Bled Workshops in Physics Vol. 9, No. 1 p. 71 A(1405) and X(3872) as multiquark systems S. Takeuchi Japan College of Social Work, Kiyose, Tokyo 204-8555, Japan Abstract. We have investigated the effects of (qq) pairs on the baryons and mesons by employing two examples: A(1405) and X(3872). The A( 1405J resonance is treated as a q3-qq scattering which couples to the q3 orbital (0s)2 0p state by the one-gluon exchange interaction. Due to the coupling of this q3 state, we find that a peak appears at around 1405 MeV. We also investigate the system by employing a baryon-meson model with a separable interaction. By simplifying the model, we can clarify the mechanism and condition to form a peak. As for the X(3872), we investigate qqcc isospin 1 and 0 systems with the orbital correlation. For the isospin 0 system, we also consider its coupling to the cc state. The results show that there can be a bound state of qqcc with Jp c = 1+ +, which is a coupled state of the J/F-p(or w) and D-D* molecules with a multiquark configuration in the short range region. Both of the two examples indicate that an extra (qq) pair may play important roles especially in the excited hadrons. 1 A(1405) by a quark model1 Properties of the A(1405) is hard to understand; the conventional quark picture, which assumes the q3 (0s)2(0p) configuration, cannot give the observed A(1405) light mass, nor the large splitting between A(1405) and A(1520) [2]. Moreover, since A(1405) has a large width, the mixing between this q3 state and the continuum should not be neglected. To describe A(1405) as a peak in the baryon-meson scattering, we have investigated q3-qq scattering system with a q3 pole [1]. The scattering is solved by employing the Quark Cluster Model (QCM). The pole, which we assume the flavor-singlet q3 (0s)2(0p) state, is treated as a bound state embedded in the continuum (BSEC). In the present model, the effective quark interaction consists of the one-gluon exchange (OGE) and the instanton-induced interaction (Ins) as well as the linear confinement potential. With a parameter set which reproduces both of the observed S-wave flavor-octet baryon and meson mass spectra, we perform the Itt-NK coupled channel QCM. We found that the peak energy can be 1405 MeV, namely by about 30 MeV below the NK threshold in the spin j isospin 0 channel even if the mass of the q3 pole without the coupling is taken to be the conventional quark model value, which is above the threshold by about 55 MeV. The peak disappears when the 1 This work has been done in collaboration with Kiyotaka Shimizu (Sophia University) [1]. 1 2 .c .c 0 1 1 _ -Mass spectrum "--- 8 (Sp) „-----.... 8 (NK) a / M q3 pole energy / ] iNK threshold i i r b 0 0 100 200 300 C.M.Energy above Sp threshold [MeV] Fig.l. Mass spectrum and the phase shift (6) of the Ln and NK coupled channel QCM. 3 2 1 coupling to the q3 pole is switched off. The obtained peak width agrees with the experiments reasonably well. The NK scattering length is roughly half of the observed value [3]. For details, please check our paper [1]. 2 A(1405) by a baryon-meson model2 Recently, it was reported that a baryon-meson model with the chiral unitary approach can reproduce the A(1405) peak without the help of an quark pole [4,5]. Then a new question arises: there should be the flavor-singlet q3 state, which is supposed to affect the baryon-meson scattering in this energy region. To investigate the mechanism and condition to form the peak, we employ a simple baryon-meson model with the semi-relativistic kinematics. T = V + VG(0)T (1) G(0) gp d4q M (2n)4 a Etot - q0 - a + ie q02 _ + ie d3 q mM 1 1 (2n)3 ^a 2m Etot _ q0 _ a + ie (2) where M[m] is the baryon [meson] mass, Cl= y^W + q2, and cu = \Jm2 + q2. The model also includes BSEC, which can be considered as the flavor-singlet q3 pole, or more accurately, as a pole not originated from the baryon-meson degrees of freedom. We divide the model space into P (the baryon-meson space) and Q (the BSEC space). Because