Dynamics of Electroweak Phase Transition In Singlet-Scalar Extension of the Standard Model
DDynamics of Electroweak Phase Transition In Singlet-Scalar Extension of theStandard Model
Gowri Kurup, Maxim Perelstein
Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA
An addition to the Standard Model of a real, gauge-singlet scalar field, coupled via a Higgsportal interaction, can reopen the possibility of a strongly first-order electroweak phase transition(EWPT) and successful electroweak baryogenesis (EWBG). If a discrete symmetry that forbidsdoublet-singlet mixing is present, this model is notoriously difficult to test at the Large HadronCollider. As a result, it emerged as a useful benchmark for evaluating the capabilities of proposedfuture colliders to conclusively test EWPT and EWBG. In this paper, we evaluate the bubblenucleation temperature throughout the parameter space of this model where a first-order transitionis expected. We find that in large parts of this parameter space, bubbles in fact do not nucleateat any finite temperature, eliminating these models as viable EWBG scenarios. This constrainteliminates most of the region where a “two-step” phase transition is naively predicted, while the“one-step” transition region is largely unaffected. In addition, expanding bubble walls must notreach relativistic speeds during the transition for baryon asymmetry to be generated. We show thatthis condition further reduces the parameter space with viable EWBG.
I. INTRODUCTION
In today’s Universe electroweak symmetry is broken,but at very high temperatures, which prevailed im-mediately after the Big Bang, the symmetry was re-stored. The transition between the symmetric and bro-ken phases, the Electroweak Phase Transition (EWPT),occurred when the Universe was about a nanosecond old.Understanding the nature of this transition is an interest-ing question in its own right. It also has profound impli-cations for understanding the origin of matter-antimatterasymmetry in the Universe: one of the most compellingexplanations of this asymmetry, the Electroweak Baryo-genesis (EWBG) scenario, is only possible if the EWPTis strongly first-order [1]. (For a review, see [2].)While it is at present not possible to recreate theEWPT in the lab, it has been suggested that measure-ments of properties of the Higgs boson can provide in-direct information about the EWPT dynamics. In theStandard Model (SM), the EWPT is an adiabatic cross-over transition [3–6]. A first-order transition is only pos-sible in the presence of Beyond-the-SM (BSM) physicsat the weak scale, with significant couplings to the Higgssector. As a result, many models with first-order EWPTpredict significant deviations of the Higgs couplings togluons, photons, weak gauge bosons, and fermions thatcan already be tested at the Large Hadron Collider(LHC). In particular, supersymmetric models with stop-catalyzed first-order EWPT are already strongly dis-favored [7–9] (but not completely ruled out [10]). Abroader variety of models will be probed by increasinglyprecise measurements of the Higgs couplings at the LHCand the proposed e + e − Higgs factories [11–14]. A partic-ularly direct and powerful probe of the EWPT dynam-ics is provided by the Higgs cubic self-coupling, whichis predicted to have significant ( > ∼
20% or more) devi-ations from the SM in most models with a first-orderEWPT [15–17]. This prediction can be conclusively tested at the 1 TeV upgrade of the International Lin-ear Collider (ILC) [18–20] and the 100 TeV proton col-lider [21]. In addition, a strongly first-order EWPT mayproduce a potentially observable gravitational wave sig-nature [22]. Complementarity between collider and grav-itational wave signatures has been explored in Refs. [23–27].One of the simplest extensions of the SM in which afirst-order EWPT is possible is a model with an addi-tional gauge-singlet scalar field, S , coupled to the SMvia a Higgs portal interaction, V int = κ | H | S . (1)This is the only renormalizable interaction of S with theSM which is invariant under a Z symmetry, S → − S .This symmetry renders the S particle stable, and it mayplay the role of dark matter [28–31]. It has been shownthat this simple model can exhibit a strongly first-orderEWPT [32–39]. The Z prohibits mixing between thedoublet and singlet scalars, so that the 125 GeV