#  Calculation of $1/m^{2}_{b}$ corrections to
$\langle\Lambda_{b}{(v,s)}|\bar{b}\gamma^{\lambda}\gamma_{5}b|\Lambda_{b}(v,s)\rangle$
for polarized $\Lambda_{b}$ in the Bethe-Salpeter equation approach

L. Zhang1111<EMAIL_ADDRESS>X.-H. Guo1222 Corresponding author.
<EMAIL_ADDRESS>1 College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, People’s Republic of China

###### Abstract

The heavy baryon $\Lambda_{Q}~{}(Q=b$ or $c)$ can be regarded as composed of a
heavy quark and a scalar light diquark which has good spin and isospin quantum
numbers. In this picture we establish the Bethe-Salpeter (BS) equation for
$\Lambda_{Q}$ to second order in the $1/m_{Q}$ expansion. With the kernel
containing both the scalar confinement and the one-gluon-exchange terms we
solve the BS equation numerically. The value of the spin-dependant form factor
for the matrix element
$\langle\Lambda_{b}{(v,s)}|\bar{b}\gamma^{\lambda}\gamma_{5}b|\Lambda_{b}(v,s)\rangle$,
$\epsilon_{b}$, which is non-zero at order $1/m^{2}_{b}$, is obtained
numerically from our model.

Bethe-Salpeter equation, Heavy quark effective theory, Heavy baryons,
$\epsilon_{b}$

###### pacs:

11.10.St, 12.39.Hg, 14.20.Mr, 14.20.Lq

## I. Introduction

In the past decades there has been much progress in heavy flavor physics due
to the discovery of the new flavor and spin symmetries $SU(2)_{f}\times
SU(2)_{s}$ in the heavy quark limit and the establishment of the heavy quark
effective theory (HQET) isg89 . The Large Hadron Collider (LHC) will provide
much more data for heavy hadrons, and hence it will be able to test the
standard model (SM) more accurately. One may expect more precise measurement
of Cabibbo-Kobayashi-Maskawa matrix elements such as $V_{ub}$ in the near
future.

There have been extensive studies in literature on inclusive semileptonic
decays of bottom hadrons, $H_{b}\rightarrow Xe\bar{\nu}_{e}$ ($X$ represents
all hadrons in the final states) Chay90 -Ura05 , especially since the
establishment of HQET. These studies include corrections to the leading order
results both from perturbative QCD ($\alpha_{s}(m_{b})$) terms and from
nonperturbative terms which are suppressed by powers of $m_{b}$. It was
pointed out that there are no $1/m_{b}$ corrections to leading order in the
$1/m_{b}$ expansion for the differential decay widths of inclusive
semileptonic decays of bottom hadrons, $d\Gamma/dq^{2}dE_{e}$, where $q$ is
the total momentum of the electron and the neutrino and $E_{e}$ is the
electron energy in the $\Lambda_{b}$’s rest frame Chay90 . Bigi et al. studied
$1/m^{2}_{b}$ corrections to the decay width $d\Gamma/dE_{e}$ Big93 . Manohar
and Wise analyzed extensively $1/m^{2}_{b}$ corrections to
$d\Gamma/dq^{2}dE_{e}$ for unpolarized bottom hadron $H_{b}$ and for polarized
$\Lambda_{b}$ Man94 . In recent years, theoretical calculations for the
inclusive semileptonic decay widths and for the moments of inclusive
observables have been carried out to order $1/m^{3}_{b}$ and
$\alpha^{2}_{s}\beta_{0}~{}(\beta_{0}=11-2n_{f}/3,$ $n_{f}$ is the number of
quark flavors) Fal98 -Ura05 .

At leading order in $\alpha_{s}(m_{b}$) the polarized differential
semileptonic decay rate for $\Lambda_{b}\rightarrow X_{u}e\bar{\nu}_{e}$
(where $X_{u}$ represents all hadrons containing an up quark) is

$\displaystyle\frac{d\Gamma}{dq^{2}dE_{e}dE_{\nu}d\cos\theta}$
$\displaystyle=$
$\displaystyle\frac{|V_{jb}|^{2}G^{2}_{F}}{2\pi^{3}}\Big{[}W_{1}q^{2}+W_{2}\Big{(}2E_{e}E_{\nu}-\frac{1}{2}q^{2}\Big{)}+W_{3}q^{2}(E_{e}-E_{\nu})\Big{]}$
(1)
$\displaystyle+\frac{|V_{jb}|^{2}G^{2}_{F}}{4\pi^{3}}\cos\theta\Bigg{\\{}\Big{[}G_{1}q^{2}+G_{2}\bigg{(}2E_{e}E_{\nu}-\frac{1}{2}q^{2}\bigg{)}+G_{3}q^{2}|(E_{e}-E_{\nu})|\Big{]}\bigg{(}E_{e}+E_{\nu}-\frac{q^{2}}{2E_{e}}\bigg{)}$
$\displaystyle+G_{6}(q^{2}-4E_{e}E_{\nu})-G_{8}q^{2}-G_{9}q^{2}\bigg{(}E_{e}-E_{\nu}+\frac{q^{2}}{2E_{e}}\bigg{)}\Bigg{\\}},$

where the electron mass is neglected, $j=u,c$, $\theta$ is the angle between
the electron three-momentum and the spin vector of $\Lambda_{b}$ in the
$\Lambda_{b}$’s rest frame, $E_{\nu}$ is the neutrino energy in the rest frame
of $\Lambda_{b}$, $q^{2}=(p_{e}+p_{\nu})^{2}$ is the invariant mass of the
lepton pair, the kinematic variables are to be integrated over the region
$q^{2}\leq 4E_{e}E_{\nu}$, $W_{1}$, $W_{2}$, and $W_{3}$ are the form factors
in spin-independent hadronic tensor $W^{\mu\nu}$, $G_{1}$, $G_{2}$, $G_{3}$,
$G_{6}$, $G_{8}$, and $G_{9}$ are the form factors in spin-dependent hadronic
tensor $W^{\mu\nu}_{S}$. $W^{\mu\nu}$ and $W^{\mu\nu}_{S}$ have the following
forms Man94 :

$\displaystyle
W^{\mu\nu}=-g^{\mu\nu}W_{1}+v^{\mu}v^{\nu}W_{2}-i\epsilon^{\mu\nu\alpha\beta}v_{\alpha}q_{\beta}W_{3}+q^{\mu}q^{\nu}W_{4}+(q^{\mu}v^{\nu}+q^{\nu}v^{\mu})W_{5},$
(2)

and

$\displaystyle W^{\mu\nu}_{S}$ $\displaystyle=$ $\displaystyle-q\cdot
s[-g^{\mu\nu}G_{1}+v^{\mu}v^{\nu}G_{2}-i\epsilon^{\mu\nu\alpha\beta}v_{\alpha}q_{\beta}G_{3}+q^{\mu}q^{\nu}G_{4}+(q^{\mu}v^{\nu}+q^{\nu}v^{\mu})G_{5}]$
(3)
$\displaystyle+(s^{\mu}v^{\nu}+s^{\nu}v^{\mu})G_{6}+(s^{\mu}q^{\nu}+s^{\nu}q^{\mu})G_{7}+i\epsilon^{\mu\nu\alpha\beta}v_{\alpha}s_{\beta}G_{8}+i\epsilon^{\mu\nu\alpha\beta}q_{\alpha}s_{\beta}G_{9}),$

where $s^{\mu}$ is the spin vector of $\Lambda_{b}$ and $v^{\mu}$ is the
velocity of $\Lambda_{b}$.

