Diagrammatic representation of scalar QCD and sign problem at nonzero chemical potential

We consider QCD at strong coupling with scalar quarks coupled to a chemical potential. Performing the link integrals we present a diagrammatic representation of the path integral weight. It is based on mesonic and baryonic building blocks, in close analogy to fermionic QCD. Likewise, the baryon loops are subject to a manifest conservation of the baryon number. The sign problem is expected to disappear in this representation and we do confirm this for three flavors, where a scalar baryon can be built and, thus, a dependence on the chemical potential occurs. For higher flavor number, we analyze examples for a potential sign problem in the baryon sector and conjecture that all weights are positive upon exploring the current conservation of each flavor.

Figure 1

Example of a diagram on a $6\times 6$ lattice (with periodic boundary conditions). Unoriented single and double lines denote bonds with unit occupation of the mesonic building blocks $X$ and $Y$, i.e., $j_{\nu }(x)=1$ and $k_{\nu }(x)=1$. Oriented bonds (due to the current conservation discussed in Sec. 3) stand for baryon building blocks $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{*}$: arrows upwards and to the right denote $n_{\nu }(x)=1$ (on one bond $n_{\nu }(x)=2$) and arrows downwards and to the left denote ${\overline{n}}_{\nu }(x)=1$. The baryon loop winds once (actually in both directions) and thus obtains a fugacity factor $e^{3\mu /T}$ in the weight.

Figure 2

Two simple examples, where a closed baryon loop is connected to a single meson line at two sites $x$ and $y$. As discussed in the text, the weights of both diagrams are positive (in a nontrivial way).