#  The double charm decays of $B_{c}$ Meson in the Perturbative QCD Approach

Zhou Rui 1,2 Zou Zhitian 1 Cai-Dian Lü1<EMAIL_ADDRESS>1 Institute of High
Energy Physics and Theoretical Physics Center for Science Facilities, Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China  2 School of
Science, Hebei United University, Tangshan, Hebei 063009, People’s Republic of
China

###### Abstract

We study the double charm decays of $B_{c}$ meson, by employing the
perturbative QCD approach based on $k_{T}$ factorization. In this approach, we
include the non-factorizable emission diagrams and W annihilation diagrams,
which are neglected in the previous naive factorization approach. The former
are important in the color-suppressed modes; while the latter are important in
most $B_{c}$ decay channels due to the large Cabibbo-Kobayashi-Maskawa matrix
elements. We make comparison with those previous naive factorization results
for the branching ratios and also give out the theoretical errors that
previously missed. We predict the transverse polarization fractions of
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0},D^{*+}_{(s)}D^{*0}$ decays for the
first time. A large transverse polarization contribution that can reach
$50\%\sim 60\%$ is predicted in some of the $B_{c}$ meson decays.

###### pacs:

13.25.Hw, 12.38.Bx, 14.40.Nd

##  I Introduction

Since the $B_{c}$ meson is the lowest bound state of two different heavy
quarks with open flavor, it is stable against strong and electromagnetic
annihilation processes. The $B_{c}$ meson therefore decays weakly.
Furthermore, the $B_{c}$ meson has a sufficiently large mass, thus each of the
two heavy quarks can decay individually. It has rich decay channels, and
provides a very good place to study nonleptonic weak decays of heavy mesons,
to test the standard model and to search for any new physics signals iiba .
The current running LHC collider will produce much more $B_{c}$ mesons than
ever before to make this study a bright future.

Within the standard model (SM), for the double charm decays of $B_{u,d,s}$
mesons, there are penguin operator contributions as well as tree operator
contributions. Thus the direct CP asymmetry may be present. However, the
double charm decays of $B_{c}$ meson are pure tree decay modes, which are
particularly well suited to extract the Cabibbo-Kobayashi-Maskawa (CKM) angles
due to the absented interference from penguin contributions. As was pointed
out in ref. plb286160  and further elaborated in ref. prd62057503 ; jpg301445
; plb555189 ; prd65034016 , the decays $B_{c}\rightarrow
D_{s}^{+}D^{0},D_{s}^{+}\bar{D}^{0}$ are the gold-plated modes for the
extraction of CKM angle $\gamma$ though amplitude relations because their
decay widths are expected to be at the same order of magnitude. But this needs
to be examined by faithful calculations.

Although many investigations on the decays of $B_{c}$ to double-charm states
have been carried out jpg301445 ; plb555189 ; prd73054024 ; prd564133 ; pan67
; prd61034012 ; prd62014019 ; prd493399  in the literature, there are
uncontrolled large theoretical errors with quite different numerical results.
In fact, all of these old calculations are based on naive factorization
hypothesis, with various form factor inputs. Most of them even did not give
any theoretical error estimates because of the non-reliability of these
models. Recently, the theory of non-leptonic B decays has been improved quite
significantly. Factorization has been proved in many of these decays, thus
allow us to give reliable calculations of the hadronic B decays. It is also
shown that the non-factorizable contributions and annihilation type
contributions, which are neglected in the naive factorization approach, are
very important in these decays cheng .

The perturbative QCD approach (pQCD) prl744388  is one of the recently
developed theoretical tools based on QCD to deal with the non-leptonic B
decays. Utilizing the $k_{T}$ factorization instead of collinear
factorization, this approach is free of end-point singularity. Thus the
Feynman diagrams including factorizable, non-factorizable and annihilation
type, are all calculable. Phenomenologically, the pQCD approach successfully
predict the charmless two-body B decays plb5046 ; prd63074009 . For the decays
with a single heavy $D$ meson in the final states (the momentum of the $D$
meson is $\frac{1}{2}m_{B}(1-r^{2})$, with $r=m_{D}/m_{B}$), it is also proved
factorization in the soft-collinear effective theory scet1 .
Phenomenologically the pQCD approach is also demonstrated to be applicable in
the leading order of the $m_{D}/m_{B}$ expansion 09101424 ; 0512347  for this
kind of decays. For the double charm decays of $B_{c}$ meson, the momentum of
the final state $D$ meson is $\frac{1}{2}m_{B_{c}}(1-2r^{2})$, which is only
slightly smaller than that of the decays with a single D meson final state.
The prove of factorization here is thus trivial. The pQCD approach is
applicable to this kind of decays. In fact, the double charm decays of
$B_{u,d,s}$ meson have been studied in the pQCD approach successfully dd1 ;
dd2 , with best agreement with experiments. In this paper, we will extend our
study to these $B_{c}$ decays in the pQCD approach, in order to give
predictions on branching ratios and polarization fractions for the experiments
to test. Since this study is based on QCD and perturbative expansion, the
theoretical error will be controllable than any of the model calculations.

Our paper is organized as follows: We review the pQCD factorization approach
and then perform the perturbative calculations for these considered decay
channels in Sec.II. The numerical results and discussions on the observables
are given in Sec.III. The final section is devoted to our conclusions. Some
details of related functions and the decay amplitudes are given in the
Appendix.

##  II  Framework

For the double charm decays of $B_{c}$, only the tree operators of the
standard effective weak Hamiltonian contribute. We can divide them into two
groups: CKM favored decays with both emission and annihilation contributions
and pure emission type decays, which are CKM suppressed. For the former modes,
the Hamiltonian is given by:

$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{cb}^{*}V_{uq}[C_{1}(\mu)O_{1}(\mu)+C_{2}(\mu)O_{2}(\mu)],$
$\displaystyle O_{1}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\beta}\otimes\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\alpha},$
$\displaystyle O_{2}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})c_{\alpha}\otimes\bar{u}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\beta},$
(1)

while the effective Hamiltonian of the latter modes reads

$\displaystyle\mathcal{H}_{eff}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}V_{ub}^{*}V_{cq}[C_{1}(\mu)O^{\prime}_{1}(\mu)+C_{2}(\mu)O^{\prime}_{2}(\mu)],$
$\displaystyle O^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\beta}\otimes\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\alpha},$
$\displaystyle O^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\bar{b}_{\alpha}\gamma^{\mu}(1-\gamma_{5})u_{\alpha}\otimes\bar{c}_{\beta}\gamma_{\mu}(1-\gamma_{5})q_{\beta},$
(2)

where $V(q=d,s)$ are the corresponding CKM matrix elements. $\alpha$, $\beta$
are the color indices. $C_{1,2}$ are Wilson coefficients at renormalization
scale $\mu$. $O_{1,2}$ and $O^{\prime}_{1,2}$ are the effective four-quark
operators.

The factorization theorem allows us to factorize the decay amplitude into the
convolution of the hard subamplitude, the Wilson coefficient and the meson
wave functions, all of which are well-defined and gauge invariant. It is
expressed as

$\displaystyle C(t)\otimes
H(x,t)\otimes\Phi(x)\otimes\exp[-s(P,b)-2\int^{t}_{1/b}\frac{d\mu}{\mu}\gamma_{q}(\alpha_{s}(\mu))],$
(3)

where $C(t)$ are the corresponding Wilson coefficients of effective operators
defined in eq.(II,II). Since the transverse momentum of quark is kept in the
pQCD approach, the large double logarithm $\ln^{2}(Pb)$ (with P denoting the
longitudinal momentum, and b the conjugate variable of the transverse
momentum) to spoil the perturbative expansion. A resummation is thus needed to
give a Sudakov factor $\exp[-s(P,b)]$ npb193381 . The term after Sudakov is
from renormalization group running with $\gamma_{q}=-\alpha_{s}/\pi$ the quark
anomalous dimension in axial gauge and $t$ the factorization scale. All non-
perturbative components are organized in the form of hadron wave functions
$\Phi(x)$ (with x the longitudinal momentum fraction of valence quark inside
the meson), which can be extracted from experimental data or other non-
perturbative methods. Since the universal non-perturbative dynamics has been
factored out, one can evaluate all possible Feynman diagrams for the hard
subamplitude $H(x,t)$ straightforwardly, which include both traditional
factorizable and so-called “non-factorizable” contributions. Factorizable and
non-factorizable annihilation type diagrams are also calculable without end-
point singularity.

###  II.1  Channels with both emission and annihilation contributions

Figure 1: Feynman diagrams for $B_{c}\rightarrow D^{+}\bar{D}^{0}$ decays.

At leading order, there are eight kinds of Feynman diagrams contributing to
this type of CKM favored decays according to eq.(II). Here, we take the decay
$B_{c}\rightarrow D^{+}\bar{D}^{0}$ as an example, whose Feynman diagrams are
shown in Fig.1. The first line are the emission type diagrams, with the first
two contributing to the usual form factor; the last two so-called “non-
factorizable” diagrams. In fact, the first two diagrams are the only
contributions calculated in the naive factorization approach. The second line
are the annihilation type diagrams, with the first two factorizable; the last
two non-factorizable. The decay amplitude of factorizable diagrams (a) and (b)
in Fig.1 is

$\displaystyle\mathcal{F}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(4)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+2r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$

where $r_{b}=m_{b}/M_{B}$, $r_{i}=m_{i}/M_{B}(i=2,3)$ with $m_{2},m_{3}$ are
the masses of the recoiling charmed meson and the emitting charmed meson,
respectively; $C_{F}=4/3$ is a color factor; $f_{3}$ is the decay constant of
the charmed meson, which emitted from the weak vertex. The factorization
scales $t_{a,b}$ are chosen as the maximal virtuality of internal particles in
the hard amplitude, in order to suppress the higher order corrections
prd074004 . The function $h_{e}$ and the Sudakov factor $\exp[-S]$ are
displayed in the Appendix B. $D$ meson distribution amplitude $\phi(x)$ are
given in Appendix C. The factor $S_{t}(x)$ is the jet function resulting from
the threshold resummation, whose definitions can be found in epjc45711 .

The formula for non-factorizable emission diagrams Fig. 1 (c) and (d) contain
the kinematics variables of all the three mesons. Its expression is:

$\displaystyle\mathcal{M}_{e}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})\times$
(5)
$\displaystyle\\{[1-x_{1}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[1-x_{1}-x_{2}+x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$

Generally, for charmless decays of B meson, the non-factorizable contributions
of the emission diagrams are small due to the cancelation between Fig. 1 (c)
and (d). While for double charm decays with the light meson replaced by a
charmed meson, since the heavy $\bar{c}$ quark and the light quark is not
symmetric, the non-factorizable emission diagrams ought to give remarkable
contributions. This has been shown in the pQCD calculation of $B\to D\pi$
decays for a very large branching ratios of color-suppressed modes dpi  and
proved by the B factory experiments.