When the electron mass is ignored, the form factors $W_{4}$, $W_{5}$, $G_{4}$,
$G_{5}$, and $G_{7}$ in Eqs.(2)(3) do not contribute to the differential
inclusive semleptomic decay rate of polarized $\Lambda_{b}$ Man94 . The
expressions for other form factors, $W_{1}$, $W_{2}$, $W_{3}$, $G_{1}$,
$G_{2}$, $G_{3}$, $G_{6}$, $G_{8}$, and $G_{9}$, were obtained to second order
in the $1/m_{b}$ expansion in Ref. Man94 . It was shown that $1/m^{2}_{b}$
corrections were characterized by two parameters, $\mu^{2}_{\pi}$ and
$\epsilon_{b}$, for polarized $\Lambda_{b}$ decays (the spin energy is zero
for $\Lambda_{b}$). $\mu^{2}_{\pi}$ is the kinetic energy which is defined as

$\displaystyle\mu^{2}_{\pi}=-\frac{\langle\Lambda_{b}|\bar{h}_{v}(iD_{\bot})^{2}h_{v}|\Lambda_{b}\rangle}{2m_{b}},$
(4)

where $h_{v}$ denotes the field of $b$ quark in HQET,
$D^{\mu}_{\bot}=D^{\mu}-v^{\mu}v\cdot D$ with $D^{\mu}$ being the covariant
derivative. The value of $\mu^{2}_{\pi}$ was calculated for the first time in
Ref. Guo07 . $\epsilon_{b}$ is defined as

$\displaystyle\langle\Lambda_{b}(v,s)|\bar{b}\gamma^{\lambda}\gamma_{5}b|\Lambda_{b}(v,s)\rangle$
$\displaystyle=$
$\displaystyle(1+\epsilon_{b})\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)$
(5) $\displaystyle=$ $\displaystyle(1+\epsilon_{b})s^{\lambda},$

where $u_{\Lambda_{b}}(v,s)$ is the Dirac spinor of $\Lambda_{b}$ with
helicity $s$. $\epsilon_{b}$ is equal to zero at leading order in the
$1/m_{b}$ expansion due to the heavy quark symmetry. It is also zero at the
first order in $1/m_{b}$ expansion because of the current conservation Luk90 .
It is the aim of the present paper to calculate the value of $\epsilon_{b}$ at
order $1/m^{2}_{b}$. This will be important for the precise measurement of
$V_{ub}$ in polarized $\Lambda_{b}$ decays.

When the quark mass is very heavy compared with the QCD scale $\Lambda_{QCD}$,
the light degrees of freedom in a heavy baryon $\Lambda_{Q}$ ($Q=b$ or $c$)
become blind to the flavor and spin quantum numbers of the heavy quark because
of the $SU(2)_{f}\times SU(2)_{s}$ symmetries. Therefore, the angular momentum
and flavor quantum numbers of the light degrees of freedom become good quantum
numbers which can be used to classify heavy baryons, and $\Lambda_{Q}$
corresponds to the state in which the angular momentum of the light degrees of
freedom is zero. So it is natural to regard the heavy baryon as composed of
one heavy quark and a light diquark. When $1/m_{Q}$ corrections are taken into
account, since the isospin of $\Lambda_{Q}$ is zero, the isospin of the light
degrees of freedom is also zero. Therefore, the spin of the light degrees of
freedom should also be zero in order to guarantee that the total wave function
of $\Lambda_{Q}$ is anti-symmetric. Hence the spin and isospin of the light
degrees of freedom are still fixed even when $1/m_{Q}$ corrections are taken
in account. Therefore, we still treat $\Lambda_{Q}$ as composed of a heavy
quark and a light diquark.

Based on the above picture, the three-body system is simplified to a two-body
system. We will establish the Bethe-Salpeter (BS) equation for $\Lambda_{Q}$
in this picture to order $1/m^{2}_{Q}$. Then, we will solve this equation
numerically by assuming that the kernel contains the scalar confinement and
the one-gluon-exchange terms. Furthermore, we will express $\epsilon_{b}$ in
terms of the BS amplitude of $\Lambda_{b}$ in order to calculate the value of
$\epsilon_{b}$ to second order in the $1/m_{b}$ expansion.

The remainder of this paper is organized as follows. In Sec. II we will
establish the BS equation for $\Lambda_{Q}$ to second order in the $1/m_{Q}$
expansion and discuss the form of its kernel. Then, we will solve the equation
numerically. In Sec. III we will express $\epsilon_{b}$ in terms of the BS
amplitude of $\Lambda_{b}$ and calculate the value of $\epsilon_{b}$ for
different model parameters. Finally, Sec. IV is reserved for summary and
discussion.

## II. Bethe-Salpeter equation for $\Lambda_{Q}$ to second order in the
$1/m_{Q}$ expansion

As we discussed in Introduction, $\Lambda_{Q}$ is regarded as a bound state of
a heavy quark and a light scalar diquark. Hence we can define the BS amplitude
of $\Lambda_{Q}$ as the following:

$\displaystyle\chi(x_{1},x_{2},P)=\langle
0|T\psi(x_{1})\varphi(x_{2})|\Lambda_{Q}(P)\rangle,$ (6)

where $\psi(x_{1})$ and $\varphi(x_{2})$ are the field operators of the heavy
quark at position $x_{1}$ and the light scalar diquark at position $x_{2}$,
respectively, $P=m_{\Lambda_{Q}}v$ is the momentum of $\Lambda_{Q}$, and $v$
is its velocity. Let $m_{Q}$ and $m_{D}$ represent the masses of the heavy
quark and the light diquark in the baryon,
$\lambda_{1}=\frac{m_{Q}}{m_{Q}+m_{D}}$,
$\lambda_{2}=\frac{m_{D}}{m_{Q}+m_{D}}$, and $p$ represents the relative
momentum of the two constituents. Then, the BS amplitude in momentum space is
defined as

$\displaystyle\chi(x_{1},x_{2},P)=e^{iPX}\int\frac{d^{4}p}{(2\pi)^{4}}e^{ipx}\chi_{P}(p),$
(7)

where $X=\lambda_{1}x_{1}+\lambda_{2}x_{2}$ is the coordinate of the center of
mass and $x=x_{1}-x_{2}$.

