# W-exchange contribution to the decays
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ using
light-cone sum rules

Yu-Ji Shi1 111Email<EMAIL_ADDRESS>Zhen-Xing Zhao 2 222Email:
<EMAIL_ADDRESS>Ye Xing 3 333Email<EMAIL_ADDRESS>and
Ulf-G. Meißner 1,4,5 444Email<EMAIL_ADDRESS>1 Helmholtz-Institut
für Strahlen- und Kernphysik and Bethe Center
for Theoretical Physics, Universität Bonn, 53115 Bonn, Germany
2 School of Physical Science and Technology, Inner Mongolia University, Hohhot
010021, China
3 School of Physics, China University of Mining and Technology, Xuzhou 221000,
China
4 Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center
for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany
5 Tbilisi State University, 0186 Tbilisi, Georgia

###### Abstract

We calculate the W-exchange contribution to the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay using light-cone sum rules.
The two-particle light-cone distribution amplitudes of the pion are used as
non-perturbative input for the sum rules calculation, and the perturbative
kernel is calculated at the leading order. We obtain the corresponding decay
branching fractions by combining our W-exchange amplitudes with the
factorizable amplitudes given by various theoretical methods from the
literature. It is shown that with the factorizable amplitudes from heavy quark
effective theory, we obtain the branching fraction ratio ${\cal
B}(\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+})/{\cal
B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})=1.42\pm 0.78$, which is consistent with
the experimental value of $1.41\pm 0.17\pm 0.1$.

##  I Introduction

The conventional quark model predicted the existence of doubly-heavy baryons
consisting of two heavy quarks (bottom or charm quarks) Gell-Mann:1964ewy ;
Zweig:1964jf ; DeRujula:1975qlm ; Jaffe:1975us ; Ponce:1978gk ; Fleck:1988vm .
After pursuing the doubly charmed baryons for decades, in 2017 the LHCb
collaboration announced the observation of the lowest-lying state
$\Xi_{cc}^{++}$ with mass $3620.6\pm 1.5(\text{stat})\pm 0.4(\text{syst})\pm
0.3(\Xi_{c}^{+})\text{MeV}/c^{2}$ LHCb:2017iph . This new baryon was observed
via the decay channel $\Xi_{cc}^{++}\to\Lambda_{c}^{+}K^{-}\pi^{+}\pi^{+}$
which is consistent with the prediction given by Ref. Yu:2017zst . One year
later, in 2018, a two-body decay channel $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$
was observed LHCb:2018pcs , which further confirms the existence of this
doubly charmed baryon. Recently, the LHCb collaboration has observed a similar
decay channel, $\Xi_{cc}^{++}\to\Xi_{c}^{+{\prime}}\pi^{+}$, and measured the
branching fraction ratio LHCb:2022rpd :

$\displaystyle\frac{{\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+})}{{\cal
B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})}\equiv\frac{{\cal B}^{\prime}}{{\cal
B}}=1.41\pm 0.17\pm 0.1,$ (1)

which means that the branching fraction of the decay into $\Xi_{c}^{+\prime}$
is larger than that into $\Xi_{c}^{+}$.

Figure 1: W-emission diagram (left) and W-exchange diagram (right) for the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay.

Theoretically, the weak decay $\Xi_{cc}^{++}\to\Xi_{c}^{+({\prime})}\pi^{+}$
receives contributions from the two topological diagrams as shown in Fig. 1,
the W-emission diagram (left) and the W-exchange diagram (right). Generally,
according to the naive factorization Bjorken:1988kk ; Wirbel:1985ji ;
Bauer:1986bm , the W-emission diagram is approximately factorizable so that it
can be calculated by considering the baryon transformation matrix element and
the pion matrix element independently. For the decay of the $\Xi_{cc}^{++}$,
the transformation matrix element is parameterized by form factors which have
been evaluated in various theoretical works based on QCD sum rules (QCDSR)
Shi:2019hbf , light-cone sum rules (LCSR) Shi:2019fph ; Hu:2019bqj ;
Hu:2022xzu ; Aliev:2022tvs ; Aliev:2022maw , QCD factorization (QCDF)
Sharma:2017txj ; Gerasimov:2019jwp , diquark effective theory (DiET)
Shi:2020qde , light-front quark model (LFQM) Wang:2017mqp ; Zhao:2018mrg ;
Cheng:2020wmk ; Ke:2019lcf ; Ke:2022gxm ; Hu:2020mxk , constituent quark model
(CQM) Gutsche:2018msz ; Gutsche:2019iac , non-relativistic quark model (NRQCD)
and heavy quark effective theory (HQET) Sharma:2017txj ; Dhir:2018twm .
Further, the pion matrix element can be simply expressed by the pion decay
constant.

Unlike the weak decays of mesons, the W-exchange diagrams in the baryon decays
are generally non-factorizable, and this difficulty increases when we are
facing the decays of doubly heavy baryons. Nowadays, except an SU(3) symmetry
analysis Wang:2017azm ; Shi:2017dto  and a phenomenological study Li:2017ndo ,
there is no QCD based or model-independent study on the W-exchange
contribution in doubly charmed baryon decays. The mostly used approach to
evaluate such a contribution is the pole-model Sharma:2017txj ;
Gerasimov:2019jwp ; Cheng:2020wmk ; Gutsche:2018msz ; Dhir:2018twm . Recently,
the combination of the factorizable contribution and the non-factorizable
contribution from the pole-model leads to the ratio ${\cal B}^{\prime}/{\cal
B}$ between 0.81 and 0.83 Sharma:2017txj ; Gerasimov:2019jwp , which means
that the branching fraction of the decay into $\Xi_{c}^{+\prime}$ is smaller,
which is obviously contrary to the experimental result. Furthermore, if one
includes the interference between the W-emission and the W-exchange
contributions, the ${\cal B}^{\prime}/{\cal B}$ will become much larger,
namely $6.74$ Cheng:2020wmk .

This deviation of the theoretical prediction from the experimental measurement
implies that a more precise theoretical calculation for the W-exchange
contribution in the $\Xi_{cc}^{++}\to\Xi_{c}^{+({\prime})}\pi^{+}$ decay is
necessary. In this work, we will use the method of LCSR to solve this problem.
LCSR were firstly proposed to study the transition form factors of the
radiative or semi-leptonic hadron decays Balitsky:1989ry ; Braun:1988qv ;
Chernyak:1990ag . In the framework of LCSR, the required transition matrix
element can be extracted from a suitable correlation function at the hadron
level. The quark-hadron duality enables us to relate this correlations
function with the one at the quark-gluon level, where it can be calculated by
the operator-product-expansion (OPE), and all the non-perturbative
contributions come from the light-cone distribution amplitudes (LCDAs) of a
certain hadron in the decay. After decades of development, a new technique of
LCSR was proposed to study the non-leptonic decay of the $B$ mesons into two
light mesons Khodjamirian:2000mi ; Khodjamirian:2003eq ; Khodjamirian:2017zdu
. This new technique of LCSR can be extended to the case of heavy or doubly
heavy baryon decays, and in this work we will use it to calculate the
W-exchange contribution in the $\Xi_{cc}^{++}\to\Xi_{c}^{+({\prime})}\pi^{+}$
decay.

This paper is organized as follows. In Sec. II, we introduce a suitable
correlation function to extract the decay amplitude of
$\Xi_{cc}^{++}\to\Xi_{c}^{+({\prime})}\pi^{+}$. In Sec. III, we perform the
hadron level calculation for the correlation function and extract the required
decay amplitude. In Sec. IV, we perform the quark-gluon level calculation for
the correlation function with the use of two-particle LCDAs of the pion. In
Sec. V, we give the numerical results on the decay amplitudes and branching
fractions of $\Xi_{cc}^{++}\to\Xi_{c}^{+({\prime})}\pi^{+}$ and compare our
results with those from the literature. Sec. VI contains a brief summary of
this work.

##  II The correlation function in LCSR

In this section, we give a suitable correlation function for the study of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay in the framework of the
LCSR. The relevant effective Hamiltonian for this decay is

$\displaystyle{\cal H}_{\rm eff}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V_{cs}V_{ud}^{*}\left(C_{1}{\cal
O}_{1}+C_{2}{\cal O}_{2}\right),$ $\displaystyle{\cal O}_{1}$
$\displaystyle=\bar{s}\gamma_{\mu}(1-\gamma_{5})c\
\bar{u}\gamma^{\mu}(1-\gamma_{5})d,$ $\displaystyle{\cal O}_{2}$
$\displaystyle=\bar{s}_{a}\gamma_{\mu}(1-\gamma_{5})c_{b}\
\bar{u}_{b}\gamma^{\mu}(1-\gamma_{5})d_{a},$ (2)

where the $C_{1},C_{2}$ are Wilson coefficients, and the subscripts $a,b$ are
color indices. Generally, the transition matrix element of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ induced by ${\cal O}_{1,2}$ can
be parameterized as

$\displaystyle\langle\Xi_{c}^{+(\prime)}(p-q)\pi^{+}(q)|{\cal
O}_{i}(0)|\Xi_{cc}^{++}(p)\rangle=i\
\bar{u}(p-q)(A^{(\prime)i}+B^{(\prime)i}\gamma_{5})u(p).$ (3)

Since the initial and final states are on-shell, $A^{(\prime)i}$ and
$B^{(\prime)i}$ are just constants. In this work, our main task is to obtain
the W-exchange contribution to $A^{(\prime)i}$ and $B^{(\prime)i}$, denoted as
$A^{(\prime)i}_{\rm WE}$ and $B^{(\prime)i}_{\rm WE}$ in what follows.

