Normalizing Flows for Fragmentation andHadronization
Ahmed Youssef

Phil Ilten

Tony Menzo

Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
youssead@ucmail.uc.edu

Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
philten@cern.ch

Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
menzoad@mail.uc.edu

Stephen Mrenna

Manuel Szewc

Michael K. Wilkinson

Fermilab
Batavia, IL 60510
mrenna@fnal.gov

Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
manuelsz24@gmail.com

Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
michael.k.wilkinson@gmail.com

Jure Zupan
Dept. of Physics
University of Cincinnati
Cincinnati, Oh 45221
zupanje@ucmail.uc.edu

Abstract
Hadronization is an important step in Monte Carlo event generators, where quarks
and gluons are bound into physically observable hadrons. Previous work has
demonstrated first steps towards a machine-learning (ML) based simulation of
the hadronization process. However, the presented architectures are limited to
producing only pions as hadron emissions. In this work we use normalizing flows
to overcome this limitation. We use masked autoregressive flows as a generator for
the kinematic distributions in the hadronization pipeline. We condition normalizing
flows (NFs) on different hadron masses and initial configuration energies, whichallows for the emission of hadrons with arbitrary masses. The NF generated
kinematic distributions match the P YTHIA generated ones well. In this paper we
present our preliminary results.

1

Introduction

Particle collisions in collider experiments are simulated with Monte Carlo event generators, which
can be factorized into three steps: (i) generation of the hard process (the particle collision); (ii) parton
shower (the evolution from high energy to low energy, in which quarks and gluon are created); and(iii) hadronization (the combination of quarks and gluons to observable particles, hadrons). While thefirst two steps are perturbative in their nature, and thus under good theoretical control with significant
efforts devoted to improving the precision even further [1–4], the hadronization step is inherently
non-perturbative, i.e. it does not follow first principle and thus can not be described theoretically.
Therefore, phenomenological models with many free parameters are typically used instead, which
oftentimes do not cover the entire underlying physics well.
The two main models used in simulating hadronization are the Lund string model [5–7], where
partons are connected via QCD color strings with a linear potential, which are iteratively split into
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

hadrons, and the cluster model [8–10], where partons are pre-confined into proto-clusters, which thendecayed into hadrons via sequential two-body decays. Both models have limitations. The string model
requires over O(20) parameters to describe the hadronization, and has some challenges describing
baryon production. The cluster model has fewer parameters, but the decays of large clusters can leadto phenomenological problems such as predicting heavy baryon distributions which do not match datawell. Widely used event generators which simulate the above steps are P YTHIA [11], H ERWIG [12],and S HERPA [13].
In this project we propose a machine-learning (ML) based simulation for hadronization, aiming to
replace or complement the common used phenomological models used in event generators. Generative
Adversial Networks (GANs) [14], Variational Auto-Encoders (VAEs) [15–17] and Normalizing Flows(NFs) [18] have demonstrated the ability for ML to generate convincing physical observables. In
addition, conditional generative models provide more flexibility and control of the output [19, 20].
Previous work demonstrated that GANs - used for the cluster model - [21] and different versions of
VAE-like sliced-Wasserstein Auto Encoders (SWAEs) - used for the Lund string model - [22] havethe potential to simulate parts of the hadronization process well. However, the presented models
were limited to the emission of only pions as a final state particle, and the training of the kinematic
distributions was performed separately with no correlation between the pz and pT distribution.
In this paper we present an updated version of ML HAD [22] utilizing NFs as a generator for the
kinematic distribution, which overcomes these limitations. This new architecture is able to learncorrelations between the kinematic distributions and is not limited to the emission of only pions, butrather can produce the emission of hadrons with arbitrary mass. We demonstrate these capabilities bytraining it on specially prepared P YTHIA hadronization outputs with an explicit infra-red (IR) cut-off.
In sec. 4 we discuss possible extensions and planned future work.

2

Architecture and method

Figure 1: An illustration of using NF as a generator in the hadronization chains. G is the generator,
which is in our case the NF. In principle any generative model, which generates the kinematics well
can be used.
We consider a simplified Lund string model to demonstrate the performance of our ML architecture.
Hadronization describes the combination of quarks and gluons into hadrons at the non-perturbative
scale. In the Lund string model, the quarks are connected via a QCD color string/tube; in particular
we are looking at a qi q̄i fragmentation event in the center-of-mass frame with quark-flavor index i
and initial energy E. The string breaks iteratively and emits a hadron h per break. In each step thequark flavors and kinematics are conserved. Fig. 1 illustrates the architecture we are using to describe
the Lund string model.
FS is the simplified P YTHIA flavor selector, which takes as input the initial string quark-flavors1 and returns the emitted hadron flavor h1 and the new string-end flavor s2 . G is the generator,
which samples the kinematics of the hadron and due to energy-momentum conservation, updates thefour momentum of the new string end. Here we can use NFs as the kinematics generator G. This
process repeats until the energy of the string falls below a predetermined cut-off value, Ecut , which
is set to 5 GeV in our case. Before each hadron emission, the string fragments are boosted to their2

center-of-mass frame using a Lorentz transformation Λ. The architecture is flexible such that we canexchange components easily. In this section we discuss conditional NFs as a generative model for the
hadron kinematic.
2.1

