Cautionary remarks on the moduli space metric for multidyon simulations

We perform a detailed numerical investigation of the approximate moduli space metric proposed by Diakonov and Petrov [Phys. Rev. D 76, 056001 (2007)] for a confining model of dyons. Our findings strongly indicate that this metric is positive definite (and, therefore, a valid moduli space metric) throughout a considerable part of configuration space only for a small number of dyons at sufficiently low density. This poses strong limitations on results obtained by an unrestricted integration over collective coordinates in this model. It also indicates that strong correlations between collective coordinates will be essential for the physical content of a dyon model, which could be exhibited by a suitable simulation algorithm.

Figure 1

Histograms obtained from 100 000 independently chosen dyon configurations representing the spectral density of the approximate moduli space metric $G$. Left column: $n_D=200$. Right column: $n_D=800$.

Figure 2

Histograms obtained from 100 000 independently chosen dyon configurations representing the probability distribution $P^{-}(n_{-})$ of the number of negative eigenvalues (as function of $n_{-}/n_D$). Left column: $\rho =1/{\mathrm{fm}}^3$. Right column: $\rho =1/125{\mathrm{fm}}^3$.

Figure 3

Integrated weights $W_G^{-}(n_{-})$ (shown as functions of $n_{-}/n_D$) for various dyon numbers $n_D$ and a fixed density $\rho =1/{\mathrm{fm}}^3$.

Figure 4

Histograms obtained from 100 000 independently chosen dyon configurations representing the probability distribution $P^{-}(n_{-})$ of the number of negative eigenvalues (as function of $n_{-}/n_D$). Left column: $n_D=200$. Right column: $n_D=800$.

Figure 5

Integrated weights $W_G^{-}(n_{-})$, shown as functions of $n_{-}/n_D$, as obtained for a number $n_D=200$ of dyons and various densities.

Figure 6

The free energy of a pair of static charges $F_{Q\overline{Q}}$ as a function of the separation $R$ for various dyon densities $\rho$.