# TOWARDS DOMAIN GENERALISATION IN ASR WITH ELITIST SAMPLING
AND ENSEMBLE KNOWLEDGE DISTILLATION

###### Abstract

Knowledge distillation (KD) has widely been used for model compression and
domain adaptation for speech applications. In the presence of multiple
teachers, knowledge can easily be transferred to the student by averaging the
models output. However, previous research shows that the student do not adapt
well with such combination. This paper propose to use an elitist sampling
strategy at the output of ensemble teacher models to select the best-decoded
utterance generated by completely out-of-domain teacher models for
generalizing unseen domain. The teacher models are trained on AMI, LibriSpeech
and WSJ while the student is adapted for the Switchboard data. The results
show that with the selection strategy based on the individual model’s
posteriors the student model achieves a better WER compared to all the
teachers and baselines with a minimum absolute improvement of about 8.4%.
Furthermore, an insights on the model adaptation with out-of-domain data has
also been studied via correlation analysis.

Index Terms—  Knowledge distillation, ASR, teacher-student training

##  1 Introduction

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Knowledge distillation (KD) [1] is a popular technique to distill the
knowledge from either single or multiple cumbersome teacher models to one
student model. The knowledge from the teacher model is transferred at the
output [1] or intermediate feature level [2]. A common technique is to
transfer the knowledge using teachers’ posterior distribution [3]. Usually,
the student model is trained with the teacher’s posteriors (soft labels) along
with the original training labels (hard labels). Therefore, the total loss is
the weighted sum of supervised and KD losses. KD is used for many tasks such
as domain adaptation [4, 5, 6], domain generalisation [7, 8, 9], and model
compression [10, 11, 12].

In automatic speech recognition, knowledge distillation is performed over
either frame-level [13] or sequence-level [14] span. Following the sequential
nature of the ASR task, sequence-level transfer is shown to be a better
approach in [13]. Sequence level KD was first proposed in [15], where the
teacher models provide sequence-level probability distribution over the whole
sample space for better knowledge transfer. In the case of multiple teacher
models, it is crucial to optimise the sampling strategy to obtain the best
possible output. For example, [16] proposed three types of selection
strategies at the output of teachers, i.e. weighted, Top-1 and Top-K. The
error metric is used as a selection criterion in all of these strategies.
Therefore, the labelled data is assumed to be available. However, this
approach is not helpful in those scenarios where the goal is to adapt unseen
out-of-domain (OOD) data. Hence, an unsupervised sampling strategy is
necessary to reduce the uncertainty of multiple teacher outcomes and select
the best output.

Furthermore, the underlying teacher model is important for the generalised
representation of the acoustic samples. Various models such as BERT[17],
BLSTMs [18] with different context windows have been used in ASR knowledge
distillation tasks. Self-supervised training paved the path to learn the
general data representation through unsupervised pre-training. Such a model is
trained with masked spans for generalised contextual latent representation of
speech [19]. These self-supervised models have been observed to be quickly
adaptable to new domains or cross-domain tasks [20].

The generalisation problem in KD is two-fold. The first problem is optimising
the teacher model’s learned representation, and the second is choosing the
appropriate distribution for a student from the teacher model. These problems
become challenging when dealing with OOD data between teacher and student. In
this paper, the first problem is tackled with a pre-trained wav2vec teacher
models where each model is fine-tuned with an in-domain corpora. The second
problem is tackled with posterior based elitist sampling strategy which
selects the best utterance decoded by the teachers. In summary, an ensemble of
teacher models are trained on completely OOD data compared to the data which
student needs to adapt. An inference is run on the unlabelled OOD data to
generate the soft labels from the teachers. Finally, an elitist sampling
strategy based on the output posteriors is used to select the best decoded
utterance from the teachers to train a student model. More specifically,
dataset from read speech: WSJ (LDC catalog LDC93S6A, LDC94S13A), LibriSpeech
(LS) [21] and meeting: AMI [22] are used to train three state-of-the-art
teacher models. These models are used to decode the Switchboard (SB) [23]
corpus, and a student model is trained. The results show that with such
selection on the ensemble of teacher outputs, the student model performs
better compared to the baselines and all the individual teacher models.

##  2 distillation using ctc loss

In teacher-student training, knowledge is usually transferred from the teacher
model in terms of the posterior distribution. The teacher model is already
trained on given labelled dataset. While training the student, its output
distribution is tried to become closer to the teacher’s posterior
distribution. This is usually achieved by using Kullback–Leibler (KL)
divergence [24] loss as:

$KL(\hat{y}_{tea}||\hat{y}_{stu})=\sum_{i=1}^{N}\hat{y}_{tea}^{i}\log{\frac{\hat{y}_{tea}^{i}}{\hat{y}_{stu}^{i}}}$
(1)

where $\hat{y}_{tea}$ and $\hat{y}_{stu}$ represent the teacher and student
output posteriors, and $N$ is the total number of examples. Frame-wise KL
divergence is usually calculated to match the two distributions. The total
loss for the student model is weighted sum of supervised and KL loss:

$\mathcal{L}_{total}=\alpha\mathcal{L}_{sup}+(1-\alpha)\mathcal{L}_{KL}$ (2)

In sequential tasks such as ASR, the studies show that knowledge can be
transferred effectively via sequence-level information instead of frame-level.
Therefore, CTC [25] loss has been used in this study. The teacher and student
models for KD are based on wav2vec2 [19] pre-trained with libriVox (LV-60K).
Dense neural network layers are connected at the output of wav2vec2 and CTC
loss is applied. The raw input to the model is represented by sequence
$\mathbf{x}=[x_{1},...x_{T_{n}}]\in X$ with each sequence of length $T_{n}$
and $X$ represents all the training sequences. The wav2vec2 model outputs the
context features represented by $\mathbf{c}=[c_{1},...c_{T_{n}}]$. Context
vector $\mathbf{c}$ given to the fully connected dense layers produces the
output represented by $\mathbf{h}=[h_{1},...h_{T_{y}}]$, where $T_{y}$ is the
output length with the vocabulary size of $G=[g_{1},...g_{z}]$, where $z$
represents grapheme. Teacher models are trained with original labels using CTC
loss as:

$\mathcal{L}_{CTC}=-\sum_{B}\log p(\mathbf{y}|\mathbf{h})$ (3)

where $\mathbf{y}$ is the output label sequence and $B=\\{X,Y\\}$ is the
training dataset.

In a standard distillation setting, the student model is trained with the
original (hard) labels and soft labels provided by the teacher model. The
total loss for the student model is the weighted sum of hard and soft label
losses as presented in equation 2. However, in our scenario, we assume that
original labels of the target domain is not available. The OOD teacher models
run the inference on the target data and provide the soft labels to train the
student. Hence, the value of $\alpha$ in equation 2 becomes zero.

For the distillation loss, we followed the work of [12, 14] which proposed to
use sequence-level loss rather than frame-level for the ASR task. Therefore,
$\mathcal{L}_{KL}$ in equation 2 is replaced by soft CTC-KD loss. This is
represented as follows:

$\mathcal{L}_{CTC-KD}=-\sum_{B}p_{tea}(\mathbf{\hat{y}}|\mathbf{h_{tea}})\log
p_{stu}(\mathbf{\hat{y}}|\mathbf{h_{stu}})$ (4)

where $p_{tea}(\mathbf{\hat{y}}|\mathbf{h_{tea}})$ and
$p_{stu}(\mathbf{\hat{y}}|\mathbf{h_{stu}})$ are posteriors from teacher and
student models. In this case, the teacher posteriors comes from the selection
criteria from multiple teachers which is explained in Section 3.

##  3 utterance selection strategies

This work utilizes multiple teacher models to generalise the out-of-domain
data for the student model. It is known that the ensemble models usually
provide better predictions compared to the single model. One of the most
straightforward approaches to distill the knowledge from the teacher models is
to average the frame-wise posterior of the teachers following the work of [10,
26] as:

$\mathbf{p_{tea}}=\frac{1}{K}\sum_{k}\mathbf{p_{k}}$ (5)

were $K$ is the total number of teachers, and $\mathbf{p_{k}}$ is the
posteriors of the whole utterance for teacher $k$. Resulting
$\mathbf{p_{tea}}$ is the frame-wise average posteriors of each utterance.
However, this strategy may not perform well because if some teachers perform
poorly then the overall average become worse. Therefore, a sampling technique
is devised which selects the best-decoded utterances from the teacher models.

In [16], authors proposed three distillation strategies, i.e. weighted, Top-1
and Top-K. The authors proposed to use error-rate as a metric of
selection/weightage in each strategy. For example, in the weighted technique,
the weights for each teacher are determined with the average error on the
mini-batch, which can only be calculated if labelled data is present.
Moreover, in their technique the student learns on the same data which
teachers have already been trained on. However, this is not the case in our
scenario, where the student learns on the data unseen by the teachers and
without original labels. Eventually, this technique cannot be directly
adapted.

