# A conceptual predator-prey model with super-long transients

Misha Chai  Corresponding author. Email<EMAIL_ADDRESS>Holger Kantz

###### Abstract

Drawing on the understanding of the logistic map, we propose a simple
predator-prey model where predators and prey adapt to each other, leading to
the co-evolution of the system. The special dynamics observed in periodic
windows contribute to the coexistence of multiple time scales, adding to the
complexity of the system . Typical dynamics in ecosystems , such as the
persistence and coexistence of population cycles and chaotic behaviors, the
emergence of super-long transients, regime shifts, and the quantifying of
resilience, are encapsulated within this single model. The simplicity of our
model allows for detailed analysis, including linear analysis, reinforcing its
potential as a conceptual tool for understanding ecosystems deeply.
Additionally, our results suggest that longer lifetimes in ecosystems might
come at the expense of reduced populations due to limited resources.

There is a growing recognition that long-term or asymptotic behavior is rare,
and that focusing on transients might be a more effective approach to
understanding the complexity in ecosystems [1, 2, 3, 4, 5, 6]. Moreover, many
models and observations suggest that transients may persist over a super-long
period of time [1, 2, 7, 8], during which cyclic and chaotic behaviors appear
repeatedly. These cyclic dynamics are one of the most notable phenomena in
population biology, particularly in predator-prey systems where the predator
and prey coexist in recurring cyclic patterns over indefinitely long periods
of time. The Lotka-Volterra model, a cornerstone in mathematical biology and
ecology, provides a fundamental framework for understanding cyclic dynamics in
predator-prey interaction. Further simplification can be achieved by the
discretization of time. Logistic map [9], for example, a well-known discrete-
time model, has been used to describe the population of a single species
influenced by carrying capacity, showcasing a spectrum of behaviors from
stable equilibrium to periodic oscillations and chaos determined by its growth
rate. A much more complex behavior can be achieved by introducing competition
models [7, 10, 11]. It has been extensively used to analyze population
dynamics and to understand the biodiversity[12, 13, 14] in ecosystems. One of
the most classic topics is the study of the complexity and biodiversity of
plankton species [15, 10, 11, 16], aimed at understanding the famous “paradox
of the plankton” [17]. In the other ecosystems, such as Forage fish [18],
insects [19], grass community [20], and Dungeness crabs [21], various methods
have also been used to understand different types of systems. However, due to
the complexity and high dimensionality of these models, unraveling the
complicated dynamics in ecosystems is challenging, let alone conducting linear
analysis. Therefore, a simple conceptual model that contains most of typical
features of real-world population, while still being amenable to theoretical
or linear analysis, is extremely important!

Fig. 1: The cattle-grass system. There are three generations of cattle and
grass. The grass consumed by the cattle in each generation is represent by
$(x,y)$\- equations. New cattle (yellow) and the leave of cattle (red) happen
at the end of the year. Fig. 2: The $(x,y)$-plane of Eq. (1). A,
$r_{n}=const$(i.e., $a=0$). B, $r_{n}\neq const$(in this case, $a=0.005$). The
orange curve indicates points move upward; the purple curve indicates points
move downward. The zoom-in region near $y=0$ also has been shown.

### Predator-prey model

Here based on the logistic map $x_{n+1}=rx_{n}(1-x_{n})$, where $x_{n}\geq 0$
is a dimensionless measure of the population in the $n$th generation and
$r\geq 0$ is the intrinsic growth rate—reflecting population changes under
ideal conditions without external factors—we propose a simple predator-prey
model in which the prey responds to predation. Thus, the evolution of the prey
can influence predator dynamics, which in turn affects prey evolution. These
dynamics can lead to the continuous co-evolution of predators and prey in
response to each other’s adaptations. It displays rich dynamical complexity,
such as the persistence and coexistence of population cycles and chaotic
behaviors[1, 7, 8], the emergence of super-long transient[2, 4, 6], and regime
shift (a sudden change that usually results in the extinction of species and
the loss of biodiversity)[5]. It can help us understand the complexity of
realistic ecosystems.

Fig. 3: The survival probability with varying $a$. Background colors green,
pink, and blue indicate three different Regime I, II and III correspondingly.
Their survival probabilities respectively follow the exponential decay with a
high escape rate (the zoom-in graph in A), the power law decay with an
exponent of $\alpha=-0.5011$ (the zoom-in graph and red dash line in B), and
the exponential decay with a smaller escape rate (the zoom-in graph in C).
Especially Regime III, the survival probability has different escape rates
with varying $a$. The bigger $a$ with bigger escape rates are in A; the
smaller $a$ with smaller escape rates are in B. In C, the changes between them
have been shown. In D, lifetime gets bigger with the decrease of $a$.
Especially, when $a$ is even smaller, lifetime grows exponentially with an
exponent $\eta=-7073$.

The constancy of growth rate in the logistic map does describe the properties
of some simple systems or toy models well, for example, a single-species algae
population with one limiting resource [22, 23]. However, in realistic systems,
the introduction of new species (e.g., through invasion from other parts of
the globe or by mutation) and the extinction of species both change the
dimension of the phase space and affect the quality of the dynamics. Thus, the
growth rate cannot be constant anymore but changes along with the system’s
evolution. This motivates us to introduce $r+y_{n}$ as the new growth rate.
Here, $r$ represents the intrinsic and fixed birth and death rate of the
predator under limiting resources and we fixed $r=3$ at the point where the
bifurcation occurs in the paper. The term $y_{n}$ reflects the changes of
limiting resources, which directly affects and adapts to changes in the
predator population, as seen in plant-consumer and host-parasite systems. Here
we use plant-consumer systems as an example, but the dynamics can be applied
to most systems where resources are the prey, and predators and prey in
response to each other’s adaptation directly.

Assume that there are $x_{n-1}$ cattle and $y_{n-1}$ grass in generation
$n-1$, and $x_{n}$ and $y_{n}$ in generation $n$, as shown in Fig. 1. The new
cattle (yellow) and the leave of cattle (red) occur at the end of the year,
and every individual is identical. As we can see, $x_{n-1}$ of cattle consumed
$y_{n}-y_{n-1}$ grass in generation $n-1$. Since the same amount of cattle
consume the same amount of grass, $x_{n}$ can be rewritten as
$x_{n}=x_{n-1}+(x_{n}-x_{n-1})$. The $x_{n}$ is divided into two parts:
$x_{n-1}$, where the corresponding grass consumption is $y_{n}-y_{n-1}$; and
$x_{n}-x_{n-1}$, which reflects the population changes with the corresponding
grass consumption being $a(x_{n}-x_{n-1})$. The control parameter $a$ scales
the relationship between predator and prey. In the cattle-grass system, this
can be understood as the weight of grass each cattle consumes. Thus in
generation $n+1$, the grass is
$y_{n}-(y_{n}-y_{n-1})-a(x_{n}-x_{n-1})=y_{n-1}-a(x_{n}-x_{n-1})$. The
generalized logistic map hence reads:

$\displaystyle x_{n+1}$ $\displaystyle=(3+y_{n})x_{n}(1-x_{n})$ $\displaystyle
y_{n+1}$ $\displaystyle=y_{n-1}-a(x_{n}-x_{n-1}).$ (1)