Using the LCSR to calculate a transition matrix element, one begins with an
appropriate correlation function which will be calculated both at the hadron
and the quark-gluon level. In our case, the correlation function corresponding
to the $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay is chosen as

$\displaystyle\Pi^{{\cal O}_{i}}(p,q,k)=i^{2}\int d^{4}x\ e^{-i(p-q)\cdot
x}\int d^{4}y\ e^{i(p-k)\cdot y}\langle
0|T\left\\{J_{\Xi_{c}^{(\prime)}}(y){\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)\right\\}|\pi^{-}(q)\rangle,$ (4)

where the hadron currents are defined as Shi:2019hbf

$\displaystyle J_{\Xi_{c}}$
$\displaystyle=\frac{1}{\sqrt{2}}\varepsilon_{abc}\left(u_{a}^{T}C\gamma_{5}s_{b}-s_{a}^{T}C\gamma_{5}u_{b}\right)Q_{c},$
$\displaystyle J_{\Xi_{c}^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}\varepsilon_{abc}\left(u_{a}^{T}C\gamma^{\mu}s_{b}+s_{a}^{T}C\gamma^{\mu}u_{b}\right)\gamma_{\mu}\gamma_{5}Q_{c},$
$\displaystyle J_{\Xi_{cc}}$
$\displaystyle=\varepsilon_{abc}\left(Q_{a}^{T}C\gamma^{\mu}Q_{b}\right)\gamma_{\mu}\gamma_{5}u_{c}.$
(5)

Following Ref. Khodjamirian:2000mi , we have temporarily included the pion in
the initial state instead of the final state as in the real decay. The
advantage of this is to enable us to factorize out the matrix element $\langle
0|J_{\Xi_{c}^{\prime}}|\Xi_{c}^{+(\prime)}\rangle$ without any ambiguity after
inserting the $\Xi_{c}^{+(\prime)}$ state into the right-hand side of
$J_{\Xi_{c}^{\prime}}$. Otherwise, the final pion state has to be moved to the
right over $J_{\Xi_{c}}$ firstly before this factorization can be performed.
Actually, during the calculation at the hadron level, by suitable analytic
continuation this initial pion state will be moved to the final state. We have
also introduced an auxiliary momentum $k$ in the correlation function. This
momentum is unphysical and should be set to zero at the end of the
calculation. The reason why it must be introduced is closely related to the
analytic continuation mentioned above, which will be explained in the next
section.

##  III Hadron level calculation in the LCSR

In this section, we derive the calculation of the correlation function in Eq.
(4) at the hadron level. For simplicity, we take the case of the $\Xi_{c}^{+}$
as example, the derivation for the decays into $\Xi_{c}^{+\prime}$ is similar.
We insert a complete set of states with the same quantum numbers as
$J_{\Xi_{c}}(y)$ into the correlation function,

$\displaystyle\Pi_{H}^{{\cal O}_{i}}(p,q,k)_{\rm WE}=$ $\displaystyle\
i^{2}\int d^{4}xd^{4}y\ e^{-i(p-q)\cdot x}e^{i(p-k)\cdot y}$
$\displaystyle\times\sum_{\pm^{\prime},\sigma^{\prime}}\int\frac{d^{3}\vec{l}}{(2\pi)^{3}}\frac{1}{2E_{l}}\langle
0|J_{\Xi_{c}}(y)|l,\sigma^{\prime},\pm^{\prime}\rangle\langle
l,\sigma^{\prime},\pm^{\prime}|{\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)|\pi^{-}(q)\rangle$
$\displaystyle+\int_{s_{\Xi_{c}}}^{\infty}ds^{\prime}\frac{\rho_{\Xi_{c}}(s^{\prime},(p-q)^{2},P^{2})}{s^{\prime}-(p-k)^{2}},$
(6)

where the integration over $\rho_{\Xi_{c}}$ represents the contribution from
the continuous spectrum, $\pm^{\prime}$ corresponds to the positive or
negative parity of the $\Xi_{c}$ states, namely $\Xi_{c}(\frac{1}{2}^{\pm})$,
and $\sigma^{\prime}$ denotes the spin of the $\Xi_{c}$. The momentum of the
one-particle state in the complete set $l$ should be on-shell,
$l^{2}=m_{\Xi_{c}}^{\pm^{\prime}2}$. This means that the first matrix element
in Eq. (6) can be simply parameterized by the $\Xi_{c}$ decay constants
$\lambda_{\Xi_{c}}^{\pm}$:

$\displaystyle\langle 0|J_{\Xi_{c}}(y)|l,\sigma^{\prime},+\rangle$
$\displaystyle=\lambda_{\Xi_{c}}^{+}u(l,\sigma^{\prime})e^{-il\cdot y},$
$\displaystyle\langle 0|J_{\Xi_{c}}(y)|l,\sigma^{\prime},-\rangle$
$\displaystyle=\lambda_{\Xi_{c}}^{-}i\gamma_{5}u(l,\sigma^{\prime})e^{-il\cdot
y},$ (7)

while the second matrix element in Eq. (6) is a function of $p^{2},\ q^{2},\
k^{2},\ (p-q)^{2}$ and $P^{2}=(p-k-q)^{2}$. Generally speaking, a matrix
element like this should also depend on $l^{2}$, however, the on-shell
condition has reduced such a dependence. Further, since one can replace the
$d^{3}\vec{l}$ integration by a four-momentum integration with the use of
residue theorem,

$\displaystyle\int\frac{d^{3}\vec{l}}{(2\pi)^{3}}\frac{1}{2E_{l}}|l\rangle\langle
l|\ e^{-il\cdot y}=\int\frac{d^{4}l}{(2\pi)^{4}}e^{-il\cdot
y}\frac{1}{l^{2}-m_{\Xi_{c}}^{\pm^{\prime}2}}|l\rangle\langle l|,$ (8)

after integrating over $d^{4}y$, one arrives at

$\displaystyle\Pi_{H}^{{\cal O}_{i}}(p,q,k)_{\rm WE}=$ $\displaystyle\
i^{3}\int d^{4}x\ e^{-i(p-q)\cdot
x}\sum_{\pm^{\prime},\sigma^{\prime}}\frac{1}{(p-k)^{2}-m_{\Xi_{c}}^{\pm^{\prime}2}}$
$\displaystyle\times\lambda_{\Xi_{c}}^{\pm^{\prime}}u^{\pm^{\prime}}(p-k,\sigma^{\prime})\langle
p-k,\sigma^{\prime},\pm^{\prime}|{\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)|\pi^{-}(q)\rangle$
$\displaystyle+\int_{s_{\Xi_{c}}}^{\infty}ds^{\prime}\frac{\rho_{\Xi_{c}}(s^{\prime},(p-q)^{2},P^{2})}{s^{\prime}-(p-k)^{2}},$
(9)

where we have defined $u^{+}=u,\ u^{-}=i\gamma_{5}u$. To simplify the
calculation we have chosen $p^{2}=k^{2}=0$ and $q^{2}=m_{\pi}^{2}\approx 0$.
Now the correlation function depends on three Lorentz invariants $(p-k)^{2}$,
$(p-q)^{2}$ and $P^{2}$, while the matrix element $\langle
p-k,\sigma^{\prime},\pm^{\prime}|{\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)|\pi^{-}(q)\rangle$ only depends on $(p-q)^{2}$
and $P^{2}$.

Further, the same correlation function can be calculated at the quark-hadron
level by the OPE in the deep Euclidean region, $(p-k)^{2}\sim(p-q)^{2}\sim
P^{2}\ll 0$, which will be explicitly done in the next section. In principle,
the expression of the same correlation function at these two levels should be
equivalent:

$\displaystyle\Pi_{H}^{{\cal O}_{i}}(p,q,k)_{\rm WE}=\Pi_{QCD}^{{\cal
O}_{i}}(p,q,k)_{\rm
WE}=\frac{1}{\pi}\int_{(m_{c}+m_{s})^{2}}^{\infty}ds^{\prime}\frac{{\rm
Im}\Pi_{QCD}^{{\cal O}_{i}}(s^{\prime},(p-q)^{2},P^{2})_{\rm
WE}}{s^{\prime}-(p-k)^{2}},$ (10)

where the $(p-k)^{2}$ dependence of $\Pi_{QCD}^{{\cal O}_{i}}$ has been
described as in the form of a dispersion integral. We have omitted the $u$
quark mass and $(m_{c}+m_{s})^{2}$ is the quark level threshold to produce a
$\Xi_{c}$ baryon. According to the quark-hadron duality, the integration over
the continuous spectrum at the hadron level is canceled by the corresponding
integration above a certain threshold $s_{\Xi_{c}}$ at the quark-gluon level,
which leads to the equation:

$\displaystyle i^{3}\int d^{4}x\ e^{-i(p-q)\cdot
x}\sum_{\pm^{\prime},\sigma^{\prime}}\frac{1}{(p-k)^{2}-m_{\Xi_{c}}^{\pm^{\prime}2}}\lambda_{\Xi_{c}}^{\pm^{\prime}}u^{\pm^{\prime}}(p-k,\sigma^{\prime})\langle
p-k,\sigma^{\prime},\pm^{\prime}|{\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)|\pi^{-}(q)\rangle_{\rm WE}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}\frac{{\rm
Im}\Pi_{QCD}^{{\cal O}_{i}}(s^{\prime},(p-q)^{2},P^{2})_{\rm
WE}}{s^{\prime}-(p-k)^{2}}.$ (11)

Generally, this threshold parameter should be slightly larger than the mass
squared of the corresponding hadron state. Here, it is chosen to be the same
as in Ref. Shi:2019hbf , where a QCDSR was used to study the semi-leptonic
decay of the $\Xi_{cc}\to\Xi_{c}$. It should be mentioned that in principle
$s_{\Xi_{c}}$ is a universal parameter which is process-independent so that
this procedure should be reasonable.

