[Paper Review] Non Perturbative One Gluon Exchange Potential from Dyson-Schwinger Equations

P. González, V. Vento+1 more|arXiv (Cornell University)|Jul 18, 2012

Particle physics theoretical and experimental studies4 references4 citations

TL;DR

This paper derives a non-perturbative one-gluon exchange (OGE) potential from Dyson-Schwinger equations (DSEs) in quenched QCD, using a momentum-dependent gluon mass to model infrared freezing of the running coupling. The resulting potential accurately reproduces charmonium spectroscopy, showing one-to-one agreement with experimental states—particularly higher excitations where conventional Cornell potentials fail—suggesting DSE-based OGE as a superior framework for heavy quarkonia dynamics.

ABSTRACT

Recent progress in the solution of Dyson-Schwinger equations of QCD allows for a non perturbative evaluation of the One Gluon Exchange (OGE) interaction. We calculate the interquark static potential for heavy mesons by assuming that it is given by this OGE interaction and we apply it to the description of charmonium.

Motivation & Objective

• To derive a non-perturbative one-gluon exchange (OGE) potential from Dyson-Schwinger equations (DSEs) in quenched QCD.
• To model the infrared behavior of QCD by incorporating a momentum-dependent gluon mass that induces freezing of the running coupling.
• To compare the resulting DSE-based OGE potential with lattice-derived potentials and conventional phenomenological models like Cornell and screened potentials.
• To test the predictive power of the DSE-OGE potential in describing the charmonium spectrum, especially for higher radial excitations.
• To assess whether non-perturbative corrections in the DSE framework can naturally account for confinement without introducing ad hoc linear terms.

Proposed method

• Solve the Dyson-Schwinger equations using a pinch-technique and background-field method (PT-BFM) truncation in the quenched approximation.
• Model the gluon propagator as a massive Euclidean form: $\Delta^{-1}(q^2) = q^2 + m^2(q^2)$, with $m^2(q^2)$ given by a logarithmic running ansatz.
• Define the non-perturbative running coupling as $\alpha(q^2) = 4\pi / \left[\beta_0 \ln\left(\frac{q^2 + \rho m^2(q^2)}{\Lambda^2}\right)\right]$, which freezes in the infrared.
• Compute the static interquark potential via the Fourier transform of the time-time component of the full gluon propagator: $V(\mathbf{r}) = -C_F \int \frac{d^3\mathbf{k}}{(2\pi)^3} \frac{4\pi\alpha(\mathbf{k}^2)}{\mathbf{k}^2 + m^2(\mathbf{k}^2)} e^{i\mathbf{k} \cdot \mathbf{r}}$.
• Fit parameters ($m_0 \sim 360-480$ MeV, $\rho \sim 1-4$, $\delta = 1/11$, $\Lambda = 300$ MeV) to match lattice gluon propagators.
• Apply the potential to solve the Schrödinger equation for charmonium, using $m_c = 1400$ MeV and $C_F = 4/3$.

Experimental results

Research questions

• RQ1 Can a non-perturbative OGE potential derived from DSEs reproduce the charmonium spectrum with better accuracy than conventional models?
• RQ2 How does the inclusion of a momentum-dependent gluon mass affect the long-range behavior of the interquark potential?
• RQ3 Does the DSE-based OGE potential naturally account for confinement without introducing an explicit linear term?
• RQ4 Why do conventional Cornell potentials fail to describe higher radial excitations like $\psi(4415)$, while the DSE-OGE potential succeeds?
• RQ5 What role do non-perturbative vertex corrections play in generating the effective linear potential observed in lattice QCD?

Key findings

• The DSE-based OGE potential exhibits a perturbative $-1/r$ behavior at short distances and a non-perturbative, confining-like behavior at large distances, with the potential flattening asymptotically to zero.
• The potential shows a one-to-one correspondence between calculated states and experimental charmonium resonances, including higher radial excitations such as $\psi(4040)$, $\psi(4160)$, and $\psi(4415)$, with deviations within 60 MeV.
• The Cornell potential fails to accommodate several known resonances like $X(4260)$ and $X(4360)$, which are well described by the DSE-OGE model.
• The DSE potential with $m_0 = 345.7$ MeV, $\rho = 1$, and $n_f = 4$ yields a ground state mass of 3151 MeV, close to the experimental $\Upsilon(1S)$ at 3096.916 MeV.
• The model suggests that non-perturbative vertex corrections may be responsible for the effective linear confining term, potentially explaining why unquenched DSE solutions could yield similar results without explicit confinement.
• The Sommer procedure removes divergent self-energy contributions, resulting in a potential that is not negative-definite and resembles a screened potential, indicating possible cancellation of non-perturbative corrections in the full solution.