The study of doubly charmed pentaquark $c c \bar qqq$ with the SU(3) symmetry

We study the masses and lifetimes of doubly charmed pentaquark $P_{cc\bar qqq}(q=u,d,s)$ primarily. The operation of masses carried out by the doubly heavy triquark-diquark model, whose results suggests the existence of stable states $cc\bar s ud$ with the parity $J^P=\frac{1}{2}^-$. The roughly calculation about lifetimes show the short magnitudes, $(4.65^{+0.71}_{-0.55})\times 10^{-13}s $ for the parity $J^P=\frac{1}{2}^-$ and $(0.93^{+0.14}_{-0.11})\times 10^{-12} s $ for $J^P=\frac{3}{2}^-$. Since the pentaquark $cc\bar s ud$ is interpreted as the stable bound states against strong decays, then we will focus on the production and possible decay channels of the pentaquark in the next step, the study would be fairly valuable supports for future experiments. For completeness, we systematically studied the production from $\Omega_{ccc}$ and the decay modes in the framework SU(3) flavor symmetry, including the processes of semi-leptonic and two body non-leptonic decays. Synthetically, we make a collection of the golden channels.

understanding of quantum chromodynamics. Besides, the two charmed quarks in pentaquark mean more scales, which might be an ideal probe to study the interplay between perturbative QCD and non-perturbative QCD, eventually leading the development of multi scale physics, especially the factorization approach. The underlying idea, therefore, is the discussion on the nature of doubly charmed pentaquark in the paper.
Doubly charmed pentaquark P (ccqqq) can be dealt with the compact pentaquark interpretation.
The diquark with two charmed quarks cc forms a color triplet3 c spin-1 state, as suggested by perturbative arguments. While the light diquark qq forms a color triplet3 c spin-0 state, as the hypothesis of "good" diquark. At this stage, we employ the triquark-diquark model [11][12][13], which heavy diquark [cc]3 combines with antiquark q3 to form a triquark system [[cc]3q3] 3 , and then, merging with light diquark [qq ′ ]3 to form the doubly charmed pentquark state. Once applying the effective Hamiltonian of mass spectrum, the masses of pentaquark would be achieved. In addition, the lifetimes can be estimated with the implement of operator product expansion (OPE) approach [14,15]. Upon the heavy quark expanding and optical theorem, we can directly determine the lifetimes at the next-to-leading order of P (ccqqq) with the different parities.
The flavor SU(3) symmetry is a convincing tool to analyze production and decay behaviours of hadrons, which has been successfully applied to the meson or baryon system [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. Though the SU(3) breaking effects in charm quark transition might be sizable, the results can still describe the experimental data well in a global viewpoint. The doubly charmed pentaquark can be produced from triply charmed baryon Ω ccc with one charmed quark weak decays, c → qqq. Moreover, the decays of pentaquark similarly give priority to the charmed quark decays. Therefore, we will force on the charmed quark weak decays both in the production and decays processes. To be more explicit, one can write down the Hamiltonian at the hadron level with hadron fields and transition operators. Some limited amount of input parameters will be introduced to describe the non-perturbative transitions. With the SU(3) amplitudes, one can obtain relations between decay widths of different processes, which can be examined in experiment. Such an analysis is also helpful to identify the decay modes that will be mostly useful to discover the doubly charmed pentaquark state P (ccqqq). Since the SU(3) analysis is based on the light quark symmetry, thus the analytical results can work well in all states with P (ccqqq) flavor constituents, for instance, the states with molecular or diquark picture.
The rest of this paper is organized as follows. In Sec. II, we discuss the mass spectrums and lifetimes of doubly charmed pentaquark. Section III is devoted to discuss the production and decays behaviours, which including two body production processes, mesonic semi-and non-leptonic decays.
In section V, we present a collection of the golden channels. We make a short summary in the end.