For the unlabelled data each teacher model may make different mistakes while
transcribing the same utterance, and one teacher would perform better than the
others. Therefore, we propose to select the utterance from that particular
teacher model which best transcribes it. This could be achieved by ranking the
decoded posteriors from the teachers. In our selection strategy, we propose to
take the average of the posteriors of the whole utterance from each teacher as
follows:

$q_{k}=\frac{1}{T_{y}}\sum_{i=1}^{T_{y}}p_{ik}$ (6)

where $T_{y}$ is the length of utterance and $k$ is the teacher. This average
is calculated for each utterance which is decoded by the teachers. The best
teacher is selected with maximum average posterior as:

$b=\arg\max_{k}q_{k}$ (7)

With such a selection strategy, each utterance is selected from the best
performing teacher $b$, and its posteriors are used in equation (4) to train
the student model. The intuition behind the selection strategy is elitist,
i.e., assuming if a group of graphemes projects the highest confidence as a
group, they are also best fitted as individuals. The grapheme-phoneme
correspondence will be optimal when we maximise the confidence score in
graphemes as a group. If the posteriors are higher for each individual label,
it refers to the model’s confidence. Taking the average of all the frames
would provide the model’s confidence for the whole utterance. So the best
utterance among the three models would have the highest average posterior.

##  4 Internal Representation Correlation

Understanding model learning from a core point of view is crucial to
establishing the relationship between input acoustic data and the target
labels. In ASR, the transcription often depends on the general spelling and
linguistic perception of the transcribers’ language dialect or country.
Statistical correlation analysis is used to understand the relationships
between representations in neural network layers [27]. The activation outputs
are extracted among layers or model outputs across model training/test,
enabling observation of shared representations among representations. SVCCA
[28] uses singular value decomposition (SVD) and canonical correlation (CC)
analysis for obtaining the shared relationships across neural representations.
The algorithm attempts to find bases $v,\;s$ that maximise the correlation
between two matrices (in this case, sets of neural layers) when the original
matrices are projected onto the bases:

$\frac{v^{T}\sum_{XY}s}{\sqrt{v^{T}\sum_{XX}v}\sqrt{s^{T}\sum_{YY}s}}$ (8)

Here $\sum_{XX},\;\sum_{XY},\;\sum_{YY}$ are the cross-covariance and
covariance of the matrices respectively. The coefficiency of the correlations
is assessed in a layer-wise approach for $\mathcal{N}$ data points, projected
views of $l_{1}$ and $l_{2}$ are compiled, where
$l_{1}=\\{z_{1}^{l_{1}},...,z_{N_{1}}^{l_{1}}\\}$ and
$l_{2}=\\{z_{2}^{l_{2}},...,z_{N_{2}}^{l_{2}}\\}$. SVD is used to prune and
preserve the top 99% of representative dimensions, which forms subspaces
$l_{1}^{\prime}\subset l_{1}$ and $l_{2}^{\prime}\subset l_{2}$. CC analysis
is used to find vectors $v$ and $s$, which maximises the correlation $\rho$
between projections of $l_{1}^{\prime}$ and $l_{2}^{\prime}$. In this
scenario, the value of $\rho$ correlates to the information encoded in the
neural representations:

$\rho=\frac{\langle
v^{T}l_{1}^{\prime},s^{T}l_{2}^{\prime}\rangle}{||v^{T}l_{1}^{\prime}||\;||s^{T}l_{2}^{\prime}||}$
(9)

In this paper, two transformer (12 encoders, 12 decoder layers) models have
been trained with the original Switchboard transcription and the teacher-
ensemble generated pseudo transcription. The layer-wise embedding
representations have been analysed with SVCCA and discussed in Section 6.1.

##  5 experiments

###  5.1 Data

Four different corpora have been used in the experiments comprising read and
conversational speech. AMI, LS and WSJ are used to train the teacher models,
while SwithBoard (SB) is for student. LS, WSJ and SB are recorded by US native
English speakers, while AMI is recorded in three different institutes,
including Edinburgh, Idiap and TNO, also covering non-native English speakers.
Among these corpora, AMI and SB belong to the conversational speech and WSJ,
and LS belongs to the read speech. Belonging to the same category does not
reflect that they are similar. All of these corpora have different
distributions in terms of recording conditions, speaking style etc.

The total duration of LS, WSJ, AMI and SB are 960h, 272h, 100h and 300h. As LS
is the largest in duration, we only considered the 360h subset referred to as
LS360. Moreover, all datasets have a sampling rate of 16Khz except SB. Hence,
SB is up-sampled from 8KHz to 16KHz rate. To analyze the performance of the
models, we considered the standard test sets from each corpus. LS-test-clean,
LS-test-other are clean and noisy test sets of LS. One test from WSJ is
considered, i.e. WSJ-Eval92-20K, which consists of 20K words vocabulary. From
AMI, the full corpus ASR test set AMI-FCASC is taken. From SB, we considered
two test sets; eval00-CallHome (LDC97S42) represented by CH and
eval00-Switchboard (SB). The eval00 combines both of these CH and SB test
sets.

###  5.2 Experimental details

The base model for teachers and student is wav2vec2-large pre-trained with
LV-60k. The output of wav2vec2 is connected with two fully connected layers,
and CTC loss is applied. During training, wav2vec2 is also fine-tuned for both
teachers and the student. Moreover, wav2vec2 requires audio to be sampled at a
16KHz rate, so only SB audio is upsampled. The grapheme-based tokeniser is
used for all the models.

In the experiments, independent teacher models are first trained on AMI, WSJ
and LS360 datasets. These models are evaluated on multiple test sets including
in-domain and out-of-domain sets as listed in table 1. This evaluation helps
to understand the performance of the model on completely out-of-domain
distribution to analyze cross-domain adaptation. The language model (LM) used
in the decoding is a 3-gram model which is trained on the training transcripts
of all the teacher models.

Teacher models are used to decode the SB data and based on the selection
strategy discussed in section 3 the best decoded-utterance is selected to
train the student model. While training the student model, it does not know
the original training labels considering that they are not available. Hence,
the student only generalises based on the posteriors from the teachers.
Moreover, similar to the acoustic models the LM is also an out-of-domain model
for the SB data. This is selected to maintain the consistency in this study to
just evaluate the unseen domain adaptation. The results of the proposed
technique are compared with individual teacher models’ and two baseline. For
the first baseline, frame-wise average of the posteriors of all the teachers
are computed and resulting posteriors are used to train the student model. For
the second baseline, we find the maximum frame-wise posteriors among the
teachers and select it as resulting distribution. The resulting posteriors are
then used to train the student. The frame-wise maximum value of posterior
represents the confidence of the model for that particular label. Hence,
frame-wise high confident soft-labels are selected. All of these results are
discussed in Table 2 and 3, where first baseline is represented by Tea_avg
(teachers averaging method) and second one as Fw_max (frame-wise maximum
method) .

Table 1: WER(%) evaluation of the teacher models on in-domain and out-of-domain test sets with and without LM. The three individual teacher models are trained on AMI, LS360 and WSJ datasets. | AMI | LS360 | WSJ
---|---|---|---
Test sets LM |  ✗ |  ✓ |  ✗ |  ✓ |  ✗ |  ✓
AMI-FCASC |  15.85 |  14.41 |  51.69 |  47.42 |  60.60 |  57.32
LS-test-clean |  14.61 |  12.41 |  3.51 |  3.04 |  10.14 |  8.39
LS-test-other |  29.69 |  26.32 |  10.76 |  9.51 |  27.19 |  24.43
WSJ-Eval92-20K |  15.87 |  13.11 |  9.60 |  7.14 |  2.39 |  1.47
eval00 |  52.04 |  49.21 |  45.86 |  42.59 |  66.43 |  64.20
Table 2: Evaluation of WER(%) of student and teacher models on swithboard test sets without language model. | Teacher models | Student models
---|---|---
| (w/o LM) | (w/o LM)
|  |  Tea_avg |  Fw_max |  ours
sets |  AMI |  LS360 |  WSJ | SWBD
eval00 |  52.04 |  45.86 |  66.43 |  58.11 |  51.71 |  37.38
CH |  56.74 |  50.55 |  73.48 |  62.68 |  54.63 |  41.15
SB |  47.09 |  40.96 |  59.07 |  53.33 |  48.67 |  33.45
Table 3: Evaluation of WER(%) of student and teacher models on swithboard test sets with a 3-gram language model trained on AMI, LS360 and WSJ transcripts. | Teacher models | Student models
---|---|---
| (with OOD LM) | (with OOD LM)
|  |  Tea_avg |  Fw_max |  ours
sets |  AMI |  LS360 |  WSJ | SWBD
eval00 |  49.21 |  42.59 |  64.20 |  46.53 |  38.09 |  32.00
CH |  54.03 |  47.18 |  71.70 |  51.84 |  42.16 |  35.72
SB |  44.17 |  37.80 |  56.37 |  40.99 |  33.84 |  28.13

##  6 Results & Discussion

The teacher models trained in individual corpora were initially evaluated on
in-domain and out-of-fomain test sets. Table 1 shows the WER of the individual
test set used for the evaluation. Each column (AMI, LS360 and WSJ) represents
the model trained on that specific corpus and each test set is evaluated with
and without LM. It can be seen that each model produces state-of-the-art
results on its own test sets represented by shaded cells. For both LS test
sets the best cross-corpora WER was produced by WSJ and then AMI. This is due
to the fact that WSJ is also read speech data and has native English speakers.
However, both WSJ and LS have different recording and environmental
conditions. Therefore, the WSJ model produced more than 2.5 times worse WER
for LS-test-clean and LS-test-other with and without LM. On the other hand,
AMI is a conversational speech and also has non-native English speakers.
Additionally, the WSJ and LS are relatively more clean speech compared to AMI
as it has many environmental and human-generated noises. Similar behaviour is
being noticed for the WSJ-Eval92-20K test set. For the AMI test set, LS360 and
WSJ produced worse results compared to the AMI model itself. Again the reason
is the speaking styles and short spontaneous utterances present in the
conversational speech. Finally, the eval00 test, which belongs to the SB data,
is best transcribed by LS360. This is due to the large training data (360h)
and more speakers compared to the AMI and WSJ. One can argue that the AMI
should perform better due to the conversational speech. However, the AMI model
is only trained on 86h and has both native and non-native English speakers.
While SB only has native English speakers.

