Imaging baryon number density within the proton

This paper utilizes Fourier-Bessel transformations of exclusive meson production cross sections from photon-proton collisions to reveal that the proton's baryon number is concentrated in a transverse radius of 0.33–0.53 fm, which is significantly smaller than the proton's charge and mass radii.

The Gist

Imagine the proton not as a solid, tiny marble, but as a bustling, invisible city. For decades, physicists have been trying to map this city, but they've been using different kinds of flashlights, and each flashlight reveals a different part of the landscape.

Some flashlights show us the electric charge (the "citizens" with a specific badge). Others show us the mass (the heavy buildings). And others show us the mechanical pressure (how hard the city pushes back if you try to squeeze it).

But there was one thing about this city that remained a mystery: Where exactly is the "Baryon Number" living?

In the world of particle physics, "Baryon Number" is like the city's official census count. It's the fundamental rule that says, "This is a proton; it must contain three specific 'valence' quarks." The big question was: Is this census count spread evenly across the whole city, or is it huddled tightly in the very center?

The Experiment: A Game of Cosmic Pinball

To answer this, the researchers didn't just look at the proton; they played a game of high-speed pinball with it.

They fired photons (particles of light) at protons. Usually, when a photon hits a proton, it bounces off gently, like a ball hitting a wall and rolling away slowly. This is called "forward production," and it tells us about the outer edges of the proton.

But the researchers were looking for something different. They waited for a rare, violent collision where the photon hits the proton so hard that the proton essentially recoils backward, like a billiard ball hit squarely in the center. This is called "backward production."

The Analogy:
Imagine a large, fluffy cloud (the proton) floating in the air.

• If you throw a gentle breeze at it (forward production), the whole cloud wobbles. You learn about the cloud's overall size.
• If you throw a heavy rock at the exact center of the cloud (backward production), only the dense, heavy core at the center gets knocked backward. The fluffy edges just ripple.

By studying these "backward recoils," the scientists were effectively shining a spotlight specifically on the dense core where the baryon number (the three valence quarks) lives.

The Method: Turning Speed into a Map

The team looked at four different types of "bounces" (producing different particles like pions and rho mesons). They measured how fast these particles were flying sideways (transverse momentum).

Here is the magic trick they used:

1. The Speed Map: They noticed that the faster the particles flew sideways, the smaller the area they came from.
2. The Fourier Transform: This is a mathematical tool (like a special lens) that takes a list of speeds and turns it into a picture of size. It's like taking the sound of a drumbeat and figuring out the size of the drum just by listening to the pitch.

They used this math to create a 2D map of the proton's interior, specifically looking for the "Baryon Number" zone.

The Discovery: The Core is Tiny!

The results were surprising and clear:

• The "Whole" Proton: When you measure the proton's electric charge or its mass, it looks like a fuzzy ball with a radius of about 0.7 to 0.8 femtometers (a femtometer is a quadrillionth of a meter).
• The "Baryon" Core: When they looked specifically for the baryon number (the three valence quarks), they found it was confined to a much smaller, tighter zone. The radius was only 0.33 to 0.53 femtometers.

The Metaphor:
Think of the proton as a dandelion seed head.

• The charge and mass are like the entire fluffy white sphere. It looks big and spread out.
• The baryon number is like the tiny, hard seed in the very center.

The study proves that while the proton's "aura" (charge and mass) spreads out widely, the actual "identity" of the proton (the baryon number) is packed tightly into a small, dense core in the middle.

Why Does This Matter?

This helps us understand the "glue" that holds the universe together.

• We know that protons are made of quarks and gluons.
• The study suggests that the gluons (the glue) and the "sea" of extra quarks spread out to make the proton look big.
• But the three "star" quarks that give the proton its identity are huddled together in the center.

It's like realizing that in a crowded party (the proton), the host (the baryon number) is standing in the center of the room, while the guests (gluons and sea quarks) are mingling all the way out to the walls.

The Future

The researchers are excited because this is just the beginning. They hope that future machines, like the Electron-Ion Collider, will act like high-definition cameras to take even sharper pictures of this tiny core, helping us understand exactly how nature builds the building blocks of matter.

In short: The proton is bigger on the outside than we thought, but its "soul" (baryon number) is packed into a surprisingly small, dense heart.

Technical Summary

Here is a detailed technical summary of the paper "Imaging baryon number density within the proton" by Klein et al.

1. Problem Statement

The spatial extent of the proton is a fundamental quantity in nuclear physics, yet its measured size varies significantly depending on the physical characteristic being probed:

• Charge radius: Sensitive to quark distributions ($\sim 0.84$ fm).
• Mass/Mechanical radius: Sensitive to energy/mass distributions and gluons ($\sim 0.9$ fm).
• Baryon number distribution: Historically unconstrained. It remains an open question whether baryon number (carried by net valence quarks) is uniformly distributed throughout the proton volume or concentrated in a specific region (e.g., the center).