the Q-space contains only one state, we can safely This work has been done in collaboration with Kiyotaka Shimizu (Sophia University). set Vqq = 0. Using P + Q = 1, we obtain the T-matrix as: T = T(P) + (1 + VppGp)VpqGqVqp(1 + GpVpp) , (3) where T(P) is the T-matrix solved within the P-space. The potential we employ is separable: Vpp=^fij^exp[-la2(p2+p'2)] (4) i rl IS .1 ^ Pole energy without coupling 1 '' \ \ ' IV ' 0 100 200 300 C.M. Energy above Sp threshold [MeV] on 2 i i -Mass spectrum ---- o(Sp) - I ! \ Single channel - If 1 1 I \ ' i Calculation I / \ 1 f / If i \ Pole energy - // A without coupling \ / —i v v / n. \/ - 2 0 100 200 300 C.M. Energy above Sp threshold [MeV] 4 4 3 3 2 Fig.2. Mass spectrum and the phase shift (6) given by the baryon-meson model with the FF-type (left figure) or the CM-type (right figure) potential. It is found that the FF-type model can reproduce the peak without introducing an extra pole if the cutoff energy of the baryon-meson interaction is rather high. This situation is similar to the chiral unitary approach. One of the key points here is that the green function, eq. (2), contains the m/w factor, which suppresses the strong attraction in the nL channel. This picture corresponds to the condition (A) mentioned above. When one uses the form factor which corresponds to the baryon and meson sizes in the quark model, however, the effective cutoff becomes lower, and the interaction becomes weaker. In such a case, the model requires an extra pole, which can be considered as the flavor-singlet q3 pole, to reproduce the observed peak (Figure 2). The situation corresponds to the condition (B). By assuming c1 =0 and and cp=0 (ci -type in the Figure 2), the peak actually becomes broad. The NK scattering length becomes —1.68+i0.42, which also agrees well with the experimental value, —(1.70±0.07)+i(0.68±0.04) [3]. When we employ the AAo-CT-type interaction, we find that the model reproduces a peak similar to the original one by introducing the q3 pole. The situation also corresponds to the condition (B). Here, we use the c1 -type for the simplicity, though both of the c1 and cp have nonzero values in the quark model picture, which can be obtained by keeping the center of mass momentum of the quark system equal to zero. Table 2. Matrix elements of the interactions between qq pairs. The color-magnetic interaction, — ((A • A)(ff • ff)}, is denoted as CMI, the pair-annihilating term of OGE (OGE-a), the spin-color part of the instanton induced interaction (Ins), and estimate value by a typical parameter set, E. color spin flavor CMI OGE-a Ins E[MeV] States 1 0 1 -16 0 12 84 r| 1 0 8 -16 0 -6 -327 n,K 1 1 1 16/3 0 0 63 w 1 1 8 16/3 0 0 63 P 8 0 1 2 0 3/4 41 8 0 8 2 0 -3/8 15 8 1 1 -2/3 9/2 9/4 97 8 1 8 -2/3 0 -9/8 -34 ccqq with Jpc =0+ + ,l+-,l++,2+ + We argue that both of the pole originated from the quark degrees of freedom and the baryon-meson continuum play important roles to form the A(1405) resonance[6]. 3 X(3872)3 After the discovery and the confirmations of the peak X(3872) and enhancement X(3941) in the n+n- J/ty channel [8,9], many works on these peaks have been reported. The peak X(3872) does not seem a simple cc state, as was summarized in, e.g., Ref. [10]. The fitting of the nn mass spectrum of the experiment suggests that the peak X(3872) is p + J/ty with JPC = 1 ++ [11]. Many theoretical works have also been performed. It was suggested that this peak is a higher partial wave of the charmonium state, a DD* molecule, a qqcc multiquark state, or the bound state of the charmonium with a glue-ball, ccg. The situation is summarized, e.g., in ref. [12]. One of the most promising explanations is that the peak is a qqcc state. The width of the X(3872) is narrow, less than 2.3 MeV [9]; namely, its decay to the DD channel should be forbidden. This restricts the spin-parity of the state. It seems that 1 ++ state is the strongest candidate [12]. In this work, the qqcc systems are investigated by a quark model with the