The Electroweak Phase Transition: A Collider Target

Determining the thermal history of electroweak symmetry breaking (EWSB) is an important challenge for particle physics and cosmology. Lattice simulations indicate that EWSB in the Standard Model (SM) occurs through a crossover transition, while the presence of new physics beyond the SM could alter this thermal history. The occurrence of a first order EWSB transition would be particularly interesting, providing the needed pre-conditions for generation of the cosmic matter-antimatter asymmetry and sources for potentially observable gravitational radiation. I provide simple arguments that if such an alternate thermal history exists, the new particles involved cannot be too heavy with respect to the SM electroweak temperature, nor can they interact too feebly with the SM Higgs boson. The corresponding quantitative expectations for masses and interaction strengths imply that their effects could in principle be observed (or ruled out) by prospective next generation high energy colliders. The simple arguments are quantitatively consistent with results obtained in a wide range of explicit model studies.


I. INTRODUCTION
What is the thermal history of electroweak symmetry-breaking (EWSB)? While this question has been the subject of theoretical investigation for more than four decades, the discovery of the Standard Model-like Higgs boson puts it squarely in the spotlight for particle physics and cosmology. Results of non-perturbative lattice computations indicate that within the minimal Standard Model (SM), EWSB occurs through a crossover transition at a temperature T ∼ 100 GeV. Had the Higgs-like scalar been lighter than ∼ 70 − 80 GeV, the transition would have been a first order phase transition [1][2][3][4][5][6]. The situation is analogous to what occurs in quantum chromodynamics (QCD). For sufficiently small baryon chemical potential µ B , as pertains to a purely SM early universe, the transition to the confined phase of QCD at T ∼ 100 MeV was also of a crossover character. This theoretical result is consistent with experimental studies of heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC). Going forward, the RHIC beam energy scan hopes to identify the onset of a first order transition (or critical point) at non-zero µ B (for recent overviews, see Refs. [7,8]).
In the case of EWSB, there exists strong motivation to consider the possibility of an alternate thermal history as compared to the crossover transition of the minimal SM, even with the mounting evidence that the observed 125 GeV scalar is, indeed, the expected SM Higgs boson. Experimentally, LHC results to date have yet to preclude the possibility that the SM-like Higgs exists within an extended scalar sector, whose interactions could modify the thermal history of EWSB. Theoretically, a plethora of well-motivated particle physics models place the SM-like Higgs within such a beyond the Standard Model (BSM) setting. The resulting patterns of early universe EWSB could be considerably richer than in the minimal SM, a possibility suggested by Weinberg [9] and others [10][11][12][13][14][15]. Perhaps, most compellingly, open problems in cosmology could find their solutions through such extended scalar sectors. In particular, the presence of a first order electroweak phase transition (EWPT) could provide the necessary conditions for generation of the cosmic baryon asymmetry through electroweak baryogenesis (for a review and references, see Ref. [16]). If a first order EWPT were sufficiently strong, the resulting distortions of spacetime in the early universe would have produced relic gravitational waves (GWs) that one might observe in the LISA mission or future GW detectors (for a review, see Ref. [17]). In addition, additional neutral scalars could comprise part of the observed dark matter relic density.
In this context, it is interesting to ask: What would it take experimentally to probe exhaustively the possibilities for the thermal history of EWSB and/or confirm that -for all intents and purposes -EWSB did in fact occur through a smooth crossover? In what follows, I present simple arguments for the vital role to be played by the LHC and future high energy colliders under active consideration. Importantly, the known properties of the SM electroweak interaction, together with the masses of the SM-like Higgs boson and the top quark, set the temperature scale for EWSB, or T EW . The arguments below imply that if there exist new bosons whose interactions substantially modify the thermal history of EWSB, their masses must be at most a few times T EW , making their possible existence a target for present and prospective colliders. For strongly coupled scenarios -manifested as higher dimension operators in the Higgs potential -the associated mass scale is similarly bounded. Specific, model-dependent studies performed to date are consistent with these general expectations. I also discuss the implications for precision measurements of Higgs boson properties. In this case, I obtain approximate lower bounds on the magnitude of deviations from SM Higgs properties that would follow from a significant departure from the SM EWSB thermal history. Although the arguments in the latter case are less airtight than those germane to the mass scale, they nevertheless provide a clear benchmark for precision Higgs boson studies that may be achievable with the LHC and prospective future colliders. In both cases (mass and precision), a wide range of EWPT studies involving specific model realizations are consistent with the general arguments provided below.
The discussion of these arguments is organized as follows. In Section II, I review the basic features of SM EWSB at non-zero temperature, using a simple perturbative framework in the high temperature effective theory. I then consider mechanisms by which new bosonic degrees of freedom having renormalizable interactions may modify this thermal history. The quantitative implications for BSM mass scale and Higgs boson properties are considered in Sections III and IV, respectively. Section V gives a brief overview of the relevant mass reach and precision at the LHC and prospective future colliders. In Section VI, I discuss scenarios with non-renormalizable operators in the Higgs potential and scenarios involving new light degrees of freedom. In VI C, I return to the consideration of T EW , discussing possible modifications of this scale due to BSM scalar interactions. Some perspective and outlook appear in Section VII.

