# Mutation Models:
Learning to Generate Levels by Imitating Evolution

Ahmed Khalifa<EMAIL_ADDRESS>Institute of Digital Games
University of MaltaMsidaMalta ,  Michael Cerny Green<EMAIL_ADDRESS>Game
Innovation Lab
New York UniversityBrooklynNYUSA and  Julian Togelius<EMAIL_ADDRESS>Game Innovation Lab
New York UniversityBrooklynNYUSA

###### Abstract.

Search-based procedural content generation (PCG) is a well-known method for
level generation in games. Its key advantage is that it is generic and able to
satisfy functional constraints. However, due to the heavy computational costs
to run these algorithms online, search-based PCG is rarely utilized for real-
time generation. In this paper, we introduce mutation models, a new type of
iterative level generator based on machine learning. We train a model to
imitate the evolutionary process and use the trained model to generate levels.
This trained model is able to modify noisy levels sequentially to create
better levels without the need for a fitness function during inference. We
evaluate our trained models on a 2D maze generation task. We compare several
different versions of the method: training the models either at the end of
evolution (normal evolution) or every 100 generations (assisted evolution) and
using the model as a mutation function during evolution. Using the assisted
evolution process, the final trained models are able to generate mazes with a
success rate of $99\%$ and high diversity of $86\%$. The trained model is many
times faster than the evolutionary process it was trained on. This work opens
the door to a new way of learning level generators guided by an evolutionary
process, meaning automatic creation of generators with specifiable constraints
and objectives that are fast enough for runtime deployment in games.

Neural Networks, Evolution, Data Augmentation, Surrogate Models, Procedural
Content Generation, Expressive Range Analysis, Level Generation

††copyright: none††conference: ; ; ††ccs: Applied computing Computer
games††ccs: Theory of computation Evolutionary algorithms††ccs: Computing
methodologies Neural networks

##  1\. Introduction

Very coarsely, we can construct content generators in two different ways. We
can either create a generator that constructs an artifact in a finite (often
fixed) number of steps without testing during the construction process. The
other way is to perform a search or optimization process where either a whole
artifact or part of it is tested repeatedly during generation to guide it
forward (Togelius et al., 2011). In general, constructive generators are much
faster and therefore better suited to real-time generation than search-based
generators. On the other hand, the lack of quality checking during the
construction process means that the expressive spaces of the generator may
need to be restricted in order to guarantee that the content is playable (not
functionally broken). Creating a good constructive generator is hard, and
often requires expert human effort for each use case. Generators based on
search and optimization, on the other hand, can guarantee playability but are
much slower; depending on the cost of content evaluation, they can be
unworkably slow.

Could these advantages be combined? Could we create constructive generators
that are fast yet have functionality guarantees and a wide expressive range?
Could these be created automatically? It stands to reason that one could use
machine learning to somehow learn generators. While the computational costs of
learning a generator may be large, using the learned model as a constructive
generator is computationally cheap; in other words, computation is front-
loaded. Various self-supervised approaches have been advanced, building on
GANs (Volz et al., 2018; Torrado et al., 2020) or autoencoders (Jain et al.,
2016; Sarkar and Cooper, 2021), that learn from existing content to generate
new content. Another recent approach, the Path of Destruction, turns existing
content into sequences of repair actions that can be modeled (Siper et al.,
2022). Obviously, self-supervised approaches require a decently-sized library
of content, such as levels, to learn from (Summerville et al., 2018; Liu et
al., 2021).

Figure 1. System diagram of the mutation models framework. The system consists
of two main parts: evolution loop and training loop. The evolution loop is the
usual evolutionary algorithm which evolves chromosomes using a fitness
function. The training loop converts the evolution history into a dataset then
trains a machine learning model on it. This trained machine learning model can
be used to assist evolution by acting as a mutator or only trained at the end
of evolution. If the model is being used as a mutator, the training loop
happens every $I$ generations to improve the accuracy of the machine learning
model. Later when the evolution is done, the trained model can be used as a
level generator as it learned to imitate the successful evolution
trajectories.

Alternatively one could use evolution (Kerssemakers et al., 2012; Khalifa and
Togelius, 2020; Earle et al., 2022) or reinforcement learning (Khalifa et al.,
2020; Shu et al., 2021; Mahmoudi-Nejad et al., 2021) (PCGRL) to learn
generators. This does not require existing content, but it does necessitate
the existence of a good reward or fitness function. Specifying a good
fitness/reward function for generating content generators can be much harder
than specifying a good fitness/reward function for “simply” generating
content. The intuition behind this is that we more or less know what good
content looks like, but not necessarily what a good content _generator_ looks
like. To take a concrete issue, a good fitness/reward function for a generator
will probably need to reward appropriately diverse output from the generator.

In this paper, we propose another way of generating competent content
generators. Much like PCGRL, we are not reliant on existing content examples.
However, our method also does not need a way to evaluate content generator
quality; we only need a way to evaluate good content. The goal is to make it
possible to create fast content generators that are as reliable, controllable,
and easy to specify as search-based generators. In a nutshell, our approach
entails building a search-based generator, and training a neural network to
take the actions the evolutionary algorithm would take. The sequence of
changes that are made by evolution can be seen as training data for a
supervised learning process, and the resulting model will take those actions
that the evolutionary process would. This can be seen as imitation learning on
evolutionary trajectories; the approach has similarities with offline
reinforcement and surrogate models in evolution.

##  2\. Background

In the following subsections, we review related background work in regards to
procedural content generation (PCG), focusing on search-based and machine
learning based methods. We also review the concept of surrogate modeling for
evolutionary systems.

###  2.1. Search-based PCG

Search-based PCG defines a family of PCG techniques powered by search methods
to generate content (Togelius et al., 2011). Evolutionary algorithms are often
used, as they can be applied to many problems like level generation in video
games. Search-based PCG has been applied to many frameworks and games, such as
the General Video Game AI framework (Perez-Liebana et al., 2019), PuzzleScript
(Khalifa and Fayek, 2015), Cut the Rope (ZeptoLab, 2010) (Shaker et al.,
2013), and Mazes (Ashlock et al., 2011). Search-based techniques have been
used to generate levels (Ashlock, 2010; Khalifa and Fayek, 2015; Bhaumik et
al., 2020), game rules (Browne and Maire, 2010; Khalifa et al., 2017; Cook et
al., 2013) and level generators (Kerssemakers et al., 2012; Khalifa and
Togelius, 2020; Earle et al., 2022).

###  2.2. PCGML

Procedural content generation via machine learning (PCGML) is a process of
generating content using ML algorithms based on input example. These methods
are not often used outside of the research community because of their reliance
on large datasets, long training times, and little control of the generated
output. There are exceptions: for example, Caves of Qud (Grinblat, 2016) uses
PCGML to generate books and other aesthetic elements. Research applications
with PCGML have used many different techniques including Markov Chains
(Snodgrass and Ontanón, 2016), N-Grams (Dahlskog et al., 2014), GANs (Volz et
al., 2018; Torrado et al., 2020), Autoencoders (Jain et al., 2016; Sarkar and
Cooper, 2021), and LSTMs (Summerville et al., 2016).

Most related to this work is a project by Siper et al. in which a network is
trained to repair levels (Siper et al., 2022), modeling a so-called Path of
Destruction. This can be done by taking a series of “goal levels” and
proceeding to randomly destroy them. A network is then trained on the repair
trajectories to learn how to convert these destroyed levels back to the goal
levels. The difference between Path of Destruction and most of the recent
PCGML work is the network generates the content iteratively (similar to Wave
Function Collapse (Karth and Smith, 2017) and Markov Random Fields (Snodgrass
and Ontanón, 2016)) while for example GAN-based or autoencoder-based
generators generate content in one shot (one pass). The approach proposed is
similar to Path of Destruction in that we are training a networks to
iteratively improve randomized levels, but instead of training it on reversed
paths of destruction we train it by imitating the successful mutation
operation of evolution.

