# KGConv, a Conversational Corpus grounded in Wikidata

Quentin Brabant2, Gwenole Lecorve2, Lina M. Rojas-Barahona2, Claire Gardent1
1CNRS/LORIA, Nancy, France
2Orange Innovation, Lannion, France
<EMAIL_ADDRESS>
{quentin.brabant, gwenole.lecorve<EMAIL_ADDRESS>

(July 2022)

###### Abstract

We present KGConv, a large, conversational corpus of 71k conversations where
each question-answer pair is grounded in a Wikidata fact. Conversations
contain on average 8.6 questions and for each Wikidata fact, we provide
multiple variants (12 on average) of the corresponding question using
templates, human annotations, hand-crafted rules and a question rewriting
neural model. We provide baselines for the task of Knowledge-Based,
Conversational Question Generation. KGConv can further be used for other
generation and analysis tasks such as single-turn question generation from
Wikidata triples, question rewriting, question answering from conversation or
from knowledge graphs and quiz generation.

##  1 Introduction

Different from open domain and task-based dialogs, information seeking dialogs
are driven by the desire to acquire or evaluate knowledge. These dialogs are
central for instance, in educational (tutoring a student about a given topic
by asking her a set of questions about that topic) and entertainment (quizzes)
settings. As large knowledge graphs such as
Wikidata111https://www.wikidata.org/ have started to emerge, recent years have
seen an increasing interest in developing datasets and conversational question
answering models that can support such information seeking interactions by
grounding dialogs in factual data. They often focus on question answering Saha
et al. (2018) or provide datasets of restricted size and variety however
Christmann et al. (2019); Lecorvé et al. (2022).

In this paper, we focus on information seeking dialogs where as illustrated in
Table 1, each question-answer turn is grounded in a single fact.

Our contribution is two fold. First, we make available the KGConv dataset
where, as illustrated in Table 1, each question-answer pair is grounded in a
Wikidata fact222We will make our code, data and pretrained models available
online after publication.. To create a diverse, large scale dataset (70k
conversations), we develop dialogs for eight different topics (Country, Food,
Person, Religion/Ideology, Space Object, Taxon333A taxon is a population, or
group of populations of biological organisms, e.g. lions or dinosaurs.,
Molecular Entity, and Historical Event). We also include a set of
conversations containing at least one Wikidata property that does not appear
in any of this eight themes to study generalisation. The dataset include
$604$K distinct question-answer pairs covering 458 Wikidata properties
verbalised by 3,879 templates. For each Wikidata triple, there are 12 question
variants in average. The dialogs consist of 8.7 question-answer turns in
average and for each question in a dialog, there are three variants: (i) an
out-of-context form, whose interpretation is independent of the previous
turns, (ii) an in-context form, where pronouns have been inserted using a rule
based approach and (iii) a synthetic in-context form obtained by rewriting in-
context forms using a T5 model trained to rewrite questions.

Our second contribution is to establish some baselines for Knowledge-Based,
Conversational Question Generation (CQG), the task of generating a question
given both a KG fact and a dialog context. While much previous work has
focused on Knowledge-Based, conversation Question Answering Saha et al.
(2018); Perez-Beltrachini et al. (2023) or on context-independent, knowledge-
based question generation Bordes et al. (2015); Elsahar et al. (2018); Han et
al. (2022), we provide a first investigation of how knowledge-based question
generation interacts with conversational context. We report results using both
automatic metrics and human evaluation.

#1 | Triple |  (NGC 4833, part of, Milky Way)
---|---|---
Question variants | OOC |  NGC 4833 is part of what astronomical object?
IC |  -
SIC |  -
OOC |  Where is NGC 4833 located?
IC |  -
SIC |  -
Answer |  Milky Way
#2 | Triple |  (NGC 4833, discoverer or inventor, Nicolas Louis de Lacaille)
Question variants | OOC |  Who was behind the discovery of NGC 4833?
IC |  -
SIC |  Who was behind the discovery?
OOC |  What was the name of the discoverer of NGC 4833?
IC |  -
SIC |  Who discovered this object?
OOC |  Who found NGC 4833?
IC |  -
SIC |  Who found this object?
Answer |  Nicolas Louis de Lacaille
#3 | Triple |  (Nicolas Louis de Lacaille, religion or worldview, Catholic Church)
Question variants | OOC |  What was Nicolas Louis de Lacaille’s religion?
IC |  What was his religion?
SIC |  -
OOC |  What faith did Nicolas Louis de Lacaille follow?
IC |  What faith did he follow?
SIC |  -
Answer |  Catholic Church
Table 1: Excerpt of a question-answer conversation along with the related
triples. The root entity is NGC 4833, from the theme “space object”. Redundant
variants are omitted for readability.

##  2 Related Work

Question generation from RDF triples is adressed in Bordes et al. (2015);
Elsahar et al. (2018); Han et al. (2022) and from small KGs depicting multi-
hop questions Serban et al. (2016); Kumar et al. (2019); Bi et al. (2020) and
recently in LC-QuAD 2.0 Dubey et al. (2019) and ParaQA Kacupaj et al. (2021).
However, these works are limited to the generation of isolated questions, thus
no conversational context is under consideration.

CoQA Reddy et al. (2019) and QuAC Choi et al. (2018) are conversational QA
corpora, in which answers are extracted from paragraphs, instead of KGs.
Similar to KGConv, ConvQuestions Christmann et al. (2019) and CSQA Saha et al.
(2018) are conversational corpora based-on structured knowledge. However, the
former does not provide the triples for the questions and contains only $315$
distinct conversations. The latter, despite being a very large dataset
covering a wide range of questions types444Single or multiple triples,
entity/numeric/boolean answers, comparative questions, contains rather
unnatural questions with a proprietary formalism which does not directly
correspond to Wikidata triples. Recent work proposes to generate questions
from SPARQL queries, especially to express complex questions Lecorvé et al.
(2022); Perez-Beltrachini et al. (2023). However, manually annotating
questions SPARQL queries is difficult and the authors turn on a small set of
$350$ reference questions. In contrast, our dataset focuses on triples and
simple questions in order to enable a large set of reference questions,
obtained through manually written verbalization templates.

Our work capitalizes on two existing corpora which provide a correspondence
between triples and questions, namely SimpleQuestions Bordes et al. (2015) and
ZeroShot Relation Extraction Levy et al. (2017). In comparison, the KGConv
corpus extends the question templates from these corpora by proposing 3,879
new templates, and focusing on 458 properties. Likewise, they focused on
isolated questions or short follow-up turns, up to three turns, while KGConv
contains $8.7$ turns per conversation in average.

##  3 Overview of the Dataset

In KGConv, each conversation is focused on a given _root_ entity. As
illustrated in Table 1, the first question bears directly on this root entity,
while further questions explore new facts about any entity discovered during
the conversation (including the root entity itself). Hence, a conversation can
be seen as a small evolving KG, where each turn expands the conversation graph
with a new entity and the property which connects it to the existing graph.

For each root entity, three conversations are derived from Wikidata in order
to increase the diversity of the dataset. The corpus covers eight themes:
Country, Food, Person, Religion/Ideology, Space Object, Taxon555A taxon is a
population, or group of populations of biological organisms, e.g. lions or
dinosaurs., Molecular Entity, and Historical Event. The theme of a
conversation corresponds to the Wikidata class associated to the root entity
e.g., Person corresponds to the Q215627 class in Wikidata. We use Taxon and
Space Object as unseen data, data not seen at training time. There is also a
set of conversations containing at least one Wikidata property that does not
appear in any of the eight regular themes. Table 2 summarizes the size of the
dataset for each theme. The number of questions in a conversation is at least
5, at most 19 and 8.6 on average. We provide in total $70,596$ conversations
with $603,905$ questions from $63,345$ Wikidata entities and $458$ properties.
To enable links with Wikidata and further extensions, Wikidata IDs are
provided for all entities and properties along with their natural language
labels.

In the conversations, each question has several paraphrases (up to 10, 6.8 on
average), and each paraphrase has three versions:

  * •

Out-of-Context (OOC): this version of a question is produced semi-
automatically from a KG fact independent of the conversation context.

  * •

In-Context (IC): this version is derived from the OOC version taking the
context into account and using rules to substitute repetitions with anaphoric
forms.

  * •

Synthetic-In-Context (SIC): this version is derived from the IC version by
applying a generative model trained to rewrite questions.

| entities | properties | triples | conv. | number of question-turns | templates | references
---|---|---|---|---|---|---|---
| (number in all conversations) | train | dev | test | total | (avg. per question-turn)
person | 32k | 327 | 72k | 26k | 185k | 29k | 11k | 226k | 7.2 | 12.5
country | 2k | 171 | 3k | 0.7k | 5k | 0.8k | 0.2k | 6k | 5.5 | 9
ideology | 1k | 169 | 2k | 0.4k | 3k | 0.6k | 0.2k | 4k | 6.6 | 11.4
space object | 3k | 116 | 6k | 6k | 0 | 0 | 50k | 50k | 7.3 | 12.4
molecular entity | 18k | 151 | 38k | 23k | 155k | 25k | 10k | 189k | 6.4 | 11.7
historical event | 5k | 189 | 8k | 5k | 35k | 6k | 2k | 43k | 5.6 | 10.2
food | 3k | 166 | 4k | 2k | 15k | 2k | 1k | 18k | 6.1 | 10.4
taxon | 3k | 215 | 5k | 2k | 0 | 0 | 16k | 16k | 7.9 | 13.6
with unseen prop. | 14k | 404 | 24k | 6k | 0 | 0 | 52k | 52k | 6.9 | 12.1
whole dataset | 63k | 458 | 143k | 71k | 398k | 63k | 143k | 604k | 6.8 | 12
Table 2: Quantitative summary of KGConv. Entities, properties and triples can
appear in several conversations and several themes but are only counted once.
The two last columns show the average number of templates used in a single
question-turn, and the number of distinct references (including OOC, IC and
SIC versions of all template-based verbalizations).

##  4 Data Collection

The process for creating the corpus is summarized in Figure 1 and elicited in
the following subsections.

Figure 1: Data flow diagram of the dataset generation.

###  4.1 Sequence Extraction

In this step, we extract sequences of triples from Wikidata which will be used
to ground the dialogs.

We first extract a set of triples from Wikidata such that (i) the subject has
at least one English label and the object either has an English label or is a
literal (string, number, date, etc.) and (ii) the property belongs to a pre-
defined set of Wikidata properties which excludes “uninteresting” properties.
In particular, we excluded properties whose prefix is not “wdt:” (to avoid
triples that provide meta-information), and properties which link entities to
their ID in some other database. For any triple $(s,p,o)$ in this set, we then
create the reversed triple $(o,-p,s)$ where $-p$ denotes the inverse of
property $p$. In this way, if our subgraph contains (France, capital, Paris),
it also contains (Paris, capital, France) which permits creating questions
about both the subject $s$ and the object $o$ e.g., “What is the capital of
France” and “Paris is the capital of which country?”. We call the resulting
Wikidata subgraph $W$ for World.

