A On-Shell Renormalization of the T=0 Potential

Dynamics of Non-renormalizable Electroweak Symmetry Breaking



Saclay T07/141

Dynamics of Non-renormalizable

[.2cm] Electroweak Symmetry Breaking

Cédric Delaunay, Christophe Grojean, James D. Wells

  CERN, Theory Division, CH 1211, Geneva 23, Switzerland

  Service de Physique Théorique, CEA Saclay, F91191, Gif-sur-Yvette, France

  Michigan Center for Theoretical Physics (MCTP)

Physics Department, University of Michigan, Ann Arbor, MI 48109

We compute the complete one-loop finite temperature effective potential for electroweak symmetry breaking in the Standard Model with a Higgs potential supplemented by higher dimensional operators as generated for instance in composite Higgs and Little Higgs models. We detail the resolution of several issues that arise, such as the cancellation of infrared divergences at higher order and imaginary contributions to the potential. We follow the dynamics of the phase transition, including the nucleation of bubbles and the effects of supercooling. We characterize the region of parameter space consistent with a strong first-order phase transition which may be relevant to electroweak baryogenesis. Finally, we investigate the prospects of present and future gravity wave detectors to see the effects of a strong first-order electroweak phase transition.

1 Introduction

The baryon asymmetry of the universe remains a mystery. Many ideas have been formulated in the literature, yet much uncertainty remains as to how the baryon asymmetry could arise. It is not even clear at what scale the initial asymmetry is produced. The Sakharov conditions for baryogenesis are baryon number violation, and violation, and a departure from equilibrium. The Standard Model (SM) does not exhibit these conditions at nearly the strength required to produce the observed asymmetry given our standard cosmological assumptions, and thus it is expected that we must go beyond the SM in order to explain the asymmetry.

The last Sakharov condition, departure from equilibrium, implies the necessity of a strong first-order phase transition. Since we know that the electroweak symmetry must be broken it is tempting to assume that the corresponding phase transition can satisfy this condition. As noted above, the SM is inadequate, but how far beyond the SM must one go to find the necessary out of equilibrium dynamics? This question has been addressed by a number of authors (e.g., see [1] for studies of the dynamics of the electroweak phase transition in various recent models). In Ref. [2] it was shown that if the Higgs potential is augmented merely by a operator, it can generate a strong first-order electroweak phase transition. As one can intuit, the scale suppressing this non-renormalizable operator must be in the neighborhood of the electroweak scale in order to generate a substantive effect on the phase transition dynamics. A tree-level analysis of this theory was conducted in [2], with some further refinements in [3], and it was concluded that a strong first order phase transition is possible even with a Higgs boson as massive as 200 GeV. Of course, for the presence of this operator to be compatible with electroweak (EW) precision data, a higher scale should suppress other dimension six operators, in particular those leading to oblique corrections. The analysis of [4] shows that the low energy effective theories of strongly interacting models, where a light composite Higgs emerges as a pseudo-Goldstone boson, have precisely this structure and single out as one of the dominant dimension six operators 1, being suppressed by the decay constant of the strong sector, parametrically lighter than the cutoff scale of the model. For a fixed value of the strong decay constant, the compatibility with precision EW data is ensured by pushing the masses of vector resonances above 2.5 TeV [4]. In that case a decay constant as low as 300 GeV would be compatible with precision measurements. Our analysis should also apply to study the dynamics of electroweak symmetry breaking in Little Higgs models, and to the more general cases where the operator is generated by integrating out a heavy massive scalar field. However, some extra fine-tunings might be needed in that case to evade EW precision data.

In this publication we extend the results of [2, 3] in several ways. First, we re-analyze the theory using the full finite temperature effective (nonrenormalizable) Higgs potential at one-loop. Second, we study the nucleation of broken phase bubbles and consider the effects of supercooling on the electroweak phase transition (EWPT) within this more complete analysis. This is an important dynamical consideration of the phase transition that can in principle have dramatic consequences to when (and if) the phase transition happens. Finally, we investigate whether or not the gravitational waves emitted at the nucleation time can be detected by present and future interferometry experiments, which would provide another way to study the origin of EW symmetry breaking and another way to test the composite nature of the Higgs. We consider each of these points in the following three sections, and then make some concluding remarks.

