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

Using high-statistics lattice QCD simulations with $N_f=2+1+1$ twisted mass fermions at physical quark masses, the authors determine non-zero nucleon strange electromagnetic form factors and radii in the continuum limit while finding the charm electromagnetic form factors consistent with zero.

The Gist

The Invisible Ghosts Inside the Proton

Imagine a proton (the heart of an atom) not as a solid ball, but as a bustling, chaotic city. Inside this city, there are three main "citizens" that make up the proton's identity: two Up quarks and one Down quark. These are the "valence" citizens; they are the ones you can see on the surface, and they determine the proton's basic charge and weight.

But, according to the laws of Quantum Physics, this city is also filled with a "sea" of virtual particles. These are pairs of quarks and anti-quarks that pop in and out of existence for a split second. Most of these are heavy or rare, but there is a specific pair of "ghosts" that scientists have been trying to catch: the Strange quarks.

These strange quarks are like invisible ghosts living in the proton's basement. They don't have a net charge (they cancel each other out), so they don't change the proton's overall electricity. However, they do have a magnetic personality. The big question for decades has been: Do these strange ghosts have a magnetic field? And if so, how strong is it?

The Challenge: Seeing the Unseeable

Detecting these ghosts is incredibly hard. In a real-world experiment, you have to shoot electrons at protons and look for tiny, weird wobbles in the data. It's like trying to hear a whisper in a hurricane. The signal is so faint that for years, experiments could only say, "Maybe they are there, maybe they aren't," with huge margins of error.

This paper is a team of scientists (led by Constantia Alexandrou and colleagues) who decided to stop guessing and start simulating the universe on a supercomputer.

The Supercomputer "Lattice"

Instead of building a giant particle accelerator, they built a digital one. They created a 4D grid (a "lattice") that represents space and time. On this grid, they programmed the rules of the universe (Quantum Chromodynamics, or QCD) and simulated protons made of quarks.

Here is what made their simulation special:

1. The Physical Setting: Many previous simulations used "fake" settings where the particles were too heavy or the universe was too small. This team tuned their simulation to match the real world perfectly. They used the actual mass of the pion (a particle related to the proton) and the actual size of the universe they simulated.
2. The Four Cameras: They didn't just take one picture. They ran the simulation on four different "resolutions" (lattice spacings). Think of this like taking a photo with a low-resolution camera, a medium one, a high one, and a 4K camera. By comparing the results, they could mathematically remove the "pixelation" (errors caused by the grid) to see the perfect, smooth picture. This is called taking the continuum limit.
3. The Noise Filter: The "strange" quarks are disconnected from the main proton structure in the math. It's like trying to hear a ghost's voice while a rock concert is playing next door. The team used advanced math tricks (called "hierarchical probing" and "spin-color dilution") to filter out the noise and isolate the ghost's voice. They ran this simulation millions of times to get a clear signal.

The Discovery: The Ghosts Have a Voice

After crunching the numbers, the team found something definitive:

• The Strange Electric Charge: The strange quarks inside the proton have zero net electric charge. This makes sense, as they cancel each other out.
• The Strange Magnetic Moment: This is the big news. The strange quarks do have a magnetic personality. They found a small, but non-zero magnetic value.
  • The Analogy: Imagine the proton is a spinning top. The main Up and Down quarks make it spin one way. The strange quarks are like tiny, invisible magnets inside the top that are spinning the other way, slightly slowing it down. The team measured exactly how much they slow it down.

They also looked for Charm quarks (even heavier ghosts). They found that these are so heavy and rare that their effect is effectively zero within their measurement precision.

Why Does This Matter?

You might ask, "So what? It's a tiny number."