Based on this set of triples, we then create sequences of triples as follows.
Each conversation will focus on triples in the neighborhood ${\cal N}(r)$ of a
root entity $r$ in $W$. This neighborhood is defined as the subgraph of $W$
containing $r$ and all nodes (i.e. entities) that are 1 or 2 edges (i.e.
properties) away from $r$; in other words, ${\cal N}(r)$ contains all triples
of the form $(r,p,o_{1})$ and their successors of the form $(o_{1},q,o_{2})$.
Roots were sampled from instances of the 8 themes in Table 2, with the
condition that their neighborhood is large enough (at least 20 triples) to
generate 3 reasonably long conversations with enough differences.

For each root, 3 triple sequences of the form $(t_{0},t_{1},\dots,t_{n})$ were
built iteratively by picking triples from ${\cal N}(r)$ in a greedy stochastic
process. At each step of the process, the subject of the chosen triple is
either the root (i.e. $s_{i}=r$) or an entity, either subject or object, from
the previous triple ($s_{i}=s_{i-1}$ or $s_{i}=o_{i-1}$). Additionally, a
triple cannot appear twice in the sequence. The decision to stop or continue
the process is made at each time step $i$ following a probability that
increases with $i$: $\Pr_{stop}(i)=0.06i-0.18$.

###  4.2 Verbalization

Questions are generated using templates like “What is the capital of ___?”,
where the slot is to be filled by the subject of a triple. Each template
$\tau$ is applicable for a given property $p_{\tau}$ (e.g.,“capital of”),
given required types $\mathcal{S}_{\tau}$ for the subject slot and required
types $\mathcal{O}_{\tau}$ for the object, which will be the answer. This
applicability condition on $\tau$ is denoted as
$C(\tau)=(p_{\tau},\mathcal{S}_{\tau},\mathcal{O}_{\tau})$. Then, a triple
$(s,p,o)$ satisfies $C(\tau)$ if: $p_{\tau}=p$,
$\mathcal{S}_{\tau}\subset\mbox{types}(s)$, and
$\mathcal{O}_{\tau}\subset\mbox{types}(o)$.

To create a large number of diverse questions for all properties in $W$, we
gathered templates from three sources. Table 3 summarizes the number of
templates and their sources. The following sections provide details on the
methods used to get templates from each source.

####  4.2.1 Zero-Shot templates

The Zero-Shot dataset Levy et al. (2017) contains 1,192 question templates
spanning 120 Wikidata properties. Each template $\tau$ was originally created
to ask a question for a given property $p$, without taking into account the
type of the subject and object. Templates grounded on properties that were no
longer in Wikidata were discarded. For the remained templates, we defined the
applicability condition as $C(\tau)=(p,\emptyset,\emptyset)$.

####  4.2.2 Simple questions v2 templates

We automatically extracted templates from the SimpleQuestions_v2 dataset
Bordes et al. (2015), which contains 108k triple-question pairs, involving
131k distinct entities and 1,837 properties.

As SimpleQuestions_v2 is based on the Freebase KG, we translated entities and
properties into their Wikidata counterpart: we relied both on the Wikidata
property P646, that links Wikidata entities to their Freebase counterpart, and
on an available mapping between Freebase and Wikidata
properties666https://www.wikidata.org/wiki/Wikidata:WikiProject_Freebase/Mapping.
This allowed us to get Wikidata counterparts for 83,447 entities and 142
properties.

We then extracted templates from question-triple pairs whose triple could be
translated to Wikidata. For each such triple-question pair $((s,p,o),q)$, we
created a template $\tau$ by replacing in $q$ the label of $s$ by an empty
slot. The applicability condition of $\tau$ was then defined as
$C(\tau)=(p,\mbox{types}(s),\mbox{types}(o))$, where $\mbox{types}(x)$ is the
set of all types of $x$ in Wikidata. Since only a small subset of Freebase
entity and triples could be translated into a Wikidata counterpart, many
triples were filtered out, so the extracted templates only cover 77 of our
properties.

####  4.2.3 New templates

We manually created additional templates in three steps: (1) extracting
applicability conditions, (2) writing templates corresponding to these
conditions, (3) validating written templates.

##### Step 1: Extracting applicability conditions.

From the neighbourhoods of potential roots of all themes, we gathered a set
$\\{(s_{i},p_{i},o_{i})\\}^{N}_{i=0}$ of Wikidata triples for which $p_{i}$ we
did not have any template. From this set of triples we generated a set of
applicability conditions
$\\{(p_{i},\mbox{types}(s_{i}),\mbox{types}(o_{i}))\\}^{N}_{i=0}$ to be
annotated with corresponding templates. This resulted in many applicability
conditions, with overly specific subject and object types for each property.
We solved this problem by iteratively merging conditions with the same
property, via semi-automated conceptual agglomerative clustering process.
Merging two conditions $(p,\mathcal{S}_{i},\mathcal{O}_{i})$ and
$(p,\mathcal{S}_{j},\mathcal{O}_{j})$ consists in replacing them with a new
one $(p,\mathcal{S}_{i}\cap S_{j},\mathcal{O}_{i}\cap\mathcal{O}_{j})$, which
necessarily is matched by more triples than the two original ones. At the end
of this process, applicability conditions that were met by less than 5 triples
were discarded.

##### Step 2: Template writing.

The next step was to write, for each applicability condition $C$, templates
that would apply to any triple matching $C$.

Three students of an NLP Master program were hired to annotate the question
templates. They were native English speakers hired on a short term contract to
perform various annotation tasks for NLP. They were paid slightly above the
national minimum wage. We provided them with an annotation tool, in which one
applicability condition at a time was displayed, along with 5 examples of
matching triples. While annotator were writing templates, their results on
these triple were displayed. To speed up this process, artificially generated
templates were also proposed to the annotators, who could accept or reject
them. Afterward, accepted artificial templates were treated in the same way as
those written by a human annotator.

##### Step 3: Template validation.

To ensure the quality of the resulting templates, all of them were manually
filtered by the authors.

| Properties | Templates | Tpl. per prop.
---|---|---|---
|  |  | min / avg /max.
SimpleQuestions |  77 |  5,817 | 1 / 76 / 453
ZeroShot RE |  75 |  771 | 1 / 10 / 31
New templates | 413 |  3,879 | 1 / 9 / 80
Total | 474 | 10,355 | 1 / 22 / 453
Table 3: Statistics on properties and templates for each sources of templates.

###  4.3 Contextualization

Applying the templates from Section 4.2 to the triple sequences from Section
4.1 yields conversations where questions are Out Of Context (OOC) i.e.,
questions which are independent of the conversational context. To improve the
naturalness of the conversations, we derived two in-context versions from
these OOC conversations. The first one, _IC_ , is obtained by applying hand-
crafted rules to introduce coreferences and correct some errors produced
during verbalization with templates. The second version (_SIC_ , Synthetic-In-
Context), results from rewriting IC questions with a T5 model trained to
rewrite questions.

####  4.3.1 Post-Processing Dialogs (IC Variants)

##### Referring Expressions.

In Wikidata, an entity can have several labels: one of those is called
“preferred label” and is meant to be used by default. In the OOC version,
entities are always referred by their preferred label. This step introduces
variability by replacing some of the preferred labels with other available
labels from Wikidata, according to the following rules: (1) the first
reference to an entity in the conversation is the preferred label, or contains
it as a substring; (2) further mentions are labels that are substrings of the
first reference. For instance, the entity Q9592 has the preferred label
$l_{1}=$ “Catholic Church”, alternative labels $l_{2}=$ “Roman Catholic
Church” and $l_{3}=$ “Roman Apostolic Catholic Church”; if $l_{2}$ is used as
the first reference to the entity, next references will use either $l_{1}$ or
$l_{2}$ but not $l_{3}$, since it is not a substring of $l_{2}$. Whenever the
subject is a person with a name and surname, we include its surname in the set
of available labels.

##### Determiners.

Deciding which label should be preceded by “the” is not trivial. For example,
“United Kingdom” and “Republic of China” require it, while “France” and
“China” do not. This step handles this problem by asking a BERT language
model777https://huggingface.co/bert-base-uncased to fill a mask token inserted
before the label; when “the” was predicted with a probability at least $0.92$,
it was inserted before the label.

##### Tense.

We noticed that most templates are written in present tense, while many
triples describe facts that are no longer true or concern dead people, past
events, etc. Questions were rewritten in the past
tense888https://github.com/bendichter/tenseflow if the corresponding triple
had an “end time” qualifier in Wikidata, or if its subject or object was a
dead person.

##### Rule-based introduction of pronouns.

Subject mentions are pronominalised using a rule-based approach: a pronoun is
used only if the subject also appears in the triple of the previous question
and if its gender differs from the gender of the object of this triple (to
avoid ambiguous pronouns); further rules are used to determine the kind of
pronoun to use (for example, if the subject reference is followed by a
possessive “s”, a possessive pronoun should be used, etc.). The
pronominalisation rules are given in Appendix A..

###  4.4 Model-based rewritings (SIC Variants)

To further increase the conversationality degree, a T5-based question
rewriting model999From the base version on HuggingFace:
https://huggingface.co/t5-base. was fine-tuned on a training set derived from
2 conversational machine reading QA datasets, namely CANARD Elgohary et al.
(2019) and CoQAR Brabant et al. (2022). This training set is made of $142$K
instances. For each instance, the input is a question $q_{i}$, along with its
conversation history $[q_{0},a_{0},\dots,q_{i-1},a_{i-1}]$, while the output
is a semantically equivalent question whose form is expected to be natural in
a conversation, denoted by $q_{i}^{*}$. In some instances, $q_{i}$ and
$q_{i}^{*}$ have respectively an OOC and a contextualized form. In others
instances, $q_{i}$ and $q_{i}^{*}$ are equal; these instances correspond to
cases where either $q_{i}$ already has an IC form, or there is no natural way
to rewrite it without losing information or bringing ambiguity. Including such
cases to the training set enables the model to learn when it should rewrite
the input question or not.

At inference time, the 20 best hypotheses are generated by the model for each
instance. Then, they are classified into three authorized categories, using a
set of expert conditions: (1) coreference with a pronoun (e.g., “In which
country is Kyoto located?” rewritten as “In which country is it located?”),
(2) coreferences with a demonstrative noun phrase (e.g., “In which country is
this city located?”), and (3) ellipses (e.g., “In which country?”). Those that
do not belong to any category are filtered out; moreover, to limit possible
ambiguities, we prohibit two consecutive reformulations of the same category.
Finally, if some hypothesis remain, the one with the highest probability is
selected as the rewritten form.

