Does the crossover from perturbative to nonperturbative physics in QCD become a phase transition at infinite N ?

Abstract

We present numerical evidence that, in the planar limit, four dimensional Euclidean Yang-Mills theory undergoes a phase transition on a finite symmetrical four-torus when the length of the sides $l$ decreases to a critical value $l_c$. For $l>l_c$ continuum reduction holds so that at leading order in $N$, there are no finite size effects in Wilson and Polyakov loops. This produces the exciting possibility of solving numerically for the meson sector of planar QCD at a cost substantially smaller than that of quenched SU(3).

Figures (5)

• Figure 1: History of the variable $p(\tilde{P}_\mu )$ for each direction. We see the evolution from a state where one of the four $Z(N)$ factors is broken to one in which all four are preserved.
• Figure 2: History of the variable $p(\tilde{P}_\mu )$ for each direction. We see the evolution from a state where all four $Z(N)$ factors are preserved to one where one factor is broken. During the first fifty passes (before the first measurement) Polyakov loops in direction 3 have acquired some structure but, ultimately, direction 2 is selected for breakdown and the Polyakov loops in the other three directions converge to a symmetric state.
• Figure 3: Here we show the difference between the distributions of the largest inter-angle spacing for smeared Polyakov loops in different directions in the phase where exactly one $Z(N)$ factor is broken. (At other couplings, where no $Z(N)$ factor is broken, all four distributions look like the three unbroken ones here).
• Figure 4: Angle distributions in four directions in the 1c phase. There are twenty seven periods in the superposed oscillations. The peaks, except close to the gap associated with direction 3, are equally spaced.
• Figure 5: The transition ranges compared to possible two loop renormalization group curves with tadpole improvement.