# Prompt to GPT-3: Step-by-Step Thinking Instructions for Humor Generation

Yuetian Chen, Bowen Shi, Mei Si
Rensselaer Polytechnic Institute
110 8th street, Troy, New York 12180
{cheny63, shib5<EMAIL_ADDRESS>

###### Abstract

> Artificial intelligence has made significant progress in natural language
> processing, with models like GPT-3 demonstrating impressive capabilities.
> However, these models still have limitations when it comes to complex tasks
> that require an understanding of the user, such as mastering human comedy
> writing strategies. This paper explores humor generation using GPT-3 by
> modeling human comedy writing theory and leveraging step-by-step thinking
> instructions. In addition, we explore the role of cognitive distance in
> creating humor.

## Introduction

Artificial Intelligence (AI) advancements have made significant progress in
various fields, particularly natural language processing. In particular, GPT-3
has demonstrated impressive capabilities in many language generation tasks
(?). This paper explores using GPT3 to generate jokes in an explainable and
controlled way. As the demand for personalized and engaging content in the
entertainment industry grows, there is an increasing need for AI models that
can produce humorous content that resonates with human audiences. Therefore,
it is crucial to develop AI models that can generate humor in a way that is
compatible with human preferences.

Figure 1: An overview of Joe Toplyn’s theory in creating monologue jokes

In the field of humor generation, there are two main approaches: template-
based and neural network-based methods. Template-based methods, such as those
used by He et al.(?) and Castro et al.(?), rely on predefined structures that
are filled with appropriate words or phrases to create jokes. While these
methods are simple and easy to implement, they are limited by the predefined
templates and the availability of suitable words or phrases.

On the other hand, neural network-based methods, such as those used by Zhang
et al.(?), and Akbar et al.(?), utilize machine learning techniques. Zhang et
al. used a neural network model to generate humorous captions for images by
incorporating relevant knowledge from external sources. Akbar et al. fine-
tuned a large pre-trained language model (GPT-2) on a joke dataset to generate
jokes. These methods have the advantage of being able to learn from data and
generate more diverse and original humor. However, the generation process is
opaque to human users, and there is no way for people to understand how the
model came up with the joke or instruct the model to generate jokes in a
particular way. This may lead to the production of inappropriate or offensive
humor.

This paper investigates how to enhance GPT-3’s ability to generate humor in an
explainable fashion by integrating Joe Toplyn – a famous late-night show
writer’s comedy writing theory (?) through step-by-step thinking instructions
to GPT3. We also explore the role cognitive distance plays in creating a humor
effect.

## Toplyn’s Theory for Writing Jokes

Toplyn is a renowned comedy writer who has worked on shows like “Late Night
with David Letterman” and “The Tonight Show with Jay Leno.” He outlines his
process for creating humorous content in his book “Comedy Writing for Late-
Night TV: How to Write Monologue Jokes, Desk Pieces, Sketches, Parodies,
Audience Pieces, Remotes, and Other Short-Form Comedy.” (?) According to
Toplyn, there are four steps to crafting a joke:

  1. 1.

Create a topic sentence based on a news item: A news article is typically used
as a starting point. The first step involves generating a topic sentence
highlighting the news item’s main point. The aim is to create a sentence that
is engaging and appropriate for humorous commentary, while still being
factually accurate and not inherently funny.

  2. 2.

Identify handles and associations: Handles refer to interesting or peculiar
words or phrases found within the topic sentence, and Toplyn usually
identifies two handles for each joke. After identifying handles, in the next
step, associations are created. Associations are the concepts or ideas related
to each handle. Toplyn generates a list of associations for each handle, which
are used in the subsequent step to develop a punchline.

  3. 3.

Develop a punchline by combining associations: A punchline connects one
association from the first handle’s list with another from the second handle’s
list. This combination should be perceived as true by most people and evoke a
negative emotion towards the first major entity in the topic. The negative
emotion is essential for generating humor in the monologue joke.

  4. 4.

Connect the topic and punchline with an angle: The angle is a sentence or
phrase that smoothly transitions the audience from the topic to the punchline.
It ensures that the joke has a coherent structure and flows naturally from
beginning to end.

The following is an example of a joke written by Joe: “At Lax, customs
inspectors seized a live shipment of 67 Giant African snails. Instead of
destroying them, officials gave them jobs at the DMV.” Joe Toplyn created an
automated joke generation system based on his theory by prompting GPT-3 (?).
This work is in the same spirit. We built upon Toplyn’s approach by
introducing a new factor - the cognitive distance between associations for
creating punchlines - to generate more effective jokes. This involves
identifying seemingly unrelated associations to craft punchlines, resulting in
better jokes.

## Implementation and Examples

In our method, we adopt step-by-step thinking instructions that combine a
chain of thought reasoning (?) with Toplyn’s theory to guide GPT-3 in
generating jokes. This approach includes a set of well-produced prompts for
each stage of the process, allowing the model to focus on individual aspects
of the joke sequentially. The chain of thought reasoning and iterative report
generation during each prompt help to ensure a coherent and consistent joke
creation process. The examples of intermediate results and final jokes are
provided in Table 1.

#### Creating the Topic Sentence from News Article

The instruction we used is as follows. It defines what a topic sentence is
with examples:

  * “We will generate Monologue joke topics for a late-night TV show by crafting succinct sentences based on actual news items. A Monologue joke comprises three parts, and our objective is to produce the first part - the topic. The topic should be founded on a real news event that grabs people’s attention and enables amusing commentary. It need not be intentionally funny but must be factually accurate. For instance, ‘Carl’s Jr. is selling a foot-long burger’ or ‘Bernie Madoff’s underpants were sold at an auction’ are suitable topics. During our conversation, I will provide a news article, and you will create a single sentence that fulfills these criteria. If you believe the news article is inappropriate for Monologue jokes, please inform me.”

Based on our experience, medium-sized articles (500 to 800 words) work best
for generating jokes. This is because they provide enough context to write a
relevant and engaging joke, while not being so long that they become
overwhelming. Articles that are too short may lack enough context for a joke,
while articles that are too long may contain extraneous information that
dilutes the comedic focus.

#### Identifying Handles and Associations

Handles are potential humor generators and can be people, places, things, or
actions. In the prompt, we provide examples to ensure accuracy in identifying
handles for a richer punchline.

  * “To further develop this monologue joke for a late-night TV show based on the provided topic sentence, we need to create the punchline, which is the surprise element at the end of the joke. To do this, follow these steps:

    * –

Determine two handles in the topic, which are interesting words or phrases.
Handles can include people, places, things, or actions.

    * –

Brainstorm a list of associations for each handle and then create two separate
lists of associations related to each handle.

Now, based on the topic you provided, identify handles and associations in the
format given by the example.”

Component |  Description
---|---
Topic |  Microsoft introduces a new AI-powered Copilot for their 365 apps, making Clippy’s ghost proud.
Handles |  AI-powered Copilot, Clippy’s ghost
Associations for AI-powered Copilot |  Artificial intelligence, Clippy 2.0, Microsoft 365, Productivity, GPT-4, Virtual assistant, Automated tasks, Office apps, Innovative technology
Associations for Clippy’s ghost |  Nostalgia, Old technology, Revolutionary technology, Paperclip, Annoying assistant, Pop-up help, Microsoft Office, 90s tech, Failed innovation
Punchline (Negative) |  Automated tasks \+ Annoying assistant: Now, it can automatically annoy you with its help.
Punchline (Positive) |  Clippy 2.0+Revolutionary technology: Clippy’s cool cousin has arrived!
Angle (Negative) |  In the spirit of Clippy,
Angle (Positive) |  It turning your office into a futuristic workspace, with one chatbot at a time -
Final Joke (Negative) |  Microsoft introduces a new AI-powered Copilot for their 365 apps, making Clippy’s ghost proud. In the spirit of Clippy, now it can automatically annoy you with its help.
Final Joke (Postive) |  Microsoft introduces a new AI-powered Copilot for their 365 apps, making Clippy’s ghost proud. It turning your office into a futuristic workspace, with one chatbot at a time - Clippy’s cool cousin has arrived!
Table 1: Example of joke generation using the proposed method with positive
and negative emotion instruction

#### Developing the Punchline

The prompt provided below emphasizes the significance of eliciting a negative
feeling towards the first major entity in the topic sentence. This is
important because it creates a target for the joke, which allows the audience
to laugh at the joke more easily.

  * “Pair an association from one list with an association from the other list. Choose a combination that most people would perceive as true to create the punchline. It is important to evoke a negative emotion towards the first major entity in the topic for the monologue joke to be humorous. Now, based on the association lists you provided, provide the punchline as shown in the example.”

Our study demonstrates that the inclusion of negative emotion in a joke plays
a crucial role in its humor. Table 1 provides two instances of jokes
addressing the same topic but with different sentiment prompts. When the
sentiment in the prompt is changed from “Negative” to “Positive,” the joke may
lose its humor. This can be attributed to the fact that the joke prompted with
the “Negative” sentiment keyword contains more negative emotion, resulting in
greater contrast between the subjects involved.

#### Connecting the Topic and Punchline with an Angle

In the final step, we guide GPT-3 to create a smooth link between the topic
sentence and the punchline. Due to the lengthiness of the complete prompt, it
is not possible to provide it here. Essentially, the previously generated
content is used as input to provide context. Then, the instruction for forming
the punchline is then given as follows:

  * “Next, craft an angle to smoothly transition the audience from the topic to the punchline.”

The example included in the prompt provides guidance on how to design an
effective angle. Specifically, given the topic and the punchline as follows:

Topic: The FBI is warning people about the dangers of charging devices in
public areas, especially hotel lobbies. Punchline: Your device might end up
with more than just bed bugs.

One of the angles could be:

…If you are using a USB port in a hotel lobby, …

Again, the example within the prompt reinforces the model’s understanding of
how to effectively weave together the various elements of the joke,
transforming the topic, punchline, and associations into a cohesive and
entertaining monologue joke. By combining the output generated in each stages,
we can get a final report shown in Algorithm 1.

