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Rescattering mechanism of weak decays ofdouble-charm baryons *

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Rescattering mechanism of weak decays of double-charm baryons*

Jia-Jie Han(韩佳杰)1,2† Hua-Yu Jiang(蒋华玉)1,3‡ Wei Liu(刘薇)1§ Zhen-Jun Xiao(肖振军)2 Fu-Sheng Yu(于福升)1,4,5,6♯

1School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China2Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing 210023, China3Theoretische Physik 1, Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany

4Lanzhou Center for Theoretical Physics, Lanzhou University, Lanzhou 730000, China5Frontier Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000, China

6Center for High Energy Physics, Peking University, Beijing 100871, China

Ξ++ccΞ++cc → Λ+c K−π+π+ Ξ+c π

+

Bcc→BcPBcc = (Ξ++cc ,Ξ

+cc,Ω

+cc) Bc = (B3,B6)

P = (π,K,η1,8)

Abstract: The doubly charmed baryon was recently observed by LHCb via the decay processes of and . These discovery channels were successfully predicted in a framework in which the

short-distance contributions are calculated under the factorization hypothesis and the long-distance contributions areestimated using the rescattering mechanism for the final-state-interaction effects. In this paper, we illustrate theabove framework in detail by systematic studies on the two-body baryonic decays involving the doublycharmed baryons , the singly charmed baryons and the light pseudoscalarmesons .

Keywords: double-charm baryon, rescattering mechanism, final-state interaction

DOI: 10.1088/1674-1137/abec68

I. INTRODUCTION

The doubly heavy baryons with two heavy flavorquarks (the b or c quark) were predicted by the quarkmodel and quantum chromodynamics (QCD) several dec-ades ago [1-3]. Their structure, analogous to a heavydouble-star system with an attached light planet [4], isvery different from single-heavy-flavor baryons and lightbaryons. Furthermore, research on the doubly heavy bary-ons is a powerful tool in investigations of doubly andfully heavy tetraquark states [5, 6]. Therefore, the doublyheavy baryons open a new window for research on theproperties of QCD [7].

Ξ+ccΞ+cc→ Λ+c K−π+

However, there have been many twists and turns inthe history of experimental searches for doubly heavy ba-ryons. In 2002, was first reported as observed by theSELEX collaboration via the mode of [8].None of the following measurements by FOCUS [9],BaBar [10], Belle [11], and LHCb [12] found signatures

Ξ++cc → Λ+c K−π+π+ Ξ+c π+

Ξ++cc → Λ+c K−π+π+

Ξ++cc → Ξ+c π+

using the same decay mode. Actually, the production rateof the doubly charmed baryons was large enough at thebeginning of LHCb running [13, 14]. The remainingproblem is the decay properties, i.e. which decay pro-cesses have the largest branching fractions and finalparticles which can easily be detected in the experiments[15]. In 2017, a theoretical analysis of all the decay pro-cesses found that and are themost favorable for the discovery of doubly charmed bary-ons [16]. Subsequently, the LHCb collaboration ob-served the doubly charmed baryon for the first time via

[17], following the theoretical sugges-tions, and confirmed the discovery via in2018 [18]. It is clear that theoretical studies of the decayproperties play an important role in experimental searchesfor the doubly heavy baryons.

There are two difficulties in theoretical calculations ofthe dynamics of doubly charmed baryon decays: charmdecay with large non-perturbative contributions, and ba-

Received 23 February 2021; Accepted 8 March 2021; Published online 14 April 2021 * Supported by the National Natural Science Foundation of China (11775117, U1732101, 11975112), and National Key Research and Development Program ofChina (2020YFA0406400) † E-mail: hanjiajie1020@163.com ‡ E-mail: jianghy15@lzu.edu.cn § E-mail: liuw18@lzu.edu.cn E-mail: xiaozhenjun@njnu.edu.cn ♯ E-mail: yufsh@lzu.edu.cn, Corresponding author

Chinese Physics C Vol. 45, No. 5 (2021) 053105

Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must main-tain attribution to the author(s) and the title of the work, journal citation and DOI. Article funded by SCOAP3 and published under licence by Chinese Physical Societyand the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Pub-lishing Ltd

053105-1

ryon decay with a three-body problem. For charm decays,QCD-inspired methods do not work well at scales ofaround 1 GeV. In the charmed meson decays, there aresignificant non-perturbative contributions to the topolo-gical amplitudes which are extracted from the experi-mental data of the decay branching fractions [19-23].However, the topological diagrammatic approach cannotbe directly used in doubly charmed baryon decays, sincethere are no available data. The charmed baryon decaysare even more complicated [24-50]. As a first attempt tostudy non-leptonic doubly charmed baryon decays, inRef. [16], the factorizable contributions were calculatedunder the factorization hypothesis, and the non-factoriz-able contributions calculated considering the rescatteringmechanism of the final-state-interaction (FSI) effects. Inthis work, we will systematically illustrate this theoretic-al framework in the decays of doubly charmed baryonsinto a charmed baryon and a light pseudoscalar meson.

Λ+c

The success of the factorization and rescatteringmechanism in suggesting the discovery channels of thedoubly charmed baryons shows that the above frame-work roughly describes the correct dynamics of doublycharmed baryon decays. There is no doubt that the factor-ization approach works well for the short-distance tree-emitted diagrams [20, 21, 23, 51, 52]. The problem ishow to calculate the long-distance contributions, whichare usually considered as FSI effects. Much work hasbeen done to calculate the FSI effects of weak decays ofheavy-flavor mesons [53-60]. Before Ref. [16], there wasno work on the long-distance contributions of doublyheavy baryon decays, and only a few works on the short-distance factorizable contributions [61, 62]. The non-per-turbative contributions are very important in charm de-cays. The rescattering mechanism of the FSI effect wasfirst investigated for decays in Ref. [60], and firststudied for doubly charmed baryon decays in Ref. [16].

The theoretical framework of the rescattering mech-anism is as follows. The doubly charmed baryon decaysvia a short-distance tree-emitted process into one baryonand one meson, which scatter with each other by exchan-ging one particle as a long-distance effect into the finalstates. It forms a triangle diagram at the hadron level. Theshort-distance and long-distance contributions are separ-ated to avoid the double-counting problem. In this work,the calculation techniques follow Ref. [58], in which thecutting rules are used to compute the imaginary part ofthe triangle diagram. There is a basic difference betweenour framework and those of Refs. [54, 58]. The trianglediagrams of the rescattering mechanism are taken as anindependent method by the calculation of hadron-levelFeynman diagrams, while in Refs. [54, 58] the trianglediagrams are calculated corresponding to the quark topo-logical diagrams. The problem of the latter method is dis-cussed elsewhere [63].

FSI calculations suffer large theoretical uncertainties.The branching fractions could be changed by one order of

ηΛ

magnitude with the variation of the non-perturbativeparameters like in the form factor of the cutting rules orthe cut-off in the loop calculation. The parameters arealways determined by the measured results of the branch-ing fractions [54, 58]. However, in the case of doublycharmed baryons, there is no available data to determinethe non-perturbative parameters. Therefore, the biggestproblem in the FSI calculations is how to control the the-oretical uncertainties. The innovation of our method is tocalculate the ratios of branching fractions, which are notsensitive to the non-perturbative parameters. The uncer-tainties of the ratios are thus well under control. That iswhy Ref. [16] could correctly and reliably predict themodes with the largest branching fractions.

|C||T | ∼

|C′||C| ∼

|E1||C| ∼

|E2||C| ∼ O

ΛhQCD

mc

Λ+c

Ξ++cc → Λ+c K−π+π+ Ξ++cc → Σ++c K∗0

Λ+c → pϕ

S U(3)

Ξ++cc

Knowledge of the relative sizes of the topological dia-grams of heavy baryon decays leads to important implica-tions for the predictions of the most favorable modes todiscover the doubly charmed baryons. In the soft-collin-ear effective theory [64, 65], the power counting rules of

are obtained. Theserelations are manifested by the most precise measure-ments of the decays performed by the BESIII collab-oration [66]. In addition, the discovery channel of

dominated by can bedirectly related to the result of [67] by exactlythe same topological diagram, with an interchange of aspectator quark. In this work, we will calculate the topo-logical diagrams by the rescattering mechanism to test theabove relations. The flavor symmetry and itsbreaking effects will also be discussed. It has to bestressed that the weak decays of doubly charmed baryonshave been widely studied [61, 62, 68-94], especially afterthe work of Ref. [16] and the experimental observation of

. The clarification of our framework will be helpful tounderstand the dynamics and nature of doubly charmedbaryons.

Ξ++cc → Ξ+c π+

This paper is arranged as follows. In Sec. II, we intro-duce the theoretical framework of the rescattering mech-anism and demonstrate the calculation details with

as an example. Then the parameter inputs,numerical results of branching fractions and relevant dis-cussions are presented in Sec. III. At the end, we give abrief summary. The effective hadron-strong-interactionLagrangians and corresponding strong coupling con-stants are collected in Appendix A. The expressions ofthe decay amplitudes for all the decay modes consideredare gathered in Appendix B.

II. THEORETICAL FRAMEWORK

A. Effective Hamiltonian and topological diagramsThe exclusive non-leptonic weak decays of doubly

charmed baryons are induced by the charge currents of

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-2

charm decays at the tree level. The penguin contributionsare safely neglected in the branching fractions of charmdecays, due to the smallness of the corresponding CKMmatrix elements. The effective Hamiltonian is given by

Heff =GF√

2

∑q′=d,s

V∗cq′Vuq[C1(µ)O1(µ)

+C2(µ)O2(µ)]+h.c.. (1)

with the four-fermion operators of

O1 =(uαqβ)V−A(q′βcα)V−A,

O2 =(uαqα)V−A(q′βcβ)V−A, (2)

q(′) = (s,d) α β Vcq′ Vuq

GF = 1.166×10−5 GeV−2

C1,2(µ)

µ = MW µ = mc

Bcc→BcP

where , and are color indices, and are the Cabibbo-Kobayashi-Maskawa (CKM) matrix ele-ments, and is the Fermi con-stant. denote the Wilson coefficients, which in-clude the short-distance QCD dynamics scaling from

to . To obtain the amplitudes of decays, one needs to evaluate the next had-

ronic matrix element of the effective Hamiltonian:

⟨BcP|Heff |Bcc⟩ =GF√

2V∗cq′Vuq

∑i=1,2

Ci⟨BcP|Oi|Bcc⟩. (3)

C′

The general tree-level topological diagrams of doublycharmed baryons decaying into a singly charmed baryonand a light meson are displayed in Fig. 1. These dia-grams can be sorted by their different topologies. Eachdiagram contains both short-distance and long-distancecontributions. T describes the color-allowed external W-emission diagram, while C and can both be used torepresent color-suppressed internal W-emission diagrams.The C diagram is the one with both the quark and anti-

C′

E1 E2 E1

E2

quark of the final light meson state coming from the weakvertex, while for only the antiquark is generated fromthe weak vertex and the quark is directly transferred fromthe light spectator of the doubly charmed baryon. Thereare also two different types of W-exchange diagrams,labeled and . In , the light quark produced by thecharmed quark decay is absorbed into the light mesons.In , the light quark produced from the charmed quarkdecay is absorbed into a singly heavy baryon. The pos-sible quark loop (penguin) diagrams relevant to the tree-level topological diagrams also have a non-negligible im-pact on the estimation of the long-distance contributionsin our framework.

a2(mc) =C1(mc)+C2(mc)/Nc

C′ E1 E2

In the calculation of these topological diagrams, it hasbeen demonstrated that T is dominated by factorizablecontributions [95] and can be calculated under the factor-ization hypothesis. However, this factorizable contribu-tion of the C diagram is heavily suppressed by the colorfactor at charm scale, with the effective Wilson coeffi-cient . The factorizable short-distance contributions are therefore negligible, but thelong-distance dynamics of C can play an important role[16]. The short-distance amplitudes of topological dia-grams , and are also expected to be suppressedat least by one order [95], while the long distance dynam-ics is more important at the scale of the charm quarkmass.

Ξ++cc → Ξ+c π+

T +C′

C′

C′

C′

In the following sections, we take the second discov-ery channel [16, 18] as an example to intro-duce our framework in detail. This decay contains twocontributions of topological amplitude, i.e. , inwhich T is dominated by the short-distance dynamics,while is dominated by the non-factorizable long-dis-tance dynamics. T usually plays the central role com-pared with (mainly due to the colour suppression).However, from our calculation, it can be seen that thenon-factorizable contributions of may have a signific-ant impact on the total amplitude.

Bcc = (Ξ++cc ,Ξ+cc,Ω

+cc) Bc P = (π,K,η1,η8)

Fig. 1. (color online) Tree-level topological diagrams for two-body non-leptonic decays of the doubly charmed baryons to a singly heavy baryon with a light pseudoscalar meson .

