One Gluon Exchange Process in QCD

• The one gluon exchange process is defined as the leading-order QCD mechanism mediating short-range interactions between quarks and gluons.
• It integrates perturbative methods with nonperturbative techniques like Dyson-Schwinger equations and lattice inputs to model effective mass generation.
• Corrections beyond the leading order, such as two-loop p⁴ terms, are crucial for producing a linear confining potential and matching hadronic spectra.

The one gluon exchange process is a central component in the theoretical treatment of the strong interaction, embodying the short-distance forces between colored objects in Quantum Chromodynamics (QCD). At its core, this process involves the exchange of a single gluon between quarks or between color sources, and is the leading-order mechanism in perturbative QCD for phenomena such as quark-quark, quark-antiquark, and gluon-gluon scattering. Its ramifications extend from the infrared structure of Yang-Mills theory, through heavy quarkonium phenomenology, to the effective interactions in hadronic and nuclear matter, and to collective phenomena in dense QCD matter.

1. Theoretical Structure of the One Gluon Exchange Mechanism

The one gluon exchange (OGE) amplitude arises from the leading term in the QCD perturbative expansion, reflecting gluon-mediated interactions among quarks. In Landau gauge, the gluon propagator takes the form

$$D_{\mu\nu}^{ab}(p) = \delta_{ab} \left[ \eta_{\mu\nu} - \frac{p_\mu p_\nu}{p^2} \right]\Delta(p),$$

where $\Delta(p)$, the scalar part, is often modeled variably depending on the regime of interest. In the infrared limit of Yang-Mills theory (Frasca, 2011), nonperturbative techniques yield a $\Delta(p)$ expressible as a sum over simple poles,

$$\Delta(p) = \sum_n \frac{B_n}{p^2 - m_n^2 + i\epsilon}.$$

This structure allows the propagator to be interpreted as a sum of Yukawa-type propagators, where mass gaps $m_n$ and weights $B_n$ reflect strong-coupling non-perturbative effects, e.g., via instanton backgrounds.

The interquark static potential resulting from OGE, relevant for heavy quarkonia and derived via a Fourier transform of the temporal gluon propagator, adopts (in its canonical perturbative limit) the form

$$V_{\text{pert}}(r) = -\frac{4}{3}\frac{\alpha_s}{r},$$

where the $4/3$ is the $SU(3)$ Casimir factor for the fundamental quark representation (Cucchieri et al., 2017).

Nonperturbative generalizations (e.g., via lattice or Dyson-Schwinger inputs) modify the propagator denominator, incorporating dynamical mass generation and possible quartic momentum corrections, as discussed below.

2. Limitations and Extensions Beyond Leading Order

A recurring theme in the paper of OGE is its inability, when treated at leading order, to produce a linearly confining potential in the infrared—contradicting lattice evidence and phenomenology for QCD confinement. Specifically, at the OGE level, the interquark potential is a sum over Yukawa potentials,

$$V_{\text{OGE}}(r) \propto -\sum_n B_n \frac{e^{-m_n r}}{r},$$

which manifestly do not grow linearly at large separation (Frasca, 2011). This screening persists even when the underlying gluon propagator is nonperturbatively dressed (e.g., from lattice simulations, (Cucchieri et al., 2017)), and is a robust statement across approaches that rely solely on OGE.

The emergence of a linear confining term requires next-to-leading order (NLO) corrections. A central result in (Frasca, 2011) is the demonstration that a natural quartic momentum term, $p^4$ in the denominator of the effective gluon propagator, arises from two-loop ("sunrise") diagrams: $$\Delta_R(p^2) \sim \left[ p^2 + \tfrac{\text{const}}{\lambda}\frac{p^4}{\mu^2} + m_n^2 \right]^{-1},$$ with $\lambda$ the gauge coupling, and $\mu$ a dynamically-induced cut-off. The $p^4$ correction is responsible, after Fourier transformation, for generating a linearly rising potential, in line with both the Gribov-Zwanziger scenario of confinement and lattice QCD findings. The two-loop integral that supplies this term, and its low-momentum expansion, are crucial, as the one-gluon level alone is structurally insufficient.

3. Nonperturbative Approaches and Model Realizations

Dyson-Schwinger Equation Formalism

The Dyson-Schwinger (DSE) equations provide a nonperturbative tool for calculating the OGE process with dynamical mass and coupling (Gonzalez et al., 2012, Biernat et al., 2018). In this approach, the gluon propagator is modeled as

$\Delta^{-1}(q^2) = q^2 + m^2(q^2),$

where $m^2(q^2)$ is a momentum-dependent mass function, often chosen to reflect lattice constraints through logarithmic running. The OGE static potential is then

$$V(r) = -C_F \int \frac{d^3k}{(2\pi)^3} \frac{4\pi\alpha(k^2)}{k^2 + m^2(k^2)} e^{i\mathbf{k}\cdot \mathbf{r}},$$

where $C_F$ is the Casimir, and $\alpha(k^2)$ is the running coupling. DSE-calculated OGE potentials flatten at large $r$, underscoring again that OGE alone does not produce a confining linear term.