Note that since ${\rm Im}\Pi_{QCD}^{{\cal
O}_{i}}(s^{\prime},(p-q)^{2},P^{2})_{\rm WE}$ is an analytic function of
$P^{2}$, the left-hand side of Eq. (11) shares this property. Therefore one
can extend $P^{2}$ to the physical region, namely $P^{2}>0$, which allows us
to replace the initial state $|\pi^{-}(q)\rangle$ of the matrix element on the
left-hand side by a final state $\langle\pi^{+}(-q)|$. After that, a Borel
transformation for $(p-k)^{2}$ on both sides of Eq. (11) is performed, and we
obtain

$\displaystyle-i^{3}\int d^{4}x\ e^{-i(p-q)\cdot
x}\sum_{\pm^{\prime},\sigma^{\prime}}e^{-m_{\Xi_{c}}^{\pm^{\prime}2}/T^{\prime
2}}\lambda_{\Xi_{c}}^{\pm^{\prime}}u^{\pm^{\prime}}(p-k,\sigma^{\prime})\langle
p-k,\sigma^{\prime},\pm^{\prime};\pi^{+}(-q)|{\cal
O}_{i}(0)\bar{J}_{\Xi_{cc}}(x)|0\rangle_{\rm WE}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}e^{-s^{\prime}/T^{\prime
2}}{\rm Im}\Pi_{QCD}^{{\cal O}_{i}}(s^{\prime},(p-q)^{2},P^{2})_{\rm WE}.$
(12)

Next, we insert another complete set of states with the same quantum numbers
as ${\bar{J}}_{\Xi_{cc}}$ into the left-hand side of Eq. (12), which becomes

$\displaystyle-$
$\displaystyle\sum_{\pm^{\prime},\pm,\sigma^{\prime},\sigma}e^{-m_{\Xi_{c}}^{\pm^{\prime}2}/T^{\prime
2}}\frac{1}{(p-q)^{2}-m_{\Xi_{cc}}^{\pm
2}}\lambda_{\Xi_{c}}^{\pm^{\prime}}\lambda_{\Xi_{cc}}^{\pm}$
$\displaystyle\times\ u^{\pm^{\prime}}(p-k,\sigma^{\prime})\langle
p-k,\sigma^{\prime},\pm^{\prime};\pi^{+}(-q)|{\cal
O}_{i}(0)|p-q,\sigma,\pm\rangle_{\rm WE}\ \bar{u}^{\pm}(p-q.\sigma)$
$\displaystyle+\int_{s_{\Xi_{cc}}}^{\infty}ds\frac{\rho_{\Xi_{cc}}(s,P^{2})}{s-(p-q)^{2}}.$
(13)

For the same reason as discussed before, the matrix element $\langle
p-k,s^{\prime},\pm^{\prime};\pi^{+}(-q)|{\cal O}_{i}(0)|p-q,s,\pm\rangle_{\rm
WE}$ only depends on $P^{2}$. The $(p-q)^{2}$ dependence on the right-hand
side of Eq. (12) can be further expressed as a dispersion integral,

$\displaystyle\frac{1}{\pi^{2}}\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}e^{-s^{\prime}/T^{\prime
2}}\int_{4m_{c}^{2}}^{\infty}ds\ \frac{1}{s-(p-q)^{2}}{\rm
Im}^{2}\Pi_{QCD}^{{\cal O}_{i}}(s^{\prime},s,P^{2})_{\rm WE}~{}.$ (14)

Now using the quark-hadron duality again to cancel out the $ds^{\prime}$
integration above a certain threshold $s_{\Xi_{cc}}$ corresponding to the
lowest $\Xi_{cc}$ state, and performing the Borel transformation for
$(p-q)^{2}$, we arrive at

$\displaystyle\sum_{\pm^{\prime},\pm,\sigma^{\prime},\sigma}e^{-m_{\Xi_{c}}^{\pm^{\prime}2}/T^{\prime
2}-m_{\Xi_{cc}}^{\pm
2}/T^{2}}\lambda_{\Xi_{c}}^{\pm^{\prime}}\lambda_{\Xi_{cc}}^{\pm}$
$\displaystyle\times u^{\pm^{\prime}}(p-k,\sigma^{\prime})\langle
p-k,\sigma^{\prime},\pm^{\prime};\pi^{+}(-q)|{\cal
O}_{i}(0)|p-q,\sigma,\pm\rangle_{\rm WE}\ \bar{u}^{\pm}(p-q,\sigma)$
$\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}\int_{4m_{c}^{2}}^{s_{\Xi_{cc}}}ds\
e^{-s^{\prime}/T^{\prime 2}}e^{-s/T^{2}}{\rm Im}^{2}\Pi_{QCD}^{{\cal
O}_{i}}(s^{\prime},s,P^{2}).$ (15)

Due to the existence of the auxiliary momentum $k$, unlike Eq. (3), the matrix
element appearing in Eq. (15) must be parameterized by four terms:

$\displaystyle\langle p-k,\sigma^{\prime},\pm^{\prime};\pi^{+}(-q)|{\cal
O}_{i}(0)|p-q,\sigma,\pm\rangle_{\rm WE}$ $\displaystyle=$ $\displaystyle\ i\
\bar{u}^{\pm^{\prime}}(p-k,\sigma^{\prime})\left[A_{1,i}^{\pm^{\prime}\pm}(P^{2})+B_{1,i}^{\pm^{\prime}\pm}(P^{2})\gamma_{5}+A_{2,i}^{\pm^{\prime}\pm}(P^{2})\frac{\not{q}}{m_{\Xi_{cc}}^{\pm}}+B_{2,i}^{\pm^{\prime}\pm}(P^{2})\frac{\not{q}\gamma_{5}}{m_{\Xi_{cc}}^{\pm}}\right]u^{\pm}(p-q,\sigma).$
(16)

The form factors $A_{1,2,i}^{\pm^{\prime}\pm}\ \text{and}\
B_{1,2,i}^{\pm^{\prime}\pm}$ are functions of $P^{2}$. Using the sum rules
equation given in Eq. (15), we can extract these four functions. Summing up
the spin indices, we have

$\displaystyle i\ e^{-m_{\Xi_{c}}^{+2}/T^{\prime
2}-m_{\Xi_{cc}}^{+2}/T^{2}}\lambda_{\Xi_{c}}^{+}\lambda_{\Xi_{cc}}^{+}(\not{p}_{2}+m_{\Xi_{c}}^{+})\left[A_{1,i}^{++}+B_{1,i}^{++}\gamma_{5}+A_{2,i}^{++}\frac{\not{q}}{m_{\Xi_{cc}}^{\pm}}+B_{2,i}^{++}\frac{\not{q}\gamma_{5}}{m_{\Xi_{cc}}^{\pm}}\right](\not{p}_{1}+m_{\Xi_{cc}}^{+})$
$\displaystyle+$ $\displaystyle i\ e^{-m_{\Xi_{c}}^{-2}/T^{\prime
2}-m_{\Xi_{cc}}^{+2}/T^{2}}\lambda_{\Xi_{c}}^{-}\lambda_{\Xi_{cc}}^{+}(\not{p}_{2}-m_{\Xi_{c}}^{+})\left[A_{1,i}^{-+}+B_{1,i}^{-+}\gamma_{5}+A_{2,i}^{-+}\frac{\not{q}}{m_{\Xi_{cc}}^{\pm}}+B_{2,i}^{-+}\frac{\not{q}\gamma_{5}}{m_{\Xi_{cc}}^{\pm}}\right](\not{p}_{1}+m_{\Xi_{cc}}^{+})$
$\displaystyle+$ $\displaystyle i\ e^{-m_{\Xi_{c}}^{+2}/T^{\prime
2}-m_{\Xi_{cc}}^{-2}/T^{2}}\lambda_{\Xi_{c}}^{+}\lambda_{\Xi_{cc}}^{-}(\not{p}_{2}+m_{\Xi_{c}}^{+})\left[A_{1,i}^{+-}+B_{1,i}^{+-}\gamma_{5}+A_{2,i}^{+-}\frac{\not{q}}{m_{\Xi_{cc}}^{\pm}}+B_{2,i}^{+-}\frac{\not{q}\gamma_{5}}{m_{\Xi_{cc}}^{\pm}}\right](\not{p}_{1}-m_{\Xi_{cc}}^{-})$
$\displaystyle+$ $\displaystyle i\ e^{-m_{\Xi_{c}}^{-2}/T^{\prime
2}-m_{\Xi_{cc}}^{-2}/T^{2}}\lambda_{\Xi_{c}}^{-}\lambda_{\Xi_{cc}}^{-}(\not{p}_{2}-m_{\Xi_{c}}^{+})\left[A_{1,i}^{--}+B_{1,i}^{--}\gamma_{5}+A_{2,i}^{--}\frac{\not{q}}{m_{\Xi_{cc}}^{\pm}}+B_{2,i}^{--}\frac{\not{q}\gamma_{5}}{m_{\Xi_{cc}}^{\pm}}\right](\not{p}_{1}-m_{\Xi_{cc}}^{-})$
$\displaystyle=$
$\displaystyle\frac{1}{\pi^{2}}\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}\int_{4m_{c}^{2}}^{s_{\Xi_{cc}}}ds\
e^{-s^{\prime}/T^{\prime 2}}e^{-s/T^{2}}{\rm Im}^{2}\Pi_{QCD}^{{\cal
O}_{i}}(s^{\prime},s,P^{2})_{\rm WE},$ (17)

