⚛️ nuclear theory

Bjorken sum rule down to the elastic region from the lattice

This paper presents a lattice QCD determination of the Bjorken sum rule by calculating the lowest moment of the nucleon's polarized structure function $g_1$ across a wide range of momentum transfers ( $0.5\lesssim Q^2 \lesssim 8$ GeV $^2$ ) using the Feynman-Hellmann method, thereby enabling the study of higher-twist effects and providing an alternative hadronic evaluation of the strong coupling constant $\alpha_s(Q^2)$ .

Original authors: J. A. Crawford, A. Hannaford-Gunn, R. Horsley, P. E. L. Rakow, T. G. Schar, G. Schierholz, H. Stüben, R. D. Young, J. M. Zanotti

Published 2026-10-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: J. A. Crawford, A. Hannaford-Gunn, R. Horsley, P. E. L. Rakow, T. G. Schar, G. Schierholz, H. Stüben, R. D. Young, J. M. Zanotti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep inside every atom, protons and neutrons form the core of matter, but their internal lives are governed by a force so powerful it defies our everyday intuition. This force, known as the strong interaction, binds quarks together to form these particles. To understand how this force works, scientists study the "spin" of these particles, a fundamental property similar to how a spinning top rotates on its axis. By firing high-energy beams of electrons at protons and neutrons and measuring how they scatter, researchers can map out how the spin is distributed among the tiny quarks inside. A specific mathematical relationship, known as the Bjorken sum rule, predicts how the difference in spin between a proton and a neutron should behave as the energy of the collision changes. This rule acts as a strict test for our theories of the strong force, bridging the gap between the high-energy world where particles behave like free-floating points and the low-energy world where they are tightly bound together.

For decades, physicists have been able to verify this rule at very high energies, where the math is relatively straightforward. However, the transition to lower energies, where the strong force becomes dominant and the particles' internal structure matters most, has remained a difficult puzzle. In this lower-energy regime, the simple rules of free particles break down, and complex interactions between quarks and the gluons that bind them create a messy, non-linear environment. Understanding this transition is crucial because it reveals how the strong force confines quarks within protons and neutrons, a phenomenon that remains one of the deepest mysteries in physics. Until now, calculating these low-energy effects from first principles has been incredibly difficult, often requiring approximations that might miss subtle but vital details.

A team of researchers from the QCDSF Collaboration has now taken a significant step forward by using a supercomputer to simulate the strong force directly, without relying on those approximations. They performed a complex calculation on a digital grid that mimics the fabric of space-time, allowing them to observe the behavior of protons and neutrons as if they were real particles. Instead of trying to measure the spin difference directly in a single snapshot, which is computationally impossible, they used a clever mathematical trick known as the Feynman-Hellmann method. This technique involves gently "perturbing" the simulated system with a virtual field, much like pressing a finger against a spring to see how it reacts, and then measuring how the energy of the proton changes in response. By analyzing these tiny energy shifts, the team could reconstruct the properties of the proton's spin structure across a wide range of energy scales.

The researchers focused their simulation on a specific range of energy levels, covering values from 0.5 to 8 GeV². This span is critical because it includes both the high-energy region where standard theories work well and the lower-energy region where new, complex effects are expected to appear. Their calculations covered the entire journey from the high-energy limit down to the point where the proton behaves almost like a solid, elastic object. To ensure their results were accurate, they had to carefully separate the signal coming from the internal quark structure from the background noise caused by the proton bouncing off itself, a process known as the elastic contribution. They successfully isolated the true signal, revealing how the spin difference evolves as the energy drops.

The results of this study show that at high energies, the simulated data aligns well with the predictions of standard theory, confirming that the team's method is sound. However, as the energy levels dropped into the middle range, the team observed a distinct deviation. The calculated values began to drift away from the simple predictions, rising sharply as the energy decreased. This behavior is exactly what one would expect if the complex, higher-order interactions between quarks and gluons were becoming significant. The team found that once they accounted for the elastic bounce-back effect, the remaining difference pointed toward these subtle, higher-twist contributions. While the data is not yet precise enough to pin down the exact size of these effects due to the limitations of the current computer grid, the trend is clear and matches the physical intuition that the strong force becomes more complicated at lower energies.

This work demonstrates that it is possible to calculate these difficult properties directly from the fundamental laws of physics using modern supercomputing techniques. The team did not just confirm what was already known; they successfully extended the calculation into a region where the physics is much harder to predict. Their findings suggest that the "messy" interactions between quarks are indeed playing a major role in the low-energy behavior of the proton's spin. While the study was performed on a single set of simulated conditions with a specific grid size, the method they used is robust and can be refined. By improving the resolution of the simulation and using lighter quarks that more closely resemble reality, future calculations could provide a precise measurement of these elusive effects. This opens a new window into understanding the strong force, offering a path to determine the strength of the interaction between quarks directly from the properties of the proton itself, rather than relying on indirect measurements.

Technical Summary: Lattice Determination of the Bjorken Sum Rule via Feynman-Hellmann

Problem Statement
The Bjorken sum rule, defined as the difference between the first moments of the proton and neutron polarized structure functions ( $g_1$ ), is a fundamental quantity for understanding nucleon spin structure and testing perturbative QCD (pQCD). While experimentally validated down to $Q^2 \approx 1 \text{ GeV}^2$ and mapped to the elastic region ( $Q^2 \to 0$ ) by recent experiments, the theoretical transition between the perturbative regime and the non-perturbative elastic/Gerasimov-Drell-Hearn (GDH) regime remains poorly understood.

