Understanding X(3862) , X(3872) , and

Z. Zhou

¹

,

Zhiguang Xiao

²

Abstract

We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of usual quark potential model and the interaction part is described by the quark pair creation model. In a one-parameter calculation, the X(3862), X(3872), and X(3930) state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, χc2(2P ), χc1(2P ), χc0(2P ) predicted in the Godfrey-Isgur model to the DD, DD * , D * D * continuum states. At the same time, we predict that the hc(2P ) state is at about 3902 MeV with a pole width of about 54 MeV. In this calculation, the X(3872) state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.PACS numbers: 12.39. Jh, 13.25.Gv, 13.75.Lb, 11.55.Fv As regards the charmonium spectrum above the openflavor thresholds, general discrepancies between the predicted masses in the quark potential model and the observed values have been highlighted for several years. Typically, among the P-wave n 2s+1 L J = 2 3 P 2 , 2 3 P 1 , 2 3 P 0 , and 2 1 P 1 states, the X(3930), discovered by the Belle Collaboration [1], is now assigned to χ c2 (2P ) charmonium state though its mass is about 50 MeV lower than the prediction in the quark potential model [2][3][4]. The properties of the other P-wave states have not been firmly determined yet. The X(3872) was first observed in the B ± → K ± J/ψπ + π − by the Belle Collaboration in 2003 [5]. Although its quantum number is 1 ++ , the same as the χ c1 (2P ), the pure charmonium interpretation was soon given up for the difficulties in explaining its decays. The pure molecular state explanation of X(3872) also encounters difficulties in understanding its radiative decays. So its nature remains to be obscure up to now. As for the χ c0 (2P ) state, the X(3915) is assigned to it several years ago, but this assignment is questioned for the mass splitting between χ c2 (2P ) and χ c0 (2P ), and its dominant decay mode [6,7]. In Ref.[8], analyses of the angular distribution of X(3915) to the final leptonic and pionic states also support the possibility of being a 2 ++ state, which means that it might be the same tensor state as the X(3930). Very recently, the Belle Collaboration announced a new result about the signal of X(3862) which could be a candidate for the χ c0 (2P ) [9]. The 2 1 P 1 state has not been discovered yet. These puzzles have been discussed exhaustively in the literatures (see refs. [10][11][12] for example), but a consistent description is still missing.In this paper, we adopt the idea of Gamow states and the solvable extended Friedrichs model developed recently [13][14][15], usually discussed in the pure mathematical physics literatur...

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