⚛️ lattice

Nucleon sigma terms in lattice QCD at physical pion mass and in the continuum

Using four $N_f=2+1+1$ twisted-mass lattice ensembles simulated at physical pion masses and multiple lattice spacings, this study determines the nucleon's light, strange, and charm sigma terms in the continuum limit, yielding values of $\sigma_{\pi N}=53.0(7.4)(1.6)$ MeV, $\sigma_s=58.2(17.0)(2.1)$ MeV, and $\sigma_c=112(28)(25)$ MeV while accounting for excited-state contamination.

Original author: Constantia Alexandrou, Christos Iona, Giannis Koutsou, Yan Li, Srijit Paul, Ferenc Pittler

Published 2026-11-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original author: Constantia Alexandrou, Christos Iona, Giannis Koutsou, Yan Li, Srijit Paul, Ferenc Pittler

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

At the heart of every atom lies the nucleus, a dense cluster of protons and neutrons that forms the scaffolding of visible matter. For decades, physicists have understood that these particles are not solid, indivisible spheres, but rather complex systems made of smaller, more fundamental constituents called quarks. These quarks are bound together by the strong force, a powerful interaction that behaves differently from gravity or magnetism, creating a dynamic environment where energy and mass are inextricably linked. A central question in this field is how much of the mass of a proton or neutron comes from the intrinsic mass of its constituent quarks, and how much arises from the energy of the forces holding them together. This distinction is crucial because the mass of the quarks themselves is a property that changes depending on the type of quark, and understanding this balance helps scientists predict how these particles interact with the rest of the universe, including potential invisible particles that might make up dark matter.

To answer this, researchers look for specific values known as sigma terms. These numbers act as a measure of sensitivity, telling us how much the mass of a nucleon would change if the mass of a specific type of quark inside it were altered. There are three main types of quarks involved in this calculation: the light up and down quarks that form the bulk of the particle, the heavier strange quark, and the even heavier charm quark. Determining these values precisely is difficult because the quantum world is noisy; the nucleon is constantly bubbling with virtual particles and excited states that can obscure the true signal. If these fleeting, noisy contributions are not accounted for correctly, the final calculation of the mass sensitivity can be significantly skewed, leading to conflicting results between different research groups.

A team of physicists has now tackled this challenge by performing a massive, high-precision simulation of the strong force using a method called lattice quantum chromodynamics. Instead of observing particles in a physical laboratory, they constructed a digital grid of space and time, filling it with the rules that govern quarks and gluons. They ran these simulations on four different versions of this grid, each with a different level of detail, ranging from a coarse mesh to a very fine one. Crucially, they tuned the simulation so that the virtual pions—the lightest particles made of quarks—had the exact same mass as the real pions found in nature. This allowed them to bypass the need for complex mathematical corrections that often introduce errors in other studies. By simulating the universe at this physical point, they could observe the nucleon exactly as it exists, rather than as an approximation.

The researchers focused on extracting the sigma terms for the light, strange, and charm quarks. A major hurdle in previous attempts was the interference from "excited states," which are temporary, higher-energy configurations of the nucleon that appear in the data and can be mistaken for the stable particle. The team developed a sophisticated analysis technique that allowed these excited states to have their own distinct energy gaps, rather than forcing them to match the energy of the stable nucleon. This flexibility was key to separating the true signal from the background noise. They analyzed the data across all four grid resolutions simultaneously, fitting the results together to see how the values changed as the grid became finer, and then projected those trends forward to what the answer would be on a grid with infinite detail, known as the continuum limit.

The study yielded precise values for the three sigma terms. For the light quarks, which determine the bulk of the nucleon's mass sensitivity, the team found a value of 53.0 MeV, with a small margin of error. For the strange quark, the value was determined to be 58.2 MeV, and for the heavy charm quark, it was 112 MeV. These results are significant because they resolve a long-standing disagreement in the field. Some previous simulations had suggested a lower value for the light quark contribution, while others had suggested a higher one. This new work, by rigorously accounting for the excited states and using physical pion masses, lands squarely in the middle of these conflicting views, providing a result that aligns well with experimental data derived from scattering experiments.

The findings have immediate implications for our understanding of the universe beyond the standard model of particle physics. The values for the strange and charm quarks are essential inputs for calculating how dark matter might interact with ordinary matter. If dark matter particles exist and collide with atomic nuclei, the strength of that interaction depends heavily on the sigma terms calculated in this study. The team's results provide a solid foundation for experiments designed to detect dark matter, helping to translate the limits set by current detectors into constraints on how dark matter couples to quarks. Furthermore, the value for the charm quark serves as a test for the theory of heavy-quark decoupling, confirming that the heavy quark behaves as predicted by the standard model, although the precision is not yet high enough to provide a definitive, stringent test of that theory.

In the end, this research represents a triumph of computational physics, demonstrating that by simulating the strong force with enough detail and by carefully untangling the complex noise of the quantum vacuum, scientists can extract fundamental properties of nature with high confidence. The team did not just find a number; they provided a clear, consistent picture of how the mass of the nucleon is distributed among its quark constituents. Their work bridges the gap between theoretical predictions and experimental reality, offering a reliable reference point for future discoveries in particle physics and cosmology. The results stand as a testament to the power of combining advanced computing with rigorous statistical analysis to reveal the hidden architecture of matter.

