# GLaM: Fine-Tuning Large Language Models for Domain Knowledge Graph Alignment
via Neighborhood Partitioning and Generative Subgraph Encoding

Stefan Dernbach1, Khushbu Agarwal1,
Alejandro Zuniga1, Michael Henry1, Sutanay Choudhury1

###### Abstract

Integrating large language models (LLMs) with knowledge graphs derived from
domain-specific data represents an important advancement towards more powerful
and factual reasoning. As these models grow more capable, it is crucial to
enable them to perform multi-step inferences over real-world knowledge graphs
while minimizing hallucination. While large language models excel at
conversation and text generation, their ability to reason over domain-
specialized graphs of interconnected entities remains limited. For example,
can we query a LLM to identify the optimal contact in a professional network
for a specific goal, based on relationships and attributes in a private
database? The answer is no – such capabilities lie beyond current methods.
However, this question underscores a critical technical gap that must be
addressed. Many high-value applications in areas such as science, security,
and e-commerce rely on proprietary knowledge graphs encoding unique
structures, relationships, and logical constraints. We introduce a fine-tuning
framework for developing Graph-aligned LAnguage Models (GLaM) that transforms
a knowledge graph into an alternate text representation with labeled question-
answer pairs. We demonstrate that grounding the models in specific graph-based
knowledge expands the models’ capacity for structure-based reasoning. Our
methodology leverages the large-language model’s generative capabilities to
create the dataset and proposes an efficient alternate to retrieval-augmented
generation styled methods.

## Introduction

Large language models (LLMs) have recently demonstrated disruptive potential
with their ability to generate text and answer questions with human-like
language proficiency. However, their reasoning remains limited by a reliance
solely on textual training data, lacking integration with structured knowledge
graphs encoding intricate real-world constraints and relationships. Bridging
this divide by aligning LLMs with multi-relational graphs can enable grounded,
factual inferences vital for applications driven by graph-structured data.

Past work on LLM-graph integration has predominantly focused on harnessing LLM
knowledge to improve graph neural network performance on tasks like node
classification and link prediction (Jin et al. 2023). The alternate direction
of augmenting or “fine-tuning” LLMs to reason over graphs has remained
relatively unexplored. For instance, existing techniques still treat knowledge
bases as external retrievable stores (Lewis et al. 2020), rather than
integrating them into model parameters. Using the LLM as an encoder to
transform text-based node and edge labels in a graph, and then fusing the LLM
and GNN-derived representations has been the dominant approach for diverse
applications ranging from product recommendation (Choudhary et al. 2022) to
biomedical question-answering in a multiple-choice setting (Yasunaga et al.
2022).

Our work is the first study on incorporating domain-specific knowledge graphs
directly into LLM representations via fine-tuning, targeting accuracy
improvements on open ended question answering(QA), a more complex task than
the multiple choice setting explored in previous works. By encoding both
schema and entities within specialized graphs like those in biomedical
repositories, recommendation systems and social networks, we can enhance
multi-hop reasoning grounded by real-world constraints. This addresses the
challenge of factual hallucinations in free-form reasoning, while retaining
versatile text handling strengths (Touvron et al. 2023; Nori et al. 2023).

Figure 1: Motivating examples for aligning foundational models with domain-
specific knowledge graphs. The left figure demonstrates a query where a LLM
needs to be integrated with a knowledge graph derived from a social network.
The right figure demonstrates a need where a LLM needs to be integrated with a
patient-profiles to disease network extracted from an electronic healthcare
records database.

### Problem Definition

Our work targets natural language question answering (QA) on graph data. We
define $f_{LLM}:\mathcal{V}\rightarrow\mathcal{V}$ as a functional
representation of a large language model that accepts a sequence of high-
dimensional discrete tokens from a vocabulary $\mathcal{V}$ as input and
produces an output sequence drawn from the same space. Given a natural
language question $Q$ (also referred to as a prompt), $f_{LLM}(\cdot)$
tokenizes $Q$ into a sequence of tokens from $\mathcal{V}$ and returns an
answer $A=f_{LLM}(\cdot)(Q)$.

Next, we introduce a graph dataset $G=(V,E)$, where $V$ is the set of vertices
and $E$ is the set of edges. Importantly, we assume that $G$ was not included
in the training data for $f_{LLM}(\cdot)$. Figure 1 describes real-world use
cases that motivate these graph QA workloads, such as social or professional
network-based recommendations or patient-specific clinical hypothesis
generation.

Our goal is to introduce a new function $f_{GLM}$ that utilizes information
from $G$ to answer question $Q$. Formally, $A=f_{GLM}(G,Q)$. In this paper, we
systematically explore following three query classes, in both open ended
question answering and multiple choice setting:

  1. 1.

Fact recall: Evaluates GLM’s ability to recall domain facts seen during
training (e.g. answering ”What are possible treatments for diabetes?” after
seeing ”Diabetes is treated with insulin and metformin”).

  2. 2.

Inverse fact recall: Assesses handling of relationship directionality, which
recent work shows standard LMs struggle with (”A is B” does not imply ”B is
A”) (Berglund et al. 2023). This is a key facet of graphs not previously
explored for LLM-graph models.

  3. 3.

Chain-of-Reasoning: Complex queries such as Figure 1 (left) that necessitate
appropriately employing graph structure knowledge.

### Technical Approach and Related Work

Exploring the intersection of large language models and knowledge graphs has
strong significant interest over the past few years. We begin by outlining key
design paradigms from literature for answering complex reasoning queries on
knowledge graphs by aligning graph data with large language models (LLMs) and
refer the reader to a collection of excellent survey articles for a detailed
overview of this emerging sub-field (Pan et al. 2023; Liu et al. 2023; Jin et
al. 2023). Any approach must address two questions: 1) how to encode graph $G$
into the LLM’s knowledge representation, and 2) how query $Q$ is executed.

Delegation to a GNN A common approach uses a graph neural network as encoder.
Given a natural language query $Q$ this requires extracting entities and
relations from the query and integrate GNN and LLM representations. Such
integration can be done via learning a joint-model coupling the LLM and GNN
representations (Saxena, Tripathi, and Talukdar 2020; Yasunaga et al. 2022) or
using a soft prompting approach that inserts a GNN-derived vector embedding
into the LLM prompt (Tian et al. 2023).

Retrieval Augmented Generation Retrieval augmented generation (RAG) approaches
follow a similar path of implementation. The difference here is that instead
of delegating to a GNN, an external graph database (Jiang et al. 2023) or a
vector database (Tian et al. 2023) containing node and/or relation embeddings
is queried . In both approaches, the LLM is used as an routing interface to a
native graph database or machine-learning model, and the answers from the
appropriate graph-based component is fed back to user with LLM serving as
generative layer that produces the final answer.

Few-Shot Prompting In this approach subgraphs relevant to $Q$ are extracted
and inserted into the prompt with examples (Fatemi, Halcrow, and Perozzi
2023). While promising, this approach faces potential drawbacks requiring
encoding full graphs in LLM prompts or performing multi-hop subgraph
retrievals for each question.

Motivation for Fine-Tuning Irrespective of their differences, all of the above
approaches potentially face a fundamental limitation - they cannot
contextually integrate symbolic constraints to shape intermediate reasoning.
Approaches retrieving embeddings or triples solely based on the initial query
overlook multifaceted execution where dynamic routing with mixture-of-experts
(Shazeer et al. 2017; Zhou et al. 2022), planning (Hao et al. 2023; Yao et al.
2023) and heuristics search (Sprueill et al. 2023; Sun et al. 2023) steps
modify information needs in every reasoning step. Fixed retrieval precludes
dynamically folding graph structure into each decision point.

In contrast, fine-tuning instills domain knowledge into model parameters and
representations a priori, rather than treating the graph as an external add-
on. By encoding constraints and dependencies directly into the knowledge
substrate, fine-tuning allows contextual graph influence at each step of
modeled cognition. Rather than acting as a static look-up, the graph becomes
an integral inference component - shaping complex reasoning in a tighter, more
fine-grained fashion.

#### Our Approach and Contributions

We introduce an algorithm to iteratively partition and encode the neighborhood
subgraph around each node into textual sentences for fine-tuning data. This
transforms graph structure into a format that large language models can ingest
and fine-tune. We explore encoding strategies on two graphs: 1) UMLS - a
biomedical knowledge base, and 2) DBLP - an academic publication network.

Our work makes the following contributions.

  1. 1.

Our neighborhood partitioning and encoding scheme accommodate real-world graph
properties like skewed size distributions and sparsity. Our approach opens up
future experimental possibilities where encoding is tuned for LLMs by setting
context size limits based on cost-accuracy tradeoffs.

  2. 2.

We propose and assess five encoding approaches leveraging the LLM’s innate
summarization and text generation strengths. For example, we evaluate
neighborhood summaries produced by the LLM. Encouragingly, our results align
with similar methods from concurrent work (Fatemi, Halcrow, and Perozzi 2023),
confirming the promise of this direction.

  3. 3.

We developed a new domain question answering dataset based on two above graphs
with a suite of evaluation tasks capturing link prediction to multi-hop
reasoning queries. The code and datasets will be released as open source upon
acceptance.

## Methods

Task Definition We propose methods for transforming a knowledge graph into a
corresponding text-based fine-tuning dataset for language models. Our goal is
to produce pairs of (context, question-answer) (Ouyang et al. 2022; Wei et al.
2021) that require neural-graph reasoning to answer open domain questions
involving relational or multi-hop reasoning.

We begin with describing a generic algorithm (Algorithm 1) that encodes a
node’s k-hop neighborhood into such a context and QA-pair through a
composition of multiple operator functions. We discuss the implementation of
these operators in finer detail in the later half of the section.

Require: Graph $G$ with nodes set $V$ and edges set $E$, context subgraph node
limit $N_{max}$

Fine-tuning dataset $D\leftarrow\emptyset$

for  _each $v\in V(G)$_ do

$G_{context}(v,k)=f_{aggr}(G,v,k)$

$partitions=f_{partition}(G_{context}(v,k),N_{max}$)

for  _each $g_{sub}\in partitions$_ do

$X_{context}=f_{enc}(g_{sub})$

$X_{qa}=f_{qa}(g_{sub})$

$append(D,concat([X_{context},X_{qa}])$

end for

end for

return $D$

Algorithm 1 Fine-tuning dataset generation.

### Optimal Generation of Subgraph Contexts

For every node $v\in V(G)$, we transform the $k$-hop neighborhood of $v$ into
a set of pairs of the form: $(f_{enc}(G,v),f_{qa}(G,v))$. Algorithm 1
describes a step-by-step in which we iterate over every node in the graph and
encode it’s k-hop neighborhood subgraph, denoted as $G_{context}(v,k)$ into
the alternate text-based representation.

  1. 1.

We retrieve the k-hop neighborhood subgraph as $G_{context}(v,k)$ using a
query function denoted as $f_{aggr}(\cdot)$.

  2. 2.

$f_{enc}$ encodes $G_{context}(v,k)$ or its partitioned subgraph into text.

  3. 3.

$f_{qa}(G,v)$ generates QA pairs requiring reasoning on $G_{context}(v,k)$.
Same subgraph is used to drive the inputs for $f_{enc}(G,v)$ and
$f_{qa}(G,v)$.

  4. 4.

The concatenated output of $f_{enc}(G,v)$ and $f_{qa}(G,v)$ is a text sequence
of discrete tokens $X_{v}$ drawn from $\mathcal{V}$, the vocabulary of the LLM
function $f_{LLM}(\cdot)$ mentioned previously.

  5. 5.

Any LLM function $f_{LLM}(\cdot)$ needs to operate within a maximum token
limit constraint (denoted as $T_{max}$). We partition $G_{context}(v,k)$ to
respect LLM token limits $T_{max}$ such that $len(X_{v})<T_{max}$.

