#  Evolutionary Deep Reinforcement Learning for Dynamic Slice Management in
O-RAN ††thanks:  This material is based upon work supported by the Air Force
Office of Scientific Research under award number FA9550-20-1-0090 and the
National Science Foundation under Grant Numbers CNS-2232048. ††thanks: F.
Lotfi was with Department of Electrical and Computer Engineering, University
of Colorado, Colorado Springs, CO, USA (e-mail: flotfi@uccs.edu), and now is
with Department of Electrical and Computer Engineering, Clemson University,
Clemson, SC, USA (flotfi@clemson.edu). O. Semiari is with Department of
Electrical and Computer Engineering, University of Colorado, Colorado Springs,
CO, USA (osemiari@uccs.edu). F. Afghah is with Department of Electrical and
Computer Engineering, Clemson University, Clemson, SC, USA
(fafghah@clemson.edu).

Fatemeh Lotfi, Omid Semiari, and Fatemeh Afghah

###### Abstract

The next-generation wireless networks are required to satisfy a variety of
services and criteria concurrently. To address upcoming strict criteria, a new
open radio access network (O-RAN) with distinguishing features such as
flexible design, disaggregated virtual and programmable components, and
intelligent closed-loop control was developed. O-RAN slicing is being
investigated as a critical strategy for ensuring network quality of service
(QoS) in the face of changing circumstances. However, distinct network slices
must be dynamically controlled to avoid service level agreement (SLA)
variation caused by rapid changes in the environment. Therefore, this paper
introduces a novel framework able to manage the network slices through
provisioned resources intelligently. Due to diverse heterogeneous
environments, the intelligent machine learning approaches require sufficient
exploration to handle the harshest situations in a wireless network and
accelerate convergence. To solve this problem, a new solution is proposed
based on evolutionary-based deep reinforcement learning (EDRL) to accelerate
and optimize the slice management learning process in the radio access
network’s (RAN) intelligent controller (RIC) modules. To this end, the O-RAN
slicing is represented as a Markov decision process (MDP) which is then solved
optimally for resource allocation to meet service demand using the EDRL
approach. In terms of reaching service demands, simulation results show that
the proposed approach outperforms the DRL baseline by $62.2\%$.

††publicationid: pubid:  978-1-6654-5975-4/22/$31.00 © 2022 IEEE

##  I Introduction

The next-generation wireless networks must be capable of managing a broad
spectrum of wireless technologies with heterogeneous quality-of-service (QoS)
and quality-of-experience (QoE) requirements. Among such technologies include
emerging applications such as holographic telepresence, the Internet of
everything, drone-based applications, and collaborative robots [1, 2, 3, 4].
To meet strict QoS and QoE requirements of these applications, the new open
radio access network (O-RAN) has been recently introduced which addresses the
demand for virtual and programmable components, as well as intelligent, data-
driven, and closed-loop control of the RAN.

The key advantages of O-RAN are the possibility for operators to mix and match
components, the use of open interfaces, the fact that it is software-centric
and scalable, and the potential to improve network performance using machine
learning (ML) approaches. Furthermore, all of these factors increase the
flexibility of the network design [5]. The disaggregation of RAN functions
into different units is a key feature of O-RAN that makes the network
adaptable. The third generation partnership project (3GPP) splits base
stations (BSs) into an open central unit (O-CU), an open distributed unit
(O-DU), and an open radio unit (O-RU) [6]. Moreover, these distributed units
with open interfaces link to RAN’s intelligent controllers (RIC) to manage and
control the network in near real-time and non-real-time controlling loops [7,
5]. The new generation networks need a controlling approach with fast
convergence, between $10$ms and $1$s for near real-time controllers, to obtain
their required QoS [8]. Furthermore, to maintain the network QoS in the face
of dynamic changes and heterogeneous requirements, O-RAN slicing is being
explored as a viable solution.

Different network slices need to be managed carefully to prevent service level
agreement (SLA) diversity. While artificial intelligence (AI) and ML
approaches are usable in network slicing, they face challenges, including the
requirement for a vast amount of diverse data and sufficient exploration to
accelerate convergence and train an ML model that can effectively generalize
to different situations without impacting the actual RAN performance [5, 9].
Given the difficulty of gaining access to this amount and variety of data,
making large-scale decisions involving several O-DUs will be challenging. The
O-RAN slicing challenge to manage and control diverse and dynamic service
requirements has recently been studied in several works [10, 11, 12, 13, 14].
Two main ML categories that have been used in the literature are supervised
deep learning and deep reinforcement learning (DRL). To avoid SLA violations,
the works in [10] and [11] proposed supervised ML-based resource provisioning
approaches for network slicing by using predictions on traffic and the number
of active users in the network. The works in [12, 13, 14] proposed DRL-based
approaches to achieve online training and dynamic resource allocation in
O-RAN. In particular, the authors in [12] provided a performance comparison
between two on-policy and off-policy models for delay-tolerant and delay-
sensitive services, and the work in [13] utilized auto-encoders to minimize
unnecessarily high-dimensional input for the DRL agent and improve DRL
controller performance in the presence of unreliable real-time wireless data.
Furthermore, the authors in [14] demonstrated the feasibility of RAN
scheduling control over real-time RIC by collecting data at the network edge
in a DRL method.

Although the prior art in [10, 11, 12, 13, 14] is effective in a set of use
cases, it has a number of limitations including slow convergence and lack of
generalizability. For example, the DRL methods in [12] require near 20k
training time steps before they converge, and thus, they can lead to
inefficient resource allocation in some delay-sensitive systems. In addition,
while the work in [13] aims to overcome the limitations of DRL approaches in
real-time wireless network scenarios, it has not been completely resolved,
especially in real-time controlling scenarios. While DRL algorithms are
effective in complex tasks, their training procedures are slow in the face of
unreliable real-time wireless data, causing delays in the O-RAN controlling
mechanism. Furthermore, DRL methods suffer from a lack of sufficient
exploration, particularly in dynamic heterogeneous environments such as the
one studied in [14]. These challenges make DRL approaches insufficient for the
O-RAN slicing scenario and mandate new solutions that can cope with the demand
for broad exploration of dynamic wireless networks.

The main contribution of this work is to utilize the opportunity provided by
O-RAN to create new experiences based on disaggregated modules. To this end,
we formulate a problem that aims to minimize the probability of SLA violation
while considering some constraints on the total required resources. To solve
this problem, we propose utilizing a joint DRL and population-based strategy
as an evolutionary-based DRL (EDRL) approach in the O-RAN slicing scenario to
accelerate the learning process in the RIC modules. By leveraging population
actors, the EDRL provides enough exploration to stabilize convergence
properties and make the learning process more robust. To do that, we consider
the O-RAN slicing issue and model the O-RAN slicing managing problem as a
Markov decision process (MDP). Then, to solve the MDP problem and find the
best allocation policy for O-RAN slicing, we offer an EDRL algorithm.
Simulation results show a $62.2$% improvement in the network performance
compared to the DRL baseline method. _To the best of our knowledge, this is
the first work that utilizes a hybrid method of evolutionary algorithms and
DRL to achieve efficient and dynamic slice management in future O-RANs_.

The rest of this paper is organized as follows. Section II presents the system
model and the problem formulation for the O-RAN slicing. Section III presents
the proposed EDRL algorithm to solve the MDP problem. Simulation results are
presented in Sec. IV and conclusions are drawn in Sec. V.

Figure 1: The proposed EDRL in Open RAN network.

##  II System model

Consider an O-RAN architecture for a wireless network with $N$ heterogeneous
users in a set $\mathcal{N}$ served by different network slices. By
considering the O-RAN architecture as a dynamic slice optimization, we have
resource management in RIC modules for O-RAN slicing as Fig 1. In this system,
there are $L=3$ types of network slices in a set $\mathcal{L}$ with specific
QoS requirements as $Q_{l},l\in\mathcal{L}$, defined as follows: enhanced
mobile broadband (eMBB), machine-type communications (MTC), and ultra-reliable
low latency communications (URLLC). Each of the slice $l\in\mathcal{L}$ serves
$N_{l}$ user equipment (UE). Furthermore, the slices must share a pool of $K$
available resources to meet the QoS requirements of their assigned UEs. To
guarantee that the SLA is satisfied, dynamic slice optimization will be
performed in the RT-RIC module to optimize slice management. In addition, as a
part of the slicing, the medium access control (MAC) layer should allocate
resources following the radio resource management (RRM) strategy provided by
the slice management. To address this challenge, next, we present the wireless
model and, accordingly, formulate the slice management problem cognizant of
wireless resource constraints.

###  II-A Wireless communication model for O-RAN network slicing

According to the O-RAN architecture shown in Fig. 1, different network slices
server different UEs with different QoS criteria. The RIC module is also in
charge of managing these network slices and the resources assigned to them.
Due to the stochastic nature of the wireless channel, static resource
management would be ineffective. As a result, dynamic resource management is
considered in order to dynamically re-assign resources to slice networks in
each frame based on UEs’ network and channel changes. As a result, while this
aids in adapting to dynamic changes in the wireless channel, it also
complicates resource assignment.

Besides, the QoS criteria $Q_{l}$ can also be defined particularly in terms of
throughput, capacity, and latency for $\mathcal{L}$ slices. To define the
slice $l$’s achievable QoS, we consider orthogonal frequency-division multiple
access (OFDMA) schemes. The slice $l$ achievable data rate can be written as

$\displaystyle c_{l}=\lim_{\tau\to\infty}\frac{B}{\tau}\sum_{t=1}^{\tau}$
$\displaystyle\sum_{n=1}^{N_{l}}\sum_{k=1}^{K}e_{n,k}b_{l,k}$
$\displaystyle\times\log\Big{(}1+\frac{p_{u}d_{n}(t)^{-\eta}h_{n,k}(t)}{I_{n,k}(t)+\sigma^{2}}\Big{)},$
(1)

where $e_{n,k}\in\\{0,1\\}$ is a binary variable that shows the resource block
(RB) allocation indicator in the RB $k$ of user $n$, and $b_{l,k}\in\\{0,1\\}$
is a binary variable that shows the RB allocation indicator in the RB $k$ of
slice $l$. Also, $B$ represents the RB bandwidth and $K$ is the total
available RBs for downlink communications. In addition, $p_{u}$ is the O-RU
transmit power per RB, and $d_{n}(t)$ is the user $n$ distance from its
assigned O-RU. Moreover, $\eta$ represents the path loss exponent, and
$h_{n,k}(t)$ is the time-varying Rayleigh fading channel gain. In (II-A),
$I_{n,k}(t)$ denotes the downlink interference from the neighboring O-RUs
transmitting over RB $k$, and $\sigma^{2}$ represents the variance of the
additive white Gaussian noise (AWGN).