NF as kinematic generator

NFs are ML architectures which perform a bijective transformation between two spaces, by firstsampling a random vector u from a simple distribution (usually a Gaussian) u ∼ π(u). The sample
is transformed by a neural network f to obtain a data point x from the desired and usually more
complicated distribution p(x), where f is designed to be invertable. We can calculate the density of x
by finding the corresponding random vector u by a change of variables:p(x) = π(u) det

∂f
∂u

 −1
,

(1)

where u = f −1 (x). To use NFs as a generative model, one starts with a sample u from the basedistribution and uses eq. 1 to map to the target distribution. Autoregressive flows use a transformationf (xi ), where xi depends only on the previous coordinates x1 , ..., xi−1 , which leads to a triangular
Jacobian matrix, and subsequently results in the more efficient computation of the determinant.
To ensure the ability to learn high dimensional transformations with a complex target distribution,
multiple layers of these bijective transformations are used; in our case we use five. One can choose
different transformations as long as they are sufficiently expressive. In the results presented in thispaper we use masked autoregressive flows (MAF) [23] with an affine [24] transformation. However,to decrease the sampling time one can use inverse autoregressive flows (IAF) [25].
2.2

Training Data

The training data is a set of P YTHIA generated first-hadron emissions for different initial stringenergies. We can reduce our process to a two variable problem by aligning the z axis of the coordinate
system with the direction of the initial string and assuming axial symmetry in P YTHIA. We can thenreconstruct the complete four momentum of a particle with the two momentum components pz andpT .
The NF is trained simultaneously on the pz and pT , which allows the NF to learn correlations
between the two variables, such that the training data set contains samples from a 2D distribution′(i) (i)of the transverse momentum and z component of the momentum xi = {pz,k , pT,k }. In addition we
condition in each step on the initial string energy and the hadron mass. This allows us to generate
the kinematics for emissions with all possible hadron masses. The energy and mass conditions aretransformed in the range 0 < c < 1 as demonstrated in [22]. The loss function is the negative log
likelihood of the learned probability distribution p(x), which is minimized during the training process.
We use the nflows package1 [26] for the implementation and train for 5 × 106 iterations.
To demonstrate the performance, we train the NF on data for masses ranging from 0.1 GeV to 1 GeV
and energies E = 100, 400, 700, 1000 GeV, such that the model can be provided with both a massand energy label. This can be extended to larger mass ranges, 0.1 GeV to 20 GeV, over which one
can cover any possible hadron emission.

3

Results

As one can see in fig. 11 the NF trained model is able to accurately reproduce first-emission kinematics
of P YTHIA for a hadronized quark anti-quark system in the center-of-mass frame of the string. In
addition, the NF is able to interpolate well for mass labels which were not included in the training, butfall between the training labels, see app. A.2. The NF is also able to extrapolate to mass labels outside
the training range. In app. A.1 one can see the results for different labels we trained on and in app. A.2
are the results for mass labels that were not trained on, either interpolated or extrapolated. Fig. 3
shows the heat map of pz and pT for P YTHIA and NF generated kinematics, which demonstrates that
the NFs are able to learn correlations between the two variables.
1

Corresponding github repo and licence can be found in: https://github.com/bayesiains/nflows30.0175

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Figure 2: The NF generated pz and pT distributions for first-hadron emissions with string energies of(label 1) E = 100, 400, 700, 1000 GeV, and hadron masses of (label 2) m = 0.78 GeV, which the
NF was trained with, compared to the P YTHIA generated target distribution (grey), as well as theratios of the NF generated to P YTHIA generated distributions.

Figure 3: Heat map illustrating the correlation between pz and pT for the P YTHIA (left) and NF(right) generated kinematics.

The model can be trained on mass labels corresponding to hadron masses ranging from 0.1 to 20 GeV.
With this range we can sample the kinematics for all physical hadrons that can be emitted since forthe kinematics only the mass of the hadron is needed to distinguish them.
In fig. 4 we show the average number of hadron multiplicities as a function of initial string energy,
using P YTHIA or ML HAD with NFs fragmentation chain. The updated version of ML HAD with NFs
shows greater agreement than the previous version utilizing the cSWAE architecture presented in [22](fig. 12).

