Possible Tetraquark Explanation for the Proposed Zcs(3985)−

The recently proposed Z cs (3985) − structure is investigated using a first-order tetraquark mass formula. This mass relationship is based on weakly bound D s− (c-bar s) + D *0 (2007)(c u-bar) and D s*− (c-bar s) + D 0 (c u-bar) meson clusters and provides a reasonable prediction (within about 3%) of the measured Z cs (3985) − mass. and The proposed Z − tetraquark is assigned a spin and parity of + with a resonance mass of 3985.2 MeV/c 2 at 5.3σ over the pure contributions from the conventional charmed mesons 1 .


Introduction
The BESIII Collaboration 1 recently reported the first candidate for a charged hidden-charm tetraquark with

Model and Formulation
Zel'dovich and Sakharov 8,9 proposed a semiempirical mass formula that provides a prediction of mesons and baryons in terms of effective quark masses. Within this formulation, quark wave functions are assumed to reside in their lowest 1S state. These mass formulas are used as the basis for deriving a first-order tetraquark mass formula. In particular, the model proposed in this paper assumes the tetraquark is partitioned into two clusters with the interaction between the clusters providing a minimal contribution to the tetraquark mass.
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 9 .
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, +1/4 for pseudoscalar and vector mesons, respectively 9 .
In formulating the tetraquark mass formula, effective quark masses provided by Griffiths 10 are utilized.
These six quarks are arranged in three generations: 11 . The three generations are specified by the square brackets and the quark charges are given within parentheses.

First-Order Mass Formula for the Z cs (3985) −
The spin of a tetraquark within the first order mass formula is determined by coupling the two meson clusters where the first-order mass formula assumes a minimally interacting L=0 configuration 2,3 between the meson clusters. The J π = 1 + assignment 1 for the Z cs (3985) − follows naturally from the coupling structure of Eq. The first-order mass formula used in this paper partitions the tetraquark into two meson clusters. The first cluster is a scalar meson (sm) (e.g., D 0 or D s − ) and the remaining cluster is a vector meson (vm) (e. g., D *0 (2007) or D s *− ). These simplifications are incorporated to minimize model complexity which is consistent with an initial first-order formulation. In addition, the general tetraquark mass formula is assumed to have the form 2,3

M = M sm + M vm + Φ
where Φ defines the interaction between the meson clusters. Within the scope of this mass formula, the meson-meson cluster interaction is assumed to be weak and sufficiently small to be ignored. Accordingly, Eq. 3 Qeios, CC-BY 4.0 · Article, July 27, 2021 Qeios ID: GLTEU2 · https://doi.org/10.32388/GLTEU2 2/4 represents a quasimolecular four quark system whose basic character is a weakly bound meson-meson system.

Results and Discussion
The angular momentum coupling from Eq. Although these results are encouraging, they are based on a first-order mass formula with a number of uncertainties including the assumed quark masses 10 and the magnitude of the meson-meson cluster interaction. However, the model does provide an initial description of the Z cs (3985) − and its J π = 1 + assignment in terms of a tetraquark structure .

Conclusions
The first-order mass formula predicts two possible tetraquark configurations for the Z cs (3985) − . Both weakly bound D s − (c-bar s) + D *0 (2007)(c u-bar) and D s *− (c-bar s) + D 0 (c u-bar) meson clusters lead to predicted mass values that are within about 3% of the measured value 1 . The predicted J π = 1 + assignment is also in agreement with data 1 .