Baryon bags in strong coupling QCD

We discuss lattice QCD with one flavor of staggered fermions and show that in the path integral the baryon contributions can be fully separated from quark and diquark contributions. The baryonic degrees of freedom (d.o.f.) are independent of the gauge field, and the corresponding free fermion action describes the baryons through the joint propagation of three quarks. The nonbaryonic dynamics is described by quark and diquark terms that couple to the gauge field. When evaluating the quark and diquark contributions in the strong coupling limit, the partition function completely factorizes into baryon bags and a complementary domain. Baryon bags are regions in space-time where the dynamics is described by a single free fermion made out of three quarks propagating coherently as a baryon. Outside the baryon bags, the relevant d.o.f. are monomers and dimers for quarks and diquarks. The partition sum is a sum over all baryon bag configurations, and for each bag, a free fermion determinant appears as a weight factor.

Figure 1

Example for the decomposition of a lattice into baryon bags ${\mathcal{B}}_i$ and the complementary region $\overline{\mathcal{B}}$. The baryon bags are marked with the thinner blue lines on the links they contain or by a blue square if they consist of only a single site. The links in the complementary region $\overline{\mathcal{B}}$ are marked with thick magenta lines. The configuration we show has six baryon bags ${\mathcal{B}}_i$, two of which consist of only a single site. Inside each baryon bag, the Grassmann integral is saturated by baryon monomers, baryon dimers, and baryon loops. In the complementary region $\overline{\mathcal{B}}$, the Grassmann integral is saturated by monomers and dimers for quarks and diquarks.

Figure 2

Example for the saturation of the Grassmann integral in the complementary domain $\overline{\mathcal{B}}$. We show two (top and bottom plots) strings of neighboring sites, with the first site labeled by $x$ and the second one labeled by $x+\widehat{\nu }$. At each site, we have three layers for the three colors. The quark monomers ${\overline{\psi }}_{x,a}{\psi }_{x,a}$ are represented by a small square around the site $x$ in the layer for color $a$, and the diquark monomers ${\overline{\psi }}_{x,a}{\psi }_{x,a}{\overline{\psi }}_{x,b}{\psi }_{x,b}$ ($a\ne b$) are represented by a rectangle around $x$ in both layers $a$ and $b$. A quark dimer ${\overline{\psi }}_{x,a}{\psi }_{x,a}{\overline{\psi }}_{x+\widehat{\nu },c}{\psi }_{x+\widehat{\nu },c}$ is represented by a single magenta line connecting $x$ in layer $a$ with $x+\widehat{\nu }$ in layer $c$, while diquark dimers ${\overline{\psi }}_{x,a}{\psi }_{x,a}{\overline{\psi }}_{x,b}{\psi }_{x,b}{\overline{\psi }}_{x+\widehat{\nu },c}{\psi }_{x+\widehat{\nu },c}{\overline{\psi }}_{x+\widehat{\nu },d}{\psi }_{x+\widehat{\nu },d}$ are represented by two magenta lines that connect $x$ and $x+\widehat{\nu }$ in the corresponding layers. For graphical reasons, the two examples shown have the structure of chains, but of course, the patterns for the saturation of the Grassmann integral can branch in all four directions.