Few-Body Syst (2013) 54:1023–1026DOI 10.1007/s00601-013-0629-x

S. Yasui · S. Ohkoda · Y. Yamaguchi · K. Sudoh · A. Hosaka

Doubly Charmed Exotic Mesons

Received: 30 September 2012 / Accepted: 11 January 2013 / Published online: 31 January 2013© Springer-Verlag Wien 2013

Abstract We discuss the mass spectrum of doubly charmed mesons as hadronic molecules composed byD and D∗ meson. Considering the heavy quark symmetry and chiral symmetry, we introduce the one-boson(π, ρ, ω) exchange potential between D and D∗ meson. For all possible quantum numbers I (J P) with isospinI , total angular momentum J (≤ 2) and parity P , we solve the fully coupled channel Schrödinger equation.We find that in many quantum numbers the bound and resonant states composed by D or D∗ meson can existnear the DD, DD∗ and D∗D∗ thresholds.

1 Introduction

Recent experimental discoveries, such as X, Y, Z in the charm sector and Yb and Zb in the bottom sector,have motivated many researchers to study the properties of exotic hadrons. The nature of exotic hadrons areintimately related the fundamental problems of the strong interaction, color confinement and chiral symmetrybreaking in vacuum and so on. New candidates of exotic hadrons, doubly charmed tetraquarks Tcc with quarkcontent ccud (two charm quarks c’s and two light antiquarks u and d), are particularly interesting. Because thereis no quark–antiquark pair annihilation, Tcc has a genuine tetraquark content. In literatures, it was discussedthat the mass of Tcc with quantum number I (J P) = 0(1+) (isospin I , total angular momentum J and parityP) can lie below the lowest threshold by D and D∗ mesons. There, the driving force to stabilize Tcc is suppliedfrom a strong attraction in ud (so-called “good”) (anti)diquark. Many calculation of four quark problem in thequark model, as well as the diquark model, have reached the conclusion that Tcc is a bound state [1–4].

However, the tetraquark structure by ccud is not the only possible form of Tcc. Suppose the situation thatccud is separated to cu and cd subsystems, and the distance between cu and cd is larger than the typical size

This work is supported in part by Grant-in-Aid for Scientific Research on Priority Areas “Elucidation of New Hadrons witha Variety of Flavors (E01: 21105006)” (S.Y. and A.H.) and by “Grant-in-Aid for Young Scientists (B) 22740174” (K.S.), fromthe ministry of Education, Culture, Sports, Science and Technology of Japan.

Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20–25 August, 2012, Fukuoka, Japan.

S. Yasui (B)KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization,1-1, Oho, Ibaraki 305-0801, JapanTel: +81-29-8796097Fax: +81-29-8796101E-mail: yasuis@post.kek.jp

S. Ohkoda, Y. Yamaguchi, A. HosakaResearch Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka 567-0047, Japan

K. SudohNishogakusha University, 6-16, Sanbancho, Chiyoda, Tokyo 102-8336, Japan

1024 S. Yasui et al.

of the hadrons, say 1 fm. Then, the degrees of freedom (d.o.f.) will be changed to D and D∗ mesons, whichmay be called hadronic molecule picture [5–8]. The change of d.o.f. introduces interesting new properties ofthe dynamics governing the D∗ and D∗ systems in model independent way [8]. First, the masses of D andD∗ mesons are regarded to be degenerate in the heavy quark mass limit. There, the spin symmetry (the heavyquark symmetry) is a good symmetry. We denote D(∗) for D or D∗ meson. Second, because D(∗) mesons havean isospin charge, there appears pion exchange between D(∗) mesons. The pion exchange is the most importantinteraction in the long range force between hadrons, as a nature of the Nambu–Goldstone mode. From thosetwo points, we will see that D(∗)D(∗) systems have sufficient attraction to form D(∗) bound and/or resonantstates [8] (see [9–11] for other exotic heavy systems).

2 Mass Spectrum of D(∗)D(∗) Systems

We consider the heavy quark symmetry and chiral symmetry for D(∗) meson systems [8]. As an interactionbetween two D(∗) mesons, we introduce the one-boson (π, ρ, ω) exchange potentials. The vertices of theexchanged bosons and D(∗) meson are given by the Lagrangians,

Lπ H H = g trHa Hbγνγ5 Aνba, (1)

LvH H = −iβtrHa Hbvμ(ρμ)ba + iλtrHa Hbσμν Fμν(ρ)ba , (2)

where the multiplet field H with a scalar field P and a vector P∗ field, each of which stands for D and D∗mesons, respectively, is defined by Ha = 1+/v

2

[P∗aμγ μ − Paγ5

]with the four velocity vμ of the heavy mesons.

The conjugate field is defined by Ha = γ0 H†a γ0, and the index a denotes up and down flavors. The axial current

is given by Aμ � ifπ

∂μπ where fπ = 135 MeV is the pion decay constant. The coupling constant |g| = 0.59for π P P∗ is determined with reference to the observed decay width � = 96 keV for D∗ → Dπ . The vector (ρand ω) meson field is defined by ρμ = i(gV /

√2)ρμ and its field tensor by Fμν(ρ) = ∂μρν − ∂νρμ + [ρμ, ρν]

where gV = 5.8 is the coupling constant for ρ → ππ decay. From those vertices, we can obtain the π , ρ, ωexchange potential by introducing the form factors at each vertex (see Ref. [8] for more details).