While Quantum Chromodynamics (QCD) suggests baryon number is carried by three valence quarks, non-perturbative effects involving gluon fields (specifically the "baryon junction" model) suggest the baryon number carrier might be a distinct configuration localized differently than the charge or mass distributions.

2. Methodology

The authors investigate the transverse spatial distribution of baryon number by analyzing exclusive backward meson photoproduction reactions ($\gamma p \to p M$ or $\gamma p \to n M$).

Reaction Channels

The study focuses on four exclusive channels:

1. $\gamma p \to p \rho^0$
2. $\gamma p \to p \omega$
3. $\gamma p \to n \pi^+$
4. $\gamma p \to p \pi^0$

Theoretical Framework

• Backward vs. Forward Production: Unlike forward production (dominated by Pomeron exchange/gluons, small momentum transfer $t$), backward production involves large momentum transfer to the baryon (large $|u|$, small $|t|$).
• Regge Theory & TDAs: Backward production is modeled using $u$-channel Regge exchanges or Transition Distribution Amplitudes (TDAs). In these processes, the momentum transfer probes the target's internal structure at high $x$ (Bjorken scaling variable), potentially isolating the baryon number carrier.
• Fourier-Bessel Transform: To extract the spatial distribution, the authors analyze the transverse momentum ($p_T$) dependence of the differential cross-section ($d\sigma/dX$, where $X=u$ for backward and $t$ for forward).
  • They assume the scattering amplitude $M$ depends primarily on $p_T$.
  • They perform a 2D Fourier-Bessel transform to convert momentum space to impact parameter space ($b$):
    $$F(b) = \int_0^\infty p_T dp_T J_0(bp_T) \sqrt{\frac{d\sigma}{dX}}$$
  • The resulting $F(b)$ represents the spatial distribution of the scattering source.

Data Handling & Systematics

• Finite Range Effects: Experimental data covers a finite range of $p_T$, while the Fourier transform requires integration from $0$to$\infty$. The authors employ two methods to handle this "windowing" effect:
  1. No Extrapolation: Treats the unmeasured tail as a systematic uncertainty (incompleteness error).
  2. Extrapolation: Fits the high-$p_T$ tail with an exponential function and extrapolates to the kinematic limit ($p_{T, max}^{kin}$).
• Metric: The size of the distribution is quantified by the Half-Width at Half-Maximum (HWHM) of the $F(b)$ distribution, which is then converted to a 2D RMS radius for comparison with other measurements.

3. Key Contributions

• First Direct Imaging of Baryon Number: This work provides the first experimental constraints on the transverse spatial distribution of baryon number within the proton.
• Methodological Innovation: It successfully applies Fourier-Bessel transforms to backward photoproduction data (specifically $u$-channel processes) to extract spatial radii, distinguishing them from standard forward ($t$-channel) measurements.
• Comparative Analysis: The study systematically compares backward production (sensitive to baryon number) with forward production (sensitive to gluons/charge) and other historical data (electron scattering, muonic hydrogen) to isolate the specific size of the baryon number carrier.

4. Results

The analysis yields distinct transverse radii for different proton characteristics:

• Baryon Number Radius (Backward Production):

  • The HWHM for $u$-channel processes (backward production) is consistently small, ranging from 0.28 to 0.45 fm.
  • Converted to RMS radii, the baryon number is confined to a transverse radius of 0.33 – 0.53 fm.
  • This size is largely independent of the specific meson produced ($\pi, \rho, \omega$) in the backward channel.

• Charge and Mass Radii (Forward Production & Other Methods):

  • Forward production ($t$-channel) involving gluon exchange (Pomeron) yields larger radii: 0.67 – 0.77 fm (for $\pi$) and 0.86 – 0.99 fm (for vector mesons like $\rho^0, J/\psi$).
  • Electron scattering and muonic hydrogen measurements yield charge radii of 0.69 – 0.72 fm (2D RMS).

• Key Finding: The baryon number is concentrated in the central region of the proton. The transverse radius of the baryon number distribution ($\sim 0.4$ fm) is significantly smaller than the proton's charge or mass radius ($\gtrsim 0.67$ fm).

5. Significance

• Validation of Baryon Junction Models: The result supports theoretical models (e.g., Kharzeev's baryon junction) where baryon number is carried by a localized configuration of gluon fields and quarks at the center of the proton, rather than being uniformly smeared across the entire proton volume.
• Resolution of the "Proton Size" Paradox: The paper clarifies that the "size" of a proton is not a single scalar value but a property dependent on the probe. The proton appears "larger" when probed for charge or mass (gluons) and "smaller" when probed for net baryon number.
• Future Directions: The authors highlight the potential of the future Electron-Ion Collider (EIC) to perform systematic measurements of backward production across a wide range of energies and $Q^2$. This will allow for further differentiation between models (e.g., diquark exchange vs. baryon junction) and potentially probe heavier mesons ($\phi, J/\psi$) to test the universality of the baryon number localization.

In conclusion, the paper establishes that baryon number is confined to a compact core within the proton, distinct from the more diffuse distributions of charge and mass.