Extending the LCSR method to the electromagnetic pion form factor at low momenta using QCD renormalization-group summation

TL;DR

This work extends the light-cone sum rule approach to the electromagnetic pion form factor by incorporating renormalization-group summation, which, together with dispersion relations, converts the perturbative expansion into a fractional analytic perturbation theory (FAPT) framework. The method is applied to twist-2 as well as twist-4 (and twist-6) contributions, yielding an emFF description that remains well-behaved at low momentum transfers ($Q^2\sim0.5$ GeV$^2$) due to Landau-pole-free couplings. Using corrected twist-2 DA coefficients and nonperturbative twist-4/6 inputs, the RG-summed predictions agree with JLab data for $Q^2\lesssim 1$ GeV$^2$, and a hybrid NLO comparison clarifies the relative size of fixed-order corrections. The approach shows promise for extracting the leading-twist pion DA from data once the complete NLO corrections are available, highlighting the interplay between perturbative refinements and nonperturbative DA inputs in shaping the emFF. Overall, the paper demonstrates that RG-summed LCSR combined with FAPT and dispersion relations provides a robust, low-$Q^2$ description of $F_\pi$, with a clear path toward quantitative determinations of the pion's internal structure.

Abstract

We obtain the electromagnetic pion form factor (emFF) $F_π$ for spacelike mid-range of momentum transfer in QCD. We use renormalization group (RG) summation within the light cone sum rules (LCSRs) to obtain the QCD radiative corrections to the $F_π$ and involve contributions of the leading twist 2 and, twists 4, 6. The additional conditions to apply here this RG summation are discussed in details. The strong coupling constants in this approach are free of Landau singularities, which allows one to go down to the lower transferred momentum $Q^2$. The prediction of the calculations performed reproduces the experimental data below/around $Q^2= 1$~GeV$^2$ significantly better than analogous predictions based on a fixed-order power-series expansion in the standard QCD.

Figures (7)

• Figure 1: The upper dashed line connected blocks show the way of the standard FOPT LCSR, here $\mu^2=\mu^2_F=\mu^2_R$ is renormalization/factorization scale, $M^2$ is the Borel parameter. The lower solid line illustrates our approach to the LCSR with the preliminary renormalization group summation. The crucial elements of our approach are presented in blocks "RG summation" and "Dispersion relation", which are taken in a frame.
• Figure 2: Solid (black) line is the result of RG sum $F^\text{(4)LCSR}_\pi$ in Eq.(\ref{['eq:F4b']}); the dashed (blue) line is the standard result $F^{(4)}_\pi$ in Eq.(\ref{['eq:standF4']}) with the value of $\delta_\text{tw-4}^2(\mu^2_0)$ in Bijnens:2002mg, both curves are taken at $M^2=1.2$ GeV$^2$.
• Figure 3: Different twist contributions of the LO RG summation: the upper widening, blue strip is twist-2 $F^\text{(tw2)LCSR,LO}_\pi$; the thin, dashed-dotted grey line is twist-4 $F^\text{(tw4)LCSR}_\pi$; the red narrowing strip with dots is twist-6 $F^\text{(tw6)}_\pi$ .
• Figure 4: Predictions for LO LCSR, $F^{\text{LCSR}}_{\pi}$ in Eq.(\ref{['eq:F-RGtot']}) for DA$^{bf}$, for the bunch of BMSmod, and for Asy DAs are presented with dotted red line, blue and grey strips - uncertainties in LO and NLO respectively in both panels. The red discs with error bars are the experimental data from PhysRevC.78.045203; the open boxes with bars are the recent lattice predictions Ding:2024lfj. Left:Dashed blue line, and in addition to them the standard NLO LCSR Bijnens:2002mg corrections -- upper solid black line (named hybrid) are based on DA$^{bf}$ from item (iv). Right: The same curve designations for $F^{\text{LCSR}}_{\pi}$ and NLO LCSR corrections based on the bunch of BMSmod DAs.
• Figure 5: The predictions for the (partial N)LO $F^{\text{LCSR}}_{\pi}$, the notations are the same as in Fig.\ref{['fig:EMFF-RG']} in both panels. The red discs with their error bars -- experimental data from PhysRevC.78.045203; the open boxes with bars -- recent lattice results Ding:2024lfj. Left: Based on DA$^{bf}$ (item (iv)), dashed blue line presents LO, upper solid black line, named "Partial", presents the (partial N)LO corrections in Eq.(\ref{['eq:F_1aLCSR']}). Right:The same designations are for the LO and the (partial N)LO to emFF, based on the bunch of BMSmod DAs (item (i)).