###  II-B Problem formulation

Due to the restricted resources shared between slices with heterogeneous
services and dynamic UEs, our goal is to minimize the probability of SLA
violation $\mathbbm{P}(\mathinner{\\!\left\lvert
Q_{l}(\boldsymbol{e},\boldsymbol{b})-\lambda_{l}\right\rvert}\geq\epsilon_{l})$,
$\forall l\in\mathcal{L}$, while considering some constraints on the total
required resources. Also, $\boldsymbol{e}$ and $\boldsymbol{b}$ represent
vectors of $e_{n,k}$ and $b_{l,k}$ as resource allocation indicators at MAC
and RIC, respectively. Therefore, we formulate the following optimization
problem to find an optimal allocation policy for O-DUs and distribute the
shared resources among $N_{l}$ heterogeneous UEs in each O-RAN slice $l$.

$\displaystyle\operatorname*{argmin}_{\boldsymbol{b},\boldsymbol{e}}$
$\displaystyle\hskip 14.22636pt\mathbbm{P}(\mathinner{\\!\left\lvert
Q_{l}(\boldsymbol{e},\boldsymbol{b})-\lambda_{l}\right\rvert}\geq\epsilon_{l}),$
(2a) s.t., $\displaystyle\hskip 14.22636pt\forall l\in\mathcal{L},\,\,\forall
n\in\mathcal{N},$ (2b) $\displaystyle\hskip
14.22636pt\sum_{l=1}^{L}\sum_{n=1}^{N_{l}}\sum_{k=1}^{K}b_{l,k}e_{n,k}\leq K,$
(2c) $\displaystyle\hskip 14.22636pt\sum_{l}b_{l,k}\leq 1,\,\,$ (2d)

where $\lambda_{l}$ and $\epsilon_{l}$ represent, respectively, the desired
threshold and margin values in QoS required by slice $l$. Constraint (2c) and
(2d) represent the feasibility conditions on allocated resources to slices and
UEs. Problem (2a) indicates that the $\mathcal{L}$ slices achieve their
demanded QoS by minimizing the probability of SLA violation. The proposed
problem is an NP-hard mixed-integer stochastic optimization problem which is
challenging to solve. Markov decision process (MDP) provides a mathematical
framework for decision-making and optimization problems involving partially
random situations. As a result, it is advantageous to model the (2a) as an MDP
and solve it using dynamic methods such as DRL approaches.

###  II-C Stochastic game-based optimization problem

By considering the RT-RIC module as an intelligent agent that makes decisions
to manage the O-RAN slicing, the other components, i.e., O-RU, O-DU, and O-CU
will act as the agent’s environment that is influenced by the agent’s actions,
as shown in Fig 1. Thus, the decision process of the mentioned O-RAN slicing
controlling the problem is represented as an MDP with tuples
$\langle\mathcal{S},\mathcal{A},T,\gamma,r\rangle$, where $\mathcal{S}$,
$\mathcal{A}$, and $T$ represent the state space, action space, and transition
probability from current state to the next state, respectively. The MDP tuples
are described as follows:

####  II-C1 State

$s_{t}\in\mathcal{S}$ represents the O-RAN status in each step of time which
contains the achievable QoS of each slice $Q_{l}$, UEs’ density in each slice
$N_{l}(t)$, and resource allocation history as the previous action $a_{t-1}$.
Therefore, the observation of the intelligent agent in time $t$ is as follows:

$s_{t}=\\{Q_{l},N_{l},a_{t-1}\mid\forall l\in\mathcal{L}\\}.$ (3)

####  II-C2 Action

$a_{t}\in\mathcal{A}$ is defined as a vector of the number of required
resources for the O-RAN slices and UEs. Thus, in each time $t$, the agent,
based on its policy, decides to perform the action as
$a_{t}=\\{\boldsymbol{e},\boldsymbol{b}\\}$.

####  II-C3 Reward

$r_{t}$ characterizes summation of the complement probability of SLA violation
in (2a), which relies on the incoming traffic of each slice and the radio
condition of the connected UEs. The desired reward value is described as:

$r_{t}=\sum_{l=1}^{\mathinner{\\!\left\lvert\mathcal{L}\right\rvert}}\Big{(}1-\mathbbm{P}(\mathinner{\\!\left\lvert
Q_{l}(\boldsymbol{e},\boldsymbol{b})-\lambda_{l}\right\rvert}\geq\epsilon_{l})\Big{)}.$
(4)

Therefore, the procedure makes best use of the available bandwidth to meet the
QoS demands of all slices. Then, the defined MDP model can be investigated
using a DRL approach. The main task of a DRL agent is to find an optimal
policy $\pi^{*}(a_{t}|s_{t};\theta_{p})$ as a mapping from the state space to
the action space that maximizes its expected average discounted reward
$\mathbb{E}_{\pi}[R(t)]$, where $R(t)=\sum_{i=0}^{\infty}\gamma^{i}r_{i,t}$.
Given a policy $\pi$, the state-value and action-value functions are defined
as $V(s_{t})=\mathbb{E}_{\pi}[R(t)\mid s_{t}]$ and
$Q(s_{t},a_{t})=\mathbb{E}_{\pi}[R(t)\mid s_{t},a_{t}]$, respectively. Due to
continuous states and actions, the Deep Deterministic Policy Gradient (DDPG)
has been used as a model-free and off-policy algorithm. As part of the actor-
critic technique, the RT-RIC agent simultaneously learns an ideal policy to
assign the resources that optimize long-term reward. The policy network
parameterized by $\boldsymbol{\theta}_{p}$ is updated using the gradient
defined as follows with $e$ random samples transitions:

$\displaystyle\nabla_{\theta_{p}}J=\frac{1}{e}\sum_{i=1}^{e}\nabla_{a}Q\big{(}s_{i},a_{i};\theta_{v}\big{)}\nabla_{\theta_{p}}\pi\big{(}a_{i},s_{i};\theta_{p}\big{)}.$
(5)

Also, the value network parameterized by $\boldsymbol{\theta}_{v}$ will be
updated by minimizing the following loss as:

$\displaystyle\min_{\theta_{v}}\frac{1}{e}\sum_{i=1}^{e}\bigg{(}r_{i}+\gamma
Q_{\pi_{i+1}}(s_{i+1},a_{i+1};\theta_{v})-Q_{\pi_{i}}(s_{i},a_{i};\theta_{v})\bigg{)}^{2}.\vspace{-0.1cm}$
(6)

Training one agent that interacts with the environment can be very time-
consuming. However, with O-RAN, we may leverage experience from all of the
disaggregated modules to guide the agent in the training procedure. For
instance, different O-DUs may experience similar network instances (i.e.,
network traffic, QoS requirements, etc.) while being deployed at different
locations across the network. Accordingly, sharing their experiences increase
the generality of the resource assignment task. The prior works in [10, 12,
13, 11, 14] utilize the supervised deep learning and DRL approaches for
network management. The method used in these works has limitations such as the
need for large-scale data, lack of enough exploration, and unstable
convergence of the supervised deep learning and DRL approaches in O-RAN
slicing. Hence, inspired by [15] we employ a hybrid strategy to solve the
problem (2a). This hybrid strategy combines the evolutionary algorithm (EA)
optimization method with the DDPG algorithm of the DRL approach to better
utilize experience samples and provide a more effective performance in less
time than DRL alone. EA provides a diverse set of samples representing a wide
range of services and traffic requirements to improve DRL learning
performance. In response, DRL injects gradient information into the EA
population. Injecting DRL gradient information into EA, augments EA’s ability
to select samples that force the DRL in policy space toward the regions with
higher reward.

##  III Evolutionary DRL method

In fact, EDRL is a hybrid method combining population-based EA and high sample
efficiency DRL approaches. The EDRL uses diverse EA experiences to train the
DRL, while DRL injects gradient information into the EA population.
Accordingly, it makes them powerful to converge faster and thus are suitable
for real-time applications [8]. The employment of a fitness metric that
aggregates returns throughout a whole episode makes EAs beneficial in
environments with reward-less states where the reward is only specified and
known for a few states and is resistant to long time horizons [15].
Accordingly, the EDRL addresses the delayed reward issue, which is obvious in
network slicing since the network requires to experience diverse policies to
offer a different reward. In general, the flow of the EDRL algorithm is
divided into three interacting phases; the population phase, the DRL phase,
and the interaction between them which are explained in the following
paragraphs.

###  III-A Population phase

The population actors $\pi_{\textit{EA},i}$ _evaluate_ in one episode of
interaction with the environment during the population phase, as illustrated
in Fig. 1. During the evaluation episode, they measure the fitness score as a
cumulative sum of return value in each interaction $t$ as
$F_{i}=\sum_{t}r_{i,t}$, and save the actors experience in replay buffer
$\mathcal{B}$ as tuple $\langle s_{t},a_{t},s_{t+1},r_{t}\rangle$. As is
clear, the measured fitness score is determined by achieved QoS of each slice
as $Q_{l}$ which depends on the wireless aspects of the environments. Then,
based on the value of the fitness scores, the population actors get sorted for
_selection_ part. Consequently, the results will be used in the _mutation_ and
_crossover_ sections to create the next generation using the elite individuals
of the population. Here, the O-RAN allows providing new experiences through
disaggregated modules in different geographical locations, such as leveraging
different populations with separate environments. As a result, the network
will experience numerous wireless communication traffics, which is crucial to
improve the generalization ability of the network’s dynamic management system.