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-3

B. Short-distance amplitudes under the

factorization hypothesisIn this section, we discuss how to calculate the factor-

izable short-distance contributions of the topologicalamplitudes T and C. The feasible approach is the factoriz-

⟨BcM|Oi|Bcc⟩ation approach, with the matrix elements inEq. (3) factorized into the product of two parts. One isparameterized as the decay constant of the emittedmesons and the other is expressed as the transition formfactors. The factorizable contribution of the T diagram isexpressed as:

⟨BcM|Heff |Bcc⟩TS D =GF√

2V∗cq′Vuqa1(µ)⟨M|uγµ(1−γ5)q|0⟩⟨Bc|q′γµ(1−γ5)c|Bcc⟩, (4)

while the factorizable C diagram is given by

⟨BcM|Heff |Bcc⟩CS D =GF√

2V∗cq′Vuqa2(µ)⟨M|q′γµ(1−γ5)q|0⟩⟨Bc|uγµ(1−γ5)c|Bcc⟩, (5)

a1(a2)a1(µ) =C1(µ)+C2(µ)/3 a2(µ) =C2(µ)+C1(µ)/3

C1(µ) = 1.21 C2(µ) = −0.42µ = mc

where represents the effective Wilson coefficients, and , with

the Wilson coefficients and atthe scale of charm decays [21]. The meson M rep-resents both the pseudoscalar and vector mesons, since,as will be seen in the next subsections, the vector mesoncontributes to the long-distance dynamics as an interme-diate state.

In both Eqs. (4) and (5), the first hadronic matrix ele-ment is parameterized in the same way, as:

⟨P(p)|uγµ(1−γ5)q|0⟩ = −i fP pµ, (6)

⟨V(p)|uγµ(1−γ5)q|0⟩ = mV fVϵ∗µ. (7)

fP fVϵµ

where and are the corresponding decay constants ofthe pseudoscalar and vector mesons, and denotes thepolarization of the vector meson. The second matrix ele-ment is usually defined as:

⟨Bc(p′, s′z)|q′γµ(1−γ5)c|Bcc(p, sz)⟩ =u(p′, s′z)[γµ f1(q2)+iσµν

qν

MBcc

f2(q2)+qµ

MBcc

f3(q2)]u(p, sz)− u(p′, s′z)

×[γµg1(q2)+ iσµν

qν

MBcc

g2(q2)+qµ

MBcc

g3(q2)]γ5u(p, sz). (8)

q = p− p′ MBcc

fi gi

with , the mass of the doubly charmed ba-ryon, and , the heavy-light transition form factors,which can only be extracted from non-perturbative ap-proaches.

Bcc→BcPBcVIn general, the weak decay amplitudes of

and have the following parametrization form:

A(Bcc→BcP) = iuBc(A+Bγ5)uBcc

, (9)

A(Bcc→BcV) =ϵ∗µuBc

[A1γµγ5+A2

pµ(Bc)MBcc

γ5

+B1γµ+B2pµ(Bc)MBcc

]uBcc. (10)

A,B A1,2,B1,2The formulas of the parameters and underthe factorization approach are:

A =λ fP(MBcc−MBc

) f1(m2),

B =λ fP(MBcc+MBc

)g1(m2), (11)

A1 =−λ fVm(g1(m2)+g2(m2)

MBcc−MBc

MBcc

),

A2 =−2λ fVmg2(m2), (12)

B1 =λ fVm(

f1(m2)− f2(m2)MBcc+MBc

MBcc

),

B2 =2λ fVm f2(m2). (13)

λ =GF√

2VCKMa1,2(µ)where , and m is the mass of the

pseudoscalar or vector meson.

C. Long-distance contributions from therescattering mechanism

The long-distance contributions are important but noteasy to evaluate. As is done in Ref. [16], we calculate theFSI effects by the rescattering of two intermediateparticles at the hadron level using the hadronic strong-in-teraction effective Lagrangian. The diagram descriptionof the rescattering mechanism is shown in Fig. 2. Theweak vertex displayed in hadron-level diagrams only in-

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-4

t/u

k2 ∼ m2Ξ+cc

mΞ+cc∼ 3.6

mΞ+cc

t/u

volves the short-distance contributions and thus can beevaluated under the factorization hypothesis. The sub-sequent scattering process could in principle be either a s-channel resonant-state process or a -channel one. Thedominant contribution of the s-channel diagram comesfrom the momentum region , which demandsthat the mass of the exchanged particle (a singly charmedbaryon in this case) approaches GeV. However,the heaviest observed singly charmed baryons to date aremuch lighter than . Therefore, the s-channel diagramis usually highly suppressed by the off-shell effect andcan be safely neglected. In our calculations, the main con-tribution will be the -channel triangle diagram, asshown in Fig. 2.

Pn n = i,1,2,3,4,k(i)

(1,2) (3,4)(k)

pn

The particles in the triangle diagram are labeled as, where the subscripts represent the par-

ent doubly charmed particle , two intermediate had-rons , two final hadron states and the ex-changed hadron , respectively, as seen in Fig. 2. Thecorresponding momenta are assigned as . In general,there are several methods to calculate the amplitude of atriangle diagram [53-59]. The main difference betweenthem is the method of dealing with the hadronic loop in-tegration.

Pi→ P3P4

We adopt the optical theorem and Cutkosky cuttingrule, as in Ref. [58]. The absorptive part of the amplitudeof is a product of two distinct parts, the decay

Pi→ P1P2 P1P2 → P3P4

P1 P2

P1P2

of and the rescattering of ,with the internal particles ( and ) being on-shell. Ac-cording to the optical theorem, the absorptive amplitudeshould sum over all possible on-shell intermediate states

with a phase integration. It can be expressed as

Abs[M(Pi→ P3P4)] =12

∑P1P2

∫d3 p1

(2π)32E1

∫d3 p2

(2π)32E2

× (2π)4δ4(p3+ p4− p1− p2)×M(Pi→P1P2)T ∗(P3P4→P1P2).

(14)2 Based on the argument in Refs. [96, 97], the -body n-

body rescattering is negligible. In this approach, the loopintegration is transferred into the dispersive part, whichcan be calculated via the dispersion relation

Dis[M(m21)] =

1π

∫ ∞

s

Abs[M(s′)]s′−m2

1

ds′. (15)

M(s′)

η

However, it suffers from large ambiguities, since we can-not reliably describe for the whole region. On theother hand, in the charmed meson decays, the large strongphases of the topological diagrams [20, 21] indicate thatthe absorptive (imaginary) part is dominant. Therefore,we will only calculate the absorptive part and neglect thedispersive part, as done in Ref. [58]. In a phenomenolo-gical analysis, the non-negligible dispersive contribu-tions can be effectively absorbed into the varying of theparameter , which will be introduced in the following.

Ξ++cc → Ξ+c π+

Ξ++cc → Ξ+c (Ξ′+c )π+(ρ+)→ Ξ+c π+

Next we express the amplitude of the decay mode as an example. All the rescattering dia-

grams are represented in Fig. 3, which can be summar-ized as . The intermediateparticles can be either light pseudoscalar or vector

Fig. 2. Diagram description of the rescattering mechanism atthe hadron level.

Ξ++cc → Ξ+c π+Fig. 3. Long-distance rescattering contributions to manifested at hadron level.

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-5

M(P1,P2; Pk)P1 P2 Pk

mesons, or anti-triplet or sextet singly charmed baryons.We use the symbol to denote a triangleamplitude with the intermediate states of , and .

Ξ+c π+Ξ′+c π+

Figure 3(a) contains two different triangle diagrams,with the intermediate two-particle states of and

, respectively. The amplitudes of the weak vertex

Ξ++cc → Ξ+c π+ Ξ++cc → Ξ′′c π+

Ξ+c π+→ Ξ+c π+

Ξ′+c π+→ Ξ+c π+

and are taken from Eq. (9) inthe factorization approach for the short-distance contribu-tions. The rescattering amplitudes of and

can simply be found from the hadronicstrong Lagrangian in Appendix A. Then the absorptivepart is written as:

Abs[M(π+,Ξ+c /Ξ′+c ;ρ0)] =

12

∑s1,λk

∫d3 p1

(2π)32E1

d3 p2

(2π)32E2(2π)4δ4(pi− p1− p2)

× iu(p4, s4)(

f1ρ0(Ξ+c /Ξ′+c )→Ξ+c γν−i f2ρ0(Ξ+c /Ξ′+c )→Ξ+c

2mΞ+cσµνkµ

)u(p2, s2)ϵν(k,λk)

×F2(t,mρ)

t−m2ρ+ imρΓρ

(−igπ+→ρ0π+ )ϵ∗α(k,λk)(p1+ p3)α iu(p2, s2)(A+Bγ5)u(pi, si)

=

∫ |p1|sinθdθdϕ32π2mΞ++cc

i2(−igπ+→ρ0π+ )F2(t,mρ)

t−m2ρ+ imρΓρ

u(p4, s4) f1ρ0(Ξ+c /Ξ′+c )→Ξ+c

−p/1− p/3+k · (p1+ p3 ) k

m2ρ

+

f2ρ0(Ξ+c /Ξ′+c )→Ξ+c2mΞ+c

(− k(p/1+ p/3)+ k · (p1+ p3))]· (p/2+m2)(A+Bγ5)u(pi, si).

(16)

t = p2k = (p3− p1)2

Ξ++cc 3 p3

θ

ϕ p1

gρππ f1Ξ+cΞ+c ρ0 f2Ξ+cΞ+c ρ0

ρ0

ρ0 F(t,mρ)

where . In the center-of-mass frame of, the -momentum of the final-state baryon is

defined in the positive z axis direction, with two angles and standing for the polar and azimuthal angles of inthe spherical coordinate system. , and are the relevant strong coupling constants, which are usu-ally extracted or calculated in the on-shell condition.However, the exchanged is generally off-shell, so thatthe strong coupling constants are not exactly correct. Toinclude the off-shell effect of , a form factor [58] is introduced as

F(t,mρ) =

Λ2−m2ρ

Λ2− t

n

. (17)

This form factor is normalized to unity in the on-shell

t = p2k = m2

ρ Λsituation . The cutoff has the form of

Λ = mρ+ηΛQCD, (18)

ΛQCD = 330 MeVη

ηη

1 2

n = 1

with for the charm decays. The para-meter cannot be calculated from the first-principlesQCD method, and usually needs to be determined phe-nomenologically by the experimental data. The results arealways sensitive to the value of . More discussionsabout can be found in Sec. III. Usually, the exponentialfactor n in Eq. (17) can be taken as or , as a monopoleor dipole behavior. For the B meson decays in Ref. [58],the resultant branching ratios are almost the same for bothchoices. Hence in our work, we choose for conveni-ence, following Ref. [58].

In the same way, the absorptive amplitudes of the re-maining triangle diagrams in Fig. 3 are:

Abs[M(ρ+,Ξ+c /Ξ′+c ;π0)] =

∫ | p1|sinθdθdϕ32π2mΞ++cc

(−igρ+→π0π+ )(igπ0(Ξ+c /Ξ′+c )→Ξ+c )u(p4, s4)iγ5( p2+m2)

× F2(t,mπ0 )t−m2

π0 + imπ0Γπ0

(−2 p3+2p3 · p1 p1

m21

(A1γ5+B1)

+−2m2

1 p3 · p2+2p3 · p1 p1 · p2

m21mΞ++cc

(A2γ5+B2))u(pi, si), (19)

Abs[M(π+,Ξ+c /Ξ′+c ;Ξ0

c)] =∫ | p1|sinθdθdϕ

32π2mΞ++cc

igπ+Ξ0c→Ξ+c g(Ξ+c /Ξ′+c )→Ξ0

cπ+ u(p3, s3)γ5 (k+mk)

×γ5 (p2+m2)(A+Bγ5)u(pi, si)F2(t,mΞ0

c)

t−m2Ξ0

c+ imΞ0

cΓΞ0

c

, (20)

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-6

Abs[M(ρ+,Ξ+c /Ξ′+c ,Ξ

0c)] =

∫ | p1|sinθdθdϕ32π2mΞ++cc

i3u(p3, s3)(

fρ+Ξ0c→Ξ+c γν−

i f2ρ+Ξ0c→Ξ+c

mk +m3σµνp

µ1

)g(Ξ+c /Ξ′+c )→Ξ0

cπ+ ( k+mk)

×F2(t,mΞ0

c)

t−m2Ξ0

c+ imΞ0

cΓΞ0

c

γ5

−gνα+pν1 pα1m2

1

(p2+m2)(A1γαγ5+A2

p2α

mΞ+cc

γ5+B1γα+B2p2α

mΞ+cc

)u(pi, si),

(21)

Abs[M(π+,Ξ+c /Ξ′+c ;Ξ′0c )] =

∫ | p1|sinθdθdϕ32π2mΞ++cc

igπ+Ξ′0c →Ξ+c g(Ξ+c /Ξ′+c )→Ξ′0c π+ u(p3, s3)γ5( k+mk)

×γ5 (p2+m2)(A+Bγ5)u(pi, si)F2(t,mΞ0

c)

t−m2Ξ0

c+ imΞ0

cΓΞ0

c

, (22)

Abs[M(ρ+,Ξ+c /Ξ′+c ,Ξ

′0c )] =

∫ | p1|sinθdθdϕ32π2mΞ++cc

i3u(p3, s3)(

fρ+Ξ′0c →Ξ+c γν−i f2ρ+Ξ′0c →Ξ+cmk +m3

σµνpµ1

)×g(Ξ+c /Ξ′+c )→Ξ′0c π+ (k+mk)

F2(t,mΞ0c)

t−m2Ξ0

c+ imΞ0

cΓΞ0

c

γ5

−gνα+pν1 pα1m2

1

× ( p2+m2)

(A1γαγ5+A2

p2α

mΞ+cc

γ5+B1γα+B2p2α

mΞ+cc

)u(pi, si). (23)

Ξ++cc → Ξ+c π+Collecting all the pieces together, the amplitude of the decay can be written in the following form:

A(Ξ++cc → Ξ+c π+) = TS D(Ξ++cc → Ξ+c π+)+ iAbs[M(π+,Ξ+c /Ξ

′+c ;ρ0)+M(ρ+,Ξ+c /Ξ

′+c ;π0)+M(π+,Ξ+c /Ξ

′+c ;Ξ0

c)

+M(ρ+,Ξ+c /Ξ′+c ,Ξ

0c)+M(π+,Ξ+c /Ξ

′+c ;Ξ′0c )+M(ρ+,Ξ+c /Ξ

′+c ,Ξ

′0c )

].

