Electromagnetic deflection effects in the integrated luminosity measurement at the CEPC

This paper quantifies the impact of electromagnetic deflection effects from incoming bunches on both initial and final state particles for integrated luminosity measurements at the CEPC's Z⁰ pole, while discussing their simulation-based characterization and potential experimental correction methods to achieve a relative precision of 10⁻⁴.

The Gist

Imagine the CEPC (Circular Electron Positron Collider) as a massive, ultra-precise racetrack where tiny particles—electrons and positrons—zoom around at nearly the speed of light. The scientists' goal is to smash these particles together to study the fundamental building blocks of the universe. To do this, they need to know exactly how many times the particles collide. This count is called integrated luminosity, and it's like a "scorecard" for the experiment. If the scorecard is off by even a tiny bit, the physics results could be wrong.

This paper is about a sneaky problem: invisible magnetic forces that mess up this scorecard.

The Setup: A Crowded Dance Floor

At the CEPC, particles aren't just single runners; they travel in tight, dense groups called "bunches." Imagine two lines of dancers (one line of electrons, one of positrons) rushing toward each other to meet in the middle. Because there are so many of them packed so tightly, they generate their own powerful electromagnetic fields, like a crowd of people pushing against each other.

The paper identifies two specific ways these "crowd pushes" ruin the measurement:

1. The "Head-Butt" Effect (EMD1)

The Analogy: Imagine two runners sprinting toward each other on a track. As they get close, they feel a magnetic pull from the other runner's group. This pull tugs them slightly off their straight path before they even meet.

• What happens: Instead of colliding head-on at a perfect angle, the runners are nudged slightly inward. This changes the angle of their collision.
• The Consequence: When they bounce off each other (creating new particles), those new particles fly off at slightly different angles than expected. The detector, which is like a camera trying to count these bounces, misses some of them because they flew just outside its "lens."
• The Fix: The authors suggest that if we can measure the exact angle of the collision very precisely (using a different type of particle collision called "di-muon production"), we can mathematically correct the scorecard. It's like realizing the runners were nudged, calculating how much they were nudged, and adjusting the final count accordingly.

2. The "Magnet Trap" Effect (EMD2)

The Analogy: Now imagine the runners have already collided and are bouncing away. As they fly off, they pass right next to the other group of runners (the ones they didn't hit, but who are still rushing by). The magnetic field of that passing group acts like a giant magnet, pulling the bouncing particles toward the center of the track.

• What happens: The particles get "focused" or squeezed toward the center line.
• The Consequence: The detector has a specific "window" (a safe zone) where it counts particles. If the magnetic pull squeezes the particles too hard, some of them get pushed out of the counting window, or they get pushed so close to the edge that the detector gets confused. This leads to a loss of count.
• The Status: This paper calculates exactly how many particles get lost this way (about 0.36% to 0.4%). However, the authors admit they don't have a perfect "fix" for this yet. They are currently working on a new method using Machine Learning (computer algorithms that learn patterns) to figure out how to correct for this loss in the future.

The Big Picture

The paper is essentially a "safety check." The scientists are saying:

1. We found a problem: The magnetic fields of the particle bunches will cause us to miss about 0.4% to 0.6% of our collision events.
2. Why it matters: The goal is to be accurate to within 0.01% (10⁻⁴). Missing 0.4% is 40 times too big an error!
3. How stable is it? They checked if changing the size or speed of the particle bunches would make the problem worse. They found that even if the bunches vary by 10%, the error doesn't get much worse, which is good news.
4. Other factors: They also looked at other things like radiation (particles losing energy like a car slowing down) and found these add a small amount of extra error, but the magnetic "nudges" and "traps" are the main culprits.

The Conclusion

This paper is the first time anyone has calculated these specific magnetic effects for the CEPC. It proves that while the effect is real and significant, it is understandable and quantifiable.

• For the first effect (the nudge), we can fix it by measuring the collision angle.
• For the second effect (the trap), we are currently developing a computer-based solution.

Without these corrections, the CEPC's "scorecard" would be wrong, potentially leading scientists to draw the wrong conclusions about the universe. With these corrections, the machine can achieve its goal of extreme precision.

Technical Summary

Technical Summary: Electromagnetic Deflection Effects in the Integrated Luminosity Measurement at the CEPC

Problem Statement
To achieve a relative precision of $10^{-4}$ in the integrated luminosity measurement at the $Z^0$ pole of the Circular Electron Positron Collider (CEPC), systematic effects must be rigorously quantified. The CEPC Conceptual Design Report (CDR) proposes increasing the number of particles per bunch from $8 \cdot 10^{10}$ to $14 \cdot 10^{10}$, resulting in intensified electromagnetic fields. These fields induce collective electromagnetic deflection (EMD) on both the initial state particles (electrons and positrons before collision) and the final state particles (Bhabha scattering products). If uncorrected, these deflections alter the four-momenta of the final state particles, modifying the count of Small-Angle Bhabha Scattering (SABS) events within the luminometer's fiducial volume and introducing a bias in the integrated luminosity ($L_{int}$) measurement.