The states composed by two D(∗) mesons are classified by quantum number I (J P) with isospin I , totalangular momentum J and parity P . Here it is important to note that D and D∗ mesons can be mixed dueto the approximate mass degeneracy. For example, the wave function for 0(1+) state contains four channels;

1√2

(DD∗ − D∗D) (3S1), 1√2

(DD∗ − D∗D) (3 D1), D∗D∗(3S1) and D∗D∗(3 D1). It is also important to note thatboth S and D waves are contained in the wave functions. Indeed, the tensor force from one-pion exchangepotential induces a strong attraction by mixing the S and D waves. This is analogous to the binding mechanismof the deuteron, where one-pion exchange potential plays the most dominant role. For other I (J P) states, seeTable 1.

We show the obtained mass spectrum for D(∗)D(∗) systems in Fig. 1. We find that there are bound andresonant states in several quantum numbers I (J P)’s. For example, in 0(1+), D(∗)D(∗) bound stats with mass3813 MeV can exist. This is a bound state, because the lowest threshold is given by DD∗ (see also Table 1).

Table 1 Possible channels of D(∗)D(∗)(2S+1 L J ) for I and J P (J ≤ 2) (Ref. [8]).

0− 1√2(DD∗ + D∗D)(3 P0)

1+ 1√2

(DD∗ − D∗D) (3S1),1√2

(DD∗ − D∗D) (3 D1), D∗D∗(3S1), D∗D∗(3 D1)

0 1− DD(1 P1),1√2

(DD∗ + D∗D) (3 P1), D∗D∗(1 P1), D∗D∗(5 P1), D∗D∗(5 F1)

2+ 1√2

(DD∗ − D∗D) (3 D2), D∗D∗(3 D2)

2− 1√2

(DD∗ + D∗D) (3 P2),1√2

(DD∗ + D∗D) (3 F2), D∗D∗(5 P2), D∗D∗(5 F2)

0+ DD(1S0), D∗D∗(1S0), D∗D∗(5 D0)

0− 1√2

(DD∗ − D∗D) (3 P0), D∗D∗(3 P0)

1 1+ 1√2

(DD∗ + D∗D) (3S1), 1√2

(DD∗ + D∗D) (3 D1), D∗D∗(5 D1)

1− 1√2

(DD∗ − D∗D) (3 P1), D∗D∗(3 P1)

2+ DD(1 D2),1√2

(DD∗ + D∗D) (3 D2), D∗D∗(1 D2), D∗D∗(5S2), D∗D∗(5 D2), D∗D∗(5G2)

2− 1√2

(DD∗ − D∗D) (3 P2),1√2

(DD∗ − D∗D) (3 F2), D∗D∗(3 P2), D∗D∗(3 F2)

Doubly Charmed Exotic Mesons 1025

Fig. 1 Mass spectrum for D(∗)D(∗) systems with various quantum numbers I (J P ). The solid lines indicate the position of themass and the numbers above the solid lines stand for E − i�/2 with E the mass and � the decay width. The grey solid linesindicate the strongly bound state (Ref. [8])

The binding energy is 62 MeV which is measured from the DD∗ thresholds. The bound state also appearsin 0(0−) and 0(2−) (the threshold is DD∗). Here we note that the bound states in 0(1+) and 0(0−) have thestrong binding energy, and the radii between D and D∗ mesons is rather small, 0.8 fm [8]. On the other hand,the bound state in 0(2−) has the small binding energy 4.3 MeV, and the radius is 1.6 fm [8]. Therefore, the0(2−) state will be regarded as a bound state of D and D∗ mesons, while the other bound states may not. Itis interesting to find several resonant states. Many of them (except for 0(2+)) have small decay width, hencethey are regarded as narrow states.

We will briefly mention that the mass spectroscopy of the D(∗)D(∗) bound/resonant states are qualitativelydifferent from that of the compact tetraquark. First, in the D(∗)D(∗) bound/resonant state, the ground state hasnegative parity (0(0−)), while the ground state in the compact tetraquark is positive parity (0(1+)). Second,many resonant states can exist in the D(∗)D(∗) systems, while no resonant state is found in the compact tet-raquark. Third, isotriplet state can exist in the D(∗)D(∗) system in 1(0−), while no isotriplet state is seen incompact tetraquark. Although it is tempting to seek for more details in two pictures, it is left for future works.

3 Summary

We have discussed the possible existence of doubly charmed mesons as bound or resonant states of D(∗)D(∗)

mesons. Using the heavy quark symmetry and chiral symmetry, we have obtained the one-boson (π , ρ, ω)exchange potentials between two D(∗) mesons. By solving the Schrödinger equation with fully coupled chan-nels, we have found several bound and/or resonant states of D(∗)D(∗) systems. The obtained mass spectrum willbe useful to research doubly charmed mesons in future experiments, such as in heavy ion collisions [12,13],electron-positron collisions [14] and so on.

Acknowledgments We thank to fruitful discussions with Prof. S. Takeuchi and Prof. M. Takizawa.

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