(24)

TS D

The short-distance contribution of this decay mode islabeled by . The analytical expressions for all the oth-er channels are given in Appendix B.

III. NUMERICAL RESULTS AND DISCUSSION

Bcc→BcPThe width of a two-body decay is

Γ(Bcc(λi)→Bc(λ f )P)=| p|

8πmBcc

12

∑λi,λ f

∣∣∣∣A(Bcc(λi)→Bc(λ f )P)∣∣∣∣2,

(25)

where

| p| =√

[m2Bcc− (mBc

+mP)2][m2Bcc− (mBc

−mP)2]/(2mBcc

),

λi λ fand , are the corresponding spin polarizations.

A. Inputs

Ξ++cc

All the inputs used in this work are clarified here. Themass of has been well measured by LHCb [17, 18].Many theoretical papers have studied the masses of the

Ξ++cc

doubly charmed baryons [4, 7]. For the ground states,there should be not much difference between the theoret-ical predictions on the masses, benefitting from the meas-urement of the mass of . We adopt the results fromRef. [98], as shown in Table 1.

BRi = Γi ·τ

Ξ++cc

The lifetimes of the doubly charmed baryons play anessential role in theoretical predictions and experimentalsearches [15]. The branching fractions are proportional tothe lifetimes, . Besides, the longer lifetime willbe helpful for experiments to reject the large back-grounds at the decay vertex. However, the theoretical pre-dictions of the lifetimes of the doubly charmed baryonshave large ambiguities due to the non-perturbative contri-butions at the charm scale [62, 68, 100, 101]. Currently,the lifetime of has been well measured by LHCb[102]. After this measurement, a phenomenological ana-lysis considering the contributions of dimension-7 operat-

Table 1. Masses and lifetimes of the doubly charmed bary-ons used in this work.

baryon Ξ++cc Ξ+cc Ω+cc

mass (GeV) 3.621 [17, 18] 3.621 [98] 3.738 [98]

lifetime 256 [102] 140 [99] 180 [99]

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-7

τ(Ξ++cc ) = 298 τ(Ξ+cc) = 44 τ(Ω+cc) = 206τ(Ξ+cc) = 140

τ(Ω+cc) = 180Ξ+cc

Ω+cc

ors gives fs, fs and fs [90]. Another work predicts fs and

fs [99]. It can be seen that there are stilllarge uncertainties in understanding the lifetimes of and . Since it is just an overall factor of the branchingfractions of individual initial particles, the values of thelifetimes could easily be changed in our results. No oneresult is preferred, but we have to use one of them; ourchoices are shown in Table 1.

Bcc→Bc

Γ(ρ0→ π+π−)Γ(K∗+→ K0π+) gρππ gρ0π+π− gρ±π±π0 gK∗+K0π+

S U(3) gK∗+K0π+ gρππ

gsK∗+K0π+ =

1√

2gρππ = 4.28

∼ ms−mu,d

ΛQCDS U(3)

u,dgK∗+K0π+ = 4.6 VPP

gK∗+K+η8=

3√

6gK∗+K0π+ = 5.63

The masses and decay constants of all the final statehadrons come from Refs. [4, 103, 104]. The heavy-lighttransition form factors of have been calculatedusing several methods. The results of form factors fromthe light-front quark model [72] have been successfullyused in the prediction of the discovery channels in Ref.[16], and are also used in this work. The strong couplingconstants of the various hadrons are also important in-puts. Some of these can be extracted from the experi-mental decay widths, such as from and

: ( , ) and are re-spectively determined as 6.05 and 4.6 [58]. Under the fla-vor symmetry, can be related to , with

. It deviates from the extracted

value of 4.6 by about 7% , which is the flavor breaking effect, mainly caused by the mass differ-

ence of the s and quarks. In our calculation, we take and relate any other coupling with

strange mesons participating to this value, e.g.

. For coupling constants of

the singly heavy baryons coupled with light mesons, weadopt the theoretical calculation results from Refs. [105-110]. A simple calculation of the uncertainty of the strongcoupling constants is about 30%, caused by the QCD sumrules [105, 110]. All the strong coupling constants whichappear are gathered in Appendix A.

ηB. Dependence on and its cancellation by ratio ofbranching fractions

The form factor in Eq. (17) is introduced into the

ΛQCDn = 1 ΛQCD = 330

η F(k2,m2exe,η) k2 mexe

η

η

η

amplitude of triangle diagrams, to describe the off-shelleffect of the exchanged particles. When the exponentialfactor and are fixed with specified values (in ourcase, we take and MeV), the formfactor is a function of three variables, i.e. the momentumsquared, the mass of the exchanged particle, and the para-meter , . (varying in a range) and are definite for any individual diagram. The remainingparameter, , however, is a process-dependent parameterwhich cannot be calculated from first-principles QCDmethods. The value of is usually determined by experi-mental data, as in Ref. [58]. In the case of the doublycharmed baryon decays, without any available data, it canbe expected that will cause large uncertainties in thetheoretical predictions of the branching fractions.

η Ξ++cc → Σ++c K0

Ξ+cc→ Λ+c K0

η η

1.0 2.0

In Fig. 4(a), we display the dependence of the branch-ing fractions on the parameter , taking and as examples, which are both dominatedby the long-distance dynamics. It is clear that the branch-ing fractions are very sensitive to the value of . With varying in the range between and , the branchingfractions could be changed by nearly one order of mag-nitude. This is a well-known feature of FSI effects, whichhave very large theoretical uncertainties.

η

Ξ++cc → Σ++c K0

Ξ+cc→ Λ+c K0

η

η

η

Ξ++cc → Σ++c K0

0.59×10−3 4.06×10−3 η

Ξ+cc→ Λ+c K0

0.29×10−3 2.15×10−3

(2.33±1.74)×10−3 (1.22±0.93)×10−3

The problem of large uncertainty induced by varyingthe value of can be solved by the ratio of branchingfractions, first proposed in Ref. [16]. From Fig. 4(a), wecan easily find that the dependences of the branching ra-tios of and on have similarline shapes. That means that the dependence of the ratioof the two branching fractions on can be mostly can-celed. Therefore, the ratio of branching fractions will beinsensitive to the value of , as Fig. 4(b) shows. FromFig. 4(a), the branching fraction of changesfrom to with the parameter vary-ing from 1.0 to 2.0. Similarly, for , it changesfrom to . The branching fractionsof both processes vary by nearly one order of magnitude.Taking the central values of the above results, they are

and , respectively,with the uncertainties around 75%. However, the ratio of

Ξ++cc → Σ++c K0

Ξ+cc→ Λ+c K0

η

Fig. 4. (color online) (a) Theoretical predictions for the branching ratios of and in logarithmic coordinates;(b) ratio of branching fractions with varying from 1.0 to 2.0.

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-8

0.48 ∼ 0.530.51±0.03

η

Ξ++cc → Σ++c K0

Ξ+cc→ Λ+c K0

S U(3)η

Ξ++cc

η

the above branching fractions is , seen in Fig.4(b). Written as , the uncertainty of the ratio is5%. Therefore, almost 70% of the uncertainties of the ab-solute branching fractions are cancelled by the ratio ofbranching fractions. Similar cancelation behavior alsohappens between all other decay channels. The reason forthis cancelation is as follows. The difference in of thetriangle diagrams for the two modes of and

mainly stems from the strong coupling con-stants and the masses of the particles in the rescatteringdiagrams. The corresponding quantities for different de-cay modes can be related under the flavor sym-metry. The dependence on should be similar for boththese modes, and can thus be mostly cancelled in the ra-tio of branching fractions. In this way, the theoretical un-certainties can be brought under control. That is why wecould successfully predict the discovery channels of in Ref. [16]. In this work, we will present the results ofbranching fractions with different values of , to showthe absolute uncertainty of any individual mode. Thereader can take a ratio between any two processes to get amore reliable result.

Note that the above cancellation can be broken downby the so-called triangle singularity, in which all the in-

η

η

B→ Dπ B→ ππ

η

termediate particles are on-shell. We will not tackle thisissue in this work. Besides, as discussed in Ref. [58], thevalues of are not identical for different classes of decaymodes. For example, the values of are different for

and decays, to satisfy the constraintsfrom the experimental data. This can be easily under-stood in that the exchanged particles are different in dif-ferent decay modes. Therefore, we should be careful toconsider the values of for different classes of decayprocesses.

C. Numerical results of branching fractions

Bcc→BcP

c→ sudV∗csVud

c→ dud c→ susV∗cdVud V∗csVus

c→ dus

We list all the branching ratios of the decay modes in Tables 2-5. The short-distance dynamics

dominant channels (with T topology) are presented inTable 2. For the long-distance-dominated processes, thenumerical results are classified into three groups accord-ing to the CKM matrix elements: (a) the Cabibbo-favored(CF) decays induced by (with the CKM element

) are listed in Table 3; (b) the singly Cabibbo-sup-pressed (SCS) decays induced by or (with the CKM element or ) are listed inTable 4; and (c) the doubly Cabibbo-suppressed (DCS)decays induced by (with the CKM element

Table 2. Branching ratios for the short-distance dynamics dominated modes. "CF", "SCS" and "DCS" represent CKM favored, singlyCKM suppressed and doubly CKM suppressed processes, respectively.

Particle Decay mode Topology BRTS D (%) BRη=1.0 (%) BRη=1.5 (%) BRη=2.0 (%) CKM

Ξ++cc → Ξ+c π+ λsd(T +C′) 6.76 7.11 8.48 10.75 CF

→ Ξ′+c π+ 1/√

2λsd(T + C′

) 4.71 4.72 4.72 4.74 CF

→ Σ+c π+ 1/√

2λd(T + C′

) 0.248 0.251 0.255 0.261 SCS

→ Λ+c π+ λd(T +C′) 0.386 0.390 0.393 0.396 SCS

→ Ξ′+c K+ 1/√

2λs(T + C′

) 0.303 0.304 0.304 0.305 SCS

→ Ξ+c K+ λs(T +C′) 0.538 0.538 0.538 0.538 SCS

→ Λ+c K+ λds(T +C′) 0.028 0.029 0.030 0.031 DCS

→ Σ+c K+ 1/√

2λds(T + C′

) 0.016 0.016 0.018 0.021 DCS

Ξ+cc → Ξ0cπ+ λsd(T −E2) 4.08 4.34 4.58 4.74 CF

→ Ξ′0c π+ 1/√

2λsd(T + E2

) 2.84 2.84 2.84 2.84 CF

→ Σ0cπ+ λd(T + E2) 0.31 0.32 0.33 0.35 SCS

→ Ξ′0c K+ 1/√

2(λsT +λd E2

) 0.18 0.18 0.18 0.18 SCS

→ Ξ0c K+ λsT +λdE2 0.32 0.32 0.33 0.33 SCS

→ Σ0c K+ λdsT 0.02 DCS

Ω+cc →Ω0cπ+ λsdT 6.09 CF

→ Ξ0cπ+ −λdT −λsE2 0.23 0.23 0.23 0.23 SCS

→ Ξ′0c π+ 1/√

2(λdT +λsE2

) 0.16 0.16 0.17 0.17 SCS

→Ω0c K+ λs

(T + E2

) 0.38 0.40 0.40 0.40 SCS

→ Ξ0c K+ λds(−T +E2) 0.019 0.091 0.091 0.091 DCS

→ Ξ′0c K+ 1/√

2λds(T + E2

) 0.011 0.011 0.011 0.011 DCS

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-9

λsdTable 3. Branching ratios for the long-distance-dominated Cabibbo-favored ( ) modes. For the channels involving internal W-emis-sion contributions, the short-distance factorizable contributions are also listed in the fourth column, for comparison.

