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

This paper presents a high-precision lattice QCD determination of the nucleon iso-vector scalar and tensor charges at the physical point by employing a novel "blending" method to suppress excited state contamination across 15 $N_f=2+1$ ensembles, yielding the most accurate predictions to date with comprehensive systematic error analysis.

The Gist

Imagine the proton and neutron (the building blocks of our atoms) as tiny, complex machines. Inside these machines, quarks are buzzing around, held together by the strong force. Physicists want to know exactly how these machines behave when they are hit by different types of "probes" (like particles in an accelerator).

To understand this, scientists need to measure specific "charges" or properties of the proton. Two of the most important, but hardest to measure, are the Scalar Charge and the Tensor Charge. Think of these as the proton's "weight" in a specific direction and its "twistiness" or resistance to being spun.

This paper is a report from a team of scientists (the CLQCD Collaboration) who have just built the most precise "ruler" ever created to measure these charges. Here is how they did it, explained in everyday terms.

1. The Problem: The "Echo" in the Room

When scientists try to measure these charges using supercomputers (simulating the universe on a grid), they face a major noise problem.

Imagine you are in a large concert hall trying to hear a single violinist (the proton's true signal). But, every time the violinist plays, the sound bounces off the walls, creating echoes (these are called excited states).

• If you listen too soon, you hear a messy mix of the violin and the loud echoes.
• If you wait too long for the echoes to die down, the violinist gets so quiet that the background noise of the crowd (statistical noise) drowns them out.

For years, scientists struggled to find the perfect moment to listen where the signal was clear and the noise was low.

2. The Solution: The "Blending" Trick

The team used a new, clever method called the "Blending Method."

Think of it like this: Instead of just listening to the violinist, they created a special microphone that can hear the violinist and the specific echoes simultaneously.

• Old Way: They tried to guess how to cancel out the echoes by waiting longer.
• New Way (Blending): They realized that if they mix the signal from the "pure" violinist with a signal from a "distorted" version of the violinist (one that includes the echoes), they can mathematically cancel out the noise.

It's like noise-canceling headphones, but for the fundamental particles of the universe. This allowed them to get a crystal-clear signal much faster than before, without needing to wait for the "echoes" to fade naturally.

3. The "Magic" Interpolation Field

To make this work, they invented a new way to "ask" the proton a question.

• Normally, they ask the proton, "What is your charge?" using a standard question.
• In this paper, they asked a hybrid question. They combined the standard question with a "current-involved" question (a question that specifically targets the parts of the proton that cause the echoes).

By mixing these two questions together in a very specific, tiny ratio (like adding a single drop of blue paint to a bucket of white), they were able to silence the echoes almost completely. This is like finding the exact frequency to cancel out a hum in a room.

4. The Results: A New Standard

Using this super-precise method on 15 different "universes" (simulations with different sizes and grid densities), they calculated the two charges with unprecedented accuracy:

• The Tensor Charge ($g_T$): They found it to be 1.026.
• The Scalar Charge ($g_S$): They found it to be 1.106.

The best part? Their measurements are twice as precise as the best previous attempts. It's like going from measuring a person's height with a ruler that has centimeter marks, to one that has millimeter marks.

5. Why This Matters: The "New Physics" Hunt

Why do we care about these numbers?

• The Standard Model: Our current best theory of physics (The Standard Model) makes predictions about these charges.
• The Hunt for New Physics: If our measurements don't match the theory, it means there is a hidden force or a new particle we haven't discovered yet (like a "ghost" in the machine).

Because this paper provides such a precise "ruler," scientists can now test the Standard Model much more strictly. If the universe is hiding a new secret, this new measurement is sharp enough to catch it.

6. A Surprising Discovery: The "Volume" Effect

While measuring, they found something interesting about how the size of their simulation box affects the results.

• Old Theory: Scientists thought the "echoes" from the box walls would shrink very quickly (like a specific type of sound dampening).
• New Finding: The data showed that the echoes actually shrink at a different, simpler rate.

This is a big deal because it means previous studies might have been slightly wrong about how to correct for the size of their simulation boxes. The authors are now telling the rest of the physics community: "Hey, we need to update our correction formulas to match this new reality."

Summary

In short, this paper is a triumph of clever math and computing power.

1. They built a better "microphone" (Blending Method) to hear the proton clearly.
2. They used a "hybrid question" to cancel out the background noise.
3. They measured the proton's properties with record-breaking precision.
4. They discovered that the rules for how simulation size affects results need to be updated.

This gives physicists a much sharper tool to hunt for the next great discovery in the universe.

Technical Summary

1. Problem Statement

The nucleon iso-vector scalar ($g_S$) and tensor ($g_T$) charges are fundamental matrix elements required for precision tests of the Standard Model and searches for new physics (e.g., probing TeV-scale physics and calculating the proton-neutron mass difference). While the axial charge ($g_A$) is well-constrained by neutron decay, $g_S$ and $g_T$ remain challenging to calculate with high precision using Lattice QCD due to two primary competing difficulties:

• Excited-State Contamination (ESC): At small source-sink separations ($t_f$), the signal is dominated by contributions from excited states.
• Statistical Noise: The signal-to-noise ratio decays exponentially with increasing $t_f$, making it difficult to reach the ground state without prohibitive computational costs.
• Systematic Uncertainties: Previous calculations suffer from uncontrolled systematic errors arising from chiral extrapolations, finite-volume effects (FVE), and discretization errors. Specifically, there is ambiguity regarding the functional form of FVEs (e.g., whether they follow Heavy Baryon Chiral Perturbation Theory (HB$\chi$PT) predictions or phenomenological forms).

