Backward-angle electroproduction of $η'$ mesons off protons at $W=2.13~\text{GeV}$ and $Q^{2}=0.46~\left(\text{GeV}/c\right)^{2}$

This paper reports the first experimental measurement of backward-angle $η'$ meson electroproduction off protons at $W=2.13~\text{GeV}$ and $Q^{2}=0.46~(\text{GeV}/c)^{2}$, yielding a differential cross section significantly lower than real-photoproduction values and providing new constraints on nucleon resonance couplings within an isobar model framework.

The Gist

Imagine the atomic nucleus as a bustling construction site. Inside, tiny building blocks called quarks are glued together by a mysterious force to form particles like protons and neutrons. For decades, physicists have had a blueprint (the "Quark Model") that explains the basic houses (ground-state particles) on this site. But they know there are also "ghost houses"—excited states of protons that should exist according to the math, but no one has ever seen them clearly. These are the "missing resonances."

This paper is like a team of detectives using a high-powered flashlight to find one of these ghost houses. Here is the story of their investigation, explained simply.

1. The Mystery: The Heavy "Eta-Prime" Particle

The detectives decided to look for a specific type of particle called the $\eta'$ (eta-prime) meson.

• The Analogy: Think of the $\eta'$ as a very heavy, exotic brick. It's much heavier than the standard bricks (like pions) usually used in these experiments. Because it's heavy, it requires a lot of energy to create.
• The Clue: This heavy brick has a special property: it only likes to hang out with protons that have a specific "spin" (a quantum property). If a proton is spinning the wrong way, the brick won't stick. This makes the $\eta'$ a perfect detective tool because if we see it, we know exactly which "ghost house" (resonance) it came from.

2. The Experiment: Shooting a "Virtual Bullet"

The team went to the Thomas Jefferson National Accelerator Facility (JLab), which is essentially a giant particle accelerator.

• The Setup: They fired a beam of high-speed electrons at a tank of hydrogen gas (which is just protons).
• The Trick: Usually, when an electron hits a proton, it bounces off. But sometimes, the electron acts like a gun firing a "virtual photon" (a flash of energy that exists only for a split second) at the proton.
• The Reaction: This virtual photon hits the proton, exciting it so much that it spits out the heavy $\eta'$ brick. The proton recoils (bounces back), and the $\eta'$ flies off.
• The Challenge: They didn't catch the $\eta'$ directly (it's too hard to see). Instead, they caught the recoiling proton and the scattered electron. By measuring exactly how these two flew away, they could calculate the "missing mass"—essentially, weighing the invisible $\eta'$ brick by seeing what was missing from the equation.

3. The Discovery: Finding the Ghost

After collecting millions of collisions, they looked at their data like a detective looking at a fingerprint.

• The Result: They saw a distinct "bump" (a peak) in their data exactly where the $\eta'$ brick should be. They had successfully created and measured the production of this heavy particle off a proton.
• The Surprise: They measured how often this happened (the "cross-section") and found it was surprisingly low—about one-sixth of what happens when you hit the proton with a real photon (light) instead of a virtual one.
• The Metaphor: Imagine trying to knock a heavy door open. If you hit it with a solid hammer (real photon), it swings wide. If you hit it with a "ghost hammer" (virtual photon) that has a bit of "wiggle room" (momentum transfer), it barely opens. The experiment showed that this "ghost hammer" is much less effective at knocking out this specific heavy brick than a real one.

4. The Theory: Testing the Blueprint

The team didn't just stop at finding the brick; they wanted to know why it happened. They used a theoretical tool called an Isobar Model.

• The Analogy: Think of this model as a complex video game simulation. The physicists programmed the game with different lists of "ghost houses" (resonances) they thought might exist. They ran the simulation to see which list of ghosts would produce the exact same result as their real-life experiment.
• The Test: They tried four different lists (Model I, II, III, and IV).
  • Some lists included only the most famous ghosts.
  • Others included more obscure ones.
• The Verdict: The simulation that included a specific set of heavy, excited protons (around 2100 MeV of energy) matched the real data best. This suggests that these specific "ghost houses" are the ones responsible for creating the $\eta'$ brick.