Higgsparticle has couplings to fermions and gauge bosons thatare identical to the SM. Moreover, if m S > m h / h → SS is kinematically forbidden, the Higgswidth is also unaffected. As a result, this model presentsa difficult case (sometimes dubbed a “nightmare sce-nario”) for tests of EWBG at future colliders, and itbecame an important benchmark for gauging their capa-bilities in this regard [40]. Some interesting recent workon this benchmark model includes suggestions for addi-tional observables that can help cover the relevant param-eter space at a 100 TeV proton collider [40, 41], and animproved calculation of the thermal scalar potential [42].An important aspect of the EWPT in this model,which has not yet been systematically taken into accountin existing studies of future collider capabilities, is the dy-namics of bubble nucleation during the transition. In thispaper, we aim to fill this gap. In particular, we evaluatethe bubble nucleation temperature, T N , throughout the a r X i v : . [ h e p - ph ] M a y parameter space relevant for EWSB and future collid-ers. We find that T N is often significantly lower than thecritical temperature T c . In fact, in large regions of theparameter space, in particular those with a “two-step”EWPT (meaning that the S field acquires a vev beforethe Higgs field does), we find that bubbles do not nucle-ate at finite temperature at all, eliminating these regionsas viable EWBG scenarios. In addition, if T N (cid:28) T c ,the large difference in the vacuum energies at the stableand metastable vacua can result in “runaway” behaviorof the bubble walls, which become highly relativistic [43].This behavior is incompatible with the EWBG scenario.We identify the region of the parameter space (again pri-marily in the two-step regime) which suffers from thisproblem. The net result of the analysis is a significantreduction of the parameter space with viable EWBG.We then comment on the implications of these additionalconstraints for the experimental probes of EWBG at fu-ture colliders.
II. SETUP
We supplement the SM with a real scalar field S , un-charged under any of the SM gauge groups, and impose a Z discrete symmetry, under which S → − S and all otherfields are unchanged. The tree-level scalar potential hasthe form V ( H ; S ) = − µ | H | + λ | H | + 12 m S + η S + κS | H | , (2)where H is the SM Higgs doublet. If µ <
0, there isan electroweak symmetry-breaking (EWSB) minimum atzero temperature, with (cid:104) H (cid:105) = (0 , v/ √
2) and (cid:104) S (cid:105) = 0.Depending on parameters, the potential may also havean electroweak symmetry-preserving local minimum at (cid:104) H (cid:105) = 0 and (cid:104) S (cid:105) = s . There are no stable minima withboth (cid:104) H (cid:105) and (cid:104) S (cid:105) non-zero for any model parameters.The model is phenomenologically viable if the vacuumwith (cid:104) H (cid:105) (cid:54) = 0 is the global minimum of the potential, v ≈
246 GeV and m h ≈
125 GeV in this vacuum, andwe restrict our attention to such parameters. This leavesthree undetermined (but constrained) parameters, m , η ,and κ .The EWPT dynamics is determined by the effectivefinite-temperature potential V eff ( T ), where T is temper-ature. Physically, V eff is the free energy density of spacefilled with constant, spatially homogeneous scalar fields: H bg = (0 , ϕ √ , S bg = s , (3) For recent studies of bubble-wall dynamics in this and similarmodels, see e.g.
Refs. [44–47]. and all other fields set to zero. The effective potentialhas the form V eff ( ϕ, s ; T ) = V ( H bg , S bg )+ V ( ϕ, s )+ V T ( ϕ, s ; T ) , (4)where V is the Coleman-Weinberg potential, and V T isthe thermal potential [48, 49]. Both can be computed inperturbation theory. At the one-loop order, V ( ϕ, s ) = (cid:88) i g i ( − F i π (cid:104) m i ( ϕ, s ) log m i ( ϕ, s ) m i ( v, − m i ( ϕ, s ) + 2 m i ( ϕ, s ) m i ( v, (cid:105) ; (5) V T ( ϕ, s ; T ) = (cid:88) i g i T ( − F i π × (cid:90) ∞ dxx log (cid:34) − ( − F i exp (cid:32)(cid:114) x + m i ( ϕ, s ) T (cid:33)(cid:35) (6)where the sum runs over all SM and BSM particles inthe theory, and g i , F i and m i ( ϕ, s ) are the multiplic-ity, fermion number, and the mass (in the presence ofbackground fields) of the particle i . The countertermsincluded in Eq. (5) ensure that the tree-level Higgs massand vev in the