When $x_{n}-x_{n-1}>0$, indicating an increase in the predator population,
this can lead to a decrease in the prey population, i.e., $y_{n+1}$, which in
turn results in a subsequent decline trend in the predator population in next
generation, denoted as $x_{n+2}$. Correspondingly, if $x_{n}-x_{n-1}<0$,
indicating a decrease in the predator population, this can lead to an increase
in the prey population $y_{n+1}$, resulting in a subsequent rise trend of
$x_{n+2}$. If the predator population remains constant, i.e.,
$x_{n}-x_{n-1}=0$, the prey population also remains constant, i.e.,
$y_{n+1}=y_{n-1}$. Since this map involves $x_{n-1}$ and $y_{n-1}$, suggesting
it resembles a delay map. Consequently, its phase space would appear to be
4-dimensional. Nonetheless, Eq. (1) can actually be derived from a
2-dimensional system, which includes an additional variable
$sgn:sgn_{n}=(-1)^{n}$ and an instantaneous update $y_{n+1}=y_{n}+sgn_{n}\cdot
a(x_{n}-0.5)$. The $x$-dynamics remain consistent with Eq. (1). Therefore,
given $x_{1}$ and $y_{1}$ as initial conditions, the whole trajectory is
uniquely determined. This representation of our map reveals that its phase
space is the direct product of the $(x,y)$-plane and the set $\\{-1,1\\}$. In
the graphical representations of the attractor, we will focus solely on
projections onto the $(x,y)$-plane.

When $y_{n}=const$ (achieved in the limit $a\to 0$), Eq. (1) becomes a one-
dimensional logistic map. The bifurcation diagram then reveals the transitions
from stable equilibrium through period-doubling bifurcations and eventually
into chaotic dynamics as $y$ increases, shown in Fig. 2A. In the whole paper,
initial conditions are chosen randomly. One of the most intriguing features of
the bifurcation diagram is the occurrence of period-$p$ windows that contain
the critical point $x_{c}$, the maximum of the function $f(x)=rx(1-x)$. The
periodic-$p$ orbit contains $x_{c}$ satisfies

${(f_{r}^{p})^{\prime}}(x_{c})=0$

, making the orbit super-attracting (super stable), as exemplified by the
well-known period-3 window, where occurs near $3.8284\ldots\leq r\leq
3.8415\ldots$ [24].

When $y_{n}\neq const.$, i.e., $a\neq 0$, the system maintains similar
dynamics at each value of the growth rate. Hence, for small $a$ (slow change
of growth rate), the resulting phase space diagram strongly resembles the
bifurcation diagram of the logistic map, shown in Fig. 2B. But the essential
difference is that Eq. (1) is an intermittent system, exhibiting various
intermittency behaviors on different time scales at different values of the
growth rates, and that all trajectories become transient. The latter is a
consequence of the fact that $3+y_{n}$ is not bound to be smaller or equal to
4, and if $3+y_{n}>4$, then the $x$-dynamics map points outside of the
interval $[0,1]$, causing them to escape towards infinity ($\infty$), which,
as noted by May[9], implies that the population becomes extinct. When this
happens, we stop the iteration, considering the trajectory as having escaped.

Since all trajectories will eventually escape from the finite phase space, we
now focus on the survival probability, which is related to the distribution of
lifetimes. Initiating our trajectories with random values for $x_{1}$ and
setting $y_{1}=1$, they exhibit quite different behaviors as a function of the
lifetime.

Regime I. A certain fraction of them moves near $y=1$. These escape after a
relatively short time in an exponential way. It is highlighted by the green
background, and the zoom-in graph is shown in Fig. 3A.

Regime II. From those which stay longer but do not drop into the period-$p$
window, their lifetimes are characterized by a power-law distribution. This
distribution has an exponent close to $-1/2$, $P(\tau>T)\propto T^{-1/2}$,
where $\tau$ represents the time until the escape of an individual trajectory.
The exponent 1/2 can be easily explained: In the equation for $y_{n}$ as shown
in Eq.(1), the increments $x_{n}-x_{n-1}$ behave similarly to white noise, as
long as the $x$-dynamics for the given $y$ value is chaotic. Actually, if we
disregard the nonlinear dependence of $x_{n}$ on $x_{n-1}$ and assume that
they are independent, then the probability distribution for the difference
$x_{n}-x_{n-1}$ will be symmetric around 0, even if the distribution of
$x_{n}$ is not. Consequently, the mean value of these increments is 0, so that
for strongly chaotic $x$, they do not impose a systematic drift on $y$. But if
the increments behave like white noise, then $y_{n}$ behaves like a Brownian
path. For a Brownian path, it is well known that the probability of crossing a
specific value (in this case, $y=1$) in the next step drops like $t^{-3/2}$,
where $t$ is the time since the last crossing of this value. Since we start
from $y_{1}=1$, the time till reaching $y_{n}>1$ asymptotically then follows
the law $t^{-1/2}$. In our numerical experiments, where the empirical value of
the exponent is $\alpha=-0.5011$ (the red dash line shown in Fig. 3B), we well
reproduce the power close to $1/2$. The slight deviation of this numerical
value from 1/2 is attributed to some trajectories being stuck in period-$p$
windows for a super-long period of time. It is highlighted by the pink
background, and the zoom-in graph of the survival probability is shown in Fig.
3B.

Regime III. The most significant behavior occurs near those $y$-values where
the stable period-3 window happens in the logistic map. Specifically, once
trajectories come to the period-3 window, they first move along it until
reaching its boundary, where there is a possibility for the trajectories to
escape outside of the window for an extremely short time of chaos. But there
is also a high possibility that it will be quickly attracted back to the
period-3 window again, repeating similar dynamics for a super-long period of
time, resulting in the occurrence of super-long transients. The smaller the
$a$ is, the slower the motion in the $y$-direction. Consequently, many
suitable $y_{n}$ have been generated near the period-3 window. Even with a
small probability, $y_{n}$ could move upward or downward, leaving the window.
However, due to the slow movement, $y_{n}$ remains extremely close to the
period-3 window. Then, because of the stability of the period-3 window,
$y_{n}$ is attracted back to the period-3 window again. As a result,
trajectories spend an exceptionally long period of time near the period-3
window, which results in a significant impact on the global properties of the
system.

Fig. 4: The $(t,x)$-plane of Eq. (1) with varying $a$. When a is big (i.e.,
$a=0.010$ and $a=0.008$), cyclic and chaotic behaviors frequently recur until
the escape of trajectories. With the decrease of $a$, chaotic behavior becomes
rarer, and the persistence of cyclic behavior becomes more significant.
Consequently. trajectories cannot escape in a short period anymore (for
instance, $a=0.004$ and $a=0.002$), indicating the occurrence of the super-
long transient.

The ability of the system bounce back to the period-3 window after chaotic
behavior is known as "resilience"[25, 26, 27], which has been defined as the
capacity to tolerate perturbations without collapsing. Particularly under
environmental changes, measuring, quantifying, and maintaining the resilience
becomes critically important. In our model, $a$ as a control parameter,
decides the resilience of the system.