5. Why Does This Matter?

This paper is a big deal for a few reasons:

1. First Time: This is the first time anyone has measured this specific reaction (creating an $\eta'$ with a virtual photon) at these angles. It's like opening a door to a room no one has ever entered.
2. New Rules: The fact that the reaction is so much weaker with virtual photons tells us something new about how the "glue" holding the quarks together behaves when it's being stretched or twisted.
3. Finding the Missing: By narrowing down which "ghost houses" are involved, they are helping physicists finally solve the mystery of the missing resonances. They are effectively saying, "We know exactly which room in the haunted mansion the ghost is hiding in."

Summary

In short, a team of scientists used a high-tech electron beam to "kick" a proton hard enough to spit out a heavy, exotic particle. By catching the debris, they proved the particle was there and measured how often it happened. They then used computer simulations to figure out which invisible, excited states of the proton were responsible. The result is a new, clearer map of the subatomic world, helping us understand the fundamental building blocks of our universe.

Technical Summary

1. Problem and Motivation

The study addresses the incomplete understanding of the excited baryon spectrum, specifically the "missing resonances" predicted by the quark model but not yet experimentally confirmed. The $\eta'$ meson is a unique probe for this research due to three key characteristics:

• Heavy Mass: Its mass ($\approx 958$ MeV/$c^2$) creates a high energy threshold, limiting the available phase space near threshold and making the reaction dominated by low-order partial waves.
• Quark Content: It contains an $s\bar{s}$ component, potentially revealing hidden strangeness in excited baryon resonances.
• Isospin Selection: The $\eta' p$ final state has total isospin $I=1/2$, meaning it couples exclusively to $N^*$ resonances ($I=1/2$) and forbids contributions from $\Delta$ resonances ($I=3/2$). This simplifies the extraction of resonance parameters.

While $\eta'$ photoproduction ($Q^2=0$) has been studied extensively, there is a significant lack of data in extreme angular regions (forward and backward) and no prior experimental data or theoretical calculations for $\eta'$ electroproduction ($Q^2 > 0$). This paper aims to fill this gap by measuring the electroproduction of $\eta'$ mesons at backward angles and comparing the results with a newly developed isobar model.

2. Methodology

Experimental Setup:

• Facility: The experiment was conducted at the Thomas Jefferson National Accelerator Facility (JLab) in Hall A during the JLab E12-17-003 run (Oct–Nov 2018).
• Beam: A continuous electron beam with energy $E_e = 4.318$ GeV.
• Target: A cryogenic liquid hydrogen ($^1$H) gas target.
• Spectrometers: Two High-Resolution Spectrometers (HRS-L and HRS-R) were used.
  • HRS-L: Detected scattered electrons ($e'$).
  • HRS-R: Detected recoil protons ($p_{scat}$).
• Kinematics: The setup was optimized for $\Lambda$ hypernuclei spectroscopy but incidentally allowed for $\eta'$ detection at ultra-backward angles relative to the virtual photon beam.
  • Center-of-mass energy: $W = 2.13$ GeV.
  • Four-momentum transfer: $Q^2 = 0.46$ (GeV/$c$)$^2$.
  • Scattering angle: $\cos \theta_{CM}^{\gamma^*\eta'} \approx -1$ (backward angle).