The decay amplitude of factorizable annihilation diagrams Fig. 1 (e) and (f)
involve only the two final states charmed meson wave functions, shown as

$\displaystyle\mathcal{F}_{a}$ $\displaystyle=$ $\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(6)
$\displaystyle\\{[1-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[1-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\}.$

For the non-factorizable annihilation diagrams Fig. 1 (g) and (h), the decay
amplitude is

$\displaystyle\mathcal{M}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(7)
$\displaystyle\times\\{[x_{1}+x_{3}-1-r_{c}]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{b}-x_{2}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$

where $r_{c}=m_{c}/M_{B}$, with $m_{c}$ the mass of c quark in $B_{c}$ meson.
Finally, the total decay amplitude for $B_{c}\rightarrow D^{+}\bar{D}^{0}$ can
be given by

$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}\bar{D}^{0})$ $\displaystyle=$
$\displaystyle
V_{cb}^{*}V_{ud}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}],$
(8)

with the combinations of Wilson coefficients $a_{1}=C_{2}+C_{1}/3$ and
$a_{2}=C_{1}+C_{2}/3$, characterizing the color favored contribution and the
color-suppressed contribution in the naive factorization, respectively. The
total decay amplitudes of $B_{c}\rightarrow D_{s}^{+}\bar{D}^{0}$,
$B_{c}\rightarrow D^{+}\bar{D}^{*0}$ and $B_{c}\rightarrow
D_{s}^{+}\bar{D}^{*0}$ can be obtained from eq.(8) with the following
replacement:

$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}\bar{D}^{0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{us}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{D^{+}\rightarrow
D^{+}_{s}},$ $\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}\bar{D}^{*0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{ud}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{\bar{D}^{0}\rightarrow\bar{D}^{*0}},$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}\bar{D}^{*0})$
$\displaystyle=$ $\displaystyle
V_{cb}^{*}V_{us}[a_{2}\mathcal{F}_{e}+C_{2}\mathcal{M}_{e}+a_{1}\mathcal{F}_{a}+C_{1}\mathcal{M}_{a}]|_{D^{+}\rightarrow
D^{+}_{s},\bar{D}^{0}\rightarrow\bar{D}^{*0}}.$ (9)

Comparing our eq.(8,II.1) with the formulas of previous naive factorization
approach jpg301445 ; plb555189 ; prd73054024 ; prd564133 ; pan67 ; prd61034012
, it is easy to see that only the first term appearing in eq.(8,II.1) are
calculated in the previous naive factorization approach. The second, third and
fourth terms in these equations, are the corresponding non-factorizable
emission type contribution, factorizable and non-factorizable annihilation
type contributions, respectively, which are all new calculations.

In $B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays, the two vector mesons
in the final states have the same helicity due to angular momentum
conservation, therefore only three different polarization states, one
longitudinal and two transverse for both vector mesons, are possible. The
decay amplitude can be decomposed as

$\displaystyle\mathcal{A}=\mathcal{A}^{L}+\mathcal{A}^{N}\epsilon_{2}^{T}\cdot\epsilon_{3}^{T}+i\mathcal{A}^{T}\epsilon_{\alpha\beta\rho\sigma}n^{\alpha}v^{\beta}\epsilon_{2}^{T\rho}\epsilon_{3}^{T\sigma},$
(10)

where $\epsilon_{2}^{T},\epsilon_{3}^{T}$ are the transverse polarization
vectors for the two vector charmed mesons, respectively. $\mathcal{A}^{L}$
corresponds to the contributions of longitudinal polarization;
$\mathcal{A}^{N}$ and $\mathcal{A}^{T}$ corresponds to the contributions of
normal and transverse polarization, respectively. And the total amplitudes
$\mathcal{A}^{L,N,T}$ have the same structures as eq.(8,II.1). The
factorization formulae for the longitudinal, normal and transverse
polarizations are listed in Appendix A.

For $B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{0}$ decays, only the longitudinal
polarization of $D^{*+}_{(s)}$ meson will contribute, due to the angular
momentum conservation. We can obtain their decay amplitudes from the
longitudinal polarization amplitudes for the $B_{c}\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays with the replacement
$\bar{D}^{*0}\rightarrow\bar{D}^{0}$.

###  II.2  Channels with pure emission type decays

Figure 2: Color-suppressed emission diagrams contributing to the
$B_{c}\rightarrow D^{+}D^{0}$ decays.

Figure 3: Color-favored emission diagrams contributing to the
$B_{c}\rightarrow D^{+}D^{0}$ decays.

There are also eight kinds of Feynman diagrams contributing to
$B_{c}\rightarrow D_{(s)}^{(*)+}D^{(*)0}$ decays according to eq.(II), but all
are emission type. Taking the decay $B_{c}\rightarrow D^{+}D^{0}$ as an
example, Fig. 2 are the color-suppressed emission diagrams while Fig. 3 are
the color-favored emission diagrams. We mark the subscript 2 and 3 to denote
the contributions from Fig. 2 and Fig. 3, respectively. The decay amplitude of
factorization emission diagrams $\mathcal{F}_{e2}$, coming from Fig. 2 (a,b),
is similar to eq.(4), but with the replacement $\bar{D}^{0}\rightarrow D^{0}$.
While the decay amplitude of non-factorization emission diagram
$\mathcal{M}_{e2}$, coming from Fig. 2 (c,d), is different from eq.(5), since
the heavy c quark and the light anti-quark are not symmetric. The expression
of the non-factorizable emission diagram is

$\displaystyle\mathcal{M}_{e2}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
(11)
$\displaystyle\times\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$

By exchanging the two final states charmed mesons in Fig. 2, one can obtain
the corresponding decay amplitudes formulae $\mathcal{F}_{e3}$ and
$\mathcal{M}_{e3}$ for Fig. 3. The total decay amplitude of $B_{c}\rightarrow
D^{+}D^{0}$ decay can be written as

$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}].$
(12)

If the final recoiling meson is the vector $D^{*}$ meson, the decay amplitudes
of factorization emission diagrams and non-factorization emission diagrams are
given as

$\displaystyle\mathcal{F}^{*}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(13)
$\displaystyle\times\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{*}_{e2}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
$\displaystyle\times\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}+r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$

The total decay amplitudes for other pure emission type decays are then

$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{+}D^{*0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}^{*}_{e3}+C_{1}\mathcal{M}^{*}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D^{*+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cd}[a_{2}\mathcal{F}^{*}_{e2}+C_{2}\mathcal{M}^{*}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{+}D^{*0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}_{e2}+C_{2}\mathcal{M}_{e2}+a_{1}\mathcal{F}^{*}_{e3}+C_{1}\mathcal{M}^{*}_{e3}],$
$\displaystyle\mathcal{A}(B_{c}\rightarrow D_{s}^{*+}D^{0})$ $\displaystyle=$
$\displaystyle
V_{ub}^{*}V_{cs}[a_{2}\mathcal{F}^{*}_{e2}+C_{2}\mathcal{M}^{*}_{e2}+a_{1}\mathcal{F}_{e3}+C_{1}\mathcal{M}_{e3}].$

The $B_{c}\rightarrow D^{*+}_{(s)}D^{*0}$ decays have a similar situation to
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$, their factorization formulae are
also listed in Appendix.A.

##  III NUMERICAL RESULTS

In this section, we summarize the numerical results and analysis in the double
charm decays of the $B_{c}$ meson. Some input parameters needed in the pQCD
calculation are listed in Table 1.

###  III.1 The Form Factors

Table 1: Parameters we used in numerical calculation npp37

Mass(GeV) | $M_{W}=80.399$ | $M_{B_{c}}=6.277$ | $m_{b}=4.2$ | $m_{c}=1.27$
---|---|---|---|---

CKM | $|V_{ub}|=(3.47^{+0.16}_{-0.12})\times 10^{-3}$ | $|V_{ud}|=0.97428^{+0.00015}_{-0.00015}$ | $|V_{us}|=0.2253^{+0.0007}_{-0.0007}$
---|---|---|---
$|V_{cs}|=0.97345^{+0.00015}_{-0.00016}$ | $|V_{cd}|=0.2252^{+0.0007}_{-0.0007}$ | $|V_{cb}|=0.0410^{+0.0011}_{-0.0007}$

Decay constants(MeV) | $f_{B_{c}}=489$ | $f_{D}=206.7\pm 8.9$ | $f_{D_{s}}=257.5\pm 6.1$
---|---|---|---

Lifetime | $\tau_{B_{c}}=0.453\times 10^{-12}\text{s}$
---|---

Table 2: The form factors for $B_{c}\rightarrow D^{(*)}_{(s)}$ at $q^{2}=0$ evaluated in the pQCD approach. The uncertainties are from the hadronic parameters. For comparison, we also cite the theoretical estimates of other models. | This work | Kiselev jpg301445  111 The non-bracket (bracketed) results are evaluated in sum rules (potential model) | IKP plb555189  | WSL prd7905402  | DSV jpg35085002  | DW prd391342 222We quote the result with $\omega=0.7\text{GeV}$
---|---|---|---|---|---|---
$F^{B_{c}\rightarrow D}$ | $0.14^{+0.01}_{-0.02}$ | 0.32 [0.29] | 0.189 | 0.16 | 0.075 | 0.255
$F^{B_{c}\rightarrow D_{s}}$ | $0.19^{+0.02}_{-0.01}$ | 0.45 [0.43] | 0.194 | 0.28 | 0.15 | –
$A_{0}^{B_{c}\rightarrow D^{*}}$ | $0.12^{+0.02}_{-0.01}$ | 0.35 [0.37] | 0.133 | 0.09 | 0.081 | 0.257
$A_{0}^{B_{c}\rightarrow D_{s}^{*}}$ | $0.17^{+0.01}_{-0.01}$ | 0.47 [0.52] | 0.142 | 0.17 | 0.16 | –

The diagrams (a) and (b) in Fig.1 or Fig.3 give the contribution for
$B_{c}\rightarrow D^{(*)}_{(s)}$ transition form factor at $q^{2}=0$ point.
Our predictions of the form factors are collected in Table 2. The error is
from the combined uncertainty in the hadronic parameters: (1) the shape
parameters: $\omega_{B}=0.60\pm 0.05$ for $B_{c}$ meson wave function,
$a_{D}=(0.5\pm 0.1)\text{GeV}$ for $D^{(*)}$ meson and $a_{D_{s}}=(0.4\pm
0.1)\text{GeV}$ for $D_{s}^{(*)}$ meson wave function dd1 ; (2) the decay
constants in the wave functions of charmed mesons, which are given in Table 1.
Since the uncertainties from decay constants of $D_{(s)}$ and the shape
parameters of the wave functions are very small, the relevant uncertainties to
the form factors are also very small. We can see that the $SU(3)$ symmetry
breaking effects between $B_{c}$ to $D^{(*)}$ and $B_{c}$ to $D^{(*)}_{s}$
form factors are large, as the decay constant of $D_{s}$ is about one-fifth
larger than that of the $D$ meson.

In the literature there are already lots of studies on $B_{c}\rightarrow
D^{(*)}_{(s)}$ transition form factors jpg301445 ; prd7905402 ; jpg35085002 ;
plb555189 ; prd391342 , whose results are collected in Table 2. Our results
are generally close to the covariant light-front quark model results of
prd7905402  and the constituent quark model results of plb555189 . However,
other results collected in Table 2, especially for the QCD sum rules (QCDSR)
jpg301445  and the Bauer, Stech and Wirbel (BSW) model jpg35085002  deviate a
lot numerically. The predictions of QCDSR jpg301445  are larger than those in
other works prd7905402 ; jpg35085002 ; plb555189 ; prd391342 . The reason is
that they have taken into account the $\alpha_{s}/v$ corrections and the form
factors are enhanced by 3 times due to the Coulomb renormalization of the
quark-meson vertex for the heavy quarkonium $B_{c}$. The results of BSW model
jpg35085002  are quite small due to the less overlap of the initial and final
states wave functions. Although, the included flavor dependence of the average
transverse quark momentum in the mesons can enhance the form factors for
$B_{c}\rightarrow D^{*}_{(s)}$ transitions, their predictions are still
smaller than other models. The large differences in different models can be
discriminated by the future LHC experiments.