II. THE ELECTROWEAK TEMPERATURE
The starting point for this discussion is the SM Higgs potential At non-zero T , the dynamics of EWSB are driven by the finite-temperature effective potential, expressed as a function of the h = √ 2Re(H 0 ), where h ≡ v = 246 is the T = 0 Higgs vacuum expectation value (vev): where V CW (h) is the zero temperature Coleman-Weinberg potential and where V (h, T ) SM is generated by thermal loops (for a pedagogical introduction, see, e.g., Ref. [18]). For present purposes it is convenient to consider the latter in the high-temperature limit at leading order (LO) and to neglect (momentarily) the V CW contribution 1 , leading to where we have truncated at second order in the EW gauge couplings at one-loop in order to avoid complications associated with gauge-dependence that are not germane to the present discussion (see Refs. [19,20] and references therein). We have also dropped terms that depend logarithmically on T or that fall off as inverse powers of the temperature. The quantity D is calculable in perturbation theory, while DT 2 0 = µ 2 /2. The T = 0 EWSB minimization conditions allow one to express µ 2 in terms of v and the Higgs quartic self coupling, yielding Here, we have retained only the dominant, leading contributions to the temperature T 0 . From the experimental values for the quantities on the RHS of Eq. (4) we obtain the EW temperature At this stage, two points merit emphasis. First, the foregoing result has been obtained in a simple, perturbative framework. In this context, the nature of the EWSB transition implied by Eq. (3) is a second order phase transition.
The critical temperature for the latter is T 0 , below which the quadratic term becomes negative and the minimum of energy lies away from the symmetric phase (h = 0). Lattice studies indicate that transition is not a true second order phase transition, as there exists no evidence for diverging correlation lengths when the Higgs vev becomes non-zero. Nevertheless, for lighter values of m H consistent with a first order transition, the lattice values for the transition temperature are consistent with those obtained using Eq. (5) (see, e.g., Ref. [2]). Thus, it suffices for our purposes to take T EW ≈ 140 GeV (we return to a more detailed discussion in Section VI C). Second, the measured values of the weak scale (equivalently, the muon lifetime), Higgs boson and top quark masss -yielding λ and y t , respectively, and EW gauge couplings are all decisive inputs in setting the value of T EW . It would be a misconception to identify T EW with v, even though they are numerically commensurate. Had the experimental Higgs and/or top quark masses in particularly taken on different values, one might have obtained an electroweak temperature appreciably different from v. In short, one should consider T EW and v to be distinct scales.
We now ask: how might the presence of new particles and interactions or strong dynamics modify the foregoing picture? And what are the implications for the corresponding mass scale M ?
The impact of new particles and interactions occurs via one or more for the following avenues: • Thermal loops • Modification of the tree-level T = 0 vacuum structure • Loop induced modification of the T = 0 vacuum We consider the first two of these possibilities in turn, and return to the third in Section VI C. For purposes of concrete illustration, we introduce an additional scalar field φ that can transform either as a singlet under S(3) C ×SU(2) L ×U(1) Y or as a non-trivial representation of one or more of these factors. For non-singlet representations, consider the following gauge invariant, renormalizable T = 0 potential with In general φ may be complex. When φ is a SM gauge singlet, it may be either real or complex. In this case, both V 0 (φ) and V 0 (H, φ) may contain terms that are odd under φ → −φ. For φ transforming as a singlet or triplet under SU(2) L , one may also include cubic interaction terms of the form H † φH. We return to these possibilities in Section IV. We also note that for higher dimensional representations of the SM gauge groups, the interactions in Eqs. (7,8) give a simplified, symbolic expression for the full set of allowed operators. For a complete discussion in the case of higher dimensional representations of SU(2) L ×U(1) Y , see Ref. [21]. For our present purposes of illustrating the impact on phase transition dynamics and the collider reach, the simplified model forms in Eqs. (7,8) suffice. With this simple framework in mind, it is useful to consider various possible thermal histories of EWSB as illustrated in Fig. 1. Panels (a) and (b) are most relevant to the form of the H-φ interaction given in Eq. (8) whereas the trajectory of panel (c) is of interest when additional terms containing odd powers of φ are included (see below). Scenario (a) is, perhaps, the most familiar one, corresponding to a single-step transition to the present pure Higgs vacuum from the symmetric phase at T = T EW . Scenario (b) illustrates a two-step transition history, wherein the universe first lands in a phase with a non-zero φ vacuum expectation value at T = T φ > T EW followed by a transition to the pure Higgs vacuum at T EW . For scenario (c) the final vacuum is one in which both the Higgs and neutral component of φ obtain vevs. The transition shown occurs in a singlet step, though it is also possible for a multi-step transition to lead to the same vacuum.

