Tomography of light hadron structures: From LCDAs to iTMDs

Technical Summary: Tomography of Light Hadron Structures

Problem Statement
The paper addresses a specific limitation in the $k_T$ factorization formalism used for exclusive QCD processes, particularly in the intermediate-energy region. While the standard formalism successfully resums large logarithms ($\ln k_T^2$) into a Sudakov exponent to handle hard transverse dynamics, it overlooks "soft transverse dynamics." In the conventional approach, the renormalization scale is typically set at the hard factorization scale, which effectively defines the transverse separation of partons within the hard interaction region. Consequently, the soft radiation in the transverse plane occurring outside this central potential field is absent from the definition of Light-Cone Distribution Amplitudes (LCDAs). This omission leads to a gap in describing the probability amplitude for hadron states where partons undergo gentle oscillations in the transverse plane while moving rapidly along the longitudinal axis.

Methodology
To resolve this, the authors propose a framework that introduces Intrinsic Transversal Momentum Distribution functions (iTMDs) to complement the existing LCDAs.

• Factorization Improvement: The authors replace the standard LCDAs in the pion electromagnetic form factor calculation with a soft pion wave function, $\psi(u, k_T)$, defined as the product of the LCDA, $\phi(u, \mu_r)$, and the iTMD, $\Sigma(u, k_T)$.
• Parametrization: The iTMDs are parametrized using a Gaussian function derived from a transversal harmonic oscillator model, preserving rotational invariance. This function depends on a transversal size parameter, $\beta^2$.
• Chiral Mass Determination: To ensure high accuracy in the perturbative QCD (pQCD) calculation, particularly for the dominant twist-3 contributions, the authors employ a modular dispersion relation. This method relates spacelike and timelike form factors without resonant model dependence, utilizing precise timelike measurements and high-precision pQCD inputs to extract the chiral mass ($m_0$).
• Scope of Calculation: The framework is systematically applied to calculate electromagnetic and transition form factors for light pseudoscalar mesons ($\pi, K, \eta, \eta', \eta_c$). The calculations include Next-to-Leading Order (NLO) QCD corrections for leading and subleading twist LCDAs and Leading Order (LO) for twist-four LCDAs. While the proton's Dirac form factor is discussed in the context of endpoint enhancement and $k_T$ resummation, the detailed application of the iTMD framework to the proton is noted as a study in progress.

Key Contributions and Results

1. Pion Electromagnetic Form Factor: The iTMDs-improved pQCD predictions show good agreement with experimental data (NA7, JLab, BABAR) and lattice QCD results in both spacelike and timelike regions. The study confirms that higher-twist contributions, specifically subleading-twist LCDAs, are indispensable for explaining the scaling behavior of the pion form factor at intermediate momentum transfers ($\sim 5$ GeV$^2$).
2. Kaon Form Factors: The inclusion of soft transverse dynamics via iTMDs is essential for explaining precise timelike kaon form factor data. However, a discrepancy remains between the spacelike predictions and lattice results. The authors suggest this may stem from $SU(3)$ flavor breaking, which introduces an additional $O(m_s)$ term in the twist-3 LCDAs of the kaon.
3. Pion Transition Form Factor: The analysis of the pion transition form factor ($\pi \gamma \gamma^*$) reveals that the iTMD effect is significant in the small-to-intermediate $Q^2$ region (few GeV$^2$) but less dominant in the intermediate region. The improved predictions support the asymptotic QCD limit ($Q^2 F_{\pi \gamma \gamma^*} \to \sqrt{2}f_\pi$) and align more closely with Belle collaboration measurements than previous predictions.
4. $\eta$ and $\eta'$ Mesons: By decomposing the transition form factors into orthogonal flavor bases ($\eta_q, \eta_s, \eta_c$), the framework favors a small mixing angle $\phi = (37.7 \pm 0.7)^\circ$.
5. $\eta_c$ Form Factor: The iTMD-improved prediction yields a larger normalization value for $F_{\eta_c \gamma \gamma^*}(0)$ compared to Light-Front Quantization (BLFQ) and Bethe-Salpeter Equation (BSE) models. This larger value is consistent with recent BESIII measurements of the $\Gamma(\eta_c \to \gamma \gamma)$ width.
6. Proton Form Factor: The paper highlights that for the proton's Dirac form factor, unexpectedly large higher-power contributions from subleading-twist LCDAs arise from endpoint enhancement. The $k_T$ resummation tames this enhancement, which is crucial for reproducing the approximate scaling of $Q^4 F_1(Q^2)$ data at intermediate $Q^2$ ($\sim 10$ GeV$^2$). However, the specific study of iTMD effects in proton form factors is explicitly noted as being in progress.

Significance
The paper claims that the introduction of iTMDs completes the $k_T$ factorization formalism for exclusive processes by accounting for the soft transverse dynamics previously overlooked. The primary significance lies in demonstrating that:

• Subleading-twist LCDAs are not merely corrections but are critical for generating the correct scaling behavior of form factors at intermediate momentum transfers.
• The iTMD framework significantly extends the validity of pQCD predictions down to the few GeV$^2$ region, providing a more accurate description of the form factor lineshape where soft dynamics dominate.
• The approach offers a precision test for flavor symmetry breaking (specifically $SU(3)$ breaking in kaons) and provides a theoretical basis for interpreting recent experimental data from BESIII and Belle regarding meson transition form factors.

The authors conclude that while the study of iTMD effects in proton form factors is ongoing, the current results for light mesons establish the necessity of combining LCDAs with intrinsic transverse momentum distributions to fully capture hadron structure.