide stringent constraints on the nucleon’s strange form factors. This high-precision theoretical input is crucial for the interpretation of ongoing and future precision experiments, such as those planned at the MESA facility in Mainz, which aim to measure the proton’s weak charge and test the Standard Model. The work successfully bridges non-perturbative QCD dynamics, hadron structure, and precision electroweak physics.