Does vacuum saturation work for the higher dimensional vacuum condensates in the QCD sum rules?

TL;DR

This paper investigates the validity of the vacuum saturation approximation in QCD sum rules for tetraquark states by analyzing higher-dimensional vacuum condensates up to dimension-10 and examining their effects on mass predictions.

Contribution

It introduces a parameter to quantify deviations from vacuum saturation and demonstrates that smaller deviations improve the convergence of the operator product expansion.

Findings

Vacuum condensates of dimensions 6, 8, 10 have similar expressions but opposite signs.

Smaller deviation from vacuum saturation improves OPE convergence.

Results can be tested against future experimental data.

Abstract

In the QCD sum rules for the tetraquark (molecular) states, the higher dimensional vacuum condensates play an important role in extracting the tetraquark masses. We carry out the operator product expansion up to the vacuum condensates of dimension-10 and observe that the vacuum condensates of dimensions $6$, $8$ and $10$ have the same expressions but opposite signs for the $C\gamma_5\otimes \gamma_\mu C$-type and $C\otimes \gamma_\mu C$ type four-quark currents, which make their influences distinguishable, and they are excellent channels to examine the vacuum saturation approximation. We introduce a parameter $\kappa$ to parameterize the derivation from the vacuum saturation or factorization approximation, and choose two sets parameters to examine the influences on the predicted tetraquark masses, which can be confronted to the experimental data in the future. In all the channels, smaller value of the $\kappa$ leads to better convergent behavior in the operator product expansion, which favors the vacuum saturation approximation.