2 One-Loop Finite Temperature Effective Potential

Once non-renormalizable interactions are allowed in the theory, as in our case, complete renormalization requires that the infinite set of higher-order operators be considered. However, one is able to truncate the list of needed operators in a perturbative expansion of the inverse cutoff scale. To study the effect of new physics on the Higgs potential in this effective field theory context, it is sufficient to work at the order where is the cutoff scale suppressing the effective operators. Higher dimensional operators will be sufficiently irrelevant to our problem and can be ignored.

Our analysis is focussed on operators that affect the Higgs self-interactions. These effective interactions parametrize the new physics responsible for EW symmetry breaking that become fully dynamical at about the scale . Thus they can be used to generically constrain beyond-the-SM physics affecting the Higgs sector. Though EW precision measurements put severe constraints on the set of operators affecting the weak bosons’ polarization tensors, the effective Higgs self-interactions are almost completely free parameters since the Higgs sector has not yet been probed directly by experiment. Thus the scale suppressing the operator we will focus on can be significantly lower than the cutoff scale of the (strongly coupled) model. This is in particular the case of composite Higgs models when the Higgs emerges from a strongly-interacting sector as a light pseudo-Goldstone boson [4]. The scale suppressing the operator is then , the decay constant of the strong sector, a quantity 4 smaller that the cutoff scale.

We start with the following classical effective potential for the SM Higgs [5]:


where which develops a vacuum expectation value (VEV) equal to . is identified at tree level with the decay constant of the strong sector – the details of this identification at one-loop are described later. We choose a vacuum configuration where only the real part of the neutral component has a constant background value: . The physical Higgs boson is , and we use the traditional background field method [6] to evaluate the quantum potential for at one-loop. We focus on the main relevant contributions coming from the gauge bosons, the top quark, and the Higgs and Goldstone scalars.

As we briefly review in Appendix C, the quantum potential for the background value up to one-loop order at finite temperature in the Landau gauge (where ghosts decouple) is




where is the euclidean loop 4-momentum, are the Matsubara frequencies in the imaginary time formalism, where for bosons (periodic on the euclidean time circle) and for fermions (anti-periodic on the euclidean time circle). The numbers of degrees of freedom for the relevant fields are . We include the fermion-loop minus sign in the definition of .

Note that in the Landau gauge one must count all three degrees of freedom of each massive vector boson and the one degree of freedom of each Goldstone scalar. This may be qualitatively understood be recalling that the Goldstone fields are independent quantum fluctuations away from the zero-temperature minimum. We present a quantitative argument showing this is not double counting in Appendix C.

We obtain the background-dependent masses appearing in (4) by expanding the theory about the background value and reading off the quadratic terms for the various quantum fluctuations. In our dimension-six model the masses are


where , and are the , and top Yukawa couplings respectively. At the zero-temperature minimum one recovers and . Note that the expressions for the masses of the weak bosons (from the Higgs kinetic term) and the top quark (from the Yukawa coupling) are unchanged compared to the SM, and (7) are written to confirm our conventions.

The one-loop correction (4) splits into a zero-temperature part and a -dependent part [7, 8] which vanishes as :




is precisely the ordinary zero temperature effective potential, as it must be to be consistent since as . The part, being UV-divergent, will be considered first in order to properly determine the renormalized parameters of the quantum theory. The finite temperature corrections will be treated afterwards.

2.1 Zero Temperature Corrections

At zero temperature the correction (4) reduces to the first term of (8),


which has been regularized in dimensions, () for gauge bosons (scalars and fermions) and .

We work in the scheme to renormalize and evaluate our potential (see the Appendix A for an alternative, but ultimately equivalent, on-shell scheme approach). The full one-loop effective potential is


where the parameters of this potential () are bare parameters, but an implicit will cancel their infinite pieces, leaving the finite pieces as the renormalized parameters.

To determine the parameters of the lagrangian in terms of physical quantities, we must impose renormalization conditions at some chosen scale . The renormalization conditions are


The left side of each equation is the theory computation, and depends on the parameters of the theory (). The right side of each equation is a measurement ( and ) or related to a measurement ( is a requirement that the potential is at a minimum which recovers the correct boson mass). The VEV depends on the choice of scale as well. We define to be equal to the VEV of the Higgs field in the Landau gauge at such that the mass is recovered. Performing our computations with the latest electroweak precision measurements [9], we find to a good approximation for a Higgs mass in our range of interest (). This Higgs VEV is close to the value in [10].