Table 2 shows the WER of the individual teacher models and student models on
SB test sets without LM. Among the student models the Tea_avg do not seems be
a good strategy for ensemble models as the WER is much worse compared two
teachers (AMI and LS360) and also other student models. The main reason is
that one of the teacher i.e. WSJ is performing worse, thus deteriorating the
average of the teachers’ posterior which eventually affected the student
models training. Whereas, Fw_max is better than majority of the teachers and
Tea_avg due to the reason that in this strategy maximum posterior is selected
among the teachers which corresponds to the high confident prediction. The
downside of this technique is that it affects the sequential nature of the
output because it selects the posteriors from different teacher models.

Furthermore, other than the proposed the best model among the teachers and
students which produced lowest WER for all the test sets is LS360. However, it
cannot be concluded that the LS360 model decodes all the utterances of the SB
better than AMI and WSJ to train the student. This is clearly depicted by
evaluation of proposed student model which has lowest WER compared to all the
teachers and baseline student models. For the CH test set, WER of proposed
student model is 41.15%, which is almost 9.4% better than the LS360, 21.5%
better than Tea_avg and 13.4% better than Fw_max in an absolute difference.
Similarly, SB achieved minimum improvement of about 7.5% compared to the LS360
model. Collectively, for eval00 the proposed student model is almost 8.4%
better than the best performing model LS360.

Moreover, Table 3 show that better results are achieved when OOD LM is used
for both teachers and students models. Compared to the results in Table 2 the
proposed student model is improved upto 5.3% in absolute difference using the
OOD LM making WER of eval00 to 32.00%. It has also been observed that the
results of Tea_avg and Fw_max models has improved significantly eventually
making Fw_max better than all the teachers and Tea_avg. This shows that frame-
wise posterior sampling seems to be a better candidate if decoding is applied
with an LM. Overall, results show the significance of the ensemble of teachers
and selection strategy to train the student. As the teacher models are trained
on completely out-of-domain data, the knowledge can still be distilled
effectively from ensemble models to adapt better student. Therefore, the
student performs much better than the baselines and individual teachers.

###  6.1 Representation Analysis

Fig. 1: SVCCA correlation coefficients through time as performance converges
within Transformer layers 1 to 12 model trained with original Switchboard
transcripts (left) and Transformer layers 1 to 12 model trained with pseudo
label transcripts (right)

Fig. 2: Difference in layer-wise SVCCA correlation between Switchboard model
trained with original labels vs Switchboard model trained with pseudo labels
through time within Transformer layers 1-6 (left) and layers 7-12( (right).

Figure 1 shows the correlation coefficiency through training of a model
trained with Switchboard data and a model trained with Switchboard pseudo-
labels. The convergence pattern shows during the training that the model
layers first maximise the mutual information between the acoustic input and
the layer latent representation and then minimise the mutual information
between the layer representation and the target graphemes [29, 30]. Figure 2
shows the difference in correlation coefficiency of the layers of a model
trained with Switchboard original transcript and a model trained with
Switchboard pseudo-label transcript. Figure 2 shows that the layers with
closer proximity to acoustic input have very little to no difference among the
Switchboard model trained with original transcripts vs the Switchboard model
trained with pseudo label transcripts. However, the encoders’ last three
layers (10-12) show a considerable difference, which justifies the difference
among the labels. It is safe to say that until the final few layers, both the
models are learning similar representations. Thus the generated pseudo labels
are not entirely different from the original label. Figures 1 & 2 clearly show
that, however, the middle layers learn similar representations, but the layers
next to the target categorical distribution (graphemes) are sensitive and less
stable than the original switchboard transcription model. This phenomenon is
attributed to the confusion in pseudo label generation, mainly for acoustic
boundary learning with hesitation before words and small interjections.
Different pseudo labels were generated for identical utterances due to
hesitation, laughing and other conversational nuances in those utterances,
which made the model confused while training. The in-domain read-speech
ensemble ASR models were not equipped with all these conversational speech
nuances. These issues can be alleviated while pseudo label generation using
language models similar to the unseen data domain.

##  7 CONCLUSION

To generalise the unseen out-of-domain data, this paper proposed to use
ensemble knowledge distillation with an elitist sampling technique. The
sampling is performed based on output posteriors, and the best utterance is
selected from the teachers to train a student model. With the proposed
technique the results show that the student model generalises well on the out-
of-domain data compared to the teacher models and baselines. Only out-of-
domain language model is used to study the effectiveness of cross-domain
acoustic model variability. Furthermore, intermediate neural representations
are analysed across different models and layers to understand the relationship
between acoustic data and transcription. In future work, one can improve the
sampling strategy and pseudo label correction using iterative pseudo labeling
and in-domain language models.

##  8 ACKNOWLEDGEMENTS

This work was conducted at the Voicebase/Liveperson Centre of Speech and
Language Technology at the University off Sheffield which is supported by
Liveperson, Inc.

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# Study of exclusive nonleptonic decay of $B_{s}^{0}\to J/\psi K_{s}$

Lopamudra Nayak<EMAIL_ADDRESS>National Institute of Science Education
and Research, An OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha,
India  Sonali Patnaik<EMAIL_ADDRESS>National Institute of
Science Education and Research, An OCC of Homi Bhabha National Institute,
Bhubaneswar, Odisha, India  Priyanka Sadangi<EMAIL_ADDRESS>National Institute of Science Education and Research, An OCC of Homi Bhabha
National Institute, Bhubaneswar, Odisha, India  Sanjay Kumar Swain
<EMAIL_ADDRESS>National Institute of Science Education and Research, An
OCC of Homi Bhabha National Institute, Bhubaneswar, Odisha, India

###### Abstract

We study the exclusive two-body nonleptonic $B_{s}^{0}\to J/\psi K_{s}$ decay
by considering the tree-level diagram within the framework of relativistic
independent quark (RIQ) model based on a flavor-independent interaction
potential in scalar-vector harmonic form. Using the factorization
approximation, we calculate the weak form factors from the overlapping
integrals of meson wave functions obtained in the RIQ model and predict the BF
for $B_{s}^{0}\to J/\psi K_{s}$ decays in reasonable agreement with existing
experimental result. Branching fraction of this decay is found to be within
the detection ability of the LHCb Collaboration and can also be compared with
the future theoretical predictions. We predict the ratio: ${\cal
R}\big{(}\frac{{\cal B}(B_{s}^{0}\to J/\psi K_{s})}{{\cal B}(B_{d}^{0}\to
J/\psi K_{s})}\big{)}$ which is in broad agreement with the data from LHCb
Collaborations. This results indicate that the present approach works well in
the description of exclusive nonleptonic $B_{s}^{0}\to J/\psi K_{s}$ decays
within the framework of the RIQ model.

††preprint:

##  I Introduction

The CDF Collaboration reported the first observation [1] of the Cabibbo
suppressed decay $B_{s}\to J/\psi K_{s}$ using a sample derived from an
integrated luminosity of $5.9fb^{-1}$ of proton-antiproton collisions at
center-of-mass energy, $\sqrt{s}=1.96$ TeV produced at the Fermilab’s Tevatron
in 2011. In 2012 a more precise measurement of this branching fraction (BF) is
presented by LHCb experiment [2] using integrated luminosity of a
$0.41fb^{-1}$ of proton proton collision at $\sqrt{s}=7$, presented
measurements of the ratio of branching fractions

$\displaystyle\frac{{\cal B}(B_{s}^{0}\to J/\psi K_{s})}{{\cal B}(B_{d}^{0}\to
J/\psi K_{s})}=0.042$ $\displaystyle\pm 0.0049\;(stat)\pm 0.0023\;(sys)$
$\displaystyle\pm 0.0033\;(\frac{f_{s}}{f_{d}}).$

where the mentioned uncertainties include statistical, systematic, and those
arising from the uncertainty in $f_{s}/f_{d}$, respectively. Using the world
average BF of $B^{0}_{d}\to J/\psi K_{s}$, LHCb gets the time integrated BF of
$B_{s}\to J/\psi K_{s}$ as:

$\displaystyle{\cal B}(B_{s}\to J/\psi K_{s})=[1.83$ $\displaystyle\pm
0.21(stat)\pm 0.10(sys)\pm 0.14\;(\frac{f_{s}}{f_{d}})$ $\displaystyle\pm
0.07\;(B^{0}\to J/\psi K_{s})]\times 10^{-5}$

The decay $B_{s}^{0}\to J/\psi K^{0}_{s}$ is particularly interesting because
it’s related to the decay $B^{0}_{d}\to J/\psi K^{0}_{s}$ through the U-spin
symmetry of the strong interaction. The decay $B_{d}^{0}\to J/\psi K^{0}_{s}$
is considered as the ”golden mode” for measuring the Cabibbo–Kobayashi–Maskawa
(CKM) angle $\sin{2\beta}$. The similarity between these decays allows us to
determine the CKM angle $\gamma$ by measuring the time-dependent CP violation
in $B^{0}_{s}\to J/\psi K^{0}_{s}$ decay, using normalization from
$B_{d}^{0}\to J/\psi K^{0}_{s}$ decay. The first step toward achieving these
goals is measuring the BF of $B^{0}_{s}\to J/\psi K^{0}_{s}$.

