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Figure 9. Continuum extrapolation of the interface tension.

Figure 9 Continuum extrapolation of the interface tension.

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Table 1. Input parameters for our benchmark points. In BM1, the combination A345 = A3+A4+A5 corresponds to a dark matter portal coupling in the IDM [39], and up = \/—/?, represents the soft Z2-breaking parameter. Masses are assumed to be the physical pole masses, while the remaining parameters are input directly in the MS scheme at the initial renormalization scale Ag. -1 2HDM scenarios to be studied on the lattice
Table 1. Input parameters for our benchmark points. In BM1, the combination A345 = A3+A4+A5 corresponds to a dark matter portal coupling in the IDM [39], and up = \/—/?, represents the soft Z2-breaking parameter. Masses are assumed to be the physical pole masses, while the remaining parameters are input directly in the MS scheme at the initial renormalization scale Ag. -1 2HDM scenarios to be studied on the lattice
Table 2. Renormalized parameters corresponding to the input parameters in table 1. The recipe for obtaining these is described in the main text and in appendix A. In BM1, we have set the U(1) coupling to zero for the sake of comparison with the results of ref. [45]. The SU(3) coupling gz is fixed at tree level.
Table 2. Renormalized parameters corresponding to the input parameters in table 1. The recipe for obtaining these is described in the main text and in appendix A. In BM1, we have set the U(1) coupling to zero for the sake of comparison with the results of ref. [45]. The SU(3) coupling gz is fixed at tree level.
Figure 1. Gauge-invariant condensates of the two doublets as measured on the lattice with fixed volume and 8g, converted to MS quantities using the relations in ref. [90]. In both cases, the doublet ¢; is heavy and almost inert at the phase transition. Instead, its fluctuations become more constrained as (¢hg2) changes due to the 3 term in the action. In BM2, (od) increases smoothly towards its J’ = 0 value as governed by our choice of tan @.
Figure 1. Gauge-invariant condensates of the two doublets as measured on the lattice with fixed volume and 8g, converted to MS quantities using the relations in ref. [90]. In both cases, the doublet ¢; is heavy and almost inert at the phase transition. Instead, its fluctuations become more constrained as (¢hg2) changes due to the 3 term in the action. In BM2, (od) increases smoothly towards its J’ = 0 value as governed by our choice of tan @.
Figure 2. Unnormalized probability distributions of the expectation value (¢h¢0)3a in BM1, measured on the lattice with varying Sg and converted to the corresponding continuum quantity. The histograms have been obtained by reweighting in TJ’ and minimizing the integrated probability difference between the peaks. As Gq increases, the phases become more separated and thus the transition grows stronger. Configurations between the two phases are suppressed exponentially by the free energy carried by the phase interface, with the suppression increasing with the area of the interface.
Figure 2. Unnormalized probability distributions of the expectation value (¢h¢0)3a in BM1, measured on the lattice with varying Sg and converted to the corresponding continuum quantity. The histograms have been obtained by reweighting in TJ’ and minimizing the integrated probability difference between the peaks. As Gq increases, the phases become more separated and thus the transition grows stronger. Configurations between the two phases are suppressed exponentially by the free energy carried by the phase interface, with the suppression increasing with the area of the interface.
Figure 3. Continuum extrapolation of the critical temperature. All volumes are plotted here anc are indistinguishable from each other at the same Bg. Table 4. Nonperturbatively determined critical temperature, latent heat, order parameter discon- tinuity and surface tension of the phase transition. The errors shown here are statistical errors of least square polynomial fitting.
Figure 3. Continuum extrapolation of the critical temperature. All volumes are plotted here anc are indistinguishable from each other at the same Bg. Table 4. Nonperturbatively determined critical temperature, latent heat, order parameter discon- tinuity and surface tension of the phase transition. The errors shown here are statistical errors of least square polynomial fitting.
Figure 4. Volume dependence of the order parameter discontinuity 2A(oh¢0) 3a/T for different lattice spacings. Bq dependence of the infinite-volume results is plotted in figure 5. In BM1, a least- squares quadratic extrapolation fits nicely to the data points, but we have also included a linear fit. The difference between the two extrapolations is roughly 3% in the continuum limit, with the quadratic fit resulting in a slightly larger error. We shall use the quadratic extrapolation from now on.
Figure 4. Volume dependence of the order parameter discontinuity 2A(oh¢0) 3a/T for different lattice spacings. Bq dependence of the infinite-volume results is plotted in figure 5. In BM1, a least- squares quadratic extrapolation fits nicely to the data points, but we have also included a linear fit. The difference between the two extrapolations is roughly 3% in the continuum limit, with the quadratic fit resulting in a slightly larger error. We shall use the quadratic extrapolation from now on.
Figure 5. Continuum extrapolations of the order parameter discontinuity in the infinite-volume limit. Both quadratic (green) and linear (blue) fits are shown in BM1. A reliable higher-order polynomial fit in BM2 would require the use of very large Gg — and consequently extremely large lattices — due to the very massive scalar degrees of freedom.
Figure 5. Continuum extrapolations of the order parameter discontinuity in the infinite-volume limit. Both quadratic (green) and linear (blue) fits are shown in BM1. A reliable higher-order polynomial fit in BM2 would require the use of very large Gg — and consequently extremely large lattices — due to the very massive scalar degrees of freedom.
