The electromagnetic form factors of heavy-light pseudo-scalar and vector mesons

Electromagnetic form factors (EFFs) encode the response of mesons to electromagnetic probes and provide key insights into their internal structure in QCD. While light and slightly flavor-asymmetric mesons (π, K, ρ) have been widely studied, heavy-light systems (e.g., ūc, ūb, c̄s) possess strong flavor asymmetry and can reveal additional information about confinement and dynamical chiral symmetry breaking (DCSB). However, experimental data are scarce and theoretical treatments are more challenging for these systems. This work addresses this gap by computing, within a non-perturbative, Poincaré-covariant Dyson-Schwinger/Bethe-Salpeter equations (DSEs/BSEs) framework, the EFFs of heavy-light pseudo-scalar and vector mesons, extracting charge radii, magnetic moments, and quadrupole moments, and comparing to other approaches and available data.

Heavy-light meson EFFs have been investigated with several methods, including light-front frameworks (LFF), constituent quark models (CQM), contact interaction models (CI), algebraic models (AM), extended Nambu–Jona-Lasinio (ENJL), lattice QCD (LQCD), and others. Results often agree for light or moderately asymmetric systems but diverge for extreme flavor asymmetry (e.g., ūb). Within DSEs/BSEs, prior EFF studies focused on flavor-symmetric or mildly asymmetric mesons (π, K, ρ). Recent developments include flavor-dependent kernels and beyond-rainbow-ladder (RL) truncations aimed at better describing heavy-light systems. The present study extends DSEs/BSEs EFF analyses systematically to heavy-light mesons and benchmarks with prior theoretical results and available experimental/lattice inputs.

• Framework: Euclidean-space Dyson-Schwinger equations (DSEs) for dressed quark propagators and Bethe-Salpeter equations (BSEs) for meson bound-state amplitudes, using rainbow-ladder (RL) truncation as a baseline.
• Quark propagators: Solve the gap equation S^{-1}(p) = i γ·p A(p^2) + B(p^2) with a mass-independent MOM renormalization at ξ = 19 GeV. The effective gluon interaction is the Qin–Chang model with infrared Gaussian term plus perturbative tail; parameters D and ω are chosen flavor-dependently (slight tweaks for c, b) to reproduce meson observables.
• Bethe-Salpeter amplitudes (BSAs): Pseudoscalar and vector mesons expanded in standard covariant bases; canonical normalization applied to extract decay constants and ensure current conservation.
• Quark-photon vertex: Obtained from the inhomogeneous BSE and constrained by the vector Ward–Green–Takahashi identity (WGTI) to preserve electromagnetic current conservation.
• Flavor-asymmetric kernel: For heavy-light systems, employ an effective weight-averaged RL kernel (“weight-RL”) with a channel-dependent weight factor η chosen to reproduce the pseudoscalar meson mass; with η fixed, vector meson masses and decay constants are predicted. This extends standard RL to account effectively for strong flavor asymmetry while maintaining WGTI compliance.
• Form-factor calculation: Use generalized impulse approximation with dressed propagators, BSAs, and quark-photon vertex. Work in a moving frame to avoid interpolation/extrapolation of BSAs/vertex. Decompose total meson form factors into contributions from lighter and heavier (anti)quarks, enabling flavor-separated analyses. For vectors, extract GE, GM, GQ and identify GE(0)=charge, GM(0)=μ, GQ(0)=Q.
• Kinematics and numerical domain: Compute for spacelike momentum transfers Q^2 < 2 GeV^2 (except where quark-propagator pole locations restrict access, notably in some B channels). Validate by comparing π and K form factors with NA7 and JLab data.
• Benchmark observables: Masses and decay constants for light, heavy, and heavy-light mesons compared with experiment and lattice; then compute charge radii for PS and VC, and μ, Q for VC.

