Evaluation of the beam-induced depolarization of the HJET target at the EIC

This paper quantitatively evaluates beam-induced depolarization of the Polarized Atomic Hydrogen Gas Jet Target (HJET) at the future Electron-Ion Collider (EIC) and concludes that, even with significantly higher beam currents than at RHIC, the resulting depolarization is negligibly small ($\lesssim 0.01\%$) and well within the accuracy requirements for EIC polarization measurements.

The Gist

Imagine you are trying to keep a group of tiny, spinning tops (hydrogen atoms) perfectly aligned in a specific direction. This is the job of a "target" in a giant particle accelerator called the Electron-Ion Collider (EIC). Scientists use these spinning tops to help measure the spin of a high-speed proton beam, much like using a compass to check the direction of a wind.

However, there is a problem. The proton beam zipping past these spinning tops is not a smooth, steady stream; it's more like a train of very fast, very short carriages (bunches) passing by. As these carriages zoom past, they create a wiggling magnetic field, like a rapidly shaking magnet.

The Big Fear
Some scientists were worried that this "wiggling magnet" from the proton train would knock the spinning tops out of alignment, causing them to lose their polarization (their "spin"). If this happened, the measurements would be wrong. A previous study suggested this loss of alignment would be huge, potentially ruining the experiment.

The New Investigation
This paper is like a detailed physics detective story. The author, A. A. Poblaguev, decided to re-run the numbers using a more precise, step-by-step simulation of how a single hydrogen atom moves through this chaotic magnetic environment. He treated the atom as a four-level system (like a four-story building where the atom can live on different floors) and tracked exactly how the shaking magnetic field from the proton beam tried to push the atom from one floor to another.

The Findings: The Tops Stay Put
The results of this new, careful calculation are very reassuring:

1. The "Wiggle" is Too Weak: The magnetic shake from the proton beam is actually very weak at the specific frequencies needed to knock the atoms off course. It's like trying to knock a heavy door off its hinges by gently tapping it with a feather. The tap just isn't strong enough.
2. The "Resonance" is Rare: For the atoms to get knocked over, the shaking would need to match the exact rhythm of the atom's natural spin (a concept called resonance). The paper shows that even if the shaking matches the rhythm, the "feather tap" is so short and weak that the atom barely notices.
3. The Result: The amount of polarization lost is incredibly tiny—less than 0.01%. To put that in perspective, if you had a million spinning tops, fewer than 1,000 would be slightly nudged, and even then, the effect is so small it's practically invisible.

Why the Previous Study Was Wrong
The paper explains that the earlier study that predicted a disaster made a math error. They essentially counted the total "shaking" energy of the beam as if it were all happening at the perfect frequency to knock the atoms over. In reality, the shaking is spread out over many different frequencies, and only a tiny, tiny fraction of it is at the "dangerous" frequency. It's like assuming that because a crowd is making a lot of noise, everyone is shouting the exact same word at the exact same time to break a glass. The author shows that the noise is actually a mix of many different sounds, so the glass (the atoms) remains safe.

What About Changes?
The author also checked what would happen if the proton beam got stronger or the "carriages" got shorter. Even if the beam parameters changed significantly (like increasing the current five times), the loss of alignment would still be well within the safety limits required for the experiment.

The Bottom Line
The paper concludes that the "wiggling magnet" from the proton beam at the future EIC will not significantly disturb the hydrogen target. The spinning tops will stay aligned, and the scientists can proceed with their measurements with high confidence. The fear of beam-induced depolarization is unfounded for the planned operating conditions.

Technical Summary

Technical Summary: Evaluation of the beam-induced depolarization of the HJET target at the EIC

Problem Statement
The Polarized Atomic Hydrogen Gas Jet Target (HJET) is a critical component for the absolute calibration of proton beam polarization at the Relativistic Heavy Ion Collider (RHIC) and is planned for the future Electron–Ion Collider (EIC). While RHIC operations have achieved systematic uncertainties of $\sim0.5\%$, the EIC requires even tighter control ($\sigma_{syst}/P \lesssim 1\%$). The EIC operating parameters differ significantly from RHIC, featuring a beam current increased by a factor of three, while bunch spacing and bunch length are reduced by nearly an order of magnitude. These changes raise concerns regarding beam-induced depolarization of the hydrogen jet target caused by the time-dependent magnetic fields of the circulating proton bunches. A recent study (Ref. [12]) claimed that unavoidable, large depolarization would occur at the EIC if the standard RHIC holding field of 120 mT were used. This paper presents an alternative, rigorous analysis to reassess these claims and evaluate the depolarization risk under nominal EIC conditions.

