Advancing Hadronization Modeling with Deep Learning

Hadronization is an important process in high-energy particle physics experiments. It involves changing quarks and gluons, which are not directly observable, into hadrons, which are the particles we can detect. Despite its significance, we do not fully understand the physics behind hadronization. As a result, the models we currently use to simulate this process have many parameters that scientists adjust based on experimental data.

Traditionally, scientists have relied on various techniques to model hadronization. However, deep generative models, which are more flexible, could potentially improve the accuracy of these models. Previous studies have shown how to train neural networks to mimic specific hadronization models by using relationships between inputs and outputs from classical techniques. The challenge arises when we try to work with actual data, as we often do not have clear connections between the observed hadrons and the underlying partons.

In this study, we propose a new method for fitting a deep generative hadronization model in a more realistic scenario where we only have access to observed hadrons. Our approach uses a type of neural network called a Generative Adversarial Network (GAN), which can create new data similar to a given dataset while maintaining a set-based structure. We demonstrate that this setup can effectively match the hadronization patterns seen in existing models.

The relationship between theory and experiment is crucial in physics. Hadronization connects the fundamental aspects of particle physics with the observable particles we can measure. However, since we lack a complete understanding of hadronization, we rely on models that are inspired by physical principles but are also quite flexible. We aim to replace these crafted models with deep learning methods that could potentially increase precision and work well with advanced computing tools like GPUs.

There are two main hadronization models used widely in the field: the Cluster Model and the string model. The cluster model is the default choice for programs like Herwig and Sherpa, while the string model is typically used in Pythia. Previous research has shown that deep generative models can effectively imitate these models when given specific data. These works mark essential progress but represent early steps toward creating a comprehensive machine learning-based model for hadronization.

While earlier studies have demonstrated the capability of neural networks to replicate existing hadronization models, our long-term goal is to fit these models directly to experimental data. A significant challenge in this fitting process is that hadronization affects the partons locally, but we only observe non-local information about hadrons. In practice, we measure a collection of hadrons without clear associations to their original partons, making it difficult to generate hadrons from them.

To overcome these challenges, we utilize a unique GAN approach. In our model, the generator, which creates hadron data from parton information, does not require a straightforward way to define the probability of how data connects. The Discriminator, which distinguishes between real and generated hadron data, can work at a different level, allowing us to impose rules on how to group hadrons based on their sets.

We embed the GAN-based hadronization model into a comprehensive fitting framework. Our generator takes individual clusters of partons as input and produces pairs of hadrons. The resulting pairs are then transformed into a format suitable for the laboratory frame. We specifically focus on the cluster model and Pions to tackle the fitting challenges. Future work may explore more complex decays and additional hadron types.

This paper is structured to first introduce the fitting framework, followed by numerical examples, and concludes with outlooks on future research.

#Statistical Approach

The objective of our model is to learn how to generate hadrons based on the characteristics of clusters. We define a generator function that maps these cluster properties onto hadron properties. Instead of defining hadrons one by one, our generator can output the angles of the produced hadrons in a specific frame.

In our initial approach, we matched hadrons directly with clusters. However, in real data analysis, we can only access individual hadron properties. To adapt our model for actual data, we modify the discriminator function to operate on sets of observed hadron properties from the same event. This change allows for a more generalized approach that can handle different lengths of hadron sets without losing information.

The implementation of our model uses deep neural networks for both the generator and discriminator. It operates effectively with PyTorch, allowing us to train our model using modern machine learning techniques. The generator is designed to produce outputs that fall within a specific range. We also ensure that the training data collected is suitable for fitting our models, focusing on events from high-energy collisions.

In our analysis, we use data from a widely known simulation tool called Herwig. This tool allows us to generate events and collect information required for our training dataset. We narrow our focus to specific types of decays for simplicity, initially examining cases with only two pions.

Additionally, we create variations in our training datasets to test the model under different conditions. By adjusting parameters in our model, we can evaluate the robustness of our fitting process.

#Fitted Models

Our training process exhibits expected trends where the results improve over time. We monitor losses for both the discriminator and generator, ensuring they converge toward appropriate values. To validate our model's performance, we employ metrics like the Wasserstein distance, which measures how well the generated data matches the real data over time.

We visualize the direct inputs and outputs of our model, demonstrating how the generator produces outputs in the expected spherical coordinates. Initially trained models show clear differences from untrained models, indicating successful learning. The outcomes align well with the known distributions from the simulation, signaling that our GAN effectively captures the underlying physics of hadronization.

Moreover, we investigate derived quantities that can be measured. This includes counting the number of produced hadrons and evaluating the energy distribution among them. Since our model’s focus is on specific types of decays, we anticipate the number of hadrons to align with this expectation, showing how well our model can mimic real-world conditions.

#Advantages of the Approach

One of the main strengths of our fitting protocol is its ability to work with large sets of unbinned inputs. Unlike traditional methods, which often require binned histograms, our model can accommodate complex, high-dimensional data. This flexibility could allow for better representation of data without arbitrary limitations from binning choices.

To quantify the information captured by our model, we use auxiliary classifiers. By comparing the model's ability to distinguish between variants of our data, we can evaluate how much information the discriminator can leverage. The results show that our neural network can extract significantly more useful information than simpler metrics derived from the observed data alone.

#Conclusions and Future Directions

In this work, we have introduced a method for fitting deep generative models for hadronization based on data. We tackled the challenge of missing connections between partons and hadrons by implementing a two-level GAN setup. Our model shows promise in fitting variations of a widely used hadronization model, demonstrating its capability to reproduce known results.

As part of our future work, we aim to expand the model's capabilities. Currently, we focus on specific types of particles, like pions, but future models should encompass a broader spectrum of hadrons. We also recognize the need for our models to account for complex decays, which may involve combinations of more than two hadrons.

A complete model might require us to rethink our approach, potentially modeling the direct mappings from partons to hadrons, moving beyond the constraints of the cluster model. We also need to consider how best to utilize data for fitting, possibly adapting current methods to suit our expanded parameter space.

Ultimately, our program is well-founded and motivated. Enhanced machine learning techniques may lead to more precise measurements in hadronization, contributing to the ongoing research effort in nuclear physics. By refining these models, we hope to gain deeper insights into the processes that govern particle behavior in high-energy environments.