###  III-B DRL phase

On the other hand, in the DRL phase, a critic network that is parameterized by
$\theta_{v}$ will be updated using a random batch of replay buffer
$\mathcal{B}$ samples by gradient descent (GD) manner as (6). Then, the critic
network trains a DRL actor $\pi_{\text{RL}}$ by sampled policy gradient (5).
The EA simply uses samples in the fitness score and then leaves information.
However, by utilizing them in a replay buffer and continually applying
powerful gradient-based actor-critic algorithms, they extract more information
from data while maintaining high sample efficiency.

###  III-C Interaction between populations and DRL

The most crucial phase will then be performed, which include an interaction
between the EA and DRL algorithms. During this phase, the $\pi_{\text{RL}}$
weights are copied to the worst-performing individuals in the population
actors $\pi_{\text{EA},i}$ and cause leverage of the learned information from
DRL and help to stabilize learning. Thus, they learned from the experience of
episodes as well as fitness scores by taking this approach. Also, a
synchronization period $\omega$ governs how frequently the RL-actors
information shares with the EA population. Furthermore, following the
selection of the elites, the $\pi_{\text{RL}}$ is updated by the best
performers of $\pi_{\text{EA},i}$ to accelerate convergence.

While the population actors explore in parameter space, and the RL actor
explores in action space, they complement each other and lead to effective
policy space exploration. Besides, the critic network is updated with samples
from the EA population’s policy, which may or may not be used in the DRL
agent’s next action. As a result, this hybrid method behaves as an on-policy
method in each synchronization period, $\omega$, and as an off-policy method
other times, providing benefits from both methods [16]. The effectiveness of
EA is determined by the selection, the crossover, and the mutation parts. It
means that the appropriate choice of these parts is critical. The parents will
be chosen in the selecting part to generate the next generation. As is
obvious, selecting for higher fitness scores will improve overall quality in
each generation. On the other hand, selection by diversity avoids to stuck in
a local extremum. The selection would be determined by taking into account the
environment. In our system model, the tournament selection [17] would have the
optimum performance while considering the O-RAN environment as follows:

$\pi_{\text{EA},p1},\pi_{\text{EA},p2}=\text{tournament}(\pi_{\text{EA}}),$
(7)

where $p1$ and $p2$ represents the selected parents to generate the next
generation. In the O-RAN environment, the populations are actor-networks that
are located in the RT-RIC modules and have interaction with distinct O-DU/O-RU
environments in different areas as Fig. 1. Following parent selection, the
crossover and mutation sections inject additional randomness into the system,
resulting in more exploration and generalization. To ensure the transfer of
parents’ valuable properties to the next generation, we use the average
function in the crossover section as follows:

$\hat{\theta}_{\text{EA}}=\text{avg}(\theta_{\text{EA},p1},\theta_{\text{EA},p2}).$
(8)

Then in the mutation part, the population would be perturbed to create new
features for the next generation as follow [15]:

$\displaystyle\tilde{\theta}_{\text{EA}}=\begin{cases}\hat{\theta}_{\text{EA}}+\mathcal{N}(0,100\xi),&\kappa\leq
q_{\text{super-mut}},\\\ \mathcal{N}(0,1),&\kappa\leq q_{\text{reset}},\\\
\hat{\theta}_{\text{EA}}+\mathcal{N}(0,\xi),&\text{otherwise},\end{cases}$ (9)

where $\mathcal{N}(0,1)$ represents a Gaussian noise with zero mean and unit
variance. Also, $\xi$ shows the mutation rate, and $\kappa$ is a random
variable in $[0,1]$. Moreover, the $q_{\text{super-mut}}$ and
$q_{\text{reset}}$ are super mutation probability and reset probability,
respectively.

Input: $N_{g}$, $N_{p}$, $N_{e}$, $\theta_{\text{EA},i},\forall
i\in[0,N_{p}]$, $\kappa$, $\omega$, $\theta_{p}$, $\theta_{v}$.
for  _iteration $g=1\mathrel{\mathop{\mathchar 58\relax}}N_{g}$_ do

for  _actor $i=1\mathrel{\mathop{\mathchar 58\relax}}N_{p}$_ do

$F_{i}=\text{evaluate}(\pi_{\text{EA},i})$.
$\mathcal{B}\leftarrow\langle s_{t},a_{t},s_{t+1},r_{t}\rangle$.

end for

elite individuals
$\mathcal{E}=\\{\pi_{\text{EA},i}\mid\max_{i}(F_{i},N_{e})\\}$.
$\text{Evolution}\big{(}\mathcal{E},\eqref{select},\eqref{cross},\eqref{mut}\big{)}$.
Update $\theta_{v}$ using (6) and a random batch of $\mathcal{B}$.
Update $\theta_{p}$ using (5).
if  _$g\mod{\omega}=0$_ then

Copy the DRL actor weight $\theta_{p}$ into the weakest individual in
population $\theta_{\text{EA},i}\leftarrow\theta_{p}$.

end if

if  _$\theta_{p}$ by (5) is converged_ then

Break.

end if

end for

Output: $\theta_{p}$, $\theta_{v}$.

Algorithm 1 The EDRL algorithm

###  III-D The proposed EDRL algorithm

In Algorithm 1, we summarize the EDRL method to solve the optimization problem
in (2a)-(2d). The input variables of the algorithm are the number of
generation $N_{g}$, the population size $N_{p}$, the number of elites $N_{e}$,
and the population actors’ network weights as $\theta_{\text{EA},i}$ which are
initialized with random weights. Also, the DRL agent’s actor and critic
network weights as $\theta_{p}$ and $\theta_{v}$ are initialized randomly.
Moreover, a random variable $0\leq\kappa\leq 1$ and a synchronization period
$\omega$ are considered as input variables in Algorithm 1. The algorithm
proceeds to output the trained policy of the DRL agent. In each generation
loop $g$, the population actors’ fitness score $F_{i}$ is measured, and each
population actor experience is stored in the replay buffer $\mathcal{B}$.
Then, in the "Evolution" section of the Algorithm, the $N_{e}$ of the best
population actors based on the measured $F_{i}$ are selected as elites
$\mathcal{E}$ to generate the next generation of population actors based on
the parent selection (7), crossover (8) and mutation (9). In this step, the
DRL agent updates its actor and critic networks by using (5), (6), and the
replay buffer random samples. The algorithm terminates once the DRL agent
policy network is converged or after $N_{g}$ maximum generation.

TABLE I:  Simulation parameters  Parameter |  Value
---|---
Subcarrier spacing |  $15$ kHz
Total bandwidth |  $10$ MHz
RB bandwidth |  $200$ kHz
$p_{u}$ |  $56$ dBm
$K$ |  50
$h$ |  10-tap Rayleigh fading channel
$\sigma^{2}$ |  $-173$ dBm
DDPG batch size |  128
Replay buffer size |  $1e6$
$q_{\text{super-mut}}$ |  $0.05$
$q_{\text{reset}}$ |  $0.1$
Mutation probability |  $0.9$
$\omega$ |  $10$
$N_{g}$ |  $100$
$N_{p}$ |  $10$

##  IV simulation results

In the simulation part, we investigate an O-RAN architecture with three slices
(i.e., eMBB, MTC, URLLC). We consider a $10$ MHz bandwidth as $50$ RBs which
can be dynamically assigned to different slices. The slices serve a total of
$30$ users which are randomly and uniformly distributed in the network and
divided among the slices as $5$, $20$, and $5$. In the EDRL part, we consider
the $10$ actor as the population, with elite fraction $0.2$ and crossover and
mutation batch size $128$ and $256$, respectively. In the DRL part, we
implement the DDPG algorithm using Pytorch with three fully-connected layers
with $128$, $256$, $256$ neurons for actor and critic networks and a tanh
activation function. For all the models, the learning rate $10^{-4}$ and Adam
optimizer are considered. Table I summarizes the simulation parameters.

Figure 2: Performance comparison of EDRL and DRL algorithm.

Figure 2 compares the performance of the proposed EDRL algorithm with the DRL
algorithm as a baseline method. The cumulative rewards in Fig. 2 are measured
with $\gamma=0.95$. The results are displayed in each episode, for comparison
with conventional DRL and are averaged over a sufficient number of runs. The
results in Fig. 2 reveal that the EDRL approach can give up to a $62.2\%$
greater final return value than the DRL method and prove the efficacy of the
proposed EDRL system over the wireless environment. Fig. 2 also illustrates
that the suggested approach achieves faster convergence than the baselines,
that is due to the employment of a population-based method with $10$ actors,
who supply many valuable experiences of diverse modules to use in the training
of RL-based actors. Furthermore, as Fig. 2 indicates, the DRL method which has
better sample efficiency shows better performance at the early episodes of the
training process, where limited samples are available. After some episodes,
the joint DRL and population-based method EDRL, which has experienced more
samples in comparison to early episodes, provide more generality in the
training process and thus outperforms the DRL.

Figure 3 shows the Cumulative Distribution Functions (CDFs) of the achieved
QoS in each slice through the EDRL and DRL training process. The slices are
assumed to be the eMBB slice, MTC slice, and URLLC slice, in that order. Each
slice meets the service demands by considering a distinct QoS for specific
services, such as the average data rate in the eMBB slice in Fig. 3(a), the
capacity of the MTC slice in Fig. 3(b), and the maximum delay in URLLC slice
in Fig. 3(c). Considering an average data rate QoS metric in eMBB slices
guarantees a stable service for connected UEs. Similarly, the capacity as a
QoS parameter in MTC slices ensures that a high connection density is
supported. Furthermore, considering an exponential random variable with an
average size of $10$ Kb as packet length for URLLC slice and selecting maximum
delay as a QoS criterion ensures the lowest possible value for maximum latency
in these delay-sensitive services.