Algorithm 1 Report for the new ”1080p Premium” feature on YouTube

1:Topic: “YouTube experiments with a new “1080p Premium” option, offering
higher-quality video for Premium subscribers.”

2:Associations for “1080p Premium”:

  * •

“Premium subscribers”

  * •

“YouTube revenue”

  * •

“Exclusive content”

  * •

“Higher price”

  * •

“Upselling”

  * •

“Better experience”

3:Associations for “higher-quality video”:

  * •

“HD resolution”

  * •

“Better image quality”

  * •

“More pixels”

  * •

“Higher bitrate”

  * •

“Sharper image”

  * •

“More data”

  * •

“Cinematic experience”

4:Punchline: “Upselling their way to a sharper disappointment.”

5:Angle: “Now, viewers can pay more to witness their favorite cat videos in
stunning clarity.”

6:Summary: “YouTube experiments with a new “1080p Premium” option, offering
higher-quality video for Premium subscribers. Now, viewers can pay more to
witness their favorite cat videos in stunning clarity – upselling their way to
a sharper disappointment.”

### Example Outputs

In this section, we present additional examples of jokes generated using our
proposed method111GitHub repository: https://github.com/Stry233/Joe-Toplyn-s-
Monologue-Joke-generation-using-ChatGPT.

The first example’s topic is Nintendo’s decision to skip E3 2023, leaving
gamers curious about what they have in store. The punchline creatively links
“surprise announcements” and “childhood memories,” by replacing the former
with the game ”hide-and-seek.” The resulting joke is: “Nintendo decides to
skip E3 2023, leaving gamers wondering what’s up their sleeve. Turns out,
their new game plan is to play hide-and-seek with our childhood memories.”

The second example involves Meta’s new AI language generator, LLaMA. The topic
centers on Meta’s hope that LLaMA will generate solutions instead of just
spitting. By connecting the associations of “AI language generator” and
“LLaMA,” the punchline is crafted: “Meta releases a new AI language generator
called LLaMA, hoping it’ll spit out solutions instead of just spitting.
Instead of taking after its namesake, it’s a problem solver that spits out
answers, not saliva.”

The third example, as shown in Table 2, makes connections among three
associations. It involves a Japanese spacecraft that’s set to make a historic
moon landing. Its cargo includes the UAE’s rover and a lunar robot made by a
Japanese toy maker. By combining the associations of “historic moon landing,”
“UAE’s rover,” and “Japanese toy maker,” we arrived at the punchline: “As the
Japanese spacecraft lands on the moon, carrying the UAE’s rover and a lunar
robot from a Japanese toy maker, Neil Armstrong’s famous quote gets a cosmic
update: ‘One giant leap for UAE’s rover, one small step for anime-kind’.”

Component |  Description
---|---
Topic |  A Japanese spacecraft is attempting a historic moon landing, delivering the UAE’s rover and a lunar robot from a Japanese toy maker.
Handles |  historic moon landing, UAE’s rover, Japanese toy maker
Associations for historic moon landing |  Neil Armstrong, Moonwalk, One small step for man, Apollo, Space race, Lunar surface
Associations for UAE’s rover |  United Arab Emirates, Space exploration, Desert, Arabian Nights, Sand dunes
Associations for Japanese toy maker |  Anime, Gundam, Hello Kitty, Action figures, Remote control toys, Collector’s items
Punchline |  Neil Armstrong’s famous quote is updated: “One giant leap for UAE’s rover, one small step for anime-kind.
Angle |  In a cosmic twist,
Final Joke |  A Japanese spacecraft is attempting a historic moon landing, delivering the UAE’s rover and a lunar robot from a Japanese toy maker. In a cosmic twist, Neil Armstrong’s famous quote is updated: “One giant leap for UAE’s rover, one small step for anime-kind.”
Table 2: Example of joke generation using the proposed method

### Create Punchlines using Unrelated Concepts

Although not mentioned in Toplyn’s theory, we found that the selection of
associations is a critical factor in creating engaging and humorous jokes.
Selecting associations that are less obviously related can lead to more
unexpected and intriguing punchlines. These types of punchlines often have a
stronger comedic effect, which enhances the humor of the joke. In contrast, if
we were to select associations that are more closely related, the resulting
punchline may not be surprising enough to provoke laughter or amusement. In
such cases, the joke might feel predictable or mundane instead.

For example, let’s say we have two handles - “Space Travel” and “Fast Food.”
We retrieve associations for each handle as follows:

Space Travel: Mars, freeze-dried meal, astronaut
Fast Food: burger, fries, drive-thru

When picking two associations for creating the punchline, we find that the
pair Mars and burger are mostly irrelevant to each other. Using this pair, we
can create a punchline that more likely surprises the audience and creates
humor. For example:

Why did the astronaut bring a burger to Mars? Because he heard it was a great
place for a ‘space’cial!

Alternatively, if we choose to use freeze-dried meal and burger as a word-pair
which has a relatively higher semantic relevance score.

Why did the astronaut prefer a freeze-dried meal over a burger? Because it’s
easier to pack for space travel!

This punchline lacks humor because the connection between the two concepts is
easily understandable and lacks an element of surprise. The audience can
quickly identify the relationship between a freeze-dried meal for astronauts
and a burger, which reduces the effectiveness of the punchline in creating
laughter.

## Discussion and Future Work

Our work demonstrates that it is possible to use step-by-step instructions to
guide GPT-3 in following human comedy writing theory, resulting in the
generation of configurable and explainable jokes. The jokes are configurable
because, although the process can be fully automated, humans can intervene and
modify the intermediate results in the joke creation process. This advancement
opens up new possibilities for AI-driven humor and entertainment and
contributes to the growing body of research in the field of AI and humor
studies.

The next step in our research involves incorporating additional comedy writing
theories and techniques. This includes exploring various types of humor like
irony, satire, and sarcasm, as well as more advanced techniques such as
wordplay, puns, and misdirection. We also plan to examine the potential of
integrating user feedback into the system, where users can rate and provide
feedback on the generated jokes, which will be used by reinforcement learning
process to improve the overall quality and effectiveness of the joke
generation process.

We also plan to further study the relationship between the level of difficulty
in creating connections between associations and the resulting perceived
humor. Specifically, we want to investigate whether the level of cognitive
effort required to make sense of seemingly unrelated associations enhances or
detracts from the overall comedic effect. This could provide valuable insights
into how to optimize the joke generation process for maximum comedic impact.

Finally, we want to personalize the system by considering users’ humor
preferences. By gathering data on users’ preferred humor styles, comedians,
and amusing joke types, we can train the system to create jokes that match
their sense of humor. Additionally, we can try creating jokes for specific
events or holidays to enhance their relevance and humor value.

## References

  * [Akbar et al. 2021] Akbar, N. A.; Darmayanti, I.; Fati, S. M.; and Muneer, A.  2021\.  Deep learning of a pre-trained language model’s joke classifier using gpt-2.  Journal of Hunan University Natural Sciences 48(8).
  * [Brown et al. 2020] Brown, T. B.; Mann, B.; Ryder, N.; Subbiah, M.; Kaplan, J.; Dhariwal, P.; Neelakantan, A.; Shyam, P.; Sastry, G.; Askell, A.; Agarwal, S.; Herbert-Voss, A.; Krueger, G.; Henighan, T.; Child, R.; Ramesh, A.; Ziegler, D. M.; Wu, J.; Winter, C.; Hesse, C.; Chen, M.; Sigler, E.; Litwin, M.; Gray, S.; Chess, B.; Clark, J.; Berner, C.; McCandlish, S.; Radford, A.; Sutskever, I.; and Amodei, D.  2020\.  Language models are few-shot learners.
  * [Castro et al. 2016] Castro, S.; Cubero, M.; Garat, D.; and Moncecchi, G.  2016\.  Is this a joke? detecting humor in spanish tweets.  In Advances in Artificial Intelligence-IBERAMIA 2016: 15th Ibero-American Conference on AI, San José, Costa Rica, November 23-25, 2016, Proceedings 15, 139–150.  Springer.
  * [He, Peng, and Liang 2019] He, H.; Peng, N.; and Liang, P.  2019\.  Pun generation with surprise.  arXiv preprint arXiv:1904.06828.
  * [Toplyn 2014] Toplyn, J.  2014\.  Comedy Writing for Late-Night TV: How to Write Monologue Jokes, Desk Pieces, Sketches, Parodies, Audience Pieces, Remotes, and Other Short-Form Comedy.
  * [Toplyn 2023] Toplyn, J.  2023\.  Witscript 3: A hybrid ai system for improvising jokes in a conversation.  arXiv preprint arXiv:2301.02695.
  * [Wei et al. 2023] Wei, J.; Wang, X.; Schuurmans, D.; Bosma, M.; Ichter, B.; Xia, F.; Chi, E.; Le, Q.; and Zhou, D.  2023\.  Chain-of-thought prompting elicits reasoning in large language models.
  * [Zhang et al. 2020] Zhang, H.; Liu, D.; Lv, J.; and Luo, C.  2020\.  Let’s be humorous: Knowledge enhanced humor generation.  arXiv preprint arXiv:2004.13317.


# Generation of narrow beams of ultrarelativistic positrons (electrons) in the
resonant strong electromagnetic field-assisted Breit-Wheeler process

S.P. Roshchupkin  Peter the Great St. Petersburg Polytechnic University,
195251, St-Petersburg, Russian Federation, Russia  V.D. Serov  Peter the Great
St. Petersburg Polytechnic University, 195251, St-Petersburg, Russian
Federation, Russia  V.V. Dubov  Peter the Great St. Petersburg Polytechnic
University, 195251, St-Petersburg, Russian Federation, Russia

###### Abstract

The resonant external field-assisted Breit-Wheeler process (Oleinik
resonances) for strong electromagnetic fields with intensities less than the
critical Schwinger field has been theoretically studied. The resonant
kinematics has been studied in detail. The case of high-energy initial gamma
quanta and emerging ultrarelativistic electron-positron pairs is studied. The
resonant differential cross section is obtained. The generation of narrow
beams of ultrarelativistic positrons (for Channel A) and electrons (for
Channel B) is predicted with a probability significantly exceeding
corresponding to the non-resonant process.

