Strong decays of the 1P and 2D doubly charmed states

We perform a systematical investigation of the strong decay properties of the low-lying $1P$- and $2D$-wave doubly charmed baryons with the $^3P_0$ quark pair creation model. The main predictions include: (i) in the $\Xi_{cc}$ and $\Omega_{cc}$ family, the $1P~\rho$ mode excitations with $J^P=1/2^-$ and $3/2^-$ should be the fairly narrow states. (ii) For the $1P~\lambda$ mode excitations, $|^2P_{\lambda}\frac{3}{2}^-\rangle$ and $|^4P_{\lambda}\frac{3}{2}^-\rangle$ have a width of $\Gamma\sim150$ MeV, and mainly decay into the $J^P=3/2^+$ ground state. Meanwhile, $|^2P_{\lambda}\frac{1}{2}^-\rangle$ and $|^4P_{\lambda}\frac{5}{2}^-\rangle$ are the narrow states with a width of $\Gamma\sim40$ MeV, and mainly decay into the ground state with $J^P=1/2^+$. (iii) The $2D_{\rho\rho}$ states mainly decay via emitting a heavy-light meson if their masses are above the threshold of $\Lambda_cD$ or $\Xi_cD$, respectively. Their strong decay widths are sensitive to the masses and can reach several tens MeV. (iv) The $2D_{\lambda\lambda}$ states may be broad states with a width of $\Gamma>100$ MeV. It should be emphasized that the states with $J^P=3/2^+$ and $5/2^+$ mainly decay into the ground state with $J^P=3/2^+$ plus a light-flavor meson, while the states with $J^P=1/2^+$ and $7/2^+$ mainly decay into the ground state with $J^P=1/2^+$ plus a light-flavor meson.

In the past score years, the properties of the doubly heavy baryons were extensively explored with various theoretical methods and models including the mass spectra [20][21][22][23][24][25][26][27][28][29][30] and semi-leptonic decays [6,[31][32][33][34][35][36][37][38][39][40][41][42]. However, only a few discussions on the decay behavior exist in literature [17][18][19][43][44][45]. In our previous work [17], we first systematically investigated the both strong and radiative transitions of the low-lying 1Pwave doubly heavy baryons with chiral and constituent quark model. In this work, we shall perform a systematic analysis of the two-body Okubo-Zweig-Iizuka (OZI) allowed strong decays of the 1P and 2D doubly charmed states with the quark pair creation(QPC) model, which may provide more information of their inner structures. The quark model classification, predicted masses [20], and OZI allowed decay modes [46] are * E-mail: lyxiao@pku.edu.cn † E-mail: lvqifang@hunnu.edu.cn ‡ E-mail: zhusl@pku.edu.cn summarized in Table I. For the low-lying 1P and 2D doubly charmed baryons, their masses are large enough to allow the decay channels containing a heavy-light flavor meson. Thus, it is suitable to apply the QPC strong decay model. Meanwhile, for further understanding the strong decays of the doubly charmed baryons, it is necessary to make a comparison of the theoretical predictions with QPC model to the results with the chiral quark model [17].
The QPC strong decay model as a phenomenological method has been employed successfully in the description of the hadronic decays of the mesons [47][48][49][50] and singly charmed baryons [51][52][53][54][55]. Systematical study of the lowlying 1P and 2D doubly charmed states with the QPC model has not been performed yet. In the framework of the QPC model, we find that (i) our results of the decay patterns of the 1P states are highly comparable with those in our previous work [17]; (ii) the 2D ρρ states mainly decay via emitting a heavy-light meson if their masses are above the threshold of Λ c D or Ξ c D, respectively; (iii) although the 2D λλ states may be broad states with a width of Γ > 100 MeV, they still have the opportunity to be discovered via their main decay channels in future experiments. This paper is structured as follows. In Sec. II we give a brief review of the QPC model. We present our numerical results and discussions in Sec. III and summarize our results in Sec. IV.
