[Paper Review] $QQ^{\prime} \bar u \bar d$ bound state in the Bethe-Salpeter equation approach

Gao Feng, X. H. Guo+1 more|arXiv (Cornell University)|Sep 30, 2013

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper investigates $QQ'\bar{u}\bar{d}$ tetraquark states composed of a heavy diquark ($QQ'$, with $Q,Q' = b$ or $c$) and a light antiquark diquark ($\bar{u}\bar{d}$) using the Bethe-Salpeter equation in the heavy quark limit. It employs a covariant instantaneous approximation with a scalar confinement and one-gluon-exchange kernel, including form factors to model diquark structure. The results show that $bb\bar{u}\bar{d}$, $bc\bar{u}\bar{d}$, and $cc\bar{u}\bar{d}$ bound states exist below relevant meson thresholds, suggesting their potential stability and detectability in future experiments.

ABSTRACT

In the heavy quark limit, we establish the Bethe-Salpeter equations for the ground state $QQ^{\prime} \bar u \bar d$ containing one heavy diquark $QQ^{\prime}$ ($Q, Q^{\prime}=b$ or $c$) and one light antidiquark $\bar u \bar d$. We solve the Bethe-Salpeter equations numerically in the covariant instantaneous approximations with the kernels containing a scalar confinement term and a one-gluon-exchange term. Numerical solutions for the Bethe-Salpeter wave functions are presented. The results show that the masses of $bb \bar u \bar d$, $bc \bar u \bar d$, and $cc \bar u \bar d$ bound states lie below the threshold of $\bar{B}^{*0}\,B^-$ or $B^0\,{B^*}^-$, $B^-D^+$ or $\bar{B}^{0} D^0$, and $D^+\,{D^*}^0$ or ${D^*}^+\,D^0$ mesons, respectively. The ground states $QQ^{\prime} \bar u \bar d$ may exist and we expect the forthcoming experimental data to confirm them.

Motivation & Objective

• To investigate the existence of $QQ'\bar{u}\bar{d}$ tetraquark states with $J^P = 1^+$ in the heavy quark limit.
• To model these states as bound states of a heavy diquark $QQ'$ and a light $\bar{u}\bar{d}$ diquark.
• To establish and solve the Bethe-Salpeter equations for these states using a kernel with scalar confinement and one-gluon exchange terms.
• To examine the dependence of bound state solutions on model parameters such as $\kappa$, $\alpha_s^{\rm eff}$, and diquark masses.
• To determine whether the masses of $bb\bar{u}\bar{d}$, $bc\bar{u}\bar{d}$, and $cc\bar{u}\bar{d}$ lie below relevant meson thresholds, indicating possible stability.

Proposed method

• Formulates the Bethe-Salpeter equation for $QQ'\bar{u}\bar{d}$ states in momentum space using a BS wave function $\chi_P^\mu(p)$ with relative momentum $p$ and total momentum $P$.
• Assumes a kernel composed of a scalar confinement term and a one-gluon-exchange term, motivated by potential models.
• Introduces form factors at the diquark-gluon vertices to account for the finite size and internal structure of diquarks.
• Solves the BS equation numerically in the covariant instantaneous approximation, allowing for relativistic and covariant treatment of the bound state.
• Varying parameters such as $\kappa$ (0.02–0.1 GeV³), $\alpha_s^{\rm eff}$, and diquark masses within physical ranges to explore solution stability.
• Compares the calculated bound state masses to thresholds of $\bar{B}^{*0}B^{-}$, $B^0{B^*}^{-}$, $B^{-}D^{+}$, $\bar{B}^{0}D^{0}$, $D^{+}{D^*}^0$, and ${D^*}^+D^0$ mesons to assess binding.

Experimental results

Research questions

• RQ1 Do $bb\bar{u}\bar{d}$, $bc\bar{u}\bar{d}$, and $cc\bar{u}\bar{d}$ tetraquark states form bound states below the thresholds of their corresponding two-meson decays?
• RQ2 What is the role of the scalar confinement and one-gluon-exchange interaction in stabilizing these tetraquark states?
• RQ3 How do the form factors modeling diquark structure affect the existence and mass of the bound states?
• RQ4 Are the solutions of the Bethe-Salpeter equation stable under variation of key parameters like $\kappa$, $\alpha_s^{\rm eff}$, and diquark masses?
• RQ5 Can the $J^P = 1^+$ quantum numbers of the $QQ'\bar{u}\bar{d}$ states be consistently described within the BS framework using the chosen kernel and approximations?

Key findings

• The $bb\bar{u}\bar{d}$ bound state mass lies below the $\bar{B}^{*0}B^{-}$ or $B^0{B^*}^{-}$ threshold, with masses ranging from 10.39 GeV to 10.60 GeV.
• The $bc\bar{u}\bar{d}$ bound state mass lies below the $B^{-}D^{+}$ or $\bar{B}^{0}D^{0}$ threshold, with masses ranging from 6.93 GeV to 7.14 GeV.
• The $cc\bar{u}\bar{d}$ bound state mass lies below the $D^{+}{D^*}^0$ or ${D^*}^+D^0$ threshold, with masses ranging from 3.66 GeV to 3.87 GeV.
• Numerical solutions of the Bethe-Salpeter wave functions are obtained and remain stable for binding energies between -10 MeV and -100 MeV across varied parameter sets.
• The existence of $bb\bar{u}\bar{d}$, $bc\bar{u}\bar{d}$, and $cc\bar{u}\bar{d}$ bound states is confirmed for reasonable values of $\kappa$, $\alpha_s^{\rm eff}$, and diquark masses in the model.
• The results suggest that these tetraquark states may be stable and detectable in future high-energy experiments.