In Fig. 4, the transient $x$-dynamics have been shown in the time domain. And
for different values of $a$ we show exemplarily typical trajectories $x_{n}$.
As we can see from the first 3 panels, when $a$ is bigger, cyclic and chaotic
behavior (similar to perturbations in realistic systems) appear repeatedly
until regime shift suddenly changes the dynamics and lead trajectories go to
infinity, which indicates the extinction of species. In this process, even
though trajectories can be quickly attracted to the period-3 window after
perturbation, the frequent occurrence of perturbation significantly increases
the probability of the escaping of trajectories, resulting in their faster
leaving the phase space, which also means the loss of resilience. When $a$
gets smaller, as shown in the last 2 panels, cyclic behavior becomes more
pronounced and perturbations become increasingly rare, which indicates the
period-3 window becomes more attractive. As a result, it is difficult for
trajectories to escape from the period-3 window, even under perturbation,
which indicates stronger resilience of the system. Consequently, trajectories
are stuck in a period-3 window for a super-long period of time, indicating the
occurrence of super-long transients.

Another quite interesting dynamic in our model is the coexistence of multiple
time scales. As we know, the stability of the period-3 window causes
trajectories to spend a super-long period of time on it, which is one of the
significant time scales. However, there are other periodic windows as well.
For example, as $a$ gets smaller, trajectories may also pass through the
period-5 window. Although it is less attractive than the period-3 window, it
can still capture points for a long time of period. As $a$ continues to
decrease, more periodic windows become significant, contributing to systems’
complexity.

The survival probabilities of Regime III with variable control parameter $a$
are shown in Fig. 3. It is highlighted by the blue background, and the zoom-in
graph of the survival probability is shown in Fig. 3C. In Fig. 3A and Fig. 3B,
we observed that the escape rate gets smaller with the decrease of $a$.
Starting from a critical point $a_{c}\in(0.0050,0.0060)$, as $a$ becomes even
smaller, the escape rate becomes super sensitive to the $a$, as seen in Fig.
3D. The lifetime ($lifetime=1/escape\ rate$) follows an exponential law with
the variation of $a$. And the exponent is $\eta=-7073$. This is a kind of
super-transient behavior, previously observed in other systems[9]: The
transition occurs near stable dynamics without escape, and lifetimes have been
observed to depend exponentially on a control parameter. Here, we are unable
to obtain the escape rate or lifetime numerically in the limit of $a\to 0$,
but we can see from Fig. 3D that the lifetime grows to extremely huge as $a\to
0$, which indicates the existence of super-long transient. In Fig. 3C, by
choosing parameters in the interval $a\in(0.0050,0.0060)$, the emergence of
the super-long transient tail has been shown with the decrease of $a$.

In Fig. 3B, additional fascinating dynamics are observed that align well with
trade-offs[28] in ecosystems. As seen, with the decrease in $a$, the escape
rate decreases, indicating a longer lifetime. However, at the same time, the
population size also diminishes. This suggests that it is might not possible
to maximize both lifetime and population size simultaneously. A longer
lifetime of ecosystems might have to come at the expense of a reduced
population. This reminds us of the evolutionary trade-offs: a trait increases
in fitness at the expense of decreased fitness in another trait due to limited
resources. The balance between different traits contributes to the success of
natural systems.

Moreover, the survival probability here closely aligns with the numerical
result on plankton species richness that Michael J. Behrenfeld illustrates in
Fig. 1 of his paper[16]. In the paper, neutral theory[29] as a method aimed to
explain the diversity and abundance of species in ecosystems, has been used to
explore the role of stochastic processes. In Fig. 1 [16], the survival
probability initially exhibits an exponential decay, then undergoes a
stochastic process, resulting in different tails (population size) based on
the strength of immigration (external factors). This closely resembles our
results. The only difference is that our data show different transient tails
based on variations in parameter $a$. This can be explained as follows: in our
model, prey (resources) respond to predation, while changes in resources are
caused by external factors. The control parameter $a$, as a scale between
resource changes and predation, also indirectly reflects the strength of
external factors. This confirms that our results align closely with
Behrenfeld’s findings, further reinforcing our confidence that our model can
serve as a conceptual tool to help ecologists and physicists understand the
complexities in ecosystems.

After trajectories leave the period-3 window and are not attracted back
quickly, they either move upward towards $y=1$ which have the possibility to
escape in a short time, or move downward towards smaller $y$-values. Once they
move downward within the periodic regime, the values of $y$ systematically
decrease, causing the trajectory to follow an inverse period-doubling process
along the purple curve in Fig. 2B until it reaches $3+y=3$ which acts as a
reflecting boundary. Subsequently, the trajectory will be bounced back to the
period-3 window or chaotic region again, following the orange curve. All those
processes have the chance to repeat again and again until the trajectory
eventually leaves the phase space through $3+y>4$. The upper endpoint of the
orange curve is dependent on the control parameter $a$. The larger the value
of $a$ is, the further the curve extends. By initializing $2<3+y_{n}<3$, in
Fig. 2B, all trajectories following the orange curve move upwards into the
region from which they have the chance to escape through $3+y>4$. For
trajectories with $3+y_{n}<2$, they move to minus infinity. Our paper mainly
focuses on trajectories that leave the phase space through $3+y>4$.

### Conclusion

The evolution of predators is influenced by their prey, meanwhile, prey adapts
to predators. This continuous co-evolution of predators and prey contributes
to the complicated dynamics in our system. Among these, the most intriguing
dynamics occur during periodic windows: i) The existence of super-long
transients. For example, trajectories spend a long period of time on the
period-3 window, which contributes to the occurrence of super-long transient,
but at the same time, also indicates that the time scale on the period-3
window differs from others. ii) The coexistence of multiple time scales. There
are many periodic windows, such as the period-3 and the period-5 window. Since
trajectories spend different amounts of time on each, this indicates the
multiple time scales in our system when $a$ gets even smaller. The cyclic
behaviors on periodic windows are similar to cyclic population dynamics in
ecosystems. As we know, with the fading out of population cycles in ecosystems
becoming increasingly common, the collapse of these cycles has become a very
interesting topic and attracts a lot of attention[3]. In our model, $a$ as the
control parameter, scaling the relationship between predators and prey,
determines the persistence of population cycles. This promotes us to ask
whether the scale between predators and prey in real-world systems might
influence the persistence of population cycles. Moreover, due to the
simplicity of our model, this gives us the chance to further explore what
contributes to the persistence of population cycles? Until a sudden regime
shift broke the population cycles and led the system to extinction, the
question arises: Does the cumulative behavior of the system lead to the
occurrence of a regime shift[26, 30]? And are there any early warning signals
for a regime shift[31, 32]? Additionally, the coexistence with chaotic
behaviors makes us think about the role chaos plays in transient
behaviors[33]. Our model gives us the chance to analyze all those topics or
even conduct linear analysis.

Furthermore, the existence of the evolutionary trade-off in our model further
supports its credibility. More importantly, the survival probability shown in
our model aligns with the findings of Michael J. Behrenfeld [16].

All of those indicate that our model, as a conceptual framework, combines most
of the dynamic features of ecosystems. This integration might provide us with
the opportunity to deeply understand, analyze, and unify these topics within a
single model.

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## Acknowledgements

We are grateful for stimulating discussions with Christian Beck, Peter
Grassberger, and Jin Yan.

## Author Contributions

Both authors developed the model system together. Misha Chai performed the
numerical simulations and created the figures. All authors contributed to the
interpretation of the results and to writing the manuscript.

## Competing Interests

The authors declare that they have no competing financial interests.