This process was applied on all questions of the IC version, leading to the
SIC (Synthetic IC) version. More details can be found in Appendix B..

| IC | SIC $\neq$ IC
---|---|---
| alt. label | pronoun
person | 13.8 | 23.7 | 41.7
country | 4.1 | 8.4 | 47.0
ideology | 6.4 | 12.3 | 56.7
space object | 1.6 | 13.7 | 60.4
molecular entity | 10.8 | 6.0 | 63.7
historical event | 6.7 | 8.4 | 66.0
food | 7.6 | 9.4 | 60.5
taxon | 2.0 | 14.2 | 57.2
unseen properties | 12.6 | 20.2 | 45.5
Table 4: Percentage of questions using an alternative label and a pronoun in
IC questions; percentage of SIC questions that differ from the IC version.

##  5 Conversational Question Generation

Knowledge-Based, Conversational Question Generation extends Question
Generation from Knowledge Graph triples Elsahar et al. (2018); Han et al.
(2022) to a conversational setting: instead of generating a question only from
a triple, we generate a question from both a triple and the preceding
conversational context. This raises the additional challenge of generating
questions in contextually appropriate forms e.g., using appropriate referring
expressions and ellipses. Leveraging the multimodal text/graph nature of our
dataset, we explore four ways of representing the context: (1) no contextual
information at all (Empty), (2) the sequence of previous questions and answers
(NL) (3) the sequence of triples underlying the questions and answers (KG) and
(4) the sequence of questions and answers with their corresponding triples
(NL+KG).

For each of the four variants, we trained a baseline by fine-tuning a T5-small
model on the three versions of questions in KGConv (OOC, IC and SIC). In the
train and dev sets, all themes are mixed together. The number of epochs for
training is determined via early stopping.

##  6 Automatic Evaluation

We evaluate each model on the test set using Google-BLEU and BERT-score taking
as references all questions associated with the input triples (in OOC, IC, and
SIC forms), around 12 references on average. The results are presented in
Table 11.

Table 5: Results of the automatic evaluation. Seen themes are those with a
non-empty training set (see Table 2), unseen themes are space object and
taxon. The scores are obtained by macro-averaging over themes. The best score
is in bold; lower scores that do not differ significantly from the best
one111111$p>0.05$ in a Mann-Whitney U test are adorned with (*).

##### Seen vs. unseen.

Unsurprisingly, all models obtain lower scores on unseen themes. Similarly,
average scores are lower on unseen properties because the verbalization highly
depends on the property of the triple.

##### OOC vs. IC vs. SIC training data.

In term of GLEU, models trained on SIC generally do better than their IC and
OOC counterparts. However, this tendency is not confirmed with BERT-scores.
Moreover, these better GLEU scores might just be an artifact of the
experimental design. Indeed references contain OOC, IC and SIC question
versions, and models trained on SIC are the only one to have be trained to
generate all three versions. Thus, SIC models might have better GLEU simply
because their training data is more in-line with the references.

##### Conversational context format.

Adding conversational context to the models trained on IC and SIC questions
consistently improves GLEU and BERT-scores. Looking at GLEU scores, it also
seems that providing the context in the form of triples (or triples and text)
provides better improvement than context in the form of text.Since
conversational context is not required to generate OOC questions, models
trained on OOC questions tend to perform better when no conversational context
is given, except for unseen properties.

##  7 Human Evaluation

The human evaluation assesses both the dataset and the two baselines: IC
(KG+NL) and SIC (KG+NL). We sample conversations from the test set. Then, for
each of them, we created 4 alternative versions by choosing a specific version
of the questions: in two versions, the questions are chosen, respectively,
among the IC and the SIC questions of the dataset. The other two versions turn
on the two IC and SIC baselines used to generate the questions from the triple
and the context. are fed to an evaluation interface, through which they are
rated by human evaluators. As an illustration, the four versions of a piece of
conversation is given in Appendix C. Table 6 gives the number of conversations
rated by evaluators depending on theme and version.

|  | Ref. | Ref. | Model | Model
---|---|---|---|---|---
|  | IC | SIC | IC | SIC
(seen) | person | 10 | 10 | 9 | 8
food | 10 | 11 | 10 | 10
(unseen) | taxon | 8 | 8 | 8 | 9
space object | 6 | 8 | 9 | 8
Table 6: Number of rated conversations. |  |  | # of | # of | Faithfulness to the triple | Linguistic | Semantic | Conversat.
---|---|---|---|---|---|---|---|---
|  |  | conv. | quest. | % yes↑ | % no↓ | % quite↓ | % idk | correctness↑ | clearness↑ | naturalness↑
Seen | Reference from | IC | 20 | 184 | 0.91 | 0.05 | 0.03 | 0.01 | 4.61 | 4.60 | 3.85
SIC | 21 | 183 | 0.77 | 0.15 | 0.08 | 0.01 | 4.64 | 4.09 | 3.52
Baseline trained on | IC | 19 | 178 | 0.84 | 0.10 | 0.04 | 0.02 | 4.67 | 4.65 | 3.63
SIC | 18 | 161 | 0.80 | 0.12 | 0.07 | 0.01 | 4.73 | 4.21 | 3.39
Unseen | Reference from | IC | 14 | 128 | 0.88 | 0.06 | 0.02 | 0.03 | 4.59 | 4.57 | 3.93
SIC | 16 | 140 | 0.64 | 0.19 | 0.11 | 0.06 | 4.51 | 3.79 | 3.00
Baseline trained on | IC | 17 | 157 | 0.71 | 0.14 | 0.13 | 0.03 | 4.53 | 4.47 | 3.65
SIC | 17 | 153 | 0.76 | 0.10 | 0.12 | 0.02 | 4.76 | 4.05 | 2.88
All | Reference from | IC | 34 | 312 | 0.90 | 0.06 | 0.03 | 0.02 | 4.60 | 4.59 | 3.88
SIC | 37 | 323 | 0.71 | 0.17 | 0.09 | 0.03 | 4.59 | 3.96 | 3.30
Baseline trained on | IC | 36 | 335 | 0.78 | 0.12 | 0.08 | 0.02 | 4.60 | 4.56 | 3.64
SIC | 35 | 314 | 0.78 | 0.11 | 0.10 | 0.02 | 4.75 | 4.13 | 3.14
Table 7: Scores from human evaluation for conversations about seen, unseen or
all themes.

###  7.1 Evaluation Setup

The ratings were provided by 15 evaluators from the authors’ research center
(the authors themselves are not included). Each evaluator could evaluate up to
50 conversations. They were told that conversations are automatically
generated, but were provided no information about the method employed.
Conversations were presented one by one to the evaluator; for each question-
answer pair, the corresponding triple was displayed, and the evaluator had to
(1) rate the linguistic correctness of the question on a 5 point scale, (2)
evaluate whether the question-answer pair expresses the information of the
triple (“yes”, “quite”, “no”, “I don’t know”). In addition, the evaluator had
to rate the naturalness of the whole conversation flow.

A second round of evaluation on 61 of the already rated conversations was
performed to assess the consistency of ratings among annotators. This
evaluation utilizes the IC and SIC versions of the questions (i.e.,
conversations using baselines were not included in the second round). This
evaluation was performed by 3 annotators who did not participate in the first
one.

###  7.2 Results

Results are provided in Table 7. We use the Mann-Whitney U and the $\chi^{2}$
test to assess significance. The former was used for correctness, clearness
and naturalness scores, since those are evaluated on an ordered scale. The
latter was used for faithfulness scores, since these form a scale that is not
completely ordered (because of the “I don’t know” answer).

##### IC vs SIC data.

Comparing the scores of IC and SIC references in the _All_ block of Table 7,
we see that, while linguistic correctness is roughly the same, SIC references
are less clear (clearness 4.59 vs 3.96, p=2e-12), less faithful to the triples
(0.90 yes vs 0.71, p=8e-8) and less natural overall (3.88 vs 3.30, p=0.006).

##### IC vs SIC baseline.

Baseline (SIC) seems a bit more linguistically correct (4.60 vs 4.75,
p=0.001), but baseline (IC) seems clearer (4.56 vs 4.13, p=4e-5) and more
natural overall, although it might be due to chance (3.64 vs 3.14, p=0.066).
Faithfulness seems to be the same.

##### Data vs baselines.

Now let us compare the baselines to the data they were trained on. For the IC
data, the baseline is less faithful to triples (0.90 yes vs 0.78, p=0.00014),
otherwise we observe no significant difference. For the SIC data, the baseline
is better on every measure, although we obtain low p-values only for clearness
(3.96 vs 4.13, p=0.012) and correctness (4.59 vs 4.75, p=0.0012).

##### Seen vs. unseen.

Scores obtained on unseen themes tend to be lower that those obtained on seen
themes. Although some of these differences are significant, they happen both
for the baseline (IC) and for the SIC references. This suggests that the
differences are due to the difficulty of themes rather that the fact that they
were seen during the training of baselines or not (more details in Appendix
D.).

##### Inter-rater agreement.

We computed Cohen’s kappa for each metric (faithfulness, correctness,
clearness, naturalness) and obtained, respectively: 0.23, 0,10, 0.22, and
0.14121212 The confusion matrices used for computing kappas can be found in
Appendix E.. Low kappas seem due to two factors: (1) the intrinsically
subjective nature of the task, which can explain that raters disagree by
giving different but close rates, (2) genuine mistakes made by raters (for
example, when faithfulness is rated at $yes$ and $no$ by two different
raters). It is also possible that differences in raters’ fluency had an impact
on agreement. Despite the low agreement, we observed the interesting
regularities reported previously in this section.

##  8 Conclusion

We make available KGConv, a new conversational dataset grounded in Wikidata
where each question-turn in the conversation contains three variants: the out-
of-context questions generated from manually annotated and existing templates;
the in-context question automatically generated by rules and the synthetic in-
context questions generated by a T5 model. Although SIC questions have more
diverse forms than IC questions, the results of human evaluation suggest that
IC questions are more reliable than SIC questions.

We also presented several baselines for the task of question generation and
found that generating questions from unseen properties is challenging for
these baselines. An interesting perspective would be to investigate methods
for tackling this particular zero-shot task.

As it provides a large number of references for each question in a dialog,
KGConv is well suited for other tasks besides Conversational Question
Generation such as in particular, single-turn question generation from facts,
question rewriting and generation of sequence of question-answer pairs from a
Knowledge graph (KG) or vice-versa.

##  9 Limitations

This corpus has been generated semi-automatically, although human annotations
were involved in the question templates, the conversations were generated
automatically from the KG. As a consequence, in some cases the flow of the
conversation may be unnatural, because humans do not usually talk in that way.
This might be specially true when conversations involve complex content (e.g.
molecular entities, space objects or historical events) that may be difficult
to be understood by non experts.

##  10 Ethics Statement

The construction of this corpus involved manual annotation of the question
templates. Therefore, we hire three students of an NLP Master program. They
were native English speakers hired on a short term contract to perform various
annotation tasks for NLP. They were paid slightly above the national minimum
wage and they had the right to the social security benefits.

