Possible Hexaquark Explanation for the 4406.9 MeV/c2 Resonance in Electron-Positron Collisions at Center-of-Mass Energies from 4.23 to 4.70 GeV

Abstract


Introduction
In Ref. 1, the energy dependent production cross section was measured for the e + e¯ → π + π¯ Ψ(3823) reaction at center-of-mass energies from 4.23 to 4.70 GeV, and resonance structures were observed in the cross section with significances exceeding 5σ.The results of the fit to the e + e¯ → π + π¯ Ψ(3823) distribution, yield two resonances 1 .The first resonance (R 1 ) has a mass of 4406.9MeV/c 2 and is based on a π + π¯ Ψ(3823) structure.A second (R 2 ) is based on a π + π¯ γ η c1 structure with a mass of 4647.9MeV/c 2 .The spin and parity of these states was not provided in Ref. 1.As noted in Ref. 1, these particles are not readily accommodated in the spectrum of conventional charmonium states, and are considered to be potential candidates for QCD exotic hadrons.In this paper, the general first-order weakly coupled meson-meson-meson hexaquark mass formulas of Refs. 2 -5 are applied to evaluate the possible mass and J π value of the candidate R 1 .The R 1 mass relationship is specifically based on a weakly coupled π + π¯ Ψ(3823) structure

Model and Formulation
Zel'dovich and Sakharov 6,7 proposed a semiempirical mass formula that provides a prediction of meson and baryon masses in terms of effective quark masses.Within this formulation, quark wave functions are assumed to reside in their lowest 1S state.These meson mass formulas are used as the basis for deriving a first-order hexaquark mass formula.In particular, the model proposed in this paper assumes the hexaquark is partitioned into three meson clusters with the interaction between the clusters providing a minimal contribution to its mass.
The meson mass (M m ) formula of Refs.6 and 7 is: where m 1 (m 2 ) are the mass of the first (second) quark comprising the meson, m o is the average mass of a first generation quark ,8,9 , and the σ i (i = 1 and 2) are the spin vectors for the quarks incorporated into the meson.The parameters δ m and b m are 40 MeV/c 2 and 615 MeV/c 2 , respectively 7 .
The last term in Eq. 1 represents the spin-spin interaction of the quarks and σ 1 •σ 2 is the scalar product of the quark spin vectors.σ 1 •σ 2 has the value -3/4 and +1/4 for pseudoscalar and vector mesons, respectively 7 .
In formulating the hexaquark mass formula, effective quark masses provided by Griffiths 8 are utilized.These effective masses for d, u, s, c, b, and t quarks are 340, 336, 486, 1550, 4730, and 177000 MeV/c 2 , respectively.The effective masses are utilized in Eq. 1.

First-Order Mass Formula
The spin and parity of a hexaquark within the first-order mass formula, summarized in Section 2.0, is determined by coupling the three meson clusters where the first-order mass formula assumes a minimally interacting L(12) = L(23) = 0 configuration ,2-5 between the clusters.Considering that L(12) = L(23) = 0, Eq. 3 simplifies the weakly coupled three meson system to the form Eq. 4 provides a primitive J π assignment using the three meson clusters comprising R 1 .The π + and π¯ have a 0ā ssignment, and the ψ(3823) is a 2¯ state.Using Eq. 4, the R 1 has the following assignment The R 1 mass is obtained using the methodology of Refs.2-5
Since the ψ(3823) is not the ground state, Eq. 6 is modified following the approach of Refs.where Δ(M(ψ(3823)) -M sm (ψ(1S) ) ) is the mass difference between the two states.

Results and Discussion
The angular momentum coupling from Eq. 5 and the first-order hexaquark mass formula of Eqs. 2 and 7 are used to construct the π + π¯ Ψ(3823) state.As noted previously, the spin and parity assignment for the R 1 is determined to be 2¯ .
No spin assignment was provided in Ref. 1.
Eqs. 2 and 7 lead to a predicted R 1 mass of 4383.6 MeV/c 2 .This result is about 1% smaller than the experimental value 1 4406.9MeV/c 2 .Although these results are encouraging, they are based on a first-order mass formula with a number of uncertainties including the assumed quark masses 8 and the magnitude of the meson-meson cluster interaction.However, the model does provide an initial description of the R 1 state that is reasonable agreement with the

Conclusions
The recently proposed R 1 structure 1 is investigated using a first-order hexaquark mass formula.This mass relationship is based on weakly coupled π+ π¯ψ(3823) meson clusters.The assumed π + π¯ Ψ(3823) configuration leads to a mass result that is within about 1% of the measured R 1 value 1 .A J π value of 2¯ is also predicted.These results suggest the R 1 structure 1 is consistent with a weakly bound π+ π¯ψ(3823) meson cluster system.
three generations are specified by the square brackets and the quark charges [in elementary charge units (e)] are given within parentheses.The hexaquark mass (M) is modeled in terms of three weakly interacting meson clusters M m (i)