III. MASS REACH
Thermal loops. The most significant, finite T loop effects of the new interactions given above are additional contributions to the Higgs boson thermal mass (the D coefficient) and the possible introduction of a barrier between the broken and unbroken vacua, opening up the possibility of a first order EWPT. We focus here on this possibility in the context of the scenario in Fig. 1(a). The relevant contribution to V (h, T ), obtained after performing the "daisy resummation" (see [18] for a discussion and references), is The first term inside the RHS gives the thermal mass-squared of the φ arising from its interactions with H and itself and from possible gauge loops (∆ g ). the The second and third terms will give the square of the T = 0 mass of the φ when h → v. Consider now the possibility that the first two terms cancel for T ∼ T EW : One may ensure tree-level stability of the full potential by choosing a 2 > 0, b 4 > 0, and b 2 < 0. In this case, one has The cubic dependence on h constitutes a barrier between the symmetric and broken phases along the Higgs direction, implying that the transition to the broken phase will be a first order EWPT. We now consider the implications for the φ mass at T = 0. It is useful to first consider the b 4 → 0 and ∆ g → 0 limit: We observe that the scale of M φ (T = 0) depends crucially on the difference between v and T EW / √ 3 and on the magnitude of the Higgs portal coupling. Varying a 2 between unity and five leads to the following range for the φ mass: Including a non-zero value for b 4 reduces the value of M φ (T = 0), so the upper end of the range is, in some sense, maximal for the foregoing inputs. That being said, there exist several reasons for relaxing this upper bound somewhat: • The criteria for defining perturbative values of the coupling is somewhat ambiguous. Use of the naïve perturbativity bounds a 2 < 2π and b 4 < 2π/3 leads to ∼ 10% larger maximum M φ (T = 0). According to the work of Ref. [22], which considered the perturbativity constraints on the Higgs quartic self-coupling, these naïve T = 0 perturbativity bounds may be too restrictive, implying that larger values of the couplings may be consistent with perturbativity. On the other hand, the importance of higher-dimensional non-renormalizable operators in the finite-T effective potential will also grow with the values of the couplings. Any analysis performed in this region of parameter space would need to include the effects of these operators before making definitive statements.
• As discussed further below, including the contribution of the cross-quartic interaction will introduce a linear dependence on a 2 in the denominator of Eq. (4) at one-loop order, thereby lowering T EW and increasing M φ (T = 0).
• A first order transition may arise even if the cancellation in Eq. (11) is not exact.
With these considerations in mind, I will take 700 GeV to be a conservative upper bound for the new scalar mass.
Tree-level vacuum structure. A second possibility for changing the thermal history of EWSB involves a multi-step transition into the Higgs phase, as in Fig. 1(b). During the first step the universe goes to a vacuum in which φ 0 = 0, followed by a second step to the Higgs phase. We will refer to the corresponding critical temperatures as T φ and T h , respectively. In this case, the operator φ † φH † H creates a barrier between the φ and Higgs vacua if the coefficient The two step history requires T φ > T h . We will again take T h ≈ T EW . The value of T φ is determined from the quadratic term in the part of the potential independent of H: where, as before, we have retained only the leading T -dependent terms. Setting the coefficient of the quadratic term to zero implies that where the inequality follows from the requirement that T φ > T h ≈ T EW . Now compute the mass T = 0 φ mass: For illustration, consider b 4 = 0.3, neglect ∆ g , and vary a 2 between one and five. Doing so leads to the range 160 GeV < ∼ M φ (T = 0) < ∼ 360 GeV, commensurate with the range associated with the thermal loop effect. Increasing the value of b 4 will lead to lower values of M φ (T = 0), while taking the unphysical b 4 → 0 limit will lead only to a slight increase. As in the previous argument, I will take 700 GeV as a conservative upper bound, taking into account the ambiguities in defining the requirements of perturbativity the restriction to renormalizable interactions.
It is worth noting that the if φ carries electroweak charge, then EWSB will occur twice in this scenario. The transition to the φ phase may itself be either a true phase transition (first or second order) or crossover. On the other hand, for φ being either a SM gauge singlet or electroweak multiplet, the EWSB to the Higgs phase will be first order due to the presence of the tree-level barrier. Each possibility opens up one or more new avenues for electroweak baryogenesis and gravitational wave generation.

IV. HIGGS BOSON PROPERTIES
In addition to performing direct searches for φ one may discern its interactions with the Higgs boson indirectly through precision measurements of Higgs boson properties. Here, we consider two categories of effects: (a) loopinduced modifications of Higgs properties arising from the cross-quartic interaction in Eq. (8) and (b) tree-level modifications arising from φ-H interactions that break the Z 2 symmetry of the potential in (7,8).

A. Z2-symmetric interactions
Higgs di-photon decays. From the discussion in Section III, it is evident that a 2 plays a decisive role in both generating the presence of a barrier in the finite-T potential and in determining the magnitude of M φ (T = 0). In order to test the validity of the foregoing arguments and more firmly establish the presence of the barrier, would be of interest to have additional information on a 2 and M φ . In this context, a precision measurement of the rate for the Higgs-like scalar to decay to two photons provides an interesting sensitivity to these two parameters when φ carries electroweak charge. Assuming, for example, that one determines M φ by other means, such as direct production (see below), the Γ(h → γγ) provides a probe of a 2 .
For the general Higgs portal interaction of Eq. (8), the relative change in Γ(h → γγ) is given to a good approximation by where N c = 3, q t and q W are top quark and W-boson charges (in units of e); q Sj is the charge of the j-th charged element of the φ mulitplet; F 0,1/2,1 (τ S ) are well-known loop functions [23] of the parameter and where we have included only the top quark and W-boson contributions to Γ(h → γγ) SM . The invariant amplitude for the SM and φ contributions carry opposite (same) signs for positive (negative) a 2 . How large might one expect the magnitude of ∆Γ(h → γγ)/Γ(h → γγ) SM to be? In Fig. 2, we give ∆Γ(h → γγ)/Γ(h → γγ) SM as a function of M φ for representative values of a 2 . Two important features emerge. First, the presence of a barrier driven by the cross-quartic Higgs portal interaction will reduce the di-photon decay rate relative to its SM value. Second, we observe that ∆Γ/Γ ∼ O(0.01) for a 2 ∼ O(1) and for M φ near the upper end of the EWPT-viable mass range for M φ . For lighter masses (consistent with the LEP bounds), the effect can be on the order of ten percent.