present, zero-temperature Universe areunchanged at one loop. The dominant contributions to V and V T typically arise from loops of the Higgs and sin-glet scalar themselves. In this case, the masses m i ( h, s )are obtained by diagonalizing the scalar mass matrix: m , = 12 (cid:16) m − µ + ( κ + 3 λ ) ϕ + ( κ + 3 η ) s ± ∆ (cid:17) , (7)where ∆ = (( m + µ + ( κ − λ ) ϕ + (3 η − κ ) s ) +16 κ ϕ s ) / . We also include contributions of the SMtop quark and the electroweak gauge bosons, but ignoreloops of other SM particles due to their small couplingsto the Higgs. It is well known that light scalar- and gaugeboson-loop contributions to V T suffer from an IR diver-gence. Certain classes of higher-loop contributions (so-called “daisy diagrams”) need to be resummed to obtaina good approximation for this object at T (cid:29) m , where m is the boson mass [50, 51]. This is achieved by employ-ing the “ring-improved” version of V T , which is obtainedfrom Eq. (6) by replacing the zero-temperature masses m i ( ϕ, s ) with thermal masses, m i → m i + Π i ( T ), whereΠ i is the one-loop two-point function at finite tempera-ture. Recently, Ref. [42] argued that in certain regionsof parameter space, further classes of diagrams may needto be resummed. We do not include these effects in thepresent study, leaving such improvement for future work.At high temperature, thermal loops generate positivemass-squared for both H and S fields, and the energet-ically favored configuration has zero background fields,( ϕ, s ) = (0 , ϕ, s ) = ( v, , → (0 , s ) → ( v, V eff coexist over a range of tempera-tures. At high temperatures, the “symmetric” minimumis energetically preferred over the “broken” minimum.(In the case of the first step of a two-step transition,“symmetric” and “broken” refer to vacua with s = 0 and s (cid:54) = 0, both of which have unbroken electroweak symme-try.) The two minima become degenerate at the criticaltemperature, T c . As the Universe continues to cool, bub-bles of the broken-minimum phase are nucleated. Nucle-ation probability per unit time per unit volume at tem-perature T is given by [52] P ∼ T exp( − S /T ) , (8)where S is the action of a critical bubble. We use the CosmoTransitions code [53] to evaluate S numericallyas a function of temperature. Nucleation temperature T N is the temperature at which the nucleation probabilityper Hubble volume becomes of order one; for electroweakphase transition, this corresponds to [52] S /T N ≈ . (9)In this paper, we use this criterion to estimate T N explic-itly throughout the model parameter space. (We assume T N = T c for second-order transitions, since there is nometastable phase in that case.) Moreover, if a minimumwith s (cid:54) = 0 develops, we evaluate the nucleation temper-atures for both (0 , → (0 , s ) and (0 , → ( v,
0) tran-sitions, to determine which one occurs first. This pro-vides robust discrimination between one-step and two-step transitions. If the transition to EW-breaking vac-uum is first-order, EWBG scenario is viable only if thebaryon asymmetry created at the expanding bubble wallis not washed out by sphalerons inside the broken phase.This requires v ( T N ) T N > , (10)where v ( T N ) is the Higgs vev at the minimum of theeffective potential at the temperature T N , i.e. at thetime of the phase transition. There is some uncertaintyas to the precise numerical criterion for baryon numberpreservation (see e.g. [54, 55]), with v/T thresholds be-tween 0.6 and 1.4 quoted in the literature. Varying theEWBG criterion within this range has no noticeable ef-fect on the conclusions of our study, such as the phasediagrams presented below.Another potentially important aspect of a first-orderEWPT is the velocity of the expanding bubble wall. Thewall experiences outward pressure due to the differencein energy densities of the symmetric and broken vacua, V vac (sym) − V vac (br), where V vac = V + V . It also expe-riences pressure P from the thermal plasma of particlesthat it is moving through; since the particles are heav-ier in the broken phase than in the symmetric one, theeffect of this pressure is to slow the wall down. Thebalance between these two forces determines whetherthe wall reaches a non-relativistic terminal velocity, orcontinues to accelerate until it becomes highly relativis-tic. In the latter case, electroweak