## Correspondence

Correspondence and requests for materials should be addressed to
<EMAIL_ADDRESS>


$(0,0,0,1,0,0,0,1)$ | $56$ | $\frac{1}{2}(1,0,0,1,0,1,1,0)$ | $56$ | $\frac{1}{4}(2,1,1,0,1,2,2,2)$ | $56$
$\frac{1}{2}(1,0,0,1,0,1,1,1)$ | $58$ | $\frac{1}{2}(2,1,1,0,0,0,1,2)$ | $60$ | $\frac{1}{2}(2,0,0,1,0,1,0,2)$ | $60$
$\frac{1}{4}(4,1,1,0,2,0,1,4)$ | $60$ | $(1,0,0,0,1,0,0,1)$ | $62$ | $\frac{1}{2}(1,1,1,0,0,1,1,1)$ | $62$
$(0,0,0,1,0,0,1,0)$ | $64$ | $\frac{1}{2}(1,1,1,0,1,0,1,1)$ | $68$ | $(1,0,0,0,0,1,1,1)$ | $72$
$\frac{1}{3}(1,1,1,1,1,1,1,1)$ | $72$ | $\frac{1}{4}(3,0,0,1,0,3,4,4)$ | $72$ | $\frac{1}{4}(2,1,1,0,1,2,4,4)$ | $72$
$(0,0,0,1,0,0,1,1)$ | $74$ | $\frac{1}{2}(1,0,0,1,0,1,2,2)$ | $74$ | $\frac{1}{2}(0,1,1,0,1,0,2,2)$ | $74$
$\frac{1}{2}(1,1,0,1,0,1,1,2)$ | $76$ | $\frac{1}{3}(1,1,1,1,1,1,1,3)$ | $78$ | $(1,0,0,1,0,0,1,0)$ | $84$
$\frac{1}{2}(2,1,1,0,1,1,0,1)$ | $84$ | $\frac{1}{2}(1,1,1,0,1,1,1,1)$ | $86$ | $\frac{1}{2}(2,1,1,0,1,0,2,2)$ | $92$
$(1,0,0,1,0,0,1,1)$ | $94$ | $(1,0,0,1,0,1,0,1)$ | $104$ | $\frac{1}{2}(1,1,1,1,1,1,1,1)$ | $112$
$(1,0,0,1,0,1,1,1)$ | $126$ | $\frac{1}{2}(2,1,1,0,1,2,2,2)$ | $126$ | $\frac{1}{2}(1,1,1,1,1,1,2,2)$ | $128$
$(1,1,1,0,1,0,1,1)$ | $148$ | $(1,1,1,0,1,1,1,1)$ | $182$ | $(1,1,1,1,1,1,1,1)$ | $240$

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# Prediction of various observables for $B_{s}^{0}\to
D_{s}^{(*)-}\ell^{+}\nu_{\ell}$ within covariant confined quark model

J. N. Pandya<EMAIL_ADDRESS>Department of Physics, Sardar Patel
University, Vallabh Vidyanagar 388120, Gujarat, India.  P. Santorelli
<EMAIL_ADDRESS>Dipartimento di Fisica “E. Pancini”, Università
di Napoli Federico II - Complesso Universitario di Monte S. Angelo Edificio 6,
via Cintia, 80126 Napoli, Italy  INFN sezione di Napoli - Complesso
Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy
N. R. Soni<EMAIL_ADDRESS><EMAIL_ADDRESS>INFN sezione di Napoli -
Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126
Napoli, Italy  Department of Physics, Faculty of Science, The Maharaja
Sayajirao University of Baroda, Vadodara 390002, Gujarat, India

###### Abstract

In 2020, the LHCb collaboration reported the exclusive branching fractions for
the channels $B_{s}^{0}\to D_{s}^{(*)-}\mu^{+}\nu_{\mu}$ for the very first
time. In view of these observations, we have recently reported the form
factors and branching fraction computations for these channels employing the
covariant confined quark model. As different other channels corresponding to
$b\to c\ell\nu_{\ell}$ have provided the hint for New Physics, the analysis of
observables such as forward-backward asymmetry, longitudinal and transverse
polarizations across the lepton flavours can serve as one of the important
probes for the search for possible New Physics. In present work, we compute
these observables for all the lepton flavours and compare our predictions with
the other theoretical approaches.

##  I Introduction

For last many years, $b\to c\ell\nu_{\ell}$ has served as a very precise probe
for the search of new physics. As both the quarks involved in this transition
are heavy, it has great phenomenological implications within and beyond the
standard model. Experimentally, precise results are available for the channels
$B\to D^{(*)}\ell\nu_{\ell}$ through different facilities such as BABAR, BELLE
and LHCb collaborations. Precise lattice results are also available
corresponding to these transitions Bailey  _et al._ (2015); Na  _et al._
(2015). Different heavy flavor anomalies corresponding to these transitions
are reported in Ref. Li and Lü (2018); Albrecht  _et al._ (2021); London and
Matias (2022) and references therein. Similarly, $B_{s}\to
D_{s}^{(*)}\ell\nu_{\ell}$ can also serve as a prominent channel for
understanding the heavy flavor dynamics and possibly the anomalies. On
experimental side, LHCb have reported the branching fractions for the channels
$B_{s}^{0}\to D_{s}^{(*)-}\mu\nu_{\mu}$ for the very first time Aaij  _et al._
(2020) and determined the ratio of the branching fractions $B_{s}^{0}\to
D_{s}^{-}\mu^{+}\nu_{\mu}$ to $B_{s}^{0}\to D_{s}^{*-}\mu^{+}\nu_{\mu}$.
Additionally, they also determined the ratios of these branching fractions
relative to $B\to D$ namely

$\displaystyle\frac{\mathcal{B}(B_{s}^{0}\to
D_{s}^{-}\mu^{+}\nu_{\mu})}{\mathcal{B}(B^{0}\to D^{-}\mu^{+}\nu_{\mu})}$
$\displaystyle=$ $\displaystyle 1.09\pm 0.05_{\mathrm{stat}}\pm
0.06_{\mathrm{syst}}\pm 0.05_{\mathrm{ext}}$
$\displaystyle\frac{\mathcal{B}(B_{s}^{0}\to
D_{s}^{*-}\mu^{+}\nu_{\mu})}{\mathcal{B}(B^{0}\to D^{*-}\mu^{+}\nu_{\mu})}$
$\displaystyle=$ $\displaystyle 1.06\pm 0.05_{\mathrm{stat}}\pm
0.07_{\mathrm{syst}}\pm 0.05_{\mathrm{ext}}.$

The ratios of the decay widths from tau mode to electron mode for
$D_{s}^{(*)}$ ($R(D_{s})$ and $R(D_{s}^{*})$) are yet to be measured
experimentally so far. However, lattice results are available for these ratio
in the Refs. McLean  _et al._ (2020); Harrison and Davies (2022). New Physics
studies have also been reported using these lattice form factors in Ref.
Penalva  _et al._ (2023). Recently, the ratios $R(D_{s}^{(*)})$ have been
computed using unitarity and lattice QCD approach Martinelli  _et al._ (2022)
where the transition form factors are computed in the entire momentum transfer
range using the dispersive matrix approach. The transition form factors,
branching fractions and ratios $R(D_{s}^{(*)})$ are also computed using three
point QCD sum rules Azizi (2008); Azizi and Bayar (2008) and also using light
cone QCD sum rules in the framework of heavy quark effective theory Bordone
_et al._ (2020); Zhang  _et al._ (2022). Transition form factors, branching
fractions and other observables have also been computed recently within the
framework of relativistic quark model (RQM) based on the quasi-potential
approach in QCD Faustov and Galkin (2013); Faustov  _et al._ (2022). The
transition form factors are also computed in both space- and time-like
momentum transfer range within the constituent quark model framework Heger
_et al._ (2021),employing the next-to-leading-order QCD corrections Cui  _et
al._ (2023) and also employing light front quark model (LFQM) Verma (2012).
Heavy to heavy and heavy to light semileptonic transitions form factors are
also computed employing the symmetry preserving vector-vector contact
interactions (SCI) Xing  _et al._ (2022). The detailed description of
transition form factors and branching fractions are available in literature,
however the detailed analysis of other physical observables such as forward
backward asymmetry, different polarizations observables are yet to be
explored. As there is no indication of these observables from experimental
side, they may also serve as crucial probes for search for the Physics beyond
the standard model. In theoretical side, there are very few references in
which these observables are studied, few of them are lattice results Harrison
and Davies (2022), RQM Faustov and Galkin (2013) and light cone QCD sum rules
Bordone  _et al._ (2020). These observables are also studied considering the
new physics scenario Dutta and Rajeev (2018); Das and Dutta (2020, 2022).