## References

  * Bi et al. (2020) Sheng Bi, Xiya Cheng, Yuan-Fang Li, Yongzhen Wang, and Guilin Qi. 2020.  Knowledge-enriched, type-constrained and grammar-guided question generation over knowledge bases.  In _Proceedings of the International Conference on Computational Linguistics (CICLing)_ , pages 2776–2786.
  * Bordes et al. (2015) Antoine Bordes, Nicolas Usunier, Sumit Chopra, and Jason Weston. 2015.  Large-scale Simple Question Answering with Memory Networks.  ArXiv:1506.02075 [cs].
  * Brabant et al. (2022) Quentin Brabant, Gwénolé Lecorvé, and Lina M. Rojas Barahona. 2022.  CoQAR: Question rewriting on CoQA.  In _Proceedings of the Thirteenth Language Resources and Evaluation Conference_ , pages 119–126, Marseille, France. European Language Resources Association.
  * Choi et al. (2018) E. Choi, H. He, M. Iyyer, M. Yatskar, W. Yih, Y. Choi, P. Liang, and L. Zettlemoyer. 2018.  QuAC: Question Answering in Context.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 2174–2184, Brussels, Belgium. Association for Computational Linguistics.
  * Christmann et al. (2019) Philipp Christmann, Rishiraj Saha Roy, Abdalghani Abujabal, Jyotsna Singh, and Gerhard Weikum. 2019.  Look before you Hop: Conversational Question Answering over Knowledge Graphs Using Judicious Context Expansion.  In _Proceedings of the ACM International Conference on Information and Knowledge Management_ , pages 729–738. Association for Computing Machinery.
  * Dubey et al. (2019) Mohnish Dubey, Debayan Banerjee, Abdelrahman Abdelkawi, and Jens Lehmann. 2019.  LC-QuAD 2.0: A Large Dataset for Complex Question Answering over Wikidata and DBpedia.  In _Proceedings of the The Semantic Web (ISWC)_ , pages 69–78. Springer International Publishing.
  * Elgohary et al. (2019) Ahmed Elgohary, Denis Peskov, and Jordan Boyd-Graber. 2019.  Can you unpack that? learning to rewrite questions-in-context.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 5918–5924, Hong Kong, China. Association for Computational Linguistics.
  * Elsahar et al. (2018) Hady Elsahar, Christophe Gravier, and Frederique Laforest. 2018.  Zero-shot question generation from knowledge graphs for unseen predicates and entity types.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ , pages 218–228, New Orleans, Louisiana. Association for Computational Linguistics.
  * Han et al. (2022) Kelvin Han, Thiago Castro Ferreira, and Claire Gardent. 2022.  Generating questions from Wikidata triples.  In _Proceedings of the Thirteenth Language Resources and Evaluation Conference_ , pages 277–290, Marseille, France. European Language Resources Association.
  * Kacupaj et al. (2021) Endri Kacupaj, Barshana Banerjee, Kuldeep Singh, and Jens Lehmann. 2021.  ParaQA: A Question Answering Dataset with Paraphrase Responses for Single-Turn Conversation.  In _Proceedings of the The Semantic Web (ISWC)_ , pages 598–613. Springer International Publishing.
  * Kumar et al. (2019) Vishwajeet Kumar, Yuncheng Hua, Ganesh Ramakrishnan, Guilin Qi, Lianli Gao, and Yuan-Fang Li. 2019.  Difficulty-Controllable Multi-hop Question Generation from Knowledge Graphs.  In _Proceedings of The Semantic Web (ISWC)_ , pages 382–398. Springer International Publishing.
  * Lecorvé et al. (2022) Gwénolé Lecorvé, Morgan Veyret, Quentin Brabant, and Lina M. Rojas-Barahona. 2022.  SPARQL-to-text question generation for knowledge-based conversational applications.
  * Levy et al. (2017) Omer Levy, Minjoon Seo, Eunsol Choi, and Luke Zettlemoyer. 2017.  Zero-Shot Relation Extraction via Reading Comprehension.  In _Proceedings of the 21st Conference on Computational Natural Language Learning (CoNLL 2017)_ , pages 333–342, Vancouver, Canada. Association for Computational Linguistics.
  * Perez-Beltrachini et al. (2023) Laura Perez-Beltrachini, Parag Jain, Emilio Monti, and Mirella Lapata. 2023.  Semantic parsing for conversational question answering over knowledge graphs.  In _Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics_ , pages 2507–2522, Dubrovnik, Croatia. Association for Computational Linguistics.
  * Reddy et al. (2019) S. Reddy, D. Chen, and C.D. Manning. 2019.  CoQA: A Conversational Question Answering Challenge.  _Transactions of the Association for Computational Linguistics_ , 7:249–266.
  * Saha et al. (2018) Amrita Saha, Vardaan Pahuja, Mitesh Khapra, Karthik Sankaranarayanan, and Sarath Chandar. 2018.  Complex Sequential Question Answering: Towards Learning to Converse Over Linked Question Answer Pairs with a Knowledge Graph.  _Proceedings of the AAAI Conference on Artificial Intelligence_ , 32(1).
  * Serban et al. (2016) Iulian Vlad Serban, Alberto García-Durán, Çaglar Gülçehre, Sungjin Ahn, Sarath Chandar, Aaron C Courville, and Yoshua Bengio. 2016.  Generating factoid questions with recurrent neural networks: The 30m factoid question-answer corpus.  In _Proceedings of the Annual Meeting of the Association of Computational Linguistics (ACL)_.

## Appendices

###  A. Pronoun insertion process

Here we describe how pronouns where inferred and insert in place of explicit
subject mentions.

We consider an OOC question from a conversation, and the triple from which it
was generated. If the subject of the triple is in the previous turn’s triple,
we try to insert a pronoun with the following steps:

  * •

We infer the gender of the subject (male, female, or neutral):

    * –

if the subject has not "human" (Q5) or "fictional character" (Q95074) in his
types, then gender is neutral;

    * –

otherwise, we use the “sex or gender” (P21) wikidata property, if available,
to assign male or female to the subject;

    * –

if the two previous steps fail, no gender is assigned and the pronoun
insertion procedure stops.

  * •

A part of speech tagging is done on the question.

  * •

We try to insert a pronoun by applying the following rules:

    * –

If the subject’s mention is directly preceded by “a” or “of”, no pronoun is
inserted;

    * –

otherwise, if the subject’s mention is directly followed by a possessive
marker, that marker is removed and the subject mention is replaced by “her”,
“his” or “its”.

    * –

otherwise, if the subject’s mention is directly preceded by a proposition or
conjunction other than “of” and “that” or by a non auxiliary verb, the subject
mention is replaced by “her”, “him” or “it”,

    * –

otherwise, the subject’s mention is replaced by “he”, “she” or “it”.

Type |  Input |  Output | OK?
---|---|---|---
Coref. w/ pronoun |  Which location is Switzerland a component of? |  Which location is it a component of? | $\checkmark$
What was the cause of death of Uriella? |  What was her cause of death? | $\checkmark$
What title was held by Martin of Tours? |  What title was held by him? | $\checkmark$
Who is in charge of the government of Warsaw? |  Who is in charge of the government there? | $\checkmark$
Pierre Chaunel was who’s spouse? |  Was Chaunel his spouse? | ✕
Coref. w/ demonstrative noun phrase |  With which country would you associate Gyeonggi Province? |  With which country would you associate this province? | $\checkmark$
Which reference work outlined Albigensian Crusade? |  Which reference work outlined this conflict? | $\checkmark$
Where are World Council of Churches’s headquarters? |  Where are these headquarters? | ✕
Ellipsis |  What is the public holiday associated with Switzerland? |  What is the public holiday? | $\checkmark$
What is the zenith of Eritrea? |  What is the zenith? | $\checkmark$
In what geographic region is Eurasia located? |  In what geographic region? | $\checkmark$
Table 8: Examples of rewritten questions at inference time.

###  B. Question rewritting from SIC

This appendix provides details the automatic question rewritting step of the
SIC variant for each type of rewritting. Below is the details of the experts
rules for each transformation type, along with examples in Table 8.

##### Coreference with a pronoun

(e.g., “In which country is Kyoto located?” rewritten as “In which country is
it located?”) A candidate rewritten question is considered an acceptable if it
contains one of the following words: “he”, “it”, “she”, “they”, “his”, “its”,
“her”, “their”, “them”, “hers”, “theirs”, “there”. I should also be close
enough from the original question. This is handled by removing the entities
from previous turns from both the original and candidate questions, and then
aligning the two word sequenecs using a Levenshtein distance. If only one word
is different or less than $25\%$ of the candidate question words are
different, the candidate question is validated.

##### Coreference with a demonstrative noun phrase

(e.g., “In which country is this city located?”) A candidate question is
considered as acceptable if it contains one of the article “this” or “these”
followed by a word. To avoid ill-formed questions like “which region is this
region in?" or "who is this located in?”, an additional constraint makes it
sure that the word after the demonstrative pronoun is not duplicated in the
question, nor too short.

##### Ellipsis

(e.g., “In which country?”) A candidate rewritten question is considered as an
ellipsis if (1) it is a prefix of the original question to make sure the
target of the questions remains the same, and (2) it does not include any
entity from the previous question and answer turns.

Only qualitative evaluations were performed but, as illustrated in Table 8,
the rewritten questions are OK in a large majority of cases.

###  C. Example of conversation

Table 9: Piece of data from the human evaluation. The four versions of one
conversation are shown together in the table.

###  D. Statistical analyses of human evaluation results

Table 10 provides the p-values resulting from statistical tests aiming to
assess whether the differences in scores between seen (person and food) and
unseen themes (taxon and space) was due to chance (scores tend to be higher in
the seen category). Several p-values are low, which tends to show that unseen
themes are indeed more difficult. However, it does not show that they are more
difficult because they were not seen during training. Indeed, the distinction
between seen and unseen is only relevant for baselines; yet, SIC questions
from the dataset obtain significantly lower scores on unseen themes. It is
therefore reasonable to assume that the difference between seen and unseen
theme is due to the fact that some themes are intrinsically harder to handle
for a pretrained T5.

|  | clearness | naturalness | correctness | faithfulness
---|---|---|---|---|---
Reference | IC | 0.103 | 0.262 | 0.172 | 0.321
from | SIC | 0.013 | 0.062 | 0.091 | 0.007
Baseline | IC | 0.054 | 0.493 | 0.028 | 0.01
trained on | SIC | 0.221 | 0.09 | 0.328 | 0.539
Table 10: $p$-values of seen/unseen differences:

###  E. Inter-rater agreement

Computing the Cohen’s kappa of ratings from Table 7 gives poor scores.
However, the confusion matrices suggest that, although the exact rate given to
a question has a high degree of subjectivity, raters tend to give close
ratings.

Table 11: Confusion matrices of human ratings.