We introduce a hyperparameter $N_{max}$ to partition $G_{context}(v,k)$ into
subgraphs within node count $N_{max}$. This prevents tokenized sequence
lengths from exceeding $T_{max}$. Choosing an optimal $N_{max}$ is key because
degree distributions in $G_{context}(v,k)$ can be highly skewed. Given cost
constraints associated with $T_{max}$, we want to pick $N_{max}$ and encoding
strategies that maximize context lengths for the LLM’s capabilities.

### Neighborhood Encoding Functions

Figure 2: Illustration of Graph Encodings in GLaM: Top left box shows
”Encoding via triples”, where each line represents an edge mapped to one
training sample. The bottom left box shows graph encoding when given a node
and relation, all relevant entities are collated into single training sample.
The bottom right box shows when all relations/edges corresponding to a node
are coalesced into single training sample. and top right box demonstrates the
impact of summarization on the training sample. Summarizing helps to 1) map
unwieldy node labels into human interpretable form, 2) reduce redundant terms,
and 3) reduce overfitting to frequent node and edge labels. Collectively this
leads to better semantic alignment betweeen the knowledge graph and LLM’s
vocabulary and improves resulting model performance in all graph tasks.

The purpose of a neighborhood encoding function is to translate the
neighborhood subgraph $G_{context}(v,k)$ centered around a node $v$ into a
textual representation that can be effectively processed by a large language
model (LLM). This process is crucial for enabling the LLM to perform higher-
order reasoning and answer complex questions about the graph.

There are two main factors that influence the choice of a neighborhood
encoding function:

  1. 1.

Communicating Graph Structure and Higher-Order Reasoning Requirements to the
LLM: The encoding function should effectively capture the structural
relationships between nodes in the subgraph, as well as any higher-order
logical dependencies that may exist. This can be achieved by incorporating
information about the edges and their types, as well as the relationships
between multiple nodes.

  2. 2.

Semantic Alignment with the LLM’s Internal Knowledge Representation: The
encoding should represent the nodes and relations in the graph in a way that
is consistent with how the LLM stores and interprets information. This can
involve using natural language labels for nodes and edges, or generating
descriptive labels using a node’s neighborhood when node labels are not
recognizable to a LLM (such as an academic network), while ensuring that the
encoded representation preserves the semantic meaning of the graph elements.

Encoding via Triples: A simple approach to neighborhood encoding is to
translate the edge data into (source, relation, target) triples. This provides
the LLM with basic information about the relationships between nodes, but it
is limited to representing only single edges per training sample and has
limited context size.

Encoding via Adjacency List/Relational Groups To enable the LLM to perform
more complex reasoning tasks, we update the neighborhood encoding to include
information about multiple nodes in the subgraph. We experiment with two
different options: including the entire adjacency list of the central node
$v$, and by partitioning the neighbors into subsets based on their relation
types. We observe that more sophisticated approaches, such as sampling
techniques are relevant for large neighbor lists but are not implemented in
current work.

Encoding via Summarization Next, we focus on the semantic alignment objective
and use prompting methods to rewrite the encoding from above methods into more
coherent representations (Figure 2).

  * •

The promptimg allows us to map unwieldy node labels to human understandable
terms: For example,“Insulin human, rDNA origin” is mapped by LLM to “Insulin
therapy from recombinant DNA” allowing for better interpretation during fine-
tuning.

  * •

It reduces redundant text from similarly labeled nodes: “Diabetes mellitus,
Type 1 diabetes mellitus, Type 2 diabetes mellitus” is mapped to “diabetes,
including Type 1 and Type 2 diabetes.”

  * •

Introduces additional knowledge/synonyms into training: “Hypoinsulinaemia” is
mapped to “low insulin levels (hypoinsulinaemia),” and “rDNA” is expanded to
“recombinant DNA.”

  * •

Prompt-based rewriting also helps reduces to address overfitting training to
only a few relation labels, by mapping them to different phrases. Examples of
such overfitting was observed with the “may treat” relationship, where the
high number of occurrence of this phrase in a specific pattern causes the LLM
to generate answers incorrectly filled with too many occurrences of the “may
treat” phrase.

Encoding via Node Descriptors The previous encoding step leveraged the LLM’s
understanding of specific entities (such as “rDNA”) to rewrite with maximal
semantic alignment. However, training on new graph data can include unfamiliar
terms to the LLM, i.e. words or phrases that appear rarely or do not occur at
all in initial training. A common example of this problem involves encoding
the names of people not common to standard LLM training datasets. Also, we do
not want to map a person based on their name, but account for their profile
attributes or k-hop connectivity in the network. We generalize this need by
transforming the k-hop context subgraph ($G_{context}(v,k)$) into a set of
text-based node descriptors by leveraging the LLM’s zero-shot capabilities.
Typically, this is a step where an alternate implementation would have
retrieved a GNN representation. For example, to expand on the information
about authors in the DBLP dataset, we prompt the LLM to extract the topic
areas of paper abstracts and construct a list of topics the author has
published on from their paper history.

Generating Question-Answer Pairs Finally, given a text context generated from
a subgraph $G_{context}(v,k)$, we generate a set of question-answer pairs via
prompting the text context for different tasks (fact recall, inverse fact
recall, multi-hop question answering). Each of the questions are also mapped
into two style of answers: 1) open-domain question-answering, and 2) multiple-
choice questions. For example, given a (head, relation, tail) triple as the
subgraph context, its multiple choice answer candidates are generated by
including one of the tail entities and a random selection of other nodes in
the graph to form a set of possible answers to the question.

## Experiments

| Fact Recall | Reverse Recall | Multi-hop Reasoning
---|---|---|---
| P | R | F1 | P | R | F1 | P | R | F1
Llama 7B Chat | $0.594$ | $0.631$ | $0.608$ | $0.382$ | $0.519$ | $0.439$ | $0.595$ | $0.631$ | $0.609$
GLaM (Triples) | $0.683$ | $0.597$ | $0.627$ | $\mathbf{0.431}$ | $0.533$ | $\mathbf{0.474}$ | $0.677$ | $0.589$ | $0.621$
GLaM (Relational Grouping) | $0.679$ | $0.678$ | $0.673$ | $0.403$ | $\mathbf{0.537}$ | $0.459$ | $0.662$ | $0.663$ | $0.657$
GLaM (LLM Summarization) | $\mathbf{0.724}$ | $\mathbf{0.725}$ | $\mathbf{0.720}$ | $0.386$ | $0.527$ | $0.445$ | $\mathbf{0.689}$ | $\mathbf{0.696}$ | $\mathbf{0.688}$
Llama 13B Chat | $0.699$ | $0.623$ | $0.652$ | $0.396$ | $0.529$ | $0.451$ | $0.695$ | $0.623$ | $0.650$
GLaM 13B (LLM Summarization) | $0.708$ | $0.730$ | $0.714$ | $0.395$ | $0.534$ | $0.453$ | $0.675$ | $0.697$ | $0.681$
Table 1: UMLS Results comparing the baseline Llama LLM with three versions of a refined GLaM on questions generated from the UMLS knowledge graph. Each version corresponds to an encoding strategy described in the Methods section. Precision (P), Recall (R), and F1 scores are reported using Bert scores. | Fact Recall | Reverse Recall | Multi-hop Reasoning
---|---|---|---
| P | R | F1 | P | R | F1 | P | R | F1
Llama 7B Chat | $0.174$ | $0.177$ | $0.175$ | $0.168$ | $0.173$ | $0.170$ | $0.168$ | $0.171$ | $0.169$
GLaM (Triples) | $0.105$ | $0.103$ | $0.104$ | $0.103$ | $0.102$ | $0.102$ | $0.100$ | $0.099$ | $0.099$
GLaM (Relational Grouping) | $0.259$ | $0.261$ | $0.259$ | $0.259$ | $0.264$ | $0.260$ | $0.256$ | $0.259$ | $0.257$
GLaM (Adjacency List) | $0.255$ | $0.258$ | $0.255$ | $0.247$ | $0.252$ | $0.249$ | $0.251$ | $0.253$ | $0.251$
GLaM (Node Descriptors) | $0.313$ | $0.312$ | $0.312$ | $0.309$ | $0.314$ | $0.311$ | $0.318$ | $0.316$ | $0.316$
GLaM -7B | $\mathbf{0.424}$ | $\mathbf{0.426}$ | $\mathbf{0.424}$ | $\mathbf{0.401}$ | $\mathbf{0.407}$ | $\mathbf{0.402}$ | $\mathbf{0.409}$ | $\mathbf{0.410}$ | $\mathbf{0.408}$
Llama 13B Chat | $0.150$ | $0.155$ | $0.152$ | $0.144$ | $0.151$ | $0.147$ | $0.153$ | $0.159$ | $0.155$
GLaM -13B | $0.446$ | $0.446$ | $0.445$ | $0.381$ | $0.385$ | $0.382$ | $0.398$ | $0.398$ | $0.397$
Table 2: DBLP Results comparing the baseline Llama LLM with five versions of GLaM on questions generated from the DBLP citation graph. GLaM-7B/13B represents a combination of strategies: aggregation of node descriptors, utilizing adjacency lists as context and performing summarization. Precision (P), Recall (R), and F1 scores are reported using Bert scores. | UMLS | DBLP
---|---|---
| Fact | Reverse | Multi-hop | Fact | Reverse | Multi-hop
| Recall | Fact Recall | Reasoning | Recall | Fact Recall | Reasoning
Llama 7B Chat | $61.23$ | $57.21$ | $61.1$ | $35.26$ | $36.19$ | $27.99$
GLaM -7B-MC | $100$ | $59.71$ | $91.93$ | $78.62$ | $75.68$ | $73.34$
Table 3: Multiple Choice Results Comparison of LLM and GLaM accuracy for fact
recall, reverse fact recall, and fact inference on the UMLS and DBLP graphs.

In this section, we address following research questions (RQ) through
experimental analysis:

  1. 1.

RQ-1 Does finetuning using graph encoding improve an LLM’s ability to recall
the facts?

  2. 2.

RQ-2 Does finetuning an LLM with graph encoding improve its ability to answer
open-domain natural language questions through performing multi-hop reasoning
on the graph domain?

  3. 3.

RQ-3 Which strategies for encoding the subgraph context yields maximal
semantic alignment of the original LLM and the target graph?

### Datasets

We present the results of training GLaMs on two graph datasets, DBLP (Tang et
al. 2008) and UMLS (Bodenreider 2004), with diverse applications and coverage
in LLMs to demonstrate the response improvement over the baseline language
models.

Unified Medical Language System111https://www.nlm.nih.gov/research/umls (UMLS)
(Bodenreider 2004) is a medical knowledge graph. We use a processed version of
the knowledge graph from Yasunaga et al. (Yasunaga et al. 2022) consisting of
$297,927$ concepts, $98$ relation types and $1,212,586$ edges that capture
relationships across a breadth of medical concepts. For GLaM training, we
select a subgraph that captures relationships between different diseases,
symptoms and medications. This results in a reduction to 4 different relation
types: “cause of”, “may cause”, “risk factor of”, and “may treat” totalling
$126,149$ triples.

DBLP222https://www.aminer.org/citation (Tang et al. 2008) is a citation graph
dataset extracted from DBLP, ACM, MAG, and other sources. The dataset includes
paper citations, abstracts, authors, publication years, venues, and titles.
For training the GLaM we focus on the set papers containing titles, abstracts,
venues, and $2$ or more authors, leading to $19,577$ unique papers.