with $p_{1}=p-q$ and $p_{2}=p-k$. Note that there are 16 independent spinor
structures, and this number is equal to that of the form factors
$A_{1,2}^{\pm^{\prime}\pm}\ \text{and}\ B_{1,2}^{\pm^{\prime}\pm}$. This
matching enables us to solve all of these form factors from the equation
above. Finally, what we really care about is the transition matrix element of
positive parity baryons with $k=0$ and $P^{2}=m_{\Xi cc}^{+2}$. Therefore,
considering the $k\to 0$ limit, and using the equation of motion, the matrix
element in Eq. (16) can be simplified as

$\displaystyle\langle p-k,\sigma^{\prime},+;\pi^{+}(-q)|{\cal
O}_{i}(0)|p-q,\sigma,+\rangle\mid_{k\to 0,P^{2}=m_{\Xi cc}^{+2}}$
$\displaystyle=$ $\displaystyle\ i\
\bar{u}^{+}(p-k,\sigma^{\prime})\left\\{\left[A_{1,i}^{++}(m_{\Xi
cc}^{+2})+\left(1-\frac{m_{\Xi_{c}}^{+}}{m_{\Xi_{cc}}^{+}}\right)A_{2,i}^{++}(m_{\Xi
cc}^{+2})\right]\right.$ $\displaystyle\left.+\left[B_{1,i}^{++}(m_{\Xi
cc}^{+2})-\left(1+\frac{m_{\Xi_{c}}^{+}}{m_{\Xi_{cc}}^{+}}\right)B_{2,i}^{++}(m_{\Xi
cc}^{+2})\right]\gamma_{5}\right\\}u^{+}(p-q,\sigma)$ $\displaystyle\equiv$
$\displaystyle\ i\ \bar{u}^{+}(p,\sigma^{\prime})\left(A_{\rm WE}^{i}+B_{\rm
WE}^{i}\gamma_{5}\right)u^{+}(p-q,\sigma).$ (18)

Now the four unknown form factors are reduced to two constants $A_{\rm
WE}^{i}$ and $B_{\rm WE}^{i}$ ($A_{\rm WE}^{\prime i}$ and $B_{\rm WE}^{\prime
i}$ for the case of $\Xi_{c}^{\prime}$), which has the same form as Eq. (27).

##  IV Quark-Gluon Level

Figure 2: Feynman diagram of the W-exchange contribution to the correlation
function in Eq. (4). The gray bubble denotes the pion LCDAs, the black dots
denote the baryon currents and the white double crossed dot represents the
four-fermion interaction vertex from ${\cal O}_{1,2}$. The dashed lines with
arrows denote the external momentum flows.

In this section, we use the OPE to calculate the correlation function in Eq.
(4) at the quark-gluon level. In the deep Euclidean region,
$(p-k)^{2}\sim(p-q)^{2}\sim P^{2}\ll 0$, the W-exchange contribution to the
correlation function can be expressed as a convolution of the perturbative
kernel and a nonperturbative matrix element of the pion:

$\displaystyle\Pi_{QCD}^{{\cal O}_{1}}(p,q,k)_{\alpha\sigma}$
$\displaystyle=-\Pi_{QCD}^{{\cal O}_{2}}(p,q,k)_{\alpha\sigma}$
$\displaystyle=-2\sqrt{2}\varepsilon_{abc}\varepsilon_{ebc}\int d^{4}xd^{4}y\
e^{-i(p-q)\cdot x}e^{i(p-k)\cdot y}$
$\displaystyle\times\left[S_{Q}(y-x)\gamma^{\nu}CS_{Q}^{T}(-x)C(1-\gamma_{5})\gamma_{\mu}CS_{s}^{T}(y)C\gamma_{5}S_{u}(y)\gamma^{\mu}(1-\gamma_{5})\right]_{\alpha\beta}(\gamma_{\nu}\gamma_{5})_{\rho\sigma}$
$\displaystyle\times\langle
0|\bar{u}_{e}^{\rho}(x)d_{a}^{\beta}(0)|\pi^{-}(q)\rangle,$ (19)

where $\alpha,\beta,\rho,\sigma$ are spinor indices and $S_{Q,u,s}$ are the
free propagators of the $c,u,s$ quarks. The superscript ”$T$” denotes the
transposed in spinor space. The last matrix element can be expressed by the
Light-Cone Distribution Amplitudes (LCDAs) of the pion. Fig. 2 shows the
Feynman diagram of the W-exchange effect in the correlation function, where
the gray bubble denotes the pion LCDAs. The contribution from two-particle
LCDAs of pion up to the twist-3 order are defined as Duplancic:2008ix ;
Khodjamirian:2020mlb

$\displaystyle\langle
0|\bar{u}_{e}^{\rho}(x)d_{a}^{\beta}(0)|\pi^{-}(q)\rangle$
$\displaystyle=-\frac{i}{12}\delta_{ae}f_{\pi}\int_{0}^{1}du\
e^{-i\bar{u}q\cdot
x}\Big{[}(\not{p}\gamma_{5})_{\beta\rho}\varphi_{\pi}(u)+(\gamma_{5})_{\beta\rho}\mu_{\pi}\phi_{3\pi}^{p}(u)$
$\displaystyle+\frac{1}{6}(\gamma_{5}\sigma_{\mu\nu})_{\beta\rho}q^{\mu}x^{\nu}\mu_{\pi}\phi_{3\pi}^{\sigma}(u)\Big{]},$
(20)

where

$\displaystyle\varphi_{\pi}(u)$
$\displaystyle=6u\bar{u}\left(1+a_{2}C_{2}^{3/2}(u-\bar{u})+a_{4}C_{4}^{3/2}(u-\bar{u})\right),$
$\displaystyle\phi_{3\pi}^{p}(u)$
$\displaystyle=1+30\frac{f_{3\pi}}{\mu_{\pi}f_{\pi}}C_{2}^{1/2}(u-\bar{u})-3\frac{f_{3\pi}\omega_{3\pi}}{\mu_{\pi}f_{\pi}}C_{4}^{1/2}(u-\bar{u}),$
$\displaystyle\phi_{3\pi}^{\sigma}(u)$
$\displaystyle=6u(1-u)\left(1+5\frac{f_{3\pi}}{\mu_{\pi}f_{\pi}}\left(1-\frac{\omega_{3\pi}}{10}\right)C_{2}^{3/2}(u-\bar{u})\right),$
(21)

are the twist-2, twist-3p and twist-$3\sigma$ LCDAs respectively.
$a_{2}=0.27$, $a_{4}=0.179$, $\mu_{\pi}=2.87$ GeV, $f_{3\pi}=0.0045$ GeV2,
$\omega_{\pi}=-1.5$ and $f_{\pi}=0.13$ GeV Duplancic:2008ix . $C^{\alpha}_{n}$
are the Gegenbauer polynomials. Now the derivation becomes straightforward,
and we take the twist-2 LCDA as an example. Its contribution to the
correlation function is

$\displaystyle\Pi_{QCD}^{{\cal O}_{1}}(p,q,k)_{(2)}$ $\displaystyle=$
$\displaystyle-4\sqrt{2}N_{c}\left(-\frac{i}{12}\right)f_{\pi}\int_{0}^{1}du\
\varphi_{\pi}(u)\int
d^{4}xd^{4}y\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\frac{d^{4}k_{2}}{(2\pi)^{4}}\frac{d^{4}k_{3}}{(2\pi)^{4}}\frac{d^{4}k_{4}}{(2\pi)^{4}}$
$\displaystyle\times e^{-i\bar{u}q\cdot x}e^{-ik_{1}\cdot(y-x)}e^{ik_{2}\cdot
x}e^{-ik_{3}\cdot y}e^{-ik_{4}\cdot y}e^{-i(p-q)\cdot x}e^{i(p-k)\cdot
y}\frac{1}{(k_{1}^{2}-m_{c}^{2})(k_{2}^{2}-m_{c}^{2})(k_{3}^{2}-m_{s}^{2})k_{4}^{2}}$
$\displaystyle\times\left[(\not{k}_{1}+m_{c})\gamma^{\nu}(\not{k}_{2}-m_{c})(1-\gamma_{5})\gamma_{\mu}(\not{k}_{3}-m_{s})\gamma_{5}\not{k}_{4}\gamma^{\mu}(1-\gamma_{5})\not{q}\gamma_{5}\gamma_{\nu}\gamma_{5}\right]~{},$
(22)

with $N_{c}$ the number of colors.

