[Paper Review] The anti-quark--quark potential from Bethe-Salpeter amplitudes on lattice

Yoichi Ikeda, Hideaki Iida|arXiv (Cornell University)|Nov 12, 2010

Quantum Chromodynamics and Particle Interactions4 references3 citations

TL;DR

This paper derives the quark-antiquark potential from Bethe-Salpeter amplitudes in quenched lattice QCD using the LKM formalism, extracting finite-mass $π$- and $\rho$-meson-like potentials. It finds that both pseudo-scalar and vector channel potentials exhibit a Coulomb-plus-linear form, with string tension increasing and Coulomb coefficient decreasing with rising quark mass.

ABSTRACT

Potentials of quark--anti-quark pairs are studied from the anti-q--q Nambu-Bethe-Salpeter (NBS) wave functions in quenched lattice QCD. With the use of a method which has been recently developed in the derivation of nuclear forces from QCD, we derive the anti-q--q potentials with finite quark masses from the NBS wave functions. We calculate the anti-q--q NBS wave functions in pseudo-scalar and vector channels for several quark masses. The derived potentials in both channels reveal linear confinement plus Coulomb potentials. We also discuss the quark-mass and channel dependence of the anti-q--q potentials.

Motivation & Objective

• To extract finite-mass quark-antiquark potentials from lattice QCD using the Nambu-Bethe-Salpeter formalism.
• To investigate the quark mass and spin-parity channel dependence of the $π$- and $\rho$-meson-like potentials.
• To validate the applicability of the LKM formalism for light quark-antiquark systems in the absence of asymptotic states.
• To determine whether the potential exhibits Coulomb and linear confinement behavior consistent with quark models.

Proposed method

• Compute equal-time Nambu-Bethe-Salpeter (NBS) amplitudes using source operators for pseudo-scalar and vector channels on the lattice.
• Extract spatial wave functions $\phi(\vec{r})$ from the NBS amplitudes at large time separations.
• Apply the LKM formalism to derive an effective Schrödinger equation from the Bethe-Salpeter equation with constant quark masses.
• Use the relation $V(r) = \frac{1}{2\mu} \frac{\nabla^2 \phi(r)}{\phi(r)} + E$ to extract the potential from the NBS wave function.
• Fit the extracted potentials to the functional form $V(r) = -A/r + \sigma r + C$ to extract string tension $\sigma$ and Coulomb coefficient $A$.
• Determine quark masses as half the vector meson mass ($m_q = m_V/2$) to simulate finite-mass quark-antiquark systems.

Experimental results

Research questions

• RQ1 Does the quark-antiquark potential derived from NBS amplitudes in lattice QCD exhibit Coulomb and linear confinement behavior?
• RQ2 How does the string tension $\sigma$ and Coulomb coefficient $A$ depend on the quark mass in the pseudo-scalar and vector channels?
• RQ3 Is the potential form robust under variations in lattice volume and cutoff, indicating reliable extraction?
• RQ4 Can the LKM formalism be successfully applied to light quark-antiquark systems despite the absence of asymptotic states?

Key findings

• The $π$- and $\rho$-meson-like potentials in both pseudo-scalar and vector channels exhibit a Coulomb-plus-linear form, $V(r) = -A/r + \sigma r + C$, consistent with quark model expectations.
• The string tension $\sigma$ increases with quark mass: it reaches 950 MeV/fm in the pseudo-scalar channel and 1011 MeV/fm in the vector channel at the heaviest quark mass ($m_{\rm PS} = 2.53$ GeV).
• The Coulomb coefficient $A$ decreases with increasing quark mass, with values ranging from 329 MeV·fm at lightest mass to 155 MeV·fm at heaviest mass in the pseudo-scalar channel.
• The quark mass dependence of the string tension is stronger in the pseudo-scalar channel than in the vector channel.
• The potential form and parameter values are stable under checks for volume and cutoff dependence, indicating robustness of the extraction method.
• The results confirm the viability of using NBS amplitudes and the LKM formalism to extract finite-mass quark-antiquark potentials in lattice QCD.