### Training and Inference setup

For both UMLS and DBLP, the extracted natural language questions and answers
were split into 70% training (fact recall) and 30% test (multi-hop reasoning).
We used Microsoft Deepspeed framework (Rasley et al. 2020) for supervised
prompt and response fine-tuning. A grid-search was performed over training
hyper-parameters using Llama-7b-chat-hf as the base model for training GLaM. A
learning rate of $1e-5$ and a cosine learning rate scheduler were used with
the fused Adam optimizer with bfloat16 precision. The maximum sequence length
was set to $256$ and a maximum of $4$ training epochs were used for all
models. A cluster of 8 A100 GPUs with 80GB of GPU memory each were used for
training with a per-device batch size of 4 questions resulting in a total
training batch size of 32. We use Llama-2-7b-chat-hf and Llama-2-13b-chat-hf
(Touvron et al. 2023) models from Hugging Face as the baseline models for
training. Training the 7b model on UMLS takes approximately 9 minutes and 16
minutes for the 13b model. For DBLP, training time is approximately 11 and 21
minutes respectively.

### Evaluation Tasks

Fact recall: This task is equivalent to question answering tasks in language
models and test GLaM’s ability to remember domain level facts seen during
training. For example, given a training sentence such as ‘‘Diabetes is treated
with insulin and metformin” (from UMLS), the model is queried for “What are
possible treatment of diabetes?”. Similiarly, for the DBLP dataset given a
sentence such as “[Students learn CS in different ways: insights from an
empirical study] was written by Anders Berglund.”, the model is queried with
“[Students learn CS in different ways: insights from an empirical study] was
written by whom?” The UMLS question set for fact recall contains $7710$
questions and the DBLP set contains $13,704$.

Inverse Fact Recall: This task is equivalent to reverse question answering
tasks (Berglund et al. 2023) in language models and test GLaM ’s ability to
infer reverse relationships from the domain level facts seen during training.
For example, given the above training statement, the model is queries for
“Which disease can be treated with insulin?” There are $11130$ questions in
the UMLS reverse fact recall question set and $13704$ in the DBLP set.

Multi-hop Reasoning: This task mirrors the link prediction task in a GNN
setting and tests the GLaM’s ability to infer new facts (graph edges) by
reasoning over facts seen during training. The UMLS question set for multi-hop
reasoning contains $3347$ questions and the DBLP set contains $5873$. A common
style of question we explore for DBLP is that of recommending authors to
collaborate with. Using the DBLP question referred to in the fact recall task
as example, a multi-hop reasoning question would ask: “Anders Berglund would
like to write a paper titled [Students learn CS in different ways: insights
from an empirical study] to publish in Proceedings of Australasian computing
education. Who should they work with and why?”

Multiple-choice: Each evaluation task: fact recall, inverse fact recall, and
multi-hop reasoning, are reformatted as multiple choice questions. Question
includes the correct answer and four additional incorrect options randomly
selected from the graphs respectively. Note that this is a much easier task
than open ended question answering setting, requiring models to only pick the
most likely answer out of given options.

### Evaluation Metrics

To account for the inherent text variability in LLM or GLaM generated answers,
we use the BERTScore metric (Zhang et al. 2019) for open-ended domain QA
setting and accuracy for multiple choice questions.

Bert Score: Compares text similarity between the model generated response to
the expected response. The microsoft/deberta-xlarge-mnli model (He et al.
2020) is used for calculating BertScore for it’s strong performance in natural
language understanding (NLU) tasks. We report precision (P), recall (R), and
F1 scores across the evaluation set.

Accuracy: We use the standard accuracy measure to evaluate a model’s ability
to identify the correct answer out of 5 possible choices in a multiple choice
setting.

### Results

Results for training GLaM are presented in Table 1 and Table 2. We discuss the
results on the individual datasets and then provide overall conclusions.

UMLS graph experiment results are given in Table 1. For both fact recall and
inference, using LLM based summarization encoding to rewrite the statements
exhibits the best performance across precision, recall and F1 scores. However,
for reverse fact recall, using the simpler training approaches leads to a
slight improvement in scores. All fine-tuned GLaM versions outperform the
baseline LLM showing that even naive training strategies offer some
improvement over the baseline LLM. While the 13b version of Llama outperforms
its 7b counterpart, once trained, there is negligible difference between the
13b and 7b GLaM .

DBLP citation graph experimental results are given in Table 2. The GLaM
version with complete adjacency and LLM based summarization achieves the best
results across all tasks. Unsurprisingly, the untrained LLM did only
moderately better than random guessing for the multiple choice task because of
the number of unfamiliar names in the dataset. There is also a general trend
of improved performance as neighborhood information is collated into the
training, with the exception of adding the venue of the publication not having
a noticeable affect,likely due to title being sufficient to capture a
publications context. There is a slight improvement of the 13b version of GLaM
over the 7b version for fact recall but the 7b version slightly outperforms
the larger GLaM on the reverse fact recall and fact inference tasks. This
combined with similar findings on UMLS indicate that the smaller LLM is
sufficient for fact retention and inference when fine-tuned for the domain.

Multiple Choice results for both UMLS and DBLP are provided in Table 3. Across
all tasks, GLaM outperforms the unrefined LLM, with the smallest difference
being on the reverse facts for UMLS where GLaM noticeably does not learn to
infer the inverse relationships from training. For UMLS fact recall GLaM
demonstrates $100\%$ accuracy on recalling the answers to the training set and
similarly performs extremely well on the multi-hop reasoning questions. We
hypothesize that the even larger gap between LLM and GLaM on the multiple
choice results compared to the difference on the open ended question results
comes from GLaM learning to differentiate the good answers from poor ones even
if it does not explicitly know the correct answer.

Graph Aligned Language Models Significantly Improve Domain Knowledge Retrieval
Tasks. Large language models are extraordinary tools for general knowledge but
can not produce answers to many domain specific questions modeled in complex
networks. This is evidenced by GLaM outperforming LLM across all domain level
tasks, including simple fact retrieval questions.

Increasing the Node Neighborhood Context During Training Improves Inference
Performance. Both the UMLS (Table 1) and DBLP (Table 2) cases demonstrate that
incorporating multiple edges into each training instance improves the language
models recall and reasoning. This is evident as GLaM training evolves from
single triple samples, to relations with multiple targets, and further to
include additional neighborhood information such as the topic areas an author
publishes in.

Node Context Summarization Using LLM Improves Learning. Using a LLM to rewrite
or summarize statements produced from the node neighborhood encoding improves
GLaM’s fact recall and multi-hop reasoning as shown on Table 1. The LLM
summarized version for the UMLS graph encoding outperforms other GLaM versions
even if the same information is present in training. We postulate that
variation in word choice, mapping of node labels to more interpretable names
helps significantly improve the learning process.

## Conclusions and Future Work

We demonstrate an effective approach to integrate domain-specific knowledge
graphs into large language models via fine-tuning. Empirically, this technique
yields significant gains in multi-hop reasoning ability over the base LLM. Our
proposed fine-tuning method encodes graph structure and it’s semantic
knowledge into the LLM, by maximally leveraging the original LLM’s strengths -
textual understanding, commonsense knowledge and generative capabilities.

In particular, quantitative experiments verify F1 score improvements of 18% on
fact recall and 13% on complex inference queries requiring multi-hop reasoning
on the UMLS domain for which the LLM already has some knowledge, and 142% and
141% respectively on DBLP’s social network structure which represents novel
information for the LLM. Given the importance of directionality of
relationships in a graph, we also measure the improvement of recalling inverse
facts by the resulting model. Overall, our experiments while preliminary in
nature, confirm that integration via fine-tuning instills more reliable
reasoning capacity based on graphs containing specialized entities and
relationships, and it enables tighter coupling of structured symbolic
knowledge with learned representations. Evaluating the effectiveness of the
partitioning and encoding schemes across a wider range of larger-scale graphs
with highly uneven connectivity distributions are candidate for future work.

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aainstitutetext: Bethe Center for Theoretical Physics and Physikalisches
Institut, Universität Bonn
Nussallee 12, 53115 Bonn, Germanybbinstitutetext: PRISMA+ Cluster of
Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg
University, 55099 Mainz, Germany

# Parameter Space of Leptogenesis in Polynomial Inflation

Manuel Drees b and Yong Xu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Polynomial inflation is a very simple and well motivated scenario. A potential
with a concave “almost” saddle point at field value $\phi=\phi_{0}$ fits well
the cosmic microwave background (CMB) data and makes testable predictions for
the running of the spectral index and the tensor to scalar ratio. In this work
we analyze leptogenesis in the polynomial inflation framework. We delineate
the allowed parameter space giving rise to the correct baryon asymmetry as
well as being consistent with data on neutrino oscillations. To that end we
consider two different reheating scenarios. $(i)$ If the inflaton decays into
two bosons, the reheating temperature can be as high as $T_{\text{rh}}\sim
10^{14}$ GeV without spoiling the flatness of the potential, allowing vanilla
$N_{1}$ thermal leptogenesis to work if $T_{\text{rh}}>M_{1}$ where $N_{1}$ is
the lightest right–handed neutrino and $M_{1}$ its mass. Moreover, if the
dominant decay of the inflaton is into Higgs bosons of the Standard Model, we
find that rare three–body inflaton decays into a Higgs boson plus one light
and one heavy neutrino allow leptogenesis even for $T_{\text{rh}}<M_{1}$ if
the inflaton mass is of order $10^{12}$ GeV or higher; in the polynomial
inflation scenario this requires $\phi_{0}\gtrsim 2.5~{}M_{P}$. This novel
mechanism of non–thermal leptogenesis is quite generic, since the coupling
leading to the three–body final state is required in the type I see–saw
mechanism. $(ii)$ If the inflaton decays into two fermions, the flatness of
the potential implies a lower reheating temperature. In this case inflaton
decay to two $N_{1}$ still allows successful non–thermal leptogenesis if
$\phi_{0}\gtrsim 0.1~{}M_{P}$ and $T_{\text{rh}}\gtrsim 10^{6}$ GeV.

MITP-24-001

January 2024

##  1 Introduction

Monomial inflationary scenarios have been ruled out by recent Planck and
BICEP/Keck 2018 measurements Planck:2018jri ; BICEP:2021xfz . We have recently
shown that a simple polynomial of degree four with a concave “almost” saddle
point at field value $\phi=\phi_{0}$ can nevertheless fit well the CMB data
Drees:2021wgd ; Drees:2022aea . The model predicts the running of the spectral
index to be $\mathcal{O}(10^{-3})$, which may be testable by the next
generation of cosmic microwave background (CMB) experiments in combination
with data on structure formation at smaller scales Munoz:2016owz . The
predicted tensor to scalar ratio $r$ (in the large field scenario) can
saturate the current bound.

A full model of the very early Universe also has to describe reheating, i.e.
transition to a thermal stage. We have considered two simple scenarios of
perturbative reheating where the inflaton decays via renormalizable couplings,
either to a pair of bosons [e.g. the standard model (SM) Higgs] or to
vector–like fermions [e.g. right-handed-neutrinos (RHNs)]. We have obtained
the full allowed parameter space of the model by $(i)$ considering the lower
bound on $T_{\text{rh}}$ from Big Bang nucleosynthesis (BBN) as well as the
stability of the inflaton potential against radiative corrections which leads
to an upper bound on $T_{\text{rh}}$ for given $\phi_{0}$; and $(ii)$
confronting the model with the latest CMB data, which leads to an upper bound
on $\phi_{0}$ and hence on the inflaton mass. In combination we find that the
inflection point has to satisfy $3\times 10^{-5}\lesssim\phi_{0}/M_{P}\lesssim
21.5$, which also allows us to derive the allowed range of reheating
temperature. It can be as large as $T_{\text{rh}}\sim 10^{14}$ GeV for bosonic
inflaton decays.

