[Paper Review] Renormalon Cancellation in Heavy Quarkonia and Determination of m_b, m_t

Y. Sumino|ArXiv.org|Apr 10, 2000

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper demonstrates that renormalon ambiguities cancel in heavy quarkonium systems like bottomonium and toponium, enabling precise extraction of the $ύb$ and $τ$ quark masses using perturbative QCD. By analyzing the large-$\beta_0$ approximation and renormalon cancellation in the binding energy, the authors show that the residual ambiguity is suppressed by $\Lambda_{\rm QCD}^2/(\alpha_s \overline{m})^2$, allowing high-precision determination of $m_b$ and $m_t$ from quarkonium spectra.

ABSTRACT

This is an elementary introduction to the recent significant theoretical progress in the field of heavy quarkonium physics. We show how renormalon cancellation takes place in the heavy quarkonium system, such as bottomonium and (remnant of) toponium resonance, and how this notion is useful in extracting the MSbar masses of the bottom and top quarks.

Motivation & Objective

• To resolve the theoretical ambiguity in quark mass extraction from heavy quarkonium spectra due to renormalon singularities.
• To demonstrate that renormalon contributions cancel in the binding energy of heavy quarkonia such as $\Upsilon(1S)$ and toponium.
• To establish a reliable framework for determining $\overline{\rm MS}$ masses $m_b$ and $m_t$ from quarkonium mass spectra with high accuracy.
• To address the conceptual challenge of pole mass ambiguities in top quark mass measurements at future colliders.

Proposed method

• Using the non-relativistic Schrödinger equation with a Hamiltonian expanded in $1/c$, where $c$ is the speed of light.
• Constructing the effective potential from perturbative QCD, including running coupling effects via the $\beta_0$-approximation.
• Analyzing the Fourier transform of the running coupling to identify the renormalon singularity at $q \sim \Lambda_{\rm QCD}$.
• Applying the generating function method to evaluate the asymptotic behavior of the $n$-th order perturbative contribution, revealing factorial growth $n!$.
• Demonstrating cancellation of leading renormalon ambiguity between $\alpha_s^n$ and $\alpha_s^{n+1}$ terms in the binding energy.
• Rewriting the mass spectrum as a single series in $\alpha_s$ after identifying and removing the divergent renormalon contributions.

Experimental results

Research questions

• RQ1 How do renormalon singularities affect the perturbative expansion of quarkonium binding energies?
• RQ2 Can the leading renormalon ambiguity be canceled in heavy quarkonium systems like bottomonium and toponium?
• RQ3 What is the residual theoretical uncertainty after renormalon cancellation in the mass spectrum of heavy quarkonia?
• RQ4 How can the $\overline{\rm MS}$ quark masses $m_b$ and $m_t$ be extracted with high precision from quarkonium spectra?
• RQ5 Is the pole mass of the top quark physically measurable, or is it plagued by $\mathcal{O}(300~{\rm MeV})$ ambiguities?

Key findings

• The leading renormalon ambiguity in the binding energy of heavy quarkonia is canceled between successive orders in the perturbative expansion.
• The residual ambiguity after cancellation is estimated as $\delta M_{1S}^{(0)} \sim \Lambda \times (\Lambda / (\alpha_s \overline{m}))^2$, which is $\ll \Lambda$.
• The perturbative series for the $1S$ state mass exhibits asymptotic behavior with coefficients growing as $n!$, but the renormalon divergence is canceled.
• The cancellation allows the $\overline{\rm MS}$ masses $m_b$ and $m_t$ to be extracted from quarkonium spectra with high accuracy, limited only by $\mathcal{O}(\alpha_s^4 \overline{m})$ corrections.
• The method provides a reliable theoretical framework to extract $m_b$ and $m_t$ from the $\Upsilon(1S)$ and (future) toponium resonance masses.
• The analysis suggests that renormalon cancellation also occurs in the top quark invariant mass distribution, though the theoretical calculation of such distributions remains challenging.