Lattice study of the critical bubble in $\mathrm{SU(8)}$ deconfinement transition

Abstract

Strongly coupled theories are of phenomenological interest, for example as dark matter candidates. Theories that can undergo first order thermal phase transitions are particularly appealing as potential sources of a stochastic gravitational wave background. Determining the expected gravitational wave signal from a first order phase transition requires accurate information on the bubble nucleation rate, but thus far for strongly coupled models these have relied on semiclassical methods. As a first step towards determining the nucleation rate, in this paper we study the confinement-deconfinement phase transition in a 4D SU(8) pure gauge model, using multicanonical Monte Carlo. Resolving the critical bubble for the first time in a pure Yang-Mills model, we determine the critical bubble probability and compare it to results from thin wall calculations. We also compare the effectiveness of different lattice pseudo-order parameters at resolving the condensation transition between the metastable phase and critical bubble branch, and point out the choice of order parameter is crucial to accurately resolve the critical configurations.

Figures (9)

• Figure 1: Schematic of the probability distribution $P(\mathcal{O})$ of order parameter $\mathcal{O}$. Only the metastable phase and the mixed phase configurations separating the phases are shown. The order parameter value corresponding to the critical bubble is marked $\mathcal{O}_c$.
• Figure 2: Schematic illustration of different topological regimes predicted by the thin-wall approximation in a box with periodic boundaries. The left-hand figure is at the critical temperature, whereas the right-hand figure is above the critical temperature, where the stable phase peak is higher than the metastable phase peak. Regions where the different geometrical configurations are stable are separated by dashed and dotted lines. Dotted lines separate the bubble regime from the cylinder regime, and dashed lines the cylinder regime from the slab regime. The topologies can remain metastable beyond these lines, indicated by the free energy curves continuing beyond the dashed/dotted lines. The curves are shown with respect to the volume fraction of the deconfined, stable phase, which can be obtained from a normalized, intensive order parameter. The two peaks are schematic and represent the Gaussian bulk fluctuations in the two phases; for large lattices, these bulk phase fluctuations can be sufficiently broad as to cover the curve corresponding to the bubble branch.
• Figure 3: Set of three example configurations from a $\beta = 44.742, N_s = 60$ run where the order parameter was $\mathcal{O} = l_{\theta}$ from Eq. (\ref{['eq:theta_order_parameter']}). Shown are the isosurfaces of the absolute value of smeared Polyakov loop $l_{s}^{(48)}$, as defined in Eq. (\ref{['eq:smeared_loop_field']}) , recorded at each spatial site. Note that the configurations appear three dimensional even though our simulation itself is four dimensional, because we are plotting a Polyakov loop-based order parameter. For reference, the deconfinement phase peak value is around 0.41. Configurations are numbered from left to right as 1, 2 and 3, for comparison with Figures \ref{['fig:three_histo']} and \ref{['fig:2dhist']}.
• Figure 4: Histograms for $\beta = 44.742, N_s = 60$, where the order parameter $\mathcal{O}$ used in the multicanonical weighting was, from left to right, $|l_p|$, $l_{\theta}$ or $l_{\sigma}$, as defined in Eqs. (\ref{['eq:old_order_parameter']}), (\ref{['eq:theta_order_parameter']}) and (\ref{['eq:sigma_order_parameter']}). Order parameter values corresponding to the configurations shown in Figure \ref{['fig:configs']} are marked. Histograms are normalized so that the metastable bulk peak height is one; for our purposes only the difference between the maximum and minimum of the histogram matters. The Gaussian bulk phase peak is significantly wider for $\mathcal{O} = |l_p|$ than for the other two choices.
• Figure 5: Two dimensional histogram from $\beta = 44.742, N_s = 60, \mathcal{O} = l_{\theta}$ runs, with the order parameter $l_{\theta}$ shown together with the non-smeared Polyakov loop volume average $|l_p|$. Again, the parameter values corresponding to the configurations shown in Figure \ref{['fig:configs']} are marked. One value of $|l_p|$ can be seen to correspond to a wide range of $l_{\theta}$ values both in the bulk phase and the mixed phase. The mixed phase branch can be seen at an angle to the bulk phase branch. Isocontour lines of relative bin frequency are shown to guide the eye. The contours are obtained after one nearest-neighbour averaging of the bin weights. Note that the two observables, $l_{\theta}$ and $|l_p|$, are correlated.