DNN predictions for pp reference $p_\mathrm{T}$ spectra at unmeasured $\sqrt{s}$

This paper presents a deep neural network-based approach trained on ALICE data from LHC Runs 1 and 2 to interpolate and extrapolate proton-proton transverse-momentum spectra for unmeasured center-of-mass energies relevant to LHC Run 3 and beyond.

The Gist

Imagine you are trying to understand what happens when two giant, super-hot fireballs collide. In the world of particle physics, these are heavy-ion collisions that create a "soup" of fundamental particles called Quark-Gluon Plasma. To understand this soup, scientists need a control group: they need to know what happens when two simple particles (protons) collide under the exact same conditions, but without the "soup" forming. This is called a proton-proton (pp) reference.

The problem is that the Large Hadron Collider (LHC) is a machine that can be tuned to different energy levels. Sometimes, scientists run experiments at an energy level where they have measured the proton-proton collisions. Other times, they run at a new, unmeasured energy level. When they don't have a direct measurement for that specific energy, they have to guess what the proton-proton data would look like.

Traditionally, scientists guessed using two methods:

1. The Theoretical Guess: Using complex math formulas (like pQCD) that work well for very fast particles but get shaky for medium-speed ones.
2. The "Connect the Dots" Guess: Drawing a smooth line between two existing measurements. This works if you assume the line follows a specific, simple shape (like a straight line or a curve), but the real data might be wiggly and complex.

The New Solution: A "Smart Predictor"
This paper introduces a new way to make that guess using a Deep Neural Network (DNN). Think of this DNN as a super-smart student who has studied a massive textbook of proton collision data.

• The Training: The student (the DNN) was fed data from the ALICE experiment at the LHC, covering five different energy levels (2.76, 5.02, 7, 8, and 13 TeV). It learned the patterns of how particle production changes as the energy changes.
• The Trick: Instead of just memorizing the numbers, the student learned the shape of the data. The researchers taught the student to look at the data in a special way (using logarithms) so that the huge differences in particle counts didn't confuse it.
• The Test: Before using it on real data, the team tested the student with "fake" data generated by two different computer simulations (PYTHIA and EPOS LHC). The student performed excellently, accurately predicting data for energies it had never seen before, both lower and higher than what it studied.

What the Student Can Do Now
Once the student proved it was reliable, the team trained it on the real ALICE data. Now, the DNN can act as a universal translator for energy levels.

• Filling the Gaps: If scientists run an experiment at 9.62 TeV (a new energy), the DNN can predict exactly what the proton-proton reference should look like, even though no one has measured it directly.
• The "Ratio" Magic: To make these predictions useful, the DNN doesn't just guess the raw numbers; it calculates the ratio between a known energy (like 5.02 TeV) and the new energy. This is like saying, "If the collision at Energy A produces 100 particles, Energy B will produce 120," regardless of the total size of the experiment.
• Comparison: The paper shows that this "Smart Predictor" agrees with the best theoretical math at high speeds, matches the simple "connect the dots" methods at low speeds, and bridges the gap in the middle where other methods struggle.

Why It Matters
With this tool, scientists can now calculate the "Nuclear Modification Factor" ($R_{AA}$) for new experiments (like those in LHC Run 3) without waiting years to get a direct proton-proton measurement. It provides a continuous, smooth map of particle behavior across a wide range of energies, removing the need to assume the data follows a specific, rigid mathematical shape.

In short, the paper presents a machine learning tool that learns from past proton collisions to accurately predict what will happen in future collisions at energies we haven't measured yet, acting as a reliable reference for studying the hottest matter in the universe.

Technical Summary

Technical Summary: DNN Predictions for pp Reference $p_T$ Spectra at Unmeasured $\sqrt{s}$

Problem Statement
In the study of Quark-Gluon Plasma (QGP) via high-energy heavy-ion collisions, the nuclear modification factor ($R_{AA}$) is a critical observable. Its calculation requires a baseline proton–proton (pp) reference measurement at the same center-of-mass energy per nucleon pair ($\sqrt{s_{NN}}$) as the heavy-ion collision. However, suitable pp measurements are often unavailable for specific energies relevant to current or future heavy-ion runs. Traditionally, these references are constructed by scaling existing spectra using theoretical energy dependencies (reliable only at high $p_T$) or by interpolating between experimental points using assumed functional forms (e.g., power laws or $x_T$-scaling). These classical approaches rely on specific assumptions about the energy dependence of particle production, which may not hold across all kinematic regions, particularly at intermediate $p_T$ or for unmeasured energies.