It is straightforward to prove that the BS equation for $\Lambda_{Q}$ has the
following form in momentum space:

$\displaystyle\chi_{P}(p)=S_{F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K(P,p,q)\chi_{P}(q)S_{D}(p_{2}),$
(8)

where $p_{1}=\lambda_{1}P+p$, $p_{2}=-\lambda_{2}P+p$ are the momenta of the
heavy quark and the light scalar diquark, respectively, $K(P,p,q)$ is the
kernel which is defined as the sum of two particle irreducible diagrams,
$S_{F}(p_{1})$ and $S_{D}(p_{2})$ are propagators of the heavy quark with
momentum $p_{1}$ and the light diquark with momentum $p_{2}$,

$\displaystyle S_{F}(p_{1})=\frac{i}{\not{p}_{1}-m_{Q}+i\varepsilon},$ (9)
$\displaystyle S_{D}(p_{2})=\frac{1}{p_{2}^{2}-m_{D}^{2}+i\varepsilon}.$ (10)

In order to solve the BS equation for $\Lambda_{Q}$ to second order in the
$1/m_{Q}$ expansion, we rewrite Eq. (8) as the following:

$\displaystyle\chi_{0P}(p)+\frac{1}{m_{Q}}\chi_{1P}(p)+\frac{1}{m_{Q}^{2}}\chi_{2P}(p)$
$\displaystyle=$
$\displaystyle\bigg{(}S_{0F}(p_{1})+\frac{1}{m_{Q}}S_{1F}(p_{1})+\frac{1}{m_{Q}^{2}}S_{2F}(p_{1})\bigg{)}\int\frac{d^{4}q}{(2\pi)^{4}}\bigg{(}K_{0}(P,p,q)$
(11)
$\displaystyle+\frac{1}{m_{Q}}K_{1}(P,p,q)+\frac{1}{m_{Q}^{2}}K_{2}(P,p,q)\bigg{)}\times\Big{(}\chi_{0P}(q)+\frac{1}{m_{Q}}\chi_{1P}(q)$
$\displaystyle+\frac{1}{m_{Q}^{2}}\chi_{2P}(q)\Big{)}S_{D}(p_{2}),$

where we have expanded the BS amplitude, the propagator of the heavy quark,
and the kernel to $1/m^{2}_{Q}$.

It is easy to show that $S_{D}$ remains unchanged in the $1/m_{Q}$ expansion.
Then, by comparing the two sides of Eq. (11) at each order in $1/m_{Q}$, we
have following equations:

to leading order in the $1/m_{Q}$ expansion:

$\displaystyle\chi_{0P}(p)=S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{0P}(q)S_{D}(p_{2}),$
(12)

to first order in the $1/m_{Q}$ expansion:

$\displaystyle\chi_{1P}(p)$ $\displaystyle=$ $\displaystyle
S_{1F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{0P}(q)S_{D}(p_{2})+S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{1}(P,p,q)\chi_{0P}(q)S_{D}(p_{2})$
(13)
$\displaystyle+S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{1P}(q)S_{D}(p_{2}),$

to second order in the $1/m_{Q}$ expansion:

$\displaystyle\chi_{2P}(p)$ $\displaystyle=$ $\displaystyle
S_{2F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{0P}(q)S_{D}(p_{2})+S_{1F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{1}(P,p,q)\chi_{0P}(q)S_{D}(p_{2})$
(14)
$\displaystyle+S_{1F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{1P}(q)S_{D}(p_{2})+S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{2}(P,p,q)\chi_{0P}(q)S_{D}(p_{2})$
$\displaystyle+S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{1}(P,p,q)\chi_{1P}(q)S_{D}(p_{2})+S_{0F}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{0}(P,p,q)\chi_{2P}(q)S_{D}(p_{2}).$

In the following we will use the variables $p_{l}=v\cdot
p-\lambda_{2}m_{\Lambda_{Q}}$, $p_{t}=p-(v\cdot p)v$. Then, in leading order,
first order and second order in the $1/m_{Q}$ expansion, we have the following
equations for the heavy quark propagator using the relation
$p_{1}=\lambda_{1}P+p$ Guo96  Guo00 :

$\displaystyle
S_{0F}(p_{1})=\frac{i(1+\not{v})}{2(p_{l}+E_{0}+m_{D}+i\varepsilon)},$ (15)
$\displaystyle
S_{1F}(p_{1})=-\frac{i(1-\not{v})}{4}+\frac{i\not{p}_{t}}{2(p_{l}+E_{0}+m_{D}+i\varepsilon)}-\frac{i(2E_{1}-p^{2}_{t})(1+\not{v})}{4(p_{l}+E_{0}+m_{D}+i\varepsilon)^{2}},$
(16) $\displaystyle S_{2F}(p_{1})$ $\displaystyle=$
$\displaystyle\frac{i(p_{l}+E_{0}+m_{D})(1+\not{v})}{8}-\frac{i\not{p}_{t}}{4}-\frac{ip^{2}_{t}}{4(p_{l}+E_{0}+m_{D}+i\varepsilon)}$
(17)
$\displaystyle-\frac{2iE_{2}(1+\not{v})+i(2E_{1}-p^{2}_{t})\not{p}_{t}}{4(p_{l}+E_{0}+m_{D}+i\varepsilon)^{2}}+\frac{i(2E_{1}-p^{2}_{t})^{2}(1+\not{v})}{8(p_{l}+E_{0}+m_{D}+i\varepsilon)^{3}},$

where $E_{0}$, $E_{1}$, and $E_{2}$ represent the binding energies, which
satisfy the following relation:

$\displaystyle
m_{\Lambda_{Q}}=m_{Q}+m_{D}+E_{0}+\frac{1}{m_{Q}}E_{1}+\frac{1}{m_{Q}^{2}}E_{2}.$
(18)

$S_{D}(p_{2})$ remains unchanged in the $1/m_{Q}$ expansion, and we rewrite it
as the following using the relation $p_{2}=-\lambda_{2}P+p$:

$\displaystyle S_{D}(p_{2})=\frac{i}{p^{2}_{l}-W^{2}_{p}+i\varepsilon},$ (19)

where $W^{2}_{p}=\sqrt{p^{2}_{t}+m^{2}_{D}}$.

Figure 1:  Diquark-gluon-diquark vetex, $\mu$ and $\alpha$ are the Lorentz and
color indices of the gluon, respectively.

We assume the kernel has the following form:

$\displaystyle-iK_{0}=1\otimes
1V_{1}+v^{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2},$ (20) $\displaystyle-
iK_{1}=1\otimes 1V_{3}+\gamma^{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{4},$
(21) $\displaystyle-iK_{2}=1\otimes
1V_{5}+\gamma^{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{6},$ (22)

where the first terms on the right hand sides of Eqs. (20), (21), and (22)
arise from the scalar confinement and the second ones are from the one-gluon-
exchange diagram, $p_{2}$ and $p^{\prime}_{2}$ are the momenta of the light
diquark attached to the gluon. $v^{\mu}$ and $\gamma^{\mu}$ in Eqs. (20)-(22)
are from the vertex of the heavy quark and the gluon. It is noted that
$\gamma^{\mu}$ in Eq. (20) becomes $v^{\mu}$ because of heavy quark symmetry
(see FIG. 1). The forms of $V_{1}$ and $V_{2}$ were given before Guo96 . In
this work, we assume that $V_{3}$ and $V_{4}$, $V_{5}$ and $V_{6}$ have the
same forms as $V_{1}$ and $V_{2}$, respectively. However, $V_{3}$ and $V_{4}$,
$V_{5}$ and $V_{6}$ are suppressed with respect to $V_{1}$ and $V_{2}$ by
order $\Lambda_{QCD}/m_{Q}$ and $\Lambda^{2}_{QCD}/m^{2}_{Q}$, respectively.
Note that $v^{\mu}$ in Eq. (20) is replaced by $\gamma^{\mu}$ in Eqs. (21) and
(22) since there is no heavy quark spin symmetry when we include $1/m_{Q}$
corrections.