1. Testing the Universe's Blueprint: This result is a "stress test" for our Standard Model of physics. If the simulation says "zero" but the real world says "one," our understanding of the universe is broken. Since their result is precise, it helps physicists rule out bad theories.
2. Helping Real Experiments: Real-world experiments (like the Qweak experiment) are trying to measure the "weak charge" of the proton to test the forces of nature. The strange quarks are a background noise in those experiments. By knowing exactly how much "noise" the strange quarks make, experimentalists can subtract it and get a clearer picture of the fundamental forces.
3. The "Intrinsic Charm" Mystery: While they found the charm contribution was zero in their data, other recent studies suggest there might be a tiny bit of "intrinsic charm" in the proton. This paper sets a very strict limit on how big that could be.

The Bottom Line

This paper is a triumph of "digital archaeology." By simulating the universe with unprecedented precision, the team has finally mapped the magnetic footprint of the invisible strange quarks inside the proton.

They found that the proton isn't just a simple trio of particles; it's a complex, swirling cloud where even the "ghosts" in the basement have a magnetic personality. And thanks to their supercomputer, we finally know exactly how strong that personality is.

Technical Summary

1. Problem Statement

The internal structure of the nucleon (proton and neutron) is governed by Quantum Chromodynamics (QCD). While valence quarks ($u, d$) dominate the nucleon's properties, the "sea" of virtual quark-antiquark pairs also contributes. The strange quark is the lightest non-valence quark, making its contribution to nucleon electromagnetic form factors a critical probe of non-perturbative QCD dynamics.

• Experimental Context: Strange form factors are difficult to measure experimentally. They are currently determined indirectly via parity-violating electron scattering asymmetries (e.g., SAMPLE, A4, HAPPEX, G0 experiments). These measurements suffer from large systematic uncertainties and correlations between electric and magnetic form factors.
• Theoretical Gap: Previous lattice QCD calculations of strange form factors often relied on:
  1. Heavier-than-physical pion masses, requiring chiral extrapolations.
  2. Single ensembles at the physical point, lacking continuum limit extrapolations.
  3. Limited statistics, particularly for the computationally expensive disconnected quark loops required for strange (and charm) quarks.

The goal of this work is to provide a first-principles, high-precision calculation of the nucleon strange electromagnetic form factors, including the strange electric radius ($\langle r^2_E \rangle_s$), magnetic radius ($\langle r^2_M \rangle_s$), and magnetic moment ($\mu_s$), directly at the physical pion mass and in the continuum limit.

2. Methodology

Lattice Setup

• Fermion Action: The authors use $N_f = 2+1+1$ twisted mass clover-improved fermions. This setup includes two degenerate light quarks, one strange, and one charm quark.
• Ensembles: Four distinct gauge ensembles were analyzed with lattice spacings ($a$) of 0.080 fm, 0.068 fm, 0.057 fm, and 0.049 fm.
• Physical Point: Quark masses were tuned to their physical values, resulting in pion masses ($m_\pi$) ranging from 135 to 141 MeV.
• Volume: The ensembles have similar physical volumes with $m_\pi L \approx 3.6 - 3.9$, ensuring finite-volume effects are controlled.
• Continuum Limit: The use of four lattice spacings allows for a direct extrapolation to the continuum limit ($a \to 0$) without the need for chiral extrapolations.

Correlation Functions & Statistics

• Disconnected Loops: Strange and charm contributions arise from disconnected fermion loops. These were evaluated using stochastic estimation with:
  • Spin-color dilution: Full dilution ($n_{dil}=12$) in color and spin.
  • Hierarchical probing: Using 512 Hadamard vectors ($n_{had}=512$) to suppress off-diagonal noise.
  • High Statistics: Approximately $10^5$ two-point functions and $10^3$ stochastic vectors per configuration were generated across the ensembles.
• Momentum Transfer: By utilizing boosted frames (non-zero sink momenta), the authors accessed $\sim 300$ distinct $Q^2$ values in the range $Q^2 \in [0, 1.3) \text{ GeV}^2$, significantly improving the $Q^2$-dependence analysis compared to previous works.

