Observations in 2022 - 2024

Effect of $\Delta p/p = 2\times10^{-4}$ ($\beta^\ast=0.3\,\mathrm{m}$) for beam 1 in 2024.

• ~10% change in measured $\beta$ beating after 2.5 weeks, with identical machine configuration ¹.
• Optics shift traced to energy error $\Delta p/p \approx 10^{-4}$.
• Average orbit corrector strength changes shifted the reference energy.
• Energy shift → modified quadrupole focusing, amplified at low $\beta^\ast$ by triplets.

Expectations for High Luminosity

Horizontal beta beating: effect of Beam 1;
$\Delta p/p = 2\times10^{-4}$ at $\beta^\ast=0.15\,\mathrm{m}$ (round).

Courtesy of S. Horney.²

• Simulations predict up to 50% beta beating.
• Current compensation technique mitigates by using a known baseline average of the orbit correctors.
• With HL-LHC, there is no baseline → we need to be able to measure energy offset directly.
• Operation requires <20% beating; current LHC routinely achieves ~10%.

Phase Response Method

Linearity Verification - ACD Simulation

• Phase response in simulation shows a highly linear behaviour.
• BPM noise at $1\times10^{-4}\, m$ has a negligible impact on the reconstruction.

2024 Measurement Results

We compared measurements taken in 2024 at 30cm before and after applying a trim in $\delta p$.

The response matrix reproduced the applied change to within 10%, confirming its accuracy.

2025 Results – 18 cm

Beam 1, 18 cm: Measurement vs Simulation

Phase errors
applied in each arc = mrad +5 +10 -15 -10 +15 +20 -20 -5 Phase errors
applied in each arc = mrad -2 -3 +1 0 -2 0 +3 +3

The phase errors were simulated by matching the main quadrupoles in each arc to the blue numbers shown in the plots.

Effect of Random Phase Shifts in Each Arc

Random phase errors were simulated by adding $\mathcal{N}(\mu=0,\,\sigma=3\,\text{mrad})$ shifts to the arc phase advance of the main quadrupoles — a level smaller than measured values.

2024 - 30 cm optics

2025 - 18 cm optics
More sensitive!

Random Phase Shifts in Each Arc - Single Case

A single case of the 2025 18cm optics illustrates that the phase errors introduce a systematic shift in the calculated dpp

Where is the 0 point?

DLMN - Gradient Descent for Accelerators

Parameters = magnet strengths k    Loss = Σ(BPM_meas - BPM_sim)²    Gradients = ∂Loss/∂k

• Gradient descent: adjusts parameters step-by-step in the direction that most reduces the loss function.
• Adam: Adaptive learning rate optimisation algorithm used to update model weights
• The network: We consider the magnet strengths as the network weights to be optimised

DLMN - Gradient Descent for Accelerators

Parameters = magnet strengths k    Loss = Σ_n(Δ)²    Gradients = ∂Loss/∂k

• Measured initial state seeds the simulation at the start BPM.
• Compare BPM measured vs simulated positions (Δ).
• Magnet strengths & $\delta p/p$ are the adjusted parameters to minimise the loss.
• Gradients are taken directly from MAD-NG.

Phase Sensitivity

Same random phase errors as in the phase-based method were applied.

18 cm optics - with random phase shifts
DLMN is far more robust, as it relies on BPM measurements (x,y) rather than derived phase advances.

Dipole Error Sensitivity

• Introduced random dipole errors (1 unit) to each main bending magnet in the ring.
• Ran the DLMN for 8 individual arcs to solve only for Δp/p (18 cm optics).
• Used BPM x,y data equivalent to 5 AC-Dipole kicks (or 1 ADT-AC-Dipole kick) measurement.

• Energy offset enters model via $x=\sqrt{2J\,\beta}\cos(\phi+\psi)+\eta\,\delta$.
• Known orbit-corrector kicks and unknown dipole errors both drive the orbit.
• Fit can trade $\delta$ against dipole, so per-arc best-fit Δp/p ≠ true value.
• Typical ring-level resolution is about ±6×10⁻⁵. Resolution from real data - to be seen.
• Systematic shift exists if the mean of the dipole errors is non-zero.

Effect of random dipole errors on DLMN energy reconstruction.

Effect of realistic dipole errors on phase-based energy reconstruction.

DLMN vs Phase-Based Method

Step 1/3 — Random phase shifts

DLMN: robust to random phase shifts.

Phase-based: noticeably more sensitive to random phase shifts.

DLMN vs Phase-Based Method

Step 2/3 — Random dipole errors

DLMN: increased error bars but trend largely maintained.

Phase-based: essentially unaffected by dipole errors.

DLMN vs Phase-Based Method

Step 3/3 — Phase + dipole combined

DLMN: still robust — slight spread from dipole/phase but trend largely maintained.

Phase-based: combined phase + dipole increases sensitivity and degrades linearity.