Renormalization-scheme variation of a QCD perturbation expansion with tamed large-order behavior

The renormalization-scheme and scale dependence of the truncated QCD perturbative expansions is one of the main sources of theoretical error of the standard model predictions, especially at intermediate energies. Recently, a class of renormalization schemes, parametrized by a single real number $C$, has been defined and investigated in the frame of the standard perturbation expansions in powers of the coupling. In the present paper we investigate the $C$-scheme variation of a Borel-improved QCD perturbation series, which implements information about the large-order divergent character of perturbation theory by means of an optimal conformal mapping of the Borel plane. In the new expansions, the powers of the strong coupling are replaced by a set of expansion functions with properties which resemble those of the expanded correlators, having in particular a singular behavior at the origin of the complex coupling plane. On the other hand, the new expansions have a tamed increase at high orders, as demonstrated by previous studies in the $\overline{\mathrm{MS}}$ renormalization scheme. Using as examples the Adler function and the hadronic decay width of the $\tau$ lepton, we investigate the properties of the Borel-improved expansions in the $C$-scheme, in comparison with the standard expansions in the $C$-scheme and the expansions in $\overline{\mathrm{MS}}$. The variation with the renormalization scale and the prescription for the choice of an optimal value of the parameter $C$ are discussed. The good large-order behavior of the Borel-improved expansions is proved also in the $C$-scheme, which is a further argument in favor of using them in applications of perturbative QCD at intermediate energies.

Figure 1

The coupling ${\widehat{a}}_{\mu }$ found by solving Eq. (13) for the scale $\mu =0.61m_{\tau }$ and $a_{\mu }=0.442(20)/\pi$ in the $\overline{\mathrm{MS}}$ scheme, as a function of C. The yellow band corresponds to the ${\alpha }_s$ uncertainty.

Figure 2

Borel-improved expansions in the $C$-scheme of the Adler function $\widehat{D}$ at $s=-m_{\tau }^2$. The central line is the expansion to order ${\alpha }_s^5$. The yellow band is obtained by either removing or doubling the last term. Marked in red are the points where the last term of the expansion vanishes and the magnitude of the last nonvanishing term is shown.

Figure 3

Borel-improved expansions to order ${\alpha }_s^5$ in the $C$-scheme of the Adler function $\widehat{D}$ at $s=-m_{\tau }^2$, for the scales $\mu =m_{\tau }$, $\mu =0.61m_{\tau }$ and $\mu =1.28m_{\tau }$, as functions of $C$. We show in each case the optimal points where the last term of the expansion vanishes and the magnitude of the last nonvanishing term.

Figure 4

Borel-improved expansion of ${\delta }_{\mathrm{FO}}^{(0)}$ to order $O({\alpha }_s^5)$ in the $C$ scheme as a function of $C$. The yellow band arises from either removing or doubling the last term in the expansion. In red are marked the points where the last term in the expansion vanishes, and the magnitude of the previous one is indicated.

Figure 5

Borel-improved expansion of ${\delta }_{\mathrm{CI}}^{(0)}$ to $O({\alpha }_s^5)$ in the $C$ scheme as a function of $C$. The yellow band arises from either removing or doubling the fifth-order term. In red is marked the point where the last term in the expansion vanishes, and the magnitude of the previous term is indicated.

Figure 6

Variation with $C$ of the Borel-improved expansions in the $C$-scheme of the Adler function $\widehat{D}$ at $s=-m_{\tau }^2$, for various truncation orders $N$.

Figure 7

Real part (left) and imaginary part (right) of the Adler function in the model [27] and its truncated Borel-improved expansions in the $C$-scheme along the circle $|s|=m_{\tau }^2\exp (i\varphi )$.