##  1 Introduction

Over the past several decades, there has been significant interest in studying
the processes of quantum electrodynamics (QED) in external electromagnetic
fields (see, for example, reviews [1]-[7], monographs [8]-[10] and articles
[11]-[53]). This is mainly associated with the appearance of lasers with high
radiation intensities and beams of small transverse dimensions [11]-[18].

An important feature of high-order by the fine structure constant QED
processes in an external field is the potential for their resonant occurrence,
where virtual intermediate particles enter the mass shell. Such resonances
were first considered by Oleinik [19, 20]. Under resonance conditions, the
conservation laws of energy and momentum are satisfied for intermediate
particles in an external field. As a result, second-order processes by the
fine structure constant effectively reduce into two sequential first-order
processes. A detailed discussion of resonant processes is presented in reviews
[2, 4], monographs [8, 9, 10], as well as recent articles [29]-[34]. It is
important to note that the probability of resonant processes can significantly
exceed the corresponding probabilities of non-resonant processes.

The process of electron-positron pair production by two gamma quanta was first
considered by Breit and Wheeler [35]. Currently, there is a significant number
of works devoted to the study of the Breit-Wheeler process in an external
electromagnetic field (see, for example, [36]-[47]). It should be noted that a
distinction should be made between the external field-stimulated Breit-Wheeler
process (a first-order process with respect to the fine structure constant)
and the external field-assisted Breit-Wheeler process (a second-order process
with respect to the fine structure constant). In this paper, Oleinik’s
resonances for the external strong field-assisted Breit-Wheeler process will
be investigated. It should be noted that in a weak field, this process was
considered in the article [47]. It is important to note that under the
conditions of resonance and the absence of interference between different
reaction channels, the original second-order process effectively reduces to
two first-order processes: the external field-stimulated Breit-Wheeler process
and the external field-stimulated Compton effect [47].

The main parameter for describing the Breit-Wheeler process in the field of a
plane electromagnetic wave is the classical relativistic-invariant parameter

$\eta=\frac{eF\lambdabar}{mc^{2}},$ (1)

numerically equal to the ratio of the work of the field on the wavelength to
the rest energy of the electron. Here $e$ and $m$ are the charge and mass of
the electron, $F$ and $\lambdabar=c/\omega$ are the electric field strength
and wavelength, $\omega$ is the frequency of the wave [1].

In this paper, we consider the resonant strong electromagnetic field-assisted
Breit-Wheeler process for high-energy gamma quanta with energies
$\hbar\omega_{1,2}\lesssim 10^{2}$ GeV. Therefore, we will consider high-
energy gamma quanta in the following, ensuring that the produced electron-
positron pair in a field of the wave is ultrarelativistic

$\hbar\omega_{1,2}\gg mc^{2},\quad E_{\pm}\gg mc^{2}.$ (2)

Here $\hbar\omega_{1,2}$ and $E_{\pm}$ are energies of the initial gamma
quanta and final positron or electron. Therefore, we will assume that the
magnitude of the classical parameter $\eta$ is upper bounded by the condition:

$\eta\ll\eta_{max},\quad\eta_{max}=\mathrm{min}\left(\frac{E_{\pm}}{mc^{2}}\right).$
(3)

Let’s estimate the maximum intensity of the electric field in the wave. For
electron-positron pair energies $E_{\pm}\lesssim 10^{2}$ GeV, it follows from
equation (3) that $\eta\ll\eta_{max}\sim 10^{5}$, or for the field strength we
have $F\ll F_{max}\sim 10^{15}$ Vcm-1 ($I\ll I_{\max}\sim 10^{28}$ Wcm-2).
Thus, the problem will consider sufficiently large intensities of the
electromagnetic wave. However, these fields must be smaller than the Schwinger
critical field $F_{*}\approx 1.3\cdot 10^{16}$ Vcm-1 [5, 50].

In the following, the relativistic system of units is used: $c=\hbar=1$.

##  2 Amplitude of the process

Let us consider this process in the field of a plane circularly polarized wave
propagating along the z axis:

$A(\varphi)=\frac{F}{\omega}\left(e_{x}\cos{\varphi}+\delta
e_{y}\sin{\varphi}\right),\quad\varphi=(kx)=\omega(t-z),\quad\delta=\pm 1.$
(4)

Here $e_{x},e_{y}$ are the polarization 4-vectors of the external field that
have the following properties: $e_{x}=(0,\mathbf{e_{x}}),\quad
e_{y}=(0,\mathbf{e_{y}}),\quad e_{x}e_{y}=0,\quad(e_{x})^{2}=(e_{y})^{2}=-1.$
The external field-assisted Breit-Wheeler process is characterized by two
Feynman diagrams (Fig.1).

Figure 1: Feynman diagrams of electron–positron-pair production by two gamma
quanta in an external field. External double lines correspond to Volkov
functions of electron or positron, wavy lines correspond to wave functions of
initial gamma quanta, and internal double lines correspond to Green’s electron
function in the field of a plane electromagnetic wave.

The amplitude of the considered process is written as follows

$S_{if}=ie^{2}\iint
d^{4}x_{1}d^{4}x_{2}\overline{\Psi}_{p_{-}}(x_{1}|A)\hat{A}_{1}(x_{1};k_{1})G(x_{2}x_{1}|A)\hat{A}_{2}(x_{2};k_{2})\Psi_{-p_{+}}(x_{2}|A)\,+\,(k_{1}\leftrightarrow
k_{2}),$ (5)

where $k_{1,2}=(\omega_{1,2},\mathbf{k_{1,2}})$ — 4-momenta of the initial
gamma quanta, $p_{\pm}=(E_{\pm},\mathbf{p_{\pm}})$ — 4-momenta of the final
electron and positron. Here and further, the notation for the convolution of a
4-vector with the Dirac gamma matrices is used:
$\hat{A}_{1,2}\equiv\gamma_{\mu}A^{\mu}_{1,2}\quad\mu=0,1,2,3$. The
4-potentials of the initial gamma quanta $A_{j}$ in expression (5) are
determined by the functions

$A_{j}(x;k_{j})=\sqrt{\frac{2\pi}{\omega_{j}}}\varepsilon_{j}e^{-ik_{j}x},\quad
j=1,2,$ (6)

where $\varepsilon_{j}$ — 4-vectors of the polarization of the initial gamma
quanta.

In the amplitude (5), the electron-positron pair corresponds to the Volkov
functions [48, 49]:

$\Psi_{p}(x|A)=\mathfrak{J}_{p}(x)\frac{u_{p}}{\sqrt{2E}},\quad\mathfrak{J}_{p}(x)=\left[1+\frac{e}{2(pk)}\hat{k}\hat{A}(kx)\right]e^{iS_{p}(x)},$
(7)
$S_{p}(x)=-(px)-\frac{e}{(kp)}\int_{0}^{kx}d\varphi[pA(\varphi)-\frac{e}{2}A^{2}(\varphi)],$
(8)

where $u_{p}$ is the Dirac bispinor. The intermediate state in the amplitude
(5) corresponds to the Green’s function of the electron in the field of a
plane wave $G(x_{2}x_{1}|A)$ [51]:

$G(x_{2}x_{1}|A)=\int\frac{d^{4}p}{(2\pi)^{4}}\mathfrak{J}_{p}(x_{2})\frac{\hat{p}+m}{p^{2}-m^{2}}\overline{\mathfrak{J}}_{p}(x_{1}).$
(9)

After simple transformations, the amplitude (5) can be represented as follows:

$S_{if}=\sum_{l=-\infty}^{+\infty}S_{l},$ (10)

where the partial amplitude $S_{l}$ corresponds to the absorption or emission
of $|l|$ photons of the external wave. For the Channel A, the partial
amplitude can be represented in the following form:

$S_{l}=\frac{i\pi
e^{2}(2\pi)^{4}e^{-id}}{\sqrt{\widetilde{E}_{-}\widetilde{E}_{+}\omega_{1}\omega_{2}}}\left[u_{p_{-}}M_{l}v_{p_{+}}\right]\delta^{(4)}\left(k_{1}+k_{2}-\widetilde{p}_{-}-\widetilde{p}_{+}-lk\right).$
(11)

Here $d$ is the phase, independent of the summation indices, $M_{l}$ — the
matrix determined by the expression

$M_{l}=\varepsilon_{1\mu}\varepsilon_{2\nu}\sum_{r=-\infty}^{+\infty}K^{\mu}_{l+r}(\widetilde{p}_{-},\widetilde{q}_{-})\frac{\hat{q}_{-}+m}{\widetilde{q}_{-}^{2}-m_{*}^{2}}K^{\nu}_{-r}(\widetilde{q}_{-},-\widetilde{p}_{+}),\quad\mu,\nu=0,1,2,3.$
(12)

In relation (12), the functions $K^{\mu}_{l+r}$ and $K^{\nu}_{-r}$ have the
following form:

$K^{\mu^{\prime}}_{n}(\widetilde{p}^{\prime},\widetilde{p})=a^{\mu^{\prime}}L_{n}(\widetilde{p}^{\prime},\widetilde{p})+b_{-}^{\mu^{\prime}}L_{n-1}+b_{+}^{\mu^{\prime}}L_{n+1}.$
(13)

Here, the matrices $a^{\mu^{\prime}}$ and $b^{\mu^{\prime}}_{\pm}$ have the
following form:

