Systematic uncertainties of hadron parameters obtained with QCD sum rules

TL;DR

This paper investigates the uncertainties in extracting hadron parameters using QCD sum rules, demonstrating that modeling the continuum introduces significant, uncontrollable systematic errors, as shown through an oscillator model example.

Contribution

It provides a detailed analysis of the limitations and uncertainties in QCD sum rule methods using an exactly solvable model, highlighting the challenges in controlling systematic errors.

Findings

Effective continuum threshold can reproduce the polarization operator exactly within a range.

Extracted parameter R is heavily influenced by the modeled hadron continuum.

Systematic uncertainties in sum rules are difficult to control without precise continuum knowledge.

Abstract

We study the uncertainties of the determination of the ground-state parameters from Shifman-Vainshtein-Zakharov (SVZ) sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator $\Pi(\mu)$, which allows one to obtain both the OPE to any order and the spectrum of states. We start with the OPE for $\Pi(\mu)$ and analyze the extraction of the square of the ground-state wave function, $R\propto|\Psi_0(\vec r=0)|^2$, from an SVZ sum rule, setting the mass of the ground state $E_0$ equal to its known value and treating the effective continuum threshold as a fit parameter. We show that in a limited ``fiducial'' range of the Borel parameter there exists a solution for the effective threshold which precisely reproduces the exact $\Pi(\mu)$ for any value of $R$ within the range $0.7 \le R/R_0 \le 1.15$ ($R_0$ is the known exact value). Thus, the value of $R$ extracted from the sum rule is determined to a great extent by the contribution of the hadron continuum. Our main finding is that in the cases where the hadron continuum is not known and is modeled by an effective continuum threshold, the systematic uncertainties of the sum-rule procedure cannot be controlled.