Hadron Structure from lattice QCD in the context of the Electron-Ion Collider

This paper reviews recent advances in lattice QCD calculations of hadron structure, specifically for pions, kaons, and nucleons, highlighting how these theoretical developments provide essential insights to inform the scientific agenda of the Electron-Ion Collider.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks and gluons. When you snap these bricks together, they form larger structures called hadrons (like protons, neutrons, pions, and kaons). These are the building blocks of everything you see, touch, and are made of.

For decades, scientists have been trying to take a picture of these Lego bricks to see exactly how they are arranged inside. But there's a problem: these bricks are glued together by the "Strong Force" (Quantum Chromodynamics, or QCD), which is so powerful that you can't just pull them apart to look at them. It's like trying to see the gears inside a watch that is running at the speed of light while being squeezed by a hydraulic press.

This paper is a report card on how scientists are using a super-powerful mathematical tool called Lattice QCD to solve this puzzle, and how this work is perfectly timed to help a massive new machine called the Electron-Ion Collider (EIC) that is being built in New York.

Here is the breakdown of the paper in simple terms:

1. The Two Ways to Look at the Lego Bricks

The paper explains that scientists are using two different strategies to understand the inside of these particles:

• Strategy A: The "Snapshot" (Lattice QCD): Imagine you want to know how a car engine works. Instead of taking it apart, you build a perfect, tiny digital model of it inside a supercomputer. You run simulations to see how the parts move. This is what Lattice QCD does. It creates a 3D grid (a "lattice") of space and time and simulates the laws of physics to calculate exactly how quarks and gluons behave.
  • The Progress: The paper says this digital modeling has gotten incredibly good. We can now calculate specific numbers (like the "weight" or "spin" of the particles) with such high precision that they match real-world experiments.
• Strategy B: The "Flash Photo" (The EIC): The Electron-Ion Collider (EIC) is a giant particle accelerator. Think of it as a high-speed camera that fires electrons at ions (heavy atoms) to take a "flash photo" of the inside of a proton. It will see the quarks and gluons moving around in real-time.
  • The Connection: The EIC is the camera, but Lattice QCD is the instruction manual. The manual tells the camera operators exactly what to look for and helps them interpret the blurry photos.

2. What Are We Trying to See? (The 3D Map)

The paper focuses on three main things scientists want to map out:

• The "Shape" (Form Factors): Just like a human has a height and width, protons have a "charge radius" (how big they are). The paper shows that Lattice QCD can now calculate the size of these particles so accurately that it matches what we see in experiments.
• The "Traffic Report" (Parton Distribution Functions - PDFs): Inside a proton, quarks and gluons are zooming around. Some are heavy and slow; some are light and fast. The "PDF" is a traffic report that tells us: "At any given moment, what percentage of the proton's speed is carried by an up-quark? A down-quark? A gluon?"
  • The Breakthrough: In the past, we could only guess the average speed of the traffic. Now, Lattice QCD can calculate the entire traffic report, showing us the distribution from slow to fast. This is crucial for the EIC to understand the "sea" of particles inside the proton.
• The "Internal Pressure" (Generalized Parton Distributions - GPDs): This is the most exciting part. It's not just about speed; it's about where the particles are.
  • The Analogy: Imagine a spinning top. You know it's spinning, but where is the weight concentrated? Is it in the middle or the edges? GPDs give us a 3D tomography (like a medical CT scan) of the proton. They show us the quarks and gluons in 3D space, revealing the internal pressure and shear forces holding the proton together.

3. The "Heavy Lifting" (Specific Particles)

The paper highlights three specific particles that are getting a lot of attention:

• The Proton: The main building block of matter. We know a lot about it, but we still don't fully understand its "spin" (why it spins the way it does).
• The Pion and Kaon: These are lighter, unstable particles. They are like the "test subjects" for the Strong Force. Because they are simpler, they are easier to simulate on a computer. The paper shows that Lattice QCD is now giving us very precise maps of these particles, which helps us understand the rules that govern the heavier protons.