OCU-PHYS 539

NITEP 102

Extranatural Flux Inflation

Takuya Hirosea and Nobuhito Marua,b,

aDepartment of Mathematics and Physics, Osaka City University,
Osaka 558-8585, Japan

bNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),
Osaka City University, Osaka 558-8585, Japan

We propose a new inflation scenario in flux compactification, where a zero
mode scalar field of extra components of the higher dimensional gauge field is
identified with an inflaton. The scalar field is a pseudo Nambu-Goldstone
boson of spontaneously broken translational symmetry in compactified spaces.
The inflaton potential is non-local and finite, which is protected against the
higher dimensional non-derivative local operators by quantum gravity
corrections thanks to the gauge symmetry in higher dimensions and the shift
symmetry originated from the translation in extra spaces. We give an explicit
inflation model in a six dimensional scalar QED, which is shown to be
consistent with Planck 2018 data.

##  1 Introduction

The hierarchy problem has been considered as one of the guiding principles to
study the physics beyond the Standard Model (SM) of particle physics. In the
SM, the quantum corrections to the mass of Higgs field are sensitive to the
square of the ultraviolet cutoff scale of the theory, which is typically
Planck scale or the scale of grand unified theory. Since the cutoff scale is
much larger than the 125 GeV Higgs mass, solving the hierarchy problem
requires an unnatural fine-tuning of parameters or a new physics beyond the SM
around TeV scale. Although the latter approaches have been mainly studied so
far, no signature of new physics has been discovered, which is likely to
increase a discrepancy between the new physics scale and the Higgs mass.
Therefore, it seems to be desirable such that the Higgs mass vanishes at the
classical level and is generated at the quantum effects by the dynamics
different from the new physics one.

As one of the approaches to the hierarchy problem, the higher dimensional
theory with magnetic flux compactification recently attracts attention.
Originally, the magnetic flux compactification has been studied in string
theory [1, 2] and many attractive properties has been known even in the field
theory: attempt to explain the number of the generations of the SM fermion
[3], computation of Yukawa coupling hierarchy [4, 5, 6]. Remarkably, it has
recently been shown that the quantum corrections to the masses of the scalar
zero-mode field originated from extra components of higher dimensional gauge
field (called Wilson-line (WL) scalar field) are canceled [7, 8, 9, 10, 11,
12]. The physical reason of the cancellation is that the WL scalar field
becomes a Nambu-Goldstone boson of the spontaneously broken translational
symmetry in compactified spaces and the shift symmetry from the translation in
compactified spaces forbids the non-derivative terms as well as the mass term
of the WL scalar field. Therefore, the massless nature of the WL scalar field
is expected to be valid at any order of perturbation.

In order to apply the scenario of the flux compactification where the WL
scalar field is identified with Higgs field to the hierarchy problem, we need
some mechanism to generate an explicit breaking term of the translational
symmetry in compactified space and the WL scalar field must be a pseudo NG
boson such as pion. In our previous paper [13], we have studied a possibility
of realizing nonvanishing WL scalar mass in flux compactification. By
generalizing loop integrals of the quantum correction to WL scalar mass at
one-loop, we derived the conditions for the loop integral and mode sum to be
finite. We further classified the four-point and three-point interaction terms
explicitly breaking the translational symmetry in compactified space and
generating the finite WL scalar mass at one-loop. Using the simplest
interactions of them, we have explicitly shown the finite WL scalar mass at
one-loop in a six dimensional scalar QED.

Inflation is a very attractive scenario to solve many problems in the standard
Big Bang cosmology and its existence is supported by observations of
cosmological parameters[14]. Inflation has been considered to happen by a
scalar field called as inflaton. Although many models of inflation has been
proposed so far, there is still no compelling model of inflation. In a slow-
roll scenario of the inflation, the scalar potential is required to be flat
and stable under quantum gravity corrections, which usually causes an
unnatural fine-tuning of parameters of the theory unless we have some dynamics
or symmetry to control the inflaton dynamics. For instance, the inflaton in
natural inflation [15] is identified with the pseudo Nambu-Goldstone boson of
some global symmetry. In extranatural inflation [16], the inflaton is
identified with the WL scalar field of the gauge field in higher dimensions
without magnetic flux. In [17], the inflaton and the curvaton are identified
with the WL scalar fields in a six dimensional gauge theory.

In this paper, we propose a new scenario of inflation in flux compactification
where the WL scalar field is identified with an inflaton. Since this is a
model of “extranatural inflation” [16] with magnetic flux in compactified
space, we refer to our scenario as “extranatural flux inflation”.111The
extranatural inflation is an inflation model where the idea of gauge-Higgs
unification is applied. For the discussion of UV insensitivity for the WL
scalar mass and potential, see [18] In constructing a slow-roll inflation
model, it is important to realize a flat inflaton potential stable against
quantum corrections and quantum gravity corrections since the slow-roll
conditions typically require the large field value larger than the Planck
scale for the inflaton. Unless this stability is guaranteed by some physical
reasons, we cannot discuss the inflation by an effective field theory
description because it is beyond the range of their applicability. Similar to
the extranatural inflation [16], the inflaton potential in our scenario is
controlled by the gauge symmetry in higher dimensions. The non-derivative
local operators of the WL scalar fields are forbidden by the gauge symmetry,
and by the shift symmetry originated from the translational symmetry in
compactified spaces in a context of flux compactification. Therefore, the
higher dimensional operators suppressed by the Planck scale, which is expected
to be generated by the black holes, are forbidden. The quantum corrections to
the local operators (by non-gravity interactions) are also forbidden. The
remarkable nature is that the non-local Wlison line operators of the WL scalar
field are generated by quantum effects and become finite inspite of the non-
renormalizable theory. This is a great advantage which is absent in other
scenarios.

As a simplest model, we consider a six dimensional scalar QED which was
discussed in [13]. In this model, the WL scalar field is identified with the
inflaton and the one-loop Coleman-Weinberg potential for the WL scalar field
is calculated. The potential is found to be finite as expected. Then, the
slow-roll parameters, the spectral index, the scalar-to-tensor ratio, and the
e-folding are numerically computed and some viable parameter sets are found.

This paper is organized as follows. In the next section, our model is
introduced and one-loop effective potential is calculated. Identifying the WL
scalar field with an inflaton, we discuss our inflation scenario by applying
the one-loop effective potential to the inflaton potential in section 3. In
section 4, our numerical analysis is found and some viable parameter sets
consistent with Planck 2018 data are shown. A final section gives our summary.

##  2 Our setup

In this section, we introduce our model and calculate the one-loop effective
potential.

###  2.1 Flux compactifiaction

We consider a six-dimensional U(1) gauge theory with a constant magnetic flux
couples to a scalar field in the bulk. The six-dimensional spacetime is a
product of four-dimensional Minkowski spacetime $M^{4}$ and two-dimensional
torus $T^{2}$. The Lagrangian we consider is

$\displaystyle\mathcal{L}=-\frac{1}{4}F_{MN}F^{MN}-(D_{M}\Phi)^{*}D^{M}\Phi+\kappa_{6}(\bar{\phi}\overline{\Phi}\Phi+\phi\overline{\Phi}\Phi),$
(1)

where the spacetime index is given by
$M,N=0,1,2,3,5,6,~{}\mu,\nu=0,1,2,3,~{}m,n=5,6$ respectively and we follow the
metric convention as $\eta_{MN}=(-1,+1,\cdots,+1)$. The field strength and the
covariant derivative of U(1) gauge field $A_{M}$ are defined by
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M},~{}D_{M}=\partial_{M}-ig_{6}A_{M}$
with a gauge coupling constant $g_{6}$ of mass dimension $-1$. $\Phi$ is a
bulk scalar field. $\phi$ is a scalar field related to the complex combination
of $A_{5,6}$ and will be defined in detail later. $\kappa_{6}$ is a
dimensionless coupling constant. The third term in (1) is introduced to
generates nonvanishing quantum corrections to the mass of $\phi$ at one-loop
[13]. In the context of this paper, this term is also crucial to obtain a
nonvanishing one-loop effective potential of $\phi$ as well as the mass term.
This is due to the property that the scalar field $\phi$ is the Nambu-
Goldstone boson of the translational symmetry in compactified spaces, which
forbids non-derivative terms such as the mass and potential terms and then the
explicit breaking terms for the translational symmetry are required.

Let us introduce the magnetic flux in our model. The magnetic flux is given by
the nontrivial background (or vacuum expectation value (VEV)) of fifth and
sixth component of the gauge field $A_{5,6}$, which must satisfy their
classical equation of motion $\partial^{m}\braket{F_{mn}}=0$. In our flux
compactification, the background of $A_{5,6}$ is chosen as

$\displaystyle\braket{A_{5}}=-\frac{1}{2}fx_{6},~{}~{}~{}\braket{A_{5}}=\frac{1}{2}fx_{5},$
(2)

which introduces a constant magnetic flux density $\braket{F_{56}}=f$ with a
real number $f$. Note that this solution breaks an extra-dimensional
translational invariance spontaneously. Integrating over $T^{2}$, the magnetic
flux is quantized as follows

$\displaystyle\frac{g_{6}}{2\pi}\int_{T^{2}}dx_{5}dx_{6}\left\langle
F_{56}\right\rangle=\frac{g_{6}}{2\pi}L^{2}f=N\in\mathbb{Z},$ (3)

where $L^{2}$ is an area of two-dimensional torus.

It is useful to define $\partial,z,$ and $\phi$ as

$\displaystyle\partial\equiv\partial_{z}=\partial_{5}-i\partial_{6},~{}~{}~{}z=\frac{1}{2}(x_{5}+ix_{6}),~{}~{}~{}\phi=\frac{1}{\sqrt{2}}(A_{6}+iA_{5}).$
(4)

Note that the VEV of $\phi$ is given by
$\braket{\phi}=(\braket{A_{6}}+i\braket{A_{5}})/\sqrt{2}=f\bar{z}/\sqrt{2}$.
We expand $\phi$ around the flux background $\phi=\braket{\phi}+\varphi$,
where $\varphi$ is a quantum fluctuation. To distinguish $\varphi$ from an
introduced bulk scalar $\Phi$, we call $\varphi$ Wilson line (WL) scalar
field.

