Study of X(6900) with unitarized coupled channel scattering amplitudes
Shi-Qing Kuang, Qi Zhou, Di Guo, Qin-He Yang, L. Dai
Abstract
Abstract In this paper, we study the resonant state X (6900). The scattering amplitudes of coupled channels, $$J/\psi J/\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:mrow></mml:math> - $$J/\psi \psi (2S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:mi>ψ</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>S</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> - $$J/\psi \psi (3770)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:mi>ψ</mml:mi><mml:mo>(</mml:mo><mml:mn>3770</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> , are constructed with the interaction of four vector mesons described by effective Lagrangians. The amplitudes are calculated up to one loop, decomposed by partial wave projection, and unitarized by Pad $$\acute{e}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>e</mml:mi><mml:mo>´</mml:mo></mml:mover></mml:math> approximation. These amplitudes are fitted to the latest experimental data sets of di- $$J/\psi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi></mml:mrow></mml:math> and $$J/\psi \psi (2S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>J</mml:mi><mml:mo>/</mml:mo><mml:mi>ψ</mml:mi><mml:mi>ψ</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mi>S</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> invariant mass spectra of LHCb, CMS, and ATLAS. High-quality solutions are obtained. With these partial wave amplitudes, we extract the pole parameters of the X (6900). Its quantum number is likely to be $$0^{++}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mn>0</mml:mn><mml:mrow><mml:mo>+</mml:mo><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math> . According to the pole counting rule as well as analysis of the phase shifts of the partial waves, it supports our previous conclusion that the X (6900) prefers to be a compact tetra-quark.