orbital correlations. The model hamiltonian has the long-range n- and a-meson exchange between quarks in addition to OGE and Ins. The wave function of the qqcc systems consists of the color, flavor, spin, and orbital parts. The flavor part is taken to be qqcc. The spin of the qq, as well as that of cc, is taken to be 1, so that the C-parity is kept positive within this part. The total spin is also taken to be 1. The orbital correlation is fully taken into account by performing the the stochastic variational approach. The color part has two 3 This work has been partially done in collaboration with Amand Faessler, Thomas Gutsche, Valery E. Lyubovitskij (X(3872), Inst. für Theo. Physik, Universitat Tübingen) and published in Ref. [7]. 0.5 R [fm] Fig.3. Density distribution of the qqcc bound state in the T=1 JPC =1 + + channel. Table 3: Binding energies of the qqcc state. IJP^ 11++ 01++ Parameter set A 26 MeV 5 MeV Parameter set B 5 MeV Not bound Parameter set A + cc-pole 26 MeV -25 MeV Density distribution -J/y-p color-singlet — D-D* color-singlet ----J/y-p color-octet ---- D-D* color-octet 0 1 2 3 components: the one where the cc pair is color-singlet, (T/4>p)n, and the color-octet one, (J/fyp)s8. Since the hyperfine interaction between the quarks is inversely proportional to mquark, properties of this system depend mainly on the interaction between the light quark-antiquark pair. In Table 2, we show the matrix elements of relevant interactions: the color-magnetic interaction (CMI), the pair-annihilating term of OGE (OGE-a), Ins, and an estimate by a typical parameter set used for a quark model. The most attractive pair is the color-singlet, spin 0, flavor-octet, which exists, e.g. in the pion. There is another weak, but still attractive pair: the color-octet, spin 1, flavor-octet one. Such a pair is found in the qqcc isospin T=1 systems. T=0 pairs may also be attractive if OGE-a and Ins are weak, whose size is not well known in these channels. By using a parameter set which gives correct baryon and meson spectrum, we find a JPC = 1 ++ bound state for each of the T=1 and 0 channels (Table 3). The absolute value of the binding energy, however, depends on the strength of the c-meson exchange: we can also find a parameter set which gives equally good hadron mass spectrum, but gives a bound state only for the T=1 state. In Figure 3, the density distribution of the (J/1p)n and (J/1p)88 components in the T=1 bound state is shown as a function of relative distance between J/1 and p. The (J/1p)n component, having a long tail, looks like a J/1-p molecule. (J/1p)88, in which the confinement keeps the two color-octet mesons close, has large overlap to (DD*)n. So, we also show the the density distribution of the (DD*)n and (DD*)88 components as a function of relative distance between D and D* in the figure. The (DD*)n component has also a long tail, which looks again like a molecule. The obtained bound state, however, is not a simple two-meson molecule. The multiquark component, where quarks in different color-singlet mesons are also correlated, is found to be important; suppose the orbital wave function is re- stricted to ^(Rj/pp) and ^D^D*^(RDD*) without inter-meson quark cor- relation, the binding energy reduces by 17 MeV. As for the T=0 channel, there should be a mixing between the qqcc state and the cc excited state. We assume that it occurs by OGE, as we did in A(1405), and that the mass of cc state is 3950 MeV, which corresponds to the value calculated by Godfrey et al. [13]. When this coupling is switched on, we find that the binding energy increases by about 20 MeV (the precise value depends on the parameters). Namely, masses of the isospin 1 state and 0 state can be close to each other, which may cause a rather large mixing between these states. Since the isospin symmetry of this system is broken as seen from mD± -mDo = 4.78 MeV, X(3872) may be a superposition of the above two bound states. 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