$a^{\mu^{\prime}}=\gamma^{\mu^{\prime}}+\frac{m^{2}\hat{k}}{2(k\widetilde{p^{\prime}})(k\widetilde{p})}k^{\nu},\quad
b^{\mu^{\prime}}_{\pm}=\frac{1}{4}\eta
m\left(\frac{\hat{e}_{\pm}\hat{k}\gamma^{\mu^{\prime}}}{(k\widetilde{p^{\prime}})}+\frac{\gamma^{\mu^{\prime}}\hat{k}\hat{e}_{\pm}}{(k\widetilde{p})}\right),$
(14) $e_{\pm}\equiv e_{x}\pm ie_{y},\quad\mu^{\prime}=\mu,\nu,\quad
n=l+r,-r,\quad\widetilde{p}=-\widetilde{p}_{+},\widetilde{q}_{-},\quad\widetilde{p^{\prime}}=\widetilde{q}_{-},\widetilde{p}_{-}.$
(15)

In relations (12), (13) there are special functions $L_{n}$ [3], which in the
case of circular polarization of the wave can be represented using Bessel
functions with integer indices

$L_{n}(\widetilde{p}^{\prime},\widetilde{p})=\exp(-in\chi_{\widetilde{p}^{\prime}\widetilde{p}})J_{n}(\gamma_{\widetilde{p}^{\prime}\widetilde{p}}),$
(16)

where is denoted

$\gamma_{\widetilde{p}^{\prime}\widetilde{p}}=m\eta\sqrt{-Q^{2}_{\widetilde{p}^{\prime}\widetilde{p}}},\quad\tan\chi_{\widetilde{p}^{\prime}\widetilde{p}}=\delta\frac{(Q_{\widetilde{p}^{\prime}\widetilde{p}}e_{y})}{(Q_{\widetilde{p}^{\prime}\widetilde{p}}e_{x})},\quad
Q_{\widetilde{p}^{\prime}\widetilde{p}}=\frac{\widetilde{p}^{\prime}}{(p^{\prime}k)}-\frac{\widetilde{p}}{(pk)}.$
(17)

In the expressions (11) and (12)
$\widetilde{p}_{\pm}=(\widetilde{E}_{\pm},\mathbf{\widetilde{p}_{\pm}})$ and
$\widetilde{q}_{-}$ are the 4-quasimomenta of the electron (positron) and
intermediate electron, $m_{*}$ is the effective mass of the electron in the
field of a circularly polarized wave (4) [32]:

$\widetilde{q}_{-}=k_{2}+rk-\widetilde{p}_{\pm},$ (18)
$\widetilde{p}_{\pm}=p_{\pm}+\eta^{2}\frac{m^{2}}{2(kp_{\pm})}k,\quad\widetilde{q}_{-}=q_{-}+\eta^{2}\frac{m^{2}}{2(kq_{-})}k,$
(19) $\widetilde{p}_{\pm}^{2}=m_{*}^{2},\quad m_{*}=m\sqrt{1+\eta^{2}}.$ (20)

##  3 The resonant kinematics

Under resonance conditions, both an electron and a positron can be
intermediate particles. Therefore, instead of two Feynman diagrams in the non-
resonant case (see Fig. 1), under resonance conditions we will have 4 Feynman
diagrams (see Fig. 2): Channels A and B, as well as Channels A’ and B’, which
are obtained from channels A and B by rearranging the initial gamma quanta
$(k_{1}\leftrightarrow k_{2})$. Each channel in the resonance conditions
effectively decays into two first-order processes by the fine structure
constant: the external field-stimulated Breit-Wheeler process (EFSBWP) and the
external field-stimulated Compton effect (EFSCE) with intermediate electrons
and positrons entering the mass shell:

$\widetilde{q}_{-}^{2}=m_{*}^{2},\quad\widetilde{q}_{+}^{2}=m_{*}^{2}.$ (21)

Figure 2: Feynman diagrams of the resonant electron–positron-pair production
by two gamma quanta in an external field, Channels A and B; for Channels A’
and B’ $k_{1}\leftrightarrow k_{2}$.

Further consideration will be carried out for resonant Channels A and B (see
Fig. 2). It is important to emphasize that the laws of conservation of energy-
momentum for intermediate processes of resonant Channels A and B have the
form:

$\mathrm{EFSBWP}:\qquad k_{2}+rk=\widetilde{q}_{\mp}+\widetilde{p}_{\pm}\quad
r=1,2,3\ldots;$ (22) $\mathrm{EFSCE}:\qquad
k_{1}+\widetilde{q}_{\mp}=\widetilde{p}_{\mp}+r^{\prime}k\quad
r^{\prime}=1,2,3\ldots\quad(r^{\prime}=l+r).$ (23)

Since the problem considers high-energy initial gamma quanta and
ultrarelativistic energies of the final electron-positron pair (2), under such
conditions, the momenta of the initial and final particles should lie within a
narrow angle cone, which should be far away from the direction of wave
propagation:

$\theta_{j\pm}\equiv\angle(\mathbf{k}_{j},\mathbf{p}_{\pm})\ll
1,\quad\theta_{i}\equiv\angle(\mathbf{k}_{1},\mathbf{k}_{2})\ll 1,$ (24)
$\theta\equiv\angle(\mathbf{p}_{\pm},\mathbf{k})\sim
1,\quad\theta_{j}\equiv\angle(\mathbf{k}_{j},\mathbf{k})\sim 1,\quad
j=1,2;\quad\theta\approx\theta_{1}\approx\theta_{2}.$ (25)

Let us note that under conditions (2), (3), the expression for the positron
(electron) quasienergy can be simplified:

$\widetilde{E}_{\pm}=E_{\pm}\left[1+\frac{1}{4\sin^{2}\frac{\theta_{\pm}}{2}}\left(\frac{m\eta}{E_{\pm}}\right)^{2}\right]\approx
E_{\pm}.$ (26)

Let us determine the resonance energy of the positron (electron) for the
second vertex (see Fig. 2). Taking into account relations (2), (3), (21), (24)
from the conservation of 4-momentum law (22) for the external field-stimulated
Breit-Wheeler process, we obtain the resonance energies of the positron (for
Channel A) or electron (for Channel B) in units of the total energy of the
initial gamma quanta:

$x_{j^{\prime}(r)}=\frac{\omega_{2}}{2\omega_{i}(\varepsilon_{2BW(r)}+\delta^{2}_{2j^{\prime}})}\left[\varepsilon_{2BW(r)}\pm\sqrt{\varepsilon_{2BW(r)}(\varepsilon_{2BW(r)}-1)-\delta^{2}_{2j^{\prime}}}\right],\quad
j^{\prime}=+,-.$ (27)

Here it is indicated:

$x_{\pm(r)}=\frac{E_{\pm}(r)}{\omega_{i}},\quad\omega_{i}=\omega_{1}+\omega_{2},\quad\delta_{2\pm}=\frac{\omega_{2}}{2m_{*}}\theta_{2\pm}.$
(28)

In this case, the ultrarelativistic parameter $\delta_{2\pm}$, which
determines the outgoing angle of the positron or electron, is contained within
the interval

$0\leq\delta^{2}_{2+}\leq\delta^{2}_{2+max},\quad\delta^{2}_{2+max}=\varepsilon_{2BW(r)}(\varepsilon_{2BW(r)}-1).$
(29)

It is important to emphasize that in equation (27), the quantity
$\varepsilon_{2BW(r)}$ is bounded from below by unity

$\varepsilon_{2BW(r)}=r\varepsilon_{2BW}\geq
1,\quad\varepsilon_{2BW}=\frac{\omega_{2}}{\omega_{BW}},$ (30)

where $\omega_{BW}$ is the characteristic quantum energy of the external
field-stimulated Breit-Wheeler process:

$\omega_{BW}=\frac{m_{*}^{2}}{\omega\sin^{2}\frac{\theta}{2}}=\left\\{\begin{array}[]{rcl}174\mbox{GeV}&\mbox{if}&\omega=3\mbox{eV},I=1.675\cdot
10^{19}\mbox{Wcm}^{-2}\\\
5.22\mbox{GeV}&\mbox{if}&\omega=0.1\mbox{keV},I=1.861\cdot
10^{22}\mbox{Wcm}^{-2}\\\
52.2\mbox{MeV}&\mbox{if}&\omega=10\mbox{keV},I=1.861\cdot
10^{26}\mbox{Wcm}^{-2}\end{array}\right.$ (31)

When estimating the value of the characteristic energy, frequencies of
electromagnetic waves in the optical and X-ray ranges were used in equation
(31), as well as values of parameters $\eta=1$ and $\theta=\pi$. It is worth
noting that the ratio between the initial energy of the gamma quantum and the
characteristic energy $\omega_{BW}$ determines the value of parameter
$\varepsilon_{2BW}$ (30), which can be either greater or less than unity. This
significantly affects the number of photons absorbed in the EFBWP.
Specifically, if the initial energy of the gamma quantum is less than the
characteristic energy, then from equations (30) and (31) it follows that this
process occurs if the number of absorbed wave photons is above a certain
minimum $r_{min}$ value, which is greater than unity:

$r\geq
r_{min}=\lceil\varepsilon_{2BW}^{-1}\rceil\quad(\omega_{2}<\omega_{BW}).$ (32)

If the initial energy of the gamma quantum is greater than the characteristic
energy, then this process takes place already when one photon of the wave is
absorbed:

$r\geq 1\quad(\omega_{2}\geq\omega_{BW}).$ (33)

Thus, the resonant energy of a positron (for Channel A) or an electron (for
Channel B) is determined by two parameters: the corresponding outgoing angle
of the positron ($\delta^{2}_{2+}$) or electron ($\delta^{2}_{2-}$), and the
parameter $\varepsilon_{2BW(r)}$. At the same time, with a fixed parameter
$\varepsilon_{2BW(r)}$, for each outgoing angle of the positron or electron,
there are two possible energies (see equation (27)).

