Valence quark distribution of the pion inside a medium with finite baryon density: A Nambu--Jona-Lasinio model approach

This paper employs a two-flavor Nambu--Jona-Lasinio model coupled with a light-cone quark model to calculate and analyze the in-medium valence quark distribution, electromagnetic form factor, and distribution amplitude of the pion at finite baryon density, comparing the resulting parton distribution functions and their Mellin moments with experimental data, lattice QCD, and theoretical predictions after NLO DGLAP evolution.

The Gist

Imagine the universe is built out of tiny, invisible Lego bricks called quarks. Usually, when we study these bricks, we look at them floating freely in a vacuum, like a single Lego piece sitting on a table. But in the real world, especially inside the cores of stars or during massive particle collisions, these bricks are packed tightly together in a crowded room. This paper asks: What happens to a specific Lego structure (a pion) when it's squeezed into this crowded room?

Here is a simple breakdown of what the researchers did and what they found, using everyday analogies.

The Main Characters

1. The Pion: Think of this as a small, bouncy ball made of two smaller pieces glued together: a quark and an anti-quark. It's the lightest "ball" in the particle world.
2. The Medium (The Crowd): This is the "finite baryon density" mentioned in the title. Imagine a crowded subway car. The "density" is how many people are packed in there. In this paper, the scientists are studying what happens to the pion when it's inside a very crowded "subway car" of nuclear matter.
3. The Tools:
   • The NJL Model: This is like a rulebook that tells the scientists how the "crowd" affects the weight of the individual Lego bricks (quarks).
   • The Light-Cone Quark Model: This is a high-speed camera that takes pictures of how the two pieces of the pion are moving and sharing space.

The Experiment: Squeezing the Pion

The researchers used a two-step process to simulate this crowded environment:

1. Step 1: Changing the Weight of the Bricks.
   In a vacuum (empty space), the quarks inside the pion have a certain "effective weight" (mass). The scientists used their rulebook (the NJL model) to calculate what happens to this weight when the pion is squeezed into a dense crowd.

   • The Result: As the crowd gets denser, the "weight" of the quarks gets lighter. It's as if the pressure of the crowd makes the bricks feel less heavy. This is a sign of "chiral symmetry restoration," a fancy way of saying the rules of how these particles hold themselves together are changing under pressure.

2. Step 2: Taking New Pictures.
   With these new, lighter weights, they used their high-speed camera (the Light-Cone model) to take new pictures of the pion. They looked at three specific things:

   • How the pieces share momentum (Distribution Amplitude): Imagine the two pieces of the pion are running a relay race. In empty space, they share the running duties somewhat evenly. In the crowded room, the researchers found the race becomes more chaotic. The pieces are less likely to be in the "middle" of the track and more likely to be at the very start or very end. The distribution gets "flatter."
   • How it reacts to a probe (Electromagnetic Form Factor): If you poke the pion with a magnet, how does it push back? In the crowd, the pion becomes "softer" or more spread out. Its "charge radius" (how big it looks from the outside) gets bigger as the crowd density increases. It's like a sponge expanding when you squeeze it in a specific way.
   • Where the pieces are found (Parton Distribution Function): This is a map showing where you are most likely to find a quark inside the pion. In the crowd, the map changes. The "peak" of where you find the quark shifts slightly toward the faster end of the spectrum.

The Evolution: Fast-Forwarding Time

The scientists didn't just look at the pion at one speed. They used mathematical equations (called DGLAP evolution) to "fast-forward" their results from a slow, low-energy view to a super-fast, high-energy view (like zooming in with a powerful microscope).

• The Finding: At low speeds (the model scale), the effects of the crowded room are very obvious. The pion looks very different. But when they fast-forwarded to high speeds, the differences between the "crowded" pion and the "empty space" pion became much smaller. The crowd's influence fades away when you look at the particle moving at extreme speeds.

The Bottom Line

The paper concludes that when a pion is trapped in a dense nuclear medium (like inside a star or a heavy-ion collision):

• Its internal building blocks (quarks) become lighter.
• The pion itself becomes slightly larger and "fluffier."
• The way its internal parts share energy changes, becoming less uniform.
• However, if you look at the pion moving at very high speeds, these changes become much less noticeable.

The researchers compared their "crowded room" predictions with existing data from particle accelerators and computer simulations (Lattice QCD) and found that their model matches the known vacuum data well, giving them confidence in their predictions for the "crowded" scenarios. They did not claim to have found a new material or a medical application; they simply mapped out how the rules of the subatomic world change when things get crowded.

Technical Summary

Technical Summary: Valence Quark Distribution of the Pion in a Finite Baryon Density Medium

Problem Statement
While the internal structure of hadrons in a vacuum is a central pursuit in nuclear and particle physics, understanding the modification of these properties within a medium of finite baryon density is equally critical. This necessity arises from two primary considerations: first, experimental evidence (such as the EMC effect and pion decay constant measurements in bound atoms) confirms that nuclear binding alters partonic distributions; second, in relativistic heavy-ion collisions, hadrons formed at later evolution stages propagate through a medium with finite temperature and baryon density, making their decay and transport properties dependent on in-medium modifications. Although various theoretical frameworks have explored in-medium hadronic properties, there is a need for a consistent approach that links the medium-modified constituent quark mass to the specific valence quark distributions of the pion.

