Timelike Vector Form Factors

• Timelike vector form factors are Lorentz-invariant functions that describe hadronic responses to external vector currents at positive q², highlighting resonance behavior and complex phase structures.
• Finite-volume lattice QCD techniques, via the Lüscher formalism, enable precise extraction of form factors like the pion’s, linking discrete energy levels to infinite-volume amplitudes.
• Phenomenological models such as vector meson dominance integrate resonance physics and dispersion relations to explain the observed oscillatory structures and phase differences in experimental data.

A timelike vector form factor is a Lorentz-invariant function that encodes the response of hadrons to external vector currents in the kinematic regime where the squared momentum transfer $q^2$ is positive ($q^2 > 0$). In this region, typically probed in particle–antiparticle production processes such as $e^+e^- \to \text{hadrons}$ and heavy meson decays, these form factors characterize the creation or annihilation of hadron–antihadron pairs via a virtual vector boson and exhibit nontrivial analytic structure, resonance behavior, and complex phases. They are central objects in hadron structure studies, enabling access to resonance parameters, phase information relevant for polarization observables, and contributions to precision Standard Model observables like the hadronic vacuum polarization in $(g-2)_\mu$.

1. Formal Definitions and Analytic Structure

Timelike vector form factors appear in the matrix elements of vector currents between the QCD vacuum and hadron–antihadron states, or in weak/EM transitions between hadrons. For a baryon (e.g., the nucleon) the electromagnetic current between $N\bar N$ states is expressed as: $$\left\langle N(p,s) \bar N(\bar p, \bar s) | J^\mu_{\mathrm{em}} | 0 \right\rangle = \bar u(p,s) \left[ \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu} q_\nu}{2 M_N} F_2(q^2) \right] v(\bar p, \bar s)$$ where $q = p + \bar p$ and $F_1, F_2$ are the Dirac and Pauli form factors.

For mesons, e.g. the pion, the electromagnetic form factor $F_\pi(s)$ appears in: $$\left\langle \pi^+ \pi^- | J^\mu_{\mathrm{em}} | 0 \right\rangle = (p_+ - p_-)^\mu F_\pi(s)$$ with $s = q^2 = (p_+ + p_-)^2$.

In the timelike region ($q^2 > 0$), physical thresholds ($q^2 \geq (m_1 + m_2)^2$ for hadrons of masses $m_1, m_2$) correspond to the on-shell production of particle–antiparticle pairs. The form factors are complex analytic functions with cuts along the real $q^2$ axis due to unitarity (final-state rescattering), as encapsulated by dispersion relations: $$F(q^2) = \int_{s_\mathrm{thr}}^\infty \frac{ds'}{\pi} \frac{\mathrm{Im}\, F(s')}{s' - q^2 - i\epsilon}$$ Their modulus and phase are accessible experimentally via cross sections and, for polarized or sequential decays, via angular distributions.

2. Extraction in Lattice QCD: Finite-Volume Formalism

Lattice QCD operates in Euclidean space, creating a fundamental challenge for accessing timelike processes. Recent strategies exploit the discrete energy spectrum of two-hadron states in finite volume, relating measured quantities to infinite-volume amplitudes via the Lüscher formalism. For the pion form factor, the methodology is as follows (Meyer, 2011, Feng et al., 2014, Ortega-Gama et al., 30 Jul 2024):

• Two-pion energy levels in a finite box obey the quantization condition:

$$\delta_1(k) + \phi(q) = n\pi,\qquad q = \frac{k L}{2\pi},\quad E_{\pi\pi}= 2\sqrt{m_\pi^2 + k^2}$$

where $\delta_1(k)$ is the $P$-wave $\pi\pi$ phase shift and $\phi(q)$ encodes finite-volume kinematics.

• Matrix elements of the vector current between the vacuum and two-pion finite-volume states are measured:

$$A_\psi = L^{3/2} \langle \psi^a_\sigma | j^b_{\sigma'}(0) | 0 \rangle$$

• The timelike form factor modulus is extracted via a master formula:

$$|F_\pi(E)|^2 = \left[q \phi'(q) + k \frac{\partial\delta_1}{\partial k}\right] \cdot \frac{3\pi E^2}{2 k^5} |A_\psi|^2$$

This direct relation circumvents the need for analytic continuation from Euclidean to Minkowski signature (ill-posed for generic spectral observables).

