Dual representation of lattice QCD with worldlines and worldsheets of Abelian color fluxes

We present a new dual representation for lattice QCD in terms of wordlines and worldsheets. The exact reformulation is carried out using the recently developed Abelian color flux method where the action is decomposed into commuting minimal terms that connect different colors on neighboring sites. Expanding the Boltzmann factors for these commuting terms allows one to reorganize the gauge field contributions according to links such that the gauge fields can be integrated out in closed form. The emerging constraints give the dual variables the structure of worldlines for the fermions and worldsheets for the gauge degrees of freedom. The partition sum has the form of a strong coupling expansion, and with the Abelian color flux approach discussed here all coefficients of the expansion are known in closed form. We present the dual form for three cases: pure SU(3) lattice gauge theory, strong coupling QCD and full QCD, and discuss in detail the constraints for the color fluxes and their physical interpretation.

Figure 1

Graphical representation of the (1,2,3,3)-ACC, which explicitly is given by $U_{x,\mu }^{12}{U}_{x+\widehat{\mu },\nu }^{23}{U}_{x+\widehat{\nu },\mu }^{33\star }{U}_{x,\nu }^{13\star }$. This ACC closes around the plaquette $(x,\mu \nu )$ running through the sequence (1,2,3,3) of color indices at the corners of the plaquette. In the graphical representation the color degrees of freedom are shown as three distinct layers of the space-time lattice, labeled with 1,2, and 3 on the lhs of the plot. Each of the matrix elements $U_{x,\mu }^{ab}$ constituting the ACC is represented by an arrow along the corresponding link $(x,\mu )$ connecting color $a$ with color $b$. For complex conjugate matrix elements the link is run through with negative orientation.

Figure 2

Graphical illustration of the sum $-p_{x,\rho \mu }^{1ss2}$ contributing to the flux $J_{x,\mu }^{12}$.

Figure 3

Schematic representation of the three color conservation constraints (28)–(30) (top three plots) and the color exchange constraint (31) (bottom plot). They impose relations between the fluxes $J_{x,\mu }^{ab}$ and admissible configurations of the cycle occupation numbers $p_{x,\mu \nu }^{abcd}$ have to respect these constraints.

Figure 4

Graphical representation of the dual variables for the fermions. As before we use a lattice with three layers to represent the color indices. In the first column of diagrams we show the monomers $s_x^a$, while the arrows in the other columns represent the dual variables $k_{x,\nu }^{ab}$ and ${\overline{k}}_{x,\nu }^{ab}$ for the forward and backward hopping, respectively. With these link variables it is possible to build dimers and oriented loops which, together with the monomers, constitute the admissible configurations for fermions.

Figure 5

Baryon loop elements in the strong coupling limit. Only the six combinations shown here are admissible and propagating fluxes in the strong coupling limit. The elements with an odd number of color flux crossings come with an explicit minus sign. For the negative direction the same fluxes are admissible and have the same signs. The corresponding diagrams are obtained by reverting the arrows.

Figure 6

Closed, nonpropagating one-link loops at strong coupling. All of these loops come with a positive weight. Also the opposite orientation is possible, which is obtained by reverting all arrows.

Figure 7

Examples of simple strong coupling baryon loops with different connectivity properties.