The double imaginary part of the correlation function is related to its double
discontinuity, which can be extracted by the cutting rules. Setting the
momentum of each propagator on-shell, we have

$\displaystyle{\rm Im}^{2}\Pi_{QCD}^{{\cal
O}_{1}}(s^{\prime},s,P^{2})_{(2)}=\frac{1}{(2i)^{2}}{\rm
Disc^{2}}\Pi_{QCD}^{{\cal O}_{1}}(s^{\prime},s,P^{2})_{(2)}$ $\displaystyle=$
$\displaystyle-4\sqrt{2}N_{c}\left(-\frac{i}{12}\right)f_{\pi}(-2\pi
i)^{4}\frac{1}{(2i)^{2}}\frac{1}{(2\pi)}\int_{0}^{1}du\ \varphi_{\pi}(u)\int
dm_{34}^{2}\int d\Phi_{\Delta}(P_{1}^{2},p_{2}^{2})\int d\Phi_{2}(m_{34}^{2})$
$\displaystyle\times\left[(\not{k}_{1}+m_{c})\gamma^{\nu}(\not{k}_{2}-m_{c})(1-\gamma_{5})\gamma_{\mu}(\not{k}_{3}-m_{s})\gamma_{5}\not{k}_{4}\gamma^{\mu}(1-\gamma_{5})\not{q}\gamma_{5}\gamma_{\nu}\gamma_{5}\right].$
(23)

Here we have introduced an extra integration on $m_{34}^{2}=(k_{3}+k_{4})^{2}$
to express the double discontinuity of the correlation function as a
convolution of a two-body phase space integration $\int d\Phi_{2}(m_{34}^{2})$
and a triangle diagram integration $\int d\Phi_{\Delta}(P_{1}^{2},p_{2}^{2})$,

$\displaystyle\int d\Phi_{2}(m_{34}^{2})=$ $\displaystyle\
\int\frac{d^{3}k_{3}}{(2\pi)^{3}}\frac{1}{2E_{k_{3}}}\frac{d^{3}k_{4}}{(2\pi)^{3}}\frac{1}{2E_{k_{4}}}\delta^{4}(k_{34}-k_{3}-k_{4}),\
\ \ \ k_{34}^{2}=m_{34}^{2},$ $\displaystyle\int
d\Phi_{\Delta}(P_{1}^{2},p_{2}^{2})=$ $\displaystyle\ \int
d^{4}k_{34}d^{4}k_{1}d^{4}k_{2}\delta(k_{1}^{2}-m_{c}^{2})\delta(k_{2}^{2}-m_{c}^{2})\delta(k_{34}^{2}-m_{34}^{2})$
$\displaystyle\times\delta^{4}(P_{1}-k_{1}-k_{2})\delta^{4}(p_{2}-k_{1}-k_{34}),$
(24)

where $P_{1}=p-uq=p_{1}+\bar{u}q$, and $p_{1}^{2}=s,\ p_{2}^{2}=s^{\prime 2}$.
This factorization for the momentum flows can also be understood intuitively
by Fig. 2. It is seen that the momentum flowing into the left-lower corner of
the triangle diagram is not $p_{1}$ itself but $P_{1}=p-uq=p_{1}+\bar{u}q$
instead, and these momenta are related as $P_{1}^{2}=up_{1}^{2}=us$.

For the contribution from the twist-$3p$ and twist-$3\sigma$ LCDAs the
calculation is similar. The only difference is that the twist-$3\sigma$ LCDA
contains a term proportional to the coordinate $x^{\nu}$. Note that from Eq.
(22) there is an exponential term ${\rm exp}(iuq\cdot x)$ in the correlation
function. One can use it to express $x^{\nu}$ as $(-i/u)(\partial/\partial
q_{\nu}){\rm exp}(iuq\cdot x)$ so that the calculation can still be done in
momentum space.

##  V Numerical Results

In this section, we first give the numerical results for the amplitudes of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decays, namely the two constants
$A_{\rm WE}^{(\prime)i}$ and $B_{\rm WE}^{(\prime)i}$ in Eq. (18). In this
work, we use the $\overline{\rm MS}$ masses for the quarks, $m_{c}(\mu)=1.27$
GeV and $m_{s}(\mu)=0.103$ GeV with $\mu=1.27$ GeV ParticleDataGroup:2020ssz .
The masses of the $u$ quark and the pion are neglected. The masses and decay
constants of the charm baryons with positive or negative parity are listed in
Table 1, where the decay constants are defined as in Eq. (7).

Table 1: The masses and decay constants of the charmed baryons with positive or negative parity. The decay constants are defined as in Eq. (7). Baryon | $\Xi_{cc}(\frac{1}{2}^{+})$ | $\Xi_{cc}(\frac{1}{2}^{-})$ | $\Xi_{c}(\frac{1}{2}^{+})$ | $\Xi_{c}(\frac{1}{2}^{-})$ | $\Xi_{c}^{\prime}(\frac{1}{2}^{+})$ | $\Xi_{c}^{\prime}(\frac{1}{2}^{-})$
---|---|---|---|---|---|---
Mass ${\rm[GeV]}$ |  $3.62$ ParticleDataGroup:2018ovx  |  $3.77$ Wang:2010it  |  $2.47$ ParticleDataGroup:2018ovx  |  $2.79$ Roberts:2007ni  |  $2.58$ ParticleDataGroup:2018ovx  |  $2.87$Wang:2010it
$\lambda$ ${\rm[GeV^{3}]}$ |  $0.109$ Hu:2017dzi ; 1902.01092 |  $0.159$Wang:2010it  |  $0.038$ Hu:2017dzi ; 1902.01092 |  $0.042$ Wang:2010fq  |  $0.076$ Hu:2017dzi ; 1902.01092 |  $0.084$ Wang:2010it

On the other hand, the LCSR contains two kinds of extra parameters, the
thresholds $s_{\Xi_{cc}},s_{\Xi_{c}},s_{\Xi_{c}^{\prime}}$ and the Borel
parameters $T^{2},T^{\prime 2}$. For the threshold parameters, we have argued
that they are process-independent and will be taken from Ref.
Khodjamirian:2000mi , $s_{\Xi_{cc}}=(4.1\pm 0.1)^{2}$ GeV2,
$s_{\Xi_{c}}=(3.2\pm 0.1)^{2}$ GeV2 and $s_{\Xi_{c}^{\prime}}=(3.3\pm
0.1)^{2}$ GeV2. These are about $0.5^{2}$ GeV2 larger than the squared masses
of the corresponding baryons. Since this difference to the squared mass is
only an empirical value proposed in Ref. Wang:2012kw , we will consider the
uncertainty of the thresholds when evaluating the error of numerical results.

Figure 3: The Borel parameter dependence of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ decay amplitudes
$A^{(2)}(B^{(2)}),A^{(3p)}(B^{(3p)})$ and $A^{(3\sigma)}(B^{(3\sigma)})$ from
the twist-2, twist-3p and twist-3$\sigma$ LCDAs respectively. In each diagram,
the blue band denotes the uncertainty from the error of the threshold:
$s_{\Xi_{cc}}=(4.1\pm 0.1)^{2}$ GeV2 and $s_{\Xi_{c}}=(3.2\pm 0.1)^{2}$ GeV2.
The upper and lower red bands denote the uncertainty from the error of Monte-
Carlo integrations.

Figure 4: The Borel parameter dependence of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$ decay amplitudes
$A^{\prime(2)}(B^{\prime(2)}),A^{\prime(3p)}(B^{\prime(3p)})$ and
$A^{\prime(3\sigma)}(B^{\prime(3\sigma)})$ from the twist-2, twist-3p and
twist-3$\sigma$ LCDAs respectively. In each diagram, the blue band denotes the
uncertainty from the error of the threshold: $s_{\Xi_{cc}}=(4.1\pm 0.1)^{2}$
GeV2 and $s_{\Xi_{c}^{\prime}}=(3.3\pm 0.1)^{2}$ GeV2. The upper and lower red
bands denote the uncertainty from the error of Monte-Carlo integrations.

