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

Strong decays of the and doubly charmed states

Li-Ye Xiao 111E-mail: lyxiao@pku.edu.cn, Qi-Fang Lü 222E-mail: lvqifang@hunnu.edu.cn, and Shi-Lin Zhu 333E-mail: zhusl@pku.edu.cn 1) School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China 2) Center of High Energy Physics, Peking University, Beijing 100871, China 3) Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China 4) Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China 5) Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
Abstract

We perform a systematical investigation of the strong decay properties of the low-lying - and -wave doubly charmed baryons with the quark pair creation model. The main predictions include: (i) in the and family, the mode excitations with and should be the fairly narrow states. (ii) For the mode excitations, and have a width of MeV, and mainly decay into the ground state. Meanwhile, and are the narrow states with a width of MeV, and mainly decay into the ground state with . (iii) The states mainly decay via emitting a heavy-light meson if their masses are above the threshold of or , respectively. Their strong decay widths are sensitive to the masses and can reach several tens MeV. (iv) The states may be broad states with a width of MeV. It should be emphasized that the states with and mainly decay into the ground state with plus a light-flavor meson, while the states with and mainly decay into the ground state with plus a light-flavor meson.

pacs:

I Introduction

Fifteen years ago, the SELEX Collaboration announced a doubly charmed baryon with mass 35191 MeV Mattson:2002vu (). One year later, another doubly charmed baryon was reported at 3770 MeV by the same collaboration Moinester:2002uw (). Unfortunately, those two signals and were not confirmed by other collaborations. Recently, the LHCb Collaboration discovered a doubly charmed baryon in the mass spectrum Aaij:2017ueg (). Its mass was measured to be 3621.400.720.27 MeV. The newly observed may provide an access point for the study of doubly heavy baryons and has attracted significant attention from the hadron physics community n1 (); n2 (); Wang:2017mqp (); n4 (); n5 (); n6 (); n7 (); n8 (); Li:2017 (); Yu:2017 (); Hu:2005gf (); Hu:2017dzi (); Xiao:2017udy (); Chen:2017jjn (); Chen:2017sbg (); Lu:2017meb ().

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 Roncaglia:1995az (); Gershtein:2000nx (); Itoh:2000um (); Ebert:2002ig (); Roberts:2007ni (); Yoshida:2015tia (); Shah:2016vmd (); Gershtein:1998sx (); Zhang:2008rt (); Wang:2010hs (); Brown:2014ena () and semi-leptonic decays Faessler:2001mr (); Albertus:2006ya (); Roberts:2008wq (); Albertus:2009ww (); Faessler:2009xn (); Onishchenko:2000wf (); Kiselev:2001fw (); White:1991hz (); SanchisLozano:1994vh (); Hernandez:2007qv (); Guo:1998yj (); Ebert:2004ck (); Wang:2017mqp (). However, only a few discussions on the decay behavior exist in literature Hackman:1977am (); Branz:2010pq (); Bernotas:2013eia (); Li:2017 (); Xiao:2017udy (); Lu:2017meb (). In our previous work Xiao:2017udy (), we first systematically investigated the both strong and radiative transitions of the low-lying -wave 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 and 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 Ebert:2002ig (), and OZI allowed decay modes Zhong:2007gp () are summarized in Table 1.

For the low-lying and 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 Xiao:2017udy ().

The QPC strong decay model as a phenomenological method has been employed successfully in the description of the hadronic decays of the mesons Godfrey:2015dva (); Godfrey:2004ya (); Godfrey:2015dia (); Godfrey:2016nwn () and singly charmed baryons Chen:2007xf (); Chen:2017gnu (); Zhao:2016qmh (); Ye:2017dra (); Ye:2017yvl (). Systematical study of the low-lying and 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 states are highly comparable with those in our previous work Xiao:2017udy (); (ii) the states mainly decay via emitting a heavy-light meson if their masses are above the threshold of or , respectively; (iii) although the states may be broad states with a width of 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
Wave function Mass Ebert:2002ig () Strong decay channel Mass Ebert:2002ig () Strong decay channel
3620 3778
3727 3872
3838 4002
3959 4102
4136 4271
4196 4325
4053 4208
4101 4252
4155 4303
,
 
Table 1: Masses and possible two body strong decay channels of the and doubly charmed baryons ( denoted by ), where =  Zhong:2007gp (). The masses (MeV) are taken from the relativistic quark model Ebert:2002ig ().

