Quartic Mass Corrections to $R_{had}$

TL;DR

The authors address quartic mass corrections to hadronic production and $Z$ decays by expanding in $m^2/s$ and $ abla$, deriving $ ext{${ m O}( abla m^4/s^2)$}$ and $ ext{${ m O}( abla^2 m^4/s^2)$}$ terms using the operator product expansion and RG methods. They obtain explicit expressions for the vector and axial correlators $R_V$ and $R_A$, including logarithmic terms that cannot be absorbed into a running mass at $ ext{${ m O}( abla^2)$}$, and discuss how these corrections behave near charm and bottom thresholds. The results extend perturbative QCD predictions to lower energies and provide guidance for interpreting data around heavy-quark thresholds, with only tiny effects on the $Z o qar{q}$ rates. Overall, quartic-mass corrections are small for $Z$ decays but become increasingly relevant in the few-GeV region above heavy-quark thresholds, thanks to mass-running improvements.

Abstract

The influence of nonvanishing quark masses on the total cross section in electron positron collisions and on the $Z$ decay rate is calculated. The corrections are expanded in $m^2/s$ and $\as$. Methods similar to those applied for the quadratic mass terms allow to derive the corrections of order $\as m^4/s^2$ and $\as^2m^4/s^2$. Coefficients which depend logarithmically on $m^2/s$ and which cannot be absorbed in a running quark mass arise in order $\as^2$. The implications of these results on electron positron annihilation cross sections at LEP and at lower energies in particular between the charm and the bottom threshold and for energies several GeV above the $b\bar b$ threshold are discussed.

Figures (4)

• Figure 1: Comparison between the complete ${\cal O}(\alpha_s)$ correction function (solid line) and approximations of increasing order (dashed lines) in $m^2$ for vector (upper Figure) and axial vector current (lower Figure) induced rates.
• Figure 2: Diagrams giving rise to nonzero vacuum expectation values of the composite operators $\overline{\psi} \psi$ and $G^2_{\mu\nu}$ in the lowest orders of perturbation theory.
• Figure 3: Contributions to $R^V$ from $m^4$ terms including successively higher orders in $\alpha_s$ (order $\alpha_s^0$/ $\alpha_s^1$/ $\alpha_s^2$ corresponding to dotted/ dashed/ solid lines) as functions of $2m_{\rm pole}/\sqrt{s}$.
• Figure 4: Predictions for $R^V$ including successively higher orders in $m^2$.