Very recently, we have studied the transition form factors, branching
fractions of these channels namely $B_{s}\to D_{s}^{(*)}\ell\nu_{\ell}$ with
$\ell=e,\mu$ and $\tau$ within the quark model framework Soni  _et al._
(2023). In this work, we have reported detailed analysis of the branching
fractions with the recent LHCb data and it is observed that our results are in
the excellent agreement with them along with lattice and other theoretical
models. In this paper, we provide much detailed description of the form
factors with comparison to the lattice and other theoretical approaches. We
also provide the detailed plots for the different physical observables such as
forward-backward asymmetries, longitudinal and transverse polarization, lepton
and hadron side convexity parameter and so on. Together with Soni  _et al._
(2023), this work will provide complete description of the dynamics of
semileptonic $B_{s}\to D_{s}^{(*)}$ decays.

Rest of the paper is organised in the following way. After the brief
introduction and some very recent literature survey, for computation of the
transition form factors, we provide very short description to our theoretical
model that is covariant confined quark model in Sec. II and provide the form
factors in the double pole approximation. We also compare our results with
other theoretical approaches. Next in sec III, we give the relations for the
computation of the different physical observables. We also provide the plots
of these observables as well as the expectation values of these observables.
In sec IV, we discuss about all the results obtained for the semileptonic
decay for the channels $B_{s}^{0}\to D_{s}^{(*)-}\ell^{+}\nu_{\ell}$ for
$\ell=e,~{}\mu,~{}\tau$. Finally, we conclude the present work in sec V.

##  II Form factors

Table 1: Form factors and double pole parameters appeared in Eq. 3 Soni  _et al._ (2023). $F$ | $F(0)$ | $a$ | $b$ | $F$ | $F(0)$ | $a$ | $b$
---|---|---|---|---|---|---|---
$F_{+}^{B_{s}\to D_{s}}$ | $0.770\pm 0.066$ | 0.837 | 0.077 | $F_{-}^{B_{s}\to D_{s}}$ | $-0.355\pm 0.029$ | 0.855 | 0.083
$A_{+}^{B_{s}\to D_{s}^{*}}$ | $0.630\pm 0.025$ | 0.972 | 0.092 | $A_{-}^{B_{s}\to D_{s}^{*}}$ | $-0.756\pm 0.031$ | 1.001 | 0.116
$A_{0}^{B_{s}\to D_{s}^{*}}$ | $1.564\pm 0.065$ | 0.442 |  $-$0.178 | $V^{B_{s}\to D_{s}^{*}}$ | $0.743\pm 0.030$ | 1.010 | 0.118

Figure 1: Form factors in comparison with LFQM Verma (2012), RQM Faustov and
Galkin (2013), LQCD McLean  _et al._ (2020) and SCI Xing  _et al._ (2022).

Within the framework of standard model, the matrix element for the
semileptonic transition can be in the general form as

$\displaystyle\mathcal{M}(M_{1}\to
M_{2}^{(*)}\ell^{+}\nu_{\ell})=\frac{G_{F}}{\sqrt{2}}V_{\mathrm{CKM}}\langle
M_{2}^{(*)}|\bar{q_{2}}O^{\mu}q_{1}|M_{1}\rangle[\ell^{+}O_{\mu}\nu_{\ell}],$
(1)

This transition matrix element can also be parametrized in terms of form
factors as

$\displaystyle\langle M_{2}(p_{2})$ $\displaystyle|$
$\displaystyle\bar{q_{2}}O^{\mu}q_{1}|M_{1}(p_{1})\rangle=F_{+}(q^{2})P^{\mu}+F_{-}(q^{2})q^{\mu},$
$\displaystyle\langle M_{2}^{*}(p_{2},\epsilon_{\nu})$ $\displaystyle|$
$\displaystyle\bar{q_{2}}O^{\mu}q_{1}|M_{1}(p_{1})\rangle=\frac{\epsilon_{\nu}^{{\dagger}}}{m_{M_{1}}+m_{M_{2}}}\left[-g^{\mu\nu}P\cdot
qA_{0}(q^{2})\right.$ (3)
$\displaystyle\left.+P^{\mu}P^{\nu}A_{+}(q^{2})+q^{\mu}P^{\nu}A_{-}(q^{2})+i\varepsilon^{\mu\nu\alpha\beta}P_{\alpha}q_{\beta}V(q^{2})\right],$

In the above equations, $G_{F}$ is the Fermi coupling constant and
$O^{\mu}=\gamma_{\mu}(1-\gamma_{5})$ is the weak Dirac matrix. Also
$P=p_{1}+p_{2}$, $q=p_{1}-p_{2}$ and $\epsilon_{\nu}$ to be the polarization
vector. Further the on-shell conditions demands $p_{1}^{2}=m_{M_{1}}^{2}$ and
$p_{2}^{2}=m_{M_{2}}^{2}$. The form factors appearing in these equations are
computed employing the effective field theoretical approach of covariant
confined quark model (CCQM) originally developed by M. A. Ivanov and G. V.
Efimov Efimov and Ivanov (1989, 1993); Ivanov and Santorelli (1999); Branz
_et al._ (2010); Ivanov  _et al._ (2012); Gutsche  _et al._ (2012); Ivanov
_et al._ (2019). The hadronic interaction Lagrangian can be written as

$\displaystyle\mathcal{L}_{\mathrm{int}}$ $\displaystyle=$ $\displaystyle
g_{M}M(x)\int dx_{1}\int
dx_{2}F_{M}(x;x_{1},x_{2})\bar{q}_{2}(x_{2})\Gamma_{M}q_{1}(x_{1})+\mathrm{H.c}.$
(4)

Above Lagrangian describes the interaction of meson with constituents. Here
the Dirac matrix $\Gamma$ is the gamma matrix according the meson.
$F_{M}(x;x_{1},x_{2})$ is the vertex function which essentially describes the
distribution of quarks inside the hadron. In CCQM, for computation of
different observables, we consider the Gaussian form of the vertex function.
It is important to note here that there are other forms of the vertex function
also, but it is observed that the physical observables are not dependent on
the detailed structure of the vertex functions. Further, Gaussian form also
makes the analytical work more convenient Faessler  _et al._ (2003); Ivanov
and Lyubovitskij (1997). In Eq. 4, $g_{M}$ is the coupling constant of meson
computed employing the Compositeness conditions Salam (1962); Weinberg (1963).
The computation of $g_{M}$ include the renormalization of self energy diagram.
This condition essentially confirms that the final mesonic state does not
contain any free quark and all the quarks are confined within the hadron.
Further, the matrix element of the meson mass operator and semileptonic
transition form factors are described by the convolution of the Feynman
propagators and Gaussian vertex functions. In order to have the loop
integration in more efficient way, we use the Fock Schwinger representation of
Feynman propagator. Finally, to overcome any ultraviolet divergences appearing
in the diagrams, we introduce the infrared cutoff parameter $\lambda=0.181$
GeV. Note here that we take $\lambda$ to be same for all the physical
processes studied using the CCQM.

