Analyticity properties of three-point functions in QCD beyond leading order

The removal of unphysical singularities in the perturbatively calculable part of the pion form factor—a classic example of a three-point function in QCD—is discussed. Different analytization procedures in the sense of Shirkov and Solovtsov are examined in comparison with standard QCD perturbation theory. We show that demanding the analyticity of the partonic amplitude as a whole, as proposed before by Karanikas and Stefanis, one can make infrared finite not only the strong running coupling and its powers, but also cure potentially large logarithms (that first appear at next-to-leading order) containing the factorization scale and modifying the discontinuity across the cut along the negative real axis. The scheme used here generalizes the analytic perturbation theory of Shirkov and Solovtsov to noninteger powers of the strong coupling and diminishes the dependence of QCD hadronic quantities on all perturbative scheme and scale-setting parameters, including the factorization scale.

Figure 1

(a) Results for the analyticized logarithmic term, ${\mathcal{L}}_2^{(2)}(Q^2)$, using three different analytization procedures: the one-loop approximate KS “analytization,” ${\mathcal{L}}_2^{(1)\mathrm{approx}}(Q^2)$ (red dash-dotted line), the two-loop approximate KS “analytization,” ${\mathcal{L}}_2^{(2)\mathrm{approx}}(Q^2)$ (green dashed line), and the exact two-loop BMKS analytization ${\mathcal{L}}_2^{(2)}(Q^2)$ (blue solid line). For comparison, we also show here the corresponding maximal analytization curve (dotted line). (b): Results are shown for the corresponding analyticized contributions ${\Delta }_2^{(2)}(Q^2)$, using the same three analytization procedures for ${\mathcal{L}}_2^{(2)}(Q^2)$ as in panel (a).

Figure 2

Results for the ratio of the factorized pion form factors, using two different analytization procedures: KS analytization and “maximal analytization,” $F_{\pi }^{\mathrm{KS}}(Q^2)/F_{\pi }^{\mathrm{Max}\mathrm{\text{−}}\mathrm{an}}(Q^2)$. The designations are: red dash-dotted line—one-loop approximation of the KS logarithmic term (${\mathcal{L}}_2^{(1)\mathrm{approx}}(Q^2)$); green dashed line—two-loop approximation (${\mathcal{L}}_2^{(2)\mathrm{approx}}(Q^2)$); blue solid line—exact two-loop KS analytization (${\mathcal{L}}_2^{(2)}(Q^2)$). Left panel: default scale setting ($\lambda =1$); middle panel: BLM scale setting; right panel: ${\alpha }_V$ scheme. The factorization scale ${\mu }_{\mathrm{F}}^2$ is set equal to $5.76{\mathrm{GeV}}^2$ [57].

Figure 3

Results for the factorized pion form factors, using two different analytization procedures: KS analytization (blue solid line) and “maximal analytization” (red dotted line) for different values of the factorization scale: in the left panel ${\mu }_{\mathrm{F}}^2=1{\mathrm{GeV}}^2$, in the middle one—${\mu }_{\mathrm{F}}^2=1{\mathrm{GeV}}^2$, and in the right one—${\mu }_{\mathrm{F}}^2=10{\mathrm{GeV}}^2$. For all panels we show results for the BLM scale setting.

Figure 4

Results for the factorized pion form factor, scaled with $Q^2$, and assuming the default scale setting (${\mu }_{\mathrm{R}}^2=Q^2$) in standard perturbation theory and APT. The latter is implemented in terms of two different analytization procedures: naive analytization and maximal “analytization.” The designations are: blue dashed line—standard perturbation theory; green dash-dotted line—naive APT; red solid line—maximal APT. The prediction obtained with the KS analytization is too close to that found with the maximal one to differentiate these curves graphically. The factorization scale ${\mu }_{\mathrm{F}}^2$ is set equal to $5.76{\mathrm{GeV}}^2$.