State Ξ cc Ω cc N 2S +1 L σ J P Wave function Mass [20] Strong decay channel Mass [20] Strong decay channel this model assumes that a pair of quark qq is created from the vacuum and then regroups with the quarks from the initial hadron to produce two outing hadrons. The created quark pair qq shall carry the quantum number of 0 ++ and be in a 3 P 0 state. Thus the QPC model is also known as the 3 P 0 model. This model has been extensively employed to study the OZIallowed strong transitions of hadron systems. Here, we adopt this model to study the strong decays of the ccq system. According to the quark rearrangement process, any of the three quarks in the initial baryon can go into the final meson. Thus three possible decay processes are take into account as shown in Fig. 1. Now, we take the Fig. 1(a) decay process A(the initial baryon)→ B(the final baryon)+C(the final meson) as an example to show how to calculate the decay width. In the nonrelativistic limit, the transition operator under the 3 P 0 model is given by where p i (i=4, 5) represents the three-vector momentum of the ith quark in the created quark pair. ω 45 0 = δ i j and ϕ 45 0 = (uū + dd + ss)/ √ 3 stand for the color singlet and flavor function, respectively. The solid harmonic polynomial Y m 1 (p) ≡ |p|Y m 1 (θ p , φ p ) corresponds to the momentum-space distribution, and χ 45 1,−m is the spin triplet state for the created quark pair. The creation operator a † 4i b † 5 j denotes the quark pair-creation in the vacuum. The pair-creation strength γ is a dimensionless parameter, which is usually fixed by fitting the well measured partial decay widths. According to the definition of the mock state [61], the spacial wave functions of the baryon and meson read, respectively, The p i (i = 1, 2, 3 and a, b) denotes the momentum of quarks in hadron A and C. P A (P C ) are the momentum of the hadron A(C). The 3 P 0 model gives a good description of the decay properties of many observed mesons with the simple harmonic oscillator space-wave functions, which are adopted to describe the spatial wave function of both baryons and mesons in the present work. The spatial wave function of a baryon without the radial excitation is The ground state spatial wave function of a meson is where the p ab stands for the relative momentum between the quark and antiquark in the meson. Then, we can obtain the partial decay amplitude in the center of mass frame, (p) stands the spatial integral and more detailed information is presented in the Appendix A and B. The A,B,C denotes the Clebsch-Gorden coefficients for the quark pair, initial and final hadrons, which come from the couplings among the orbital, spin, and total angular momentum. Its expression reads Finally, the decay width In the equation, p is the momentum of the daughter baryon in the center of mass frame of the parent baryon A |p| = In the present calculation, we adopt m u = m d = 220 MeV, m s = 419 MeV, and m c = 1628 MeV [49] for the constituent quark masses. The masses of the baryons and mesons involved in our calculations, listed in Table II, are from the Particle Data Group [62] except for the doubly charmed baryons, which is from Ref. [20]. The value of the harmonic oscillator strength R is 2.5 GeV −1 , for all light flavor mesons while it is R = 1.67GeV −1 for the D meson and R = 1.54GeV −1 for the D s meson [49]. The parameter α ρ of the ρ-mode excitation between the two charm quarks is taken as α ρ = 0.66 GeV [17], while α ρ between the two light quarks is taken as α ρ = 0.4 GeV. Another harmonic oscillator parameter α λ is obtained with the relation: For the strength of the quark pair creation from the vacuum, we take the same value as in Ref. [49], γ = 6.95. For the strange quark pair ss creation, we use γ ss = γ/ √ 3 [60].

III. CALCULATIONS AND RESULTS
For the P-wave doubly charmed states, the masses are adopted from Ref. [20] (showed in Table I) due to a good agreement with the mass of the lowest doubly charmed baryon Ξ ++ cc (3621) observed by the LHCb collaboration. However, there is no prediction for the masses of the D-wave states. So the masses of the D-wave baryons are varied in a rough range when their decay properties are investigated.
A. The P-wave doubly charmed states Within the quark model, there are two 1P ρ doubly heavy baryons with J P = 1 2 − and J P = 3 2 − , respectively. Their masses are above the threshold of Ξ cc π or Ξ cc K. However, the OZI-allowed two body strong decays are forbidden since the spatial wave functions for the 1P and 0S states are adopted with the simple harmonic oscillator wave functions which are orthogonal. In this work, we focus on the strong decays of the 1P λ states.
We analyze the decay properties of the 1P λ states in the Ξ cc and Ω cc family, and collect their partial strong decay The comparison of the partial decay widths of the 1P λ states from the QPC model and the chiral quark model [17]. Γ total stands for the total decay width and B represent the ratio of the branching fractions Γ[Ξ cc π/K]/Γ[Ξ * cc π/K]. The unit is MeV.  [17]. The dominant decay modes are Ξ cc π and Ξ * cc π with the partial decay ratio This value is about 2.5 times of the ratio in Ref. [17].
This ratio may be a useful distinction between |Ξ cc 2 P λ − may be a narrow state with a total decay width around Γ ∼60 MeV, which is about one half of that in Ref. [17]. This state decays mainly through the Ξ cc π channel. The predicted partial width ratio between Ξ cc π and Ξ * cc π is which can be tested in future experiments.