###  III.2 Branching Ratios

With the decays amplitudes $\mathcal{A}$ obtained in Sec.II, the branching
ratio $\mathcal{BR}$ reads as

$\displaystyle\mathcal{BR}=\frac{G_{F}\tau_{B_{c}}}{32\pi
M_{B}}\sqrt{(1-(r_{2}+r_{3})^{2})(1-(r_{2}-r_{3})^{2})}|\mathcal{A}|^{2}.$
(16)

As stated in Sec II, the contributions from the penguin operators are absent,
since the penguins add an even number of charmed quarks, while there is
already one from the initial state. There should be no CP violation in these
processes. We tabulate the branching ratios of the considered decays in Table
3 and 4. The processes (1)-(4) in Table 3 have a comparatively large branching
ratios ($10^{-5}$) with the CKM factor $V_{cb}^{*}V_{ud}\sim\lambda^{2}$.
While the branching ratios of other processes are relatively small due to the
CKM factor suppression. Especially for the processes (1)-(4) in Table 4, these
channels are suppressed by CKM element $V_{ub}/V_{cb}$ and $V_{cd}/V_{ud}$.
Thus their branching ratios are three order magnitudes smaller.

Table 3: Branching ratios ($10^{-6}$) of the CKM favored decays with both emission and annihilation contributions, together with results from other models. The errors for these entries correspond to the uncertainties in the input hadronic quantities, from the CKM matrix elements, and the scale dependence, respectively. | channels | This work | Kiselevjpg301445  | IKPplb555189  | IKSprd73054024  | LCprd564133  | CFprd61034012
---|---|---|---|---|---|---|---
1 | $B_{c}\rightarrow D^{+}\bar{D}^{0}$ | $32^{+6+1+2}_{-6-1-4}$ | 53 | 32 | 33 | 86 | 8.4
2 | $B_{c}\rightarrow D^{+}\bar{D}^{*0}$ | $34^{+7+2+3}_{-6-1-3}$ | 75 | 83 | 38 | 75 | 7.5
3 | $B_{c}\rightarrow D^{*+}\bar{D}^{0}$ | $12^{+3+1+0}_{-3-0-1}$ | 49 | 17 | 9 | 30 | 84
4 | $B_{c}\rightarrow D^{*+}\bar{D}^{*0}$ | $34^{+9+2+0}_{-8-1-0}$ | 330 | 84 | 21 | 55 | 140
5 | $B_{c}\rightarrow D_{s}^{+}\bar{D}^{0}$ | $2.3^{+0.4+0.1+0.2}_{-0.4-0.1-0.2}$ | 4.8 | 1.7 | 2.1 | 4.6 | 0.6
6 | $B_{c}\rightarrow D_{s}^{+}\bar{D}^{*0}$ | $2.6^{+0.4+0.1+0.1}_{-0.6-0.1-0.2}$ | 7.1 | 4.3 | 2.4 | 3.9 | 0.53
7 | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{0}$ | $0.7^{+0.1+0.0+0.0}_{-0.2-0.0-0.0}$ | 4.5 | 0.95 | 0.65 | 1.8 | 5
8 | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$ | $2.8^{+0.7+0.1+0.1}_{-0.6-0.1-0.0}$ | 26 | 4.7 | 1.6 | 3.5 | 8.4
Table 4: Branching ratios ($10^{-7}$) of the CKM suppressed decays with pure emission contributions, together with results from other models. The errors for these entries correspond to the uncertainties in the input hadronic quantities, from the CKM matrix elements, and the scale dependence, respectively. | channels | This work | Kiselevjpg301445  | IKPplb555189  | IKSprd73054024
---|---|---|---|---|---
1 | $B_{c}\rightarrow D^{+}D^{0}$ | $1.0^{+0.2+0.1+0.0}_{-0.1-0.0-0.0}$ | 3.2 | 1.1 | 3.1
2 | $B_{c}\rightarrow D^{+}D^{*0}$ | $0.7^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$ | 2.8 | 0.25 | 0.52
3 | $B_{c}\rightarrow D^{*+}D^{0}$ | $0.9^{+0.1+0.1+0.0}_{-0.2-0.0-0.0}$ | 4.0 | 3.8 | 4.4
4 | $B_{c}\rightarrow D^{*+}D^{*0}$ | $0.8^{+0.2+0.1+0.2}_{-0.1-0.0-0.0}$ | 15.9 | 2.8 | 2.0
5 | $B_{c}\rightarrow D_{s}^{+}D^{0}$ | $30^{+5+3+1}_{-4-2-1}$ | 66 | 25 | 74
6 | $B_{c}\rightarrow D_{s}^{+}D^{*0}$ | $19^{+3+2+0}_{-3-1-1}$ | 63 | 6 | 13
7 | $B_{c}\rightarrow D_{s}^{*+}D^{0}$ | $25^{+4+2+0}_{-3-2-1}$ | 85 | 69 | 93
8 | $B_{c}\rightarrow D_{s}^{*+}D^{*0}$ | $24^{+3+2+1}_{-3-2-1}$ | 404 | 54 | 45

For comparison, we also cite other theoretical results jpg301445 ; plb555189 ;
prd73054024 ; prd564133 ; prd61034012  for the double charm decays of $B_{c}$
meson in Tables 3 and 4. In general, the results of the various model
calculations are of the same order of magnitude for most channels. However the
difference between different model calculations is quite large. This is
expected from the large difference of input parameters, especially the large
difference of form factors shown in Table 2. As stated in the introduction,
all the calculations of these $B_{c}$ to two D meson decays in the literature
use the same naive factorization approach. Their difference relies only on the
input form factors and decay constants. Therefore the comparison of results
with any of them is straightforward. Larger branching ratios come always with
the larger form factors. As stated in the previous subsection, our results of
form factors are comparable with the relativistic constituent quark model
(RCQM) plb555189 ; prd73054024 , thus our branching ratios in Table 3 are also
comparable with theirs except for the processes $B_{c}\rightarrow
D^{*+}\bar{D}^{*0}$ and $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$. Due to the
sizable contributions of transverse polarization amplitudes, our branching
ratios are larger than those in RCQM model, whose transverse contribution is
negligible.

Since all the previous calculations in the literature are model calculations,
it is difficult for them to give the theoretical error estimations. In our
pQCD approach, the factorization holds at the leading order expansion of
$m_{D}/m_{B}$. At this order, we can do the systematical calculation, so as to
the error estimations in the tables. The first error in these entries is
estimated from the hadronic parameters: (1) the shape parameters:
$\omega_{B}=0.60\pm 0.05$ for $B_{c}$ meson, $a_{D}=(0.5\pm 0.1)\text{GeV}$
for $D^{(*)}$ meson and $a_{D_{s}}=(0.4\pm 0.1)\text{GeV}$ for $D_{s}^{(*)}$
meson dd1 ; (2) the decay constants in the wave functions of charmed mesons,
which are given in Table 1. The second error is from the uncertainty in the
CKM matrix elements, which are also given in Table 1. The third error arises
from the hard scale t varying from $0.75t$ to $1.25t$, which characterizing
the size of next-to-leading order QCD contributions. The not large errors of
this type indicate that our perturbative expansion indeed hold. It is easy to
see that the most important uncertainty in our approach comes from the
hadronic parameters. The total theoretical error is in general around 10% to
30% in size.

The eight CKM favored channels (proportional to $|V_{cb}|$) in Table 3 receive
contributions from both emission diagrams and annihilation diagrams. From
Fig.1, one can find that the contributions from the factorizable emission
diagrams are color-suppressed. The naive factorization approach can not give
reliable predictions due to large non-factorizable contributions fac . As was
pointed out in Sec.II, the non-factorizable emission diagrams give large
contributions in pQCD approach because the asymmetry of the two quarks in
charmed mesons. Thus, the branching ratios of these decays are dominated by
the non-factorizable emission diagrams.

The eight CKM suppressed channels (proportional to $|V_{ub}|$) in Table 4 can
occur only via emission type diagrams. There are two types of emission
diagrams in these decays, one is color-suppressed, one is color favored. It is
expected that the color-favored factorizable amplitude $\mathcal{F}_{e3}$
dominates in eq.(II.2). However, the non-factorizable contribution
$\mathcal{M}_{e2}$, proportional to the large $C_{2}$, is enhanced by the
Wilson coefficient. Numerically it is indeed comparable to the color-favored
factorizable amplitude. This large non-factorizable contribution has already
been shown in the similar $B\to D\pi$ decays theoretically and experimentally
dpi . In all of these channels the non-factorizable contributions play a very
important role, therefore the branching ratios predicted in table 3 and 4 are
not like the previous naive factorization approach calculations jpg301445 ;
plb555189 ; prd73054024 ; prd564133 ; prd61034012 . They are not simply
proportional to the corresponding form factors any more, but with a very
complicated manner, since we have also additional annihilation type
contributions.

From Table III and IV, one can see that as it was expected the magnitudes of
the branching ratios of the decays $B_{c}\rightarrow D^{+}_{s}\bar{D}^{0}$ and
$B_{c}\rightarrow D^{+}_{s}D^{0}$ are very close to each other. In our
numerical results, the ratio of the two decay widths is estimated as
$\frac{\Gamma(B_{c}\rightarrow D_{s}^{+}D^{0})}{\Gamma(B_{c}\rightarrow
D_{s}^{+}\bar{D}^{0})}\approx 1.3$. They are very suitable for extracting the
CKM angle $\gamma$ though the amplitude relations. Hopefully they will be
measured in the experiments soon. However, the decays $B_{c}\rightarrow
D^{+}\bar{D}^{0},D^{+}D^{0}$ are problematic from the methodic point of view
for $\mathcal{BR}(B_{c}\rightarrow D^{+}D^{0})\ll\mathcal{BR}(B_{c}\rightarrow
D^{+}\bar{D}^{0})$. The corresponding ratio in $B_{c}\rightarrow
D^{+}D^{0},D^{+}\bar{D}^{0}$ decays is $\frac{\Gamma(B_{c}\rightarrow
D^{+}D^{0})}{\Gamma(B_{c}\rightarrow D^{+}\bar{D}^{0})}\sim 10^{-3}$, which
confirm the latter decay modes are not useful to determine the angle $\gamma$
experimentally.

Table 5: The transverse polarizations fractions ($\%$) for $B_{c}\rightarrow VV$. The errors correspond to the uncertainties in the hadronic parameters and the scale dependence, respectively.  | $B_{c}\rightarrow D^{*+}\bar{D}^{*0}$ | $B_{c}\rightarrow D_{s}^{*+}\bar{D}^{*0}$ | $B_{c}\rightarrow D^{*+}D^{*0}$ | $B_{c}\rightarrow D_{s}^{*+}D^{*0}$
---|---|---|---|---
$\mathcal{R}_{T}$ | $58^{+3+1}_{-3-0}$ | $68^{+2+1}_{-2-1}$ | $4^{+1+1}_{-1-1}$ | $6^{+1+2}_{-0-1}$

For the $B_{c}$ decays to two vector mesons, the decays amplitudes
$\mathcal{A}$ are defined in the helicity basis

$\displaystyle\mathcal{A}=\sum_{i=0,+,-}|\mathcal{A}_{i}|^{2},\quad$ (17)

where the helicity amplitudes $\mathcal{A}_{i}$ have the following
relationships with $\mathcal{A}^{L,N,T}$

$\displaystyle\mathcal{A}_{0}=\mathcal{A}^{L},\quad\mathcal{A}_{\pm}=\mathcal{A}^{N}\pm\mathcal{A}^{T}.$
(18)

We also calculate the transverse polarization fractions $\mathcal{R}_{T}$ of
the $B_{c}\to D_{(s)}^{*}D^{*}$ decays, with the definition given by

$\displaystyle\mathcal{R}_{T}=\frac{|\mathcal{A}_{+}|^{2}+|\mathcal{A}_{-}|^{2}}{|\mathcal{A}_{0}|^{2}+|\mathcal{A}_{+}|^{2}+|\mathcal{A}_{-}|^{2}}.$
(19)

This should be the first time theoretical predictions in the literature, which
are absent in all the naive factorization calculations. According to the power
counting rules in the factorization assumption, the longitudinal polarization
should be dominant due to the quark helicity analysis. Our predictions for the
transverse polarization fractions of the decays $B_{c}\rightarrow
D^{*+}_{(s)}D^{*0}$, which are given in Table 5, are indeed small, since the
two transverse amplitudes are down by a power of $r_{2}$ or $r_{3}$ comparing
with the longitudinal amplitudes. However, for $B_{c}\rightarrow
D^{*+}_{(s)}\bar{D}^{*0}$ decays, the most important contributions for these
two decay channels are from the non-factorizable tree diagrams in Fig. 1(c)
and 1(d). With an additional gluon, the transverse polarization in the non-
factorizable diagrams does not encounter helicity flip suppression. The
transverse polarization is at the same order as longitudinal polarization.
Therefore, we can expect the transverse polarizations take a larger ratio in
the branching ratios, which can reach $\sim 60\%$. The fact that the non-
factorizable contribution can give large transverse polarization contribution
is also observed in the $B^{0}\to\rho^{0}\rho^{0}$, $\omega\omega$ decays
rhorho  and in the $B_{c}\rightarrow D_{s}^{*+}\omega$ decay 11121257 .