Direct lattice QCD calculations of higher-twist contributions (power corrections of order $1/Q^2$ ) are notoriously difficult due to operator mixing with lower-dimensional operators, which induces power-divergent mixing. Consequently, there is a need for alternative first-principles methods to access these moments and the associated higher-twist effects without the complications of operator renormalization. Furthermore, the Bjorken sum rule offers a potential avenue for a non-perturbative definition of the strong coupling constant, $\alpha_s$ , across all momentum scales.

Methodology
The authors employ the Feynman-Hellmann (FH) theorem to calculate the polarized forward Compton amplitude, thereby avoiding the computationally expensive and noisy direct calculation of four-point correlation functions.

Theoretical Framework:
- The polarized Compton amplitude is related to the structure function $g_1$ via a dispersion relation. On the lattice, access is restricted to the unphysical region $|\omega| < 1$ , where the amplitude can be expanded in terms of Cornwall-Norton moments of $g_1$ . The lowest moment ( $n=1$ ) corresponds directly to the Bjorken sum rule.
- Two external perturbing fields are introduced to the QCD action, coupling to vector currents. The second derivative of the nucleon energy with respect to these couplings yields the Compton amplitude.
- To isolate the spin-dependent part, the authors analyze the energy splitting between spin-up and spin-down nucleon states in the presence of the background fields.
Lattice Setup:
- Calculations were performed on a single gauge ensemble generated by the QCDSF Collaboration with $2+1$ dynamical flavors (stout-smeared, non-perturbatively $O(a)$ -improved Wilson fermions).
- The ensemble has a lattice spacing $a \approx 0.068$ fm, volume $48^3 \times 96$ , and is tuned to the $SU(3)$ -flavor-symmetric point with a pion mass $m_\pi \approx 412$ MeV.
- The nucleon is kept at rest ( $p=0$ ), and the momentum transfer $q$ is varied to cover the range $0.5 \lesssim Q^2 \lesssim 8 \text{ GeV}^2$ .
Technical Innovations:
- Diagonal vs. Non-Diagonal Perturbations: The authors address cases where the perturbation is diagonal in spin space (allowing extraction via simple ratios) and cases where it is non-diagonal (introducing spin mixing). For the latter, they implement a Generalized Eigenvalue Problem (GEVP) to diagonalize the correlator matrix and extract the energy eigenvalues.
- LOPE Improvement: To mitigate discretization errors arising from broken rotational symmetry, the authors apply a Lattice Operator Product Expansion (LOPE) improvement, replacing kinematic factors with lattice-specific trigonometric forms.
- Elastic Subtraction: Since the Compton amplitude includes both elastic and inelastic contributions, the authors calculate the elastic pole contribution using electromagnetic form factors (derived from three-point functions) and subtract them to isolate the inelastic moments required for the sum rule.

Key Contributions and Results

First-Principles Calculation: The paper presents a lattice determination of the Bjorken sum rule moments over a broad $Q^2$ range ( $0.5 \text{ to } 8 \text{ GeV}^2$ ), covering both perturbative and non-perturbative regions.
Validation of Methodology: The study successfully demonstrates the feasibility of using the FH technique with GEVP to handle non-diagonal spin perturbations, providing a complementary framework to direct four-point function calculations.
Comparison with Theory:
- High $Q^2$ ( $> 4 \text{ GeV}^2$ ): The calculated moments show qualitative agreement with perturbative QCD predictions (N3LO accuracy) and the quark-parton model, though a systematic offset due to discretization artifacts is observed.
- Mid $Q^2$ ( $0.5 - 4 \text{ GeV}^2$ ): The lattice results deviate from pure perturbative predictions as $Q^2$ decreases. Upon adding the calculated elastic contributions, the lattice data moves closer to the theoretical curve, but a residual discrepancy remains.
Higher-Twist Effects: The residual discrepancy between the lattice moments (after elastic subtraction) and the perturbative prediction is interpreted as a signal of higher-twist contributions. However, the authors note that due to the remaining discretization artifacts at large $Q^2$ , a rigorous quantitative extraction of these higher-twist terms is not yet possible.

Significance and Claims
The paper claims that this work demonstrates the feasibility of determining non-singlet Deep Inelastic Scattering (DIS) moments using the Feynman-Hellmann method on the lattice. The primary significance lies in:

Providing a first-principles approach to access higher-twist matrix elements without the complications of power-divergent operator mixing.
Offering a pathway to study the transition between perturbative and non-perturbative regimes in the Bjorken sum rule.
Establishing a foundation for future determinations of the strong coupling $\alpha_s$ from hadronic observables, provided that continuum and chiral extrapolations are performed and higher-twist effects are controlled.

The authors remain modest regarding the current precision, acknowledging that the single lattice spacing and heavy pion mass limit the ability to definitively separate higher-twist effects from discretization errors. They conclude that further work with multiple lattice spacings and lighter quark masses is required to reliably estimate the $1/Q^2$ power corrections and perform a direct determination of $\alpha_s$ .

Technical Summary: Lattice Determination of the Bjorken Sum Rule via Feynman-Hellmann

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