Neutral and charged pion Form Factors in the intermediate-energy region from double-dilaton HQCD model

TL;DR

This work investigates neutral and charged pion form factors in the intermediate-energy, space-like region using a non-perturbative running coupling derived from a double-dilaton holographic QCD model. Form factors are expressed through pion distribution amplitudes and expanded in Gegenbauer polynomials, with an infrared-fixed-point coupling $\hat{\alpha}_s(Q^2)$ that smoothly matches to perturbative QCD at $Q_0=3.79$ GeV. By fitting high-energy data with the coefficients $c_n$ in $Q^2F_0(Q^2)$ and $Q^2F_{\pi}(Q^2)$ and employing Padé Approximants to interpolate with low-energy information, the study obtains $c_0$ and $c_1$ values (e.g., $c_0\approx3.616$, $c_1\approx-3.20$ for $F_0$; $c_0\approx3.075$, $c_1\approx-2.664$ for $F_{\pi}$) and identifies matching scales around a few GeV$^2$. The analysis also derives a simple isospin-breaking form factor $F_{Iso}(Q^2)$ from the DA difference, giving $F_{Iso}(0)=0.055(5)$ and translating to $\Delta m_{\pi}^2\approx (1.1\pm0.1)\times10^{-3}$ GeV$^2$, close to the experimental value. Overall, the results indicate non-perturbative effects extend into the intermediate regime and suggest a larger effective $\Lambda_{QCD}$, motivating further refinements including resummation and more detailed isospin analyses.

Abstract

We compute the Form Factors of both neutral and charged pion using a non-perturbative running of the strong coupling constant $α_s$ obtained using a double-dilaton Holographic QCD model. These form factors remain poorly understood in the intermediate-energy region, which marks the transition between low- and high-energy physics. In particular, experimental data for the neutral pion Form Factor exhibits a deviation from the expected asymptotic behavior, and the charged pion form factor remains comparatively less explored. To address these issues, we employ the pion distribution amplitude formalism to investigate the Form Factor behavior in this intermediate regime. Our results suggests that non-perturbative physics of the strong interaction is relevant even at energy scales traditionally considered perturbative, implying that the perturbative regime could occur at higher energies than previously thought. Finally, our approach allows us to study isospin-breaking effects through the quadratic pion mass difference.

Figures (5)

• Figure 1: Comparison between experimental data of $Q^2F_0(Q^2)$ from CELLO Col. CELLO, purple triangles, CLEO Col. CLEO, red stars, BABAR Col. BaBar, black triangles, and BELLE Col. Belle, green dots; and the model prediction either fitted to data using Eq.(\ref{['hybrid']}) (solid blue line) or without fitting to them using Eq.(\ref{['eqn:neutralFFExpansion']}) with $n=1$ (dotted blue line). Dashed blue vertical line indicates the matching $Q^2_0$ point.
• Figure 2: Charged pion form factor. The solid blue curve is the result of using Eq.(\ref{['eqn:chargedFFExpansion']}) with $n=1$ at high energies with a matching procedure described in the text at low energies, see Sec.(\ref{['subsec:3.2']}). Experimental data in green dots (JLab JLAB1, JLAB2), red stars (NA7 NA7), purple triangles (Fermilab Dally1, Dally2, Dally3) and black triangles (Wilson Sync. Lab. Bebek74, Bebek76 and Bebek78). Dashed blue vertical line points the matching condition. Black dashed curve emulates the pQCD result.
• Figure 3: Normalized neutral (charged) pion distribution amplitude in discontinuous (solid) blue at $Q^2=1 \text{ GeV}^2$, in green at $Q^2=10^6 \text{ GeV}^2$, and the asymptotic limit $Q^2\to \infty$ in orange.
• Figure 4: Difference between neutral and charged pion DAs, giving the size of isospin breaking effects at $Q^2=1 \text{ GeV}^2$ (blue), at $Q^2=10^6 \text{ GeV}^2$ (green) and the asymptotic limit $Q^2\to\infty$ (orange); this last one is identically zero since neutral and charged pion DAs have the same asymptotic DA.
• Figure 5: Isospin breaking form factor $F_{\rm Iso}(Q^2)$ from Eq.(\ref{['eqn:isospinFF']}).