Particles Decay modes Topology BRTS D (×10−3) BRη=1.0(×10−3) BRη=1.5(×10−3) BRη=2.0(×10−3)

Ξ++cc → Σ++c K0

C 0.015 0.59 1.91 4.06

Ξ+cc →Ω0c K+ E2 0.049 0.15 0.29

→ Σ+c K0 1/

√2(C+ E1

) 0.009 1.55 4.72 9.95

→ Λ+c K0 −C+E1 0.017 0.29 0.98 2.15

→ Σ++c K− E1 0.075 0.26 0.55

→ Ξ′+c π0 1/2(−C′ + E2) 0.33 0.98 1.97

→ Ξ′+c η1 1/√

6(C′ + E1 + E2) 0.57 1.73 3.50

→ Ξ′+c η8 1/2√

3(C′ −2E1 + E2) 0.22 0.66 1.37

→ Ξ+c π0 −1/√

2(C′ +E2) 3.24 10.2 21.0

→ Ξ+c η1 1/√

3(C′ +E1 −E2) 0.18 0.57 1.20

→ Ξ+c η8 1/√

6(C′ −2E1 −E2) 0.11 0.35 0.66

Ω+cc → Ξ′+c K0 1/

√2(C+ C′) 0.010 1.10 3.38 6.84

→ Ξ+c K0 −C+C′ 0.017 0.73 2.30 4.38

Table 4. Same as Table 3 but for the long-distance-dominated singly Cabibbo-suppressed modes.

Particles Decay modes Topology BRTS D (×10−5) BRη=1.0(×10−5) BRη=1.5(×10−5) BRη=2.0(×10−5)

Ξ++cc → Σ++c π0 −1/

√2C 0.062 3.39 11.5 25.4

→ Σ++c η1 1/√

3(λd +λs)C 0.022 1.18 2.43 4.35

→ Σ++c η8 1/√

6(λd −2λs)C 0.059 3.24 8.62 15.3

Ξ+cc → Σ+c π0 1/2λd(− C− C′ + E1 + E2

) 0.038 3.85 12.1 25.0

→ Σ+c η1 1/√

6[λd

(C+ C′ + E1 + E2

)+λsC

] 0.026 2.52 7.69 16.4

→ Σ+c η8 1/2√

3[λd

(C+ C′ + E1 + E2

)−2λsC] 0.016 1.64 4.52 8.70

→ Λ+c π0 1/√

2λd(C−C′ −E1 −E2) 0.057 1.53 4.76 9.79

→ Λ+c η1 1/√

3[λd

(C′ −C+E1 −E2

)−λsC] 0.031 0.84 1.90 4.07

→ Λ+c η8 1/√

6[λd

(C′ −C+E1 −E2

)+2λsC

] 0.020 0.55 0.93 1.59

→ Σ++c π− λd E1 0.38 1.18 2.46

→ Ξ′+c K0 1/√

2(λsC′ +λd E1

) 1.17 3.79 8.00

→ Ξ+c K0 λsC′ +λdE1 2.77 8.75 18.1

Ω+cc → Σ+c K0 1/

√2(λdC′ +λsE1

) 0.26 0.75 1.48

→ Λ+c K0 λdC′ +λsE1 0.52 1.66 3.49

→ Σ++c K− λsE1 0.23 0.77 1.69

→ Ξ′+c π0 1/2(−λdC+λsE2

) 0.38 1.48 3.61

→ Ξ′+c η1 1/√

6[λs

(C+ C′ + E1 + E2

)+λdC

] 0.49 1.71 3.45

→ Ξ′+c η8 1/2√

3[λs

(E2 −2C−2C′ −2E1

)+λdC

] 2.04 6.26 13.0

→ Ξ+c π0 1/√

2(λdC−λsE2) 4.50 13.2 26.0

→ Ξ+c η1 1/√

3[λs

(C′ −C+E1 −E2

)−λdC] 11.3 37.3 77.3

→ Ξ+c η8 1/√

6[λs

(2C−2C′ +2E1 −E2

)−λdC] 6.66 21.9 45.5

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-10

V∗cdVus

tilde T

) are given in Table 5. The topological amplitudesfor the channels with the sextet single charmed baryonsare distinguished from the anti-triplet baryons by addinga , e.g. .

C′ E2η 2.0

C′ E2

CS D

a2(µ) a2(mc) = −0.017a1(mc) = 1.07

In Table 2, it is clear that the factorizable short-dis-tance contributions of diagram amplitude T are dominantrelative to the long-distance contributions of and .When the parameter tends to , the long-distance con-tributions of and also have a visible or comparableeffect. On the other hand, from Tables 3-5, the long-dis-tance dynamics dominates the decay modes in Table 3 to5, since the only calculable short-distance amplitude is heavily suppressed by the effective Wilson coefficient

at the charm mass scale, i.e. ismuch smaller than . The latter is used at theweak decay vertex of the triangle diagram in our calcula-tions.

D. Discussions on the topological diagrams

|C||T | ∼

|C′||C| ∼

|E1||C| ∼

|E2||C| ∼ O

ΛhQCD

mQ

ΛhQCD/mc ∼ 1

The topological diagrams have the relations of

in heavy baryon de-

cays, manifested by the soft-collinear effective theory[64, 65]. These relations are important in phenomenolo-gical studies on the searches for double-heavy-flavor ba-ryons, and give us more hints on the dynamics of heavybaryon decays. From the above relations, all the tree-leveltopological diagrams are at the same order in charmed ba-ryon decays, due to . It would be very use-ful to numerically test these relations in our framework.

Ξ++cc → Σ++c K0 (

C)

Ξ++cc → Ξ′+c π+(1/√

2(T + C′

))CLD/TS D C′LD/CLD

From the amplitudes of and , we obtain the ratios

and :

|CLD||TS D|

=|A(Ξ++cc → Σ++c K

0)|LD√

2|A(Ξ++cc → Ξ′+c π+)|S D= 0.21 ∼ 0.55, (26)

|C′LD||CLD|

=

√2|A(Ξ++cc → Ξ′+c π+)|LD

|A(Ξ++cc → Σ++c K0)|LD

= 1.33 ∼ 1.45. (27)

C′LD/CLD

Ξ++cc → Σ++c π0 (−1/

√2C

)Ξ++cc → Σ+c π+(

1/√

2(T + C′

))Ξ++cc → Ξ′+c K+

(1/√

2(T + C′

))Ξ++cc →

Σ++c K0 (C)Ξ++cc → Σ+c K+

(1/√

2(T + C′

))The ratio can also be calculated by the de-

cay channels , , and

, as:

|C′LD||CLD|

=|A(Ξ++cc → Σ+c π+)|LD

|A(Ξ++cc → Σ++c π0)|LD= 0.72 ∼ 0.88, (28)

|C′LD||CLD|

=|A(Ξ++cc → Ξ′+c K+)|LD

|A(Ξ++cc → Σ++c π0)|LD= 0.69 ∼ 0.85, (29)

|C′LD||CLD|

=

√2|A(Ξ++cc → Σ+c K+)|LD

|A(Ξ++cc → Σ++c K0)|LD= 0.98 ∼ 1.24. (30)

E1 E2

In the following, to consider the relations between C, and , it would be more convenient to study the pro-

cesses with a pure topological diagram. The advantage ofavoiding interference between diagrams is that they couldbe directly determined by experimental data in the future.

Ξ+cc→ Σ++c K−(E1

)Ξ+cc→Ω0

c K+(E2

)Ξ++cc → Σ++c K

0(C)

E1LD E2LD CLD

In Table 3, by the three single amplitude channels , and

, the ratios among the amplitudes , and can easily be calculated, giving:

Table 5. Same as Table 3 but for the long-distance-dominated doubly Cabibbo-suppressed modes.

Particles Decay modes Topology BRTS D (×10−6) BRη=1.0(×10−6) BRη=1.5(×10−6) BRη=2.0(×10−6)

Ξ++cc → Σ++c K0 C 0.043 1.31 4.69 10.75

Ξ+cc → Σ+c K0 1/√

2(C+ C′

) 0.035 4.52 16.0 36.4

→ Λ+c K0 −C+C′ 0.05 2.39 8.88 21.0

Ω+cc → Σ+c π0 1/2(−E1 + E2) 0.07 0.23 0.52

→ Σ+c η1 1/√

6(C′ + E1 + E2

) 0.14 0.45 0.89

→ Σ+c η8 1/2√

3(−2C′ + E1 + E2

) 0.09 0.30 0.66

→ Λ+c π0 −1/√

2(E1 +E2) 0.19 0.61 1.34

→ Λ+c η1 1/√

3(C′ +E1 −E2) 0.59 1.89 4.13

→ Λ+c η8 −1/√

6(2C′ −E1 +E2) 0.28 0.89 1.97

→ Σ0cπ+ E2 0.14 0.45 0.91

→ Σ++c π− E1 0.16 0.59 1.31

→ Ξ′+c K0 1/√

2(C+ E1

) 0.028 9.68 31.9 68.0

→ Ξ+c K0 −C+E1 0.045 0.82 2.84 6.54

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-11

|E1LD||CLD|

=|A(Ξ+cc→ Σ++c K−)|LD

|A(Ξ++cc → Σ++c K0)|LD

= 0.45 ∼ 0.46, (31)

|E2LD||CLD|

=|A(Ξ+cc→Ω0

c K+)|LD

|A(Ξ++cc → Σ++c K0)|LD

= 0.38 ∼ 0.40, (32)

|E1LD||E2LD|

=|A(Ξ+cc→ Σ++c K−)|LD

|A(Ξ+cc→Ω0c K+)|LD

= 1.12 ∼ 1.24. (33)

E1LD/CS D E2LD/CS D

E1LD

CS D

Ξ++cc → Σ++c π0 (−1/

√2C

)Ξ+cc→Σ++c π

− (E1)

Ω+cc→ Σ++c K−(E1

)From above, the ratios and have

similar values, at first order. The ratios between and can also be calculated from Table 4, i.e. from the

modes , and:

|E1LD||CLD|

=|A(Ξ+cc→ Σ++c π

−)|LD√2|A(Ξ++cc → Σ++c π0)|LD

= 0.28 ∼ 0.26, (34)

|E1LD||CLD|

=|A(Ω+cc→ Σ++c K−)|LD√2|A(Ξ++cc → Σ++c π0)|LD

= 0.29 ∼ 0.30. (35)

Ξ++cc → Σ++c K0 (C)Ω+cc→ Σ0

cπ+ (E2

)Ω+cc→ Σ++c π

− (E1)We can verify the results using the data from Table 5.

By the three channels , and , we get the ratios as:

|E2LD||CLD|

=|A(Ω+cc→ Σ0

cπ+)|LD

|A(Ξ++cc → Σ++c K0)|LD= 0.34 ∼ 0.37, (36)

|E1LD||CLD|

=|A(Ω+cc→ Σ++c π

−)|LD

|A(Ξ++cc → Σ++c K0)|LD= 0.40 ∼ 0.41. (37)

From Eqs. (26-37), considering the relatively largeparameter uncertainty, all these results are consistent withthe relations found in [64, 65],

|C||T | ∼

|C′||C| ∼

|E1||C| ∼

|E2||C| ∼ O

ΛhQCD

mc

∼ O(1). (38)

S U(3)Some results in Eqs. (26-37) are different from each

other. This can be understood by the flavor break-ing effects, shown in the following subsection.

Ξ+cc→ Σ++c π−

S U(3)

|Ad | = 2.6×10−7 |As| = 1.0×10−7 |Ad+s| =1.5×10−7

S U(3)

Λc

As mentioned in Sec. II A, the long-distance quark-loop diagrams are also taken into account in our calcula-tions. Taking as an example, the quark-looptopological diagrams and the corresponding hadronic tri-angle diagrams are shown in Fig. 5. It can be expectedthat any individual loop diagram is as large as the treediagrams. However, the d-quark and s-quark loop dia-grams are mostly cancelled due to the GIM mechanism.Therefore, the total contribution of quark loop diagramscomes from the flavor symmetry breaking effects.Numerically, the magnitudes of the d-quark loop, s-quarkloop and the sum of the both loop diagrams are

GeV, GeV, GeV, respectively. It can be seen that the long-

distance dynamics contributes the relatively large breaking effect. It is difficult to test this effect in thedoubly charmed baryons in experiments. We will invest-igate the long-distance quark-loop contributions in the Dmeson and decays in a future study.

S U(3)E. Discussions on flavor symmetryand its breaking

S U(3)

S U(3)

Flavor symmetry is very significant in theweak decays of heavy hadrons. In terms of a few SU(3)irreducible amplitudes, a number of relations between thewidths of doubly charmed baryon decays are obtained[73]. It is important to numerically test the flavor symmetry and its breaking effects.

S U(3)S U(3)

In the following, we show our numerical results forthe ratios of decay widths. In the limit, they shouldbe unity. Any deviation would indicate breakingeffects.

Γ(Ξ++cc → Λ+c π+

)/Γ

(Ξ++cc → Ξ+c K+

)= |λd(T +C′)|2/|λs(T +C′)|2 = 0.73 ∼ 0.74, (39)

Γ(Ξ+cc→ Ξ+c K0)/Γ(Ω+cc→ Λ+c K

0)= |λsC′+λdE1|2/|λdC′+λsE1|2 = 6.67 ∼ 6.85, (40)

Γ(Ω+cc→ Ξ0

cπ+)/Γ(Ξ+cc→ Ξ0

c K+)= | −λdT −λsE2|2/|λsT +λdE2|2 = 0.54 ∼ 0.56, (41)

Ξ+cc→ Σ++c π−

Fig. 5. Quark-loop topological diagrams and their corres-ponding hadronic triangle diagrams for .

