11institutetext: School of Biomedical Engineering & Imaging Sciences, King’s
College London, UK 22institutetext: Oxford University Clinical Research Unit,
Ho Chi Minh City, Vietnam 33institutetext: Mahidol Oxford Research Unit,
Thailand

# B-line Detection in Lung Ultrasound Videos: Cartesian vs Polar
Representation

Hamideh Kerdegari 11 Phung Tran Huy Nhat 1122 Angela McBride 22 Luigi Pisani
33 Reza Razavi 11 Louise Thwaites 22 Sophie Yacoub 22 Alberto Gomez 11

###### Abstract

Lung ultrasound (LUS) imaging is becoming popular in the intensive care units
(ICU) for assessing lung abnormalities such as the appearance of B-line
artefacts as a result of severe dengue. These artefacts appear in the LUS
images and disappear quickly, making their manual detection very challenging.
They also extend radially following the propagation of the sound waves. As a
result, we hypothesize that a polar representation may be more adequate for
automatic image analysis of these images. This paper presents an attention-
based Convolutional+LSTM model to automatically detect B-lines in LUS videos,
comparing performance when image data is taken in Cartesian and polar
representations. Results indicate that the proposed framework with polar
representation achieves competitive performance compared to the Cartesian
representation for B-line classification and that attention mechanism can
provide better localization.

###### Keywords:

Lung ultrasound B-line classification Temporal model Cartesian representation
Polar representation

##  1 Introduction

Recently, lung ultrasound (LUS) imaging has increased in popularity for rapid
lung monitoring in patients in the intensive care units (ICU). Particularly
for dengue patients, LUS can capture image artefacts such as B-lines that
indicate a pulmonary abnormalities such as oedema and effusions [1]. B-lines
are bright lines extending from the surface of the lung distally following the
direction of propagation of the sound wave (shown in Figure 1). LUS imaging is
useful for assessing lung abnormalities though the presence of B-lines.
However, these lines become visible randomly during respiratory cycle in the
affected area only [2]; therefore, manually detecting these artefacts becomes
challenging for inexperienced sonographers, and particularly in low and middle
income countries with higher prevalence of these diseases where training
opportunities and experise are scarce.

Figure 1: Examples of LUS B-line frames. B-line artefacts are presented as
bright lines that develop from the surface of the lung.

In order to provide an automatic solution to the LUS B-line detection problem,
recent studies proposed classification, segmentation and localization of
B-line artefacts in individual LUS frames. For example, a convolutional neural
network (CNN) followed by a class activation map was proposed by Sloun et al.
[3] to classify B-lines and produce a segmentation map of them, respectively.
A weakly supervised localization of B-lines was proposed using a spatial
transformer network [4]. In another study [5], a single-shot CNN was used to
localize B-lines with bounding boxes. Previous work by Kerdegari et al. [6]
showed that employing temporal information can improve B-line detection task
in LUS, leveraging a temporal attention mechanism to localize B-line frames
within LUS videos.

Furthermore, attention mechanisms have been used widely for spatial
localization of lung lesions particularly in CT and x-ray lung images. For
instance, a residual attention U-Net for multi-class segmentation of CT images
[7] and x-ray images [8] was proposed. A lesion-attention deep neural network
(LA-DNN) was presented by [9] to do two tasks of B-line classification and
multi-label attention localization of five lesions. All these studies employed
spatial attention mechanism for lung lesion localization.

LUS images are usually used in a standard Cartesian coordinate representation
(i.e., scan-converted). In this representation, the B-lines commonly appear
densely in the middle of frustum. Therefore, data preprocessing techniques
such as downsampling might cause information loss with Cartesian
representation. Additionally, the radial direction that B-lines follow is
known but this information is not exploited. In this paper, we propose to use
a polar representation to, first, reduce information loss when downsampling
the data, and second, leverage prior knowledge about line formation by having
one dimension aligned with the lines.

To this end, we compare the performance of the temporal attention-based
convolutional+LSTM model proposed by [6] when using Cartesian and polar
representations. In summary, the contribution of this paper is investigating
the effect of using LUS polar coordinate representation on the B-line
detection and localization performance. Also, we evaluate the effect of
different downsampling factors of LUS video with polar and Cartesian
representations for B-line detection and localization tasks.

##  2 Model Architecture

This paper employs a model that combines a deep visual feature extractor such
as a CNN with a long short-term memory (LSTM) network that can learn to
recognize temporal dynamics of videos; and a temporal attention mechanism to
learn where to pay more attention in the video. Figure 2 shows the core of our
model. This model works by passing each frame from the video through our CNN
model (The architecture details are explained in Figure 2, right) to produce a
fixed length feature vector representation. The outputs of our CNN are passed
into a bidirectional LSTM (16 hidden units, tanh activation function) network
as a recurrent sequence learning model. Then, the LSTM outputs are passed to
the attention network [10] to produce an attention score ($e_{t}$) for each
attended frame ($h_{t}$): $e_{t}=h_{t}w_{a}$, where $w_{a}$ represents
attention layer weight matrix. From $e_{t}$, an importance attention weight
($a_{t}$) is computed for each attended frame:
$a_{t}=\frac{\exp(e_{t})}{\sum_{i=1}^{T}\exp(e_{i})}$. To learn which frame of
the video to pay attention to, $a_{t}$s are multiplied with the LSTM output.
Finally, the output of LUS video classification is generated by averaging the
attention weighted temporal feature vector over time and passing to a fully
connected layer for video classification.

Figure 2: Overview of the proposed framework, Left: LUS videos are first
processed through CNN layers, then a bidirectional LSTM is used to extract
temporal information and finally an attention mechanism is applied to localize
B-line frames within LUS videos. Right: Detailed architecture of our CNN.

##  3 Experimental Setup

###  3.1 Dataset and Preprocessing

The dataset used in the experiments was collected at the Hospital of Tropical
Diseases in Ho Chi Minh City, Vietnam. It includes about 5 hours of lung
ultrasound videos collected from 60 dengue patients. These videos were
collected using a Sonosite M-Turbo machine (Fujifilm Sonosite, Inc., Bothell,
WA) with a low-medium frequency (3.5-5 MHz) convex probe. The Kigali ARDS
protocol [11], as a standardised operating procedure was applied at 6 points
(2 anterior, 2 lateral and 2 posterolateral) on each side of the chest to
perform LUS exams.

The four-second LUS video clips have been resized from original size of
$640\times 480$ pixels to $64\times 64$ pixels for training, and fully
anonymised through masking. A qualified sonographer annotated these clips
using the VGG annotator tool [12]. During the annotation procedure, each video
clip was annotated by being assigned either a B-line or non-B-line label.
Further, B-line frames and B-line regions in the B-line video clips were
annotated to be used as annotations for temporal and spatial B-line
localization task later.