baryogenesis cannotoccur, since there is not enough time to generate thebaryon-antibaryon asymmetry in the region in front ofthe advancing bubble wall. Thus, to find viable modelsof EWBG one must not only require a strongly first-orderEWPT, but also demand that the bubble wall does notreach v wall ∼ V vac (sym) − V vac (br) − P > , (11)where the pressure P is calculated assuming v wall ∼ P ≈ (cid:88) i (cid:0) m i (br) − m i (sym) (cid:1) g i T N π ˜ J i (cid:18) m i (sym) T N (cid:19) , (12)where ˜ J i ( x ) = (cid:90) ∞ y dy (cid:112) y + x e √ y + x + ( − F i . (13)We will apply the Bodeker-Moore (BM) criterion ,Eq. (11), to further constrain the viable parameter spacefor EWBG. Note that Ref. [43] argued that if the BMcriterion is satisfied, the walls will exhibit “runaway” be-havior, continuing to accelerate indefinitely once they arerelativistic. Very recently, the analysis has been refinedto include the effect of transition radiation by chargedparticles crossing the bubble wall, with the result thatthe wall velocities are limited [56]. However, the newlyestablished speed limit, γ ∼ /α , is still highly relativis-tic, so that the conclusions regarding viability of EWBGare unaffected. III. RESULTS
We performed a comprehensive scan of the model pa-rameter space, ( m , κ, η ). For each point in the scanwith viable zero-temperature vacuum structure, we de-termine the transition history (one-step or two-step);critical temperature and transition order (for each step,in the case of two-step transition); nucleation tempera-ture, for each first-order transition; and, in the case offirst-order EWSB transition, whether or not the BM cri-terion is satisfied. The results are summarized in a seriesof two-dimensional slices through the parameter space,Figs. 1, 4, 5, 6. For clarity, we trade the scalar mass pa-rameter m for the physical mass of the singlet scalar, m S = (cid:0) ∂ V vac ( v, /∂S (cid:1) / , in these plots. FIG. 1: Phase transition dynamics in the m S − κ plane, with η = η min + 0 .
1. Region I (green): one-step strongly first-order transition; Region II (yellow): two-step transition withstrongly first-order electroweak-symmetry breaking step; Re-gion III (red): no thermal phase transition (a would-be two-step transition, but bubbles fail to nucleate); Region IV (pur-ple): same as red, with a would-be one-step transition; Re-gion V (blue): second-order transition; Region VI (gray): noviable EWSB at zero temperature; Region VII (white): non-perturbative regime ( η >
The main new result is that in large parts of the pa-rameter space where a naive criterion used in previousstudies suggests a strongly first-order electroweak phasetransition, bubble nucleation in fact does not occur at anyfinite temperature, so there is no thermal phase transi-tion at all. Instead, the system becomes trapped in themetastable state with unbroken electroweak symmetry,either at the origin or at (0 , s ). Eventually, it may tran-sition to the stable EW-breaking vacuum by tunnelingat T = 0, and such models may be viable descriptionsof today’s Universe; however, they do not provide viablescenarios for electroweak baryogenesis. Any discussion ofcollider experiments required to test EWBG must takethis constraint into account.A striking example is provided by Fig. 1. FollowingRef. [40], in this plot we fixed the singlet quartic couplingat η = η min + 0 .
1, where η min = λm /µ is the minimumvalue for which ( v,
0) is the global minimum of the tree-level potential. Essentially the entire region where a two-step transition would be expected is eliminated due tofailure to nucleate bubbles at any temperature. A two-step thermal phase transition can only occur in a verynarrow sliver of parameter space at the bottom of thisregion, shown in yellow in Fig. 1. The reason is thatin the two-step region, a large potential barrier betweenthe EW-preserving and EW-breaking vacua is present at
20 40 60 80 100 1200200400600800
FIG. 2: Ratio S /T , where S is the critical bubble action, for m S = 300 GeV and κ = 1 .
55 (red) and 1.54 (yellow). For bothpoints, a two-step first-order transition is naively expected. Infact, thermal transition does not occur at κ = 1 . FIG. 3: Thermal potential at the critical temperature, alongthe line in field space connecting the EW-symmetric and bro-ken vacua, for m S = 300 GeV, κ = 1 .