(a) Qos in slice 1

(b) Qos in slice 2

(c) Qos in slice 3

Figure 3: CDF of achieved QoS in each slice through EDRL and DRL training
process.

Figure 4 shows the per-user throughput in each slice of the simulation O-RAN
environment. The results presented in Fig. 4 were obtained during $100$
generation iterations by distributing $5$, $20$, and $5$ users to the slices,
respectively. As seen in Fig. 4, the network users follows the QoS demand in
Fig. 3. Furthermore, the results indicate that the Algorithm 1 was successful
in keeping the users’ throughput within the intended range.

(a) Slice 1 UEs

(b) Slice 2 UEs

(c) Slice 3 UEs

Figure 4: CDF of achieved throughput per users through EDRL and DRL training
process.

##  V Conclusion

In this paper, we have developed a novel O-RAN slicing framework over an
evolutionary-based DRL approach. To this end, we have formulated an
optimization problem that efficiently allocates the shared available RBs to
each O-RAN slice. To solve this problem, we have modeled the optimization
problem as an MDP, then by employing the desegregated modules in O-RAN
architecture, we develop a new EDRL algorithm to find an optimal policy for
allocating the available resources to distinct O-RAN slices. Accordingly,
utilizing the population experiences in the DRL training, the trained policy
is more general and robust in different traffic situations over the wireless
networks. The simulation results have shown up to $62.2\%$ improvements in
maximum rewards compared to the DRL baseline method. Further, the results have
highlighted the importance of utilizing different experiences and
generalization in policy training in dynamic wireless networks, and
demonstrated the efficiency of the proposed algorithm in presence of wireless
bandwidth constraints.

## References

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# Signatures of complex new physics in $b\to c\tau\bar{\nu}$ transitions

Suman Kumbhakar111Presently at Centre for High Energy Physics, Indian
Institute of Science Bangalore 560012<EMAIL_ADDRESS>Indian Institute of
Technology Bombay, Mumbai 400076, India

###### Abstract

The anomalies in the measurements of $R_{D}$ and $R_{D^{*}}$ continue to
provide motivation for physics beyond the Standard Model. In this work, we
assume the new physics Wilson coefficients to be complex and find their values
by doing a global fit to the present $b\rightarrow c\tau\bar{\nu}$ data. We
find that the number of allowed solutions depend on the choice of the upper
limit on $Br(B_{c}\rightarrow\tau\bar{\nu})$. We find that the forward-
backward asymmetries in $B\rightarrow(D,D^{*})\tau\bar{\nu}$ decays have the
capability to distinguish between different solutions. Further we calculate
the maximum values of CP violating triple product asymmetries in $B\to
D^{*}\tau\bar{\nu}$ decay allowed the current data. We observe that only one
of the three CP asymmetries can be enhanced up to a maximum value of $\sim
2-3\%$ whereas the other asymmetries remain smaller.

##  I Introduction

The heavy meson decays, in particular the $B$ meson decays, are a very fertile
ground to probe possible physics beyond the Standard Model (SM). In the past
few years, several measurements by BaBar, Belle and LHCb in the $B$ meson
decays show significant deviations from their SM predictions. One such class
of decays occurs through the charged current $b\rightarrow c\tau\bar{\nu}$
transition which is a tree level process in the SM. In this sector, two
interesting observables are

$R_{D}=\frac{\mathcal{B}(B\rightarrow
D\tau\bar{\nu})}{\mathcal{B}(B\rightarrow D\\{e/\mu\\}\bar{\nu})}\,,\quad
R_{D^{*}}=\frac{\mathcal{B}(B\rightarrow
D^{*}\tau\bar{\nu})}{\mathcal{B}(B\rightarrow D^{*}\\{e/\mu\\}\bar{\nu})}\,.$
(1)

These flavor ratios are consecutively measured by BaBar Lees:2012xj ;
Lees:2013uzd , Belle Huschle:2015rga ; Sato:2016svk ; Hirose:2016wfn ;
Abdesselam:2019dgh and LHCb Aaij:2015yra ; Aaij:2017uff ; Aaij:2017deq
collaborations. The SM predicts $R_{D}$ to be $0.299\pm 0.003$ whereas the
present experimental world average is $0.340\pm 0.027\,(\rm stat.)\pm
0.013\,(\rm syst.)$. For $R_{D^{*}}$, the SM prediction is $0.258\pm 0.005$
and the experimental world average is $0.295\pm 0.011\,(\rm stat.)\pm
0.008\,(\rm syst.)$. The SM predictions and the world averages are noted down
from Heavy Flavor Averaging Group Amhis:2019ckw . The present average values
of $R_{D}$ and $R_{D^{*}}$ exceed the SM predictions by $1.4\sigma$ and
$2.5\sigma$ respectively. Including the correlation of $-0.38$, the tension
between the measurements and the SM predictions is at the level of
$3.1\sigma$. This discrepancy is an indication of lepton flavor universality
(LFU) violation between $\tau$ and $\mu/e$ leptons.

In addition, the LHCb collaboration measured another flavor ratio
$R_{J/\psi}=\Gamma(B_{c}\rightarrow
J/\psi\tau\bar{\nu})/\Gamma(B_{c}\rightarrow J/\psi\mu\bar{\nu})$ whose value
is $0.71\pm 0.17\,(\rm stat.)\pm 0.18\,(\rm syst.)$ Aaij:2017tyk . Eventhough
the uncertainties are quite large, it is $1.7\sigma$ higher than its SM
prediction $0.289\pm 0.010$ Dutta:2017xmj . This is an additional hint of LFU
violation in the $b\rightarrow c\ell\bar{\nu}$ sector. These deviations could
be due to presence of new physics (NP) either in $b\rightarrow c\tau\bar{\nu}$
or in $b\rightarrow c\\{\mu,e\\}\bar{\nu}$ transition. However, it has been
shown in Refs. Alok:2017qsi ; Iguro:2020cpg that the latter possibility is
ruled out by other measurements. Therefore, we assume the presence of NP only
in $b\rightarrow c\tau\bar{\nu}$ transition.

Apart from these, Belle collaboration has measured two angular observables in
the $B\to D^{*}\tau\bar{\nu}$ decay $-$ (a) the $\tau$ polarization
$P^{D^{*}}_{\tau}$ and (b) the $D^{*}$ longitudinal polarization fraction
$F^{D^{*}}_{L}$. The measured values of these two quantities are
Hirose:2016wfn ; Abdesselam:2019wbt

$\displaystyle P^{D^{*}}_{\tau}$ $\displaystyle=$ $\displaystyle-0.38\pm
0.51\,(\rm stat.)^{+0.21}_{-0.16}\,(\rm syst.),$ (2) $\displaystyle
F^{D^{*}}_{L}$ $\displaystyle=$ $\displaystyle 0.60\pm 0.08\,(\rm stat.)\pm
0.04\,(\rm syst.).$ (3)

The measured value of $P_{\tau}^{D^{*}}$ is consistent with its SM prediction
of $-0.497\pm 0.013$ Tanaka:2012nw whereas for $F^{D^{*}}_{L}$ it is
$1.6\sigma$ higher than the SM prediction of $0.46\pm 0.04$ Alok:2016qyh .

The anomalies in $b\rightarrow c\tau\bar{\nu}$ transition have been studied in
various model independent techniques Jung:2018lfu ; Bhattacharya:2018kig ;
Hu:2018veh ; Alok:2019uqc ; Asadi:2019xrc ; Murgui:2019czp ; Bardhan:2019ljo ;
Blanke:2019qrx ; Shi:2019gxi ; Becirevic:2019tpx ; Sahoo:2019hbu ;
Cheung:2020sbq ; Cardozo:2020uol . The Wilson coefficients (WCs) of the NP
operators are determined by doing a fit to the data available in this sector
along with the constraint on the branching ratio of $B_{c}\to\tau\bar{\nu}$
decay. In Ref. Alok:2019uqc , it has been shown that the NP Lorentz structure
in form of $(V-A)\times(V-A)$ is the only one operator solution allowed by the
present data.

In this paper we do a global fit to of all present data on $b\rightarrow
c\tau\bar{\nu}$ transition by starting with a most general effective
Hamiltonian. Assuming the NP WCs to be complex, we find the allowed NP
solutions with their corresponding WCs. We show that one/two/three NP
solution(s) is (are) allowed if we make the three different choices on the
upper limits of $10\%$/$30\%$/$60\%$ on the branching ratio of
$B_{c}\rightarrow\tau\bar{\nu}$. We calculate the predictions of angular
observables in $B\rightarrow(D,D^{*})\tau\bar{\nu}$ decays and comment on
their ability to distinguish between the allowed solutions. Further, we
compute the predictions of the CP violating triple product asymmetries in
$B\to D^{*}\tau\bar{\nu}$ decay for the three NP solutions. We show that only
one of these three asymmetries can be enhanced up to a maximum of $\sim 2-3\%$
in the presence the allowed NP scenarios.

The paper is organized as follows. In Section II, we describe our methodology
for calculation and present our fit results. In this section, we calculate the
predictions of the angular observables of $B\rightarrow(D,D^{*})\tau\bar{\nu}$
decays and discuss their distinguishing capabilities. In section III, we
determine the maximum possible CP violating triple product asymmetries in
$B\to D^{*}\tau\bar{\nu}$ decay allowed by the current data. We present our
conclusions in section IV.

