[Paper Review] Quartic mass corrections to $R_{had}$ at order $\\alpha_{s}^{3}$

K.G. Chetyrkin, Robert V. Harlander+1 more|arXiv (Cornell University)|May 15, 2000

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper computes quartic mass corrections (proportional to m⁴/s²) in perturbative QCD up to order α₃ₛ for vector and axial-vector current-induced hadronic cross sections in e⁺e⁻ annihilation. It demonstrates that α₃ₛ corrections are sizeable and scale-dependent, but stability improves significantly when the running mass is replaced by the scheme-invariant pole mass, enabling reliable predictions from high to low energies down to threshold for charm, bottom, and top quark production.

ABSTRACT

The total cross section for the production of massive quarks in electron positron annihilation can be predicted in perturbative QCD. After expansion in m^2/s the quartic terms, i.e. those proportional to m^4/s^2, are calculated up to order \\alpha_s^3 for vector and axial current induced rates. Predictions relevant for charm, bottom and top quarks production are presented. The \\alpha_s^3 corrections are shown to be comparable to terms of order \\alpha_s and \\alpha_s^2. As a consequence, the predictions exhibit a sizeable dependence on the renormalization scale. The stability of the prediction is improved and, at the same time, the relative size of the large order terms decreases by replacing the running mass \\bar{m}(\\mu) with the scheme independent invariant one \\hat{m}. By combining these results with the prediction for massless case and the quadratic mass terms the cross section for massive quark production at electron positron colliders is put under control in order \\alpha_s^3 from the high energy region down to fairly low energies.

Motivation & Objective

• To compute next-to-next-to-leading order (NNLO) QCD corrections involving m⁴/s² terms in the cross section for massive quark production in e⁺e⁻ annihilation.
• To address the large renormalization scale dependence of α₃ₛ corrections in the running mass scheme (MS̄) for heavy quarks.
• To improve theoretical stability and reduce uncertainty in Rhad predictions by replacing the running mass with the scheme-invariant pole mass ̂m.
• To complete the perturbative QCD description of Rhad up to α₃ₛ, including mass effects, from high energies down to near-threshold production.
• To provide a consistent, high-precision prediction for Rhad valid across the full energy range of e⁺e⁻ colliders, including LEP and future linear colliders.

Proposed method

• Uses the operator product expansion (OPE) and renormalization group equations (RGEs) to systematically construct logarithmic terms in the polarization function Π(q²) up to O(α₃ₛ) and O(m⁴/s²).
• Applies the RGEs for the strong coupling β-function and quark mass anomalous dimension γm to resum logarithms and determine the structure of m⁴/s² terms.
• Performs calculations using massless propagators and massive tadpole integrals, both at most in three-loop order, within the MS̄ scheme at scale µ² = s.
• Replaces the running mass m(µ) with the invariant mass ̂m via the perturbative integral relation m(µ) = ̂m exp(∫ da γm(a)/β(a)), reducing scale dependence.
• Combines the new quartic terms with known massless and quadratic (m²/s) contributions to reconstruct the full Rhad up to O(α₃ₛ).
• Validates results numerically by comparing predictions across different renormalization scales and confirms improved stability with the invariant mass scheme.

Experimental results

Research questions

• RQ1 How do quartic mass corrections (m⁴/s²) at O(α₃ₛ) affect the hadronic cross section Rhad in e⁺e⁻ annihilation?
• RQ2 Why do α₃ₛ corrections to m⁴/s² terms exhibit strong dependence on the renormalization scale in the MS̄ scheme?
• RQ3 Can replacing the running mass m(µ) with the invariant mass ̂m reduce the scale dependence and improve the stability of O(α₃ₛ) predictions?
• RQ4 To what extent do the O(α₃ₛ) corrections to m⁴/s² terms affect the overall uncertainty in Rhad for charm, bottom, and top quark production?
• RQ5 Can a consistent, high-precision prediction for Rhad be achieved across the full energy range—from high-energy asymptotic regimes to near-threshold production—by combining massless, quadratic, and quartic terms at O(α₃ₛ)?

Key findings

• The O(α₃ₛ) corrections to the m⁴/s² terms are comparable in magnitude to αs and α²s corrections, leading to significant renormalization scale dependence in the MS̄ scheme.
• For charm quarks at √s = 6 GeV, the uncertainty in the m⁴/s² term due to scale variation (µ from √s/2 to 2√s) is ±0.0005.
• For bottom quarks at √s = 14 GeV, the scale-induced uncertainty in the m⁴/s² term is ±0.0016, which is still manageable.
• For top quarks, the m⁴/s² term shows negligible scale dependence, indicating reduced sensitivity to higher-order corrections.
• Replacing the running mass m(µ) with the invariant mass ̂m reduces the scale dependence of the O(α₃ₛ) prediction and suppresses the relative size of large-order terms.
• At µ = √s, the O(α₃ₛ) prediction for the m⁴/s² term in the invariant mass scheme varies between −0.13·10⁻² and −0.20·10⁻², compared to −0.10·10⁻² to −0.20·10⁻² in the MS̄ scheme, showing improved stability.