Two further ingredients are required for a realistic cosmological model: it
has to explain the existence of Dark Matter (DM), and it has to generate an
excess of baryons over antibaryons. Recently it has been shown Bernal:2021qrl
that in polynomial inflation a rather large parameter space exists for either
thermal or non–thermal DM production after inflation. In this work we show
that the model can also produce the baryon asymmetry. To this end, we focus on
the most simple “vanilla leptogenesis” in the framework of the type I seesaw
mechanism. Here a lepton asymmetry is generated (solely) from the decay of the
lightest right–handed neutrino $N_{1}$, and the model parameters are chosen
such that the measured squared mass differences and mixing angles of the light
neutrinos are reproduced. $N_{1}$ can be produced either thermally
Fukugita:1986hr ; Buchmuller:2004nz ; Covi:1996wh ; Fong:2012buy  or
non–thermally (via inflaton decay) Lazarides:1990huy ; Giudice:1999fb ;
Asaka:1999yd ; Asaka:1999jb ; Senoguz:2003hc ; Hahn-Woernle:2008tsk ;
Antusch:2010mv ; Croon:2019dfw , depending on its mass and on the reheating
temperature.

In the framework of non–thermal leptogenesis, so far only the inflaton two
body decay to RHN, $\phi\rightarrow N_{1}N_{1}$, has been considered in the
literature, see e.g. Refs. Lazarides:1990huy ; Giudice:1999fb ; Asaka:1999yd ;
Asaka:1999jb ; Senoguz:2003hc ; Hahn-Woernle:2008tsk ; Antusch:2010mv ;
Croon:2019dfw . Here we point out that non–thermal leptogenesis is possible
even if the dominant inflaton decay is into two SM Higgs bosons $H$; this is
the simplest, and hence best motivated, bosonic inflaton decay mode. Type I
seesaw requires the existence of $HN\nu$ couplings, so inflaton decays
$\phi\to H\ell N_{1}$ will occur, albeit with suppressed rate, as long as
$m_{\phi}>M_{1}$. In this work we will show that such three body decays can
also give rise to successful non–thermal leptogenesis even without introducing
any direct couplings between the inflaton and RHNs. We analyze this scenario
again for polynomial inflation; however, the only relevant inflationary
parameters are the reheat temperature and the inflaton mass, so this mechanism
obviously can be applied to other models of inflation as well.

The remainder of this paper is organized as follows. In Sec. 2 we introduce
the Lagrangian of our model. In Sec. 3 we briefly review polynomial inflation
Drees:2021wgd ; Drees:2022aea . Reheating is reviewed in Sec. 4, while the
masses and mixing angles of the light neutrinos are discussed in Sec. 5, with
focus on normal ordering (i.e. an effectively massless lightest neutrino). In
Sec. 6 we investigate both thermal and non–thermal channels for leptogenesis.
We sum up our findings in Sec. 7. The calculation of the branching ratio for
three–body inflaton decays is described in the Appendix.

##  2 The Model Setup

Applying Occam’s razor Occam:1495nn , we minimally extend the Standard Model
(SM) with a scalar inflaton field $\phi$ and three right-handed Majorana
neutrinos (RHNs) $N_{I}$ with $I=1,2,3$. Insisting that all non–gravitational
interactions are renormalizable leads to the following action:

$S=\int d^{4}x\,\sqrt{-g}\left(\mathcal{L}_{\text{Gravity}}+\mathcal{L}_{\rm
SM}+\mathcal{L}_{\phi}+\mathcal{L}_{N}+\mathcal{L}_{H\phi}\right)\,.$ (1)

Here $g$ is the determinant of the metric; we assume the latter to have the
Friedmann–Lemaître–Robertson–Walker (FLRW) form, defined by
$g_{\mu\nu}=\text{diag}(1,-a^{2},-a^{2},-a^{2})$, with $a$ denoting the scale
factor. Moreover, $H$ in eq.(1) denotes the SM Higgs doublet, and
$\mathcal{L}_{\text{Gravity}}$ is the Einstein–Hilbert Lagrangian:

$\mathcal{L}_{\text{Gravity}}=\frac{M_{P}^{2}}{2}\,R\,,$ (2)

with $R$ being the Ricci scalar and $M_{P}\simeq 2.4\times 10^{18}$ GeV the
reduced Planck mass.

Furthermore, $\mathcal{L}_{\rm SM}$ in eq.(1) is the usual SM Lagrangian,
while $\mathcal{L}_{\phi}$ describes the inflaton sector:

$\mathcal{L}_{\phi}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\,.$
(3)

We consider the most general renormalizable inflaton potential $V(\phi)$:111A
possible linear term can be absorbed by shifting the field $\phi$, and a
possible additive constant should be tiny in order to produce an (almost)
vanishing cosmological constant after inflation.

$V(\phi)=b\,\phi^{2}+c\,\phi^{3}+d\,\phi^{4}.$ (4)

The SM and inflation sectors can couple via

$\mathcal{L}_{H\phi}=-V(H,\phi)=-\lambda_{12}\,\phi\,H^{\dagger}H-\frac{1}{2}\lambda_{22}\,\phi^{2}\,H^{\dagger}H\,;$
(5)

in particular, the coupling $\lambda_{12}$ allows perturbative reheating via
$\phi\rightarrow HH$ decays.

The final ingredient of the action (1) is the piece of the Lagrangian
describing the RHNs, including possible interactions with the inflaton. It is
given by:

$\displaystyle\mathcal{L}_{N}=i\,\overline{N_{I}}\,\gamma^{\mu}\,\partial_{\mu}N_{I}-\left(\frac{1}{2}M_{I}\overline{N^{c}_{I}}N_{I}+h.c.\right)\,-\left(Y_{\alpha
I}\,\bar{\ell}_{\alpha}\tilde{H}N_{I}+h.c.\right)-\left(\frac{y_{I}}{2}\,\phi\overline{N^{c}_{I}}N_{I}+h.c.\right)\,.$
(6)

We have worked in a basis with diagonal Majorana mass matrix for the $N_{I}$,
the $M_{I}$ denoting its eigenvalues. Moreover,
$\ell_{\alpha},\;\alpha\in\\{1,2,3\\}$ are the lepton doublets,
$\tilde{H}\equiv i\sigma_{2}H^{\star}$, and $Y_{\alpha I}$ and $y_{I}$ are
Yukawa couplings; the latter have been normalized such that $y_{I}$ appears as
$\phi N_{I}N_{I}$ vertex factor. This coupling might allow $\phi\rightarrow
N_{I}N_{I}$ decays, a second possibility for perturbative reheating. We note
that in full generality we should also allow off–diagonal $\phi N_{I}N_{K}$
couplings, since the coupling matrix to the inflaton and the $N_{I}$ mass
matrix in general cannot be diagonalized simultaneously. However, in our
application we will be interested in the lightest RHN in a hierarchical
scenario, $M_{1}\ll M_{2},\,M_{3}$; here essentially only the $\phi
N_{1}N_{1}$ coupling is relevant.

Before closing this section, we would like to comment on the relation between
the inflaton couplings to Higgs bosons and RHNs. Once one of these couplings
is introduced, the other one will be generated by one–loop diagrams.
Specifically, if $\lambda_{12}\neq 0$, a triangle diagram with two Higgs
bosons and a lepton doublet in the loop will generate a finite coupling of the
inflaton to two RHNs,
$y^{\text{eff}}_{I}\sim\lambda_{12}\,M_{I}|Y|^{2}/(16\pi^{2}m_{\phi}^{2})$,
where $m_{\phi}^{2}$ comes from the denominators of the propagators and
$M_{I}$ appears after utilizing the Dirac equation for the lepton momentum in
the numerator. Conversely, if $y_{I}\neq 0$, a triangle diagram with two RHNs
and one lepton doublet in the loop will generate a divergent inflaton
couplings to two SM Higgs bosons, $\lambda_{12}^{\text{eff}}\propto
y_{I}M_{I}|Y|^{2}$.

The loop–induced $y^{\text{eff}}_{I}$ will generate a partial width for
$\phi\rightarrow NN$ decays which is suppressed by a loop factor
$|Y|^{2}/(16\pi^{2})$ relative to the width for $\phi\rightarrow HN\ell$
decays that we will discuss in sec.6.1.2; it will thus generally not be of
great phenomenological importance. On the other hand, the fact that $y_{I}\neq
0$ generates a divergent contribution to $\lambda_{12}$ means that a theory
with nonvanishing coupling of the inflaton to RHNs but no inflaton coupling to
SM Higgs bosons is, strictly speaking, not renormalizable. Choosing
$\lambda_{12}\neq 0$ but $y_{I}=0$ is therefore in some sense “more minimal”,
since it needs fewer fundamental parameters in the Lagrangian. However, the
fact that a finite $y_{I}^{\text{eff}}$ will be generated means that this
“more minimal” set–up cannot be ensured by a (conserved) symmetry.

##  3 Inflation

In this section we review the polynomial inflation formalism presented
recently in Refs. Drees:2021wgd ; Drees:2022aea . The potential in eq. (4)
features an inflection point at $\phi=\phi_{0}=-\frac{3}{8}\frac{c}{d}$ if
$b=\frac{9}{32}\frac{c^{2}}{d}$. However, in order to match the CMB data the
potential has to have a finite (positive) slope rather than an exact
inflection point (where both the first and second derivatives of the potential
vanish). Moreover, a potential with a concave shape is favored by the Planck
2018 data Planck:2018vyg . We thus reparametrize the potential as

$\displaystyle V(\phi)$
$\displaystyle=d\left[\phi^{4}+\frac{c}{d}(1-\beta)\phi^{3}+\frac{9}{32}\left(\frac{c}{d}\right)^{2}\phi^{2}\right]$
$\displaystyle=d\left[\phi^{4}+A(1-\beta)\phi^{3}+\frac{9}{32}A^{2}\phi^{2}\right]\,,$
(7)

where $A\equiv\frac{c}{d}\equiv-\frac{8}{3}\phi_{0}$ determines the location
of the flat region of the potential and $\beta$ is introduced in order to
produce a finite slope near $\phi_{0}$; i.e. $\beta=0$ leads to a true
inflection point at $\phi_{0}$. Successful inflation results for $0<\beta\ll
1$.

We define the traditional potential slow roll (SR) parameters Lyth:2009zz :

$\displaystyle\epsilon_{V}\equiv\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\,;\
\ \eta_{V}\equiv\frac{V^{\prime\prime}}{V}\,;\ \
\xi_{V}^{2}\equiv\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}\,.$ (8)

They do not depend on the overall normalization of the potential $d$, and must
be small during inflation, i.e. $\epsilon_{V}$, $|\eta_{V}|$ and
$|\xi_{V}^{2}|\ll 1$.