###  2.3. PCGRL

Procedural content generation via reinforcement learning (PCGRL) (Khalifa et
al., 2020) is a process of generating content using RL algorithms. PCGRL
transforms the generation process to a game playing process where an agent can
take actions at several states and get a reward based on that. The work
proposes three different methods to achieve that: narrow, turtle, and wide
representations. Narrow representation transforms the generation into a
process of asking the agent about each tile in the level if it needs to change
or not and if yes what is the value of change. On the other hand, turtle
representation has more control over the location by allowing the agent to
either modify the current tile (like narrow) or move to a neighboring tile.
Finally, wide representation provides the agent with maximum control by
allowing the agent to select the location freely and the modification value
for it. In this work, we will be using the narrow representation as it
provides a small action space for the agent and has comparable results to the
other two (Khalifa et al., 2020).

PCGRL techniques have been used in level/experience generation (Chen et al.,
2018; Nam and Ikeda, 2019; Werneck and Clua, 2020; Earle et al., 2021; Zakaria
et al., 2022; Mahmoudi-Nejad et al., 2021; Shu et al., 2021). It is difficult
to transform content generation into a reinforcement environment, thus there
is a few PCGRL examples in either academia or industry compared to other
methods. Similar to Path of Destruction, PCGRL generators are _iterative_ ,
meaning that they produce content using multiple modification steps, action by
action. This grants certain advantages to PCGRL: for example, it is often
easier to build mixed-initiative systems around them (Guzdial et al., 2018,
2019; Delarosa et al., 2021).

###  2.4. Surrogate Modeling

Surrogate models are models of computationally expensive processes, which can
then be used in lieu of the process itself (Ong et al., 2003). Surrogate
models are meant to be easier to evaluate than the process it models, which
translates to time and computational improvements. Surrogates are typically
constructed using a data-driven, bottom-up approach, typically by training a
network on a distribution of intelligently selected data points.

Within evolutionary computation, it is common to use machine learning to build
surrogate models of the fitness function (Jin, 2011). A fitness surrogate
model takes a genome as input and returns a fitness value, just as the fitness
function would; the advantage is that the trained model is much faster than
the actual fitness function. In evolutionary algorithm applications where the
fitness function commonly dominates the computation time, surrogate modeling
is very useful. These surrogate models can be trained during an individual
evolutionary run or over several such runs. While it would be possible in
principle to learn surrogate models of all parts of an evolutionary algorithm,
they are predominantly applied to the fitness function. The only attempt we
know to create surrogate-based mutation is built on a surrogate of the fitness
function rather than the mutation function itself (Abboud and Schoenauer,
2001).

##  3\. Mutation Models Framework

In this paper, we train a machine learning model to imitate the evolutionary
process. The model takes a chromosome (in this case, level) as input and
outputs a modification to it. The end result is a model that can generate
content without the need for an evaluation function. Figure 1 shows the full
system diagram of the mutation models framework, which consists of two parts:
the evolution loop and the training loop. The evolution loop is a standard
evolutionary algorithm which evolves content by trying to increase fitness.
Within the training loop, data is extracted from the evolution loop to train a
machine learning model to perform successful mutations (mutations that leads
to higher fitness) like the evolutionary process, thus the name _mutation
models_. The next two subsections explain both loops in detail.

###  3.1. Evolution Loop

The evolution loop is a normal, simple evolutionary algorithm. In this work,
we use the standard $\mu+\lambda$ evolution strategy without self-adaptation
(Beyer and Schwefel, 2002). The process as shown in Figure 1 is simple: it
optimizes levels to maximize their fitness. Unlike most evolutionary methods,
our method retains the evolution history of every chromosome by recording the
process of mutations made to get to its current form. The following steps
explain the evolution loop used in this work:

  1. (1)

Generate a random population of levels of size $\mu+\lambda$.

  2. (2)

Pick the top $\mu$ levels based on the fitness function to be parents for the
next generation.

  3. (3)

Generate $\lambda$ new levels by mutating the parent chromosomes ($\mu$
levels).

  4. (4)

Repeat the above steps for $G$ generations or until convergence.

We are not restricted to only use a $\mu+\lambda$ evolution strategy
algorithm. Other evolutionary algorithms would work well as long as they do
not require crossover and allow us to keep track of the evolutionary history
of the chromosomes across generations (for example, in a continuous domain it
would straightforward to use the high-performing CMA-ES algorithm). However,
the selected mutation function will determine how the machine learning
algorithm imitates evolution. In this work, the mutations are small, they only
change one tile at a time. For example: the system picks a random x,y
coordinates (location) in the level and swaps the current tile value with a
new tile value or decide not change at all (action). The action does not
always need to be random: this could be sampled from a machine learning model
trained to perform successful mutations (mutations that improve the fitness
function). This will be discussed in the next subsection.

(a) Mutation location (x=0, y=0)

(b) Mutation Location (x=6, y=6)

Figure 2. Examples of transforming a level and a mutation location to a
cropped $8x8$ level. This $8x8$ cropped level act as the input for our trained
machine learning model.

(a) Fitness over time

(b) Dataset size over time

(c) Evolution history similarity over time

Figure 3. Fitness, dataset size, and dataset similarity of the top 10
chromosomes of each generation over the course of “Normal” evolution and
“Assisted” evolution. The shaded area shows the 95% confidence interval over 3
runs.

###  3.2. Training Loop

The training loop is responsible for training a machine learning algorithm to
learn the evolutionary process. The idea is to train a model on the successful
mutations which cause the top $X$ chromosomes from the population to have a
high fitness. Doing so allows the model to learn mutations that are likely to
lead to improvement of the level’s fitness. The following steps explain the
training loop used in this work:

  1. (1)

Pick top $X$ chromosomes from the population.

  2. (2)

Extract the trajectory data (evolution history) from these chromosomes to
build a dataset of levels and mutations.

  3. (3)

Train a machine learning model on the extracted dataset.

The network’s training loop can occur at the end of evolution on the top $X$
evolved maps, which we call the “Normal” method. It can also occur every $I$
generations, allowing the network to be used as the mutation operator during
the evolution loop, which we call the “Assisted” method. The “Assisted” method
is similar in idea to on-policy reinforcement learning: a network creates its
own dataset, which it then trains on. In contrast to traditional reinforcement
learning, the evolutionary process here is helping filter the training dataset
by only keeping the best data points.

In order to build the training dataset, we need to convert the evolution
history of a chromosome into inputs and outputs for the machine learning
model. We found inspiration in the “Narrow” representation in the PCGRL
(Khalifa et al., 2020) environment. In the narrow representation, the input is
the mutation location and level state at a certain point in history, while the
output is the mutation value (no change or the new tile value). In preliminary
experiments, we realized that using the entire level state does not help the
machine learning model generalize. We follow the work by Siper et al. (Siper
et al., 2022) and Ye et al. (Ye et al., 2020) in cropping the level state
around the mutation location, as shown in Figure 2.

##  4\. Experiments

To demonstrate each method’s capabilities, we apply them on a simple domain in
which they generate 2D maze layouts. We test the two methods of training
explained in section 3.2: “Normal” and “Assisted”. The “Normal” method trains
the model at the end after evolution is done, while “Assisted” method retrains
the model from scratch every $100$ generations and uses the freshly-trained
model as a mutation function in the evolution loop. All our experiments are
repeated $3$ times (the full evolution loop including training a neural
network either at the end of evolution in the case of “Normal” or during
evolution in the case of “Assisted”), and we show the average max fitness,
dataset size, and dataset similarity across these runs and the 95% confidence
interval. After evolution is done, the final trained models from each method
are tested for their capabilities on generating content without the need for a
fitness function.

###  4.1. Domain

We use a 2D maze layout of size $14x14$ (excluding the borders) as our proof-
of-concept test-bed. The goal is to generate a 2D layout of solid and empty
tiles such that all the empty tiles are accessible from any other empty tile
and the longest path in that layout is maximized. Figure 4 shows the top 4
evolved maps by the evolutionary algorithm showcasing full connectivity and
long winding paths.

###  4.2. Evolution Loop

As discussed in section 3.1, we are using $\mu+\lambda$ evolution strategy
with $\mu=\lambda=50$. The starting levels are initialized using a uniform
distribution with $50\%$ probability for each tile to be either empty or
solid. The mutation operator picks a random location ($x$ and $y$ position)
and uses either a random mutation value for the “Normal” method, or samples
the value from the trained machine learning model for the “Assisted” method.
To guarantee diversity in the “Assisted” method, there is a $25\%$ chance to
sample a random action instead from the trained model. We run the evolution
process for $2000$ generations with a cascading fitness function which tries
to satisfy the connectivity constraint first then maximizes path length as
shown in equation 1:

(1) $f=\begin{cases}0.5\cdot(1-\frac{n}{20})&\text{if }n>1\text{ and }n<20\\\
0.5+0.5\cdot\frac{p}{98}&\text{if }n==1\\\ 0&otherwise\\\ \end{cases}$

where $n$ is the number of regions in the current level and $p$ is the length
of the longest path length in the map.

