Efficient calculation of exclusive diffractive cross sections at the EIC and LHeC with the Sartre event generator

This paper introduces a significantly optimized numerical calculation for the Sartre event generator that accelerates lookup table production by 3–4 orders of magnitude while eliminating numerical instabilities, thereby enabling efficient simulation of exclusive diffractive cross sections for diverse processes at the EIC, LHeC, RHIC, and LHC.

The Gist

Imagine you are trying to predict the outcome of a very complex game of billiards, but instead of a table, you have a giant, fuzzy cloud of tiny particles (protons and nuclei) smashing into each other at nearly the speed of light. Physicists want to know exactly how these particles bounce off each other, especially when they don't break apart but instead "glance" off one another in a specific way called exclusive diffraction.

To do this, they use a computer program called Sartre. Think of Sartre as a super-advanced weather forecast for particle collisions. It doesn't just guess; it calculates the odds of every possible outcome so scientists can simulate millions of "what-if" scenarios before they even turn on the real particle accelerators (like the EIC or LHeC).

Here is the problem the paper solves, explained simply:

The Old Problem: The "Library of Doom"

In the past, Sartre worked like a librarian trying to write a book for every single possible scenario.

• The Task: To make a prediction, the computer had to calculate a massive, 4-dimensional math problem (involving size, speed, angle, and position) for thousands of different situations.
• The Bottleneck: To get a smooth, accurate prediction, the computer had to repeat this calculation about 500 times for every single scenario to account for the random jiggling of particles inside the target.
• The Result: It took a supercomputer farm (a massive cluster of computers) years of non-stop work just to create the "lookup tables" (the pre-calculated answer keys) for one specific type of collision.
• The Glitch: Because the math involved rapidly shaking waves (like a guitar string vibrating), the computer often got confused, leading to "numerical glitches"—sudden, weird spikes in the data that made the predictions look broken.

The New Solution: The "Magic Shortcut"

The authors, Tobias Toll and his team, found a way to speed this up by 3,000 to 10,000 times. They didn't just make the computer faster; they changed how it did the math.

1. The Fourier Transform Trick (The "Recipe" Analogy)
Imagine you are trying to figure out the ingredients of a soup by tasting it. The old way was to taste the soup, guess the ingredients, taste it again, and repeat this thousands of times to get it right.
The new way is like realizing that the soup's flavor is actually a Fourier Transform of its ingredients. In math terms, this means the pattern of how particles scatter is directly related to a "mirror image" of their positions.

• Instead of calculating the answer for every single angle one by one, the new method uses a Fast Fourier Transform (FFT). This is like using a magic sieve that sorts all the answers at once.
• The Analogy: If the old method was walking through a forest to count every tree one by one, the new method is taking a helicopter photo and counting them all in a single second.

2. Pre-Cooking the Ingredients
The team realized that many parts of the calculation were the same for every single scenario.

• The Analogy: Imagine baking 1,000 cakes. The old way was to mix the flour, eggs, and sugar from scratch for every single cake. The new way is to mix a giant batch of batter once, and then just pour it into different cake pans. This saves a massive amount of time.

3. Smoothing Out the Bumps
Because the new method calculates the data on a perfect mathematical grid, it naturally avoids the "glitches" and spikes that plagued the old method. The data comes out smooth and clean, like a perfectly paved road instead of a bumpy dirt track.

The Result: From Supercomputers to Laptops

Before this paper, you needed a "computing farm" (a room full of servers) and years of time to generate the data for a specific experiment.

• Now: A single laptop can generate the same data in a few hours.
• Why it matters: This means scientists can now instantly create predictions for any combination of particles they want to study. They don't have to choose which experiments to simulate anymore; they can simulate them all.

What They Predicted

Using this new, super-fast version of Sartre, the authors made fresh predictions for upcoming experiments:

• At the EIC (Electron-Ion Collider): They showed how light particles (like rho mesons) behave, proving that the new tool can handle the complex "non-linear" physics where particles interact strongly.
• At the LHeC (Large Hadron-electron Collider): They predicted how heavy particles (like the Upsilon meson) scatter. Because these heavy particles are tiny, they act like high-resolution microscopes, allowing scientists to see the "hotspots" (tiny sub-structures) inside protons that were previously invisible.

Summary

The paper presents a massive upgrade to a particle physics tool. By using a mathematical shortcut (Fourier transforms) and smart pre-calculation, they turned a process that took years on a supercomputer into a process that takes hours on a laptop. This removes the "bottleneck," allowing physicists to explore every possible scenario for future particle collisions without waiting for a computer farm to finish its work.

