# Smaller Language Models are Better
Black-box Machine-Generated Text Detectors

Fatemehsadat Mireshghallah1, Justus Mattern2, Sicun Gao1
Reza Shokri4, Taylor Berg-Kirkpatrick1
1 University of California San Diego, 2 RWTH Aachen
3 National University of Singapore
[fatemeh<EMAIL_ADDRESS>
justus.mattern<EMAIL_ADDRESS>
###### Abstract

With the advent of fluent generative language models that can produce
convincing utterances very similar to those written by humans, distinguishing
whether a piece of text is machine-generated or human-written becomes more
challenging and more important, as such models could be used to spread
misinformation, fake news, fake reviews and to mimic certain authors and
figures. To this end, there have been a slew of methods proposed to detect
machine-generated text. Most of these methods need access to the logits of the
target model or need the ability to sample from the target. One such black-box
detection method relies on the observation that generated text is locally
optimal under the likelihood function of the generator, while human-written
text is not. However, in reality, we usually have very limited knowledge of
the generator, let alone access to it. As such, in this paper we set out to
explore whether models other than the generator can be used to differentiate
between machine-generated and human-written text. We find that overall,
smaller and partially-trained models are better universal text detectors: they
can more precisely detect text generated from both small and larger models.
Interestingly, we find that whether the detector and generator were trained on
the same data is not critically important to the detection success. For
instance the OPT-125M model has an AUC of 0.81 in detecting ChatGPT
generations, whereas a larger model from the GPT family, GPTJ-6B, has AUC of
0.45.

##  1 Introduction

Figure 1: Experimental methodology of our work: We want to study how models
can cross-detect, as in distinguish between human-written and machine-
generated text that is not necessarily generated by them. To this end, we
create a target pool consisting of human-written machine-generated text ,
created by prompting the generative model with the first 20 tokens of the
human-written text. We then generate perturbations of each target sequence
using a perturbation model. We find the loss of the target pool and
perturbations under a detector model, to estimate the local optimality of the
likelihood function around the target and use that to determine if a sequence
is machine generated or not.

With the rapid improvement in fluency of the text generated by large language
models (LLMs), these system are being adopted more and more broadly in a wide
range of applications, including chatbots, writing assistants, and
summarizers. Generations from these models are shown to have human-like
fluency Liang et al. (2022); Yuan et al. (2022), making it difficult for human
readers to differentiate machine-generated text from human-written text. This
can have significant ramifications, as such LLM-based tools can be abused for
unethical purposes like phishing, astroturfing, and generating fake news He et
al. (2023). As such, we need to be able to reliably and automatically detect
machine generated text.

Previous work has found that identifying local optima (curvature) in the
likelihood surface of a trained language model allows for detection of
training utterances Mattern et al. (2023), and generations for a given target
model Mitchell et al. (2023). Specifically, the approximate measure of local
optimality, dubbed curvature, is formed by comparing the loss of a target
sequence to the loss of its perturbations, under the target model. The
intuition in both prior works is that this measure of curvature is larger
around training samples/generations from the model, compared to unseen human-
written text and can therefore be used to determine if a given sequence is
part of the training data or not (by Mattern et al.) or a generation of the
target model or not (by Mitchell et al.).

In practice, however, in many cases where we want to distinguish between
machine-generated text and human-written text we do not know what models could
have been used to generated a sequence, or even if we do know the model, we
might not have access to its loss on a given sequence (e.g. ChatGPT), or
access might be behind a paywall (e.g. GPT3). Therefore, in this paper we set
out to explore the detection of machine-generated text without having
knowledge about the generative model. We do this by exploring whether the same
curvature measure can be used to cross-detect text generated by models other
than the target generative model, and under what conditions such cross-
detection performs best. We use surrogate detector models, whose loss
functions we do have access to. Then, we run the curvature test using the
surrogate (see Figure 1) and compare detection power with the same test, but
using the true generator’s loss.

To this end, we conduct experiments on a slew of models with different sizes
(from tens of millions to billions of parameters), architectures (GPTs, OPTs,
Pythias) and pre-training data (Webtext and the Pile) and also from different
training stages (ranging from the first thousand steps of training to full
training– 143k steps). Our main finding is that cross-detection can come very
close to self-detection in terms of distinguishablity, and that there are
universal cross-detectors with high average distinguishablity performance,
meaning they perform well in terms of detecting generations from a wide-range
of models, regardless of the architecture or training data. More specifically,
we find that smaller models are better universal detectors. For instance the
OPT-125M model comes within $0.07$ area under the ROC curve of self-detection,
on average (see Figure 4). And for models where we don’t have self-detection,
such as ChatGPT, the AUC is $0.81$, whereas OPT 6.7B’s AUC is $0.58$.

We also find that partially trained models are better detectors than the fully
trained ones, and this gap is bigger for larger models (see Figure 7). We then
further investigate some possible reasons for this phenomenon by analyzing
curvature and log-likelihood of the different models, and find that larger
models are more conservative in terms of the likelihood and curvature they
assign to generations from other models. Smaller models, however, assign
higher likelihood to generations of models their size or larger, therefore
they can be used to cross-detect on a broader range of models so the smaller
model is the best universal detector.

##  2 Methodology

Figure 1 shows the methodology of our work, and how we conduct our
experiments: For a given target pool of sequences, the task is to determine if
each sequence is human-written or machine-generated by running a curvature
(local optimality) test over a surrogate detector model that is different from
the generator model, as our main assumption is that we have no information
about the generator model. In the rest of this section we delve deeper into
the details of each component in the setup.

#### Target pool.

The pool of sequences for which we want to conduct the machine-generated text
detection. We form this pool such that there is a 50%/50% composition of
machine-generated/human-written text. The machine-generated text is created by
prompting the generator model with the first $20$ tokens of each human-written
sequence.

#### Generator model.

This is the target model the generations of which we are trying to distinguish
from human-written text. We do not always have full access to this model. In
fact, in most cases we may not even know what model generated the text. This
scenario is what we are actually interested in, i.e. we want to know how can
we detect text generated by unknown models?

#### Curvature (local optimality) test.

The method we use to distinguish between machine-generated and human-written
text relies on the local optimality (curvature) of the target sequence,
building on the intuition that generations are likelier to be locally optimal,
and unseen human written text is not Mitchell et al. (2023); Mattern et al.
(2023). To visualize the local neighborhood of the target sequence, we
generate perturbations of it and have the target generative model evaluate
their loss. As such, the curvature is then calculated as:

$d(x)=\log\ p_{\theta}(x)-\frac{1}{k}\sum_{i}\log\ p_{\theta}(\tilde{x_{i}})$
(1)

Where $x$ is the target sequence, $\theta$ is the parameterization of the
target model, $\tilde{x_{i}}$ is the $i$th perturbation of sample $x$ (i.e.
the $i$th neighbor) out of the overall $k$ perturbations. The perturbed
sequences are generated by masking $15\%$ of $x$ and filling the mask using a
perturbation model. The curvature is thresholded to make the machine-
generated/human-written text decision.

#### Perturbation Model

This model helps generate neighbors by filling in randomly selected spans of
the target sequences in the pool and perturbing them. We use T5-3B for this
purpose in our experiments.

#### Detector model.

This model is used as a surrogate for the target model, to help us detect
generations when using the curvature test. The pool of sequences and their
neighbors are fed to the detector model, and their loss under the detector
model is measured and used to calculate curvature and to distinguish between
generations and human written text.

#### Success metric.

We evaluate the success of the detector by measuring the area under the ROC
curve (AUC), i.e. the false positive vs. true positive rate curve. The higher
the AUC, the more distinguishing power the detection mechanism has.

#### Evaluation strategy.

The results we report in the paper fall into two main categories: (1) using a
model to detect its own generations, which is the main goal of Mitchell et al.
(2023) as well. In this setup, the target and detector models are the same, we
call this self-detection. (2) using a model different from the generator of
the text to detect the generations. In this setup, what we are basically doing
is acting as if a surrogate model has generated the text. In other words, we
want to see how well a model would claim another model’s generation as its
own. Here, the target and detector models are not the same. We call this
cross-detection. This second setup represents the black-box case where we not
only do not have full access to the target model that generated the text, we
also do not know what model it was or what architecture/size it had, so we are
trying to find the best universal detector that would correctly classify it.