4

Conclusion and Outlook

The updated version of ML HAD utilizing NF, appears to be well suited for modeling the nonperturbative process of hadronization. This was demonstrated by training an NF architecture on
435

Pythia
MLhad

30

40
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25
25
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20
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15

10

10

5

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Figure 4: Average number of hadrons produced in the hadronization chain as function of initial stringenergy using ML HAD (orange) and P YTHIA (blue) for 2 × 104 fragmentation chains and only pions
as emitted hadrons.
a simplified P YTHIA hadronization model, limited to string ends made out light quark flavors. It
overcomes the limitation of allowing only pions as emitted hadrons, presented in [22, 21], and isable to simulate all possible hadron emissions. In addition, our NF is able to be trained on pz and pT
simultaneously, such that possible correlations between these two variables can be learned.
Due the architecture of the hadronization pipeline, see fig. 1, the NF model can be easily implemented
in the fragmentation chain. In fig 4 we generate a hadronization chain and compare the average
hadron multiplicity of P YTHIA with the ML model. The updated ML HAD version utilizing NFs
shows a better performance than the previous version. However, despite these encouraging resultsfrom masked autoregressive flows (MAF), we may switch to a different NF in the future due to theslow sampling speed of the MAF.
Another advantage of the presented architecture is that the exact probability distribution of the drawn
samples is known. Given the sample, one can compute the exact likelihood that would be generated
by the model. Future work will include uncertainty estimation of hadronization models.
Looking forward, the training of the ML architecture presented in this work will be performed on
physically accessible observables. Possible candidates can be found here [27–32]. Many of theseobservable will be accessible due to open-data efforts from different collaborations. The presented
architecture will be used in a pipeline to perform training on experimental data. The code for ML HAD
can be found in https://gitlab.com/uchep/mlhad.

Acknowledgment
We thank Mike Williams for the suggestion of using normalizing flows. AY, JZ, and TM acknowledge
support in part by the DOE grant de-sc0011784 and NSF OAC-2103889. PI and MW are supported
in part by NSF OAC-2103889. SM is partially supported by the DOE/HEP QuantISED program
grant “HEP Machine Learning and Optimization Go Quantum”, identification number 0000240323.

5

Broader Impact
In high energy particle physics, characterizing the uncertainty of phenomenological models is criticalfor interpreting results. To date, common hadronization models have not included uncertainty
estimates. This NF architecture will allow us to explore such uncertainties with a well-defined
methodology. While the resulting hadronization model may not be applicable to the larger community,
the methodology for interpreting the estimated uncertainty will provide an important benchmark
across a wide range of ML applications. Additionally, we will work towards interpretation of
the model for a specific physical process. Such interpretation and uncertainty estimation of ML
architectures will play a crucial role in understanding the ethical implications of broader-ranging ML
applications such as self-driving cars, parole determination, or healthcare.

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7

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A

Plots for different labels

A.1

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In fig. 5 to 8 one can see the results on labels, which the flow was not trained trained on.

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E = 700.0 GeV
E = 1000.0 GeV

0.020

2.0

PDF

PDF

0.015

Target, m = 0.2 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

2.5

0.010

1.5
1.0

0.005

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0

200

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600

800

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Ratio

Ratio

0.000

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pz

0.0

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1.2

0.6

0.8

1.0

1.2

pT

Figure 910Target, m = 0.4 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

0.01750.0150

2.0

PDF

PDF

0.0125

Target, m = 0.4 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

2.5

0.01000.00751.5
1.0

0.00500.5

0.0025
0

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200

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800

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1000

Ratio

Ratio

0.0000

0

200

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pz

0.6

0.8

1.0

1.2

0.6

0.8

1.0

1.2

pT

Figure 100.0175

Target, m = 0.65 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

0.01500.0100

2.0

PDF

PDF

0.0125

Target, m = 0.65 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

2.5

0.00751.5
1.0

0.00500.5

0.0025
0

1.5
1.0
0.5

200

400

600

800

0.0

1000

Ratio

Ratio

0.0000

0

200

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1.0
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0.4

pz

0.6

0.8

1.0

0.6

0.8

1.0

1.2

pT

Figure 110.020

Target, m = 1.2 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

2.0

PDF

PDF

0.015

Target, m = 1.2 GeV
E = 100.0 GeV
E = 400.0 GeV
E = 700.0 GeV
E = 1000.0 GeV

2.5

0.010

1.5
1.0

0.005
0.5

1.5
1.0
0.5

0

200

400

600

800

0.0

1000

Ratio

Ratio

0.000

0

200

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1.5
1.0
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pz

0.0

0.2

0.4

0.0

0.2

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0.8

1.0

0.6

0.8

1.0

pT

Figure 12111.2