II. MASS AND LIFETIME OF P (ccqqq) STATE (i). Mass
We study the mass spectrum of S-wave pentaquark states P ccqqq (q = u, d, s) in the framework of non-relativity doubly heavy triquark-diquark model [11][12][13]. Under the picture of diquark consisted of two light quarks qq ′ in color3 state and triquark consisted of doubly heavy diquark cc in color3 plus a light anti-quarkq in color3, which labeled as [[cc] c3 [q] c3 ] c3 [qq ′ ] c3 , the effective Hamiltonian of mass spectrum can then be written as with the constituent mass of diquark and triquark is M 0 , in addition, the interaction of triquark  3 . We respectively give the forms as follows.
In the doubly heavy triquark-diquark system, the suggested spin of doubly heavy charmed diquark is S cc = 1. Similarily, the spin of "good" light diquark in the S-wave pentaquark state P ccqqq is chosen as S qq ′ = 0 [32]. Accordingly, we can write directly the possible configuration of S-wave pentaquark P ccqqq , signed as |S cc , S t , L t ; S qq ′ , L qq ′ ; S, L .
Here, the light diquark qq ′ can be any one of the constituents (ud, du, us, su, ds, sd). Except for the spin of doubly heavy diquark S cc and light diquark S qq ′ , the orbital angular momentum L t = L qq ′ = L = 0, the spin of triquark ([cc]q) S t turn out to be 1 2 or 3 2 . The two states with parity J P = 1 2 − and J P = 3 2 − , sandwiching the effective mass Hamiltonian Eq. 1, then yield the mass spectrum matrix of S-wave pentaquark P ccqqq : In particular, the determination of spin-spin interaction between three spins inside the triquark, i.e., S c · Sq, can be drawn by the Wigner 6j-symbols. Further more, for the interaction between triquark and light diquark, such as, S c · S q and S q · Sq, it is convenient to utilize the Wigner 9jsymbols to describe the recouplings. The remaining step is the numerical operation. With regard to the work, we choose the values of spin-spin coupling given as [33]: (K cc )3 = 57 MeV, (K qq ′ )3 = 98 MeV, K sd/ū = 200 MeV. We adopt the mass of quark and diquark [33], for instance, m u/d = 362 MeV, m s = 540 MeV, m c = 1.667 GeV, m cc ∼ 2m c , m ud = 576 MeV, m sq = 800 MeV. Certainly, one should consider the uncertainty in these couplings and masses. We assign the couplings to be 10% of each value, so as the mass of heavy charmed diquark.
We diagonalize the mass matrix and obtain the split mass of pentaquark P ccqqq shown in Tab. I.
As a contrast, we show the results from chiral effective theory(ChEFT), quark model with colormagnetic interaction(CMI) and QCD sum rule(QCDSR). Moreover, the lowest strong thresholds are placed at the end of the table. From which we may find that the center mass of pentaquark P ccsud with parity 1 2 − is higher than the lowest strong threshold Ξ ccqK about 23 MeV. Therefore, it is possible a stable bound state against the strong interaction. The conclusion is consistent with the Refs [34,35]. Still, it is worth noting, the difference value much smaller than the uncertainties in our work, this may play an increasingly role at the final conclusion. In addition, the mass of  In the part, we roughly study the lifetimes of pentaquark P ccqqq with the parity under the framework of operator product expansion(OPE) technique. As always, the decays widths can be expressed as therein, m P , λ and p µ P are the mass, spin and four-momentum of pentaquark P ccqqq respectively.
where G F is the Fermi constant, V CKM is the CKM element, the coefficients c 3,c is the perturbative coefficient of HQE. Further more, the heavy quark matrix element is corresponding with charm number of pentaquark state.
Consequently, we reach the decay widths and lifetimes under the leading and next-to-leading order, In this work, the heavy quark mass m c = 1.4 GeV, the perturbative short-distance coefficient have been determined as c 3,c = 6.29 ± 0.72 at the leading order(LO), and c 3,c = 11.61 ± 1.55 at the next-to-leading order(NLO) [14].