There are only two other model parameters namely quark mass and size parameter
which are fixed by fitting with some basic properties such as leptonic decay
widths with the available experimental data or lattice simulations. The
parametrization is also achieved in such a way that the deviation in the decay
widths computation remains minimum. The uncertainties in the fitted size
parameters are determined from the differences in the predicted and
experimental data. It is worth interesting to note here that the observed
uncertainness are found to be less than 5% for all the mesonic case. These
uncertainties are then transported in the computation of all the observables
such as form factors and then branching fractions and other physical
observables. It is observed that the uncertainties are less than 5% at the
$q^{2}=0$ and less than 10% for $q^{2}=q^{2}_{\mathrm{max}}$. The propagation
of uncertainty in any observables can be computed using the most general
technique. For instance, the uncertainty propagation in the forward backward
asymmetry for the transition corresponding to the channel $B_{s}^{0}\to
D_{s}^{-}$ can be written as Soni  _et al._ (2020)

$\displaystyle\Delta(A_{\mathrm{FB}}(q^{2}))=\sqrt{\sum_{i=+,-}\left(\frac{\partial(A_{\mathrm{FB}}(q^{2}))}{\partial
F_{i}}\Delta F_{i}\right)^{2}}.$ (5)

Here $\Delta F_{i}$ is the uncertainty in the form factor. For all the finer
details regarding the computational techniques, we suggests the readers to
refer Refs. Branz  _et al._ (2010); Lyubovitskij  _et al._ (2003).

We also present the form factors Eq. (3) in the double pole approximation
form. The relation reads

$\displaystyle
F(q^{2})=\frac{F(0)}{1-a\left(\frac{q^{2}}{m_{M_{1}}^{2}}\right)+b\left(\frac{q^{2}}{m_{M_{1}}^{2}}\right)^{2}}.$
(6)

The parameters $a$ and $b$ and the form factor $F(0)$ are listed in Tab.
LABEL:tab:form_factor. We also transform of our form factors with the Bauer-
Stech-Wirbel form factors Wirbel  _et al._ (1985) so that we can have the
brief comparison of our form factors in the entire $q^{2}$ range with
different other theoretical approaches such as light front quark model Verma
(2012), relativistic quark model Faustov  _et al._ (2022) and lattice
simulations McLean  _et al._ (2020). The transformed (primed) form factors
reads

$\displaystyle F_{0}^{\prime}$ $\displaystyle=$ $\displaystyle
F_{+}+\frac{q^{2}}{m_{M_{1}}^{2}-m_{M_{2}}^{2}}F_{-},\ \ \ \ \ \
F_{+}^{\prime}=F_{+}$ $\displaystyle A_{0}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{m_{M_{1}}-m_{M_{2}}}{2m_{M_{2}}}\left(A_{0}-A_{+}-\frac{q^{2}}{m_{M_{1}}^{2}-m_{M_{2}}^{2}}A_{-}\right),$
(7) $\displaystyle A_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{m_{M_{1}}-m_{M_{2}}}{m_{M_{1}}+m_{M_{2}}}A_{0},\ \ \ \ \ \
A_{2}^{\prime}=A_{+},\ \ \ \ \ \ V^{\prime}=V.$

Note that in order to avoid confusion, we will not use prime from now on. In
Fig. 1, we compare transition form factors $F_{+}(q^{2})$ and $F_{0}(q^{2})$
with light front quark model, relativistic quark model and also the lattice
simulations.

CCQM is a very versatile model and can be employed for studying not only
mesons and baryons, but also mutiquark states as well. In last few years, we
have employed CCQM for studying the semileptonic decays of charmed mesons Soni
and Pandya (2017); Soni  _et al._ (2018); Ivanov  _et al._ (2019); Soni  _et
al._ (2020); Soni and Pandya (2019a, b, 2021), bottom-strange mesons Soni  _et
al._ (2023) and rare semileptonic decays of bottom mesons Soni  _et al._
(2022, 2021); Issadykov  _et al._ (2022). Ivanov et. al., have also utilised
CCQM for computation of various decay properties of $B$, $B_{s}$ and $B_{c}$
mesons in Ref. Ivanov  _et al._ (2001); Faessler  _et al._ (2002); Ivanov  _et
al._ (2005, 2006, 2012); Dubnicka  _et al._ (2017); Issadykov and Ivanov
(2018); Dubnička  _et al._ (2019, 2022); Ivanov  _et al._ (2023).

Table 2: Averages of different physical observables for $B_{s}^{0}\to D_{s}^{-}\ell^{+}\nu_{\ell}$ channel. Observable | $-\langle A_{FB}^{e}\rangle\times 10^{-6}$ | $-\langle A_{FB}^{\mu}\rangle$ | $-\langle A_{FB}^{\tau}\rangle$ | $-\langle C_{F}^{e}\rangle$ | $-\langle C_{F}^{\mu}\rangle$ | $-\langle C_{F}^{\tau}\rangle$
---|---|---|---|---|---|---
Present | $1.156\pm 0.295$ | $0.015\pm 0.004$ | $0.362\pm 0.113$ | $1.500\pm 0.395$ | $1.455\pm 0.384$ | $0.260\pm 0.079$
RQM Faustov  _et al._ (2022) | $0.97$ | $0.013$ | $0.36$ | $1.5$ | $1.46$ | $0.3$
Observable | $\langle P_{L}^{e}\rangle$ | $\langle P_{L}^{\mu}\rangle$ | $-\langle P_{L}^{\tau}\rangle$ | $-\langle P_{T}^{e}\rangle\times 10^{-3}$ | $-\langle P_{T}^{\mu}\rangle$ | $-\langle P_{T}^{\tau}\rangle$
Present | $1.000\pm 0.263$ | $0.958\pm 0.253$ | $0.337\pm 0.119$ | $1.115\pm 0.294$ | $0.205\pm 0.054$ | $0.839\pm 0.264$
RQM Faustov  _et al._ (2022) | 1.000 | 0.960 | $0.27$ | $1.02$ | $0.19$ | $0.85$
Table 3: Averages of different physical observables for $B_{s}^{0}\to D_{s}^{-*}\ell^{+}\nu_{\ell}$ channel. Observable | $-\langle A_{FB}^{e}\rangle$ | $-\langle A_{FB}^{\mu}\rangle$ | $-\langle A_{FB}^{\tau}\rangle$ | $-\langle C_{F}^{e}\rangle$ | $-\langle C_{F}^{\mu}\rangle$ | $-\langle C_{F}^{\tau}\rangle$
---|---|---|---|---|---|---
Present | $0.195\pm 0.024$ | $0.201\pm 0.025$ | $0.284\pm 0.037$ | $0.452\pm 0.089$ | $0.436\pm 0.086$ | $0.060\pm 0.012$
RQM Faustov  _et al._ (2022) | $0.26$ | $0.27$ | $0.32$ | $0.35$ | $0.33$ | $0.040$
Observable | $\langle P_{L}^{e}\rangle$ | $\langle P_{L}^{\mu}\rangle$ | $\langle P_{L}^{\tau}\rangle$ | $-\langle P_{T}^{e}\rangle\times 10^{-3}$ | $-\langle P_{T}^{\mu}\rangle$ | $-\langle P_{T}^{\tau}\rangle$
Present | $1.000\pm 0.148$ | $0.985\pm 0.145$ | $0.504\pm 0.067$ | $0.322\pm 0.063$ | $0.058\pm 0.011$ | $0.126\pm 0.034$
RQM Faustov  _et al._ (2022) | 1.000 | 0.990 | 0.530 | $0.23$ | $0.040$ | $0.035$
Observable | $\langle F_{L}^{e}\rangle$ | $\langle F_{L}^{\mu}\rangle$ | $\langle F_{L}^{\tau}\rangle$ |  |  |
Present | $0.534\pm 0.088$ | $0.534\pm 0.088$ | $0.458\pm 0.070$ |  |  |
RQM Faustov  _et al._ (2022) | 0.49 | 0.49 | 0.42 |  |  |
LQCD Harrison and Davies (2022) | – | – | 0.440 (16) |  |  |

