# EchoVest: Real-Time Sound Classification and Depth Perception Expressed
through Transcutaneous Electrical Nerve Stimulation

Jesse Choe  Siddhant Sood  and Ryan Park

###### Abstract

Over 1.5 billion people worldwide live with hearing impairment [17]. Despite
various technologies that have been created for individuals with such
disabilities, most of these technologies are either extremely expensive or
inaccessible for everyday use in low-medium income countries. In order to
combat this issue, we have developed a new assistive device, EchoVest, for
blind/deaf people to intuitively become more aware of their environment.
EchoVest transmits vibrations to the user’s body by utilizing transcutaneous
electric nerve stimulation (TENS) based on the source of the sounds. EchoVest
also provides various features, including sound localization, sound
classification, noise reduction, and depth perception. We aimed to outperform
CNN-based machine-learning models, the most commonly used machine learning
model for classification tasks, in accuracy and computational costs. To do so,
we developed and employed a novel audio pipeline that adapts the Audio
Spectrogram Transformer (AST) model, an attention-based model, for our sound
classification purposes, and Fast Fourier Transforms for noise reduction. The
application of Otsu’s Method helped us find the optimal thresholds for
background noise sound filtering and gave us much greater accuracy. In order
to calculate direction and depth accurately, we applied Complex Time
Difference of Arrival algorithms and SOTA localization. Our last improvement
was to use blind source separation to make our algorithms applicable to
multiple microphone inputs. The final algorithm achieved state-of-the-art
results on numerous checkpoints, including a 95.7% accuracy on the ESC-50
dataset for environmental sound classification.

##  1 Introduction

According to the World Health Organization, if a person’s hearing thresholds
are below 20 dB, they are said to have hearing loss [17]. The consequences of
this condition can vary in their severity and may include difficulties in
communication, leading to social isolation for older individuals, decreased
academic performance in children, and limited job opportunities for adults in
areas without adequate accommodations for those with hearing loss. Currently,
1.5 billion people globally, or one in every five people, live with hearing
loss and this is projected to increase to 2.5 billion people, or 25%, of the
world population by 2050. The majority of individuals with hearing loss, 80%,
live in low and middle-income countries. Despite this, a significant amount of
hearing loss goes unaddressed, costing governments around the world nearly
$980 billion annually, with the majority of these costs incurred in low and
middle-income countries. This high cost is attributed to the expensive nature
of hearing impairment devices such as hearing aids and cochlear implants,
which range in cost from $2,000 to $7,000 for hearing aids [16] and $30,000 to
$50,000 for cochlear implants [10].

We aim to address the problem of unaddressed hearing impairment by creating
EchoVest, a cost-effective wearable alternative to current hearing impairment
solutions. EchoVest utilizes sound localization and depth perception through
the use of TENS pads and sound classification through Audio Spectrogram
Transformers (ASTs). To implement sound localization and depth perception,
EchoVest selectively activates TENS pads [14] with amplified signals based on
the distance and location of the sound source. With a total cost of
manufacturing of $98.90, EchoVest is a significantly more affordable and
effective option than existing hearing impairment technologies.

##  2 Materials

The primary objective was to create an inexpensive, durable, and wearable
device for the user’s day to day activities. We used a mesh vest as the base
with wiring interwoven throughout the mesh. A mesh vest is ideal for contact
between skin and the output nodes. On the back of the mesh, we employed a
Raspberry Pi 3B+ as our central computer in order to control all of the output
devices. The Machine-Learning libraries and other built-in software that we
integrated into our algorithms all required a 64-bit system and the Raspberry
Pi 3B+ was the cheapest processor on the market that we were able to obtain.
In order to record sounds, we used a ReSpeaker 4-mic Array due to the fact
that we could get 4 different streams of audio input. The continuous stream of
4 mics made it possible for us to calculate the direction and distance of
sounds. The last piece of significant hardware that we used were TENS
electrodes. TENS is a service that delivers mild electrical currents through
electrodes placed on the body. By applying TENS to our vest, we are able to
directly stimulate the user’s nerves and leave them with a multi-dimensional
feeling. A full materials list breakdown with all other assorted materials can
be seen in the Figure 1 below. Our essential pieces of hardware and their
application can be seen in Figure 2.

(a) Figure 1: Materials List with Costs

(b) Figure 2: Materials Diagram on Vest

##  3 Methods

We designed EchoVest to determine the relative location and distance of a
sound source from each microphone in our vest for audio spatial awareness. We
were able to triangulate the relative location of the sound source in real
time and determine the sound’s arrival angle using the Open embeddeD Auditory
System’s (ODAS) built-in sound localization algorithms [4]. We utilized Time
Difference of Arrival (TDoA) with Generalized Cross-Correlation with Phase
Transforms (GCC-PHAT) to calculate the distance from the sound source to the
microphones in real time. After recording the audio signals coming from each
microphone, we calculated the cross-correlation function by sliding one signal
in relation to the other for each time step to see how similar their waveforms
are to one another. The time delay between the signals, or TDoA value, between
the two microphones is represented by the time step with the highest cross-
correlation value [8]. We were able to select the appropriate pad based on the
angle of sound arrival and alter the strength of our electrical pad signals in
response to the distance from the microphone, calculated using the distance-
rate-time equation.

In order to enhance EchoVest’s sound localization and depth perception, we
utilized a Blind Source Separation (BSS) approach that combined Principal
Component Analysis (PCA), Non-Negative Matrix Factorization (NMF), and
Independent Component Analysis (ICA) to separate the combined sound file from
the microphone array into individual sound files for each microphone. We first
reduced the dimensionality of the sound input with PCA and NMF. NMF factorized
the sound input into two smaller matrices, as seen in Figure 3, with non-
negative elements [15], while PCA transformed the sound data into a new
coordinate system with axes along the directions of maximum variance [7]. ICA
then separated the mixed sound signal into subcomponents by assuming that only
one component was Gaussian and that the components were independent from each
other [4]. The cross-correlation matrix and TDOA values from sound
localization allowed us to identify each sound with its corresponding
microphone, resulting in enhanced sound localization and depth perception.

Figure 2: Figure 3: The original sound (top) was separated into two separate
sounds (left and right)

We employed a signal processing approach that combined Fast Fourier Transforms
(FFTs) with Otsu’s method to effectively remove background noise from our
sound input and enhance the sound classification accuracy. First, we converted
the sound input into its frequency domain using FFT, an efficient algorithm
that computes the discrete Fourier transform in real-time. We then implemented
Otsu’s Method, a commonly used image processing algorithm, to denoise the
sound input by selecting an optimal noise threshold. Otsu’s Method classifies
pixels into background and foreground based on their intensity levels and was
applied to the audio by converting it into a frequency histogram from the
Fourier transform [2]. This effectively filtered out white noise from the
input signals and improved the sound classification accuracy. Figures
comparing the sound before and after applying Fast Fourier Transform (FFT)
with Otsu’s Method are presented below in Figure 4.

Figure 3: Figure 4: The sound waves before and after noise reduction (left)
and the audio spectrograms before and after noise reduction (right)

We implemented a Sound Spectrogram Transformer that was trained on the ESC-50
dataset, a dataset for Environmental Sound Classification which consists of
2000 environmental audio recordings suitable for benchmarking methods of
environmental sound classification, using 5-fold cross-validation to prevent
overfitting. Additionally, this dataset was used in order to limit the amount
of semantical classes to only classify sounds frequently associated with real-
time environments. As shown in Figure 5, this transformer takes the audio
waveform input of T seconds, outputted from Otsu’s Method, and is converted
into a 128x100t spectrogram using log Mel filterbank features computed with a
25ms Hamming window every 10ms. The model outputs a Transformer encoder’s
[CLS] token, which serves as the audio spectrogram representation for
classification. The corresponding label is matched by using a linear layer
with sigmoid activation.

To assess the accuracy of the classification data produced by the transformer
model in a real-time environment, the resultant semantic class was paired with
a timestamp associated with the live sound data. This live sound data was
simulated by playing a variety of sounds around the mic array with white noise
that included people talking, laughter, and the air conditioner running. The
aligned time series data were then used to calculate errors and determine
accuracy of the real-time classification system. The sample capture rate for
the audio input was 0.205 seconds.

Figure 4: Figure 5: Audio Spectrogram Transformer (AST) Model Architecture

The only constraint for the electrical system was that we needed to provide an
equal amount of current and voltage to each of the output nodes located on the
vest. The intensity of the electrical current expelled from the TENS
electrodes is directly related to the current from the Raspberry Pi, which
outputs a maximum current of 50 hertz through the 5V pin-outs. By applying two
strategic parallel circuits, as demonstrated in Figure 6, we were able to
ensure that each output node only revised a maximum current of 12.5 HZ. 12.5
HZ is under-powered for the typical TENS electrode (50 Hertz), the current is
still high enough to be felt and also guarantees user safety.

Figure 5: Figure 6: Electrical Diagram

The process for setting up the OS and driver was challenging due to many
libraries being outdated and the Re-Speaker 4 Mic Array Drivers being meant
for a 32-bit version of Raspberry Pi. We downloaded specific versions of
packages and downgraded our Raspberry Pi OS to 64-bit Raspberry Pi OS Bullseye
11.0 Debian Release. Lastly, the Re-Speaker 4-Mic Array Drivers was written
for a 32-bit system with Linux Kernel Version 4.9.80+ while the earliest
release of Raspberry Pi 64-bit OS had 5.10.40+ Kernels. We addressed this
issue by modifying the ReSpeaker Driver Scripts and creating overlay
diversions for every component of the driver.

##  4 Results

Figure 6: Figure 7 : Prototype Development

Referencing Figure 7, Prototype 1 had a simple design which served as a basic
setup on a breadboard, comprising the ReSpeaker, LEDs, and the Raspberry Pi.
This prototype allowed us to test our ML algorithms and hardware, which
enabled accurate node activations and the assessment of our model’s accuracy
in the presence of live background noise. The next stage of prototyping
involved integrating our breadboard design with the mesh vest, to test its
practical functionality and identify any issues with our original mesh design.
Prototype 3 was focused on making the vest portable by incorporating a battery
pack, streamlining the wiring, and correcting node positioning. Additionally,
the position of the ReSpeaker was adjusted to eliminate significant vest
feedback that was observed in the previous prototype. As a result, the device
is now easy to use, simply by putting on the vest and switching on the battery
pack. We tested the device with LEDS by having a test subject put on the vest
and had a person playing sounds on a phone from 2 meters away. By visually
being able to see the changes in luminosity of the LEDS, we could confirm that
our depth and direction algorithms were working.

As shown in Table 1, the implementation of Fast Fourier Transforms with Otsu’s
Method outperformed three other noise reduction techniques with a peak signal-
to-noise ratio (PSNR) of 57.5 dB. FFT with Otsu’s Method is an uncommon
technique for noise reduction, but it proved to be more effective than each of
the other algorithms due to its higher PSNR value, which indicates the
algorithm’s ability to reduce noise. Referencing Table 2, our Audio
Spectrogram Transformer model outperformed many traditional sound
classification models in terms of accuracy, with higher accuracies on the
ESC-50 dataset and higher mean average precisions (mAPs) on the AudioSet
dataset. Specifically, the Audio Spectrogram Transformer achieved an accuracy
of 95.7% on the ESC-50 dataset and a 0.485 mAP on the AudioSet.

Table 1: Comparison of PSNR Values for Various Noise Reduction Algorithms

Algorithm | PSNR (dB)
---|---
Wiener Filtering | 36.791
Spectral Gating | 55.235
Spectral Subtraction | 57.116
FFT with Otsu’s Method | 57.529

Note: Peak signal-to-noise ratio (PSNR) measures the amount of noise present
in a signal and how it affects the quality of the signal. It is expressed in
decibels (dB), and a higher PSNR value indicates a lower amount of noise and a
higher quality signal.

Table 2: Table 2: Accuracy Comparison of Various Sound Classification Models on the ESC-50 Dataset Model | Accuracy (%)
---|---
auDeep | 64.3
WSNet | 66.25
SoundNet | 74.2
Human | 81.3
AclNet | 85.65
AVID | 89.2
EAT-S | 95.25
AST | 95.7

Also, we conducted a small test of the electrical stimulation of the TENS
electrodes on the vest. We had 10 different human volunteers (55-65 years old)
report the amount of stimulation that they received on a scale from 0-10 (with
0 being no electrical stimulation and 10 being the stimulation with the
maximum current output) when all the TENS were activated. This process was
then repeated in 3 different environments.

Table 3: The Effect of Different Current levels from a TENS Electrode and
Changes of Environment on the Physical Stimulation Received (0-10)

Testing Volunteer # Environment Indoors at 68°F (control) Outdoors at 68°F
Outdoors at 32°F Current (mA) 0 6 12.5 0 6 12.5 0 6 12.5 1 0 5 10 0 6 10 0 5
10 2 0 6 10 1 6 10 1 6 10 3 0 5 10 0 5 10 1 5 10 4 0 5 10 0 5 10 0 5 10 5 0 6
10 0 5 10 1 5 10 6 0 5 10 0 5 9 0 6 10 7 0 6 10 0 5 10 0 5 10 8 0 6 10 0 5 10
0 6 9 9 0 6 10 0 5 10 0 6 10 10 0 6 10 0 5 10 0 6 10

##  5 Discussion

Our implementation of the Fast Fourier Transform (FFT) algorithm using Otsu’s
Method for noise thresholding effectively removed white noise from our audio
input, which preserved the model’s sound classification accuracy of 95.7%.
This confirmed that our preprocessing pipeline was efficient in accurate sound
classification in real-time environments. This implementation was much more
effective than other methods, since the peak signal-to-noise ratio (PSNR) was
higher than other methods without requiring machine learning, which allowed
our algorithm to process under the limited computational powers. We tested the
different noise reduction algorithms in Table 1 by calculating the PSNR (using
the PSNR formula) of the original and denoised sounds for each of the four
different noise reduction algorithms we considered.