Ii model

The QPC model was first proposed by Micu Micu:1968mk (), Carlitz and Kislinger Carlitz:1970xb (), and further developed by the Orsay group LeYaouanc:1972vsx (); LeYaouanc:1988fx (); LeYaouanc:1977fsz (). For the OZI-allowed strong decays of hadrons, this model assumes that a pair of quark is created from the vacuum and then regroups with the quarks from the initial hadron to produce two outing hadrons. The created quark pair shall carry the quantum number of and be in a state. Thus the QPC model is also known as the model. This model has been extensively employed to study the OZI-allowed strong transitions of hadron systems. Here, we adopt this model to study the strong decays of the 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 (the initial baryon) (the final baryon)+(the final meson) as an example to show how to calculate the decay width. In the nonrelativistic limit, the transition operator under the model is given by

where (=4, 5) represents the three-vector momentum of the th quark in the created quark pair. and stand for the color singlet and flavor function, respectively. The solid harmonic polynomial corresponds to the momentum-space distribution, and is the spin triplet state for the created quark pair. The creation operator denotes the quark pair-creation in the vacuum. The pair-creation strength is a dimensionless parameter, which is usually fixed by fitting the well measured partial decay widths.

Figure 1: Doubly charmed baryons decay process in the model.

According to the definition of the mock state Hayne:1981zy (), the spacial wave functions of the baryon and meson read, respectively,

(2)
(3)

The denotes the momentum of quarks in hadron and . are the momentum of the hadron . The 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

(4)

The ground state spatial wave function of a meson is

(5)

where the 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,

(6)

Here, stands the spatial integral and more detailed information is presented in the Appendix A and B. The 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

(7)

Finally, the decay width reads

(8)

In the equation, is the momentum of the daughter baryon in the center of mass frame of the parent baryon

(9)
State Mass State Mass State Mass
3621.00 2453.97 134.977
3727.00 2518.41 139.570
3778.00 2452.90 493.677
3872.00 2517.50 547.862
2286.46 2467.93 957.780
2695.20 2575.70 1864.83
2765.90 2645.90 1869.58
1968.27
Table 2: Masses (MeV) of the baryons and mesons in the decays Aaij:2017ueg (); Olive:2016xmw (); Ebert:2002ig ().

In the present calculation, we adopt MeV, MeV, and MeV Godfrey:2015dva () for the constituent quark masses. The masses of the baryons and mesons involved in our calculations, listed in Table 2, are from the Particle Data Group Olive:2016xmw () except for the doubly charmed baryons, which is from Ref. Ebert:2002ig (). The value of the harmonic oscillator strength is 2.5 , for all light flavor mesons while it is for the meson and for the meson Godfrey:2015dva (). The parameter of the -mode excitation between the two charm quarks is taken as GeV Xiao:2017udy (), while between the two light quarks is taken as GeV. Another harmonic oscillator parameter is obtained with the relation:

(10)

For the strength of the quark pair creation from the vacuum, we take the same value as in Ref. Godfrey:2015dva (), . For the strange quark pair creation, we use  LeYaouanc:1977fsz ().

Total
State Mass This work Ref Xiao:2017udy () This work Ref Xiao:2017udy () This work Ref Xiao:2017udy () This work Ref Xiao:2017udy ()
4136 21.9 15.6 18.6 33.9 40.5 49.5 1.18 0.46
4196 13.7 21.6 117 101 131 123 0.18 0.21
4053 200 133 0.60 1.22 201 134 333 110
4101 4.43 7.63 127 84.6 131 92.2 0.03 0.09
4155 45.9 75.3 12.6 22.8 58.5 98.1 3.64 3.30
Total
State Mass This work Ref Xiao:2017udy () This work Ref Xiao:2017udy () This work Ref Xiao:2017udy () This work Ref Xiao:2017udy ()
4271 49.3 33.1 1.53 2.36 50.8 35.5 32.2 14.0
4325 8.50 11.4 199 174 208 185 0.04 0.06
4208 378 323 378 323
4252 2.02 3.08 154 137 156 140 0.01 0.02
4303 29.1 41.5 2.62 4.38 31.7 45.9 11.1 9.47
Table 3: The comparison of the partial decay widths of the states from the QPC model and the chiral quark model  Xiao:2017udy (). stands for the total decay width and represent the ratio of the branching fractions . The unit is MeV.

Iii Calculations and Results

For the -wave doubly charmed states, the masses are adopted from Ref. Ebert:2002ig () (showed in Table 1) due to a good agreement with the mass of the lowest doubly charmed baryon observed by the LHCb collaboration. However, there is no prediction for the masses of the -wave states. So the masses of the -wave baryons are varied in a rough range when their decay properties are investigated.

iii.1 The -wave doubly charmed states

Within the quark model, there are two doubly heavy baryons with and , respectively. Their masses are above the threshold of or . However, the OZI-allowed two body strong decays are forbidden since the spatial wave functions for the and states are adopted with the simple harmonic oscillator wave functions which are orthogonal. In this work, we focus on the strong decays of the states.

We analyze the decay properties of the states in the and family, and collect their partial strong decay widths in Table 3. In the family, the total decay width of is about MeV, which is compatible with the result in Ref. Xiao:2017udy (). The dominant decay modes are and with the partial decay ratio

(11)

This value is about 2.5 times of the ratio in Ref. Xiao:2017udy ().