###  3.2 Polar Coordinate Representation

Like other common applications of ultrasound imaging, lung ultrasound images
are normally presented in Cartesian coordinates (shown in Figure 3, left). In
this case, the information particularly B-line artefacts are presented densely
in the centre of the frustum to some extend. Therefore, when we downsample the
LUS videos as input to our network some information are lost. To overcome this
limitation, we transform each video clip into its associated polar coordinate
representation (shown in Figure 3, right). With polar coordinate
representation, information are expanded along the degree axes of polar data;
therefore, less information are missed during downsampling of the data.
Additionally, there is not much information in the left and right up corner
(black areas) of Cartiesian coordinate representation. As a result, when these
areas are removed in the polar coordinate representation, the network can
concentrate on the areas of each frame where more useful information are
exist.

Figure 3: Examples of a B-line frame in Cartesian coordinate (left) and polar
coordinate (right) representation.

Polar representation is carried out by the following reparameterization, used
to resample the Cartesian images into a polar grid using bilinear
interpolation:

$\begin{array}[]{rl}x=&r\sin(\alpha)\\\ y=&r\cos(\alpha)\end{array}$ (1)

Where $r$ is the depth, or radius (distance form the beam source to a pixel
location) and $\alpha$ is the angle measured from the y axis.

###  3.3 Implementation Details

Our network was implemented using Keras library with a Tensorflow backend. The
network was optimised using Adam optimizer with the learning rate (lr) set to
$10^{-6}$. Batch normalization was used for both CNN and LSTM parts of the
network. Batch size of 25, dropout of 0.2 and $L2=10^{-5}$ for regularization
were employed. Data augmentation was applied to the training data by adding
horizontally-flipped frames. 5-fold cross validation was used and the network
converged after 60 epochs. The class imbalance was addressed by weighting the
probability to draw a sample by its relative class occurrence in the training
set.

##  4 Experiments and Results

In order to investigate the potential benefit of employing polar
representation and various video resolutions in B-line detection task, we
trained our model with Cartesian and polar representations using various input
video sizes of $64\times 64$, $32\times 32$ and $16\times 16$ resolution.
Furthermore, we reduced the depth size of polar data to 32 and 16 samples,
while keeping the number of angular elements to 64 (hence maintaining angle
resolution), therefore having the video size of $64\times 32$ and $64\times
16$ resolution for training.

To assess the classification performance of the model, the harmonic mean of
precision and recall expressed as $F1=2\times\frac{Precision\times
Recall}{Precision+Recall}\times 100$ score (%) was utilised. The
classification performance for Cartesian and polar data are presented in
Figure 4. An alpha value of 0.05 was selected as the statistical significance
threshold. Shapiro-Wilk test showed that all data were normally distributed.

Figure 4: B-line classification performance (F1 score) of Cartesian and polar
representations with various video resolutions.

Our baseline video resolution ($64\times 64$) received the highest performance
for both polar and Cartesian representations. Also, a paired t-test revealed
that the performance of polar data (83.5%) is significantly higher than
Cartesian data (81%) (t=2.776, p=0.017) in all cases with the same number of
pixels. This demonstrates that the model can extract more information from a
polar representation. When we decreased the video resolution into $32\times
32$ and $16\times 16$, the performance dropped compared to the baseline video
resolution, although the drop was less significant in polar images. For video
resolutions of $32\times 32$, paired t-test showed significant difference
between Cartesian and polar representation in B-line detection task (t=1.035 ,
p=0.028). However, this difference is not significant for video resolution of
$16\times 16$ (t=-1.104, p=0.165), probably because the downsampling is too
aggressive and B-lines become barely distinguishable in any representation.
Furthermore, we decreased the depth size of polar data into 32 and 16 to
evaluate the contribution of depth information in B-line detection. Compared
to the depth size of 64 in baseline resolution, the performance decreased
significantly for both depth sizes of 32 (t=2.835 , p=0.008) and 16 (t=1.503 ,
p=0.018).

Additionally, we investigated the impact of downsampling along scan-lines and
along angles. To do this, we compared two video resolutions that had the same
amount of pixels: $32\times 32$ and $64\times 16$. Results showed that video
resolution of $64\times 16$ (64 along the angle dimension) has significantly
higher performance which shows that preserving information along the angle
dimension helps in this specific task where artefacts are aligned along
constant-angle lines (t=2.43, p=0.03).

We further evaluated B-line temporal localization accuracy using both data
representations. We calculated intersection over union (IoU) of predicted
temporal localized frames with their ground truth annotations. Results are
presented in Table 1. With polar representation, the model is able to localize
B-line frames temporally with higher performance than Cartesian
representation. Additionally, the attention weights for true B-line frames are
higher in polar representation and for the non B-line frames lower, compared
to Cartesian representation, further suggesting that the network learns to
differentiate B-line and non B-line frames better in polar representation.

Table 1: IoU values showing B-line localisation accuracy (%) for various video
resolutions of Cartesian and polar representations. Video Resolution
---
| 64*64 | 32*32 | 16*16 | 64*32 | 64*16
Cartesian | 67.1 | 56.3 | 42.2 | — | —
Polar | 73.2 | 62.5 | 43.1 | 67.7 | 65.1

##  5 Conclusion

This paper investigates the effect of employing ultrasound polar coordinate
representation on LUS B-line detection and localization tasks. We employed an
attention-based convloutional+LSTM model capable of extracting spatial and
temporal features from LUS videos and localizing B-line frames using a
temporal attention mechanism. We evaluated B-line classification and
localization with this architecture using Cartesian and polar coordinate
representations with different resolutions. Using our LUS video dataset,
results showed that polar representation consistently outperforms Cartesian in
terms of classification accuracy and temporal localization accuracy.

Our future work will explore an spatiotemporal attention mechanism that is
able to detect the B-line artefacts and localize them both spatially and
temporally within LUS videos in polar coordinates. B-line spatial localization
may help clinicians to quantify the severity of the disease. Overall, these
findings will assist management of ICU patients with dengue particularly in
low- and middle-income countries where ultrasound operator expertise is
limited.