As shown in Table 3, the TENS stimulation is largely unaffected by changes in
temperature and is largely unaffected by environmental factors. Due to a lack
of professional expertise and knowledge of wiring and Raspberry Pi current
control, we could not guarantee the safety of the user due to the possible
rapid changes in current. Thus, we replaced the TENS with LEDs, which were the
closest alternative to the TENS as they could describe the distance and depth
through visual aid with the brightness of the LED indicating distance and the
specific placement of the LED representing direction. As a result, EchoVest
serves as a proof-of-concept that the various components and pipeline
accomplish the task.

Currently, we are creating an app that hosts the classification model and
sound preprocessing on the cloud because of potential for customizable
features. However, EchoVest will still be completely localized and will not
require the app to function. The app serves as a means to increase
functionality with wifi and bluetooth capabilities by giving the user the
ability to not be notified of certain sounds and change the various strengths
of the TENS to the person’s suitability. Furthermore, our future goal includes
a partnership with companies like Ring that would allow us to utilize and
implement our sound classification and preprocessing pipeline to their system.
Their systems are outdated and do not take out sufficient white noise and,
therefore, limit the functionality of their doorbell system. In addition to
company partnerships, EchoVest can be directly applied to search and rescue
operations as it gives personnel a heightened sense of their surroundings.
EchoVest has multiple functionalities, which makes EchoVest a multi-purpose
solution to a variety of real world problems.

##  6 Conclusion

In this paper, we developed an accessible, cost-efficient wearable product for
localizing and classifying sounds. We further demonstrate that our
preprocessing pipeline, consisting of Otsu’s Method and FFT, sufficiently
preserves sound classification accuracy in a real-time environment. EchoVest
utilizes optimized machine learning to efficiently and effectively lower
computation costs, thereby reducing product costs. EchoVest costs a maximum of
$98 per unit to manufacture and is easily accessible to the general public due
to the lack of customization needed. Traditional hearing aid devices require
numerous prerequisites, such as hearing test, medical clearance, and hearing
aid evaluation. Constrastingly, EchoVest will be available to the public
without any specialization or pre-requisites because its only variable would
be the size of the vest.

## References

  * [1] Bisgaard, N., Zimmer, S., Laureyns, M., & Groth, J. (2021).  A model for estimating hearing aid coverage world-wide using historical data on hearing aid sales.  _International Journal of Audiology_ , 61(10), 841–849.  https://doi.org/10.1080/14992027.2021.1962551
  * [2] Chen, H., & Gururajan, R. (2012).  Otsu’s Threshold Selection Method Applied in De-noising Heart Sound of the Digital Stethoscope Record.  _Lecture Notes in Electrical Engineering_ , 239–244.  https://doi.org/10.1007/978-3-642-26001-8_31
  * [3] Gong, Y., Chung, Y.-A., & Glass, J. (2021).  AST: Audio Spectrogram Transformer.  _ArXiv:2104.01778_ https://arxiv.org/abs/2104.01778
  * [4] Grondin, F., & Michaud, F. (2019).  Lightweight and optimized sound source localization and tracking methods for open and closed microphone array configurations.  _Robotics and Autonomous Systems, 113, 63–80._ https://doi.org/10.1016/j.robot.2019.01.002
  * [5] Health, C. for D. and R. (2019, April 24).  Hearing Aids. FDA.  https://www.fda.gov/medical-devices/consumer-products/hearing-aids
  * [6] Hyvärinen, A., & Oja, E. (2000).  Independent Component Analysis: Algorithms and Applications.  _Neural Networks, 13(45), 411–430._ https://www.cs.helsinki.fi/u/ahyvarin/papers/NN00new.pdf
  * [7] Jolliffe, I. T., & Cadima, J. (2016).  Principal component analysis: a review and recent developments.  _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374(2065), 20150202._ https://doi.org/10.1098/rsta.2015.0202
  * [8] Kwon, B., Park, Y., & Park, Y. (2010, October 1).  Analysis of the GCC-PHAT technique for multiple sources.  _IEEE Xplore._ https://doi.org/10.1109/ICCAS.2010.5670137
  * [9] McKee, M. M., Choi, H., Wilson, S., DeJonckheere, M. J., Zazove, P., & Levy, H. (2018).  Determinants of Hearing Aid Use Among Older Americans With Hearing Loss. The Gerontologist.  https://doi.org/10.1093/geront/gny051
  * [10] National Institute on Deafness and other Communication Disorders. (2018, June 15).  Cochlear Implants.  _NIDCD._ https://www.nidcd.nih.gov/health/cochlear-implants
  * [11] Nunez, K. (2020, February 27).  Cochlear Implant: Cost, Pros, Cons, Risks, How It Works.  _Healthline._ https://www.healthline.com/health/cochlear-implant
  * [12] Papers with Code - ESC-50 Benchmark (Audio Classification). (n.d.).  Paperswithcode.com.  _Retrieved February 2, 2023, from_ https://paperswithcode.com/sota/audio-classification-on-esc-50
  * [13] Rev, & Rev. (2021, May 17).  Exploring Your Speech-to-text Options: Advantages and Disadvantages of Speech Recognition Software. Rev.  https://www.rev.com/blog/speech-to-text-technology/advantages-and-disadvantages-of-speech-recognition-software
  * [14] Transcutaneous electrical nerve stimulator (TENS). (2018, April 4).  University of Iowa Hospitals & Clinics.  https://uihc.org/health-topics/transcutaneous-electrical-nerve-stimulator-tens
  * [15] Wang, Y.-X., & Zhang, Y.-J. (2013).  Nonnegative Matrix Factorization: A Comprehensive Review.  _IEEE Transactions on Knowledge and Data Engineering, 25(6), 1336–1353._ https://doi.org/10.1109/tkde.2012.51
  * [16] Watson, A. M. (2022, August 16).  How Much Do Hearing Aids Cost?  _GoodRx._ https://www.goodrx.com/health-topic/ear/hearing-aid-cost
  * [17] World Health Organization: WHO. (2019, September 18).  Hearing loss. Who.int; World Health Organization: WHO.  https://www.who.int/health-topics/hearing-loss#tab=tab_1


[a]Yannick Meurice

###### Abstract

We discuss recent progress in Tensor Lattice Field Theory and economical,
symmetry preserving, truncations suitable for quantum
computations/simulations. We focus on spin and gauge models with continuous
Abelian symmetries such as the Abelian Higgs model and emphasize noise-robust
implementations of Gauss’s law. We discuss recent progress concerning the
comparison between field digitizations and character expansions, symmetry
breaking in tensor language, wave-packet preparation and possible new
implementations of Abelian models using Rydberg atoms.

##  1 Introduction

Tensor network methods are playing an increasingly important role in lattice
gauge theory. The annual lattice conferences certainly helped fostering
contacts between the communities involved in the early developments [1, 2, 3,
4, 5, 6, 7]. The number of contributions has grown steadily with the years. In
2021, the keyword “tensor network" appears 24 times in the book of abstracts
and the keyword “tensor renormalization" 12 times.

The tensor network approaches can be divided roughly into two categories. The
first one relies on Hamiltonian formulations and often makes use of powerful
computational tools developed in the condensed matter community, such as the
Matrix Product States (MPS) in 1+1 dimensions and higher-dimensional
extensions such as Projected Entangled Pair States (PEPS). For recent general
reviews see [8, 9]; for reviews focused on lattice gauge theory see [10, 11];
for recent use of PEPS in gauge theories see [12]. For Hamiltonian
formulations of non-Abelian gauge theories at this conference, see [13]. For
Hamiltonian studies of the Abelian Higgs and $O(2)$ models at this conference
see [14, 15].

The other category of approaches is based on the conventional Lagrangian
lattice formulation and relies on various types of dualities and character
expansions. It is sometimes called Tensor Lattice Field Theory (TLFT). For
recent reviews that provide a roadmap resembling the one followed by standard
review articles [16, 17] on lattice gauge theory (the “Kogut sequence") see
[18, 19]. When the field variables are continuous and compact, character
expansions provide discrete indices that fit naturally the needs of quantum
computing. This is the main theme of these proceedings.

TLFT provides reformulations that are exactly equivalent to the original
lattice models. One of the original motivations for this approach was the
possibility of doing local coarse-graining [20, 21, 18]. This involves
approximations (“truncations") that can be checked in regimes where the usual
Monte Carlo sampling in configuration space is possible. This is often called
the Tensor Renormalization Group (TRG). For examples in this conference see
for instance [22, 23, 24, 25, 26]. Another possibility is to sample the tensor
configurations appearing in the reformulations as in the worm algorithm [27,
28, 29, 7, 30, 31]. These approaches are completely Lagrangian based. However,
it is possible to connect with the Hamiltonian approach by using the transfer
matrix and taking the time-continuum limit [18, 19, 8, 30].

Basic aspects of TLFT are reviewed in Sec. 2. Because of the equivalence with
the standard formulation, symmetries of the action lead to known identities or
theorems which need to be recast in the new formulation. In particular, local
gauge invariance is manifest at every step of the reformulation. In addition,
truncations preserve symmetries [32, 33]. These questions and economical
implementation of Gauss’s laws are briefly discussed in Sec. 3. The simple
case of the Abelian Higgs model in 1+1 dimension and a possible implementation
with Rydberg-dressed atoms in optical lattice [34, 35] are reviewed in Sec. 4.
This inspired proposals to implement the same model with configurable arrays
of Rydberg atoms [36, 37] which were presented at the conference. This is
discussed in Sec. 5. More details can be found in a recent publication [38].
Finally, related results presented at this conference [39, 26, 15, 14] are
briefly summarized in Sec. 6.

##  2 Tensor Lattice Field Theory for Quantum Computing

Quantum computing for quantum field theory requires a complete discretization.
Discretization of space is well understood from the success of lattice QCD.
Discretization of field integration for continuous field variables can be
achieved with various methods. In most lattice simulations, the variables of
integration are compact and tensor formulations use character expansions (such
as Fourier series) to re-express the partition function as discrete sums of
contracted tensors. The “hard" integrals are done exactly and field
integrations provide Kronecker deltas that encode the symmetries. For
continuous field variables, the sums are infinite, but truncations to finite
sums do not break symmetries [32, 33]. Our general expectation is that the
correct large distance physics can be obtained with “a few qubits per unit
cells". Comparison with other methods based on field discretization or quantum
links are highly desirable [40, 41, 42, 31].

As a simple illustration of the discrete reformulation, we consider the
lattice non-linear $O(2)$ model. Its partition function reads:

$Z_{O(2)}=\prod_{x}\int_{-\pi}^{\pi}\frac{d\varphi_{x}}{2\pi}e^{\beta\sum\limits_{x,\mu}\cos(\varphi_{x+\hat{\mu}}-\varphi_{x})}.$
(1)

Using the character expansion, which in this case amounts to a Fourier series,
we expand the Boltzmann weights as

$e^{\beta\cos(\varphi_{x+\hat{\mu}}-\varphi_{x})}=\sum_{n_{x,\mu}=-\infty}^{\infty}e^{in_{x,\mu}(\varphi_{x+\hat{\mu}}-\varphi_{x})}I_{n_{x,\mu}}(\beta),$
(2)

and obtain

$Z_{O(2)}={\rm Tr}\prod_{x}T^{(x)}_{n_{x-\hat{1},1},n_{x,1},\dots,n_{x,D}}.$
(3)

with a local tensor $T^{(x)}$ with $2D$ indices. The explicit form is

$T^{(x)}_{n_{x-\hat{1},1},n_{x,1},\dots,n_{x-\hat{D},D},n_{x,D}}=\sqrt{I_{n_{x-\hat{1},1}}I_{n_{x,1}},\dots,I_{n_{x-\hat{D},D}}I_{n_{x,D}}}\times\delta_{n_{x,\text{out}},n_{x,\text{in}}},$
(4)

Schematically, we replace integrations over circles located at the sites by
discrete sums over indices located on links:

${\color[rgb]{0,0,1}\prod_{x}\int_{-\pi}^{\pi}d\varphi_{x}}\Longrightarrow{\color[rgb]{1,0,0}\sum_{\\{n\\}}}$
(5)

This can be ilustrated graphically for $D=2$ as:

##  3 Truncations, symmetries and Gauss’s law

Tensorial truncations are compatible with the general identities derived from
global and local symmetries [32, 33]. Symmetries are encoded in tensor
selection rules and not in the numerical values of the tensor elements.
Universal properties of these models can be reproduced with highly simplified
formulations desirable for implementations with quantum computers or for
quantum simulations experiments ("few qubits per unit cells"). In tensorial
reformulations Noether theorem can be stated as: for each symmetry there is a
corresponding tensor redundancy (and we can "fix" the corresponding
integration variable). This applies not only to continuous but also discrete
symmetries. Note however, that truncations may affect the type of phase
transitions [43, 14, 15].

Noise robust enforcement of Gauss’s law is easy for bosonic matter. For the
Abelian Higgs model, the gauge quantum numbers determine the matter quantum
numbers completely. This is a discrete version of $\partial_{j}E^{j}=\rho$. In
the pure gauge case, one can associate new quantum numbers with the space-
space plaquettes [33] in a discrete version of $E^{k}=\partial_{j}C^{jk}$. See
also [44, 45, 46, 47] for related discussions.

##  4 Optical lattice implementation of the Abelian Higgs model

One of the simplest non-trivial gauge theories is the Abelian Higgs model. See
[34] for the TLFT formulation. After decoupling the Higgs model and taking the
time-continuum limit in 1+1 dimensions, we obtain an Hamiltonian that can be
implemented on an optical lattice [35] as illustrated in the figure below for
a spin-2 approximation.

$H={\color[rgb]{1,0,0}\frac{U}{2}\sum_{i}\left(L^{z}_{(i)}\right)^{2}}+{\color[rgb]{0,0,1}\frac{Y}{2}\sum_{i}(L^{z}_{(i)}-L^{z}_{(i+1)})^{2}}-{\color[rgb]{0,1,0}X\sum_{i}U^{x}_{(i)}}.$
(6)

The color coded explanations [19] for an optical lattice ladder with one atom
per rung are: tunneling along the vertical direction ($\bar{L}^{z}=\pm 2,\ \pm
1,\ 0$, green), no tunneling in the the horizontal direction but short range
attractive (Rydberg-dressed) interactions (blue). A parabolic potential is
applied in the spin (vertical) direction (red). See [34, 35] for details.

