${\widehat{\tau}}^{\mathrm{(X,nv)}}_{k}(\boldsymbol{x})=\frac{\overline{r}(t_{k},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}{\widehat{\tau}}^{(k)}(\boldsymbol{x})+\frac{\overline{r}(t_{0},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}{\widehat{\tau}}^{(0)}(\boldsymbol{x}).$
(102)

where the estimators ${\widehat{\tau}}^{(k)}$ and ${\widehat{\tau}}^{(0)}$ are
built respectively on $\mathbf{S}_{t_{k}}$ and $\mathbf{S}_{t_{0}}$ by
regressing
$({D}_{i}^{(k)})_{i\in\mathbf{S}_{t_{k}}}=(Y_{i}(t_{k})-\widehat{\mu}_{t_{0}}(\boldsymbol{X}_{i}))_{i\in\mathbf{S}_{t_{k}}}$
and
$({D}_{i}^{(0)})_{i\in\mathbf{S}_{t_{0}}}=(\widehat{\mu}_{t_{k}}(\boldsymbol{X}_{i})-Y_{i}(t_{0}))_{i\in\mathbf{S}_{t_{0}}}$
on $\boldsymbol{X}$. In the following, we denote
${\widehat{\tau}}^{(k)}(\boldsymbol{x})=\boldsymbol{f}(\boldsymbol{x})^{\top}\widehat{\boldsymbol{\beta}}^{(k)}$
and
${\widehat{\tau}}^{(0)}(\boldsymbol{x})=\boldsymbol{f}(\boldsymbol{x})^{\top}\widehat{\boldsymbol{\beta}}^{(0)}$.
Here, $\widehat{\boldsymbol{\beta}}^{(k)}$ denotes the OLS estimator of
${\widehat{\tau}}^{(k)}$ and is given by:

$\displaystyle\widehat{\boldsymbol{\beta}}^{(k)}$
$\displaystyle=\big{(}\mathbf{H}_{k}^{\top}\mathbf{H}_{k}\big{)}^{-1}\mathbf{H}_{k}^{\top}\big{(}Y_{\mathrm{obs,i}}-\widehat{\mu}_{t_{0}}(\boldsymbol{X}_{i})\big{)}_{i\in\mathbf{S}_{t_{k}}}$
(103)
$\displaystyle=\big{(}\mathbf{H}_{k}^{\top}\mathbf{H}_{k}\big{)}^{-1}\mathbf{H}_{k}^{\top}\big{(}Y_{\mathrm{obs,i}}-\boldsymbol{f}(\boldsymbol{X}_{i})^{\top}\widehat{\boldsymbol{\beta}}_{t_{0}}\big{)}_{i\in\mathbf{S}_{t_{k}}}$
$\displaystyle=\big{(}\mathbf{H}_{k}^{\top}\mathbf{H}_{k}\big{)}^{-1}\mathbf{H}_{k}^{\top}\boldsymbol{y}_{k}-\big{(}\mathbf{H}_{k}^{\top}\mathbf{H}_{k}\big{)}^{-1}\mathbf{H}_{k}^{\top}\mathbf{H}_{k}\widehat{\boldsymbol{\beta}}_{t_{0}}$
$\displaystyle=\widehat{\boldsymbol{\beta}}_{t_{k}}-\widehat{\boldsymbol{\beta}}_{t_{0}}={\widehat{\boldsymbol{\beta}}}_{k},$

where
${\widehat{\boldsymbol{\beta}}}_{k}={\widehat{\boldsymbol{\beta}}}_{t_{k}}-{\widehat{\boldsymbol{\beta}}}_{t_{0}}$
is the T-learner OLS estimator as given in (97).

By similar calculus, we show that

$\displaystyle\widehat{\boldsymbol{\beta}}^{(0)}$
$\displaystyle=\big{(}\mathbf{H}_{0}^{\top}\mathbf{H}_{0}\big{)}^{-1}\mathbf{H}_{0}^{\top}\big{(}\widehat{\mu}_{t_{k}}(\boldsymbol{X}_{i})-Y_{\mathrm{obs,i}}\big{)}_{i\in\mathbf{S}_{t_{0}}}$
(104)
$\displaystyle=\big{(}\mathbf{H}_{0}^{\top}\mathbf{H}_{0}\big{)}^{-1}\mathbf{H}_{0}^{\top}\big{(}\boldsymbol{f}(\boldsymbol{X}_{i})^{\top}\widehat{\boldsymbol{\beta}}_{t_{k}}-Y_{\mathrm{obs,i}}\big{)}_{i\in\mathbf{S}_{t_{0}}}$
$\displaystyle=\big{(}\mathbf{H}_{0}^{\top}\mathbf{H}_{0}\big{)}^{-1}\mathbf{H}_{0}^{\top}\mathbf{H}_{0}\widehat{\boldsymbol{\beta}}_{t_{k}}-\big{(}\mathbf{H}_{0}^{\top}\mathbf{H}_{0}\big{)}^{-1}\mathbf{H}_{0}^{\top}\boldsymbol{y}_{0}$
$\displaystyle=\widehat{\boldsymbol{\beta}}_{t_{k}}-\widehat{\boldsymbol{\beta}}_{t_{0}}={\widehat{\boldsymbol{\beta}}}_{k}.$

It results that

$\displaystyle{\widehat{\tau}}^{\mathrm{(X,nv)}}_{k}(\boldsymbol{x})$
$\displaystyle=\frac{\overline{r}(t_{k},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}\boldsymbol{f}(\boldsymbol{x})^{\top}\widehat{\boldsymbol{\beta}}^{(k)}+\frac{\overline{r}(t_{0},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}\boldsymbol{f}(\boldsymbol{x})^{\top}\widehat{\boldsymbol{\beta}}^{(0)}$
(105)
$\displaystyle=\Big{(}\frac{\overline{r}(t_{k},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}+\frac{\overline{r}(t_{0},\boldsymbol{x})}{\overline{r}(t_{k},\boldsymbol{x})+\overline{r}(t_{0},\boldsymbol{x})}\Big{)}\boldsymbol{f}(\boldsymbol{x})^{\top}{\widehat{\boldsymbol{\beta}}}_{k}=\boldsymbol{f}(\boldsymbol{x})^{\top}{\widehat{\boldsymbol{\beta}}}_{k}$

In the end, the naive X-learner is no more than a simple T-learner, the
error’s upper bound of the naive X-learner is given therefore by:

$\sigma^{2}\Big{(}\frac{1}{\rho(t_{k})}+\frac{1}{\rho(t_{0})}\Big{)}.$ (106)

##  Appendix C Discussion about the binarized R-learner.

Another alternative to R-learning to continuous treatments is proposed by
(Kaddour et al., 2021). The approach considers both Assumptions 5.1 and 5.2 on
the outcome
$Y(t)=\boldsymbol{f}(\boldsymbol{X})^{\top}\boldsymbol{\beta}_{t}+\varepsilon(t)$,
then establishes the binarized (Robinson, 1988) decomposition such that

$Y_{\mathrm{obs}}-m(\boldsymbol{X})=\boldsymbol{f}(\boldsymbol{X})^{\top}(\boldsymbol{\beta}_{T}-e^{\boldsymbol{\beta}}(\boldsymbol{X}))+\epsilon,$
(107)

where $\epsilon=\varepsilon(T)$,
$m(\boldsymbol{x})=\mathbb{E}(Y_{\mathrm{obs}}\mid\boldsymbol{X}=\boldsymbol{x})$
and
$e^{\boldsymbol{\beta}}(\boldsymbol{x})=\mathbb{E}(\boldsymbol{\beta}_{T}\mid\boldsymbol{X}=\boldsymbol{x})$.

Considering the mean squared error of $\epsilon$ as a loss function and
minimizing it allows us to identify the optimal $\widehat{\boldsymbol{\beta}}$
and therefore CATEs. Given two nuisance estimators $\widehat{m}$ and
$\widehat{e}^{\boldsymbol{\beta}}$ of $m$ and ${e}^{\boldsymbol{\beta}}$, one
needs to solve the following problem:

$\widehat{\boldsymbol{\beta}}=\operatorname{argmin}_{\boldsymbol{\beta}\in\mathcal{F}}\
\frac{1}{n}\sum_{i=1}^{n}\Big{[}\left(Y_{\mathrm{obs},i}-\widehat{m}(\boldsymbol{X}_{i})\right)-\boldsymbol{f}(\boldsymbol{X}_{i})^{\top}\big{(}\boldsymbol{\beta}_{T_{i}}-\widehat{e}^{\boldsymbol{\beta}}(\boldsymbol{X}_{i})\big{)}\Big{]}^{2},$
(108)

where $\mathcal{F}$ is the space of candidate models $\boldsymbol{\beta}$. The
previous problem corresponds to a classical OLS estimator and has, therefore,
a unique solution.