We define the end of inflation at a field value $\phi_{\text{end}}$ such that
$\epsilon_{V}(\phi_{\text{end}})=1$. The total number of e–folds between the
time when the CMB pivot scale $k_{\star}=0.05\ \rm{Mpc}^{-1}$ first crossed
out the horizon till end of inflation can be computed analytically in this
model Drees:2022aea :

$\displaystyle N_{\text{CMB}}$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{1}{\sqrt{2\epsilon_{V}}}d\phi$
$\displaystyle\simeq\frac{1}{24}\left\\{3\phi^{2}-4\phi\phi_{0}+15\phi_{0}^{2}-\phi_{0}^{2}\sqrt{\frac{2}{\beta}}\arctan\left(\frac{\phi_{0}-\phi}{\sqrt{2\beta}\phi_{0}}\right)-\phi_{0}^{2}\ln\left[(\phi_{0}-\phi)^{2}\right]\right\\}\Bigg{|}^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\,.$
(9)

Here $\phi_{\text{CMB}}$ is the field value when $k_{\star}$ first crossed out
of the horizon. The predictions for the normalization $\mathcal{P}_{\zeta}$ of
the power spectrum, its spectral index $n_{s}$, the running $\alpha$ of the
spectral index, and the tensor to scalar ratio $r$ during SR inflation are
given by Lyth:2009zz

$\displaystyle\mathcal{P}_{\zeta}$
$\displaystyle=\frac{V}{24\pi^{2}\epsilon_{V}}\,;$ (10) $\displaystyle n_{s}$
$\displaystyle=1-6\epsilon_{V}+2\eta_{V}\,;$ (11) $\displaystyle\alpha$
$\displaystyle=16\epsilon_{V}\eta_{V}-24\epsilon_{V}^{2}-2\xi_{V}^{2}\,;$ (12)
$\displaystyle r$ $\displaystyle=16\epsilon_{V}\,.$ (13)

The recent Planck 2018 measurements plus results on baryonic acoustic
oscillations (BAO) at the pivot scale $k_{\star}=0.05\ \rm{Mpc}^{-1}$ imply
Planck:2018vyg :

$\mathcal{P}_{\zeta}=(2.1\pm 0.1)\cdot 10^{-9}\,;\ n_{s}=0.9659\pm 0.0040\,;\
\alpha=-0.0041\pm 0.0067\,.$ (14)

The most stringent constraint on $r$ comes from BICEP/Keck 2018 (together with
Planck data) BICEP:2021xfz :

$r_{0.05}<0.035\,\quad\text{at 95\% \text{C.L.}}\,.$ (15)

##  4 Reheating

In this section, we focus on two reheating scenarios, where the inflaton
primarily decays either into fermions (specifically, into RHNs) or into Higgs
bosons. The decay rate for inflaton decays to a pair of $N_{1}$ is given by

$\Gamma_{\phi\to\bar{N}_{1}N_{1}}=\frac{y_{1}^{2}\,m_{\phi}}{16\pi}\left(1-\frac{4M_{1}^{2}}{m_{\phi}^{2}}\right)^{3/2}\,;$
(16)

for the bosonic channel one has

$\Gamma_{\phi\to
H^{\dagger}H}=\frac{\lambda_{12}^{2}}{8\pi\,m_{\phi}}\sqrt{1-\frac{4m_{H}^{2}}{m_{\phi}^{2}}}\,.$
(17)

The reheating temperature $T_{\text{rh}}$, i.e. the highest temperature of the
radiation dominated era, can be defined via
$H(T_{\text{rh}})=\frac{2}{3}\Gamma_{\phi}$, where $H(T_{\text{rh}})$ denotes
the Hubble parameter at $T=T_{\text{rh}}$. This gives

$T_{\text{rh}}=\sqrt{\frac{2}{\pi}}\left(\frac{10}{g_{\star}}\right)^{1/4}\sqrt{M_{P}\,\Gamma_{\phi}}\,,$
(18)

where $g_{\star}(T)$ denotes the number of relativistic degrees of freedom
contributing to the SM energy density $\rho_{R}$; in the SM,
$g_{\star}=106.75$ for temperatures much higher than the electroweak scale. In
order to allow successful BBN, the reheating temperature has to satisfy
$T_{\text{rh}}\gtrsim 4~{}\text{MeV}$ Kawasaki:2000en ; Hannestad:2004px . On
the other hand, the inflaton potential at $\phi_{0}$ will be destabilized by
one–loop corrections if the inflaton has too large couplings. Radiative
stability implies Drees:2021wgd :

$\left|\left(\frac{\lambda_{12}}{\phi_{0}}\right)^{2}\ln\left(\frac{\lambda_{12}}{\phi_{0}}\right)-\left(\frac{\lambda_{12}}{\phi_{0}}\right)^{2}\right|<64\pi^{2}d\,\beta\simeq
6.1\times 10^{-19}\left(\frac{\phi_{0}}{M_{P}}\right)^{6}\,,$ (19)

and

$\left|y_{1}^{4}-3y_{1}^{4}\ln\left(y_{1}^{2}\right)\right|<128\pi^{2}d\,\beta\simeq
1.2\times 10^{-18}\left(\frac{\phi_{0}}{M_{P}}\right)^{6}\,.$ (20)

The approximate equalities hold for $\phi_{0}\lesssim 3M_{P}$ where fixing
$\phi_{0}$ essentially determines $\beta$ and $d$.

Altogether, $\phi_{0}$ can lie in the rather wide range

$3\times 10^{-5}\,M_{P}\lesssim\phi_{0}\lesssim 21.5\,M_{P}\,.$ (21)

The lower bound comes from the combination of the lower bound
$T_{\text{rh}}\gtrsim 4~{}\text{MeV}$ with the upper bounds of eq.(19) and
eq.(20) Drees:2021wgd ; note that the latter immediately imply upper bounds on
$T_{\text{rh}}$ via eq.(18). The upper bound on $\phi_{0}$ in (21) is
essentially due to the upper bound (15) on $r$ Drees:2022aea .

##  5 Neutrino Masses

A simple method to derive the see–saw expression for the mass matrix of the
light neutrinos is to integrate out the RHNs via their Euler–Lagrange
equation,
$\partial_{\mu}\frac{\partial\mathcal{L}}{\partial(\partial_{\mu}\bar{N}_{I})}-\frac{\partial\mathcal{L}}{\partial\bar{N}_{I}}=0$.
Solving for $N_{I}$ and plugging the result back into eq.(6), one finds (after
electroweak symmetry breaking) a Majorana mass term for the light active
neutrinos $\nu_{\alpha}$:

$\mathcal{L}\supset\bar{\nu}_{\alpha}\left(-2\frac{v^{2}Y_{\alpha
I}Y^{T}_{\alpha I}}{M_{I}}\right)\nu_{\alpha}^{c}\,,$ (22)

where $v=174$ GeV is the vacuum expectation value (vev) of the Higgs field.
The mass matrix of the light neutrinos is therefore

$\widetilde{m}_{\nu}=-v^{2}YM_{I}^{-1}Y^{T}\,,$ (23)

where $Y$ and $M_{I}$ are matrices. $\widetilde{m}_{\nu}$ can be diagonalized
with the help of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix222We work
in the basis for the lepton doublets where the charged lepton mass matrix is
diagonal. $U$:

$U^{T}\widetilde{m}_{\nu}U=\text{diag}(m_{1},m_{2},m_{3})\equiv m_{\nu}^{\rm
diag}\,.$ (24)

In the following we assume the normal hierarchy for the light neutrinos where
$m_{1}<m_{2}<m_{3}$, and consider the current best fit Esteban:2018azc :

$\displaystyle m_{1}$ $\displaystyle=0\,;$ $\displaystyle m_{2}$
$\displaystyle=\sqrt{\Delta m_{\odot}^{2}}=(8.6\pm 0.1)\times
10^{-3}\,\mathrm{eV}\,,$ $\displaystyle m_{3}$ $\displaystyle=\sqrt{\Delta
m_{\mathrm{atm}}^{2}}=(5.02\pm 0.03)\times 10^{-2}\,\mathrm{eV}\,.$ (25)

Using the Casas–Ibarra parametrization Casas:2001sr , one can express the
Yukawa coupling matrix $Y$ in terms of the physical neutrino mass matrix
$m_{\nu}^{\rm diag}$, the PMNS matrix $U$, the (diagonal) RHN mass matrix $M$
and an undetermined complex orthogonal matrix $R$:

$Y=\frac{i}{v}U^{\star}\sqrt{m_{\nu}^{\rm diag}}R^{T}\sqrt{M}\,.$ (26)

This expression can be solved for $R$:

$\displaystyle(R^{T})_{ij}=-iv\frac{(U^{T}Y)_{ij}}{\sqrt{m_{i}}\sqrt{M_{j}}}\Longleftrightarrow
R_{ji}=-iv\frac{(U^{T}Y)_{ij}}{\sqrt{m_{i}}\sqrt{M_{j}}}\,.$ (27)

Since we assume $m_{1}=0$, see eq.(5), we actually only need two RHNs
Ibarra:2003up , i.e. we can set $M_{3}\rightarrow\infty$. Since $m_{2}\neq 0$
and $m_{3}\neq 0$, eq.(27) implies that $R_{32}$ and $R_{33}$ have to vanish.
On the other hand $R_{31}$ might not be zero since the product $m_{1}M_{3}$ is
ill defined in our setup. We can thus write $R$ as

$R=\left(\begin{array}[]{ccc}R_{11}&R_{12}&R_{13}\\\ R_{21}&R_{22}&R_{23}\\\
R_{31}&0&0\end{array}\right)\,;\ \
R^{T}=\left(\begin{array}[]{ccc}R_{11}&R_{21}&R_{31}\\\ R_{12}&R_{22}&0\\\
R_{13}&R_{23}&0\end{array}\right)\,,$ (28)

Since $RR^{T}=R^{T}R=\mathbb{1}$, one has

$\displaystyle\mathbb{1}$
$\displaystyle=\left(\begin{array}[]{ccc}R_{11}^{2}+R_{12}^{2}+R_{13}^{2}&R_{11}R_{21}+R_{12}R_{22}+R_{13}R_{23}&R_{11}R_{31}\\\
R_{11}R_{21}+R_{12}R_{22}+R_{13}R_{23}&R_{21}^{2}+R_{22}^{2}+R_{23}^{2}&R_{21}R_{31}\\\
R_{11}R_{31}&R_{21}R_{31}&R_{31}^{2}\end{array}\right)\,;$ (32)
$\displaystyle\mathbb{1}$
$\displaystyle=\left(\begin{array}[]{ccc}R_{11}^{2}+R_{21}^{2}+R_{31}^{2}&R_{11}R_{12}+R_{21}R_{22}&R_{11}R_{13}+R_{21}R_{23}\\\
R_{11}R_{12}+R_{21}R_{22}&R_{12}^{2}+R_{22}^{2}&R_{12}R_{13}+R_{22}R_{23}\\\
R_{11}R_{13}+R_{21}R_{23}&R_{12}R_{13}+R_{22}R_{23}&R_{13}^{2}+R_{23}^{2}\end{array}\right)\,.$
(36)

The $(3,3)$ element in the first eq.(32) implies $R_{31}^{2}=1$. The $(3,1)$
and $(3,2)$ elements of that equation then require $R_{11}=R_{21}=0$. The
$(1,1)$ and $(2,2)$ elements in turn imply
$R_{12}^{2}+R_{13}^{2}=R_{22}^{2}+R_{23}^{2}=1$, and the $(1,2)$ element gives
$R_{12}R_{22}+R_{13}R_{23}=0$. Turning to the second eq.(32), the above
relations suffice to satisfy the equations for the $(1,1),\ (1,2)$ and $(1,3)$
elements. The $(2,2)$ and $(3,3)$ elements imply
$R_{12}^{2}+R_{22}^{2}=R_{13}^{2}+R_{23}^{2}=1$, and the $(2,3)$ element
requires $R_{12}R_{13}+R_{22}R_{23}=0$. Hence we can write the matrix $R$ as
Ibarra:2003up

$R=\left(\begin{array}[]{ccc}0&\cos z&\sin z\\\ 0&-\sin z&\cos z\\\
1&0&0\end{array}\right)\,,$ (37)

with $z$ being a complex angle.333We remind the reader that $\cos z=\left({\rm
e}^{iz}+{\rm e}^{-iz}\right)/2,\ \sin z=\left({\rm e}^{iz}-{\rm
e}^{-iz}\right)/(2i)$ hold also for complex argument $z$, so that
$\cos^{2}z+\sin^{2}z=1$ as usual. Also, $(\cos z)^{*}=\cos(z^{*})$ and $(\sin
z)^{*}=\sin(z^{*})$.