##  3 Experimental Setup

###  3.1 Models

We want to experiment with a wide range of models, with different
architectures, parameter counts and training datasets, therefore we use the
following model families in our experiments: Facebook’s OPT (we use the 125M,
350M, 1.3B, and 6.7B models), EleutherAI’s GPT-J, GPTNeo and Pythia Biderman
et al. (2023) (we use GPTNeo-125M, GPTNeo-1.3B, GPTNeo-2.7B, GPTJ-6B and
Pythia models ranging from 70M to 2.8B parameters), and OpenAI’s GPT models
(distilGPT, GPT2-Small, GPT2-Medium, GPT2-Large, GPT2-XL, GPT-3 and ChatGPT).

We also have experiments where we use partially trained models as detectors.
For those experiments, we only use the Pythia models as they are the only ones
with available, open-source partially trained checkpoints. For each Pythia
models, there is also a de-duplicated version available, where the model is
trained on the de-duplicated version of the data, as opposed to the original
dataset. All the models we use are obtained from HuggingFace Wolf et al.
(2019).

###  3.2 Dataset

#### Evaluation dataset.

We follow Mitchell et al. (2023)’s methodology for pre-processing and feeding
the data. We use a subsample of the SQuAD dataset Rajpurkar et al. (2016),
where the original dataset sequences are used as the human-written text in the
target sequence pool. We then use the first $20$ tokens of each human-written
sequence as a prompt, and feed this to the target model, and have it generate
completions for it. We then use this mix of generations and human-written text
to create the target pool for which we do the detection. In all cases,
following the methodology from Mitchell et al. (2023), our pool consists of
$300$ human-written target samples, and $300$ machine-generated samples, so
the overall pool size is $600$.

#### Pre-training datasets for the generative models.

The ElutherAI and Facebook models (GPTJ, GPTNeo, Pythia and OPT families) are
all trained on the Pile dataset Gao et al. (2020), a curated collection of
$22$ English language datasets (consisting of web-crawled data, academic
articles, dialogues, etc.). As mentioned above there are two versions of each
Pythia model Biderman et al. (2023), one version is trained on Pile, the other
is trained on de-duplicated Pile. The de-duplicated Pile is approximately 207B
tokens in size, compared to the original Pile which contains 300B tokens.
There is limited information and access to the training data of the OpenAI
models. The GPT-2 family is reportedly trained on the WebText dataset, GPT-3
is trained on a combination of the Common Crawl 111https://commoncrawl.org,
WebText2, books and Wikipedia, and there is not any information released about
the training data of ChatGPT.

##  4 Does cross-detection work?

As mentioned before, the main goal of our paper is to study ways in which
machine-generated text could be distinguished from human-written text, without
access to any auxiliary information about the model that generated the text.
To this end, we conduct an extensive set of experiments where we use $23$
models with different sizes and architectures as detectors of text generated
by $15$ other models. We also experiment with using partially trained
checkpoints of the detector models, to see how the detection power of the
models changes as the training progresses.

Our main finding is that cross detection can perform as well as self-
detection, or come very close to it. Figures 4 and 7 show how close each
detector comes, in terms of AUC, to self-detection. We can see that on
average, OPT-125M is the best fully trained universal cross-detector, showing
on average $0.07$ lower AUC, compared to self-detection. If we look at
partially trained detector models, however, we see that the Pythia-160M comes
as close as $0.05$ AUC points, with its $5k$, $10k$ and $50k$ step trained
models (the fully trained model is trained for $143k$ steps). These models
seem to even outperform self-detection in some cases, for example for GPTJ-6B.
In the rest of this section we further elaborate on these results and draw
connections between model size and training, and detection power.

Figure 2: Summary of the cross-detection area under the ROC curve (AUC)
results for a selection of generative (the $4$ models over the X axis) and
detector (OPT-125M and OPT-6.7B) models. We can see that the smaller OPT model
is a better universal cross-detector. Full results are shown in Figure 3.
Figure 3: AUC heatmap for cross-detection, where the rows are generative
models and columns are the surrogate detector models, both sorted by model
size. We can see that smaller models are better detectors and larger models
are the worst models in terms of detection power.  Figure 4: AUC difference
between self-detection and cross-detection heatmap (to better see how close
cross-detection comes to self detection), where the rows are generative models
and columns are the surrogate detector models, both sorted by model size. This
plot is basically Figure 3, where each cell in a row is subtracted by the
self-detection AUC for that row.

###  4.1 Smaller Models Are Better Detectors

In this section we aim to find patterns in the cross-detection power of
different models, and see if it has any correlation with model family, size,
training set and detection power. To this end, we use $23$ different models
with different parameter counts, ranging from $70M$ to $6.7B$ to detect
machine-generation texts from all the models listed in Section 3.1.

(a) Pythia 70M

(b) Pythia 410M

(c) Pythia 2.8B

Figure 5: Summary of the results for cross-detection power of different
detector models trained for different number of steps. Each subfigure shows a
different detector model, and the X xis shows the training step for the
checkpoint used as a detector. We only show results for 4 generative models
here, using only 3 detector models. The results for all $15$ generative models
are shown in Figure 6. Figure 6: AUC heatmap for cross-detection, where the
rows are generative models and columns are the surrogate detector models from
the Pythia family, at different training step checkpoints ($1k$, $5k$, $10k$,
$50k$, $100k$ and $143k$), both sorted by model size. We can see that
partially trained models are better detectors.  Figure 7: AUC difference
between self-detection and cross-detection heatmap (to better see how close
cross-detection comes to self detection), here the rows are generative models
and columns are the surrogate detector models from the Pythia family, at
different training step checkpoints ($1k$, $5k$, $10k$, $50k$, $100k$ and
$143k$), both sorted by model size. This plot is basically Figure 6, where
each cell in a row is subtracted by the self-detection AUC for that row.

Figure 3 shows the results for this experiment, where the rows are the
generative models (sizing up from bottom row to top) and the columns shows the
detector models (sizing up from right to left). So each cell shows the
detection power (AUC) of the given detector model (column), on text generated
from the generative model (row). The last row is the mean, which is an overall
metric of how good of a detector that model is. Figure 4 shows how cross-
detection fares against self-detection, and it is missing the ChatGPT and
GPT-3 rows as we do not have self-detection results for them (given how we
have no access to their loss, or the losses are behind a paywall).

For both plots, we see that the bottom left has the lowest values, showing
that larger models are not good at detecting machine generated text from other
models, and they are particularly bad at it for detecting small model
generations. We can also see that smaller models are much better detectors, as
the right side of the graph has much higher AUC values.

Another observation is the correlations between the dataset and model
architecture of the generative and detector models. As the heatmap shows,
models from the same architecture family and trained on the same/overlapping
dataset are better at detecting their own text, compared to models from a
different family. For instance, for detecting text generated by OPT-6.7B the
other models from the OPT family are the best cross-detectors, with AUCs
ranging from 0.89-0.87 (OPT-6.7B self-detects with AUC 0.91). The next best
cross-detector is the smallest GPTNeo-125M with AUC 0.86. However, the OpenAI
GPT2 model of the same size has a lower AUC of 0.84 (and overall the GPT2
family has the lowest cross-detection AUC on OPT), which we hypothesize is due
to the larger gap in the training data, as the OPT and GPTNeo/GPTJ models are
all trained on the Pile dataset, but GPT2 is trained on the Webtext. All in
all, the difference due to the dataset/architecture differences is small as
most of the dataset for all these models is comprised of web-crawled data,
showing that cross-detection can be effective, regardless of how much
information we have about the target model, and how accessible similar models
are.

We have also provided an overall summary of the heatmaps in Figure 2, where we
have presented the numbers from the best overall detector with mean AUC of
0.92 (OPT-125M) and the biggest model of the same family, OPT-6.7B with
average AUC of 0.46. One noteworthy observation is that OPT-125M can detect
generations from models like GPT3 and ChatGPT with relatively high AUC (0.81),
whereas if the intuitive approach of taking another large, “similar” model
were to be taken and we were to use OPT-6.7B, we would get AUC of 0.67 and
0.58 for these models, respectively, which are both close to random (0.5).

We hypothesize that the reason behind large models being poor detectors of
text generated by other models (especially smaller ones), is that larger
models have a more refined taste, therefore they don’t attribute text
generated by other models as their own generations. Smaller models, however,
attribute any machine-generated text as their own, since they have a less
specific taste and are looser fitting models. We discuss this further in
Section 5.