If the space of candidate models $\mathcal{F}$ is separable, then the
optimization problem can be divided into the following sub-problems:

$\displaystyle\widehat{\boldsymbol{\beta}}_{t_{0}}=\operatorname{argmin}\
\frac{1}{n_{k}}\sum_{i\in\mathbf{S}_{t_{0}}}\Big{[}\left(Y_{\mathrm{obs},i}-\widehat{m}(\boldsymbol{X}_{i})\right)-\boldsymbol{f}(\boldsymbol{X}_{i})^{\top}\big{(}\boldsymbol{\beta}_{t_{0}}-\widehat{e}^{\boldsymbol{\beta}}(\boldsymbol{X}_{i})\big{)}\Big{]}^{2}$
$\displaystyle\vdots$
$\displaystyle\widehat{\boldsymbol{\beta}}_{t_{K}}=\operatorname{argmin}\
\frac{1}{n_{K}}\sum_{i\in\mathbf{S}_{t_{K}}}\Big{[}\left(Y_{\mathrm{obs},i}-\widehat{m}(\boldsymbol{X}_{i})\right)-\boldsymbol{f}(\boldsymbol{X}_{i})^{\top}\big{(}\boldsymbol{\beta}_{t_{K}}-\widehat{e}^{\boldsymbol{\beta}}(\boldsymbol{X}_{i})\big{)}\Big{]}^{2}.$

However, this approach does not consider the interactions between different
$\widehat{\boldsymbol{\beta}}_{t}$ and is computationally heavy when the
number of possible treatments $K$ becomes larger. It also requires specifying
the family of models $\mathcal{F}$ and precise the dimension $p$ for
Assumption 5.2.

There are two main differences between the generalized R-learner and the
binarized: 1) In the binarized R-learner,
$(\widehat{\boldsymbol{\beta}}_{t_{k}})_{k=1}^{K}$ may be solved separately
but using a small sample ($\mathbf{S}_{t_{k}}$ instead of
$\mathbf{D}_{\mathrm{obs}}$); 2) The solution
$(\widehat{\boldsymbol{\beta}}_{t_{k}})_{k=1}^{K}$ of the binarized R-learner
is unique and is given by the OLS estimator of the binarized R-loss function.

##  Appendix D Additional details about simulated analytical functions in
section 6.1

In this section, we consider a treatment $T$ with $K+1=10$ possible values in
$\mathcal{T}=\\{t_{k}:=\frac{k}{K},k\in\\{0,\ldots,K\\}\\}$, drawn from a
uniform distribution, and the following outcome functions.

The linear model outcome for $X\in\mathbb{R}$:

$Y(t)\mid X\sim\mathcal{N}\big{(}(1+t)X,\sigma^{2}\big{)}.$ (109)

The multivariate hazard rate (Imbens, 2000) outcome satisfies for
$\boldsymbol{X}\in\mathbb{R}^{5}$:

$Y(t)\mid\boldsymbol{X}\sim\mathcal{N}\big{(}t+\|\boldsymbol{X}\|\exp{(-t\|\boldsymbol{X}\|)}\
,\sigma^{2}\big{)}.$ (110)

We compute the exact components of each model in the following subsections:
the GPS $r$, the potential outcome models $\mu_{t}$ and the observed outcome
model $m$.

###  D.1 Computing nuisance components

#### The Generalized Propensity Score (GPS).

In the first design (RCT), we sample $n$ units such that $T$ and
$\boldsymbol{X}$ are independent. The true propensity score is known

$r(t,\boldsymbol{X})=\mathbb{P}(T=t)=1/(K+1)\text{ for }t\in\mathcal{T}.$
(111)

In the second design (observational studies), we combine $K+2$ samples in a
single sample of $n$ units. The first sample $\mathbf{D}_{K+1}$ contains
$n_{K+1}=n/2$ units where the treatment is assigned randomly: $\boldsymbol{X}$
and $T$ are independent, $\mathbb{P}(T=t)=1/(K+1)$,
$\boldsymbol{X}\sim\mathcal{N}(\boldsymbol{0},\mathbf{I}_{5})$ when the hazard
rate model is applied and $X\sim\mathcal{U}(0,1)$ when the linear model is
applied. For $k=0,\ldots,K$, the sample $\mathbf{D}_{k}$ contains
$n_{k}=n/(2(K+1))$ units and the distribution of $(\boldsymbol{X},T)$ does not
respect a RCT setting. For the linear model, the joint distribution of
$({X},T)$ is given by:

$T=\frac{k}{K}\text{ and $X$ follows a uniform distribution
$\mathcal{U}(I_{k})$ with}\
I_{k}=\Big{[}\frac{k}{K+1},\frac{k+1}{K+1}\Big{)}.$ (112)

For the hazard rate model, the joint distribution of $(\boldsymbol{X},T)$ is
given by:

$\displaystyle T=\frac{k}{K}\text{, $X_{1}$ follows a truncated standardized
normal distribution on}\
I_{k}=\big{[}q_{\frac{k}{K+1}},q_{\frac{k+1}{K+1}}\big{)}$ (113) and $X_{j}$
follow a standardized normal distribution $\mathcal{N}(0,1)$ for $2\leq j\leq
5$,

where $q_{\alpha}$ is the ${\alpha}$-quantile of the standardized normal
distribution with $q_{0}=-\infty$ and $q_{1}=+\infty$ . This strategy of
selecting preferentially only observations with certain characteristics is
called preferential selection sampling and creates thus a selection bias on
observed data.

For all $k\in\\{0,\ldots,K\\}$, the true propensity score satisfies for the
linear model:

$\displaystyle r(t_{k},x)$ $\displaystyle=\left\\{\begin{array}[]{l
l}\frac{K+2}{2(K+1)}&\quad\text{if $x\in I_{k}$,}\\\\[4.26773pt]
\frac{1}{2(K+1)}&\quad\text{otherwise.}\\\\[4.26773pt] \end{array}\right.$
(114)

and, for the hazard rate model, it satisfies for
$\boldsymbol{x}\in\mathbb{R}^{5}$:

$\displaystyle r(t_{k},\boldsymbol{x})$
$\displaystyle=\left\\{\begin{array}[]{l l}\frac{K+2}{2(K+1)}&\quad\text{if
$x_{1}\in I_{k}$,}\\\\[4.26773pt]
\frac{1}{2(K+1)}&\quad\text{otherwise.}\\\\[4.26773pt] \end{array}\right.$
(115)

###### Proof.

We show proof for the hazard rate model with normal distribution. The proof
remains the same for the linear model in a non-randomized setting.

Let $A$ be a given event, and then

$\mathbb{P}(A)=\sum_{k=0}^{K+1}\frac{n_{k}}{n}\mathbb{P}_{k}(A),$ (116)

where $\mathbb{P}$ is the observed probability distribution of the combined
sample and $\mathbb{P}_{k}$ denotes the probability measure induced by (111),
(113) and the unconfoundedness assumption 3.1.

Given the treatment $T=t_{j}$ and covariate vector
$\boldsymbol{x}=(x,x_{2},\ldots,x_{5})$, we have

$\displaystyle r(t_{j},\boldsymbol{x})$ $\displaystyle=\mathbb{P}(T=t_{j}\mid
X_{1}=x)$ (117) $\displaystyle=\lim_{\delta\rightarrow
0}\mathbb{P}(T=t_{j}\mid X_{1}\in[x,x+\delta])$
$\displaystyle=\lim_{\delta\rightarrow
0}\frac{\mathbb{P}\left(T=t_{j},X_{1}\in[x,x+\delta]\right)}{\mathbb{P}\left(X_{1}\in[x,x+\delta]\right)}.$

On the one hand,

$\displaystyle\mathbb{P}(T=t_{j},X_{1}\in$
$\displaystyle[x,x+\delta])=\sum_{k=0}^{K+1}\frac{n_{k}}{n}\mathbb{P}_{k}(T=t_{j},X_{1}\in[x,x+\delta])$
(118) $\displaystyle=\frac{n_{j}}{n}\
\mathbb{P}_{j}(T=t_{j},X_{1}\in[x,x+\delta])+\frac{n_{K+1}}{n}\
\mathbb{P}_{K+1}(T=t_{j},X_{1}\in[x,x+\delta])$
$\displaystyle=\frac{n_{j}}{n}\
\mathbb{P}_{j}(X_{1}\in[x,x+\delta])+\frac{n_{K+1}}{n}\
\mathbb{P}_{K+1}(T=t_{j})\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])$
$\displaystyle=\frac{1}{2(K+1)}\
\mathbb{P}_{j}(X_{1}\in[x,x+\delta])+\frac{1}{2(K+1)}\
\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta]).$

For $x\in\mathbb{R}$, there exists a unique $j_{0}$ such that $x\in
I_{j_{0}}$. For $\delta$ small enough, we have $[x,x+\delta]\subset I_{j_{0}}$
and, consequently, $[x,x+\delta]\cap I_{j}=\emptyset$ for all $j\neq j_{0}$.
This implies:

$\mathbb{P}_{j}(X_{1}\in[x,x+\delta])=\frac{\mathbb{P}_{K+1}\left(X_{1}\in[x,x+\delta],X_{1}\in
I_{j}\right)}{\mathbb{P}_{K+1}\left(X_{1}\in
I_{j}\right)}=\frac{\mathbb{P}_{K+1}\left(X_{1}\in[x,x+\delta]\right)}{\mathbb{P}_{K+1}\left(X_{1}\in
I_{j}\right)}\mathbf{1}\\{j=j_{0}\\}.$ (119)

Therefore,

$\displaystyle\mathbb{P}(T=t_{j},X_{1}\in[x,x+\delta])$
$\displaystyle=\frac{1}{2(K+1)}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])\
(\frac{\mathbf{1}\\{j=j_{0}\\}}{\mathbb{P}_{K+1}(X_{1}\in I_{j_{0}})}+1)$
(120)
$\displaystyle=\Big{(}\frac{1}{2}\mathbf{1}\\{j=j_{0}\\}+\frac{1}{2(K+1)}\Big{)}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta]).$