Having parameterized $R$, we can use eq.(26) and $m_{1}=0$ to derive

$(Y^{\dagger}Y)_{ij}=\frac{\sqrt{M_{i}M_{j}}}{v^{2}}\left(m_{2}R^{\star}_{i2}R_{j2}+m_{3}R^{\star}_{i3}R_{j3}\right)\,.$
(38)

In particular, we’ll need the following two quantities in the following:

$\displaystyle(Y^{\dagger}Y)_{11}=\sum_{\alpha=1}^{3}Y^{\star}_{\alpha
1}Y_{\alpha 1}$ $\displaystyle=\frac{M_{1}}{v^{2}}\left(m_{2}|\cos
z|^{2}+m_{3}|\sin z|^{2}\right)\,;$ (39) $\displaystyle(Y^{\dagger}Y)_{21}$
$\displaystyle=\frac{\sqrt{M_{1}\,M_{2}}}{v^{2}}\left[-m_{2}\sin z^{\star}\cos
z+m_{3}\sin z\cos z^{\star}\right]\,.$ (40)

##  6 Baryogenesis via Leptogenesis

As noted earlier, we focus on the simplest version of leptogenesis, where the
asymmetry is generated solely in the decay of the lightest RHN. The heavier
RHNs cannot be produced either thermally or non–thermally if $M_{3}\gg
M_{2}>T_{\text{rh}},\ m_{\phi}$. The yield of both thermal and non–thermal
leptogenesis is then proportional to the CP asymmetry parameter Fong:2012buy

$\epsilon_{1}=\frac{1}{8\pi}\frac{1}{\left(Y^{\dagger}Y\right)_{11}}\operatorname{Im}\left[\left(Y^{\dagger}Y\right)_{21}^{2}\right]g\left(\frac{M_{2}^{2}}{M_{1}^{2}}\right)\,.$
(41)

This parameter describes the (loop–induced) difference between the branching
ratios into CP–conjugate states, i.e. between final states containing a lepton
and those containing an antilepton. The loop function $g$ in eq.(41) is given
by

$\displaystyle g(x)$
$\displaystyle=\sqrt{x}\left[\frac{1}{1-x}+1-(1+x)\ln\left(\frac{1+x}{x}\right)\right]$
$\displaystyle\simeq-\frac{3}{2}\left(\frac{1}{x}\right)^{1/2}-\frac{5}{6}\left(\frac{1}{x}\right)^{3/2}+\mathcal{O}\left(\frac{1}{x}\right)^{5/2}\quad\text{for}\quad
x\gg 1\,.$ (42)

From eq.(40) we have

$\displaystyle\left(Y^{\dagger}Y\right)_{21}^{2}$
$\displaystyle=\frac{M_{1}\,M_{2}}{v^{4}}\left[-m_{2}\sin z^{\star}\cos
z+m_{3}\sin z\cos z^{\star}\right]^{2}$
$\displaystyle=\frac{M_{1}\,M_{2}}{v^{4}}\left[m_{2}^{2}\,\sin^{2}z^{\star}\,\cos^{2}z+m^{2}_{3}\sin^{2}z\cos^{2}z^{\star}-2m_{2}m_{3}|\sin
z|^{2}|\cos z|^{2}\right]\,,$ (43)

and therefore

$\displaystyle\text{Im}\left[\left(Y^{\dagger}Y\right)_{21}^{2}\right]$
$\displaystyle=\frac{M_{1}\,M_{2}}{v^{4}}\text{Im}\left[m_{2}^{2}\,\sin^{2}z^{\star}\,\cos
z^{2}+m^{2}_{3}\sin z^{2}\cos^{2}z^{\star}\right]$
$\displaystyle=\frac{M_{1}\,M_{2}}{v^{4}}(m_{3}^{2}-m_{2}^{2})\,\text{Im}\left(\sin^{2}z\right)\,;$
(44)

In the last step we used $\cos^{2}z=1-\sin^{2}z$ and
$\text{Im}\left(\sin^{2}z\sin^{2}z^{*}\right)=0$. Now define
$\cos^{2}z=x+i\,y$, hence $\sin^{2}z=1-\cos^{2}z=1-x-i\,y$, with
$x,y\in\mathbb{R}$. For $M^{2}_{2}\gg M^{2}_{1}$ we can then derive a simple
upper bound on $\epsilon_{1}$:

$\displaystyle\epsilon_{1}$
$\displaystyle=\frac{1}{8\pi}\frac{v^{2}}{M_{1}\left(m_{2}|\cos
z|^{2}+m_{3}|\sin
z|^{2}\right)}\operatorname{Im}\left[\left(Y^{\dagger}Y\right)_{21}^{2}\right]g\left(\frac{M_{2}^{2}}{M_{1}^{2}}\right)$
$\displaystyle=\frac{1}{8\pi}\frac{v^{2}}{M_{1}}\frac{M_{1}\,M_{2}}{v^{4}}\frac{(m_{3}^{2}-m_{2}^{2})\,\text{Im}\left(\sin^{2}z\right)}{\left(m_{2}|\cos
z|^{2}+m_{3}|\sin
z|^{2}\right)}\left(-\frac{3}{2}\frac{M_{1}}{M_{2}}\right)\left[1+\mathcal{O}\left(\frac{M_{1}^{2}}{M_{2}^{2}}\right)\right]$
$\displaystyle\simeq-\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}^{2}-m_{2}^{2})\frac{\text{Im}\left(\sin^{2}z\right)}{\left(m_{2}|\cos
z|^{2}+m_{3}|\sin z|^{2}\right)}$
$\displaystyle=-\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}^{2}-m_{2}^{2})\,\frac{-y}{\left(m_{2}\sqrt{x^{2}+y^{2}}+m_{3}\sqrt{(1-x)^{2}+y^{2}}\right)}$
$\displaystyle\lesssim\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}^{2}-m_{2}^{2})\,\frac{y}{\left(m_{2}\sqrt{1+y^{2}}+m_{3}\sqrt{y^{2}}\right)}$
$\displaystyle\lesssim\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}-m_{2})\simeq
10^{-5}\left(\frac{M_{1}}{10^{11}\ \text{GeV}}\right)\,.$ (45)

In the fifth step we have assumed $\epsilon_{1}>0$, which is needed for a
positive baryon asymmetry, and used $m_{3}>m_{2}$, so that the denominator is
minimized, and hence $\epsilon_{1}$ is maximized, for $x=1$. The penultimate
step follows because the $y-$dependent fraction reaches its maximal value
$1/(m_{2}+m_{3})$ for $y\rightarrow\infty$, and in the last step we have used
the numerical values from eq.(5).

The Baryon Asymmetry of the Universe (BAU) today is given by Kolb:1990vq

$\eta_{B}\equiv\frac{n_{B}}{n_{\gamma}}=\left(\frac{s}{n_{\gamma}}\right)_{0}\left(\frac{n_{B}}{n_{B-L}}\right)\frac{n_{B-L}}{s}\simeq
7\times\frac{28}{79}Y_{B-L}\,.$ (46)

Here $n_{\gamma}=2\zeta(3)T^{3}/\pi^{2}$ and
$s=2\pi^{2}\,g_{\star\,s}T^{3}/45$ denote the photon number and entropy
densities, respectively, $g_{\star\,s}$ being a measure of the number of
relativistic degrees of freedom. The subscript $0$ refers to the current time,
when $T=T_{0}\simeq 2.73$ K and $g_{\star\,s}\simeq 3.9$ in the absence of
“dark radiation”. Finally, the factor $28/79$ results from the transfer of the
primarily generated $B-L$ asymmetry to the baryon asymmetry by electroweak
sphaleron processes. Using the BAU value reported by Planck 2018, one has

$\displaystyle\eta_{B}\simeq 6\times 10^{-10}\Rightarrow Y_{B-L}\simeq
10^{-10}\,.$ (47)

Hence one needs a $B-L$ yield $Y_{B-L}\gtrsim 10^{-10}$ in order to generate
the observed BAU. In the subsequent subsections we will explore how a $B-L$
asymmetry of this magnitude can be generated in the framework of polynomial
inflation via either thermal or non–thermal $N_{1}$ production.

###  6.1 Leptogenesis with Bosonic Reheating

In this section, we focus on the bosonic reheating scenario, where the
inflaton dominantly decays to a pair of SM Higgs bosons. We will discuss
thermal and non–thermal $N_{1}$ production in turn.

####  6.1.1 Thermal Leptogenesis

In the conceptually simplest case $N_{1}$ was dominantly generated by
scattering reactions involving SM particles in the thermal plasma after
reheating. The $B-L$ number yield $Y_{B-L}$ is then given by Buchmuller:2004nz
; Fong:2012buy :

$Y_{B-L}=Y^{\text{eq}}_{N_{1}}\kappa_{f}\epsilon_{1}=\frac{45}{\pi^{4}g_{\star
s}}\epsilon_{1}\kappa_{f}\,.$ (48)

In this case $Y_{B-L}$ is mainly controlled by the scaled $N_{1}$ equilibrium
density, given by the numerical factor in eq. (48) (valid for
$T_{\text{rh}}>M_{1}$); the CP asymmetry factor $\epsilon_{1}$; and an
efficiency factor $\kappa_{f}$ parameterizing wash–out effects from inverse
decays. The latter can be estimated as Buchmuller:2004nz

$\kappa_{f}\simeq(2\pm 1)\times
10^{-2}\left(\frac{0.01\mathrm{eV}}{\tilde{m}_{1}}\right)^{1.1}\,,$ (49)

where $\tilde{m}_{1}=v^{2}\left(Y^{\dagger}Y\right)_{11}/M_{1}\geq m_{2}\simeq
0.0086$ eV, so $\kappa_{f}\lesssim 10^{-2}$.

Inserting the upper bound on $\epsilon_{1}$ given by (6) and
$\kappa_{f}\lesssim 10^{-2}$ into eq.(48) leads to a lower bound on $M_{1}$:

$M_{1}\gtrsim 10^{10}~{}\text{GeV}\,.$ (50)

Thermal leptogenesis only works if the reheating temperature
$T_{\text{rh}}>M_{1}\gtrsim 10^{10}~{}\text{GeV}$, as originally discussed in
Ref. Davidson:2002qv . Specifically, (inverse) decays
$N_{1}\leftrightarrow\ell_{\alpha}H$ attain equilibrium at a temperature of a
few $M_{1}$ (assuming the universe ever was this hot, of course), but will
freeze out for $T$ slightly below $M_{1}$; note that the coupling involved is
quite small, see (39). For bosonic inflaton decay in our model,
$T_{\text{rh}}\geq 10^{10}$ GeV is possible if $\phi_{0}\geq 0.4M_{P}$
Drees:2021wgd ; for $\phi_{0}\geq 10M_{P}$, a reheating temperature as high as
$10^{14}$ GeV can be achieved Drees:2022aea . On the other hand, if the
inflaton predominantly decays fermionically (into fermions other than
$N_{1}$), $T_{\text{rh}}>10^{10}$ GeV requires $\phi_{0}>6M_{P}$, and the
maximal reheat temperature consistent with radiative stability of the inflaton
potential is about $10^{11}$ GeV Drees:2022aea . Thermal leptogenesis (at
least in this simple form) is therefore far more plausible for bosonic
inflaton decays in our model.

####  6.1.2 Non–thermal Leptogenesis via Inflaton Three–Body Decay

Having shown that standard thermal leptogenesis is possible in our scenario,
here we point out that, at least in the large field version of the model,
non–thermal leptogenesis is also possible, even if the inflaton primarily
decays bosonically, specifically into pairs of SM Higgs bosons. As far as we
are aware this is a novel variant of leptogenesis.