##  II Fit Methodology and Results

We start with the most general effective Hamiltonian for $b\rightarrow
c\tau\bar{\nu}$ transition which contains all possible Lorentz structures.
This is expressed as Freytsis:2015qca

$\mathcal{H}_{\rm
eff}=\frac{4G_{F}}{\sqrt{2}}V_{cb}\left[\mathcal{O}_{V_{L}}+\frac{\sqrt{2}}{4G_{F}V_{cb}}\frac{1}{\Lambda^{2}}\left\\{\sum_{i}\left(C_{i}\mathcal{O}_{i}+C^{{}^{\prime}}_{i}\mathcal{O}^{{}^{\prime}}_{i}+C^{{}^{\prime\prime}}_{i}\mathcal{O}^{{}^{\prime\prime}}_{i}\right)\right\\}\right],$
(4)

where $G_{F}$ is the Fermi coupling constant and $V_{cb}$ is the Cabibbo-
Kobayashi-Maskawa (CKM) matrix element. Here we assume that the neutrino is
left chiral. We also assume the new physics scale $\Lambda=1$ TeV. The five
unprimed operators

$\mathcal{O}_{V_{L}}=(\bar{c}\gamma_{\mu}P_{L}b)(\bar{\tau}\gamma^{\mu}P_{L}\nu)\
,\quad\mathcal{O}_{V_{R}}=(\bar{c}\gamma_{\mu}P_{R}b)(\bar{\tau}\gamma^{\mu}P_{L}\nu)\
,\quad$
$\mathcal{O}_{S_{L}}=(\bar{c}P_{L}b)(\bar{\tau}P_{L}\nu),\quad\mathcal{O}_{S_{R}}=(\bar{c}P_{R}b)(\bar{\tau}P_{L}\nu),\quad\mathcal{O}_{T}=(\bar{c}\sigma_{\mu\nu}P_{L}b)(\bar{\tau}\sigma^{\mu\nu}P_{L}\nu)\
,$ (5)

form the complete set of operators consistent with global baryon number and
lepton number conservation. The primed and double primed operators
$\mathcal{O}^{{}^{\prime}}_{i}$ and $\mathcal{O}^{{}^{\prime\prime}}_{i}$ only
arise in different Leptoquark models Freytsis:2015qca depending on their spin
and charge. A more rigorous discussion on all possible Leptoquarks can be
found in Ref. Dorsner:2016wpm . The Lorentz structures of all these operators
are described in Ref. Freytsis:2015qca . In particular,
$\mathcal{O}^{{}^{\prime}}_{i}$ and $\mathcal{O}^{{}^{\prime\prime}}_{i}$
operators can be expressed in terms of five unprimed operators using Fierz
identities. The constants $C_{i}$, $C^{{}^{\prime}}_{i}$ and
$C^{{}^{\prime\prime}}_{i}$ are the respective WCs of the NP operators in
which NP effects are hidden. In this analysis, we assume these NP WCs to be
complex.

Using the effective Hamiltonian given in Eq. (4), we calculate the expressions
of measured observables $R_{D}$, $R_{D^{*}}$, $R_{J/\psi}$, $P^{D^{*}}_{\tau}$
and $F^{D^{*}}_{L}$ as functions of the NP WCs. To obtain the values of NP
WCs, we do a fit of these expressions to the measured values of the
observables. In doing the fit, we take only one NP operator at a time. We
define the $\chi^{2}$ function as follows

$\chi^{2}(C_{i})=\sum_{R_{D},R_{D^{*}},R_{J/\psi},P_{\tau}^{D^{*}},F^{D^{*}}_{L}}\left(O^{th}(C_{i})-O^{exp}\right)\mathcal{C}^{-1}\left(O^{th}(C_{i})-O^{exp}\right),$
(6)

where $O^{th}(C_{i})$ are NP predictions of each observable and $O^{exp}$ are
the corresponding experimental central values. The $\mathcal{C}$ denotes the
covariance matrix which includes both theory and experimental correlations.

The $B\rightarrow(D,D^{*})\ell\bar{\nu}$ decay distributions depend upon
hadronic form-factors. The determination of these form-factors can be
calculated with the HQET techniques which are presently known at
$O(1/m_{b},1/m^{2}_{c},\alpha_{s})$. In this work we use the HQET form factors
in the form parametrized by Caprini et al. Caprini:1997mu . The parameters for
$B\rightarrow D$ decay are determined from the lattice QCD Aoki:2019cca
calculations and we use them in our analyses. For $B\rightarrow D^{*}$ decay,
the HQET parameters are extracted using data from Belle and BaBar experiments
along with the inputs from lattice. In this work, the numerical values of
these parameters are taken from refs. Bailey:2014tva and Amhis:2019ckw . The
form factors for $B_{c}\rightarrow J/\psi$ transition and their uncertainties
from ref. Wen-Fei:2013uea are used in the calculation of $R_{J/\psi}^{th}$.
These form factors are calculated in perturbative QCD framework.

To obtain the values of NP WCs, we minimize the $\chi^{2}$ function by taking
non-zero value of one NP WC at a time. While doing so, we set other
coefficients to be zero. This minimizations is performed by the CERN $\tt
MINUIT$ library James:1975dr ; James:1994vla . We find that the values of
$\chi^{2}_{\rm min}$ fall into two disjoint ranges $\lesssim 4.5$ and $\gtrsim
9$. We keep only those NP WCs which satisfy $\chi^{2}_{\rm min}\leq 4.5$. The
central values of these allowed WCs of NP solutions are listed in Table 1. We
do not provide the errors of individual best fit values because of the
correlation between the real and imaginary parts. In stead, we show the
$1\sigma$ allowed regions for theses NP solutions in Fig. 1.

NP type | Best fit value(s) | $\chi^{2}_{\rm min}$ | pull
---|---|---|---
$C_{V_{L}}$ | $0.10\pm 0.12\,i$ | $4.55$ | $4.1$
$C^{\prime}_{S_{L}}$ | $0.25\pm 0.86\,i$ | $4.50$ | $4.2$
$C^{\prime\prime}_{T}$ | $0.06\pm 0.09\,i$ | $3.45$ | $4.3$
$C_{S_{L}}$ | $-0.82\pm 0.45\,i$ | $2.50$ | $4.4$
Table 1: Best fit values of NP WCs at $\Lambda=1$ TeV for the measurements of $R_{D}$, $R_{D^{*}}$, $R_{J/\psi}$, $P^{D^{*}}_{\tau}$ and $F^{D^{*}}_{L}$. We list the central values of the NP solutions with $\chi^{2}_{\rm min}\leq 4.5$. For the SM, we have $\chi^{2}_{\rm SM}=21.80$. The pull values are calculated using pull = $\sqrt{\chi^{2}_{\rm SM}-\chi^{2}_{\rm min}}$. |
---|---
|
Figure 1: The allowed $1\sigma$ regions for the complex NP WCs listed in Table
1. For each plot, the blue colored region corresponds to the $1\sigma$
parameter space whereas the red dots represents the best fit values of NP WCs.

The purely leptonic decay $B_{c}\rightarrow\tau\bar{\nu}$ plays a crucial role
to constrain the NP solutions in this sector. This decay is subject to
helicity suppression in the SM whereas this suppression is removed for the
pseudo-scalar operators. Therefore, these NP operators are highly constrained
by this observable. Within the NP framework, the branching fraction of
$B_{c}\rightarrow\tau\bar{\nu}$ can be expressed as

$\displaystyle Br(B_{c}\rightarrow\tau\bar{\nu})$ $\displaystyle=$
$\displaystyle\frac{|V_{cb}|^{2}G^{2}_{F}f^{2}_{B_{c}}m_{B_{c}}m^{2}_{\tau}\tau^{exp}_{B_{c}}}{8\pi}\left(1-\frac{m^{2}_{\tau}}{m^{2}_{B_{c}}}\right)^{2}\times$
(7)
$\displaystyle\left|1+C_{V_{L}}-C_{V_{R}}+\frac{m^{2}_{B_{c}}}{m_{\tau}(m_{b}+m_{c})}(C_{S_{R}}-C_{S_{L}})\right|^{2},$

where the decay constant $f_{B_{c}}=434\pm 15$ MeV Colquhoun:2015oha and the
measured lifetime $\tau^{exp}_{B_{c}}=0.510\pm 0.009$ ps Tanabashi:2018oca .
Here $m_{b}$ and $m_{c}$ are the running quark masses evaluated at the
$\mu_{b}=m_{b}$ scale. The SM predicts this branching fraction to be $\sim
2.15\times 10^{-2}$.

In Ref. Akeroyd:2017mhr , the upper limit on this branching ratio is set to be
$10\%$ from the LEP data which are admixture of $B_{c}\to\tau\bar{\nu}$ and
$B_{u}\to\tau\bar{\nu}$ decays at $Z$ peak. To extract the
$Br(B_{c}\to\tau\bar{\nu})$, one needs to know the ratio of fragmentation
functions of $B_{c}$ and $B_{u}$ mesons defined as $f_{c}/f_{u}$. The value of
this ratio is obtained from the data of Tevatron Abe:1998fb ; Abulencia:2006zu
and LHCb Aaij:2014jxa . On the other hand, the authors of Ref. Alonso:2016oyd
obtained this upper limit to be $30\%$ by making use of the lifetime of
$B_{c}$ meson. This is estimated by considering that the
$B_{c}\to\tau\bar{\nu}$ decay rate does not exceed the fraction of the total
width which is allowed by the calculation of the lifetime in the SM. In Ref.
Blanke:2019qrx , the authors have argued that these two different upper limits
are too conservative and these could be over-estimated. However, taking all
uncertainties into account the decay width of $B_{c}$ meson can be relaxed up
to $60\%$ which is not that much conservative. Therefore, we consider these
three different upper limits on branching ratio of
$B_{c}\rightarrow\tau\bar{\nu}$ to constrain the NP parameter space. In this
analysis, the NP WCs are defined at a scale $\Lambda$= 1 TeV. However, all
these physical processes happen at $m_{b}$ scale. Therefore, we include the
renormalization group (RG) effects in the evolution of the WCs from the scale
of $1$ TeV to the $m_{b}$ scale Gonzalez-Alonso:2017iyc . In particular, these
effects are important for the scalar and tensor operators.

|
---|---
|
Figure 2: The $1\sigma$ regions allowed by $b\to c\tau\bar{\nu}$ data (blue)
and parameter spaces for three different upper limits $10\%$ (green), $30\%$
(yellow), $60\%$ (violet) of $Br(B_{c}\rightarrow\tau\bar{\nu})$ for each
complex NP WC listed in Table 1. In each plot, the red dots represent the best
fit points.