We can invert these equations to obtain the theory parameters as a function of measurements:


Note, the parameters have scale dependence, and we have defined , etc.

Up to now we have glossed over some important subtleties. The physical Higgs mass must be defined at , whereas the one-loop effective potential is constructed for . To take account of this, and retain the label for the physical Higgs boson mass, we need to rewrite the renormalization condition as


where is the two-point function of the Higgs boson (numerically, we used the LoopTools software [11] to evaluate this two-point function). This approach has the added benefit that the IR singularity in as the Goldstone mass goes to zero is canceled by the IR singularity in . We discuss these IR singularity issues in more detail in the Appendix B.

The physical parameter is not a unique choice for how to parametrize the measured tri-Higgs coupling, and we wish to rewrite it in a more convenient manner. First, like the Higgs mass, the Higgs tri-scalar coupling has IR divergences at when the Goldstone bosons become massless. These IR divergences are also not dangerous because they are matched by the IR divergences of , and cancel in measured cross-sections. Thus, it is convenient to separate out this IR divergence when parametrizing the tri-Higgs coupling observable: , where contains IR sensitive Goldstone terms2. Furthermore, since the tri-Higgs coupling in the SM is fixed once the Higgs mass is known, we would like our convention to reflect this manifestly in the decoupling limit of ,


For finite values of , the deviations of from can be defined by convention to be


This convention (i.e., the factor of 6) ensures that can be identified directly as the decay constant of the strong sector, , at tree level. Putting these elements together, we can now rewrite the third renormalization condition as


We emphasize that eq. (22) is merely a reparametrization of the tri-Higgs physical observable in terms of the decay constant, , rather than for the benefits described above, and that is a computable function of .

Following the prescription provided above, all the parameters of our Higgs potential () can now be written in terms of physical observables (). Thus, we are now able to analyze the potential using physical observables as inputs.

2.2 Finite temperature corrections

From the splitting (8) of the full one-loop effective potential into a part and a part, we get that the latter finite temperature component is:


where the upper (lower) sign stands for bosons (fermions). In the high-temperature regime (), the function expansions are


with and . Note that in [2] only the first terms in (24) and (25) were retained, which leads to the following approximate thermal one-loop correction:


with .

The dominant contributions gathered in (26) are simply a (positive) thermal mass which (meta)-stabilizes the origin of the potential at high temperature. This approximation was sufficient in [2], and further refined in [3], to demonstrate the possibility of a strong first order PT within an effective extension of the SM. Fig. (1) shows the discrepancy between the complete thermal correction and the high-temperature expansion around the critical temperature, illustrating the worthwhileness of using the integrals of (23) for the more detailed analysis.

Figure 1: Different potentials close to the critical temperature for  GeV and  GeV ( is the decay constant of the strong sector the Higgs emerges from). The dashed curve is the potential of [2] which includes only the thermal mass term of the Higgs, while the solid and dotted ones represent the full one-loop potential with (solid) and without (dotted) the ring diagram contributions. In blue, we have also plotted the imaginary part of the full one-loop potential with the ring contributions (solid blue) as well as the imaginary part of the ring contributions alone (dashed blue). This illustrates the cancelation of the large imaginary parts between the ring and the one-loop contributions, while there still exists an additional and smaller imaginary part for some values of due to a negative quartic coupling (see the discussion in Section 2.3.2 for details). An imaginary part of the potential can be interpreted as a decay rate of some quantum states of the scalar fields to some others but the imaginary part of the full potential is always tiny compared to the real part around the transition temperature and the system is stable enough throughout the entire time of the transition.