Analysis Techniques

• Renormalization: The local vector current was renormalized non-perturbatively using the RI'/MOM scheme. A joint fit across all lattice spacings was performed to determine the renormalization factor $Z_V$.
• Ground State Extraction: Plateau fits were performed on the ratio of three-point to two-point correlation functions. Source-sink separations up to $2.6$ fm were used to suppress excited state contamination.
• Form Factor Extraction: A Singular Value Decomposition (SVD) approach was used to disentangle the electric ($G_E$) and magnetic ($G_M$) form factors from the correlation ratios, handling the overconstrained system of equations arising from multiple momentum combinations.
• Continuum Extrapolation & Model Averaging:
  • Two approaches were tested: a "two-step" method (fit $Q^2$ then extrapolate $a$) and a "one-step" method (simultaneous fit of $Q^2$ and $a^2$).
  • Parameterizations: Dipole, Galster-like (for $G_E$), and $z$-expansion were used.
  • Model Averaging: To quantify systematic errors arising from fit ansatz and momentum cuts ($Q^2_{cut}$), the authors employed Akaike Information Criterion (AIC) based model averaging. This combines results from various fits weighted by their statistical probability.

3. Key Contributions

1. First Continuum Limit at Physical Point: This is the first calculation of strange nucleon form factors that simultaneously achieves the physical pion mass and the continuum limit using four lattice spacings.
2. Unprecedented Statistics: The high statistics on disconnected loops significantly reduced the stochastic noise, which is typically the dominant error source for strange quark contributions.
3. Robust Systematic Error Control: The use of model averaging across different parameterizations (Dipole, Galster, $z$-expansion) and momentum cuts provides a rigorous estimate of systematic uncertainties, separating them from statistical errors.
4. Charm Contribution: The study also computed charm electromagnetic form factors, finding them consistent with zero, providing a benchmark for intrinsic charm studies.

4. Results

The final results, obtained via model averaging, are:

• Strange Electric Charge Radius:
  $$\langle r^2_E \rangle_s = -0.00545(49)_{stat}(26)_{sys} \text{ fm}^2$$
• Strange Magnetic Radius:
  $$\langle r^2_M \rangle_s = -0.01212(280)_{stat}(72)_{sys} \text{ fm}^2$$
• Strange Magnetic Moment:
  $$\mu_s = -0.01792(195)_{stat}(18)_{sys}$$

Key Observations:

• Non-Zero Values: Both electric and magnetic form factors are found to be non-zero, confirming the presence of strange quark content in the nucleon's electromagnetic structure.
• Precision: The results are significantly more precise than previous lattice calculations and experimental extractions.
• Charm Quarks: The charm electric and magnetic form factors ($G^c_E, G^c_M$) and radii were found to be consistent with zero within statistical precision, as expected due to the heavy charm mass suppressing sea contributions at low $Q^2$.

5. Significance

• Constraint on Weak Charge: The precise determination of strange form factors provides crucial input for the interpretation of the Qweak experiment, which measures the proton's weak charge. Strange quark contributions are a major source of uncertainty in extracting the weak mixing angle from parity-violating scattering data.
• Validation of Lattice QCD: The agreement of these results with other recent lattice studies (e.g., LHPC, Mainz, $\chi$QCD) and the reduction of uncertainties validate the $N_f=2+1+1$ twisted mass approach and the hierarchical probing techniques for disconnected loops.
• Experimental Benchmark: The paper provides a high-precision theoretical benchmark that is more accurate than current experimental extractions (which rely on parity-violating asymmetry reanalyses). This helps resolve discrepancies in the interpretation of experimental data regarding the strange content of the nucleon.
• Intrinsic Charm: While the charm contribution was found to be zero in this setup, the methodology establishes a robust framework for future studies of intrinsic charm, which has been a topic of recent interest in global PDF analyses (e.g., NNPDF).

In conclusion, this work represents a milestone in lattice QCD, delivering the most precise determination to date of the strange electromagnetic structure of the nucleon, free from chiral and continuum extrapolation ambiguities.