8, and two representa-tive values of η , 2.0 (red) and 2.5 (yellow). For both points,a two-step first-order transition is naively expected. In fact,thermal transition does not occur at η = 2 . any temperature, down to T = 0. As a result, the criticalbubble action S is limited from below, and if this limitis sufficiently large, the bubble-nucleation criterion (9) isnever satisfied. This is illustrated in Fig. 2. In contrast,in the one-step region, there is no EW-preserving vacuumat T = 0 at tree level. This guarantees bubble nucleationat finite temperature, unless the couplings are very strongand loop corrections become important. Consequently, There is some uncertainty as to the precise numerical value ofthe right-hand side in Eq. (9). We use 100 in Figs. 1-6. Wehave checked that varying this threshold by 20% does not have asignificant effect on the phase diagrams. The reason is clear fromFig. 2: at the boundary between the regions with and withoutthermal phase transition, small changes in model parameters leadto large changes in the critical bubble action.
FIG. 4: Phase transition dynamics in the m S − κ plane, with η = η min + 2 .
5. Same labeling and color code as in Fig. 1.
FIG. 5: Phase transition dynamics in the κ − η plane, with m S = 300 GeV. Same labeling and color code as in Fig. 1. most of the one-step region survives this constraint.The shape of the potential, and hence dynamics of bub-ble nucleation, depend on the singlet quartic coupling η as well as m S and κ . We find that for larger η , it is easierto find points in the two-step region where the thermalEWPT does occur, and is strongly first-order. The rea-son is that as η is increased, the critical temperature ofthe transition between the EW-symmetric and brokenvacua increases, and both the height and the width ofthe potential barrier decrease; see Fig. 3. This makestunneling between the two vacua easier, allowing a ther-mal phase transition to occur. The effect of varying η onthe viable parameter space is illustrated in Figs. 4 and 5.Note, however, that even at large η , most of the two-step region is eliminated by the requirement of bubblenucleation at non-zero temperature. I n c o r r e c t T = V a c u u m N o T h e r m a l P T S e c o n d O r d e r P T A B
FIG. 6: Phase transition dynamics in the κ − η plane, with m S = 300 GeV. In region B (red) bubble walls accelerate torelativistic speeds and EWBG cannot occur, while in regionA (blue) EWBG is possible. Even if this requirement is satisfied, models in whichthe nucleation temperature T N significantly below thecritical temperature T c are likely to fail the BM crite-rion for relativistic bubble wall motion. This is becausein this case, the symmetry-breaking vacuum would typ-ically have a significantly lower vacuum energy at T N compared to the symmetric vacuum, resulting in a strongoutward pressure on the bubble wall. To check this, weimplemented the BM criterion, Eq. (11), in our scans.The result, shown in Fig. 6, is consistent with expec-tations. The BM criterion eliminates a region borderingthat where no thermal EWPT occurs, since by continuitythis is the region where T N is the lowest. This extra con-straint must also be taken into account in the discussionof collider probes of EWBG. IV. DISCUSSION
We re-considered the dynamics of EWPT in a modelwith a singlet scalar field S coupled to the SM via a Z -symmetric Higgs portal, Eq. (1). We found that therequirements of thermal EWPT (bubble nucleation atnon-zero temperature) and non-relativistic bubble wallmotion eliminate much of the parameter space that waspreviously thought to provide viable EWBG models. Inparticular, most of the parameter space where a two-stepphase transition was thought to occur, is now eliminated.The effect of the new requirements in the region where aone-step transition was expected is less significant.The model studied here has recently emerged as a use-ful benchmark for planning the physics program at fu-ture colliders. While absence of mixing between dou-blet and singlet states makes this model challenging toprobe at the LHC, Ref. [40] argued that the proposedfuture facilities will be able to probe the EWBG sce-nario in this model conclusively. This can be achievedwith a combination of Higgs cubic coupling measure-ments [16], direct