On the other hand,

$\displaystyle\mathbb{P}(X_{1}\in[x,x+\delta])$
$\displaystyle=\sum_{k=0}^{K+1}\frac{n_{k}}{n}\mathbb{P}_{k}(X_{1}\in[x,x+\delta])$
(121)
$\displaystyle=\frac{1}{2(K+1)}\sum_{k=0}^{K}\frac{\mathbb{P}_{K+1}\left(X_{1}\in[x,x+\delta],X_{1}\in
I_{k}\right)}{\mathbb{P}_{K+1}\left(X_{1}\in
I_{k}\right)}+\frac{1}{2}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])$
$\displaystyle=\frac{1}{2(K+1)}\
\frac{\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])}{\mathbb{P}_{K+1}(X_{1}\in
I_{j_{0}})}+\frac{1}{2}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])$
$\displaystyle=\frac{1}{2}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])+\frac{1}{2}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])$
$\displaystyle=\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])$

Finally,

$\displaystyle r(t_{j},\boldsymbol{x})$ $\displaystyle=\lim_{\delta\rightarrow
0}\frac{\mathbb{P}\left(T=t_{j},X_{1}\in[x,x+\delta]\right)}{\mathbb{P}\left(X_{1}\in[x,x+\delta]\right)}$
(122) $\displaystyle=\lim_{\delta\rightarrow
0}\frac{\Big{(}\frac{1}{2}\mathbf{1}\\{j=j_{0}\\}+\frac{1}{2(K+1)}\Big{)}\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])}{\mathbb{P}_{K+1}(X_{1}\in[x,x+\delta])}$
$\displaystyle=\frac{1}{2}\mathbf{1}\\{j=j_{0}\\}+\frac{1}{2(K+1)}$
$\displaystyle=\frac{(K+1)\mathbf{1}\\{j=j_{0}\\}+1}{2(K+1)}$
$\displaystyle=\left\\{\begin{array}[]{l l}\frac{K+2}{2(K+1)}&\quad\text{if
$x\in I_{j}$,}\\\\[4.26773pt]
\frac{1}{2(K+1)}&\quad\text{otherwise.}\\\\[4.26773pt] \end{array}\right.$

∎

#### The potential outcome models.

The potential outcome models are given directly by the conditional mean. For
the linear model, $\mu_{t}$ satisfies for all $t\in\mathcal{T}$ and
$x\in[0,1]$:

$\mu_{t}({x})=(1+t){x}.$ (123)

For the hazard rate model, $\mu_{t}$ is given by:

$\mu_{t}(\boldsymbol{x})=t+\|\boldsymbol{x}\|\exp{(-t\|\boldsymbol{x}\|)}.$
(124)

#### The observed outcome models.

For the linear model, the observed outcome model $m$ can be computed as:

$\displaystyle m({x})$ $\displaystyle=\mathbb{E}(Y_{\mathrm{obs}}\mid{X}={x})$
(125) $\displaystyle=\mathbb{E}((1+T)\boldsymbol{X}\mid{X}={x})$
$\displaystyle=(1+\mathbb{E}(T\mid{X}={x})){x}$
$\displaystyle=\big{(}1+\sum_{k=0}^{K}{r}(t_{k},{x})t_{k}\big{)}{x},$

where $r$ is given by (114).

and, for the hazard rate model, $m$ can be computed as:

$\displaystyle m(\boldsymbol{x})$
$\displaystyle=\mathbb{E}(\mathbb{E}(Y_{\mathrm{obs}}\mid\boldsymbol{X},T)\mid\boldsymbol{X}=\boldsymbol{x})$
(126)
$\displaystyle=\mathbb{E}(T+\|\boldsymbol{X}\|\exp{(-T\|\boldsymbol{X}\|)}\mid\boldsymbol{X}=\boldsymbol{x})$
$\displaystyle=\mathbb{E}(T\mid\boldsymbol{X}=\boldsymbol{x})+\|\boldsymbol{x}\|\
\mathbb{E}(\exp{(-T\|\boldsymbol{X}\|)}\mid\boldsymbol{X}=\boldsymbol{x})$
$\displaystyle=\sum_{k=0}^{K}{r}(t_{k},\boldsymbol{x})t_{k}+\sum_{k=0}^{K}\|\boldsymbol{x}\|\
r(t_{k},\boldsymbol{x})\exp{(-t_{k}\|\boldsymbol{x}\|)},$

where $r$ is given by (115).

###  D.2 Additional numerical results and plots.

In this section, we present the results of different simulations and scenarios
for linear and hazard rate models with $K+1=10$, $n=2000$ for the linear
model, and $n=10000$ for the Hazard rate model. In the randomized setting, the
sample $\mathbf{D}_{\mathrm{obs}}$ is sampled randomly, and the propensity
score is given by (111). In a non-randomized setting, the sample
$\mathbf{D}_{\mathrm{obs}}$ is given by preferential selection as described in
Section D and the GPS is given by (114). When we say that ”the models’
nuisance components are exact”, we replace the expression of the estimators
$\widehat{\mu}_{t},\widehat{m}$ or $\widehat{r}$ with the expressions obtained
in Section D.

### Linear model in a randomized setting.

Table 5: mPEHE for three different ML base-learners; Case where nuisance components are exact. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 2.269 | 2.09 | 0.029
DR-Learner | 0.159 | 0.134 | 7.59 $10^{-3}$
X-Learner | 0.022 | 0.029 |  1.68 $\mathbf{10^{-3}}$
RLin-Learner | 7.78 $10^{-3}$
Table 6: mPEHE for three different ML base-learners; Case where nuisance
components are well-specified.

Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
T-Learner | 0.061 | 0.037 | 7.81 $10^{-3}$
S-Learner | 0.029 | 0.039 |  4.25 $\mathbf{10^{-3}}$
NvX-Learner | 0.056 | 0.033 | 7.81 $10^{-3}$
M-Learner | 1.24 | 1.16 | 0.230
DR-Learner | 0.063 - 0.055 | 0.060 - 0.060 | 7.55 - 4.40 $\mathbf{10^{-3}}$
X-Learner | 0.059 - 0.030 | 0.041 - 0.073 | 7.81 - 4.44 $10^{-3}$
RLin-Learner | 0.137 | 0.127 | 0.050

  * •

For the DR and X-learners: $\mu_{t}$ are estimated by T-learning (left value)
or S-learning (right value).

Table 7: mPEHE for three different ML base-learners; Case where the propensity score is misspecified. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 3.83 | 3.57 | 1.41
DR-Learner | 0.128 | 0.240 | 0.012
X-Learner | 0.030 | 0.073 | 4.42 $10^{-3}$
RLin-Learner | 0.336 | 0.333 | 0.341
Table 8: mPEHE for three different ML base-learners; Case where the outcome models are misspecified. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 1.24 | 1.16 | 0.230
DR-Learner | 0.741 | 0.803 | 0.261
X-Learner | 0.308 | 0.307 | 0.271
RLin-Learner | 0.053
Table 9: mPEHE for three different ML base-learners; Case where nuisance components are misspecified. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 3.83 | 3.57 | 1.41
DR-Learner | 1.73 | 1.95 | 0.771
X-Learner | 0.308 | 0.307 | 0.278
RLin-Learner | 0.285

### Linear model in non-randomized setting

Table 10: mPEHE for three different ML base-learners; Case where nuisance components are exact. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 3.70 | 2.67 | 0.2200
DR-Learner | 0.288 | 0.214 | 9.94 $10^{-3}$
X-Learner | 0.023 | 0.030 |  1.61 $\mathbf{10^{-3}}$
RLin-Learner | 9.79 ${10^{-3}}$
Table 11: mPEHE for three different ML base-learners; Case where nuisance
components are well-specified.

Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
T-Learner | 0.068 | 0.042 | 9.56 $10^{-3}$
RegT-Learner | 0.062 | 0.042 | 0.012
S-Learner | 0.035 | 0.050 |  4.24 $\mathbf{10^{-3}}$
NvX-Learner | 0.062 | 0.038 | 9.56 $10^{-3}$
M-Learner | 1.43 | 1.15 | 0.218
DR-Learner | 0.068 - 0.072 | 0.068 - 0.092 | 0.012 - 6.14 $10^{-3}$
X-Learner | 0.062 - 0.036 | 0.045 - 0.073 | 0.013 - 4.34 $10^{-3}$
RLin-Learner | 0.162 | 0.140 | 0.079

  * •

For the DR and X-learners: $\mu_{t}$ are estimated by T-learning (left value)
or S-learning (right value).

### Hazard rate model in randomized setting

Table 12: mPEHE for three different ML base-learners; Case where nuisance components are exact. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 4.26 | 4.22 | 0.52
DR-Learner | 0.126 | 0.139 | 0.110
X-Learner | 0.045 | 0.087 | 0.110
RLin-Learner | 0.111
Table 13: mPEHE for three different ML base-learners; Case where nuisance
components are well-specified.

Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
T-Learner | 0.171 | 0.271 | 0.113
S-Learner | 0.154 | 0.265 | 0.667
NvX-Learner | 0.174 | 0.322 | 0.113
M-Learner | 1.52 | 1.75 | 0.814
DR-Learner | 0.163 - 0.154 | 0.288 - 0.282 | 0.115 - 0.472
X-Learner | 0.161 - 0.149 | 0.287 - 0.285 |  0.113 \- 0.648
RLin-Learner | 0.230 | 0.241 | 0.71

  * •

For the DR and X-learners: $\mu_{t}$ are estimated by T-learning (left value)
or S-learning (right value).