The study of exclusive nonleptonic decays of heavy meson $B_{s}$ has provided
a precise and consistent picture of the flavor sector of the Standard Model
(SM) over the past decade [3]. Several of these channels provide techniques
for investigating CP violation and determining the CKM matrix angles within
the unitarity triangle. They also serve as avenues to test various QCD-
inspired models and explore potential deviations from the physics beyond SM
(BSM). Among the possibilities of nonleptonic $B_{s}$ decay modes, the CKM-
suppressed mode induced by quark level transition $b\to c\bar{c}d$ that
involve a charmonium meson in final state are of particular interest.
Specially, the charmonium vector meson $J/\psi$ is of great experimental
interest because of its clean signal reconstruction
$J/\psi\to\mu^{+}\mu^{-}$[3]. This is the case of the vector-pseudoscalar (VP)
mode, $B_{s}\to J/\psi K_{s}$ where, it is the most sensitive probe to measure
the complex phase $\phi_{s}$ associated with the $B_{s}-\bar{B_{s}}$ mixing
process, which is extracted from the angular analysis of the time-dependent
differential decay width[4]. Using the data from CDF and the $e^{+}e^{-}$
B-factories as a guideline, ref.[5] explore the sensitivity for $\gamma$ and
the penguin parameters. They highlight the significance of the mixing phase
$\phi_{s}$ between $B^{0}_{s}$ and $\bar{B}^{0}_{s}$, which is approximately
-$2^{\circ}$ within the SM but could be amplified by new physics (NP), as a
crucial factor in these investigations. In Belle II and experiments involved
in the high-luminosity phase of the Large Hadron Collider (HL-LHC) [6, 7], the
study of the $B_{s}$ meson will hold considerable importance. This stems from
the anticipation of a substantial yield of $B_{s}$ mesons in the ongoing
operations, offering an opportunity for in-depth investigation with remarkable
statistical accuracy. These inquiries will not only serve as a pivotal
confirmation of B meson findings but will also actively seek indications of
NP.

Describing nonleptonic decay is notoriously challenging due to the strong
influence of confining color forces and the involvement of matrix elements of
local four-quark operators within the non-perturbative Quantum Chromodynamics
(QCD) approach, a mechanism still not thoroughly understood within the SM
framework. When excluding weak annihilation contributions, the nonleptonic
transition amplitudes are conveniently approximated using the so-called naive
factorization method [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21].
This approach works reasonably well in two-body nonleptonic $B_{c}$ decays,
particularly where the quark-gluon sea is suppressed within heavy
quarkonium[22, 23, 24, 25, 26]. Bjorken’s original argument on color
transparency [27], theoretical advancements based on QCD in the limit of large
number of colors ($\frac{1}{N_{c}}$) [28], and the Heavy Quark Effective
Theory (HQET) [29] provide rationale for this approximation. In this study,
our focus lies solely on the impact of current-current operators [30] in
calculating the dominant tree-level diagram for these decays. While the
penguin diagram’s influence may be notable in assessing CP-violation and
probing for NP or Beyond Standard Model (BSM) physics, its significance to the
decay amplitudes under consideration here is deemed less pronounced. Indeed,
previous research has demonstrated that the contribution of QCD and
electroweak penguin operators [31, 32, 33, 34] is negligible compared to that
of current-current operators in these decays, primarily due to significant
suppression of CKM elements. Given the very small Wilson coefficients
associated with penguin operators, their contribution to weak decay amplitudes
is only pertinent in rare decay scenarios. Such instances arise when the tree-
level contribution is heavily CKM-suppressed, as in $\bar{B}\to\bar{K}^{\pi}$,
or when matrix elements of current-current operators are entirely absent, as
observed in $\bar{B}\to\bar{K}\gamma$ and $\bar{B}^{0}\to\bar{K}^{0}\phi$
[30].

The relativistic independent quark(RIQ) model, has been applied in wide-
ranging hadronic sector describing the static properties of hadrons [35, 36,
37] and their decay properties in the radiative, weak radiative, rare
radiative [38, 39, 40, 41, 42, 43]; leptonic, weak leptonic, radiative
leptonic [43, 44, 45, 46, 47] and semileptonic [48, 49, 50] decays of mesons.
In our recent analysis we predict the magnetic dipole and electromagnetic
transitions of $B_{c}$ and $B_{c}^{*}$ mesons in their ground as well as
excited states[51, 52];the exclusive semileptonic $B_{c}$-mesons decays to the
charmonium ground states in the vanishing [53] and non-vanishing [54, 55, 56,
57] lepton mass limit. In this model our group have predicted [11, 12, 58, 59,
60, 61] the exclusive two body nonleptonic decays of heavy flavored meson to
the charmonium and charm mesons in their ground as well as radially excited
states. This has greatly inspired us to broaden the utilization of the RIQ
model to examine the two-body nonleptonic decay of $B_{s}\to J/\psi K_{s}$.

The rest of the paper is organized as follows. In section II, we present the
effective Hamiltonian and factorization for two-body nonleptonic decay modes
of $B_{s}$ mesons induced by $b\to c\bar{c}d$ transitions at the quark level.
Section III elaborates on the weak decay form factors, which represent the
hadronic amplitudes, and are computed from the overlap integral of meson wave-
functions within the framework of the RIQ model. Section IV is dedicated to
the presentation of numerical results and subsequent discussion, while Section
V provides a concise summary and conclusion. Additionally, the Appendix offers
a brief overview of the RIQ model framework, the wave packet representation of
meson states, and the momentum probability amplitudes of constituent quarks
within the meson bound-state.

##  II Effective Hamiltonian and factorization approximation

The effective weak Hamiltonian for two body nonleptonic $B_{s}$ decays,
induced by $b\to c\bar{c}d$ at the quark level, (as shown in Fig.1) consists
of a sum of local four quark operators ${\cal O}_{i}$ multiplied by short
distance coefficients ${\cal C}_{i}$ and products of elements of the quark
mixing matrix $\lambda_{c}=V_{cb}V_{cd}^{*}$, can be written as:

$\displaystyle{\cal H}_{eff}=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\lambda_{c}\sum{\cal C}_{i}{\cal O}_{i}$
$\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}V_{cd}^{*}\big{[}{\cal
C}_{1}(\mu)(\bar{c}b)(\bar{d}c)+{\cal
C}_{2}(\mu)(\bar{d}b)(\bar{c}c)\big{]}+H.c.$

Here $G_{F}$ is the Fermi coupling constant $V_{ij}$ are CKM factors,
$(\bar{q}_{\alpha}q_{\beta})$ is a short notation for $V-A$ current
$q_{\alpha}\gamma^{\mu}(1-\gamma_{5})q_{\beta}$, and ${\cal C}_{1,2}$ are the
Wilson coefficients. As the decays of neutral $B_{s}$ mesons into $J/\psi
K_{s}$ final state are dominated by a tree-level amplitude proportional to
$V_{cb}V_{cd}^{*}$, so we consider here only the tree-level current-current
operators (${\cal C}_{1,2}$) by neglecting the penguin diagrams.

In the framework of naive factorization, the nonleptonic decay amplitude is
approximated by the product of two matrix elements of quark currents as:

$\langle M_{1}M_{2}|{\cal O}|B_{s}\rangle_{i}=\langle
M_{1}|J^{\mu}|0\rangle\langle M_{2}|J_{\mu}|B_{s}\rangle+(M_{2}\leftrightarrow
M_{1})$ (2)

where $J_{\mu}$ is the weak current and $M_{i}$ is a pseudoscalar($P$) or a
vector($V$) meson. One of these is the matrix element for the $B_{s}$
transition to one final mesons state, while the other matrix element
corresponds to the transition from the vacuum to other final meson state. The
latter is given by the corresponding meson decay constant. In this way the
hadronic matrix element of four-quark operators can be expressed as the
product of decay constant and invariant weak form factors [62, 63, 64, 65, 66,
67, 68, 69, 70, 71, 72].