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-12

Γ(Ξ++cc → Σ++c π

0)/13Γ(Ξ++cc → Σ++c η8

)= | − 1

√2

C|2/| − 1√

6(λd −2λs)C|2 = 3.14 ∼ 4.98, (42)

Γ(Ξ++cc → Σ+c π+

)/Γ

(Ξ++cc → Ξ′+c K+

)= | 1√

2λd(T + C′)|2/| 1

√2λs(T + C′)|2 = 1.28 ∼ 1.30, (43)

Γ(Ξ+cc→ Σ++c π

−)/Γ(Ω+cc→ Σ++c K−)= |λd E1|2/|λsE1|2 = 1.87 ∼ 2.12, (44)

Γ(Ξ+cc→ Σ0

cπ+)/Γ(Ω+cc→Ω0

c K+)= |λd(T + E2)|2/|λs(T + E2)|2 = 1.03 ∼ 1.13, (45)

Γ(Ξ+cc→ Ξ′+c K0)/Γ(Ω+cc→ Σ+c K

0)= | 1√

2(λsC′+λd E1)|2/| 1

√2

(λdC′+λsE1)|2 = 5.79 ∼ 6.95, (46)

Γ(Ω+cc→ Ξ′0c π+

)/Γ

(Ξ+cc→ Ξ′0c K+

)= | 1√

2(λdT ′+λsE2|2/|

1√

2(λsT ′+λd E2)|2 = 0.69 ∼ 0.74. (47)

S U(3)

Γ(Ξ+cc→ Ξ+c K0)/

Γ(Ω+cc→ Λ+c K

0)= 6.67 ∼ 6.85

Ξ+cc→ Ξ+c K0

M(K+,Ξ0c ;ρ+) Ω+cc→ Λ+c K

0

M(π+,Ξ0c ; K∗+)

f1(Ξ+c → Ξ0cρ+) = 8.5

f2(Ξ+c → Ξ0cρ+) = 10.6 f1(Ξ0

c → Λ+c K∗−) = −4.6f2(Ξ0

c → Λ+c K∗−) = −6 Γ(Ξ+cc→ Ξ+c K0)

Γ(Ω+cc→ Λ+c K

0)Γ(Ξ+cc→ Ξ′+c K0)/

Γ(Ω+cc→ Σ+c K

0)= 5.79 ∼ 6.95

M(K+,Ξ0c ;ρ+) M(π+,Ξ0

c ; K∗+)f1(Ξ′+c → Ξ0

cρ+) = 2.1

f2(Ξ′+c → Ξ0cρ+) = 116 f1(Σ+c → Ξ0

c K∗+) = −2.2f2(Σ+c → Ξ0

c K∗+) = −13

From the above numerical results, it can be found thatthe long-distance final-state interactions can contribute tothe large breaking effect. It stems from the ex-changed particles, hadronic strong coupling constants,transition form factors, decay constants, and interferencebetween different diagrams. In our calculation of Eq. (40)and Eq. (46), the large values mainly stem from thestrong coupling constants. Taking

as an example, a factor of2.5 in the amplitude could account for the large ratio. Thedecay mode is dominated by the triangle dia-gram , while is dominated by

. To evaluate these two triangle diagrams,we need strong coupling constants ,

[105] and , [105]. Therefore, is

much larger than . As for , with the dominant triangle

diagrams and , the relevantstrong coupling constants are ,

[105] and , [105], respectively, which leads to

the large value in Eq. (46). All the values of strong coup-lings are taken from the literature. These non-perturbat-ive quantities actually have large theoretical uncertainties,and need to be carefully studied in the future.

IV. SUMMARY

Bcc→BcPBcc = (Ξ++cc ,Ξ

+cc,Ω

+cc)

Bc = (B3,B6)

In this work, we have introduced the whole theoretic-al framework of the rescattering mechanism by investig-ating the forty-nine two-body baryon decays ,where are the doubly charmed bary-ons, are the singly charmed baryons and

P = (π,K,η1,8)

S U(3)

are the light pseudoscalar mesons. It hasbeen interpreted in detail for the physical foundation ofthe rescattering mechanism at the hadron level. Further-more, as a self-consistent test of the rescattering mechan-ism, the relations of topological diagrams and flavor

symmetry have been discussed. The main pointsare the following:

(1)Bcc→BcP

η

We have provided theoretical predictions for thebranching ratios of all the decays considered,and discussed the dependence on the parameter .

(2) The numerical results of the branching ratiosshow the same conclusion as the charm meson decays:the non-factorizable long-distance contributions play animportant role in doubly charmed baryon decays.

(3)

|C||T | ∼

|C′||C| ∼

|E1||C| ∼

|E2||C| ∼ O

ΛhQCD

mc

∼ O(1)

We have obtained the same counting rules as theanalysis in SCET for the topological amplitudes in charm

decays, that is ,

which will be significant guidance for further studies ofcharmed baryon decays.

(4) S U(3)S U(3)

Large symmetry breaking effects are ob-tained in our method. More studies on the break-ing effects of the doubly charmed baryon decays areneeded in the future.

ACKNOWLEDGEMENTS

We are grateful to all our collaborators in this seriesof studies on the doubly heavy baryons. Especially, weare grateful to Cai-Dian Lü, Run-Hui Li, Wei Wang,Zhen-Xing Zhao and Zhi-Tian Zou for collaboration ontheoretical work, and to Yuan-Ning Gao, Ji-Bo He andYan-Xi Zhang for discussions on the experimentalsearches.

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-13

APPENDIX A: EFFECTIVE LAGRANGIANS

The effective Lagrangians used in the rescattering

mechanism are those given in Refs. [105-110]:

LVPP =igVPP√

2Tr[Vµ[P,∂µP]], (A1)

LVVV =igVVV√

2Tr[(∂νVµVµ−Vµ∂νVµ)Vν], (A2)

LPB6 B6= gPB6 B6

Tr[B6iγ5PB6], (A3)

LPB3 B3= gPB3 B3

Tr[B3iγ5PB3], (A4)

LPB6 B3= gPB6 B3

Tr[B6iγ5PB3]+h.c., (A5)

LVB6 B6= f1VB6 B6

Tr[B6γµVµB6]

+f2PB6 B6

2m6Tr[B6σµν∂

µVνB6], (A6)

LVB3 B3= f1PB3 B3

Tr[B3γµVµB3]

+f2PB3 B3

2m3Tr[B3σµν∂

µVνB3], (A7)

LVB6 B3= f1VB6 B3

Tr[B6γµVµB3]

+f2VB6 B3

m6+m3Tr[B6σµν∂

µVνB3]+h.c., (A8)

P(JP = 0−) =

π0

√2+η8√

6π+ K+

π− − π0

√2+η8√

6K0

K− K0 −√

23η8

+

1√

3

η1 0 00 η1 00 0 η1

, (A9)

V(JP = 1−) =

ρ0

√2+ω√

2ρ+ K∗+

ρ− − ρ0

√2+ω√

2K∗0

K∗− K∗0 ϕ

,

(A10)

B6

(JP =

12

+)=

Σ++cΣ+c√

2

Ξ′+c√2

Σ+c√2Σ0

cΞ′0c√

2

Ξ′+c√2

Ξ′0c√2Ω0

c

,

B3

(JP =

12

+)=

0 Λ+c Ξ+c

−Λ+c 0 Ξ0c

−Ξ+c −Ξ0c 0

. (A11)

Strong coupling constants are collected in Tables A1,A2 and A3.

APPENDIX B: EXPRESSIONS OF AMPLITUDES

Bcc→BcPThe expressions of amplitudes for all forty-seven

decays considered in this paper are as follows:

Table A1. Strong coupling constants of VPP and VVV vertices.

Vertex g Vertex g Vertex g Vertex g Vertex g

ρ+→ π0π+ 6.05 ρ0→ π+π− 6.05 ρ+→ K+K0 4.60 ρ0→ K0K

0 −3.25 ρ0→ K+K− 3.25

ϕ→ K−K+ 4.60 K∗0→ η8K

0 5.63 K∗0→ K−π+ 4.60 K

∗0→ K0π0 −3.25 K∗+→ K+π0 3.25

K∗+→ η8K+ 5.63 K∗+→ π+K0 4.60 K∗0→ π−K+ 4.60 K∗0→ K0η8 5.63 K∗0→ π0K0 −3.25

ω→ K+K− 3.25 ϕ→ K0K0 4.60 ω→ K0K

0 3.25

ρ+→ ρ0ρ+ 7.38 ρ0→ ρ−ρ+ 7.38 ρ+→ K∗+K∗0 5.22 ρ0→ K∗+K∗− 3.69 ω→ K∗+K∗− 3.69

K∗0→ ϕK

∗0 5.22 K∗0→ K

∗0ρ0 −3.69 K

∗0→ K∗0ω 3.69 K∗+→ ρ+K∗0 5.22 K∗+→ ϕK∗+ 5.22

K∗+→ ωK∗+ 3.69 K∗0→ ρ0K∗0 −3.69 K∗0→ ωK∗0 3.69 K∗0→ K∗0ϕ 5.22 ϕ→ K∗−K∗+ 5.22

ω→ K∗0K∗0 3.69 ϕ→ K

∗0K∗0 5.22 ρ0→ K∗0K

∗0 −3.69 K∗+→ K∗+ρ0 3.69 K∗0→ K∗−ρ+ 5.22

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-14

PB3B3 PB3B6 PB6B6Table A2. Strong coupling constants of , and vertices.

Vertex g Vertex g Vertex g Vertex g Vertex g

Ξ+c → Λ+c K0 0.9 Λ+c → Ξ+c K0 0.9 Ξ+c → Ξ+c η8 −0.7 Λ+c → Λ+c η8 0.81 Ξ0

c → Λ+c K− −0.9

Ξ0c → Ξ0

cη8 −0.7 Ξ0c → Ξ+c π− 0.99 Ξ+c → Ξ0

cπ+ 0.99 Ξ0

c → Ξ0cπ

0 −0.7 Ξ+c → Ξ+c π0 0.7

Σ+c → Ξ+c K0 −5.0 Ξ+c → Σ++c K− −7.1 Σ++c → Ξ+c K+ −7.1 Ξ+c → Ξ′+c η8 5.4 Ξ′+c → Ξ+c η8 5.4

Σ+c → Λ+c π0 6.5 Λ+c → Σ++c π− −6.5 Σ++c → Λ+c π+ −6.5 Λ+c → Σ0

cπ+ 6.5 Σ0

c → Λ+c π− 6.5

Ξ′+c → Λ+c K0 −4.6 Λ+c → Ξ′0c K+ 4.6 Ξ′0c → Λ+c K− 4.4 Ξ0

c → Σ+c K− −5.0 Σ+c → Ξ0c K+ −5.0

Σ0c → Ξ0

c K0 −7.1 Ξ0c → Ξ′0c η8 5.4 Ξ′0c → Ξ0

cη8 5.4 Ξ0c →Ω0

c K0 6.5 Ω0c → Ξ0

c K0 6.5

Ξ′+c → Ξ0cπ+ 4.4 Ξ0

c → Ξ′0c π0 3.1 Ξ′0c → Ξ0cπ

0 3.1 Ξ+c →Ω0c K+ 6.5 Ω0

c → Ξ+c K− 6.5

Ξ′+c → Ξ+c π0 3.1 Ξ+c → Ξ′0c π+ 4.4 Ξ′0c → Ξ+c π− 4.4 Ξ′+c → Σ+c K0 6.4 Σ+c → Ξ′+c K0 6.4

Σ++c → Ξ′+c K+ 9.0 Ξ′+c → Ξ′+c η8 −2.3 Σ+c → Σ+c η8 4.6 Σ+c → Σ++c π− 8.0 Σ++c → Σ+c π+ 8.0

Σ++c → Σ++c π0 8.0 Ξ′0c → Σ+c K− 6.4 Σ+c → Ξ′0c K+ 6.4 Ξ′0c → Σ0

c K0 9.0 Σ0

c → Ξ′0c K0 9.0

Ω0c →Ω0

cη8 −10.4 Ω0c → Ξ′+c K− 9.0 Ξ′+c →Ω0

c K+ 9.0 Ω0c → Ξ′0c K

0 9 Ξ′−b →Ω0c K0 9

Σ+c → Σ0cπ+ 8.0 Σ0

c → Σ0cη8 4.6 Σ0

c → Σ0cπ

0 −8.0 Ξ′0c → Ξ′+c π− 5.7 Ξ′+c → Ξ′0c π+ 5.7

Ξ′+c → Ξ′+c π0 4.0 Λ+c → Ξ0c K+ −0.9 Ξ+c → Σ+c K

0 −5.0 Λ+c → Σ+c π0 6.5 Λ+c → Ξ′+c K0 −4.6

Ξ0c → Σ0

c K0 −6.5 Ξ0

c → Ξ′+c π− 4.4 Ξ+c → Ξ′+c π0 3.1 Ξ′+c → Σ++c K− 9.0 Σ++c → Σ++c η8 4.6

Ξ′0c → Ξ′0c η8 −2.3 Σ0c → Σ+c π− 6.5 Ξ′0c → Ξ′0c π0 −4.0 Ξ+c → Ξ+c η1 0.07 Λ+c → Λ+c η1 0.75

Ξ0c → Ξ0

cη1 0.07 Ξ′+c → Ξ′+c η1 2.6 Σ+c → Σ+c η1 2.6 Σ++c → Σ++c η1 2.6 Ξ′0c → Ξ′0c η1 2.6

Ω0c →Ω0

cη1 11.0 Σ0c → Σ0

cη1 2.6

VB3B3 VB3B6 VB6B6Table A3. Strong coupling constants of , and vertices.