Methodology
The authors employ a hybrid framework combining the Light-Cone Quark Model (LCQM) with the two-flavor Nambu–Jona-Lasinio (NJL) model.

1. Medium-Modified Constituent Quark Mass (NJL Model):
   To determine the constituent quark mass ($m^*$) as a function of baryon density ($\rho_B$), the authors utilize the NJL model formulated for symmetric nuclear matter, following the approach of Bentz and Thomas. The model treats nucleons as quark-diquark bound states propagating in a background of constituent quark vacuum. The Lagrangian includes scalar, pseudoscalar, and vector interaction channels. The density-dependent mass is obtained by minimizing the grand canonical potential ($\Omega$), which accounts for vacuum quark loops, mean scalar and vector fields, and the nucleonic Fermi sea. The model parameters are fixed to reproduce vacuum observables (pion mass, decay constant, constituent quark mass) and the empirical saturation point of nuclear matter. Proper-time regularization is employed to handle ultraviolet and infrared divergences.

2. Pion Structure (LCQM):
   The pion is described as a superposition of Fock states, with the analysis restricted to the leading valence quark-antiquark ($|q\bar{q}\rangle$) component. The light-cone wave functions (LCWFs) are constructed using the constituent quark mass derived from the NJL model. The momentum-space wave function adopts the Brodsky-Huang-Lepage prescription, while the spin structure incorporates Melosh-Wigner rotation. The harmonic scale parameter ($\beta$) is fixed in the vacuum to reproduce the experimental pion decay constant ($f_\pi \approx 130$ MeV) and is assumed to remain unchanged in the nuclear medium. Consequently, the in-medium modifications are driven solely by the density dependence of $m^*$.

3. Calculations and Evolution:
   Using the density-dependent LCWFs, the authors calculate:

   • The Distribution Amplitude (DA) and the in-medium pion decay constant.
   • The Electromagnetic Form Factor (EMFF) and the charge radius.
   • The Parton Distribution Function (PDF) and its Mellin moments.

   The in-medium PDFs and Mellin moments, initially calculated at the model scale ($Q^2 = 0.20 \text{ GeV}^2$), are evolved to a perturbative scale ($Q^2 = 25 \text{ GeV}^2$) using Next-to-Leading Order (NLO) Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations.

Key Contributions and Results

• Constituent Quark Mass: The study confirms that the constituent quark mass decreases monotonically with increasing baryon density, eventually becoming negligible at asymptotically large densities. This behavior is consistent with the partial restoration of chiral symmetry in dense baryonic matter.
• Distribution Amplitude (DA): In the medium, the pion DA flattens (spreads out). Specifically, the distribution is suppressed at intermediate longitudinal momentum fractions ($x \sim 0.3\text{--}0.7$) and enhanced at low and high $x$. The in-medium decay constant ($f^*_\pi$) decreases monotonically with density, reaching approximately 95% of its vacuum value at nuclear saturation density ($\rho_0$).
• Electromagnetic Form Factor (EMFF): The in-medium EMFF is suppressed relative to the vacuum value across the momentum transfer range $Q^2 \approx 0\text{--}10 \text{ GeV}^2$. The charge radius of the pion increases rapidly with baryon density at lower densities ($\rho_B/\rho_0 \sim 0\text{--}2$) and slowly saturates around $0.6 \text{ fm}$ at higher densities ($\rho_B > 3\rho_0$).
• Parton Distribution Functions (PDFs): At the model scale, increasing baryon density shifts the peak of the valence quark PDF toward higher longitudinal momentum fractions ($x$), indicating an increased probability of finding the quark at higher momentum in a dense medium. However, after NLO DGLAP evolution to $25 \text{ GeV}^2$, the medium effects on the PDF become marginal, suggesting that dense medium effects are more pronounced in the non-perturbative regime than in the perturbative regime.
• Mellin Moments: The moments $\langle \xi^n \rangle$ and the inverse moment $\langle x^{-1} \rangle$ increase with baryon density. The total valence quark momentum fraction ($2\langle x \rangle$) at high scales ($Q^2 = 49 \text{ GeV}^2$) is found to be approximately 40%, consistent with lattice QCD and global fit extractions, implying that gluons and sea quarks carry the remaining 60% of the momentum.

Significance
The paper claims to provide a self-consistent description of pion structure in nuclear matter by bridging the gap between effective field theories for dense matter (NJL) and light-cone phenomenology (LCQM). By explicitly linking the reduction of the constituent quark mass (a signature of chiral symmetry restoration) to the modification of the pion's valence quark distributions, the work offers a microscopic explanation for in-medium hadronic properties. The results, particularly the suppression of the EMFF and the modification of the DA, are compared with available experimental data and lattice QCD studies, showing consistency with vacuum benchmarks while quantifying the specific deviations induced by finite baryon density. The study highlights that while medium effects are significant at the non-perturbative scale, they diminish upon evolution to perturbative scales, a crucial insight for interpreting high-energy scattering data involving pions in nuclear environments.