Detailed Analysis of the Three Quark Potential in SU(3) Lattice QCD

TL;DR

This study uses SU(3) lattice QCD to analyze the three-quark potential, confirming the Y-ansatz with a three-body linear confinement term and demonstrating universality of the string tension across quark systems.

Contribution

First detailed lattice QCD analysis confirming the Y-ansatz for three-quark potential with universality of string tension and Coulomb coefficients.

Findings

3Q potential well fitted by Y-ansatz with Coulomb and linear terms

String tension in 3Q system matches Q-Qbar system within a few percent

Y-ansatz supported over other potential models

Abstract

The static three-quark (3Q) potential is studied in detail using SU(3) lattice QCD with $12^3 \times 24$ at $\beta=5.7$ and $16^3 \times 32$ at $\beta=5.8$, 6.0 at the quenched level. For more than 300 different patterns of the 3Q systems, we perform the accurate measurement of the 3Q Wilson loop with the smearing method, which reduces excited-state contaminations, and present the lattice QCD data of the 3Q ground-state potential $V_{\rm 3Q}$. We perform the detailed fit analysis on $V_{\rm 3Q}$ in terms of the Y-ansatz both with the continuum Coulomb potential and with the lattice Coulomb potential, and find that the lattice QCD data of the 3Q potential $V_{\rm 3Q}$ are well reproduced within a few % deviation by the sum of a constant, the two-body Coulomb term and the three-body linear confinement term $\sigma_{\rm 3Q} L_{\rm min}$, with $L_{\rm min}$ the minimal value of the total length of color flux tubes linking the three quarks. From the comparison with the Q-$\bar {\rm Q}$ potential, we find a universality of the string tension as $\sigma_{\rm 3Q} \simeq \sigma_{\rm Q \bar Q}$ and the one-gluon-exchange result for the Coulomb coefficients as $A_{\rm 3Q} \simeq \frac12 A_{\rm Q \bar Q}$. We investigate also the several fit analyses with the various ans\"atze: the Y-ansatz with the Yukawa potential, the $\Delta$-ansatz and a more general ansatz including the Y and the $\Delta$ ans\"atze in some limits. All these fit analyses support the Y-ansatz on the confinement part in the 3Q potential $V_{\rm 3Q}$, although $V_{\rm 3Q}$ seems to be approximated by the $\Delta$-ansatz with $\sigma_\Delta \simeq 0.53 \sigma$.