In Fig. 2, we have shown the parameter space which span $1\sigma$ region
allowed by present $b\rightarrow c\tau\bar{\nu}$ data and by the three
different upper limits on the branching ratio of
$B_{c}\rightarrow\tau\bar{\nu}$. The best fit point for each solution listed
in Table 1 is also plotted within the $1\sigma$ allowed region. Only the
$\mathcal{O}_{V_{L}}$ solution falls within the allowed space constrained by
$Br(B_{c}\rightarrow\tau\bar{\nu})<10\%$. The allowed $1\sigma$ regions for
$\mathcal{O}^{\prime}_{S_{L}}$ and $\mathcal{O}^{\prime\prime}_{T}$ solutions
fall into the regions allowed by the constraints
$Br(B_{c}\rightarrow\tau\bar{\nu})$ $<30\%$ and $<60\%$ respectively. The best
fit NP WCs of $\mathcal{O}_{S_{L}}$ solution do not fall into the region
allowed by the constraint $Br(B_{c}\rightarrow\tau\bar{\nu})<60\%$ whereas a
small fraction of the $1\sigma$ region overlaps with the region allowed by
$30\%<Br(B_{c}\rightarrow\tau\bar{\nu})<60\%$. Hence we can reject the mildly
allowed $\mathcal{O}_{S_{L}}$ solution. We list the final three allowed NP
solutions in Table 2.

Using the best fit values of the allowed solutions, we provide the predicted
central values of the quantities used in the fit, i.e.,
$R_{D},R_{D^{*}},R_{J/\psi}$, $P^{D^{*}}_{\tau}$ and $F_{L}^{D^{*}}$, for each
solution. This will allow us to see how close are the predictions of NP
solutions to the experimental measurements. We note the following observations
by looking at the predictions in Table 2:

  * •

The predictions of $R_{D}$, $R_{D^{*}}$ and $P^{D^{*}}_{\tau}$ for the three
solutions are within 1$\sigma$ of the respective experimental averages.

  * •

The predicted values of $R_{J/\psi}$ and $F^{D^{*}}_{L}$ for the three
solutions are within $\sim 1.6\sigma$ of the experimental measurements. The
Lorentz structure of $\mathcal{O}^{\prime}_{S_{L}}$ is different from that of
$\mathcal{O}_{V_{L}}$. But the prediction of $F^{D^{*}}_{L}$ for
$\mathcal{O}^{\prime}_{S_{L}}$ solution is the same as that of
$\mathcal{O}_{V_{L}}$ solution because the value of WC is quite small.
However, these two NP solutions fall in two different ranges of
$Br(B_{c}\to\tau\bar{\nu})$ because the helicity suppression is lifted in
presence of the $\mathcal{O}^{\prime}_{S_{L}}$ solution.

NP type | Best fit value(s) | $R_{D}$ | $R_{D^{*}}$ | $R_{J/\psi}$ | $P^{D^{*}}_{\tau}$ | $F^{D^{*}}_{L}$
---|---|---|---|---|---|---
SM | $C_{i}=0$ | $0.297\pm 0.008$ | $0.253\pm 0.002$ | $0.289\pm 0.008$ | $-0.499\pm 0.004$ | $0.457\pm 0.005$
$C_{V_{L}}|_{10\%}$ | $0.10\pm 0.12\,i$ | $0.364\pm 0.010$ | $0.294\pm 0.005$ | $0.334\pm 0.010$ | $-0.499\pm 0.005$ | $0.443\pm 0.007$
$C^{\prime}_{S_{L}}|_{30\%}$ | $0.25\pm 0.86\,i$ | $0.336\pm 0.009$ | $0.295\pm 0.005$ | $0.339\pm 0.011$ | $-0.419\pm 0.006$ | $0.443\pm 0.007$
$C^{\prime\prime}_{T}|_{60\%}$ | $0.06\pm 0.09\,i$ | $0.333\pm 0.010$ | $0.296\pm 0.006$ | $0.344\pm 0.009$ | $-0.375\pm 0.005$ | $0.420\pm 0.006$
Table 2: Central values of best fit NP WCs at $\Lambda=1$ TeV by making use of data of $R_{D}$, $R_{D^{*}}$, $R_{J/\psi}$, $P^{D^{*}}_{\tau}$ and $F^{D^{*}}_{L}$. Here we allow only those solutions for which $\chi^{2}_{\rm min}\leq 4.5$ as well as for three different upper limits $10\%$, $30\%$ and $60\%$ of $Br(B_{c}\rightarrow\tau\bar{\nu})$. We also provide the predictions of each observables which are taken into the fit. NP type | Best fit value(s) | $P^{D}_{\tau}$ | $A^{D}_{FB}$ | $A^{D^{*}}_{FB}$
---|---|---|---|---
SM | $C_{i}=0$ | $0.325\pm 0.001$ | $0.360\pm 0.002$ | $-0.063\pm 0.005$
$C_{V_{L}}|_{10\%}$ | $0.10\pm 0.12\,i$ | $0.325\pm 0.002$ | $0.360\pm 0.002$ | $-0.063\pm 0.006$
$C^{\prime}_{S_{L}}|_{30\%}$ | $0.25\pm 0.86\,i$ | $0.420\pm 0.001$ | $0.212\pm 0.003$ | $0.0001\pm 0.005$
$C^{\prime\prime}_{T}|_{60\%}$ | $0.06\pm 0.09\,i$ | $0.414\pm 0.002$ | $0.100\pm 0.004$ | $0.009\pm 0.006$
Table 3: Average values of angular observables $P^{D}_{\tau}$, $A^{D}_{FB}$
and $A^{D^{*}}_{FB}$ for the SM and three solutions listed in Table 2.

We consider other angular observables in $B\to(D,D^{*})\tau\bar{\nu}$ decay
which are yet to be measured. In particular, we are interested in the
following three observables Alok:2018uft

  * •

The polarization of $\tau$ lepton in $B\to D\tau\bar{\nu}$ decay,
$P^{D}_{\tau}$

  * •

The forward-backward asymmetry in $B\to D\tau\bar{\nu}$ decay, $A^{D}_{FB}$
and

  * •

The forward-backward asymmetry in $B\to D^{*}\tau\bar{\nu}$ decay,
$A^{D^{*}}_{FB}$.

|
---|---

Figure 3: The predictions of angular observables $P^{D}_{\tau}$, $A^{D}_{FB}$
and $A^{D^{*}}_{FB}$ as a function of $q^{2}$ (GeV2) for the SM and three
solutions listed in Table 3. The color code for each case is shown in each
plot.

We compute the average values of these three angular observables for the
allowed NP solutions. The predicted values are listed in Table 3. For
completeness, we also plot these observables as a function of
$q^{2}=(p_{B}-p_{D^{(*)}})^{2}$, where $p_{B}$ and $p_{D^{(*)}}$ are the
respective four momenta of $B$ and $D^{(*)}$ mesons. These are shown in Fig.
3. From Table 3 and Fig. 3, we observe the following features

  * •

The predictions of all three observables for the $\mathcal{O}_{V_{L}}$
solution are exactly same as those of the SM. This is because the Lorentz
structure of $\mathcal{O}_{V_{L}}$ operator is same as the SM.

  * •

The $P^{D}_{\tau}$ has very poor discriminating capability.

  * •

The predictions of $A^{D}_{FB}$ and $A^{D^{*}}_{FB}$ for the
$\mathcal{O}^{\prime}_{S_{L}}$ and $\mathcal{O}^{\prime\prime}_{T}$ solutions
are markedly different. These two solutions can be distinguished by forward-
backward asymmetries.

##  III CP violating triple product asymmetries

If the hints of LFU violation in $b\to c\tau\bar{\nu}$ sector is indeed due to
new physics, then it is very likely that the new physics will contain
additional phases which can lead to some signatures of CP violation in the
relevant decay modes. In this section, we discuss about the possible CP
violation in $B\to D^{*}\tau\bar{\nu}$ decay. The simplest possible CP
violating observable, which one could think of, is the direct CP asymmetry
between the decay and its CP conjugate mode. In order to have a non-zero value
of direct CP asymmetry, we need strong phase difference between the amplitudes
besides the weak phase. For $B\to D^{*}\tau\bar{\nu}$ decay, there is no
strong phase difference in the SM because of unique final state of the decay
and its CP conjugate mode. In Ref. Aloni:2018ipm , the authors suggested a
mechanism where this strong phase difference could arise due to interference
between the higher resonances of $D^{*}$ meson. They have shown that the CP
violation could be as large as $\sim 10\%$ only for the tensor NP. However,
the tensor NP is now ruled out by the Belle measurement on $F^{D^{*}}_{L}$.