4. The Future: A Perfect Partnership

The paper concludes with a big picture view:

• The EIC will be the ultimate microscope, taking millions of photos of the inside of atoms.
• Lattice QCD will be the super-computer brain, providing the theoretical "ground truth" to check if the photos make sense.

The Bottom Line:
For a long time, theoretical physics (the math) and experimental physics (the machines) were running on parallel tracks. This paper says they are finally merging. The math is now precise enough to tell the machines exactly what to expect, and the machines are about to provide data that will force the math to get even better.

Together, they will finally answer the ultimate question: How does the invisible glue of the universe hold the visible world together? By combining the "digital simulation" with the "real-world camera," we are about to get the first true, 3D, high-definition map of the building blocks of reality.

Technical Summary

1. Problem Statement

The Electron-Ion Collider (EIC) aims to provide a three-dimensional (3D) tomographic image of hadrons (pions, kaons, and nucleons) by probing their internal quark and gluon structure. To fully exploit the EIC's experimental capabilities, theoretical inputs with percent-level precision are required.

• The Challenge: Light-cone matrix elements (such as Parton Distribution Functions - PDFs, and Generalized Parton Distributions - GPDs) cannot be directly computed in Euclidean lattice QCD (LQCD) due to the sign problem.
• The Gap: While LQCD has historically provided precise results for low-order Mellin moments (integrals of PDFs) and form factors, extracting the full $x$-dependence (Bjorken-$x$) and higher-twist effects has been difficult. Furthermore, achieving the precision required to match EIC data necessitates controlling systematic uncertainties (discretization, finite volume, chiral extrapolation) and incorporating QED effects.

2. Methodology

The paper reviews state-of-the-art LQCD methodologies used to bridge the gap between theoretical calculations and EIC observables.

A. Traditional Approach: Mellin Moments

• Operator Product Expansion (OPE): Light-cone matrix elements are expanded into a tower of local operators.
• Local Operators: The $n$-th moment of a PDF is computed via matrix elements of local operators involving covariant derivatives (e.g., $\bar{\psi}\gamma_{\{\mu_1} i\overleftrightarrow{D}_{\mu_2} \dots i\overleftrightarrow{D}_{\mu_n\}} \psi$).
• Limitations: This method is limited to low-order moments ($n \leq 4$) due to operator mixing and increasing statistical noise at higher orders.

B. Direct Determination Approaches (LaMET and SDF)

To access the full $x$-dependence, the paper highlights two modern frameworks:

1. Large-Momentum Effective Theory (LaMET / Quasi-PDFs):
   • Computes space-like matrix elements of non-local operators with a boosted hadron state ($P_z \sim 1.5\text{--}3$ GeV).
   • Relates the "quasi-distribution" to the light-cone PDF via a perturbative matching kernel ($C$) and a power correction expansion in $\Lambda_{QCD}^2 / P_z^2$.
2. Short-Distance Factorization (SDF / Pseudo-PDFs):
   • Computes the ratio of space-like matrix elements (Ioffe-time distributions) to cancel renormalization factors.
   • Matches the coordinate-space matrix element to the light-cone PDF using a perturbative kernel, valid for small $z^2$.

C. Generalized Parton Distributions (GPDs)

• Calculated by allowing different initial and final boosted hadron states.
• Breit Frame vs. Asymmetric Frame: While early calculations used the Breit frame (expensive), recent work utilizes the asymmetric (lab) frame, expanding matrix elements into gauge-invariant functions to extract GPDs at multiple momentum transfers simultaneously.

D. Simulation Parameters

• Ensembles: Simulations use $N_f = 2+1$ and $N_f = 2+1+1$ dynamical fermions (Wilson, Staggered, Domain Wall, Twisted Mass).
• Physical Point: Calculations are performed at or near the physical pion mass ($m_\pi \approx 140$ MeV).
• Systematics: Continuum extrapolations ($a \to 0$) and infinite volume limits ($L \to \infty$) are performed using ensembles with lattice spacings as small as $a \approx 0.05$ fm.