###  4.3. Training Loop

The training loop is responsible about two things, creating a training dataset
for the model and training the model. For the dataset generation, we pick the
top $10$ chromosomes from the population and extract their evolution history.
The evolution history consists of the current level, the mutation location,
and the mutation action at every generation in a chromosome’s history. We
transform the level and mutation location into an single image by adopting the
narrow representation from the PCGRL (Khalifa et al., 2020) framework. We crop
the level around the mutation location to be of size $8x8$ as shown in Figure
2. This cropped level is used as the input for the machine learning model. The
output of the model is the mutation action that corresponds to the cropped
level. In this domain, there are 3 different mutation actions: “No Change”,
“Change to Empty”, and “Change to Solid”.

The machine learning model is a small convolutional neural network (224,771
parameters) with a similar architecture to the Atari Deep-Q network (Mnih et
al., 2013). The network consists of $3$ convolutional layers (with $3x3$
filters and kernel size of $32$, $64$, and $128$ respectively) followed by $2$
fully connected layers (with $256$ and $3$ neurons respectively). We use 2 max
pooling layers, each size 2, after the first and second convolution layers to
decrease the input space size. All convolutional layers use same padding to
make sure the size of the input remains constant. All activation functions are
Relu except for the last layer, where we use softmax. The neural network is
trained using an Adam optimizer with learning rate of $10^{-4}$, a minibatch
size of $32$, and categorical cross entropy loss. For the “Normal” method, we
train a new network for $2$, $4$, and $8$ epochs to explore how different
epochs impact network results. For the “Assisted” method, we only train the
network for $2$ epochs due to the computation costs. The “Assisted” method
network is retrained from scratch (with random weights and a reset learning
rate) every $100$ generations. This decision was decided as preliminary
experiments showed that models trained continuously had a higher chance to
overfit on earlier data.

###  4.4. Model Inference

During the evolution/training step, both the “Normal” and “Assisted” methods
utilize a fitness function. The evolutionary algorithm creates a dataset which
is used to train a neural network to imitate the evolutionary process. After
the model is trained (during _inference_), the trained model does not use any
fitness function in either method. The inference step is the same regardless
of which evolution/training procedure is used. We use the final trained
network from each experiment to generate $100$ levels starting from noise
($50\%$ solid and $50\%$ empty levels). The network actions are sampled using
the probability distribution from the last softmax layer. The trained networks
are run sequentially like a scanline over the entire map multiple times, as
proposed in (Khalifa et al., 2020). The networks stop iterating if they reach
one of two cases: success (all the empty tiles are fully connected) or failure
(iterated on the whole map for $196$ times). We record the success rate and
the iteration number that the model requires to reach a successful level. We
also record the path length of each of the successful generated levels with
the number of empty tiles, which we use to calculate the diversity of the
generated content. This diversity is shown both as an expressive range
analysis as well as a percentage of generated levels having different
combinations of path lengths and empty tiles.

##  5\. Results

In this section, we explore the results from our system. We start showing the
evolution results in Section 5.1. These results are collected from the
evolutionary generator which uses an evaluation function to create levels. We
analyze the effect of using the “Normal” vs “Assisted” methods on the
evolution loop. In contrast, Section 5.2 shows the results of the model
inference. The results are collected from the final trained models, which do
not use a fitness function during inference.

###  5.1. Evolution Results

(a) Top 4 generated maps from “Assisted” Evolution

(b) Top 4 generated maps from “Normal” Evolution

Figure 4. Top four maps at generation 2000, from two different evolutionary
runs (“Assisted” evolution and “Normal” evolution). All the four maps look
almost identical with few tile differences as evolutionary strategy tends to
converge when most chromosomes in the population become highly similar to each
other.

Figure 3 displays different evolution metrics across the $2000$ generations.
The usage of “Assisted” method does not change the fitness performance of the
evolutionary algorithm with respect to the “Normal” method. The only
noticeable difference is the size of the extracted dataset (Figure 3(b)) where
the “Assisted” method manages to create a slightly larger dataset than the
“Normal” method. This could be an effect of the network refusing to change
some chromosomes early (using the“No Change” action) when the mutation
location is not good enough. This would lead to these chromosomes having a
longer history than usual as they survive for multiple generations.

The last metric we observe is the similarities between the evolution histories
of the top $10$ chromosomes. We notice that the top $10$ begin with very
different trajectories but end up having almost identical trajectories with
only slight differences (Figure 3(c)). This was expected as evolutionary
strategy tends to converge when most chromosomes in the population become
highly similar to each other as shown in Figure 4. In future work, this might
be solved by using a more complex evolutionary algorithm or a quality
diversity algorithm such as MAP-Elites (Mouret and Clune, 2015) which could
ensure diverse trajectories. The similarity metric that we use is based on the
Ratcliff-Obershelp algorithm (Black, 2021).

###  5.2. Model Inference Results

Method | Success | Diversity | Iterations
---|---|---|---
Assisted-2 | 99.67% ± 0.49% | 86.83% ± 3.8% | 18.21 ± 18.57
Normal-2 | 30.17% ± 32.7% | 28.5% ± 30.62% | 61.7 ± 47.22
Normal-4 | 66.83% ± 49.22% | 59.33% ± 43.69% | 23.55 ± 48.59
Normal-8 | 65.17% ± 48.37% | 60% ± 44.59% | 31.78 ± 23.84

Table 1. The average and 95% confidence interval of 100 generated levels from
3 runs for success rate, diversity, and number of iterations by the final
trained network from “Assisted” and “Normal” evolution. The “-#” after the
name of an experiment designates the number of epochs the final network used
for training.

In this section, we compare the results between the final trained network from
the “Assisted” method using $2$ epochs ($Assisted-2$) and the final trained
networks from the “Normal” method using $2$, $4$, and $8$ epochs ($Normal-2$,
$Normal-4$, and $Normal-8$ respectively). Table 1 displays the success rate
and diversity of the generated levels as well as the number of iterations each
model needs to reach success (which we will refer from now on as
“iterations”). The diversity and iterations are only calculated on the
successfully generated levels, and not for failures.

(a) Assisted-2

(b) Normal-2

(c) Normal-4

(d) Normal-8

Figure 5. Four successful maps generated from the different trained model via
inference. None of these maps are generated using a fitness function. Each
subcaption specifies the name of the experiment, divided into 2 parts: name
and number. The name reflects the evolution process that created the dataset
for this model to be trained on (“Normal” or “Assisted”). The number after the
name of the experiment designates the number of epochs the network used for
training.

(a) Assisted-2

(b) Normal-2

(c) Normal-4

(d) Normal-8

Figure 6. Expressive range analysis of the generated maps during inference
using the trained networks from the 4 different experiments. The expressive
range plots all the successful generated maps on a 2D space, organized by the
number of empty tiles and longest path length. Like Figure 5, the name of
experiment is divided into 2 parts: name and number. The name reflects the
evolution process that created the dataset for this model to be trained on
(“Normal” or “Assisted”). The ‘number after the name of the experiment
designates the number of epochs the network used for training.

All of the “Normal” trained models appear sensitive to the extracted dataset
and varied widely in performance and diversity. We believe that using random
distributions of mutations in the “Normal” method to build the evolution
trajectories produces a noisy and unstable dataset to train on. However,
trajectories built using the output of the trained network used by the
“Assisted” method seem to produce a more consistent dataset that is easier to
train a policy on. This is similar to on-policy reinforcement learning where
the network is trying to learn from it own experience using rewards. In this
work, the reward comes in the form of dataset filtering through evolution:
only successful mutations are kept, and these mutations are the ones that the
network will be trained on. It is not surprising that training for more epochs
improves the performance of “Normal” method networks. However, this
performance plateaus after only 4 epochs which confirms that the “Assisted”
method helps training more than we expected. All of the networks take many
iterations to converge on a successful level with minimum average of $18$ and
maximum average of $61$. We believe that the “Assisted-2” may take less
iterations if it is trained for more epochs with a result similar to what
happens for the “Normal” methods.