Figure 3: The energy of the positron (Channel A) or electron (Channel B)
(27)-(30) for the external field-stimulated Breit-Wheeler process with
absorption of one and two photons of the wave at different frequencies and
intensities of the electromagnetic wave (31). Solid lines correspond to the
”+” and dashed lines correspond to the ”–” signs before the square root in
(27). The energies of the initial gamma quanta: Fig.3a) –
$\omega_{1}=10\mbox{GeV},\omega_{2}=180\mbox{GeV}$; Fig.3b) –
$\omega_{1}=0.5\mbox{GeV},\omega_{2}=7\mbox{Gev}$; Fig.3c) –
$\omega_{1}=10\mbox{MeV},\omega_{2}=80\mbox{MeV}$.

Figure 3 shows the dependence of the energy of the positron (for Channel A) or
electron (for Channel B) (see equations (27)-(30)) for the external field-
stimulated Breit-Wheeler process with absorption of one and two photons of the
wave at different frequencies, intensities of the electromagnetic wave
(equation (31), and various initial gamma quanta energies. From this figure,
it follows that the interval for the outgoing angle of the positron (electron)
significantly depends on the number of absorbed photons of the wave.
Additionally, for the same outgoing angle, there are two possible particle
energies, except for the maximum outgoing angle.

Now let’s determine the resonant electron (positron) energy at the first
vertex (see Fig. 2). Taking into account equations (2), (3), (21), and (24),
from the conservation law of the 4-momentum (equation (23) of the external
field-stimulated Compton effect, we obtain the resonant energies of the
electron (for Channel A) or the positron (for Channel B) in terms of the total
energy of the initial gamma quanta:

$x_{\mp(r^{\prime})}=\frac{\omega_{1}}{2\omega_{i}(\varepsilon_{1C(r^{\prime})}-\delta^{2}_{1\mp})}\left[\varepsilon_{1C(r^{\prime})}+\sqrt{\varepsilon_{1C(r^{\prime})}^{2}+4(\varepsilon_{1C(r^{\prime})}-\delta^{2}_{1\mp})}\right].$
(34)

Here is denoted:

$x_{\mp(r^{\prime})}=\frac{E_{\mp}(r^{\prime})}{\omega_{i}},\quad\delta_{1\mp}=\frac{\omega_{1}}{m_{*}}\theta_{1\mp}.$
(35)
$\quad\varepsilon_{1C(r^{\prime})}=r^{\prime}\varepsilon_{1C},\quad\varepsilon_{1C}=\frac{\omega_{1}}{\omega_{C}},\quad\omega_{C}=\frac{1}{4}\omega_{BW}.$
(36)

Here $\omega_{C}$ s the characteristic quantum energy of the external field-
stimulated Compton effect. This energy is four times less than the
characteristic energy for the external field-stimulated Breit-Wheeler process.
Additionally, it should be noted that the ultrarelativistic parameter
$\delta^{2}_{1\mp}$, which determines the outgoing angle of the electron or
positron, should not take values close to $\varepsilon_{1C(r^{\prime})}$, in
order to satisfy the condition $x_{\mp(r^{\prime})}<1$ (see equation (34)). It
should also be noted that there are no limitations on the parameter
$\varepsilon_{1C(r^{\prime})}$ for the external field-stimulated Compton
effect. Therefore, this process occurs for any number of emitted photons of
the wave $r^{\prime}\geq 1$.

Furthermore, we will assume that the energies of the initial gamma quanta,
within the framework of conditions (2), satisfy the additional conditions:

$\omega_{2}>\omega_{BW},\quad\omega_{1}\ll\omega_{BW}.$ (37)

Conditions (37) mean that parameter $\varepsilon_{2BW}>1$, and parameter
$\varepsilon_{1BW}\ll 1$ (see equations (30) and (31)). Therefore, in Channels
A and B, the external field-stimulated Breit-Wheeler process occurs with a
number of absorbed photons of the wave $r\geq 1$, and for the exchange
resonant diagrams A’ and B’ the number of absorbed photons is $r\geq
r_{min}=\lceil\varepsilon_{1BW}^{-1}\rceil\gg 1$. Thus, within the framework
of conditions (37), resonant Channels A’ and B’ will be suppressed, and we
will only consider two resonant Channels A and B (see Fig. 2). It is also
important to consider that for Channel A, the resonant energy of the positron
is determined by its outgoing angle relative to the momentum of the second
gamma quantum in the EFBWP, while the resonant energy of the electron is
determined by its outgoing angle relative to the momentum of the first gamma
quantum in the EFSCE. For Channel B, we have the opposite situation, where the
energy of the electron is determined by its outgoing angle relative to the
momentum of the second gamma quantum, and the energy of the positron is
determined by its outgoing angle relative to the momentum of the first gamma
quantum (see Fig. 2). Therefore, Channels A and B are distinguishable and do
not interfere with each other.

It is important to note that under resonance conditions (21), the resonant
energies of the positron and electron for each reaction channel are determined
by different physical processes: the external field-stimulated Breit-Wheeler
process (27) and the Compton external field-stimulated effect (34). At the
same time, the energies of the electron-positron pair are related to each
other by the general law of conservation of energy

$x_{+}+x_{-}\approx 1\quad(x_{\pm}=\frac{E_{\pm}}{\omega_{i}}).$ (38)

It should be noted that in equation (38) we have neglected a small correction
term $|l|\omega/\omega_{i}\ll 1$. Taking into account equations (27) and (34),
as well as the law of conservation of energy (38) for Channels A and B, we
obtain the following equations relating the outgoing angles of the positron
and electron:

$\delta^{2}_{1\mp}=\varepsilon_{1C(r^{\prime})}-\frac{(\omega_{1}/\omega_{i})}{(1-x_{\pm(r)})}\left[\varepsilon_{1C(r^{\prime})}+\frac{(\omega_{1}/\omega_{i})}{(1-x_{\pm(r)})}\right].$
(39)

Here the upper (lower) sign corresponds to Channel A (B). In equation (39),
the left side represents the ultrarelativistic parameter associated with the
outgoing angle of the electron (positron) relative to the momentum of the
first gamma quantum, and the right side is the function of the
ultrarelativistic parameter $\delta_{2\pm}$, associated with the outgoing
angle of the positron (electron) relative to the momentum of the second gamma
quantum. Under given parameters $\varepsilon_{1C(r^{\prime})}$ and
$\varepsilon_{2BW(r)}$, equation (39) uniquely determines the outgoing angles
of the electron and positron, and therefore their resonant energies (see Fig.
3 and Fig. 4).

Figure 4: The dependence of the energy of the electron (Channel A) or positron
(Channel B) (34), (39) for the external field-stimulated Compton effect at
different frequencies, intensities of the electromagnetic wave (31), and
initial gamma quanta energies under the condition of energy conservation in
the first and second vertices (38). Solid lines correspond to the ”+” and
dashed lines correspond to the ”–” signs before the square root in expressions
(27), (34), (39). The energies of the initial gamma quanta: Fig.4a), 4a’) –
$\omega_{1}=10\mbox{GeV},\omega_{2}=180\mbox{GeV}$; Fig.4b), 4b’) –
$\omega_{1}=0.5\mbox{GeV},\omega_{2}=7\mbox{Gev}$; Fig.4c), 4c’) –
$\omega_{1}=10\mbox{MeV},\omega_{2}=80\mbox{MeV}$.

Figure 4 presents the dependence of the energy of the electron (for channel A)
or positron (for channel B) (34), (39) for the external field-stimulated
Compton effect at different frequencies, intensities of the electromagnetic
wave (31), and initial gamma quanta energies under the condition of energy
conservation in the first and second vertices (38). The graphs are given for
different numbers of absorbed ($r$) and emitted ($r^{\prime}$) photons of the
wave.

It is also worth noting the important case when the quantum parameter
$\varepsilon_{2BW(r)}\gg 1$. In this case, from the expression (27) with the
”+” sign before the square root, the energy of the positron (Channel A) or
electron (Channel B) approaches the energy of the highly energetic second
gamma quantum:

$E_{\pm}\approx\omega_{2}\left[1-\frac{(1+\delta^{2}_{2\pm})}{4\varepsilon_{2BW(r)}}\right]\longrightarrow\omega_{2}\quad(\delta^{2}_{2\pm}\ll\varepsilon_{2BW(r)}).$
(40)

The expression with the ”–” sign before the square root in equation (27) leads
to the minimum energy of the positron or electron
$E_{\pm}\sim\omega_{2}/\varepsilon_{2BW(r)}\ll\omega_{2}$. However, this case
is unlikely. Similarly, for the first gamma quantum, when the quantum
parameter $\varepsilon_{1C(r^{\prime})}\gg 1$, we obtain the energy of the
electron (Channel A) or positron (Channel B) approaching the energy of the
first gamma quantum:

$E_{\mp}\approx\omega_{1}\left[1-\frac{(1+\delta^{2}_{1\mp})}{\varepsilon_{1C(r^{\prime})}}\right]\longrightarrow\omega_{1}\quad(\delta^{2}_{1\mp}\ll\varepsilon_{1C(r^{\prime})}).$
(41)

Thus, if the quantum parameters $\varepsilon_{1C(r^{\prime})}$ and
$\varepsilon_{2BW(r)}$ take large values, the resonant energies of the
positron and electron tend towards the energies of the corresponding initial
gamma quanta.

