REVIEW
Electroweak Phase Transition in Two Higgs Doublet Models
High Energy Physics - Phenomenology · Submitted 3 Sep 1996 · 29 references
Pith's verdict: A careful, honest one-loop study of the 2HDM electroweak phase transition; the 'persists at higher orders' clause is the softest part and rests on a heuristic ε criterion that deserves a closer look at figure 3.
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Referee report
The paper
The paper claims
A second Higgs doublet can make the electroweak phase transition strong enough to preserve a baryon asymmetry, while supersymmetry cannot.
The reading
What carries the argument
The load-bearing object is the one-loop finite-temperature effective potential with ring-diagram resummation, $V_{ring}$, evaluated for the CP-even Higgs direction. Three pieces do the work. First, an infrared-safe treatment of the Goldstone bosons: because they are massless at $phi = v$ in Landau gauge but have nonzero $M^{2prime}(v)$, the naive renormalization condition diverges, so the authors replace $log M^2$ by the pole-mass integral $F(M^2, m_{h^0}^2) = log m_{h^0}^2 - 2 + dots$ of eq. (5), which is finite and lets the Goldstone contributions be included consistently. Second, an analytic approximation to the thermal integral that joins the small-$M/T$ expansion to the large-$M/T$ asymptotic series at a matching point chosen so that values and derivatives agree, with a separate matching for negative $M^2$; this keeps the calculation valid where the conventional $M/T$ expansion fails, which is precisely where the interesting region lies. Third, the comparison of two resummation prescriptions, the Parwani method (replace $M^2(phi) to M^2(phi,T)$ everywhere) and the Arnold-Espinosa method (resum only the cubic term), whose spread is used as a proxy for the size of missing higher orders.
Extended reading notes
Core claim
The paper's central claim is that in a two-Higgs doublet model with the discrete symmetry that forces the two doublets to share a common vacuum expectation value (tan $beta = 1$), there is a significant region of the plane of the light Higgs mass $m_{h^0}$ against the common heavy Higgs mass $m_{A^0}=m_{H^0}=m_{H^pm}$ where the ratio $phi_c/T_c$ of the Higgs field value to the critical temperature exceeds one, the threshold required for electroweak baryogenesis. The authors argue this identification should survive higher-order corrections, on the evidence that their estimated perturbative expansion parameter $epsilon$ of eq. (19) is anticorrelated with $phi_c/T_c$: the region where the transition is strong is the region where the loop expansion looks best behaved. The result depends on the soft mass parameter $mu_3^2$ in the potential term $mu_3^2 Phi_1^dagger Phi_2$: the window closes as $mu_3^2$ grows, and the authors explicitly fail to reproduce an earlier claim that raising $mu_3^2$ generally strengthens the transition, attributing the discrepancy to a region where the potential develops a spurious second minimum. In the minimal supersymmetric standard model, by contrast, the tree-level relation $mu_3^2 = m_{A^0}^2/2$ plus the constrained quartic couplings prevent the extra Higgs bosons from strengthening the transition at all, and the authors find that using the naive critical temperature $T_0$ (where the second derivative at the origin vanishes) rather than the degeneracy temperature $T_1$ substantially overestimates the effect.
Load-bearing premise
The whole conclusion rests on the one-loop ring-resummed effective potential being quantitatively trustworthy in the region where $phi_c/T_c > 1$, and the authors check this only with a heuristic power-counting estimate of the expansion parameter rather than an actual two-loop computation; they say plainly in the conclusions that perturbation theory may still fail despite such safeguards.
Editorial extensions
If this is right
- Electroweak baryogenesis remains viable in a two-Higgs doublet model with tan $beta = 1$ for light Higgs masses up to roughly 300 GeV, provided the heavy Higgs bosons sit in a band above the light one and $mu_3^2$ is not too large.
- The sphaleron constraint cannot be met by the extra Higgs bosons of the minimal supersymmetric standard model at tan $beta = 1$, so supersymmetric baryogenesis must rely on other ingredients such as stop-sector effects or a different $tan beta$ regime.
- Using the temperature where the two minima become degenerate, rather than the temperature where the curvature at the origin vanishes, changes whether a given parameter point passes the sphaleron bound, and the naive choice systematically overestimates the strength of the transition.