##  III Different physical observables

After computation of transition form factors, we compute various other
physical observables such as forward-backward asymmetry, lepton side convexity
parameter, longitudinal and transverse polarization. In Ref. Soni  _et al._
(2023), we have computed the semileptonic branching fractions and made a
detailed comparison with the LHCb data and here in present work, we compute
some other physical observables that are yet to be identified by the
experimental facilities worldwide. These observables are forward backward
asymmetry, convexity parameter, longitudinal and transverse polarization of
charged leptons, and longitudinal polarization fractions of daughter vector
meson. In semileptonic decays, these observables are dependent on the lepton
masses and therefore these observables are very important probes to understand
the effects of lepton masses. In present work, we use the same notations used
for studying the semileptonic $D$ and $D_{s}$ decays Ivanov  _et al._ (2019).
We compute the following physical observable in the entire $q^{2}$ range and
their relations are defined as Ganbold  _et al._ (2015); Gutsche  _et al._
(2015); Zhang  _et al._ (2021); Faustov  _et al._ (2022):

  1. 1.

Forward-backward asymmetry:

$\displaystyle
A_{\mathrm{FB}}(q^{2})=\frac{3}{4}\frac{\mathcal{H}_{P}-2\frac{m_{\ell}^{2}}{q^{2}}\mathcal{H}_{SL}}{\mathcal{H}_{tot}}$
(8)

  2. 2.

Lepton side convexity parameter:

$\displaystyle
C_{F}^{\ell}=\frac{3}{4}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)\frac{\mathcal{H}_{U}-2\mathcal{H}_{L}}{\mathcal{H}_{tot}}$
(9)

  3. 3.

Longitudinal polarization of charged leptons:

$\displaystyle
P_{L}^{\ell}=\frac{(\mathcal{H}_{U}+\mathcal{H}_{L})\left(1-\frac{m_{\ell}^{2}}{2q^{2}}\right)-\frac{3m_{\ell}^{2}}{2q^{2}}\mathcal{H}_{s}}{\mathcal{H}_{tot}}$
(10)

  4. 4.

Transverse polarization of charged leptons:

$\displaystyle P_{T}^{\ell}=-\frac{3\pi
m_{\ell}}{8\sqrt{q^{2}}}\frac{\mathcal{H}_{P}+2\mathcal{H}_{SL}}{\mathcal{H}_{tot}}$
(11)

  5. 5.

Longitudinal polarization fraction for the final vector meson:

$\displaystyle
F_{L}(q^{2})=\frac{\mathcal{H}_{L}\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)+\frac{3m_{\ell}^{2}}{2q^{2}}\mathcal{H}_{S}}{\mathcal{H}_{tot}}$
(12)

Using the above Eq. (12), one can also compute the transverse polarization
vector of the final vector mesons via relation $F_{T}=1-F_{L}$. In these
relations, $\mathcal{H}^{\prime}s$ are the bilinear combinations of the
helicity components of the hadronic tensors related to the helicity amplitudes
($H$) as Ganbold  _et al._ (2015); Gutsche  _et al._ (2015); Zhang  _et al._
(2021); Faustov  _et al._ (2022),

$\displaystyle\mathcal{H}_{U}$ $\displaystyle=$
$\displaystyle|H_{+}|^{2}+|H_{-}|^{2},\ \ \ \ \ \
\mathcal{H}_{P}=|H_{+}|^{2}-|H_{-}|^{2},$ $\displaystyle\mathcal{H}_{L}$
$\displaystyle=$ $\displaystyle|H_{0}|^{2},\ \ \ \ \ \
\mathcal{H}_{S}=|H_{t}|^{2},$ $\displaystyle\mathcal{H}_{SL}$ $\displaystyle=$
$\displaystyle Re(H_{0}H_{t}^{\dagger}).$ (13)

Here the helicity amplitudes are related to the transition form factors
computed using CCQM. The relations read

  1. 1.

For $B_{s}^{0}\to D_{s}^{-}$ transitions

$\displaystyle H_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{q^{2}}}(PqF_{+}+q^{2}F_{-}),$ $\displaystyle
H_{\pm}$ $\displaystyle=$ $\displaystyle 0\ \ \ {\textrm{a}nd}\ \ \
H_{0}=\frac{2m_{M_{1}}|\bf{p_{2}}|}{\sqrt{q^{2}}}F_{+},$ (14)

  2. 2.

For $B_{s}^{0}\to D_{s}^{-*}$ transitions

$\displaystyle H_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{M_{1}}+m_{M_{2}}}\frac{m_{M_{1}}|{\bf
p_{2}}|}{m_{M_{2}}\sqrt{q^{2}}}\left((m_{M_{1}}^{2}-m_{M_{2}}^{2})(A_{+}-A_{-})+q^{2}A_{-}\right)$
$\displaystyle H_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{M_{1}}+m_{M_{2}}}(-(m_{M_{1}}^{2}-m_{M_{2}}^{2})A_{0}\pm
2m_{M_{1}}|{\bf p_{2}}|V)$ $\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{m_{M_{1}}+m_{M_{2}}}\frac{1}{2m_{M_{2}}\sqrt{q^{2}}}(-(m_{M_{1}}^{2}-m_{M_{2}}^{2})(m_{M_{1}}^{2}-m_{M_{2}}^{2}-q^{2})A_{0}+4m_{M_{1}}^{2}|{\bf
p_{2}}|^{2}A_{+}).$

Here the form factors are defined in Eq. (1). Also the $|{\bf
p_{2}}|=\lambda^{1/2}(m_{M_{1}}^{2},m_{M_{2}}^{2},q^{2})/2m_{M_{1}}$ is the
momentum of the $D_{s}^{(*)-}$ meson in the rest frame of $B_{s}^{0}$ mesons
and $\lambda$ is the Källen function. Further, in all these equations $M_{1}$
is the parent meson ($B_{s}^{0}$) and $M_{2}$ is the daughter mesons
($D_{s}^{(*)-}$). Using these relations we compute the physical observables
and listed in Tab. LABEL:tab:Obs_Bs_Ds and LABEL:tab:Obs_Bs_Dsv. We also
compare our findings with the relativistic quark model predictions Faustov
_et al._ (2022). Note here that in order to compute the expectation values of
these observables, one has to multiply with the phase factor $|{\bf
p_{2}}|(q^{2}-m_{\ell}^{2})q^{2}$ to the numerator and denominator and
integrate separately. The CCQM model parameters used for the present
computations are quark masses $m_{b}=5.05$ GeV, $m_{c}=1.672$ GeV,
$m_{s}=0.428$ GeV and meson size parameters $\Lambda_{B_{s}}=2.05\pm 0.014$
GeV, $\Lambda_{D_{s}}=1.75\pm 0.035$ GeV and $\Lambda_{D_{s}^{*}}=1.56\pm
0.014$ GeV Soni  _et al._ (2023).