3. Key Contributions and Results

A. Mellin Moments and Form Factors

• Pion and Kaon:
  • Precise determination of vector, scalar, and tensor charges and form factors ($F_V, F_S, F_T$).
  • Momentum Fractions: ETMC calculated quark and gluon momentum fractions ($\langle x \rangle$) for pions and kaons, including disconnected diagrams. Results show agreement with phenomenology, though some tension exists for the gluon fraction in the pion.
  • Higher Moments: Computed up to the 4th moment ($\langle x^3 \rangle, \langle x^4 \rangle$). Ratios of $u$-to-$s$ quark moments in the kaon indicate that the strange quark PDF is supported at larger $x$ values than the up quark, and SU(3) symmetry breaking is more pronounced for higher moments.
• Nucleon:
  • Charges: Isovector axial ($g_A$) and tensor ($g_T$) charges are computed with high precision, agreeing with FLAG2024 averages.
  • Form Factors: Electromagnetic (EM) form factors (electric and magnetic) for proton, neutron, and strange quarks are reproduced with unprecedented accuracy, including disconnected contributions. LQCD results for the neutron electric form factor are more precise than current experimental data.
  • Spin Decomposition: Preliminary results show gluons contribute $\approx 45\%$ to the nucleon spin and momentum. The $D$-term (related to pressure/shear forces) has been computed, providing insights into the mechanical structure of the nucleon.

B. Direct PDF and GPD Calculations

• PDFs:
  • Valence Distributions: LaMET and SDF approaches have successfully extracted valence PDFs for pions, kaons, and nucleons.
  • Gluon PDFs: First direct computations of gluon PDFs for pions, kaons, and nucleons using the pseudo-distribution approach. Results are compatible with phenomenological fits (e.g., JAM, CT18).
  • Impact: Including LQCD inputs in global analyses (e.g., JAM) significantly reduces uncertainties in isovector and strange quark distributions.
• GPDs:
  • First extractions of nucleon and pion GPDs (unpolarized, helicity, transversity) in both quasi- and pseudo-distribution frameworks.
  • 3D Imaging: Using zero-skewness GPDs, the authors constructed impact parameter space distributions, visualizing the 3D spatial structure of hadrons.
  • GFFs: Generalized Form Factors (GFFs) extracted from GPDs agree with direct local operator calculations, validating the method.

C. Higher Twist and TMDs

• Twist-3: Progress in computing twist-3 PDFs (e.g., $g_T(x)$, $e(x)$) and the $d_2$ moment, which are crucial for understanding quark-gluon correlations.
• TMDs: While not covered in depth, the paper notes that exploratory studies using staple-shaped Wilson lines are laying the groundwork for Transverse-Momentum-Dependent distributions.

4. Significance

• Synergy with EIC: The paper demonstrates that LQCD has matured to a point where it can provide ab initio inputs with controlled systematic uncertainties that match the precision targets of the EIC.
• Validation of Phenomenology: LQCD results serve as critical cross-checks for phenomenological extractions from experimental data, particularly in regions where data is sparse (e.g., large $x$, gluon distributions, strange quark content).
• 3D Structure: The ability to compute GPDs and reconstruct 3D images (impact parameter space) directly from QCD allows for a fundamental understanding of hadron mass, spin, and mechanical properties (pressure/shear forces).
• Future Directions: The field is moving toward:
  • Calculations at higher boosts ($P_z \gtrsim 3$ GeV) to reduce power corrections.
  • Inclusion of QED effects for percent-level precision.
  • Extension to higher moments and higher-twist distributions.
  • Utilization of machine learning and improved algorithms (e.g., multilevel methods) to enhance statistical precision.

In conclusion, the paper argues that the combination of high-precision EIC experiments and advanced LQCD calculations will lead to a quantitative, multidimensional understanding of hadron structure rooted directly in Quantum Chromodynamics.