Methodology
The analysis treats the ground-state hydrogen atom as a four-level hyperfine system governed by the Breit-Rabi Hamiltonian in an external holding magnetic field ($B_{hold}$). The depolarization process is modeled as a time-dependent quantum mechanical evolution driven by the magnetic field generated by the proton bunches.

1. Field Characterization: The proton beam current is modeled as a train of Gaussian bunches. The resulting time-dependent magnetic field is decomposed into a Fourier series of harmonic components. The amplitude of these harmonics is modulated by a Gaussian spectral envelope determined by the bunch length ($\sigma_t$).
2. Quantum Evolution: The interaction between the hydrogen atom and the oscillating magnetic field is described using the Schrödinger equation in the interaction picture. The system of coupled differential equations for the probability amplitudes is solved numerically.
3. Trajectory Tracking: Unlike simplified estimates, this study performs numerical tracking of hydrogen atoms through the beam region. It accounts for the specific transverse density profile of the jet, the velocity distribution of the atoms, and the spatial variation of the beam-induced magnetic field as the atom traverses the scattering chamber.
4. Transition Analysis: The study evaluates both resonant transitions (where a beam harmonic frequency matches a hyperfine transition frequency) and non-resonant transitions. It distinguishes between the depolarization relevant to the Breit-Rabi Polarimeter (BRP) measurement ($w_{brp}$) and the effective depolarization relevant to the recoil spectrometer ($w_{jet}$).

Key Contributions

• Re-evaluation of Ref. [12]: The paper identifies a critical normalization error in the harmonic amplitudes used in Ref. [12]. It demonstrates that the previous study overestimated the Rabi frequency by a factor of $\sim60$ and the depolarization probability by a factor of $\sim3600$ by failing to account for the Gaussian spectral envelope suppression of high-frequency harmonics and by neglecting the large detuning ($\Delta\omega$) inherent in the spatial field variation.
• Comprehensive Numerical Benchmarking: The paper provides a complete numerical solution to the time-dependent quantum mechanical problem, benchmarking the results against analytical limits and previous benchmarks (e.g., LHCspin project).
• Parameter Stability Analysis: The study systematically evaluates the stability of the depolarization results against plausible variations in EIC beam parameters, including reduced bunch length, increased beam current, and elliptical beam profiles.
• Cross-Experiment Comparison: The methodology is applied to the HERMES experiment to verify the model against known experimental observations of depolarization, confirming the importance of low-velocity atomic components in that specific context.

Results

• Negligible Depolarization: For the nominal EIC flattop beam parameters and a holding field of 120 mT (or the optimized 129 mT), the calculated beam-induced depolarization of the jet target is $\lesssim 0.01\%$. This is well below the systematic uncertainty requirements for EIC polarimetry.
• Resonant Transitions: While certain holding field values can theoretically satisfy resonance conditions for specific hyperfine transitions (e.g., $|2\rangle \to |3\rangle$), the high-frequency harmonics required to drive these transitions are exponentially suppressed by the short bunch length. Consequently, even at exact resonance, the transition probabilities remain negligible ($< 10^{-4}$).
• Non-Resonant Transitions: The dominant contribution comes from non-resonant transitions ($|1\rangle \leftrightarrow |2\rangle$ and $|3\rangle \leftrightarrow |4\rangle$). However, the calculated probability for these transitions is also extremely small ($\sim 10^{-5}$ to $10^{-4}$).
• Parameter Robustness:
  • Reducing the bunch length to 0.1 ns (half the nominal value) increases depolarization but allows for safe operation if the holding field is maintained within a specific window ($129 \pm 0.35$ mT).
  • Increasing the beam current by a factor of five (or reducing the beam radius) increases depolarization by a factor of $\sim25$, yet the resulting value ($\sim 2 \times 10^{-3}$) remains within acceptable limits.
  • The elliptical nature of the EIC beam actually reduces the induced magnetic field compared to a round beam approximation, further lowering the depolarization risk.
• HERMES Context: The model successfully reproduces the magnitude of depolarization observed at HERMES when accounting for the specific low transverse velocities of atoms in a storage cell, validating the physical approach used.

Significance and Claims
The paper concludes that the claim of "unavoidably large depolarization" in Ref. [12] is incorrect due to methodological errors in calculating the driving field amplitudes. The rigorous numerical evaluation presented here demonstrates that beam-induced depolarization of the HJET target at the EIC is negligibly small ($\lesssim 0.01\%$) under nominal operating conditions. This result confirms that the HJET, utilizing the same fundamental principles as the successful RHIC program, can meet the stringent polarization accuracy requirements of the EIC without requiring fundamental changes to the target design or holding field strategy, provided the holding field is chosen to avoid specific narrow resonance windows. The study reinforces the reliability of the HJET as a primary polarimeter for the EIC hadron program.