What concerns the Borel parameters, to simplify the problem we apply the
following equation to relate the two Borel parameters corresponding to the $s$
and $s^{\prime}$ channels Ball:1991bs :

$\displaystyle\frac{T^{2}}{T^{\prime
2}}\approx\frac{M_{1}^{2}-m_{1}^{2}}{M_{2}^{2}-m_{2}}\equiv\frac{1}{\kappa},$
(25)

where $M_{1(2)}$ is the mass of the initial (final) baryon and $m_{1(2)}$ is
the mass of the quark before (after) the weak decay. In Fig. 3 and Fig. 4 we
present the Borel parameter dependence of the
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$ decay amplitudes
$A^{(\prime)(2)}(B^{(\prime)(2)}),A^{(\prime)(3p)}(B^{(\prime)(3p)})$ and
$A^{(\prime)(3\sigma)}(B^{(\prime)(3\sigma)})$ contributed from the twist-2,
twist-3p and twist-3$\sigma$ LCDAs, respectively. In each diagram, the blue
band denotes the uncertainty from the error of the threshold. The upper and
lower red bands denote the uncertainty from the error of the Monte Carlo
integrations. Generally, the window of Borel parameters is chosen to satisfy
three requirements. The first one is that they must be small enough so that
the contribution from the continuous spectrum can be suppressed, which
determines their upper bound. The second one is that they must be large enough
to ensure the OPE convergence, which determines their lower bound. The last
one is that the result must be stable in this window. To determine the upper
bound of the Borel parameter, we require that the pole contribution must be
larger than the continuous spectrum contribution, namely:

$\displaystyle\frac{\int_{(m_{c}+m_{s})^{2}}^{s_{\Xi_{c}}}ds^{\prime}\int_{4m_{c}^{2}}^{s_{\Xi_{cc}}}ds\
e^{-s^{\prime}/\kappa T^{2}}e^{-s/T^{2}}{\rm Im}^{2}\Pi_{QCD}^{{\cal
O}_{i}}(s^{\prime},s,P^{2})}{\int_{(m_{c}+m_{s})^{2}}^{\infty}ds^{\prime}\int_{4m_{c}^{2}}^{\infty}ds\
e^{-s^{\prime}/\kappa T^{2}}e^{-s/T^{2}}{\rm Im}^{2}\Pi_{QCD}^{{\cal
O}_{i}}(s^{\prime},s,P^{2})}>0.5.$ (26)

The numerator represents the pole contribution, which is just the integral on
the right-hand side of Eq. (15). The denominator is the same integral but the
upper limits of $s$ and $s^{\prime}$ are extended to infinity, so that it
contains both pole and continuous spectrum contributions. This requirement
shows that the upper bound of the Borel parameter is around $T^{2}=7$ GeV2 for
$\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ and $T^{2}=6$ GeV2 for
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$.

The lower bounds of the Borel parameters are determined in principle by the
ratio between the contribution from the leading order and next-to-leading
order QCD corrections to the perturbative kernel of OPE. However, in this work
we have only considered the leading order contribution so that this method
cannot be used. From Fig. 3 and Fig. 4, it can be seen that the upper bounds
given above are in a relatively stable region. Therefore, although we cannot
determine the lower bound quantitatively, we can take a range below the upper
bound of the Borel parameter. Here, we set the window as $6<T^{2}<8$ GeV2 for
$\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ and $5<T^{2}<7$ GeV2 for
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$. The amplitudes and the
corresponding errors from the uncertainties of
$s_{\Xi_{cc}},s_{\Xi_{c}^{\prime}}$ and $T^{2}$ are listed in Table 2. Note
that most of the contributions from the twist-3 LCDAs are larger than that
from twist-2. This feature is common in the LCSR studies on the heavy-to-light
decays. For example, in the LCSR calculation of the $B\to\pi$ form factors
Duplancic:2008ix ; Rusov:2017chr , the contribution of twist-3 is generally of
the same order or even larger than that of twist-2. The highly suppressed
contributions should come from the LCDAs of twist-4 or higher.

Table 2: Decay amplitudes of $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ from the W-exchange contribution. The Borel parameters are set in the region $6<T^{2}<8$ GeV2 for $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ and $5<T^{2}<7$ GeV2 for $\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$. $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ | Twist-$2$ | Twist-$3p$ | Twist-$3\sigma$ | Total
---|---|---|---|---
$A_{{\rm WE}}$ | $0.0084\pm 0.0024$ | $-0.077\pm 0.01$ | $-0.056\pm 0.002$ | $-0.124\pm 0.011$
$B_{{\rm WE}}$ | $-0.064\pm 0.01$ | $0.052\pm 0.01$ | $0.165\pm 0.025$ | $0.153\pm 0.029$
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$ | Twist-$2$ | Twist-$3p$ | Twist-$3\sigma$ | Total
$A_{{\rm WE}}^{\prime}$ | $0.0027\pm 0.0005$ | $0.0089\pm 0.002$ | $-0.018\pm 0.0003$ | $-0.0062\pm 0.002$
$B_{{\rm WE}}^{\prime}$ | $0.0023\pm 0.0006$ | $0.052\pm 0.016$ | $0.011\pm 0.003$ | $0.066\pm 0.016$
Table 3: Comparison of the decay amplitudes of $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ from this work with those from the literature. All the amplitudes below are in unit $10^{-2}G_{F}$ GeV2. Here we list the W-exchange amplitudes from the pole model (PM) Cheng:2020wmk ; Sharma:2017txj ; Dhir:2018twm  and three-loop quark model (3LQM) Gutsche:2018msz . We also list the factorizable amplitudes from QCD sum rules (QCDSR) Shi:2019hbf , light-front quark model (LFQM)Cheng:2020wmk , nonrelativistic quark model (NRQM) Sharma:2017txj ; Dhir:2018twm  and heavy quark effective theory (HQET) Sharma:2017txj ; Dhir:2018twm . The notation LFQM + PM for example in the first column means the factorizable amplitude is from LFQM while the W-exchange amplitude is from PM, otherwise both of them are from the same theoretical approach.  $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ | $A^{{\rm fac}}$ | $A^{{\rm nf}}$ | $A^{{\rm tot}}$ | $B^{{\rm fac}}$ | $B^{{\rm nf}}$ | $B^{{\rm tot}}$
---|---|---|---|---|---|---
This work | $--$ | $-16.67\pm 1.41$ | $--$ | $--$ | $20.47\pm 3.89$ | $--$
QCDSR Shi:2019hbf  | $-8.74\pm 2.91$ | $--$ | $--$ | $16.76\pm 5.36$ | $--$ | $--$
LFQM + PM Cheng:2020wmk  | $7.40$ | $-10.79$ | $-3.83$ | $15.06$ | $-18.91$ | $3.85$
3LQM Gutsche:2018msz  | $-8.13$ | $10.50$ | $3.37$ | $-12.97$ | $18.53$ | $5.56$
NRQM + PM Sharma:2017txj ; Dhir:2018twm  | $7.38$ | $0$ | $7.38$ | $16.77$ | $24.95$ | $41.72$
HQET + PM Sharma:2017txj ; Dhir:2018twm  | $9.52$ | $0$ | $9.52$ | $19.45$ | $24.95$ | $44.40$
$\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$ | $A^{\prime{\rm fac}}$ | $A^{\prime{\rm nf}}$ | $A^{\prime{\rm tot}}$ | $B^{\prime{\rm fac}}$ | $B^{{\rm\text{$\prime$}nf}}$ | $B^{{\rm\prime tot}}$
This Work | $--$ | $-0.83\pm 0.28$ | $--$ | $--$ | $8.86\pm 2.16$ | $--$
QCDSR Shi:2019hbf  | $-3.55\pm 0.68$ | $--$ | $--$ | $34.13\pm 11.6$ | $--$ | $--$
LFQM + PM Cheng:2020wmk  | $4.49$ | $-0.04$ | $4.45$ | $48.50$ | $-0.06$ | $48.44$
3LCQM Gutsche:2018msz  | $-4.34$ | $-0.11$ | $-4.45$ | $-37.59$ | $-1.37$ | $-38.96$
NRQM + PM Sharma:2017txj ; Dhir:2018twm  | $4.29$ | $0$ | $4.29$ | $53.65$ | $0$ | $53.65$
HQET + PM Sharma:2017txj ; Dhir:2018twm  | $5.10$ | $0$ | $5.10$ | $62.37$ | $0$ | $62.37$

Table 3 shows the comparison of our results with those from the literature.
Here, we have unified the definition of the amplitudes from all of these
works. The amplitudes presented in the table are the parameters of the matrix
element induced by the effective Hamiltonian from Eq. (2) instead of ${\cal
O}_{1,2}$:

$\displaystyle\langle\Xi_{c}^{+(\prime)}(p-q)\pi^{+}(q)|{\cal H}_{\rm
eff}(0)|\Xi_{cc}^{++}(p)\rangle_{\rm fac,nf}=i\
\bar{u}(p-q)\big{[}A^{(\prime)\rm fac,nf}+B^{(\prime)\rm
fac,nf}\gamma_{5}\big{]}u(p),$ (27)

where “fac” means the factorizable or W-emission contribution while “nf”
denotes the non-factorizable or equivalently the W-exchange contribution, and
$(A,B)^{(\prime)\rm tot}=(A,B)^{(\prime)\rm fac}+(A,B)^{(\prime)\rm nf}$. All
the amplitudes in Table 3 are in units of $10^{-2}G_{F}$ GeV2. Here we have
listed the W-exchange amplitudes from the pole model (PM) Cheng:2020wmk ;
Sharma:2017txj ; Dhir:2018twm  and three-loop constituent quark model (3LCQM)
Gutsche:2018msz . We have also listed the corresponding factorizable
amplitudes from QCDSR Shi:2019hbf , LFQM Cheng:2020wmk , nonrelativistic quark
model (NRQM) Sharma:2017txj ; Dhir:2018twm  and heavy quark effective theory
(HQET) Sharma:2017txj ; Dhir:2018twm . The notation LFQM + PM for example in
the first column means that the factorizable amplitude is from LFQM while the
W-exchange amplitude is from PM. On the other hand, 3LCQM alone means that the
factorizable and non-factorizable amplitudes are both from 3LCQM. From the
comparison in Table 3, one can find that our results for the W-exchange
amplitudes of $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ are consistent with most of
the results from literatures. However, in terms of the W-exchange amplitudes
of $\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+}$, our results are much larger
than those from the literature, which provides a possibility to explain the
anomaly of ${\cal B}^{\prime}/{\cal B}$. Two issues deserve further
discussion::

  * •

We have only used the two-particle LCDAs of the pion for the calculation up to
the leading order. Although the contribution from the higher-twist LCDAs or
QCD loop corrections is expected to be suppressed, in principle they are still
necessary for improving the accuracy of the W-exchange contribution, which
will be included in a future study.