- The anticorrelation between the estimated expansion parameter $epsilon$ and $phi_c/T_c$ means the parameter points that pass the baryogenesis bound are, on the authors' own diagnostic, the ones least likely to be spoiled by infrared problems.
- Any conclusion about the strength of the transition drawn from a truncated $M/T$ expansion is unreliable in exactly the mass range where two-Higgs doublet models become interesting.
Reading between the lines
- Because the authors restrict to tan $beta = 1$ for tractability, the natural next step is the same scan at general tan $beta$; the top Yukawa coupling then splits between the two doublets, which changes the fermionic contribution to the potential and could either widen or close the allowed band.
- The comparison between the two resummation schemes is used as an error bar but is not an error bar: a genuine two-loop calculation of the ring-resummed potential, or a lattice mapping onto the three-dimensional effective theory, would convert the authors' plausibility argument into a bound.
- The failure to confirm the earlier claim that larger $mu_3^2$ strengthens the transition, traced to a spurious double-minimum region, suggests that some of the earlier parameter-space maps were reading off an artifact of the one-loop potential rather than a physical first-order transition.
- The same machinery applied to the MSSM at larger tan $beta$, where the light Higgs mass is lifted by loop corrections and the stop contributions enter, would test whether the negative result here is a property of the extra Higgs bosons or of the tan $beta = 1$ slice.
Original abstract
We reexamine the strength of the first order phase transition in the electroweak theory supplemented by an extra Higgs doublet. The finite-temperature effective potential, $V_{eff}$, is computed to one-loop order, including the summation of ring diagrams, to study the ratio $\phi_c/T_c$ of the Higgs field VEV to the critical temperature. We make a number of improvements over previous treatments, including a consistent treatment of Goldstone bosons in $V_{eff}$, an accurate analytic approximation to $V_{eff}$ valid for any mass-to-temperature ratios, and use of the experimentally measured top quark mass. For two-Higgs doublet models, we identify a significant region of parameter space where $\phi_c/T_c$ is large enough for electroweak baryogenesis, and we argue that this identification should persist even at higher orders in perturbation theory. In the case of the minimal supersymmetric standard model, our results indicate that the extra Higgs bosons have little effect on the strength of the phase transition.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We reexamine the strength of the first order phase transition in the electroweak theory supplemented by an extra Higgs doublet. The finite-temperature effective potential, $V_{eff}$, is computed to one-loop order, including the summation of ring diagrams, to study the ratio $\phi_c/T_c$ of the Higgs field VEV to the critical temperature. We make a number of improvements over previous treatments, including a consistent treatment of Goldstone bosons in $V_{eff}$, an accurate analytic approximation to $V_{eff}$ valid for any mass-to-temperature ratios, and use of the experimentally measured top quark mass. For two-Higgs doublet models, we identify a significant region of parameter space where $\phi_c/T_c$ is large enough for electroweak baryogenesis, and we argue that this identification should persist even at higher orders in perturbation theory. In the case of the minimal supersymmetric standard model, our results indicate that the extra Higgs bosons have little effect on the strength of the phase transition.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We reexamine the strength of the first order phase transition in the electroweak theory supplemented by an extra Higgs doublet. | contribution 0.92 | 0.86 | 0.04 | unclear 0.79 |
| s1 | The finite-temperature effective potential, $V_{eff}$, is computed to one-loop order, including the summation of ring diagrams, to study the ratio $\phi_c/T_c$ of the Higgs field VEV to the critical temperature. | method 0.99 | 0.95 | 0.04 | asserted 0.98 |
| s2 | We make a number of improvements over previous treatments, including a consistent treatment of Goldstone bosons in $V_{eff}$, an accurate analytic approximation to $V_{eff}$ valid for any mass-to-temperature ratios, and use of the experimentally measured top quark mass. | contribution 0.73 | 0.95 | 0.04 | asserted 1.00 |
| s3 | For two-Higgs doublet models, we identify a significant region of parameter space where $\phi_c/T_c$ is large enough for electroweak baryogenesis, and we argue that this identification should persist even at higher orders in perturbation theory. | result 0.94 | 0.95 | 0.84 | asserted 0.89 |
| s4 | In the case of the minimal supersymmetric standard model, our results indicate that the extra Higgs bosons have little effect on the strength of the phase transition. | result 1.00 | 0.93 | 0.09 | hedged possibility 0.99 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/9609240 by the authors.