Methodology
The study utilizes the GuineaPig C++ V.1.2.2 software to simulate beam-beam interactions, modeling the collective electromagnetic fields of the bunches. The simulation assumes Gaussian beam profiles with parameters updated from the CEPC post-CDR design (Table 1), including a crossing angle of 33 mrad.

• Initial State (EMD1): The simulation tracks the deflection of incoming particles as they traverse the collective field of the opposite bunch. The impact on the collision frame is analyzed by associating these modified initial states with SABS events generated by the BHLUMI V4.04 generator. The final state four-momenta are adjusted via event-by-event boosts and rotations to reflect the modified collision geometry.
• Final State (EMD2): Outgoing Bhabha electrons and positrons are tracked through the electromagnetic fields of the opposite-charge outgoing bunches to determine the deflection of their trajectories.
• Radiative Effects: The study incorporates Initial State Radiation (ISR), Final State Radiation (FSR), and beamstrahlung (BS) to assess their combined impact with EMD.
• Validation: Results are compared against standard GuineaPig outputs and estimates from other colliders (FCC-ee, ILC) to ensure intrinsic consistency. The stability of results against variations in beam parameters (±10%) and simulation slice counts is also tested.

Key Contributions and Results

1. EMD1 (Initial State Deflection):

   • Mechanism: The collective fields attract initial particles toward the $z$-axis, effectively reducing the crossing angle by approximately 140 $\mu$rad (70 $\mu$rad per beam).
   • Impact on SABS: This reduction does not bias the mean polar angle of the final state but causes a smearing (broadening) of the polar angle distribution with an RMS of ~83 $\mu$rad.
   • Luminosity Loss: Due to the steep dependence of the Bhabha cross-section on the polar angle ($d\sigma/d\theta \sim \theta^{-3}$), this broadening near the luminometer's inner acceptance edge results in a relative count loss of $\sim 4 \cdot 10^{-3}$.
   • Stability: Variations in bunch parameters (up to ±10%) induce relative deviations in the Bhabha count of no more than $(-5 \cdot 10^{-4}, +2 \cdot 10^{-4})$. The effect is most sensitive to the bunch length ($\sigma_z$) due to the large Piwinski angle regime.

2. EMD2 (Final State Deflection):

   • Mechanism: Outgoing particles are focused toward the $z$-axis by the fields of the opposite bunches. This produces a mean polar-angle bias rather than just smearing.
   • Magnitude: For particles at the inner edge of the luminometer (53 mrad), the mean deflection is 53.4 $\mu$rad.
   • Luminosity Loss: This focusing causes a relative count loss of $\sim 3.6 \cdot 10^{-3}$.
   • Stability: Similar to EMD1, the effect is relatively insensitive to ±10% beam parameter variations, with count deviations not exceeding $(-0.2, +0.8)$ per mille.

3. Radiative Effects:

   • The inclusion of ISR, FSR, and BS enhances the luminosity loss. ISR is the dominant radiative contributor, creating asymmetric low-energy tails that boost effective collision frames and increase acollinearity.
   • Combined with EMD2, the total relative luminosity loss increases to 3.8‰ (Table 2).

Significance and Proposed Corrections
This paper provides the first estimate of electromagnetic deflection effects at the CEPC $Z^0$ pole. The combined uncorrected loss of SABS counts is estimated at $\Delta L_{int}/L_{int} \lesssim 6 \cdot 10^{-3}$, which significantly exceeds the target precision of $10^{-4}$.

• Correction for EMD1: The authors propose that the crossing angle can be measured with a precision of ~1 $\mu$rad using central tracker reconstruction of di-muon production ($\sim 70$ pb$^{-1}$, approx. 10 minutes of run time). Once the crossing angle is measured, the EMD1-induced bias can be corrected. With this correction, residual uncertainties from bunch parameter variations are reduced to the $5 \cdot 10^{-4}$ level.
• Correction for EMD2: A fully established experimental correction method for EMD2 is not yet available. The authors note that dedicated studies using machine learning and probability regression are ongoing to develop a correction strategy.
• Non-Gaussian Beams: While the simulation assumes Gaussian profiles, the authors qualitatively discuss that non-Gaussian features (tails, core-halo structures) would likely introduce uncertainties of only a few times $10^{-4}$, similar to radiative effects.

Conclusion
The study concludes that while EMD1 and EMD2 are significant sources of systematic error, they are quantifiable. EMD1 can be corrected via precise crossing angle measurements, while EMD2 requires further methodological development. The paper emphasizes that without these corrections, the integrated luminosity measurement at CEPC would fail to meet its precision goals.