B. Z2-breaking interactions
We now consider the possible inclusion of renormalizable operators that break the Z 2 -symmetry of Eqs. (7,8): where φ may be an SM gauge singlet (real or complex) or a real triplet that transforms as (1, 3, 0). The "h.c."is unnecessary for real φ and the b 3 term vanishes for the real triplet. Note that in the Z 2 -symmetric limit, the neutral component of φ may acquire a vacuum expectation value, thereby spontaneously breaking fo the Z 2 symmetry. For gauge singlets, this situation results in the existence of cosmic domain walls, which can be problematic [24][25][26]. For the real triplet, the constraints from the electroweak ρ parameter [27] imply that φ 0 /v < ∼ 0.01. For both the singlet and real triplet, the presence of a non-vanishing a 1 implies that the vev of φ cannot vanish. Given the ρ parameter constraints, one immediately concludes that a 1 is at most of order a few GeV for the real triplet. For the gauge singlet case, no such constraints exist. In what follows, then, we will consider only scenarios with explicit Z 2 breaking and consider the consequences for collider phenomenology.
Mixing angle. When φ 0 ≡ x 0 = 0, the interactions proportional to a 1 and a 2 will lead to mixing between the neutral components of H and φ, leading to two mass eigenstates h 1,2 : with the mixing angle given by Note that the contribution from a 2 depends on its product with x 0 . The corresponding impact on the mixing angle can be vanishingly small for sufficiently small x 0 , even for a 2 ∼ O(1). In contrast, the impact of the dimensionful parameter a 1 carries no such suppression, though in the limit of small x 0 one must also have a small a 1 due to the conditions for minimizing the scalar potential. As discussed above, at T > 0, the cross-quartic operator proportional to a 2 can induce a barrier between the origin and the Higgs broken phase vev through thermal loops or between a high-temperature φ 0 vacuum and the Higgs vev as a tree-level vev. Here, we concentrate on the role played by non-vanishing a 1 . When a 1 < 0, Its presence also implies the existence of a tree-level barrier between the origin in field space and the vaccum located at . A direct transition to this vacuum from the origin at T > 0 -as illustrated in Fig. 1(c)will be first order owing to this tree-level barrier.
At this stage, we possess no quantitative guidance for the values of a 1 , a 2 x 0 , and sin 2θ other than from the electroweak ρ parameter in the case of the real triplet. An additional consideration, however, may be drawn from the requirements for successful electroweak baryogenesis (EWBG) and/or the generation of observable gravitational waves (GW). Both rely on a first order EWPT that is sufficiently "strong". The first order EWPT proceeds via bubble nucleation. In EWBG, the baryon asymmetry is generated when CP-violating interactions at the bubble walls induce a non-vanishing density of left-handed fermions in the unbroken phase (bubble exterior). The latter, in turn, biases the rapid electroweak sphaleron (EWS) processes into generating a net B+L asymmetry that diffuses inside the expanding bubbles. Preservation of this asymmetry in the Higgs phase implies that the sphaleron processes inside the bubbles must be sufficiently quenched. The rate is given by (see Ref. [18] for a pedagogical discussion and references) where E sph is the sphaleron energy and A(T ) is an in principle calculable prefactor. In the high-T effective theory, one has at leading order wherev(T ) is the Higgs vev in the LO high-temperature theory 2 . The relevant temperature in this case is the bubble nucleation temperature, T N , which lies below T EW (typically just below). As a rough estimate, the requirement that the initial bayon (B+L) asymmetry be preserved implies that Relatingv(T EW ) to the parameters of the scalar potential then yields, to a good approximation, In the limit of negligible a 2 x 0 , then, one has To estimate the corresponding lower bound on the magnitude of | sin θ| we take m 1 to be the observed Higgs-like boson mass and m 2 to be given conservatively by twice the upper bounds on mass range resulting from our earlier arguments 3 , or m 2 ≈ 700 GeV. We then obtain Eq. (29) sets a scale for precision Higgs studies, although the foregoing arguments are not as air tight as those leading to the upper bound on mass scale. The presence of non-vanishing a 2 x 0 may lead to cancellations between the two terms in the numerator of Eq. (23), leading to smaller values of | sin θ|, and the value of T EW can be a factor of a few smaller than given in Eq. (5) due to additional contributions from the new scalars. Indeed, explicit studies [28] indicate that the individual terms in the numerator of Eq. (23) can be larger in magnitude than implied by Eq. (27), while leading to a small mixing angle. Nevertheless, these studies also indicate that -assuming a flat prior for the choice of scalar potential parameters -the typical magnitude of | sin θ| lies well above 0.01 for the vast majority of cases. Thus, it appears that the arguments leading to Eq. (29) do yield a robust guide to the scale of precision needed to see the impact of a strong first order EWPT associated with Z 2 -breaking.
Higgs boson self-coupling. The Higgs boson self-coupling is a key parameter in the dynamics of EWSB. The presence of a non-vanishing sin θ would imply a change in the strength of the Higgs boson triple self coupling [13] λ hhh → λ 111 = λ cos θ 3 + 1 4v (a 1 + 2a 2 x 0 ) cos 2 θ sin θ + · · · where the + · · · indicate higher order terms in sin θ that are negligible for purposes of our discussion. The magnitude of this change, relative to the SM prediction λ hhh = λ is given by Consider now the small x 0 regime in which the 2a 2 x 0 term is negligible. In this case, one has where we have used the inequalities in Eqs. (29) and (27). The bound in Eq. (32) is again not airtight but consistent with the result of explicit model studies. As in the case of the mixing angle, these results indicate that considerably larger magnitudes for the shift in the triple self-coupling are favored.