##  5 Rydberg atom simulators

Arrays of ${}^{87}Rb$ atoms separated by adjustable distances, homogeneously
coupled to the excited Rydberg state $\mathinner{|{r}\rangle}$ with detuning
$\Delta$ [36] offer new possibilities for quantum simulations [48, 49, 50].
Each atom has 2 states: $\mathinner{|{g}\rangle}$, the ground state, and
$\mathinner{|{r}\rangle}$ the Rydberg state. We use $n$ to denote the
occupation of $\mathinner{|{r}\rangle}$; in the “qubit" picture:
$\mathinner{|{g}\rangle}\rightarrow\mathinner{|{0}\rangle}$ and
$\mathinner{|{r}\rangle}\rightarrow\mathinner{|{1}\rangle}$. The Hamiltonian
reads

$H=\frac{\Omega}{2}\sum_{i}(\mathinner{|{g_{i}}\rangle}\mathinner{\langle{r_{i}}|}+\mathinner{|{r_{i}}\rangle}\mathinner{\langle{g_{i}}|})-\Delta\sum_{i}n_{i}+\sum_{i<j}\Omega
R_{b}^{6}/R_{ij}^{6}n_{i}n_{j},$ (7)

with $i$ and $j$ run over atoms in configurable arrays.

We will make use of the Rydberg blockade which means that
$\mathinner{|{rr}\rangle}$ is suppressed if two atoms are too close.
Programming means assembling atom arrays with tweezers. This setup was used to
identify a rich phase diagram in the (distance between atoms, $\Delta/\Omega$)
plane [37].

We now turn to proposals for spin-2 and spin-1 truncations presented at the
conference and discussed in more details in [38]. A simple adaptation of the
spin-2 optical lattice construction of Sec. 4 is illustrated below. As
discussed below, the $X$ term can be induced by the $\Omega$ interactions.

In view of the currently available technology, we suggested to start with a
simpler construction for spin-1:

Each set of three atoms can be seen as a qutrit (for motivations see [51,
52]). The $X$-term (tunneling in the optical lattice formulation) is induced
by transitions to heavy states at second order in perturbation theory [38].

An even more economical version with only two atoms per links is given below:

For the one spin system we can compare the simulator with the target model for
spin-1:

$H=\frac{U}{2}(L^{z})^{2}-XU^{x}.$ (8)

$H$ is invariant under charge conjugation: we use $\mathcal{C}$ eigenstates

$\mathinner{|{\pm}\rangle}\equiv\frac{1}{\sqrt{2}}(\mathinner{|{1}\rangle}\pm\mathinner{|{-1}\rangle}).$
(9) $\mathcal{C}\mathinner{|{\pm}\rangle}=\pm\mathinner{|{\pm}\rangle}.$ (10)

They are also eigenstates of $(L^{z})^{2}$ with eigenvalue 1, and
$\mathcal{C}\mathinner{|{0}\rangle}=\mathinner{|{0}\rangle}$. There is only
one $\mathcal{C}$-odd state which is $\mathinner{|{-}\rangle}$.

$U^{x}\mathinner{|{-}\rangle}=0;\
H\mathinner{|{-}\rangle}=\frac{U}{2}\mathinner{|{-}\rangle}.$ (11)

The two atom setup discussed above provides a very simple implementation of
the one-spin system. Its Hamiltonian reads

$\displaystyle H^{2R}=$
$\displaystyle-\Delta(n_{+1}+n_{-1})+V_{0}n_{+1}n_{-1}$
$\displaystyle+\frac{\Omega}{2}\sum_{\pm 1}(\mathinner{|{g_{\pm
1}}\rangle}\mathinner{\langle{r_{\pm 1}}|}+\mathinner{|{r_{\pm
1}}\rangle}\mathinner{\langle{g_{\pm 1}}|}).$

The correspondence between the states of the simulator and the target is given
below:

state | ket | target | energy ($\Omega=0$)
---|---|---|---
$\begin{matrix}\bullet\\\ \circ\end{matrix}$ | $\mathinner{|{rg}\rangle}$ | $\mathinner{|{1}\rangle}$ | $-\Delta$
$\begin{matrix}\circ\\\ \circ\end{matrix}$ | $\mathinner{|{gg}\rangle}$ | $\mathinner{|{0}\rangle}$ | 0
$\begin{matrix}\circ\\\ \bullet\end{matrix}$ | $\mathinner{|{gr}\rangle}$ | $\mathinner{|{-1}\rangle}$ | $-\Delta$
$\begin{matrix}\bullet\\\ \bullet\end{matrix}$ | $\mathinner{|{rr}\rangle}$ | - | $-2\Delta+V_{0}$

The matching between the target and the two-atom simulator is simply
$\Delta=-\frac{U}{2}$ and $\Omega=-X$. Except for possible transitions to
$\mathinner{|{rr}\rangle}$, the correspondence is exact and the linear formula
applies for arbitrary values of $X$. A comparison between the target and the
simulator is given in Fig. 1. More details for setups with 3 (the other spin-1
proposal), and 4 or 6 atoms with spin-spin interactions can be found in [38].

Figure 1: $|\mathinner{\langle{m}|}U(t)\mathinner{|{m=1}\rangle}|^{2}$, one
site with exact Hamiltonian $U=1,\ X=0.5$ (solid lines) and Rydberg
Hamiltonian with $\Omega=-0.5$, $\Delta=-0.5$ and $V_{0}=64|\Omega|=32$ (empty
symbols) as in [38].

##  6 Related results

In this section, we give a brief summary of related results presented at this
conference.

###  6.1 Phase shifts from real time evolution

The quantum Ising model provides a nice connection between the TLFT
formulation of the Ising model and quantum computing. In [39], methods to
construct wavepackets and measure phase shifts using IBMQ and trapped ions
facilities [53] were presented. A renormalized reflection probability that can
be measured in the early stages of a collision led to estimates of a time
delay related to the phase shift by the Wigner formula:

$\Delta t^{\star}=\delta^{\prime}(k)/(\partial E/\partial k).$ (12)

###  6.2 Critical behavior of Spin truncations in the $O(2)$ limit

In [14, 15], the effect of truncations on the critical behavior were discussed
for the $O(2)$ model and the Abelian Higgs model. For instance [43], in the
case of the charge representation of the $O(2)$ model

$\hat{H}_{charge}=\frac{Y}{2}\sum_{l=1}^{L+1}(\hat{S}_{l}^{z})^{2}-\frac{X}{2}\sum_{i=1}^{L}(\hat{U}_{l}^{+}\hat{U}_{l+1}^{-}+\hat{U}_{l}^{-}\hat{U}_{l+1}^{+}),$
(13)

energy gaps $\Delta E_{V=\infty}$ for spin truncations $S=1,2,3,4$ were
considered. For $S\geq 2$ a behavior of the type $A\exp(-b/\sqrt{Y-Y_{c}})$ as
in regular Kosterlitz Thouless (KT) was found to be accurate. However for
$S=1$, the parametrization $A\sqrt{Y-Y_{c}}\exp\left[-b/(Y-Y_{c})\right]$
which corresponds to a $SU(2)$ symmetry on the KT separatrix was found to be
more accurate.

###  6.3 Interpolation among $Z_{q}$ clock models

In [54, 26] a $O(2)$ model with symmetry breaking :

$\Delta S=\gamma\sum_{x}\cos(q\varphi_{x}),$ (14)

was considered. In the limit of large $\gamma$, the angles $\varphi=\frac{2\pi
k}{q}\ k=0,1,..,\lfloor q\rfloor$ are strongly favored. The study of the
symmetry breaking in tensor language is quite interesting and give accurate
results at small $\beta$. For integer $q$, we have a $Z_{q}$ symmetry but for
non-integer $q$ there is only a $Z_{2}$ symmetry. The schematic phase diagram
from [54] is shown below. The finite $\gamma$ region remains to be explored
and could be related to the Rydberg arrays phase diagram in [37].

##  7 Conclusions

Tensor Lattice Field Theory is a generic tool to discretize path integral
formulations of lattice models. For gauge theories, the reformulation can be
obtained by a complete integration over the gauge configurations and is
manifestly gauge-invariant. Microscopic truncations respect symmetries and
have interesting critical properties. Noether theorem can be re-expressed for
any kind (continuous, discrete or finite) Abelian symmetry group: for each
symmetry, there is a corresponding tensor redundancy. Noise-robust economical
implementation of Gauss’s law for pure gauge models. Rydberg atom
implementations of scalar QED are expected in the near future. The tensor
formulation provides a space-time picture that makes clear the building blocks
of the quantum algorithms and suggest ways to implement them with NISQ
devices.

Acknowledgements: This research was supported in part by the Dept. of Energy
under Award Numbers DE-SC0010113 and DE-SC0019139. EG was partially supported
by the DOE through the Fermilab QuantiSED program in the area of
"Intersections of QIS and Theoretical Particle Physics." This manuscript has
been authored by Fermi Research Alliance, LLC under Contract No. DE-
AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office
of High Energy Physics.

