⚛️ phenomenology

Pion Distribution Amplitudes from Functional QCD

This paper presents the first functional QCD calculation of the pion distribution amplitude using the functional renormalisation group and large-momentum effective theory, yielding a second-order moment of 0.267 that is significantly smaller than existing lattice results and consistent with other nonperturbative approaches.

Original authors: Lei Chang, Wei-jie Fu, Chuang Huang, Jan M. Pawlowski, Yang-yang Tan

Published 2026-07-17

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CC BY 4.0

Original authors: Lei Chang, Wei-jie Fu, Chuang Huang, Jan M. Pawlowski, Yang-yang Tan

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

Imagine the universe is built from tiny, invisible LEGO bricks called quarks. These bricks snap together to form protons and neutrons, the core of every atom in your body. But quarks are shy; they never hang out alone. They are glued together by a force so strong it's like a rubber band that gets tighter the further you pull it. This glue is called the strong force, and the "glue" itself is made of particles called gluons.

One of the most famous LEGO structures is the pion. It's a lightweight particle that acts like a messenger, helping protons and neutrons stick together inside an atom's nucleus. But here's the mystery: even though we know pions exist, we don't fully understand how the quarks inside them share their energy. Do they split the energy equally, like two friends splitting a pizza 50-50? Or does one hog most of the slice? This "sharing pattern" is called the Distribution Amplitude (DA). Figuring this out is crucial because the pion is a special kind of particle born from a cosmic event called "chiral symmetry breaking," which is the reason most of the visible mass in the universe exists. If we can't map out how the pion's quarks share their momentum, we're missing a key piece of the puzzle for how matter gets its weight.

For a long time, scientists have been trying to take a "snapshot" of this sharing pattern using two different methods: one involves smashing particles together in giant computers (lattice QCD), and the other involves solving complex math equations that describe how particles interact (functional QCD). The problem is, these two methods have been giving conflicting answers. It's like two photographers taking pictures of the same object, but one sees a wide, flat shape while the other sees a narrow, sharp peak. They couldn't agree on what the pion actually looks like.

Now, a team of researchers has stepped in with a fresh approach to settle the score. They used a powerful mathematical toolkit called the Functional Renormalization Group (fRG) to calculate the pion's distribution from the ground up, using only the fundamental rules of the strong force and the masses of the quarks as their starting point. No guessing, no fudging the numbers with extra "adjustment knobs."

To get a clear picture, they had to do something tricky. In the real world, quarks zip around at the speed of light, making them impossible to photograph directly. So, the scientists used a clever trick called Large-Momentum Effective Theory (LaMET). Think of it like trying to take a photo of a speeding race car. If you take a picture while the car is moving slowly, it looks blurry and distorted. But if you can somehow calculate what the car would look like if it were zooming at an incredibly high speed, you can mathematically "zoom out" to see its true shape.

The team pushed their calculations to a record-breaking speed, simulating the pion with a momentum of 4.5 GeV. This is more than double the speed previous computer simulations could handle. At this high speed, the "blur" disappeared. The data became fully saturated, meaning the picture stopped changing no matter how much faster they went. This allowed them to extrapolate the result to the "light-cone limit"—the theoretical perfect snapshot of the pion's internal structure.

What did they find? The pion's quarks don't split the energy in a narrow, sharp spike, nor do they spread it out in a double-humped shape. Instead, the pion has a broad, smooth, single-hump shape, like a gentle hill. The researchers calculated a specific number to describe this shape, called the second-order moment, and found it to be 0.267.

This number is significant because it settles a long-standing debate. It is noticeably smaller than the value of 0.300 found in recent high-speed computer simulations (lattice-LaMET), suggesting those earlier simulations might have been slightly off due to not reaching high enough speeds. However, the new result of 0.267 fits perfectly with other non-computer methods, like QCD sum rules and other mathematical models.

In short, this paper suggests that the pion is a broad, smooth distribution of energy, not a sharp spike. By pushing the simulation speeds higher than ever before, the team provided a clearer, more reliable picture that aligns with other theories, resolving a conflict that has puzzled physicists for years. They didn't just guess; they calculated it from first principles, showing that when you look at the pion with enough speed and precision, its true, broad nature finally comes into focus.