In both this and the previous section, we observe that the quantities a 2 and x 0 remain the least constrained by EWPT considerations, at least at this simple level of perturbative analysis. For the Z 2 -symmetric case, the mass of the new scalar is fixed by T EW and a 2 (and to a lesser extent b 4 ) once we require the presence of a first order EWPT. In the presence of explicit Z 2 -breaking, the sign and magnitude of a 2 will determine whether the expected lower bounds | sin θ| and |∆λ/λ| are given by Eqs. (29) and (32), respectively, or whether cancellations imply their circumvention. At present, we have no generic probe of x 0 as an independent parameter, though detailed study of φ decays can provide indirect information.
A first order EWPT in the Z 2 symmetric case requires a 2 > 0, implying a relative decrease in Higgs diphoton decay rate. In the presence explicit of Z 2 -breaking, a positive value for a 2 -implying a decrease in the di-photon decay rate -would allow for cancellations in the quantity a 1 + 2a 2 x 0 that governs the mixing angle and triple self-coupling, thereby allowing for one to circumvent the bounds in Eqs. (29) and (32). On the other hand, a negative value for this parameter -implying an increase in the diphoton decay rate -would preclude the possibility of such cancellations. Here again, an ultra-precise determination of the Higgs di-photon decay rate would determine the sign of a 2 and thereby potentially help solidify our expectations for the magnitude of sin θ.

V. COLLIDER PHENOMENOLOGY: THE LHC AND BEYOND
The foregoing discussion provides concrete, benchmark mass and precision targets for present and prospective future colliders. One may now ask: What capabilities would be required to reach these benchmarks? Are these capabilities within the realm of the LHC or next generation colliders? In what follows, I provide simple estimates for the mass and precision reach of prospective future colliders as they bear on these benchmarks. One should bear in mind that these estimates are intended to be indicative of what may be possible in the future and that they do not constitute definitive studies. The latter, which require detailed simulations of signal and background events, detector capabilities, etc. go beyond the scope of the present work.
I first consider the mass reach. If the new scalars are charged under SU(3) C , then present LHC exclusion limits on various observables implies severe constraints for masses below one TeV (for a discussion, see, e.g., Ref. [30]). Consequently, I will focus on φ being an SU(3) C singlet.
Electroweak pair production.. In the case of electroweak multiplets, scalars may be pair produced through electroweak Drell-Yan processes, such as e + e − → φ + φ − or pp → φ + φ 0 X. In either case, the leading order (LO) partonic cross section for the process f 1f2 → V * → φ 1 φ 2 mediated by a virtual gauge boson V = γ, Z, or W ± with mass M V iŝ Here, where g is the gauge coupling; g V (g A ) is the vector (axial vector) coupling of the parton pair f 1f2 to V ; g φ is the corresponding coupling to the φ 1 φ 2 pair; v = 246 GeV is the Higgs vacuum expectation value; and withŝ being the parton center of mass energy. Here, we have not included the vector boson decay width Γ V , though one could easily do so by replacing the V propagator-squared by the appropriate Breit-Wigner formula. For 2M φ >> M Z as implied by LEP limits, the impact of including Γ V will not be appreciable. We have also normalized the function F V and prefactor G V so that the former is dimensionless and the latter has the dimensions of a cross section. To set the scale, one has for a process mediated by a virtual W boson G W ≈ 980 fb. Focusing first on prospective e + e − colliders, we discuss three options under consideration: the International Linear Collider (ILC) [31]; a circular e + e − collider as proposed for either the Circular Electron-Positron Collider (CEPC) in China [32] or the CERN Future Circular Collider (FCC) in the ee mode [33]; and the Compact Linear Collider proposed for CERN [34,35]. The center of mass energies √ s are set at specific values for these facilities. I take the following: √ s = 500 GeV (ILC); 240 GeV (CEPC/FCC-ee); 340 GeV (FCC-ee); and 1.5 TeV and 3 TeV (CLIC), where the latter give the middle and highest value of the three center of mass energy options under study. It is worth noting that due to the fixed beam energies, the different facilities would have greatest sensitivity to φ pair production for Having scaled the parton center of mass (CM) energy by 2M φ , we observe a universal behavior, with a maximum occurring at √ŝ /2M φ ≈ 1.7 for all values of M φ but with the magnitude of F Z dropping by about an order of magnitude for each representative choice of M φ . Thus, for a given e + e − CM energy E CM , the maximal sensitivity will be for a scalar mass ∼ E CM /3.4. To be concrete, the CLIC 1.5 TeV option would be best suited to M φ ≈ 440 GeV, while a 500 GeV ILC would having maximum sensitivity to a mass roughly 150 GeV. Similarly, the FCC-ee with E CM = 340 GeV would be ideally suited to probing a 100 GeV new scalar. For M φ near the upper end of our conservative EWPT-viable range, the optimal CM energy is roughly 2.4 TeV. The degradation in sensitivity by going to higher energy, such as the CLIC 3 TeV option, is modest. Note, however, that for a given beam energy, the cross section drops quickly with increasing M φ , going to zero as M φ → E CM /2.
