One-Gluon-Exchange Potentials

• One-gluon-exchange effective potentials are fundamental constructs in QCD that model color charge interactions via single gluon exchange, capturing the essence of asymptotic freedom.
• They arise from both perturbative and nonperturbative approaches, with applications ranging from quarkonium spectroscopy and nucleon interactions to quark-gluon plasma thermodynamics.
• Incorporating higher-order corrections and lattice-inspired modifications, these potentials address spin-dependent splittings, singularity regularization, and effective density-dependent interactions.

One-gluon-exchange (OGE) effective potentials are fundamental constructs in quantum chromodynamics (QCD) and related quantum field theories that describe the interaction between color charges via the exchange of a single gluon. These potentials arise naturally in both perturbative and nonperturbative treatments and are widely used for modeling hadronic bound states, analyzing nucleon-nucleon forces, exploring quark-gluon plasma thermodynamics, and investigating high-energy scattering processes. Their structure, approximations, associated singularities, and extension beyond leading order vary significantly with context and physical regime. The following sections provide a comprehensive analysis of OGE effective potentials as they manifest in a broad spectrum of methods and applications.

1. Field-Theoretic Origin and Diagrammatic Structure

In QCD, the leading-order interaction between color sources is mediated by a single gluon, represented diagrammatically by the exchange of a gluon between quark lines. The resulting OGE potential takes a form analogous to the Coulomb potential in QED but incorporates the color structure: $V_{OGE}(r) = -\frac{4}{3}\frac{\alpha_s}{r}$ where $\alpha_s$ is the strong coupling constant, and the $4/3$ factor arises from the Casimir operator in the fundamental representation of SU(3). This short-distance potential captures the essential antiscreening feature of QCD responsible for asymptotic freedom.

Beyond tree level, effective vertices and propagators encode the relevant dynamics. For instance, the effective QCD Lagrangian in the effective action approach can be written as $$\mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{QCD}}(V + A) + \text{induced terms}$$, with interactions local in rapidity (Braun et al., 2011). When gluon reggeization is important, as in high-energy QCD, the OGE vertex structure becomes more intricate, involving reggeon fields and requiring careful treatment of longitudinal momentum singularities: $$V_{1} = i[f^{db_1c}f^{cb_2a}]\frac{(q - q_1)^2}{(q - q_1)^2 + i0} \left\{ 2 q_{+}(e{\cdot}q)_\perp - \frac{q_\perp^2}{q_{1 -}}\left[(e, q - q_1)_\perp - \frac{(q - q_1)^2}{p_\perp^2}(e{\cdot}p)_\perp \right] \right\}$$ with explicit $1/q_{1-}$ pole singularities whose integration requires principal value prescription (Braun et al., 2011).

2. Nonperturbative and Lattice-Inspired OGE Potentials

Treatments incorporating nonperturbative effects commonly employ modified gluon propagators featuring dynamical mass generation or infrared “freezing” of the coupling. In the Dyson-Schwinger equation (DSE) framework, the gluon propagator adopts the form: $\Delta(q^2) = \frac{1}{q^2 + m^2(q^2)}$ with $m^2(q^2)$ a running gluon mass function, e.g.,

$$m^2(q^2) = m_0^2 \left[ \frac{\ln \left[(q^2 + \rho m_0^2)/\Lambda^2 \right]}{\ln (\rho m_0^2 / \Lambda^2 ) } \right]^{-1 - \delta}$$

The corresponding OGE potential is then obtained through a Fourier transform of the time–time component of this dressed propagator, often resulting in a flattened (“screened”) potential at large $r$, contrasted to the linearly rising Cornell potential from lattice QCD (Gonzalez et al., 2011, Cucchieri et al., 2017, Gonzalez et al., 2012). The inclusion of nonperturbative lattice propagators into the effective potential (called $V_{\text{LGP}}$ in (Cucchieri et al., 2017)) yields improved agreement with bound-state spectroscopy, although a strictly linear confining term is absent unless explicitly added.

3. OGE Potentials in Bound-State and Many-Body Systems

Quarkonium and Heavy Mesons

In heavy quarkonia (charmonium, bottomonium), OGE effective potentials—either from perturbative QCD, DSEs, or lattice inputs—serve as the kernel for Schrödinger-type equations: $$\left[ -\frac{\nabla^2}{2\mu} + V(r)\right]\psi(r) = E\psi(r)$$ Combined with a confining long-range potential, these OGE-derived forms are successful in predicting level spacings and relative ordering, especially after accounting for one-loop corrections producing additional spin-dependent terms (tensor and spin-orbit) essential for the accurate description of P-wave splittings (Capelo-Astudillo et al., 24 Jan 2025).

Relativistic and Nonrelativistic Baryonic Systems

In nucleon-nucleon (NN) interactions, the OGE potential's short-range components, particularly the hyperfine color magnetic interaction,

$$V^{\text{c.m.}}_{OGE} \sim (\sigma_i \cdot \sigma_j) \delta^{(3)}(\mathbf{r}_i - \mathbf{r}_j)$$

are critical for the repulsive core. Smearing the $\delta$-function with Gaussian profiles to account for constituent quark size further influences the repulsive/attractive balance and channel-dependent splitting (Shastry et al., 2018, Nilakanthan et al., 2018, S. et al., 2019). Incorporating confined gluon propagators (“COGEP”) modifies both color electric and color magnetic terms, introducing new intermediate-range attractions and connecting the modeling to the structure of confinement (Nilakanthan et al., 2018, S. et al., 2019).