In this work, we focus on CP violating triple product asymmetries (TPA) in
$B\to D^{*}\tau\bar{\nu}$ decay. The full angular distribution of quasi-four
body decay $B\to D^{*}(\to D\pi)\tau\bar{\nu}$ can be described by four
independent parameters $-$ (a) $q^{2}=(p_{B}-p_{D^{*}})^{2}$ where $p_{B}$ and
$p_{D^{*}}$ are respective four momenta of $B$ and $D^{*}$ meson, (b)
$\theta_{D}$ the angle between $B$ and $D$ mesons where $D$ meson comes from
$D^{*}$ decay, (c) $\theta_{\tau}$ the angle between $\tau$ momenta and $B$
meson, and (d) $\phi$ the angle between $D^{*}$ decay plane and the plane
defined by the $\tau$ and $\nu$ momenta Duraisamy:2014sna . The triple
products (TP) are obtained by integrating the full decay distribution in
different ranges of the polar angles $\theta_{D}$ and $\theta_{\tau}$. These
are following Alok:2011gv ; Duraisamy:2013kcw ; Bhattacharya:2019olg ;
Bhattacharya:2020lfm

$\displaystyle\frac{d^{2}\Gamma^{(1)}}{dq^{2}d\phi}$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}\int^{1}_{-1}\frac{d^{4}\Gamma}{dq^{2}d\cos\theta_{\tau}d\cos\theta_{D}d\phi}d\cos\theta_{\tau}d\cos\theta_{D}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{d\Gamma}{dq^{2}}\left[1+\left(A^{(1)}_{C}\,\cos
2\phi+A^{(1)}_{T}\,\sin 2\phi\right)\right],$
$\displaystyle\frac{d^{2}\Gamma^{(2)}}{dq^{2}d\phi}$ $\displaystyle=$
$\displaystyle\int^{1}_{-1}d\cos\theta_{\tau}\left[\int^{1}_{0}-\int^{0}_{-1}\right]\frac{d^{4}\Gamma}{dq^{2}d\cos\theta_{\tau}d\cos\theta_{D}d\phi}d\cos\theta_{D}$
(9) $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{d\Gamma}{dq^{2}}\left[A^{(2)}_{C}\,\cos\phi+A^{(2)}_{T}\,\sin\phi\right],$

and

$\displaystyle\frac{d^{2}\Gamma^{(3)}}{dq^{2}d\phi}$ $\displaystyle=$
$\displaystyle\left[\int^{1}_{0}-\int^{0}_{-1}\right]d\cos\theta_{\tau}\left[\int^{1}_{0}-\int^{0}_{-1}\right]\frac{d^{4}\Gamma}{dq^{2}d\cos\theta_{\tau}d\cos\theta_{D}d\phi}d\cos\theta_{D}$
(10) $\displaystyle=$
$\displaystyle\frac{2}{3\pi}\frac{d\Gamma}{dq^{2}}\left[A^{(3)}_{C}\,\cos\phi+A^{(3)}_{T}\,\sin\phi\right].$

The coefficients $A^{(i)}_{C}$ of $\cos\phi$ and $\cos 2\phi$ are even under
CP transformation and hence we are not interested in these. However, the
angular coefficients $A^{(i)}_{T}$ of $\sin\phi$ and $\sin 2\phi$ are odd
under the CP transformation which leads to these quantities to be CP violating
observables. These three TPs are defined as follows Duraisamy:2013kcw :

$A^{(1)}_{T}(q^{2})=\frac{4V^{T}_{5}}{A_{L}+A_{T}},\quad
A^{(2)}_{T}(q^{2})=\frac{V^{0T}_{3}}{A_{L}+A_{T}},\quad
A^{(3)}_{T}(q^{2})=\frac{V^{0T}_{4}}{A_{L}+A_{T}},$ (11)

where $V$’s are the angular coefficients and $A_{L}$ and $A_{T}$ are the
longitudinal and transverse amplitudes respectively. The expressions for these
quantities are given in Appendix A and also can be found in Ref.
Duraisamy:2014sna . The SM predictions of these TPs are almost zero.
Therefore, the complex NP WCs can predict a non-zero value for these
quantities. Thus these TPs provide a new degree of freedom to test beyond SM
physics. For the CP conjugate decay, the definitions in Eq. (11) take the
following forms

$\bar{A}^{(1)}_{T}(q^{2})=-\frac{4\bar{V}^{T}_{5}}{\bar{A}_{L}+\bar{A}_{T}},\quad\bar{A}^{(2)}_{T}(q^{2})=\frac{\bar{V}^{0T}_{3}}{\bar{A}_{L}+\bar{A}_{T}},\quad\bar{A}^{(3)}_{T}(q^{2})=-\frac{\bar{V}^{0T}_{4}}{\bar{A}_{L}+\bar{A}_{T}}.$
(12)

Using Eqs. (11) and (12), three asymmetries can be defined between the
corresponding TPs of the decay and its CP conjugate. These TPAs are defined as
follows

$\displaystyle\langle A^{(1)}_{T}(q^{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(A^{(1)}_{T}(q^{2})+\bar{A}^{(1)}_{T}(q^{2})\right),$
$\displaystyle\langle A^{(2)}_{T}(q^{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(A^{(2)}_{T}(q^{2})-\bar{A}^{(2)}_{T}(q^{2})\right),$
$\displaystyle\langle A^{(3)}_{T}(q^{2})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(A^{(3)}_{T}(q^{2})+\bar{A}^{(3)}_{T}(q^{2})\right).$
(13)

|
---|---

Figure 4: The TPAs are plotted as a function of $q^{2}$ (GeV2) for the SM and
three best fit NP WCs listed in Table 2. The color code for each plot is shown
in figure.

First we calculate the predictions of these TPAs for the SM and the three best
fit NP solutions listed in Table 2 as a function of $q^{2}$. These predictions
are shown in Fig. 4. From this figure, we make the following observations

  * •

The TPAs $\langle A^{(1)}_{T}(q^{2})\rangle$ and $\langle
A^{(3)}_{T}(q^{2})\rangle$ depend only on the $\mathcal{O}_{V_{L}}$ and
$\mathcal{O}_{V_{R}}$ operators. The $\mathcal{O}_{V_{L}}$ has the same
Lorentz structure as the SM. Therefore, the $\mathcal{O}_{V_{L}}$ solution
predicts these two asymmetries to be zero for whole $q^{2}$ range. For other
two NP solutions, the predictions are zero because these two asymmetries do
not depend on those NP WCs.

  * •

The TPA $\langle A^{(2)}_{T}(q^{2})\rangle$ depends on $\mathcal{O}_{V_{L}}$,
$\mathcal{O}_{V_{R}}$, $\mathcal{O}_{S_{L}}$, $\mathcal{O}_{S_{R}}$ and
$\mathcal{O}_{T}$ operators. The $\mathcal{O}_{V_{L}}$ operator has the same
Lorentz structure as the SM. Hence, the prediction of this TPA is zero for the
$\mathcal{O}_{V_{L}}$ solution for whole $q^{2}$ range. The
$\mathcal{O}^{\prime}_{S_{L}}$ and $\mathcal{O}^{\prime\prime}_{T}$ operators
are linear combinations of $\mathcal{O}_{S_{L}}$ and $\mathcal{O}_{T}$.
Therefore, we get some non-zero value of this TPA for these two solutions. For
the $\mathcal{O}^{\prime}_{S_{L}}$ solution, $\langle
A^{(2)}_{T}(q^{2})\rangle$ reaches a maximum value of $\sim 0.7\%$ at
$q^{2}\simeq 6$ GeV2 and decreases to zero at $q^{2}_{\rm max}$. For the
$\mathcal{O}^{\prime\prime}_{T}$ solution, $\langle A^{(2)}_{T}(q^{2})\rangle$
reaches a maximum value of $\sim 1.7\%$ at $q^{2}\simeq 5.4$ GeV2 and
decreases to zero at $q^{2}_{\rm max}$.

|
---|---
Figure 5: The second TPA is plotted as a function of $q^{2}$ (GeV2) for three
benchmark NP WCs $C^{\prime}_{S_{L}}=0.24+i$ (blue curve),
$C^{\prime\prime}_{T}=0.06+0.098i$ (black curve) and $C_{S_{L}}=-0.35-0.60i$
(red curve).

Our next aim is to compute the maximum CP violation allowed by the present
$b\to c\tau\bar{\nu}$ data. To calculate this, we choose a benchmark point
from the $1\sigma$ allowed parameter space of each NP solution. From Fig. 4,
we have learned that for any complex value of $C_{V_{L}}$ three TPAs lead to
zero. Only the second TPA $\langle A^{(2)}_{T}(q^{2})\rangle$ is non-zero for
the $\mathcal{O}^{\prime}_{S_{L}}$ and $\mathcal{O}^{\prime\prime}_{T}$
solutions. Therefore, we pick a benchmark points from Fig 1 for each of these
two solutions. These points are $C^{\prime}_{S_{L}}=0.24\pm i$ and
$C^{\prime\prime}_{T}=0.06+0.098i$ , which can lead to the maximum value of
the TPA $\langle A^{(2)}_{T}(q^{2})\rangle$ in $B\to D^{*}\tau\bar{\nu}$
decay. In the left panel of Fig. 5, we plot the TPA $\langle
A^{(2)}_{T}(q^{2})\rangle$ as a function of $q^{2}$ for these two benchmark
points of $\mathcal{O}^{\prime}_{S_{L}}$ and $\mathcal{O}^{\prime\prime}_{T}$
solutions. From this plot, we observe that it has almost same features which
are obtained from the plot of $\langle A^{(2)}_{T}(q^{2})\rangle$ in Fig 4. We
have not got much larger value of TPA $\langle A^{(2)}_{T}(q^{2})\rangle$ than
what we got for the best fit NP solutions.

As per discussion in Sec II, the $\mathcal{O}_{S_{L}}$ solution listed in
Table 1 is marginally disfavored because the best fit values of $C_{S_{L}}$
does not satisfy the constraint of $Br(B_{c}\to\tau\bar{\nu})<60\%$. However,
a small fraction of the $1\sigma$ region of this solution falls on the region
spanned by the constraint $30\%<Br(B_{c}\to\tau\bar{\nu})<60\%$. For
completeness, we calculate the predictions of TPAs for this solution. We can
get a allowed value of $C_{S_{L}}$ which can give to maximum possible TPA for
the $\langle A^{(2)}_{T}(q^{2})\rangle$. We choose a benchmark point
$C_{S_{L}}=-0.35-0.60i$ from the allowed region and calculate the second TPA.
In right panel of Fig. 5, we plot $\langle A^{(2)}_{T}(q^{2})\rangle$ as a
function of $q^{2}$ for the benchmark point of $C_{S_{L}}$. From this plot, we
observe that the second TPA reaches a maximum value of $\sim 2.6\%$ at
$q^{2}\simeq 5$ GeV2 and decreases to zero at $q^{2}_{\rm max}$. In fact, this
is the maximum value of $\langle A^{(2)}_{T}(q^{2})\rangle$ predicted by the
scalar operator solution among all the predictions made by allowed NP
solutions.

