QCD analytic perturbation theory: From integer powers to any power of the running coupling

We propose a new generalized version of the QCD analytic perturbation theory of Shirkov and Solovtsov for the computation of higher-order corrections in inclusive and exclusive processes. We construct nonpower series expansions for the analytic images of the running coupling and its powers for any fractional (real) power and complete the linear space of these solutions by constructing the index derivative. Using the Laplace transformation in conjunction with dispersion relations, we are able to derive at the one-loop order closed-form expressions for the analytic images in terms of the Lerch function. At the two-loop order we provide approximate analytic images of products of powers of the running coupling and logarithms—typical in higher-order perturbative calculations and when including evolution effects. Moreover, we supply explicit expressions for the two-loop analytic coupling and the analytic images of its powers in terms of one-loop quantities that can strongly simplify two-loop calculations. We also show how to resum powers of the running coupling while maintaining analyticity, a procedure that captures the generic features of Sudakov resummation. The algorithmic rules to obtain analytic-coupling expressions within the proposed fractional analytic perturbation theory from the standard QCD power-series expansion are supplied ready for phenomenological applications and numerical comparisons are given for illustration.

Figure 1

(a) Comparison of different curves for ${\mathcal{A}}_{\nu }^{(1)}(L)$ as functions of the index $\nu \le 0$, corresponding to various values of $L$, ranging from $L=-3$ to $L=+3$. The blue broken lines correspond to $L<0$, whereas the red thick broken lines correspond to $L>0$. The two dotted lines denote the results for $L=\pm 1$, with the line associated with the value $L=1$ being closer to the zero line. (b) The same comparison for $\nu \ge 0$ using the same line designations, but for values of $L$, ranging from $L=-10$ to $L=+10$ (dashed lines). The dotted lines here correspond to $L=\pm 2$, whereas the short-dashed lines represent the result for $L=\pm 5$. The blue solid lines in both panels show ${\mathcal{A}}_{\nu }^{(1)}(0)$.

Figure 2

Comparison of different curves for ${\mathcal{A}}_m^{(1)}(L)$ as functions of $L$, corresponding to various values of the index $m$, ranging from $m=2$ to $m=5$. To show all the details in a single plot, higher couplings are multiplied by numerical factors in order to normalize them to the scale of ${\mathcal{A}}_2^{(1)}(L)$. The solid line shows ${\mathcal{A}}_2^{(1)}(L)$, the blue dotted line corresponds to $8·{\mathcal{A}}_3^{(1)}(L)$, the green short-dashed line to $60·{\mathcal{A}}_4^{(1)}(L)$, and the red long-dashed line to $480·{\mathcal{A}}_5^{(1)}(L)$.

Figure 3

(a) The dotted line corresponds to ${\Delta }_2^{\mathrm{FAPT}}(L)$ and the solid line to ${\Delta }_3^{\mathrm{FAPT}}(L)$. (b) The dashed line corresponds to ${\Delta }_3^{\mathrm{PT}}(L)$ and the solid line to ${\Delta }_3^{\mathrm{FAPT}}(L)$.

Figure 4

(a) Comparison of different results for ${\mathcal{A}}_2^{(2)}(L)$. The solid line corresponds to ${\mathcal{A}}_2^{(2);\mathrm{FAPT}}(L)$, computed analytically via Eq. (3.29); ${\mathcal{A}}_2^{(2);\mathrm{num}}(L)$ (dashed line) is derived by means of a numerical integration. The dotted line represents the available results of the numerical procedure of Magradze in [23]. (b) Comparison of FAPT and standard QCD PT with respect to the fractional index (power) of the coupling. The solid line represents ${\mathcal{A}}_{0.62}^{(2);\mathrm{FAPT}}(L)$, computed analytically via Eq. (3.29), while the dashed line stands for $a_{(2)}^{0.62}(L)$.

Figure 5

Analytic images ${\mathcal{A}}_{\nu }^{(2)}(L)$ at the two-loop level for some characteristic index values, calculated with NNLO FAPT. The solid line corresponds to $\nu =0$, the dash-dotted one to $\nu =1$, and the lowest line to $\nu =2$. All other lines correspond to $\nu =k/4$ with $k=1,\dots ,7$.

Figure 6

(a) Functions $F_{\mathrm{S}}^{(\mathrm{an})}(x,L)$ calculated for different values of $x$ using the one-loop FAPT. The horizontal solid line corresponds to $x=0$, while the broken lines represent the specific $x$ values shown above them. (b) Comparison of the toy Sudakov function, calculated within the present framework (solid line), with the result of the naive analytization [2, 36, 37] (dashed line), and that from conventional QCD perturbation theory (dotted line). Note the different scale for the ordinate relative to panel (a).