Higgs portal searches in channels suchas pp → V SS, qqSS [40, 41], and a very precise measure-ment of σ ( e + e − → Zh ) at electron-positron Higgs fac-tories [11, 57, 58]. The new constraints considered herereduce the parameter space with viable EWBG, whichshould in principle make the colliders’ task easier. How-ever, comparing the predictions for collider observablesin Ref. [40] with the new constraints presented here in-dicates that the newly eliminated parts of the parame-ter space are the ones with the strongest collider signals.This should not be surprising, since for a given m S , ourconstraints place an upper bound on κ , while all colliderobservables deviate from the SM more with growing κ .(Note that T = 0 collider observables are independent of η up to the one-loop order, since η does not enter V vac inthe present vacuum at one loop). Thus, the sensitivitygoals established by previous studies as benchmarks forfuture colliders remain unchanged.Our findings seem to indicate that in this model, a vi-able two-step first-order transition occurs only in a ratherspecial, narrow region of the parameter space, in effectrequiring some degree of tuning between the model pa-rameters. This may appear to make this scenario “un- likely”. However, it is important to remember that theparameters of the model may emerge from a more fun-damental theory at higher energy scales, which may infact correlate parameters that we treat as independent.Therefore, it would be incorrect to interpret our resultsas in any way reducing the motivation for an experimen-tal program that will address the viability of EWBG inthis model. Acknowledgments
We are grateful to Andrey Katz for discussions whichinitiated this investigation; to Nicolas Rey-Le Lorier forcollaboration at the early stages of the project; to Car-roll Wainwright for valuable communications regarding
CosmoTransitions ; to Yu-Dai Tsai for helpful discus-sions; and to Michael Baker for pointing out a typograph-ical error in the original version of this preprint. Thiswork is supported by the U.S. National Science Founda-tion grant PHY-1316222, and by the Simons Foundationgrant [1] V. A. Kuzmin, V. A. Rubakov, and M. E. Shaposhnikov,
On the Anomalous Electroweak Baryon Number Noncon-servation in the Early Universe , Phys. Lett.
B155 , 36(1985).[2] D. E. Morrissey and M. J. Ramsey-Musolf,
Electroweakbaryogenesis , New J.Phys. , 125003 (2012), 1206.2942.[3] Y. Aoki, F. Csikor, Z. Fodor, and A. Ukawa, The End-point of the first order phase transition of the SU(2)gauge Higgs model on a four-dimensional isotropic lat-tice , Phys. Rev.
D60 , 013001 (1999), hep-lat/9901021.[4] F. Csikor, Z. Fodor, and J. Heitger,
Endpoint of thehot electroweak phase transition , Phys. Rev. Lett. , 21(1999), hep-ph/9809291.[5] M. Laine and K. Rummukainen, What’s new with theelectroweak phase transition? , Nucl. Phys. Proc. Suppl. , 180 (1999), hep-lat/9809045.[6] M. Gurtler, E.-M. Ilgenfritz, and A. Schiller, Where theelectroweak phase transition ends , Phys. Rev.
D56 , 3888(1997), hep-lat/9704013.[7] T. Cohen, D. E. Morrissey, and A. Pierce,
ElectroweakBaryogenesis and Higgs Signatures , Phys. Rev.
D86 ,013009 (2012), 1203.2924.[8] D. Curtin, P. Jaiswal, and P. Meade,
Excluding Elec-troweak Baryogenesis in the MSSM , JHEP , 005(2012), 1203.2932.[9] A. Katz, M. Perelstein, M. J. Ramsey-Musolf, andP. Winslow, Stop-Catalyzed Baryogenesis Beyond theMSSM , Phys. Rev.
D92 , 095019 (2015), 1509.02934.[10] S. Liebler, S. Profumo, and T. Stefaniak,
Light Stop MassLimits from Higgs Rate Measurements in the MSSM: IsMSSM Electroweak Baryogenesis Still Alive After All? ,JHEP , 143 (2016), 1512.09172.[11] A. Katz and M. Perelstein, Higgs Couplings andElectroweak Phase Transition , JHEP , 108 (2014), 1401.1827.[12] S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, Singlet Higgs phenomenology and the electroweak phasetransition , JHEP , 010 (2007), 0705.2425.[13] P. H. Damgaard, D. O’Connell, T. C. Petersen, andA. Tranberg, Constraints on New Physics from Baryoge-nesis and Large Hadron Collider Data , Phys. Rev. Lett. , 221804 (2013), 1305.4362.[14] S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright, andP. Winslow,
Singlet-catalyzed electroweak phase transi-tions and precision Higgs boson studies , Phys. Rev.