## ACKNOWLEDGMENT

The VITAL Consortium: OUCRU: Dang Trung Kien, Dong Huu Khanh Trinh, Joseph
Donovan, Du Hong Duc, Ronald Geskus, Ho Bich Hai, Ho Quang Chanh, Ho Van Hien,
Hoang Minh Tu Van, Huynh Trung Trieu, Evelyne Kestelyn, Lam Minh Yen, Le
Nguyen Thanh Nhan, Le Thanh Phuong, Luu Phuoc An, Nguyen Lam Vuong, Nguyen
Than Ha Quyen, Nguyen Thanh Ngoc, Nguyen Thi Le Thanh, Nguyen Thi Phuong Dung,
Ninh Thi Thanh Van, Pham Thi Lieu, Phan Nguyen Quoc Khanh, Phung Khanh Lam,
Phung Tran Huy Nhat, Guy Thwaites, Louise Thwaites, Tran Minh Duc, Trinh Manh
Hung, Hugo Turner, Jennifer Ilo Van Nuil, Sophie Yacoub. Hospital for Tropical
Diseases, Ho Chi Minh City: Cao Thi Tam, Duong Bich Thuy, Ha Thi Hai Duong, Ho
Dang Trung Nghia, Le Buu Chau, Le Ngoc Minh Thu, Le Thi Mai Thao, Luong Thi
Hue Tai, Nguyen Hoan Phu, Nguyen Quoc Viet, Nguyen Thanh Nguyen, Nguyen Thanh
Phong, Nguyen Thi Kim Anh, Nguyen Van Hao, Nguyen Van Thanh Duoc, Nguyen Van
Vinh Chau, Pham Kieu Nguyet Oanh, Phan Tu Qui, Phan Vinh Tho, Truong Thi
Phuong Thao. University of Oxford: David Clifton, Mike English, Heloise
Greeff, Huiqi Lu, Jacob McKnight, Chris Paton. Imperial College London:
Pantellis Georgiou, Bernard Hernandez Perez, Kerri Hill-Cawthorne, Alison
Holmes, Stefan Karolcik, Damien Ming, Nicolas Moser, Jesus Rodriguez Manzano.
King’s College London: Alberto Gomez, Hamideh Kerdegari, Marc Modat, Reza
Razavi. ETH Zurich: Abhilash Guru Dutt, Walter Karlen, Michaela Verling, Elias
Wicki. Melbourne University: Linda Denehy, Thomas Rollinson.

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# Search for $e\to\tau$ Charged Lepton Flavor Violation at the EIC with the
ECCE Detector

J.-L. Zhang S. Mantry J. K. Adkins Y. Akiba A. Albataineh M. Amaryan I. C.
Arsene C. Ayerbe Gayoso J. Bae X. Bai M.D. Baker M. Bashkanov R. Bellwied F.
Benmokhtar V. Berdnikov J. C. Bernauer F. Bock W. Boeglin M. Borysova E. Brash
P. Brindza W. J. Briscoe M. Brooks S. Bueltmann M. H. S. Bukhari A. Bylinkin
R. Capobianco W.-C. Chang Y. Cheon K. Chen K.-F. Chen K.-Y. Cheng M. Chiu T.
Chujo Z. Citron E. Cline E. Cohen T. Cormier Y. Corrales Morales C. Cotton J.
Crafts C. Crawford S. Creekmore C.Cuevas J. Cunningham G. David C. T. Dean M.
Demarteau S. Diehl N. Doshita R. Dupré J. M. Durham R. Dzhygadlo R. Ehlers L.
El Fassi A. Emmert R. Ent C. Fanelli R. Fatemi S. Fegan M. Finger M. Finger Jr
J. Frantz M. Friedman I. Friscic D. Gangadharan S. Gardner K. Gates F. Geurts
R. Gilman D. Glazier E. Glimos Y. Goto N. Grau S. V. Greene A. Q. Guo L. Guo
S. K. Ha J. Haggerty T. Hayward X. He O. Hen D. W. Higinbotham M. Hoballah T.
Horn A. Hoghmrtsyan P.-h. J. Hsu J. Huang G. Huber A. Hutson K. Y. Hwang C. E.
Hyde M. Inaba T. Iwata H.S. Jo K. Joo N. Kalantarians G. Kalicy K. Kawade S.
J. D. Kay A. Kim B. Kim C. Kim M. Kim Y. Kim Y. Kim E. Kistenev V. Klimenko S.
H. Ko I. Korover W. Korsch G. Krintiras S. Kuhn C.-M. Kuo T. Kutz J. Lajoie D.
Lawrence S. Lebedev H. Lee J. S. H. Lee S. W. Lee Y.-J. Lee W. Li W.B. Li X.
Li X. Li X. Li X. Li Y. T. Liang S. Lim C.-H. Lin D. X. Lin K. Liu M. X. Liu
K. Livingston N. Liyanage W.J. Llope C. Loizides E. Long R.-S. Lu Z. Lu W.
Lynch D. Marchand M. Marcisovsky C. Markert P. Markowitz H. Marukyan P.
McGaughey M. Mihovilovic R. G. Milner A. Milov Y. Miyachi A. Mkrtchyan P.
Monaghan R. Montgomery D. Morrison A. Movsisyan H. Mkrtchyan A. Mkrtchyan C.
Munoz Camacho M. Murray K. Nagai J. Nagle I. Nakagawa C. Nattrass D. Nguyen S.
Niccolai R. Nouicer G. Nukazuka M. Nycz V. A. Okorokov S. Orešić J.D. Osborn
C. O’Shaughnessy S. Paganis Z. Papandreou S. F. Pate M. Patel C. Paus G.
Penman M. G. Perdekamp D. V. Perepelitsa H. Periera da Costa K. Peters W.
Phelps E. Piasetzky C. Pinkenburg I. Prochazka T. Protzman M. L. Purschke J.
Putschke J. R. Pybus R. Rajput-Ghoshal J. Rasson B. Raue K.F. Read K. Røed R.
Reed J. Reinhold E. L. Renner J. Richards C. Riedl T. Rinn J. Roche G. M.
Roland G. Ron M. Rosati C. Royon J. Ryu S. Salur N. Santiesteban R. Santos M.
Sarsour J. Schambach A. Schmidt N. Schmidt C. Schwarz J. Schwiening R. Seidl
A. Sickles P. Simmerling S. Sirca D. Sharma Z. Shi T.-A. Shibata C.-W. Shih S.
Shimizu U. Shrestha K. Slifer K. Smith D. Sokhan R. Soltz W. Sondheim J. Song
J. Song I. I. Strakovsky P. Steinberg P. Stepanov J. Stevens J. Strube P. Sun
X. Sun K. Suresh V. Tadevosyan W.-C. Tang S. Tapia Araya S. Tarafdar L.
Teodorescu D. Thomas A. Timmins L. Tomasek N. Trotta R. Trotta T. S. Tveter E.
Umaka A. Usman H. W. van Hecke C. Van Hulse J. Velkovska E. Voutier P.K. Wang
Q. Wang Y. Wang Y. Wang D. P. Watts N. Wickramaarachchi L. Weinstein M.
Williams C.-P. Wong L. Wood M. H. Wood C. Woody B. Wyslouch Z. Xiao Y.
Yamazaki Y. Yang Z. Ye H. D. Yoo M. Yurov N. Zachariou W.A. Zajc W. Zha J.-X.
Zhang Y. Zhang Y.-X. Zhao X. Zheng P. Zhuang

###### Abstract

The recently approved Electron-Ion Collider (EIC) will provide a unique new
opportunity for searches of charged lepton flavor violation (CLFV) and other
new physics scenarios. In contrast to the $e\leftrightarrow\mu$ CLFV
transition for which very stringent limits exist, there is still a relatively
large discovery space for the $e\to\tau$ CLFV transition, potentially to be
explored by the EIC. With the latest detector design of ECCE (EIC
Comprehensive Chromodynamics Experiment) and projected integral luminosity of
the EIC, we find the $\tau$-leptons created in the DIS process $ep\to\tau X$
are expected to be identified with high efficiency. A first ECCE simulation
study, restricted to the 3-prong $\tau$-decay mode and with limited statistics
for the Standard Model backgrounds, estimates that the EIC will be able to
improve the current exclusion limit on $e\to\tau$ CLFV by an order of
magnitude.

