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Light scalar quarkonia from QCD Laplace sum rule at higher order

R. M. Albuquerque, S. Narison, D. Rabetiarivony

TL;DR

The paper addresses the ambiguous nature of light scalar mesons by applying relativistic Laplace QCD spectral sum rules with perturbative corrections up to $N5LO$ and an operator product expansion truncated at dimension $D=6$ to estimate masses and couplings of light scalar $ar{q}q$, di-hadron molecules, and tetraquark states. Through stability analyses under the constraint $R_{P/C}\ge 1$ and a LO-to-N5LO perturbative treatment of the spectral functions, the authors derive mass–coupling predictions and show that an on-shell mass near $500$–$600$ MeV is disfavored for certain continuum thresholds. They report a $\pi^+\pi^-$-like molecule mass around $1.02$ GeV with a sizable coupling, examine radial excitations and SU(3) breaking, and present extensive results for various tetraquark configurations. The conclusions indicate that scalar mesons likely involve a mixed picture of quarkonium, tetraquark, and gluonic components, and that distinguishing these scenarios requires improved hadronic and $\gamma\gamma$ coupling data.

Abstract

We review our estimations on the light scalar $\bar{q}q$, $(\bar{q}q')(\bar{q'}q)$ and $\overline{qq'}qq'$ ($q,q'\equiv u,d,s$) states from relativistic Laplace sum rule (LSR) within stability criteria and including higher order perturbative (PT) corrections up to the (estimated) N5LO. We evaluate the QCD spectral functions at Lowest Order (LO) of PT QCD and up to the $D=6$ dimension of quark and gluon condensates. Using stability criteria and the constraint: Pole contribution is larger than the QCD continuum one ($R_{P/C}\geqslant 1$) our results exclude an on-shell mass around $(500-600)$ MeV obtained for values of the QCD continuum threshold $t_c \leqslant(1\sim 1.5)$ GeV$^2$. The complete results for the different scalar states are given in Tables 1 to 3. We conclude from the complete analysis that the assignement of the nature of the scalar mesons is not crystal clear and needs further studies.

Light scalar quarkonia from QCD Laplace sum rule at higher order

TL;DR

The paper addresses the ambiguous nature of light scalar mesons by applying relativistic Laplace QCD spectral sum rules with perturbative corrections up to and an operator product expansion truncated at dimension to estimate masses and couplings of light scalar , di-hadron molecules, and tetraquark states. Through stability analyses under the constraint and a LO-to-N5LO perturbative treatment of the spectral functions, the authors derive mass–coupling predictions and show that an on-shell mass near – MeV is disfavored for certain continuum thresholds. They report a -like molecule mass around GeV with a sizable coupling, examine radial excitations and SU(3) breaking, and present extensive results for various tetraquark configurations. The conclusions indicate that scalar mesons likely involve a mixed picture of quarkonium, tetraquark, and gluonic components, and that distinguishing these scenarios requires improved hadronic and coupling data.

Abstract

We review our estimations on the light scalar , and () states from relativistic Laplace sum rule (LSR) within stability criteria and including higher order perturbative (PT) corrections up to the (estimated) N5LO. We evaluate the QCD spectral functions at Lowest Order (LO) of PT QCD and up to the dimension of quark and gluon condensates. Using stability criteria and the constraint: Pole contribution is larger than the QCD continuum one () our results exclude an on-shell mass around MeV obtained for values of the QCD continuum threshold GeV. The complete results for the different scalar states are given in Tables 1 to 3. We conclude from the complete analysis that the assignement of the nature of the scalar mesons is not crystal clear and needs further studies.

Paper Structure

This paper contains 8 sections, 13 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Laplace sum rule
  3. Stability criteria
  4. The ordinary $\bar{q}q$ states
  5. $\pi^+\pi^-\,,K^+K^-\,,K^+\pi^-\,,\eta\pi^0$ states
  6. The tetraquark $\overline{ud}ud\,,\overline{ud}us\,,\overline{us}ds$ states
  7. Nearby radial excitations effects
  8. Comments and Conclusion

Figures (5)

  • Figure 1: The mass and coupling of $\pi^+\pi^-$ as a function of $\tau$ at N3LO for # values of $t_c$.
  • Figure 2: $t_c$ behaviour of the optimal results of $\pi^+\pi^-$.
  • Figure 3: Behaviour of the optimal results versus the truncation of the PT series for $\pi^+\pi^-$ molecule. We take $t_c=2.31\,\hbox{GeV}^2$ which corresponds to the central value of the mass and the one of the coupling. $\#7$ in the loops axis corresponds to the tachyon gluon mass contribution.
  • Figure 4: The Finite width effect on the mass for $t_c=2.31\,\,\hbox{GeV}^2$ corresponding to the central value of $M_{\pi^+\pi^-}=1017\,\,\hbox{MeV}$ in a NWA.
  • Figure 5: The $\tau-$behaviour of the mass and coupling of the $\bar{u}d$ ground state within a "two resonance $\oplus$ QCD continuum" versus the LSR variable $\tau$ and for different values of $t_c$.