[Paper Review] Heavy- and light-flavor symmetry partners of the $T_{cc}^+(3875)$, the $X(3872)$ and the $X(3960)$ from light-meson exchange saturation

Fang‐Zheng Peng, Mao-Jun Yan+1 more|arXiv (Cornell University)|Apr 26, 2023

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper investigates the molecular nature of the $T_{cc}^+(3875)$, $X(3872)$, and $X(3960)$ using a saturation model of light-meson exchanges ($\sigma$, $\rho$, $\omega$) to derive effective contact-range interactions. It finds a tension between molecular descriptions: if $X(3872)$ is molecular, $T_{cc}^+$ cannot be purely molecular, and vice versa, suggesting the $X(3872)$ is more likely molecular than $T_{cc}^+$. The model predicts a $T_{cc}^{*+}(4015)$ state as a key signature of a purely molecular $T_{cc}^+$.

ABSTRACT

The spectrum of the charmed meson-(anti)meson system is a fundamental tool for disentangling the nature of a few exotic hadrons, including the recently discovered $T_{cc}^+(3875)$ tetraquark, the $X(3960)$, or the $X(3872)$, the nature of which is still not clear after almost two decades of its discovery. Here we consider that the charmed meson-(anti)meson short-range interaction is described by the exchange of light-mesons ($σ$, $ρ$, $ω$). The effects of light-meson exchanges are recast into a simple contact-range theory by means of a saturation procedure, resulting in a compact description of the two-hadron interaction. From this, if the $T_{cc}^+$ were to be an isoscalar $D^* D$ molecule, then there should exist an isoscalar $J=1$ $D^* D^*$ partner, as constrained by heavy-quark spin symmetry. Yet, within our model, the most attractive two charmed meson configurations are the isovector $J=0$ $D^* D^*$ molecule and its sextet $D_s^* D^*$ and $D_s^* D_s^*$ flavor partners. Finally, we find a tension between the molecular descriptions of the $T_{cc}^+$ and that of the $X(3872)$ and $X(3960)$, where most parameter choices suggest that if the $T_{cc}^+$ is purely molecular then the $X(3872)$ overbinds (or conversely, if the $X(3872)$ is a molecule the $T_{cc}^+$ does not bind). This might be consequential for determining the nature of these states.

Motivation & Objective

• To assess whether the $T_{cc}^+(3875)$, $X(3872)$, and $X(3960)$ can be consistently described as molecular states using a unified set of parameters.
• To determine the existence and properties of heavy- and light-flavor symmetry partners of these exotic states within a saturation model of light-meson exchange.
• To resolve the tension between molecular descriptions of $X(3872)$ and $T_{cc}^+$ by identifying which state is more likely to be molecular based on model consistency.
• To predict new molecular states, particularly the $I=0$, $J=1^+$ $D^*D^*$ partner of $T_{cc}^+$, as a signature of its molecular nature.

Proposed method

• The short-range two-hadron interaction in charmed meson-(anti)meson systems is modeled via exchange of light mesons ($\sigma$, $\rho$, $\omega$), which are then saturated into an effective contact-range theory.
• The saturation procedure maps the non-local light-meson exchange potential into a local contact interaction, with coupling constants determined by fitting to a reference state (here, the $X(3872)$).
• The model uses isospin and spin symmetry to predict flavor and spin partners of the $T_{cc}^+$, $X(3872)$, and $X(3960)$, including $D^*D^*$, $D_s^*D^*$, and $D_s^*D_s^*$ molecules.
• The interaction Hamiltonian is derived from vector and scalar meson exchanges, with matrix elements computed in the contact-range approximation to simplify the two-body scattering problem.
• The binding energy and scattering behavior are evaluated using the Bethe-Salpeter equation in the ladder approximation, with solutions analyzed for bound and virtual states.
• The model compares the binding strength of $D^*\bar{D}$ (for $X(3872)$) and $D^*D$ (for $T_{cc}^+$) systems, highlighting the role of vector meson exchange in generating isospin-dependent attraction.

Experimental results

Research questions

• RQ1 Can the $T_{cc}^+(3875)$, $X(3872)$, and $X(3960)$ be consistently described as molecular states within a single effective field theory framework?
• RQ2 What are the heavy- and light-flavor symmetry partners of the $T_{cc}^+$, $X(3872)$, and $X(3960)$, and what are their quantum numbers and masses?
• RQ3 Is there a fundamental tension between the molecular descriptions of $X(3872)$ and $T_{cc}^+$, such that both cannot be purely molecular with the same parameters?
• RQ4 Does the $T_{cc}^+$ have a $J=1^+$ $D^*D^*$ partner state, and if so, what is its predicted mass and significance?
• RQ5 Which of the $X(3872)$ or $T_{cc}^+$ is more likely to be molecular, based on the model's consistency and binding energy predictions?

Key findings

• A tension exists between the molecular descriptions of $X(3872)$ and $T_{cc}^+$: if $X(3872}$ is purely molecular, the $T_{cc}^+$ cannot be purely molecular with the same parameters, and vice versa.
• The $X(3872)$ is more likely to be molecular than $T_{cc}^+$, as the model favors binding in the $D^*\bar{D}$ system due to stronger vector meson exchange attraction.
• The most attractive two-charm meson configurations are the isovector $J=0^+$ $D^*D^*$ molecule and its sextet partners $D_s^*D^*$ and $D_s^*D_s^*$, which are predicted as virtual states near threshold.
• If the $T_{cc}^+$ is purely molecular, a $I=0$, $J=1^+$ $D^*D^*$ partner state — the $T_{cc}^{*+}(4015)$ — must exist at approximately 4015 MeV, providing a key experimental test.
• The $X(3872)$-based saturation model predicts a $2^{++}$ hidden-charm partner state below threshold, consistent with the $X(3872)$'s $1^{++}$ quantum numbers.
• The $X(3960)$ is predicted as a $0^{++}$ $D_s\bar{D}_s$ virtual state near threshold, consistent with its observed mass and width, and with lattice QCD results.