[AANL]organization=A. Alikhanyan National Laboratory, city=Yerevan,
country=Armenia

[AcademiaSinica]organization=Institute of Physics, Academia Sinica,
city=Taipei, country=Taiwan

[AUGIE]organization=Augustana University, city=Sioux Falls, state=SD,
country=USA

[BNL]organization=Brookhaven National Laboratory, city=Upton, state=NY,
country=USA

[BrunelUniversity]organization=Brunel University London, city=Uxbridge,
country=UK

[CanisiusCollege]organization=Canisius College, city=Buffalo, state=NY,
country=USA

[CCNU]organization=Central China Normal University, city=Wuhan, country=China

[Charles]organization=Charles University, city=Prague, country=Czech Republic

[CIAE]organization=China Institute of Atomic Energy, Fangshan, city=Beijing,
country=China

[CNU]organization=Christopher Newport University, city=Newport News, state=VA,
country=USA

[Columbia]organization=Columbia University, city=New York, state=NY,
country=USA

[CUA]organization=Catholic University of America, city=Washington DC,
country=USA

[CzechTechUniv]organization=Czech Technical University, city=Prague,
country=Czech Republic

[Duquesne]organization=Duquesne University, city=Pittsburgh, state=PA,
country=USA

[Duke]organization=Duke University, cite=Durham, state=NC, country=USA

[FIU]organization=Florida International University, city=Miami, state=FL,
country=USA

[GeorgiaState]organization=Georgia State University, city=Atlanta, state=GA,
country=USA

[Glasgow]organization=University of Glasgow, city=Glasgow, country=UK

[GSI]organization=GSI Helmholtzzentrum fuer Schwerionenforschung GmbH,
city=Darmstadt, country=Germany

[GWU]organization=The George Washington University, city=Washington, DC,
country=USA

[Hampton]organization=Hampton University, city=Hampton, state=VA, country=USA

[HUJI]organization=Hebrew University, city=Jerusalem, country=Isreal

[IJCLabOrsay]organization=Universite Paris-Saclay, CNRS/IN2P3, IJCLab,
city=Orsay, country=France

[CEA]organization=IRFU, CEA, Universite Paris-Saclay, cite= Gif-sur-Yvette,
country=France

[IMP]organization=Chinese Academy of Sciences, city=Lanzhou, country=China

[IowaState]organization=Iowa State University, city=Iowa City, state=IA,
country=USA

[JazanUniversity]organization=Jazan University, city=Jazan, country=Sadui
Arabia

[JLab]organization=Thomas Jefferson National Accelerator Facility,
city=Newport News, state=VA, country=USA

[JMU]organization=James Madison University, city=Harrisonburg, state=VA,
country=USA

[KobeUniversity]organization=Kobe University, city=Kobe, country=Japan

[Kyungpook]organization=Kyungpook National University, city=Daegu,
country=Republic of Korea

[LANL]organization=Los Alamos National Laboratory, city=Los Alamos, state=NM,
country=USA

[LBNL]organization=Lawrence Berkeley National Lab, city=Berkeley, state=CA,
country=USA

[LehighUniversity]organization=Lehigh University, city=Bethlehem, state=PA,
country=USA

[LLNL]organization=Lawrence Livermore National Laboratory, city=Livermore,
state=CA, country=USA

[MoreheadState]organization=Morehead State University, city=Morehead,
state=KY,

[MIT]organization=Massachusetts Institute of Technology, city=Cambridge,
state=MA, country=USA

[MSU]organization=Mississippi State University, city=Mississippi State,
state=MS, country=USA

[NCKU]organization=National Cheng Kung University, city=Tainan, country=Taiwan

[NCU]organization=National Central University, city=Chungli, country=Taiwan

[Nihon]organization=Nihon University, city=Tokyo, country=Japan

[NMSU]organization=New Mexico State University, city=Las Cruces, state=NM,
country=USA

[NRNUMEPhI]organization=National Research Nuclear University MEPhI,
city=Moscow, country=Russian Federation

[NRCN]organization=Nuclear Research Center - Negev, city=Beer-Sheva,
country=Isreal

[NTHU]organization=National Tsing Hua University, city=Hsinchu, country=Taiwan

[NTU]organization=National Taiwan University, city=Taipei, country=Taiwan

[ODU]organization=Old Dominion University, city=Norfolk, state=VA, country=USA

[Ohio]organization=Ohio University, city=Athens, state=OH, country=USA

[ORNL]organization=Oak Ridge National Laboratory, city=Oak Ridge, state=TN,
country=USA

[PNNL]organization=Pacific Northwest National Laboratory, city=Richland,
state=WA, country=USA

[Pusan]organization=Pusan National University, city=Busan, country=Republic of
Korea

[Rice]organization=Rice University, city=Houston, state=TX, country=USA

[RIKEN]organization=RIKEN Nishina Center, city=Wako, state=Saitama,
country=Japan

[Rutgers]organization=The State University of New Jersey, city=Piscataway,
state=NJ, country=USA

[CFNS]organization=Center for Frontiers in Nuclear Science, city=Stony Brook,
state=NY, country=USA

[StonyBrook]organization=Stony Brook University, city=Stony Brook, state=NY,
country=USA

[RBRC]organization=RIKEN BNL Research Center, city=Upton, state=NY,
country=USA

[SDU]organizaton=Shandong University, city=Qingdao, state=Shandong,
country=China

[Seoul]organization=Seoul National University, city=Seoul, country=Republic of
Korea

[Sejong]organization=Sejong University, city=Seoul, country=Republic of Korea

[Shinshu]organization=Shinshu University, city=Matsumoto, state=Nagano,
country=Japan

[Sungkyunkwan]organization=Sungkyunkwan University, city=Suwon,
country=Republic of Korea

[TAU]organization=Tel Aviv University, city=Tel Aviv, country=Israel

[USTC]organization=University of Science and Technology of China, city=Hefei,
country=China

[Tsinghua]organization=Tsinghua University, city=Beijing, country=China

[Tsukuba]organization=Tsukuba University of Technology, city=Tsukuba,
state=Ibaraki, country=Japan

[CUBoulder]organization=University of Colorado Boulder, city=Boulder,
state=CO, country=USA

[UConn]organization=University of Connecticut, city=Storrs, state=CT,
country=USA

[UNGeorgia]organization=University of North Georgia, cite=Dahlonega, state=GA,
country=USA

[UH]organization=University of Houston, city=Houston, state=TX, country=USA

[UIUC]organization=University of Illinois, city=Urbana, state=IL, country=USA

[UKansas]organization=Unviersity of Kansas, city=Lawrence, state=KS,
country=USA

[UKY]organization=University of Kentucky, city=Lexington, state=KY,
country=USA

[Ljubljana]organization=University of Ljubljana, Ljubljana, Slovenia,
city=Ljubljana, country=Slovenia