Vertex f1 f2 Vertex f1 f2 Vertex f1 f2 Vertex f1 f2

Λ+c → Λ+cω 4.9 6 Λ+c → Ξ+c K∗0 4.6 6 Ξ+c → Λ+c K∗0 4.6 6 Ξ+c → Ξ+c ϕ 4.6 6

Ξ0c → Λ+c K∗− −4.6 −6 Ξ0

c → Ξ0cϕ 4.6 16 Ξ0

c → Ξ+c ρ− 8.5 10.6 Ξ+c → Ξ0cρ+ 8.5 10.6

Ξ0c → Ξ0

cρ0 −6 −7.5 Ξ+c → Ξ+cω 5.5 7.5 Ξ+c → Ξ+c ρ0 6 7.5 Λ+c → Σ+c ρ0 2.6 16

Λ+c → Σ++c ρ− −2.6 −16 Σ++c → Λ+c ρ+ −2.6 −16 Λ+c → Ξ′+c K∗0 −2.3 −14 Ξ′+c → Λ+c K

∗0 −2.3 −14

Σ+c → Ξ+c K∗0 −2.2 −13 Ξ+c → Σ++c K∗− −3.1 −18.4 Σ++c → Ξ+c K∗+ −3.1 −18.4 Ξ+c → Ξ′+c ϕ −2.1 −13

Λ+c → Σ0cρ+ 2.6 16 Σ0

c → Λ+c ρ− 2.6 16 Λ+c → Ξ′0c K∗+ 2.3 14.1 Ξ′0c → Λ+c K∗− 2.3 14.1

Σ+c → Ξ0c K∗+ −2.2 −13 Ξ0

c → Σ0c K∗0 −2.2 −13 Σ0

c → Ξ0c K∗0 −2.2 −13 Ξ0

c → Ξ′0c ϕ −2.1 −13

Ξ0c →Ω0

c K∗0 3.3 20 Ω0c → Ξ0

c K∗0 3.3 20 Ξ0

c → Ξ′+c ρ− 2.1 115.6 Ξ′+c → Ξ0cρ+ 2.1 115.6

Ξ′0c → Ξ0cω 1.2 8 Ξ0

c → Ξ′0c ρ0 −1.5 −11 Ξ′0c → Ξ0cρ

0 −1.5 −11 Ξ+c →Ω0c K∗+ 3.3 20

Ξ+c → Ξ′0c ρ+ 2.1 15.6 Ξ′−b → Ξ+c ρ− 2.1 15.6 Ξ+c → Ξ′+c ω 1.5 11 Ξ′+c → Ξ+cω 1.5 11

Ξ′+c → Ξ+c ρ0 1.5 11.0 Σ+c → Σ+cω 3.5 24 Σ+c → Σ′+b ρ− 4.0 27.0 Σ++c → Σ+c ρ+ 4 27

Ξ′+c → Σ+c K∗0 3.5 21.2 Σ++c → Σ++c ω 3.5 24 Σ++c → Σ++c ρ

0 4 27 Σ++c → Ξ′+c K∗+ 5 30

Ξ′+c → Ξ′+c ϕ 4 21 Ω0c →Ω0

cϕ 11 52 Ω0c → Ξ′+c K∗− 7 35 Ξ′+c →Ω0

c K∗+ 7 35

Ξ′0c →Ω0c K∗0 7 35 Σ0

c → Σ+c ρ− 4 27 Σ+c → Σ0cρ+ 4 27 Σ0

c → Σ0cω 3.5 24

Σ0c → Ξ′0c K∗0 5 30 Ξ′0c → Σ0

c K∗0 5 30 Σ+c → Ξ′0c K∗+ 3.5 21.2 Ξ′0c → Σ+c K∗− 3.5 21.2

Ξ′0c Ξ′+c ρ− 3.5 22.6 Ξ′+c → Ξ′0c ρ+ 3.5 22.6 Ξ′0c → Ξ′0c ω 2.4 15 Ξ′0c → Ξ′0c ρ0 −2.5 −16

Ξ′+c → Ξ′+c ρ0 2.5 16 Λ+c → Ξ0c K∗+ −4.6 −6 Ξ0

c → Ξ0cω 5.5 7.5 Σ+c → Λ+c ρ0 2.6 16

Ξ+c → Σ+c K∗0 −2.2 −13 Ξ′+c → Ξ+c ϕ −2.1 −13 Ξ0

c → Σ+c K∗− −2.2 −13 Ξ′0c → Ξ0cϕ −2.1 −13

Ξ0c → Ξ′0c ω 1.2 8 Ω0

c → Ξ+c K∗− 3.5 20 Ξ+c → Ξ′+c ρ0 1.2 8 Σ+c → Ξ′+c K∗0 3.5 21.2

Ξ′+c → Σ++c K∗− 5.0 30.0 Ω0c → Ξ′0c K

∗0 5 30 Σ0c → Σ0

cρ0 −4 −27 Ξ′0c → Ξ′0c ϕ 4 21

Ξ′+c → Ξ′+c ω 2.4 15

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-15

A(Ξ++cc → Λ+c K+) =T (Ξ++cc → Λ+c K+)+M(K+,Λ+c ;ω)+M(K+,Σ+c ;ρ0)+M(K∗+,Λ+c ;η8)+M(K∗+,Σ+c ;π0)

+M(K+,Λ+c ;Ξ0c)+M(K+,Λ+c ;Ξ′0c )+M(K+,Σ+c ;Ξ0

c)+M(K+,Σ+c ;Ξ′0c )+M(K∗+,Λ+c ;Ξ0c)

+M(K∗+,Λ+c ;Ξ′0c )+M(K∗+,Σ+c ;Ξ0c)+M(K∗+,Σ+c ;Ξ′0c ), (B1)

A(Ξ++cc → Λ+c π+) =T (Ξ++cc → Λ+c π+)+M(π+,Σ+c ;ρ0)+M(ρ+,Σ+c ;π0)+M(K+,Ξ+c ; K∗0)+M(K+,Ξ′+c ; K∗0)

+M(K∗+,Ξ+c ; K0)+M(K∗+,Ξ′+c ; K0)+M(π+,Λ+c ;Σ0c)+M(π+,Σ+c ;Σ0

c)+M(ρ+,Λ+c ;Σ0c)

+M(ρ+,Σ+c ;Σ0c)+M(K+,Ξ+c ;Ξ0

c)+M(K+,Ξ+c ;Ξ′0c )+M(K+,Ξ′+c ;Ξ0c)+M(K+,Ξ′+c ;Ξ′0c )

+M(K∗+,Ξ+c ;Ξ0c)+M(K∗+,Ξ+c ;Ξ′0c )+M(K∗+,Ξ′+c ;Ξ0

c)+M(K∗+,Ξ′+c ;Ξ′0c ), (B2)

A(Ξ++cc → Ξ+c K+) =T (Ξ++cc → Ξ+c K+)+M(K+,Ξ+c ;ρ0)+M(K+,Ξ+c ;ω)+M(K+,Ξ′+c ;ρ0)+M(K+,Ξ′+c ;ω)

+M(K∗+,Ξ+c ;π0)+M(K∗+,Ξ+c ;η8)+M(K∗+,Ξ′+c ;π0)+M(K∗+,Ξ′+c ;η8)+M(π+,Λ+c ; K∗0

)

+M(π+,Σ+c ; K∗0

)+M(ρ+,Λ+c ; K0)+M(ρ+,Σ+c ; K

0)+M(K+,Ξ+c ;ϕ)+M(K+,Ξ′+c ;ϕ)

+M(K∗+,Ξ+c ;η8)+M(K∗+,Ξ′+c ;η8)+M(K+,Ξ+c ;Ω0c)+M(K+,Ξ′+c ;Ω0

c)+M(K∗+,Ξ+c ;Ω0c)

+M(K∗+,Ξ′+c ;Ω0c)+M(π+,Λ+c ;Ξ0

c)+M(π+,Λ+c ;Ξ′0c )+M(π+,Σ+c ;Ξ0c)+M(π+,Σ+c ;Ξ′0c )

+M(ρ+,Λ+c ;Ξ0c)+M(ρ+,Λ+c ;Ξ′0c )+M(ρ+,Σ+c ;Ξ0

c)+M(ρ+,Σ+c ;Ξ′0c ), (B3)

A(Ξ++cc → Ξ+c π+) =T (Ξ++cc → Ξ+c π+)+M(π+,Ξ+c ;ρ0)+M(π+,Ξ′+c ;ρ0)+M(ρ+,Ξ+c ;π0)+M(ρ+,Ξ′+c ;π0)

+M(π+,Ξ+c ;Ξ0c)+M(π+,Ξ+c ;Ξ′0c )+M(π+,Ξ′+c ;Ξ0

c)+M(π+,Ξ′+c ;Ξ′0c )+M(ρ+,Ξ+c ;Ξ0c)

+M(ρ+,Ξ+c ;Ξ′0c )+M(ρ+,Ξ′+c ;Ξ0c)+M(ρ+,Ξ′+c ;Ξ′0c ), (B4)

A(Ξ+cc→ Λ+c K0) =CS D(Ξ+cc→ Λ+c K

0)+M(π+,Ξ0

c ; K∗+)+M(π+,Ξ′0c ; K∗+)+M(ρ+,Ξ0c ; K+)+M(ρ+,Ξ′0c ; K+)

+M(π+,Ξ0c ;Σ0

c)+M(π+,Ξ′0c ;Σ0c)+M(ρ+,Ξ0

c ;Σ0c)+M(ρ+,Ξ′0c ;Σ0

c), (B5)

A(Ξ+cc→ Λ+c K0) =CS D(Ξ+cc→ Λ+c K0)+M(K+,Σ0c ;ρ+)+M(K∗+,Σ0

c ;π+)+M(K+,Σ0c ;Ξ0

c)+M(K+,Σ0c ;Ξ′0c )

+M(K∗+,Σ0c ;Ξ0

c)+M(K∗+,Σ0c ;Ξ′0c ), (B6)

A(Ξ+cc→ Λ+c π0) =CS D(Ξ+cc→ Λ+c π0)+M(π+,Σ0c ;ρ+)+M(ρ+,Σ0

c ;π+)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)

+M(K∗+,Ξ0c ; K+)+M(K∗+,Ξ′0c ; K+)+M(π+,Σ0

c ;Σ0c)+M(ρ+,Σ0

c ;Σ0c)+M(K+,Ξ0

c ;Ξ0c)

+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c ), (B7)

A(Ξ+cc→ Λ+c η1) =CS D(Ξ+cc→ Λ+c η1)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c )

+M(π+,Σ0c ;Σ0

c)+M(ρ+,Σ0c ;Σ0

c), (B8)

A(Ξ+cc→ Λ+c η8) =CS D(Ξ+cc→ Λ+c η8)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)+M(K∗+,Ξ′0c ; K+)

+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)

+M(K∗+,Ξ0c ;Ξ′0c )+M(K∗+,Ξ′0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c )+M(π+,Σ0c ;Σ0

c)+M(ρ+,Σ0c ;Σ0

c), (B9)

A(Ξ+cc→ Ξ+c K0) =CS D(Ξ+cc→ Ξ+c K0)+M(K+,Ξ0c ;ρ+)+M(K+,Ξ′0c ;ρ+)+M(K∗+,Ξ0

c ;ρ+)+M(K∗+,Ξ′0c ;ρ+)

+M(K+,Ξ0c ;Ω0

c)+M(K+,Ξ′0c ;Ω0c)+M(K∗+,Ξ0

c ;Ω0c)+M(K∗+,Ξ′0c ;Ω0

c)+M(π+,Σ0c ;Ξ0

c)

+M(π+,Σ0c ;Ξ′0c )+M(ρ+,Σ0

c ;Ξ0c)+M(ρ+,Σ0

c ;Ξ′0c ), (B10)

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-16

A(Ξ+cc→ Ξ0c K+) =T (Ξ+cc→ Ξ0

c K+)+M(π+,Σ0c ; K

∗0)+M(ρ+,Σ0

c ; K0)+M(K+,Ξ0

c ;ϕ)+M(K+,Ξ′0c ;ϕ)

+M(K∗+,Ξ0c ;η8)+M(K∗+,Ξ′0c ;η8), (B11)

A(Ξ+cc→ Ξ0cπ+) = T (Ξ+cc→ Ξ0

cπ+)+M(π+,Ξ0

c ;ρ0)+M(π+,Ξ′0c ;ρ0)+M(ρ+,Ξ0c ;π0)+M(ρ+,Ξ′0c ;π0), (B12)

