Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2002-04-16
JHEP 0208 (2002) 044
Physics
High Energy Physics
High Energy Physics - Theory
12 pages. Longer version, with additional clarifications. Results and conclusions are unchanged
Scientific paper
10.1088/1126-6708/2002/08/044
We study the $\theta$ dependence of four-dimensional SU($N$) gauge theories, for $N\geq 3$ and in the large-N limit. We use numerical simulations of the Wilson lattice formulation of gauge theories to compute the first few terms of the expansion of the ground-state energy $F(\theta)$ around $\theta=0$, $F(\theta)-F(0) = A_2 \theta^2 (1 + b_2 \theta^2 + ...)$. Our results support Witten's conjecture: $F(\theta)-F(0) = {\cal A} \theta^2 + O(1/N)$ for sufficiently small values of $\theta$, $\theta < \pi$. Indeed we verify that the topological susceptibility has a nonzero large-N limit $\chi_\infty=2 {\cal A}$ with corrections of $O(1/N^2)$, in substantial agreement with the Witten-Veneziano formula which relates $\chi_\infty$ to the $\eta^\prime$ mass. Furthermore, higher order terms in $\theta$ are suppressed; in particular, the $O(\theta^4)$ term $b_2$ (related to the $\eta^\prime - \eta^\prime$ elastic scattering amplitude) turns out to be quite small: $b_2=-0.023(7)$ for N=3, and its absolute value decreases with increasing $N$, consistently with the expectation $b_2=O(1/N^2)$.
Debbio Luigi Del
Panagopoulos Haralambos
Vicari Ettore
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