Figure 5 displays four successful maps from each experiment. The “Assisted”
generated maps (Figure 5(a)), have features similar to the original final
maps. They contain long paths which contain long vertical lines. The
“Normal-2” generated maps are more open in the upper left corner. We believe
the small amount of epochs allows the network to learn a very simple policy
which erases tiles to have full connectivity. “Normal-4” and “Normal-8” models
also have more open space compared to “Assisted” but appeared more structured
than “Normal-2”. Often, the easiest way to reach full connectivity is to
remove tiles from the level, as this will naturally “open it up.” We believe
that “Normal-2” levels appear more open is because of the cascaded fitness
function, where the evolution trajectory starts by connecting all the empty
tiles then later improve path length. These evolution trajectories tend to
have more “Change to Empty” actions at the beginning of evolution. We believe
that the trained policy might have learned this behavior when the level is
noisy. It learns to fully connect the map first, then later increase path
length. This doesn’t happen in the “Assisted-2” maps since the evolution is
guided by a neural network that can take different actions which can be
adjusted later. And since we terminate generation when everything is
connected, “Normal” networks might achieve full connectivity just as
generation ends, and before it begins to add solid tiles, giving the feeling
of open space.

Finally, Figure 6 displays the expressive range analysis of the trained models
across 3 runs. Although “Assisted-2” generated maps have winding paths, they
use on average more empty tiles compared to the rest of the models. Although
it appears that the “Normal” method uses more empty tiles due to the openness
of the maps, this is not always the case. For example, “Normal-2” maps seems
more empty but a lot of the bottom left corner is solid. This is the same with
“Normal-4” and “Normal-8” where most of the walls are clustered in the center.
“Normal” maps give the feeling of openness while containing more solid tiles
compared to the “Assisted” generated maps.

##  6\. Discussion

We can contrast the mutation models method proposed in this paper to
surrogate-assisted evolutionary computation. From a reinforcement learning
perspective, fitness surrogate models can be thought of as models of the
V-function, or in other words state value estimators. By using such a model to
test multiple actions (mutations/crossover) the best action (surviving genome)
can be chosen. The mutation models learned here are instead models of direct
action selection. They are similar to models of the Q-function (action value
estimators), but instead of returning the value of a particular mutation, they
simply return the mutation to make (similar to actor networks).

Imitating mutation may not have the same “speedup effect” on evolution as
surrogate modeling fitness functions as we still use the fitness function
during evolution (which is computationally intensive) and the fitness value
increases in the same rate as normal evolution (according to figure 3(a)).
However, they can provide expressive range where constructive generation
methods fail to do so. Figure 6 shows that the “Assisted-2” method especially
has a lot of promise in this area. Table 1 shows that the “Assisted-2” results
in the highest map diversity. We use the top $10$ chromosomes of each
generation to train with. Perhaps by expanding this, we might find even
greater expressive range.

Method | Success | Diversity | Wall Time (secs)
---|---|---|---
Network | 99.67% ± 0.49% | 86.83% ± 3.8% | 0.6612 ± 2.3874
Evolution | 100% | 96% | 12.6957 ± 2.2571

Table 2. The average and 95% confidence interval of 100 generated levels by 3
networks trained using “Assisted” evolution for success rate, diversity, and
wall clock time compared to the average for success rate, diversity, and wall
clock time of running 100 normal evolutionary runs till the evolution find a
fully connected level.

Search-based PCG methods are computationally expensive and are thus not
commonly used for online generation. We ran an experiment to measure the wall
clock time of normal evolution till it find a fully connected level or 2000
generations reached. The evolution took roughly 20x longer than mutation model
inferencing as seen in table 2. This difference would be even greater if we
had used a more expensive fitness function during evolution, such as gameplay
simulation. In other words, the speed advantage of Mutation Models will
increase as the generation task gets harder. Learning generators negates the
need for an evaluation function like fitness, thus speeding up the overall
generation process during inference. Evaluation is what typically takes most
of the computational time for search-based generators. Through the use of our
method, developers might be able to benefit from the advantages that search-
based PCG provides without having to bear the cost of online computation.
Although evolution guarantees 100% success rate and almost 100% diversity as
seen in table 2, the “Assisted” trained networks have comparable success rate
and diversity with drastically less time needed to generate the content.

We restricted the evolutionary algorithm to only have mutation to simplify the
process of extracting trajectories. Removing crossover usually hinders
evolution and in some problems the evolution might get stuck in local optima.
This was not the case in our problem due to its simplicity. There is multiple
different methods to extract trajectories while crossover is happening. These
methods can be categorised into two main directions:

  * •

Converting the crossover into small mutation steps. When we extract
trajectories whenever a crossover switch a big amount of tiles, we compare
these tiles to the state before crossover and unpack these changes one a time
in a random order (similar to path of destruction (Siper et al., 2022)).

  * •

Consider mutation as a new starting point for a new trajectory. When we
extract trajectories whenever a crossover happen we consider the current game
level as new level and there is no dependency on the previous level.

This work introduces the opportunity to use an evolutionary algorithm as an
assisted process for reinforcement learning, similar to Go-Explore algorithm
(Ecoffet et al., 2019) and offline reinforcement learning (Levine et al.,
2020). There is a lot to be explored and studied about this new paradigm. For
example, what will happen if the evolution trajectories between the selected
chromosomes are different? If we use a quality diversity algorithm such as
MAP-Elites (Mouret and Clune, 2015), can we possibly learn a conditional
mutator that can change the content towards a certain area in the generative
space? We only discussed a very simple test bed problem, do the results from
this test bed generalize to more complex games or even generic optimization
problems? Also, what about the training algorithm, we only used basic
supervised learning on trajectories but one could try using Backward algorithm
similar to the one used in Go-Explore to get better results (Ecoffet et al.,
2019). Finally, we only explored creating the dataset from the evolution
history, what if we can create the dataset using a similar technique to the
Path of Destruction (Siper et al., 2022) where the evolution process is used
as a method to generate the goal set. We believe that using something other
than evolutionary history can free us to use more advanced evolutionary
methods and/or indirect representation.

##  7\. Conclusion

This paper proposes Mutation Models, a new method of generating content by
building a machine learning model that imitates evolution. We use evolution
trajectories to train a machine learning models in order to imitate mutation.
This allows us to automatically create content generators by only specifying
the fitness of the content, which is relatively domain-agnostic. The end
result is an iterative content generator that is fast and does not need a
fitness function during inference. Model building consists of two main loops
(as shown in Figure 1): the evolution loop and training loop. The evolution
loop is a normal evolutionary algorithm that generates content, while the
training loop is responsible for extracting a training dataset from the top
$X$ chromosomes. The training dataset is constructed by converting the
evolution history into state-action pairs, on which a traditional supervised
learning model is trained. We propose two methods for the framework: the
“Normal” method (the machine learning model is trained after the evolution
ends) and the “Assisted” method (the machine learning model is trained every
$I$ generations and used to assist evolution). The results show that although
the “Assisted” method does not help the evolution to be faster or more
efficient, it does stabilize the trained networks, which learn better policies
to imitate evolution in comparison to “Normal” method. Both methods can infer
on randomly initialized maps and repair them without use of an evaluation
function.

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# Bottom hadro-chemistry in high-energy hadronic collisions

Min He1 and Ralf Rapp2 1Department of Applied Physics, Nanjing University of
Science and Technology, Nanjing 210094, China  2Cyclotron Institute and
Department of Physics and Astronomy, Texas A&M University, College Station,
Texas 77843-3366, U.S.A.