###  4.2 Partially-Trained Models are Better Detectors

Our approach in this section is very similar to the previous one, except here
we aim to find correlations between how far along in the training process a
model is, and its cross-detection power. To this end, we take different
training checkpoints of the Pythia models Biderman et al. (2023) at different
steps (steps $1000$, $5000$, $10000$, $50000$, $100000$ and $143000$) with
different sizes (2.8B, 410M, and 70M), and use them as detectors of
generations from the 4 target models. Figure 5 shows the results for this
experiment (Figures 6 and 7 show entire heatmaps of this experiment, similar
to what was presented in the previous section). For each model we can see that
the final checkpoint is consistently the worst one in terms of machine-
generated text detection, and it is one of the middle checkpoints that has the
best performance.

Our hypothesis for this is similar to that of Section 4, where we believe that
partially trained models have not yet fit to the training data tightly, so
they over claim other models’ generations as their own, whereas the longer a
model is trained, the sequences it ranks higher as its own narrow down.

(a) Curvature: OPT-125M as Detector

(b) Curvature: OPT-350M as Detector

(c) Curvature: OPT-6.7B as Detector

(d) Loglikelihood: OPT-125M as Detector

(e) Logliklihood: OPT-350M as Detector

(f) Logliklihood: OPT-6.7B as Detector

Figure 8: Breakdown of curvature and log likelihood values (mean and standard
deviation) for the best universal detector (OPT-125M), a medium sized detector
(OPT-350M) and a larger detector from the same family (OPT-6.7B), to see the
difference detection powers.  Figure 9: AUC of the three cross-detectors from
Figure 8

##  5 Analysis: Curvature and Log-likelihood Breakdown

To help shed light on why smaller models are better detectors and larger
models are not good at detecting machine generated text, we plot a breakdown
of the curvature metric (Section 2) and log-likelihood values for the best
universal detector (OPT-125M), a medium sized detector of the same family
(OPT-350M) and a largest one from the same family (OPT-6.7B) from Section 4,
shown in Figure 8. The Y axis is the curvature/log likelihood of the target
generations (from the $15$ models from Section 3.1) under the detector models
(OPT-125M, 350M or 6.7B). The X axis is the number of parameters for the
generative models (we do not know how many parameters ChatGPT has, so we
plotted it as the last dot in the plots, after GPT-3 with 540B parameters).
Figure 9 plots the AUCs for detection under the three models, for the $15$
generative models.

We can see that for the smaller detector model (Figures 8(a) and 8(d)), the
mean curvature and log-likelihood values for the generated text are
consistently higher than the curvature for the human-written text. However,
for the larger model (Figure 8(c) and 8(f)), the curvature and log-likelihood
values for the machine-generated text is in most cases smaller than or around
the same value as the human written text. The curvature and log-likelihood
values for human written text for both graphs are stable since the text is the
same and doesn’t depend on the target model.

We can also see that overall the curvature and likelihood values for the
larger model are higher, especially for the original text, than those of the
smaller model, and the values for text generated by the other models have
lower curvature and likelihood value. This shows that the larger model places
higher likelihood on the human written text and fits it better. The smaller
model, however, assigns lower curvature and likelihood to the human-written
text compared to generations by a large gap, and the assigned values are
overall lower than those of the large model.

Broadly we observe that all all detectors are behaving similarly, as in all
models respond similarly to machine generated text from other models, so long
as the other model is same size or bigger. In other words, they place high
likelihood on text from larger models. However, for models smaller than
themselves, they place lower likelihood and curvature. As such, smaller models
are better universal detectors, as the size of the set of sequences they
assign higher likelihood and curvature to is bigger than it is for large
models, and this higher curvature is much higher than the curvature assigned
to the human written text.

Also, another thing we should keep in mind is that our estimation of
“curvature” hinges upon generating numerous perturbations (neighbors) and
comparing their loss with that of a target point, therefore if these perturbed
neighbors are not in fact “neighbors”, as in they are farther from the target
point, our measure of curvature is not accurate (the closer the perturbed
points are, the more accurate estimation of curvature we achieve). The spikes
in all the sub-figures of Figure 8 graphs are for the detector model detecting
its own text.

##  6 Ablating The Perturbation Generation

The perturbation generation method directly impacts the size and shape of the
neighborhood we create around a target point, and use to determine the shape
of the loss function around it and test its local optimality. If the generated
perturbations are too far from a target point, they will have lower likelihood
and create inaccurately high curvature estimates.

As mentioned in Section 4, one hypothesis we have for why small models are
better machine-generated text detectors is that they have flatter, looser
fitting loss functions whereas larger models have higher curvatures, are
sharper and more compressed. As such, for better analysis of the shape of a
function around a target point on a larger model, one needs to generate
perturbations closer to that point to magnify the local neighborhood where we
test optimality, since we hypothesize that the function is more spiked and
changes fast, as opposed to a smaller model that is smoother. To further
explore this hypothesis, we look into different perturbation generation
methods to change the size of the neighborhoud we look at, and see how the
curvature and detection power of the models change.

We investigate two different methods for changing the distance of the
generated perturbations: (1) we change the mask filling model size, by
experimenting with T5-Small, T5-Large and T5-3B Wolf et al. (2019); Raffel et
al. (2020) to test the intuition that larger mask-filling models, generate
semantically closer neighbors than a smaller model. (2) We change the
percentage of the tokens that get masked and replaced by the mask-filling
model, as the more tokens we mask and replace, the farther the generated
perturbations would be.

(a) T5-3B

(b) T5-Large

(c) T5-Small

(d) AUCs for different perturbation (masking) models

Figure 10: The effect of changing the perturbation (masking) model on
curvature values and self-detection power of different models with different
sizes (AUC).

###  6.1 Mask Filling Model

Figure 10 shows the curvature numbers for each model trying to detect its own
generations, so for each model the generator is also the detector. We
experiment with three perturbation generating models, with three different
sizes: (1) T5-small ($60$ million parameters) (2) T5-Large ($770$ million
parameters) (3) T5-3B (3 billion parameter). The intuition behind using three
model sizes is to see the effect of having a better replacement model on the
measured curvatures and the detection power of the detector models.

We can see that as the masking model sizes down (going from top to the bottom
subfigures), the overall curvature values for both human-written and machine-
generated text increases (going from 0.2 maximum in Figure 10(a) to 0.6
maximum in Figure 10(c)), and the two sets of texts become less
distinguishable. T5-Small produces low-quality (low-fluency) neighbors that
are assigned lower likelihoods by the detector model, resulting in high
curvature numbers for both human and machine generated text, making them
indistinguishable. As we improve the mask filling model, however, the
generated neighbors become of higher quality (and semantically closer to the
target point), thereby creating a more accurate estimate of the curvature and
providing better distinguishablity, as shown by the AUC numbers in Figure
10(d).

(a) $90\%$ Masking

(b) $50\%$ Masking

(c) $15\%$ Masking

(d) $2\%$ Masking

(e) $1\%$ Masking

(f) AUCs for different masking percentages

Figure 11: The effect of changing the masking percentage on curvature values
and self-detection power of different models with different sizes (AUC).

###  6.2 Masking Percentage

Figure 11 shows the results for the experiment where we change the percentage
of tokens that are masked, to produce the neighbors. In all previous
experiments, we used $15\%$ masking with mask span length of $2$ tokens
following the experimental setup in Mitchell et al. (2023).

In this section, however, we change the percentage of the masked tokens (and
we set the masking to be contiguous) to see how it affects the curvature mean
and standard deviation values, and the AUCs. We can see that as the masking
percentage decreases (from $90\%$ to $2\%$), the AUCs and the self-detection
power of models increase rather consistently. When we go to $1\%$, however, we
see the AUC drop. If we look at Figure 11(e) which depicts the curvature
measures for the $1\%$ masking, we see that the curvatures overlap between
machine-generated and human-written text, which we hypothesize is because the
perturbations are all too close to the original sequences, and as such do not
define the neighborhood well.

Its noteworthy that the slight discrepancy between the results for $15\%$
masking in this section and the previous section is that there, the mask span
length was $2$ (as it is the optimal span length found by us and Mitchell et
al.), so the masked portion of the sequence is not contiguous. In this
experiment, however, to have better control, we set the mask span length to
the maximum possible (full sequence length), so we get contiguous masking.