Combining Eqs. (12)-(22), we have the BS equations for $\Lambda_{Q}$ in
leading order, first order, and second order in the $1/m_{Q}$ expansion:

$\displaystyle\chi_{0P}(p)$ $\displaystyle=$
$\displaystyle\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i\big{[}1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}\big{]}\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon},$
(23) $\displaystyle\chi_{1P}(p)$ $\displaystyle=$
$\displaystyle\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{3}+\gamma_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{4}]\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
(24)
$\displaystyle+\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}]\chi_{1P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
$\displaystyle+i\bigg{[}\frac{(-2E_{1}+p^{2}_{t})(1+\not{v})}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}}-\frac{1-\not{v}}{4}+\frac{\not{p}_{t}}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
i\int\frac{d^{4}q}{(2\pi)^{4}}[1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}]\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon},$
$\displaystyle\chi_{2P}(p)$ $\displaystyle=$
$\displaystyle\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}]\chi_{2P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
(25)
$\displaystyle+\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{3}+\gamma_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{4}]\chi_{1P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
$\displaystyle+\frac{i(1+\not{v})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{5}+\gamma_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{6}]\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
$\displaystyle+i\bigg{[}\frac{(-2E_{1}+p^{2}_{t})(1+\not{v})}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}}-\frac{1-\not{v}}{4}+\frac{\not{p}_{t}}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}]\chi_{1P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
$\displaystyle+i\bigg{[}\frac{(-2E_{1}+p^{2}_{t})(1+\not{v})}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}}-\frac{1-\not{v}}{4}+\frac{\not{p}_{t}}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)}\bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{3}+\gamma_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{4}]\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}$
$\displaystyle+i\bigg{[}\frac{(p_{l}+m_{D}+E_{0})(1+\not{v})}{8}-\frac{\not{p}_{t}}{4}+\frac{-p^{2}_{t}}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)}-\frac{2E_{2}(1+\not{v})}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}}$
$\displaystyle~{}~{}-\frac{(2E_{1}-p^{2}_{t})\not{p}_{t}}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}}+\frac{(2E_{1}-p^{2}_{t})^{2}(1+\not{v})}{8(p_{l}+m_{D}+E_{0}+i\varepsilon)^{3}}\bigg{]}\int\frac{d^{4}q}{(2\pi)^{4}}i[1\otimes
1V_{1}+v_{\mu}\otimes(p_{2}+p^{\prime}_{2})^{\mu}V_{2}]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\chi_{0P}(q)\frac{i}{p^{2}_{l}-W^{2}_{P}+i\varepsilon}.$

It is noted that Eqs. (23) and (24) are just the equations obtained before in
the heavy quark limit Guo96 , and to first order in the $1/m_{Q}$ expansion
Guo00 .

In general, $\chi_{P}(p)$ can be expanded as

$\chi_{P}(p)=(A+B\not{v}+C\not{p}+D\not{v}\not{p})u_{\Lambda_{Q}}(v,s),$ (26)

where $A$, $B$, $C$, $D$ are Lorentz scalar functions. It is easy to prove
that

$\displaystyle\not{v}\chi_{0P}(p)=\chi_{0P}(p),$ (27)

hence we have Guo96

$\displaystyle\chi_{0P}(p)=\phi_{0P}(p)u_{\Lambda_{Q}}(v,s),$ (28)

where $\phi_{0P}(p)$ is a scalar function.

However, $\not{v}\chi_{1P}(p)$ and $\not{v}\chi_{2P}(p)$ do not equal
$\chi_{1P}(p)$ and $\chi_{2P}(p)$ respectively, so we define

$\displaystyle\chi^{+}_{1P}(p)=\frac{1+\not{v}}{2}\chi_{1P}(p),$
$\displaystyle\chi^{-}_{1P}(p)=\frac{1-\not{v}}{2}\chi_{1P}(p),$
$\displaystyle\chi^{+}_{2P}(p)=\frac{1+\not{v}}{2}\chi_{2P}(p),$
$\displaystyle\chi^{-}_{2P}(p)=\frac{1-\not{v}}{2}\chi_{2P}(p).$ (29)

It is easy to see that

$\displaystyle\not{v}\chi^{+}_{1,2P}(p)=\chi^{+}_{1,2P}(p),$ (30)
$\displaystyle\not{v}\chi^{-}_{1,2P}(p)=-\chi^{-}_{1,2P}(p).$ (31)

Hence, like $\chi_{0P}(p)$, $\chi^{+}_{1,2P}(p)$ can be expressed as

$\displaystyle\chi^{+}_{1,2P}(p)=\phi^{+}_{1,2P}(p)u_{\Lambda_{Q}}(v,s),$ (32)

where $\phi^{+}_{1P}(p)$ and $\phi^{+}_{2P}(p)$ are scalar functions.

By using Eq. (26) and (31), we can see that

$\displaystyle\chi^{-}_{1,2P}(p)=\not{p}_{t}\phi^{-}_{1,2P}(p)u_{\Lambda_{Q}}(v,s),$
(33)

where again $\phi^{-}_{1P}(p)$ and $\phi^{-}_{2P}(p)$ are scalar functions.
Then we obtain equations for all the scalar functions:

$\displaystyle\phi_{0P}(p)=\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q),$
(34) $\displaystyle\phi^{+}_{1P}(p)$ $\displaystyle=$
$\displaystyle\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{3}+(p_{l}+q_{l})V_{4}]\phi_{0P}(q)$
(35)
$\displaystyle+\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi^{+}_{1P}(q)$
$\displaystyle+\frac{-i(-2E_{1}+p^{2}_{t})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q),$
$\displaystyle\phi^{-}_{1P}(p)$ $\displaystyle=$
$\displaystyle\frac{-i}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q),$
(36) $\displaystyle\phi^{+}_{2P}(p)$ $\displaystyle=$
$\displaystyle\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi^{+}_{2P}(q)$
(37)
$\displaystyle+\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{3}+(p_{l}+q_{l})V_{4}]\phi^{+}_{1P}(q)$
$\displaystyle+\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}(p_{t}\cdot
q_{t}-q^{2}_{t})V_{4}\phi^{-}_{1P}(q)$
$\displaystyle+\frac{-i}{(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{5}+(p_{l}+q_{l})V_{6}]\phi_{0P}(q)$
$\displaystyle+\frac{-i(-2E_{1}+p^{2}_{t})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi^{+}_{1P}(q)$
$\displaystyle+\frac{-i}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]p_{t}\cdot
q_{t}\phi^{-}_{1P}(q)$
$\displaystyle+\frac{-i(-2E_{1}+p^{2}_{t})}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{3}+(p_{l}+q_{l})V_{4}]\phi_{0P}(q)$
$\displaystyle+\Big{[}\frac{ip^{2}_{t}}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}+\frac{iE_{2}}{(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{-i(2E_{1}-p^{2}_{t})^{2}}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{3}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\Big{]}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q)$
$\displaystyle+\frac{-i}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[p_{t}\cdot
q_{t}-p^{2}_{t}]V_{4}\phi_{0P}(q),$ $\displaystyle\phi^{-}_{2P}(p)$
$\displaystyle=$
$\displaystyle\frac{i}{2(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\frac{p_{t}\cdot
q_{t}}{p_{t}\cdot p_{t}}\phi^{-}_{1P}(q)$ (38)
$\displaystyle+\frac{-i}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi^{+}_{1P}(q)$
$\displaystyle+\frac{i}{2(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}\bigg{(}1+\frac{p_{t}\cdot
q_{t}}{p_{t}\cdot p_{t}}\bigg{)}V_{4}\phi_{0P}(q)$
$\displaystyle+\frac{-i}{2(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{3}+(p_{l}+q_{l})V_{4}]\phi_{0P}(q)$
$\displaystyle+\frac{i}{4(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q)$
$\displaystyle+\frac{i(2E_{1}-p^{2}_{t})}{4(p_{l}+m_{D}+E_{0}+i\varepsilon)^{2}(p^{2}_{l}-W^{2}_{P}+i\varepsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}[V_{1}+(p_{l}+q_{l})V_{2}]\phi_{0P}(q).$

From Eqs. (34) and (36) it is easy to see that Guo00

$\displaystyle\phi^{-}_{1P}(p)=\frac{1}{2}\phi_{0P}(p).$ (39)

The numerical solutions for $\phi_{0P}(p)$ and $\phi^{+}_{1P}(p)$ can be
obtained by discretizing the integration region into $n$ pieces (with $n$
sufficiently large). In this way, the integral equations become matrix
equations and the BS scalar functions $\phi_{0P}(p)$ and $\phi^{+}_{1P}(p)$
become $n$ dimensional vectors. Thus $\phi_{0P}(p)$ is the solution of the
eigenvalue equation $(A-I)\phi_{0}=0$, where $A$ is an $n\times n$ matrix
corresponding to the right hand side of Eq. (34). In order to have a unique
solution for the ground state, the rank of $(A-I)$ should be $n-1$. From Eq.
(35), $\phi^{+}_{1P}(p)$ is the solution of $(A-I)\phi_{1}=B$, where $B$ is an
$n$ dimensional vector corresponding to the first and third integral terms on
the right hand side of Eq. (35). In order to have solutions for
$\phi^{+}_{1P}(p)$, the rank of the augmented matrix $(A-I,B)$ should be equal
to that of $(A-I)$, i.e., $B$ can be expressed as linear combination of the
$n-1$ linearly independent columns in $(A-I)$. This is difficult to guarantee
if $B\neq 0$, since the way to divide $(A-I)$ into $n$ columns is arbitrary.
Therefore, following Ref. Guo00  we demand the following condition in order to
have solutions for $\phi^{+}_{1P}(p)$

$\displaystyle\int\frac{d^{4}q}{(2\pi)^{4}}\Bigg{[}\bigg{(}V_{3}+(p_{l}+q_{l})V_{4}\bigg{)}+\frac{-E_{1}+\frac{p^{2}_{t}}{2}}{p_{l}+m_{D}+E_{0}+i\varepsilon}\bigg{(}V_{1}+(p_{l}+q_{l})V_{2}\bigg{)}\Bigg{]}\phi_{0P}(q)=0.$
(40)

The simplest forms for $V_{3}$ and $V_{4}$ satisfiying Eq. (40) are

$\displaystyle
V_{3}=\frac{E_{1}-\frac{p^{2}_{t}}{2}}{p_{l}+m_{D}+E_{0}+i\varepsilon}V_{1},$
(41) $\displaystyle
V_{4}=\frac{E_{1}-\frac{p^{2}_{t}}{2}}{p_{l}+m_{D}+E_{0}+i\varepsilon}V_{2}.$
(42)

With Eq. (40), $\phi^{+}_{1P}(p)$ satisfies the same eigenvalue equation as
$\phi_{0P}(p)$. Therefore, we have

$\displaystyle\phi^{+}_{1P}(p)=\sigma\phi_{0P}(p),$ (43)

where $\sigma$ is a constant of proportionality with mass dimension. It was
determined to be zero by applying Luke’s theorem Luk90  at the zero-recoil
point in HQET Guo00 . Therefore, $\phi^{+}_{1P}(p)$ does not contribute.

As to the forms of $V_{5}$ and $V_{6}$, we simply assume that

$\displaystyle V_{5}=\delta^{2}V_{1},$ $\displaystyle V_{6}=\delta^{2}V_{2},$
(44)

where $\delta$ is a parameter which is of order $\Lambda_{QCD}$.

From Eq. (38) we can see that $\phi^{-}_{2P}(p)$ is only related to
$\phi_{0P}(p)$ and $\phi^{+}_{1P}(p)$, so our goal is to solve the BS
equations of $\phi_{0P}(p)$ and $\phi^{+}_{2P}(p)$ numerically.

In the covariant instantaneous approximation,
$\tilde{V_{i}}=V_{i}|_{p_{l}=q_{l}},i=1,2$ Guo96 -Jin92 . In the meson case,
one has Jin92

$\displaystyle\tilde{V}_{1}|_{meson}=\frac{8\pi\kappa^{\prime}}{[(p_{t}-q_{t})^{2}+\mu^{2}]^{2}}-(2\pi)^{3}\delta^{3}(p_{t}-q_{t})\int\frac{d^{3}k}{(2\pi)^{3}}\frac{8\pi\kappa^{\prime}}{(k^{2}+\mu^{2})^{2}},$
(45)
$\displaystyle\tilde{V}_{2}|_{meson}=-\frac{16\pi}{3}\frac{\alpha_{seff}}{(p_{t}-q_{t})^{2}+\mu^{2}},$
(46)

where $\kappa^{\prime}$ and $\alpha_{seff}$ are coupling parameters related to
the scalar confinement and the one-gluon-exchange terms, respectively. The
parameter $\mu$ is introduced to avoid the infrared divergence in numerical
calculations, and in the end we will take the limit $\mu\rightarrow 0$. In the
baryon case, since scalar confinement term is still due to scalar interaction,
the form of $\tilde{V_{1}}$ need not be changed. Only the parameter
$\kappa^{\prime}$ in the meson case has to be replaced by $\kappa$ which
describes the scalar confinement interaction between the heavy quark and the
diquark. However, the diquark is not a pointlike object, there should be a
form factor in $\tilde{V_{2}}$, $F(Q^{2})(Q=p_{2}-p^{\prime}_{2})$, to
describe the structure of the diquark (see FIG.1) Ans87 ,