Technical Summary: Pion Distribution Amplitudes from Functional QCD

Problem Statement
The pion light-front parton distribution amplitude (PDA), $\phi_\pi(x)$ , describes the longitudinal momentum sharing between valence quarks and antiquarks. As a pseudo-Goldstone boson of dynamical chiral symmetry breaking (DCSB), the pion's PDA offers a direct probe of DCSB mechanisms. However, the shape of the pion PDA remains a subject of significant controversy. Different nonperturbative approaches yield conflicting results, particularly regarding the second-order moment $\langle \xi^2 \rangle_\pi$ , which ranges from 0.23 to 0.30 across various studies. Resolving this tension is critical for understanding how DCSB manifests in hadron structure. Existing lattice QCD calculations utilizing Large-Momentum Effective Theory (LaMET) have been limited by accessible longitudinal momenta ( $P_z \lesssim 2.15$ GeV), where the quasi-PDA has not yet saturated, leading to potentially large extrapolation errors to the light-cone limit.

Methodology
This work presents the first functional QCD calculation of the pion PDA using the Functional Renormalisation Group (fRG) approach combined with LaMET. The methodology relies on a parameter-free, first-principles framework:

Functional QCD Framework: The calculation utilizes the 2+1 flavor functional QCD setup developed in prior works [1–3]. The quantum effective action is decomposed into glue, glue-matter, and pure matter sectors. Crucially, the approximation includes the complete set of dominant and subdominant four-quark operators and quark-gluon tensor structures necessary for describing spontaneous chiral symmetry breaking.
Inputs and Parameters: The only inputs are the strong coupling and current quark masses, fixed by the physical ratios $m_\pi/f_\pi$ and $m_K/f_\pi$ . No phenomenological parameters or model assumptions are introduced. The quark two-point functions and pion Bethe-Salpeter (BS) amplitudes are obtained self-consistently.
LaMET Implementation: To access the full $x$ $x$ -dependence without model-dependent moment reconstructions, the light-cone PDA is connected to a Euclidean quasi-PDA computed at large longitudinal momentum $P_z$ $P_{z}$ .
- High Momentum Reach: The authors push the pion momentum up to $P_z = 4.5$ GeV, more than double the limit of current lattice calculations.
- Analytic Continuation: To handle the complex plane requirements of the quasi-LFWF integral at high $P_z$ , the authors employ a deformed integration contour and Taylor expansions of the quark mass function and BS amplitude in the complex $p_0$ plane (up to fourth order). This allows for the inclusion of analytic information necessary to reach $P_z \sim 4.5$ GeV while maintaining numerical stability.
Extraction: The quasi-light-front wave function (quasi-LFWF) is constructed from the quark propagator and resonant four-quark correlation functions. The quasi-PDA is obtained by integrating the quasi-LFWF over transverse momentum. The physical light-cone PDA is then extracted by extrapolating the quasi-PDA to the limit $P_z \to \infty$ .

Key Contributions and Results

Saturation of Quasi-PDA: By reaching $P_z = 4.5$ GeV, the study demonstrates that the quasi-PDA becomes fully saturated. No sizable residual effects are observed in this regime, rendering the extrapolation errors to the light-cone limit negligible.
Pion PDA Shape: The resulting light-cone PDA exhibits a broad, concave, unimodal shape without a double-humped structure. It shows a flat behavior in the intermediate- $x$ region, indicating a strong impact of DCSB.
Second-Order Moment: The calculated second-order moment is $\langle \xi^2 \rangle_\pi = 0.267$ . This value is significantly smaller than the lattice-LaMET determination ($0.300$) and aligns closely with other nonperturbative approaches, including lattice OPE, QCD sum rules, and DSE/BSE calculations.
Higher-Order Moments: The fourth and sixth-order moments ( $\langle \xi^4 \rangle_\pi = 0.139$ , $\langle \xi^6 \rangle_\pi = 0.088$ ) are also consistent with sum rules and DSE/BSE results, further validating the approach.
Endpoint Extrapolation: The authors perform extrapolations in the endpoint regions ( $x \to 0, 1$ ) to handle limitations of Euclidean inputs. Varying the fitting range for these endpoints yields only marginal changes in the final moments, indicating small uncertainties in this procedure.

Significance
The paper claims to resolve a long-standing conflict between lattice-QCD LaMET results and other nonperturbative approaches regarding the variation of pion parton distributions. The authors attribute the discrepancy in the second-order moment to the access to larger $P_z$ values in their functional approach, which allows for a more complete inclusion of relevant QCD dynamics at higher scales and a more faithful incorporation of DCSB effects.

The work demonstrates that the functional QCD framework can provide first-principles, parameter-free predictions for parton distributions. By achieving saturation of the quasi-PDA and reducing extrapolation errors to negligible levels, this study opens a new avenue for studying hadron structure, offering a robust alternative to lattice methods for calculating light-cone observables.

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