Breakdown of perturbation theory and ring diagrams

In thermal quantum field theory, the traditional perturbative expansion in terms of small coupling constants breaks down due to IR-divergences (inherent in massless models) generated by long-range fluctuations appearing as soon as one moves to finite temperature [12]. For instance, taking massless theory at finite temperature, one can show that the self-energy, which goes like at first order, receives a subleading correction and not as one would expect [13]. For our case, in the high-temperature expansion, or equivalently small mass expansion, of the thermal bosonic corrections (24), we also see a sign of this perturbation theory breakdown through the emergence of a monomial term of order . The main consequence is that, as it stands, we cannot trust the completeness of the one-loop result (23) because there are some higher-loop corrections of the same order [7], as if the effect of temperature is to “dilute” the one-loop correction to some multi-loop orders in the IR. Furthermore the leading part of these multi-loop corrections is all contained in the so-called ring (or daisy) diagrams shown in Fig. (2). They are -loop diagrams where of them are “ring attached” to a main one. Since this “loop-dilution” is a finite temperature effect, the ring diagrams only need to be resummed in the IR-limit of vanishing momenta running in their petals [7]. It is also well-known that they can be taken into account by using propagators resummed in the IR [14]. By solving a Dyson-like equation, this turns out to simply shift the bosonic masses by a -dependent constant as , where is the self-energy of the (bosonic) field in the IR limit, , known as a Debye mass ( is labeled as in [14]).

The higher-loop ring diagrams are needed due to IR divergences (i.e., ). On the other hand, the one-loop result is trustworthy for massive (i.e., ) particles, because the long-range fluctuations arising at finite temperature will never hit an IR mass-pole in such cases. Hence the ring diagrams will only contribute significantly at high-temperature () where the particles can be approximated as nearly massless. Also, this allows us to understand why only the bosonic degrees of freedom feel the breakdown of the perturbative expansion3. The reason is that only bosonic fields have a vanishing Matsubara frequency, recalling that equals for bosons and for fermions. Only this particular (zero-)mode will behave as a massless degree of freedom and generate IR-divergences at high-temperature, while the other (non zero-)modes act as a mass of order and thus lead to negligible contributions. Therefore the fermionic propagators need not be resummed, because fermions do not have pole-mass in the IR.

Figure 2: Some generic examples of ring diagrams where each solid line may represent either a scalar, a fermion or a gauge field. The small loops correspond to thermal loops in the IR limit. They are all separately IR divergent, but their sum is IR finite.

Applying the techniques of [14] to our theory, we compute the finite temperature mass shifts (Debye masses) that are needed in the ring diagram resummation:


Note that these ’s are computed in the high-temperature limit of the unbroken phase which is justified by the ring diagrams being irrelevant for as we have discussed. At high temperature the photon and are not mass eigenstates, but one can treat them as mass eigenstates in this computation with the above-given Debye masses and obtain the correct resummed potential.

Incorporating the ring corrections

The traditional way the ring diagrams are implemented in the literature consists in shifting all the Matsubara modes for the bosonic fields. This is the so-called self-consistent method [15] where the potential (4) is replaced by


The thermal shift of the gauge masses only for the longitudinal polarizations is understood, and is simply zero. However, when applying this approach the UV divergent part becomes -dependent through the and requires -dependent counter-terms to be made finite. Indeed after doing to (31) the same splitting procedure we did to get (8), and after dimensionally regularizing the UV-divergent part, we get the following result:


where the factor depends on . This standard technique clashes with physical intuition since it would mean that the UV behavior of the theory depends on the IR dynamics. Although this mixing is not introducing any calculational errors to our working approximation, one can avoid it by simply shifting only the Matsubara modes which carry the leading contribution from the ring diagrams relevant at one-loop order.

As argued above, the dilution of the one-loop correction happens only for massless modes. Hence all the corrections we seek within the ring diagrams are gathered when resumming only the zero-mode of the propagator in the IR. Doing so, (4) is to be replaced by


where the prime means that the zero modes are excluded from the sum. We can easily extract the ring part from the last expression and we find


where an irrelevant (infinite) constant has been ignored in the second line, and . Notice that includes a monomial of order which proves a posteriori the existence of a perturbation theory breakdown in evaluating the Higgs potential. Furthermore, these extra corrections modify the cubic term in , which partly controls4 the strength of the first order phase transition. Thus, the addition of these terms is critical for our analysis of the electroweak phase transition.

In summary, the full -dependent renormalized effective potential at one-loop is


where definitions of all terms are given above. This is the potential we analyze for the remainder of the article.

2.3 Reality of the quantum potential

As the scalar masses become negative, the various contributions we obtained for the quantum potential develop some imaginary parts which we discuss below for both the and cases.

