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The Baryon Wilson Loop Area Law in QCD

  • Cornwall, John M.
Published Article
Publication Date
May 17, 1996
Submission Date
May 16, 1996
DOI: 10.1103/PhysRevD.54.6527
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There is still confusion about the correct form of the area law for the baryonic Wilson loop (BWL) of QCD. Strong-coupling (i.e., finite lattice spacing in lattice gauge theory) approximations suggest the form $\exp [-KA_Y]$, where $K$ is the $q\bar{q}$ string tension and $A_Y$ is the global minimum area, generically a three-bladed area with the blades joined along a Steiner line ($Y$ configuration). However, the correct answer is $\exp[-(K/2)(A_{12}+A_{13}+A_{23})]$, where, e.g., $A_{12}$ is the minimal area between quark lines 1 and 2 ($\Delta$ configuration). This second answer was given long ago, based on certain approximations, and is also strongly favored in lattice computations. In the present work, we derive the $\Delta$ law from the usual vortex-monopole picture of confine- ment, and show that in any case because of the 1/2 in the $\Delta$ law, this law leads to a larger value for the BWL (smaller exponent) than does the $Y$ law. We show that the three-bladed strong-coupling surfaces, which are infinitesimally thick in the limit of zero lattice spacing, survive as surfaces to be used in the non-Abelian Stokes' theorem for the BWL, which we derive, and lead via this Stokes' theorem to the correct $\Delta$ law. Finally, we extend these considerations, including perturbative contributions, to gauge groups $SU(N)$, with $N>3$.

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