2. Methodology

The CLQCD Collaboration employed a novel combination of advanced techniques to overcome these challenges:

A. The "Blending" Method

To address the statistical noise and ESC simultaneously, the authors utilized the recently proposed blending method.

• Mechanism: This method extends the distillation technique by projecting full quark propagators onto a subspace spanned by low-lying eigenvectors of the Laplacian (distillation) combined with a set of stochastic random vectors (blending).
• Advantage: It provides an unbiased, high-precision stochastic estimate of the all-to-all fermion propagator. This allows for the calculation of correlation functions with arbitrary source-sink separations and operator insertions without the need for additional, costly propagator inversions.
• Efficiency: The paper demonstrates that this method achieves the same statistical precision as previous studies (e.g., PNDME, ETMC) using 17 to 877 times fewer quark propagator inversions.

B. Current-Involved Interpolation Fields

To specifically suppress ESC, the authors introduced a new operator strategy based on "current-involved" interpolation fields.

• Concept: They constructed a linear combination of the standard nucleon interpolator ($N$) and a new operator ($N_X \equiv N O_X$), where $O_X$ is the current operator itself.
• Physics: The operator $N_X$ couples strongly to specific excited states enhanced by the current (e.g., disconnected diagrams scaled by volume $V$). By including $N_X$ in the basis, these dominant ESCs can be canceled out.
• Implementation: The ground state matrix element is extracted via a joint three-state fit to both the standard ratio and the ratio using the improved interpolator ($N + c_X N_X$). The coefficient $c_X$ is optimized to minimize ESC.

C. Simulation Setup

• Ensembles: The calculation utilized 15 $N_f = 2+1$ lattice ensembles generated by the CLQCD collaboration.
• Parameters: These ensembles cover 5 different lattice spacings ($a \approx 0.052 - 0.105$ fm), multiple volumes, and three ensembles at the physical pion mass ($m_\pi \approx 135$ MeV).
• Action: Tadpole-improved tree-level Symanzik gauge action and Clover fermion action.

3. Key Contributions & Innovations

1. First Direct Physical Point Calculation: This is the first time such high-precision $g_S$ and $g_T$ results were obtained directly at the physical pion mass, eliminating the need for aggressive chiral extrapolation.
2. Resolution of Finite-Volume Effects (FVE): The high-precision data revealed that the FVEs for $g_S$ are not well-described by the HB$\chi$PT ansatz ($m_\pi^2 e^{-m_\pi L}/\sqrt{m_\pi L}$). Instead, the data strongly favors a phenomenological exponential form ($e^{-m_\pi L}$). Using the HB$\chi$PT form would introduce a systematic bias.
3. Demonstration of ESC Suppression: The study provides robust numerical evidence that the linear combination of standard and current-involved interpolators universally suppresses ESC across different pion masses, volumes, and lattice spacings.
4. Computational Efficiency: The blending method is shown to be a highly efficient pathway for high-precision hadron structure calculations, enabling extensive data reuse from a single set of generated propagators.

4. Results

The authors report the most precise lattice QCD predictions to date for the iso-vector scalar and tensor charges at the $\overline{\text{MS}}$ scale of 2 GeV:

• Tensor Charge:
  $$g_T^{\text{QCD}} = 1.0264(77)_{\text{tot}}(53)_{\text{stat}}(13)_a(46)_{\text{FV}}(01)_\chi(28)_{\text{ex}}(04)_{\text{re}}$$

• Scalar Charge:
  $$g_S^{\text{QCD}} = 1.106(43)_{\text{tot}}(31)_{\text{stat}}(03)_a(28)_{\text{FV}}(01)_\chi(08)_{\text{ex}}(08)_{\text{re}}$$

• Uncertainty Breakdown: The total uncertainty is reduced by a factor of two compared to previous FLAG averages. The statistical precision is improved by a factor of three or more.

• Systematic Control:

  • Continuum Limit: Linear $a^2$ dependence observed; higher-order corrections are negligible.
  • Chiral Extrapolation: Minimal uncertainty due to the inclusion of physical point data.
  • FVE: The uncertainty is dominated by the ambiguity in the FVE ansatz (phenomenological vs. HB$\chi$PT), which the authors estimate conservatively.

5. Significance and Implications

• New Physics Constraints: The precise values of $g_S$ and $g_T$ provide tighter constraints on Beyond Standard Model (BSM) physics, particularly in the context of nucleon decay and electric dipole moments.
• Proton-Neutron Mass Difference: Using the new $g_S$ value, the authors predict the neutron-proton mass difference as:
  $$m_n - m_p = 1.60(23) \text{ MeV}$$
  This result agrees with the experimental value ($1.293$ MeV) within $1.3\sigma$. The paper highlights that the discrepancy depends heavily on the QED correction used, underscoring the need for direct Lattice QCD+QED calculations.
• Methodological Benchmark: The work establishes the "blending method" combined with current-involved interpolators as a new standard for high-precision lattice calculations, offering a scalable solution for future studies of parton distribution functions (PDFs) and other complex matrix elements.
• Reassessment of Systematics: The finding that $g_S$ FVEs follow a simple exponential rather than the chiral-suppressed HB$\chi$PT form suggests that previous lattice studies relying on the latter ansatz may need to reassess their systematic uncertainties.

In conclusion, this paper represents a major leap forward in lattice QCD precision, resolving long-standing issues with excited-state contamination and finite-volume effects to deliver definitive values for nucleon charges at the physical point.