Figure 6. Infinite-volume extrapolation of the latent heat.
Figure 6. Infinite-volume extrapolation of the latent heat.
Figure 7. Continuum extrapolation of the latent heat. The behavior is similar to that of Av/T, but with slightly larger statistical errors. In BM1, we again show both linear and quadratic fits. Surface tension. The final equilibrium quantity we measure is the tension a of the phase boundary separating the symmetric and broken phases. The interface tension reduces the likelihood of mixed-phase configurations, visible in the probability distributions of figure 2 as a suppressed “valley” between the two phases. The suppression is proportional to exp(—a A/T), with A being the area of the phase boundary, and can be measured from the probability distributions using the histogram method [108]. Specifically, the quantity
Figure 7. Continuum extrapolation of the latent heat. The behavior is similar to that of Av/T, but with slightly larger statistical errors. In BM1, we again show both linear and quadratic fits. Surface tension. The final equilibrium quantity we measure is the tension a of the phase boundary separating the symmetric and broken phases. The interface tension reduces the likelihood of mixed-phase configurations, visible in the probability distributions of figure 2 as a suppressed “valley” between the two phases. The suppression is proportional to exp(—a A/T), with A being the area of the phase boundary, and can be measured from the probability distributions using the histogram method [108]. Specifically, the quantity
Figure 8. Extrapolation of the surface tension to an infinite surface area. Statistical errors are substantial in the Gg = 32 case and could be improved with more simulations.
Figure 8. Extrapolation of the surface tension to an infinite surface area. Statistical errors are substantial in the Gg = 32 case and could be improved with more simulations.
Figure 10. 2-loop topologies contributing to the 3d Veg, in Landau gauge. Sunset-type diagrams on the first row are divergent in the UV and exactly determine the RG evolution of the mass parameters in the 3d EFT.
Figure 10. 2-loop topologies contributing to the 3d Veg, in Landau gauge. Sunset-type diagrams on the first row are divergent in the UV and exactly determine the RG evolution of the mass parameters in the 3d EFT.
Table 5. Comparison of thermodynamic quantities as obtained either with the EFT approach, or with the resummed 1-loop potential in the full theory. Error bars indicate sensitivity to the RG- scale as described in the text. For the lattice results we show the statistical uncertainty. Numbers in the last column correspond to central values without scale uncertainties.
Table 5. Comparison of thermodynamic quantities as obtained either with the EFT approach, or with the resummed 1-loop potential in the full theory. Error bars indicate sensitivity to the RG- scale as described in the text. For the lattice results we show the statistical uncertainty. Numbers in the last column correspond to central values without scale uncertainties.
Figure 11. Location of the global minimum of the perturbative effective potential as a function of the temperature. At high temperature, the minimum is at the origin and the electroweak sym- metry is restored. From left to right: 2-loop potential in the 3d EFT, l-loop Veg with Parwani resummation, 1-loop Veg with Arnold-Espinosa resummation. Values for two different choices of the RG scale are shown, and the coloured band in between illustrates the perturbative uncertainty as described in the text. In BM1, the inert doublet ¢, does not develop a VEV due to unbroken Zz symmetry.
Figure 11. Location of the global minimum of the perturbative effective potential as a function of the temperature. At high temperature, the minimum is at the origin and the electroweak sym- metry is restored. From left to right: 2-loop potential in the 3d EFT, l-loop Veg with Parwani resummation, 1-loop Veg with Arnold-Espinosa resummation. Values for two different choices of the RG scale are shown, and the coloured band in between illustrates the perturbative uncertainty as described in the text. In BM1, the inert doublet ¢, does not develop a VEV due to unbroken Zz symmetry.
fixes the electromagnetic fine-structure constant
fixes the electromagnetic fine-structure constant
the two prescriptions is formally of higher order in perturbation theory. In the IDM limit (u? = 0 and v, = 0), the calculation proceeds analogously, but instead of tan and cos(G — a) we input the self-coupling ;(A) and the combination (A3(A) + Aq(A) + A5(A)) /2. Detailed formulas in the IDM case (neglecting g’ contributions to the self energies) can be found in ref. [45]. In ref. [45], it is also discussed how higher- order corrections to eqs. (A.6)-(A.15) can be partially resummed by solving the equations self-consistently” , without performing the linearization. In BM1, we adopt their approach and solve the parameters iteratively, dropping g’ terms inside the loop corrections. This ensures that our study of thermal effects in BM1 is directly comparable to the 2-loop results of ref. [45].
the two prescriptions is formally of higher order in perturbation theory. In the IDM limit (u? = 0 and v, = 0), the calculation proceeds analogously, but instead of tan and cos(G — a) we input the self-coupling ;(A) and the combination (A3(A) + Aq(A) + A5(A)) /2. Detailed formulas in the IDM case (neglecting g’ contributions to the self energies) can be found in ref. [45]. In ref. [45], it is also discussed how higher- order corrections to eqs. (A.6)-(A.15) can be partially resummed by solving the equations self-consistently” , without performing the linearization. In BM1, we adopt their approach and solve the parameters iteratively, dropping g’ terms inside the loop corrections. This ensures that our study of thermal effects in BM1 is directly comparable to the 2-loop results of ref. [45].
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