## References

  * [1] M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, “Matrix Product States for Lattice Field Theories,” _PoS_ , vol. LATTICE2013, p. 332, 2014.
  * [2] Y. Meurice, A. Denbleyker, Y. Liu, T. Xiang, Z. Xie, J.-F. Yu, J. Unmuth-Yockey, and H. Zou, “Comparing Tensor Renormalization Group and Monte Carlo calculations for spin and gauge models,” _PoS_ , vol. LATTICE2013, p. 329, 2014.
  * [3] K. Van Acoleyen, B. Buyens, J. Haegeman, and F. Verstraete, “Matrix product states for Hamiltonian lattice gauge theories,” _PoS_ , vol. LATTICE2014, p. 308, 2014.
  * [4] Y. Shimizu and Y. Kuramashi, “Grassmann Tensor Renormalization Group Study of Lattice QED with Theta Term in Two Dimensions,” _PoS_ , vol. LATTICE2014, p. 303, 2014.
  * [5] H. Saito, M. C. Bañuls, K. Cichy, J. I. Cirac, and K. Jansen, “The temperature dependence of the chiral condensate in the Schwinger model with Matrix Product States,” _PoS_ , vol. LATTICE2014, p. 302, 2014.
  * [6] J. F. Unmuth-Yockey, Y. Meurice, J. Osborn, and H. Zou, “Tensor renormalization group study of the 2d O(3) model,” _PoS_ , vol. LATTICE2014, p. 325, 2014.
  * [7] Y. Meurice, Y. Liu, J. Unmuth-Yockey, L.-P. Yang, and H. Zou, “Sampling versus Blocking,” _PoS_ , vol. LATTICE2014, p. 319, 2014.
  * [8] J. Haegeman and F. Verstraete, “Diagonalizing transfer matrices and matrix product operators: A medley of exact and computational methods,” _Annual Review of Condensed Matter Physics_ , vol. 8, no. 1, pp. 355–406, 2017.
  * [9] S.-J. Ran, E. Tirrito, C. Peng, X. Chen, L. Tagliacozzo, G. Su, and M. Lewenstein, _Tensor Network Contractions_ , 2020, vol. 964.
  * [10] M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and S. Kühn, “Tensor Networks and their use for Lattice Gauge Theories,” _PoS_ , vol. LATTICE2018, p. 022, 2018.
  * [11] M. C. Bañuls _et al._ , “Simulating Lattice Gauge Theories within Quantum Technologies,” _Eur. Phys. J. D_ , vol. 74, no. 8, p. 165, 2020\.
  * [12] P. Emonts, M. C. Bañuls, I. Cirac, and E. Zohar, “Variational Monte Carlo simulation with tensor networks of a pure $\mathbb{Z}_{3}$ gauge theory in (2+1)d,” _Phys. Rev. D_ , vol. 102, no. 7, p. 074501, 2020.
  * [13] Z. Davoudi, I. Raychowdhury, and A. Shaw, “Search for efficient formulations for Hamiltonian simulation of non-Abelian lattice gauge theories,” _Phys. Rev. D_ , vol. 104, no. 7, p. 074505, 2021.
  * [14] J. Zhang, this conference, 2021.
  * [15] S.-W. Tsai, this conference, 2021.
  * [16] J. B. Kogut, “An introduction to lattice gauge theory and spin systems,” _Rev. Mod. Phys._ , vol. 51, pp. 659–713, Oct 1979. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.51.659
  * [17] J. B. Kogut, “The lattice gauge theory approach to quantum chromodynamics,” _Rev. Mod. Phys._ , vol. 55, pp. 775–836, Jul 1983. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.55.775
  * [18] Y. Meurice, R. Sakai, and J. Unmuth-Yockey, “Tensor lattice field theory with applications to the renormalization group and quantum computing,” 10 2020.
  * [19] P. Y. Meurice, _Quantum Field Theory_. IOP, 2 2021.
  * [20] Y. Meurice, “Accurate exponents from approximate tensor renormalizations,” _Phys. Rev. B_ , vol. 87, no. 6, p. 064422, 2013.
  * [21] Y. Liu, Y. Meurice, M. P. Qin, J. Unmuth-Yockey, T. Xiang, Z. Y. Xie, J. F. Yu, and H. Zou, “Exact Blocking Formulas for Spin and Gauge Models,” _Phys. Rev. D_ , vol. 88, p. 056005, 2013.
  * [22] D. Kadoh, this conference, 2021.
  * [23] S. Akiyama, Y. Kuramashi, and Y. Yoshimura, “Quantum Field Theories with Tensor Renormalization Group,” in _38th International Symposium on Lattice Field Theory_ , 11 2021.
  * [24] J. Bloch, R. Lohmayer, S. Schweiss, and J. Unmuth-Yockey, “Effective $\mathbb{Z}_{3}$ model for finite-density QCD with tensor networks,” in _38th International Symposium on Lattice Field Theory_ , 10 2021.
  * [25] R. Sakai, this conference, 2021.
  * [26] L. Hostetler, J. Zhang, R. Sakai, J. Unmuth-Yockey, A. Bazavov, and Y. Meurice, “Clock model interpolation and symmetry breaking in O(2) models,” _PoS_ , vol. LATTICE2021, p. 353, 2021.
  * [27] N. Prokof’ev and B. Svistunov, “Worm Algorithms for Classical Statistical Models,” _Phys. Rev. Lett._ , vol. 87, p. 160601, 2001.
  * [28] T. Korzec and U. Wolff, “A worm-inspired algorithm for the simulation of Abelian gauge theories,” _PoS_ , vol. LATTICE2010, p. 029, 2010.
  * [29] D. Banerjee and S. Chandrasekharan, “Finite size effects in the presence of a chemical potential: A study in the classical nonlinear $o(2)$ sigma model,” _Phys. Rev. D_ , vol. 81, p. 125007, Jun 2010. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.81.125007
  * [30] H. Zou, Y. Liu, C.-Y. Lai, J. Unmuth-Yockey, L.-P. Yang, A. Bazavov, Z. Y. Xie, T. Xiang, S. Chandrasekharan, S.-W. Tsai, and Y. Meurice, “Progress towards quantum simulating the classical $\text{O}(2)$ model,” _Phys. Rev. A_ , vol. 90, p. 063603, Dec 2014. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevA.90.063603
  * [31] H. Singh and S. Chandrasekharan, “Qubit regularization of the O(3) sigma model,” _Phys. Rev. D_ , vol. 100, p. 054505, Sep 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.100.054505
  * [32] Y. Meurice, “Examples of symmetry-preserving truncations in tensor field theory,” _Phys. Rev. D_ , vol. 100, no. 1, p. 014506, 2019.
  * [33] Y. Meurice, “Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories,” _Phys. Rev. D_ , vol. 102, no. 1, p. 014506, 2020.
  * [34] A. Bazavov, Y. Meurice, S.-W. Tsai, J. Unmuth-Yockey, and J. Zhang, “Gauge-invariant implementation of the Abelian Higgs model on optical lattices,” _Phys. Rev. D_ , vol. 92, no. 7, p. 076003, 2015.
  * [35] J. Zhang, J. Unmuth-Yockey, J. Zeiher, A. Bazavov, S. W. Tsai, and Y. Meurice, “Quantum simulation of the universal features of the Polyakov loop,” _Phys. Rev. Lett._ , vol. 121, no. 22, p. 223201, 2018.
  * [36] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, “Probing many-body dynamics on a 51-atom quantum simulator,” _Nature_ , vol. 551, no. 7682, pp. 579–584, Nov. 2017.
  * [37] A. Keesling, A. Omran, H. Levine, H. Bernien, H. Pichler, S. Choi, R. Samajdar, S. Schwartz, P. Silvi, S. Sachdev, P. Zoller, M. Endres, M. Greiner, Vuletić, V. , and M. D. Lukin, “Quantum Kibble-Zurek mechanism and critical dynamics on a programmable Rydberg simulator,” _Nature_ , vol. 568, no. 7751, pp. 207–211, Apr. 2019\.
  * [38] Y. Meurice, “Theoretical methods to design and test quantum simulators for the compact Abelian Higgs model,” _Phys. Rev. D_ , vol. 104, no. 9, p. 094513, 2021.
  * [39] P. Dreher, this conference, 2021.
  * [40] N. Klco and M. J. Savage, “Digitization of scalar fields for quantum computing,” _Phys. Rev. A_ , vol. 99, no. 5, p. 052335, 2019.
  * [41] H. Lamm, S. Lawrence, and Y. Yamauchi, “General Methods for Digital Quantum Simulation of Gauge Theories,” _Phys. Rev. D_ , vol. 100, no. 3, p. 034518, 2019.
  * [42] R. C. Brower, D. Berenstein, and H. Kawai, “Lattice Gauge Theory for a Quantum Computer,” _PoS_ , vol. LATTICE2019, p. 112, 2020.
  * [43] J. Zhang, Y. Meurice, and S.-W. Tsai, “Truncation effects in the charge representation of the O(2) model,” _Phys. Rev. B_ , vol. 103, no. 24, p. 245137, 2021.
  * [44] J. Bender and E. Zohar, “Gauge redundancy-free formulation of compact QED with dynamical matter for quantum and classical computations,” _Phys. Rev. D_ , vol. 102, no. 11, p. 114517, 2020.
  * [45] J. F. Haase, L. Dellantonio, A. Celi, D. Paulson, A. Kan, K. Jansen, and C. A. Muschik, “A resource efficient approach for quantum and classical simulations of gauge theories in particle physics,” _Quantum_ , vol. 5, p. 393, 2021.
  * [46] J. Stryker, “Compiling Quantum Gauge Theories for Quantum Computation,” Ph.D. dissertation, U. Washington, Seattle (main), 2020.
  * [47] J. F. Unmuth-Yockey, “Gauge-invariant rotor hamiltonian from dual variables of 3d $u\mathbf{(}1\mathbf{)}$ gauge theory,” _Phys. Rev. D_ , vol. 99, p. 074502, Apr 2019. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevD.99.074502
  * [48] A. Celi, B. Vermersch, O. Viyuela, H. Pichler, M. D. Lukin, and P. Zoller, “Emerging Two-Dimensional Gauge Theories in Rydberg Configurable Arrays,” _Physical Review X_ , vol. 10, no. 2, p. 021057, Apr. 2020.
  * [49] F. M. Surace, P. P. Mazza, G. Giudici, A. Lerose, A. Gambassi, and M. Dalmonte, “Lattice Gauge Theories and String Dynamics in Rydberg Atom Quantum Simulators,” _Physical Review X_ , vol. 10, no. 2, p. 021041, Apr. 2020.
  * [50] S. Notarnicola, M. Collura, and S. Montangero, “Real-time-dynamics quantum simulation of (1 +1 )-dimensional lattice QED with Rydberg atoms,” _Physical Review Research_ , vol. 2, no. 1, p. 013288, Mar. 2020.
  * [51] E. Gustafson, “Prospects for Simulating a Qudit Based Model of (1+1)d Scalar QED,” _Phys. Rev. D_ , vol. 103, no. 11, p. 114505, 2021.
  * [52] A. Ciavarella, N. Klco, and M. J. Savage, “Trailhead for quantum simulation of SU(3) Yang-Mills lattice gauge theory in the local multiplet basis,” _Phys. Rev. D_ , vol. 103, no. 9, p. 094501, 2021.
  * [53] E. Gustafson, Y. Zhu, P. Dreher, N. M. Linke, and Y. Meurice, “Real-time quantum calculations of phase shifts using wave packet time delays,” _Phys. Rev. D_ , vol. 104, no. 5, p. 054507, 2021.
  * [54] L. Hostetler, J. Zhang, R. Sakai, J. Unmuth-Yockey, A. Bazavov, and Y. Meurice, “Clock model interpolation and symmetry breaking in O(2) models,” _Phys. Rev. D_ , vol. 104, no. 5, p. 054505, 2021.


# Weak Radiative Decays of Anti-triplet Bottomed Baryons in Light-Front Quark
Model

Chao-Qiang Geng, Chia-Wei Liu, Zheng-Yi Wei and Jiabao Zhang  School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, China

###### Abstract

We study the weak radiative decays of ${\bf B}_{b}\to{\bf B}_{n}\gamma$ with
${\bf B}_{b(n)}$ the anti-triplet-bottom (octet) baryons in the light front
quark model. We obtain that ${\cal B}(\Lambda_{b}\to\Lambda\gamma)=(7.1\pm
0.3)\times 10^{-6}$, which agrees well with the current experimental value of
$(7.1\pm 1.7)\times 10^{-6}$. We predict that ${\cal
B}(\Xi_{b}^{0}\to\Xi^{0}\gamma)=(1.0\pm 0.1)\times 10^{-5}$ and ${\cal
B}(\Xi_{b}^{-}\to\Xi^{-}\gamma)=(1.1\pm 0.1)\times 10^{-5}$, which are
consistent with the latest upper limits set by the LHCb collaboration. In
addition, we find that the $SU(3)_{F}$ flavor symmetry breaking effects for
the modes related to the $b\to d\gamma$ transition can be as large as $20\%$.

##  I Introduction

It is known that in the weak radiative decays associated with the $b\to
s\gamma$ transition, the photons are purely left-handed in the standard model
(SM) to ${\cal O}(m_{s}^{2}/m^{2}_{b})$ precision. Clearly, a signal of the
right-handed photons in the experiment would be a smoking gun of new physics
Pati:1974yy ; Mohapatra:1974gc ; Mohapatra:1974hk ; Senjanovic:1975rk ;
Senjanovic:1978ev ; Mohapatra:1980yp ; Lim:1981kv ; Everett:2001yy ;
Atwood:1997zr . However, as the photon polarizations can not be measured
directly at the current experimental $b$-facilities, such as LHCb, we have to
analyze the cascade decays of hadrons to extract the polarization information
LHCb:2020dof ; LHCb:2014vnw ; Geng:2021sxe ; Liu:2021rvt ;
GarciaMartin:2019bxm . In addition, since the two-body radiative decays are
factorizable, the processes have a clean background for the theoretical
computation.

Recently, the LHCb collaboration has reported the following decay branching
ratios LHCb:2021hfz ; LHCb:2019wwi :

${\cal B}(\Lambda_{b}\to\Lambda\gamma)=(7.1\pm 1.7)\times
10^{-6}\,,\quad\mathcal{B}(\Xi_{b}^{-}\to\Xi^{-}\gamma)<1.3\times 10^{-4}\,,$
(1)

which are the same sizes as the charmless nonleptonic two-body decays.
Remarkably, the LHCb collaboration has also measured the lifetimes of the
anti-triplet bottom baryons (${\bf B}_{b}$) with high precision LHCb:2014chk ;
LHCb:2014jst , and carried out a full angular analysis for $\Lambda_{b}\to
J/\psi\Lambda$ LHCb:2020iux . These results in the baryon decays clearly
provide great opportunities to test the SM. On the theoretical side, the
radiative bottom decays of ${\bf B}_{b}\to{\bf B}_{n}\gamma$ with ${\bf
B}_{n}$ the the low-lying octet baryons have been studied with many
approaches, such as the heavy quark effective theory Singer:1996xh ,
perturbative QCD He:2006ud , $SU(3)_{F}$ flavor symmetry Wang:2020wxn , light-
cone sum rule (LCSR) Wang:2008sm ; Olamaei:2021eyo , Bethe-Salpeter equation
(BSE) Liu:2019rpm , quark model (QM) Chatley:1982gw ; Gutsche:2013pp ;
Faustov:2017wbh ; Faustov:2017ous , and effective Lagrangian PhysRevD.51.1199
. In this paper, we adopt the light front quark model (LFQM), where the quark
spins and the center-of-mass motions of hadrons can be treated in a consistent
and fully relativistic manner, as the wave functions of the baryons are
manifestly boost invariant.

The LFQM has been extensively studied in the mesonic processes Schlumpf:1992vq
; Zhang:1994ti ; PhysRevD.57.5697 ; PhysRevD.59.114002 ; PhysRevD.62.074017 ;
PhysRevD.64.114024 ; Geng_2003 ; PhysRevD.69.074025 ; Geng:2016pyr ;
Cheng:2017pcq ; Chang:2019obq ; Shi:2016gqt ; Shen:2013oua  as well as the
baryon semileptonic and nonleptonic ones Zhao:2018mrg ; Zhao:2018zcb ;
PhysRevD.98.056002 ; PhysRevD.99.014023 ; PhysRevD.100.034025 ; Geng:2020fng ;
Geng:2021nkl . However, in the LFQM, the transition form factors can only be
calculated in the non-timelike region. To obtain the form factors in the
timelike region, where the semileptonic and nonleptonic decays occur, certain
$q^{2}$-dependencies must be assumed, reducing the predicting power of the
LFQM. In contrast, such a drawback does not exist in ${\bf B}_{b}\to{\bf
B}_{n}\gamma$, allowing the LFQM to be tested more rigorously.

As a complement, we will also show the results from the $SU(3)_{F}$ flavor
symmetry similar to those in Ref. Wang:2020wxn , which works well in the
bottomed meson He:1998rq ; PhysRevD.64.034002 ; PhysRevD.69.074002 ;
PhysRevD.93.114002 ; PhysRevD.50.4529 ; PhysRevD.52.6356 ; Zhou:2016jkv ;
PhysRevD.91.014011 ; PhysRevLett.75.1703  and baryon decays Dery:2020lbc ;
HE201582 ; PhysRevD.92.036010 ; PhysRevD.91.115003 ; SINGER1996202 ;
Wang:2017azm  as well as the charmed meson Grossman:2012ry ;
PIRTSKHALAVA201281 ; PhysRevD.86.014014 ; SAVAGE1991414 ; PhysRevD.92.014004
and baryon decays PhysRevD.42.1527 ; ALTARELLI1975277 ; PhysRevD.93.056008 ;
GENG2018265 ; PhysRevD.97.073006 ; Geng:2017mxn ; GENG2019214 ; Wang:2017azm ;
Wang:2019dls ; Wang:2017gxe , and compare them with our evaluations from the
LFQM.

This paper is organized as follows. In Sec. II we present the formalisms for
the decay widths and the tensor form factors. The numerical results and
discussions are given in Sec. III. We conclude in Sec. IV.