Of course there is difficulty inherent in such an approach because the
Wilson’s coefficients, which include the short distance QCD effect between
$\mu=m_{N}$ and $\mu=m_{b}$ are $\mu$-scale and renormalization scheme
dependent, while $\langle{\cal O}\rangle_{i}$ are $\mu$-scale and
renormalization scheme independent. As a result, the physical amplitude
depends on the $\mu$ scale. However, the naive factorization disentangles the
long distance effects from the short distance sector assuming that the matrix
element $\langle{\cal O}\rangle$ at $\mu$ scale, contain non-factorizable
contributions in order to cancel the $\mu$ dependence and scheme dependence of
$c_{i}(\mu)$[68, 18, 69, 20, 70, 71, 73]. In general, it works in some two-
body nonleptonic decays of heavy mesons in the limit of a large number colors.
It is expected that the factorization scheme works reasonably well in two-body
nonleptonic $B_{s}$ decays with charmonium meson in the final states[74, 75].

We also neglect here the so-called $W$-exchange and annihilation diagram,
since in the limit $M_{W}\to\infty$, they are connected by Fiertz
transformation and are doubly suppressed by the kinematic factor of the order
$(\frac{M_{i}^{2}}{M_{W}^{2}})$. We also discard the color octet current which
emerge after the Fiertz transformation of color-singlet operators. Clearly,
these currents violate factorization since they can not provide transitions to
the vacuum states. Taking into account the Fiertz reordered contribution, the
relvant coefficients are not $c_{1}(\mu)$ and $c_{2}(\mu)$ but the
combination:

$a_{1,2}(\mu)=c_{1,2}(\mu)+\frac{1}{N_{c}}c_{2,1}(\mu)$ (3)

where, $N_{c}$ is color factor. Assuming large $N_{c}$ limit to fix the QCD
coefficients $a_{1}\approx c_{1}$ and $a_{2}\approx c_{2}$ at $\mu\approx
m_{b}^{2}$, nonleptonic decays of heavy mesons have been analyzed in Ref.[62,
76, 77, 78, 79, 80, 79].

The matrix elements corresponding to the transition from vacuum to one of the
final state pseudoscalar or vector meson are covariantly expanded in terms of
the meson decay constant $f_{P,V}$ as:

$\displaystyle\langle
P|\bar{q}_{i}^{{}^{\prime}}\gamma^{\mu}\gamma_{5}q_{j}|0\rangle=if_{P}p_{P}^{\mu}$
$\displaystyle\langle
V|\bar{q}^{{}^{\prime}}_{i}\gamma^{\mu}q_{j}|0\rangle=e^{*\mu}f_{V}m_{V}$ (4)

The covariant decomposition of matrix elements of the weak current $J_{\mu}$
between initial and final pseudoscalar meson state is:

$\displaystyle\langle P(p_{P})|\bar{q}_{c}\gamma_{\mu}q_{b}|B_{s}(p)\rangle$
$\displaystyle=\big{[}(p+p_{P})_{\mu}-\frac{M^{2}-m_{P}^{2}}{q^{2}}q_{\mu}\big{]}F_{1}(q^{2})+\frac{M^{2}-m_{P}^{2}}{q^{2}}q_{\mu}F_{0}(q^{2})$
$\displaystyle=(p+p_{P})_{\mu}f_{+}(q^{2})+(p-p_{P})_{\mu}f_{-}(q^{2})$ (5)

where

$\displaystyle f_{+}(q^{2})=$ $\displaystyle F_{1}(q^{2})$ (6) $\displaystyle
f_{-}(q^{2})=$
$\displaystyle\frac{M^{2}-m^{2}_{P}}{q^{2}}\big{[}F_{0}(q^{2})-F_{1}(q^{2})\big{]}$
(7)

For transition to the vector meson final state, corresponding matrix element
is parametrized as :

$\langle
V(p_{V})|\bar{q}_{c}\gamma_{\mu}q_{b}|B_{s}(p)\rangle=\frac{2V(q^{2})}{M+m_{V}}i\epsilon_{\mu\nu\rho\sigma}\
e^{*\nu}p^{\rho}p_{V}^{\sigma}$ (8)

and

$\displaystyle\langle
V(p_{V})|\bar{q}_{c}\gamma_{\mu}\gamma_{5}q_{b}|B_{s}(p)\rangle$
$\displaystyle=(M+m_{V})e^{*}_{\mu}A_{1}(q^{2})-\frac{A_{2}(q^{2})}{M+m_{V}}(e^{*}.q)(p+p_{V})_{\mu}$
$\displaystyle-2m_{V}\frac{e^{*}.q}{q^{2}}q_{\mu}A_{3}(q^{2})+2m_{V}\frac{e^{*}.q}{q^{2}}q_{\mu}A_{0}(q^{2})$
(9)

where

$A_{3}(q^{2})=\frac{M+m_{V}}{2m_{V}}A_{1}(q^{2})-\frac{M-m_{V}}{2m_{V}}A_{2}(q^{2})$
(10)

Here $p,p_{P,V}$ stand for the four momentum of the initial and final state
meson, respectively. M is the mass of decaying $B_{s}$ and $m_{P}$ and $m_{V}$
stand for the mass of the pseudoscalar and vector mesons, respectively, in the
final state. $q=p-p_{P,V}$ denotes the four momentum transfer and
$\hat{e}^{*}$: the polarization of the final state vector meson.
In order to cancel the poles at $q^{2}=0$, invariant weak form factors:
$F_{0}(q^{2}),F_{1}(q^{2}),A_{0}(q^{2})$ and $A_{3}(q^{2})$ satisfy following
conditions:

$F_{0}(0)=F_{1}(0)\ \ \ \ and\ \ \ A_{0}(0)=A_{3}(0)$ (11)

The decay rate for nonleptonic transition $M\to PV(VP)$ is expressed in terms
of the decay amplitude $A(B_{s}\to PV(VP))$ as:

$\Gamma(B_{s}\to PV,VP)=\frac{|\vec{k}|^{3}}{8\pi m_{V}^{2}}|A(B_{s}\to
PV,VP)|^{2}$ (12)

where $\vec{k}$ is the magnitude of three-momentum of the final state meson.
In the parent meson rest frame it is given by

$|\vec{k}|=\frac{1}{2M}\biggl{\\{}\big{[}M^{2}-(m_{P,V}+m_{V,P})^{2}\big{]}\big{[}M^{2}-(m_{P,V}-m_{V,P})^{2}\big{]}\biggr{\\}}^{1/2}$
(13)

The relevant decay amplitude (2) can either be expressed in the form:

$\displaystyle A=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}(CKMfactor)(QCDfactor)2m_{V}f_{V}F_{1}^{B_{s}\to
P}(q^{2})$ $\displaystyle or$ $\displaystyle A=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}(CKMfactor)(QCDfactor)2m_{V}f_{P}A_{0}^{B_{s}\to
V}(q^{2})$

for $B_{s}\to PV$ and $B_{s}\to VP$ decay, respectively. The factorized
amplitudes (2) expressed in terms of meson decay constants $(f_{P,V})$ and
weak form factors $F_{1}^{B_{s}\to P},\ A_{0}^{B_{s}\to V}$. It is then
straight forward to predict the decay rate for different decay processes in
the RIQ model framework.

##  III transition amplitude and weak decay form factors in the relativistic
independent quark model.

We study two-body nonleptonic $B_{s}$-decays of category: $B_{s}\to VP(PV)$,
where P and V stand for pseudoscalar and vector meson final states,
respectively. The decay amplitude is calculated here from relevant tree-level
”class-II” type diagram as shown in Fig. 1. The ”class-II” decay is
characterized by internal $W$-emission, where the decay amplitude is
proportional to the QCD-factor $a_{2}(\mu)$. As described above we consider
the two-body nonleptonic $B_{s}$-decays induced by $b\to c\bar{c}d$ transition
at quark level, with $s$-quark remaining a spectator. In the present study we
restrict our discussion to class II $B_{s}\to J/\psi K_{s}$ decay mode.