Analysis Techniques:

• Missing Mass Spectroscopy: The $\eta'$ meson was not detected directly. Instead, the reaction $e + p \to e' + p + \eta'$ was reconstructed by measuring the scattered electron and recoil proton. The missing mass ($m_X$) was calculated using energy and momentum conservation. A peak at $m_X \approx 958$ MeV/$c^2$ indicates $\eta'$ production.
• Event Selection:
  • Vertex Cut: Selected events originating from the hydrogen gas target (excluding aluminum target walls) using $Z$-vertex correlation between the two spectrometers.
  • Particle ID: Protons were identified in HRS-R using coincidence time ($T_{coin}$) measurements, distinguishing them from pions and kaons.
• Simulation: A Monte Carlo simulation (SIMC) was used to model the spectrometer acceptance, target geometry, and background processes (specifically multi-pion production). The simulation determined the solid angle and the functional form of the background under the missing mass peak.
• Cross Section Calculation: The differential cross section was derived using the One-Photon-Exchange Approximation (OPEA), separating the electron scattering flux from the virtual photoproduction cross section.

3. Key Contributions

1. First Measurement of $\eta'$ Electroproduction: This paper reports the first-ever experimental measurement of $\eta'$ meson electroproduction off protons.
2. Backward-Angle Data: It provides crucial data at backward angles ($\cos \theta \approx -1$), a region previously unmeasured for $\eta'$, which is sensitive to high-angular-momentum resonance states.
3. Theoretical Framework Extension: The authors developed a new isobar model calculation extending the BS model (originally for $K^+\Lambda$) to the $\eta'N$ channel. This model incorporates:
   • Effective Lagrangians with $s, t, u$-channel tree-level contributions.
   • Electromagnetic form factors (Extended Vector Meson Dominance model) to handle the $Q^2$ dependence.
   • Hadronic form factors (dipole/multi-dipole) to ensure consistency with photoproduction data.
4. Model Comparison: The study systematically compares the new data against four different resonance sets (Model I–IV) using the Akaike Information Criterion (AIC) to determine the most probable resonance contributions.

4. Results

Experimental Results:

• A clear peak corresponding to the $\eta'$ mass was observed in the missing mass spectrum.
• The differential cross section for the reaction $\gamma^* p \to \eta' p$ at $W=2.13$ GeV, $Q^2=0.46$ (GeV/$c$)$^2$, and backward angles is:
  $$\frac{d\sigma}{d\Omega}_{CM} = 4.4 \pm 0.8 \text{ (stat)} \pm 0.4 \text{ (sys)} \text{ nb/sr}$$
• Comparison with Photoproduction: This value is approximately one-sixth of the cross section measured for real photoproduction ($Q^2=0$) at similar energies and backward angles (LEPS data).

Theoretical Analysis:

• $Q^2$ Dependence: Theoretical calculations without electromagnetic form factors predicted a rapid increase in cross section with $Q^2$, which contradicts the data. Calculations including electromagnetic form factors successfully reproduced the observed suppression (quenched behavior) at $Q^2 > 0$.
• Resonance Models:
  • Model I (containing $N(1895)1/2^-$, $N(1900)3/2^+$, $N(2100)1/2^+$, $N(2120)3/2^-$) was statistically the best fit to existing photoproduction data based on AIC.
  • However, regarding the $W$-dependence of the new electroproduction data, Model II (which adds $N(1880)1/2^+$ and $N(2060)5/2^-$) provided the most favorable description of the rising trend near $W=2.13$ GeV.
  • The data suggests that resonances near $W \sim 2100$ MeV (specifically $N(2100)1/2^+$ in Model II and $N(2130)3/2^-$ in Model I) play a dominant role in the $\eta' p$ coupling.

5. Significance

• Validation of Theory: The successful application of the One-Photon-Exchange Approximation and the isobar model with electromagnetic form factors validates the theoretical framework for connecting real photoproduction ($Q^2=0$) and virtual electroproduction ($Q^2>0$).
• Constraints on Missing Resonances: The new data imposes strict constraints on the coupling strengths between the $\eta' p$ final state and nucleon resonances. It specifically highlights the importance of higher-mass resonances around 2100 MeV, which were previously less constrained.
• Future Directions: The results demonstrate that backward-angle electroproduction is a powerful tool for isolating specific resonance contributions. The paper calls for further measurements with wider $W$ coverage and improved statistical precision to fully resolve the nature of the "missing" resonances in the baryon spectrum.