[Paper Review] Doubly hidden $0^{++}$ molecules and tetraquarks states from QCD at NLO

R.M. Albuquerque, Stéphan Narison+3 more|arXiv (Cornell University)|Feb 16, 2021

Particle physics theoretical and experimental studies4 references4 citations

TL;DR

This paper uses relativistic QCD Laplace Sum Rules at Next-to-Leading Order (NLO) with non-perturbative condensates to predict masses and couplings of fully heavy $0^{++}$ molecules and tetraquarks, finding good agreement with LHCb's broad (6.2–6.7 GeV) and narrow (6.9 GeV) structures. The narrow state is best interpreted as a $ar{\chi}_{c0}\chi_{c0}$ molecule or its tetraquark counterpart.

ABSTRACT

Motivated by the LHCb-group discovery of exotic hadrons in the range (6.2 $\sim$ 6.9) GeV, we present new results for the masses and couplings of $0^{++}$ fully heavy $(\bar{Q}Q)(Q\bar{Q})$ molecules and $(QQ)(\overline{QQ})$ tetraquaks states from relativistic QCD Laplace Sum Rule (LSR) within stability criteria where Next-to-Leading Order (NLO) Factorized (F) Perturbative (PT) corrections is included. As the Operator Product Expansion (OPE) usually converges for $d\leqslant 6-8$, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to $\langle G^3 angle$. We also emphasize the importance of PT radiative corrections for heavy quark sum rules in order to justify the use of the running heavy quark mass value in the analysis. We compare our predictions in Table 3 with the ones from ratio of Moments (MOM). The broad structure arround (6.2 $\sim$ 6.9) GeV can be described by the $\overlineη_cη_c$, $\overline{J/ψ}J/ψ$ and $\overlineχ_{c1}χ_{c1}$ molecules or/and $\overline{S}_c S_c$, $\overline{A}_c A_c$ and $\overline{V}_c V_c$ tetraquarks lowest mass ground states. The narrow structure at (6.8 $\sim$ 6.9) GeV if it is a $0^{++}$ state can be a $\overlineχ_{c0}χ_{c0}$ molecules or/and its analogue $\overline{P}_c P_c$ tetraquark. The $\overlineχ_{c1}χ_{c1}$ predicted mass is found to be below the $χ_{c1}χ_{c1}$ threshold while for the beauty states, all of the estimated masses are above the $η_b η_b$ and $Υ(1S)Υ(1S)$ threshold.

Motivation & Objective

• To interpret the LHCb-observed broad structure around 6.2–6.7 GeV and narrow peak at 6.9 GeV as $0^{++}$ fully heavy tetraquark or molecular states.
• To improve mass and coupling predictions for $0^{++}$ states using QCD Laplace Sum Rules (LSR) at NLO with non-perturbative contributions.
• To assess the role of perturbative radiative corrections and non-perturbative condensates ($\langle G^3 \rangle$, $\langle \alpha_s G^2 \rangle$) in heavy quark sum rules.
• To compare LSR predictions with those from the Ratio of Moments (MOM) method and evaluate consistency across methods.
• To determine whether molecular or tetraquark states better describe the observed structures, particularly the narrow 6.9 GeV state.

Proposed method

• Applies the finite-energy Laplace Sum Rule (LSR) formalism to four-quark current correlators for $0^{++}$ states with interpolating currents for both molecules and tetraquarks.
• Uses the Minimal Duality Ansatz (MDA) to model the spectral function with a single resonance peak and a QCD continuum threshold $t_c$.
• Evaluates the correlator at Leading Order (LO) of perturbative QCD up to dimension-eight operators, including $\langle G^3 \rangle$ and $\langle \alpha_s G^2 \rangle$ condensates.
• Incorporates NLO perturbative corrections via factorized diagrams, with negligible contribution from non-factorized diagrams.
• Imposes stability criteria on the LSR variable $\tau$, continuum threshold $t_c$, and subtraction scale $\mu$ to extract robust predictions.
• Compares results with the Ratio of Moments (MOM) method and uses systematic error estimation from inclusion/exclusion of $\langle G^3 \rangle$.

Experimental results

Research questions

• RQ1 Can $0^{++}$ fully heavy tetraquark and molecular states explain the broad (6.2–6.7 GeV) and narrow (6.9 GeV) structures observed by LHCb?
• RQ2 How do NLO perturbative corrections and non-perturbative condensates affect the mass predictions of these exotic states?
• RQ3 What is the relative contribution of factorized vs. non-factorized diagrams in the NLO QCD corrections?
• RQ4 Are the LSR predictions consistent with those from the Ratio of Moments (MOM) method?
• RQ5 Can the LSR method distinguish between molecular and tetraquark states with identical quantum numbers?

Key findings

• The broad structure in the 6.2–6.7 GeV range is best described by $\bar{\eta}_c\eta_c$, $\bar{J}/\psi J/\psi$, and $\bar{\chi}_{c1}\chi_{c1}$ molecules or $\bar{S}_cS_c$, $\bar{A}_cA_c$, and $\bar{V}_cV_c$ tetraquarks.
• The narrow structure at 6.9 GeV is best interpreted as a $\bar{\chi}_{c0}\chi_{c0}$ molecule or its $\bar{P}_cP_c$ tetraquark analogue.
• The predicted mass of $\bar{\chi}_{c1}\chi_{c1}$ is below the $\chi_{c1}\chi_{c1}$ threshold, while all beauty-state masses lie above the $\eta_b\eta_b$ and $\Upsilon(1S)\Upsilon(1S)$ thresholds.
• LSR predictions for charm and $\bar{P}_bP_b$ states agree well with LO results from [39], but discrepancies appear for $\bar{S}_bS_b$, $\bar{A}_bA_b$, and $\bar{V}_bV_b$ states, possibly due to methodological differences.
• The LSR and MOM methods yield consistent results within errors for $\bar{\chi}_{c0}\chi_{c0}$ and $\bar{S}_cS_c$, with LSR predictions of $6471 \pm 67$ MeV and $19717 \pm 118$ MeV respectively.
• The method cannot clearly disentangle molecular from tetraquark states with identical quantum numbers due to overlapping spectral functions.