Problem
The nucleon sigma terms ( $\sigma_q$ ), which quantify the sensitivity of the nucleon mass to the quark mass of a specific flavor ( $q$ ), are critical parameters for understanding nucleon mass decomposition and for phenomenological searches for physics beyond the Standard Model (BSM). Specifically, the light-quark sigma term ( $\sigma_{\pi N}$ ) is essential for interpreting dark matter direct detection limits (e.g., from LZ, XENONnT, and PandaX-4T) and Higgs-mediated $\mu \to e$ conversion, while the strange ( $\sigma_s$ ) and charm ( $\sigma_c$ ) terms contribute to the Higgs-nucleon coupling.

Current lattice QCD determinations of $\sigma_{\pi N}$ exhibit significant discrepancies. The FLAG average for $N_f = 2+1$ results ( $42.2(2.4)$ MeV) lies significantly below the phenomenological value derived from pion-nucleon scattering ( $\approx 56.0$ MeV). Furthermore, previous lattice calculations often relied on single lattice spacings or joint extrapolations in lattice spacing and pion mass, and many studies struggled with systematic uncertainties arising from excited-state contamination, particularly from low-lying multihadron states (e.g., $N\pi\pi$ ) that are strongly excited by the scalar current but less visible in nucleon mass determinations.

Methodology
This work determines the light ( $\sigma_{\pi N}$ ), strange ( $\sigma_s$ ), and charm ( $\sigma_c$ ) sigma terms using four $N_f = 2+1+1$ twisted-mass gauge ensembles generated by the Extended Twisted Mass Collaboration (ETMC). The study is distinguished by the following methodological features:

Physical Point and Continuum Limit: Simulations are performed at physical pion masses with four distinct lattice spacings ranging from $0.080$ fm to $0.049$ fm, enabling a robust extrapolation to the continuum limit.
Excited-State Treatment: The analysis explicitly accounts for excited-state contamination by allowing the energy gap in the three-point function ( $\Delta E_{3pt}$ ) to differ from that of the nucleon two-point function ( $\Delta E_{2pt}$ ). This approach avoids the bias introduced by constraining excited-state energies solely based on nucleon mass data. The analysis identifies a gap close to the $N\pi\pi$ threshold ( $\approx 360\text{--}450$ MeV), which is significantly lower than the Roper resonance gap observed in two-point functions.
Global Fitting Strategy: The authors perform two complementary analyses:
1. Per-ensemble fits: Extracting sigma terms for each lattice spacing individually before extrapolating to the continuum.
2. Global fits: Simultaneously fitting all four ensembles with a continuum ansatz ( $\sigma(a) = \sigma + b a^2$ ) for the sigma terms and energy gaps. This method treats the correlation between the sigma term and the energy gap explicitly.
Renormalization: Using twisted-mass fermions at maximal twist, the product of the bare quark mass and the bare scalar matrix element is renormalization-group invariant, eliminating the need for explicit renormalization factors.

Key Contributions

Systematic Control: The study provides a comprehensive analysis that isolates the impact of multiparticle states without imposing external constraints on their energy gaps, addressing a known source of bias in previous extractions.
Consistency Check: By comparing results from per-ensemble fits against global fits, the work validates the stability of the continuum extrapolation and the treatment of excited states.
Heavy-Quark Decoupling: The determination of $\sigma_c$ allows for a test of the heavy-quark expansion at the charm mass scale.

Results
The authors report the following values for the sigma terms in the continuum limit, with statistical and systematic uncertainties respectively:

Light-quark sigma term: $\sigma_{\pi N} = 53.0(7.4)(1.6)$ MeV.
Strange sigma term: $\sigma_s = 58.2(17.0)(2.1)$ MeV.
Charm sigma term: $\sigma_c = 112(28)(25)$ MeV.

These results are obtained after converting the phenomenological Roy–Steiner value to the isospin-symmetric convention used in the lattice calculation. The lattice results for $\sigma_{\pi N}$ and $\sigma_s$ are compatible with most recent lattice determinations (except for a specific JLQCD result) and align with the phenomenological value once converted to the same isospin convention. The value for $\sigma_c$ agrees with other lattice determinations and the leading-order heavy-quark expansion prediction ( $\approx 61$ MeV), though the current precision ( $\sim 34\%$ ) does not yet provide a stringent test of heavy-quark decoupling.

Significance
The paper claims that these results provide valuable input for phenomenological searches for physics beyond the Standard Model. Specifically, the derived values for $\sigma_{\pi N}$ and $\sigma_s$ allow for the calculation of the Higgs-nucleon coupling at leading order in the heavy-quark matching, yielding $f_N = 0.314(16)$ . This value is in agreement with previous determinations where the uncertainty was dominated by $\sigma_s$ . The work demonstrates that a rigorous treatment of excited-state contamination and a global fit across multiple physical-mass ensembles can resolve discrepancies in the lattice determination of nucleon sigma terms, bringing them into agreement with phenomenological expectations.

More like this