Realization of all-to-all fermion propagator for the first principle high accuracy strong interaction prediction

Problem Statement

Lattice Quantum Chromodynamics (LQCD) is a primary tool for calculating hadron structure from first principles, such as the nucleon's axial vector coupling ($g_A$). However, achieving sub-percent precision requires addressing two major challenges:

1. Computational Cost of All-to-All Propagators: Calculating $N$-point correlation functions (where $N \ge 4$), which are essential for QED corrections, isospin symmetry breaking (ISB), and disconnected quark loops, requires all-to-all fermion propagators. Exact computation scales as $O(V^2)$ with lattice volume $V$, making it prohibitively expensive.
2. Excited-State Contamination (ESC): Extracting ground-state matrix elements is complicated by contamination from excited states (e.g., Roper-like states, multi-particle states like $\pi N$). While the distillation method (projecting onto low-momentum modes) effectively suppresses ESC for hadron spectra, it discards high-momentum modes essential for local current insertions, limiting its utility for $N$-point functions and matrix elements.

Existing techniques like sequential sources, stochastic approaches with low-mode substitution (LMS), and field sparsening offer partial solutions but struggle to simultaneously handle high-precision $N$-point functions, disconnected diagrams, and ESC suppression without incurring massive computational costs.

Methodology: The "Blending" Algorithm

The authors propose a novel stochastic method called the "blending" algorithm to approximate all-to-all fermion propagators efficiently and unbiasedly.

• Blending Space Construction: The method constructs a "blending space" for a fixed time slice by combining two subspaces:
  1. Low-Frequency Modes (LFM): The subspace $L_1$ spanned by the $N_e$ lowest-lying eigenvectors of the gauge-covariant spatial Laplacian (similar to distillation). This captures long-distance physics and allows for the construction of spatially extended hadron states.
  2. High-Frequency Modes (SHFM): The orthogonal complement $L_2$ is approximated stochastically using $N_{st}$ random vectors ($\eta_j$) uniformly sampled from $L_2$.
• Unbiased Estimation: The identity operator on the full space is approximated by a weighted sum of projectors onto these basis vectors.
  • The low-mode projectors retain their exact weight (1).
  • The stochastic high-mode projectors are assigned a weight of $\dim(L_2)/N_{st}$ to ensure an unbiased estimate of the identity on the complement subspace.
  • For operators involving products of fields (e.g., in $N$-point functions), a weight tensor $\Omega^{(2)}_{ij}$ is applied to correct for double counting and ensure the estimator remains unbiased, particularly for diagonal stochastic terms.
• Compression: The all-to-all quark propagator $S$ is compressed by projecting it onto this blending space. This reduces storage requirements from $O((4N_c N_L^3 N_T)^2)$ to $O((4N_T(N_e + N_{st}))^2)$, enabling flexible computation of correlation functions with arbitrary interpolating operators and current insertions.
• Excited-State Control: The method facilitates the use of Generalized Eigenvalue Problem (GEVP) analysis with an enriched basis of interpolators (including "current-involved" operators like $N f_1$ scattering states) without additional inversion costs, as the compressed propagators can be reused for multiple operator combinations.

Key Contributions and Results

The authors demonstrate the efficiency and accuracy of the blending method through several applications on two CLQCD ensembles: C24P29 ($m_\pi \approx 292$ MeV, $a \approx 0.105$ fm) and F64P13 ($m_\pi \approx 134$ MeV, $a \approx 0.077$ fm).

1. Validation of Unbiasedness:

   • The method was tested using the conserved vector current matrix element of the pion ($\pi^+$). Results showed that using only low modes ($N_{st}=0$) yielded a biased result ($\approx 0.64$), whereas including stochastic modes ($N_{st} \ge 50$) restored the result to the theoretical expectation of 1.000(29).
   • Consistency checks were performed for the pion electric form factor ($f_{\pi\pi}(Q^2)$) and charge radius using both 3-point and 4-point correlation functions. The results were consistent within uncertainties, validating the 4-point function calculation.

2. Nucleon Axial Coupling ($g_A$):

   • Iso-vector ($g_A^{u-d}$): Using a joint 3-state fit across various low-mode counts ($\bar{N}_e$) and source-sink separations, the authors obtained $g_A^{u-d} = 1.2337(84)$ at the physical pion mass. This result eliminates the ambiguity associated with selecting a specific $\bar{N}_e$ and shows excellent agreement with other high-precision studies.
   • Singlet and Flavor Decomposition: The method enabled the calculation of disconnected quark loops for the singlet axial charge ($g_A^{u+d+s}$) and individual flavor contributions.
     • Using a GEVP analysis with a basis including $N$, $N f_1$, and $N f'_1$ interpolators, they suppressed excited-state contamination effectively.
     • Final results at the physical point ($a=0.077$ fm) are:
       • $g_A^u = 0.8408(86)$
       • $g_A^d = -0.3929(86)$
       • $g_A^s = -0.0381(57)$
       • $g_A^{u+d+s} = 0.410(20)$

3. Computational Efficiency:

   • The blending method significantly reduces the number of required Dirac operator inversions compared to traditional sequential source or stochastic methods.
   • For the isovector $g_A$, the cost was reduced by approximately one order of magnitude compared to recent high-precision studies (e.g., ETM, RQCD, PNDME) targeting similar precision.
   • For the singlet $g_A$ (involving disconnected loops), the method achieved a 4% precision with 0.70 million inversions, compared to 1.37–34.1 million inversions in comparable previous works. This efficiency stems from the ability to reuse the compressed propagator for multiple interpolators and current insertions without new inversions.

Significance and Claims

The paper claims that the blending method provides a robust framework for high-precision LQCD calculations of hadron structure, specifically addressing the dual challenges of computational cost and excited-state contamination.

• Systematic Control: By allowing the construction of correlation functions with multiple interpolating operators sharing the same quantum numbers, the method facilitates robust extraction of ground-state matrix elements via variational analysis (GEVP), effectively suppressing ESC.
• Scalability: At the physical pion mass, the method makes sub-percent precision predictions for nucleon matrix elements attainable with significantly reduced computational resources.
• Versatility: The approach is applicable to various $N$-point functions, including those with disconnected diagrams, QED corrections, and ISB effects, which are crucial for testing the Standard Model.
• Limitations: The authors note that the computational cost remains prohibitive for Overlap fermions due to their higher inversion costs, suggesting a need for further optimizations for broader applicability.

In summary, the blending method represents a significant step forward in making high-precision, first-principle calculations of complex hadronic observables feasible, enabling future studies of multi-body scattering and parton physics with reduced systematic and statistical uncertainties.