[NewHampshire]organization=University of New Hampshire, city=Durham, state=NH,
country=USA

[Oslo]organization=University of Oslo, city=Oslo, country=Norway

[Regina]organization= University of Regina, city=Regina, state=SK,
country=Canada

[USeoul]organization=University of Seoul, city=Seoul, country=Republic of
Korea

[UTsukuba]organization=University of Tsukuba, city=Tsukuba, country=Japan

[UTK]organization=University of Tennessee, city=Knoxville, state=TN,
country=USA

[UVA]organization=University of Virginia, city=Charlottesville, state=VA,
country=USA

[Vanderbilt]organization=Vanderbilt University, city=Nashville, state=TN,
country=USA

[VirginiaTech]organization=Virginia Tech, city=Blacksburg, state=VA,
country=USA

[VirginiaUnion]organization=Virginia Union University, city=Richmond,
state=VA, country=USA

[WayneState]organization=Wayne State University, city=Detroit, state=MI,
country=USA

[WI]organization=Weizmann Institute of Science, city=Rehovot, country=Israel

[WandM]organization=The College of William and Mary, city=Williamsburg,
state=VA, country=USA

[Yamagata]organization=Yamagata University, city=Yamagata, country=Japan

[Yarmouk]organization=Yarmouk University, city=Irbid, country=Jordan

[Yonsei]organization=Yonsei University, city=Seoul, country=Republic of Korea

[York]organization=University of York, city=York, country=UK

[Zagreb]organization=University of Zagreb, city=Zagreb, country=Croatia

##  1 Charged-Lepton Flavor Violation and Leptoquarks

The discovery of neutrino oscillations provided conclusive evidence of lepton
flavor violation. Lepton flavor violation in the neutrino sector also results
in charged lepton flavor violation (CLFV) through loop-suppressed processes
such as $\mu\to e\gamma$. However, the resulting predicted CLFV rates are
highly suppressed due to the small neutrino masses – Br$(\mu\to
e\gamma)<10^{-54}$ – and are far beyond the reach of any current or planned
experiments. On the other hand, many Beyond Standard Models (BSM) scenarios
predict CLFV rates that are both much larger and within reach of ongoing or
near-future experiments. For example, supersymmetry-based models predict rates
as high as Br$(\mu\to e\gamma)\sim 10^{-15}$ [1], while the current
experimental limit on the $\mu\to e\gamma$ process already reached Br$(\mu\to
e\gamma)<4.2\times 10^{-13}$ [2]. On the other hand, while there have been
extensive searches for CLFV processes between the first and second lepton
generations, denoted as CLFV(1,2) for brevity, the constraints on CLFV(1,3)
processes that involve $e\leftrightarrow\tau$ transitions are weaker by
several orders of magnitude. These constraints on CLFV(1,3) [3, 4] were
obtained through searches for $e+p\to\tau+X$, $\tau\to e\gamma$, and $p+p\to
e+\tau+X$ at HERA [5, 6], BaBar [7], and the LHC [8] respectively. However,
there are many BSM scenarios such as grand unified theories with leptoquarks
and R-parity violating supersymmetry that predict CLFV(1,3) rates that are
enhanced compared to CLFV(1,2) processes, motivating continued searches
dedicated for $e\leftrightarrow\tau$ transitions.

We carry out the simulation analysis based on the design of the ECCE Detector
(recommended as Detector 1 by the EIC Detector Proposal Advisory Panel [9]),
for determining the sensitivity to the CLFV(1,3) process $e+p\to\tau+X$ in the
leptoquark framework [3, 4], though such analysis could also be performed in
the SMEFT framework [4]. Leptoquarks are color triplet bosons that carry both
lepton ($L$) and baryon ($B$) numbers, coupling leptons to quarks and
mediating the $e+p\to\tau+X$ CLFV(1,3) process at tree-level, as shown in Fig.
1. The leptoquarks are classified into 14 types [10] based on their fermion
number $F=3B+L$ ($F=0$ or $|F|=2$), spin (scalar or vector), chiral couplings
to leptons (left-handed or right-randed), $SU(2)_{L}$ representation (singlet,
doublet, or triplet), and $U(1)_{Y}$ hypercharge.

In the region where the leptoquark mass $M_{\rm LQ}$ is much larger than the
characteristic energy scale of the experiment, represented by the center-of-
mass energy $\sqrt{s}$ and the four-momentum transfer $\sqrt{Q^{2}}$, the
CLFV(1,3) process $e+p\to\tau+X$ is mediated by a contact interaction and the
tree-level cross section for $F=0$ or $|F|=2$ leptoquarks takes the form:

$\displaystyle\sigma_{F=0}=\sum_{\alpha,\beta}\frac{s}{32\pi}{\left[\frac{\lambda_{1\alpha}\lambda_{3\beta}}{M_{LQ}^{2}}\right]}^{2}\times$
$\displaystyle\hskip 22.76228pt\int dx\int
dy\>\big{\\{}x\bar{q}_{\alpha}(x,xs)f(y)+xq_{\beta}(x,-u)g(y)\big{\\}},$
$\displaystyle\sigma_{|F|=2}=\sum_{\alpha,\beta}\frac{s}{32\pi}{\left[\frac{\lambda_{1\alpha}\lambda_{3\beta}}{M_{LQ}^{2}}\right]}^{2}\times$
(1) $\displaystyle\hskip 22.76228pt\int dx\int
dy\>\big{\\{}xq_{\alpha}(x,xs)f(y)+\bar{q}_{\beta}(x,-u)g(y)\big{\\}},$

where $u=x(y-1)s$ with $x$ the Bjorken scaling variable and $y$ the fractional
energy loss of the electron in the proton-rest frame. The kinematic
$y$-dependent functions $f(y),g(y)$ are $f(y)=1/2,g(y)=(1-y)^{2}/2$ and
$f(y)=2(1-y)^{2},g(y)=2$ for scalar and vector leptoquarks, respectively. The
quantity $\lambda_{1\alpha}\lambda_{3\beta}/(M_{LQ}^{2})$ characterizes the
strength of the contact interaction. The $\lambda_{ij}$ parameters, assumed to
be real numbers for this analysis, denote the leptoquark couplings between the
$i$-th lepton generation and $j$-th quark generation.

Figure 1: From [3]: Representative Feynman diagrams for $e\to\tau$ scattering
processes via one-leptoquark mediator. The fermionic number $F$ is assumed to
be conserved, as in the BRW effective model [10]. The partonic cross section
is convoluted with the PDF of the initial state (anti)quark of each diagram,
and depends on the parameter $\lambda_{1\alpha}\lambda_{3\beta}/M_{LQ}^{2}$.

