Extending the application of the light-cone sum rules method to low momenta using QCD renormalization-group summation: Theory and phenomenology

We show that using renormalization-group summation to generate the QCD radiative corrections to the $\pi -\gamma$ transition form factor, calculated with light-cone sum rules (LCSR), renders the strong coupling free of Landau singularities while preserving the QCD form-factor asymptotics. This enables a reliable applicability of the LCSR method to momenta well below $1{\mathrm{GeV}}^2$. This way, one can use the new preliminary BESIII data with unprecedented accuracy below $1.5{\mathrm{GeV}}^2$ to fine tune the prefactor of the twist-six contribution. Using a combined fit to all available data below $3.1{\mathrm{GeV}}^2$, we are able to determine all nonperturbative scale parameters and a few Gegenbauer coefficients entering the calculation of the form factor. Employing these ingredients, we determine a pion distribution amplitude with conformal coefficients $(b_2,b_4)$ that agree at the $1\sigma$ level with the data for $Q^2\le 3.1{\mathrm{GeV}}^2$ and fulfill at the same time the lattice constraints on $b_2$ at ${\mathrm{N}}^3\mathrm{LO}$ together with the constraints from QCD sum rules with nonlocal condensates. The form-factor prediction calculated herewith reproduces the data below $1{\mathrm{GeV}}^2$ significantly better than analogous predictions based on a fixed-order power-series expansion in the strong coupling constant.

Figure 1

Results of the two-step fitting procedure described in the text. The following notations are used: BMS DA—black cross [18]; platykurtic DA—black/white cross [31, 32]; DA shown as a grey triangle selected from the BMS set of DAs to be inside the $1\sigma$ error ellipse (innermost red line) while fulfilling the ${\mathrm{N}}^3\mathrm{LO}$ (vertical solid lines) lattice constraints on $b_2$ from lattice QCD [17]. Left: first step of this procedure in which we determine the admissible regions of the higher-twist parameters ${\delta }_{\mathrm{tw}\text{−}4}^2$ and ${\delta }_{\mathrm{tw}\text{−}6}^2$. The larger rectangle denotes the range of values with ${\delta }_{\mathrm{tw}\text{−}6}^2=(1.84_{-0.24}^{+0.84})\times {10}^{-4}{\mathrm{GeV}}^6$, obtained in [33], while the smaller rectangle corresponds to the estimate ${\delta }_{\mathrm{tw}\text{−}6}^2=(1.76\pm 0.13)\times {10}^{-4}{\mathrm{GeV}}^6$. This estimate and the red point ${\delta }_{\mathrm{tw}\text{−}6}^2=(1.61\pm 0.26)\times {10}^{-4}{\mathrm{GeV}}^6$ were obtained in this work. Right: results of the fitting procedure for the twist-two conformal coefficients $b_2$, $b_4$ with fixed higher-twist parameters. The two rectangles along the lower diagonal denote the range of $(b_2,b_4)$ determined within the BMS approach [18] for two different values of ${\lambda }_q^2=0.4{\mathrm{GeV}}^2$ (larger shaded rectangle) and $0.45{\mathrm{GeV}}^2$ (transparent rectangle), where the BMS DA [18] is represented by . The smaller shaded rectangle encloses the range of $(b_2,b_4)$ coefficients associated with DAs having a platykurtic profile [32], like the model proposed in [31]. The dashed-dotted, dashed, and solid vertical lines show the lattice results for $b_2$ from [17] for the NLO (0.109(37)), NNLO (0.139(32)), and ${\mathrm{N}}^3\mathrm{LO}$ (${0.159}_{-0.027}^{+0.025}$), respectively.

Figure 2

Theoretical predictions for the scaled ${\gamma }^{*}\gamma {\pi }^0$ transition form factor $Q^2{F}_{\mathrm{FAPT}}^{\gamma \pi }(Q^2)$ [GeV] using different DAs discussed in the text in comparison with various data up to $Q^2<5.5{\mathrm{GeV}}^2$ with labels as indicated in the figure. The grey and the red solid lines were obtained with the DA denoted by using the FAPT/LCSR and FOPT/LCSR scheme, respectively. The black dashed line represents the FAPT result obtained with the pk-DA [31], while the green strip shows the theoretical uncertainties of the BMS DAs calculated with QCD sum rules with nonlocal condensates [18]. The displayed FAPT/FOPT TFF results employ the best-fit nonperturbative higher-twist parameters ${\delta }_{\mathrm{tw}\text{−}4}^2=0.19{\mathrm{GeV}}^2$ and ${\delta }_{\mathrm{tw}\text{−}6}^2=1.61\times {10}^{-4}{\mathrm{GeV}}^6$.