References
How later papers used it
88 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.
Uses its method or tool 3 of 12 papers with this label · 1 by the same authors
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Used the extrapolation method introduced in this paper for evaluating the one-loop correction.
Gravitational Techniwaves (2009) · 1.00 · paraphrased: the licence does not allow quoting
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Used the method of this paper to interpolate between the high and low temperature asymptotics of the temperature-dependent one-loop term.
Electroweak phase transition in technicolor (2010) · 1.00 · paraphrased: the licence does not allow quoting
By the same authors
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Used the approximation described in this paper, in which the high-temperature and low-temperature expansions are smoothly joined together at some large value of the mass-to-temperature ratio.
Electroweak Phase Transition and LHC Signatures in the Singlet Majoron Model (2009) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
84 arXiv papers cite this in 104 sentences
Papers citing it · 65 sentences without a confident label
Uses its method or tool 11 papers
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We follow otherwise the conventions of [ref] but use a slightly different method for estimating the temperature dependent one-loop correction, which involves a combination of the high temperature and low temperature asymptotic series [ref] (see Appendix [ref] for details).
Gravitational Techniwaves · Effective potential for MWT · 0.97
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[ref] , in which the high-$T$ and low-$T$ expansions are smoothly joined together at some large value of $m_i/T$.
Electroweak Phase Transition and LHC Signatures in the Singlet Majoron Model · The potential · 1.00
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Following previous analyses [ref] , we compensate possible radiative shifts of the EW VEV and address the problematic Goldstone contributions with a set of counterterms specified in [equation] where the coefficients are determined by the renormalisation conditions [equation] where VEV here corresponds to the zero-temperature vacuum $(h,H)=(v,0)$ and $m_IR$ is the infrared cutoff used for the regularisation of Goldstone contributions, which we set to the Higgs boson mass, $m_IR = m_h$ [ref] .[Further improvements can be obtained by using the effective potential in a dimensionally reduced field theory [ref] .]
Multi-step phase transitions and gravitational waves in the inert doublet model · Coleman-Weinberg correction to the potential · 0.97
Uses its data or values 1 paper
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However, the current experimental limit of $m_H$ is well above these values, at $m_H > 114$ GeV [ref] , thus turning the Standard Model an inadequate theory for baryogenesis.] Supersymmetric extensions of the Standard Model have been the most intensively studied [ref] , but it is also possible to get a strong transition from more generic two-Higgs doublet models [ref] , from technicolor theories [ref] , etc.
Primordial magnetogenesis · Magnetogenesis in phase transitions · 0.81
Compares with it 2 papers
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The scheme dependence of the thermal resummation is relatively small, which is in stark contrast to cases of EWPT in two Higgs doublet model [ref] .
Electroweak phase transition in a complex singlet extension of the Standard Model with degenerate scalars · Results · 0.94
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[ref] and [ref] are quite similar to those of the two Higgs doublet models [ref] .
The Electroweak Phase Transition in Nearly Conformal Technicolor · Explanation of Results · 1.00
Cites it as background 24 papers
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The electroweak phase transition (EWPT) in the 2HDM context has been extensively studied for both the CP conserving case [ref] and with the source of CP-violation [ref] .
A new insight into the phase transition in the early Universe with two Higgs doublets · Introduction · 0.99
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The Standard Model (SM) predicts a smooth crossover [ref] , but new states at the electro-weak scale can alter this picture dramatically (see, e.g. [ref] , for a review see [ref] ).
The Electro-Weak Phase Transition at Colliders: Discovery Post-Mortem · Introduction · 0.85
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The possibility of accommodating a strong FOEWPT in the 2HDM has been studied abundantly in the past (see e.g. [ref] ).
First shot of the smoking gun: probing the electroweak phase transition in the 2HDM with novel searches for $A \to ZH$ in $\ell^+ \ell^- t \bar{t}$ and $\nu \nu b \bar{b}$ final states · Introduction · 0.96
Discussion
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