##  7 Related Work

The problem of machine-generated text detection has already been studied for
multiple years using a variety of different approaches (Ippolito et al., 2020;
Jawahar et al., 2020; Uchendu et al., 2020, 2021): Both Gehrmann et al. (2019)
and Dugan et al. (2022) have found that humans generally struggle to
distinguish between human- and machine-generated text, hereby motivating the
development of automatic solutions. Among those, some methods aim to detect
machine-generated text by training a classifier in a supervised manner
(Bakhtin et al., 2019; Uchendu et al., 2020), while others perform detection
in a zero-shot manner (Solaiman et al., 2019; Ippolito et al., 2020). There is
also a line of work that relies on bot detection through question answering
Wang et al. (2023); Chew and Baird (2003), which is outside the scope of this
paper.

Most recently, Mitchell et al. (2023) introduced the zero-shot method
DetectGPT, which is based on the hypothesis that texts generated from a LLM
lie on local maxima, and therefore negative curvature, of the model’s
probability distribution. Thus, minor rewrites of machine-generated texts,
which are in practice obtained through word replacements suggested by a
separate model such as T5 (Raffel et al., 2020), are consistently assigned
lower probabilities than the original text, whereas rewrites of human-written
texts can have both higher or lower probabilities assigned to them.

Beyond the approaches discussed in this paper, other strategies have been
proposed to enable the detection of machine-generated text in the wild.
Particularly through efforts on the side of the LLM provider, more powerful
detection methods can be devised. One such method is watermarking, which
injects algorithmically detectable patterns into the released text while
ideally preserving the quality and diversity of language model outputs.
Watermarks for natural language have already been proposed by Atallah et al.
(2001) and have since been adapted for outputs of neural language models (Fang
et al., 2017; Ziegler et al., 2019). Notable recent attempts for transformer
based language models include work by Abdelnabi and Fritz (2021), who propose
an adversarial watermarking transformer (AWT). While this watermarking method
is dependent on the model architecture, Kirchenbauer et al. (2023) propose a
watermark that can be applied to texts generated by any common autoregressive
language model. As a strategy more reliable than watermarking, Krishna et al.
(2023) suggest a retrieval-based approach: By storing all model outputs in a
database, LLM providers can verify whether a given text was previously
generated by their language model. In practice, this would however require
storage of large amounts of data and highly efficient retrieval techniques in
order to provide fast responses as the number of generated texts grows.

#### Evasion of Detectors

As detecting machine-generated text is becoming a topic of high interest,
researchers are also aiming to study the limits of machine-generated text
detectors. The broad literature of text-based adversarial attacks demonstrates
that text classifiers such as e-mail spam filters, and therefore most likely
also machine-generated text detectors, can be fooled using minor perturbations
that largely preserve fluency and semantics of the original texts (Alzantot et
al., 2018; Jin et al., 2020; Li et al., 2020, 2021). Recent work has also
studied attacks specifically designed to fool machine-generated text detectors
(Sadasivan et al., 2023) and found that classifiers can be evaded through
simple paraphrases and many watermarking techniques can be recreated by
humans. This, along with the outlook that language models will most likely
become more powerful and human-like, raises the question if it will ever be
possible to detect machine-generated text reliably.

##  8 Conclusion

As LLMs are becoming more ubiquitous and embedded in different user-facing
services, it is important to be able to distinguish between human written text
and machine-generated text, so as to be able to verify the authenticity of
news articles, product reviews, etc. As such, we set out to explore the
possibilities of using existing models to detect generations from unknown
sources, and distinguish them from human written text. We find that when using
zero-shot detection methods that rely on local optimality, smaller models are
overall better at detecting generations, and larger models are poor detectors.
We hypothesize that this has to do with the shape of the loss function for
these different types of models, and how well they fit their training data.
However, further analysis of the loss landscape is needed to fully verify this
claim.

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# Expected sensitivity on the anomalous quartic neutral gauge couplings in
$\gamma\gamma$ collisions at the CLIC

E<EMAIL_ADDRESS>Department of Physics, Sinop University,
Turkey.
M<EMAIL_ADDRESS>Department of Physics, Sivas Cumhuriyet
University, 58140, Sivas, Turkey. A. Gutiérrez-
<EMAIL_ADDRESS>Facultad de Física, Universidad Autónoma
de Zacatecas
Apartado Postal C-580, 98060 Zacatecas, México.
M. A<EMAIL_ADDRESS>Unidad Académica de Ciencias
Químicas, Universidad Autónoma de Zacatecas
Apartado Postal C-585, 98060 Zacatecas, México.
V<EMAIL_ADDRESS>Department of Physics, Ankara University,
Turkey.

###### Abstract

The presence of multi-boson self-interactions is implied by the non-Abelian
gauge structure of the Standard Model (SM). Precise measurements of these
interactions allow not only testing the nature of the SM but also new physics
contribution arising from the beyond SM. The investigation of these
interactions can be approached in a model-independent manner using an
effective theory approach, which forms the main motivation of this study. In
this paper, we examine the anomalous neutral quartic gauge couplings through
the process $\gamma\gamma\rightarrow ZZ$ at the Compact Linear Collider (CLIC)
with the center-of-mass energy of $\sqrt{s}=3$ TeV, integrated luminosities of
${\cal L}=5$ $\rm ab^{-1}$. The anomalous neutral quartic gauge couplings is
implemented into FeynRules to generate a UFO module inserted into Madgraph to
generate both background and signal events. These events are then passed
through Pythia 8 for parton showering and Delphes to include realistic
detector effects. We obtain that the sensitivities on the anomalous quartic
neutral gauge couplings with $95\%$ Confidence Level are given as:
$f_{T0}/\Lambda^{4}=[-1.06;1.08]\times 10^{-3}$ ${\rm TeV^{-4}}$,
$f_{T1}/\Lambda^{4}=[-1.06;1.08]\times 10^{-3}$ ${\rm
TeV^{-4}}$,$f_{T2}/\Lambda^{4}=[-1.06;1.08]\times 10^{-3}$ ${\rm
TeV^{-4}}$,$f_{T0}/\Lambda^{4}=[-1.06;1.08]\times 10^{-3}$ ${\rm TeV^{-4}}$,
$f_{T5}/\Lambda^{4}=[-4.08;4.08]\times 10^{-4}$ ${\rm TeV^{-4}}$ and
$f_{T8}/\Lambda^{4}=[-1.10;1.10]\times 10^{-4}$ ${\rm TeV^{-4}}$. Our results
on the anomalous quartic neutral gauge couplings are set more stringent
sensitivity with respect to the recent experimental limits.

###### pacs:

12.60.-i, 14.70.Hp, 14.70.Bh
Keywords: Electroweak interaction, Models beyond the Standard Model, Anomalous
couplings.

##  I Introduction

In order to address deficiencies in the Standard Model (SM), such as the
asymmetry between matter and antimatter, the strong CP problem, the origin of
mass, and the nature of dark energy and dark matter, the new physics beyond
the Standard Model (BSM) is needed. This exploration involves examining BSM
physics models through various processes at existing and upcoming colliders.
One effective approach to search for BSM physics involves investigating
anomalous gauge boson couplings, including triple gauge couplings (TGC) and
quartic gauge couplings (QGC), which are described by the electroweak
$SU(2)_{L}\times U(1)_{Y}$ gauge symmetry of the SM. The measurements of the
gauge bosons self-couplings, triple $VVV$ and quartic $VVVV$ with
$V=\gamma,Z,W^{\pm}$, provide a fundamental test of the non-Abelian structure
of the electroweak theory of the SM and can offer essential information about
new physics effects at high energies. Additionally, knowledge of the
electroweak sector of the SM can only be improved through unique direct
measurements of the TGC and the QGC. However, the QGC predicted by the SM are
$(W^{+}W^{-}W^{+}W^{-},W^{+}W^{-}ZZ,W^{+}W^{-}\gamma\gamma,W^{+}W^{-}Z\gamma)$,
while quartic neutral gauge couplings (QNGC)
$(ZZZZ,ZZZ\gamma,ZZ\gamma\gamma,Z\gamma\gamma\gamma,\gamma\gamma\gamma\gamma)$
are absent. These couplings can contribute to the production of multiple
bosons in colliders, and their precise measurements can either confirm the
validity of the SM or lead to the discovery of BSM physics.