With this information in hand, it is straightforward to determine the number of produced φ pairs for a given M φ , E CM , and integrated luminosity. In Table I, we give this information for each prospective collider, choosing M φ in each case to given the maximum cross section. For purposes of illustration, we will assume the scalar multiplet is a real electroweak triplet and that the final state consists of a φ + φ − pair. We take as projected design integrated luminosities as given n the fourth column of Table I. The anticipated numbers of signal events are shown in the final column.
In general, it is evident that even for new scalars at the upper end of the conservative EWPT mass range, the various e + e − colliders will yield 10, 000 or more signal events. Given the clean environment for these colliders, observation of a signal should in principle be feasible. Obtaining concrete projections will require more detailed information about the expected signature, detector resolution, efficiency and other experimental details. For example, in the absence of Z 2 -breaking interactions, the neutral component of φ may be stable. Electroweak radiative corrections will increase the mass of the components of charge Q with resect to the neutral state by M Q − M 0 ≈ Q 2 ∆M , with ∆M = (166 ± 1) MeV [36]. The φ ± will thus decay to the φ 0 plus a soft lepton pair or soft pion that is difficult to detect, yield a disappearing charged track (DCT) [37]. The detectability of the DCT will depend on the φ ± lifetime, detector resolution, and trigger. Assuming these issues are addressed, the upper limit φ ± mass reach will depend on the collider CM energy.
We now turn to the corresponding analysis for pp collisions. In this case, while the beam energy is fixed, the parton CM energy is not. Instead, one must integrate over the parton distribution functions (pdfs), leading to the following expression for the cross section σ(pp → φ 1 φ 2 X): where the sum is over all partons a and b in the colliding protons, √ŝ 0 = 2M φ , and dL ab /dŝ is the parton luminosity   function constructed from the pdfs, suitably evolved to the energy scale of the partonic sub-process. We consider the charged current (CC) process pp → W + * → φ + φ 0 as the factor G W is larger than the corresponding factors for the neutral current pair production . For purposes of comparing different collider options, it is useful to plot dL ab /dŝ for CC processes as a function ofŝ for three different CM energies: 14 TeV, 27 TeV, and 100 TeV, corresponding respectively to the HL-LHC [38], HE-LHC [39], and either the FCC-hh [40] or SppC. Recalling that for a given M φ the optimal parton CM energy is ∼ 3.4M φ , we see that for a 700 GeV particle, a 100 TeV pp collider will have roughly 60 times more signal events than the LHC, assuming the same integrated luminosity. Given the proposed FCC-hh integrated luminosity of 30 ab −1 , the total number of signal events would be 600 times greater than for the HL-LHC. To make this comparison more concrete, I provide in Table II the cross sections and expected number of signal events for representative values of M φ , assuming the design integrated luminosities for the LHC, HE-LHC, and FCC-hh. Note that the results shown are based on LO cross sections, computed independently using the parton luminosity functions obtained with CTEQ pdfs in the package ManeParse [41] and directly using the pdf set cteql6 [42]. The corresponding K-factor for the LHC with √ s = 13 TeV is the same as for slepton pair production, as both the scalars φ and sleptons carry only electroweak quantum numbers 4 . The resulting values are modest: K = 1.18 at NLO [44], with small corrections of order a percent arising from next-to-leading logarithm (NNL) and next-to-next-to-leading logarithmic (NNLL) resummations matched to approximate next-to-next-to-leading order (aNNLO) QCD corrections [45]. To my knowledge, the corresponding computations do not exist for the HL-LHC or a 100 TeV pp collider, so for purposes of comparison among the different collider options I do not apply a K-factor correction for the HL-LHC results.
Singlet-like scalar production. For SM gauge singlets, DY pair production rates will be highly suppressed by four powers of the small singlet-doublet mixing angle. On the other hand, production of one or more singlet-like scalars may occur at appreciable rates via the following mechanisms: (i) Single scalar production through mixing. The production cross sections for production of one singlet-like scalar having mass m 2 [see Eq. (22)] will go as sin 2 θ times the cross section for production of a single purely SM Higgs boson with mass m 2 . For m 2 < 2m 1 , the h 2 decay branching ratios will be identical to those of a SM Higgs boson of the same mass. For heavier m 2 , the decay h 2 → h 1 h 1 is kinematically allowed, and the corresponding branching ratios to the SM Higgs decay final states and the di-Higgs state will depend in detail on the model parameters.
(ii) Pair production through the Higgs portal. In the limit of vanishing mixing angle, the h 1 h 2 h 2 coupling remains non-zero and is proportional to a 2 . Thus, one may consider the process pp → h ( * ) 1 → h 2 h 2 in the absence of mixing (see, e.g., Ref. [46]) and pp → h ( * ) 2 → h 2 h 1 for non-zero sin θ. In the former instance, for m 2 < m 1 /2, the intermediate Higgs-like scalar may be on-shell, leading to "exotic Higgs decay" modes.
In what follows, I consider the simpler case of single scalar production (i) and comment briefly on the other cases below. Our focus here will suffice to illustrate the potential mass reach of the LHC and prospective future colliders.
To that end, we study the associated production mechanism e + e − → Z * → Zh 2 and the gluon-gluon fusion (ggF) production mechanism that gives the largest "heavy Higgs" production cross section in pp collisions.
Associated heavy Higgs production in e + e − annihilation. The LO partonic associated production cross section for a purely SM Higgs boson is given byσ where k is the Higgs boson momentum in the partonic CM frame and s W (c W ) is the sine (cosine) of the weak mixing angle. In the presence of h-φ 0 mixing, the cross section for associated production of the h 1 state will be the same as in Eq. (37) but multiplied by cos 2 θ. For | sin θ| given by the lower bound in Eq. (29), the resulting decrease in the SM associated production cross section will be far too small to be observable in the proposed leptonic Higgs factories.