In the Ω cc family, the |Ω cc 2 P λ The decay width of the state |Ω cc MeV. Meanwhile, its strong decays are governed by the Ξ cc K channel. In this case, the |Ω cc 4 P λ  Since we adopt the simple harmonic oscillator spatial wave functions in present work, the strong decays of 2D ρρ doubly charmed states via emitting a light-flavor meson are forbidden due to the orthogonality of the spatial wave functions. So, we focus on the decay processes via emitting a heavy-light flavor meson. Due to the lack of the mass predictions for the Dwave doubly charmed states, we investigate the strong decay properties as the functions of the masses in a possible range.
First of all, we conduct systematic research on the strong decays of 2D ρρ states in the Ξ cc family in Fig. 4. For the state |Ξ cc 2 D ρρ MeV, the predicted branching ratio is So, this state is most likely to be observed in the Λ c D channel. Then, we analyze the decay properties of the 2D ρρ states in the Ω cc family, and plot the partial decay widths and total decay width as functions of the masses in Fig. 5.
To investigate the decay properties of the |Ω cc 2 D ρρ In brief, the 2D ρρ states of Ξ cc and Ω cc can decay through emitting a heavy-light meson when their masses are above the threshold of Λ c D and Ξ c D, respectively. Their total decay widths maybe reach several tens MeV if their masses are large enough. However, most of those states may lie below the threshold of Λ c D or Ξ c D, respectively.

λ-mode excitations
As emphasized in our previous work [17], the λ-mode orbitally excited state has relatively larger mass than a ρ-mode orbitally excited state for the doubly charmed baryons. The 2D λλ states should be heavier than the 2D ρρ states with the same J P . Thus, many other decay modes are allowed when we study the strong decay properties of 2D λλ states.
In the Ξ cc family, we estimate the mass of the |Ξ cc 2 D λλ 3 2 + in the range of (4.50-4.90) GeV, and then investigate its strong decay properties as a function of the mass in Fig. 6. The decay width of the state |Ξ cc 2 D λλ The main decay channel is Ξ * cc π and the predicted branching ratio is On the other hand, the partial decay width of Γ[|Ξ cc 2 D λλ is sizable. The partial width ratio between Σ * c D and Ξ * cc π is  + on the mass is plotted in Fig. 6 as well.
According to the figure, we can see that the state has a predicted width of Γ ≃ (150 − 590) MeV, and mainly decays into Ξ cc π and Ξ * cc π. The predicted partial width ratio is Meanwhile, the role of the Σ c D channel becomes more and more important as the mass increases. The branching ratio is We estimate the mass of |Ξ cc 4 D λλ 1 2 + in the range of (4.20-4.60) GeV and calculate its strong decay widths, which are shown in Fig. 6. From the figure, the state |Ξ cc 4 D λλ 1 2 + is a moderate state with a width of Γ ≃ (65 − 118) MeV, and its strong decays are governed by the Ξ cc π channel. The predicted branching ratio is It should be pointed out that if the decay channel Ω cc K is opened, which is sensitive to the mass, the branching ratio of this decay channel may reach 41%. Since the predicted width of |Ξ cc 4 D λλ  110) MeV, and its strong decays are dominated by the Ξ cc π and Ξ * cc π channels. However, the partial width of Ξ cc π decreases dramatically with the mass. So, the predicted branching ratio of the Ξ cc π channel varies in a wide range of The branching ratio of the Ξ * cc π channel is stable, which is This state has good potential to be discovered in the Ξ cc π and Ξ * cc π channels. For the state |Ξ cc 4 D λλ 5 2 + , we plot its partial decay widths and total widths as a function of the mass in the range of (4.30-4.70) GeV. From Fig. 6, its total decay width is about Γ ≃ (59 − 260) MeV. The partial decay width ratio of the main two decay channels Ξ cc π and Ξ * cc π is Γ[|Ξ cc 4 D λλ Meanwhile, from the Fig. 6 we notice that the strong decays of the state |Ξ cc 4 D λλ 7 2 + are dominated by the Ξ cc π and Ξ * cc π channels as well, when the mass lies in the range of (4.35-4.75) GeV. But the total decay width of |Ξ cc 4 D λλ In addition, we extract the strong decays of the 2D λλ states in the Ω cc family, and plot their decay properties as functions of the masses in Fig. 7. Usually, the mass of the Ω cc resonances is about 150 MeV larger than that of the Ξ cc resonances [19,20]. Thus we estimate the mass of the state |Ω cc 2 D λλ 3 2 + might be in the range of (4.65-5.05) GeV. According to our theoretical calculations, |Ω cc 2 D λλ 3 2 + is a broad state with a width of Γ ≃ (114 − 769) MeV, and Ξ * cc K almost saturates its total decay widths.
