Monopole Loop Distribution and Confinement in SU(2) Lattice Gauge Theory

The abelian-projected monopole loop distribution is extracted from maximal abelian gauge simulations. The number of loops of a given length falls as a power nearly independent of lattice size. This power increases with β = 4/g, reaching five around β = 2.85, beyond which loops any finite fraction of the lattice size vanish in the infinite lattice limit, suggesting the continuum theory lacks confinement. PACS:11.15.Ha, 11.30.Qc. A strong correlation has been established between confinement and abelian monopoles extracted in the maximal abelian gauge. The entire SU(2) string tension appears to be due to the monopole portion of the projected U(1) field[1, 2]. Confinement appears to require the presence of large loops of monopole current of order the lattice size, possibly in a percolating cluster. As β = 4/g is raised, the finite lattice theory undergoes a deconfining phase transition which is coincident with the disappearance of large monopole loops[3]. This transition is interpreted as a finite-temperature phase transition, one which exists if one of the four lattice dimensions is kept finite as the others become infinite, but which disappears in the 4-d symmetric infinite lattice limit. In the U(1) lattice gauge theory itself, monopoles have also been identified as the cause of the phase transition[4]. However, in this theory, monopoles are interpreted as strong coupling lattice artifacts which do not survive the continuum limit. The continuum limit is non-confining. In order for the SU(2) theory to confine in the continuum limit, it is necessary for some abelian monopoles to survive this limit, i.e. to exist as physical objects. Evidence has been presented of scaling of the monopole density which would make this so[5]. However, it is not sufficient just to have some monopoles survive this limit; one needs large loops of a finite physical size to survive. Small loops of finite size on the lattice, which are by far the most abundant, will shrink to zero physical size as the lattice spacing goes to zero, becoming irrelevant. It is believed that to cause confinement, monopole loops must be at least as large as the relevant Wilson loops of nuclear size, and may need to span the entire space. Consider a large but finite universe, represented as an N lattice with lattice spacing a. The strong interactions should not care whether the universe is finite or infinite, so long as it is much larger than a hadron. One can then take the continuum limit as a → 0, N → ∞, simultaneously, holding Na constant at the universe size. It is then clear that the size of any object that is to remain of finite physical size in the continuum limit must also become infinitely large in lattice units, with linear dimension proportional to N , i.e., some finite fraction of the full lattice size. It would therefore appear that at the very least, monopole loops some finite fraction of the lattice size must survive the continuum limit if it is to be confining, and probably loops at least as large as the lattice itself. Recently, it was shown that if the plaquette is restricted to be greater than 0.5, the loop distribution function falls so fast that no monopole loops any finite fraction of the lattice size survive in the large lattice limit for any value of β[6]. Here it will be shown that for the standard Wilson action the same is true if β > 2.85. By studying the monopole loop size distribution function, one can tell how quickly the probability of finding loops of increasing size decreases with loop size. This function