By incorporating high-dimensional operators, it is possible to extend gauge
boson couplings, thereby affecting the anomalous triple gauge couplings (ATGC)
and the anomalous quartic gauge couplings (AQGC). Gauge boson operators have
been defined using either linear or non-linear effective Lagrangians. In the
nonlinear approach, the chiral Lagrangian parameterization is employed to
conserve and realize the SM gauge symmetry 1 ; 2 . Within this framework, the
ATGC and the AQGC arise from dimension-6 operators. The utilization of
dimension-6 operators is particularly convenient for analyzing the AQGC and
comparing the results obtained from experiments at the LEP 3 . Additionally,
dimension-8 operators are derived using a linear representation of the
electroweak gauge symmetry, which is broken by the conventional SM Higgs
mechanism. However, with the recent discovery of a new particle consistent
with the SM Higgs boson, investigating the AQGC based on the linear effective
Lagrangian has gained significant importance.

The CLIC is a collider that offers advantages such as high energy, high
luminosity, compact design, and a clean environment Charles ; Aicheler ; Blas
; Roloff ; CLIC-1802-2018 ; Robson ; franc . Its high energy enables the
exploration of new particles and allows for precise measurements. High
luminosity increases the production rate of rare events and enhances
sensitivity to new physics processes. The compact design makes implementation
easier and reduces construction costs. The clean environment, resulting from
$e^{-}e^{+}$ collisions, enables more precise measurements and better
identification of rare processes. Additionally, the CLIC complements the
discoveries made at the LHC and the HL-LHC, contributing to a better
understanding of the fundamental nature of the universe. The future design of
the linear collider is expected to include operation in $e\gamma$ and
$\gamma\gamma$ modes. In these modes, real photon beams can be generated by
converting the incoming $e^{-}$ and $e^{+}$ beams into photon beams using the
Compton backscattering mechanism. The maximum collision energy is anticipated
to be $90\%$ for $e\gamma$ collisions and $80\%$ for $\gamma\gamma$ collisions
relative to the original $e^{-}e^{+}$ collision energy. However, the projected
luminosities for $e\gamma$ collisions are $39\%$, while for $\gamma\gamma$
collisions, they are $15\%$ compared to the $e^{-}e^{+}$ luminosities tel . In
this study, our objective is to investigate the sensitivity of the AQGC
through the process $\gamma\gamma\rightarrow ZZ$ at the CLIC. The research
focuses on a comprehensive physics case for the CLIC with center-of-mass
energy of 3 TeV.

This paper is structured as follows: Section II provides an overview of the
dimension-8 operators in the effective field theory. Section III focuses on
the process $\gamma\gamma\to ZZ$, and discusses its relevance in the context
of the AQGC, referencing recent experimental results. Finally, Section IV
presents our conclusions.

##  II Effective field theory approach BSM for the AQGC

The Effective Field Theory (EFT) approach, described with the SM extended by
dimension-8 operators, provides a perfect tool. There are two methods to
search for physics BSM. One is to search for new particles. The other is to
search for new interactions between known particles. This last method is the
one we use to study AQGC. The EFT approach is a simple and elegant way to
treat new interactions in BSM. The effective Lagrangian with dimension-8
operators used to investigate the AQGC is given as follows NPB989-2023 ;
JPG49-2022 ; EPJC81-2021 ; JPG47-2020 ; EPJP135-2020 ,

${\cal L}_{eff}={\cal
L}_{SM}+\sum_{k=0}^{1}\frac{f_{S,k}}{\Lambda^{4}}O_{S,k}+\sum_{i=0}^{7}\frac{f_{M,i}}{\Lambda^{4}}O_{M,i}+\sum_{j=0,1,2,5,6,7,8,9}\frac{f_{T,j}}{\Lambda^{4}}O_{T,j},$
(1)

where ${\cal L}_{SM}$ denotes the renormalizable SM Lagrange and each
$O_{S,k}$, $O_{M,i}$ and $O_{T,j}$ is a gauge-invariant operator of
dimension-8 constructed from SM fields, and $\frac{f_{S,k}}{\Lambda^{4}}$,
$\frac{f_{M,i}}{\Lambda^{4}}$ and $\frac{f_{T,j}}{\Lambda^{4}}$ are the
corresponding effective Wilson coefficients. According to the structure of the
operator, there are three classes of genuine AQGC operators Degrande ; Eboli3
: independent scalar operators (S-type operators), independent mixed operators
(M-type operators), and independent transverse operators (T-type operators).

$\bullet$ Contribution of $O_{S,k}$-type operators:

$\displaystyle O_{S,0}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}(D_{\nu}\Phi)]\times[(D^{\mu}\Phi)^{\dagger}(D^{\nu}\Phi)],$
(2) $\displaystyle O_{S,1}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}(D^{\mu}\Phi)]\times[(D_{\nu}\Phi)^{\dagger}(D^{\nu}\Phi)],$
(3) $\displaystyle O_{S,2}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}(D_{\nu}\Phi)]\times[(D^{\nu}\Phi)^{\dagger}(D^{\mu}\Phi)].$
(4)

$\bullet$ Contribution of $O_{M,k}$-type operators:

$\displaystyle O_{M,0}$ $\displaystyle=$ $\displaystyle
Tr[W_{\mu\nu}W^{\mu\nu}]\times[(D_{\beta}\Phi)^{\dagger}(D^{\beta}\Phi)],$ (5)
$\displaystyle O_{M,1}$ $\displaystyle=$ $\displaystyle
Tr[W_{\mu\nu}W^{\nu\beta}]\times[(D_{\beta}\Phi)^{\dagger}(D^{\mu}\Phi)],$ (6)
$\displaystyle O_{M,2}$ $\displaystyle=$
$\displaystyle[B_{\mu\nu}B^{\mu\nu}]\times[(D_{\beta}\Phi)^{\dagger}(D^{\beta}\Phi)],$
(7) $\displaystyle O_{M,3}$ $\displaystyle=$
$\displaystyle[B_{\mu\nu}B^{\nu\beta}]\times[(D_{\beta}\Phi)^{\dagger}(D^{\mu}\Phi)],$
(8) $\displaystyle O_{M,4}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}W_{\beta\nu}(D^{\mu}\Phi)]\times
B^{\beta\nu},$ (9) $\displaystyle O_{M,5}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}W_{\beta\nu}(D^{\nu}\Phi)]\times
B^{\beta\mu}+h.c.,$ (10) $\displaystyle O_{M,7}$ $\displaystyle=$
$\displaystyle[(D_{\mu}\Phi)^{\dagger}W_{\beta\nu}W^{\beta\mu}(D^{\nu}\Phi)].$
(11)

$\bullet$ Contribution of $O_{T,k}$-type operators:

$\displaystyle O_{T,0}$ $\displaystyle=$ $\displaystyle
Tr[W_{\mu\nu}W^{\mu\nu}]\times Tr[W_{\alpha\beta}W^{\alpha\beta}],$ (12)
$\displaystyle O_{T,1}$ $\displaystyle=$ $\displaystyle
Tr[W_{\alpha\nu}W^{\mu\beta}]\times Tr[W_{\mu\beta}W^{\alpha\nu}],$ (13)
$\displaystyle O_{T,2}$ $\displaystyle=$ $\displaystyle
Tr[W_{\alpha\mu}W^{\mu\beta}]\times Tr[W_{\beta\nu}W^{\nu\alpha}],$ (14)
$\displaystyle O_{T,5}$ $\displaystyle=$ $\displaystyle
Tr[W_{\mu\nu}W^{\mu\nu}]\times B_{\alpha\beta}B^{\alpha\beta},$ (15)
$\displaystyle O_{T,6}$ $\displaystyle=$ $\displaystyle
Tr[W_{\alpha\nu}W^{\mu\beta}]\times B_{\mu\beta}B^{\alpha\nu},$ (16)
$\displaystyle O_{T,7}$ $\displaystyle=$ $\displaystyle
Tr[W_{\alpha\mu}W^{\mu\beta}]\times B_{\beta\nu}B^{\nu\alpha},$ (17)
$\displaystyle O_{T,8}$ $\displaystyle=$ $\displaystyle
B_{\mu\nu}B^{\mu\nu}B_{\alpha\beta}B^{\alpha\beta},$ (18) $\displaystyle
O_{T,9}$ $\displaystyle=$ $\displaystyle
B_{\alpha\mu}B^{\mu\beta}B_{\beta\nu}B^{\nu\alpha}.$ (19)

In Eqs. (2)-(19), the different operators are classified regarding Higgs
versus gauge boson field content. In this set of genuine AQGC operators given
by Eqs. (2)-(19), $\Phi$ stands for the Higgs doublet, and the covariant
derivatives of the Higgs field are given by
$D_{\mu}\Phi=(\partial_{\mu}+igW^{j}_{\mu}\frac{\sigma^{j}}{2}+\frac{i}{2}g^{\prime}B_{\mu})\Phi$,
and $\sigma^{j}(j=1,2,3)$ represent the Pauli matrices, while $W^{\mu\nu}$ and
$B^{\mu\nu}$ are the gauge field strength tensors for $SU(2)_{L}$ and
$U(1)_{Y}$.

