Higher Mellin Moments of the Unpolarized PDF of the Pion and the Kaon from Lattice QCD

This paper presents lattice QCD results for the first four Mellin moments of the unpolarized parton distribution functions of the pion and kaon, computed using a physical-mass $N_f=2+1+1$ twisted mass fermion ensemble to reconstruct the valence PDFs and compare them with existing theoretical and phenomenological determinations.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks and gluons. These bricks snap together to form larger structures called hadrons, like protons, neutrons, pions, and kaons.

For a long time, scientists have been trying to take a "snapshot" of how these bricks are arranged inside the pions and kaons. This snapshot is called a Parton Distribution Function (PDF). Think of the PDF as a map that tells you: "If you pick a random piece of momentum inside this particle, what is the chance it belongs to a specific quark?"

However, taking a direct photo of these particles is incredibly hard because pions and kaons are unstable; they fall apart almost instantly. You can't pin them down on a table to look at them like you can with a proton.

The "Recipe" Approach

Instead of taking a direct photo, the scientists in this paper used a clever indirect method. Imagine you have a cake, but you can't see inside it. However, you can measure the cake's total weight, its density, and how it reacts when you poke it in specific ways. From these measurements, you can work backward to guess the recipe: how much flour, sugar, and eggs were used.

In physics, these "measurements" are called Mellin Moments.

• The first moment tells you the average momentum (the "average weight" of the pieces).
• The second moment tells you how spread out the momentum is (how "fluffy" or "dense" the distribution is).
• The third and fourth moments give even more detailed clues about the shape of the distribution.

The team used a supercomputer to run a simulation of the universe's fundamental rules (Quantum Chromodynamics, or QCD). They didn't just calculate the first two clues; they calculated the third and fourth moments for both pions and kaons. This is like measuring the cake's texture and elasticity, not just its weight.

The Pion vs. The Kaon: A Tale of Two Cousins

The paper compares two very similar particles:

1. The Pion: Made of two "light" quarks.
2. The Kaon: Made of one "light" quark and one "strange" quark.

The "strange" quark is heavier, like swapping a light feather for a small stone in your Lego set. The scientists wanted to see how this extra weight changed the internal structure.

What they found:

• The Pion's Map: The momentum in the pion is spread out more evenly. It's like a smooth, fluffy cloud where the pieces are distributed broadly.
• The Kaon's Map: The momentum is more concentrated. Because the strange quark is heavier, it tends to carry more of the "load." The map shows a sharper peak, meaning the heavy quark is hogging more of the momentum at specific points.
• The Symmetry Break: In a perfect world, light and strange quarks would behave identically (like identical twins). But the results showed they are actually quite different cousins. The difference (called "SU(3) symmetry breaking") was about 30–40%, and it got even more pronounced when looking at the higher, more detailed moments.

Reconstructing the Picture

Once they had these four "clues" (the first four moments), the team used a mathematical formula to reconstruct the full map (the PDF) of how the quarks are distributed.

They tested two different shapes for this map:

1. A simple shape: Assuming the map is smooth and predictable.
2. A complex shape: Allowing for weird bumps and curves.

They found that the simple shape worked best. The reconstructed maps confirmed that the pion is "broader" (more spread out) than the kaon. The strange quark in the kaon tends to sit at a higher "speed" (momentum) than the light quarks in the pion.

Why This Matters (According to the Paper)

The paper explains that while we have some experimental data from the past (some from 40 years ago!), it's very limited. Future experiments at CERN and a new machine called the Electron-Ion Collider will try to measure these particles directly.

This paper provides a theoretical blueprint for those future experiments. By calculating these moments from first principles (using only the laws of physics and a supercomputer, without guessing), the team gives experimentalists a reliable target to aim for. It's like giving a treasure hunter a precise map before they even start digging, ensuring they know exactly what the treasure (the internal structure of the pion and kaon) should look like.

In summary: The scientists used a supercomputer to calculate detailed "fingerprints" (moments) of pions and kaons. They used these fingerprints to draw a map of how the particles' insides are organized, revealing that the heavier strange quark in the kaon creates a distinctly different internal structure compared to the lighter pion.

Technical Summary

Technical Summary: Higher Mellin Moments of the Unpolarized PDF of the Pion and the Kaon from Lattice QCD

Problem Statement
Quantum Chromodynamics (QCD) describes the strong interaction, yet the internal partonic structure of light mesons, specifically pions and kaons, remains less understood than that of the nucleon. While pions and kaons are pseudo-Goldstone bosons arising from spontaneously broken chiral symmetry, their structural differences—specifically the presence of a valence strange quark in the kaon versus only light quarks in the pion—offer a direct probe of SU(3) flavor symmetry breaking. Experimentally, extracting Parton Distribution Functions (PDFs) for these unstable particles is challenging due to the lack of fixed targets, with existing data being sparse and decades old. Theoretical models exist but often lack systematic improvability. Consequently, there is a critical need for first-principles calculations to provide precise Mellin moments and reconstructed PDFs to guide upcoming experiments like AMBER at CERN and the Electron-Ion Collider (EIC).