##  4 The resonant differential cross section

Previously, it has been shown that in conditions (2), (3), and (37), exchange
Channels A’ and B’ are suppressed. In addition, Channels A and B are
distinguishable and therefore do not interfere (see text after equation (37)).
It is also important to note that resonance processes with different numbers
of absorbed and emitted wave photons correspond to significantly different
probabilities and energies of electron-positron pair. Therefore, they do not
interfere either. Due to this, summation over all possible processes with
absorption of $r$ wave photons is not necessary in the amplitude (12):

$M_{rr^{\prime}}=\varepsilon_{1\mu}\varepsilon_{2\nu}K^{\mu}_{r^{\prime}}(\widetilde{p}_{-},\widetilde{q}_{-})\frac{\hat{q}_{-}+m}{\widetilde{q}_{-}^{2}-m_{*}^{2}}K^{\nu}_{-r}(\widetilde{q}_{-},-\widetilde{p}_{+}),\quad
r^{\prime}=l+r.$ (42)

The resonant differential cross section for Channels A and B and unpolarized
initial gamma quanta and the final electron-positron pair is obtained from the
amplitude (10), (11), (42) in a standard way [53]. After simple calculations,
we obtain:

$d\sigma_{rr^{\prime}}=\frac{2m^{6}r^{2}_{e}}{\widetilde{E}_{-}\widetilde{E}_{+}m_{*}^{2}\delta^{2}_{\eta
i}}\frac{K_{1\mp(r^{\prime})}P_{2\pm(r)}}{|\widetilde{q}_{\mp}^{2}-m_{*}^{2}|^{2}}\delta^{(4)}\left[k_{1}+k_{2}-\widetilde{p}_{-}-\widetilde{p}_{+}-(r^{\prime}-r)k\right]d^{3}\widetilde{p}_{-}d^{3}\widetilde{p}_{+}.$
(43)

Here, the upper (lower) sign corresponds to Channel A (B), $r_{e}=e^{2}/m$ is
the classical electron radius. In obtaining the resonant differential cross-
section (43), the resonant probability was divided by the flux density of the
initial gamma quanta [53]:

$j=\frac{(k_{1}k_{2})}{\omega_{1}\omega_{2}}\approx\frac{m^{2}_{*}}{2\omega_{1}\omega_{2}}\delta^{2}_{\eta
i},\quad\delta^{2}_{\eta
i}\equiv\frac{\omega_{1}\omega_{2}}{m^{2}_{*}}\theta^{2}_{i}.$ (44)

In expression (43), the function $P_{2\pm(r)}$ determines the probability of
the external field-stimulated Breit-Wheeler process [1], and the function
$K_{1\mp(r^{\prime})}$ determines the probability of the external field-
stimulated Compton effect [1]:

$P_{2\pm(r)}=J^{2}_{r}(\gamma_{2\pm(r)})+\eta^{2}(2u_{2\pm(r)}-1)\left[\left(\frac{r^{2}}{\gamma_{2\pm(r)}^{2}}-1\right)J^{2}_{r}+J^{\prime
2}_{r}\right],$ (45)
$K_{1\mp(r^{\prime})}=-4J^{2}_{r^{\prime}}(\gamma_{1\mp(r^{\prime})})+\eta^{2}\left[2+\frac{u_{1\mp(r^{\prime})}^{2}}{1+u_{1\mp(r^{\prime})}}\right](J^{2}_{r^{\prime}-1}+J^{2}_{r^{\prime}+1}-2J^{2}_{r^{\prime}}).$
(46)

The arguments of the Bessel functions for the external field-stimulated Breit-
Wheeler process (45) and the external field-stimulated Compton effect (46)
have the following form:

$\gamma_{2\pm(r)}=2r\frac{\eta}{\sqrt{1+\eta^{2}}}\sqrt{\frac{u_{2\pm(r)}}{v_{2\pm(r)}}\left(1-\frac{u_{2\pm(r)}}{v_{2\pm(r)}}\right)},$
(47)
$\gamma_{1\mp(r^{\prime})}=2r^{\prime}\frac{\eta}{\sqrt{1+\eta^{2}}}\sqrt{\frac{u_{1\mp(r^{\prime})}}{v_{1\mp(r^{\prime})}}\left(1-\frac{u_{1\mp(r^{\prime})}}{v_{1\mp(r^{\prime})}}\right)}.$
(48)

Here, the relativistic-invariant parameters are equal to:

$u_{1\mp(r^{\prime})}=\frac{(k_{1}k)}{(p_{\mp}k)}\approx\frac{(\omega_{1}/\omega_{i})}{x_{\mp(r^{\prime})}},\quad
v_{1\mp(r^{\prime})}=\frac{2r^{\prime}(q_{\mp}k)}{m^{2}_{*}}\approx\varepsilon_{1C(r^{\prime})}\left(\frac{x_{\mp(r^{\prime})}}{(\omega_{1}/\omega_{i})}-1\right),$
(49)
$u_{2\pm(r)}=\frac{(k_{2}k)^{2}}{4(p_{\pm}k)(q_{\mp}k)}\approx\frac{(\omega_{2}/\omega_{i})}{4x_{\pm(r)}\left(1-\frac{x_{\pm(r)}}{(\omega_{2}/\omega_{i})}\right)},\quad
v_{2\pm(r)}=r\frac{(k_{2}k)}{2m_{*}^{2}}\approx\varepsilon_{2BW(r)}.$ (50)

The elimination of resonant singularity in expression (43) is carried out by
the Breit-Wigner procedure [47, 52]:

$m_{*}\longrightarrow\mu_{*}=m_{*}-i\Gamma_{\mp(r)},\quad\Gamma_{\mp(r)}=\frac{\widetilde{q}_{\mp}^{0}}{2m_{*}}W_{1},$
(51)

where $W_{1}$ is the total probability (per unit of time) of the external
field-stimulated Compton effect on the intermediate electron (for Channel A)
or positron (for Channel B).

$W_{1}=\frac{\alpha m^{2}}{4\pi\widetilde{q}_{\mp}^{0}}K(\varepsilon_{1C}),$
(52)
$K(\varepsilon_{1C})=\sum_{s=1}^{\infty}\int_{0}^{s\varepsilon_{1C}}\frac{du}{(1+u)^{2}}K(u,s\varepsilon_{1C}).$
(53)

Here, $\alpha$ is the fine-structure constant, and the function
$K(u,s\varepsilon_{1C})$ is determined by the expression:

$K(u,s\varepsilon_{1C})=-4J^{2}_{s}(\gamma_{1(s)})+\eta^{2}\left[2+\frac{u^{2}}{1+u}\right](J^{2}_{s-1}+J^{2}_{s+1}-2J^{2}_{s})$
(54)
$\gamma_{1(s)}=2s\frac{\eta}{\sqrt{1+\eta^{2}}}\sqrt{\frac{u}{s\varepsilon_{1C}}\left(1-\frac{u}{s\varepsilon_{1C}}\right)}.$
(55)

Taking into account the relations (51)-(55), the resonant denominator in the
cross-section (43) takes the following form:

$|\widetilde{q}_{\mp}^{2}-m_{*}^{2}|^{2}\longrightarrow
m_{*}^{4}\frac{x^{2}_{\mp(r^{\prime})}}{(\omega_{1}/\omega_{i})^{2}}\left[\left(\delta^{2}_{1\mp(0)}-\delta^{2}_{1\mp}\right)^{2}+\Upsilon^{2}_{\mp(r^{\prime})}\right].$
(56)

Here, the ultrarelativistic parameter $\delta^{2}_{1\mp}$ is related to the
resonance energy of the electron (for Channel A) or positron (for Channel B)
by the relation (34), and the corresponding parameter $\delta^{2}_{1\mp(0)}$
can take arbitrary values unrelated to the energy of the electron (positron).
In this case, the corresponding angular width of the resonance
$\Upsilon_{\mp(r^{\prime})}$ is determined by the expression:

$\Upsilon_{\mp(r^{\prime})}=\frac{\alpha m^{2}}{4\pi
m_{*}^{2}}\frac{\omega_{1}}{\omega_{i}x_{\mp(r^{\prime})}}K(\varepsilon_{1C}).$
(57)

Considering relation (26), we can set $d^{3}\widetilde{p}_{\pm}\approx
d^{3}p_{\pm}$ and integrate the three-dimensional momentum of the electron
(positron) as well as the energy of the positron (electron) for Channel A (for
Channel B) using the delta-function in expression (43). After simple
calculations, we obtain the following expression for the resonant differential
cross-section for Channels A and B:

$R_{2\pm(rr^{\prime})}=\frac{d\sigma_{rr^{\prime}}}{d\delta^{2}_{2\pm}}=8\pi
r^{2}_{e}\left(\frac{m}{\delta_{\eta
i}\omega_{i}}\right)^{2}\frac{x_{\pm(r)}}{x_{\mp(r^{\prime})}^{3}}\left(\frac{m}{m_{*}}\right)^{4}\left(\frac{\omega_{1}}{\omega_{2}}\right)^{2}\frac{K_{1\mp(r^{\prime})}P_{2\pm(r)}}{\left[\left(\delta^{2}_{1\mp(0)}-\delta^{2}_{1\mp}\right)^{2}+\Upsilon^{2}_{\mp(r^{\prime})}\right]}.$
(58)

Here, the upper (lower) sign corresponds to Channel A (B). It should be noted
that the differential cross-section (58) has a characteristic Breit-Wigner
resonance structure [52]. Let’s determine the maximum resonant differential
cross-section when

$\left(\delta^{2}_{1\mp(0)}-\delta^{2}_{1\mp}\right)^{2}\ll\Upsilon^{2}_{\mp(r^{\prime})}.$
(59)

Under conditions (59), the resonant cross-section (58) takes its maximum
value, which is equal to:

$R_{2\pm(rr^{\prime})}^{max}=\frac{d\sigma_{rr^{\prime}}^{max}}{d\delta^{2}_{2\pm}}=r^{2}_{e}c_{\eta
i}\Psi_{\pm(rr^{\prime})}.$ (60)

Here, the function $c_{\eta i}$ is determined by the initial setup parameters

$c_{\eta
i}=\frac{2(4\pi)^{3}}{\alpha^{2}K^{2}(\varepsilon_{1C})}\left(\frac{m}{\delta_{\eta
i}\omega_{2}}\right)^{2}\sim 10^{8}\left(\frac{m}{\delta_{\eta
i}\omega_{2}}\right)^{2},$ (61)

and the functions $\Psi_{\pm(rr^{\prime})}$ determine the spectral-angular
distribution of the generated electron-positron pair:

$\Psi_{\pm(rr^{\prime})}=\frac{x_{\pm(r)}}{1-x_{\pm(r)}}K_{1\mp(r^{\prime})}P_{2\pm(r)}.$
(62)

It is important to emphasize that the magnitude of the maximum resonant
differential cross-section significantly depends on the value of the function
$c_{\eta i}$ (61). Let’s require that the function $c_{\eta i}>1$. Then, from
relation (61), we obtain a condition on the initial ultrarelativistic
parameter $\delta^{2}_{\eta i}$ (44):

$\delta^{2}_{\eta i}<\left(10^{4}\frac{m}{\omega_{2}}\right)^{2}.$ (63)

It should be noted that the corresponding Breit-Wheeler differential cross-
section without an external field in this kinematics (24) has the following
order of magnitude [35]:

$\frac{d\sigma_{BW}}{d\delta^{2}_{2\pm}}\sim
r_{e}^{2}\left(\frac{m}{\delta_{i}\omega_{i}}\right)^{2},\quad\delta_{i}=\frac{\sqrt{\omega_{1}\omega_{2}}\theta_{i}}{m}.$
(64)

From relations (60)-(62) and (64), it can be seen that the maximum resonant
cross-section significantly exceeds the corresponding Breit-Wheeler cross-
section without an external field.