###  2.2 Kaluza-Klein mass spectrum

We need to derive Kaluza-Klein mass spectrum of the bulk scalar field $\Phi$
for the calculation of effective potential. To begin, we define the covariant
derivatives in the complex coordinates $D,\bar{D}$ as

$\displaystyle D$
$\displaystyle=D_{5}-iD_{6}=\partial-\sqrt{2}g_{6}\phi=\mathcal{D}-\sqrt{2}g_{6}\varphi,$
(5) $\displaystyle\bar{D}$
$\displaystyle=D_{5}+iD_{6}=\bar{\partial}+\sqrt{2}g_{6}\bar{\phi}=\bar{\mathcal{D}}+\sqrt{2}g_{6}\bar{\varphi},$
(6) $\displaystyle\mathcal{D}$
$\displaystyle=\mathcal{D}_{5}-i\mathcal{D}_{6}=\partial-\sqrt{2}g_{6}\braket{\phi},$
(7) $\displaystyle\bar{\mathcal{D}}$
$\displaystyle=\bar{\mathcal{D}}_{5}+i\bar{\mathcal{D}}_{6}=\bar{\partial}+\sqrt{2}g_{6}\braket{\bar{\phi}}.$
(8)

Next, we regard $\mathcal{D},\bar{\mathcal{D}}$ as creation and annihilation
operators by

$\displaystyle
a=\frac{1}{\sqrt{2g_{6}f}}i\bar{\mathcal{D}},~{}~{}~{}a^{\dagger}=\frac{1}{\sqrt{2g_{6}f}}i\mathcal{D},$
(9)

which satisfy the commutation relation $[a,a^{\dagger}]=1$. This
correspondence is an analogy to the quantum mechanics in magnetic field.
Hereafter, we denote $\alpha=2g_{6}f=4\pi N/L^{2}$.

We summarize the property of creation and annihilation operators. The ground
state mode functions $\xi_{0,j},~{}\bar{\xi}_{0,j}$ are determined by
$a\xi_{0,j}=0,~{}a^{\dagger}\bar{\xi}_{0,j}=0$, where $j=0,\cdots,|N|-1$
accounts for the degeneracy of the ground state. Creation and annihilation
operators act on mode functions as

$\displaystyle
a^{\dagger}\xi_{n,j}=\sqrt{n+1}\xi_{n+1,j},~{}~{}~{}a\xi_{n,j}=\sqrt{n}\xi_{n-1,j},$
(10)

and we can construct the higher mode function $\xi_{n,j}$ in the same way as
the harmonic oscillator (in detail, see [19])

$\displaystyle\xi_{n,j}=\frac{1}{\sqrt{n!}}(a^{\dagger})^{n}\xi_{0,j},~{}~{}~{}\bar{\xi}_{n,j}=\frac{1}{\sqrt{n!}}(a)^{n}\bar{\xi}_{0,j},$
(11)

where $n=0,1,2\cdots$ is Landau level. The higher mode function satisfies an
orthonormality condition

$\displaystyle\int_{T^{2}}d^{2}x\bar{\xi}_{n^{\prime},j^{\prime}}\xi_{n,j}=\delta_{n,n^{\prime}}\delta_{j,j^{\prime}}.$
(12)

Finally, we extract the mass term from the second term in (1). Focusing on the
extra-dimensional part of the second term in (1), we obtain

$\displaystyle\mathcal{L}_{scalar~{}mass}$
$\displaystyle=-\mathcal{D}_{m}\overline{\Phi}\mathcal{D}^{m}\Phi$
$\displaystyle=-\overline{\Phi}\alpha\left(a^{\dagger}a+\frac{1}{2}\right)\Phi.$
(13)

Noting that $a^{\dagger}a$ is a number operator, the KK mass of bulk scalar
field is given by

$\displaystyle m^{2}_{scalar}=\alpha\left(n+\frac{1}{2}\right).$ (14)

###  2.3 Four-dimensional effective Lagrangian

Using the expansion of $\phi=\braket{\phi}+\varphi$, the Lagrangian (1) is
accordingly deformed as

$\displaystyle\mathcal{L}$
$\displaystyle\supset-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-D_{\mu}\overline{\Phi}D^{\mu}\Phi-m^{2}_{scalar}\bar{\Phi}\Phi$
$\displaystyle\quad-
ig_{6}\sqrt{2\alpha}\bar{\varphi}\bar{\Phi}a^{\dagger}\Phi+ig_{6}\sqrt{2\alpha}\varphi\bar{\Phi}a\Phi-2g^{2}_{6}\bar{\varphi}\varphi\overline{\Phi}\Phi$
$\displaystyle\quad+\kappa_{6}(\bar{\varphi}\overline{\Phi}\Phi+\varphi\overline{\Phi}\Phi)+\kappa_{6}(\braket{\bar{\phi}}\overline{\Phi}\Phi+\braket{\phi}\overline{\Phi}\Phi),$
(15)

where we note that the unnecessary terms in our discussion are omitted. To
derive a four-dimensional effective Lagrangian by KK reduction, we need to
expand $\Phi$ in terms of mode functions $\xi_{n,j}$

$\displaystyle\Phi=\sum_{n,j}\Phi_{n,j}\xi_{n,j}.$ (16)

Integrating over $T^{2}$, the four-dimensional effective Lagrangian is
obtained by

$\displaystyle\mathcal{L}_{4D}$
$\displaystyle=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\partial^{\mu}\bar{\varphi}\partial_{\mu}\varphi$
$\displaystyle\quad+\sum_{n,j}\left(-D_{\mu}\overline{\Phi}_{n,j}D^{\mu}\Phi_{n,j}-\alpha\left(n+\frac{1}{2}\right)\overline{\Phi}_{n,j}\Phi_{n,j}\right.$
$\displaystyle\quad-
ig_{4}\sqrt{2\alpha(n+1)}\overline{\varphi}\overline{\Phi}_{n+1,j}\Phi_{n,j}+ig_{4}\sqrt{2\alpha(n+1)}\varphi\overline{\Phi}_{n,j}\Phi_{n+1,j}-2g^{2}_{4}\bar{\varphi}\varphi\overline{\Phi}_{n,j}\Phi_{n,j}$
$\displaystyle\quad+\kappa_{4}\bar{\varphi}\overline{\Phi}_{n,j}\Phi_{n,j}+\kappa_{4}\varphi\overline{\Phi}_{n,j}\Phi_{n,j}+\kappa_{4}\braket{\phi}_{I}\overline{\Phi}_{n,j}\Phi_{n,j}+\kappa_{4}\braket{\bar{\phi}}_{I}\overline{\Phi}_{n,j}\Phi_{n,j}\Big{)},$
(17)

where $g_{4}=g_{6}/L$ is a four-dimensional gauge coupling constant and
$\kappa_{4}=\kappa_{6}/L$ is a four-dimensional coupling constant.
$\braket{\phi}_{I}$ and $\braket{\bar{\phi}}_{I}$ are expressed by

$\displaystyle\braket{\phi}_{I}=\int_{T^{2}}d^{2}x\braket{\phi}\bar{\xi}_{n,j}\xi_{n^{\prime},j^{\prime}},~{}~{}~{}\braket{\bar{\phi}}_{I}=\int_{T^{2}}d^{2}x\braket{\bar{\phi}}\bar{\xi}_{n,j}\xi_{n^{\prime},j^{\prime}}.$
(18)

When $\braket{\phi}=f\bar{z}/\sqrt{2}$, $\braket{\phi}_{I}$ and
$\braket{\bar{\phi}}_{I}$ lead to zero because of odd function with respect to
integral variables $z$ or $\bar{z}$. In the following, we omit the third and
fourth terms in the fourth line of (17).

###  2.4 One-loop effective potential

One-loop effective potential is described as

$\displaystyle
V(\varphi,\bar{\varphi})=N\sum_{n=0}^{\infty}\int\frac{d^{4}k}{(2\pi)^{4}}\ln\left(k^{2}+\alpha\left(n+\frac{1}{2}\right)+M^{2}(\varphi,\bar{\varphi})\right),$
(19)

where we have taken into account loop contributions from the bulk scalar field
$\Phi$. $N$ is a number of the degeneracy. $M^{2}(\varphi,\bar{\varphi})$ is a
field-dependent mass for the bulk scalar field $\Phi$.

As for this $M^{2}(\varphi,\bar{\varphi})$, we consider two limiting cases for
a free parameter $U(1)$ gauge coupling, namely $g_{4}\ll 1$ and $g_{4}\gg 1$.
For that purpose, we read $M^{2}(\varphi,\bar{\varphi})$ from (17) as

$\displaystyle M^{2}(\varphi,\bar{\varphi})$
$\displaystyle=-\kappa_{4}\bar{\varphi}-\kappa_{4}\varphi+2g^{2}_{4}\bar{\varphi}\varphi.$
(20)

While only the first two terms in (20) are considered in the $g_{4}\ll 1$
case, the last term proportional to $g^{2}_{4}$ in (20) is also considered in
the $g_{4}\gg 1$ case in addition to the first two terms. In the case of
$g_{4}\simeq{\cal O}(1)$, the terms linear in $g_{4}$ in (17) should be also
taken into account in $M^{2}(\varphi,\bar{\varphi})$. However, the obtained
eigenvalues of $M^{2}(\varphi,\bar{\varphi})$ become complicated and makes the
computation of the effective potential hard. Therefore, we do not discuss this
case in this paper.

We can express the effective potential by using Schwinger’s proper time as

$\displaystyle V$
$\displaystyle=-N\sum_{n=0}^{\infty}\int\frac{d^{4}k}{(2\pi)^{4}}\int_{0}^{\infty}\frac{dt}{t}e^{-k^{2}t-\alpha\left(n+\frac{1}{2}\right)t}e^{-M^{2}(\varphi,\bar{\varphi})t}$
$\displaystyle=-N\frac{1}{16\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{3}}\frac{e^{-\frac{\alpha}{2}t}}{1-e^{-\alpha
t}}e^{-M^{2}(\varphi,\bar{\varphi})t}.$ (21)

To proceed a calculation of the effective potential further, we notice an
integral representation of Hurwitz zeta function

$\displaystyle\zeta[s,a]=\frac{1}{\Gamma(s)}\int_{0}^{\infty}dt\frac{t^{s-1}e^{-at}}{1-e^{-t}},~{}~{}~{}\mathrm{Re}~{}s>1.$
(22)

Then, the effective potential and its derivatives by $\varphi$ can be
expressed by

$\displaystyle V$
$\displaystyle=-N\frac{\alpha^{2}}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon-2)\zeta\left[\epsilon-2,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right],$
(23) $\displaystyle V_{\varphi}$
$\displaystyle=-N\frac{\alpha\kappa}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon-1)\zeta\left[\epsilon-1,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right],$
(24) $\displaystyle V_{\varphi\bar{\varphi}}$
$\displaystyle=-N\frac{\kappa^{2}}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon)\zeta\left[\epsilon,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right],$
(25)

where a parameter $\epsilon$ is introduced to regularize the integral of $t$.
In particular, we can check that the $\epsilon\rightarrow 0$ limit indeed
agrees with the results in case of $M^{2}(\varphi,\bar{\varphi})=0$ obtained
in[13] by diagrammatic calculations using the dimensional regularization.

In the $g_{4}\ll 1$ case, we ignore $2g^{2}_{4}\bar{\varphi}\varphi$ in (20)
as mentioned above. For convenience, we define the dimensionless variables in
a four dimensional sense as

$\displaystyle
z=\frac{\varphi}{M_{P}},~{}~{}~{}y=M_{P}\frac{\kappa_{4}}{\alpha},$ (26)

$M^{2}(\varphi,\bar{\varphi})/\alpha$ is then expressed by

$\displaystyle\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})=-(z+\bar{z})y=-2xy,\quad\mathrm{Re}~{}z=x.$
(27)

Thus, the effective potential is rewritten by

$\displaystyle V=-N\frac{\alpha^{2}}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon-2)\zeta\left[\epsilon-2,\frac{1}{2}-2xy\right],$ (28)

and the effective potential is shown in figure 1.