###### Abstract

The hadro-chemistry of bottom quarks ($b$) produced in hadronic collisions
encodes valuable information on the mechanism of color-neutralization in these
reactions. Since the $b$-quark mass is much larger than the typical hadronic
scale of $\sim$1 GeV, $b\bar{b}$ pair production is expected to be well
separated from subsequent hadronization processes. A significantly larger
fraction of $b$ baryons has been observed in proton-proton ($pp$) and proton-
antiproton ($p\bar{p}$) reactions relative to $e^{+}e^{-}$ collisions,
challenging theoretical descriptions. We address this problem by employing a
statistical hadronization approach with an augmented set of $b$-hadron states
beyond currently measured ones, guided by the relativistic quark model and
lattice-QCD computations. Assuming relative chemical equilibrium between
different $b$-hadron yields, thermal densities are used as fragmentation
weights of $b$-quarks into various hadron species. With quark model estimates
of the decay patterns of excited states, the fragmentation fractions of
weakly-decaying $b$ hadrons are computed and found to agree with measurements
in $p\bar{p}$ collisions at the Tevatron. By combining transverse-momentum
($p_{T}$) distributions of $b$-quarks from perturbative QCD with thermal
weights and independent fragmentation toward high $p_{T}$, a fair description
of the $p_{T}$-dependent $\bar{B}_{s}^{0}/B^{-}$ and $\Lambda_{b}^{0}/B^{-}$
ratios measured in $pp$ collisions at the LHC is obtained. The observed
enhancement of $\Lambda_{b}^{0}$ production is attributed to the feeddown from
thus far unobserved excited $b$ baryons. Finally, we implement the hadro-
chemistry into a strongly-coupled transport approach for $b$-quarks in heavy-
ion collisions, utilizing previously determined $b$-quark transport
coefficients in the Quark-Gluon Plasma, to highlight the modifications of
hadro-chemistry and collective behavior of $b$ hadrons in Pb-Pb collisions at
the LHC.

###### pacs:

25.75.-q 25.75.Dw 25.75.Nq

Introduction.— The masses of charm ($c$) and especially bottom ($b$) quarks
are much greater than the nonperturbative scale of Quantum Chromodynamics
(QCD), $\Lambda_{\rm QCD}$, and therefore their production in experiment
offers valuable tests of perturbative-QCD dynamics Norrbin:2000zc ;
Mangano:1991jk . However, the heavy-quark (HQ) conversion into heavy-flavor
(HF) hadrons is an intrinsically soft process that usually requires
phenomenological modeling of nonperturbative fragmentation functions (FFs) to
describe the production yields and momentum spectra of the observed hadrons
Norrbin:2000zc ; Kartvelishvili:1977pi ; Peterson:1982ak ; Braaten:1994bz .
Fragmentation fractions of heavy quarks into weakly-decaying heavy hadrons,
which include feeddown from excited states via strong or electromagnetic
decays, provide a critical test of hadronization mechanisms, and are commonly
denoted as $f_{u}$, $f_{d}$, $f_{s}$ and $f_{\rm baryon}$, representing the
probabilities of, e.g., a $b$ quark hadronizing into a $B^{-}$, $\bar{B}^{0}$,
$\bar{B}_{s}^{0}$ meson and a $b$ baryon (or their charge-conjugate
counterparts), respectively. Precise knowledge of these fractions is also
important to improve the sensitivity of searches for physics beyond the
standard model via rare decays of $b$ hadrons CMS:2014xfa .

Traditionally, $b$-quark fragmentation has been assumed to be universal across
different colliding systems based on the notion that hadronization occurs
nonperturbatively at the scale of $\Lambda_{\rm QCD}$ Lisovyi:2015uqa
independent of the environment. This is supported, within uncertainties, by
measurements of the $f_{s}/f_{d}$ ratio that are consistent between
$e^{+}e^{-}$ collisions at the $Z^{0}$ resonance at LEP ALEPH:1995npv ;
DELPHI:2003pao  and $pp$ collisions at the LHC LHCb:2011ldp ; LHCb:2011leg ;
LHCb:2013vfg ; ATLAS:2015esn ; LHCb:2019fns ; LHCb:2019lsv ; LHCb:2021qbv ;
ALICE:2021mgk ; CMS:2022wkk . However, a substantially larger value of
$f_{\Lambda_{b}}/f_{d}$ has been observed in high-energy $b$ jets produced in
$p\bar{p}$ CDF:2008yux  and $pp$ LHCb:2014ofc ; LHCb:2015qvk ; LHCb:2019fns ,
compared to $b$ jets from $Z^{0}$ decays ALEPH:1997jlh ; DELPHI:2003pao ;
HFLAV:2019otj , thus challenging the universality assumption. Similar
discrepancies have been reported in the charm ($c$) sector ALICE:2017thy ;
ALICE:2020wfu ; ALICE:2021dhb .

In practice, FFs for $b$ or $c$ hadrons are usually inferred from $e^{+}e^{-}$
annihilation data. Employing these FFs gives a satisfactory description of
$p_{T}$-differential cross sections for $b$ and $c$ mesons in hadronic
collisions within various calculational schemes, such as Fixed Order Next-to-
Leading Logarithm (FONLL) Frixione:2007nw ; Cacciari:2012ny ; Catani:2020kkl ,
$k_{T}$-factorization Catani:1990eg ; Collins:1991ty ; Maciula:2018iuh  or
general-mass variable-flavor number scheme (GM-VFNS) Kniehl:2005mk ;
Kniehl:2011bk ; Kniehl:2020szu . However, the application of FFs for $c$
baryons substantially underestimates $\Lambda_{c}$ production, especially at
low $p_{T}$ Maciula:2018iuh ; Kniehl:2020szu , in $pp$ collisions at the LHC
ALICE:2017thy ; ALICE:2020wfu , further questioning their universality. An
extraction of the FFs for $b$ baryons is currently lacking Kramer:2018rgb .

Effects of the partonic environment on HQ hadronization have first been put
forward in elementary hadronic collisions Hwa:1994uha ; Cuautle:1997ti ;
Braaten:2002yt ; Rapp:2003wn ; Berezhnoy:2013qln ; Christiansen:2015yqa ;
He:2019vgs . Specifically, HQ hadronization may be affected through
recombination with valence quarks in the initial state Hwa:1994uha ;
Cuautle:1997ti ; Braaten:2002yt ; Rapp:2003wn ; Berezhnoy:2013qln , or multi-
parton interactions in the final state Christiansen:2015yqa . In $pp$
collisions at LHC energies, this has been pursued via a statistical
coalescence production of $c$ hadrons Andronic:2009sv ; He:2019tik , where
hadron yields are determined by the thermo-statistical weights governed by
their masses at a universal hadronization “temperature” Braun-
Munzinger:2003pwq ; Becattini:2009sc ; Andronic:2017pug . In the present work,
we generalize this approach to the bottom sector to compute the $b$ hadro-
chemistry, using a large set of $b$-hadron states that goes well beyond the
currently observed spectrum ParticleDataGroup:2020ssz . By further employing
quark model estimates of the decay systematics of excited $b$ hadrons, we are
able to predict a large set of fragmentation fractions of weakly-decaying $b$
hadrons. We also evaluate the $p_{T}$ dependence of the hadro-chemistry via a
combined recombination/fragementation scheme, which enables predictions for
the total $b\bar{b}$ cross section as well as for the $\bar{B}_{s}^{0}/B^{-}$
and $\Lambda_{b}^{0}/B^{-}$ ratios in $pp$ collisions. Finally, we implement
the new hadro-chemistry into our Langevin transport approach for heavy-ion
collisions and highlight predictions for the nuclear modification factor of
selected $b$ hadrons in 5 TeV Pb-Pb collisions.