In our paper, we examine the sensitivities on the anomalous
$f_{T,i}/\Lambda^{4}$ ($i=0,1,2,5,6,7,8,9$) couplings defined by dimension-8
operators related to the AQGC via the process $\gamma\gamma\to ZZ$ at the
CLIC. The anomalous quartic gauge boson coupling relevant to the process
$\gamma\gamma\to ZZ$ is $ZZ\gamma\gamma$. Operators for these coupling are
given by

$\displaystyle V_{ZZ\gamma\gamma,1}=F^{\mu\nu}F_{\mu\nu}Z^{\alpha}Z_{\alpha},$
(20) $\displaystyle
V_{ZZ\gamma\gamma,2}=F^{\mu\nu}F_{\mu\alpha}Z_{\nu}Z^{\alpha},$ (21)
$\displaystyle
V_{ZZ\gamma\gamma,3}=F^{\mu\nu}F_{\mu\nu}Z^{\alpha\beta}Z_{\alpha\beta},$ (22)
$\displaystyle
V_{ZZ\gamma\gamma,4}=F^{\mu\nu}F_{\nu\alpha}Z^{\nu\alpha}Z_{\mu\nu},$ (23)
$\displaystyle
V_{ZZ\gamma\gamma,5}=F^{\mu\nu}F_{\alpha\beta}Z_{\mu\nu}Z^{\alpha\beta},$ (24)
$\displaystyle
V_{ZZ\gamma\gamma,6}=F^{\mu\beta}F_{\alpha\nu}Z^{\mu\beta}Z_{\alpha\nu}.$ (25)

Here, $Z^{\mu\nu}=\partial^{\mu}Z^{\nu}-\partial^{\nu}Z^{\mu}$. The
corresponding coefficients of vertices are represented as follows

$\displaystyle\alpha_{ZZ\gamma\gamma,1}=\frac{e^{2}\bar{\nu}}{16\Lambda^{4}}\frac{1}{c_{W}s_{W}}(\frac{s_{W}}{c_{W}}f_{M,0}+2\frac{c_{W}}{s_{W}}f_{M,2}-f_{M,4}),$
(26)

$\displaystyle\alpha_{ZZ\gamma\gamma,2}=\frac{e^{2}\bar{\nu}}{16\Lambda^{4}}\frac{1}{c_{W}s_{W}}(\frac{s_{W}}{2c_{W}}f_{M,7}-\frac{s_{W}}{c_{W}}f_{M,1}-2\frac{c_{W}}{s_{W}}f_{M,3}-2f_{M,5}),$
(27)

$\displaystyle\alpha_{ZZ\gamma\gamma,3}=\frac{c_{W}^{2}s_{W}^{2}}{2\Lambda^{4}}(f_{T,0}+f_{T,1}-2f_{T,6}+4f_{T,8})+\frac{c_{W}^{2}+s_{W}^{2}}{\Lambda^{4}}f_{T,5},$
(28)

$\displaystyle\alpha_{ZZ\gamma\gamma,4}=\frac{c_{W}^{2}s_{W}^{2}}{\Lambda^{4}}(f_{T,2}+4f_{T,9})+\frac{(c_{W}^{2}-s_{W}^{2})^{2}}{2\Lambda^{4}}f_{T,7},$
(29)

$\displaystyle\alpha_{ZZ\gamma\gamma,5}=\frac{c_{W}^{2}s_{W}^{2}}{\Lambda^{4}}(f_{T,0}+f_{T,1}-2f_{T,5}+4f_{T,8})+\frac{(c_{W}^{2}-s_{W}^{2})^{2}}{2\Lambda^{4}}f_{T,6},$
(30)

$\displaystyle\alpha_{ZZ\gamma\gamma,6}=\frac{c_{W}^{2}s_{W}^{2}}{2\Lambda^{4}}(f_{T,2}-2f_{T,7}+4f_{T,9}).$
(31)

The formalism of the AQGC with dimension-8 operators has been widely discussed
in the literature with experimental and phenomenological studies t1 ; t2 ; t4
; t5 ; t6 ; t66 ; t8 ; t9 ; t10 ; t11 ; t12 ; t13 ; t14 ; t15 ; t16 ; t17 ;
t19 ; a1 ; a2 ; a3 ; a33 ; a4 ; a5 ; a6 ; a7 ; a8 ; a9 ; a10 ; a11 ; a12 ; a13
; a133 ; a14 ; a15 ; a16 ; a17 ; a18 ; a19 ; a20 ; a21 ; a22 ; a23 . However,
the best experimental limits obtained at 95$\%$ Confidence Level (C.L.) on
$f_{T,i}/\Lambda^{4}$ ($i=0,2,5,6,7,8,9$) couplings are examined through the
process $pp\to Z(\to\nu\bar{\nu})\gamma jj\to\nu\bar{\nu}\gamma jj$ at the LHC
with $\sqrt{s}=13$ TeV with $L_{int}=139$ fb-1 and the process $pp\to Z(\to
ll)\gamma jj\to ll\gamma jj$ at the LHC with $\sqrt{s}=13$ TeV with
$L_{int}=35.9$ fb-1den ; den1 . These are given as follows

$\displaystyle-9.4<f_{T,0}/\Lambda^{4}<8.4\,(\times
10^{-2}\,\text{TeV}^{-4}),$ (32)

$\displaystyle-1.2<f_{T,1}/\Lambda^{4}<1.3\,(\times
10^{-1}\,\text{TeV}^{-4}),$ (33)

$\displaystyle-2.8<f_{T,2}/\Lambda^{4}<2.8\,(\times
10^{-1}\,\text{TeV}^{-4}),$ (34)
$\displaystyle-8.8<f_{T,5}/\Lambda^{4}<9.9\,(\times
10^{-2}\,\text{TeV}^{-4}),$ (35)
$\displaystyle-4<f_{T,6}/\Lambda^{4}<4\,(\times 10^{-1}\,\text{TeV}^{-4}),$
(36) $\displaystyle-9<f_{T,7}/\Lambda^{4}<9\,(\times
10^{-1}\,\text{TeV}^{-4}),$ (37)
$\displaystyle-5.9<f_{T,8}/\Lambda^{4}<5.9\,(\times
10^{-2}\,\text{TeV}^{-4}),$ (38)
$\displaystyle-1.3<f_{T,9}/\Lambda^{4}<1.3\,(\times
10^{-1}\,\text{TeV}^{-4}).$ (39)

Table I presents a list of all the genuine AQGC modified by dimension-8
operators. In particular, the $\gamma\gamma\to ZZ$ channel is especially
sensitive to the $f_{T,i}/\Lambda^{4}$, with $i=0,1,2,5,6,7,8,9$. For this
reason, we consider only these operators in our study. In addition, the limits
obtained for the $f_{T,8}/\Lambda^{4}$ and $f_{T,9}/\Lambda^{4}$ coupling
parameters are of interest because they can be extracted only by studying the
production of electroweak neutral bosons. Therefore, we focus on the
$f_{T,i}/\Lambda^{4}$ parameters to examine AQGC couplings through the process
$\gamma\gamma\to ZZ$ at the CLIC.