##  IV  conclusion

All the previous calculations in the literature for the $B_{c}$ meson decays
to two charmed mesons are based on the very simple naive factorization
approach. The branching ratios predicted in this kind of model calculation
depend heavily on the input form factors. Since all of these modes contain
dominant or large contributions from color-suppressed diagrams, the predicted
branching ratios are also not stable due to the large unknown non-factorizable
contributions. In this paper, we have performed a systematic analysis of the
double charm decays of the $B_{c}$ meson in the pQCD approach based on $k_{T}$
factorization theorem, which is free of end-point singularities. All
topologies of decay amplitudes are calculable in the same framework, including
the non-factorizable one and annihilation type. It is found that the non-
factorizable emission diagrams give a remarkable contribution. There is no CP
violation for all these decays within the standard model, since there are only
tree operators contributions. The predicted branching ratios range from very
small numbers of $\mathcal{O}(10^{-8})$ up to the largest branching fraction
of $\mathcal{O}(10^{-5})$. Since all of the previous naive factorization
calculations did not give the theoretical uncertainty in the numerical
results, it is not easy to compare our results with theirs. The theoretical
uncertainty study in the pQCD approach shows that our numerical results are
reliable, which may be tested in the upcoming experimental measurements. We
predict the transverse polarization fractions of the $B_{c}$ decays with two
vector $D^{*}$ mesons in the final states for the first time. Due to the
cancelation of some hadronic parameters in the ratio, the polarization
fractions are predicted with less theoretical uncertainty. The transverse
polarization fractions are large in some channels, which mainly come from the
non-factorizable emission diagrams.

###### Acknowledgements.

We thank Hsiang-nan Li and Fusheng Yu for helpful discussions. This work is
partially supported by National Natural Science Foundation of China under the
Grant No. 11075168; Natural Science Foundation of Zhejiang Province of China,
Grant No. Y606252 and Scientific Research Fund of Zhejiang Provincial
Education Department of China, Grant No. 20051357.

##  Appendix A Factorization formulas for $B_{c}\rightarrow VV$

In the $B_{c}$ decays to two vector meson final states, we use the superscript
L, N and T to denote the contributions from longitudinal polarization, normal
polarization and transverse polarization, respectively. For the CKM favored
$B_{c}\rightarrow D^{*+}_{(s)}\bar{D}^{*0}$ decays, the decay amplitudes for
different polarizations are

$\displaystyle\mathcal{F}^{L}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(20)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{L}_{e}$ $\displaystyle=$
$\displaystyle-\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})\times$
(21)
$\displaystyle\\{[1-x_{1}-x_{3}+r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[1-x_{1}-x_{2}+x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{F}^{L}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(22)
$\displaystyle\\{[1-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[1-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{M}^{L}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(23)
$\displaystyle\\{[x_{1}+x_{3}-1-r_{c}]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{b}-x_{2}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$
$\displaystyle\mathcal{F}^{N}_{e}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(24)
$\displaystyle\\{[2-r_{b}+r_{2}(4r_{b}-x_{2}-1)]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{T}_{e}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(25)
$\displaystyle\\{[2-r_{b}-r_{2}(1-x_{2})]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{N}_{e}$ $\displaystyle=$
$\displaystyle-\mathcal{M}^{T}_{e}=\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{2}^{2}\omega_{B}^{2}}{2})$
(26)
$\displaystyle\times\\{[x_{1}+x_{3}-1]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{1}-x_{3}]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{F}^{N}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}r_{2}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(27)
$\displaystyle\\{[2-x_{2}]\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})-$
$\displaystyle[2-x_{3}]\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{F}^{T}_{a}$ $\displaystyle=$
$\displaystyle-8C_{F}f_{B}\pi
M_{B}^{4}r_{2}r_{3}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\times$
(28)
$\displaystyle\\{x_{2}\alpha_{s}(t_{e})h_{e}(\alpha_{a},\beta_{e},b_{2},b_{3})\exp[-S_{ef}(t_{e})]S_{t}(x_{3})+$
$\displaystyle
x_{3}\alpha_{s}(t_{f})h_{e}(\alpha_{a},\beta_{f},b_{3},b_{2})\exp[-S_{ef}(t_{f})]S_{t}(x_{2})\\},$
$\displaystyle\mathcal{M}^{N}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(29)
$\displaystyle\\{[r_{2}^{2}(x_{2}-1)+r_{3}^{2}(x_{3}-1)]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{2}^{2}x_{2}+r_{3}^{2}x_{3}-2r_{2}r_{3}r_{b}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\},$
$\displaystyle\mathcal{M}^{T}_{a}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(30)
$\displaystyle\\{[r_{2}^{2}(x_{2}-1)-r_{3}^{2}(x_{3}-1)]\alpha_{s}(t_{g})h_{e}(\beta_{g},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{g})]$
$\displaystyle-[r_{2}^{2}x_{2}-r_{3}^{2}x_{3}]\alpha_{s}(t_{h})h_{e}(\beta_{h},\alpha_{a},b_{1},b_{2})\exp[-S_{gh}(t_{h})]\\}.$

For the CKM suppressed $B_{c}\rightarrow D^{*+}_{(s)}D^{*0}$ decays, the decay
amplitudes for different polarizations are

$\displaystyle\mathcal{F}^{L}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(31)
$\displaystyle\\{[-(r_{2}-2)r_{b}+2r_{2}x_{2}-x_{2}]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle+r^{2}_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{N}_{e2}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(32)
$\displaystyle\\{[2-r_{b}+r_{2}(4r_{b}-x_{2}-1)]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{F}^{T}_{e2}$ $\displaystyle=$ $\displaystyle
2\sqrt{\frac{2}{3}}C_{F}f_{B}f_{3}r_{3}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}\int_{0}^{\infty}b_{1}b_{2}db_{1}db_{2}\phi_{2}(x_{2})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(33)
$\displaystyle\\{[2-r_{b}-r_{2}(1-x_{2})]\alpha_{s}(t_{a})h_{e}(\alpha_{e},\beta_{a},b_{1},b_{2})S_{t}(x_{2})\exp[-S_{ab}(t_{a})]$
$\displaystyle-
r_{2}\alpha_{s}(t_{b})h_{e}(\alpha_{e},\beta_{b},b_{2},b_{1})S_{t}(x_{1})\exp[-S_{ab}(t_{b})]\\},$
$\displaystyle\mathcal{M}^{L}_{e2}$ $\displaystyle=$
$\displaystyle\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})\times$
(34)
$\displaystyle\\{[2-x_{1}-x_{2}-x_{3}-r_{2}(1-x_{2})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]-$
$\displaystyle[x_{3}-x_{1}+r_{2}(1-x_{2})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\},$
$\displaystyle\mathcal{M}^{N}_{e2}$ $\displaystyle=$
$\displaystyle-\mathcal{M}^{T}_{e2}=\frac{8}{3}C_{F}f_{B}\pi
M_{B}^{4}\int_{0}^{1}dx_{2}dx_{3}\int_{0}^{\infty}b_{2}b_{3}db_{2}db_{3}\phi_{2}(x_{2})\phi_{3}(x_{3})\exp(-\frac{b_{1}^{2}\omega_{B}^{2}}{2})$
(35)
$\displaystyle\times\\{[r_{3}(x_{1}-x_{3})]\alpha_{s}(t_{c})h_{e}(\beta_{c},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{c})]+$
$\displaystyle[2r_{c}-r_{3}(1-x_{1}-x_{3})]\alpha_{s}(t_{d})h_{e}(\beta_{d},\alpha_{e},b_{3},b_{2})\exp[-S_{cd}(t_{d})]\\}.$

##  Appendix B Scales and related functions in hard kernel

We show here the functions $h_{e}$, coming from the Fourier transform of hard
kernel,

$\displaystyle h_{e}(\alpha,\beta,b_{1},b_{2})$ $\displaystyle=$
$\displaystyle h_{1}(\alpha,b_{1})\times h_{2}(\beta,b_{1},b_{2}),$
$\displaystyle h_{1}(\alpha,b_{1})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}K_{0}(\sqrt{\alpha}b_{1}),&\quad\quad\alpha>0\\\
K_{0}(i\sqrt{-\alpha}b_{1}),&\quad\quad\alpha<0\end{array}\right.$ (38)
$\displaystyle h_{2}(\beta,b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\theta(b_{1}-b_{2})I_{0}(\sqrt{\beta}b_{2})K_{0}(\sqrt{\beta}b_{1})+(b_{1}\leftrightarrow
b_{2}),&\quad\beta>0\\\
\theta(b_{1}-b_{2})J_{0}(\sqrt{-\beta}b_{2})K_{0}(i\sqrt{-\beta}b_{1})+(b_{1}\leftrightarrow
b_{2}),&\quad\beta<0\end{array}\right.$ (41)

where $J_{0}$ is the Bessel function and $K_{0}$, $I_{0}$ are modified Bessel
function with $K_{0}(ix)=\frac{\pi}{2}(-N_{0}(x)+iJ_{0}(x))$. The hard scale t
is chosen as the maximum virtuality of the internal momentum transition in the
hard amplitudes, including $1/b_{i}(i=1,2,3)$:

$\displaystyle t_{a}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{a}|},1/b_{1},1/b_{2}),\quad
t_{b}=\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{b}|},1/b_{1},1/b_{2}),$
$\displaystyle t_{c}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{c}|},1/b_{2},1/b_{3}),\quad
t_{d}=\max(\sqrt{|\alpha_{e}|},\sqrt{|\beta_{d}|},1/b_{2},1/b_{3}),$
$\displaystyle t_{e}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{e}|},1/b_{2},1/b_{3}),\quad
t_{f}=\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{f}|},1/b_{2},1/b_{3}),$
$\displaystyle t_{g}$ $\displaystyle=$
$\displaystyle\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{g}|},1/b_{1},1/b_{2}),\quad
t_{h}=\max(\sqrt{|\alpha_{a}|},\sqrt{|\beta_{h}|},1/b_{1},1/b_{2}),$ (42)

where

$\displaystyle\alpha_{e}$ $\displaystyle=$
$\displaystyle(1-x_{2})(x_{1}-r_{2}^{2})(1-r_{3}^{2})M_{B}^{2},\quad\alpha_{a}=-(1+(r_{3}^{2}-1)x_{2})(1+(r_{2}^{2}-1)x_{3})M_{B}^{2},$
$\displaystyle\beta_{a}$ $\displaystyle=$
$\displaystyle[r_{b}^{2}+(r_{2}^{2}-1)(x_{2}+r_{3}^{2}(1-x_{2}))]M_{B}^{2},\quad\beta_{b}=(1-r_{3}^{2})(x_{1}-r_{2}^{2})M_{B}^{2},$
$\displaystyle\beta_{c}$ $\displaystyle=$
$\displaystyle[r_{c}^{2}-(1-x_{2}(1-r_{3}^{2}))(1-x_{1}-x_{3}(1-r_{2}^{2}))]]M_{B}^{2},$
$\displaystyle\quad\beta_{d}$ $\displaystyle=$
$\displaystyle(1-x_{2})(1-r_{3}^{2})[x_{1}-x_{3}-r_{2}^{2}(1-x_{3})]M_{B}^{2},$
$\displaystyle\beta_{e}$ $\displaystyle=$
$\displaystyle-[1+(r_{3}^{2}-1)x_{2}]M_{B}^{2},\quad\beta_{f}=-[1+(r_{2}^{2}-1)x_{3}]M_{B}^{2},$
$\displaystyle\beta_{g}$ $\displaystyle=$
$\displaystyle[r_{c}^{2}+(1-x_{2}(1-r_{3}^{2}))(x_{1}+x_{3}-1-r_{2}^{2}x_{3})]M_{B}^{2},\quad$
$\displaystyle\beta_{h}$ $\displaystyle=$
$\displaystyle[r_{b}^{2}-x_{2}(r_{3}^{2}-1)(x_{1}-x_{3}(1-r_{2}^{2}))]M_{B}^{2}.$
(43)

The Sudakov factors used in the text are defined by

$\displaystyle S_{ab}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{1})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+\frac{5}{3}\int_{1/b_{1}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{cd}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{3})$
$\displaystyle+\frac{11}{3}\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{3}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{ef}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{3})+2\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+2\int_{1/b_{3}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$
$\displaystyle S_{gh}(t)$ $\displaystyle=$ $\displaystyle
s(\frac{M_{B}}{\sqrt{2}}x_{1},b_{1})+s(\frac{M_{B}}{\sqrt{2}}x_{2},b_{2})+s(\frac{M_{B}}{\sqrt{2}}x_{3},b_{2}),$
(44)
$\displaystyle+\frac{5}{3}\int_{1/b_{1}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu)+4\int_{1/b_{2}}^{t}\frac{d\mu}{\mu}\gamma_{q}(\mu),$

where the functions $s(Q,b)$ are defined in Appendix A of epjc45711 .
$\gamma_{q}=-\alpha_{s}/\pi$ is the anomalous dimension of the quark.

##  Appendix C Meson Wave functions

In the nonrelativistic limit, the $B_{c}$ meson wave function can be written
as prd81014022

$\displaystyle\Phi_{B_{c}}(x)=\frac{if_{B}}{4N_{c}}[(\hbox
to0.0pt{/\hss}{P}+M_{B_{c}})\gamma_{5}\delta(x-r_{c})]\exp(-\frac{b^{2}\omega_{B}^{2}}{2}),$
(45)

in which the last exponent term represents the $k_{T}$ distribution. Here, we
only consider the dominant Lorentz structure and neglect another contribution
in our calculation epjc28515 .

In the heavy quark limit, the two-particle light-cone distribution amplitudes
of $D_{(s)}/D_{(s)}^{*}$ meson are defined as prd67054028

$\displaystyle\langle D_{(s)}(P_{2})|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle$
$\displaystyle=$
$\displaystyle\frac{i}{\sqrt{2N_{c}}}\int^{1}_{0}dxe^{ixP_{2}\cdot
z}[\gamma_{5}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}})\phi_{D_{(s)}}(x,b)]_{\alpha\beta},$
$\displaystyle\langle
D_{(s)}^{*}(P_{2})|q_{\alpha}(z)\bar{c}_{\beta}(0)|0\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{\sqrt{2N_{c}}}\int^{1}_{0}dxe^{ixP_{2}\cdot z}[\hbox
to0.0pt{/\hss}{\epsilon}_{L}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}^{*}})\phi^{L}_{D_{(s)}^{*}}(x,b)$ (46)
$\displaystyle+\hbox to0.0pt{/\hss}{\epsilon}_{T}(\hbox
to0.0pt{/\hss}{P}_{2}+m_{D_{(s)}^{*}})\phi^{T}_{D_{(s)}^{*}}(x,b)]_{\alpha\beta},$

with the normalization conditions:

$\displaystyle\int^{1}_{0}dx\phi_{D_{(s)}}(x,0)=\frac{f_{D_{(s)}}}{2\sqrt{2N_{c}}},\quad\int^{1}_{0}dx\phi^{L}_{D_{(s)}^{*}}(x,0)=\int^{1}_{0}dx\phi^{T}_{D_{(s)}^{*}}(x,0)=\frac{f_{D_{(s)}^{*}}}{2\sqrt{2N_{c}}},$
(47)

where we have assumed $f_{D_{(s)}^{*}}=f^{T}_{D_{(s)}^{*}}$. Note that
equations of motion do not relate $\phi^{L}_{D_{(s)}^{*}}$ and
$\phi^{T}_{D_{(s)}^{*}}$. We use the following relations derived from HQET
hqet  to determine $f_{D^{*}_{(s)}}$

$\displaystyle
f_{D^{*}_{(s)}}=\sqrt{\frac{m_{D_{(s)}}}{m_{D_{(s)}^{*}}}}f_{D_{(s)}}.$ (48)

The distribution amplitude $\phi^{(L,T)}_{D_{(s)}^{(*)}}$ is taken as 09101424

$\displaystyle\phi^{(L,T)}_{D_{(s)}^{(*)}}=\frac{3}{\sqrt{2N_{c}}}f_{D^{(*)}_{(s)}}x(1-x)[1+a_{D^{(*)}_{(s)}}(1-2x)]\exp(-\frac{b^{2}\omega^{2}_{D_{(s)}}}{2}).$
(49)

We use $a_{D}=0.5\pm 0.1,\omega_{D}=0.1\text{GeV}$ for $D/D^{*}$ meson and
$a_{D}=0.4\pm 0.1,\omega_{D_{s}}=0.2\text{GeV}$ for $D_{s}/D_{s}^{*}$ meson,
which are determined in Ref. dd1  by fitting.

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# Color-Dipole Picture versus Hard Pomeron
in Deep Inelastic Scattering

Dieter Schildknecht  Fakultät für Physik, Universität Bielefeld
D-33501 Bielefeld, Germany
and
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)
Föhringer Ring 6, D-80805 München, Germany

###### Abstract

For photon virtualities of $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$,
the results of the (hard Pomeron) fit in the tensor-Pomeron model in terms of
the variables $Q^{2}$ and the virtual-photon-proton energy squared, $W^{2}$,
provide empirical evidence for the validity of the Color Dipole Picture (CDP).
Consistency of the CDP with the perturbative QCD (pQCD) improved parton model
implies the prediction of $C_{2}=\epsilon_{0}\cong 0.30$ for the exponent
$C_{2}$ of the energy-squared dependence in agreement with the results of the
fits. For $Q^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20GeV^{2}$, the CDP yields a parameter-free smooth
transition from $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$ to $Q^{2}=0$ photoproduction, in
distinction from the tensor-Pomeron model that relies on the additional
parameter quantifying the intercept of the soft-Pomeron trajectory.

###### pacs:

##  I Introduction

The recent article Ewerz  by Britzger, Ewerz, Glazow, Nachtmann and Schmidt
revives the two-Pomeron-plus-Reggeon approach from ref. Donnachie  by
introducing tensor couplings of the hard Pomeron, $\mathbbm{P}_{0}$, the soft
Pomeron, $\mathbbm{P}_{1}$, and the $f_{2R}$ Reggeon at the photon and proton
vertices, $\gamma^{*}\gamma^{*}(\mathbbm{P}_{0},\mathbbm{P}_{1},f_{2R})$ and
$pp(\mathbbm{P}_{0},\mathbbm{P}_{1},f_{2R})$, to fit the experimental data on
deep-inelastic lepton-nucleon scattering (DIS) at low values of the Bjorken
variable $x\simeq Q^{2}/W^{2}<0.01$, including the $Q^{2}=0$ photoproduction
limit111In standard notation, $Q^{2}$ and $W^{2}$ denote the photon virtuality
and the square of the photon-proton center-of-mass energy..

Regge theory, by generalization from hadronic interactions, assumes a power-
law $W$ dependence for the photoabsorption cross section, as
$(W^{2})^{\alpha_{j}(0)-1}$, where the three Donnachie  Regge intercepts,
$\alpha_{0}(0)=1+\epsilon_{0},\alpha_{1}(0)=1+\epsilon_{1}$ and
$\alpha_{2}(0)$, are free parameters to be extracted from a fit to the
experimental data. The form of the $Q^{2}$ dependence of the
$\gamma^{*}\gamma^{*}(\mathbbm{P}_{0},\mathbbm{P}_{1},f_{2R})$ vertex
contributions not being fixed by Regge theory, in ref. Ewerz  the $Q^{2}$
dependence is model-independently parametrized by smooth functions in terms of
splines. Altogether the fit Ewerz  contains 25 free parameters, namely the 3
Regge intercepts, 5 parameters for the
$\gamma^{*}\gamma^{*}(\mathbbm{P}_{0},\mathbbm{P}_{1},f_{2R})$ couplings at
$Q^{2}=0$, and 17 parameters for a model-independent description of the
$Q^{2}$ dependence in terms of spline functions.

In the present article, we confront the interpretation of the DIS experimental
data in Regge theory with their representation in the Color-Dipole Picture
(CDP). We proceed in three steps. In a first step, we concentrate on the
region of large $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, in Regge theory associated with the
hard Pomeron $\mathbbm{P}$ with intercept $\alpha_{0}(0)$. In the second step,
we show that consistency of the CDP and the perturbative-QCD-improved parton
model implies a successful prediction for the numerical value of the exponent
of the $W^{2}$ dependence, $\epsilon_{0}=C_{2}\cong 0.30$. In the third step,
we consider the transition to low $Q^{2}$, including the $Q^{2}=0$ limit of
photoproduction, in Regge theory associated with the soft Pomeron with
intercept $\alpha_{1}(0)$.

It will turn out that the model-independent 17-parameter fit of the $Q^{2}$
dependence in the tensor-Pomeron approach for $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$
leads to, and confirms, the parameter-free prediction of the
$Q^{2}$-dependence of the CDP, wherein the photoabsorption cross section,
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})$, depends on the low-x scaling variable
$\eta^{-1}=\Lambda^{2}_{sat}(W^{2})(Q^{2}+m^{2}_{0})^{-1}$, i.e.
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})=\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2}))$,
with $\Lambda^{2}_{sat}(W^{2})\propto(W^{2})^{C_{2}}$ and
$C_{2}=\epsilon_{0}$. Specifically, for $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$,
in the CDP,
$\sigma_{\gamma^{*}p}\propto\eta^{-1}=\Lambda^{2}_{sat}(W^{2})/Q^{2}$, as
confirmed by the tensor-Pomeron-fit result.