The ZEUS and H1 experiments at HERA placed upper limits on
$\lambda_{1\alpha}\lambda_{3\beta}/M_{LQ}^{2}$ [11, 5, 12, 6]. The HERA data
set corresponded to $\sqrt{s}=300$ and 318 GeV and a total integrated
luminosity of up to 130 pb-1. With several orders of magnitude increase in the
luminosity, the EIC has the potential to improve upon the HERA limits for both
diagonal ($\alpha=\beta$) and off-diagonal ($\alpha\neq\beta$) components, and
provide complementary information [4] to the constraints from BaBar and the
LHC.

##  2 EIC/ECCE Simulation and Vertex Identification

The EIC Comprehensive Chromodynamics Experiment (ECCE) detector concept [13]
addresses the full EIC science mission as described in the EIC community White
Paper [14] and the 2018 National Academies of Science (NAS) Report [15]. It is
simultaneously fully capable, low-risk, and cost-effective. ECCE strategically
repurposes select components of existing experimental equipment to maximize
its overall capabilities within the envelope of planned resources. For
example, the central barrel of the detector incorporates the storied 1.4 T
BaBar superconducting solenoid, and the sPHENIX barrel hadronic calorimeter
currently under construction.

The goal of this work is to study the potential for discovering the
$e^{-}p\to\tau^{-}X$ CLFV process at the EIC based on realistic detector
simulations, and identification of such events over Standard Model (SM)
backgrounds that include neutral-current (NC) deep inelastic scattering (DIS),
charged-current (CC) DIS, and photoproductions. CLFV events leading to a
final-state $\tau$ and are characterized by a high-momentum isolated $\tau$
which is balanced by a jet in the transverse plane. Since the $\tau$ will
decay into stable particles after a short flying distance of $\sim\mu$m, only
its decay products are visible in the detector. A critical requirement of CLFV
searches is thus the secondary vertex reconstruction performance which relies
on the tracking and especially the vertex detectors. The ECCE conceptual
design uses state-of-the-art technologies consisting Monolithic Active Pixel
Sensor (MAPS) based silicon vertex/tracking subsystem to achieve high
precision primary and decay vertex determination.

The three SM background processes affect the CLFV searches as follows: First,
the SM NC DIS events are very similar to leptoquark events where the $\tau$
decays to only one charged particle plus neutrinos in the final state, making
this channel very difficult to study. Secondly, due to the presence of at
least one neutrino in all $\tau$-decay channels, significant missing $p_{T}$
is expected. This feature is similar to the SM CC DIS events. The third main
background of concern is from photoproduction events, mostly due to their very
high yield. In this study, we focus the identification of CLFV candidate
events that contain a high-pT quark-initiated jet along with an isolated and
high-pT $\tau$ which replaces the scattered electrons in the typical NC DIS
events, and the rejection of events from all three SM background processes.

Based on the number of the charged particles in the final state, the dominant
$\tau$ decay modes can be categorized into “1-prong” (one charged particle)
and “3-prong” (three charged particles). The “3-prong” decay modes have a
branching ratio of $\sim$15% while the “1-prong” modes have a branching ratio
of $\sim$85%. Although the “1-prong” modes have larger branching ratio, they
are more demanding to identify. For example, the “1-prong” mode could be one
of the two leptonic decays, $\tau\to e\bar{\nu}_{e}\nu_{\tau}$ or
$\tau\to\mu\bar{\nu}_{\mu}\nu_{\tau}$. If the charged particle is an electron,
it is very similar to DIS NC events. If the charged particle is a muon, it
will require good muon identification. Another possible “1-prong” mode is
$\tau\to\nu_{\tau}\pi^{-}$, containing one charged pion, and can be studied in
the near future. In this study, we focus only on searching for the $\tau$
“3-prong” decay events.

The features of “3-prong” leptoquark events include: 1) no scattered electron
is detected; 2) a high transverse-momentum ($P_{T}$) $\tau$-jet consists of
three charged particles within a relatively small cone; 3) all three charged
particles originate from a common secondary vertex; 4) a high $P_{T}$ hadronic
jet is found back-to-back from the $\tau$-jet; and 5) a $P_{T}$-imbalance
caused by the escaped neutrinos which should be part of the candidate
$\tau$-jet. In order to simulate such candidate events and the capability to
identify them, the leptoquark quark generator LQGENEP [16] (version 1.0) is
used to produce the signal Monte-Carlo (MC) events, while Djangoh and Pythia
generators are used to produce the background DIS and photoproduction MC
events, respectively. Considering the large mass of leptoquarks, we focus on
the highest energy, 18$\times$275 GeV $ep$ collision. For the LQGENEP
simulation, the $Q^{2}$ range is set to $>10$ GeV2 and a default value of
leptoquark mass 1.9 TeV from the generator is used, see A.1 for input files.
Scanning through a series of leptoquark mass values, no visible difference in
the characteristics of these signal events was observed at EIC kinematics. The
input files for background event generation are given in A.2 for DIS NC and
CC, and A.3 for photoproduction. After passing the generated events through
the ECCE GEANT4 simulation, an analysis algorithm with preliminary selection
requirements based on the features of the signal and background events are
applied on both leptoquark and SM MC event samples.

As mentioned earlier, a precise identification capability of the interaction
vertex is essential for the secondary vertex reconstruction and $\tau$
identification. Figure 2 shows the vertex resolution for different track
multiplicities, where we see that the ECCE configuration can provide a vertex
resolution of $20-30\mu$m while the decay length of $\tau$ lepton is
$\sim$87$\mu$m. Therefore the ECCE vertex resolution is sufficient for
identifying $\tau$ decays.

Figure 2: Primary vertex resolution: difference between truth and
reconstructed information for different track multiplicity. Only the
$x$-component is shown here, while $y$\- and $z$-components show similar
characteristics but are not shown.

To reconstruct the secondary vertex, we first look for 3-$\pi$ candidate
events. In the following, the charged pion’s tracking information is from the
simulated tracking and the vertex detector responses, though particle
identification (PID) is based on generator information (namely, perfect PID is
assumed). In our algorithm, one track is matched to a second track and the
middle point at the closest approach is the candidate secondary vertex
position which will be further justified based on the topological structure.
For a given 3-$\pi$ candidate event, there are three such pair-combinations
and we can reconstruct three “intermediate” vertices and the candidate vertex
will be the average of all three. Figure 3 shows the correlation of three
intermediate vertices from the three pair combinations where the three decay
lengths $dl_{12}$, $dl_{13}$, and $dl_{23}$, as extracted by the distance from
the primary to the secondary “intermediate” vertices, are shown as the
$x$-axis, the positive half of the $y$-axis, and the negative half of the
$y$-axis, respectively. There are clearly two bands centralized at lines
$y=\pm x$ and the 3-$\pi$ vertex can be identified by requiring either or both
correlations. In fact, when combined with vector alignment cuts, coincidence
between two of the three “intermediate” vertices (either upper or lower half
of Fig. 3) is usually enough to indicate a “3-prong" secondary vertex.