##  IV Conclusions

In this work, we have done a global fit of $b\to c\tau\bar{\nu}$ data assuming
NP WCs to complex. We find that the $\mathcal{O}_{V_{L}}$ solution is the only
NP solution allowed by the constraint $Br(B_{c}\to\tau\bar{\nu})<10\%$. If we
relax the constraint to $30\%$ or $60\%$, then we get one or two additional
allowed NP solutions. We calculate the predictions of angular observables in
$B\to(D,D^{*})\tau\bar{\nu}$ decays. We find that the forward-backward
asymmetries in these two decays are quite useful to distinguish the two
solutions other than the $\mathcal{O}_{V_{L}}$ solution.

We then compute the maximum values of CP violating TPAs in $B\to
D^{*}\tau\bar{\nu}$ decay for the allowed NP solutions. These TPAs are zero in
the SM. Hence any non-zero measurement of these quantities would give a
smoking gun signal of physics beyond SM. Here we find that the predictions of
first and third TPAs are zero for all NP solutions whereas the second TPA
reaches a maximum value of $\sim 1.9\%$ for the $\mathcal{O}^{\prime}_{S_{L}}$
solution and $\sim 0.9\%$ for the $\mathcal{O}^{\prime\prime}_{T}$ solution.
The mildly favored NP solution $\mathcal{O}_{S_{L}}$ predicts a maximum value
of $\sim 2.6\%$ for the second TPA which is the maximum predicted value among
all the NP predictions.

To measure the angular observables and TPAs, the reconstruction of the $\tau$
lepton momentum is crucial. This is quite difficult because of the missing
neutrinos. The LHCb collaboration has already made a fair attempt to
reconstruct the $\tau$ lepton through
$\tau^{-}\to\pi^{-}\pi^{+}\pi^{-}(n\pi^{0})\nu_{\tau}$ decay channel
Aaij:2017uff . However, in case of Belle II, it is very hard to reconstruct
the $\tau$ momentum through leptonic decay
$\tau^{-}\to\ell^{-}\nu_{\tau}\bar{\nu_{\ell}}$ because of multiple neutrinos
in the final state. Thus, LHCb may be able to measure $\theta_{\tau}$ and
$\phi$ with a better precision than Belle II and this could lead to a null
test of the TPAs. We hope LHCb would be able to overcome this challenge in the
near future Cerri:2018ypt . Recently in Ref. Marangotto:2018pbs , the author
discussed an outline to measure the full angular distribution and the CP
violating TPAs for $B\to D^{*}\ell\bar{\nu}$ decays at the collider
experiments.

## Acknowledgements

We would like to thank the organizers of WHEPP 2019 at IIT Guwahati, where
this work had been initiated. We thank Amarjit Soni for useful suggestions at
WHEPP. We also thank S. Uma Sankar for useful discussions and for careful
reading of the manuscript.

##  Appendix A Angular Coefficients

The total longitudinal and transverse amplitudes are defined as
Duraisamy:2014sna

$A_{L}=\left(V_{1}^{0}-\frac{1}{3}V_{2}^{0}\right),\quad
A_{T}=2\left(V_{1}^{T}-\frac{1}{3}V_{2}^{T}\right).$ (14)

The longitudinal coefficients $V^{0}_{1}$ and $V^{0}_{2}$ are written as

$\displaystyle V_{1}^{0}$ $\displaystyle=$ $\displaystyle
2\left[\left(1+\frac{m^{2}_{\tau}}{q^{2}}\right)\left(|\mathcal{A}_{0}|^{2}+16|\mathcal{A}_{0T}|^{2}\right)+\frac{2m^{2}_{\tau}}{q^{2}}|\mathcal{A}_{tP}|^{2}-\frac{16m_{\tau}}{\sqrt{q^{2}}}{\rm
Re}\left[\mathcal{A}_{0T}\mathcal{A}^{*}_{0}\right]\right],$ $\displaystyle
V_{2}^{0}$ $\displaystyle=$ $\displaystyle
2\left(1-\frac{m^{2}_{\tau}}{q^{2}}\right)\left[-|\mathcal{A}_{0}|^{2}+16|\mathcal{A}_{0T}|^{2}\right],$
(15)

and the transverse coefficients $V^{T}_{1}$, $V^{T}_{2}$ and $V^{T}_{5}$ are
given by

$\displaystyle V_{1}^{T}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(3+\frac{m^{2}_{\tau}}{q^{2}}\right)\left(|\mathcal{A}_{\parallel}|^{2}+|\mathcal{A}_{\perp}|^{2}\right)+8\left(1+\frac{3m^{2}_{\tau}}{q^{2}}\right)\left(|\mathcal{A}_{\parallel
T}|^{2}+|\mathcal{A}_{\perp
T}|^{2}\right)-\frac{16m^{2}_{\tau}}{\sqrt{q^{2}}}{\rm
Re}\left[\mathcal{A}_{\parallel
T}\mathcal{A}^{*}_{\parallel}+\mathcal{A}_{\perp
T}\mathcal{A}^{*}_{\perp}\right],$ $\displaystyle V_{2}^{T}$ $\displaystyle=$
$\displaystyle\left(1-\frac{m^{2}_{\tau}}{q^{2}}\right)\left[\frac{1}{2}\left(|\mathcal{A}_{\parallel}|^{2}+|\mathcal{A}_{\perp}|^{2}\right)-8\left(|\mathcal{A}_{\parallel
T}|^{2}+|\mathcal{A}_{\perp T}|^{2}\right)\right],$ $\displaystyle V_{5}^{T}$
$\displaystyle=$ $\displaystyle 2\left(1-\frac{m^{2}_{\tau}}{q^{2}}\right){\rm
Im}\left[\mathcal{A}_{\parallel}\mathcal{A}^{*}_{\perp}\right].$ (16)

The expressions for mixed angular coefficients $V^{0T}_{3}$ and $V^{0T}_{4}$
are given by

$\displaystyle V^{0T}_{3}$ $\displaystyle=$ $\displaystyle 2\sqrt{2}\,{\rm
Im}\left[-\mathcal{A}_{\parallel}\mathcal{A}^{*}_{0}+\frac{m^{2}_{\tau}}{q^{2}}\mathcal{A}_{\perp}\mathcal{A}^{*}_{tP}+\frac{4m_{\tau}}{\sqrt{q^{2}}}\left(\mathcal{A}_{0T}\mathcal{A}^{*}_{\parallel}-\mathcal{A}_{\parallel
T}\mathcal{A}^{*}_{0}+\mathcal{A}_{\perp
T}\mathcal{A}^{*}_{tP}\right)\right],$ $\displaystyle V^{0T}_{4}$
$\displaystyle=$
$\displaystyle\sqrt{2}\left(1-\frac{m^{2}_{\tau}}{q^{2}}\,{\rm
Im}\left[\mathcal{A}_{\perp}\mathcal{A}^{*}_{0}\right]\right).$ (17)

The corresponding hadronics matrix elements are expressed as

$\displaystyle\mathcal{A}_{0}$ $\displaystyle=$
$\displaystyle\frac{m_{B}+m_{D^{*}}}{2m_{D^{*}}\sqrt{q^{2}}}\left[\left(m^{2}_{B}-m^{2}_{D^{*}}-q^{2}\right)A_{1}(q^{2})-\frac{\lambda_{D^{*}}}{(m_{B}+m_{D^{*}})^{2}}A_{2}(q^{2})\right](1+C_{V_{L}}-C_{V_{R}}),$
$\displaystyle\mathcal{A}_{\pm}$ $\displaystyle=$
$\displaystyle(m_{B}+m_{D^{*}})A_{1}(q^{2})\left(1+C_{V_{L}}-C_{V_{R}}\right)\mp\frac{\sqrt{\lambda_{D^{*}}}}{(m_{B}+m_{D^{*}})}V(q^{2})\left(1+C_{V_{L}}+C_{V_{R}}\right),$
$\displaystyle\mathcal{A}_{t}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda_{D^{*}}}}{\sqrt{q^{2}}}A_{0}(q^{2})(1+C_{V_{L}}-C_{V_{R}}),$
$\displaystyle\mathcal{A}_{P}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda_{D^{*}}}}{m_{b}+m_{c}}A_{0}(q^{2})(C_{S_{R}}-C_{S_{L}}),$
$\displaystyle\mathcal{A}_{0T}$ $\displaystyle=$
$\displaystyle\frac{C_{T}}{2m_{D^{*}}}\left[\left(m^{2}_{B}+3m^{2}_{D^{*}}-q^{2}\right)T_{2}(q^{2})-\frac{\lambda_{D^{*}}}{m^{2}_{B}-m^{2}_{D^{*}}}T_{3}(q^{2})\right],$
$\displaystyle\mathcal{A}_{\pm T}$ $\displaystyle=$ $\displaystyle
C_{T}\left[\frac{m^{2}_{B}-m^{2}_{D^{*}}}{\sqrt{q^{2}}}T_{2}(q^{2})\pm\sqrt{\frac{\lambda_{D^{*}}}{q^{2}}}T_{1}(q^{2})\right].$
(18)

Further the transversity amplitudes can be defined as

$\mathcal{A}_{\parallel(T)}=\frac{1}{\sqrt{2}}\left(\mathcal{A}_{+(+T)}+\mathcal{A}_{-(-T)}\right),\quad\mathcal{A}_{\perp(T)}=\frac{1}{\sqrt{2}}\left(\mathcal{A}_{+(+T)}-\mathcal{A}_{-(-T)}\right).$
(19)

The amplitude $\mathcal{A}_{tP}$ is a combination of $t$ and $P$ amplitudes
which is given by

$\mathcal{A}_{tP}=\left(A_{t}+\frac{\sqrt{q^{2}}}{m_{\tau}}\mathcal{A}_{P}\right)$
(20)

All the above expressions for the angular coefficients and hadronic amplitudes
are taken from the Ref. Duraisamy:2014sna . The form factors appeared in the
hadronic amplitudes $V(q^{2})$, $A_{0,1,2}(q^{2})$ and $T_{1,2,3}(q^{2})$ are
calculated in HQET parametrization Caprini:1997mu and their expressions can be
also found in Ref. Sakaki:2013bfa .

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