The main observation is that for dominant $\phi\rightarrow HH$ decays, the
inflaton can also undergo three–body decay $\phi\to H\ell N_{1}$ as long as
$M_{1}<m_{\phi}$. This can happen if one of the Higgs bosons in the final
state is (far) off-shell, and transitions into $N_{1}$ plus an SM lepton
$\ell$; see fig. 3 in the Appendix. This scenario is particularly interesting
for relatively small inflaton couplings, i.e. for $T_{\text{rh}}<M_{1}$, where
the usual thermal leptogenesis is not possible. We emphasize that the coupling
that allows this three–body decay is automatically present in the type I
see–saw model. Specifically, the branching ratio for the three–body decay is
proportional to $\sum_{\alpha}Y^{*}_{\alpha 1}Y_{\alpha 1}$, which we
evaluated in eq.(39). Note that this precisely cancels the denominator of
$\epsilon_{1}$, see eq.(6). As shown in the Appendix, the resulting $B-L$
yield is given by:

$\displaystyle Y_{B-L}$
$\displaystyle=\left[\frac{3}{4}\frac{T_{\text{rh}}}{m_{\phi}}\,\text{BR}(\phi\to\ell
N_{1}H)\right]\,\epsilon_{1}$ $\displaystyle\simeq-4.8\times
10^{-11}\left(\frac{m_{\phi}}{10^{11}\
\text{GeV}}\right)\left(\frac{T_{\text{rh}}}{10^{11}\ \text{GeV}}\right){\rm
Im}\left(\sin^{2}z\right)\cdot\sqrt{\gamma}f(\gamma)\,.$ (51)

The function $f(\gamma)$ appears in the calculation of the branching ratio for
the three–body decay; it is given by

$f(\gamma)=\sqrt{\gamma}\left[\gamma^{2}+4\gamma-5-2(1+2\gamma)\ln\gamma\right]\,,$
(52)

with $\gamma\equiv M_{1}^{2}/m_{\phi}^{2}$. The expression in square
parenthesis results from integrating the squared matrix element over the
three–body phase space, whereas the factor $\sqrt{\gamma}$ results from the
squared $N_{1}$ Yukawa coupling. Note that $0<\gamma<1$; for $\gamma\ll 1$,
the phase space factor is dominated by the term $-2\ln\gamma$. Of course,
$f(\gamma)\rightarrow 0$ for $\gamma\to 1$ or equivalently $M_{1}\to
m_{\phi}$, since this corresponds to the closing of phase space; specifically,
$f(\gamma)\simeq(1-\gamma)^{4}/6$ as $\gamma\rightarrow 1$. We note that
$\sqrt{\gamma}f(\gamma)$ reaches its maximal value of $0.095$ at
$\gamma=0.081$. Finally, in eq.(6.1.2) we have assumed that no wash–out
occurs. This is an oversimplification if $T_{\text{rh}}$ is only slightly
below $M_{1}$.

Figure 1: $T_{\text{rh}}$ as function of $M_{1}$ that yields the observed
Baryon asymmetry via inflaton three body decays (blue line) from eq.(6.1.2)
with ${\rm Im}\left(\sin^{2}z\right)=1$, for four different values of
$\phi_{0}$. In the gray shaded region at the top loop corrections dominate the
slope of the potential at $\phi_{0}$, invalidating our analysis of inflation.
In the pink shaded region at the right $\phi\rightarrow H\ell N_{1}$ decays
are kinematically forbidden. The dotted vertical red line indicates the
minimal value of $M_{1}$ allowing thermal leptogenesis; the latter is only
possible above the dotted black lines.

We saw in eq.(47) that $Y_{B-L}\gtrsim 10^{-10}$ is required in order to
reproduce the observed BAU. The blue curves in Figs. 1 show the minimal value
of $T_{\text{rh}}$ that allows to satisfy this condition, using eq.(6.1.2);
for definiteness we have taken ${\rm Im}\left(\sin^{2}z\right)=1$. The upper
left panel is for $\phi_{0}=21.5\ M_{P}$, leading to $m_{\phi}\simeq 5.7\times
10^{13}$ GeV; this large value of $\phi_{0}$, and correspondingly heavy
inflaton, saturates the current upper bound on $r$. We exclude the gray shaded
region, where $T_{\text{rh}}\gtrsim 2\times 10^{14}$ GeV, since here the Higgs
loop will spoil the flatness of inflaton potential so that the inflationary
predictions, which are based on the tree–level potential, cannot be trusted.
The black dotted line corresponds to $T_{\text{rh}}=M_{1}$, above which
$N_{1}$ particles thermalize, leading to thermal leptogenesis as investigated
above. Note that when $T_{\text{rh}}>M_{1}$ the inflaton three-body decay
could still generate $N_{1}$ particles, which however would not change their
number density if they thermalize. The red dotted line corresponds to
$M_{1}\simeq 10^{10}~{}\text{GeV}$, denoting the lower bound on $M_{1}$ for
thermal leptogenesis.

Below the black dotted line, where $M_{1}>T_{\text{rh}}$, thermal leptogenesis
is suppressed. In such case we find that the non–thermal channel via inflaton
three–body decays might still be able to source the required BAU, as depicted
by the blue line. When $M_{1}\to m_{\phi}$, the three-body decay is phase
space closed, explaining why the blue curves suddenly go up.

We also show examples with smaller $\phi_{0}$ in Fig. 1: upper right panel
$\phi_{0}=10\,M_{P}$, lower left panel $\phi_{0}=5\,M_{P}$ and lower right
panel $\phi_{0}=2.5\,M_{P}$. For $\phi_{0}\lesssim 2.5\,M_{P}$, ${\rm
Im}(\sin^{2}z)>1$ would be required, which would require some cancellation to
occur in eq.(23).

Several comments are in order before we close this section. First, non–thermal
leptogenesis via inflaton three–body decay investigated in this section is
also viable for other inflation models (e.g. the Starobinsky
Starobinsky:1980te  and $\alpha$-attractor models Kallosh:2013hoa ) as long as
the inflaton mass satisfies $m_{\phi}\gtrsim 10^{12}~{}\text{GeV}$ and
$T_{\text{rh}}\gtrsim 10^{10}$ GeV. Second, it might also work in scenarios
with a later epoch of matter domination by some very massive particle, e.g. a
modulus field Drees:2017iod  which predominantly decays into a pair of SM
Higgs bosons. The above bound on $m_{\phi}$ should then be satisfied by this
modulus field, and $T_{\text{rh}}$ refers to the temperature at the end of the
modulus matter dominated epoch. Finally, for $m_{\phi}>2M_{1}$ the inflaton
can also decay into pairs of RHNs, either via four–body decays, or via loop-
induced two-body decays as commented in the end of Sec. 2. However, these
processes have (even) smaller branching ratios than the three–body final
state, which will therefore dominate non–thermal $N_{1}$ production in our
bosonic reheating scenario.

###  6.2 Non–thermal Leptogenesis with Fermionic Reheating

Figure 2: Parameter space for non–thermal leptogenesis if the inflaton
predominantly decays into two $N_{1}$, for four different values of $\phi_{0}$
and hence of $m_{\phi}$. Above the blue lines the correct BAU can be generated
non–thermally, see eq.(53). In the gray shaded regions our tree–level analysis
of inflation is not reliable, while in the pink shaded regions
$\phi\rightarrow N_{1}N_{1}$ decays are kinematically forbidden. Finally,
below the dotted black lines $N_{1}$ does not attain thermal equilibrium.

In this section we assume that RHNs are produced via inflaton two body decays,
$\phi\to N_{1}N_{1}$ Lazarides:1990huy ; Giudice:1999fb ; Asaka:1999jb ;
Senoguz:2003hc ; Hahn-Woernle:2008tsk ; Antusch:2010mv ; these decays also
reheat the universe. We saw above that in our inflationary scenario
$T_{\text{rh}}$ is too low for thermal leptogenesis if the inflaton decays
fermionically, except for a very limited region of parameter space at large
$\phi_{0}$. Here we show that, even in our very simple type I see–saw model,
the parameter space for leptogenesis is greatly expanded by this non–thermal
$N_{1}$ production channel.

We will again assume that washout effects can be neglected if
$T_{\text{rh}}<M_{1}$. The $B-L$ number yield after reheating can then simply
be estimated as Kolb:1990vq ; Asaka:1999jb :

$Y_{B-L}=\frac{n_{B-L}}{s}=\left[\frac{3}{2}\frac{T_{\text{rh}}}{m_{\phi}}\,\text{BR}(\phi\to
N_{1}N_{1})\right]\,\epsilon_{1}\,.$ (53)

The expression between the square parentheses denotes the yield of $N_{1}$
from inflaton decay, allowing for a general branching ratio BR for
$\phi\rightarrow N_{1}N_{1}$; in our scenario of fermionic reheating with a
hierarchical RHN spectrum, this branching ratio is unity, $\text{BR}=1$.

The allowed parameter space $(T_{\text{rh}},M_{1})$ is shown in Fig. 2. Here
we consider $\phi_{0}=21.5\,M_{P}$ (upper left), corresponding to the largest
allowed value compatible with the latest CMB experiments;
$\phi_{0}=1.0\,M_{P}$ (upper right), $\phi_{0}=0.5\,M_{P}$ (lower left), and
$\phi_{0}=0.1\,M_{P}$ (lower right). The gray regions are again excluded since
radiative corrections from the $N_{1}$ loop would spoil the flatness of the
inflaton potential near $\phi_{0}$. In the pink regions, $M_{1}>m_{\phi}/2$
hence $\phi\rightarrow N_{1}N_{1}$ is kinematically impossible.444A small
sliver of this region, with $M_{1}$ only slightly above $m_{\phi}/2$, may
still be acceptable. The dominant inflaton decay mode in this region is
$\phi\rightarrow N_{1}H\ell$ via off–shell $N_{1}$ exchange. However, the
inflaton decay width, and hence $T_{\text{rh}}$, drops very quickly in this
region; eq.(53) shows that this also suppresses the generated BAU. The blue
line gives $Y_{B-L}=10^{-10}$ if the upper bound (6) on $\epsilon_{1}$ is used
in eq.(53), i.e. below these lines the generated net lepton number is always
too small. Finally, the black dotted lines again show $T_{\text{rh}}=M_{1}$,
i.e. below these lines wash–out effects are small but above these lines
$N_{1}$ thermalizes.

As remarked earlier, in a small region of parameter space with large
$\phi_{0}$ and $T_{\text{rh}}>M_{1}$ (above the dotted black lines) thermal
leptogenesis can occur. We note that the blue solid and black dotted lines
intersect at

$M_{1}^{\prime}=10^{3}\
{\text{GeV}}\;\cdot\sqrt{\frac{2}{3}\frac{m_{\phi}}{\text{GeV}}}\,.$ (54)

Non–thermal leptogenesis can only occur for $M_{1}>M_{1}^{\prime}$. We see
that for $\phi_{0}<M_{P}$ this is automatically satisfied by all points on the
blue line where radiative corrections to the inflaton potential are under
control. In this “small field” inflationary scenario, the maximal reheating
temperature $T_{\rm rh}^{\rm max}\propto\phi_{0}^{5/2}$ for fermionic
reheating Drees:2021wgd . Moreover, eqs. (53) and (6) show that $Y_{B-L}$ is
maximal for maximal $M_{1}\simeq m_{\phi}/2$, hence the maximal
$Y_{B-L}\propto T_{\rm rh}^{\rm max}\propto\phi_{0}^{5/2}$. In particular, for
$\phi_{0}\leq 0.1M_{P}$, even non–thermal leptogenesis becomes impossible in
our simple neutrino mass model.

Altogether the allowed range of $\phi_{0}$ leading to successful leptogenesis
via $\phi\rightarrow N_{1}N_{1}$ decays, compatible with both CMB observations
and neutrino oscillation data, is

$\displaystyle 0.1\,M_{P}\lesssim\phi_{0}\lesssim 21.5\,M_{P}\,.$ (55)

In addition, the reheating temperature has to lie in the range

$10^{6}~{}\text{GeV}\lesssim T_{\text{rh}}\lesssim 10^{10}~{}\text{GeV}\,,$
(56)

while the mass of the lightest RHN should satisfy

$10^{8}~{}\text{GeV}<M_{1}\lesssim 3\cdot 10^{13}~{}\text{GeV}\,.$ (57)

##  7 Summary and Conclusions

Polynomial inflation is a very simple and predictive model. It fits current
cosmic microwave background (CMB) data well, and makes testable predictions
for the running of the spectral index and the tensor to scalar ratio.