Figure 3: Coincidence among three “intermediate” vertices for 3-$\pi$ event
identification. The $x$-axis, the positive $y$ and the negative $y$-axis
(displaced and direction reversed for clarity), represent the “intermediate”
vertices from the three pair-combinations $12$, $13$ and $23$, respectively.

##  3 Event Selection

We used ten selection criteria to identify $e\to\tau$ events and to reject SM
backgrounds. Their effects are shown in Fig. 4, where the vertical axis shows
how many sample events pass each of the selection criteria, and the horizontal
axis are the progressive selections defined as follows:

  * 1.

input: initial input events. We used $10^{6}$ (1M) MC events for each of
leptoquark, DIS NC, DIS CC, and photoproduction processes;

  * 2.

PrVtx: there must be a primary vertex reconstructed;

  * 3.

Epzh: $\sum_{h}(E-p_{z})>$ 18 GeV, where $E$ and $p_{z}$ are the energy and
the $z-$component (along the beams) of the 3-momentum of the final state
particles, respectively, and the summation is over all detected hadrons;

  * 4.

misspt: 1 $<$ missing $p_{T}<$ 9 GeV, here the lower limit is to suppress
events with small missing $p_{T}$, e.g. photo-production events, and the upper
limit is to suppress DIS events with large missing $p_{T}$ caused by neutrinos
(CC) or miss-detected electrons (NC);

  * 5.

3-pion: candidate 3 charged pions are found in a $\Delta R<1.0$ cone, where
$R$ is cone radius in the azimuth($\phi$)-pseudorapidity($\eta$) space,
$\Delta R\equiv\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$;

  * 6.

away1GeV: $p_{T}$ sum of all tracks on the away-side of the candidate 3-$\pi$,
$\sum_{\Delta\phi(-\vec{p}_{3\pi})<1.0}p_{T}$, is $>1$ GeV;

  * 7.

nearIso: $p_{T}$ sum in a cone around the candidate 3-$\pi$, $\sum_{\Delta
R(\vec{p}_{3\pi})<1.0}p_{T}$, is $<3.0$ GeV;

  * 8.

3pi_pt: $p_{T}$ sum of the 3 charged-pion candidate, $p_{T(3\pi)}$, is $>3.0$
GeV;

  * 9.

30$\mu$m: candidate decay length reconstructed from any pair of the 3 charged
pions is $>30\mu$m;

  * 10.

dRsum: sum of the “distances” (in $\phi-\eta$ space) of the 3 charged pions
decay vectors, $\Delta$R1,2 \+ $\Delta$R1,3 \+ $\Delta$R2,3, is $<0.4$. Here
the decay vector is defined as starting from the primary vertex and pointing
to the secondary vertex;

  * 11.

decayL: average of the reconstructed decay lengths from three pair
combinations of the 3-$\pi$ candidate, ($dl_{12}$ \+ $dl_{13}$ \+
$dl_{23}$)/3, is $>0.5$ mm;

  * 12.

cMass:
$\sqrt{M_{3\pi}^{2}+p^{2}_{3\pi}\mbox{sin}^{2}\theta}+p_{3\pi}\mbox{sin}^{2}\theta<1.8$
GeV, where $\theta$ is the angle between the reconstructed decay direction and
the $3\pi$ momentum direction, and $M_{3\pi}$ is the mass reconstructed from
the 3-$\pi$ [17];

  * 13.

missing phi: missing $p_{T}$ is azimuthally on the near side of the candidate
3-$\pi$, that is, $\Delta\phi$ between $\vec{p_{3\pi}}$ and
$\vec{p_{T}^{miss}}$ is $<1.0$.

Figure 4: MC statistics of leptoquark (blue), DIS CC (red), DIS NC (magenta),
and photoproduction (orange) events, as ten selection criteria are
progressively applied on 1 M input events for each channel. Please see text
for details.

From Fig. 4, it can be seen that the $e\to\tau$ events can be effectively
selected with this set of preliminary cuts. In addition, selections using the
decay length are the most discriminating feature of the $\tau$-jet. We
illustrate this feature, characterized by the precision of ECCE’s vertex
detector, in Fig. 5. The left panel shows a comparison between the true decay
length from the generator and the decay length reconstructed from tracks at
the detector level, while the right panel shows a 119 $\mu$m resolution of the
decay length under ECCE configuration, capable for the $\tau$ vertex
identification.

Figure 5: Top: Reconstructed decay length from Geant4 detector simulation vs.
true decay length from generator level. Bottom: difference between
reconstructed and true decay length with a Gaussian fit.

##  4 Sensitivity to Leptoquarks

We can now deduce the sensitivity to the leptoquark signal cross section based
on simulations of the 3-prong decay mode (15% branching ratio) of the $\tau$
lepton, discussed in the last section, and considering different possible
values for the detection efficiency of the other $\tau$ decay modes. In Fig.
4, 1M MC event samples are generated for each of the four processes: the
leptoquark mediated signal process $e+p\to\tau+X$, and three background
processes, NC DIS, CC DIS, and photoproduction. After all selection cuts are
applied, including the requirement of detecting three pions corresponding to
the 3-prong tau decay mode, about 8K (7878) of the original 1M leptoquark
signal events remain. For an integrated luminosity of 100 fb -1, the 1M signal
events generated corresponds to a signal cross section of 104fb. Thus, if we
assume that only the 3-prong mode has a non-zero detection efficiency, the
minimum required cross section for detecting a single signal event in the
3-prong mode is $10^{4}~{}{\rm fb}/7878=1.3$ fb. On the other hand, if we
allow for the other tau decay modes and assume that they can be detected with
the same efficiency as the 3-prong decay mode, then the number of candidate
events that now survive selection cuts will be $7878/(15\%)=52520$. Thus, in
this case the minimum required cross section for detecting a single signal
event will be $10^{4}{\rm fb}/52520=0.19$ fb. We also consider an intermediate
scenario in which, in addition to the 3-prong mode, we allow for the muon and
single charge pion decay modes ($\sim$ 40 % branching ratio) and assume they
can be detected at half the detection efficiency of the 3-prong decay. In this
case the total number of candidate events that now survive all selection cuts
will be $7878+7878/(15\%)\times(40\%)\times 1/2=18382$. In this case, the
minimum required cross section to detect a single signal event is $10^{4}{\rm
fb}/18382=0.54$ fb.

For searches of rare events like leptoquark mediated $e\to\tau$ transitions,
the background simulation is more difficult than the signal. Among the three
types of background events, the CC DIS background is easier to estimate
because the total cross section is only $\approx 2.3\times 10^{4}$ fb, and the
1 M CC DIS MC events generated correspond to about 43% of the expected
statistics for 100 fb-1 of integrated luminosity. There are 4 CC DIS events
that pass the event selection, which can be scaled up to $4/43\%=9$ events for
100 fb-1. With more careful optimization, this number can be suppressed even
further. On the other hand, the NC DIS and photoproduction backgrounds are
much harder to evaluate because their cross sections are much larger and of
order $10^{7}$ fb. Thus, the 1M generated MC events correspond to only $\sim
10^{6}/(10^{7}{\rm fb}\times 100{\rm fb}^{-1})=0.1\%$ of the expected data
sample for 100 fb-1 of integrated luminosity. With this limited MC event
sample, zero NC and photoproduction event satisfies all selection criteria.
Thus, at this stage, it is difficult to estimate how many background events
will survive the selection if the MC sample is increased by factor $10^{3}$,
which is needed to simulate 100 fb-1 of NC DIS and photoproduction data.