##  II Formalisms

We consider the weak radiative decays of anti-triplet bottom baryons induced
by the quark transitions of $b\to f\gamma$ with $f=(s,d)$. We will ignore the
contributions from $W$-exchange diagrams since they are suppressed by the CKM
elements. The effective Hamiltonians from the transitions are given by
Buchalla:1995vs

$\mathcal{H}_{eff}(b\to
f\gamma)=-\frac{G_{F}}{\sqrt{2}}\frac{e}{4\pi^{2}}V^{*}_{tf}V_{tb}C_{7\gamma}^{eff}(\mu_{b})m_{b}[\bar{f}i\sigma^{\mu
k}(1+\gamma^{5})b]\epsilon_{\mu},$ (2)

where $\sigma^{\mu
k}=\frac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})k_{\nu}$, and
$C_{7\gamma}^{eff}(\mu_{b})$ corresponds to the effective Wilson coefficient
at the scale of $\mu_{b}$ with $C_{7\gamma}^{eff}(5.09~{}\rm{GeV})=-0.303$.
The decay amplitudes are obtained by sandwiching $\mathcal{H}_{eff}$ with the
initial and final states

$\mathcal{M}({\bf B}_{b}\to{\bf
B}_{n}\gamma)=-\frac{G_{F}}{\sqrt{2}}\frac{e}{4\pi^{2}}V^{*}_{tf}V_{tb}C_{7\gamma}^{eff}(\mu_{b})m_{b}\langle{\bf
B}_{n}|\bar{f}i\sigma^{\mu k}(1+\gamma^{5})b|{\bf
B}_{b}\rangle\epsilon_{\mu}.$ (3)

The matrix elements above can be parametrized in terms of the tensor form
factors, given by

$\displaystyle\langle{\bf B}_{n}|\bar{f}i\sigma^{\mu k}b|{\bf B}_{b}\rangle$
$\displaystyle=\overline{u}_{{\bf
B}_{n}}[f_{1}^{TV}(k^{2})(\gamma^{\mu}k^{2}-k^{\mu}\not{k})/M_{{\bf
B}_{b}}-f_{2}^{TV}(k^{2})i\sigma^{\mu k}]u_{{\bf B}_{b}},$ (4)
$\displaystyle\langle{\bf B}_{n}|\bar{f}i\sigma^{\mu k}\gamma^{5}b|{\bf
B}_{b}\rangle$ $\displaystyle=\overline{u}_{{\bf
B}_{n}}[f_{1}^{TA}(k^{2})(\gamma^{\mu}k^{2}-k^{\mu}\not{k})/M_{{\bf
B}_{b}}-f_{2}^{TA}(k^{2})i\sigma^{\mu k}]\gamma^{5}u_{{\bf B}_{b}}.$

where $u_{{\bf B}_{b(n)}}$ stands for the Dirac spinor of ${\bf B}_{b(n)}$,
$M$ denotes the baryon mass, and $k^{\mu}$ is the momentum transfer between
the initial and final states. In our case of the radiative decays, $k^{2}=0$.
We will neglect the contributions associated with the form factors of
$f_{1}^{TV}(k^{2})$ and $f_{1}^{TA}(k^{2})$ unless particularly noted in the
rest of this paper. Consequently, the decay rates are given as

$\Gamma({\bf B}_{b}\to{\bf
B}_{n}\gamma)=\frac{\alpha_{em}}{64\pi^{4}}G_{F}^{2}m_{b}^{2}M_{{\bf
B}_{b}}^{3}|V^{*}_{tf}V_{tb}|^{2}(C_{7\gamma}^{eff})^{2}\left(1-\frac{M_{{\bf
B}_{n}}^{2}}{M_{{\bf
B}_{b}}^{2}}\right)^{3}(|f^{TV}_{2}|^{2}+|f^{TA}_{2}|^{2}).$ (5)

with $\alpha_{em}$ the fine-structure constant of the electromagnetic
interaction.

Unfortunately, to calculate the form factors of $f_{2}^{TV,TA}(k^{2})$, the
baryon wave functions are required, which can not be reliably obtained from
the first principle due to the nonperturbative effect. In this work, we use
the approach in the LFQM for the baryon wave functions, in which a baryon
state with momentum $P$ and spin $(S,S_{z})$ is expressed as Zhang:1994ti ;
Schlumpf:1992vq ; Geng:2020fng ; Geng:2021nkl ; Cheng:2004cc ; Ke:2019smy ;
Ke:2007tg ; Ke:2012wa

$\displaystyle|B,P,S,S_{z}\rangle=$
$\displaystyle\sum_{\lambda_{1},\lambda_{2},\lambda_{3}}\int\left\\{d^{3}\tilde{p}_{1}\right\\}\left\\{d^{3}\tilde{p}_{2}\right\\}\left\\{d^{3}\tilde{p}_{3}\right\\}2(2\pi)^{3}\frac{1}{\sqrt{P^{+}}}\delta^{3}\left(\tilde{P}-\tilde{p}_{1}-\tilde{p}_{2}-\tilde{p}_{3}\right)$
(6)
$\displaystyle\times\Psi^{SS_{z}}\left(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3}\right)C^{\alpha\beta\gamma}F_{abc}|q_{\alpha}^{a}\left(\tilde{p}_{1},\lambda_{1}\right)q_{\beta}^{b}\left(\tilde{p}_{2},\lambda_{2}\right)q_{\gamma}^{c}\left(\tilde{p}_{3},\lambda_{3}\right)\rangle,$

where $\Psi$ represents the vertex function between the baryon and quarks,
$C^{\alpha\beta\gamma}\ (F_{abc})$ corresponds to the color (flavor) factor
with $\alpha,\beta,\gamma\ (a,b,c)$ being its indices, and $\tilde{p}_{i}$ are
the light-front 3-momenta of the $i$-th constituent quark, defined by

$p_{i}=(p_{i}^{-},p_{i}^{+},p_{i}^{1},p_{i}^{2})=(p_{i}^{-},\tilde{p}_{i})=(p_{i}^{-},p_{i}^{+},p_{i\perp}),$
(7)

with $p_{i}^{\pm}=p_{i}^{0}\pm p_{i}^{3}$ and
$p_{i}^{-}p_{i}^{+}-p^{2}_{i\perp}=m_{i}^{2}$. Here, the integration measure
and Delta function are given as

$\differential^{3}\tilde{p}_{i}\equiv\frac{dp_{i}^{+}d^{2}p_{i\perp}}{2(2\pi)^{3}},\quad\delta^{3}(\tilde{p})=\delta\left(p^{+}\right)\delta^{2}\left(p_{\perp}\right).$
(8)

respectively, along with the normalization

$\displaystyle\innerproduct{q_{\alpha^{\prime}}^{a^{\prime}}(\tilde{p}_{i}^{\prime},\lambda^{\prime})}{q_{\alpha}^{a}(\tilde{p}_{i},\lambda)}=2(2\pi)^{3}\delta^{3}\left(\tilde{p}^{\prime}_{i}-\tilde{p}_{i}\right)\delta_{\lambda^{\prime}\lambda}\delta_{\alpha^{\prime}\alpha}\delta^{a^{\prime}a}\,,$
$\displaystyle\left\langle\mathbf{B},P^{\prime},S^{\prime},S_{z}^{\prime}\mid\mathbf{B},P,S,S_{z}\right\rangle=2(2\pi)^{3}P^{+}\delta^{3}(\tilde{P}^{\prime}-\tilde{P})\delta_{S_{z}^{\prime}S_{z}}.$
(9)

The vertex function can be further decomposed as Zhang:1994ti ;
Schlumpf:1992vq ; Geng:1997ws

$\Psi^{SS_{z}}\left(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\lambda_{1},\lambda_{2},\lambda_{3}\right)=\Phi\left(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3}\right)\Xi^{SS_{z}}\left(\lambda_{1},\lambda_{2},\lambda_{3}\right),$
(10)

in which $\Phi$ is the momentum distribution function, $\Xi^{SS_{z}}$ stands
for the helicity wave function given as

$\Xi^{SS_{z}}\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)=\sum_{s_{1},s_{2},s_{3}}\prod_{i=1}^{3}\langle\lambda_{i}|R_{i}^{\dagger}|s_{i}\rangle\innerproduct{\frac{1}{2}s_{1},\frac{1}{2}s_{2},\frac{1}{2}s_{3}}{SS_{z}},$
(11)

where $R_{i}$ is the Melosh matrix, which brings the $i$-th quark from its
spin state to a helicity state, and
$\innerproduct{\frac{1}{2}s_{1},\frac{1}{2}s_{2},\frac{1}{2}s_{3}}{SS_{z}}$ is
the Clebsch-Gordan coefficient, embodied in the spin wave function.

The explicit forms of $\Phi$ and $R_{i}$ depend on the parametrization scheme
of the internal motions of the constituent quarks. In our calculation, we
choose the diquark scheme with the first two quarks being coupled to each
other, while the other coupling schemes can be easily done by the permutations
Geng:2020fng . The kinematic variables are given as

$\displaystyle\tilde{P}$
$\displaystyle=\tilde{p}_{1}+\tilde{p}_{2}+\tilde{p}_{3},\qquad\xi_{3}=\frac{p_{1}^{+}}{p_{1}^{+}+p_{2}^{+}},\qquad\eta_{3}=1-\frac{p_{3}^{+}}{P^{+}},$
(12) $\displaystyle q_{3\perp}$
$\displaystyle=(1-\xi_{3})p_{1\perp}-\xi_{3}p_{2\perp},\quad
Q_{3\perp}=(1-\eta_{3})(p_{1\perp}+p_{2\perp})-\eta_{3}p_{3\perp}.$

Note that $(\xi_{3},q_{3\perp})$ describe the internal motion within the
diquark system, while $(\eta_{3},Q_{3\perp})$ the relative motion between the
diquark and the third quark Bakker:1979eg . The invariant masses are then
given as

$\displaystyle M_{3}^{2}$
$\displaystyle=\frac{q_{3\perp}^{2}}{\xi_{3}(1-\xi_{3})}+\frac{m_{1}^{2}}{\xi_{3}}+\frac{m_{2}^{2}}{1-\xi_{3}},$
(13) $\displaystyle M^{2}$
$\displaystyle=\frac{Q_{3\perp}^{2}}{\eta_{3}(1-\eta_{3})}+\frac{M_{3}^{2}}{\eta_{3}}+\frac{m_{3}^{2}}{1-\eta_{3}}.$

In this work, we adopt the Gaussian-type momentum wave function for the ground
state baryon Schlumpf:1992vq ; Geng:2020fng ; Geng:2021nkl . In this
particular set of kinematic variables, we have that

$\phi_{3}\equiv\Phi(\xi_{3},q_{3\perp},\eta_{3},Q_{3\perp})=\mathcal{N}\sqrt{\frac{\partial
q_{3z}}{\partial\xi_{3}}\frac{\partial
Q_{3z}}{\partial\eta_{3}}}\exp(-\frac{\vec{Q}_{3}^{2}}{2\beta_{Q}^{2}}-\frac{\vec{q}_{3}\\!^{2}}{2\beta_{qq^{\prime}}^{2}}),$
(14)

with $\mathcal{N}=\left(\beta_{qq^{\prime}}\beta_{Q}\pi\right)^{-3/2}$ and

$\displaystyle q_{3z}$
$\displaystyle=\frac{\xi_{3}M_{3}}{2}-\frac{m_{1}^{2}+q_{3\perp}^{2}}{2\xi_{3}M_{3}},\quad\,\vec{q}_{3}\\!^{2}=q_{3\perp}^{2}+q_{3z}^{2},$
(15) $\displaystyle Q_{3z}$
$\displaystyle=\frac{\eta_{3}M}{2}-\frac{m_{3}^{2}+Q_{3\perp}^{2}}{2\eta_{3}M},\quad\vec{Q}_{3}^{2}=Q_{3\perp}^{2}+Q_{3z}^{2},$

where $\beta_{qq^{\prime}}$ and $\beta_{Q}$ are the confinement energy scales
within the diquak system and between the diquark and third quark,
respectively. Note that we take the shape parameters as the internal kinematic
freedoms to describe the diquark systems instead of the diquark masses.

If the integration variables $(\vec{q},\vec{Q})$ are used instead of
$(\xi_{3},q_{3\perp},\eta_{3},Q_{3\perp})$, we obtain that

$\int\differential\xi_{3}\differential\eta_{3}\differential^{2}q_{3\perp}\differential^{2}Q_{3\perp}\left|\phi_{3}\right|^{2}=\int\differential^{3}\vec{q}_{3}\differential^{3}\vec{Q}_{3}{\cal
N}^{2}\exp(-\frac{\vec{Q}_{3}^{2}}{2\beta_{Q}^{2}}-\frac{\vec{q}_{3}\\!^{2}}{2\beta_{qq^{\prime}}^{2}})=1,$
(16)

where the wave functions are clearly Gaussian. On the other hand, the angular
dependencies are embodied in $R_{i}$, given as

$\begin{array}[]{l}R_{1}=R_{M}\left(\eta_{3},Q_{3\perp},M_{3},M\right)R_{M}\left(\xi_{3},q_{3\perp},m_{1},M_{3}\right),\\\
R_{2}=R_{M}\left(\eta_{3},Q_{3\perp},M_{3},M\right)R_{M}\left(1-\xi_{3},-q_{3\perp},m_{2},M_{3}\right),\\\
R_{3}=R_{M}\left(1-\eta_{3},-Q_{3\perp},m_{3},M\right),\end{array}$ (17)

with

$R_{M}\left(\xi,q_{\perp},m,M\right)=\frac{m+\xi
M-i\vec{\sigma}\cdot(\vec{n}\times\vec{q})}{\sqrt{(m+\xi
M)^{2}+q_{\perp}^{2}}},$ (18)

where $\vec{\sigma}=(\sigma^{1},\sigma^{2},\sigma^{3})$, representing the
Pauli matrices, and $\vec{n}=(0,0,1)$. We emphasize that the explicit forms of
$R_{i}$ depend on the parametrization schemes.