Methodology
This work presents a data-driven approach using Deep Neural Networks (DNNs) to interpolate and extrapolate inclusive charged-particle transverse-momentum ($p_T$) spectra to unmeasured collision energies without assuming a specific functional form for the energy dependence.

• Dataset and Preprocessing: The model is trained on ALICE data from LHC Runs 1 and 2, covering five collision energies ($\sqrt{s} = 2.76, 5.02, 7, 8, \text{and } 13 \text{ TeV}$). The dataset comprises 46 $dN/dp_T$ values per energy within $0.15 < p_T < 10 \text{ GeV}/c$. To optimize performance, input variables ($p_T, \sqrt{s}$) and outputs ($dN/dp_T$) undergo logarithmic transformation. An additional $1/p_T$ scaling is applied to outputs to facilitate low-$p_T$ extrapolation. Systematic uncertainties are accounted for by varying data points 500 times within their bounds.
• Model Architecture and Training: A fully connected DNN implemented in TensorFlow is trained using the Nadam optimizer to minimize Mean Absolute Error (MAE). The architecture is determined via a hyperparameter scan (varying layers, nodes, activation functions, initializers, learning rates, and batch sizes) using Bayesian optimization. The scan utilizes PYTHIA (Monash 2013 tune) simulated data to identify the optimal architecture.
• Uncertainty Estimation: Total uncertainty is derived from two sources:
  1. Aleatoric uncertainty: Quantified by training an ensemble of 20 models with identical architecture but different initialization values.
  2. Epistemic uncertainty: Quantified by an ensemble of five models with different architectures (the top five from the hyperparameter scan).
     The total uncertainty is the quadratic sum of these two components.
• Validation and Application: The architecture selected via PYTHIA is validated against an independent dataset of EPOS LHC simulations. The final "ALICE-based DNN" is then trained on actual ALICE data. To improve high-$p_T$ extrapolation, the training set is extended with power-law parametrized data up to $p_T = 50 \text{ GeV}/c$.

Key Results

• Interpolation and Extrapolation: The DNN achieves an excellent description of the training data and precise interpolation within the measured energy range. Extrapolation performance degrades as the energy difference from the training data increases; deviations are most significant at the lower unmeasured energy ($\sqrt{s} = 1.5 \text{ TeV}$) and at high $p_T$ ($> 10 \text{ GeV}/c$).
• Comparison with Other Methods:
  • At low $p_T$, the DNN aligns well with PYTHIA simulations and power-law interpolations.
  • At intermediate $p_T$, the DNN remains consistent with power-law approaches up to $\approx 5 \text{ GeV}/c$.
  • At high $p_T$, the DNN predictions align with NLO pQCD calculations and $x_T$-based interpolations over the overlap region ($3 < p_T < 50 \text{ GeV}/c$).
  • Unlike power-law interpolations, which show fluctuations due to interval-by-interval fitting, the DNN provides a smooth, continuous prediction.
• Construction of Reference Spectra: The authors demonstrate the construction of pp reference $p_T$ spectra for LHC Run 3 and beyond ($\sqrt{s} = 5.36, 6.37, 9.62, \text{and } 13.6 \text{ TeV}$). This is achieved by scaling a baseline measurement ($\sqrt{s} = 5.02 \text{ TeV}$) using the ratio of DNN predictions at the target and baseline energies, corrected for the ratio of total inelastic cross sections. This enables the calculation of $R_{AA}$ for collisions where no direct pp measurement exists, such as pO collisions at $\sqrt{s} = 9.62 \text{ TeV}$.

Significance and Claims
The paper claims that the DNN-based method offers a robust, assumption-free alternative to traditional theory-driven or functional interpolation methods for constructing pp reference spectra. By avoiding assumptions about the underlying physical processes governing the energy dependence of spectra, the model provides continuous predictions across a wide $p_T$ range ($0.10 - 50 \text{ GeV}/c$) and energy range ($1.5 - 27 \text{ TeV}$). The authors note that while the model was trained on inclusive charged particles, the approach is extendable to individual particle species. The method successfully bridges the gap between existing measurements and the energy requirements of future heavy-ion programs, facilitating the analysis of QGP properties in upcoming LHC runs.