At the moment, instead of providing a specific estimate of the background, we
show in Fig. 6 the leptoquark cross-section ECCE could be sensitive to as a
function of the number of background events that survive the event selection
for 100 fb-1 of integrated luminosity, based on a simple calculation as
follows: If the number of background events is $B$, the number of leptoquark
signal events $S$ should exceed the “3$\sigma$" limit of the expected observed
background events, $S+B>B+3\sqrt{B}$, for the signal to be established. The
corresponding cross section is scaled from the value where only one leptoquark
signal event can be observed ($S=1$).

Figure 6: Cross section sensitivity for leptoquark search vs number of
residual background events for 100 fb-1 integrated luminosity. The grey line
corresponds to the scenario that only “3-prong" decay modes are detected. The
blue line corresponds to the scenario where electron and pion “1-prong" decay
modes could be detected with 50% efficiency of the “3-prong" case. And the red
line shows the scenario if all decay modes were detected at the same
efficiency as the “3-prong” case.

As a best-case scenario estimate of the sensitivity to the leptoquark signal
cross section, we do not consider any NC and photoproduction background event
since none of these events passed all the selection cuts on our limited MC
event sample. For a $5\sigma$ (99.99994% confidence level) discovery criteria
of $S/\sqrt{(}B)\geq 5$ ($S$ being signal and $B$ being background) and use
$B=9$ events from CC background, we need $S=15$ leptoquark events or a total
of $15+9=24$ events to claim $e\to\tau$ CLFV discovery. Alternatively,
detection of less than $9+9=18$ events will provide a $3\sigma$ (99.7% C.L.)
exclusion limit on the leptoquark cross section, which would be $1.3$
fb$\times 3\sqrt{9}=11.4$ fb, $0.54$ fb$\times 3\sqrt{9}=5.0$ fb, and $0.19$
fb$\times 3\sqrt{9}=1.7$ fb, for detection possibility of “3-prong only”,
“3-prong + 1-prong with 50% efficiency", and “all decay modes detected with
same efficiency as 3-prong”, respectively. The exclusion potential, expressed
in terms of $\lambda_{1\alpha}\lambda_{3\beta}/M_{LQ}^{2}$, are shown in Figs.
7 and 8 for scalar and vector leptoquark states, respectively. This is a
preliminary estimate, and different statistical methods and a larger MC event
sample to better estimate NC DIS and photoproduction backgrounds could give
rise to different estimates.

Figure 7: Limits on the scalar leptoquarks with $F=0$ $S_{1/2}^{L}$ (top) and
$|F|=2$ $\tilde{S}_{0}^{R}$ (bottom) from 100 fb-1 of $ep$ $18\times 275$ GeV
data, based on a sensitivity to leptoquark-mediated $ep\to\tau X$ cross
section of size 1.7 fb (red triangles) or 11.4 fb (grey triangles) with ECCE.
Note that due to small value of $\sqrt{s}$, EIC cannot constraint the third
generation couplings of $S_{1/2}^{L}$ to top quarks. Limits from HERA [11, 5,
12, 6] are shown as cyan solid squares, and limits from $\tau\to e\gamma$
decays [3] are shown as green solid circles.

Figure 8: Limits on the vector leptoquarks with $F=0$ $\tilde{V}_{0}^{R}$
(top) and $|F|=2$ $V_{1/2}^{L}$ (bottom) from 100 fb-1 of $ep$ $18\times 275$
GeV data, based on a sensitivity to leptoquark-mediated $ep\to\tau X$ cross
section of size 1.7 fb (red triangles) or 11.4 fb (grey triangles) from ECCE.
Note that due to small value of $\sqrt{s}$, EIC cannot constraint the third
generation couplings of $\tilde{V}_{0}^{R}$ to top quarks. Limits from HERA
[11, 5, 12, 6] are shown as cyan solid squares. Limits from $\tau\to e\gamma$
decays [3] exist but require some work to convert to the 4-fermion contact
term. This will be done in the future.

##  5 Summary

We carried out the first projection analysis for charged lepton flavor
violation in the $e\to\tau$ transition channel, using EIC simulations with the
ECCE detector configuration. More work needs to be done in the future
alongside the development of ECCE into a project detector, such as using
detector-based particle identification, study more $\tau$ decay modes, and
carry out the background study with higher statisitics. Our current study,
using the simulation and detector resources at hand, shows that the EIC will
place a more stringent limit on $e\to\tau$ CLFV mediated by leptoquarks than
the previous HERA data. The very high vertex resolution of the ECCE detector
configuration plays a critical role in our study.

##  6 Acknowledgements

We thank the EIC Silicon Consortium for cost estimate methodologies concerning
silicon tracking systems, technical discussions, and comments. We acknowledge
the important prior work of projects eRD16, eRD18, and eRD25 concerning
research and development of MAPS silicon tracking technologies.

We thank the EIC LGAD Consortium for technical discussions and acknowledge the
prior work of project eRD112.

We thank Guillelmo Gomez Ceballos Retuerto and Hubert Spiesberger for their
useful discussions and comments.

We acknowledge support from the Office of Nuclear Physics in the Office of
Science in the Department of Energy, the National Science Foundation, and the
Los Alamos National Laboratory Laboratory Directed Research and Development
(LDRD) 20200022DR.

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##  Appendix A Simulation Input Files and Other Details

###  A.1 LQGENEP Input Files for Leptoquark Signal Simulation

The input file for Leptoquark event generation using LQGENEP 1.0, for
$Q^{2}_{\mathrm{min}}=10$ GeV2, is shown in Fig. 9.

Figure 9: LQGENEP 1.0 input file for $ep$ $18\times 275$ GeV, $Q^{2}>10$ GeV2
setting.

###  A.2 Djangoh Input Files for DIS NC and CC Background Event Simulation

DIS background NC and CC events were generated with Djangoh.4.6.10 , with the
input files shown in Fig. 10.

Figure 10: Djangoh (4.6.10) input files for $ep$ $18\times 275$ GeV, DIS CC
(left) and NC (right) background simulations for the leptoquark study. The
$Q^{2}_{\mathrm{min}}$ is set at 1 GeV2 and 10 GeV2 for DIS CC and NC,
respectively.

###  A.3 Pythia Input File for Photoproduction Background Simulation

The input file for photoproduction background generation using Pythia is shown
in Fig. 11.

Figure 11: Pythia (6.428) input files for photoproduction for $ep$ $18\times
275$ GeV setting.