### Hazard rate model in non-randomized setting

Table 14: mPEHE for three different ML base-learners; Case where nuisance components are exact. Meta-learner | XGBoost | RandomForest | Linear Model
---|---|---|---
M-Learner | 6.28 | 5.54 | 0.511
DR-Learner | 0.185 | 0.171 | 0.110
X-Learner | 0.045 | 0.087 | 0.110
RLin-Learner | 0.112
Table 15: mPEHE for three different ML base-learners; Case where nuisance
components are well-specified.

Meta-learner | XGboost | RandomForest | Linear Model
---|---|---|---
T-Learner | 0.184 | 0.254 | 0.132
RegT-Learner | 0.158 | 0.257 | 0.115
S-Learner | 0.166 | 0.272 | 0.659
NvX-Learner | 0.181 | 0.308 | 0.132
M-Learner | 1.56 | 1.56 | 0.885
DR-Learner | 0.151 - 0.171 | 0.279 - 0.288 |  0.115 \- 0.507
X-Learner |  0.149 \- 0.162 | 0.274 - 0.290 | 0.118 - 0.639
RLin-Learner | 0.235 | 0.183 | 1.02

  * •

For the DR and X-learners: $\mu_{t}$ are estimated by T-learning (left value)
or S-learning (right value).

###  D.3 Asymptotic performances when $K$ increases.

In this subsection, we consider the effect of increasing $K$ on the hazard
rate function with XGBoost. For each value $K$, we sample $J=10$ different
non-randomized samples following the preferential selection as defined
previously in Appendix D.1. The mPEHE is then computed by averaging the mPEHE
over the $J=10$ samples. The results are drawn in the figure below.

Figure 1: Variation of meta-learners’ performances as functions of the number
of possible treatment values $K$ for the hazard rate function in an
observational design setting. 1: All meta-learners; 1: Without the M-learner

When we consider the effect of increasing $K$ on the hazard rate function with
a linear model (with $p=2$), we notice the expected effect of $K$ on the
M-learner: The error bound is increasing with $K$. This means that the
behaviour of the M-learner with XGBoost can be explained by the over-fitting
of the base-learner.

Figure 2: Variation of meta-learner’s performances when the number of possible
treatment values $K$ for the hazard rate function in observational design
setting with a linear model. 2: All meta-learners; 2: Focus on the T-learning
methods.

##  Appendix E Description of the semi-synthetic dataset.

### Motivation

The difficulty in evaluating a causal model’s performance in real-world
applications motivates the need to create a semi-synthetic dataset. In this
subsection, we consider a multistage fracturing Enhanced Geothermal System
(EGS).

Enhanced Geothermal Systems (EGS) are geothermal wells that generate
geothermal energy by creating fluid connectivity in low-permeability
conductive rocks through hydraulic, thermal, or chemical stimulation. The EGS
concept involves extracting heat by constructing a subsurface fracture system
to which water can be added via injection wells. Indeed, rocks are permeable
due to slight fractures and pore spaces between mineral grains, and the
injected water is heated by contact with the rock and returns to the surface
through production wells. Moreover, Enhanced geothermal systems (EGS) have a
high potential for developing and supplying renewable energy sources that are
more efficient and cheaper than traditional hydrocarbon resources.

For energy companies, the goal is to optimize the design of the geothermal
well (fracture spacing, Lateral Length etc.) to generate the maximum
geothermal energy. However, some economic and operational problems present
challenges: On the one hand, if the fractures are too small or too few, rocks
will not be exploited sufficiently. On the other hand, if the number of
fractures in a given rock is too high, the fractures may cool down faster. We
would have a costly design that will not maximize the extracted heat.

We assume that the heat extraction performance of the EGS satisfies the
following physical model:

$Q_{well}=Q_{fracture}\times{\ell_{L}}/{d}\times\eta_{d},$ (127)

where $Q_{well}$ is the heat extraction performance delivered by the well
(output), $Q_{fracture}$ is the unknown heat extraction performance from a
single fracture that can be generated using a complex seven-parameter model,
including reservoir characteristics and fracture design, $\ell_{L}$ is the
Lateral Length of the well, $d$ is the average spacing between two fractures
and $\eta_{d}$ is the stage efficiency penalizing the individual contribution
when fractures are close to each other. We refer to Figure 3 for a graphical
description of the EGS and its inputs/output.

Finally, the model in (127) respects the unconfoundedness assumption 3.1, and
we can control all its variables in the simulations. We note that, in
practice, all inputs are continuous with a given density. However, we
discretize these variables in their input space to create a full factorial
design.

### Description of the data-set

This section describes the data-generating process of our semi-synthetic
dataset simulating the heat delivered by a multistage fracturing EGS. The
process involved the creation of a conceptual reservoir model and the
modelling of multiple wells’ completion scenarios. The output (heat extraction
performance) obtained from physics-based simulation experiments was tabulated
with inputs in the semi-synthetic dataset.

The input data for the model were fabricated to ease confidentiality and non-
disclosure information issues. However, data has been selected from reliable
sources such as field observations, journals and books to be within the range
of interest. Doing so allowed the building of a plain but representative
reservoir model that would provide realistic results of an EGS.

The heat extraction performance from a single fracture (${Q}_{fracture}$) is
determined using fracture length, fracture height, fracture width, fracture
permeability, reservoir porosity, reservoir permeability and pore pressure.
Modelling and simulation work were done using preprocessor and reservoir
simulation tools PETREL and ECLIPSE.

The four physical parameters of the fracture were investigated, and the list
of values used for each parameter can be observed in Table 16. In the end,
$10\times 10\times 2\times 3=600$ fracture’s simulation cases have been
realized.

Table 16: Fracture parameters and their range of variation for simulations. Variable | Range of variation
---|---
Fracture length (ft) |  $[100,1000]$ by a step of $100$ ft
Fracture height (ft) |  $[50,500]$ by a step of 50 ft
Fracture width (in) | $\\{0.1,0.2\\}$
Fracture Permeability (md) | $\\{30000,85000,19000\\}$

To emulate distinct reservoir schemes, it was decided to vary three main
parameters; porosity, permeability and pore pressure. For porosity and
permeability, the simulator takes the minimum and maximum values and estimates
the physical properties across the reservoir. Three different multipliers were
applied to define three (Low, Base and High) scenarios. Concerning pore
pressure, three specific values were defined to simulate under-normal, normal
(base) and overpressure (high) gradient conditions. Therefore, $3\times
3\times 3=27$ possible scenarios were defined. Table 17 displays the range of
minimum and maximum values for the three reservoir parameters to be varied.

Figure 3: The Causal DAG associated with the multistage EGS. Nodes in yellowish brown represent the reservoir characteristics, they can only be simulated, but in reality, we cannot intervene in these variables. Nodes in Dark green represent the fracture design. Engineers control them, and intervening in them is possible whenever there is a need to make a new fracture in the well. Nodes in blue represent a well’s design and can be chosen arbitrarily by engineers or statisticians. Nodes in black denote the outputs. $Q_{fracture}$ is only given by the simulator, whereas $Q_{well}$ is given by the physical model in (127). Note that this graph contains nine nodes, but both $K_{\mathrm{min}}$ and $K_{\mathrm{max}}$ represent the same physical parameter $K$, and the same remark is valid for $Por_{\mathrm{min}}$ and $Por_{\mathrm{max}}$. Table 17: Reservoir parameters and their range of variation for simulations. Variable | Range of variation
---|---
$(\text{K}_{\mathrm{min}},\text{K}_{\mathrm{max}})$ (md) | $\\{(0.0054,0.0157),(0.054,0.157),(0.109,0.314)\\}$
$(\text{Por}_{\mathrm{min}},\text{Por}_{\mathrm{max}})$ (dec) | $\\{(0.0054,0.0157),(0.054,0.157),(0.109,0.314)\\}$
Pore pressure (psi) | $\\{5000,7000,9000\\}$

By combining different reservoir scenarios with single fracture simulations,
we obtained a single dataset with 16,200 possible cases for a fracture in a
reservoir then we simulated the heat extraction performance for each
experiment. Simulation’s results were tabulated in the dataset
”Single_Fracture_Simulation_Cases_16200.csv”.

The next step is to define well characteristics (lateral lengths and fracture
spacing) to evaluate the heat extraction performance of the well when
reservoir and fracture properties are not changed.

Table 18: Well parameters and their range of variation. Variable | Range of variation
---|---
Lateral length (ft) |  $[2000,14000]$ by a step of $1000$ ft
Fracture spacing (ft) |  $[100,500]$ by a step of $100$ ft

Regarding the spacing efficiency coefficient, this coefficient was used to
model interactions between fractures and penalize the heat extraction
performance of a single fracture in the presence of other close fractures,
that is, when the spacing between two fractures is small. Indeed, if the
fractures are spaced too close, there may not be enough thermal energy in the
rock to heat the water, decreasing the heat extraction efficiency. Modelling
this efficiency led to the efficiency table ”Fracture_Efficency.csv” that
describes what would be the well’s heat performance behaviour with respect to
the fracture spacing selected. Based on this table, one can interpolate the
efficiency to draw the curve (see Figure 4) and thus obtain the spacing
efficiency coefficient for any desired value fracture spacing.

Figure 4: Cross plot between fracture spacing efficiency and average stage
spacing.

The final generation of the semi-synthetic dataset ”Main_Dataset.csv” was
achieved by combining two main tables created using the R programming
language. This table allows calculating the heat performance of a well for any
lateral length and fracture spacing between 500 ft and 100 ft with the
associated spacing efficiency coefficient defined in the efficiency table,
following the physical model in (127).

The three datasets are available in the zip file in Supplementary Materials
”Semi-synthetic-EGS.zip”. They will also be shared for public use upon
acceptance of the paper.