The states of and are most likely to be the moderate states with a width of MeV, and the decay channel is their dominant decay mode. The partial decay width of is considerable. The partial decay width ratio is

(12)

This ratio may be a useful distinction between and in future experiments. These results are in good agreement with the predictions in Ref. Xiao:2017udy ().

The state has a broad width of MeV, and the decay channel almost saturates its total decay widths. This broad state may be observed in channel in future experiments.

Figure 2: The strong decay partial widths of the -wave states as a function of the mass.
Figure 3: The strong decay partial widths of the -wave states as a function of the mass.
Figure 4: The strong decay partial widths of the -wave states as a function of the mass.

From the Table 3, the state may be a narrow state with a total decay width around 60 MeV, which is about one half of that in Ref. Xiao:2017udy (). This state decays mainly through the channel. The predicted partial width ratio between and is

(13)

which can be tested in future experiments.

In the family, the and might be two narrow states with a total decay width of 40 MeV, and their strong decays are dominated by the channel.

The decay width of the state is about 380 MeV. Meanwhile, its strong decays are governed by the channel. In this case, the might be too broad to observed in experiments. However, for the states and , if their masses lie below the threshold of , they are likely to be two fairly narrow states with the total decay widths of MeV and MeV, respectively. Otherwise, they shall have a broad width of 200 MeV, and mainly decay into channel.

Considering the mass uncertainties of the states, we plot the strong decay width as a function of the mass in Figs. 23. From the Figs. 23, the partial width of dominant decay channel for most of states are sensitive to the mass. In addition, in the family, if the states are above the threshold of , they can decay via with a partial width about several MeV.

Figure 5: The strong decay partial widths of the -wave states as a function of the mass.

iii.2 The -wave doubly charmed states

iii.2.1 -mode excitations

Since we adopt the simple harmonic oscillator spatial wave functions in present work, the strong decays of 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 -wave 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 states in the family in Fig. 4. For the state , we put the mass range between the threshold ( MeV) and MeV. From Fig. 4, we can see that the state is a fairly narrow state with a width of a few MeV when its mass varies in the range. Its strong decay is dominated by .

Taking the masses of and in the range of (4.152-4.450) GeV, they are two narrow states with a width of MeV and mainly decay into if their masses are below the threshold of . However, when the channel is open, the total decay widths of those two states are sensitive to the mass and can increase up to several tenths MeV. If so, their dominant decay modes should be .

For the states and , if their masses are above the threshold of , they mainly decay into and have a width of several tens MeV.

Taking the mass of in the range of (4.20 - 4.60) GeV, we get that the decay width of this state is about MeV. Its strong decays are governed by the channel in the whole mass region considered in the present work. When we take the mass of with MeV, the predicted branching ratio is

(14)

So, this state is most likely to be observed in the channel.

Then, we analyze the decay properties of the states in the 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 , we plot its decay widths as a function of the mass in the range of GeV. From the figure, its strong decay width is around a few MeV. This state mainly decays through the channel.

For the states and , we take their masses in the range of GeV. If they lie below the threshold, the total decay widths are about MeV, and are dominated by . However, if their masses are above the threshold of , their dominant decay channels should be and their total decay widths may reach several tenths MeV.

Taking the masses of and in the range of GeV and GeV, respectively, their decay widths depend considerably on their mass and are governed by the channel.

Assuming the mass of the in the range of (4.35-4.75) GeV, this state has a width of MeV. If we take the mass of with MeV, the total decay width is about MeV, and the predicted branching ratio is

(15)

In brief, the states of and can decay through emitting a heavy-light meson when their masses are above the threshold of and , 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 or , respectively.

Figure 6: The strong decay partial widths of the -wave states as a function of the mass.
Figure 7: The strong decay partial widths of the -wave states as a function of the mass.

iii.2.2 -mode excitations

As emphasized in our previous work Xiao:2017udy (), the -mode orbitally excited state has relatively larger mass than a -mode orbitally excited state for the doubly charmed baryons. The states should be heavier than the states with the same . Thus, many other decay modes are allowed when we study the strong decay properties of states.

In the family, we estimate the mass of the 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 is about MeV. The main decay channel is and the predicted branching ratio is

(16)

On the other hand, the partial decay width of is sizable. The partial width ratio between and is

(17)

when we fix the mass of this state on GeV.

For the state , its mass might be in the range of (4.55-4.95) GeV. The dependence of the strong decay properties of 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 MeV, and mainly decays into and . The predicted partial width ratio is

(18)

Meanwhile, the role of the channel becomes more and more important as the mass increases. The branching ratio is

(19)

We estimate the mass of 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 is a moderate state with a width of MeV, and its strong decays are governed by the channel. The predicted branching ratio is

(20)

It should be pointed out that if the decay channel is opened, which is sensitive to the mass, the branching ratio of this decay channel may reach  41. Since the predicted width of