Table 1: The AQGC altered with dimension-8 operators are shown with X. Gauge-invariant operator | $WWWW$ | $WWZZ$ | $ZZZZ$ | $WW\gamma Z$ | $WW\gamma\gamma$ | $ZZZ\gamma$ | $ZZ\gamma\gamma$ | $Z\gamma\gamma\gamma$ | $\gamma\gamma\gamma\gamma$
---|---|---|---|---|---|---|---|---|---
$O_{S0}$, $O_{S1}$ | X | X | X |  |  |  |  |  |
$O_{M0}$, $O_{M1}$, $O_{M7}$ | X | X | X | X | X | X | X |  |
$O_{M2}$, $O_{M3}$, $O_{M4}$, $O_{M5}$ |  | X | X | X | X | X | X |  |
$O_{T0}$, $O_{T1}$, $O_{T2}$ | X | X | X | X | X | X | X | X | X
$O_{T5}$, $O_{T6}$, $O_{T7}$ |  | X | X | X | X | X | X | X | X
$O_{T8}$, $O_{T9}$ |  |  | X |  |  | X | X | X | X

##  III The photon-induced production process $\gamma\gamma\to ZZ$

A feature of present and future $e^{+}e^{-}$ colliders is the possibility to
transform it to a photon-collider $\gamma\gamma$ through the process of
Compton backscattering of laser light off the high energy electrons Eboli-1993
; Kingman-1993 ; Moortgat ; Ginzburg ; Telnov . For the evaluation of the
total cross-section of the $\gamma\gamma\to ZZ$ signal, we define $m_{e}$ and
$E_{e}$, as the incident electron beam mass and energy; $E_{0}$ and
$E_{\gamma}$ as the laser photon and the backscattered photon energies;
$\sqrt{s}$ as the center-of-mass energy of $e^{+}e^{-}$ collisions;
$\sqrt{\hat{s}}$ as the center-of-mass energy of the backscattered photon; and
$m_{Z}$ as the mass of the produced $Z$ gauge boson. Therefore, the
$\sigma(s)$ cross-section for the pair production of $Z$ gauge bosons at the
photon collider is obtained by convoluting the subprocess cross-section
$\sigma(\hat{s})$ with the photon luminosity at an $e^{+}e^{-}$ linear
collider as follows

$\sigma(s)=\int^{y_{max}}_{2m_{Z}}dz\frac{dL_{\gamma\gamma}}{dz}\sigma(\hat{s}),\hskip
28.45274pt\hat{s}=z^{2}s,$ (40)

where $y_{max}=E_{\gamma}/E_{e}$, and $\frac{dL_{\gamma\gamma}}{dz}$ is the
photon luminosity, which is defined as:

$\frac{dL_{\gamma\gamma}}{dz}=2s\int^{y_{max}}_{z^{2}/y_{max}}\frac{dy}{y}f_{\gamma}(y)f_{\gamma}(z^{2}/y).$
(41)

In Eq. (40), $f_{\gamma}(y)$ correspond to the energy spectrum of Compton
backscattered photons:

$\displaystyle
f_{\gamma}(y)=\frac{1}{g(\zeta)}\Bigl{[}1-y+\frac{1}{1-y}-\frac{4y}{\zeta(1-y)}+\frac{4y^{2}}{\zeta^{2}(1-y)^{2}}\Bigr{]},$
(42)

with

$\displaystyle
g(\zeta)=\Bigl{(}1-\frac{4}{\zeta}-\frac{8}{\zeta^{2}}\Bigr{)}\log{(\zeta+1)}+\frac{1}{2}+\frac{8}{\zeta}-\frac{1}{2(\zeta+1)^{2}},$
(43)

and

$\displaystyle
y=\frac{E_{\gamma}}{E_{e}},\;\;\;\;\zeta=\frac{4E_{0}E_{e}}{M_{e}^{2}},\;\;\;\;y_{max}=\frac{\zeta}{1+\zeta},$
(44)

where $y$ is the energy fraction transferred from the electron to the photon,
and the maximum value of the $y_{max}=\frac{\zeta}{1+\zeta}$ reaches 0.83 when
$\zeta=4.8$, and is when the photon conversion efficiency drops drastically
due to the $e^{+}e^{-}$ pair production from the laser photons and the photon
backscattering.

###  III.1 Generation of Signal and background events

For the computation of the total cross-section and backgrounds, we have
implemented the interaction terms in the MadGraph5_aMC@NLO MadGraph  through
Feynrules package AAlloul  as a Universal FeynRules Output (UFO) module
CDegrande . However, Monte Carlo event samples for both the signal and
relevant background processes are generated using MadGraph5_aMC@NLO, and then
passed through Pythia 8.2 pyt  to include the initial and final parton
showering and fragmentation. The detector response is simulated using Delphes
3.4.2 del , utilizing the CLIC configuration cards.

The tree-level Feynman diagram for the process $\gamma\gamma\to ZZ$ is given
in Fig. 1. $\sigma_{\gamma\gamma\to ZZ}$ is the contribution due to BSM
physics, which comes from the AQGC $ZZ\gamma\gamma$. In this article, the main
background sources arising from the SM are $\gamma\gamma\to
l^{+}l^{-}\nu\bar{\nu}$ and $\gamma\gamma\to WW\to l\nu_{l}l\nu_{l}$ which are
similar as the signal. To analyze the process $\gamma\gamma\to ZZ$, we apply a
pre-selection criterion starting from the cuts selected: at least a pair of
leptons of the same flavor with opposite charge $(N_{l}(e,\mu)>=2)$ must be
present in the event, and the transverse momentum and pseudo-rapidity of the
leading and sub-leading charged leptons must be $p_{T}^{\ell^{1}}>10$ GeV,
$p_{T}^{\ell^{2}}>10$ GeV and $|\eta^{\ell^{1}}|<2.5$,
$|\eta^{\ell^{2}}|<2.5$, respectively. The criteria include the presence of a
dilepton of the same flavor and opposite charge to reconstruct the $Z$ boson.
In addition, in Fig. 2, we observe that the event selection is optimized by
imposing the distribution of transverse momentum balance ratio
$|E^{miss}_{T}-p^{ll}_{T}|/p^{ll}_{T}<0.7$ for signal $\gamma\gamma\to ZZ$
with some couplings ($f_{T,0,2,5,8}/\Lambda^{4}=3$ ${\rm TeV^{-4}}$) and
relevant background processes. This is because the
$|E^{miss}_{T}-p^{ll}_{T}|/p^{ll}_{T}$ distribution for the signal deviates
significantly from the corresponding backgrounds for values less than 0.7.
Another sensitive and important kinematic variable for our study is the
transverse component $a^{ll}_{T}$. This variable is equal to $a^{ll}_{T}=|{\bf
p}^{ll}_{T}\times\hat{t}|$ and correspond to one of two orthogonal components
of $p^{ll}_{T}$ relative to the dilepton thrust axis $\hat{t}$ , where
$p^{ll}_{T}={\bf p}^{l_{1}}_{T}+{\bf p}^{l_{2}}_{T}$ and $\hat{t}=({\bf
p}^{l_{1}}_{T}-{\bf p}^{l_{2}}_{T})/|{\bf p}^{l_{1}}_{T}-{\bf p}^{l_{2}}_{T}|$
NIMA602-2009 ; PRD91-2015 . Therefore, we must take into account that in the
plane transverse to the beam direction, the di-lepton thrust axis is indicated
with $\hat{t}=({\bf p}^{l_{1}}_{T}-{\bf p}^{l_{2}}_{T})/|{\bf
p}^{l_{1}}_{T}-{\bf p}^{l_{2}}_{T}|$. In this case, Fig. 3 shows that by
applying $a^{ll}_{T}>50$ GeV, the $\gamma\gamma\to ZZ$ signal can be separated
from the relevant backgrounds. As is given in Fig. 4, choosing
$E^{miss}_{T}>150$ GeV provides a well seperation region between the
$\gamma\gamma\to ZZ$ signal and $\gamma\gamma\to WW\to
l^{-}\nu_{l}l^{+}\bar{\nu}_{l}$ background with $f_{T,0,2,5,8}/\Lambda^{4}=3$
${\rm TeV^{-4}}$. It is possible to suppress related backgrounds from the
$\gamma\gamma\to ZZ$ signal by limiting the distance $\Delta R(l_{1},l_{2})<3$
between leading and sub-leading charged leptons in the $(\eta-\phi)$ plane in
Fig. 5. Finally, the cuts used in our analysis are summarized in Table II.