Figure 5: The dependence of the maximum resonance differential cross-section
(60) (in units of $r_{e}^{2}$) on the positron outgoing angle (Channel A) or
electron outgoing angle (Channel B) for various frequencies and intensities,
as well as the numbers of absorbed ($r$) and emitted ($r^{\prime}$) photons.
The value of the initial ultrarelativistic parameter $\delta^{2}_{\eta
i}=10^{-4}$. The energies of the initial gamma quanta: Fig.5a) –
$\omega_{1}=10\mbox{GeV},\omega_{2}=180\mbox{GeV}$; Fig.5b) –
$\omega_{1}=0.5\mbox{GeV},\omega_{2}=7\mbox{Gev}$; Fig.5c), 5c’) –
$\omega_{1}=10\mbox{MeV},\omega_{2}=80\mbox{MeV}$.

Figure 5 shows the dependencies of the maximum resonance differential cross-
section (60) on the positron outgoing angle (for Channel A) or electron
outgoing angle (for Channel B) for various frequencies and intensities, as
well as the numbers of absorbed and emitted photons at the first and second
vertices (see Fig. 2). The study focused on the regions of optical and X-ray
frequencies of the external strong electromagnetic wave at different
sufficiently high energies of initial gamma quanta. It is important to note
that the energy of the second high-energy gamma quantum for each frequency and
intensity of the wave was chosen according to condition (33), in order for the
stimulated Breit-Wheeler process to occur with the highest probability, and
the energy of the first gamma quantum was chosen to be much lower than the
energy of the second gamma quantum (37). In this case, with increasing
frequency of the external field, the characteristic energy of the Breit-
Wheeler process decreased (see relation (31)). Therefore, energies of initial
gamma quanta were chosen to be lower for the X-ray frequency range than for
the optical frequency range. As a result, the function (61) increased, leading
to an increase in the maximum resonance cross-section. This case is shown in
Figures 5a) to 5c). However, if the energy of initial gamma quanta remains
constant and the intensity of the external field increases, then the maximum
resonance cross-section decreases (see Figures 5c) and 5c’)). Table 1 displays
the values of positron (for Channel A) and electron (for Channel B) energies,
as well as the corresponding maximum values of the resonance differential
cross-section according to their spectral-angular distribution (see Figures
5a) to 5c’)) for different frequencies and intensities of the wave, as well as
different energies of initial the gamma quanta.

| ($r,r^{\prime}$) | $\delta^{2}_{2\pm max}$ | $x_{\pm(r)}$ | $R^{max}_{2\pm(rr^{\prime})}$
---|---|---|---|---
$I=1.675\cdot 10^{19}\mbox{Wcm}^{-2}$, $\omega=3\mbox{eV}$, $\omega_{1}=10\mbox{GeV}$, $\omega_{2}=180\mbox{GeV}$ | (1,1) | 0 | 0.56 | 44
(1,2) | 0 | 3
(2,1) | 0.12 | 0.76 | 29
(2,2) | 0.07 | 0.78 | 10
$I=1.861\cdot 10^{22}\mbox{Wcm}^{-2}$, $\omega=100\mbox{eV}$, $\omega_{1}=0.5\mbox{GeV}$, $\omega_{2}=7\mbox{GeV}$ | (1,1) | 0 | 0.7 | $2.9\cdot 10^{4}$
(1,2) | 0 | $5.3\cdot 10^{3}$
(2,1) | 0.06 | 0.82 | $1.2\cdot 10^{4}$
(2,2) | 0.07 | 0.81 | $5.2\cdot 10^{3}$
$I=1.861\cdot 10^{26}\mbox{Wcm}^{-2}$, $\omega=10\mbox{keV}$, $\omega_{1}=10\mbox{MeV}$, $\omega_{2}=80\mbox{MeV}$ | (1,1) | 0 | 0.71 | $1.9\cdot 10^{8}$
(1,2) | 0 | $6.3\cdot 10^{7}$
(2,1) | 0.03 | 0.8 | $1.7\cdot 10^{8}$
(2,2) | 0.08 | 0.79 | $3.6\cdot 10^{7}$
$I=1.675\cdot 10^{27}\mbox{Wcm}^{-2}$, $\omega=10\mbox{keV}$, $\omega_{1}=10\mbox{MeV}$, $\omega_{2}=80\mbox{MeV}$ | (4,1) | 0.2 | 0.47 | $8.3\cdot 10^{6}$
(4,2) | 0.17 | 0.5 | $3.4\cdot 10^{6}$
(5,1) | 0.11 | 0.64 | $2\cdot 10^{7}$
(5,2) | 0.27 | 0.56 | $5.5\cdot 10^{6}$
Table 1: The maximum values of the resonance differential cross-section.

From Table 1, it can be observed that if the energy of one of the initial
gamma quanta slightly exceeds the characteristic Breit-Wheeler energy, the
production of electron-positron pairs occurs with a very large cross-section.
For the optical frequency range, the resonance differential cross-section can
exceed the value in magnitude by a factor of 44, while for the X-ray frequency
range, it can exceed the value by eight orders of magnitude. In this case, the
positrons (electrons) are emitted in a narrow cone and with very high energy.

## Conclusion

We considered the resonant Breit-Wheeler process modified by an external
strong electromagnetic field for high-energy initial gamma quanta when the
energy of one of them significantly exceeded the energy of the other. The
following results were obtained:

  1. 1.

The resonant kinematics of the process has been studied in detail. It was
demonstrated that the problem involves two characteristic energies: the Breit-
Wheeler energy $\omega_{BW}$ (31) and the Compton effect energy $\omega_{C}$
(36). These energies differ from each other by a factor of four. The ratios of
the initial gamma quanta energies to these characteristic energies
significantly affect the number of absorbed or emitted wave photons and,
ultimately, the probability of the process.

  2. 2.

The resonant energies of the positron and electron strongly depend on their
outgoing angles, as well as the characteristic quantum parameters
$\varepsilon_{2BW(r)}$ (30) and $\varepsilon_{1C(r^{\prime})}$ (36).
Furthermore, the outgoing angles of the electron and positron are
interdependent (39).

  3. 3.

The maximum resonant differential cross-section is achieved when the energy of
one of the initial gamma quantum slightly exceeds the characteristic Breit-
Wheeler energy. In this case, for the optical frequency range and
$\omega_{2}=180\mbox{GeV}$, the maximum resonant cross-section is
$R_{2\pm(rr^{\prime})}^{max}=44r^{2}_{e}$, whereas for the X-ray frequency
range, it is $R_{2\pm(rr^{\prime})}^{max}\sim(10^{6}\div 10^{8})r^{2}_{e}$.

The obtained results can be utilized to achieve ultrarelativistic positron
(electron) beams with a very high probability in the external field-modified
Breit-Wheeler process. Additionally, these results can be employed to explain
the fluxes of ultrarelativistic positrons (electrons) near neutron stars and
magnetars, as well as in the modeling of physical processes in laser-induced
thermonuclear fusion.

The research was funded by the Ministry of Science and Higher Education of the
Russian Federation under the strategic academic leadership program “Priority
2030” (Agreement 075-15-2023-380 dated 20.02.2023).

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# An empirical study to understand how students use ChatGPT for writing essays
and how it affects their ownership

Andrew Jelson<EMAIL_ADDRESS>Virginia TechBlacksburgVirginiaUSA24061 and
Sang Won Lee<EMAIL_ADDRESS>Virginia TechBlacksburgVirginiaUSA24061

(2018)

###### Abstract.

As large language models (LLMs) become more powerful and ubiquitous, systems
like ChatGPT are increasingly used by students to help them with writing
tasks. To better understand how these tools are used, we investigate how
students might use an LLM for essay writing, for example, to study the queries
asked to ChatGPT and the responses that ChatGPT gives. To that end, we plan to
conduct a user study that will record the user writing process and present
them with the opportunity to use ChatGPT as an AI assistant. This study’s
findings will help us understand how these tools are used and how
practitioners — such as educators and essay readers — should consider writing
education and evaluation based on essay writing.