A(Ξ+cc→ Ξ+c π0) =CS D(Ξ+cc→ Ξ+c π0)+M(π+,Ξ0c ;ρ+)+M(π+,Ξ′0c ;ρ+)+M(ρ+,Ξ0

c ;π+)+M(ρ+,Ξ′0c ;π+)

+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)

+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B13)

A(Ξ+cc→ Ξ+c η1) =M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B14)

A(Ξ+cc→ Ξ+c η8) =M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)

+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B15)

A(Ξ+cc→ Ξ′+c η1) =M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B16)

A(Ξ+cc→ Ξ′+c η8) =M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)

+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B17)

A(Ξ+cc→ Σ0c K+) = T (Ξ+cc→ Σ0

c K+), (B18)

A(Ω+cc→Ω0cπ+) = T (Ω+cc→Ω0

cπ+), (B19)

A(Ω+cc→ Λ+c η1) = CS D(Ω+cc→ Λ+c η1)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B20)

A(Ω+cc→ Λ+c η8) =CS D(Ω+cc→ Λ+c η8)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)

+M(K∗+,Ξ′0c ; K+)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )

+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ0c ;Ξ′0c )+M(K∗+,Ξ′0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B21)

A(Ω+cc→ Ξ′+c η1) =CS D(Ω+cc→ Ξ′+c η1)+M(K+,Ω0c ;Ω0

c)+M(K∗+,Ω0c ;Ω0

c)+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )

+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B22)

A(Ω+cc→ Ξ′+c η8) =CS D(Ω+cc→ Ξ′+c η8)+M(K+,Ω0c ; K∗+)+M(K∗+,Ω0

c ; K+)+M(K+,Ω0c ;Ω0

c)+M(K∗+,Ω0c ;Ω0

c)

+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′c;Ξ0c)+M(π+,Ξ0

c ;Ξ′c)+M(π+,Ξ′0c ;Ξ′0c )

+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′c;Ξ0c)+M(ρ+,Ξ0

c ;Ξ′c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B23)

A(Ω+cc→ Ξ+c η1) =CS D(Ω+cc→ Ξ+c η1)+M(K+,Ω0c ;Ω0

c)+M(K∗+,Ω0c ;Ω0

c)+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )

+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B24)

A(Ω+cc→ Ξ+c η8) =CS D(Ω+cc→ Ξ+c η8)+M(K+,Ω0c ; K∗+)+M(K∗+,Ω0

c ; K+)+M(K+,Ω0c ;Ω0

c)+M(K∗+,Ω0c ;Ω0

c)

+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ′c;Ξ0c)+M(π+,Ξ0

c ;Ξ′c)+M(π+,Ξ′0c ;Ξ′0c )

+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ′c;Ξ0c)+M(ρ+,Ξ0

c ;Ξ′c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B25)

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-17

A(Ω+cc→ Σ+c η1) = CS D(Ω+cc→ Σ+c η1)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B26)

A(Ω+cc→ Σ+c η8) =CS D(Ω+cc→ Σ+c η8)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)

+M(K∗+,Ξ′0c ; K+)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )

+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ0c ;Ξ′0c )+M(K∗+,Ξ′0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B27)

A(Ω+cc→ Λ+c K0) =CS D(Ω+cc→ Λ+c K

0)+M(π+,Ξ0

c ; K∗+)+M(π+,Ξ′0c ; K∗+)+M(ρ+,Ξ0c ; K+)+M(ρ+,Ξ′0c ; K+)

+M(π+,Ξ0c ;Σ0

c)+M(π+,Ξ′0c ;Σ0c)+M(ρ+,Ξ0

c ;Σ0c)+M(ρ+,Ξ′0c ;Σ0

c)+M(K+,Ω0c ;Ξ0

c)

+M(K+,Ω0c ;Ξ′0c )+M(K∗+,Ω0

c ;Ξ0c)+M(K∗+,Ω0

c ;Ξ′0c ), (B28)

A(Ω+cc→ Λ+c π0) =M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)+M(K∗+,Ξ′0c ; K+)

+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )

+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ0c ;Ξ′0c )+M(K∗+,Ξ′0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B29)

A(Ω+cc→ Ξ0c K+) = T (Ω+cc→ Ξ0

c K+)+M(K+,Ξ0c ;ϕ)+M(K+,Ξ′0c ;ϕ)+M(K∗+,Ξ0

c ;η8)+M(K∗+,Ξ′0c ;η8), (B30)

A(Ω+cc→ Ξ0cπ+) =T (Ω+cc→ Ξ0

cπ+)+M(π+,Ξ0

c ;ρ0)+M(π+,Ξ′0c ;ρ0)+M(ρ+,Ξ0c ;π0)+M(ρ+,Ξ′0c ;π0)

+M(K+,Ω0c ; K∗0)+M(K∗+,Ω0

c ; K0), (B31)

A(Ω+cc→ Ξ+c K0) =CS D(Ω+cc→ Ξ+c K

0)+M(π+,Ω0

c ; K∗+)+M(ρ+,Ω0c ; K+)+M(π+,Ω0

c ;Ξ0c)+M(π+,Ω0

c ;Ξ′0c )

+M(ρ+,Ω0c ;Ξ0

c)+M(ρ+,Ω0c ;Ξ′0c ), (B32)

A(Ω+cc→ Ξ+c K0) =CS D(Ω+cc→ Ξ+c K0)+M(K+,Ξ0c ;ρ+)+M(K+,Ξ′0c ;ρ+)+M(K∗+,Ξ0

c ;π+)+M(K∗+,Ξ′0c ;π+)

+M(K+,Ξ0c ;Ω0

c)+M(K+,Ξ0c′−;Ω0

c)+M(K∗+,Ξ0c ;Ω0

c)+M(K∗+,Ξ′0c ;Ω0c), (B33)

A(Ω+cc→ Ξ+c π0) =CS D(Ω+cc→ Ξ+c π0)+M(π+,Ξ0c ;ρ+)+M(π+,Ξ′0c ;ρ+)+M(ρ+,Ξ0

c ;π+)+M(ρ+,Ξ′0c ;π+)

+M(K+,Ω0c ; K∗+)+M(K∗+,Ω0

c ; K+)+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)

+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B34)

A(Ξ++cc → Σ+c K+) =T (Ξ++cc → Σ+c K+)+M(K+,Λ+c ;ρ0)+M(K+,Σ+c ;ω)+M(K∗+,Λ+c ;π0)+M(K∗+,Σ+c ;η8)

+M(K+,Λ+c ;Ξ0c)+M(K+,Λ+c ;Ξ′0c )+M(K+,Σ+c ;Ξ0

c)+M(K+,Σ+c ;Ξ′0c )+M(K∗+,Λ+c ;Ξ0c)

+M(K∗+,Λ+c ;Ξ′b)+M(K∗+,Σ+c ;Ξ0c)+M(K∗+,Σ+c ;Ξ′0c ), (B35)

A(Ξ++cc → Σ+c π+) =T (Ξ++cc → Σ+c π+)+M(π+,Λ+c ;ρ0)+M(ρ+,Λ+c ;π0)+M(K+,Ξ+c ; K∗0)+M(K+,Ξ′+c ; K∗0)

+M(K∗+,Ξ+c ; K0)+M(K∗+,Ξ′+c ; K0)+M(π+,Λ+c ;Σ0c)+M(π+,Σ+c ;Σ0

c)+M(ρ+,Λ+c ;Σ0c)

+M(ρ+,Σ+c ;Σ0c)+M(K+,Ξ+c ;Ξ0

c)+M(K+,Ξ+c ;Ξ′0c )+M(K+,Ξ′+c ;Ξ0c)+M(K+,Ξ′+c ;Ξ′0c )

+M(K∗+,Ξ+c ;Ξ0c)+M(K∗+,Ξ+c ;Ξ′0c )+M(K∗+,Ξ′+c ;Ξ0

c)+M(K∗+,Ξ′+c ;Ξ′0c ), (B36)

A(Ξ++cc → Σ++c K0) =CS D(Ξ++cc → Σ++c K

0)+M(π+,Ξ+c ; K∗+)+M(π+,Ξ′+c ; K∗+)+M(ρ+,Ξ+c ; K+)

+M(ρ+,Ξ′+c ; K+)+M(π+,Ξ+c ;Σ+c )+M(π+,Ξ+c ;Λ+c )+M(π+,Ξ′+c ;Σ+c )+M(π+,Ξ′+c ;Λ+c )+M(ρ+,Ξ+c ;Σ+c )+M(ρ+,Ξ+c ;Λ+c )+M(ρ+,Ξ′+c ;Σ+c )+M(ρ+,Ξ′+c ;Λ+c ), (B37)

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-18

A(Ξ++cc → Σ++c K0) =CS D(Ξ++cc → Σ++c K0)+M(K∗+,Λ+c ;π+)+M(K∗+,Σ+c ;π+)+M(K+,Λ+c ;ρ+)+M(K+,Σ+c ;ρ+)+M(K+,Λ+c ;Ξ+c )+M(K+,Λ+c ;Ξ′+c )+M(K+,Σ+c ;Ξ+c )+M(K+,Σ+c ;Ξ′+c )+M(K∗+,Λ+c ;Ξ+c )+M(K∗+,Λ+c ;Ξ′+c )+M(K∗+,Σ+c ;Ξ+c )+M(K∗+,Σ+c ;Ξ′+c ), (B38)

A(Ξ++cc → Σ++c π0) =CS D(Ξ++cc → Σ++c π

0)+M(π+,Λ+c ;ρ+)+M(π+,Σ+c ;ρ+)+M(ρ+,Λ+c ;π+)+M(ρ+,Σ+c ;π+)+M(π+,Λ+c ;Σ+)+M(π+,Σ+c ;Λ+)+M(ρ+,Λ+c ;Σ+)+M(ρ+,Σ+c ;Λ+)+M(K+,Ξ+c ; K∗+)+M(K+,Ξ′+c ; K∗+)+M(K∗+,Ξ+c ; K+)+M(K∗+,Ξ′+c ; K+)+M(K+,Ξ+c ;Ξ+c )+M(K+,Ξ+c ;Ξ′+c )+M(K+,Ξ′+c ;Ξ+c )+M(K+,Ξ′+c ;Ξ′+c )+M(K∗+,Ξ+c ;Ξ+c )+M(K∗+,Ξ+c ;Ξ′+c )+M(K∗+,Ξ′+c ;Ξ+c )+M(K∗+,Ξ′+c ;Ξ′+c ), (B39)

A(Ξ++cc → Ξ′+c K+) =T (Ξ++cc → Ξ′+c K+)+M(K+,Ξ+c ;ρ0)+M(K+,Ξ+c ;ω)+M(K+,Ξ′+c ;ρ0)+M(K+,Ξ′+c ;ω)

+M(K∗+,Ξ+c ;π0)+M(K∗+,Ξ+c ;η8)+M(K∗+,Ξ′+c ;π0)+M(K∗+,Ξ′+c ;η8)+M(π+,Λ+c ; K∗0

)

+M(π+,Σ+c ; K∗0

)+M(ρ+,Λ+c ; K0)+M(ρ+,Σ+c ; K

0)+M(K+,Ξ+c ;ϕ)+M(K+,Ξ′+c ;ϕ)

+M(K+,Ξ+c ;Ω0c)+M(K+,Ξ′+c ;Ω0

c)+M(K∗+,Ξ+c ;Ω0c)+M(K∗+,Ξ′+c ;Ω0

c)+M(π+,Λ+c ;Ξ0c)

+M(π+,Λ+c ;Ξ′0c )+M(π+,Σ+c ;Ξ0c)+M(π+,Σ+c ;Ξ′0c )+M(ρ+,Λ+c ;Ξ0

c)+M(ρ+,Λ+c ;Ξ′0c )

+M(ρ+,Σ+c ;Ξ0c)+M(ρ+,Σ+c ;Ξ′0c ), (B40)

A(Ξ++cc → Ξ′+c π+) =T (Ξ++cc → Ξ′+c π+)+M(π+,Ξ+c ;ρ0)+M(π+,Ξ′+c ;ρ0)+M(ρ+,Ξ+c ;π0)+M(ρ+,Ξ′+c ;π0)

+M(π+,Ξ+c ;Ξ0c)+M(π+,Ξ+c ;Ξ′0c )+M(π+,Ξ′+c ;Ξ0

c)+M(π+,Ξ′+c ;Ξ′0c )+M(ρ+,Ξ+c ;Ξ0c)

+M(ρ+,Ξ+c ;Ξ′0c )+M(ρ+,Ξ′+c ;Ξ0c)+M(ρ+,Ξ′+c ;Ξ′0c ), (B41)

A(Ξ++cc → Σ++c η1) =CS D(Ξ++cc → Σ++c η1)+M(π+,Σ+c ;Σ+c )+M(π+,Λ+c ;Λ+c )+M(ρ+,Σ+c ;Σ+c )+M(ρ+,Λ+c ;Λ+c )+M(K+,Ξ+c ;Ξ+c )+M(K+,Ξ′+c ;Ξ′+c )+M(K∗+,Ξ+c ;Ξ+c )+M(K∗+,Ξ′+c ;Ξ′+c ), (B42)