Lattice Propagator Inputs

Improved models that insert lattice-determined propagators into the OGE formula yield nonperturbative but still nonconfining static potentials (Cucchieri et al., 2017). Only upon the ad hoc addition of a linear term ("Cornell potential") does the model reproduce both short-distance and large-distance spectroscopy accurately. Thus, the OGE process, by itself, yields a potential that saturates at large $r$, reflecting the screening inherent in the nonperturbative gluon mass generation.

Nuclear and Hadronic Realizations

In nucleon-nucleon (NN) models, OGE contributions modified by confinement ("confined OGE potential" or COGEP) are implemented via confined propagators characterized by exponential damping and contact (delta-function) interactions reflecting color-magnetic spin-spin effects (Nilakanthan et al., 2018, S. et al., 2019). Exchange kernels in antisymmetrized six-quark systems are treated with the Resonating Group Method and Born-Oppenheimer approximations, yielding short-range repulsion (from color-magnetic $\delta^3(r)$ kernels) and intermediate-range attraction (from the nontrivial structure of the confined propagator). These mechanisms obviate the need for phenomenological $\sigma$ meson exchange in generating NN intermediate-range attraction.

4. Confinement, Cut-offs, and Infrared Structure

A key technical insight from (Frasca, 2011) is the necessity of an infrared cut-off $\mu$ arising naturally from the analysis of classical solutions in Yang-Mills theory and strong-coupling expansions. In evaluating the two-loop (sunrise) diagrams, this cut-off limits the relevant integration region, effectively regularizing divergent integrals and setting a physical scale for the "infrared effective theory." This is crucial for the consistency of the nonperturbative framework, as it ensures that loop corrections do not depend on extraneous ultraviolet features. The derivation's reliance on coordinate rescaling, and the isolation of the quartic $p^4$ term, connects operator expansions to observable string tension: $$\sqrt{\sigma} \propto g^2 \sqrt{ \frac{C_2(R)d(R)}{4\pi} \mu },$$ tying confinement directly to the structure of the corrected propagator.

5. Broader Physical Implications

Gluon Propagation and Vacuum Structure

The behavior of the OGE process in curved gauge backgrounds—parametrized by dimension-2 condensates and generalized $\theta$-vacua—introduces a mass gap for the gluon. The effective propagator thereby transforms from Coulombic to Yukawa-like, $D(k) \sim 1/(k^2 - m^2)$, suppressing long-range propagation and naturally explaining both confinement and the gluonic contribution to the hadronic mass budget (Lee et al., 2015). This framework generalizes to hypothesized gluon-induced dark matter if macroscopic curved gauge slices persist outside of hadrons.

Hadron and Nuclear Phenomenology

In heavy quarkonia, the OGE potential, after inclusion of nonperturbative corrections or lattice dressing, reproduces short-distance splittings but fails to capture confinement unless additional linear terms are introduced (Cucchieri et al., 2017). In nucleon systems, COGEP, when combined with instanton-induced interactions, accounts for both spin-dependent splitting and intermediate attraction, aligning with empirical potentials without invoking extraneous phenomenological exchanges (S. et al., 2019).

In quark-gluon plasma modeling, OGE-based Fermi liquid interactions provide a repulsive interquark force, modifying pressure and phase transitions and resulting in a lower deconfinement threshold in the phase diagram, relative to an ideal gas (Modarres et al., 2012).

Gauge Dependence and Covariance

In covariant frameworks (e.g., CST), the gauge dependence of the mass function induced by OGE interactions is rigorously controlled. The gauge parameter can be chosen (e.g., Yennie gauge, $\xi=3$) so that the off-shell mass function is continuous, while the constituent quark mass and gap equations remain gauge-independent, in line with physical expectations (Biernat et al., 2018).

6. Summary Table: OGE Features Across Frameworks

Aspect	OGE at Leading Order	Beyond OGE / Corrections	Confinement
Infrared Potential	Sum over Yukawa (exponential) terms	Two-loop (p⁴) term from sunrise diagrams	Linear for large r (after NLO)
Heavy Quark Systems	Coulombic ($1/r$) or screened	Flattened at large r, needs linear term	Achieved via added linear
Nuclear/Hadronic	Short-range + spin (color-magnetic)	Intermediate attraction (COGEP, III)	Reproduced at quark level
Thermodynamics (QGP)	Repulsive for massless quarks	Phase diagram sensitivity to parameters	Deconfinement threshold shifts
Vacuum/Curvature	Massless gluon in flat background	Massive gluon from condensate/curvature	Yukawa-type, mass gap

The one gluon exchange process forms the basis for a wide array of strong-interaction phenomena, with its limitations and necessary corrections deeply illuminating the emergence of color confinement, hadronic spectra, nuclear forces, and collective behaviors in QCD matter. The necessity to go beyond leading order—through dynamical mass generation in nonperturbative propagators, two-loop corrections, and consistent infrared regularization—is well established both by fundamental theory and by precise lattice and phenomenological studies.