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Woodstock ’18: ACM Symposium on Neural
Gaze Detection; June 03–05, 2018; Woodstock, NY††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††copyright: none

##  1\. Introduction

Over the past few years, HCI researchers have been looking into how we can use
Large Language Models(LLMs) as tools to enhance the creation process (Gero et
al., 2023; Zhang et al., 2023; Han et al., 2023). One side of the research
examines how AI has become relevant to writers (Gero et al., 2023; Zhang et
al., 2023). For example, ChatGPT can provide assistance in creating ideas or
proofreading an essay. Although there is concern about how students might use
these LLMs to do the work for them, many instructors view the increasing use
of these tools as inevitable and believe that students can still learn
effectively through the thoughtful use of AI coding assistants (Wang et al.,
2023). Writing assistant tools have emerged in different ways over the last
two decades, and there have been numerous papers looking into their
effectiveness on essay writing for students, especially for ESL writers (Huang
et al., 2020; Dong and Shi, 2021; Jayavalan and Razali, 2018; Karyuatry, 2018;
ONeill and Russell, 2019; Koltovskaia, 2020). While most students find
increased performance with Grammarly, for example, they do not effectively use
the tool, and only make moderate changes to their drafts. ChatGPT has also
been investigated as a tool to help students and researchers in the creation
process for natural and computer languages (Shoufan, 2023; Liu et al., 2023;
Murillo and D’Angelo, 2023; Stark et al., 2023). As LLMs serve as practical
support tools and can produce work of comparable quality to that of humans, it
can be a challenge for readers to consider these tools when they consume
written content

One particular domain that shares such challenges is writing education. As
these systems become stronger and more efficient, students might use ChatGPT
to write their papers. This presents new, unique challenges for education.
Several articles have been published in recent years on the creation of
policies and expectations of LLM assistance in education (Adams et al., 2022;
Cotton et al., 2023; Halaweh, 2023; Biswas, 2023), and most of them have come
to the conclusion that while the use of LLM will become more prevalent in the
future, instructors should prepare ways for students to use it effectively.
Other investigators have investigated the different risks and benefits of
using these LLMs in educational environments (Anders, 2023; Sok and Heng,
2023; Joyner, 2023; Warner, 2023; Mosaiyebzadeh et al., 2023). These papers
discuss the different ways in which we can implement policies to help use LLM
assistants. Other researchers have looked at the effectiveness of ChatGPT in
problem-solving process (Shoufan, 2023; Rudolph et al., 2023; Ali et al.,
2023; Moore et al., 2022). These papers have all found that while ChatGPT is
effective at replicating human work. However, we do not understand how ChatGPT
will impact how students learn essential academic skills such as writing. To
understand the impact, we first need to understand how writers may use ChatGPT
in their writing practice in an educational context.

Our user study seeks to answer the following research questions.

  * •

RQ1: In what ways do students use LLM-powered tools, in our case ChatGPT, in
essay writing?

  * •

RQ2: How does using LLM-powered tools affect students’ perceived ownership?

Understanding the answers to these questions will help us refine how we should
design writing assistant tools that integrate LLM-powered intelligence into
the tools. Furthermore, this research can be leveraged to grasp the level of
trust students place in LLMs, and the various approaches — policy and learning
activities — instructors can adopt to prepare their students to learn writing
skills and utilize these tools effectively.

Figure 1. The Essay Recording Page. We used a sample ACT question as the
prompt

##  2\. Methods

To understand the different ways students use ChatGPT, we need to track the
queries they make and the responses that ChatGPT provides. Since ChatGPT is an
independent app, we built a system that makes ChatGPT available within the
writing platform so that we can record user interactions on a large scale.
Using the tool, we plan to conduct a study asking students to write an essay
with ChatGPT assistance and collect data — query, response, and writing — for
further analysis. We outline the details of the study below.

Figure 2. The simulated ChatGPT

###  2.1. Instrument Development

Our application has two main features: the ChatGPT feature and a text editor.
Both features are developed to track how the essay was written and what kinds
of queries and responses are generated in ChatGPT, and we chose a web
application to replicate ChatGPT. We want to simulate the experience of using
ChatGPT in our design, so we use a tab feature to simulate a modern browser.

The first ’tab’ (Fig 1) of our application is a writing platform that can
record the essay writing process at keystroke levels. The participant will be
asked to answer an essay question in a text box that records inputs from the
user, tracking cursor position, insertions, deletions, cuts, and pastes
performed. We also record the timestamp so that we can see when the user made
each edit. Then, we will take this recording to observe and analyze their
writing process asynchronously. Having the time stamps allows us to see how
they alternate between the editor and the in-house ChatGPT and how they
integrated ChatGPT responses into their writing (e.g., pasted text). Later, we
plan on using the recorded scenarios for an educator to evaluate what level of
ChatGPT reliance is considered cheating or plagiarism. All of this will be
sent to a database on submission. These features were implemented using
CodeMirror 5 API and the CodeMirror-Record (Jisuanke., 2023) files.

To track how users use LLM chat tools like ChatGPT, we implement a chat-bot
with the Open AI API (model GPT3.5-turbo), shown in Fig 2. Users are allowed
to ask any question, but the bot will be pre-prompted to be an essay
assistance tool. As users use the tool, we record their query and time stamp
to see how and when ChatGPT was prompted for assistance during the creation
process. This will also be sent to the database on submission.

###  2.2. Study Details

We will ask the user to write the response to a pre-selected essay question
that we have chosen and received from a professor at our university who
teaches a junior-level ethics course, i.e., Professionalism in Computing.
Using this prompt, we will see how university students might use ChatGPT for
help with assignments.

Lastly, we will ask the user to complete an exit survey to see how users felt
about using ChatGPT when writing their essays. These questions will be focused
on how the tool impacts their writing performance and their perceived
ownership of the essay (shown in Appendix A.1). As mentioned above, we want to
simulate browser tabs for a few reasons. We want the writers to feel that
ChatGPT is available to them but not forced upon them. They can choose whether
or not to use it, and it is not always on the screen when writing as a
distraction.

For recruitment, we will collect our data from students in an ethics and
professionalism course at our university. Most of these students are in their
third year and are familiar with essay writing. We will give them a monetary
incentive to help with the recruitment process.

###  2.3. Data Analysis

We plan to analyze our data in multiple different ways. First, we will look
directly at the queries made to ChatGPT. Using open coding, we will categorize
these questions to see how users prompt the LLM. This will provide us with a
better understanding of how people use ChatGPT in the essay creation process.

Next, we look at the essay itself. As previously stated, the recording
features track the user’s inputs and store them in our database with
timestamps. With this data, we can understand how the response they received
from ChatGPT contributes to the writing process by comparing the responses
that they get and what new content is added or how the essay is revised
immediately after the responses from ChatGPT. This provides us with insight
into how the users use LLM-powered tools and their effectiveness. The example
metrics that we planned to use include the number of words copied ChatGPT and
eventually contributed to the final text, the pace of writing, and the types
of generation (e.g., keystroke, pasted text, deleted text, cut-and-pasted
text) and how that correlates to ChatGPT usage.

We also plan to look at where users made changes to their code. As ChatGPT is
available throughout the creation process, understanding where the changes in
the writing were made is important, for example, if users backtrack and use
ChatGPT as a proofreader. This could cause changes to the first line of their
writing at the end of the submission, and we will be able to visualize it.
Other users could potentially ignore the ChatGPT response and continue with
their writing task, so looking at the location will also increase our
understanding of how users utilize the LLM in their writing.

##  3\. Expected Contributions

Overall, we expect to gain insight into how users use ChatGPT when writing
essays. We will be able to identify patterns in the questions asked and to
what extent they implement ChatGPT in their writing. This will be beneficial
to instructors who intend to better understand how their students might use
ChatGPT and allow them to recognize the level at which their students might
use an LLM in their assignment or how to better integrate ChatGPT into their
course. This will also benefit software engineers or individuals who wish to
create an LLM-powered writing assistant because they can see what features and
questions are most common, leading to the development of a better tool.

## References

  * (1)
  * Adams et al. (2022) Catherine Adams, Patti Pente, Gillian Lemermeyer, Joni Turville, and Geoffrey Rockwell. 2022.  Artificial Intelligence and Teachers’ New Ethical Obligations.  _The International Review of Information Ethics_ 31, 1 (Nov. 2022).  https://doi.org/10.29173/irie483 Number: 1.
  * Ali et al. (2023) Jamal Kaid Mohammed Ali, Muayad Abdulhalim Ahmad Shamsan, Taha Ahmed Hezam, and Ahmed A. Q. Mohammed. 2023.  Impact of ChatGPT on Learning Motivation: Teachers and Students’ Voices.  _Journal of English Studies in Arabia Felix_ 2, 1 (March 2023), 41–49.  https://doi.org/10.56540/jesaf.v2i1.51 Number: 1.
  * Anders (2023) Brent A. Anders. 2023.  Is using ChatGPT cheating, plagiarism, both, neither, or forward thinking?  _Patterns (N Y)_ 4, 3 (Feb. 2023), 100694.  https://doi.org/10.1016/j.patter.2023.100694
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##  Appendix A Appendix

###  A.1. Exit Survey

Thank you for participating in our study. Please answer the following
questions as part of our exit survey.

For the following questions, please answer based on your perceived ownership
of the essay:

  1. (1)

I feel that this is my essay

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  2. (2)

I feel that this essay belongs to me

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  3. (3)

I feel a high degree of ownership towards this essay

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  4. (4)

I feel the need to protect my ideas from being used by others.

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  5. (5)

I feel that this essays success is my success

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  6. (6)

I feel this essay was written by me

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  7. (7)

I feel the need to protect the ideas written in the essay

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  8. (8)

I do not feel like anyone else wrote this essay.

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

For the following questions, please answer based on your usage of ChatGPT:

  1. (1)

I feel like ChatGPT helped me in the creation process of my writing

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  2. (2)

I feel like ChatGPT helped me with proofreading my essay

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  3. (3)

I feel like ChatGPT made my essay better

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  4. (4)

I liked using ChatGPT as an assistant during my essay writing

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

  5. (5)

My writing would have been better without ChatGPT assistance

    1. (a)

strongly agree, agree, somewhat agree, neutral, somewhat disagree, disagree,
strongly disagree

Thank you for completing our survey. Winners of the essay writing competition
will receive an email after the study is complete.