Imaginary part at

In the zero-temperature limit, the logarithm of (36) leads to the following scheme-independent imaginary part5


where is the Heaviside function which equals 1 when the field is tachyonic, and zero otherwise. The Higgs boson can obtain a negative mass squared for some values of its VEV, originating from the fact that the classical potential is not convex everywhere. Indeed, depending on the cutoff value, either the origin is unstable () or a potential barrier separates two local minima (), both of which lead to concave regions of the effective potential as a function of the VEV. A similar analysis shows that the Goldstone boson can become tachyonic for some values of the VEV as well, leading to another contribution to the imaginary part of the effective potential. However, we shall see shortly that the imaginary part (37) exactly cancels out with another contribution coming from the finite temperature corrections for the temperature range we are interested in for the phase transition.

Imaginary part at

At finite temperature both the integrals of (23) and the ring contributions (35) are spoiled by imaginary parts when scalar fields are tachyons. In the high-temperature limit, the imaginary part of (23) is (see (24)):


The first term cancels the imaginary part from the logarithm of the potential correction (37), while the second is only compensated when the ring diagrams are added, since their imaginary part is given by


as long as the temperature satisfies for all . Although somewhat more complicated algebraically to show (see Appendix C.4 for details), this cancellation occurs also for smaller temperatures of order .

Nevertheless and despite this cancelation, the potential is not everywhere real because for some values of and , and the second term of the ring correction (35) becomes imaginary. In the SM this term does not lead to an imaginary part once the temperature (meta)stabilizes the origin since the SM scalars could only become tachyonic for a negative quadratic coupling in the Higgs potential. Thus, the SM potential is real as long as the origin is (meta)stable. On the other hand, with the additional piece in the potential, the scalar masses can be negative also through a negative quartic coupling, allowing this additional imaginary part to the potential at temperature around the critical temperature.

An imaginary part of the potential can be interpreted as a decay rate of some quantum states of the scalar fields to some others [16]. Thus, one can rely on the real part of the potential as long as its imaginary part remains small enough to consider the field stable during the phase transition, in which case it can be discarded. We checked that the imaginary part of the one-loop potential is always tiny compared to the real part around the transition temperature, thanks to the previously demonstrated cancelations of large imaginary pieces. Thus, we conclude that the system is stable enough throughout the entire time of the transition, and that its dynamics is driven by the real part of the one-loop potential we computed.

3 Dynamics of the Electroweak Phase Transition

Now that we have the formalism developed for our analysis of the finite temperature Higgs potential at one loop, we are in the position to study the dynamics of the phase transition. One of our first considerations must be the analysis of when (and if) the phase transition actually occurs. This is not simply a matter of determining the temperature at which the symmetry breaking minimum becomes the global minimum. An analysis of the energetics of bubble formation must be undertaken for a more complete picture. The nucleated bubbles can then undergo collisions and the surrounding plasma experience turbulence, which generate gravity waves that could possibly be detected in experiments. We discuss these issues in this section.

Throughout this section, we report our numerical results of various relevant quantities as contour plots that scan the allowed region of the parameter space . We recall that is the physical Higgs mass while is the decay constant of the strong sector (or more generally the energy scale suppressing the operator) physically defined through the triple Higgs self-interaction as defined in the previous sections, and we work in the scheme for . The bounds delinating the region of first-order phase transition are both numerically computed using the complete one-loop potential at finite temperature. The lower one is set by requiring that EW symmetry is broken at and restores at high temperature, while above the upper bound the Higgs vacuum is likely to undergo a second-order phase transition or a smooth crossover. In general, determining the latter is not an easy task as it requires a non-perturbative analysis of the effective potential when the transition is not strongly first-order [17]. Indeed, the phase transition always appears first-order at the perturbative level, even though very weakly. Moreover, as increases one tends to recover the SM potential, which leads non-perturbatively to a continuous crossover, instead of a weak first-order transition at one-loop, for GeV [18]. We estimated the upper bound by considering that as soon as the phase transition is as weak as in the SM for GeV, it is likely to be a crossover.

3.1 The onset of nucleation and EW baryogenesis

The effective potential ensures the presence of a potential barrier at finite temperature which is a necessary ingredient to have a first-order phase transition. It proceeds by spontaneous nucleation of non-vanishing VEV bubbles into a surrounding symmetric metastable vacuum. As soon as the universe cools down to a critical temperature the symmetry-breaking vacuum becomes energetically favorable and then thermal fluctuations allow the bubbles to form. However, the temperature of the transition is not necessarily close to . Once created, a bubble needs to consume a part of the latent heat liberated in order to maintain its interface with the symmetric phase surrounding it. It turns out that for just below it is often the case that the bubbles are too small and surface tension makes them collapse and disappear. Hence the phase transition effectively starts at a smaller temperature when enough free energy is available to permit the nucleation of sufficiently large bubbles that can grow and convert the entire universe into the broken phase. This supercooling phenomenon can substantially delay the phase transition and thus modify the spectrum of gravity waves significantly, as we shall discuss shortly (important supercooling effects were also observed in some of the analyses of Ref. [1])

When does the nucleation start?