Bottom-hadron spectrum and strong decays.— The experimental effort to search
for missing resonances in the HF sector has been ongoing for decades
Crede:2013kia ; Chen:2016spr ; Charles:2016rim . The current particle data
group (PDG) listings are rather scarce especially for $b$ baryons
ParticleDataGroup:2020ssz . Many additional $b$ hadrons are predicted by quark
model studies Ebert:2009ua ; Ebert:2011kk ; Roberts:2007ni  and in good
agreement with lattice-QCD (lQCD) results Lewis:2008fu ; Brown:2014ena . We
therefore employ a statistical hadronization model (SHM) using two different
sets of $b$-hadrons as input: (a) PDG-only states ParticleDataGroup:2020ssz
and (b) a relativistic quark model (RQM) Ebert:2009ua ; Ebert:2011kk  which
additionally includes 18 $B$’s, 16 $B_{s}$’s, 27 $\Lambda_{b}$’s, 45
$\Sigma_{b}$’s, 71 $\Xi_{b}$’s, and 41 $\Omega_{b}$’s, up to meson (baryon)
masses of 6.5 (7) GeV. Since we are mostly concerned with the relative
production yields of $b$ hadrons, we use the grand-canonical version of SHM,
which works well for bulk hadron production in minimum-bias $pp$ collisions at
the LHC energies Sharma:2018jqf ; Das:2016muc  (the smallness of the total
number of $b$ hadrons requires a canonical treatment of the $b$ number when
computing the absolute yields, but the induced canonical suppression factor is
common to all $b$ hadron containing a single $b$ quark and thus cancels out in
hadron ratios Chen:2020drg ; likewise, the $b$ fugacity factor, which is fixed
by the total $b\bar{b}$ cross section, is dropped). The thermal density of a
given $b$ hadron of mass $m_{i}$ and spin-isospin degeneracy $d_{i}$ and
containing $N_{s}^{i}$ strange or antistrange quarks is then evaluated at the
hadronization temperature $T_{H}$ as

$n_{i}^{\rm
primary}=\frac{d_{i}}{2\pi^{2}}\gamma_{s}^{N_{s}^{i}}m_{i}^{2}T_{H}K_{2}(\frac{m_{i}}{T_{H}})\
,$ (1)

where $K_{2}$ is the modified Bessel function of second kind and
$\gamma_{s}$$\sim$0.6 STAR:2008med ; He:2019tik  the strangeness suppression
factor in elementary reactions. While the SHM analysis of light-hadron yields
in heavy-ion collisions at the LHC Andronic:2017pug  indicates a hadronization
temperature very comparable to the pseudo-critical chiral transition
temperature $T_{\rm pc}^{\chi}$$\sim$155 MeV determined in lQCD HotQCD:2014kol
; Borsanyi:2013bia , a higher hadronization temperature $T_{H}$$\sim$170 MeV
appears to be more appropriate for HF hadrons in elementary reactions
Andronic:2009sv . A flavor hierarchy in the effective hadronization
temperature has also been suggested based on lattice calculations of quark
flavor susceptibilities Bellwied:2013cta . In the following, we therefore use
$T_{H}$=170 MeV as the default value and $T_{H}$=160 MeV as part of our error
estimate.

Figure 1: The (a) meson and (b) baryon decay process $A\rightarrow B+C$ in the
${}^{3}P_{0}$ model.

Information about decays of excited $b$ hadrons into their weakly-decaying
ground states is very limited, even for observed states
ParticleDataGroup:2020ssz . Instead of taking rather incomplete results
available in the literature, we estimate the branching ratios ($BR$s) of the
OZI-allowed strong decays for all employed $b$ hadrons within the
${}^{3}P_{0}$ pair creation model Capstick:2000qj . As schematically shown in
Fig. 1, a pair of quarks with $J^{PC}=0^{++}$ is created from the vacuum and
regroups with the quarks within the initial $b$ hadron into the outgoing meson
or baryon. For example, in the case of a baryon decay, there are three ways of
quark regrouping Chen:2007xf , leading to a $b$-baryon plus a light meson or a
$b$-meson plus a light baryon in the final state, cf. Fig. 1(b). We do not
attempt to perform full calculations using realistic hadron wavefunctions
Roberts:1992esl ; Ferretti:2015rsa ; Yu:2022ymb , but obtain the needed $BR$s
via counting all possible diagrams of the types in Fig. 1 once a decay channel
opens up, by assuming the number of diagrams for the channel to be
proportional to its $BR$. The probability of creating a quark pair is assumed
to be $\propto e^{-2m_{q}/T_{H}}$; therefore a diagram involving creating a
$s\bar{s}$ is weighted by $e^{-2m_{s}/T_{H}}/e^{-2m_{u/d}/T_{H}}\simeq 1/3$
taking the current-quark masses $m_{u/d}$$\simeq$8 MeV and $m_{s}$$\simeq$100
MeV (this estimate is robust even if constituent-quark masses are used). We
account for direct three-body decays of $b$-baryons by counting the
regroupings resulting from creation of two $u\bar{u}$ or $d\bar{d}$ pairs and
find significant $BR$s for the $\Lambda_{b}^{0}+n\pi$ channels of the excited
$\Lambda_{b}$’s or $\Sigma_{b}$’s decays, which is supported by the measured
information on $\Lambda_{b}(5912)^{0}$ and $\Lambda_{b}(5920)^{0}$
(reminiscent of $\Lambda_{c}(2595)^{+}$ and $\Lambda_{c}(2625)^{+}$ in the $c$
sector) ParticleDataGroup:2020ssz . While low-lying excited states all end up
in their corresponding ground states (including the pure electromagnetic decay
of $B^{*}$ and $B_{s}^{*}$), higher states have significant cross-feeddowns,
e.g., $BR\sim 14$% for excited $B$’s to $\bar{B}_{s}^{0}+K$, $\sim$85-100% for
excited $B_{s}$’s to $B^{-}$$/$$\bar{B}^{0}+K$, $\sim$20-30% for excited
$\Lambda_{b}$’s ($\Sigma_{b}$’s) to $B^{-}(\bar{B}^{0})+N$, $\sim$20% ($20$%)
for excited $\Xi_{b}$’s to $\Lambda_{b}^{0}+\pi+K$
($B^{-}$$/$$\bar{B}^{0}+\Sigma$), and $\sim 75$% ($\sim 25$%) for excited
$\Omega_{b}$’s to $\Xi_{b}^{0-}+K$ ($B^{-}$$/$$\bar{B}^{0}+\Xi$). These
estimates are overall consistent with available results from a relativized
quark model Ferretti:2015rsa ; Yu:2022ymb .

Bottom-hadron fractions and ratios.— With the $BR$s as estimated above, the
total densities of the weakly-decaying ground states are obtained via

$n_{\alpha}=n_{\alpha}^{\rm primary}+\sum_{i}n_{i}^{\rm primary}\cdot
BR(i\rightarrow\alpha)\ .$ (2)

These densities are converted into fractions of the total $b$ content in Table
1, under the constraint of
$f_{u}+f_{d}+f_{s}+f_{\Lambda_{b}^{0}}+f_{\Xi_{b}^{0,-}}+f_{\Omega_{b}^{-}}=1$
when neglecting the tiny fractions of states made of two or more heavy quarks
(e.g., $B_{c}$ mesons, doubly-bottom baryons or bottomonia). When going from
the PDG to the RQM scenario, a marked transfer of the $b$ content from the
meson to the baryon sector occurs, comparable to the experimentally observed
$b$-hadron fractions in $e^{+}e^{-}$ vs. $p{\bar{p}}$ collisions HFLAV:2019otj
(and reminiscent of the charm sector ALICE:2021dhb ). Specifically, the
fraction of $B^{-}$ ($\bar{B}_{s}^{0}$) is reduced by $\sim$10(15)%, but the
fractions of $\Lambda_{b}^{0}$ and $\Xi_{b}^{0,-}$ are both enhanced by
$\sim$50% upon inclusion of additional baryons in the RQM at $T_{H}$=170 MeV,
relative to the PDG scenario (similarly at $T_{H}$=160 MeV). The weakly-
decaying $b$-hadron fractions obtained in the RQM for both $T_{H}$=170 and
$160$ MeV turn out to agree with the measurements in $p\bar{p}$ collisions at
the Tevatron within uncertainties, $f_{u}=f_{d}=0.340\pm 0.021$,
$f_{s}=0.101\pm 0.015$ and $f_{\rm baryon}=0.220\pm 0.048$ HFLAV:2019otj .

The calculated ratios of $\bar{B}^{0}$, $\bar{B}_{s}^{0}$, $\Lambda_{b}^{0}$
and $\Xi_{b}^{0,-}$ to $B^{-}$ are summarized in Table 2. The mesonic ratios
are rather stable, but baryonic ratios are more sensitive to variations in the
hadronization temperature. While an equal production of $B^{-}$ and
$\bar{B}^{0}$ always holds due to isospin symmetry, the
$\bar{B}_{s}^{0}/B^{-}$ ratio is reduced by $\sim$7% upon inclusion of
additional states in the RQM scenario. The most pronounced effect is caused by
the inclusion of missing baryons, enhancing the baryonic ratios by $\sim$60%
relative to the PDG scenario, leading to $\Lambda_{b}^{0}/B^{-}\sim 0.51$ with
$T_{H}$=170 MeV, rather comparable to
$f_{\Lambda_{b}^{0}}/(f_{u}+f_{d})=0.259\pm 0.018$ as measured by LHCb in 13
TeV $pp$ collisions LHCb:2019fns .