A further necessary condition for a successful cosmological model is that it
offers an explanation for the observed baryon asymmetry of the Universe (BAU).
In this work we therefore revisit leptogenesis in the framework of polynomial
inflation. We assume a type–I see–saw model with massless lightest neutrino,
and therefore only two very heavy right–handed neutrinos. We show that even in
this simple ansatz, in some part of the parameter space both the observed BAU
and the neutrino oscillation data can be explained.

To that end we consider two different perturbative reheating scenarios. If the
inflaton decays bosonically, the reheating temperature in our model can be as
high as $T_{\text{rh}}\sim 10^{14}~{}\text{GeV}$, compatible with “plain
vanilla” $N_{1}$ thermal leptogenesis. We also point out, for the first time,
that even for $T_{\text{rh}}<M_{1}$, where thermal leptogenesis does not work,
inflaton three–body decays of the inflaton into an SM Higgs boson, a RHN
$N_{1}$ and an SM doublet lepton can lead to successful leptogenesis. As long
as it is kinematically allowed, this decay will always occur if the primary
inflaton decay mode is into two SM Higgs bosons; the required couplings are
automatically present in the see–saw neutrino mass model, and can be expressed
in terms of neutrino masses and a single complex mixing angle. We find that
this three–body decay scenario is viable for inflaton mass $m_{\phi}\gtrsim
10^{12}~{}\text{GeV}$. In the polynomial inflation scenario this corresponds
to $\phi_{0}\gtrsim 2.5~{}M_{P}$, but this novel mechanism should also be
applicable to other models of large–scale inflation.

We also investigate the case where the inflaton primarily decays into two
$N_{1}$. This fermionic decay mode leads to somewhat lower maximal reheating
temperature. Thermal leptogenesis therefore works only in a rather small
region of parameter space, but this inflaton decay mode allows non–thermal
leptogenesis, if $\phi_{0}\gtrsim 0.1~{}M_{P}$, $M_{1}\gtrsim 3\cdot 10^{8}$
GeV and reheating temperature $T_{\text{rh}}\gtrsim 10^{6}~{}\text{GeV}$.

In conclusion, we have shown that a model with a renormalizable inflaton
potential, in combination with a simple see–saw model with hierarchical masses
of the right–handed neutrinos, allows successful thermal or non–thermal
leptogenesis if inflation occurred at a field value $\phi\simeq\phi_{0}\gtrsim
0.1M_{P}$, corresponding to an Hubble parameter during inflation $H_{I}\gtrsim
2\cdot 10^{7}$ GeV. Deviating from our assumption of a very hierarchical
spectrum of right–handed neutrinos should allow to construct viable models
operating at much smaller energy scales. For example, it is known that thermal
“resonant” leptogenesis, where two RHNs are nearly degenerate, can work at the
TeV scale Pilaftsis:2003gt . It would be interesting to investigate this
quantitatively.

## Acknowledgments

Y.X. acknowledges the support from the Cluster of Excellence “Precision
Physics, Fundamental Interactions, and Structure of Matter” (PRISMA+ EXC
2118/1) funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) within the German Excellence Strategy (Project No. 390831469).

##  Appendix A Inflaton Three–Body Decay into Right Handed Neutrino

In this appendix, we first calculate the branching ratio for inflaton
three–body decays $\phi\to H\,\ell\,N_{1}$. Then we estimate the $B-L$ yield
generated from the subsequent decay of $N_{1}$.

Figure 3: $N_{1}$ production via inflaton three–body decay.

Our basic assumption is that the main inflaton decay mode is into a pair of SM
Higgs bosons via the coupling $\lambda_{12}$. The three–body decay shown in
fig. 3 is then automatically allowed, as long as $M_{1}<m_{\phi}$. The RHN
$N_{1}$ couples to the SM Higgs and lepton doublet through the Yukawa coupling
$Y_{\alpha 1}\,\bar{\ell}_{\alpha}\tilde{H}N_{1}$, which is required for the
explanation of the light neutrino masses.

We label the momenta for particles involved as follows:

$\displaystyle\phi(p)\to H(k_{1})\,H(q)\to
H(k_{1})\,\ell(k_{2})N_{1}(k_{3})\,,$

The matrix element for a fixed lepton flavor $\alpha$ is given by

$i\mathcal{M}_{\alpha}=2i\,\lambda_{12}\,\frac{i}{q^{2}-m_{H}^{2}}\,Y_{\alpha\,1}\bar{u}(k_{2})\text{v}(k_{3})\,.$
(58)

After squaring and summing over lepton flavor and the $SU(2)$ index, this
yields:

$\displaystyle|\mathcal{M}|^{2}$
$\displaystyle=\sum_{\alpha}4\,\lambda^{2}_{12}\left(Y^{\star}_{\alpha\,1}Y_{\alpha\,1}\right)\left(4k_{2}\cdot
k_{3}-4M_{1}m_{\ell_{\alpha}}\right)\frac{1}{q^{4}}$ $\displaystyle\simeq
16\,\lambda^{2}_{12}k_{2}\cdot
k_{3}\frac{1}{q^{4}}\sum_{\alpha}\left(Y^{\star}_{\alpha\,1}Y_{\alpha\,1}\right)\,.$
(59)

In the second step we have neglected the small lepton mass. Note also that we
assume unbroken $SU(2)$, hence all four (real) components of the Higgs
doublet, and both charged and (light) neutral leptons, contribute.

The three–body phase space integral can be written as (see e.g. Appendix B of
Barger:1987nn ):

$\int
d\Pi_{3}=\frac{m_{\phi}^{2}}{128\pi^{3}}\int^{1-\gamma}_{0}dx_{1}\int^{1-\frac{\gamma}{1-x_{1}}}_{1-x_{1}-\gamma}dx_{2}\,,$
(60)

where $x_{i}=2E_{i}/m_{\phi}\,,\ i=1,2$, $E_{1}$ being the energy of the Higgs
boson and $E_{2}$ that of the charged lepton in the inflaton rest frame, and
$\gamma=M_{1}^{2}/m_{\phi}^{2}$. We have again neglected all masses in the
final state except for that of the heavy RHN $N_{1}$. The product of
$4-$vectors appearing in the squared matrix element (A) is related to $x_{1}$
via $2k_{2}\cdot k_{3}=m_{\phi}^{2}(1-x_{1})-M_{1}^{2}$.

Inserting eq. (A) into eq.(60) allows to compute the three–body decay rate:

$\displaystyle\Gamma_{\text{3-body}}\equiv\Gamma_{\phi\to\ell HN_{1}}$
$\displaystyle=\frac{1}{2m_{\phi}}\int d\Pi_{3}|\mathcal{M}|^{2}$
$\displaystyle=\frac{\lambda^{2}_{12}}{32\pi^{3}\,m_{\phi}}\int^{1-\gamma}_{0}dx_{1}\int^{1-\frac{\gamma}{1-x_{1}}}_{1-x_{1}-\gamma}dx_{2}\frac{\,(1-x_{1})-\gamma}{(1-x_{1})^{2}}\,\sum_{\alpha}\left(Y^{\star}_{\alpha\,1}Y_{\alpha\,1}\right)$
$\displaystyle=\frac{\lambda^{2}_{12}}{32\pi^{3}\,m_{\phi}}\sum_{\alpha}\left(Y^{\star}_{\alpha\,1}Y_{\alpha\,1}\right)\frac{1}{2}\left[\gamma^{2}+4\gamma-5-2(1+2\gamma)\ln\gamma\right]\,.$
(61)

The branching ratio for the channel $\phi\to\ell N_{1}H$ is therefore

$\displaystyle\text{BR}(\phi\to\ell N_{1}H)$
$\displaystyle=\frac{\Gamma_{\text{3-body}}}{\Gamma_{\text{3-body}}+\Gamma_{\phi\to
H^{\dagger}H}}\simeq\frac{\Gamma_{\text{3-body}}}{\Gamma_{\phi\to
H^{\dagger}H}}$ $\displaystyle=\frac{m_{\phi}}{8\pi^{2}v^{2}}\left(m_{2}|\cos
z|^{2}+m_{3}|\sin z|^{2}\right)\cdot f(\gamma)\,,$ (62)

where we have used $\Gamma_{\phi\to
H^{\dagger}H}=\lambda_{12}^{2}/(8\pi\,m_{\phi})$, and eq. (39) for
$\sum_{\alpha}\left(Y^{\dagger}_{\alpha\,1}Y_{\alpha\,1}\right)$. The function
$f(\gamma)$ is defined as:

$\displaystyle
f(\gamma)=\sqrt{\gamma}\left[\gamma^{2}+4\gamma-5-2(1+2\gamma)\ln\gamma\right]\,.$
(63)

It peaks at $\gamma\simeq 0.0177$ with $f(\gamma)^{\text{max}}\simeq 0.456$.

If washout effects can be neglected, i.e. for $M_{1}>T_{\text{rh}}$, the
generated $B-L$ yield is given by:555Note the suppression by a factor of $1/2$
relative to eq. (53), which is due to the fact that now only a single $N_{1}$
is produced in each three–body inflaton decay.

$\displaystyle Y_{B-L}$
$\displaystyle=\left[\frac{3}{4}\frac{T_{\text{rh}}}{m_{\phi}}\,\text{BR}(\phi\to\ell
N_{1}H)\right]\,\epsilon_{1}$
$\displaystyle=\frac{3}{4}\frac{T_{\text{rh}}}{m_{\phi}}\left[\frac{m_{\phi}}{8\pi^{2}v^{2}}\left(m_{2}|\cos
z|^{2}+m_{3}|\sin z|^{2}\right)\cdot f(\gamma)\right]$ $\displaystyle\hskip
14.22636pt\cdot\left[-\frac{3}{16\pi}\frac{M_{1}}{v^{2}}(m_{3}^{2}-m_{2}^{2})\frac{\text{Im}\left(\sin^{2}z\right)}{\left(m_{2}|\cos
z|^{2}+m_{3}|\sin z|^{2}\right)}\right]$
$\displaystyle=-\frac{9}{512\pi^{2}}\frac{M_{1}\,T_{\text{rh}}}{v^{4}}(m_{3}^{2}-m_{2}^{2})\text{Im}\left(\sin^{2}z\right)\cdot
f(\gamma)$ $\displaystyle\simeq-4.8\times
10^{-11}\left(\frac{m_{\phi}}{10^{11}\
\text{GeV}}\right)\left(\frac{T_{\text{rh}}}{10^{11}\
\text{GeV}}\right)\text{Im}\left(\sin^{2}z\right)\cdot\sqrt{\gamma}f(\gamma)\,.$
(64)

In the third line we have utilized eq. (6).

Before closing this section, we note that RHNs could also be produced in the
scattering of an inflaton and a daughter particle, $\phi H\to\ell N_{1}$ and
$\phi\,\ell\to HN_{1}$. For $m_{\phi}>M_{1}$ this is possible even if the
daughter particle in the initial state is soft, e.g. after thermalization with
low reheating temperature. The rate for these scattering reactions involves
the same product of couplings that appears in the three–body decay width.
However the rate for inflaton scattering on a thermalized daughter particle is
suppressed by a factor $T/m_{\phi}$ compared to the three–body decay rate; the
rate for inflaton scattering on daughter particles before they thermalize is
even smaller. We have therefore neglected these $2\to 2$ scattering reactions.

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