Although the probability to tunnel via the excitation of instantons is very tiny, about , the decay of the false vacuum can nonetheless proceed through thermal fluctuations which help to overcome the potential barrier. The rate per unit of space-time for this process is given in the semi-classical WKB approximation by where is the euclidean action for the Higgs VEV evaluated on the so-called bounce solution of the euclidean equation of motion [19]. For temperatures much higher than their inverse radius, the bubbles overlap in euclidean time and feel the IR breaking of Lorentz symmetry [20, 21], in which case the bounce solution is -symmetric and is the solution of


subject to the boundary conditions


The bounce solution physically represents the Higgs VEV profile of a static unstable (either expanding or shrinking) bubble, and measures the distance from the bubble center. For such a static solution of the equation of motion, the action factorizes as , with


Moreover for small temperatures of the order of the bubble size, we replace the bounce for the -symmetric solution which minimizes the action when the breaking of Lorentz symmetry is not significant. Finally we use the traditional overshooting/undershooting method to numerically solve the equation of motion.

There is a supercooling effect that can delay the onset of the first order phase transition to temperatures much smaller than . A first order phase transition can only proceed in the presence of a potential barrier separating the two vacua and the nucleation could potentially start at a temperature far below that of . This is especially likely in the case where the barrier persists down to . Since the amount of supercooling is controlled by the size of the nucleated bubble, one needs to take into account that the phase transition proceeds in an expanding universe. One can thus consider that the nucleation starts at the time when the probability of creating at least one bubble per horizon volume is of order one. This condition guarantees the percolation of bubbles in the early universe and translates into the following criterion for determining the nucleation temperature:


where is the reduced Planck mass.

The contours of constant nucleation temperature are reported in the left panel of Fig. 3. We point out that there exists a region (painted red in Fig. 3) with low and such that the criterion eq. (43) is not satisfied, meaning that the expansion of the universe does not permit the bubbles to percolate. Thus the nucleation never starts and the universe remains trapped in a symmetric vacuum. In addition, the right panel of Fig. 3 helps one to realize further the numerical significance of the supercooling effect by plotting the deviation of the nucleation temperature from . We see that, for large values of , the deviation is not significant since the potential barrier disappears at a temperature not much less than the critical one. On the another hand, as soon as one lowers , the barrier persists to lower and lower temperatures, making the supercooling delay of the phase transition important. Thus the knowledge of the nucleation temperature becomes necessary to clearly understand the dynamics of the phase transition in this region.

Figure 3: The left panel of this figure shows contours of the nucleation temperature in the allowed region for an EW symmetry-breaking first order phase transition ( is the decay constant of the strong sector the Higgs emerges from, and is the physical Higgs mass). Below the red lower bound the EW symmetry remains intact in the vacuum while above the blue upper one the phase transition is second order or not even occurs. Within the red band, the universe is trapped in a metastable vacuum since no expanding bubble is nucleated and the transition never proceeds. The contours are from left to right for . The right panel of this figure shows contours of the relative deviation of the nucleation temperature from the critical one: . This measures the degree to which the phase transition is delayed by the overcooling effect. The contours are, from above, for .

Saving the baryon-asymmetry from wash-out

Understanding the dynamics of the phase transition is a worthy endeavor on its own; however, one of the key reasons for understanding the nature of the EW phase transition is to determine if a baryon asymmetry can be produced and survive the process. Calculations in the previous sections enable us to refine some of the results of [2], where the possibility of a strong first order phase transition was first demonstrated.

So far we have computed the crucial ratio at the nucleation temperature in the cases where only the thermal masses are included and where the complete one-loop potential is used. This allows us to compare the effect on the wash-out criterion of the supercooling of the phase transition and the usefulness of the one-loop potential. The contour plots of Fig. (4) show the common fact that the lower the value of , the stronger the phase transition for a fixed Higgs mass. The qualitative result of considering the temperature delay from to is that for a given point in the parameters plane, the phase transition is generically stronger at . Indeed not only is the nucleation temperature potentially much smaller than , but also the value of the Higgs VEV grows as the universe cools down.