$f_{\alpha}$ | $B^{-}$ | $\bar{B}^{0}$ | $\bar{B}_{s}^{0}$ | $\Lambda_{b}^{0}$ | $\Xi_{b}^{0,-}$
---|---|---|---|---|---
PDG(170) | 0.3697 | 0.3695 | 0.1073 | 0.1157 | 0.03698
PDG(160) | 0.3782 | 0.3780 | 0.1094 | 0.1023 | 0.03144
RQM(170) | 0.3391 | 0.3389 | 0.09152 | 0.1737 | 0.05503
RQM(160) | 0.3533 | 0.3532 | 0.09620 | 0.1502 | 0.04565
Table 1: Fractions of ground-state $b$ hadrons (relative to total $b\bar{b}$) from the SHM with $T_{H}$=170 and 160 MeV in the PDG and RQM scenarios. $\Xi_{b}^{0,-}$ denotes the sum of two isospin states. The $\Omega_{b}^{-}$ fraction is $\sim$0.1% (not shown here). $r_{\alpha}$ | $\bar{B}^{0}/B^{-}$ | $\bar{B}_{s}^{0}/B^{-}$ | $\Lambda_{b}^{0}/B^{-}$ | $\Xi_{b}^{0,-}/B^{-}$
---|---|---|---|---
PDG(170) | 0.9995 | 0.2904 | 0.3129 | 0.1000
PDG(160) | 0.9995 | 0.2894 | 0.2706 | 0.08313
RQM(170) | 0.9994 | 0.2699 | 0.5122 | 0.1623
RQM(160) | 0.9996 | 0.2723 | 0.4250 | 0.1292
Table 2: Ratios of $\bar{B}^{0}$, $\bar{B}_{s}^{0}$, $\Lambda_{b}^{0}$ and
$\Xi_{b}^{0,-}$ to $B^{-}$ at $T_{H}$=170 and 160 MeV in the PDG and RQM
scenarios.

Bottom-hadron $p_{T}$-spectra in $pp$ collisions.— To compute the $p_{T}$
differential cross sections of ground-state $b$ hadrons, we simulate the
fragmentation and decay processes using the $b$-quark $p_{t}$ spectrum in
$\sqrt{s}=5.02$, $7$ and $13$ TeV $pp$ collisions from FONLL Frixione:2007nw ;
Cacciari:2012ny . A $b$ quark sampled from the spectrum is fragmented into $b$
hadrons via the same FF Kartvelishvili:1977pi  as implemented in FONLL,

$D_{b\rightarrow H_{b}}(z)\propto z^{\alpha}(1-z)\ ,$ (3)

where $z=p_{T}/p_{t}$ is the fraction of the $b$-hadron’s ($H_{b}$) momentum,
$p_{T}$, over the $b$-quark momentum, $p_{t}$. The fragmentation weight of
$H_{b}$ is determined by its thermal density $n_{H_{b}}^{\rm primary}$, Eq.
(1), normalized by the sum $\sum_{H_{b}}n_{H_{b}}^{\rm primary}$. Each $H_{b}$
produced from fragmentation is then decayed into the ground-state particles
with a constant matrix element, i.e., decay kinematics solely determined by
phase space and $BR$s estimated above.

Figure 2: (a): $p_{T}$-differential cross sections of ground-state $B^{-}$ in
$\sqrt{s}$=7 and 13 TeV $pp$ collisions for the RQM (solid lines) and PDG
(dashed lines) scenarios at $T_{H}$=170 MeV, in comparison with LHCb data at
$2<y<2.5$ LHCb:2017vec . (b): the same for $B^{-}$, $\bar{B}_{s}^{0}$,
$\Lambda_{b}^{0}$ and $\Xi_{b}^{0,-}$ in $\sqrt{s}$=5.02 TeV $pp$ collisions
at mid-rapidity, compared to $|y|<2.4$ CMS data CMS:2017uoy ; CMS:2018eso
scaled to $|y|<0.5$ via FONLL Frixione:2007nw ; Cacciari:2012ny .

Figure 3: $p_{T}$-dependent ratios of (a): $\bar{B}_{s}^{0}/B^{-}$, (b):
$\Lambda_{b}^{0}/B^{-}$ and (c): $\Xi_{b}^{0,-}/B^{-}$ for RQM (solid lines)
and PDG (dashed lines) scenarios in $\sqrt{s}$=13 TeV (red, $2<y<2.5$) and
5.02 TeV (green, $|y|<0.5$) $pp$ collisions, together with ratios in
$\sqrt{s_{NN}}$=5.02 TeV PbPb collisions (blue solid lines, 20-40% centrality)
at mid-rapidity, in comparison with available LHCb LHCb:2019lsv ; LHCb:2014ofc
; LHCb:2019fns  and CMS CMS:2021mzx  data. The horizontal arrow in the middle
panel indicates the LEP average for $f_{\Lambda_{b}}/f_{d}$ HFLAV:2019otj
from $Z^{0}$ decays at an average $b$-quark transverse momentum of $\langle
p_{t}(b)\rangle$$\sim$40 GeV.

The parameter $\alpha$ in Eq. (3) is tuned to fit the slope of the $p_{T}$
spectra of ground state $b$ hadrons (kinematic effects from recombination are
partially absorbed by this tune). For the RQM scenario at $T_{H}=170$ MeV, we
find that with $\alpha_{B}=45$ (for simplicity taken the same for all $B$
mesons; similarly $\alpha_{B_{s}}=25$ for all $B_{s}$ mesons and $\alpha_{\rm
baryon}=8$ for all $b$ baryons), the measured $B^{+}+B^{-}$
$p_{T}$-differential cross section at $2<y<2.5$ LHCb:2017vec  can be described
with a total $b\bar{b}$ cross section of $d\sigma^{b\bar{b}}/dy=34.55~{}\mu b$
($68.87~{}\mu b$) in $\sqrt{s}=7$ TeV (13 TeV) $pp$ collisions, cf. Fig. 2(a).
The latter is a prediction based on the computed hadro-chemistry content and
is consistent with the LHCb data for semileptonic $b$ decays LHCb:2016qpe .
For the PDG scenario, the $B^{+}+B^{-}$ data turn out to be equally well
described but with a $\sim 10\%$ smaller total $b\bar{b}$ cross section. The
decrease in the latter is due to the reduction of $b$ content in the baryon
sector, as demonstrated by the significantly smaller $p_{T}$-differential
yields of $\Lambda_{b}^{0}$ and $\Xi_{b}^{0,-}$ compared to their RQM
counterparts, cf. Fig. 2(b) for $\sqrt{s}=5.02$ TeV $pp$ collisions at mid-
rapidity, where available data for $B^{+}$ and $B_{s}^{0}$ are compared; the
value $d\sigma^{b\bar{b}}/dy=39.30~{}\mu b$ deduced from the RQM scenario is
comparable to the value measured by ALICE via non-prompt $D$ mesons
ALICE:2021mgk .

We now turn to the $p_{T}$-dependent $b$-hadron ratios. For
$\bar{B}_{s}^{0}/B^{-}$, shown in Fig. 3(a), the additional states in the RQM
scenario reduce the PDG scenario results by over 10% at low $p_{T}$, improving
the description of the LHCb data for $f_{s}/f_{d}$ which are approximately
constant with $p_{T}$. For $\Lambda_{b}^{0}/B^{-}$, the RQM scenario is
clearly favored by the LHCb data, cf. Fig. 3(b). The substantial gap between
data and the PDG scenario results is largely overcome by the feeddown of the
large set of “missing” baryons included in the RQM calculation, leading to a
fair description of the data, including its increasing trend toward low
$p_{T}$. For comparison, the LEP average of $f_{\Lambda_{b}}/f_{d}$
HFLAV:2019otj  in $Z$ decays is indicated as a horizontal arrow. Finally, our
predictions for the $\Xi_{b}^{0,-}/B^{-}$ ratio from RQM and PDG scenarios are
compared in Fig. 3(c), exhibiting similar features as in the case of
$\Lambda_{b}^{0}/B^{-}$.

