Systematic Errors in General Spin Precession in Storage Ring

This paper presents an analytical formulation for systematic errors in spin precession within storage rings, specifically addressing the requirements for measuring muon anomalous magnetic moments and electric dipole moments with high precision by determining precession frequencies up to second-order corrections.

The Gist

Imagine you are trying to measure the spin of a tiny, spinning top (a muon) as it races around a circular track at nearly the speed of light. This isn't just a game; scientists are doing this to find "new physics" that could explain mysteries the Standard Model of particle physics can't solve.

The paper by Takeshi Fukuyama is essentially a high-precision instruction manual for correcting the "wobble" in our measurements.

Here is the breakdown using simple analogies:

1. The Goal: Finding the Invisible

Scientists want to measure two tiny things about these spinning tops:

• The "G-2" (Anomalous Magnetic Moment): How much the top's magnetic personality differs from what we expect.
• The "EDM" (Electric Dipole Moment): A tiny separation of positive and negative charge inside the top. If we find this, it's a "smoking gun" for new physics.

To find these, they need to measure how fast the top spins (precesses) with extreme accuracy—down to 0.1 parts per million. That's like trying to measure the thickness of a human hair from the distance of a football field.

2. The Problem: The "Wobble" (Systematic Errors)

In the real world, the muons don't run in a perfect, flat circle like a train on a track. They are a "bunch" of particles that:

• Wiggle up and down (vertical oscillation).
• Wiggle side-to-side (radial oscillation).
• Tilt slightly as they move.

Think of a race car driver trying to stay on the center line of a track. Even if they try to drive perfectly straight, the car naturally bounces and sways. In the storage ring, these wiggles cause the magnetic and electric fields to look slightly different to the particle than they do to the scientists watching from the outside.

If you ignore these wiggles, your measurement of the spin speed will be slightly wrong. In the world of high-energy physics, "slightly wrong" means you miss the discovery entirely.

3. The Solution: The "Pitch Correction"

The paper focuses on a specific type of wobble called the "pitch correction."

The Analogy: Imagine a roller coaster car going around a loop. If the car tilts up and down as it goes around (like a pitch), the driver's view of the horizon changes.

• Old View: Scientists used a formula (by a physicist named Farley) that corrected for this tilt, but only for a very specific, simple setup.
• New View: Fukuyama says, "Real experiments are messier. The beams are wider, and there are electric fields involved."

Fukuyama developed a new, more general mathematical formula that accounts for:

1. Electric Fields: Unlike the old simple model, real storage rings use electric fields to focus the beam.
2. 3D Wiggles: The particles don't just wiggle up and down; they wiggle in all directions.
3. Higher Precision: He calculated the errors up to a very high level of detail (second-order effects), which is necessary to reach that 0.1 ppm accuracy.

4. The "Aha!" Moment

The paper addresses a debate in the physics community: Which mathematical formula is the right one to use for the observed spin?

Fukuyama proves that his new, complex formula is the correct one. He shows that if you simplify his formula to match the old, simple setup, it perfectly reproduces Farley's famous result. This gives scientists confidence that his new, more complex formula is also correct for the modern, complicated experiments (like the Muon g-2 experiment at Fermilab).

5. Why It Matters

Think of this paper as calibrating a microscope.

• Before this, scientists had a good lens, but they knew it had a slight distortion when looking at things that weren't perfectly flat.
• Fukuyama has figured out exactly how that distortion works when the object is wiggling in 3D and when electric fields are present.
• By applying his corrections, scientists can now look through the microscope and see the "true" spin of the muon without the blur of the wobble.

In summary: This paper provides the rigorous math needed to clean up the "noise" caused by the natural wiggling of particles in a storage ring. Without these corrections, the search for new physics would be like trying to hear a whisper in a hurricane; with them, the signal becomes clear.

Technical Summary

1. Problem Statement

The paper addresses the need for high-precision analytical estimation of systematic errors in the spin precession of particles (specifically muons) circulating in storage rings. This is critical for experiments like the muon $g-2$ and Electric Dipole Moment (EDM) projects (e.g., J-PARC, BNL-E821, FNAL).

• The Challenge: To measure anomalous magnetic moments ($g-2$) and EDMs with precision up to 0.1 ppm, the spin precession frequency must be determined up to the order of $O(\epsilon^2)$, where $\epsilon$ represents the small extensions of the particle's position and velocity direction ($O(10^{-3} - 10^{-4})$).
• The Controversy: There was an existing ambiguity regarding the definition of the "observed" spin precession frequency. Specifically, it was debated whether the frequency relative to the beam direction ($\Omega'$, Eq. 9) or the frequency relative to the laboratory reference orbit ($\Omega$, Eq. 10) was the correct observable for experimental setups involving betatron oscillations.
• Scope: The analysis considers particles with both anomalous magnetic dipole moments (MDM) and electric dipole moments (EDM), moving in general electromagnetic fields ($E \neq 0$) with non-zero beam spreads in both radial and vertical directions.