Table 2: List of selected cuts for the analysis of the process $\gamma\gamma\to ZZ$. |  Definitions
---|---
|  Preselection: $N_{\ell\,(e,\,\mu)}>=2$ and a pair of leptons
|  Transverse momentum and pseudo-rapidity of the leading and sub-leading
charged leptons: $p_{T}^{\ell^{1}}>10$ GeV, $p_{T}^{\ell^{2}}>10$ GeV and
$|\eta^{\ell^{1}}|<2.5$, $|\eta^{\ell^{2}}|<2.5$
|  $p_{T}$ balance: $|E_{T}^{miss}-p_{T}^{\ell\ell}|/p_{T}^{\ell\ell}<0.7$
|  Transverse component of $p_{T}^{\ell\ell}$ with respect to the thrust axis:
$a_{T}^{\ell\ell}>50$ GeV
|  Missing energy transverse: $E_{T}^{miss}>150$ GeV
|  Minimum distance between leptons: $\Delta R({\ell_{1},\ell_{2}})<3.0$

###  III.2 Expected sensitivity on $f_{T,j}/\Lambda^{4}$ via $\gamma\gamma$
colliders

We use the $\chi^{2}$ distribution to study the expected sensitivity on
$f_{T,j}/\Lambda^{4}$ at $95\%$ C. L. through $\gamma\gamma$ colliders. This
distribution is defined as follows

$\displaystyle\chi^{2}=\sum_{i}^{n_{bins}}(\frac{N_{i}^{SIG}-N_{i}^{B}}{N_{i}^{B}\Delta_{i}})^{2}.$
(45)

Here, $N_{i}^{SIG}$ and $N_{i}^{B}$ are the total number of event of the
signal (only new physics contribution of the process $\gamma\gamma\to ZZ$) and
the total backgrounds, respectively. On the other hand,
$\Delta_{i}=\sqrt{1/N_{i}^{B}}$ is the statistical uncertainty for every
single bin. Also, we assume that one of the $Z$ bosons decays pair charged
leptons $(l^{+}l^{-};l=e,\mu)$ and the other decays to neutrino pair.

Fig. 6 shows the expected sensitivities at $95\%$ C.L. on the AQGC through the
process $\gamma\gamma\to ZZ$ at the center-of-mass energy of 3 TeV and
luminosity of $5$ ${\rm ab^{-1}}$ for the third stage of the CLIC and the
latest experimental results in the literature. As illustrated in this figure,
our best limits on the AQGC improve much better than the current experimental
limits by up to two orders of magnitude. Also, to better understand the
results, Table III represents the expected sensitivities for the Wilson
coefficients $f_{T,j}/\Lambda^{4}$, $j=0-2,5-9$. Here, any couplings are
calculated while fixing the other couplings to zero. Our best sensitivity
limits on the AQGC at the CLIC might reach up to the order of magnitude
$O(10^{-4}-10^{-2})$. As can be seen from Eqs. (28) and (30), the limits of
$f_{T,0}/\Lambda^{4}$ and $f_{T,1}/\Lambda^{4}$ should be the same. This can
be easily seen in Table III. Also, $f_{T,5}/\Lambda^{4}$,
$f_{T,8}/\Lambda^{4}$ and $f_{T,9}/\Lambda^{4}$ couplings have the best
sensitivities among the AQGC. These results are two orders of magnitude more
sensitive than the latest experimental results reported in the literature.

Table 3: Expected lower and upper sensitivity at $95\%$ C.L. on the AQGC through the process $\gamma\gamma\to ZZ$ at the CLIC with $\sqrt{s}=3$ TeV, ${\cal L}=5\hskip 2.84526pt\rm ab^{-1}$ are presented. Wilson coefficient (TeV-4) | ${\cal L}=5\hskip 2.27621pt{\rm ab^{-1}}$
---|---
$f_{T0}/\Lambda^{4}$ | $[-1.06;1.08]\times 10^{-3}$
$f_{T1}/\Lambda^{4}$ | $[-1.06;1.08]\times 10^{-3}$
$f_{T2}/\Lambda^{4}$ | $[-4.76;4.76]\times 10^{-3}$
$f_{T5}/\Lambda^{4}$ | $[-4.08;4.08]\times 10^{-4}$
$f_{T6}/\Lambda^{4}$ | $[-1.30;1.18]\times 10^{-3}$
$f_{T7}/\Lambda^{4}$ | $[-1.33;1.49]\times 10^{-2}$
$f_{T8}/\Lambda^{4}$ | $[-1.10;1.10]\times 10^{-4}$
$f_{T9}/\Lambda^{4}$ | $[-1.07;0.89]\times 10^{-3}$

##  IV RESULTS AND CONCLUSIONS

The study of AQGC, as defined by the $SU(2)_{L}\times U(1)_{Y}$ gauge symmetry
within the SM, can confirm the model and provide clues to the existence of BSM
at a higher energy scale, which can be parameterized with higher-order
operators in an EFT. Additionally, future lepton colliders with high energy
and luminosity, and detection expectations can open a wide window for new
physics research. In light of these motivations, we examine the sensitivity of
AQGC at CLIC options with $\sqrt{s}=3$ TeV with the corresponding integrated
luminosity of 5 $\rm ab^{-1}$ through phenomenological analysis of
$\gamma\gamma\to ZZ$ production process taking into account charged lepton
decay channel of $Z$ boson. In the analysis, some kinematic variables are used
like angular separation $\Delta R(l_{1},l_{2})$, missing energy transverse
$E^{miss}_{T}$, $p_{T}$ balance $|E^{miss}_{T}-p^{l}_{T}|/p^{ll}_{T}$ and
$a^{ll}_{T}$ which plays an essential role for distinguishing the signals from
the relevant backgrounds. Apart from the cut-based method, a detector
simulation has been performed in this study using the Delphes card. Therefore,
our results indicate that the expected sensitivity on the Wilson coefficients
$f_{T_{j}}/\Lambda^{4}$ at $95\%$ C.L. with $\sqrt{s}=3$ TeV, ${\cal L}=5$
$\rm ab^{-5}$ are two orders better than the latest experimental limits.
Finally, we understand that $\gamma\gamma$ collisions at the CLIC will have a
great potential to probe the AQGC.

Acknowledgements

A. G. R. and M. A. H. R. thank SNI and PROFEXCE (México). The numerical
calculations reported in this paper were fully performed at TUBITAK ULAKBIM,
High Performance and Grid Computing Center (TRUBA resources).

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Figure 1:  Feynman diagram for the signal process $\gamma\gamma\to ZZ$ induced
by the effective $ZZ\gamma\gamma$ vertex. New physics represented by a black
circle in the electroweak sector can modify the QGC.

Figure 2:  Distribution of transverse momentum balance ratio
$|E^{miss}_{T}-p^{ll}_{T}|/p^{ll}_{T}$ for signal $\gamma\gamma\to ZZ$ with
some couplings ( $f_{T,0,2,5,8}/\Lambda^{4}=3$ ${\rm TeV^{-4}}$) and relevant
background processes. The dashed lines show the signal predictions for
illustrative AQGC parameters.

Figure 3:  Transverse component $a^{ll}_{T}$ (GeV) according to the thrust
axis for signal $\gamma\gamma\to ZZ$ with some couplings
($f_{T,0,2,5,8}/\Lambda^{4}=3$ ${\rm TeV^{-4}}$) and relevant background
processes. The dashed lines show the signal predictions for illustrative AQGC
parameters.

Figure 4:  Distribution of missing energy transverse $E^{miss}_{T}$ (GeV) for
signal $\gamma\gamma\to ZZ$ with some couplings (
$f_{T,0,2,5,8}/\Lambda^{4}=3$ ${\rm TeV^{-4}}$) and relevant background
processes. The dashed lines show the signal predictions for illustrative AQGC
parameters.

Figure 5:  Distance $\Delta R(l_{1},l_{2})$ between leading and sub-leading
charged leptons for signal $\gamma\gamma\to ZZ$ with some couplings
($f_{T,0,2,5,8}/\Lambda^{4}=3$ ${\rm TeV^{-4}}$) and relevant background
processes. The dashed lines show the signal predictions for illustrative AQGC
parameters.

Figure 6:  Comparison of the current experimental limits and projected
sensitivity on $f_{T_{i}}/\Lambda^{4}$ couplings ($i=0,1,2,5,6,7,8,9$) at the
CLIC with $\sqrt{s}=3$ TeV.