III. SU(3) ANALYSIS
We will discuss the possible production and decay modes of pentaquark with the quark constituent of ccqqq in this section. The production can be achieved by the study about weak decays of triply heavy baryon Ω ccc . At the stage of decay modes, we focus on the explores of stable pentaquark candidates, which give priority to the weak decays similarly.
The weak interaction of production and decays for the pentaquark states, with the transition c → qqq, can be classified by the quantities of CKM matrix elements.
• For the case of c quark semi-leptonic decays, The general electro-weak Hamiltonian can be expressed as with q = (d, s), in which the transition operator of c → qℓ + ν ℓ forms a SU(3) triplet H 3 , and • For the case of c quark non-leptonic decays, we classify the transitions into three groups.
which are Cabibbo allowed, singly Cabibbo suppressed, and doubly Cabibbo suppressed transitions respectively. The transition c → q 1q2 q 3 can be decomposed as Here, we offer the nonzero SU(3) tensor components of Cabibbo allowed transition given as As the transition ofc →ūdd andc →ūss with singly Cabibbo suppressed, the combination of tensor components are corresponding with the overall CKM factor, which defined as For the doubly Cabibbo suppressed transitionc →dsū, we have (I). Particle Multiplets in SU (3) The pentaquark with the quark constituents ccqqq contain three light quark and thus can form an SU(3)6 and an SU(3) 15, labeled as P 6 and P 15 respectively. We give the SU(3) representations P 6 as follows.
(P 6 ) In addition, the baryons with the components qqq, can form an SU(3) octet T 8 and an SU (3) decuplet T 10 . The octet has the expression and the light decuplet is given as Consistently, the singly charmed baryons cqq are expected to form a anti-triplet and a sextet, respectively as [38,39] In the meson sector, singly charmed mesons form an SU(3) triplet or anti-triplet, light mesons (a) form an octet plus singlet, all multiplets are collected as For completeness, we also draw the weight diagrams of multiplets, shown in Fig. 1 and Fig. 2.
(II). Production of P ccqqq from Ω ccc The pentaquark P ccqqq can be produced by the weak decays of triply charmed baryon Ω ccc , once one charmed quark decays c → qqq in baryon. The typical Feynman diagrams with two final states are shown in Fig. 3. In particular, the final states include the pentaquark we wanted and a light meson. Within the framework of SU(3) symmetry analysis, we then construct the Hamiltonian at the hadronic level, which straightly written as where the coefficients, such as a 1 , a 2 , b 1 , b 2 , ..., represent the non-perturbative parameters. The SU (3) representations of corresponding hadrons and transitions above, permit us to expand the Hamiltonian into diverse decay channels, revealed with combinations of non-perturbative coefficients. We forward and collect the possible production channels of P 6 (ccqqq) states in Tab. II.
Especially, the Cabibbo allowed, singly Cabibbo suppressed and doubly Cabibbo suppressed are displayed respectively. In addition, the channels corresponding with P 15 (ccqqq) states are placed into Tab. III. Referring to the amplitudes of production channels above, we reduce the relations between different channels.
(iii). Decay modes of pentaquark P 6 (ccqqq) In this part, we will study the possible decays of pentaquark P ccqqq states. Generally, the excited states P 15 can primarily decays into P 6 states. In that case, we need only to discuss the decay modes of P 6 states individually in the paper. As first step, we consider the semileptonic decays of P 6 (ccqqq) with the transition of c → d/sℓ + ν. Following the SU(3) analysis, the construction of the corresponding Hamiltonian is forward, which can be constructed as Accordingly, the relations of decay width between six channels can be deduced directly, with the effect of phase spaces ignored, given as The transition of charmed quark decay c → qqq can lead to non-leptonic decays of pentaquark ccqqq states P 6 . Immediately, it is straightforward to construct the Hamiltonian in the hadronic level under the SU(3) light quark symmetry. We raise the possible Hamiltonian of sextet pentaquark P 6 decay into charmed mesons and light baryon T 8 , T 10 as follows.
We expand the Hamiltonian and collect the possible processes, entrying into Tab. IV and Tab. V., Meanwhile, it is ready to reduce the relations of decay width between different channels. We deduced the relations as follows.
More technically, the Hamiltonian of the sextet state P 6 decay into light mesons and singly charmed baryons can be constructed below.
We expand the decay channels of pentaquark octet P 10 turning into anti-charmed mesons and light baryon, whose decay amplitudes collected in Tab.VI and Tab.VII. Moreover, the relations between different channels are given as follows.

IV. GOLDEN CHANNELS
As a collection, we will screen out some golden channels to produce and reconstruct the pentaquark P ccqqq in the section. In principle, the main considerations are the CKM elements in the transition. The amplitudes of c-quark decay transitions such as c → sdu and c → sℓ + ν ℓ will receive the largest contribution as V * cs ∼ 1. Beyond that, the detection efficiency is also a serviceable factor, generally speaking, charged particles have higher rates to be detected than neutral particles.
According to the selection schemes, one use the following criteria [39], finally, we can obtain the golden decay channels in Table VIII. • Branching fractions: one chooses the corresponding channels with the quark transition of c →sdū orc →sℓ −ν ℓ .

V. CONCLUSIONS
In the paper, we discussed the mass spectrums of doubly charmed pentaquarks P ccqqq primarily under the doubly heavy triquark-diquark model. Moreover, the lifetimes were considered at the next-to-leading order with the approach of OPE. The calculation suggested some   The golden channels of production and decays with Cabibbo allowed P (ccqqq).