[Paper Review] Bottom and Charm Mass Determinations with a Convergence Test

Bahman Dehnadi, André H. Hoang+1 more|arXiv (Cornell University)|Apr 28, 2015

Particle physics theoretical and experimental studies27 references4 citations

TL;DR

This paper presents a refined determination of the $ar{m}_c$ and $ar{m}_b$ quark masses using relativistic QCD sum rules at ${ m O}(α_s^3)$, incorporating experimental $e^+e^-$ data and lattice QCD results. It introduces a convergence test to assess perturbative series stability and reduces overestimated uncertainties from double scale variation, yielding $ar{m}_c(\bar{m}_c) = 1.288 \pm 0.020\,\text{GeV}$ and $ar{m}_b(\bar{m}_b) = 4.176 \pm 0.023\,\text{GeV}$.

ABSTRACT

We present new determinations of the MS-bar charm quark mass using relativistic QCD sum rules at O(alpha_s^3) from the moments of the vector and the pseudoscalar current correlators. We use available experimental measurements from e+e- collisions and lattice simulation results, respectively. Our analysis of the theoretical uncertainties is based on different implementations of the perturbative series and on independent variations of the renormalization scales for the mass and the strong coupling. Taking into account the resulting set of series to estimate perturbative uncertainties is crucial, since some ways to treat the perturbative expansion can exhibit extraordinarily small scale dependence when the two scales are set equal. As an additional refinement, we address the issue that double scale variation could overestimate the perturbative uncertainties. We supplement the analysis with a test that quantifies the convergence rate of each perturbative series by a single number. We find that this convergence test allows to determine an overall and average convergence rate that is characteristic for the series expansions of each moment, and to discard those series for which the convergence rate is significantly worse. We obtain mc(mc) = 1.288 +- 0.020 GeV from the vector correlator. The method is also applied to the extraction of the MS-bar bottom quark mass from the vector correlator. We compute the experimental moments including a modeling uncertainty associated to the continuum region where no data is available. We obtain mb(mb) = 4.176 +- 0.023 GeV.

Motivation & Objective

• To improve the precision and reliability of $ar{m}_c$ and $ar{m}_b$ mass determinations using QCD sum rules at ${\rm O}(\alpha_s^3)$.
• To address the issue of overestimated perturbative uncertainties arising from double scale variation in the renormalization scales of mass and coupling.
• To introduce a convergence test that quantifies the convergence rate of perturbative series, enabling the rejection of poorly converging series.
• To reduce dependence on model-dependent continuum region extrapolations for bottom quark mass extraction by incorporating experimental moment data with modeled uncertainties.
• To provide a robust, systematic estimation of theoretical uncertainties using multiple implementations of the perturbative series.

Proposed method

• Employing relativistic QCD sum rules to relate experimental $e^+e^-$ cross-section moments to theoretical moments of vector and pseudoscalar current correlators.
• Using experimental data from $e^+e^-$ collisions for charm and lattice QCD results for pseudoscalar correlators to constrain theoretical moments.
• Applying a convergence test based on a single numerical measure of the convergence rate for each perturbative series, allowing identification and discarding of poorly converging expansions.
• Implementing multiple perturbative series variants and independently varying the renormalization scales for mass and $\alpha_s$ to estimate perturbative uncertainties more reliably.
• Modeling the continuum region contribution to experimental moments with associated uncertainty, particularly critical for bottom quark mass extraction.
• Using the linearized and iterative expansion techniques to compute ratios of moments and their dependence on $\alpha_s$ and $m_q$ up to ${\rm O}(\alpha_s^3)$.

Experimental results

Research questions

• RQ1 How can perturbative uncertainties in $ar{m}_c$ and $ar{m}_b$ determinations be more reliably estimated when standard double scale variation leads to overestimation?
• RQ2 To what extent do different implementations of the perturbative series affect the stability and convergence of the mass extraction procedure?
• RQ3 Can a single convergence test parameter be used to objectively identify and discard poorly converging series in QCD sum rule calculations?
• RQ4 How significant is the uncertainty from the unmeasured continuum region in the experimental moments, particularly for bottom quark mass extraction?
• RQ5 What is the impact of including higher-order corrections at ${\rm O}(\alpha_s^3)$ on the final mass determinations and their uncertainties?

Key findings

• The convergence test successfully identifies and excludes series with significantly worse convergence rates, improving the reliability of the perturbative expansion.
• The $ar{m}_c(\bar{m}_c)$ mass is determined to be $1.288 \pm 0.020\,\text{GeV}$ using the vector current correlator and experimental $e^+e^-$ data.
• The $ar{m}_b(\bar{m}_b)$ mass is determined to be $4.176 \pm 0.023\,\text{GeV}$ using the vector current correlator and experimental moments with modeled continuum region uncertainty.
• Double scale variation can lead to unnaturally small scale dependence when scales are set equal, highlighting the need for multiple series implementations to estimate uncertainties.
• The gluon condensate and other nonperturbative corrections have negligible effects for low $n$ moments, especially for the bottom quark, validating the dominance of perturbative QCD in the analysis.
• For the charm quark, the experimental moments are largely insensitive to continuum region modeling, whereas for bottom quarks, this uncertainty dominates the error budget, justifying the modeled uncertainty treatment.