Technical Summary

Technical Summary: Efficient Calculation of Exclusive Diffractive Cross Sections with the Sartre Event Generator

Problem Statement
Exclusive diffraction at small $x_{IP}$ is a critical probe for understanding gluon saturation and spatial gluon substructure in future Electron-Ion Collider (EIC) and Large Hadron-electron Collider (LHeC) experiments. The Sartre event generator simulates these processes using the color dipole model, requiring the calculation of the first and second moments of the interaction amplitude to generate coherent and incoherent events, respectively.

Historically, the production of lookup tables for these moments has been a severe computational bottleneck. The calculation involves averaging four-dimensional integrals over approximately 500 nucleon configurations for each kinematic point. This process previously required "CPU-years" per combination of initial nuclear target and final state vector meson, necessitating large computing farms. Furthermore, the integrands contain rapidly fluctuating functions (Bessel functions and oscillating exponentials) at large momentum transfer $|t|$, leading to slow convergence in standard adaptive integration routines (such as Cuhre) and resulting in numerical glitches and spikes in the generated tables.

Methodology
The authors present a new numerical implementation that fundamentally restructures the calculation of the dipole amplitude moments. The core innovation is recognizing that the impact parameter ($\vec{b}$) integral constitutes a Fourier transform to the momentum transfer ($\vec{\Delta}$) space.

1. Fourier Transform Utilization: Instead of performing a full 4D numerical integration for every specific $t$-bin, the authors transform the $\vec{b}$ integral into a Discrete Fourier Transform (DFT) or Fast Fourier Transform (FFT).

   • By utilizing the FFT, the algorithm generates a grid of $t$-values (reciprocal to the $\vec{b}$ grid) simultaneously. This reduces the problem from a 3D calculation ($Q^2, W^2, t$) to a 2D calculation ($Q^2, W^2$), where all $t$-bins are obtained directly from the transform.
   • The authors also employ the projection-slice theorem to reduce the 2D Fourier transform to a 1D transform, further optimizing the routine.
   • For azimuthally symmetric cases, a 1D Hankel transform is used.

2. Precomputation and Optimization:

   • Common factors in the calculation, such as the kernel $K_{T,L}$ (involving the photon-vector meson wave function overlap) and the opacity $\rho(x_{IP}, r)$, are precomputed for each kinematic point. This avoids redundant calculations across the 500 nucleon configurations.
   • The opacity $\rho$, which depends on $x_{IP}$, is tabulated with a log-linear interpolation scheme to account for the small $t$-dependence of $x_{IP}$, ensuring accuracy without excessive computational cost.

3. Implementation Options: The new Sartre version offers both DFT and FFT options. The DFT allows for arbitrary user-specified $t$-binning, while the FFT fixes $t$-values to the reciprocal grid, offering different trade-offs between flexibility and speed.

Key Results

• Efficiency Gains: The new method achieves a speed-up of 3 to 4 orders of magnitude compared to the previous Cuhre-based implementation. Table generation times have been reduced from "CPU-years" to less than one CPU-day. With standard parallelization, this allows table production on a single laptop rather than requiring computing farms.
• Numerical Stability: The new calculation eliminates the numerical glitches and spikes previously observed in the incoherent cross-section variance at large $|t|$, as the FFT approach does not suffer from the convergence issues of adaptive integration on rapidly oscillating integrands.
• Validation: Comparisons between the new FFT/DFT method and the previous Cuhre method show agreement within 1–5% across various kinematic points. The slight discrepancies at higher $|t|$ are attributed to the treatment of $x_{IP}$ dependence, which is corrected in the new implementation.
• Physics Predictions: Using the new generator, the authors provide novel predictions for exclusive $\rho$, $J/\psi$, and $\Upsilon$ production at the EIC and LHeC:
  • EIC: Predictions for $\rho$ production demonstrate sensitivity to non-linear saturation effects (IPsat vs. IPnosat) but show limited sensitivity to nucleon substructure due to the large dipole size of the $\rho$ meson.
  • EIC and LHeC: Predictions for heavy vector mesons ($J/\psi$ and $\Upsilon$) reveal that these processes are excellent probes for nucleon substructure (hotspots). The results show a clear separation between models with and without substructure at large $|t|$, particularly for $\Upsilon$ production at the high energies available at the LHeC.

Significance and Claims
The paper claims that this technical advancement removes the primary barrier to simulating the full landscape of exclusive diffractive processes. Previously, researchers had to make restrictive choices regarding which processes (combinations of targets and final states) to precompute tables for. With the new efficiency, Sartre can now generate lookup tables for any process of interest in a few hours.

This capability enables comprehensive studies of:

• Non-linear saturation effects (IPsat vs. IPnosat).
• Nucleon substructure (hotspots).
• Coherent and incoherent diffraction across the full kinematic range of the EIC and LHeC.

The authors emphasize that this shift moves the production of Sartre lookup tables from the domain of large-scale computing farms to the accessibility of individual researchers' workstations, facilitating more flexible and extensive phenomenological studies for upcoming collider experiments.