$\displaystyle F(Q^{2})=\frac{\alpha_{seff}Q^{2}_{0}}{Q^{2}+Q^{2}_{0}},$ (47)

where $Q^{2}_{0}$ is a parameter which freezes $F(Q^{2})$ when $Q^{2}$ is very
small. In the high energy region the form factor is proportional to
$\frac{1}{Q^{2}}$ which is consistent with perturbative QCD calculations Lep80
. By analyzing the electromagnetic form factor for the proton, it was found
that $Q^{2}_{0}=3.2GeV^{2}$ can lead to consistent results with the
experimental data Ans87 . Based on the above analysis , the form of kernel for
BS equation in the baryon case is taken as Guo96

$\tilde{V}_{1}=\frac{8\pi\kappa}{[(p_{t}-q_{t})^{2}+\mu^{2}]^{2}}-(2\pi)^{3}\delta^{3}(p_{t}-q_{t})\int\frac{d^{3}k}{(2\pi)^{3}}\frac{8\pi\kappa}{(k^{2}+\mu^{2})^{2}},$
(48)
$\displaystyle\tilde{V}_{2}=-\frac{16\pi}{3}\frac{\alpha^{2}_{seff}Q^{2}_{0}}{[(p_{t}-q_{t})^{2}+\mu^{2}][(p_{t}-q_{t})^{2}+Q^{2}_{0}]}.$
(49)

From the BS equation solutions in the meson case it was found that the values
$m_{b}=5.02GeV$ and $m_{c}=1.58GeV$ give predictions which are in good
agreement with experiments Jin92 . Hence in the baryon case we take

$\displaystyle
m_{D}+E_{0}+\frac{1}{m_{b}}E_{1}+\frac{1}{m^{2}_{b}}E_{2}=0.62GeV.$ (50)

The dimension of $\kappa$ is three and that of $\kappa^{\prime}$ is two. This
extra dimension in $\kappa$ should be caused by nonperturbative diagrams which
include the frozen form factor $F(Q^{2})$ at low momentum region. Since
$\Lambda_{QCD}$ is the only parameter which is related to confinement, we
expect that

$\displaystyle\kappa=\delta\kappa^{\prime},$ (51)

where $\delta$ is of order $\Lambda_{QCD}$. It is noted that the two
proportionality parameters in Eqs. (44) and (51) could be different in general
although they are both of order $\Lambda_{QCD}$. However, in order to simplify
our model we assume these two parameters are the same. The value of $\kappa$
was determined to be in the region between $0.02GeV^{3}$ and $0.08GeV^{3}$ in
Ref. Guo07 . Since $\kappa^{\prime}$ is about $0.2GeV^{2}$ Jin92 , we take
$\delta$ to be in the range from $0.1GeV$ to $0.4GeV$.

One notes that $E_{1}\sim\Lambda_{QCD}E_{0}$, $E_{2}\sim\Lambda_{QCD}E_{1}$,
hence we further assume that $E_{1}=\delta E_{0}$, $E_{2}=\delta
E_{1}=\delta^{2}E_{0}$.

Table 1: Values of $\alpha_{seff}$ for different $m_{D}$ and $\kappa$. | $\kappa(GeV^{3})$ | 0.02 | 0.04 | 0.06 | 0.08
---|---|---|---|---|---
$m_{D}=650MeV$ | $\alpha_{seff}$ | 0.66 | 0.7 | 0.73 | 0.75
$m_{D}=700MeV$ | $\alpha_{seff}$ | 0.69 | 0.73 | 0.76 | 0.78
$m_{D}=750MeV$ | $\alpha_{seff}$ | 0.74 | 0.77 | 0.79 | 0.8
$m_{D}=800MeV$ | $\alpha_{seff}$ | 0.77 | 0.8 | 0.82 | 0.83

In general, $\phi_{0P}(p)$ can be the function of $p_{l}$ and $p_{t}$.
Defining $\tilde{\phi}_{0P}(p_{t})\equiv\int(dp_{l}/2\pi)\phi_{0P}$, one gets
immediately the BS equation for $\tilde{\phi}_{0P}(p_{t})$:

$\displaystyle\tilde{\phi}_{0P}(p_{t})=-\frac{1}{2(-W_{p}+m_{D}+E_{0})W_{p}}\int\frac{d^{3}q_{t}}{(2\pi)^{3}}(\tilde{V}_{1}-2W_{p}\tilde{V}_{2})\tilde{\phi}_{0P}(q_{t}),$
(52)

where we have used the residue theorem and selected the upper contour in the
$p_{l}$ plane which has a singular point $-W_{p}+i\varepsilon$ (see FIG.2).

Figure 2:  Three singular points, $-m_{D}-E_{0}-i\varepsilon$,
$-W_{p}+i\varepsilon$ and $W_{p}-i\varepsilon$, in the $p_{l}$ plane.

Substituting $\tilde{V}_{1}$ and $\tilde{V}_{2}$ into Eq. (52) we have Guo96

$\displaystyle(-W_{p}+m_{D}+E_{0})\tilde{\phi}_{0P}(p_{t})$ $\displaystyle=$
$\displaystyle-\frac{1}{2W_{p}}\Bigg{\\{}\int\frac{q^{2}_{t}dq_{t}}{4\pi^{2}}\frac{16\pi\kappa}{(p^{2}_{t}+q^{2}_{t}+u^{2})^{2}-4p^{2}_{t}q^{2}_{t}}\tilde{\phi}_{0P}(q_{t})+\frac{32\pi\alpha^{2}_{seff}Q^{2}_{0}W_{p}}{3(Q_{0}-u^{2})}$
(53)
$\displaystyle\times\int\frac{q^{2}_{t}dq_{t}}{4\pi^{2}}\frac{1}{2p_{t}q_{t}}\bigg{[}ln\frac{(p_{t}+q_{t})^{2}+u^{2}}{(p_{t}-q_{t})^{2}+u^{2}}-ln\frac{(p_{t}+q_{t})^{2}+Q^{2}_{0}}{(p_{t}-q_{t})^{2}+Q^{2}_{0}}\bigg{]}\tilde{\phi}_{0P}(q_{t})\Bigg{\\}}$
$\displaystyle+\frac{1}{2W_{p}}\int\frac{q^{2}_{t}dq_{t}}{4\pi^{2}}\frac{16\pi\kappa}{(p^{2}_{t}+q^{2}_{t}+u^{2})^{2}-4p^{2}_{t}q^{2}_{t}}\tilde{\phi}_{0P}(p_{t}).$

This is a eigenvalue equation of $\tilde{\phi}_{0P}(p_{t})$. Solving this
equation we obtain the values of the model parameter $\alpha_{seff}$
corresponding to different values of $\kappa$ and $m_{D}$ (see TABLE 1).