Another important result for the baryon-asymmetry of the universe, is that it can be saved from the wash-out through sphaleron processes, namely , for a not-so-small value of . Indeed, in order to allow baryogenesis during the EWPT in the approximation of [2] some fine-tuning might be required in some approaches without any particular dynamics to make the suppression scale of the dimension six operator in the Higgs sector relatively smaller than the TeV scale required in the gauge sector to pass EW precision measurements. But the full one-loop potential tells us that for values of the Higgs mass above the current experimental bound can be larger – as large as TeV – and the baryon-asymmetry can still freeze out.

Figure 4: Plot of the ratio characterizing the strength of the phase transition using the thermal mass approximation of [2] (left) and the complete one-loop potential (right). The contours are for from top to bottom. is the decay constant of the strong sector the Higgs emerges from, and is the physical Higgs mass.

3.2 Gravitational Waves

As a bubble expands a part of the latent heat released accelerates the bubble wall and introduces turbulent motions in the hot plasma. After bubbles collide, spherical symmetry is broken. This enables gravitational radiation to be emitted. The turbulence of the plasma after bubble collisions is another important source of gravitational radiation (see [23] for an introduction to the physics of gravity waves). In the following, we characterize the spectrum of gravitational radiation that one can expect from the first order phase transition we have detailed in this article. We compare these results with the sensitivities of current gravity wave detectors, and of proposed gravity wave detectors of the future.

Characterizing the spectrum

Previous studies [24, 25, 26] of the gravity wave spectrum culminate in showing that it can be fully characterized by the knowledge of only two parameters derived ultimately from the effective potential6. The first one is the rate of time-variation of the nucleation rate, named . Its inverse gives the duration of the phase transition, therefore defining the characteristic frequency of the spectrum. The second important parameter, , measures the ratio of the latent heat to the energy density of the dominant kind, which is radiation at the epoch considered: . They are both numerically computed from the effective action at the nucleation temperature as follows. The time-dependence of the rate of nucleation is mainly concentrated in the effective action and is defined by . Using the adiabaticity of the universe one obtain the following dimensionless parameter:


where is the expansion rate when nucleation starts. The latent energy is the sum of the amount of energy seperating the metastable vacuum to the stable one and the entropy variation between these two phases. Hence one has:


The left and right panels of Fig. 5 show contours of constant and , respectively, at the time of nucleation.

Figure 5: The panel on the left contains contours of the latent heat from top to bottom. The panel on the right draws contours of the parameter, , measuring the duration of the phase transition. From above one has . is the decay constant of the strong sector the Higgs emerges from, and is the physical Higgs mass.

Observability at interferometry experiments

Future interferometry experiments could offer us a way to observe the EWPT. A detailed analysis of the potential to directly see gravitational waves from the first-order phase transition can be compared with the sensitivity expected from the correlated third generation LIGO detector on earth and the LISA and BBO detectors in space. A general analysis that we utilize has been presented in [22], where both bubble collisions and turbulent motions were considered. Qualitatively, gravity-wave detectors will give us a better chance to observe the phase transition today if the latent heat energy released is large and the emission lasts a long time. This can be understood easily by recalling that the power spectrum is given by the square of the quadrupole moment of the source which in turns scales as the kinetic energy over the time of emission [29]. In other words, typically has to be and as small as to get a sufficiently high energy density .

Relying on our effective (nonrenormalizable) potential approach, we find that generically the dynamics of the first order EWPT beyond the SM generate too weak gravity waves to observe except for a tiny region of the parameter space. Namely, by looking closely at Figs. 5 one can see that for a Higgs mass slightly above the LEP2 bound, GeV, and a relatively low scale, GeV, we get at best and . The corresponding nucleation temperature in this region is about GeV, according to Fig 3. For such a temperature scale, only LISA and BBO will be sensitive to the emitted spectrum of gravity waves, according to the results presented in Figs. 3 and 4 of [22]. Its detectability is probably beyond the capability of LISA. This result is in qualitative agreement with the results of [30]. Indeed LISA requires at least values of for in order to see the characteristic peak from turbulence while the collision peak starts to be probed for . On the other hand, BBO should be able to observe both peaks if is around (keeping ).