This formalism has enabled precision lattice extractions of $|F_\pi(E)|^2$, the associated spectral function $\rho(s)$, and the ρ-meson decay constant (via analytic continuation to the resonance pole) (Meyer, 2011, Feng et al., 2014, Ortega-Gama et al., 30 Jul 2024).

3. Phenomenological Models and Analytic Properties

Phenomenological descriptions of timelike vector form factors are dominated by resonance physics and the requirements of analyticity, unitarity, and causality. The leading framework is Vector Meson Dominance (VMD), where the photon couples to hadrons via intermediate vector mesons ($\rho$, $\omega$, $\phi$, etc.). The extended Lomon–Gari–Krümpelmann and other VMD models for baryons and hyperons incorporate several key features (Lomon et al., 2012, Li et al., 2020, Chen et al., 2 Jul 2025, Yan et al., 2023):

• Form factors are modeled as sums over Breit–Wigner resonances, with each contribution proportional to:

$B_R(s) = \frac{m_R^2}{m_R^2 - s - i m_R \Gamma_R}$

For broad states (e.g. $\rho$), widths are momentum-dependent, e.g.

$$\Gamma_\rho(s) = \Gamma_0 \left(\frac{M_\rho^2}{s}\right) \left(\frac{s - s_0}{M_\rho^2 - s_0}\right)^{3/2},\quad s_0 = 4 M_\pi^2$$

Regularization via subtraction of spurious poles ensures physical analyticity.

• The total form factors take the schematic form (for nucleons or hyperons):

$$F_1(s) = g(s)\left[f_{1,0} + \sum_R \beta_{R} B_R(s)\right]$$

where $g(s)$ is an “intrinsic” dipole $(1 - \gamma s)^{-2}$ reflecting the three-quark core, and the sum covers all relevant (ground and excited) vector mesons.

• Coupling constants and resonance parameters are fitted to $e^+e^- \to N\bar N$, $e^+e^- \to B\bar B$, and other relevant cross section and polarization data.

Analytic continuation from spacelike to timelike momentum transfer is governed by dispersion relations. For hadrons, the asymptotic relations derived from analyticity and unitarity guarantee that at large $|q^2|$, the real parts of spacelike and timelike form factors coincide, with vanishing imaginary parts; finite-$q^2$ corrections are needed at moderate energies (Ramalho et al., 2019, Ramalho, 2020).

4. Resonances, Oscillatory Structure, and Final-State Interactions

Physical form factors in the timelike region exhibit pronounced resonance peaks (e.g., $\rho$ dominance in $F_\pi$, various vector states in baryonic FFs), nonmonotonic line shapes, and sometimes oscillatory modulations:

• Nonmonotonicity (peaks and dips) arises from the interference between ground state and excited vector mesons, as in the extended VMD fits to proton and neutron data (Yan et al., 2023), $\Sigma$ and $\Lambda$ hyperons (Chen et al., 2 Jul 2025), and multi-baryon thresholds.
• Oscillatory features in nucleon form factors above threshold have been interpreted as arising from interference between isoscalar and isovector amplitudes, with the modulation frequency set by the resonance widths. Analyses show that the oscillatory residuals in $e^+e^- \to N\bar N$ cross sections can be modeled as (Cao et al., 2021):

$$G_N^\mathrm{osc} \sim A_N e^{-B_N p} \cos(C_N p + D_N)$$

where $p$ is the nucleon momentum, and that equal amplitude modulations for $p$ and $n$ indicate isospin-pure production, while phase shifts between the two are signatures of imaginary parts from rescattering.

• Final-state interaction (FSI) models—e.g., distorted-wave Born approximation with incorporation of the Jost function—produce threshold enhancements and damped oscillatory patterns in the cross section and form factor by modeling the $N\bar N$ (or $Y\bar Y$) system as subject to a potential of Yukawa range (Qian et al., 2022, Haidenbauer et al., 2020). For S-wave, the enhancement factor is $1/|\mathcal{J}(p)|^2$.
• For hyperons, inclusion of FSI via coupled-channel meson-exchange potentials reproduces both threshold enhancement and observed nontrivial phase motion in the ratio $|G_E/G_M|$ (Haidenbauer et al., 2020).