In principle, a more promising avenue could be direct production of the state h 2 using associated production with a higher-energy lepton collider. In practice, it appears difficult to achieve sufficient statistics with any of the proposed lepton colliders. To illustrate, we give in Table III the cross sections and corresponding expected number of events for different e + e − CM energies and integrated luminosities for representative values of M φ and a | sin φ| = 0.01. Except for the lightest values of M φ at the lower CM energies, the cross section is too small to yield any signal. On the other hand, one may use the values for σ and dtL to determine the minimum | sin θ| that one might probe for a given M φ . In short, a complete probe of a H † Hφ-induced strong first order EWPT using associated production does not appear to be possible with any of the currently envisioned new lepton colliders. However, a significant portion of the relevant parameter space would still be experimentally accessible.
Gluon fusion heavy Higgs production in pp collisions. The cross sections for ggF production for a heavy SM Higgs boson for √ s = 14 TeV have been tabulated by the LHC Higgs Cross Section Working Group. To obtain the corresponding h 2 production cross section, one simply scales the SM cross sections by sin 2 θ. The corresponding cross sections for higher pp CM energies requires use of Eq. (36). Recall that the parton luminosity as a function ofŝ varies with pp CM energy, as previously illustrated for the CC DY process in Fig. 4. To gain a rough idea of the impact of the difference in parton luminosity, we plot in Fig. 5 the ratio of parton luminosities for the ggF process at 14 and 100 TeV. As an illustration, for threshold production of an on-shell h 2 with m 2 = 700 GeV, the parton luminosity at a 100 TeV collider is roughly sixty times larger than at the LHC. The corresponding gain in σ(pp → hX) will be larger due to the integral in Eq. (36). As an illustration, we give in Table IV the production cross sections for representative masses and mixing angles given in Refs. [29,47,48] after rescaling to the benchmark lower value of | sin θ| given in Eq. (29). At the upper end of the target mass range, one would expect at most several hundred signal events at the HL-LHC, implying that discovery would be challenging at best. At the higher energy and design luminosity of a 100 TeV pp collider, on the other hand, one would anticipate several hundred thousand events. In referring to the values in   [29,47,48] are considerably larger than 0.01. The results in these studies were obtained by scanning over the parameters of the potential in Eqs. (7,8,21), and requiring that the first order EWPT completes (e.g., a sufficiently large tunneling rate) and that the baryon number preservation criterion be satisfied. Hence, the benchmarks given in Table IV appear to be quite conservative.    TeV pp sensitivities to φ 0 production via the gluon fusion process, rescaling the cross sections given in Refs. [29,47,48] by the minimum | sin θ| of Eq. (29) for representative choices of M φ .

VI. OTHER CONSIDERATIONS
The foregoing discussion illustrates how dynamics that modify the thermal history of EWSB and lead to a first order EWPT cannot involve new particles that are arbitrarily heavy or interact too feebly with the SM Higgs boson. The possible signatures for collider probes generally lie well within the reach of the LHC and/or prospective future colliders under consideration. The results of detailed studies within specific models are broadly consistent with these simple, more general arguments. In fact, the requirements on mass and precision reach obtained in model realizations are generally more optimistic than those appearing above. Thus, we can be fairly confident in our primary conclusion that T EW sets a concrete, well-defined scale for new dynamics that collider studies may, in principle, probe exhaustively.
That being said, there remain a few other general considerations that one should address on this topic.
• The foregoing arguments rely on the various patterns of symmetry breaking illustrated in Fig. 1, driven by thermal loops involving the new degrees of freedom and/or tree-level barriers in the tree-level scalar potential at the renormalizable level. The presence of higher dimensional operators can play a role analogous to the tree-level barriers discussed above if the associated mass scale is not too heavy with respect to T EW .
• It is conceivable that the new particles associated with a first order EWPT are relatively light compared to T EW . It is natural, then, to ask about the collider reach for both direct and indirect searches.
• The value of T EW itself may change in the presence of new interactions, and one may wonder about the corresponding impact on the mass and precision targets discussed above. In particular, contributions from loops at either T > 0 or T = 0 can lower the transition temperature under certain conditions. These changes in T EW motivate, in part, the choice of a somewhat larger upper bound on the M φ mass range compared to the values ∼ 360 − 375 obtained from the simple arguments given above.
In what follows, I comment briefly on each of these points.

A. Non-renormalizable Interactions
The lowest-dimension non-renormalizable, gauge-invariant operators that contain only Higgs boson fields enter the Lagrangian at d = 6. Following Ref. [49], consider the corresponding Higgs potential of the form where the notationṼ 0 indicates that the leading order scalar potential is distinct from the potential in Eq. (1). In both cases, the potential minimum occurs at H 0 = v/ √ 2 and the square of the Higgs boson mass is m 2 h = 2λv 2 . Writing Eq. (38) in terms of the field h gives For Λ 2 < 3v 2 /λ = 3v 4 /m 2 h , one hasλ < 0. The presence of the negative quartic term corresponds to a barrier between the symmetric and broken phases at T = 0. Given the measured value of m h , one then requires the mass scale Λ to be less than ∼ 840 GeV. The authors of Ref. [49] find that for the EWPT transition to be strongly first order, the upper bound on Λ is reduced by roughly 5% for m h = 125 GeV. These results are consistent with what has been obtained in similar work by the authors of Ref. [50].