##  II Large Values of $\mathbf{Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20{\rm\bf
GeV}^{2}}$

. For large values of $Q^{2}$, the total photoabsorption cross section in the
Regge approach is determined Ewerz  by the hard-Pomeron contribution.
According to formula (6.1) in ref. Ewerz , the fit led to the result

$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto\hat{b}_{0}(Q^{2})(W^{2})^{\epsilon_{0}},$
(1)

where

$\epsilon_{0}=0.3008\left(+73\atop-84\right)\cong 0.30\pm 0.1,$ (2)

and, with $\eta_{0}=0.967(73)\cong 1.0$,

$\hat{b}_{0}(Q^{2})=\frac{1}{(Q^{2})^{\eta_{0}}}\cong\frac{1}{Q^{2}}.$ (3)

According to (1) and (3), in the tensor-Pomeron model, for
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20GeV^{2}$, the photoabsorption cross section fulfills
the proportionality Ewerz

$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto\frac{(W^{2})^{\epsilon_{0}}}{Q^{2}}$
(4)

with $\epsilon_{0}\cong 0.30$ from (2).

In ref. DIFF2000 , in a phenomenological fit to the photoabsorption
experimental data, for $x\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}0.1$ and $0\leq Q^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1000GeV^{2}$,
technically by assuming a piecewise linear functional dependence of the cross
section on the variable $\eta(W^{2},Q^{2})$ introduced DIFF2000  via

$\frac{1}{\eta(W^{2}Q^{2})}=\frac{\Lambda^{2}_{sat}(W^{2})}{Q^{2}+m^{2}_{0}},$
(5)

where

$\Lambda^{2}_{sat}(W^{2})\propto(W^{2})^{C_{2}},$ (6)

we found

$\sigma_{\gamma^{*}p}(W^{2},Q^{2})=\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2})).$
(7)

According to our fit, the photoabsorption cross section, including its
$Q^{2}=0$ limit, only depends on the single low-x scaling variable
$\eta(W^{2},Q^{2})$, rather than $W^{2}$ and $Q^{2}$ separately. The fit gave
DIFF2000

$C_{2}=0.28\pm 0.06$ (8)

for the exponent $C_{2}$ in (6), as well as $m^{2}_{0}=0.125\pm 0.027~{}{\rm
GeV}^{2}$ for the mass parameter $m_{0}$, with $m^{2}_{0}<m^{2}_{\rho}$, as
expected, where $m_{\rho}$ denotes the $\rho$-meson mass.

By inspection of the graphical representation of the experimental data for the
photoabsorption cross section as a function of $\eta(W^{2},Q^{2})$, one reads
off an explicit functional dependence that is approximately given by DIFF2000

$\displaystyle\sigma_{\gamma^{*}p}(W^{2},Q^{2})=$
$\displaystyle\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2}))\propto\frac{1}{\eta(W^{2},Q^{2})}=\frac{(W^{2})^{C_{2}}}{Q^{2}},$
(9) $\displaystyle(Q^{2}+m^{2}_{0}\gg\Lambda^{2}_{sat}(W^{2})),$

where $\Lambda^{2}_{sat}(W^{2})\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}7{\rm GeV}^{2}$ in
the energy range, where experimental data are available.

Comparison of (9) with (4), and of (8) with (2), reveals that for
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$ the tensor-Pomeron fit (1) from ref.
Ewerz  confirms the result of the phenomenological fit (9) of
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto 1/\eta(W^{2},Q^{2})$ from ref.
DIFF2000 .

A comment on the two-Pomeron fit of ref. Donnachie  is appropriate. In
distinction from the model-independent fit of the $Q^{2}$ dependence in ref.
Ewerz , the $Q^{2}$ dependence of the photoabsorption cross section in ref.
Donnachie  is described by parameter-dependent analytic expressions in
$Q^{2}$. The fit gave Donnachie  an intercept of $\alpha_{0}\cong 1.4$ or

$\epsilon_{0}\cong 0.40,$ (10)

significantly different from (2).

It is gratifying that the result of the tensor-Pomeron fit Ewerz  for
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, with $\epsilon_{0}\cong 0.30$,
supports and confirms the fit in ref. DIFF2000 . The tensor-Pomeron fit, under
the restriction to $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, has rediscovered the empirical
scaling law DIFF2000  of
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})=\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2}))$.

Scaling in $\eta(W^{2},Q^{2})$, including the observed specific functional
dependence (9) of $\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto
1/\eta(W^{2},Q^{2})=\Lambda^{2}_{sat}(W^{2})/Q^{2}$, valid for sufficiently
large $Q^{2}$, is a consequence of the color-dipole picture DIFF2000 . For
clarity and completeness, we elaborate on this essential point in detail.

The color dipole picture (CDP) of deep inelastic scattering (DIS) at low x is
based on the transverse position-space representation

$\displaystyle\sigma_{\gamma^{*}_{L,T}}(W^{2},Q^{2})$ $\displaystyle=\int
dz\int d^{2}\vec{r}_{\bot}|\psi_{L,T}(\vec{r}_{\bot},z(1-z),Q^{2})|^{2}$ (11)
$\displaystyle\sigma_{(q\bar{q})p}(\vec{r}^{~{}2}_{\bot},z(1-z),W^{2})$

combined with the interaction of a $q\bar{q}$ color dipole with the color
field in the nucleon Cvetic ,

$\displaystyle\sigma_{(q\bar{q})p}(\vec{r}^{~{}2}_{\bot},z(1-z),W^{2})$
$\displaystyle=\int
d^{2}\vec{l}_{\bot}\tilde{\sigma}(\vec{l}^{~{}2}_{\bot},z(1-z),W^{2})$ (12)
$\displaystyle\left(1-e^{-i\vec{l}_{\bot}\vec{r}_{\bot}}\right).$

In (11) and (12), $\vec{r}_{\bot}$ denotes the transverse interquark distance,
$z(1-z)$ with $0\leq z\leq 1$ describes the quark-antiquark $(q\bar{q})$
configuration, and $\vec{l}_{\bot}$ the transverse gluon momentum absorbed by
the $q\bar{q}$ state.

The representation (11), including the $(q\bar{q})p$ interaction (12), follows
Cvetic  from off-diagonal vector dominance Sakurai  upon taking into account
the internal structure of the vector states, $V\equiv(q\bar{q})^{J=1}$, in the
$\gamma^{*}\to V$ transition, and transforming from momentum space to
transverse position space. The minus sign in front of the exponential in (12),
in the pre-QCD era originated Sakurai , Cvetic  from the required consistency
Sakurai  of scaling behavior in the timelike region of
$e^{+}e^{-}$-annihilation, $e^{+}e^{-}\to$ hadrons, and in the spacelike
region of deep inelastic scattering (DIS) of $e^{-}p\to e^{-}+$ hadrons.

In QCD, the $(q\bar{q})^{J=1}$ states act as color-dipole states, thus
recognizing the interaction (12) as a consequence of the color-dipole nature
of the $(q\bar{q})^{J=1}$ states. In the limit of vanishing dipole size,
$\vec{r}^{~{}2}_{\bot}\to 0$, a $(q\bar{q})$-color-dipole state acts as a
color-neutral object of vanishing interaction with the color field in the
proton (“color transparency”), while for large dipole size,
$\vec{r}^{~{}2}_{\bot}\to\infty$, the cross section has to remain finite and
becomes hadronlike (“saturation”), thus leading to (12). A specific ansatz for
the $(q\bar{q})p$ cross section fulfilling the requirements of color
transparency and saturation is mentioned in (30) below.

One observes that perturbative QCD (pQCD) has not been mentioned in arriving
at the interaction cross section (12). In pQCD, the form of the interaction
(12) is understood as a consequence of the color-gauge-invariant interaction
of the $(q\bar{q})$ dipole with the color field in the nucleon via exchange of
(at least) two gluons. The form of the position-space representation (11) with
(12) from off-diagonal GVD, from the point of view of pQCD is accordingly
recognized as a consequence of the gauge-invariant interaction of a
$(q\bar{q})^{J=1}$-color-dipole state with the gluon field in the proton. The
representation of the photoabsorption cross section in (11) and (12) is
nevertheless a non-perturbative222In ref. Nachtmann , “Towards a Non-
Perturbative Derivation of the CDP”, in a very detailed field-theoretic
analysis the form of the photoabsorption cross section (11) was re-
established, by assumption excluding a dependence of the dipole cross section
on $z(1-z)$ in (11). The treatment in ref. Nachtmann , rederiving (11), while
not incorporating color-transparency and saturation, misses an essential
element of the CDP. one, otherwise it would not include and contain the smooth
transition to the $Q^{2}=0$ photoproduction cross section.

It is the structure of the $(q\bar{q})p$-interaction amplitude contained in
(12) that is responsible for the CDP prediction of the $1/Q^{2}$ dependence of
the photoabsorption cross section, $\sigma_{\gamma^{*}p}\propto 1/Q^{2}$
(given in (19) below), that, according to the fit result (1) with (3), is
confirmed by the experimental data.

Upon introducing the variables
$\vec{r}^{~{}\prime}=\sqrt{z(1-z)}\vec{r}_{\bot}$ and
$\vec{l}^{~{}\prime}_{\bot}=\vec{l}_{\bot}/\sqrt{z(1-z)}$, and the cross
sections for longitudinally and transversely polarized
$(q\bar{q})^{J=1}_{L,T}$ ($J=1$, vector) states, for the color-dipole-proton
cross section (12), one indeed finds DIFF2000 ; Kuroda  the two limiting cases
of “color transparency” for $\vec{r}^{~{}\prime 2}_{\bot}\to 0$,

$\displaystyle\sigma_{(q\bar{q})^{J=1}_{L,T}p}(r^{~{}\prime 2}_{\bot},W^{2})$
$\displaystyle=\frac{1}{4}\pi\vec{r}^{~{}\prime 2}_{\bot}\int
d\vec{l}^{~{}\prime 2}_{\bot}\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{L}p}(\vec{l}^{~{}\prime
2}_{\bot},W^{2})$ $\displaystyle\times\left\\{\begin{array}[]{l}1,\\\
\rho_{W}\end{array}\right.$ $\displaystyle\left(\vec{r}^{~{}\prime
2}_{\bot}\ll\frac{1}{\vec{l}^{~{}\prime 2}_{\bot~{}Max}(W^{2})}\right),$ (15)

where

$\rho_{W}\equiv\frac{\int d\vec{l}^{~{}\prime 2}_{\bot}\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{T}p}(\vec{l}^{~{}\prime
2}_{\bot},W^{2})}{\int d\vec{l}^{~{}\prime 2}_{\bot}\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{L}p}(\vec{l}^{~{}\prime
2}_{\bot},W^{2})},$ (16)

and “saturation” for $\vec{r}^{~{}\prime 2}_{\bot}\to\infty$,

$\displaystyle\sigma_{(q\bar{q})^{J=1}_{L,T}p}$ $\displaystyle(r^{~{}\prime
2}_{\bot},W^{2})=\pi\int d\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{L,T}p}(\vec{l}^{~{}\prime 2},W^{2})$
(17)
$\displaystyle\equiv\sigma_{L,T}^{(\infty)}(W^{2}),~{}~{}~{}~{}\left(\vec{r}^{~{}\prime
2}_{\bot}\gg\frac{1}{\vec{l}^{~{}\prime 2}_{\bot~{}Max}(W^{2})}\right).$

In (15) and (17), $\vec{l}^{~{}\prime 2}_{\bot~{}Max}(W^{2})$ denotes the
upper limit of the range of integration over $d\vec{l}^{~{}\prime 2}_{\bot}$
that yields the dominant contribution to the integrals. The parameter
$\rho_{W}$ in (16), where $W$ indicates a potential dependence of
$\rho=\rho_{W}$ on the energy $W$, determines the ratio of (the first moment
of) the cross sections for the scattering of transversely (helicity $\pm 1$)
and longitudinally (helicity $0$) polarized $(q\bar{q})^{J=1}_{L,T}$ states on
the proton. The deviation from $\rho_{W}=1$ quantifies the deviation of the
$(q\bar{q})^{J=1}$-proton cross section from helicity independence of
$(q\bar{q})^{J=1}_{L}p=(q\bar{q})^{J=1}_{T}p$.