As the baryon wave functions are given, we are now ready to calculate the form
factors. To illustrate the calculation, we take $\Lambda_{b}\to\Lambda\gamma$
as an example, while the others can be obtained with slight modifications. The
relevant ones in Eq. (5) can be extracted through the following equalities,

$\displaystyle
f_{2}^{TV}=\frac{1}{4P^{+}}\langle{\Lambda,P^{\prime},\uparrow}|\bar{f}i\sigma^{R+}b|{\Lambda_{b},P,\downarrow}\rangle,$
(19) $\displaystyle
f_{2}^{TA}=-\frac{1}{4P^{+}}\langle{\Lambda,P^{\prime},\uparrow}|\bar{f}i\sigma^{R+}\gamma_{5}b|{\Lambda_{b},P,\downarrow}\rangle,$

where $\gamma^{R}=\gamma^{1}+i\gamma^{2},\gamma^{+}=\gamma^{0}+\gamma^{3}$,
and the Dirac spinors in the light-front formalism can be found in Appendix A.
We choose $k^{+}=0$ to perform the calculation. In the LFQMs, this particular
frame is often used to avoid the zero-mode graphs Choi:1998nf ; Choi:2005fj ;
Choi:2011xm ; Choi:2012zzb ; Choi:2013ira . It has been shown that their
contributions to the vector form factors vanish at the limit of $k^{+}\to 0$
Schlumpf:1992vq ; Zhang:1994ti . In this work, we would take it as a working
assumption for the tensor ones and test it with the experimental data.

The full wave functions of $\Lambda_{b}$ and $\Lambda$ are given as

$\displaystyle\ket{\Lambda_{b}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{udb}-\ket{dub})+\phi_{2}\chi^{\rho
2}(\ket{ubd}-\ket{dbu})+\phi_{1}\chi^{\rho 1}(\ket{bud}-\ket{bdu})\right],$
(20) $\displaystyle\ket{\Lambda}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{uds}-\ket{dus})+\phi_{2}\chi^{\rho
2}(\ket{usd}-\ket{dsu})+\phi_{1}\chi^{\rho 1}(\ket{sud}-\ket{sdu})\right],$

while the others can be found in Appendix B. In our study, since diquark
clusters are viewed as effective particles, they are chosen in a way to
acquire definite angular momenta. For $\Lambda$ and $\Sigma^{0}$, the $(u,d)$
pairs would form the states with $J=0$ and $J=1$, respectively, whereas the
$(u,s)$ and $(d,s)$ pairs would be the mixtures of $J=0$ and $J=1$. Thus, we
choose the $(u,d)$ pairs to form diquark clusters instead of the others. This
way of constructing the baryon wave functions would break the $SU(3)$ flavor
symmetry by hand, as it does not allow a diquark cluster with a light quark
and a strange quark inside $\Lambda$ or $\Sigma$. Such breaking effects are
embedded in the expressions of the baryon octet, as the $\Lambda$ and $\Sigma$
baryons are taken to preserve the isospin symmetry instead of the $U-$spin or
$V-$spin symmetry.

There are 6 terms, which contribute to the transition, read as

$\displaystyle\ket{udb}\to\ket{uds},~{}\ket{dbu}\to\ket{dsu},~{}\ket{bud}\to\ket{sud},$
(21)
$\displaystyle\ket{dub}\to\ket{dus},~{}\ket{ubd}\to\ket{usd},~{}\ket{bdu}\to\ket{sdu}.$

The first one in Eq. (21) contributes to the form factors as

$\displaystyle(f_{2}^{TV})_{udb\to uds}=$
$\displaystyle\frac{1}{4P^{+}}\int\differential{\xi_{3}}\differential{\eta_{3}}\differential^{2}{q_{3\perp}}\differential^{2}{Q_{3\perp}}\phi_{3}^{\prime}\phi_{3}F_{uds}F_{udb}$
(22)
$\displaystyle\times\sum_{\lambda_{i}^{\prime},\lambda_{i}}\Xi^{\frac{1}{2},+\frac{1}{2}}(\lambda_{i}^{\prime})^{\dagger}\delta_{\lambda_{1}^{\prime}\lambda_{1}}\delta_{\lambda_{2}^{\prime}\lambda_{2}}(O_{3}^{TV})_{\lambda_{3}^{\prime}\lambda_{3}}\Xi^{\frac{1}{2},-\frac{1}{2}}(\lambda_{i}),$

with

$\displaystyle(O_{3}^{TV})_{\lambda_{3}^{\prime}\lambda_{3}}=\frac{1}{1-\eta}\bar{u}\left(\vec{p}_{3}\,^{\prime}\lambda_{3}^{\prime}\right)i\sigma^{R+}u\left(\vec{p}_{3}\lambda_{3}\right)=-4P^{+}(\sigma^{R})_{\lambda_{3}^{\prime}\lambda_{3}},$
(23)
$\displaystyle(O_{3}^{TA})_{\lambda_{3}^{\prime}\lambda_{3}}=\frac{1}{1-\eta}\bar{u}\left(\vec{p}_{3}\,^{\prime}\lambda_{3}^{\prime}\right)i\sigma^{R+}\gamma_{5}u\left(\vec{p}_{3}\lambda_{3}\right)=4P^{+}(\sigma^{R})_{\lambda_{3}^{\prime}\lambda_{3}},$

where $F_{uds}=F_{udb}=1/\sqrt{6}$, and $O_{3}^{TV,TA}$ describe the $b\to s$
transition at the quark level, note that we have noramlized $\sigma^{R}$ as
$(\sigma^{1}+i\sigma^{2})/2$.

On the other hand, the helicity wave functions for the initial and final
baryons are

$\Xi^{\frac{1}{2},+\frac{1}{2}}(\lambda_{i}^{\prime})^{\dagger}=\sum_{\chi^{\rho
3}_{\uparrow}}\prod_{i=1}^{3}\bra{s_{i}^{\prime}}R_{i}^{\prime}\ket{\lambda_{i}^{\prime}},\quad\Xi^{\frac{1}{2},-\frac{1}{2}}(\lambda_{i})=\sum_{\chi^{\rho
3}_{\downarrow}}\prod_{i=1}^{3}\bra{\lambda_{i}}R_{i}^{\dagger}\ket{s_{i}}.$
(24)

Combing them together, we get

$\displaystyle(f_{2}^{TV})_{udb\to uds}$
$\displaystyle=-\frac{1}{6}\int\differential{\xi_{3}}\differential{\eta_{3}}\differential^{2}{q_{3\perp}}\differential^{2}{Q_{3\perp}}\phi_{3}^{\prime}\phi_{3}$
(25) $\displaystyle\times$ $\displaystyle\sum_{\chi^{\rho
3}_{\uparrow},\chi^{\rho
3}_{\downarrow}}\prod_{i=1,2}\bra{s_{i}^{\prime}}R_{i}^{\prime}\cdot
R_{i}^{\dagger}\ket{s_{i}}\bra{s_{3}^{\prime}}R_{3}^{\prime}\cdot\sigma^{R}\cdot
R_{3}^{\dagger}\ket{s_{3}}.$

Similarly, $f_{2}^{TA}$ can be obtained, given as

$\displaystyle(f_{2}^{TA})_{udb\to uds}$
$\displaystyle=-\frac{1}{6}\int\differential{\xi_{3}}\differential{\eta_{3}}\differential^{2}{q_{3\perp}}\differential^{2}{Q_{3\perp}}\phi_{3}^{\prime}\phi_{3}$
(26) $\displaystyle\times$ $\displaystyle\sum_{\chi^{\rho
3}_{\uparrow},\chi^{\rho
3}_{\downarrow}}\prod_{i=1,2}\bra{s_{i}^{\prime}}R_{i}^{\prime}\cdot
R_{i}^{\dagger}\ket{s_{i}}\bra{s_{3}^{\prime}}R_{3}^{\prime}\cdot\sigma^{R}\cdot
R_{3}^{\dagger}\ket{s_{3}}.$

By the permutation symmetry of the Fermi statistics, it is straightforward to
see that the six transitions give exactly the same contributions to the form
factors. As a result, the transition form factors between the baryons can be
obtained by multiplying Eqs. (25) and (26) by 6, given by

$\displaystyle(f_{2}^{TV})_{\Lambda_{b}\to\Lambda}=(f_{2}^{TA})_{\Lambda_{b}\to\Lambda}$
$\displaystyle=-\int\differential{\xi_{3}}\differential{\eta_{3}}\differential^{2}{q_{3\perp}}\differential^{2}{Q_{3\perp}}\phi_{3}^{\prime}\phi_{3}$
(27) $\displaystyle\times\sum_{\chi^{\rho 3}_{\uparrow},\chi^{\rho
3}_{\downarrow}}\prod_{i=1,2}\bra{s_{i}^{\prime}}R_{i}^{\prime}\cdot
R_{i}^{\dagger}\ket{s_{i}}\bra{s_{3}^{\prime}}R_{3}^{\prime}\cdot\sigma^{R}\cdot
R_{3}^{\dagger}\ket{s_{3}},$

It is worth to mention that the equality

$f_{2}^{TV}=f_{2}^{TA}$ (28)

only holds at $k^{2}=0$. This result is also consistent with that in Refs.
Gutsche:2013pp ; Wang:2008sm .

The equivalence can be understood intuitively in terms of the valence quark
framework, in which the spin direction of $\Lambda$ is attributed to its
strange quark solely. As a result, it is necessary for $\Lambda$ to have the
same helicity with the strange quark, which is left-handed. A direct
consequence is that

$H_{+}=0\,,$ (29)

where $H_{+(-)}$ corresponds to the helicity amplitude with the subscript
denotes the helicity of $\Lambda$.

Without carrying out the numerical detail, Eq. (29) is sufficient for one to
analyze the decay angular distributions. For $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\gamma$, the angular dependency is given as GarciaMartin:2019bxm ;
Gutsche:2013pp

${\cal
D}(\Omega)\equiv\frac{1}{8\pi}\frac{1}{\Gamma}\frac{\partial^{3}\Gamma}{\partial\cos\theta\partial\cos\theta_{1}\partial\phi}=1-P_{b}\alpha_{\Lambda}\cos\theta\cos\theta_{1}-P_{b}P_{L}\cos\theta+\alpha_{\Lambda}P_{L}\cos\theta_{1}\,,$
(30)

where $P_{b}$ is related to the $\Lambda_{b}$ polarization, $\alpha_{\Lambda}$
is the up-down asymmetry parameter of $\Lambda\to p\pi^{-}$, which can be
determined by the experiments BESIII:2018cnd , and the definitions of the
angles are given in FIG. I , in which $\hat{n}_{\Lambda_{b}}$ is the polarized
direction of $\Lambda_{b}$. For $b\to s(d)\gamma$, the longitudinal
polarization is defined by

$P_{L}\equiv\frac{|H_{+}^{2}|-|H_{-}^{2}|}{|H_{+}^{2}|+|H_{-}^{2}|}=-1+{\cal
O}(m_{s(d)}^{2}/m_{b}^{2})\,,$ (31)

where the second equality comes from Eq. (29). It is interesting to point out
that the distribution is independent of $\phi$ due to the angular momentum
conservation. Note that in contrast to ${\bf B}_{b}\to{\bf B}_{n}P$ with $P$ a
psuedoscalar meson, the up-down asymmetry parameter $\alpha_{b}$ defined
through the equality of

$\frac{1}{\Gamma}\frac{\partial\Gamma}{\partial\cos\theta}\propto
1+P_{b}\alpha_{b}\cos\theta=1-P_{b}P_{L}\cos\theta\,,$ (32)

has an opposite sign respecting to $P_{L}$. It is attributed to that the
photon is spin-1 and transversely polarized.

Figure 1: Angles in ${\cal D}(\vec{\Omega})$ for $\Lambda_{b}\to\Lambda(\to
p\pi^{-})\gamma$.

##  III Numerical Results and Discussions

For the CKM matrix elements, we take the Wolfenstein parametrization, given as

$\lambda=0.22650\pm 0.00048,\quad
A=0.790_{-0.012}^{+0.017},\quad\rho=0.141_{-0.017}^{+0.016},\quad\eta=0.357\pm
0.011,$ (33)

where the values and uncertainties are quoted from the Particle Data Group
Zyla:2020zbs . The values of $\beta_{q_{I}q_{I}^{\prime}}$ and $\beta_{Q_{b}}$
can be found in Ref. Geng:2021nkl , in which $\beta_{Q_{b}}$ are taken to be
the same for all ${\bf B}_{b}$ due to the heavy quark symmetry. The values of
$\beta_{sq_{I}}$ and $\beta_{ss}$ are taken to be slightly larger than
$\beta_{q_{I}q_{I}^{\prime}}$, since strange quarks are heavier than $u$ and
$d$, resulting in a smaller diquark system. We note that $\beta_{sq_{I}}$ and
$\beta_{ss}$ are also used for $\beta_{Q}$ of ${\bf B}_{n}$. Taking $\Xi^{0}$
as an example, the diquark system is made of $u$ and $s$, so we have
$\beta_{qq^{\prime}}=\beta_{sq_{I}}$ and $\beta_{Q}=\beta_{ss}$. Our numerical
results of the form factors are listed in Table 2, where $\Xi_{b}$ and $\Xi$
in the second line stand for either $\Xi_{b}^{0}$ and $\Xi^{0}$, or
$\Xi_{b}^{-}$ and $\Xi^{-}$, respectively.

Table 1: Theoretical Inputs for the baryon wave functions of the LFQM in the unit of GeV $m_{u}$ | $m_{d}$ | $m_{s}$ | $m_{b}$ | $\beta_{q_{I}q_{I}^{\prime}}$ | $\beta_{sq_{I}}$ | $\beta_{ss}$ | $\beta_{Q_{b}}$
---|---|---|---|---|---|---|---
0.26 | 0.26 | 0.31 | 4.88 | 0.365 | 0.373 | 0.377 | 0.601
Table 2: Form Factors of ${\bf B}_{b}\to{\bf B}_{n}\gamma$ Channel | $f_{2}^{TV}$ | Channel | $f_{2}^{TV}$
---|---|---|---
$\Lambda_{b}\,\to\Lambda\;\gamma$ | $-0.123$ | $\Xi_{b}^{0}\;\to\Sigma^{0}\gamma$ | $-0.096$
$\Xi_{b}\,\to\Xi\gamma$ | $0.143$ | $\Xi_{b}^{-}\,\to\Sigma^{-}\gamma$ | $-0.134$
$\Lambda_{b}\>\to n\;\>\gamma$ | $0.135$ | $\Xi_{b}^{0}\;\,\to\Lambda\;\gamma$ | $-0.056$

To check our results in the LFQM, we would also like to study the decays based
on the $SU(3)_{F}$ flavor symmetry. With $SU(3)_{F}$, the wave functions among
the low-lying octet baryons share the identical spacial distribution, so as
the wave functions of the anti-triplet bottom baryons. Consequently, by
substituting $s(d)$ for $b$ in ${\bf B}_{b}$ and taking the inner products of
the spin-flavor wave functions with ${\bf B}_{n}$, the relative sizes of the
form factors among ${\bf B}_{b}\to{\bf B}_{n}\gamma$ can be determined. The
$SU(3)_{F}$ relations Wang:2020wxn  are given in Table III with
$\lambda_{s(d)}=V_{tb}V^{*}_{ts(d)}$. By taking the experimental data for the
branching ratio of $\Lambda_{b}\to\Lambda\gamma$ as a theoretical input, we
can obtain the $SU(3)_{F}$ predictions for ${\bf B}_{b}\to{\bf B}_{n}\gamma$.

