Effective Theories of MSSM at High Temperature

TL;DR

This work develops infrared-safe, three-dimensional effective theories for the MSSM electroweak phase transition via 1-loop dimensional reduction, including a bosonic 3d theory, a 3d two-Higgs-doublet model, and a 3d SU(2)+Higgs model. By integrating out heavy fields and diagonalizing the Higgs sector, the authors obtain a tractable 3d framework whose non-perturbative lattice results yield infrared-safe bounds on the lightest Higgs mass compatible with baryogenesis, contingent on a small right-handed stop mass parameter $m_U^2$ and suitable $m_A$. They also discuss a more elaborate regime with very light stops, proposing an alternative effective theory that includes an SU(3) triplet, which may be necessary to capture strong-transition physics. Overall, the paper provides a systematic perturbative construction of IR-safe 3d EFTs for MSSM thermodynamics and clarifies when simpler SU(2)+Higgs reductions suffice versus when richer theories are required for baryogenesis viability. The approach offers a controlled bridge between perturbative analysis and lattice simulations for MSSM electroweak baryogenesis studies.

Abstract

We construct effective 3d field theories for the Minimal Supersymmetric Standard Model, relevant for the thermodynamics of the cosmological electroweak phase transition. The effective theories include a 3d theory for the bosonic sector of the original 4d theory; a 3d two Higgs doublet model; and a 3d SU(2)+Higgs model. The integrations are made at 1-loop level. In integrals related to vacuum renormalization we take into account only quarks and squarks of the third generation. Using existing non-perturbative lattice results for the 3d SU(2)+Higgs model, we then derive infrared safe upper bounds for the lightest Higgs boson mass required for successful baryogenesis at the electroweak scale. The Higgs mass bounds turn out to be close to those previously found with the effective potential, allowing baryogenesis if the right-handed stop mass parameter $m_U^2$ is small. Finally we discuss the effective theory relevant for $m_U^2$ very small, the most favourable case for baryogenesis.

Figures (8)

• Figure 1: The generic types of graphs needed for dimensional reduction of (a) wave functions and masses, (b) scalar couplings and (c) the gauge coupling. Wiggly lines are vector propagators and dashed lines represent generic propagators of particle type P$=$Q, U, D, S, H, A, C, g, q, f. Here Q, U, D denote the corresponding squarks, S is a squark in general, H is a Higgs doublet, A and C are the SU(2) and SU(3) gauge fields, g is a ghost, q a third generation quark and f a general fermion. For the coupling constants, 1-loop dimensional reduction is directly related to 1-loop vacuum renormalization and hence only squarks and quarks are considered in the internal lines. For the masses, the thermal screening terms proportional to $T^2$ are not related to vacuum renormalization and hence we include all the modes with $m \hbox{$<$$\sim$} 2\pi T$ in the loops.
• Figure 2: The graphs needed for integrating out the heavy Higgs doublet from the 3d two Higgs doublet model. The solid line represents the heavy field $\theta$. Graph (a) is a contribution to the mass parameter $m_\phi^2$, graphs (b) are contributions to the scalar self-coupling $\lambda_\phi$, (c) is an induced 6-point function, and (d) is a mixing term generated at 1-loop level.
• Figure 3: The effect of the CP-even and CP-odd Higgs masses $m_h$ and $m_A$ on $x$ ($x$ is defined in eq. (\ref{['bound']})). All the numbers are in GeV. The mixing parameters have been set to zero. With the thin lines, we show as an alternative parametrization the value of $x$ as a function of $\tan\!\beta$ (at $\bar{\mu}=200$ GeV) in the present scheme.
• Figure 4: The effect of the top mass on $x$. Here the top mass is taken at tree-level, in accordance with the other uncertainties in the calculation. The thick lines are for constant $m_h=70$ GeV, the thin lines for constant $\tan\!\beta =2.0$. The mixing parameters have been set to zero.
• Figure 5: The effect of $m_U$ on $x$ for constant $m_h$ (thick lines) and $\tan\!\beta$ (thin lines). Note that at tree-level, the right-handed stop mass $m_{\tilde{t}_R}$ at zero temperature is given by $m_{\tilde{t}_R}^2 = m_U^2+ m_t^2$ for vanishing $\tilde{A}_t$ and $g'$, see (\ref{['matrLR']})--(\ref{['mtLR']}).