Bottom hadrons in PbPb collisions.— The hadro-chemistry computed above in $pp$
collisions serves as a controlled reference for studying modifications in
heavy-ion collisions. Toward this end, we employ a strongly coupled transport
approach previously developed for the $c$ sector He:2019vgs  and calculate the
hadro-chemistry and nuclear modification factor of $b$ hadrons in
$\sqrt{s_{NN}}$=5.02 TeV PbPb collisions. In this approach, the $b$ quark
diffusion in the hydrodynamically evolving QGP is simulated via relativistic
Langevin equations whose accuracy is improved compared to $c$ quarks because
of the $\sim 3$ times larger $b$-quark mass (no shadowing is put on the
initial $b$-quark spectrum). The transport coefficient is taken from lQCD-
potential based $T$-matrix computations Riek:2010fk  but amplified by the same
$K$=1.6 factor as done for the $c$ sector He:2019vgs , to mimic missing
contributions from spin-dependent forces ZhanduoTang:2023tdg  and/or radiative
energy loss. At $T_{H}$=170 MeV $b$-quark hadronization into mesons/baryons is
computed by the 4-momentum conserving resonance recombination model (RRM)
Ravagli:2007xx ; He:2019vgs . The RRM is implemented event-by-event in
combination with the Langevin diffusion based on selfconsistently determined
recombination probabilities, $P_{i}(p_{b}^{*})$ He:2019vgs . The sum of
$\sum_{i}P_{i}(p_{b}^{*})$ over all primary $b$ hadrons $i$ is renormalized to
unity at vanishing $b$ quark restframe momentum ($p_{b}^{*}$) to guarantee the
majority of low-momentum $b$ quarks hadronize through recombination, while
leftover $b$ quarks, as a result of the decreasing $\sum_{i}P_{i}(p_{b}^{*})$
toward large $p_{b}^{*}$, fragment in the same manner as in $pp$. A prominent
feature of our RRM implementation is the inclusion of space-momentum
correlations (SMCs) in the quark phase-space distributions He:2019vgs , which
augments the flow effect in the $p_{T}$-spectra of high-mass $b$ hadrons and
generally extends the reach of recombination toward higher $p_{T}$.

In practice, constituent-quark ($m_{u,d}$=0.33 GeV, $m_{s}$=0.45 GeV,
$m_{b}$=4.88 GeV) and -diquark masses (scalar $m_{[ud]}$=0.71 GeV within
$\Lambda_{b}$’s, axial-vector $m_{\\{ud\\}}$=0.909 GeV within $\Sigma_{b}$’s,
$m_{[us]}$=0.948 GeV and $m_{\\{us\\}}$=1.069 GeV within $\Xi_{b}$’s, and
$m_{\\{ss\\}}$=1.203 GeV within $\Omega_{b}$’s) are taken from RQM studies
Ebert:2009ua ; Ebert:2011kk . We use energy-dependent widths (cf. Ref.
Vovchenko:2018fmh ) with onshell values of $\Gamma^{0}\sim 0.1$ GeV in the
meson, diquark and baryon cross sections in RRM, which suppresses artificial
low-$s$ tails of the pertinent Breit-Wigner amplitudes for $\sqrt{s}$ values
far below the nominal resonance mass (thereby significantly reducing the
sensitivity of final results to variations of the width values).

Figure 4: (a): nuclear modification factors for ground-state $B^{-}$,
$\bar{B}_{s}^{0}$, $\Lambda_{b}^{0}$ and $\Xi_{b}^{0,-}$ in 20-40%
$\sqrt{s_{NN}}$=5.02 TeV PbPb collisions at mid-rapidity, together with CMS
data for $B^{+-}$ in 0-100% centrality CMS:2017uoy . (b): the same for non-
prompt $D^{0}$ and $D_{s}^{+}$ (0-10% centrality) in comparison with ALICE
data ALICE:2022tji ; ALICE:2022xrg .

Primary $b$ hadrons formed from hadronization undergo further diffusion in the
hadronic phase until kinetic freezeout using our previously calculated
$D$-meson thermalization rate He:2011yi  scaled down by the $b$-hadron mass.
These hadrons are then decayed to obtain the $p_{T}$ spectra of ground-state
$b$ hadrons. The ratios of $\bar{B}_{s}^{0}/B^{-}$, $\Lambda_{b}^{0}/B^{-}$
and $\Xi_{b}^{0,-}/B^{-}$ are shown in Fig. 3. Compared to their counterparts
in $pp$ collisions, the $\bar{B}_{s}^{0}/B^{-}$ ratio exhibits a significant
enhancement up to $p_{T}\sim$10 GeV, resulting from $b$-quark coupling to the
enhanced strangeness in QGP through recombination; an enhanced
$\Lambda_{b}^{0}/B^{-}$ ratio appears in the intermediate-$p_{T}$ region due
to a stronger flow effect on generally heavier baryons as captured by RRM with
SMCs, peaking at a higher $p_{T}\sim$6 GeV and extending to significantly
larger $p_{T}\sim$15 GeV than the corresponding ratio in the $c$ sector
He:2019vgs ; ALICE:2021bib  because of the larger $b$ quark mass. The
$\Xi_{b}^{0,-}/B^{-}$ ratio develops a more pronounced enhancement as it
combines the strange-quark and baryon features. The nuclear modification
factors, $R_{AA}$, defined as the ratio of $p_{T}$ differential yield in PbPb
collisions to the cross section in $pp$ collisions scaled by the nuclear
overlap function ALICE:2018tvk , are shown in Fig. 4(a) for ground-state
$B^{-}$ (same for $\bar{B}^{0}$), $\bar{B}_{s}^{0}$, $\Lambda_{b}^{0}$ and
$\Xi_{b}^{0,-}$ in semicentral PbPb collisions, with an expected hierarchy of
flow effects and suppression driven by their quark content. Upon weak-decay of
these hadrons into non-prompt $c$ hadrons utilizing Pythia8 Sjostrand:2014zea
, the resulting $R_{AA}$’s for non-prompt $D^{0}$ and $D_{s}^{+}$ in central
collisions show fair agreement with ALICE data, cf. Fig. 4(b).

Summary.— Employing the statistical hadronization model we have evaluated the
hadro-chemistry of $b$ hadrons in $pp$ collisions at collider energies. The
spectrum of $b$-hadrons has been taken from theoretical predictions of the
relativistic quark model which is largely supported by lattice-QCD
computations of vacuum spectroscopy. Many of the RQM states, especially in the
baryon sector, are not yet observed and are therefore much more numerous than
the current PDG listings. With strong and electromagnetic feedown estimated
from the ${}^{3}P_{0}$ model, we have performed quark model estimates of
excited $b$-hadron decays which enabled a comprehensive prediction of
fragmentation fractions of weakly-decaying $b$ hadrons for the first time;
pertinent ratios turn out to agree with measurements in $p\bar{p}$ collisions
at the Tevatron. We have further calculated $p_{T}$ differential cross
sections for ground-state $b$ hadrons using fragmentation weights of an
underlying (perturbative) $b$-quark spectrum determined by the SHM. The
resulting $p_{T}$-dependent $\bar{B}_{s}^{0}/B^{-}$ and
$\Lambda_{b}^{0}/B^{-}$ ratios agree with LHCb data. All of our results
critically depend on the excited states that the RQM predicts beyond the PDG
listings. We have furthermore deployed the new hadro-chemsitry into our
strongly-coupled HF transport model for heavy-ion collisions, to evaulate
spectral modifications in PbPb collisions. The $p_{T}$-dependent modifications
of ratios between different ground-state $b$ hadrons have been quantified,
highlighting the role of $b$ quarks as probes of the QGP – a central pillar of
experimental efforts at both RHIC Dean:2021rlo  and the LHC in the near future
Citron:2018lsq ; ALICE3:leterofintent2022 ; LHCb:2022ine ; ALICE:2022wpn .

Acknowledgments.– This work was supported by NSFC grant 12075122 and the U.S.
NSF under grant no. PHY-1913286. MH thanks Fabrizio Grosa for help with the
simulation of weak decays of $b$ hadrons.

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