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Renormalon-based resummation for spacelike and timelike QCD quantities whose perturbation expansion has general form

Cesar Ayala, Gorazd Cvetič, Reinhart Kögerler

Abstract

We present a generalisation of our previous approach of a renormalon-motivated resummation of the QCD observables. Previously it was applied to the spacelike observables whose perturbation expansion was $D(Q^2) = a(Q^2) + O(a^2)$, where $a(Q^2) \equiv α_s(Q^2)/π$ is the running QCD coupling. Now we generalise the resummation to spacelike quantities $D(Q^2) = a(Q^2)^{ν_0} + O(a^{ν_0+1})$ and timelike quantities $F(σ) = a(σ)^{ν_0} + O(a^{ν_0+1})$, where $ν_0$ is in general a noninteger number ($0<ν_0 \leq 1$). We evaluate with this approach a timelike quantity, namely the scheme-invariant factor of the Wilson coefficient of the chromomagnetic operator in the heavy-quark effective Lagrangian, and related quantities.

Renormalon-based resummation for spacelike and timelike QCD quantities whose perturbation expansion has general form

Abstract

We present a generalisation of our previous approach of a renormalon-motivated resummation of the QCD observables. Previously it was applied to the spacelike observables whose perturbation expansion was , where is the running QCD coupling. Now we generalise the resummation to spacelike quantities and timelike quantities , where is in general a noninteger number (). We evaluate with this approach a timelike quantity, namely the scheme-invariant factor of the Wilson coefficient of the chromomagnetic operator in the heavy-quark effective Lagrangian, and related quantities.

Paper Structure

This paper contains 10 sections, 158 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Resummation for the case of the spacelike observable ${\cal D}(Q^2)$
  3. Resummation for timelike observable ${\cal F}(\sigma)$
  4. The use of holomorphic QCD (AQCD) couplings in the formalism
  5. An example: Implementation of resummation, timelike case
  6. Conclusions
  7. Proof of Theorem 2
  8. Proof of Theorem 4
  9. Renormalisation scheme dependence of the expansion coefficients
  10. Mathematical and numerical aspects of the formalism

Figures (7)

  • Figure 1: ${\mathcal{A}}(Q^2)$ of 3$\delta$AQCD for positive $Q^2$ and various values of the threshold scale parameter $M_1$. For comparison, the underlying pQCD coupling $a(Q^2)$ (in the same LMM scheme) is included.
  • Figure 2: The spacelike running coupling ${\widetilde{\mathcal{A}}}_{\nu_0}(Q^2)$ for positive $Q^2$ (left-hand figure), and the timelike running coupling ${\widetilde{\mathfrak H}}_{\nu_0}(\sigma)$ (right-hand figure), in 3$\delta$AQCD, with $n_f=3$, $\alpha_s^{\overline{\rm MS}}(M_Z^2)=0.1180$, and three different values of the IR-threshold scale parameter $M_1$.
  • Figure 3: The renormalon-resummed ${\cal F}(\sigma)$, as a function of the squared timelike momentum (squared mass) $\sigma$, in 3$\delta$AQCD, for $n_f=3$, $M_1=0.150$ GeV and $\alpha_s^{\overline{\rm MS}}(M_Z^2)=0.1180$. For comparison, we include also the corresponding pQCD TPS Eq. (\ref{['TPSMSb']}) in the $\overline{\rm MS}$ and (L)MM schemes, with three terms included ($N=3$).
  • Figure 4: The resummed values of ${\cal F}(\sigma)$ as in Fig. \ref{['FigFsigCentr']}, when (a) the IR-threshold scale $M_1$ is varied and $\alpha_s^{\overline{\rm MS}}(M_Z^2)=0.1180$; (b) when $\alpha_s^{\overline{\rm MS}}(M_z^2)$ value is varied and $M_1=0.150$ GeV.
  • Figure 5: The closed contour in the complex $z$-plane for the integral (\ref{['FD1b']}) for the case $0 < t < 1$. The limits $R \to + \infty$ and $\varepsilon \to +0$ are taken.
  • ...and 2 more figures