  * •

From Table 3, we see that the various predictions on the factorizable
contribution from the literature are not totally consistent with each other.
These works assume the naive factorization which is based on color
transparency Bjorken:1988kk . It states that in the bottom hadron decays, the
$b$ quark is heavy enough so that the emitted light meson flies quickly and
decouples from other hadrons before it is caught up by the soft gluons.
However, since the charm quark is lighter, in the charmed hadron decays the
effect of soft gluon exchange may be important. Therefore, such effects need
to be worked out in the future.

Table 4: Comparison of the decay branching fractions of
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ from this work with those from
the literature. The lifetime of the $\Xi_{cc}^{++}$ is chosen as
$\tau(\Xi_{cc}^{++})=2.56\times 10^{-13}s$ Cheng:2020wmk . In the first five
lines the branching fractions are evaluated by the non-factorizable amplitudes
from this work and the factorizable amplitudes from the literature. The last
four lines present the branching fractions with both these two kinds of
amplitudes evaluated in the literature. In the last column we list the values
of ${\cal B}^{\prime}/{\cal B}$ from all the theoretical methods mentioned in
Table 3.

Method | $A^{{\rm tot}}$ | $B^{{\rm tot}}$ | ${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})$ | $A^{\prime{\rm tot}}$ | $B^{\prime{\rm tot}}$ | ${\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+\prime}\pi^{+})$ | ${\cal B}^{\prime}/{\cal B}$
---|---|---|---|---|---|---|---
QCDSR+LCSR | $-25.4\pm 4.32$ | $37.23\pm 9.25$ |  $40\pm 14$ % | $-4.38\pm 0.96$ | $42.99\pm 13.76$ |  $3.91\text{$\pm$2.5}$ % | $0.098\pm 0.14$
LFQM+LCSR | $-9.27\pm 1.41$ | $35.53\pm 3.89$ |  $7.54\pm 2.22$ % | $3.66\pm 0.28$ | $57.36\pm 2.16$ |  $5.83\pm 0.5$ % | $0.77\pm 0.42$
3LCQM+LCSR | $-24.8\pm 1.41$ | $7.5\pm 3.89$ |  $35.55\pm 4.29$ % | $-5.17\pm 0.28$ | $-28.73\pm 2.16$ |  $2.75\pm 0.35$ % | $0.08\pm 0.02$
NRQM+LCSR | $-9.29\pm 1.41$ | $37.24\pm 3.89$ |  $7.82\pm 2.25$ % | $3.46\pm 0.28$ | $62.51\pm 2.16$ |  $6.70\pm 0.54$ % | $0.85\pm 0.44$
HQET+LCSR | $-7.18\pm 1.41$ | $39.92\pm 3.89$ |  $6.22\pm 1.94$ % | $4.27\pm 0.28$ | $71.23\pm 2.16$ |  $8.85\pm 0.62$ % | $1.42\pm 0.78$
LFQM+PM | $-3.83$ | 3.85 |  $0.69$ % | $4.45$ | $48.44$ |  $4.65$ % | $6.74$
3LCQM | $3.37$ | $5.56$ |  $0.71$ % | $-4.45$ | $-38.96$ |  $3.39$ % | $4.77$
HQET+PM | $7.38$ | $41.72$ |  $6.64$ % | $4.29$ | $53.65$ |  $5.39$ % | $0.81$
NRQM+PM | 9.52 | 44.40 |  $9.19$ % | $5.1$ | $62.37$ |  $7.34$ % | $0.8$

Now we calculate the decay branching fractions by combining each factorizable
amplitude from the literature and the non-factorizable amplitudes from this
work, and make a comparison among them. The decay width is expressed as

$\displaystyle\Gamma^{(\prime)}=\frac{p_{c}}{8\pi}\left[\frac{\big{(}m_{\Xi_{cc}}+m_{\Xi_{c}^{(\prime)}}\big{)}^{2}-m_{\pi}^{2}}{m_{\Xi_{cc}}^{2}}|A^{\rm
tot(\prime)}|^{2}+\frac{\big{(}m_{\Xi_{cc}}-m_{\Xi_{c}^{(\prime)}}\big{)}^{2}-m_{\pi}^{2}}{m_{\Xi_{cc}}^{2}}|B^{\rm
tot(\prime)}|^{2}\right],$ (28)

where $p_{c}$ is the magnitude of the pion three-momentum in the rest-frame of
the $\Xi_{cc}$. The lifetime of the $\Xi_{cc}^{++}$ is chosen as
$\tau(\Xi_{cc}^{++})=2.56\times 10^{-13}s$, the Wilson coefficients are chosen
as $C_{1}=1.35$ and $C_{2}=-0.64$ Cheng:2020wmk , $G_{F}=1.166\times 10^{-5}$
GeV-2 and $V_{cs}=0.975,\ V_{ud}=0.973$ ParticleDataGroup:2020ssz . The
branching fractions are listed in Table 4. In the first five lines the
branching fractions are evaluated by the non-factorizable amplitudes from this
work and the factorizable amplitudes from the literature. The last four lines
present the branching fractions with both kinds of amplitudes being evaluated
in the literature. In the last column we list the values of ${\cal
B}^{\prime}/{\cal B}$ from all the theoretical methods mentioned above. We
note that most of the calculated ${\cal B}^{\prime}/{\cal B}$ are much larger
or smaller than the experimental value $({\cal B}^{\prime}/{\cal B})_{\rm
exp}=1.41\pm 0.17\pm 0.1$. However, the value obtained from HQET+LCSR leads to
the fraction

$({\cal B}^{\prime}/{\cal B})_{\rm HQET+LCSR}=1.42\pm 0.78~{},$ (29)

which agrees amazingly well with the experimental value. Although there exists
non-negligible uncertainty for this theoretical result, it still implies that
$({\cal B}^{\prime}/{\cal B})_{\rm exp}=1.41>1$ can be understood and realized
theoretically without introducing any physics beyond the Standard Model. The
absolute branching fraction of $\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+}$ from
HQET+LCSR is

$\displaystyle{\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})_{\rm
HQET+LCSR}=6.22\pm 1.94\ \%.$ (30)

Up to now, there is no experimental announcement on this absolute branching
fraction. Instead, in Ref. Cheng:2020wmk  the authors have given an evaluation
of it by phenomenological methods:

$\displaystyle{\cal B}(\Xi_{cc}^{++}\to\Xi_{c}^{+}\pi^{+})=1.83\pm 1.01\ \%,$
(31)

which is consistent with our result within the uncertainties. For the details
on this phenomenological evaluation we refer to the discussion around the Eq.
(38) in Ref. Cheng:2020wmk . However, due to the lack of experimental
measurements all the theoretical predictions given above are still waiting to
be tested by future experiments.

##  VI Conclusion

We have calculated the W-exchange contribution in the
$\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ decay with the use of the LCSR.
The two-particle LCDAs of pion are used as the non-perturbative inputs for the
sum rules calculation, and the perturbative kernel is calculated at the
leading order. We obtain the relative decay branching fraction ${\cal
B}^{\prime}/{\cal B}$ of $\Xi_{cc}^{++}\to\Xi_{c}^{+(\prime)}\pi^{+}$ by
combining our W-exchange amplitudes with the factorizable amplitudes from
various theoretical methods in the literature. We find that using the
factorizable amplitudes from heavy quark effective theory, we obtain the ratio
${\cal B}^{\prime}/{\cal B}=1.42\pm 0.78$, which is consistent with the
experimental value, $1.41\pm 0.17\pm 0.1$. The corresponding absolute decay
branching fraction is consistent with that evaluated in the literature, which
should be tested by future experiments.

## Acknowledgements

This work is supported in part by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 - TRR 110). The work of UGM was supported in part by the
Chinese Academy of Sciences (CAS) President’s International Fellowship
Initiative (PIFI) (Grant No. 2018DM0034) and by VolkswagenStiftung (Grant No.
93562). The work of ZXZ is supported in part by National Science Foundation of
China under Grant No. 12065020. The work of YX is supported in part by
National Science Foundation of China under Grant No. 12005294.

