Confining strings in representations with common n-ality

We study the spectrum of confining strings in SU(3) pure gauge theory, by means of lattice Monte Carlo simulations, using torelon operators in different representations of the gauge group. Our results provide direct evidence that the string spectrum is according to predictions based on n-ality. Torelon correlations in the rank-2 symmetric channel appear to be well reproduced by a two-exponential picture, in which the lowest state is given by the fundamental string σ1 = σ, the heavier string state is such that the ratio σ2/σ1 is approximately given by the Casimir ratio Csym/Cf = 5/2, and the torelon has a much smaller overlap with the lighter fundamental string state. The spectrum of confining strings in 4-d SU(N) gauge theories has been much investigated recently. Several numerical studies in the context of a lattice formulation of the theory have appeared in the literature, providing results for color sources associated with representations higher than the fundamental, see e.g. Refs. [1, 2, 3, 4, 5, 6]. General arguments show that the string tension must depend only on the n-ality, k = mod(l, N), of a representation built out of the (anti-)symmetrized tensor product of l copies of the fundamental representation. The confining string with n-ality k is usually called k-string, and σk is the corresponding string tension. Using charge conjugation, σk = σN−k. As a consequence, SU(3) has only one independent string tension determining the large distance behavior of the potential for k 6= 0. One must consider larger values of N to look for distinct k-strings. Lattice results for N = 4, 5, 6 [6, 5] show a nontrivial spectrum for the kstrings. In particular the data of Ref. [6], obtained using color sources in the antisymmetric representations of rank k, turn out to be well reproduced by the sine formula σk σ ≈ sin(kπ/N) sin(π/N) , (1) (σ ≡ σ1) within their errors. The sine formula has been suggested by several theoretical works, especially in the context of supersymmetric theories, see e.g. Refs. [7, 8] and references therein. On the other hand, numerical results for different representations with the same n-ality apparently contradict the picture that n-ality is what really matters. For example, in the SU(3) case, Monte Carlo data for the Wilson loops for several representations [2, 3] show apparently area laws up to rather large distances, approximately 1 fm, also for representations with zero n-ality, and the extracted string tensions turn out to be consistent with the so-called Casimir scaling [1]. In the lattice study of Ref. [5, 6], considering larger values of N , the k-string tensions were extracted from the torelon masses, i.e. from the exponential decay of correlations of characters of Polyakov lines. In Ref. [6], while the antisymmetric representations provided rather clean measurements of σk reproducing the sine formula, the numerical results for the symmetric representations suggested different values of the corresponding string tensions. For example, in the case of rank 2, σsym/σ ∼> 2, which is approximately the value suggested by Casimir scaling or by the propagation of two noninteracting fundamental strings. These results that apparently contradict n-ality have been recently discussed in Ref. [9]. They have been explained by arguing that standard color sources, such as Wilson loops and