Dense Quark Matter and Quark-Gluon Plasma

In descriptions of quark-gluon plasma (QGP) and dense quark matter, OGE-derived vector couplings appear as effective four-fermion interactions,

$$\mathcal{L}_{\text{int}} \sim -g_V (\bar{q} \gamma^\mu q)^2$$

with $g_V$ estimated by matching the QCD exchange energy (with nonperturbative gluon propagator and “frozen” $\alpha_s$) to the mean-field energy density: $$g_V(p_F; m_g) \approx \frac{4\pi \alpha_s/3}{9 m_g^2 + 8 p_F^2}$$ This effective repulsion is central for modeling the nuclear equation of state at neutron star densities (Song et al., 2019). In the context of the QGP, the one-gluon-exchange interaction modifies the equation of state and phase boundaries, shifting deconfinement to lower baryon densities and temperatures (Modarres et al., 2012).

4. Singularity Structure, Regularization, and Longitudinal Integration

OGE effective potentials often possess singularities requiring specific prescriptions:

• In multi-Regge effective action formalism, vertices may contain $1/q_-$ poles. The correct integration contour is the Cauchy principal value, mandated by comparison to full pQCD diagrams (Braun et al., 2011).
• Double gluon exchange diagrams, after longitudinal momentum integrations, yield $\delta$-function contributions (e.g., $\delta(q_{1-})$), which must be combined with regular pieces to reproduce the full amplitude (Braun et al., 2011).

Applications employing the quantum inverse scattering method for singular potentials indicate that effective OGE-like potentials can take forms such as $$V_{\text{eff}}(x) \sim -\frac{\lambda(\lambda-1)}{\sinh^2 x}$$ (zero-momentum Ruijsenaars-Schneider potential), with frame dependence appearing when mapping to scattering data (Elbistan et al., 2016).

5. Limitations of OGE Picture and Higher-Order Corrections

It is well established that a bare one-gluon exchange potential cannot by itself generate confinement—its Fourier transform of a Yukawa/gluon-mass pole yields a screened, non-confining potential. Next-to-leading order diagrams, such as the two-loop “sunrise” correction, introduce higher powers of momentum in the propagator denominator ($p^4$ terms), whose Fourier transform produces a linearly rising (confining) term in the static potential. This mechanism aligns with the expectation from the area law for Wilson loops and is necessary for realistic modeling of pure Yang-Mills infrared behavior (Frasca, 2011).

Recent modeling of excited charmed mesons demonstrates that including one-loop corrections to the OGE—specifically, $\mathcal{O}(\alpha_s^2)$ tensor and spin-orbit terms with logarithmic $r$-dependence—significantly improves the description of P-wave splittings, highlighting the indispensable role of higher-order interactions for precision spectroscopy (Capelo-Astudillo et al., 24 Jan 2025).

6. Extensions: Anisotropy, Holography, and High-Energy Scattering

In anisotropic QGP environments, heavy-quark OGE potentials acquire explicit angular dependence due to direction-dependent screening masses. These can be reduced to isotropic, quantum-number-dependent effective masses by projecting onto spherical harmonics, yielding 1D potentials that mimic the effect of anisotropy while preserving computational efficiency (Islam et al., 2022).

In high-energy QCD, OGE effective potentials correspond to reggeized-gluon-mediated interactions, requiring separation of rapidity sectors and the use of effective action methods, as pioneered in the Reggeon field theory approach (Braun et al., 2011).

In the context of AdS/CFT, effective OGE-like interactions appear via cubic extremal coupling terms between gluon and graviton Kaluza-Klein modes on sevenbranes, influencing four-point correlators (such as $\langle 22pp \rangle$) through graviton exchange contributions at $\mathcal{O}(1/N^2)$. The proper matching of such terms is essential for the nonperturbative determination of CFT data, particularly for extracting finite results in the presence of extremal divergences using the unmixing of single and double trace operators (Chester et al., 29 May 2025).

7. Summary Table: OGE Effective Potentials Across Methods and Applications

Framework/Regime	OGE Structure/Modification	Key Outcome
pQCD, Tree-Level	$V(r) = -\frac{4}{3}\frac{\alpha_s}{r}$	Coulomb-like, short-range force
DSE/Lattice	Dressed propagator $1/(q^2 + m^2(q^2))$, “frozen” $\alpha_s$	Screened, non-confining, improved spectra
Confined/Quark Models (NN)	Hyperfine color-magnetic terms, finite-size quark smearing, COGEP modifications	Repulsive core, varying with channel
High-Energy/Reggeon Field Theory	Effective vertices with reggeon splitting/R→RRP, singular longitudinal integration	pQCD equivalence, principal value prescription
Gauge/Chiral Dynamics	OGE + constant + linear potentials in CST; gauge dependence suppressed on-shell	Physically reasonable quark mass function
Thermodynamics/Dense Matter	OGE as density-dependent four-fermion vector interaction ($g_V$)	Vector repulsion, neutron star phenomenology
AdS Holography	Extremal gluon-graviton couplings, graviton exchange in correlation functions	Finite potentials after unmixing divergences

8. Concluding Remarks

One-gluon-exchange effective potentials provide a unifying language for describing color-mediated forces in a myriad of non-Abelian systems. Their diverse forms—ranging from simple Coulombic kernels to highly nonlocal, gauge-invariant, and nonperturbatively dressed objects—reflect the underlying physics of the regime under consideration. While OGE captures much of the perturbative dynamics and even aspects of nonperturbative phenomenology via suitable extensions, it is the interplay with confining effects, higher-order corrections, and collective modes (reggeons, diquarks, graviton exchanges) that ensures consistency with both experimental observations and lattice QCD. Across all analyses, careful attention to singularity structure, regularization, and systematic inclusion of beyond-OGE effects is essential for an accurate and rigorous theoretical description.