D91 ,035018 (2015), 1407.5342.[15] S. Kanemura, Y. Okada, and E. Senaha,
Electroweakbaryogenesis and quantum corrections to the triple Higgsboson coupling , Phys. Lett.
B606 , 361 (2005), hep-ph/0411354.[16] A. Noble and M. Perelstein,
Higgs self-coupling as a probeof electroweak phase transition , Phys.Rev.
D78 , 063518(2008), 0711.3018.[17] P. Huang, A. Joglekar, B. Li, and C. E. M. Wagner,
Prob-ing the Electroweak Phase Transition at the LHC , Phys.Rev.
D93 , 055049 (2016), 1512.00068.[18] H. Baer, T. Barklow, K. Fujii, Y. Gao, A. Hoang,S. Kanemura, J. List, H. E. Logan, A. Nomerotski,M. Perelstein, et al.,
The International Linear ColliderTechnical Design Report - Volume 2: Physics (2013),1306.6352.[19] K. Fujii et al.,
Physics Case for the International LinearCollider (2015), 1506.05992.[20] J. Tian, in
Helmholtz Alliance Linear Collider Fo-rum: Proceedings of the Workshops Hamburg, Munich,Hamburg 2010-2012, Germany , DESY (DESY, Ham-burg, 2013), pp. 224–247, URL http://inspirehep.net/record/1475534/files/1238328_224-247.pdf . [21] R. Contino et al., Physics at a 100 TeV pp collider: Higgsand EW symmetry breaking studies (2016), 1606.09408.[22] C. Grojean and G. Servant,
Gravitational Waves fromPhase Transitions at the Electroweak Scale and Beyond ,Phys. Rev.
D75 , 043507 (2007), hep-ph/0607107.[23] K. Hashino, M. Kakizaki, S. Kanemura, and T. Mat-sui,
Synergy between measurements of gravitational wavesand the triple-Higgs coupling in probing the first-orderelectroweak phase transition , Phys. Rev.
D94 , 015005(2016), 1604.02069.[24] P. Huang, A. J. Long, and L.-T. Wang,
Probing theElectroweak Phase Transition with Higgs Factories andGravitational Waves , Phys. Rev.
D94 , 075008 (2016),1608.06619.[25] K. Hashino, M. Kakizaki, S. Kanemura, P. Ko, andT. Matsui,
Gravitational waves and Higgs boson cou-plings for exploring first order phase transition in themodel with a singlet scalar field , Phys. Lett.
B766 , 49(2017), 1609.00297.[26] M. Artymowski, M. Lewicki, and J. D. Wells,
Grav-itational wave and collider implications of electroweakbaryogenesis aided by non-standard cosmology , JHEP ,066 (2017), 1609.07143.[27] A. Beniwal, M. Lewicki, J. D. Wells, M. White, andA. G. Williams, Gravitational wave, collider and darkmatter signals from a scalar singlet electroweak baryoge-nesis (2017), 1702.06124.[28] J. McDonald,
Gauge singlet scalars as cold dark matter ,Phys. Rev.
D50 , 3637 (1994), hep-ph/0702143.[29] C. P. Burgess, M. Pospelov, and T. ter Veldhuis,
TheMinimal model of nonbaryonic dark matter: A Singletscalar , Nucl. Phys.
B619 , 709 (2001), hep-ph/0011335.[30] L. Feng, S. Profumo, and L. Ubaldi,
Closing in on sin-glet scalar dark matter: LUX, invisible Higgs decays andgamma-ray lines , JHEP , 045 (2015), 1412.1105.[31] M. J. Baker and J. Kopp, The Vev Flip-Flop: Dark Mat-ter Decay between Weak Scale Phase Transitions (2016),1608.07578.[32] J. R. Espinosa and M. Quiros,
Novel Effects in Elec-troweak Breaking from a Hidden Sector , Phys. Rev.
D76 ,076004 (2007), hep-ph/0701145.[33] V. Barger, P. Langacker, M. McCaskey, M. J. Ramsey-Musolf, and G. Shaughnessy,
LHC Phenomenology ofan Extended Standard Model with a Real Scalar Singlet ,Phys. Rev.
D77 , 035005 (2008), 0706.4311.[34] J. R. Espinosa, T. Konstandin, J. M. No, and M. Quiros,
Some Cosmological Implications of Hidden Sectors , Phys.Rev.