The experimental signatures of the d = 6 interaction in the potential would be modifications of the Higgs boson properties. In particular, the authors of Ref. [51] noted that the corresponding loop-induced change in the associated production process e + e − → Zh at √ s = 250 GeV would range from ∼ 1 to ∼ 2.5% for a values of Λ leading to a strong first order EWPT. An effect of this magnitude would be well within reach of the Higgs factories presently under consideration.

B. New Light Particles
For the case of φ being a SM gauge singlet, it was observed in Ref. [13] that a strong first order EWPT may arise even when M φ may be substantially below m h . For m h > 2M φ , the h-φ interaction will then lead to new Higgs boson decay modes. For a potential having a Z 2 symmetry, these decays will be unobservable, as φ is stable and will leave no traces in the detector. In the presence of a broken Z 2 (either explicitly or spontaneously), the φ-h mixing will enable the φ to decay via all kinematically-allowed SM Higgs boson decay channels. If the latter are sufficiently prompt, one may search for these "exotic" Higgs decay modes. According to the initial analysis of Ref. [13], the corresponding exotic decay branching ratios could be signficant.
Two recent studies have analyzed this regime in detail [52,53]. The authors of Ref. [53] showed that there exists a lower bound on the exotic decay branching ratio as a function of M φ for choices of parameters yielding a strong, first order EWPT. This study considered the real singlet extension for two cases: (a) explicit Z 2 -breaking and (b) Z 2 -symmetric. For M φ > ∼ 10 GeV, the a combination of the LHC and prospective future lepton colliders appear to have the sensitivity needed to probe these scenarios. The work of Ref. [52] considered spontaneous Z 2 -breaking 5 and analyzed the reach of both collider studies and gravitational wave probes, indicating that at least some portion of the relevant parameter space would be experimentally accessible.

C. The Electroweak Temperature Revisited
Here, I consider three possible ways in which new interactions may modify T EW from the SM value of roughly 140 GeV: (a) T > 0 loops involving the new particles; (b) T = 0 loops, encoded in the Coleman-Weinberg potential; and (c) non-renormalizable interactions. In each case, it is possible that T EW can be lower than in the SM, leading to a somewhat larger value of the allowed M φ .
For a 2 = 5, T EW will be lowered to ≈ 90 GeV. The corresponding increase in the value of M φ (T = 0) for either the one-step or two-step scenario will be relatively modest. For EWGB, on the other hand, the lower temperature will enable more effective baryon number preservation via suppression of the broken phase sphaleron rate [see Eq. (24)].

T = 0 Loops: Coleman-Weinberg
The authors of Refs. [54,55] showed that for scenarios in which a first order EWPT occurs, the impact of T = 0 loops involving new particles can lower the critical temperature, thereby enhancing the degree of baryon number preservation and the magnitude of the gravitational wave signal. It is less clear whether this effect alone could induce a first order EWPT without the additional contributions to the barrier as discussed above. The change in This expression is analogous to the one given in Eq. (4), which defines T EW in the SM, but differs in several crucial ways: (i) the overall scale is now set by Λ/2 rather than v [the last factor in Eq. (48)]; (ii) the dependence on λ in the numerator is different; and (iii) the contributions of O(v 2 /Λ 2 ) modify the first two factors. From Eq. (48) we find that T 0 increases monotonically from 34 GeV to 108 GeV.

VII. OUTLOOK
As the high energy physics community considers the long-term future of the energy frontier program, it is important to bear in mind any opportunities where one may -at least in principle -anticipate reaching definitive conclusions about the laws of nature. In this discussion, I hope to have convinced the reader that the probing the thermal history of EWSB constitutes one such opportunity. In brief: nature has handed us a scale, the electroweak temperature T EW . Any physics that significantly alters the SM EWSB crossover transition at this temperature cannot be arbitrarily heavy with respect to T EW , nor can it interact too feebly with the SM Higgs boson. The corresponding mass scale of new particles and inferred modifications of SM Higgs boson properties appear to be within the reach of various future high energy colliders currently under consideration. An experimental program that includes one or more of these prospective colliders could conceivably teach us whether the EWSB transition was, for all intents and purposes, a crossover transition, or whether the preconditions existed for generating the matter-antimatter asymmetry in conjunction with EWSB, with possible associated astrophysical imprints in relic gravitational waves.
On the theoretical side, performing state-of-the art computations of the dynamics at non-zero temperature and making a robust correlation with the possible experimental signatures is vital. In this regard, the the work carried out starting over two decades ago on the SM EWSB transition, using the methods of high-temperature effective theory and lattice gauge theory simulations, provides a roadmap for the future (for a clear pedagogical discussions, see, e.g., Ref. [57]). In particular, the use of perturbation theory (PT) to analyze the T > 0 behavior of gauge theory suffers from well-known limitations (see, e.g., Refs. [58][59][60]). While reliance on PT is unavoidable when initially assessing the EWPT implications of a wide range of explicit models, and while one may turn to various strategies to improve the performance of T > 0 PT, results of non-perturbative studies are ultimately needed to gauge the reliability of perturbative studies. Thus, a combination of perturbative analyses of BSM scenarios and nonperturbative computations intended to "benchmark" PT appears warranted. In this respect, recent work involving high-T effective theory and explicit lattice simulations [61][62][63][64][65] is an encouraging sign.