Meanwhile, the state |Ω cc 2 D λλ 5 2 + is most likely to be a very broad state as well, and the total decay width is about Γ ≃ (280 − 1000) MeV with the mass in the range of (4.70-5.10) GeV. Its strong decays are governed by the Ξ cc K and Ξ * cc K channels. The predicted partial width ratio between Ξ cc K and Ξ * cc K is Γ[|Ω cc 2 D λλ 5 2 The partial width of Ξ ′ cc D is sizable as well. This broad state might be hard to be observed in experiments.
Taking the mass of |Ω cc 4 D λλ + , we plot its strong decay properties as a function of the mass in the range of (4.40-4.80) GeV in Fig. 7. The total decay width of |Ω cc 4 D λλ Its strong decays are dominated by the Ξ cc K and Ξ * cc K channels, and the predicted partial decay width ratio is The total decay width of |Ω cc 4 D λλ MeV with the mass in the range of (4.45-4.85) GeV. From the Fig 7, this state mainly decays through the Ξ * cc K channel. The branching ratio is The partial width of the Ξ cc K channel is sizable as well.
The partial decay widths of the |Ω cc 4 D λλ 7 2 + strongly depend on its mass. Taking the mass of |Ω cc 4 D λλ 7 2 + in the range of (4.50-4.90) GeV, the total decay width varies in a wide range of Γ ≃ (43 − 708) MeV. Its strong decays are governed by the Ξ cc K and Ξ * cc K channels, and the partial decay width ratio is Meanwhile, the partial decay width of Ξ c D is sizable, and the predicted partial width ratio between Ξ c D and Ξ cc K is In conclusion, in the Ξ cc and Ω cc family, the 2D λλ states with J P = 1/2 + , 7/2 + mainly decay into the ground state with J P = 3/2 + through emitting a light-flavor meson, while the 2D λλ states with J P = 3/2 + , 5/2 + mainly decay into the ground state with J P = 1/2 + plus a light-flavor meson. The states | 4 D λλ 1 2 + and | 4 D λλ 3 2 + are most likely to be the moderate states with the total widths of Γ ∼ 100 MeV, which are insensitive to their masses, and might be discovered in their dominant decay channels.

IV. SUMMARY
In the present work, we have systematically studied the strong decay properties of the low-lying 1P and 2D doubly charmed baryons in the framework of the 3 P 0 quark pair creation model. Our main results are summarized as follows.
For the 1P ρ-mode doubly charmed baryons, their decay widths should be fairly narrow because of the absence of the strong decay modes. In addition, for the 1P λ-mode excitations, the states | 2 P λ 3 2 − and | 4 P λ 3 2 − are predicted to be moderate states with a width of Γ ∼ 150 MeV. Their strong decays are governed by the Ξ * cc π or Ξ * cc K channel. However, the states | 2 P λ 1 2 − and | 4 P λ 5 2 − are most likely to be narrow states with a total decay width of Γ ∼ 40 MeV, and their strong decays are dominated by the Ξ cc π or Ξ cc K channel. Such narrow states have good potential to be observed in future experiments. Meanwhile, the dominant decay mode of the state | 4 P λ 1 2 − is Ξ cc π or Ξ cc K as well, but the total decay width of this state is about Γ > 200 MeV.
Since the strong decays of 2D ρρ doubly charmed baryons via emitting a light-flavor meson are forbidden, they mainly decay via emitting a heavy-light meson with a total decay width of several tens MeV if their masses are large enough. The partial strong decay widths of the 2D ρρ doubly charmed baryons strongly depend on their masses. The measurement of masses in the future will be helpful to understand their inner structures.
Within the range of mass we considered, the 2D λλ states with J P = 1/2 + , 7/2 + mainly decay through the Ξ * cc π or Ξ * cc K channels, respectively, while the 2D λλ states with J P = 3/2 + , 5/2 + mainly decay through the Ξ cc π or Ξ cc K channels. It should be remarked that the states | 4 D λλ where p ρ = 1 √ 2 (p 1 − p 2 ) and p λ = 1 √ 6 (p 1 + p 2 − 2p 3 ). The ground state wave function of the meson is where p ab stands for the relative momentum between the quark and antiquark in a meson. Since all the final states are in the S -wave ground states in the present work, the momentum space integration I can be further expressed as Π(l ρA , m ρA , l λA , m λA , m). Based on Fig. 1(a), the explicit form of the momentum space integration Π(l ρA , m ρA , l λA , m λA , m) are presented in the following.
For the S -wave decay, For the P-wave decay, For the D-wave decay, Here, for the above expressions and .

(A12)
Appendix B: The decay mode with a singly heavy baryon plus a heavy-light meson From Fig. 1(c), the momentum space integration Π(l ρA , m ρA , l λA , m λA , m) can be expressed in the following.