Figure 6: Example of gravity wave spectrum produced during the EW phase transition both by turbulence (left peak) and collision effects (right peak slightly emerging from the tail of the turbulence spectrum). This plot is for  GeV and  GeV where , and  GeV. Note that suitable values of to get a strong signal always imply a small nucleation temperature ( GeV) due to important overcooling effects that drag the peak below the lower bound of the space-based detectors frequency band ( Hz), making the gravity waves delicate to observe.

Thus it seems that one will have to wait until the launching of the second generation of space-based interferometers to really study the EWPT through gravity wave detectors within this framework. Moreover this would be possible only in the maximizing case where the Higgs mass is close to its current experimental bound and the composite scale of the Higgs is relatively low.

4 Conclusions

In this article we have reported on a complete computation of the one-loop finite temperature effective potential in models where the Higgs boson is composite and emerges as a light pseudo-Goldstone boson of a strongly interacting sector (our analysis could also be relevant for studying the dynamics of electroweak symmetry breaking in Little Higgs theories). These models are characterized by higher dimensional operators in the Higgs sector suppressed by the strong decay constant, , a scale parametrically smaller than the cutoff of the strong sector. Interestingly, by following the details of the phase transition dynamics, the parameter space of a strong first-order phase transition has actually grown for large value of , and shrunk for small value of cutoff, compared to the tree-level result found in [2]. It has grown at the higher end by going beyond the high temperature approximation. The parameter space has shrunk on the lower end, since we found that bubbles cannot be nucleated well enough there to overcome the effects of an expanding universe. We encountered some subtleties along the way, including infrared singularities and imaginary components to the potential, that were resolved.

It was also necessary to compute the details of the phase transition dynamics in order to investigate the possibility of detecting gravitational radiation from the first order phase transition occuring in the early universe. After bubbles are nucleated, their collisions and subsequent turbulence in the plasma give rise to gravity waves. In the assumption of a detonation regime, the effects depend on only two parameters, the latent heat and the duration of the phase transition , both of which can be determined by solving the bounce equation, and analyzing the full one-loop finite temperature effective potential at the scale of the nucleation temperature. Although LIGO and LISA are likely not sensitive to these effects, we found that BBO, a planned second generation experiment of space-based interferometers, could be sensitive to the gravity waves produced during this phase transition.


We thank J.R. Espinosa, S. Martin, M. Perelstein, M. Serone and G. Servant for helpful conversations. We also want to thank T. Hahn for his help with the LoopTools software. This work is supported in part by the Department of Energy and the Michigan Center for Theoretical Physics (MCTP), by the RTN European Program MRTN-CT-2004-503369, by the EU FP6 Marie Curie RTN “UniverseNet” (MRTN-CT-2006-035863) and by the CNRS/USA exchange grant 3503.

Appendix A On-Shell Renormalization of the Potential

The on-shell scheme identifies lagrangian parameters as physical parameter (i.e., observables). It is the scheme employed by [31], although we augment that discussion by describing a self-consistent approach with higher order operators, and describe the details of how IR divergences from massless Goldstone bosons cancel.

Renormalizing our theory in the on-shell scheme is most convenient when we begin by writing the full potential in the following form:




which has been regularized in dimensions, () for gauge bosons (scalars and fermions) and . In this parametrization of the tree-potential, the scalar -dependent masses are: and . The on-shell scheme imposes that is the vacuum expectation of the Higgs field, , and . The precise meaning of is defined below.

The counter terms, , are determined by the renormalization conditions:


where and are needed to take us from the IR-sensitive and unphysical limit of the effective potential to , where physical observables and the tri-Higgs coupling are defined. Detailed computations demonstrating the cancelation of the IR divergences in this scheme are presented in Appendix B.

We wish to have a more direct physical parameter that parametrizes deviations from the SM, and so we redefine


which constitutes the definition of the physical observable . Recall that the tri-Higgs coupling in the SM is fixed with knowledge of , and thus is determined completely by and the other parameters of the SM:


Since depends on , this expression is technically equal to the SM one only in the limit of , which is all that we need for the analysis to be self-consistent.

We are now able to invert the renormalization conditions and compute the counter terms, which depend on the various derivatives of , , and . Upon expanding the result, one can express the renormalized full one-loop potential as