Methodology
This work employs Lattice QCD to compute the Mellin moments of the unpolarized valence PDFs for the pion and kaon up to the fourth order (specifically $\langle x^2 \rangle$ and $\langle x^3 \rangle$, in addition to the previously calculated $\langle x \rangle$).

• Ensemble and Setup: The calculations utilize a single $N_f = 2 + 1 + 1$ gauge ensemble of twisted mass clover-improved fermions (labeled cB211.072.64). The quark masses are tuned to approximately their physical values, with a lattice spacing of $a \approx 0.0796$ fm and a physical pion mass ($M_\pi \approx 140$ MeV).
• Operators and Matrix Elements: The moments are extracted from the matrix elements of local operators involving higher covariant derivatives. To avoid mixing with lower-dimensional operators, the authors select Lorentz indices that are all distinct, requiring boosted meson frames. Specifically, $\langle x^2 \rangle$ is computed with momentum $p = (2\pi/L, 2\pi/L, 0)$ and permutations, while $\langle x^3 \rangle$ uses $p = (2\pi/L, 2\pi/L, 2\pi/L)$.
• Extraction: Moments are derived from the ratio of three-point to two-point correlation functions in the forward limit. The analysis employs a simultaneous fit of effective energies and ratios, utilizing a two-state fit for ground-state energy determination and model averaging based on the Akaike Information Criterion (AIC) to handle systematic uncertainties from fitting ranges.
• Renormalization: Matrix elements are renormalized non-perturbatively in the RI'-MOM scheme and subsequently converted to the $\overline{\text{MS}}$ scheme at a scale of $\mu = 2$ GeV. The renormalization factors ($Z_{VDD}$ and $Z_{VDDD}$) are computed using momentum sources in Landau gauge, with one-loop lattice artifacts subtracted to improve the continuum extrapolation.
• PDF Reconstruction: Using the calculated moments ($\langle x \rangle, \langle x^2 \rangle, \langle x^3 \rangle$) and assuming sea quark contributions are negligible (focusing on connected contributions), the valence PDFs are reconstructed by fitting to a standard functional form $q(x) = N x^\alpha (1-x)^\beta (1+\gamma x)$. Both two-parameter ($\gamma=0$) and three-parameter ($\gamma \neq 0$) fits are explored.

Key Contributions and Results

• Mellin Moments: The paper presents the first lattice QCD determination of the third and fourth Mellin moments ($\langle x^2 \rangle$ and $\langle x^3 \rangle$) for the pion and kaon using physical quark masses and local operators. The results are summarized in the $\overline{\text{MS}}$ scheme at 2 GeV:
  • Pion ($u$): $\langle x^2 \rangle = 0.1021(34)$, $\langle x^3 \rangle = 0.079(25)$.
  • Kaon ($u$): $\langle x^2 \rangle = 0.0925(11)$, $\langle x^3 \rangle = 0.0557(54)$.
  • Kaon ($s$): $\langle x^2 \rangle = 0.14312(86)$, $\langle x^3 \rangle = 0.0881(41)$.
• SU(3) Symmetry Breaking: By comparing the ratios of moments between light and strange quarks in the kaon, the authors observe clear SU(3) flavor symmetry breaking. The deviation from unity is approximately 30–40%, a magnitude that increases for higher moments. This indicates that the strange quark carries a significantly larger fraction of the kaon's momentum compared to the light quarks, consistent with the strange quark's larger mass.
• PDF Reconstruction: The reconstructed valence PDFs reveal that the pion distribution is broader than the kaon distributions. The kaon $s$-quark PDF peaks at a higher $x$ value ($x_{\text{peak}} \approx 0.45$) compared to the kaon $u$-quark ($x_{\text{peak}} \approx 0.35$) and the pion $u$-quark ($x_{\text{peak}} \approx 0.41$). The effective power $\beta_{\text{eff}}$ governing the large-$x$ fall-off is found to be slightly below unity for the pion and slightly above unity for the kaon quarks in the two-parameter fit scenario.
• Comparison: The calculated moments are compared with other lattice QCD results (including previous ETMC work, MIT, and RQCD), phenomenological global analyses (JAM, xFitter), and model calculations (Dyson-Schwinger, Light-Front). The results generally agree within errors with existing lattice data and phenomenological constraints, particularly the JAM analysis which incorporates lattice inputs.

Significance
The paper claims that these results provide significant theoretical input for the design and interpretation of future experiments at CERN (AMBER) and the EIC. By providing a first-principles determination of higher moments and reconstructed PDFs at physical quark masses, the work helps constrain global QCD analyses where experimental data for kaons is historically scarce. The demonstration of significant SU(3) breaking in the moments offers a quantitative measure of flavor symmetry breaking in the partonic structure of mesons. Furthermore, the study highlights the utility of local operators for extracting higher moments and establishes a baseline for future improvements, such as the inclusion of disconnected contributions and the calculation of higher-order moments using gradient flow or heavy-quark operator product expansion methods.