Finally, we emphasize that this study’s design methodology focused on
generating a semi-synthetic dataset using reservoir numerical simulation and
creating a new benchmarking dataset for comparing and validating causal
inference methods. Indeed, following the last step of creating the final
dataset ”Main_Dataset.csv”, any user can define different distributions (with
different values) on lateral lengths in the range $[2000,14000]$ and fracture
spacing in range $[100,500]$, pick-up the corresponding spacing efficiency
coefficients using the curve drawn in Figure 4 and generate a new semi-
synthetic dataset by extrapolating them with
”Single_Fracture_Simulation_Cases_16200.csv” dataset.

### The creation of a non-randomized biased dataset.

The idea of this step was to create a collection of biased data from the main
semi-synthetic dataset to emulate observational data found in real-world
situations. For example, geothermal wells with larger lateral lengths are
likely to have more fractures (expensive wells are located in better
geological areas). The opposite is seen for smaller wells that tend to be
associated with fewer fractures. This situation creates a discrepancy between
what engineers expect from physical models and what they observe in the field
data. The biased data, with 9,992 observations, was generated by following the
preferential selection strategy from the main dataset. Figure 5 shows the
difference between the real heat extraction performance of the EGS and the
observed heat extraction performance on the field: low (under-estimated) heat
performance for small wells and high (over-estimated) heat performance for
large wells.

Figure 5: An illustration of selection bias on the heat performance. Red line:
The heat extraction performance on the main dataset (i.e. Ground Truth Model).
Blue line: The heat performance on the biased dataset (i.e. observed
response).


# Effects of coupled channels on $c\bar{b}$ masses and decays in NRQM with
OGEP

Manjunath Bhat  Department of Physics, St Philomena college, Darbe, Puttur 574
202, India
<EMAIL_ADDRESS>Antony Prakash Monteiro  Department of Physics, St
Philomena college, Darbe, Puttur 574 202, India
<EMAIL_ADDRESS>K. B. Vijaya Kumar  Department of Physics,
Mangalore University, Mangalagangothri P.O., Mangalore - 574199, India
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

The complete spectrum of $c\bar{b}$ states is obtained in a phenomenological
non relativistic quark model(NRQM), which consists of a confinement potential
and one gluon exchange potential (OGEP) as effective quark - antiquark
potential with coupled channel effects. We make predictions for the radiative
decay (E1 and M1) widths and weak decay widths of $c\bar{b}$ states in the
framework of NRQM formalism.

###### keywords:

Mesons; Phenomenological quark models; Non relativistic quark models;
Leptonic; semileptonic; radiative decays of mesons, coupled channel effects

PACS: 14.40.-n;12.3.-x,12.39.-Jh,13.20.-v

##  1 INTRODUCTION

The investigation of masses of $c\bar{b}$ states gives us an opportunity to
obtain information on the nature of the strong interaction thereby it throws
up an interesting issue and a tantalizing problem. Since the charmed bottom
meson $c\bar{b}$ is an intermediate state of the $c\bar{c}$ and $b\bar{b}$
mesons, its analysis could give detailed information on the balance between
perturbative and non perturbative effects. There are a good number of
theoretical models that study leptonic, semi leptonic and hadronic decay
channels of $c\bar{b}$ states. Using NRQM formalism we have already studied
mass spectra and decay properties of $c\bar{b}$ meson. This work attempts to
study the effects of coupled channels on $c\bar{b}$ masses and its decays in
NRQM.

The NRQM formalism is found to provide systematic treatment of the
perturbative and non perturbative components of QCD at hadronic scale. The
masses of the $c\bar{b}$ states are predicted using NRQM whose parameters are
tuned to produce the spectra of the observed charmonium and bottomonium
states.

The paper is organized in 4 sections. In sec. 2 we give the description of our
model in the theoretical background, the framework of the coupled-channel
analysis and description of various decays. In sec. 3 we discuss the results
and the conclusions are drawn in sec. 4 with a comparison to other models.

##  2 THEORETICAL BACKGROUND

###  2.1 The Hamiltonian

In our model we use the Hamiltonian which includes includes kinetic energy,
confinement potential and one gluon potential (OGEP)[1, 2, 3, 4].

$H=K+V_{CONF}+V_{OGEP}$ (1)

where $K$ is the kinetic energy part, $V_{CONF}$ is confinement potential that
comes from the non perturbative treatment of QCD, $V_{OGEP}$ is the residual
interaction from perturbative treatment of quark-antiquark system.

###  2.2 The Linear Confinement Potential

In literature different types of confinement potentials are chosen depending
upon the unique features of the phenomenological quark model under
consideration. They can be harmonic oscillator potential ($V\sim r^{2}$) or
logarithmic potential ($V\sim ln(r)$) or linear potential ($V\sim r$). We deem
it fit to choose linear potential that represents non perturbative effect of
QCD that explains quark confinement within the color singlet system [2, 3].

$V_{CONF}(\vec{r}_{ij})=-a_{c}r_{ij}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}$
(2)

where $a_{c}$ is the confinement strength, $\lambda_{i}$ and $\lambda_{j}$ are
the generators of the color SU(3) group for the $i^{\rm th}$ and $j^{\rm th}$
quarks. Since confinement is of two body system we leave out the spin-orbit
contribution for it adds nothing practically to the interaction.

###  2.3 The Short Distance Behaviour

The one gluon exchange potential(OGEP) describes the short distance behavior.
The central part of the two-body potential due to OGEP is [4],

$V_{OGEP}(\vec{r}_{ij})=\frac{\alpha_{s}}{4}\vec{\lambda}_{i}\cdot\vec{\lambda}_{j}\left[\frac{1}{r_{ij}}-\frac{\pi}{M_{i}M_{j}}\left(\frac{M_{i}}{M_{j}}+\frac{M_{j}}{M_{i}}+\frac{2}{3}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}\right)\delta(\vec{r}_{ij})\right]$
(3)

where the first term represents the residual Coulomb energy and the second
term is the chromo-magnetic interaction leading to the hyperfine splitting.
$\sigma_{i}$ is the Pauli spin operator and $\alpha_{s}$ is the quark-gluon
coupling constant.

The spin-orbit interaction of OGEP is given by,

$V^{SO}_{OGEP}(\vec{r})=-\frac{\alpha_{s}}{4}{\bf\lambda_{i}\cdot\lambda_{j}}\left[\frac{3}{8M_{i}M_{j}}\frac{1}{r^{3}}(\vec{r}\times\vec{p})\cdot({\bf\sigma_{i}}+{\bf\sigma}_{j})\right]$
(4)

The following tensor term [5, 6] is considered,

$V^{ten}_{OGEP}(\vec{r})=-\frac{\alpha_{s}}{4}{\bf\lambda_{i}\cdot\lambda_{j}}\left[\frac{1}{4M_{i}M_{j}}\frac{1}{r^{3}}\right]\hat{S}_{ij}$
(5)

where,

$\hat{S}_{ij}=[3(\vec{\sigma}_{i}\cdot\hat{r})(\vec{\sigma}_{j}\cdot\hat{r})-\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}]$
(6)

###  2.4 Coupled Channel Effects

The coupled channel effects (hadronic loop effects) have been neglected by
most of the QCD inspired potential models in calculating the masses of mesons.
The $BD$, $B_{s}D_{s}$ etc., channels strongly couple to the $c\bar{b}$
states. These channels give rise to mass shifts both below and above BD meson
pair creation threshold. Also above threshold these effects lead to the strong
decay of $B_{c}$ meson. These effects in our calculation are introduced
explicitly through the ${}^{3}P_{0}$ pair creation model for the decay of
meson $A\to B+C$ which was proposed by Micu and developed by Le Yaouanc et al
and others[7, 8, 9, 10]. The main assumption of the model is that the strong
decay of meson $A$ takes place through the creation of a pair of quark and
anti-quark from vacuum with quantum number $J^{PC}=0^{++}$. The created quark
anti-qurak pair recombines with the quark and anti-qurak in the initial meson
state forming final meson states i.e, mesons B and C. [11, 12, 13, 14, 15, 16,
17, 7, 8, 18, 9, 10, 19, 20, 21].

In the coupled channel model, the full hadronic state is given by [8, 9, 10]

$\lvert\psi\rangle=\lvert A\rangle+\sum_{BC}\lvert BC\rangle$ (7)

for open flavour strong decay $A\to BC$. Here A, B, C denote mesons.

The wave function $\lvert\psi\rangle$ obeys the equation

$H\lvert\psi\rangle=M\lvert\psi\rangle$ (8)

The Hamiltonian H for this combined system consists of a valence Hamiltonian
$H_{0}$ and an interaction Hamiltonian $H_{I}$ which couples the valence and
continuum sectors. The matrix element of the valence-continuum coupling
Hamiltonian is given by [9, 10]

$\langle BC\rvert H_{I}\lvert
A\rangle=h_{fi}\delta(\vec{P}_{A}-\vec{P}_{B}-\vec{P}_{C})$ (9)

where $h_{fi}$ is the decay amplitude.