5. Experimental Observables: Modulus and Phase Extraction

Timelike form factors are accessed through $e^+e^-$ cross sections and angular analyses of final state decay distributions:

• The modulus $|F(q^2)|$ is extracted from the total or differential cross section:

$$\sigma_{e^+e^- \to B\bar B}(q^2) = \frac{4\pi \alpha^2 \beta C}{3q^2}\left[|G_M(q^2)|^2 + \frac{1}{2\tau} |G_E(q^2)|^2 \right], \quad \beta = \sqrt{1 - 4M_B^2/q^2}$$

with the effective form factor defined as

$$|G_\mathrm{eff}(q^2)|^2 = \frac{2\tau|G_M|^2 + |G_E|^2}{2\tau + 1}$$

• The phase between electric and magnetic form factors, $\Delta\Phi = \arg(G_E/G_M)$, is extracted through joint angular distributions of baryon and antibaryon decay products (Collaboration et al., 2023). The generic angular structure involves terms proportional to $\cos(\Delta\Phi)$ and $\sin(\Delta\Phi)$, which can be disentangled in a multidimensional likelihood fit. Experimental results for $\Sigma^+$ EMFFs demonstrate nonzero and $q^2$-evolving phase differences (e.g., $\Delta\Phi = 55^\circ$ at $\sqrt{s}=2.6454$ GeV).
• For vector mesons, production and decay angular correlations allow access not only to the modulus but to the full helicity amplitude structure and relative phases between multipole form factors $G_C$, $G_M$, $G_Q$ (Leupold, 2021). High-$q^2$ phases are tied by analyticity to spacelike zero crossings, enabling stringent QCD-based tests.

6. Role in Standard Model Precision and Dark Sector Probes

Timelike vector form factors are pivotal for several precision phenomenology programs:

• The hadronic vacuum polarization, dominated at low $q^2$ by the two-pion channel, enters as the dominant error source in the Standard Model prediction for the anomalous magnetic moment of the muon, $(g-2)_\mu$. Precise knowledge of $|F_\pi(E)|^2$ over $E \in [2m_\pi, 4m_\pi]$ is required; direct lattice QCD access via finite-volume methodology reduces theoretical uncertainties (Meyer, 2011).
• For collider searches for new light vector bosons (dark photons, $U(1)_{B-L}$, etc.), the nucleon timelike form factors control the production rate, especially in bremsstrahlung and initial state radiation processes. Models incorporating superconvergent, resonance-based expansions with correct normalization and asymptotic behavior have been developed to enable reliable phenomenology for both proton and neutron channels (Kling et al., 11 Sep 2025).
• In heavy hadron and quarkonium decays, timelike transition form factors (e.g., $V(q^2)$ in vector–pseudoscalar transitions) are essential inputs for calculations of Dalitz decays ($\psi_A \to \psi_B \ell^+ \ell^-$), and their frame and current component dependence in theoretical calculations is used to estimate truncation-induced Lorentz symmetry violations (Li et al., 2019, Li, 2020).

7. Outlook and Theoretical Consistency

The modern theoretical description of timelike vector form factors integrates lattice QCD, dispersive techniques, refined VMD models, and a systematic treatment of final-state interactions and analytic constraints:

• Lattice QCD is now able to compute these form factors in the elastic region and for coupled channels, employing finite-volume technology and dispersive parameterizations to bridge spacelike and timelike domains (Ortega-Gama et al., 30 Jul 2024).
• Global VMD-inspired fits, constrained simultaneously by spacelike and timelike data, yield unified parameterizations that respect QCD scaling, analyticity, and unitarity (Lomon et al., 2012, Yan et al., 2023).
• The analytic properties tie detailed spacelike features, such as zero crossings, to asymptotic relative phases in the timelike region, permitting joint experimental–theoretical programs aimed at testing QCD dynamics (Leupold, 2021).
• For baryons and hyperons, progress in experimental measurements of $\Delta\Phi$ and $|G_E/G_M|$, as well as in the control of nonmonotonic and oscillatory features, provides new access to the interplay between resonance structure, FSI, and fundamental compositeness.

In sum, timelike vector form factors are a critical and technically rich probe of hadron structure, connecting resonance physics, analytic field theory constraints, and high-precision Standard Model and new physics searches. The synthesis of first-principles lattice calculations, advanced modeling of analytic properties, and polarization-sensitive experiments is rapidly expanding the toolkit for their exploration in QCD and beyond.