QCD sum rule analysis of $0^{+}$ four-quark states

TL;DR

This paper uses QCD sum rules at next-to-leading order to analyze $0^{+}$ four-quark states, identifying their mass ranges and potential experimental counterparts, while addressing uncertainties in condensate factorization.

Contribution

It provides a comprehensive NLO QCD sum rule analysis of $0^{+}$ four-quark states, including mass estimates and a method for renormalizing multi-quark operators.

Findings

Nonet masses mostly around 1-2 GeV, matching observed $0^+$ mesons.

Identification of lighter nonets (~1 GeV) possibly corresponding to light $0^+$ mesons.

Heavier states (>2 GeV) are also predicted, with uncertainties from condensate factorization.

Abstract

We present a comprehensive QCD sum rules analysis at next-to-leading order for all types of $J^P=0^{+}$ four-quark states composed of $u$, $d$, and $s$ quarks. The eigenvectors of the renormalization matrix are chosen to be the renormalized four-quark operators, which can be equally interpreted as tetraquark or molecule operators. Meanwhile, the typical nonet masses given by bare tetraquark operators are lower than those given by bare molecule operators. Most of the nonet masses are around $1-2\text{GeV}$, and they can be interpreted as the $0^+$ mesons observed in experiments. We find a category of four-quark nonets with masses $\lesssim1\text{GeV}$, potentially corresponding to the light $0^+$ mesons $f_0(500)$, $K^*_0(700)$, $f_0(980)$, and $a_0(980)$. On the other hand, the possible 27-fold states are heavier than most of the nonets, with masses $\gtrsim 2\text{GeV}$. The main uncertainty arises from the factorization of high-dimensional condensates, which usually underestimates their values. To address this, we introduce deviation factors for the dimension-6, -8, and -10 condensates, and vary them over a wide range to obtain conservative estimates of the $0^+$ four-quark state masses. Some general properties of the $0^+$ light four-quark states can be derived that do not rely on precise numerical values. We also find that the ambiguity in the factorization of the dimension-8 condensate can introduce a larger discrepancy than previously estimated. As a byproduct, we propose a simple trick for renormalizing multi-quark operators at the one-loop level.