The cross section $\sigma^{(\infty)}_{L,T}(W^{2})$ in (17) is a purely
hadronic quantity that fulfills hadronic unitarity, and it is at most weakly,
logarithmically, dependent on the energy $W$.

The photoproduction cross section, as a consequence of the explicit form of
the “photon wave function” $\psi_{L,T}(\vec{r}_{\bot},z(1-z),Q^{2})$ in (11),
for sufficiently large $Q^{2}\gg 0$, receives contributions from only a finite
range of $\vec{r}^{~{}\prime 2}_{\bot}$ that is restricted by
$\vec{r}^{~{}\prime 2}_{\bot}\leq 1/Q^{2}$. For given large energy $W$ and
sufficiently large $Q^{2}$, such that $1/Q^{2}\ll 1/\vec{l}^{~{}\prime
2}_{\bot~{}Max}(W^{2})$, the photoabsorption cross section (11) is determined
by the color-dipole cross section for $\vec{r}^{~{}\prime 2}_{\bot}\ll
1/\vec{l}^{~{}\prime 2}_{\bot~{}Max}(W^{2})$, i.e. in the color-transparency
limit. Introducing the first moment of the distribution in $\vec{l}^{~{}\prime
2}_{\bot}$ for $(q\bar{q})^{J=1}_{L}p$ scattering,

$\Lambda^{2}_{sat}(W^{2})\equiv\frac{\int d\vec{l}^{~{}\prime
2}_{\bot}\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{L}p}(\vec{l}^{~{}\prime
2}_{\bot},W^{2})}{\int d\vec{l}^{~{}\prime
2}_{\bot}\bar{\sigma}_{(q\bar{q})^{J=1}_{L}p}(\vec{l}^{~{}\prime
2}_{\bot},W^{2})},$ (18)

the photoabsorption cross section (11), upon inserting (15) and employing (16)
and (17), becomes DIFF2000 ; Kuroda

$\sigma_{\gamma^{*}_{L,T}p}(W^{2},Q^{2})\propto\sigma_{L}^{(\infty)}(W^{2})\frac{\Lambda^{2}_{sat}(W^{2})}{Q^{2}}\left\\{\begin{array}[]{l}1,\\\
2\rho_{W}\end{array}\right..$ (19)

With the approximation of $\sigma_{L}^{(\infty)}(W^{2})\cong{\rm const}$, and
$\rho_{W}=\rho={\rm const}$ due to invariance under Lorentz boosts, and
specifying $\Lambda^{2}_{sat}(W^{2})\propto(W^{2})^{C_{2}}$, we indeed
recognize the result of the phenomenological fit (9) (that agrees with the
tensor-Pomeron fit (1)) as a consequence of the CDP (11) and (12). The energy
dependence of the fit (9), as
$\Lambda^{2}_{sat}(W^{2})\propto(W^{2})^{C_{2}}$, according to (19) with (18)
is recognized as the energy dependence of the effective transverse momentum of
the gluon absorbed by the color-dipole $q\bar{q}$ state in the
$\vec{r}^{~{}\prime 2}_{\bot}\to 0$ limit.

While the fitted large-$Q^{2}$ energy dependence (1) of the photoabsorption
cross section in the tensor-Pomeron model, as $(W^{2})^{\epsilon_{0}}$, is
consistent with the energy dependence (9) of the CDP, as $(W^{2})^{C_{2}}$,
its interpretation Ewerz  is entirely different. The fitted energy dependence
for $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, in the tensor-Pomeron model is
interpreted as evidence for the exchange of a hypothetical “hard-Pomeron
trajectory” with intercept $\alpha_{0}(0)=1+\epsilon_{0}$, independent
empirical evidence for this conjectured trajectory being lacking.
Independently of this interpretation, the agreement of the fit (1) with the
fit (9) now explicitly being recognized as a consequence of the CDP,
explicitly confirms our above conclusion that the fit (1) provides additional
empirical support for the previously empirically established validity DIFF2000
of the CDP of DIS at low $x\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}0.1$.

##  III Consistency of pQCD with the CDP and the numerical value of
$\mathbf{C_{2}\cong 0.30}$

For $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, in both, the fit (4) and the fit
(9), the $Q^{2}$ dependence of the photoabsorption cross section is given by

$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto\frac{1}{Q^{2}}.$ (20)

The structure function of the proton, $F_{2}(x,Q^{2})\cong
Q^{2}\sigma_{\gamma^{*}p}(W^{2},Q^{2})$, accordingly only depends on $W^{2}$,

$F_{2}(x,Q^{2})=F_{2}\left(W^{2}=\frac{Q^{2}}{x}\right).$ (21)

Explicitly, this $W$ dependence of the experimental data is shown in Fig. 1
taken from ref. Kuroda . A eye-ball two-parameter fit to the experimental data
in fig. 1 gives Kuroda

$F_{2}(W^{2})=f_{2}\left(\frac{W^{2}}{1{\rm GeV}^{2}}\right)^{C_{2}},$ (22)

where $f_{2}\cong 0.063$ and $C_{2}=0.29$.

Fig.1a Fig.1b

Figure 1: In fig.1a we show the experimental data for $F_{2}(x\cong
Q^{2}/W^{2},Q^{2})$ as a function of $1/W^{2}$, and in fig.1b, for comparison,
as a function of $x$. The theoretical result based on (22) is also shown in
fig.1a.

The observed $W^{2}$ dependence (22) of the structure function
$F_{2}=F_{2}(W^{2})$, interpreted as a consequence of the perturbative-QCD
(pQCD) improved parton model, implies a remarkable constraint Kuroda  on the
magnitude of the exponent $C_{2}$.

In the pQCD improved parton model, the longitudinal structure function
$F_{L}(x,Q^{2})=F_{2}(x,Q^{2})/(1+1/R)$, where $R$ refers to the longitudinal-
to-transverse photoabsorption-cross-section ratio, at a rescaled value
$x\to\xi_{L}x$, for a wide range of different gluon distributions, is
proportional to the gluon distribution multiplied by $\alpha_{s}(Q^{2})$,
compare ref. Martin . Similarly, the logarithmic derivative of
$F_{2}(x,Q^{2})$, at a rescaled value of $x\to\xi_{L}x$, is proportional to
the gluon distribution Prytz . Eliminating the gluon distribution, and
adopting the empirically supported dependence (22) of $F_{2}=F_{2}(W^{2})$,
one finds an evolution equation for $F_{2}(W^{2})$ that is given by Kuroda

$(2\rho_{W}+1)\frac{\partial}{\partial
lnW^{2}}F_{2}\left(\frac{\xi_{L}}{\xi_{2}}W^{2}\right)=F_{2}(W^{2}).$ (23)

In (23), we have replaced the longitudinal-to-transverse ratio of the
photoabsorption cross sections via $R=1/2\rho_{W}$, compare (19).

Inserting $F_{2}\propto(W^{2})^{C_{2}}$, see (22), from (23), we obtain the
constraint Kuroda

$(2\rho_{W}+1)C_{2}\left(\frac{\xi_{L}}{\xi_{2}}\right)^{C_{2}}=1.$ (24)

The constraint (24) connects the exponent $C_{2}$ of the power-law dependence
(22) of $F_{2}=F_{2}(W^{2})$ with the value of $\rho_{W}$ that quantifies the
deviation from helicity independence, $\rho_{W}=1$, of $(q\bar{q})^{J=1}$
scattering from the proton according to (16). Constancy of $C_{2}$, according
to (24), implies constancy of $\rho_{W}=\rho$. From the point of view of pQCD,
the constancy333A refined analysis Kuroda  of the CDP leads to a correction of
$R=1/2\rho$ that implies a decrease of $R\propto
1/\eta(W^{2},Q^{2})=\Lambda^{2}_{sat}(W^{2})/Q^{2}$ in the limit of very large
values of $Q^{2}$ that reach the limit of $x\cong Q^{2}/W^{2}\to 0.1$. of
$\rho$ implies Kuroda  a gluon distribution multiplied by $\alpha_{s}(Q^{2})$
that coincides with the $q\bar{q}$ sea distribution.

Expanding the exponential in (24) (to first order in $C_{2}$), and solving for
$C_{2}$ yields

$C_{2}\cong\frac{1}{2\rho+1}\frac{1}{\left(1-\frac{1}{2\rho+1}ln\frac{\xi_{2}}{\xi_{L}}\right)}.$
(25)

The ratio of $\xi_{2}/\xi_{L}$ in (24) and (25) is given by Martin ; Prytz
$\xi_{2}/\xi_{L}=1.25$.

Table 1: The value of $\rho$ determines the ratio of the cross sections for transversely, $(q\bar{q})^{J=1}_{T}$, and longitudinally polarized, $(q\bar{q})^{J=1}_{L}$, states on the proton, see (16). A value of $\rho\not=1$ corresponds to a deviation of $R$ from the value of $R=0.5$ corresponding to helicity independence of $(q\bar{q})^{J=1}p$ scattering. Theoretical values of $C_{2}$ from (25) are compared with experimental ones. For $C_{2}|_{Exp}=0.27\pm 0.1$, see (32) below. $\rho$ | $R=\frac{1}{2\rho}$ | $\frac{1}{2\rho+1}$ | $C_{2}$ | $C_{2}|_{Exp}$
---|---|---|---|---
1 | 0.5 | $\frac{1}{3}\cong 0.33$ | 0.360 |
|  |  |  |  $0.30\pm 0.01$ (ref.1)
$\frac{4}{3}$ | 0.375 | $\frac{3}{11}\cong 0.27$ | 0.290 |  $0.27\pm 0.01$ (ref.3)
2 | 0.25 | $\frac{1}{5}=0.20$ | 0.209 |

In Table 1444The “rigorous upper limit” of $R\leq 0.37248$ from the “standard
color-dipole model of low x DIS”Ewerz ; Nachtmann , excluding $\rho=1(R=1/2)$
and even $\rho=4/3=0.375$ in Table 1, depends Schi  on the ad hoc assumption
contained in the “standard color-dipole model” Ewerz ; Nachtmann  of excluding
Nachtmann  a $z(1-z)$ dependence of the $(q\bar{q})p$ cross section
$\sigma_{(q\bar{q})p}$ in (11), compare footnote 2. Fits to experimental data
test this underlying assumption. A violation of the “rigorous upper limit” by
experimental data can be taken care of by allowing for a dependence on
$z(1-z)$, see e.g. the ansatz (30) that implies $\rho=1$ (i.e. helicity
independence of $(q\bar{q})^{J=1}p$ scattering, compare (16)) and
$R=1/2\rho=0.5$ in violation of the bound., we show the results for $C_{2}$
from (25) for different values of $\rho$ and $R=1/2\rho$. We comment on the
results in Table 1 as follows:

  * i)

The different dependence on the configuration, $z(1-z)$, of longitudinally and
transversely polarized $(q\bar{q})^{J=1}$ states, via the uncertainty
principle, implies an enhanced transverse size of transversely polarized
relative to longitudinally polarized $(q\bar{q})^{J=1}$ states. This estimate
within the CDP implies $\rho=4/3$ Kuroda . Consistency of the CDP with pQCD,
from (25), predicts $C_{2}=0.29$, in agreement with the observed value of
$C_{2}|_{Exp}=\epsilon_{0}\cong 0.30$ from (2) and $C_{2}|_{Exp}=0.28\pm 0.06$
from (8) and $C_{2}|_{Exp}=0.27\pm 0.01$ from (32) below. We conclude that the
consistency between the CDP and pQCD is empirically established.