Figure 1: Diagram contributing to the $B_{s}\to J/\psi K_{s}$ Class II type
decay

In fact the decay process physically occurs in the momentum eigen-states of
participating mesons. Therefore, in a field-theoretic description of the decay
process, it is appropriate to represent the meson bound-state in terms of a
momentum wave-packet reflecting momentum and spin distribution between
constituent quark and antiquark in the meson-core. In RIQ model, the wave
packet corresponding to a meson bound-state
$|B_{s}(\vec{p},S_{B_{s}})\rangle$, for example, at a definite moment
$\vec{p}$ and spin state $S_{B_{s}}$ is represented by:

$\displaystyle|B_{s}(\vec{p},S_{B_{s}})\rangle=$
$\displaystyle\;\hat{\Lambda}(\vec{p},S_{B_{s}})|(\vec{p}_{\bar{b}},\lambda_{\bar{b}});(\vec{p}_{s},\lambda_{s})\rangle$
(14) $\displaystyle=$
$\displaystyle\;\hat{\tilde{b}}^{\dagger}_{b}(\vec{p}_{b},\lambda_{b})\hat{b}_{s}^{\dagger}(\vec{p}_{s},\lambda_{s})|0\rangle$

where
$|(\vec{p}_{\bar{b}},\lambda_{\bar{b}});(\vec{p}_{s},\lambda_{s})\rangle$ is
the Fock-space representation of the unbound quark and antiquark in a color-
singlet configuration with respective momentum and spin:
$(\vec{p}_{b},\lambda_{b})$ and $(\vec{p}_{s}\lambda_{s})$.
$\hat{b}^{\dagger}_{q}(p_{b},\lambda_{b})$ and
$\hat{\tilde{b}}_{s}(\vec{p}_{s},\lambda_{s})$ are, the quark and antiquark
creation operator. Here $\hat{\Lambda}(\vec{p},S_{B_{s}})$ is a bag like
integral operator taken in the form:

$\displaystyle\hat{\Lambda}(\vec{p},S_{B_{s}})=$
$\displaystyle\frac{\sqrt{3}}{\sqrt{N(\vec{p})}}\sum_{\lambda_{b},\lambda_{s}}\zeta^{B_{s}}_{b,s}(\lambda_{b},\lambda_{s})\int
d^{3}p_{b}d^{3}p_{s}$ (15)
$\displaystyle\times\;\delta^{(3)}(\vec{p}_{b}+\vec{p}_{s}-\vec{p}){\cal
G}_{B_{s}}(\vec{p}_{b},\vec{p}_{s})$

Here $\sqrt{3}$ is the effective color factor and
$\zeta^{B_{s}}_{b,s}(\lambda_{b},\lambda_{s})$ is the SU(6) spin-flavor
coefficients for $B_{s}$ meson state. Imposing the normalization condition in
the form $\langle
B_{s}(\vec{p})|B_{s}(\vec{p}^{{}^{\prime}})\rangle=\delta^{(3)}(\vec{p}-\vec{p}^{{}^{\prime}})$,
the meson state normalization $N(\vec{p})$ is obtainable in an integral form:

$N(\vec{p})=\int d^{3}\vec{p}_{b}|{\cal
G}_{B_{s}}(\vec{p}_{b},\vec{p}-\vec{p}_{b})|^{2}$ (16)

Finally ${\cal G}_{B_{s}}(\vec{p}_{b},\vec{p}-\vec{p}_{b})$ denote the
momentum distribution function for the quark and antiquark pair in the meson
core. In this model ${\cal G}_{B_{s}}(\vec{p}_{b},\vec{p}-\vec{p}_{b})$ is
taken in the form:${\cal
G}_{B_{s}}(\vec{p}_{b},\vec{p}-\vec{p}_{b})=\sqrt{G_{b}(\vec{p}_{b})G_{s}(\vec{p}-\vec{p}_{b})}$
in the straight forward extension of the ansatz of Margolis and Mendel in
their bag model description [81]; where $G_{b}(\vec{p}_{b})$ and
$G_{s}(\vec{p}-\vec{p}_{b})$ refer to individual momentum probability
amplitude of the constituent quark. Here the effective momentum distribution
function in fact embodies the bound-state character in
$|B_{s}(\vec{p},S_{B_{s}})\rangle$.

Any residual internal dynamics responsible for the decay process can,
therefore, be described at the constituent level by the otherwise unbound
quark and antiquark using usual Feynman technique. The constituent level
$S$-matrix element $S_{fi}^{b}\to c\bar{c}d$ obtained from the appropriate
Feynman diagram, when operated upon by the bag like operator
$\hat{\Lambda}(\vec{p},S_{B_{s}})$ (14) can give rise to the mesonic level
$S$-matrix in the form:

$S_{fi}^{B_{s}\to J/\psi K_{s}}\to\hat{\Lambda}(\vec{p},S_{B_{s}})S_{fi}^{b\to
c\bar{c}d}$ (17)

The decay mode: $B_{s}\to J/\psi K_{s}$, is induced by $b\to c\bar{c}d$
transition at quark level with emission of $W$-boson. The internally emitted
$W$-boson with four momentum $q$, which decays to a quark-antiquark pair,
subsequently the quark hadronize with the resulting anti-quark $c$ from
$b$-antiquark to form the charmonium$(J/\psi)$ state. While the antiquark
hadronize with the spectator quark $s$ to form the $K_{s}$. Considering the
wave packet representation of the participating meson states in the factorized
decay amplitude (3), the $S$-matrix element for $B_{s}\to J/\psi K_{s}$ can be
obtained in the general form:

$S_{fi}=(2\pi)^{4}\delta^{(4)}(p-q-k)(-i{\cal
M}_{fi})\times\frac{1}{\sqrt{V2E_{B_{s}}}}\prod_{f}\frac{1}{V2E_{f}}$ (18)

The invariant transition amplitude ${\cal M}_{fi}$ is infact extracted in the
form :

${\cal M}_{fi}=\frac{G_{F}}{\sqrt{2}}V_{bc}V_{cd}a_{2}A$ (19)

where, $A=h^{\mu}H_{\mu}$with

$h^{\mu}=\sqrt{\frac{2E_{J/\psi}}{(2\pi)^{3}}}\langle
J/\psi(\vec{p_{J}/\psi},S_{P_{2}})|J^{\mu}|0\rangle$ (20)

and

$\displaystyle H_{\mu}=$
$\displaystyle\frac{1}{\sqrt{N_{B_{s}}(\vec{p})N_{P_{1}}(\vec{k})}}\int{\frac{d^{3}{\vec{p}_{b}}{\cal
G}_{B_{s}}(\vec{p}_{b},\vec{p}-\vec{p}_{b}){\cal
G}_{K_{s}}(\vec{p}_{b}+\vec{k}-\vec{p},\vec{p}-\vec{p}_{b})}{\sqrt{E_{b}(\vec{p}_{b})E_{d}(\vec{p}_{b}+\vec{k}-\vec{p})}}}$
(21)
$\displaystyle\times\sqrt{\big{[}E_{b}(\vec{p}_{b})+E_{d}(\vec{p}-\vec{p}_{b})\big{]}\big{[}E_{d}(\vec{p}_{b}+\vec{k}-\vec{p})+E_{\bar{s}}(\vec{p}-\vec{p}_{b})\big{]}}$
$\displaystyle\langle S_{K_{s}}|J_{\mu}(0)|S_{B_{s}}\rangle$

Here, $E_{b}(\vec{p}_{b})$ and $E_{d}(\vec{p}_{b}+\vec{k}-\vec{p})$ denote the
energy of the non-spectator quark in the parent and daughter meson,
respectively. $\vec{p}_{b}$, $\vec{p}$, and $\vec{k}$ represent the momentum
vectors of the non-spectator constituent quark $b$, the parent meson $B_{s}$,
and the daughter meson $K_{s}$, respectively. $q$ stands for the four-momentum
transfer associated with the process. Symbolically, $\langle
S_{K_{s}}|J{\mu}|S_{B_{s}}\rangle$ represents the spin matrix elements of the
effective vector-axial vector current.

It is noteworthy that in the description of decay processes like $B_{s}\to
J/\psi K_{s}$ within the RIQ model framework, three-momentum conservation is
explicitly ensured through
$\delta^{(3)}(\vec{p}_{q_{1}}+\vec{p}_{\bar{q_{2}}}-\vec{p})$ in the involved
meson states. However, energy conservation in such a scheme is not explicitly
ensured. This is a common issue in potential model descriptions of mesons as
bound states of valence quarks and antiquarks interacting via some
instantaneous potential. This issue has been addressed in our prior analyses
concerning radiative leptonic decays of heavy-flavored mesons: $B$, $B_{c}$,
$D$, and $D_{s}$ [82, 37, 83, 84, 54]. In those analyses, the effective
momentum distribution function ${\cal G}_{M}(\vec{p}_{q_{1}},\vec{p}_{q_{2}})$
encapsulating the bound-state characteristics of the meson ensures energy
conservation in an average sense, satisfying $E_{M}=\langle
M(\vec{p},S_{M})|\big{[}E_{q_{1}}(\vec{p}_{q_{1}})+E_{q_{2}}(\vec{p}_{q_{2}})\big{]}|M(\vec{p},S_{M})\rangle$.
Therefore, we impose the energy conservation constraint:
$M=E_{q_{1}}(\vec{p}_{q_{1}})+E_{q_{2}}(-\vec{p}_{q_{1}})$, where $M$ denotes
the mass of the meson at rest. Along with the three-momentum conservation via
appropriate $\delta^{(3)}(\vec{p}_{q_{1}}+\vec{p}_{q_{2}}-\vec{p})$ in the
meson state, this ensures the required four-momentum conservation:
$\delta^{(4)}(p-k-q)$ at the mesonic level, which is separated from the quark
level integration to obtain the $S$-matrix element in the standard form (18).