A(Ξ++cc → Σ++c η8) =CS D(Ξ++cc → Σ++c η8)+M(π+,Σ+c ;Σ+c )+M(π+,Λ+c ;Λ+c )+M(ρ+,Σ+c ;Σ+c )+M(ρ+,Λ+c ;Λ+c )+M(K+,Ξ+c ; K∗+)+M(K+,Ξ′+c ; K∗+)+M(K∗+,Ξ+c ; K+)+M(K∗+,Ξ′+c ; K+)+M(K+,Ξ+c ;Ξ+c )+M(K+,Ξ+c ;Ξ′+c )+M(K+,Ξ′+c ;Ξ+c )+M(K+,Ξ′+c ;Ξ′+c )+M(K∗+,Ξ+c ;Ξ+c )+M(K∗+,Ξ+c ;Ξ′+c )+M(K∗+,Ξ′+c ;Ξ+c )+M(K∗+,Ξ′+c ;Ξ′+c ),

(B43)

A(Ξ+cc→Ω0c K+) =M(π+,Ξ0

c ; K∗0

)+M(π+,Ξ′0c ; K∗0

)+M(ρ+,Ξ0c ; K

0)+M(ρ+,Ξ′0c ; K

0), (B44)

A(Ξ+cc→ Σ+c K0) =CS D(Ξ+cc→ Σ+c K

0)+M(π+,Ξ0

c ; K∗+)+M(π+,Ξ′0c ; K∗+)+M(ρ+,Ξ0c ; K+)

+M(ρ+,Ξ′0c ; K+)+M(π+,Ξ0c ;Σ0

c)+M(π+,Ξ′0c ;Σ0c)+M(ρ+,Ξ0

c ;Σ0c)+M(ρ+,Ξ′0c ;Σ0

c), (B45)

A(Ξ+cc→ Σ+c K0) =CS D(Ξ+cc→ Σ+c K0)+M(K+,Σ0c ;ρ+)+M(K∗+,Σ0

c ;π+)+M(K+,Σ0c ;Ξ0

c)+M(K+,Σ0c ;Ξ′0c )

+M(K∗+,Σ0c ;Ξ0

c)+M(K∗+,Σ0c ;Ξ′0c ), (B46)

A(Ξ+cc→ Σ+c π0) =CS D(Ξ+cc→ Σ+c π0)+M(π+,Σ0c ;ρ+)+M(ρ+,Σ0

c ;π+)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)

+M(K∗+,Ξ0c ; K+)+M(K∗+,Ξ′0c ; K+)+M(π+,Σ0

c ;Σ0c)+M(ρ+,Σ0

c ;Σ0c)+M(K+,Ξ0

c ;Ξ0c)

+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)+M(K∗+,Ξ0c ;Ξ′0c )

+M(K∗+,Ξ′0c ;Ξ0c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B47)

Rescattering mechanism of weak decays of double-charm baryons Chin. Phys. C 45, 053105 (2021)

053105-19

A(Ξ+cc→ Σ+c η1) =CS D(Ξ+cc→ Σ+c η1)+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)

+M(K∗+,Ξ′0c ;Ξ′0c )+M(π+,Σ0c ;Σ0

c)+M(ρ+,Σ0c ;Σ0

c), (B48)

A(Ξ+cc→ Σ+c η8) =CS D(Ξ+cc→ Σ+c η8)+M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)+M(K∗+,Ξ′0c ; K+)

+M(K+,Ξ0c ;Ξ0

c)+M((K+,Ξ0c ;Ξ′0c )+M((K+,Ξ′0c ;Ξ0

c)+M((K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)

+M((K∗+,Ξ0c ;Ξ′0c )+M((K∗+,Ξ′0c ;Ξ0

c)+M((K∗+,Ξ′0c ;Ξ′0c )M(π+,Σ0c ;Σ0

c)+M(ρ+,Σ0c ;Σ0

c), (B49)

A(Ξ+cc→ Σ0cπ+) =T (Ξ+cc→ Σ0

cπ+)+M(π+,Σ0

c ;ρ0)+M(ρ+,Σ0c ;π0)+M(K+,Ξ0

c ; K∗0)+M(K+,Ξ′0c ; K∗0)

+M(K∗+,Ξ0c ; K0)+M(K∗+,Ξ′0c ; K0), (B50)

A(Ξ+cc→ Σ++c K−) =M(π+,Ξ0c ;Σ+c )+M(π+,Ξ0

c ;Λ+c )+M(π+,Ξ′0c ;Σ+c )+M(π+,Ξ′0c ;Λ+c )+M(ρ+,Ξ0c ;Σ+c )

+M(ρ+,Ξ0c ;Λ+c )+M(ρ+,Ξ′0c ;Σ+c )+M(ρ+,Ξ′0c ;Λ+c ), (B51)

A(Ξ+cc→ Σ++c π−) =M(π+,Σ−b ;Σ+c )+M(π+,Σ−b ;Λ+c )+M(ρ+,Σ−b ;Σ+c )+M(ρ+,Σ−b ;Λ+c )+M(K+,Ξ0

c ;Ξ+c )

+M(K+,Ξ0c ;Ξ′+c )+M(K+,Ξ′0c ;Ξ+c )+M(K+,Ξ′0c ;Ξ′+c )+M(K∗+,Ξ0

c ;Ξ+c )

+M(K∗+,Ξ0c ;Ξ′+c )+M(K∗+,Ξ′0c ;Ξ+c )+M(K∗+,Ξ′0c ;Ξ′+c ), (B52)

A(Ξ+cc→ Ξ′0c K+) =T (Ξ+cc→ Ξ′0c K+)+M(π+,Σ−b ; K∗0

)+M(ρ+,Σ−b ; K0)+M(K+,Ξ0

c ;ϕ)+M(K+,Ξ′0c ;ϕ)

+M(K∗+,Ξ0c ;η8)+M(K∗+,Ξ′0c ;η8), (B53)

A(Ξ+cc→ Ξ′0c π+) = T (Ξ+cc→ Ξ′0c π+)+M(π+,Ξ0c ;ρ0)+M(π+,Ξ′0c ;ρ0)+M(ρ+,Ξ0

c ;π0)+M(ρ+,Ξ′0c ;π0), (B54)

A(Ξ+cc→ Ξ′+c K0) =CS D(Ξ+cc→ Ξ′+c K0)+M(K+,Ξ0c ;ρ+)+M(K+,Ξ′0c ;ρ+)+M(K∗+,Ξ0

c ;π+)

+M(K∗+,Ξ′0c ;π+)+M(K+,Ξ0c ;Ω0

c)+M(K+,Ξ′0c ;Ω0c)+M(K∗+,Ξ0

c ;Ω0c)+M(K∗+,Ξ′0c ;Ω0

c)

+M(π+,Σ−b ;Ξ0c)+M(π+,Σ−b ;Ξ′0c )+M(ρ+,Σ−b ;Ξ0

c)+M(ρ+,Σ−b ;Ξ′0c ), (B55)

A(Ξ+cc→ Ξ′+c π0) =CS D(Ξ+cc→ Ξ′+c π0)+M(π+,Ξ0c ;ρ+)+M(π+,Ξ′0c ;ρ+)+M(ρ+,Ξ0

c ;π+)+M(ρ+,Ξ′0c ;π+)

+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)

+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ), (B56)

A(Ω+cc→Ω0c K+) =T (Ω+cc→Ω0

c K+)+M(π+,Ξ0c ; K

∗0)+M(π+,Ξ′0c ; K

∗0)+M(ρ+,Ξ0

c ; K0)+M(ρ+,Ξ′0c ; K

0)

+M(K+,Ω0c ;ϕ)+M(K∗+,Ω0

c ;η8), (B57)

A(Ω+cc→ Σ+c K0) =CS D(Ω+cc→ Σ+c K

0)+M(π+,Ξ0

c ; K∗+)+M(π+,Ξ′0c ; K∗+)+M(ρ+,Ξ0c ; K+)

+M(ρ+,Ξ′0c ; K+)+M(π+,Ξ0c ;Σ0

c)+M(π+,Ξ′0c ;Σ0c)+M(ρ+,Ξ0

c ;Σ0c)+M(ρ+,Ξ′0c ;Σ0

c)

+M(K+,Ω0c ;Ξ0

c)+M(K+,Ω0c ;Ξ′0c )+M(K∗+,Ω0

c ;Ξ0c)+M(K∗+,Ω0

c ;Ξ′0c ), (B58)

A(Ω+cc→ Σ+c π0) =M(K+,Ξ0c ; K∗+)+M(K+,Ξ′0c ; K∗+)+M(K∗+,Ξ0

c ; K+)+M(K∗+,Ξ′0c ; K+)

+M(K+,Ξ0c ;Ξ0

c)+M(K+,Ξ0c ;Ξ′0c )+M(K+,Ξ′0c ;Ξ0

c)+M(K+,Ξ′0c ;Ξ′0c )+M(K∗+,Ξ0c ;Ξ0

c)

+M(K∗+,Ξ0c ;Ξ′0c )+M(K∗+,Ξ′0c ;Ξ0

c)+M(K∗+,Ξ′0c ;Ξ′0c ), (B59)

Jia-Jie Han, Hua-Yu Jiang, Wei Liu et al. Chin. Phys. C 45, 053105 (2021)

053105-20

A(Ω+cc→ Σ0cπ+) =M(K+,Ξ0

c ; K∗0)+M(K+,Ξ′0c ; K∗0)+M(K∗+,Ξ0c ; K0)+M(K∗+,Ξ′0c ; K0), (B60)

A(Ω+cc→ Σ++c K−) =M(π+,Ξ0c ;Σ+c )+M(π+,Ξ0

c ;Λ+c )+M(π+,Ξ′0c ;Σ+c )+M(π+,Ξ′0c ;Λ+c )+M(ρ+,Ξ0c ;Σ+c )

+M(ρ+,Ξ0c ;Λ+c )+M(ρ+,Ξ′0c ;Σ+c )+M(ρ+,Ξ′0c ;Λ+c )+M(K+,Ω0

c ;Ξ+c )+M(K+,Ω0c ;Ξ′+c )

+M(K∗+,Ω0c ;Ξ+c )+M(K∗+,Ω0

c ;Ξ′+c ), (B61)

A(Ω+cc→ Σ++c π−) =M(K+,Ξ0

c ;Ξ+c )+M(K+,Ξ0c ;Ξ′+c )+M(K+,Ξ′0c ;Ξ+c )+M(K+,Ξ′0c ;Ξ′+c )+M(K∗+,Ξ0

c ;Ξ+c )

+M(K∗+,Ξ0c ;Ξ′+c )+M(K∗+,Ξ′0c ;Ξ+c )+M(K∗+,Ξ′0c ;Ξ′+c ), (B62)

A(Ω+cc→ Ξ′0c K+) = T (Ω+cc→ Ξ′0c K+)+M(K+,Ξ0c ;ϕ)+M(K+,Ξ′0c ;ϕ)+M(K∗+,Ξ0

c ;η8)+M(K∗+,Ξ′0c ;η8), (B63)

A(Ω+cc→ Ξ′0c π+) =T (Ω+cc→ Ξ′0c π+)+M(π+,Ξ0c ;ρ0)+M(π+,Ξ′0c ;ρ0)+M(ρ+,Ξ0

c ;π0)+M(ρ+,Ξ′0c ;π0)

+M(K+,Ω0c ; K∗0)+M(K∗+,Ω0

c ; K0), (B64)

A(Ω+cc→ Ξ′+c K0) =CS D(Ω+cc→ Ξ′+c K

0)+M(π+,Ω0

c ; K∗+)+M(ρ+,Ω0c ; K+)+M(π+,Ω0

c ;Ξ0c)+M(π+,Ω0

c ;Ξ′0c )

+M(ρ+,Ω0c ;Ξ0

c)M(ρ+,Ω0c ;Ξ′0c ), (B65)

A(Ω+cc→ Ξ′+c K0) =CS D(Ω+cc→ Ξ′+c K0)+M(K+,Ξ0c ;ρ+)+M(K+,Ξ′0c ;ρ+)+M(K∗+,Ξ0

c ;π+)+M(K∗+,Ξ′0c ;π+)

+M(K+,Ξ0c ;Ω0

c)+M(K+,Ξ′0c ;Ω0c)+M(K∗+,Ξ0

c ;Ω0c)+M(K∗+,Ξ′0c ;Ω0

c), (B66)

A(Ω+cc→ Ξ′+c π0) =CS D(Ω+cc→ Ξ′+c π0)+M(π+,Ξ0c ;ρ+)+M(π+,Ξ′0c ;ρ+)+M(ρ+,Ξ0

c ;π+)+M(ρ+,Ξ′0c ;π+)

+M(K+,Ω0c ; K∗+)+M(K∗+,Ω0

c ; K+)+M(π+,Ξ0c ;Ξ0

c)+M(π+,Ξ0c ;Ξ′0c )+M(π+,Ξ′0c ;Ξ0

c)

+M(π+,Ξ′0c ;Ξ′0c )+M(ρ+,Ξ0c ;Ξ0

c)+M(ρ+,Ξ0c ;Ξ′0c )+M(ρ+,Ξ′0c ;Ξ0

c)+M(ρ+,Ξ′0c ;Ξ′0c ). (B67)

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