With the numerical result of $\tilde{\phi}_{0P}(p_{t})$, we can solve out
$\tilde{\phi}^{+}_{2P}(p_{t})$ numerically from Eq. (37).

The normalization equation for the BS equation takes the following form Lur68
:

$\displaystyle\frac{i}{(2\pi)^{4}}\int
d^{4}qd^{4}q^{\prime}\bar{\chi}_{k}(q^{\prime})\frac{\partial}{\partial
k_{0}}[I(q^{\prime},q,k)+K(q^{\prime},q,k)]\chi_{k}(q)=2k_{0},$ (54)

where
$I(q^{\prime},q,P)=\delta^{(4)}(q^{\prime}-q)[S_{F}(p_{1})]^{-1}[S_{D}(-p_{2})]^{-1}$,
in which $p_{1}=\lambda_{1}P+q^{\prime},p_{2}=-\lambda_{2}P+q^{\prime}$, and
$k_{0}$ represents the energy of $\Lambda_{b}$, which is the mass of
$\Lambda_{b}$ in the rest frame of $\Lambda_{b}$. With this condition we can
give the plots of normalized $\tilde{\phi}_{0P}(p_{t})$ and
$\tilde{\phi}^{+}_{2P}(p_{t})$ in FIGs. 3 and 4, respectively.

Figure 3:  Plot of $\tilde{\phi}_{0P}(p_{t})$ with $m_{D}=650MeV$,
$\kappa=0.04$. Figure 4:  Plot of $\tilde{\phi}^{+}_{2P}(p_{t})$ with
$m_{D}=650MeV$, $\kappa=0.04$.

## VI. Numerical results of $\epsilon_{b}$

With the numerical results of the BS amplitudes solved out above, we can
calculate the value of $\epsilon_{b}$. The left hand side of Eq. (5) can be
expressed as the overlap integral of the BS amplitude as the following:

$\displaystyle\langle\Lambda_{b}(v,s)|\bar{b}\gamma^{\lambda}\gamma_{5}b|\Lambda_{b}(v,s)\rangle=\int\frac{d^{4}p}{(2\pi)^{4}}\bar{\chi}_{P}(p)\gamma^{\lambda}\gamma_{5}\chi_{P}(p)S^{-1}_{D}(p_{2}).$
(55)

Hence we have

$\displaystyle(1+\epsilon_{b})\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)=\int\frac{d^{4}p}{(2\pi)^{4}}\bar{\chi}_{P}(p)\gamma^{\lambda}\gamma_{5}\chi_{P}(p)S^{-1}_{D}(p_{2}).$
(56)

Expanding both sides of Eq. (56) to order $1/m^{2}_{Q}$ , we have

$\displaystyle\bigg{(}1+\epsilon_{0b}+\frac{1}{m_{b}}\epsilon_{1b}+\frac{1}{m^{2}_{b}}\epsilon_{2b}\bigg{)}\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)$
(57) $\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\bar{u}_{\Lambda_{b}}(v,s)\bigg{[}\phi_{0P}(p)+\frac{1}{m_{b}}\big{(}\phi^{+}_{1P}(p)+\phi^{-}_{1P}(p)\not{p}_{t}\big{)}+\frac{1}{m^{2}_{b}}\big{(}\phi^{+}_{2P}(p)+\phi^{-}_{2P}(p)\not{p}_{t}\big{)}\bigg{]}$
$\displaystyle\times\gamma^{\lambda}\gamma_{5}\bigg{[}\phi_{0P}(p)+\frac{1}{m_{b}}(\phi^{+}_{1P}(p)+\phi^{-}_{1P}(p)\not{p}_{t})+\frac{1}{m^{2}_{b}}\Big{(}\phi^{+}_{2P}(p)\phi^{-}_{2P}(p)\not{p}_{t}\Big{)}\bigg{]}u_{\Lambda_{b}}(v,s)S^{-1}_{D}(p_{2}),$

where we have expanded $\epsilon_{b}$ in Eq. (5) as

$\displaystyle\epsilon_{b}=\epsilon_{0b}+\frac{1}{m_{b}}\epsilon_{1b}+\frac{1}{m^{2}_{b}}\epsilon_{2b}.$
(58)

By comparing the two sides of Eq. (57) at each order in the $1/m_{Q}$
expansion, we have the following equations:

to leading order in the $1/m_{Q}$ expansion:

$\displaystyle(1+\epsilon_{0b})\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)=\int\frac{d^{4}p}{(2\pi)^{4}}\bar{u}_{\Lambda_{b}}(v,s)\phi_{0P}(p)\gamma^{\lambda}\gamma_{5}\phi_{0P}(p)u_{\Lambda_{b}}(v,s)S^{-1}_{D}(p_{2}),$
(59)

to first order in the $1/m_{Q}$ expansion:

$\displaystyle\epsilon_{1b}\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\bar{u}_{\Lambda_{b}}(v,s)[\phi_{0P}(p)\gamma^{\lambda}\gamma_{5}\phi^{+}_{1P}(p)+\phi_{0P}(p)\gamma^{\lambda}\gamma_{5}\not{p}_{t}\phi^{-}_{1P}(p)$
(60)
$\displaystyle+\phi^{+}_{1P}(p)\gamma^{\lambda}\gamma_{5}\phi_{0P}(p)+\phi^{-}_{1P}(p)\not{p}_{t}\gamma^{\lambda}\gamma_{5}\phi_{0P}(p)]u_{\Lambda_{b}}(v,s)S^{-1}_{D}(p_{2}),$

to second order in the $1/m_{Q}$ expansion:

$\displaystyle\epsilon_{2b}\bar{u}_{\Lambda_{b}}(v,s)\gamma^{\lambda}\gamma_{5}u_{\Lambda_{b}}(v,s)$
$\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\bar{u}_{\Lambda_{b}}(v,s)[\phi_{0P}(p)\gamma^{\lambda}\gamma_{5}\phi^{+}_{2P}(p)+\phi_{0P}(p)\gamma^{\lambda}\gamma_{5}\not{p}_{t}\phi^{-}_{2P}(p)+\phi^{+}_{1P}(p)\gamma^{\lambda}\gamma_{5}\phi^{+}_{1P}(p)$
(61)
$\displaystyle+\phi^{+}_{1P}(p)\gamma^{\lambda}\gamma_{5}\not{p}_{t}\phi^{-}_{1P}(p)+\phi^{-}_{1P}(p)\not{p}_{t}\gamma^{\lambda}\gamma_{5}\phi^{+}_{1P}(p)+\phi^{-}_{1P}(p)\not{p}_{t}\gamma^{\lambda}\gamma_{5}\not{p}_{t}\phi^{-}_{1P}(p)$
$\displaystyle+\phi^{+}_{2P}(p)\gamma^{\lambda}\gamma_{5}\phi_{0P}(p)+\phi^{-}_{2P}(p)\not{p}_{t}\gamma^{\lambda}\gamma_{5}\phi_{0P}(p)]u_{\Lambda_{b}}(v,s)S^{-1}_{D}(p_{2}).$