Since the axial vector current does not contribute to the decay amplitude in
the decay processes: $B_{s}\to J/\psi K_{s}$, the only non-vanishing vector
current part of (21) is simplified after calculating corresponding spin matrix
elements using usual spin algebra. The resulting time-like and space-like part
of the hadronic matrix element in the parent meson rest-frame are obtained,
respectively, as:

$\displaystyle\langle K_{s}(\vec{k})|V_{0}|B_{s}(0)\rangle=H_{0}=$
$\displaystyle\int
d\vec{p}_{b}C(\vec{p}_{b})\bigl{\\{}\big{[}E_{b}(\vec{p}_{b})+m_{b}\big{]}$
(22)
$\displaystyle\big{[}E_{d}(\vec{p}_{b}+\vec{k})+m_{d}\big{]}+\vec{p}_{b}^{2}\bigr{\\}}$

and

$\langle P_{1}(\vec{k})|V_{i}|B_{s}(0)\rangle=H_{i}=\int d\vec{p}_{b}\
C(\vec{p}_{b})\ \big{[}E_{b}(\vec{p}_{b})+m_{b}\big{]}k_{i}\ \ ,$ (23)

where

$\displaystyle C(\vec{p}_{b})$ $\displaystyle=\frac{{\cal
G}_{B_{s}}(\vec{p}_{b},-\vec{p}_{b}){\cal
G}_{K_{s}}(\vec{p}_{b}+\vec{k},-\vec{p}_{b})}{\sqrt{N_{B_{s}}(0)N_{K_{s}}(\vec{k})}}$
(24)
$\displaystyle\sqrt{\frac{\big{[}E_{b}(\vec{p}_{b})+E_{s}(-\vec{p}_{b})\big{]}\big{[}E_{d}(\vec{p}_{b}+\vec{k})+E_{s}(-\vec{p}_{b})\big{]}}{E_{b}(\vec{p}_{b})E_{d}(\vec{p}_{b}+\vec{k})\big{[}E_{b}(\vec{p}_{b})+m_{b}\big{]}\big{[}E_{d}(\vec{p}_{b}+\vec{k})+m_{d}\big{]}}}$

Now a comparison of the results (22,23,24) with the corresponding expression
of the covariant factorized amplitude (II,6,7) yields the Lorentz invariant
form factors $f_{+}(q^{2})$ in the form:

$\displaystyle F_{1}(q^{2})$ $\displaystyle=f_{+}(q^{2})=\frac{1}{2}\int
d\vec{p}_{b}C(\vec{p}_{b})$ (25)
$\displaystyle\biggl{\\{}\big{[}E_{b}(\vec{p}_{b})+m_{b}\big{]}\big{[}E_{d}(\vec{p}_{b}+\vec{k})+m_{d}\big{]}+\vec{p}_{b}^{2}+$
$\displaystyle\big{[}E_{b}(\vec{p}_{b})+m_{b}\big{]}\big{[}M-E_{P}\big{]}\biggr{\\}}$

The decay amplitude in the $B_{s}$-rest frame in such decays is obtainable in
terms of the invariant form factor $F_{1}(q^{2})$ as:

$\displaystyle\langle P(\vec{k})V(\vec{q})|{\cal H}_{eff}|B_{s}(0)\rangle\;$
$\displaystyle=i\frac{G_{F}}{\sqrt{2}}V_{bc}V_{cd}2a_{2}$ (26)
$\displaystyle\times\;m_{J/\psi}f_{J/\psi}F_{1}(q^{2})(e^{*}.p)$

Here $e^{*}$ denotes the polarization vector associated with the daughter
meson(V). The decay rate $\Gamma(B_{s}\to J/\psi K_{s})$ is expressed as

$\Gamma(B_{s}\to J/\psi K_{s})=\frac{|\vec{k}|^{3}}{8\pi
m_{J/\psi}^{2}}|A_{2}|^{2}|F_{1}(q^{2})|^{2}$ (27)

where

$|A_{2}|=\frac{G_{F}}{\sqrt{2}}V_{bc}V_{cd}2a_{2}m_{J/\psi}f_{J/\psi}$ (28)

##  IV Numerical results and discussion

To compute the two-body nonleptonic $B_{s}$ decays within the RIQ model, it is
necessary to determine the flavor-independent potential parameters
$(a,V_{0})$, quark masses $(m_{q})$, and their respective binding energies
$(E_{q})$. Notably, these parameters have been previously established in our
model to accurately reproduce experimental meson spectra across both light and
heavy flavors sector[35, 37, 36] and subsequently used in the description of a
wide ranging hadronic phenomena[38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48,
49, 50, 85, 51, 52, 53, 86, 11, 12, 58, 59] involving participating mesons in
their ground state. Consequently, the potential parameters employed in this
study are:

$(a,V_{0})=(0.17166\ GeV^{3},-0.1375\ GeV)$ (29)

The quark masses and corresponding binding energies in $GeV$ are taken as:

$\displaystyle m_{d}=$ $\displaystyle 0.07875\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
E_{d}=0.47125$ $\displaystyle m_{s}=$ $\displaystyle 0.31575\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ E_{s}=0.591$ $\displaystyle m_{c}=$ $\displaystyle 1.49275\ \ \ \
\ \ \ \ \ \ \ \ \ \ \ E_{c}=1.57951$ $\displaystyle m_{b}=$ $\displaystyle
4.77659\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E_{b}=4.76633$ (30)

For relevant CKM parameters and lifetime of $B_{s}$-meson, we take their
values from PDG [87] as:

$\displaystyle|V_{cb}|=$ $\displaystyle 0.0408\pm 0.0014,\ |V_{cd}|=0.221\pm
0.004,$ $\displaystyle\tau_{B_{s}}=$ $\displaystyle 1.521\pm 0.005\ ps$ (31)

For the masses and decay constants of the involved mesons, which are regarded
as phenomenological inputs in this analysis, we adopt their values from the
existing observed data Ref.[87] and lattice computation[88] respectively.
Accordingly the updated meson masses (MeV) and decay constant(MeV) used in the
present analysis are

$\displaystyle M_{B_{s}}=$ $\displaystyle 5366.92\pm 0.10\ \ \ \ \ \ \ \
f_{J/\psi}=418\pm 9$ $\displaystyle M_{J/\psi}=$ $\displaystyle 3096.9\pm
0.006$ $\displaystyle M_{K_{s}}=$ $\displaystyle 497.61\pm 0.013$

As mentioned earlier, our calculations utilize the potential parameters (29)
and the quark masses along with their binding energies (IV). These parameters
were previously determined at the static level application of the RIQ model by
fitting the mass spectra of ground-state mesons [35, 36]. The same parameter
set has been consistently applied in prior studies employing the RIQ model,
providing a satisfactory description of various hadronic phenomena across both
light and heavy flavor mesons in their ground state. Hence, our approach does
not involve any adjustable parameters that would require fine-tuning for
predicting a broad range of hadronic phenomena, as mentioned earlier.
Therefore, our studies involve parameter-free calculations.

Concering the QCD coefficients $(a_{1},a_{2})$, various values have been
employed in existing literature when computing the nonleptonic transitions of
$B_{s}$ mesons resulting from $b$-quark decay. For example Colangelo
$\it{etal.}$, in Ref [76] use QCD co-efficient $(a_{2}^{b})=(-0.26)$ as fixed
in Ref [89] whereas, S.Dubnicka $\it{etal.}$[90], use the QCD co-efficient
$(a_{2}^{b})=(-0.27)$ for the calculation of BF. We have used these two values
of $a_{2}$ in our calculation.

Utilizing the model parameters provided in equations (29) and (IV), our
initial investigation focuses on examining the $q^{2}$-dependence of the form
factors within the permissible kinematic region: $0<q^{2}\leq
q^{2}_{\text{max}}$, as dictated by the analytic expression (25). Employing a
self-consistent dynamic approach, we derive the form factor through the
overlapping integrals of meson wave functions, thereby inherently encoding the
$q^{2}$-dependence in the relevant expressions. The $q^{2}$-dependence of the
form factor $F_{1}(q^{2})$ for nonleptonic $B_{s}$ decays to ground states is
illustrated in Figure 2. Our analysis reveals that the form factors associated
with transitions to $1S$ meson states exhibit a consistent rise with
increasing $q^{2}$ across the entire kinematic spectrum. This observation
aligns with the understanding that the $B_{s}\to J/\psi K_{s}$ transition
displays a larger $q^{2}$-dependence, as the relevant form factor
$F_{1}(q^{2})$ exhibit an increase in magnitude with rising $q^{2}$.

Figure 2: $q^{2}$-dependence of form factor for the decay $B_{s}\to J/\psi
K_{s}$

We then compute the BF using the expressions provided in Eq. (27). Our
findings for a range of QCD coefficients $(a_{2})$ from the operator product
expansion are presented in Table 1. Notably, our results demonstrate
satisfactory agreement with experimental observations. The ratio $\frac{{\cal
B}(B_{s}^{0}\to J/\psi K_{s})}{{\cal B}(B_{d}^{0}\to J/\psi K_{s})}=0.0431\pm
0.0017\;(stat)\pm 0.0012\;(sys)\\!\pm 0.0025(\frac{f_{s}}{f_{d}})$ determined
by LHCb experiment[91], prompted us to compute the same observable within our
model framework for thoroughness. By utilizing the globally accepted decay
rate of $B_{d}^{0}\to J/\psi K_{s}$, we calculate the ratio (only the central
values) and compared it with experimental findings, revealing a strong
concordance between the two. As we are resorting to the parameter free
calculation from the model framework, therefore the dominant theoretical
errors stem solely from the uncertainties of the input parameters
$|V_{cb}|,|V_{cd}|$ and $f_{J/\psi}$ which are within a range of $\pm 10\%$.
As we embark on an era of Belle II and the LHCb upgrades [6, 7], it becomes
evident that numerous intriguing and promising advancements await us, both in
experimental and theoretical realms. With the advance of precision
measurements these endeavors may serve as a gateway to exploring physics in
$B_{s}$ systems.