2. Methodology

The author employs a rigorous analytical approach based on the Thomas-Bargmann-Michel-Telegdi (BMT) equation and relativistic kinematics.

• Coordinate System: The analysis uses cylindrical coordinates $(x, y, z)$ relative to a reference orbit, where $x$ is the tangential direction, $y$ is radial, and $z$ is vertical. The beam profile is described by small deviations $y, z$ and their derivatives $y', z'$ (betatron oscillations).
• Approximation Scheme: The paper performs a perturbation expansion up to $O(\epsilon^2)$. It assumes the velocity magnitude is constant ($E \cdot v = 0$) but allows for non-zero electric fields (unlike previous work which assumed $E=0$).
• Derivation Steps:
  1. Kinematics: Derived the velocity vector $\mathbf{v}$ and its magnitude considering the curved reference orbit and betatron oscillations (Eq. 15-16).
  2. Field Expansion: Expanded the Lorentz force and electromagnetic fields ($\mathbf{B}, \mathbf{E}$) in terms of the small deviations $y, z, y', z'$ up to second order (Eq. 20-22).
  3. Spin Precession Formulation:
     • Calculated the spin precession angular velocity in the particle rest frame ($\Omega'$).
     • Calculated the spin precession angular velocity relative to the reference orbit ($\Omega$), which accounts for the rotation of the reference frame itself.
  4. Equations of Motion: Utilized Hill's equations for betatron oscillations to describe the time evolution of $y$ and $z$, ensuring consistency up to $O(\epsilon^2)$.
  5. Time Averaging: Applied time-averaging techniques to the oscillating terms to extract the systematic frequency shifts.

3. Key Contributions

• Resolution of the Definition Controversy: The paper definitively demonstrates that Eq. (10) (the frequency relative to the reference orbit, $\Omega$) is the correct observable for the experimental setup, not Eq. (9).
• Generalization of Farley's Pitch Correction: The author derives a general formula that reproduces the famous Farley pitch correction (a systematic error due to the vertical pitch angle of the beam) as a special case.
  • The derivation shows that the pitch correction is reduced by a factor of 2 due to the contributions of the transverse components ($\Omega_x, \Omega_y$) of the precession frequency.
• Inclusion of EDM and General Fields: Unlike previous analyses limited to specific magnetic focusing or $E=0$ cases, this formulation:
  • Explicitly includes the Electric Dipole Moment (EDM) parameter ($\eta$).
  • Handles general electric and magnetic field configurations (weak focusing).
  • Accounts for beam spreads in both radial ($y$) and vertical ($z$) directions simultaneously.

4. Results

• Analytical Formulation: The paper provides explicit expressions for the spin precession frequency $\Omega$ up to $O(\epsilon^2)$ (Eq. 23). The dominant terms include the standard $aB$ term, while correction terms involve products of the pitch angle ($z'$), radial displacement ($y$), and field gradients.
• Reproduction of Farley's Result: By applying the general formula to the specific conditions of Farley's setup (weak magnetic focusing, $E=0$), the author derives the time-averaged frequency shift:
  $$\langle \dot{\phi} \rangle = \Omega_0 (1 - C)$$
  where the correction factor $C$ confirms the pitch correction is halved ($-\frac{1}{4}\psi_0^2$ instead of $-\frac{1}{2}\psi_0^2$) when considering the full vector nature of the precession.
• Impact of Beam Profile: The analysis reveals that in realistic scenarios where the initial vertical position $z_0 \neq 0$, the cancellation of terms seen in idealized cases does not occur. This implies that systematic errors depend on the specific initial beam distribution ($y_0, z_0, y'_0, z'_0$).
• EDM Sensitivity: The formulation highlights that the main contribution of the EDM appears in the $e_2$ (radial) direction, necessitating precise control of radial beam spreads and electric field gradients.

5. Significance

• Precision Physics: This work provides the necessary theoretical framework to interpret data from next-generation muon $g-2$ and EDM experiments. Without these $O(\epsilon^2)$ corrections, systematic errors could exceed the target precision of 0.1 ppm, potentially masking or mimicking new physics signals.
• Experimental Design: The results guide the design of storage rings and beam injection systems, emphasizing the need to minimize beam spreads ($y, z$) and understand the interplay between magnetic focusing and electric fields.
• Theoretical Clarity: By resolving the ambiguity between $\Omega$ and $\Omega'$, the paper ensures that experimentalists and theorists are comparing the same physical quantities, preventing misinterpretation of frequency shifts caused by magnetic field gradients or $v \times E$ effects.

In summary, Fukuyama's paper extends the theoretical understanding of spin dynamics in storage rings, providing a robust, general analytical tool to calculate systematic errors up to the second order of beam spread, which is essential for the success of high-precision searches for physics beyond the Standard Model.