Getting rid of the spinors, $\epsilon_{b}$ at each order in the $1/m_{Q}$
expansion can be expressed as the following:

$\displaystyle(1+\epsilon_{0b})=\int\frac{d^{4}p}{(2\pi)^{4}}\phi_{0P}(p)\phi_{0P}(p)S^{-1}_{D}(p_{2}),$
(62)
$\displaystyle\epsilon_{1b}=2\int\frac{d^{4}p}{(2\pi)^{4}}\phi_{0P}(p)\phi^{+}_{1P}(p)S^{-1}_{D}(p_{2}),$
(63) $\displaystyle\epsilon_{2b}$ $\displaystyle=$ $\displaystyle
2\int\frac{d^{4}p}{(2\pi)^{4}}\phi_{0P}(p)\phi^{+}_{2P}(p)(p^{2}_{l}-W^{2}_{P})-\frac{1}{3}\int\frac{d^{4}p}{(2\pi)^{4}}\phi^{-}_{1P}(p)\phi^{-}_{1P}(p)p^{2}_{t}(p^{2}_{l}-W^{2}_{P}).$
(64)

From Eq. (62) one can see immediately that $\epsilon_{0b}=0$ because the
Isgur-Wise function is normalized to 1. From Eq. (63) we have
$\epsilon_{1b}=0$ since $\phi^{+}_{1P}(p)=0$. This is consistent with the
demand of the current conservation Luk90 .

Substituting Eqs. (34), (37), and (39) into Eq. (64), using
$\tilde{\phi}(p_{t})=\int\frac{dp_{l}}{2\pi}\phi(p)$ and the residue theorem,
we obtain the value of $\epsilon_{2b}$ with the obtained numerical results of
$\tilde{\phi}_{0P}(p_{t})$ and $\tilde{\phi}^{+}_{2P}(p_{t})$. The results are
listed in TABLE II.

We can see from TABLE II that the value of $\epsilon_{2b}$ depends on our
model parameters, $m_{D}$ and $\kappa$. In the ranges of the parameters,
$\epsilon_{2b}$ is always negative. The absolute value of $\epsilon_{2b}$
increases with the increase of $m_{D}$ and $\kappa$. When $m_{D}=650MeV$,
$\sqrt{|\epsilon_{2b}|}$ changes from $0.33GeV$ to $0.56GeV$. When
$m_{D}=800MeV$, $\sqrt{|\epsilon_{2b}|}$ changes from $0.62GeV$ to $0.79GeV$.
As expected, $\sqrt{|\epsilon_{2b}|}$ is of order $\Lambda_{QCD}$. This
indicates that our results are reasonable.

Table 2: Values of $\epsilon_{2b}$ for different $m_{D}$ and $\kappa$. | $\kappa(GeV^{3})$ | 0.02 | 0.04 | 0.06 | 0.08
---|---|---|---|---|---
$m_{D}=650MeV$ |  $\epsilon_{2b}(GeV^{2})$ | -0.11 | -0.19 | -0.26 | -0.32
$m_{D}=700MeV$ |  $\epsilon_{2b}(GeV^{2})$ | -0.16 | -0.25 | -0.32 | -0.41
$m_{D}=750MeV$ |  $\epsilon_{2b}(GeV^{2})$ | -0.27 | -0.36 | -0.44 | -0.49
$m_{D}=800MeV$ |  $\epsilon_{2b}(GeV^{2})$ | -0.39 | -0.47 | -0.56 | -0.62

## IV. Summary and discussion

More and more data have been and will be collected at LHC. This makes more
precise measurement of Cabibbo-Kobayashi-Maskawa matrix elements, especially
the value of $V_{ub}$, become possible. $V_{ub}$ can be measured through the
inclusive semileptonic decays of polarized $\Lambda_{b}$. Since the decay rate
involves two parameters, $\mu^{2}_{\pi}$ and $\epsilon_{b}$, to second order
in the $1/m_{b}$ expansion, the theoretical calculation of these two
parameters is important to extract more precise $V_{ub}$ from experimental
data. $\mu^{2}_{\pi}$ was calculated in the previous work. In this paper we
focus on the theoretical calculation of $\epsilon_{b}$ in the BS equation
approach. Since $\epsilon_{b}$ is only non-zero at second order in the
$1/m_{b}$ expansion, it is necessary to establish the BS equation for
$\Lambda_{b}$ to this order.

We regard the heavy baryon $\Lambda_{b}$ as composed of a heavy $b$ quark and
a light scalar diquark based on the fact that the light degrees of freedom in
$\Lambda_{b}$ have fixed spin and isospin quantum numbers. In this picture, we
establish the BS equation for $\Lambda_{b}$ to second order in the $1/m_{b}$
expansion. Then we solve the BS equation numerically by applying the kernel
which include the scalar confinement and the one-gluon-exchange terms. To
second order in the $1/m_{b}$ expansion, we obtain numerical results for BS
scalar functions. Expressing $\epsilon_{b}$ as the overlap integral of the BS
amplitude for $\Lambda_{b}$ we obtain the numerical values for $\epsilon_{b}$.
It is found that $\epsilon_{b}$ is only non-zero at $1/m^{2}_{b}$ as expected.
This indicates that our result is consistent with the requirements from the
heavy quark symmetry and the current conservation. At $1/m^{2}_{b}$,
$\epsilon_{b}$ can be expressed as $\epsilon_{2b}/m^{2}_{b}$. We found that in
the ranges of our model parameters, $\epsilon_{2b}$ is always negative, and
varies form $-0.11GeV^{2}$ to $-0.62GeV^{2}$. We find that
$\sqrt{|\epsilon_{2b}|}$ is of order $\Lambda_{QCD}$ as expected.

There are some uncertainties in our model. Compared with the heavy meson case,
heavy baryons are much more complicated. To order $1/m_{b}$ there are two
parameters in our model, $m_{D}$ and $\kappa$ Guo00 . When we expand to second
order in the $1/m_{b}$ expansion, more uncertainties are involved, including
the forms for the kernel $V_{5}$ and $V_{6}$ and the values of binding
energies at orders $1/m_{b}~{}(E_{1})$ and $1/m^{2}_{b}~{}(E_{2})$. To
simplify our model we assume that $V_{5}=\delta^{2}V_{1}$,
$V_{6}=\delta^{2}V_{2}$, $E_{2}=\delta E_{1}=\delta^{2}E_{0}$, with the same
proportionality parameter $\delta$ which is of order $\Lambda_{QCD}$. Our
model to first order in the $1/m_{Q}$ expansion has been tested in
$\Lambda_{b}\rightarrow\Lambda_{c}$ semileptonic decay and
$\Lambda_{b}\rightarrow\Lambda_{c}\pi$ nonleptonic decay Guo96 Guo00 Ber12 .
Further tests are needed in other physical processes and for the model to
$1/m^{2}_{Q}$.

###### Acknowledgements.

This work was partially supported by Natural Science Foundation of China under
Contract No. 10975018, No. 11175020, No. 11275025 and the Fundamental Research
Funds for the Central Universities of China.

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