  * ii)

A value of $\rho=1$, assuming helicity independence of $(q\bar{q})^{J=1}p$
scattering, in the approximation of ignoring the transverse-size enhancement
discussed in i), according to the consistency condition (25) implies
$C_{2}=0.360$, which is excluded by the measured value of
$C_{2}|_{Exp}=\epsilon_{0}\cong 0.30$. Concerning the direct experimental
determination of $R=1/2\rho$, see below.

  * iii)

A value of a transverse-to-longitudinal enhancement by a factor of $\rho=2$,
or $R=1/2\rho=0.25$ requires $C_{2}=\epsilon_{0}=0.21$, and is definitely
excluded by $C_{2}|_{Exp}\cong 0.30$.

  * iv)

We add a comment on the direct measurement of the ratio $R=1/2\rho$ for
$Q^{2}\gg 0$. First results from the H1 and ZEUS collaborations H1 ; ZEUS
showed agreement Kuroda  with the prediction of

$\displaystyle F_{L}=\frac{1}{1+\frac{1}{R}}$ $\displaystyle
F_{2}=\frac{1}{1+2\rho}F_{2}=0.27F_{2},$ (26) $\displaystyle(\rho=4/3).$

More recent measurements showed a smaller value of R Andreev . The more
indirect determination of $R$ from the two-Pomeron fit to the reduced cross
sections led to values roughly between $R=0.35$ and $R=0.5$ Ewerz . It is very
unfortunate that precision measurements on this very important ratio $R$ will
not become available in the near-by future.

##  IV Low values of $\mathbf{Q^{2}}$, the photoproduction limit of
$\mathbf{Q^{2}\to 0}$

We come to the fourth part of this article, the examination of the low-$Q^{2}$
region of $Q^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, including the photoproduction limit
of $Q^{2}=0$.

In the tensor-Pomeron model, with decreasing $Q^{2}$, the contribution of the
soft Pomeron (plus $f_{2R}$ Reggeon) to the total photoabsorption cross
section becomes increasingly important. In the $Q^{2}=0$ photoproduction
limit, the hard Pomeron yields Ewerz  a vanishing contribution. The soft-
Pomeron contribution behaves as Ewerz

$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto(W^{2})^{\epsilon_{1}},~{}~{}{\rm
for}~{}(Q^{2}\to 0),$ (27)

where

$\epsilon_{1}=0.0935\left({+76\atop-64}\right)\cong 0.094,$ (28)

close to $\epsilon_{1}=0.096$ from ref. Donnachie .

In the CDP, with decreasing $Q^{2}\to 0$ or increasing $1/\eta(W^{2},Q^{2})$,
the photoabsorption cross section receives contributions from an increasingly
larger dipole size $\vec{r}_{\bot}^{~{}\prime 2}$. The photoabsorption cross
section (11) is determined by the $\vec{r}^{~{}\prime 2}_{\bot}\to\infty$
saturation limit (17) of the dipole cross section (12). A detailed
examination, based on (17), yields Kuroda

$\displaystyle\sigma_{\gamma^{*}p}(W^{2},Q^{2})$
$\displaystyle\propto\sigma_{T}^{(\infty)}(W^{2})ln\frac{\Lambda^{2}_{sat}(W^{2})}{Q^{2}+m^{2}_{0}},$
(29) $\displaystyle\left(Q^{2}+m^{2}_{0}\ll\Lambda^{2}_{sat}(W^{2})\right).$

The transition from $Q^{2}\gg\Lambda^{2}_{sat}(W^{2})$ in (19) to
$Q^{2}\ll\Lambda^{2}_{sat}(W^{2})$ in (29) occurs by transition from
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto 1/\eta(W^{2},Q^{2})$ in (19) to
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto ln\left(1/\eta(W^{2},Q^{2})\right)$
in (29). In distinction from the two-Pomeron model, no independent additional
parameter must be introduced in this smooth transition from $Q^{2}\gg 20{\rm
GeV}^{2}$ to $Q^{2}=0$. The cross section (29) fulfills Kuroda  the Froissart
bound $\sigma_{\gamma^{*}p}\propto ln^{2}W^{2}$.555The photon makes a
transition to (quasi-asymptotic) on-shell $(q\bar{q})^{J=1}$ states, see e.g.
ref. KuSchi  for a concise representation of this point. For e.g. $Q^{2}\to
0$, the range of masses of contributing $(q\bar{q})^{J=1}$ states is
essentially given by transitions $\gamma\to(q\bar{q})^{J=1}$ to low-lying
$(q\bar{q})^{J=1}$ on-shell color-dipole states representing the low-lying
vector mesons, $\gamma\to(\rho^{0},w,\phi)$. (For an early representation of
the $(\rho^{0},w,\phi)$ contributions by a massive $J=1$ continuum compare
ref. 5.) Accordingly, $\sigma_{\gamma
p}(W^{2})\propto\sigma_{(\rho^{0},w,\phi)p}(W^{2})\propto ln^{2}W^{2}$, where
the second step rests on the $ln^{2}W^{2}$ dependence of the hadronic cross
section $\sigma_{(\rho^{0},w,\phi)}(W^{2})$ in the high-energy limit.
Heisenberg . In (29) and in (30), we have $\sigma^{(\infty)}(W^{2})\propto
lnW^{2}$. Compare also ref. KuSchi  for a discussion of the $W$ dependence of
$\sigma^{(\infty)}(W^{2})$ and its experimental determination. The fact that
$\gamma^{*}$ is not an asymptotic state Ewerz  is recognized as becoming
irrelevant with respect to the high-energy limit of the photo-absorption cross
section as soon as the underlying (empirically supported DIFF2000 ; Sakurai )
dynamical mechanism of $\gamma^{*}\to(q\bar{q})^{J=1}$ with subsequent on-
shell hadronic $(q\bar{q})^{J=1}p$ scattering is taken into account. For
successful fits to the measured total cross sections for $\gamma^{*}p$, as
well as $\gamma p,\pi^{\pm}p,pp,\bar{p}p$, scattering, based on Froissart-
bound saturation $(ln^{2}W^{2})$ rather than Regge theory
$\left((W^{2})^{\alpha_{j}(0)-1}\right)$, compare ref. Block .

The considerations on the CDP so far were exclusively obtained by analysing
the general expressions (11) and (12). A detailed interpolation between the
limits of $1/\eta(W^{2},Q^{2})$ for $\eta(W^{2},Q^{2})\gg 1$, and
$ln\left(1/\eta(W^{2},Q^{2})\right)$ for $\eta(W^{2},Q^{2})\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1$, requires a
specific ansatz for the dipole cross section in (11). The fit in ref. DIFF2000
was based on the ansatz

$\displaystyle\sigma_{(q\bar{q})p}(\vec{r}_{\bot},z(1-z),W^{2})$
$\displaystyle=\sigma^{(\infty)}(W^{2})\left(1-J_{0}\left(r_{\bot}z(1-z)\Lambda^{2}_{sat}(W^{2})\right)\right),$
(30)

where $\sigma^{(\infty)}(W^{2})$ is of hadronic size and logarithmically
dependent on $W^{2}$. The ansatz (30) contains color transparency (15) and
saturation (17). The dependence on $r^{\prime}_{\bot}=r_{\bot}z(1-z)$ implies
Kuroda  $\rho=1$ and $R=1/2\rho=1/2$ for $Q^{2}\gg 0$666A refined ansatz
Kuroda  for the dipole interaction incorporates $\rho=4/3$.

The fit to the total photoabsorption cross section (11) upon insertion of (30)
gave DIFF2000

$\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}+W^{2}_{0}}{1{\rm
GeV}^{2}}\right)^{C_{2}}\simeq C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}},$ (31)

where

$\displaystyle C_{2}$ $\displaystyle=$ $\displaystyle 0.27\pm 0.01,$
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle 0.34\pm 0.05{\rm
GeV}^{2},$ (32)

as well as

$\displaystyle m^{2}_{0}$ $\displaystyle=0.16\pm 0.01{\rm GeV}^{2},$
$\displaystyle W^{2}_{0}$ $\displaystyle=882\pm 246{\rm GeV}^{2}.$ (33)

One finds $2{\rm GeV}^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}\Lambda^{2}_{sat}(W^{2})\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}7{\rm GeV}^{2}$
for the HERA energy range of $30{\rm GeV}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}W\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}300{\rm GeV}$. The
result (32) is consistent with, and improves the accuracy of the
phenomenological fit (8), and it is consistent with the tensor-Pomeron fit
(2). For the fit, the hadronic cross section $\sigma^{(\infty)}(W^{2})$ was
consistently expressed DIFF2000 ; Kuroda  in terms of a fit to the $Q^{2}=0$
photoproduction cross section. In addition to this input of a fit to the
$Q^{2}=0$ photoproduction cross section, essentially only three independent
fit parameters $C_{2},C_{1}$ and $m^{2}_{0}$, have been used for a successful
representation of the body of the experimental data on low-x DIS, including
the transition to $Q^{2}=0$ photoproduction. Compare the previous Section for
a theoretical prediction of the value of $C_{2}$.

##  V Conclusions

We end with the following conclusions:

  * i)

The hard-Pomeron-fit result Ewerz  of
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto(W^{2})^{\epsilon_{0}}/(Q^{2})^{\eta_{0}}$
for $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$, where
$\epsilon_{0}=0.3008\left({+73\atop{-84}}\right)\cong 0.30\pm 0.1$ and
$\eta_{0}=0.967(73)\cong 1.00\pm 0.04$, based on the assumption of a power-
like $W^{2}$ dependence and 17 parameters for a model-independent
parametrization of the $Q^{2}$ dependence, confirms the prediction DIFF2000
of the CDP of $\sigma_{\gamma^{*}p}(W^{2},Q^{2})\propto
1/\eta(W^{2},Q^{2})=\Lambda^{2}_{sat}(W^{2})/Q^{2}$ with the predicted
$1/Q^{2}$ dependence and the fitted $W^{2}$ dependence of
$\Lambda^{2}_{sat}(W^{2})\propto(W^{2})^{C_{2}}$, where $C_{2}=0.27\pm 0.01$.

  * ii)

Requiring consistency of the CDP and pQCD for sufficiently large
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10~{}{\rm GeV}^{2}$ yields the prediction of
$C_{2}=0.29$ for the exponent $C_{2}$ that is confirmed by the CDP fit result
of $C_{2}=0.27\pm 0.01$ and the hard-Pomeron-fit result of $\epsilon_{0}\cong
0.30\pm 0.1$.

  * iii)

In distinction from the tensor-Pomeron approach, in the CDP, the transition
from $Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$ to $Q^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}20{\rm GeV}^{2}$,
including the $Q^{2}=0$ photoproduction limit, is obtained by a smooth
transition from $\sigma_{\gamma^{*}p}\propto 1/\eta(W^{2},Q^{2})$ to
$\sigma_{\gamma^{*}p}\propto ln(1/\eta(W^{2},Q^{2}))$, without the necessity
for introducing an additional fit parameter.

  * iv)

The essential new element of DIS compared to photoproduction consists of the
additional degree of freedom of the virtuality, $Q^{2}$, of the (virtual)
photon. The $Q^{2}$ dependence of the photoabsorption cross section in the CDP
is uniquely predicted as a consequence of the mass-dispersion relation of
Generalized Vector Dominance (GVD) formulated some 50 years ago. In the
tensor-Pomeron-Regge approach this $Q^{2}$ dependence is fitted by a large
number of parameters to allow for a representation of the energy dependence of
the photoabsorption experimental data in terms of a linear superposition of a
power-law ansatz associated with, respectively, a soft and a hard Pomeron
trajectory. Additional independent empirical information on the (hard) Pomeron
trajectory is lacking so far, however.

###### Acknowledgements.

The author thanks Kuroda-san for useful remarks on the preliminary version of
this paper.

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