The d* dibaryon structure and the hidden color channel effect Jia-lun Ping, Hong-xia Huang Dept. of Physics, Nanjing Normal University Fan Wang Dept. of Physics, Nanjing University

Outline I.The Discovery of d* dibaryon II.The hidden color components explanation of the small width of d* III.The classification of 6-quark states IV.The d* structure V.summary

I. The discovery of d* dibaryon 1964 F.J. Dyson and N.H. Xuong predicted a D03(IJP=03+) 6-quark state at 2.35 GeV based on M.Gell-Mann quark model. PRL 13(1964) we predicted the IJP=03+ is an inevitable dibaryon resonance and named it as d*. T. Goldman, et al. PRC39(1989) we extended the Feshbach resonance to NN and channel coupling and obtained the right resonance mass and partial width, strongly supported WASA-at-COSY experimental search of d* dibaryon. J.L. Ping et al, arxiv: [nucl-th], PRC79(2009)

2011 WASA-at-COSY found evidence of the d* exotics M. Bashkanov et al., arxiv: [nucl-ex]; [nucl-ex]; PRL 102(2009)052301;106(2011)242302; CERN Courier 2011, Aug WASA-at-COSY confirmed the existence of the d* six-quark state. P. Adlarson et al, arXiv: [nucl-es];PRL 112(2014)202301; CERN Courier 2014, July 23.

II.The hidden color components explanation of the small width of d* dibaryon The observed width (~80 MeV) of the d* dibaryon is much smaller than the two width (~240 MeV) even after taking into account the reduction due to (~84 MeV) binding, our theoretical estimate based on the naïve phase space reduction is ~110 MeV (PRC79(2009)024001). A popular explanation of the small width of the d* dibaryon is attributed to the hidden color components of d*: M. Bashkanov, S.J. Brodsky, H. Clement, Phys. Lett. B727(2013)438; arXiv: [hep-ph]. F. Huang, et al., Chin. Phys. C39(2015)7, Y.B. Dong, et al. Phys. Rev. C91(2015)6,

Different interpretation of the hidden color component effect There are two independent components of d* within the limited two baryon Hilbert space, i.e., the d* consists of two orbital ground state baryons. One is the colorless, the other is a hidden color component. F. Huang and Y.B. Dong both assume that the hidden color channel component dominates the d* structure and it does not decay at all. BBC also assume the hidden color channel component is the dominant one but it decays through.

III.The classification of 6-quark states Nuclear physics employed both group chain classification wave functions and cluster wave functions for multi-body systems. They are also called symmetry bases and physical bases. We use color singlet quark cluster wave functions to study the NN interaction and add hidden color quark cluster wave functions to study the color van der Waals force in the beginning of 1980’s. F. Wang, Y. He, Chinese Nucl. Phys. 2(1980)261; 4(1982)176; in Contributed papers of the workshop “from collective states to quark in nuclei, p.2, Bologna, Italy, Nov , 1980.

M. Harvey introduced the transformation between physical and symmetry bases. Nucl. Phys. A352(1981) For the 6-quark cluster wave function one mainly interested in those ones where every three quarks form a ground state baryon, i.e., three quarks occupy the same lowest orbital state (s-wave state). Two baryons located in two different centers, the left one has the left centered s wave l, the right one has the right centered s wave r. The overlap is between 1 and 0 depends on the separation x of the two centers. Only in the infinite separation limit the ->0, i.e., mutual orthogonal.

Under the totally orbital symmetry assumption [3] of the individual baryon, the orbital symmetry of 6-quark system is limited to The parity of the first two are positive and the second two are negative. Because of the non-orthogonality of l and r orbit, only the totally symmetry [6] survives as a state in the x->0 limit.

In order to keep all of the four symmetry bases normalized in any separation x, Harvey nomalized every symmetry basis with a nomalization factor N([f]). In the x- >0 limit, the four orbital symmetry bases have the following limit. [6]-> [42]-> [51]-> [33]-> Harvey’s cluster bases are mutual orthogonal, but not easy to use in quark model calculation.