The mass shift of meson A due to its continuum coupling to BC can be expressed
in terms of partial wave amplitude ${\cal M}_{LS}$ [8, 10]

$\displaystyle\Delta
M_{A}^{(BC)}=\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}-i\epsilon}\int
d\Omega_{p}|h_{fi}(p)|^{2}$
$\displaystyle\quad=\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}-i\epsilon}\sum_{LS}|{\cal
M}_{LS}|^{2}$ $\Delta M_{A}^{(BC)}={\cal
P}\int^{\infty}_{0}dp\frac{p^{2}}{E_{B}+E_{C}-M_{A}}\sum_{LS}|{\cal
M}_{LS}|^{2}+i\pi\left(\frac{p*E_{B}*E_{C}}{M_{A}}\sum_{LS}|{\cal
M}_{LS}|^{2}\right)|_{E_{B}+E_{C}=M_{A}}$ (10)

The decay amplitude $h_{fi}$ can be combined with relativistic phase space to
give the differential decay rate, which is

$\frac{d\Gamma_{A\to BC}}{d\Omega}=2\pi P\frac{E_{B}E_{C}}{M_{A}}|h_{fi}|^{2}$
(11)

where in the rest frame of A, we have $\vec{P}_{A}=0$ and
$P=|\vec{P}_{B}|=|\vec{P}_{C}|$, and

$P=\sqrt{[M^{2}_{A}-(M_{B}+M_{C})^{2}][M^{2}_{A}-(M_{B}-M_{C})^{2}]}/(2M_{A})$
(12)

Finally the total decay rate is given by [8, 10]

$\Gamma_{A\to BC}=2\pi P\frac{E_{B}E_{C}}{M_{A}}\sum_{LS}|{\cal M}_{LS}|^{2}$
(13)

###  2.5 Decay Properties

####  2.5.1 Electric Dipole (E1) Transitions

The partial width for electric dipole (E1) transitions is given by

$\Gamma_{(i\to f+\gamma)}=(2J^{\prime}+1)\frac{4}{3}Q^{2}_{b}\alpha
k^{3}_{0}S_{if}^{E}\left|{\cal E}_{if}\right|^{2}$ (14)

Here $k_{0}$ is the energy of the emitted photon and it is given by
$k_{0}=\frac{m^{2}_{a}-m^{2}_{b}}{2m_{a}}$. $\alpha$ is the fine structure
constant. $Q_{b}=1/3$ is the charge of the b quark in units of $|e|$, the
statistical factor
$S_{if}^{E}=\rm{max}(l,l^{\prime})\left\\{\begin{array}[]{ccc}J&1&J^{\prime}\\\
l^{\prime}&s&l\end{array}\right\\}^{2}$, $J,~{}J^{\prime}$ are the total
angular momentum of initial and final mesons, $l,~{}l^{\prime}$ are the
orbital angular momentum of initial and final mesons and $s$ is the spin of
initial meson.

${\cal
E}_{if}=\frac{3}{k_{0}}\int^{\infty}_{0}r^{3}R_{nl}(r)R^{\prime}_{nl}(r)dr\left[\frac{k_{0}r}{2}j_{0}\left(\frac{k_{0}r}{2}\right)-j_{1}\left(\frac{k_{0}r}{2}\right)\right]$
(15)

is the radial overlap integral which has the dimension of length, with
$R_{nl}(r)$ being the normalized radial wave functions for the corresponding
states.

####  2.5.2 Magnetic Dipole (M1) Transitions

The partial decay width for M1 transitions is [22, 23, 24, 25, 26, 27, 28]

$\begin{split}\Gamma_{a\to b+\gamma}=\delta_{L_{a}L_{b}}4\alpha
k^{3}_{0}\frac{E_{b}(k_{0})}{m_{a}}\left(\frac{Q_{c}}{m_{c}}+(-1)^{S_{a}+S_{b}}\frac{Q_{\bar{b}}}{m_{\bar{b}}}\right)^{2}(2S_{a}+1)\times(2S_{b}+1)(2J_{b}+1)\\\
\left\\{\begin{array}[]{ccc}S_{a}&L_{a}&J_{a}\\\
J_{b}&1&S_{b}\end{array}\right\\}^{2}\left\\{\begin{array}[]{ccc}1&\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&S_{a}&S_{b}\end{array}\right\\}^{2}\times\left[\int^{\infty}_{0}R_{n_{b}L_{b}}(r)r^{2}j_{0}(kr/2)R_{n_{a}L_{a}}(r)dr\right]^{2}\end{split}$
(16)

where $\int^{\infty}_{0}drR_{n_{b}L_{b}}(r)r^{2}j_{0}(kr/2)R_{n_{a}L_{a}}(r)$
is the overlap integral for unit operator between the coordinate wave
functions of the initial and the final meson states, $j_{0}(kr/2)$ is the
spherical Bessel function, $m_{b}$ is the mass of bottom quark. $J_{b}$ is the
total angular momentum of final meson state.

####  2.5.3 Weak Decays

Weak decays of $B_{c}$ meson plays a special role in our understanding of
physics of the Standard Model and beyond. Various diagrams can contribute to
the weak decays at the quark level. These are mainly a) Spectator quark, b)
W-exchange, c) W-annihilation and d) Penguin diagrams. Due to the helicity and
color considerations, the W-exchange diagrams are suppressed. Penguin diagrams
are also expected to be small in strength. Hence the dominant quark level
processes seem to be the process in which one of the quarks(anti-quark) behave
like spectator and the W-annihilation[29, 30]. Using this picture, after
evaluating the contributing diagrams we get the decay widths for a hadron
containing a b quark or c quark as in eqns(18, 19 and 20).

A rough estimate of the $B_{c}$ weak decay widths can be done by treating the
$\bar{b}$-quark and $c$-quark decays independently so that $B_{c}$ decays can
be divided into three classes [31, 32]$\colon$ (i)the $\bar{b}$-quark decay
with spectator $c$-quark, (ii) the $c$-quark decay with spectator
$\bar{b}$-quark, and (iii) the annihilation $B^{+}_{c}\rightarrow
l^{+}\nu_{l}$ ($c\bar{s},~{}u\bar{s}$), where $l=e,~{}\mu,~{}\tau$. The total
decay width can be written as the sum over partial widths

$\Gamma(B_{c}\rightarrow X)=\Gamma_{1}(\bar{b}\rightarrow
X)+\Gamma_{2}(c\rightarrow X)+\Gamma_{3}(ann)$ (17)

In the spectator approximation:

$\Gamma_{1}(\bar{b}\rightarrow
X)=\frac{9G^{2}_{F}|V_{cb}|^{2}m^{5}_{b}}{192\pi^{3}}$ (18)

and

$\Gamma_{2}(c\rightarrow
X)=\frac{5G^{2}_{F}|V_{cs}|^{2}m^{5}_{c}}{192\pi^{3}}$ (19)

In the above expressions $V_{cb}$ and $V_{cs}$ are the elements of the CKM
matrix, $G_{F}=1.16637\times 10^{-5}$ is the Fermi coupling constant, $m_{c}$
and $m_{b}$ are the masses of c and b quarks respectively.

The decay of vector meson into charged leptons proceeds through the virtual
photon $(q\bar{q}\rightarrow l^{+}l^{-})$. The ${}^{3}S_{1}$ and ${}^{3}D_{1}$
states have quantum numbers of a virtual photon, $J^{PC}=1^{--}$ and can
annihilate into lepton pairs through one photon. Annihilation widths such as
$c\bar{b}\rightarrow l\nu_{l}$ are given by the expression

$\Gamma_{3}(ann)=\frac{G^{2}_{F}}{8\pi}|V_{bc}|^{2}f^{2}_{B_{c}}M_{B_{c}}\sum_{i}m^{2}_{i}\left(1-\frac{m^{2}_{i}}{M^{2}_{B_{c}}}\right)C_{i}$
(20)

where $m_{i}$ is the mass of the heavier fermion in the given decay channel.
For lepton channels $C_{i}=1$ while for quark channels
$C_{i}=3|V_{q\bar{q}}|^{2}$.
The pseudo scalar decay constant $f_{B_{c}}$ is defined by [33]

$\langle 0\rvert\bar{b}(x)\gamma^{\mu}\gamma_{5}c(x)\lvert
B_{c}(k)\rangle=if_{B_{c}}V_{cb}k^{\mu}$ (21)

where $k^{\mu}$ is the four-momentum of the $B_{c}$ meson. In the non
relativistic limit the pseudo scalar decay constant is proportional to the
wave function at the origin and is given by van Royen-Weisskopf formula [34]

$f_{B_{c}}=\sqrt{\frac{12}{M_{B_{c}}}}\psi(0)$ (22)

Here $\psi(0)$ is wavefunction at the origin.

##  3 Results and Discussion

###  3.1 Mass Spectra

The parameters used in our model are listed in Table 3.1. We have fixed the
parameters using the approach used in our earlier works[1, 3, 35]. We obtain
the parameter $b$ by minimizing the expectation value of the Hamiltonian i.e,
$\frac{\partial\langle\psi\rvert H\lvert\psi\rangle}{\partial b}=0$. The
confinement strength $a_{c}$ is fixed by the stability condition for variation
of mass of the vector meson($B_{c}^{*}$ meson) against the size parameter $b$.
We initially assume a set of values for the parameters $\alpha_{s}$, $m_{b}$,
$m_{c}$ and we consturct a $5\times 5$ matrix and diagonalize the matrix to
obtain mass of $B_{c}$ meson states. Then we tune these parameters to obtain
an agreement with the experimental value for the mass of $B_{c}$ meson.