Table 1: Results for the decay $B_{s}\to J/\psi K_{s}$ in RIQ model | Our Result | Exp. Result
---|---|---
| $a_{2}=-0.27$ | $a_{2}=-0.26$ |
${\cal B}(B_{s}\to J/\psi K_{s})$ | $2.026^{+0.321}_{-0.291}\times 10^{-5}$ | $1.879^{+0.297}_{-0.270}\times 10^{-5}$ |  $(1.92\pm 0.14)\times 10^{-5}$[87]
$\frac{{\cal B}(B_{s}^{0}\to J/\psi K_{s})}{{\cal B}(B_{d}^{0}\to J/\psi K_{s})}$ | $0.0451$ | $0.0418$ |  $0.0431$[91]

##  V Summary and conclusion

In this study, we investigate the specific case of exclusive two-body
nonleptonic decays, focusing on $B_{s}\to J/\psi K_{s}$. Specifically, we
examine the decay category $B_{s}\to PV$ within the framework of RIQ model,
employing a flavor-independent interaction potential expressed in a scalar-
vector harmonic form. Utilizing the factorization approximation, we extract
weak decay form factors, representing decay amplitudes and their dependence on
the momentum transfer ($q^{2}$), across the entire kinematic range $0\leq
q^{2}\leq q^{2}_{\text{max}}$. These form factors are derived from the
overlapping integrals of meson wave functions within the RIQ model. Our
investigation indicates that the form factors corresponding to transitions
involving $1S$ meson states consistently increase as $q^{2}$ rises throughout
the entire kinematic range. This finding is in line with the notion that the
$B_{s}\to J/\psi K_{s}$ transition demonstrates a pronounced
$q^{2}$-dependence, as evidenced by the growing magnitude of the relevant form
factors ($F_{1}(q^{2})$) with increasing $q^{2}$. Our analysis of BF for the
$B_{s}\to J/\psi K_{s}$ decay is of the order of $10^{-5}$, primarily due to
the doubly suppressed CKM matrix element. The resulting BF obtained from our
work aligns with experimental observations reported by the LHCb Collaboration.
To ensure thoroughness additionally, we compute the ratio of BF of $B_{s}$ to
$B_{d}$ for the same final state $J/\psi$, utilizing the world average decay
rate of $B_{d}^{0}\to J/\psi K_{s}$ and then compared it with the experimental
findings.

With the increasing amount of $B_{s}$ samples produced by the updating Belle
experiment, these endeavors may provide an avenue for delving into meticulous
examination within these channels. It will be intriguing to observe if these
investigations yield unexpected findings in the era of high precision B
physics and if they enable us to detect deviations from the SM, potentially
involving new CP violation sources.

##  Appendix A CONSTITUENT QUARK ORBITALS AND MOMENTUM PROBABILITY AMPLITUDES

In RIQ model a meson is picturised as a color-singlet assembly of a quark and
an antiquark independently confined by an effective and average flavor
independent potential in the form:
$U(r)=\frac{1}{2}(1+\gamma^{0})(ar^{2}+V_{0})$ where ($a$, $V_{0}$) are the
potential parameters. It is believed that the zeroth order quark dynamics
generated by the phenomenological confining potential $U(r)$ taken in equally
mixed scalar-vector harmonic form can provide adequate tree level description
of the decay process being analyzed in this work. With the interaction
potential $U(r)$ put into the zeroth order quark Lagrangian density, the
ensuing Dirac equation admits static solution of positive and negative energy
as:

$\displaystyle\psi^{(+)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{ig_{\xi}(r)}{r}\\\
\frac{{\vec{\sigma}}.{\hat{r}}f_{\xi}(r)}{r}\end{array}\;\right)U_{\xi}(\hat{r})$
(34) $\displaystyle\psi^{(-)}_{\xi}(\vec{r})\;$ $\displaystyle=$
$\displaystyle\;\left(\begin{array}[]{c}\frac{i({\vec{\sigma}}.{\hat{r}})f_{\xi}(r)}{r}\\\
\frac{g_{\xi}(r)}{r}\end{array}\;\right){\tilde{U}}_{\xi}(\hat{r})$ (37)

where, $\xi=(nlj)$ represents a set of Dirac quantum numbers specifying the
eigen-modes; $U_{\xi}(\hat{r})$ and ${\tilde{U}}_{\xi}(\hat{r})$ are the spin
angular parts given by,

$\displaystyle U_{ljm}(\hat{r})$ $\displaystyle=$
$\displaystyle\sum_{m_{l},m_{s}}<lm_{l}\;{1\over{2}}m_{s}|jm>Y_{l}^{m_{l}}(\hat{r})\chi^{m_{s}}_{\frac{1}{2}}$
$\displaystyle{\tilde{U}}_{ljm}(\hat{r})$ $\displaystyle=$
$\displaystyle(-1)^{j+m-l}U_{lj-m}(\hat{r})$ (38)

With the quark binding energy $E_{q}$ and quark mass $m_{q}$ written in the
form $E_{q}^{\prime}=(E_{q}-V_{0}/2)$, $m_{q}^{\prime}=(m_{q}+V_{0}/2)$ and
$\omega_{q}=E_{q}^{\prime}+m_{q}^{\prime}$, one can obtain solutions to the
resulting radial equation for $g_{\xi}(r)$ and $f_{\xi}(r)$in the form:

$\displaystyle g_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}(\frac{r}{r_{nl}})^{l+l}\exp(-r^{2}/2r^{2}_{nl})L_{n-1}^{l+1/2}(r^{2}/r^{2}_{nl})$
$\displaystyle f_{nl}$ $\displaystyle=$ $\displaystyle
N_{nl}(\frac{r}{r_{nl}})^{l}\exp(-r^{2}/2r^{2}_{nl})$ (39)
$\displaystyle\times$
$\displaystyle\left[(n+l-\frac{1}{2})L_{n-1}^{l-1/2}(r^{2}/r^{2}_{nl})+nL_{n}^{l-1/2}(r^{2}/r^{2}_{nl})\right]$

where, $r_{nl}=a\omega_{q}^{-1/4}$ is a state independent length parameter,
$N_{nl}$ is an overall normalization constant given by

$N^{2}_{nl}=\frac{4\Gamma(n)}{\Gamma(n+l+1/2)}\frac{(\omega_{nl}/r_{nl})}{(3E_{q}^{\prime}+m_{q}^{\prime})}$
(40)

and $L_{n-1}^{l+1/2}(r^{2}/r_{nl}^{2})$ etc. are associated Laguerre
polynomials. The radial solutions yields an independent quark bound-state
condition in the form of a cubic equation:

$\sqrt{(\omega_{q}/a)}(E_{q}^{\prime}-m_{q}^{\prime})=(4n+2l-1)$ (41)

The solution of the cubic equation provides the zeroth order binding energies
of the confined quark and antiquark for all possible eigenmodes.

In the relativistic independent particle picture of this model, the
constituent quark and antiquark are thought to move independently inside the
$B_{s}$-meson bound state with momentum $\vec{p}_{b}$ and $\vec{p}_{s}$,
respectively. Their individual momentum probability amplitudes are obtained in
this model via momentum projection of respective quark orbitals (A1) in
following forms:

For ground state mesons:($n=1$,$l=0$)

$\displaystyle G_{b}(\vec{p}_{b})$ $\displaystyle={{i\pi{\cal
N}_{b}}\over{2\alpha_{b}\omega_{b}}}\sqrt{{(E_{p_{b}}+m_{b})}\over{E_{p_{b}}}}(E_{p_{b}}+E_{b})$
$\displaystyle\times\exp{(-{{\vec{p_{b}}}^{2}\over{4\alpha_{b}}})}$
$\displaystyle{\tilde{G}}_{s}(\vec{p}_{s})$ $\displaystyle=-{{i\pi{\cal
N}_{s}}\over{2\alpha_{s}\omega_{s}}}\sqrt{{(E_{p_{s}}+m_{s})}\over{E_{p_{s}}}}(E_{p_{s}}+E_{s})$
(42) $\displaystyle\times\exp{(-{{\vec{p_{s}}}^{2}\over{4\alpha_{s}}})}$

The binding energies of constituent quark and antiquark for the ground state
of $B_{s}$ meson as well as the final meson states for $n=1$; $l=0$ can also
be obtained by solving respective cubic equations representing appropriate
bound state conditions.

###### Acknowledgements.

L.N., and S.S. duly acknowledge useful discussion from Prof. Rukmani Mohanta,
during the initial stage of this project. Authors L.N.,and S.P., are also duly
grateful and acknowledge helpful discussion and substantial contribution of
Professors N. Barik, P. C. Dash, and S. Kar in the development of the RIQ
model framework. L.N., S.P., P.S., and S.S. thanks the NISER, Department of
Atomic Energy, India for the financial support.

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