The usual cluster wave functions are easier to be used in quark cluster model calculations and so they are the commonly used ones. These cluster wave functions (after anti –symmetry) are not mutual orthogonal because the individual quark orbital wave function l and r are not orthogonal. Based on these cluster bases to count how much percentage of hidden color component within d* is meaningless. In the cluster separation x->0 limit, only the [6] symmetry basis survives (to be a state), all the other symmetry states disappear. In the x-> limit, to claim the [6] symmetry state is still a mixture of colorless and hidden color channel is meaningless because there is only one state [6] survives for the IS=03 channel. The colorless and hidden color anti-symmetry cluster bases collapse to one state.

Harvey used the group chain classified symmetry bases to express the colorless channels, but used another group chain classified symmetry bases to express the hidden color channels. J.Q. Chen et al. remedied this drawback and developed a systematic method to calculate the transformation coefficients between symmetry and cluster bases. For deuteron channel they obtained 5 hidden color channels and two colorless channels. J.Q. Chen et al., Nucl. Phys. A393(1983)

The symmetry bases and the cluster bases (including both colorless and hidden color channels) are equivalent and both can be used to diagonalize the Hamiltonian of the multi-quark systems. In fact we can also use solely the colorless or hidden color channels if one allows the orbital excited hadron clusters, because one can transform the hidden color channels to colorless channels and vice versa via the quark exchange. D. Robson, Prog. Part. Nucl. Phys., 8(1982)257; F. Wang, C.W. Wong, Nuovo Cimento, 86A(1985)283; F. Wang, Chinese Prog. Phys., 9(1989)297.

For color SU(3), the dimension of irreps [222] =1 For S(6), the dimension of irreps [222] =5 Constructing the bases of 6-quark system from two three-quark clusters Outer-product of S(6) 5 bases of [222] [111]x[111] (color singlet) color singlets are enough!

IV.The d* structure The d* should be the lowest state with quantum number IS=03 obtained from the diagonalization of the 6-quark Hamiltonian. We diagonalized two quark model Hamiltonian within the two channel space, one colorless and one hidden color channel, to obtain the eigen- states.

For Harvey’s two cluster bases our result is almost a half-half mixing. c1: color-singlet c8: color-octet x (fm) E1(MeV) c12 c82 E2(MeV) c12 c

For the usual cluster bases, the mixing coefficients are not the probability amplitude. To obtain the probability amplitude one has to orthonomalize the cluster channel wave function. We first diagonalize the 6-quark model Hamiltonian with the two anti-symmetric cluster wave function, obtain the ground state d* eigen-function, then calculate the overlap between the anti-symmetric colorless and hidden color channel wave function, then define the hidden color component as. We admit this is just one possible definition of an orthogonal component and we take it as the hidden color channel wave function.

Under this definition of hidden color channel, our result has quite small hidden color component for the ground state. x (fm) E1(MeV) c1 2 c8 2 E2(MeV) c1 2 c At least our two results show that how much hidden color component within the d* is model dependent.

To use the two quark cluster channel wave functions as the bases limit the Hilbert space of the multi-quark systems. We found this limitation can not be a good approximation when two baryon very close together even for the loosely bound deuteron d. F. Wang et al., Phys. Rev. Lett., 69(1992)2901. d* is a tightly bound system, its rms radius is about 1fm, binding energy is 84 MeV, so we relax this limitation by allowing different quark clustering components, i.e., in addition to the 3,3 quark cluster we add 4,2; 5,1; 6,0 clusters and found all of these configurations does exist in the d*.

We also do a 6-body calculation without assuming any clustering and this calculation is on going.

V. Summary d* is compact deep bound six quark state, two baryon clustering is a crude approximation. Up to now the hidden color channel wave function is just a coupling scheme, the hidden color component effect of d* is model dependent. With the two baryon clustering approximation of d* structure, the d* is always a mixing of the hidden color component and the colorless channel. The percentage of hidden color component is dependent on the definition. The colorless component directly decay but with a phase space reduction. The hidden color channel also decay to through the channel coupling.