Parameters of the model $m_{c}$ (MeV) 1525.0 $m_{b}$ (MeV) 4825.0 $b$ (fm)
0.350 $\alpha_{s}$ 0.3 $a_{c}~{}{\rm MeV~{}fm^{-1}}$ 175

Table 1: Mass shifts (in MeV). Bare $c\bar{b}$ State |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
$n~{}^{2S+1}L_{J}$ | BD | $B_{s}D_{s}$ | $B_{0}D_{0}$ | $B^{*}D$ | $B^{*}_{s}D_{s}$ | $B^{*}D^{*}$ | $B^{*}_{s}D^{*}_{s}$ | Total
$1~{}^{1}S_{0}$ | 0 | 0 | 0 | -5.661 | -5.033 | -10.434 | -9.328 | -30.456
$1~{}^{3}S_{1}$ | -2.046 | -1.805 | -2.052 | -3.955 | -3.496 | -7.293 | -6.488 | -27.135
$1~{}^{3}P_{0}$ | -57.922 | -57.406 | -57.946 | 0 | 0 | -19.088 | -18.932 | -211.294
$1~{}^{1}P_{1}$ | 0 | 0 | 0 | -18.49 | -18.393 | -37.603 | -37.901 | -112.387
$1~{}^{3}P_{1}$ | 0 | 0 | 0 | -38.390 | -38.049 | 0 | 0 | -76.439
$1~{}^{3}P_{2}$ | -40.618 | -40.314 | -40.632 | 0 | 0 | 0 | 0 | -121.557
$2~{}^{1}S_{0}$ | 0 | 0 | 0 | -1.547 | -1.361 | -2.837 | -2.523 | -8.268
$2~{}^{3}S_{1}$ | -0.546 | -0.476 | -0.548 | -1.929 | -1.711 | -1.049 | -0.920 | -7.179
$1~{}^{3}D_{1}$ | -30.675 | -30.312 | -30.682 | -15.326 | -15.146 | -3.077 | -3.044 | -128.262
$1~{}^{1}D_{2}$ | 0 | 0 | 0 | -3.147 | -3.111 | -27.643 | -27.49 | -61.391
$1~{}^{3}D_{2}$ | 0 | 0 | 0 | -27.214 | -27.552 | -69.486 | -68.957 | -193.209
$1~{}^{3}D_{3}$ | -40.753 | -40.359 | -40.772 | -54.308 | -53.783 | -20.835 | -20.606 | -230.663
$2~{}^{3}P_{0}$ | -148.72 | -146.395 | -148.828 | 0 | 0 | -48.589 | -47.903 | -540.435
$2~{}^{1}P_{1}$ | 0 | 0 | 0 | -25.081 | -24.744 | -49.343 | -48.741 | -147.909
$2~{}^{3}P_{1}$ | 0 | 0 | 0 | -98.623 | -97.088 | 0 | 0 | -195.711
$2~{}^{3}P_{2}$ | -79.114 | -77.890 | -79.171 | 0 | 0 | 0 | 0 | -236.175
Table 2: $B_{c}$ meson mass spectrum (in MeV). State |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
$n~{}^{2S+1}L_{J}$ | This work | Ref.[36] | Ref. [37] | Ref. [38] | Ref. [33] | Ref.[39] | Ref.[40] | Ref.[41]
$1~{}^{1}S_{0}$ | 6276 | 6247 | 6253 | 6260 | 6264 | 6270 | 6271 | 6286
$1~{}^{3}S_{1}$ | 6347 | 6308 | 6317 | 6340 | 6337 | 6332 | 6338 | 6341
$1~{}^{3}P_{0}$ | 6654 | 6689 | 6683 | 6680 | 6700 | 6699 | 6706 | 6701
$1P$ | 6683 | 6738 | 6717 | 6730 | 6730 | 6734 | 6741 | 6737
$1P^{\prime}$ | 6729 | 6757 | 6729 | 6740 | 6736 | 6749 | 6750 | 6.760
$1~{}^{3}P_{2}$ | 6732 | 6773 | 6743 | 6760 | 6747 | 6762 | 6768 | 6772
$2~{}^{1}S_{0}$ | 6853 | 6853 | 6867 | 6850 | 6856 | 6835 | 6855 | 6882
$2~{}^{3}S_{1}$ | 6881 | 6886 | 6902 | 6900 | 6899 | 6881 | 6887 | 6914
$1~{}^{3}D_{1}$ | 6990 |  | 7008 | 7010 | 7012 | 7072 | 7028 | 7019
$1D$ | 6985 |  | 7001 | 7020 | 7012 | 7077 | 7041 | 7028
$1D^{\prime}$ | 7010 |  | 7016 | 7030 | 7009 | 7079 | 7036 | 7028
$1~{}^{3}D_{3}$ | 7021 |  | 7007 | 7040 | 7005 | 7081 | 7045 | 7032
$2~{}^{3}P_{0}$ | 7107 |  | 7088 | 7100 | 7108 | 7091 | 7122 |
$2P$ | 7123 |  | 7113 | 7140 | 7135 | 7126 | 7145 |
$2P^{\prime}$ | 7128 |  | 7124 | 7150 | 7142 | 7145 | 7150 |
$2~{}^{3}P_{2}$ | 7136 |  | 7134 | 7160 | 7153 | 7156 | 7164 |

We evaluate the bare state masses and shifts due to $BD$, $B_{s}D_{s}$,
$B^{0}D^{0}$, $B^{*}D$, $B^{*}_{s}D_{s}$, $B^{*}D^{*}$ and
$B^{*}_{s}D^{*}_{s}$ loops (with $M_{B}=5279.26~{}\rm{MeV}$,
$M_{B_{s}}=5366.77~{}\rm{MeV}$, $M_{B^{0}}=5279.58~{}\rm{MeV}$,
$M_{B^{*}}=5324.6~{}\rm{MeV}$, $M_{B^{*}_{s}}=5415.4~{}\rm{MeV}$,
$M_{D}=1869.61~{}\rm{MeV}$, $M_{D_{s}}=1968.30~{}\rm{MeV}$,
$M_{D_{0}}=1864.84~{}\rm{MeV}$, $M_{D^{*}}=2006.96~{}\rm{MeV}$ and
$M_{D^{*}_{s}}=2112.1$ MeV).

We consider the mixing between ${}^{3}P_{1}$ and ${}^{1}P_{1}$ and also
between ${}^{3}D_{2}$ and ${}^{1}D_{2}$ eigenstates due to the spin-orbit
interaction terms. The mixing yields the $B_{c}$ mesons with $J=1$ and $J=2$
states $P1$, $P1^{\prime}$, $D2$ and $D2^{\prime}$. These states are in
general represented as

$\displaystyle\lvert nL^{\prime}\rangle=\lvert
n~{}^{1}L_{J}\rangle\cos\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\sin\theta_{nL}$ (23) $\displaystyle\lvert
nL\rangle=-\lvert n~{}^{1}L_{J}\rangle\sin\theta_{nL}+\lvert
n~{}^{3}L_{J}\rangle\cos\theta_{nL}$ (24) $~{}J=L=1,2,3,\cdots$

where $\theta_{nL}$ is a mixing angle, and the primed state has the heavier
mass. For $L=J=1$ we have mixing of P states, with mixing angles
$\theta_{1P}=0.4^{\circ}$ and $\theta_{2P}=0.05^{\circ}$. Similarly for
$L=J=2$ we have mixing of D states, with mixing angle
$\theta_{1D}=0.05^{\circ}$.

Table 2 shows the results for the masses of the $c\bar{b}$ states. The
calculated masses are compared with other theoretical models and with
available experimental data. Overall we obtain a good fit to the spectrum. The
hyperfine mass splitting of singlet and triplet states
$m(n~{}^{3}S_{1})-m({}^{1}S_{0})$ can shed light on the spin dependence of the
energy levels. We obtain a hyperfine splitting of 71 MeV which is in good
agreement with the other theoretical models. This difference is justified by
calculating the ${}^{3}S_{1}-{}^{1}S_{0}$ splitting of the ground state which
is given by

$M({}^{3}S_{1})-M({}^{1}S_{0})=\frac{32\pi\alpha_{s}|\psi(0)|^{2}}{9m_{c}m_{b}}$
(25)

We predict a mass of 6853 MeV for the first radial excitation $B_{c}$(2S)
which is in good agreement with the experimental value 6842$\pm$4$\pm$5 MeV of
$B_{c}$(2S)[42]. The first radial excitation $B_{c}$(2S) is heavier than
$B_{c}$(1S) by 577 MeV. The hyperfine splitting of $2S$ states is 28 MeV. The
difference between the $B^{*}_{c}$(2S) and $B^{*}_{c}$(1S) masses turns out to
be 534 MeV.

###  3.2 Decay Properties

The dominant multipole transitions $E1$ and $M1$ have been studied and this
helps us to extract information about new meson states and discover them.
Radiative transitions are very important and interesting because the charge
structure of the mesons and their quantum numbers can be determined through
these transitions. We consider $E1$ and $M1$ radiative transitions non
relativistically for $B_{c}$ meson states. This potential model approach
provides deatiled predictions which are further compared with experimental
data. The possible $E_{1}$ decay modes listed in Table 3 are calculated and
values of widths are given in the same. Though most of the predictions
qualitatively agree with other theoretical models, some differ. These
differences are due to different phase spaces arising from the different mass
predictions. Wavefunction effects also play a major role in determining decay
widths. The choice of ${}^{3}P_{1}-^{1}P_{1}$ mixing angles in different
models is also a cause for the significant difference between the theoretical
models in case of transitions involving $P1$ and $P1^{\prime}$ states.

The radiative M1 transition rates of $B_{c}$ meson states are calculated and
the results are presented in Table 4. The M1 decay widths for allowed
transitions (n${}^{3}S_{1}\rightarrow n^{\prime}{{}^{1}S_{0}}+\gamma$,
$n=n^{\prime}$ ) have been calculated and are compared with other non
relativistic quark models [33, 43, 41]. The decay widths of hindered
transitions($n\neq n^{\prime}$) are zero in the non relativistic limit due to
the orthogonality of the initial and final state wave functions. The hindered
$M1$ transition rates are enhanced in this model by incorporating relativistic
effects to the wavefuncion.