Table 3: Amplitude ratios of ${\bf B}_{b}\to{\bf B}_{n}\gamma$ given by $SU(3)_{F}$ Channel | $\Lambda_{b}\to\Lambda\gamma$ | $\Xi_{b}\to\Xi\gamma$ | $\Lambda_{b}\to n\gamma$ | $\Xi_{b}^{0}\to\Sigma^{0}\gamma$ | $\Xi_{b}^{-}\to\Sigma^{-}\gamma$ | $\Xi_{b}^{0}\to\Lambda\gamma$
---|---|---|---|---|---|---
Amplitude | $\lambda_{s}$ | $-\sqrt{\frac{3}{2}}\lambda_{s}$ | $-\sqrt{\frac{3}{2}}\lambda_{d}$ | $\frac{\sqrt{3}}{2}\lambda_{d}$ | $\sqrt{\frac{3}{2}}\lambda_{d}$ | $\frac{1}{2}\lambda_{d}$
Table 4: Numerical results of the branching ratios for ${\bf B}_{b}\to{\bf B}_{n}\gamma$ Quark level | Branching Ratios |  LFQM | $SU(3)_{F}$ |  $SU(3)_{F}$ Wang:2020wxn  | Other Models | Exp. Data
---|---|---|---|---|---|---
$b\to s\gamma$ | $10^{6}\mathcal{B}(\Lambda_{b}\,\to\Lambda\;\gamma)$ |  $7.1\pm 0.3$ | $7.1\pm 1.7$ | $7.1\pm 3.4$ |  $7.3\pm 1.5$ Wang:2008sm  | $7.1\pm 1.7$ LHCb:2019wwi
$4.0$ Gutsche:2013pp
$10.0$ Faustov:2017wbh
$10^{5}\mathcal{B}(\Xi_{b}^{0}\,\to\Xi^{0}\gamma)$ | $1.0\pm 0.1$ | $1.1\pm 0.3$ | $1.16\pm 0.60$ |  $1.02^{+0.60}_{-0.46}$ Olamaei:2021eyo  |
$10^{5}\mathcal{B}(\Xi_{b}^{-}\to\Xi^{-}\\!\gamma)$ | $1.1\pm 0.1$ | $1.2\pm 0.3$ | $1.23\pm 0.64$ |  $1.08^{+0.63}_{-0.49}$ Olamaei:2021eyo  |  $<13$ LHCb:2021hfz
$b\to d\gamma$ | $10^{7}\mathcal{B}(\Lambda_{b}\>\to n\;\>\gamma)$ | $4.0\pm 0.4$ | $4.9\pm 1.2$ | $5.03\pm 2.67$ |  $3.69^{+3.76}_{-1.95}$ Liu:2019rpm  |
$3.7$ Faustov:2017ous  |
$10^{7}\mathcal{B}(\Xi_{b}^{0}\;\to\Sigma^{0}\gamma)$ | $2.1\pm 0.2$ | $2.6\pm 0.6$ | $2.71\pm 1.50$ |  $5.77^{+3.16}_{-2.47}$ Olamaei:2021eyo  |
$10^{7}\mathcal{B}(\Xi_{b}^{-}\,\to\Sigma^{-}\gamma)$ | $4.4\pm 0.4$ | $5.5\pm 1.3$ | $5.74\pm 3.21$ |  $6.14^{+3.36}_{-2.63}$ Olamaei:2021eyo  |
$10^{8}\mathcal{B}(\Xi_{b}^{0}\;\,\to\Lambda\;\gamma)$ | $7.4\pm 0.7$ | $8.7\pm 2.1$ | $9.17\pm 5.10$ |  |

Our results of the branching ratios from the LFQM and $SU(3)_{F}$ are given in
Table 4, where we have also shown the $SU(3)_{F}$ evaluations in Ref.
Wang:2020wxn  and some of other theoretical predictions in the literature,
such as LCSR Wang:2008sm ; Olamaei:2021eyo , BSE Liu:2019rpm  and QM
Gutsche:2013pp ; Faustov:2017ous , as well as the current experimental data
LHCb:2019wwi ; LHCb:2021hfz . In particular, for $b\to s\gamma$ in the LFQM
approach, we find that ${\cal B}(\Lambda_{b}\to\Lambda\gamma)=(7.1\pm
0.3)\times 10^{-6}$, which agrees well with the experimental measured value.
In addition, we obtain that

${\cal B}(\Xi_{b}^{0}\to\Xi^{0}\gamma)=(1.0\pm 0.1)\times 10^{-5}\,,\quad{\cal
B}(\Xi_{b}^{-}\to\Xi^{-}\gamma)=(1.1\pm 0.1)\times 10^{-5}\,,$ (34)

which are 1.5 times larger than ${\cal B}(\Lambda_{b}\to\Lambda\gamma)$. Note
that the form factors for $\Xi_{b}^{0}\to\Xi^{0}\gamma$ and
$\Xi_{b}^{-}\to\Xi^{-}\gamma$ are exactly the same due to the isospin
symmetry, but their branching ratios slightly differ due to the lifetime
difference. Similarly, we have that

$\Gamma(\Xi_{b}^{-}\to\Sigma^{-}\gamma)=2\Gamma(\Xi_{b}^{0}\to\Sigma^{0}\gamma)$
(35)

guaranteed by the isospin symmetry.

In addition, our LFQM results in $b\to s\gamma$ agree well with both the
predictions given by $SU(3)_{F}$ and Refs. Wang:2008sm ; Olamaei:2021eyo ;
Liu:2019rpm . Note that the values based on $SU(3)_{F}$ in Ref. Wang:2020wxn
was made within 2$\sigma$ errors with respect to the experimental result of
$\mathcal{B}(\Lambda_{b}\to\Lambda\gamma)$. Furthermore, our $SU(3)_{F}$
results of the center values in Table 4 also slightly differ from those in
Ref. Wang:2020wxn . These differences arise from the long-distance
contributions of ${\bf B}_{b}\to{\bf B}_{n}\psi_{i}(\to\gamma)$ included in
Ref. [22], which modify the ratios between $b\to s\gamma$ and $b\to d\gamma$.

It is interesting to see that the decay branching ratios associated with $b\to
d\gamma$ in the LFQM are about $20\%$ smaller than those predicted by the
$SU(3)_{F}$ symmetry, which clearly show the $SU(3)_{F}$ breaking effects. In
contrast, the ones given by Ref. Olamaei:2021eyo  are larger than the
$SU(3)_{F}$ predicted values. Note that the equalities of Eqs. (28) and (35)
do not hold in Ref. Olamaei:2021eyo  . Future experimental searches on
$\Xi_{b}\to\Sigma\gamma$ could discriminate the various theoretical
approaches.

##  IV Conclusions

We have performed a systematic analysis of ${\bf B}_{b}\to{\bf B}_{n}\gamma$
based on the LFQM. We have obtained ${\cal
B}(\Lambda_{b}\to\Lambda\gamma)=(7.1\pm 0.3)\times 10^{-6}$, ${\cal
B}(\Xi_{b}^{0}\to\Xi^{0}\gamma)=(1.0\pm 0.1)\times 10^{-5}\,,$ and ${\cal
B}(\Xi_{b}^{-}\to\Xi^{-}\gamma)=(1.1\pm 0.1)\times 10^{-5}$. Our results agree
with the current experimental data and are consistent with other theoretical
values in the literature. In addition, for $b\to s\gamma$ we have found that
our results in the LFQM are in good agreement with those based on the
$SU(3)_{F}$ symmetry. Moreover, we have demonstrated that the $SU(3)_{F}$
breaking effects for $b\to d\gamma$ are as large as $20\%$.

We have also explicitly shown that $f_{2}^{TV}=f_{2}^{TA}$ at $k^{2}=0$ in the
LFQM, resulting in that $P_{L}=-\alpha=-1+{\cal O}(m_{s(d)}^{2}/m_{b}^{2})$
for $b\to s(d)\gamma$, which are independent of the theoretical input. A
dedicated experimental measurement of the angular distribution of
$\Lambda_{b}\to\Lambda(\to p\pi^{-})\gamma$ are strongly recommended for
testing the SM and probing possible effects from new physics.

###### Acknowledgements.

We want to thank Dr. Tien-Hsueh Tsai for his valuable assistance on the
numerical part of this work.

##  Appendix A Dirac Spinor in Light-Front formalism

We adopt the notation given in Ref. Zhang:1994ti  for the light front
formalism. The Dirac spinors turn out to be

$\displaystyle
u(p,\lambda)=\frac{1}{\sqrt{p^{+}}}\left(p^{+}+\beta+\bm{\alpha}_{\perp}\bm{p}_{\perp}\right)\times\begin{cases}\chi(\uparrow)&\text{
for }\lambda=+1\\\ \chi(\downarrow)&\text{ for }\lambda=-1\end{cases},$ (36)
$\displaystyle v(p,\lambda)=\frac{1}{\sqrt{p^{+}}}\left(p^{+}-\beta
m+\bm{\alpha}_{\perp}\bm{p}_{\perp}\right)\times\begin{cases}\chi(\downarrow)&\text{
for }\lambda=+1\\\ \chi(\uparrow)&\text{ for }\lambda=-1\end{cases},$

where $\beta=\gamma^{0}$ and
$\bm{\alpha}_{\perp}=(\gamma^{0}\gamma^{1},\gamma^{0}\gamma^{2})$ with
$\gamma^{\alpha}$ ($\alpha=0,1,2,3$) being the Dirac gamma matrices, while the
two $\chi$-spinors are given by

$\chi(\uparrow)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{l}1\\\ 0\\\ 1\\\
0\end{array}\right)\quad\text{ and
}\quad\chi(\downarrow)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}0\\\ 1\\\
0\\\ -1\end{array}\right),$ (37)

resulting in the helicity eigenstates

$u(p,+)=\frac{1}{\sqrt{2p^{+}}}\begin{pmatrix}p^{+}+m\\\ p^{1}+ip^{2}\\\
p^{+}-m\\\ p^{1}+ip^{2}\end{pmatrix}\mbox{\quad
and\quad}u(p,-)=\frac{1}{\sqrt{2p^{+}}}\begin{pmatrix}-p^{1}+ip^{2}\\\
p^{+}+m\\\ p^{1}-ip^{2}\\\ -p^{+}+m\end{pmatrix}\,.$ (38)

Accordingly, the relations in Eq. (19) and (23) can be verified directly.

##  Appendix B Momentum-Spin-Flavor Wave Function

The momentum-spin-flavor wave functions for ${\bf B}_{b}$ and ${\bf B}_{n}$
are given as

$\displaystyle\ket{\Xi_{b}^{-}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{dsb}-\ket{sdb})+\phi_{2}\chi^{\rho
2}(\ket{dbs}-\ket{sbd})+\phi_{1}\chi^{\rho 1}(\ket{bds}-\ket{bsd})\right],$
(39) $\displaystyle\ket{\Xi_{b}^{0}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{usb}-\ket{sub})+\phi_{2}\chi^{\rho
2}(\ket{ubs}-\ket{sbu})+\phi_{1}\chi^{\rho 1}(\ket{bus}-\ket{bsu})\right],$
$\displaystyle\ket{\Lambda_{b}^{0}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{udb}-\ket{dub})+\phi_{2}\chi^{\rho
2}(\ket{ubd}-\ket{dbu})+\phi_{1}\chi^{\rho 1}(\ket{bud}-\ket{bdu})\right],$

$\displaystyle\ket{p}$
$\displaystyle=\frac{1}{\sqrt{3}}\phi\left[\chi^{\lambda
3}\ket{uud}+\chi^{\lambda 2}\ket{udu}+\chi^{\lambda 1}\ket{duu}\right],$ (40)
$\displaystyle\ket{n}$
$\displaystyle=\frac{1}{\sqrt{3}}\phi\left[\chi^{\lambda
3}\ket{ddu}+\chi^{\lambda 2}\ket{dud}+\chi^{\lambda 1}\ket{udd}\right],$
$\displaystyle\ket{\Xi^{0}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left[\phi_{3}\chi^{\lambda
3}\ket{ssu}+\phi_{2}\chi^{\lambda 2}\ket{sus}+\phi_{1}\chi^{\lambda
1}\ket{uss}\right],$ $\displaystyle\ket{\Xi^{-}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left[\phi_{3}\chi^{\lambda
3}\ket{ssd}+\phi_{2}\chi^{\lambda 2}\ket{sds}+\phi_{1}\chi^{\lambda
1}\ket{dss}\right],$ $\displaystyle\ket{\Sigma^{+}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left[\phi_{3}\chi^{\lambda
3}\ket{uus}+\phi_{2}\chi^{\lambda 2}\ket{usu}+\phi_{1}\chi^{\lambda
1}\ket{suu}\right],$ $\displaystyle\ket{\Sigma^{-}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left[\phi_{3}\chi^{\lambda
3}\ket{dds}+\phi_{2}\chi^{\lambda 2}\ket{dsd}+\phi_{1}\chi^{\lambda
1}\ket{sdd}\right],$