## References

  * (1) M. Gell-Mann, Phys. Lett. 8, 214-215 (1964) doi:10.1016/S0031-9163(64)92001-3
  * (2) G. Zweig, CERN-TH-412.
  * (3) A. De Rujula, H. Georgi and S. L. Glashow, Phys. Rev. D 12, 147-162 (1975) doi:10.1103/PhysRevD.12.147
  * (4) R. L. Jaffe and J. E. Kiskis, Phys. Rev. D 13, 1355 (1976) doi:10.1103/PhysRevD.13.1355
  * (5) W. Ponce, Phys. Rev. D 19, 2197 (1979) doi:10.1103/PhysRevD.19.2197
  * (6) S. Fleck, B. Silvestre-Brac and J. M. Richard, Phys. Rev. D 38, 1519-1529 (1988) doi:10.1103/PhysRevD.38.1519
  * (7) R. Aaij et al. [LHCb], Phys. Rev. Lett. 119, no.11, 112001 (2017) doi:10.1103/PhysRevLett.119.112001 [arXiv:1707.01621 [hep-ex]].
  * (8) F. S. Yu, H. Y. Jiang, R. H. Li, C. D. Lü, W. Wang and Z. X. Zhao, Chin. Phys. C 42, no.5, 051001 (2018) doi:10.1088/1674-1137/42/5/051001 [arXiv:1703.09086 [hep-ph]].
  * (9) R. Aaij et al. [LHCb], Phys. Rev. Lett. 121, no.16, 162002 (2018) doi:10.1103/PhysRevLett.121.162002 [arXiv:1807.01919 [hep-ex]].
  * (10) R. Aaij et al. [LHCb], JHEP 05, 038 (2022) doi:10.1007/JHEP05(2022)038 [arXiv:2202.05648 [hep-ex]].
  * (11) J. D. Bjorken, Nucl. Phys. B Proc. Suppl. 11, 325-341 (1989) doi:10.1016/0920-5632(89)90019-4
  * (12) M. Wirbel, B. Stech and M. Bauer, Z. Phys. C 29, 637 (1985) doi:10.1007/BF01560299
  * (13) M. Bauer, B. Stech and M. Wirbel, Z. Phys. C 34, 103 (1987) doi:10.1007/BF01561122
  * (14) Y. J. Shi, W. Wang and Z. X. Zhao, Eur. Phys. J. C 80, no.6, 568 (2020) doi:10.1140/epjc/s10052-020-8096-2 [arXiv:1902.01092 [hep-ph]].
  * (15) Y. J. Shi, Y. Xing and Z. X. Zhao, Eur. Phys. J. C 79, no.6, 501 (2019) doi:10.1140/epjc/s10052-019-7014-y [arXiv:1903.03921 [hep-ph]].
  * (16) X. H. Hu and Y. J. Shi, Eur. Phys. J. C 80, no.1, 56 (2020) doi:10.1140/epjc/s10052-020-7635-1 [arXiv:1910.07909 [hep-ph]].
  * (17) X. H. Hu and Y. J. Shi, [arXiv:2202.07540 [hep-ph]].
  * (18) T. M. Aliev, S. Bilmis and M. Savci, [arXiv:2206.08253 [hep-ph]].
  * (19) T. M. Aliev, S. Bilmis and M. Savci, [arXiv:2205.14012 [hep-ph]].
  * (20) N. Sharma and R. Dhir, Phys. Rev. D 96, no.11, 113006 (2017) doi:10.1103/PhysRevD.96.113006 [arXiv:1709.08217 [hep-ph]].
  * (21) A. S. Gerasimov and A. V. Luchinsky, Phys. Rev. D 100, no.7, 073015 (2019) doi:10.1103/PhysRevD.100.073015 [arXiv:1905.11740 [hep-ph]].
  * (22) Y. J. Shi, W. Wang, Z. X. Zhao and U.-G. Meißner, Eur. Phys. J. C 80, no.5, 398 (2020) doi:10.1140/epjc/s10052-020-7949-z [arXiv:2002.02785 [hep-ph]].
  * (23) Z. X. Zhao, Eur. Phys. J. C 78, no.9, 756 (2018) doi:10.1140/epjc/s10052-018-6213-2 [arXiv:1805.10878 [hep-ph]].
  * (24) W. Wang, F. S. Yu and Z. X. Zhao, Eur. Phys. J. C 77, no.11, 781 (2017) doi:10.1140/epjc/s10052-017-5360-1 [arXiv:1707.02834 [hep-ph]].
  * (25) H. Y. Cheng, G. Meng, F. Xu and J. Zou, Phys. Rev. D 101, no.3, 034034 (2020) doi:10.1103/PhysRevD.101.034034 [arXiv:2001.04553 [hep-ph]].
  * (26) H. W. Ke, F. Lu, X. H. Liu and X. Q. Li, Eur. Phys. J. C 80, no.2, 140 (2020) doi:10.1140/epjc/s10052-020-7699-y [arXiv:1912.01435 [hep-ph]].
  * (27) H. W. Ke and X. Q. Li, Phys. Rev. D 105, no.9, 096011 (2022) doi:10.1103/PhysRevD.105.096011 [arXiv:2203.10352 [hep-ph]].
  * (28) X. H. Hu, R. H. Li and Z. P. Xing, Eur. Phys. J. C 80, no.4, 320 (2020) doi:10.1140/epjc/s10052-020-7851-8 [arXiv:2001.06375 [hep-ph]].
  * (29) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij and Z. Tyulemissov, Phys. Rev. D 99, no.5, 056013 (2019) doi:10.1103/PhysRevD.99.056013 [arXiv:1812.09212 [hep-ph]].
  * (30) T. Gutsche, M. A. Ivanov, J. G. Körner, V. E. Lyubovitskij and Z. Tyulemissov, Phys. Rev. D 100, no.11, 114037 (2019) doi:10.1103/PhysRevD.100.114037 [arXiv:1911.10785 [hep-ph]].
  * (31) R. Dhir and N. Sharma, Eur. Phys. J. C 78, no.9, 743 (2018) doi:10.1140/epjc/s10052-018-6220-3
  * (32) W. Wang, Z. P. Xing and J. Xu, Eur. Phys. J. C 77, no.11, 800 (2017) doi:10.1140/epjc/s10052-017-5363-y [arXiv:1707.06570 [hep-ph]].
  * (33) Y. J. Shi, W. Wang, Y. Xing and J. Xu, Eur. Phys. J. C 78, no.1, 56 (2018) doi:10.1140/epjc/s10052-018-5532-7 [arXiv:1712.03830 [hep-ph]].
  * (34) R. H. Li, C. D. Lü, W. Wang, F. S. Yu and Z. T. Zou, Phys. Lett. B 767, 232-235 (2017) doi:10.1016/j.physletb.2017.02.003 [arXiv:1701.03284 [hep-ph]].
  * (35) I. I. Balitsky, V. M. Braun and A. V. Kolesnichenko, Nucl. Phys. B 312, 509-550 (1989) doi:10.1016/0550-3213(89)90570-1
  * (36) V. M. Braun and I. E. Filyanov, Z. Phys. C 44, 157 (1989) doi:10.1007/BF01548594
  * (37) V. L. Chernyak and I. R. Zhitnitsky, Nucl. Phys. B 345, 137-172 (1990) doi:10.1016/0550-3213(90)90612-H
  * (38) A. Khodjamirian, Nucl. Phys. B 605, 558-578 (2001) doi:10.1016/S0550-3213(01)00194-8 [arXiv:hep-ph/0012271 [hep-ph]].
  * (39) A. Khodjamirian, T. Mannel and B. Melic, Phys. Lett. B 571, 75-84 (2003) doi:10.1016/j.physletb.2003.08.012 [arXiv:hep-ph/0304179 [hep-ph]].
  * (40) A. Khodjamirian and A. A. Petrov, Phys. Lett. B 774, 235-242 (2017) doi:10.1016/j.physletb.2017.09.070 [arXiv:1706.07780 [hep-ph]].
  * (41) A. V. Rusov, Eur. Phys. J. C 77, no.7, 442 (2017) doi:10.1140/epjc/s10052-017-5000-9 [arXiv:1705.01929 [hep-ph]].
  * (42) Z. G. Wang, Eur. Phys. J. A 49, 131 (2013) doi:10.1140/epja/i2013-13131-7 [arXiv:1203.6252 [hep-ph]].
  * (43) G. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, JHEP 04, 014 (2008) doi:10.1088/1126-6708/2008/04/014 [arXiv:0801.1796 [hep-ph]].
  * (44) A. Khodjamirian, B. Melić, Y. M. Wang and Y. B. Wei, JHEP 03, 016 (2021) doi:10.1007/JHEP03(2021)016 [arXiv:2011.11275 [hep-ph]].
  * (45) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no.8, 083C01 (2020) doi:10.1093/ptep/ptaa104
  * (46) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, no.3, 030001 (2018) doi:10.1103/PhysRevD.98.030001
  * (47) Z. G. Wang, Eur. Phys. J. A 47, 81 (2011) doi:10.1140/epja/i2011-11081-8 [arXiv:1003.2838 [hep-ph]].
  * (48) W. Roberts and M. Pervin, Int. J. Mod. Phys. A 23, 2817-2860 (2008) doi:10.1142/S0217751X08041219 [arXiv:0711.2492 [nucl-th]].
  * (49) X. H. Hu, Y. L. Shen, W. Wang and Z. X. Zhao, Chin. Phys. C 42, no.12, 123102 (2018) doi:10.1088/1674-1137/42/12/123102 [arXiv:1711.10289 [hep-ph]].
  * (50) Z. G. Wang, Eur. Phys. J. C 68, 479-486 (2010) doi:10.1140/epjc/s10052-010-1365-8 [arXiv:1001.1652 [hep-ph]].
  * (51) P. Ball, V. M. Braun and H. G. Dosch, Phys. Rev. D 44, 3567-3581 (1991) doi:10.1103/PhysRevD.44.3567