Figure 1: Schematic picture of the effective potential in the case of
$g_{4}\ll 1$. The blue and yellow lines shows $y=1.0\times 10^{0}$,
$y=1.0\times 10^{-1}$ respectively.

If $L\sim M^{-1}_{P}$, the effective potential is close to flat as $y$ (or
$\kappa_{4}$) takes smaller value. Taking into account for the consistency
with the original theory [7, 9], the small value of $y$ is favored. If $y\ll
1$, $\kappa_{4}$ is small, which is independent of $g_{4}$. This implies that
linear terms in $g_{4}$ can be neglected because we can always take
$g_{4}\ll\kappa_{4}L$.

In the $g_{4}\gg 1$ case, $M^{2}(\varphi,\bar{\varphi})/\alpha$ is expressed
by

$\displaystyle\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})$
$\displaystyle=-(z+\bar{z})y+2\frac{g^{2}_{4}M^{2}_{P}}{\alpha}|z|^{2}$
$\displaystyle=-2uy+2G(u^{2}+v^{2}),$ (29)

where $z\equiv u+iv$ and $G\equiv g^{2}_{4}M^{2}_{P}/\alpha$ are defined in
the second equality. Note that $G$ is almost an order of $g^{2}_{4}$ because
$\alpha$ is independent of $g_{4}$. Setting $u=v$ for simplicity, the
effective potential is given by

$\displaystyle V=-N\frac{\alpha^{2}}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon-2)\zeta\left[\epsilon-2,\frac{1}{2}-2uy+4Gu^{2}\right],$
(30)

which is shown in figure 2.

Figure 2: Schematic picture of the effective potential in the case of
$g_{4}\gg 1$. We take $y=1$ for simplicity. The yellow and blue lines shows
$G=1.0\times 10^{2}$, $G=1.0\times 10^{3}$ respectively.

This effective potential in the case of $g_{4}\gg 1$ behaves as
$V\propto\Gamma[\epsilon-2]\zeta[\epsilon-2,4Gu^{2}]$. Comparing with the
potential in figure 1, it seems difficult to apply the potential in figure 2
to an inflation model.

##  3 Inflationary parameters

Using the four-dimensional effective potential for the WL scalar field (23),
we propose a cosmological inflation model in flux compactifiaction, where the
WL scalar field is identified with an inflaton.

Slow-roll parameters $\epsilon$ and $\eta$ in our model are given by

$\displaystyle\epsilon$
$\displaystyle=\frac{M^{2}_{P}}{2}\left(\frac{V_{\varphi}}{V}\right)^{2}=\frac{M^{2}_{P}}{2}\left(\frac{\kappa_{4}}{\alpha}\lim_{\epsilon\rightarrow
0}\frac{\Gamma(\epsilon-1)}{\Gamma(\epsilon-2)}\frac{\zeta\left[\epsilon-1,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}{\zeta\left[\epsilon-2,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}\right)^{2},$
(31) $\displaystyle\eta$
$\displaystyle=M^{2}_{P}\frac{V_{\varphi\varphi}}{V}=M^{2}_{P}\left(\frac{\kappa_{4}^{2}}{\alpha^{2}}\lim_{\epsilon\rightarrow
0}\frac{\Gamma(\epsilon)}{\Gamma(\epsilon-2)}\frac{\zeta\left[\epsilon,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}{\zeta\left[\epsilon-2,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}\right).$
(32)

Noting that Hurwitz zeta function can be expressed by Bernoulli polynomials
$B_{n}(x)$ as follows

$\displaystyle\zeta[-n,x]=-\frac{B_{n+1}(x)}{n+1},$ (33)

we can further simplify (31) and (32),

$\displaystyle\epsilon$
$\displaystyle=\frac{y^{2}}{2}\left(-2\frac{\zeta\left[-1,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}{\zeta\left[-2,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}\right)^{2}=\frac{9y^{2}}{2}\left(\frac{B_{2}(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi}))}{B_{3}(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi}))}\right)^{2},$
(34) $\displaystyle\eta$
$\displaystyle=y^{2}\left((-1)(-2)\frac{\zeta\left[0,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}{\zeta\left[-2,\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right]}\right)=6y^{2}\frac{B_{1}(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi}))}{B_{3}(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi}))}.$
(35)

Slow-roll conditions to realize inflation require $\epsilon\ll 1,|\eta|\ll 1$.

The number of e-folding before the end of inflation is

$\displaystyle
N_{*}=\frac{1}{M^{2}_{P}}\int_{\varphi_{f}}^{\varphi_{*}}\frac{V}{V_{\varphi}}d\varphi=\frac{2}{3y}\int_{\varphi_{*}}^{\varphi_{f}}\frac{B_{3}\left(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right)}{B_{2}\left(\frac{1}{2}+\frac{1}{\alpha}M^{2}(\varphi,\bar{\varphi})\right)}d\varphi.$
(36)

To solve the horizon and flatness problems, the number of e-folding $N_{*}$
has to be at least $50<N_{*}<60$. $\varphi_{f}$ is the value of the end of
inflation determined by $\epsilon(\varphi_{f})=1$, which violates the slow-
roll conditions. $\varphi_{*}$ is determined so that the e-folding can satisfy
$50<N_{*}<60$.

The spectral index and the tensor-to-scalar ratio are given in a slow-roll
approximation as

$\displaystyle n_{s}=1-6\epsilon+2\eta,~{}~{}~{}r=16\epsilon.$ (37)

Planck 2018 data [14] gives constraints on $n_{s}=0.9649\pm 0.0042$ and
$r<0.10$.

##  4 Numerical results

In this section, our numerical results are shown.

###  4.1 $g_{4}\ll$1 case

In this case, $M^{2}(\varphi,\bar{\varphi})/\alpha$ corresponds to (27), where
the slow-roll parameters $\epsilon$ and $\eta$ are provided by

$\displaystyle\epsilon$
$\displaystyle=\frac{9y^{2}}{2}\left(\frac{B_{2}(\frac{1}{2}-2xy)}{B_{3}(\frac{1}{2}-2xy)}\right)^{2},~{}~{}~{}\eta=6y^{2}\frac{B_{1}(\frac{1}{2}-2xy)}{B_{3}(\frac{1}{2}-2xy)}.$
(38)

To compute the e-folding $N_{*}$, we need to know the value of end of
inflation $x_{f}=\varphi_{f}/M_{P}$, which is determined by the condition of
the end of inflation $\epsilon(x_{f})=1$. The number of e-folding is

$\displaystyle
N_{*}=\frac{2}{3y}\int_{x_{i}}^{x_{f}}\frac{B_{3}(\frac{1}{2}-2xy)}{B_{2}(\frac{1}{2}-2xy)}dx,$
(39)

where $x_{i}=\mathrm{Re}\varphi_{*}/M_{P}$. Sample of our numerical solutions
$x_{i},x_{f},N_{*}$ at some points of $y$ are shown in Table 1, where the
e-folding $N_{*}=50,60$ are taken. One might think that our results are not
reliable since the WL scalar field value is quite larger than the Planck
scale, which is beyond an applicability of the effective field theory. As
mentioned in introduction, the gauge symmetry in our theory is not broken by
quantum gravity effects and forbids any dangerous higher dimensional local
operators suppressed by the Planck scale as well as the non-derivative local
operators of the WL scalar field. Therefore, our results are reliable.

| $y$ | $x_{i}$ | $x_{f}$ | $N_{*}$
---|---|---|---|---
$A_{50}$ | $1.0\times 10^{-2}$ | $\quad-30.4669\quad$ | $\quad-25.3611\quad$ | $\quad 50.002\quad$
$A_{60}$ | $1.0\times 10^{-2}$ | $-31.0285$ | $-25.3611$ | 60.0098
$B_{50}$ | $5.0\times 10^{-3}$ | $-55.2516$ | $-50.3573$ | 50.0016
$B_{60}$ | $5.0\times 10^{-3}$ | $-55.7738$ | $-50.3573$ | 60.0004
$C_{50}$ | $1.0\times 10^{-3}$ | $-255.063$ | $-250.354$ | 50.0144
$C_{60}$ | $1.0\times 10^{-3}$ | $-255.549$ | $-250.354$ | 60.0162
$D_{50}$ | $5.0\times 10^{-4}$ | $-505.038$ | $-500.354$ | 50.0099
$D_{60}$ | $5.0\times 10^{-4}$ | $-505.519$ | $-500.354$ | 60.0087
$E_{50}$ | $1.0\times 10^{-4}$ | $-2505.018$ | $-2500.35$ | 50.0098
$E_{60}$ | $1.0\times 10^{-4}$ | $-2505.495$ | $-2500.35$ | 60.0078
Table 1: Sample of our numerical solutions $x_{i},x_{f},N_{*}$ at some points
of $y$.

Using the numerical solutions in Table 1, the slow-roll parameters
$\epsilon,\eta$, the spectral index $n_{s}$, and the scalar-to-tensor ratio
$r$ are calculated and shown in Table 2. Comparing our results in Table 2 with
$n_{s}$ and $r$ in Planck 2018 data, our results are found to be relatively
good agreement with the data. If $y$ is taken to be a large value such as
$y=1.0\times 10^{2}$, $n_{s}$ and $r$ cannot be satisified with Planck 2018
data.

| $\epsilon$ | $\eta$ | $n_{s}$ | $r$
---|---|---|---|---
$A_{50}$ | $\quad 0.00683107\quad$ | $\quad 0.00494671\quad$ | $\quad 0.968907\quad$ | $\quad 0.109297\quad$
$A_{60}$ | 0.00582958 | 0.00444092 | 0.973904 | 0.0932733
$B_{50}$ | 0.00594149 | 0.00271376 | 0.969779 | 0.0950639
$B_{60}$ | 0.00502904 | 0.00245613 | 0.974738 | 0.0804646
$C_{50}$ | 0.00517205 | 0.000586594 | 0.970141 | 0.0827528
$C_{60}$ | 0.00432933 | 0.000534704 | 0.975093 | 0.0692693
$D_{50}$ | 0.00507359 | 0.000296245 | 0.970151 | 0.0811775
$D_{60}$ | 0.00423959 | 0.000270297 | 0.975103 | 0.0678334
$E_{50}$ | 0.00499408 | 0.0000597248 | 0.970155 | 0.0799053
$E_{60}$ | 0.00416705 | 0.0000545352 | 0.975107 | 0.0666728
Table 2: Inflation parameters $\epsilon,\eta,n_{s},r$ obtained from our model.
Figure 3: Our results (table 2) in the $n_{s}$-$r$ plot from Planck 2018 data
[14]. Orange circles are our results, where the small and large ones represent
$N_{*}=50$ and $N_{*}=60$, respectively.