• Validation on light mesons: π and K electromagnetic form factors are in good agreement with NA7 and JLab data.
• Flavor-symmetry breaking and quark-level splitting: In heavy-light mesons, the lighter-quark contribution to the form factor steepens (larger radius) as the mass of the partner quark increases, while the heavier-quark contribution flattens (smaller radius). The total EFF emerges from the competition of these contributions.
• Vector vs pseudoscalar sizes: For the same flavor content, vector meson charge radii exceed those of pseudoscalars, indicating spin-dependent interactions expand the meson size.
• Representative pseudoscalar charge radii (sqrt<r^2> in fm; full): π ≈ 0.646; K+ ≈ 0.608; D+ ≈ 0.435; D0 ≈ 0.619; B0 ≈ 0.352; B_s ≈ 0.337i; B_c ≈ 0.219. For the extreme ūb system, the u-quark’s radius nearly saturates the meson radius (PS: ≈0.618 fm; VC: ≈0.655 fm for the u contribution), signaling an almost stationary b quark.
• Representative vector-meson observables (full): ρ: sqrt<r^2> ≈ 0.722 fm, μ ≈ 2.006 (e/2M_V), Q ≈ −0.364 (e/M_V^2); K*+: 0.679 fm, 2.121, −0.433; D*+: 0.473 fm, 2.434, −0.525; B*+: 0.657 fm, 7.880, −2.177; D_s* : 0.381 fm, 2.267, −0.461; B_s*: 0.347 fm, −2.774, 0.887; B_c*: 0.231 fm, 3.193, −0.669.
• Magnetic moments in nuclear magnetons (μ_N), comparison across models is generally good for mild asymmetry; larger spread for extreme asymmetry. Examples (this work): ρ ≈ 2.492 μ_N, K*+ ≈ 2.261 μ_N, D*+ ≈ 1.132 μ_N, D0 ≈ −0.944 μ_N, B+ ≈ 1.386 μ_N, B0 ≈ −0.589 μ_N, D_s ≈ 1.007 μ_N, B_s* ≈ −0.480 μ_N, B_c* ≈ 0.472 μ_N.
• Neutral systems show nontrivial GE, GM, GQ behavior away from Q^2=0 despite net charge zero, reflecting internal structure.
• Masses and decay constants: With η fixed via PS masses, the framework reproduces PS and V masses/decay constants across channels in good agreement with experiment and lattice benchmarks.

The study addresses how strong flavor asymmetry reshapes the electromagnetic structure of heavy-light mesons within a symmetry-preserving DSEs/BSEs framework. By enforcing WGTI and using an effective flavor-weighted kernel, the calculation preserves current conservation and yields realistic spectra and decay constants. The separation into lighter and heavier quark contributions reveals a clear mechanism: increasing partner mass compresses the heavy constituent’s spatial distribution while the light constituent’s distribution expands. Their competition governs the total EFF, charge radius, and, for vectors, magnetic and quadrupole moments. The systematic finding that vector mesons have larger radii than their pseudoscalar counterparts underscores the role of spin-dependent interactions in inflating hadron size, consistent with flavor-symmetric cases. Comparisons with experimental data (π, K) validate the approach at low Q^2; cross-model comparisons for heavy-light channels show broad consistency for mildly asymmetric systems but larger variance for extremely asymmetric cases, pointing to sensitivity to interaction modeling and the need for data. The observed correlation between larger quark-level charge radii and increased μ and |Q| in vectors further elucidates the interplay of spatial and spin structures in heavy-light mesons.

This work presents the first systematic DSEs/BSEs calculation of electromagnetic form factors for heavy-light pseudoscalar and vector mesons across us, uc, ub, sc, sb, and cb systems. Using a flavor-weighted RL kernel consistent with WGTI, the study reproduces masses and decay constants and extracts charge radii for PS and VC, along with μ and Q for VC. Key conclusions are: (1) flavor symmetry breaking splits quark-level EFFs, with light-quark distributions expanding and heavy-quark distributions contracting as asymmetry grows; (2) vector mesons exhibit larger charge radii than pseudoscalars of the same flavor content; (3) in vectors, larger quark radii correlate with larger magnetic and quadrupole moments. The predictions align with available data for light mesons and are broadly consistent with other models, especially for moderate asymmetry. Future work includes implementing systematically improved beyond-RL kernels, refining flavor-dependent interactions, extending kinematic reach, and confronting forthcoming experimental and lattice results to further constrain hadron structure in QCD.

• Kernel approximation: The flavor-weighted RL kernel is an effective model; a fully derived beyond-RL kernel with explicit vertex and gluon-dressing effects is not implemented, potentially impacting quantitative precision for extreme flavor asymmetry.
• Kinematic range: Results are limited to spacelike Q^2 < 2 GeV^2; the presence of quark-propagator poles in the complex plane restricts access, especially for B systems.
• Parameter tuning: Flavor-dependent interaction parameters (D, ω) and weight factors η are tuned (η fixed by PS masses), introducing model dependence.
• Data scarcity: Experimental measurements of heavy-light meson EFFs and vector μ, Q are sparse; validation relies on cross-model comparison and limited lattice inputs.
• Numerical aspects: Although observables are α-independent, practical choices of momentum partitioning and contour integration must avoid pole contamination, which can affect numerical stability and accessible Q^2 regions.