##  IV Results and Discussion

Having determined the model parameters namely quark masses and size
parameters, we compute the transition form factors as per Eq. (3) in the
entire $q^{2}$ range. In fig. 1 we plot the form factors and also compare our
form factors with light front quark model Verma (2012), relativistic quark
model Faustov  _et al._ (2022), SCI Xing  _et al._ (2022) as well as with the
lattice simulations McLean  _et al._ (2020). For $B_{s}^{0}\to D_{s}^{-}$ form
factors, our results are very near to other theoretical approaches including
lattice simulations. However, for $F_{0}(q^{2}<10\mathrm{GeV}^{2})$, our form
factors are significantly higher than the other approaches. Our results for
vector form factors are in very good agreement with the other approaches.

Using the transition form factors we compute different physical observables
using the relations Eq. (8 \- 12) and their expectation values are listed in
Tab. LABEL:tab:Obs_Bs_Ds and LABEL:tab:Obs_Bs_Dsv. These observables are yet
to be identified experimentally, however, for $B\to D^{(*)}$ transitions,
these observables are studied extensively by the $B$ factories. Therefore
these observables are also expected for $B_{s}\to D_{s}^{(*)-}$ transitions
from the experiments these channels are much similar to that of $B\to D^{(*)}$
except for the spectator quark. We also compare our results with the
predictions using the relativistic quark model by Faustov et. al., Faustov
_et al._ (2022) and it is observed that our results are in excellent agreement
with them for all the observables. This agreement is expected as our form
factors are also nearly matching well. In Figs. 2 \- 6, we also plot all these
observables in the whole kinematical range of momentum transfer squared along
with the spread of uncertainty. It is observed that the lepton side convexity
parameter for the channel $B_{s}^{0}\to D_{s}^{-}e^{+}\nu_{e}$ and
longitudinal polarization for the channels $B_{s}^{0}\to
D_{s}^{(*-)}e^{+}\nu_{e}$ are found to be constant throughout the whole
$q^{2}$ range and also their spread in the uncertainty is also found to be
very small and constant. Recently, HPQCD collaboration Harrison and Davies
(2022) have also provided the results on longitudinal polarization fraction
for the $D_{s}^{(*)-}$ meson and from Tab. LABEL:tab:Obs_Bs_Dsv, it is
observed that our results are in excellent agreement with them. It is
interesting to note here that the lepton side convexity parameter for the
channel $B_{s}^{0}\to D_{s}^{-}e^{+}\nu_{e}$ and longitudinal polarization of
charged leptons for the channel $B_{s}^{0}\to D_{s}^{(*)-}e^{+}\nu_{e}$ is
found to be constant through out the $q^{2}$ range. This nature is also
observed in the RQM predictions Faustov  _et al._ (2022).

In our previous study Soni  _et al._ (2023), we have reported the semileptonic
branching fractions and also provided detailed comparison with the recent LHCb
measurements as well as with the lattice simulation results. We determined the
normalised decay rates for the channel $B_{s}^{0}\to
D_{s}^{*-}\mu^{+}\nu_{\mu}$ in the recoil parameter bins and they are found to
be in very good agreement with the LHCb as well as lattice results. We also
determined the ratios $R(D_{s})=0.271\pm 0.069$ and $R(D_{s}^{*-})=0.240\pm
0.034$ and they are in very good agreement with the other theoretical
approaches and lattice simulations within the uncertainties Soni  _et al._
(2023). These ratios are also in agreement with the other channels concerning
the transition $b\to c\tau\nu_{\tau}/b\to c\mu\nu_{\mu}$ such as $B\to
D^{(*)}\ell\nu_{\ell}$ Ivanov  _et al._ (2015) as well as
$B_{c}\to(\eta_{c},J/\psi)\ell\nu_{\ell}$ Issadykov  _et al._ (2017) studied
employing CCQM. Further, the ratio $\mathcal{B}(B_{s}^{0}\to
D_{s}^{-}\mu^{+}\nu_{\mu})/\mathcal{B}(B_{s}^{0}\to
D_{s}^{*-}\mu^{+}\nu_{\mu})=0.451\pm 0.096$ is also found to be in excellent
agreement with the LHCb data Soni  _et al._ (2023). Overall, all the results
presented employing CCQM are in very good agreement with the available
experimental data and lattice simulations. The same was also observed for
studying the semileptonic decays of $D$ and $D_{s}$ mesons Soni and Pandya
(2017); Soni  _et al._ (2018); Ivanov  _et al._ (2019); Soni  _et al._ (2020).

##  V Conclusion

In this article, we have studied a very interesting channel corresponding to
the quark level transition $b\to c\ell^{+}\nu_{\ell}$. $B\to
D^{(*)}\ell\nu_{\ell}$ have gained lot of attention as some of its results
deviate from the standard model predictions, the another channel $B_{s}^{0}\to
D_{s}^{(*)}\ell^{+}\nu_{\ell}$ have started getting attention after the first
observation from the LHCb collaboration. Here, we compute the transition form
factors for the channels $B_{s}\to D_{s}^{-}$ and $B_{s}^{0}\to D_{s}^{-*}$
using the covariant confined quark model. We also compare our form factors
with different other theoretical predictions including lattice simulations. We
further compute different physical observables such as forward backward
asymmetry, lepton side convexity parameter, longitudinal and transverse
polarizations along with their expectation values. Since these observables are
yet to be examined from the experimental side, we compare only with the
available theoretical prediction relativistic quark model. In general, all our
results are in very good agreement with the available theoretical studies and
lattice predictions.

Figure 2: Forward backward asymmetries for the decay channels $B_{s}^{0}\to
D_{s}^{(*)-}\ell^{+}\nu_{\ell}$.

Figure 3: Lepton side convexity parameter for the decay channels $B_{s}^{0}\to
D_{s}^{(*)-}\ell^{+}\nu_{\ell}$.

Figure 4: Longitudinal polarization of charged leptons for the decay channels
$B_{s}^{0}\to D_{s}^{(*)-}\ell^{+}\nu_{\ell}$.

Figure 5: Transverse polarization of charged leptons for the decay channels
$B_{s}^{0}\to D_{s}^{(*)-}\ell^{+}\nu_{\ell}$.

Figure 6: Longitudinal polarization fraction for final state $D_{s}^{*-}$
meson.

## ACKNOWLEDGEMENTS

We would like to thank Prof. Mikhail A. Ivanov for useful discussions on some
aspects of this work.

## Data Availability Statement

There are no data associated with the manuscript.

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