Our results are shown in $(n_{s},r)$ plot of Figure 3 from Planck 2018 data
[14]. Orange circles are our results where small and large ones corresponds to
$N_{*}=50$ and $N_{*}=60$, respectively. As the parameter $y$ is decreased,
our results in $(n_{s},r)$ plot go downward. Our results are within a
parameter region indicating the combining data of Planck TT, TE,
EE+lowE+lensing at CL95%.

From the parameter $y$, we can estimate the value of $\kappa_{6}$, which
provides the compactification scale and the 6D Planck scale $M_{6P}$ in our
model.

|  $L$ [GeV-1]$(\kappa_{6}=10^{x}L)$ | $M_{6P}=\sqrt{M_{P}/L}$
---|---|---
$y=1.0\times 10^{-2}$ | $\quad 3.20941\times 10^{-x/2-10}\quad$ | $\quad 1.9497\times 10^{x/4+14}\quad$
$y=5.0\times 10^{-3}$ | $2.26939\times 10^{-x/2-10}$ | $2.3186\times 10^{x/4+14}$
$y=1.0\times 10^{-3}$ | $1.0149\times 10^{-x/2-10}$ | $3.46711\times 10^{x/4+14}$
$y=5.0\times 10^{-4}$ | $7.17646\times 10^{-x/2-11}$ | $4.12311\times 10^{x/4+14}$
$y=1.0\times 10^{-4}$ | $3.20941\times 10^{-x/2-11}$ | $6.16549\times 10^{x/4+14}$
Table 3: The value of $L,M_{6P}$

$\kappa_{6}L$ is determined by $y$ as follows.

$\displaystyle y=M_{P}\frac{\kappa_{4}}{\alpha}=M_{P}\frac{\kappa_{6}/L}{4\pi
N/L^{2}}=\frac{M_{P}L\kappa_{6}}{4\pi N}$ $\displaystyle\Leftrightarrow$
$\displaystyle\kappa_{6}=4\pi N\frac{y}{M_{P}L}$ (40)

where the number of degeneracy is assumed to be $N=10$. If we assume
$\kappa_{6}=m_{inflaton}L$ and $m_{inflaton}=10^{x}$ GeV, $L$ is estimated.
$\kappa_{6}L,~{}L,~{}M_{6P}$ are shown in Table 3.

Now, we discuss how small the gauge coupling $g_{4}$ is required for a
successful inflation. Although the gauge coupling itself $g_{4}$ is a free
parameter, the constraint from slow-roll parameter condition can be obtained
through the coupling constant $\kappa_{6}$, which can be derived from
$\epsilon\ll 1$,

$\displaystyle\frac{3}{\sqrt{2}}yB_{2}(1/2-2x_{i}y)\ll B_{3}(1/2-2x_{i}y),$

which implies,

$\displaystyle
y\ll\sqrt{\frac{2\sqrt{2}x_{i}+1}{4(12x_{i}^{2}+8\sqrt{2}x_{i}^{3})}}.$ (41)

In the condition $\epsilon\ll 1$, $y\ll 1$ is immediately found. Therefore, we
obtain $\kappa_{6}\ll 10^{-19}/L$, which means $\kappa_{6}\ll 1$ because the
maximum value of $10^{-19}/L$ is at $1/L\sim M_{P}$. As mentioned in section
2.4, we can always take the free parameter $g_{4}$ less than $\kappa_{6}$. For
a successful inflation in our model, we have only to take the free parameter
gauge coupling $g_{4}$ with $g_{4}\ll\kappa_{6}$ and this can be always
possible.

###  4.2 $g_{4}\gg$1 case

In this case, $M^{2}(\varphi,\bar{\varphi})$ corresponds to (29). Under $u=v$,
$\epsilon$ and $\eta$ are expressed by

$\displaystyle\epsilon=\frac{9y^{2}}{2}\left(\frac{B_{2}(\frac{1}{2}+4Gu^{2})}{B_{3}(\frac{1}{2}+4Gu^{2})}\right)^{2},~{}~{}~{}\eta=6y^{2}\frac{B_{1}(\frac{1}{2}+4Gu^{2})}{B_{3}(\frac{1}{2}+4Gu^{2})},$
(42)

where we ignore $-2uy$ because $y$ is small. The number of e-folding is

$\displaystyle
N_{*}=\frac{2}{3y}\int_{u_{i}}^{u_{f}}\frac{B_{3}(\frac{1}{2}+4Gu^{2})}{B_{2}(\frac{1}{2}+4Gu^{2})}du.$
(43)

As in the $g_{4}\ll 1$ case, We obtain the value of $u_{i}$ and $u_{f}$ for a
value of $G$. Taking $G=1,0\times 10^{3}$ as an example, we find
$u_{i}=-3.8403$ and $u_{f}=-0.7282$. Using these values, $n_{s}$ and $r$ are
$n_{s}=0.99569$ and $r=0.0206896$. $r$ is consistent with Planck 2018
constraint, but $n_{s}$ is not. Thus, comparing with the potential in
$g_{4}\ll 1$ case, the potential in the $g_{4}\gg 1$ case is not suitable for
the inflation.

###  4.3 The vacuum energy during inflation

In order for our model to be consistent with inflationary setup, the vacuum
energy during inflation should be smaller than 4D Planck scale and the
compactification scale. We verify this requirement. As you can see Table 1,
$x_{f}y$ takes 1/4 during inflation. Thus, the vacuum energy becomes

$\displaystyle V_{vac}$
$\displaystyle=\braket{V}=-N\frac{\alpha^{2}}{16\pi^{2}}\lim_{\epsilon\rightarrow
0}\Gamma(\epsilon-2)\zeta[\epsilon-2,1/2]$
$\displaystyle=-N\frac{\alpha^{2}}{16\pi^{2}}\left(\frac{1}{2}\zeta^{(1,0)}[-2,1/2]\right)$
$\displaystyle=-\frac{3N^{3}\zeta(3)}{32\pi^{2}}\frac{1}{L^{4}}$ (44)

where we take into account in the second equality that the VEV of inflaton
field is zero during inflation. Setting $N=10$, $V_{vac}$ is estimated to be
$\mathcal{O}(10)\times
L^{-4}\sim\mathcal{O}(10)\times\left(\frac{M_{6P}}{M_{P}}\right)^{4}M_{6P}^{4}$.
In large extra dimensions, 6D Planck scale is smallar than 4D Planck scale
$M_{6P}<M_{P}$, unless the compactification scale is the 4D Planck scale.
Therefore, $1/L^{4}<|V_{vac}|<M^{4}_{P}$ are satisfied as long as the
compactification scale is smaller than 4D Planck scale.

##  5 Summary

In this paper, we have proposed a new inflation model in flux
compactification, which is refered to as “extranatural flux inflation”. In
this model, the WL scalar field, which is originated from extra components of
the gauge field in higher dimensions, is identified with an inflaton field.
The great advantage of our model is that the inflation potential is protected
by the gauge symmetry of the theory from the dangerous higher dimensional
local operators by quantum gravity effects. Our inflation potential is
nonlocal and finite, which is generated by quantum corrections. Therefore, the
extranatural inflation with flux is very predictable regardless of the non-
renormalizable theory. We have considered a model of six dimensional scalar
QED in flux compactification and calculated an inflation potential. We have
shown that our model in the weak gauge coupling case is consistent with Planck
2018 data.

## Acknowledgments

We would like to thank Kazumasa Okabayashi for useful discussions and
comments.

## References

  * [1] R. Blumenhagen, B. Kors, D. Lust and S. Stieberger, Phys. Rept. 445, 1 (2007) [hep-th/0610327].
  * [2] L. E. Ibanez and A. M. Uranga, Cambridge Univ. Press (2012).
  * [3] E. Witten, Phys. Lett. 149B, 351 (1984).
  * [4] D. Cremades, L. E. Ibanez and F. Marchesano, JHEP 0405 079 (2004) [hep-th/0404229].
  * [5] H. Abe, K. -S. Choi, T. Kobayashi and H. Ohki, JHEP 0906 080 (2009) [hep-th/0903.3800].
  * [6] Y. Matsumoto and Y. Sakamura, PTEP 2016 (2016) 5, 053B06 [hep-th/1602.01994].
  * [7] W. Buchmuller, M. Dierigl, E. Dudas and J. Schweizer, JHEP 1706 (2017) 039 [hep-th/1611.03798].
  * [8] D. M. Ghilencea and H. M. Lee, JHEP 1706 039 (2017) [hep-th/1703.10418].
  * [9] W. Buchmuller, M. Dierigl, E. Dudas, JHEP 1808 151 (2018) [hep-th/1804.07497].
  * [10] T. Hirose and N. Maru, JHEP 1908 (2019) 054 [hep-th/1904.06028].
  * [11] M. Honda and T. Shibasaki, JHEP 03 (202) 031 [hep-th/1912.04581].
  * [12] T. Hirose and N. Maru, J. Phys. G 48 (2021) 5, 055005 [hep-th/2012.03494].
  * [13] T. Hirose and N. Maru, JHEP 06 (2021), 159 [hep-th/2104.01779].
  * [14] Y. Akrami et al. [Planck Collaboration], “Planck 2018 results. X. Constraints on inflation,” Astron.Astrophys. 641 (2020) A10 [astro-ph.CO/1807.06211]
  * [15] K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. 65 (1990), 3233-3236; F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. D 47 (1993), 426-455 [arXiv:hep-ph/9207245 [hep-ph]].
  * [16] N. Arkani-Hamed, H. C. Cheng, P. Creminelli, and L. Randall, Phys. Rev. Lett. 90, 221302 (2003) [hep-th/0301218]
  * [17] T. Inami, Y. Koyama, C.-M. Lin, and S. Minakami, Prog. Theor. Phys. 125 (2011) 345-358 [hep-ph/1004.5477]
  * [18] H. Hatanaka, T. Inami and C.S. Lim, Mod. Phys. Lett. A 13, 2601 (1998); I. Antoniadis, K. Benakli and M. Quiros, New J. Phys. 3, 20 (2001); G. von Gersdorff, N. Irges and M. Quiros, Nucl. Phys. B 635, 127 (2002); R. Contino, Y. Nomura and A. Pomarol, Nucl. Phys. B 671, 148 (2003); C.S. Lim, N. Maru and K. Hasegawa, J. Phys. Soc. Jap. 77, 074101 (2008); N. Maru and T. Yamashita, Nucl. Phys. B 754, 127 (2006); Y. Hosotani, N. Maru, K. Takenaga and T. Yamashita, Prog. Theor. Phys. 118, 1053 (2007);
  * [19] Y. Hamada and T. Kobayashi, Prog. Theor. Phys. 128 (2012) 903 [hep-th/1207.6867].