$\displaystyle\ket{\Sigma^{0}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\lambda
3}(\ket{uds}+\ket{dus})+\phi_{2}\chi^{\lambda
2}(\ket{usd}+\ket{dsu})+\phi_{1}\chi^{\lambda 1}(\ket{sud}+\ket{sdu})\right],$
(41) $\displaystyle\ket{\Lambda}$
$\displaystyle=\frac{1}{\sqrt{6}}\left[\phi_{3}\chi^{\rho
3}(\ket{uds}-\ket{dus})+\phi_{2}\chi^{\rho
2}(\ket{usd}-\ket{dsu})+\phi_{1}\chi^{\rho 1}(\ket{sud}-\ket{sdu})\right].$

Here, the spin wave functions are defined as

$\displaystyle\chi^{\rho
3}_{\uparrow}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow\uparrow}-\ket{\downarrow\uparrow\uparrow}),\quad\chi^{\lambda
3}_{\uparrow}=\frac{1}{\sqrt{6}}(2\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\downarrow\uparrow}-\ket{\downarrow\uparrow\uparrow}),$
(42) $\displaystyle\chi^{\rho
2}_{\uparrow}=\frac{1}{\sqrt{2}}(\ket{\uparrow\uparrow\downarrow}-\ket{\downarrow\uparrow\uparrow}),\quad\chi^{\lambda
2}_{\uparrow}=\frac{1}{\sqrt{6}}(2\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}-\ket{\downarrow\uparrow\uparrow}),$
$\displaystyle\chi^{\rho
1}_{\uparrow}=\frac{1}{\sqrt{2}}(\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\downarrow\uparrow}),\quad\chi^{\lambda
1}_{\uparrow}=\frac{1}{\sqrt{6}}(2\ket{\downarrow\uparrow\uparrow}-\ket{\uparrow\downarrow\uparrow}-\ket{\uparrow\uparrow\downarrow}).$

The definitions of momentum wave functions $\phi_{1,2}$ are given as

$\phi_{1,2}\equiv\Phi(\xi_{1,2},q_{1,2\perp},\eta_{1,2},Q_{1,2\perp})=\mathcal{N}\sqrt{\frac{\partial
q_{1,2z}}{\partial\xi_{1,2}}\frac{\partial
Q_{1,2z}}{\partial\eta_{1,2}}}\exp(-\frac{\vec{Q}_{1,2}^{2}}{2\beta_{Q}^{2}}-\frac{\vec{q}_{1,2}\\!^{2}}{2\beta_{qq^{\prime}}^{2}}),$
(43) $\displaystyle q_{1,2z}$
$\displaystyle=\frac{\xi_{1,2}M_{1,2}}{2}-\frac{m_{2,3}^{2}+q_{1,2\perp}^{2}}{2\xi_{1,2}M_{1,2}},\quad\,\vec{q}_{1,2}\\!^{2}=q_{1,2\perp}^{2}+q_{1,2z}^{2},$
(44) $\displaystyle Q_{1,2z}$
$\displaystyle=\frac{\eta_{1,2}M}{2}-\frac{m_{1,2}^{2}+Q_{1,2\perp}^{2}}{2\eta_{1,2}M},\quad\vec{Q}_{1,2}^{2}=Q_{1,2\perp}^{2}+Q_{1,2z}^{2},$

## References

  * (1) J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974) [Erratum: Phys.Rev. D 11, 703 (1975)].
  * (2) R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 2558 (1975).
  * (3) R. N. Mohapatra and J. C. Pati, Phys. Rev. D 11, 566 (1975).
  * (4) G. Senjanovic and R. N. Mohapatra, Phys. Rev. D 12, 1502 (1975).
  * (5) G. Senjanovic, Nucl. Phys. B 153, 334 (1979).
  * (6) R. N. Mohapatra and G. Senjanovic, Phys. Rev. D 23, 165 (1981).
  * (7) C. S. Lim and T. Inami, Prog. Theor. Phys. 67, 1569 (1982).
  * (8) L. L. Everett, G. L. Kane, S. Rigolin, L. T. Wang, and T. T. Wang, JHEP 01, 022 (2002).
  * (9) D. Atwood, M. Gronau, and A. Soni, Phys. Rev. Lett. 79, 185 (1997).
  * (10) R. Aaij et al. (LHCb), JHEP 12, 081 (2020).
  * (11) R. Aaij et al. (LHCb), Phys. Rev. Lett. 112, 161801 (2014).
  * (12) C. Q. Geng and C. W. Liu, JHEP 11, 104 (2021).
  * (13) C. W. Liu and C. Q. Geng, JHEP 01, 128 (2022).
  * (14) L. M. García Martín et al., Eur. Phys. J. C 79, 634 (2019).
  * (15) R. Aaij et al. (LHCb), JHEP 01, 069 (2022).
  * (16) R. Aaij et al. (LHCb), Phys. Rev. Lett. 123, 031801 (2019).
  * (17) R. Aaij et al. (LHCb), Phys. Rev. Lett. 113, 032001 (2014).
  * (18) R. Aaij et al. (LHCb), Phys. Rev. Lett. 113, 242002 (2014).
  * (19) R. Aaij et al. (LHCb), JHEP 06, 110 (2020).
  * (20) P. Singer and D. X. Zhang, Phys. Lett. B 383, 351 (1996).
  * (21) X. G. He, T. Li, X. Q. Li, and Y. M. Wang, Phys. Rev. D 74, 034026 (2006).
  * (22) R. M. Wang, X. D. Cheng, Y. Y. Fan, J. L. Zhang, and Y. G. Xu, J. Phys. G 48, 085001 (2021).
  * (23) Y. M. Wang, Y. Li, and C. D. Lu, Eur. Phys. J. C 59, 861 (2009).
  * (24) A. R. Olamaei and K. Azizi, arXiv:2109.09783 [hep-ph].
  * (25) L. L. Liu, C. Wang, X. W. Kang, and X. H. Guo, Eur. Phys. J. C 80, 193 (2020).
  * (26) P. K. Chatley and A. C. Sharma, Phys. Rev. D 25, 2351 (1982).
  * (27) T. Gutsche, M. A. Ivanov, J. G. Korner, V. E. Lyubovitskij, and P. Santorelli, Phys. Rev. D 87, 074031 (2013).
  * (28) R. N. Faustov and V. O. Galkin, Phys. Rev. D 96, 053006 (2017).
  * (29) R. N. Faustov and V. O. Galkin, Mod. Phys. Lett. A 32, 1750125 (2017).
  * (30) H. Y. Cheng, C. Y. Cheung, G. L. Lin, Y. C. Lin, T. M. Yan, and H. L. Yu, Phys. Rev. D 51, 1199 (1995).
  * (31) H. M. Choi and C. R. Ji, Phys. Rev. D 58, 071901 (1998).
  * (32) H. M. Choi and C. R. Ji, Phys. Rev. D 72, 013004 (2005).
  * (33) H. M. Choi and C. R. Ji, Phys. Lett. B 696, 518-522 (2011).
  * (34) H. M. Choi and C. R. Ji, Few Body Syst. 52, 409-414 (2012).
  * (35) H. M. Choi and C. R. Ji, Few Body Syst. 55, 435-440 (2014).
  * (36) F. Schlumpf, Phys. Rev. D 47, 4114 (1993) [Erratum: Phys.Rev.D 49, 6246 (1994)].
  * (37) W. M. Zhang, Chin. J. Phys. 32, 717 (1994).
  * (38) C. Q. Geng, C. C. Lih, and W. M. Zhang, Phys. Rev. D 57, 5697 (1998).
  * (39) C. C. Lih, C. Q. Geng, and W. M. Zhang, Phys. Rev. D 59, 114002 (1999).
  * (40) C. Q. Geng, C. C. Lih, and W -M. Zhang, Phys. Rev. D 62, 074017 (2000).
  * (41) C. Q. Geng, C. W. Hwang, C. C. Lih, and W. M. Zhang, Phys. Rev. D 64, 114024 (2001).
  * (42) C. Q. Geng and C. C. Liu, Journal of Physics G: Nuclear and Particle Physics 29, 1103 (2003).
  * (43) H. Y. Cheng, C. K. Chua, and C. W. Hwang, Phys. Rev. D 69, 074025 (2004).
  * (44) C. Q. Geng, C. C. Lih, and C. Xia, Eur. Phys. J. C 76, 313 (2016).
  * (45) H. Y. Cheng and X. W. Kang, Eur. Phys. J. C 77, 587 (2017) [Erratum: Eur.Phys. J. C 77, 863 (2017)].
  * (46) Q. Chang, L. T. Wang, and X. N. Li, JHEP 12, 102 (2019).
  * (47) Y. J. Shi, W. Wang, and Z. X. Zhao, Eur. Phys. J. C 76, 555 (2016).
  * (48) Y. L. Shen and G. Li, Eur. Phys. J. C 73, 2441 (2013).
  * (49) Z. X. Zhao, Eur. Phys. J. C 78, 756 (2018).
  * (50) Z. X. Zhao, Chin. Phys. C 42, 093101 (2018).
  * (51) Z. P. Xing and Z. X. Zhao, Phys. Rev. D 98, 056002 (2018).
  * (52) C. K. Chua, Phys. Rev. D 99, 014023 (2019).
  * (53) C. K. Chua, Phys. Rev. D 100, 034025 (2019).
  * (54) C. Q. Geng, C. C. Lih, C. W. Liu, and T. H. Tsai, Phys. Rev. D 101, 094017 (2020).
  * (55) C. Q. Geng, C. W. Liu, and T. H. Tsai, Phys. Lett. B 815, 136125 (2021).
  * (56) X. G. He, Eur. Phys. J. C 9, 443 (1999).
  * (57) X. G. He, Y. K. Hsiao, J. Q. Shi, Y. L. Wu, and Y. F. Zhou, Phys. Rev. D 64, 034002 (2001).
  * (58) H. K. Fu, X. G. He, and Y. K. Hsiao, Phys. Rev. D 69, 074002 (2004).
  * (59) Y. K. Hsiao, C. F. Chang, and X. G. He, Phys. Rev. D 93, 114002 (2016).
  * (60) M. Gronau, O. F. Hernandez, D. London, and J. L. Rosner, Phys. Rev. D 50, 4529 (1994).
  * (61) M. Gronau, O. F. Hernandez, D. London, and J. L. Rosner, Phys. Rev. D 52, 6356 (1995).
  * (62) S. H. Zhou, Q. A. Zhang, W. R. Lyu, and C. D. Lu, Eur. Phys. J. C 77, 125 (2017).
  * (63) H. Y. Cheng, C. W. Chiang, and A. L. Kuo, Phys. Rev. D 91, 014011 (2015).
  * (64) N. G. Deshpande and X. G. He, Phys. Rev. Lett. 75, 1703 (1995).
  * (65) A. Dery, M. Ghosh, Y. Grossman, and S. Schacht, JHEP 03, 165 (2020).
  * (66) X. G. He and G. N. Li, Physics Letters B 750, 82 (2015).
  * (67) M. He, X. G. He, and G. N. Li, Phys. Rev. D 92, 036010 (2015).
  * (68) S. Shivashankara, W. Wu, and A. Datta, Phys. Rev. D 91, 115003 (2015).
  * (69) P. Singer, Nuclear Physics B - Proceedings Supplements 50, 202 (1996).
  * (70) W. Wang, Z. P. Xing, and J. Xu, Eur. Phys. J. C 77, 800 (2017).
  * (71) Y. Grossman and D. J. Robinson, JHEP 04, 067 (2013).
  * (72) D. Pirtskhalava and P. Uttayarat, Physics Letters B 712, 81 (2012).
  * (73) H. Y. Cheng and C. W. Chiang, Phys. Rev. D 86, 014014 (2012).
  * (74) M. J. Savage, Physics Letters B 257, 414 (1991).
  * (75) S. Muller, U. Nierste, and S. Schacht, Phys. Rev. D 92, 014004 (2015).
  * (76) M. J. Savage and R. P. Springer, Phys. Rev. D 42, 1527 (1990).
  * (77) G. Altarelli, N. Cabibbo, and L. Maiani, Physics Letters B 57, 277 (1975).
  * (78) C. D. Lu, W. Wang, and F. S. Yu, Phys. Rev. D 93, 056008 (2016).
  * (79) C. Q. Geng, Y. K. Hsiao, Y. H. Lin, and L. L. Liu, Physics Letters B 776, 265 (2018).
  * (80) C. Q. Geng, Y. K. Hsiao, C. W. Liu, and T. H. Tsai, Phys. Rev. D 97, 073006 (2018).
  * (81) C. Q. Geng, Y. K. Hsiao, C. W. Liu, and T. H. Tsai, JHEP 11, 147 (2017).
  * (82) C. Q. Geng, C. W. Liu, T. H. Tsai, and S. W. Yeh, Physics Letters B 792, 214 (2019).
  * (83) D. Wang, Eur. Phys. J. C 79, 429 (2019).
  * (84) D. Wang, P. F. Guo, W. H. Long, and F. S. Yu, JHEP 03, 066 (2018).
  * (85) G. Buchalla, A. J. Buras, and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
  * (86) H. Y. Cheng, C. K. Chua, and C. W. Hwang, Phys. Rev. D 70, 034007 (2004).
  * (87) H. W. Ke, N. Hao, and X. Q. Li, Eur. Phys. J. C 79, 540 (2019).
  * (88) H. W. Ke, X. Q. Li, and Z. T. Wei, Phys. Rev. D 77, 014020 (2008).
  * (89) H. W. Ke, X. H. Yuan, X. Q. Li, Z. T. Wei, and Y. X. Zhang, Phys. Rev. D 86, 114005 (2012).
  * (90) C. Q. Geng, C. C. Lih, and W. M. Zhang, Phys. Rev. D 57, 5697 (1998).
  * (91) B. L. G. Bakker, L. A. Kondratyuk, and M. V. Terentev, Nucl. Phys. B 158, 497 (1979).
  * (92) M. Ablikim et al. (BESIII), Nature Phys. 15, 631 (2019).
  * (93) P. Zyla et al. (Particle Data Group), PTEP 2020, 083C01 (2020).