D78 , 123528 (2008), 0809.3215.[35] J. R. Espinosa, T. Konstandin, and F. Riva,
Strong Elec-troweak Phase Transitions in the Standard Model with aSinglet , Nucl. Phys.
B854 , 592 (2012), 1107.5441.[36] J. M. Cline and K. Kainulainen,
Electroweak baryogenesisand dark matter from a singlet Higgs , JCAP , 012(2013), 1210.4196.[37] M. Perelstein and Y.-D. Tsai,
750 GeV diphoton ex-cess and strongly first-order electroweak phase transition ,Phys. Rev.
D94 , 015033 (2016), 1603.04488.[38] V. Vaskonen,
Electroweak baryogenesis and gravitationalwaves from a real scalar singlet (2016), 1611.02073.[39] L. Marzola, A. Racioppi, and V. Vaskonen,
Phase tran- sition and gravitational wave phenomenology of scalarconformal extensions of the Standard Model (2017),1704.01034.[40] D. Curtin, P. Meade, and C.-T. Yu,
Testing Elec-troweak Baryogenesis with Future Colliders , JHEP ,127 (2014), 1409.0005.[41] N. Craig, H. K. Lou, M. McCullough, and A. Thalapillil, The Higgs Portal Above Threshold , JHEP , 127 (2016),1412.0258.[42] D. Curtin, P. Meade, and H. Ramani, Thermal Resum-mation and Phase Transitions (2016), 1612.00466.[43] D. Bodeker and G. D. Moore,
Can electroweak bubblewalls run away? , JCAP , 009 (2009), 0903.4099.[44] T. Konstandin, G. Nardini, and I. Rues,
From Boltz-mann equations to steady wall velocities , JCAP ,028 (2014), 1407.3132.[45] J. Kozaczuk,
Bubble Expansion and the Viability ofSinglet-Driven Electroweak Baryogenesis , JHEP , 135(2015), 1506.04741.[46] S. J. Huber, T. Konstandin, G. Nardini, and I. Rues, Detectable Gravitational Waves from Very Strong PhaseTransitions in the General NMSSM , JCAP , 036(2016), 1512.06357.[47] M. Chala, G. Nardini, and I. Sobolev,
Unified explanationfor dark matter and electroweak baryogenesis with directdetection and gravitational wave signatures , Phys. Rev.
D94 , 055006 (2016), 1605.08663.[48] L. Dolan and R. Jackiw,
Symmetry Behavior at FiniteTemperature , Phys.Rev. D9 , 3320 (1974).[49] S. Weinberg, Gauge and Global Symmetries at High Tem-perature , Phys.Rev. D9 , 3357 (1974).[50] P. Fendley, The Effective Potential and the CouplingConstant at High Temperature , Phys.Lett.
B196 , 175(1987).[51] M. Carrington,
The Effective potential at finite tempera-ture in the Standard Model , Phys.Rev.
D45 , 2933 (1992).[52] M. Dine, R. G. Leigh, P. Y. Huet, A. D. Linde, andD. A. Linde,
Towards the theory of the electroweak phasetransition , Phys. Rev.
D46 , 550 (1992), hep-ph/9203203.[53] C. L. Wainwright,
CosmoTransitions: Computing Cos-mological Phase Transition Temperatures and BubbleProfiles with Multiple Fields , Comput. Phys. Commun. , 2006 (2012), 1109.4189.[54] H. H. Patel and M. J. Ramsey-Musolf,
Baryon Washout,Electroweak Phase Transition, and Perturbation Theory ,JHEP , 029 (2011), 1101.4665.[55] K. Fuyuto and E. Senaha, Improved sphaleron decouplingcondition and the Higgs coupling constants in the realsinglet-extended standard model , Phys. Rev.
D90 , 015015(2014), 1406.0433.[56] D. Bodeker and G. D. Moore,
Electroweak Bubble WallSpeed Limit (2017), 1703.08215.[57] N. Craig, C. Englert, and M. McCullough,
New Probeof Naturalness , Phys. Rev. Lett. , 121803 (2013),1305.5251.[58] N. Craig, M. Farina, M. McCullough, and M. Perelstein,
Precision Higgsstrahlung as a Probe of New Physics ,JHEP03