We have calculated weak decay widths of $B_{c}$ meson. The decay widths are
calculated using $|V_{bc}|=0.044$ [44] and $|V_{cs}|=0.975$ [44]. Calculated
values of $\Gamma_{1}(\bar{b}\rightarrow X)$ is $9.628\times
10^{-4}~{}\rm{eV}$, $\Gamma_{2}(c\rightarrow X)$ is $7.712\times
10^{-4}~{}\rm{eV}$ and $\Gamma_{3}$ is $3.56\times 10^{-6}~{}\rm{eV}$. Adding
these results we get the total decay width
$\Gamma(\rm{total})=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}=18.104\times
10^{-4}~{}\rm{eV}$ corresponding to a life time of $\tau=0.364~{}\rm{ps}$. The
values of decay constant in various theoretical models are listed in Table 5
and in Table 6 we compare the life time of $B_{c}$ meson calculated in our
model with other models.

Table 3: E1 transition rates of $B_{c}$ meson. Transition | k0 | This Work | Ref. [39] | Ref. [33] | Ref. [37] | Ref.[41]
---|---|---|---|---|---|---
| MeV | keV | keV | keV | keV | keV
$1^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ | 307 | 30.7 | 75.5 | 79.2 | 65.3 | 74.2
$1P\rightarrow 1^{3}S_{1}\gamma$ | 336 | 49.4 | 87.1 | 99.5 | 77.8 | 75.8
$1P^{\prime}\rightarrow 1^{3}S_{1}\gamma$ | 382 | 74.3 | 13.7 | 0.1 | 8.1 | 26.2
$1^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ | 385 | 112.7 | 122 | 112.6 | 102.9 | 126
$1P\rightarrow 1^{1}S_{0}\gamma$ | 407 | 31.9 | 18.4 | 0 | 11.6 | 32.5
$1P^{\prime}\rightarrow 1^{1}S_{0}\gamma$ | 453 | 44.5 | 147 | 56.4 | 131.1 | 128
$2^{3}S_{1}\rightarrow 1^{3}P_{0}\gamma$ | 227 | 8.0 | 5.53 | 7.8 | 7.7 | 9.6
$2^{3}S_{1}\rightarrow 1P\gamma$ | 198 | 8.5 | 7.65 | 14.5 | 12.8 | 13.3
$2^{3}S_{1}\rightarrow 1P^{\prime}\gamma$ | 152 | 3.9 | 0.74 | 0 | 1.0 | 2.5
$2^{3}S_{1}\rightarrow 1^{3}P_{2}\gamma$ | 149 | 6.3 | 7.59 | 17.7 | 14.8 | 14.5
$2^{1}S_{0}\rightarrow 1P\gamma$ | 170 | 5.0 | 1.05 | 0 | 1.9 | 6.4
$2^{1}S_{0}\rightarrow 1P^{\prime}\gamma$ | 124 | 1.9 | 4.40 | 5.2 | 15.9 | 13.1
$2^{3}P_{0}\rightarrow 1^{3}S_{1}\gamma$ | 760 | 0 |  | 21.9 | 16.1 |
$2P\rightarrow 1^{3}S_{1}\gamma$ | 776 | 0 |  | 22.1 | 15.3 |
$2P^{\prime}\rightarrow 1^{3}S_{1}\gamma$ | 781 | 0 |  | 2.1 | 2.5 |
$2^{3}P_{2}\rightarrow 1^{3}S_{1}\gamma$ | 789 | 0 |  | 25.8 | 19.2 |
$2P\rightarrow 1^{1}S_{0}\gamma$ | 847 | 0 |  |  | 3.1 |
$2P^{\prime}\rightarrow 1^{1}S_{0}\gamma$ | 852 | 0 |  |  | 20.1 |
$2^{3}P_{0}\rightarrow 2^{3}S_{1}\gamma$ | 197 | 15.0 | 34.0 | 41.2 | 25.5 |
$2P\rightarrow 2^{3}S_{1}\gamma$ | 242 | 31.7 | 45.3 | 54.3 | 32.1 |
$2P^{\prime}\rightarrow 2^{3}S_{1}\gamma$ | 247 | 48.2 | 10.4 | 5.4 | 5.9 |
$2^{3}P_{2}\rightarrow 2^{3}S_{1}\gamma$ | 255 | 49.5 | 75.3 | 73.8 | 49.4 |
$2P\rightarrow 2^{1}S_{0}\gamma$ | 270 | 47.5 | 13.8 |  | 8.1 |
$2P^{\prime}\rightarrow 2^{1}S_{0}\gamma$ | 275 | 68.7 | 90.5 |  | 58.0 |
Table 4: M1 transition rates for the $B_{c}$ meson. Transition | $k_{0}$ | This work | Ref. [41] | Ref. [43] | Ref. [39] | Ref.[33]
---|---|---|---|---|---|---
| $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$ | $\Gamma(keV)$
$1~{}^{3}S_{1}\rightarrow 1^{1}S_{0}\gamma$ | 71 | 0.059 | 0.190 | 0.060 | 0.073 | 0.135
$2~{}^{3}S_{1}\rightarrow 2^{1}S_{0}\gamma$ | 28 | 0.0017 | 0.043 | 0.010 | 0.030 | 0.029
Table 5: Comparison of predictions for the pseudo scalar decay constant of the $B_{c}$ meson. Parameter | This work | Ref.[45] | Ref. [46] | Ref.[47] | Ref.[48]
---|---|---|---|---|---
$f_{B_{c}}$ | 439.735 | 500 | 512 | 479 | 440$\pm$20
Table 6: Comparison of life time of $B_{c}$ meson (in ps). This work | Experiment[44] | Ref.[31] | Ref.[37] | Ref.[49] | Ref. [50]
---|---|---|---|---|---
0.379 | 0.452$\pm 0.033$ | 0.47 | 0.55$\pm 0.15$ | 0.50 | 0.75

The $c\bar{b}$ states which lie below BD threshold are stable against strong
decays. However, the states which are above the BD threshold undergo two body
strong decays. We have calculated strong decay widths of $c\bar{b}$ states
which lie above the BD threshold using the equation (13). The decay widths are
calculated within the ${}^{3}P_{0}$ pair creation model. The results are
presented in Table 7.

Table 7: Strong decay widths of the $B_{c}$ meson. Transition | $\Gamma(MeV)$
---|---
$2~{}^{1}P_{1}\rightarrow B^{*}+D$ | 54.599
$2~{}^{3}P_{1}\rightarrow B^{*}+D$ | 2.145
$2~{}^{3}P_{2}\rightarrow B+D$ | 99.386
$2~{}^{3}P_{2}\rightarrow B^{0}+D^{0}$ | 108.185
$2~{}^{3}P_{2}\rightarrow B^{*}+D$ | 31.247
$1~{}^{3}D_{2}\to B^{*}+D$ | 0.198
$1~{}^{3}D_{2}\to B^{*}_{s}+D_{s}$ | 5.837
$1~{}^{3}D_{2}\to B^{*}+D^{*}$ | 2.123
$1~{}^{3}D_{2}\to B^{*}_{s}+D^{*}_{s}$ | 20.885

##  4 Conclusions

From the study of mass spectra and decay properties of $c\bar{b}$ states using
a non relativistic quark model with coupled channel effects we draw the
following conclusions

  1. [i)]

  2. (1)

Our results for mass spectra for $c\bar{b}$ states with coupled channel
effects included for ground states agree within a few MeV, when compared to
other theoretical models. For calculation of mass spectrum, the coupled
channel effects are notably visible.

  3. (2)

Our calculated value of the hyperfine splitting of the ground state vector and
pseudo scalar $c\bar{b}$ states 71 MeV, agree with the value predicted by
Penin et al, $M(B^{*}_{c})-M(B_{c})=50\pm 17(th)~{}\rm{MeV}$[51].

  4. (3)

The ground state $B_{c}$ and $B^{*}_{c}$ meson masses lie within the ranges
$6194~{}\rm{MeV}<M_{B_{c}}<6292~{}\rm{MeV}$ and
$6284~{}\rm{MeV}<M_{B^{*}_{c}}<6357~{}\rm{MeV}$ as quoted by Kwong and
Rosner[22].

  5. (4)

While calculating $M1$ hindered transition rates, we find relativistic effects
play an important role. The zero rates of hindered transitions are due to
wavefunction orthogonality. The inclusion of the relativistic effects may
increase the values of hindered transition rates.

  6. (5)

We find, our calculated $E1$ decay rates are in good agreement with the other
theoretical model calculations. The differences found in decay rates are
ascribed to differences in mass predictions, wavefunction effects and mixing
angles.

  7. (6)

Branching ratio for $b$-quark decays is 53% , for $c$-quark decays 42% and for
annihilation channel it is 5% in estimating the weak decay widths.

  8. (7)

The life time of $c\bar{b}$ state, $f_{B_{c}}$ and strong decay widths
predicted in this work are found to be in good agreement with experimental
values as well as with other theoretical predictions.

The NRQM in this study has proven successful in describing $B_{c}$ meson
properties. All the observed states can be successfully accommodated in our
model.

Acknowledgements

One of the authors (APM) is grateful to BRNS, DAE, India for granting the
project and JRF (37(3)/14/21/2014BRNS).

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