Thanks to the recent discovery of the Higgs boson at LHC [1, 2] the standard model (SM) describing the dynamics of elementary particles is now complete. However, the SM accommodates neither dark matter (DM) nor neutrinos with a finite mass. Therefore, the SM is incomplete as a theory to explain phenomena in our Universe, and consequently it has to be extended. These unsatisfactory features are the main motivations for probing both theoretically and experimentally new physics around the TeV scale.

Besides the problems mentioned above there are also problems of a more theoretical nature. One of them is the origin of the electroweak (EW) scale. Certainly, the SM cannot explain it, but a hint might exist in the SM: The Higgs mass term is the only term that breaks scale invariance at the classical level. In fact there have recently been many studies on a scale-invariant extension of the SM. There are basically two types of scenario: one [3–47] relies on the Coleman–Weinberg (CW) potential [48], while the other [49–59] is based on non-perturbative effects in non-abelian gauge theory such as dynamical chiral symmetry breaking [60–62] or condensation of the gauge-invariant scalar bilinear [63–65]. The common thinking is that a classically scale-invariant physics around TeV is responsible for the origin of the SM scale.

Along this line of thought we have suggested a new model [59], in which SM singlet scalar fields are coupled with non-abelian gauge fields in a hidden sector. Below a certain energy scale the scalar fields condensate in the form of the bilinear, i.e. , by a non-perturbative effect of the hidden sector. Because of the condensate the Higgs portal term turns to a Higgs mass term with a squared mass proportional to . However, this is too naive, because it is a non-perturbative effect, and there is a back reaction on the condensate from the Higgs through the portal. In [59] we have proposed an effective theory for the condensation of the scalar bilinear and investigated the vacuum structure in the self-consistent mean field approximation (SCMFA) [66, 67]. Furthermore, we have introduced flavors to the scalar fields and shown that realistic DM candidates, which are the excited states above the vacuum, exist in the model. Thus, the DM and EW scales have the same origin.

In this paper we will study phase transitions at finite temperature in our model. There will be EW and scale phase transitions. As is well known, a strong first-order EW phase transition is important for baryon asymmetry in the Universe [68–75]. By the scale phase transition we mean a transition between phases with a zero and nonzero condensates of the scalar bilinear. Note that (to the best of our knowledge) the scale phase transition in a non-abelian gauge theory has not been studied and therefore the nature of the phase transition is not known. Since we have an effective theory for the condensation of the scalar bilinear at hand, we will address this problem by means of the effective theory. The first sections will be used to explain the model as well as the effective theory. We expect that there exists a nontrivial interplay between the EW and scale phase transitions, because the EW scale is created by the condensate in the hidden sector. We will be able to confirm this expectation in Sect. 5. Moreover, it will turn out that the EW and scale phase transitions can be a strong first-order phase transition in a certain parameter space of the model. Section 6 will be devoted to a summary.

Our hidden sector [59] consists of strongly interacting gauge fields coupled with the scalar fields in the fundamental representation of . The hidden sector Lagrangian is given by (1) where , is the matrix-valued gauge field, the trace is taken over the color indices, and the SM Higgs doublet field is denoted by . The total Lagrangian is the sum of and , where the scalar potential of the SM part, , is (2) Note that the Higgs mass term is absent. Below a certain energy scale the gauge coupling becomes so large that the invariant scalar bilinear dynamically forms a invariant condensate [64, 65], (3) which breaks classical scale invariance. But the condensate (3) is not an order parameter, because scale invariance is broken by scale anomaly, too [76, 77]. This hard breaking by anomaly is only logarithmic, and it implies that that the coupling constants depend on the energy scale [76, 77]. Therefore, we have assumed in [59] that the non-perturbative breaking is dominant, so that we can ignore the scale anomaly in writing down an effective Lagrangian to the condensation of the scalar bilinear at the tree level. The effective Lagrangian does not contain the gauge fields, because they are integrated out, while it contains the “constituent” scalar fields . Since the effective theory should dynamically describe the condensation of the scalar bilinear, which should be the origin of the breaking of scale invariance, the effective Lagrangian has to be invariant under scale transformation: (4) where we assume that all 's are positive. This is the most general form which is consistent with the symmetry and the classical scale invariance, where the kinetic term for is included in .¹ That is, has the same global symmetry as even at the quantum level, where and are given in (1) and (2), respectively. Note that the couplings , , and in are not the same as , , and in , because the latter are effective couplings which are dressed by hidden gluon contributions.

In the SCMF approximation [66], which has proved to be a successful approximation for the Nambu–Jona-Lasinio theory [61, 62], the perturbative vacuum is Bogoliubov–Valatin (BV) transformed to , such that (5) where the real mean fields and are introduced as the excitations of the condensate . Here, (normalized as ) are the generators in the hermitian matrix representation, and and are the wave function renormalization constants of a canonical dimension 2. The unbroken flavor symmetry implies (6) where a nonzero can be absorbed into , so that we can always assume .

In the SCMF approximation, is determined in a self-consistent way as follows. One first splits up the effective Lagrangian (4) into the sum, i.e., , where is normal ordered (i.e. ), and contains at most bilinear terms of which are not normal ordered. Using the Wick theorem, (7) etc., we find (8) where (9) and the linear term in is suppressed because it will be cancelled against the corresponding tad pole correction. To the lowest order in the SCMF approximation, the “interacting” part does not contribute to the amplitudes without external 's (the mean field vacuum amplitudes). We emphasize that, in applying the Wick theorem, only the invariant bilinear product has a non-zero (BV transformed) vacuum expectation value.

Given the effective Lagrangian , we next compute an effective potential by integrating out the mean field fluctuations , where the fluctuations of the SM fields including will be taken into account later on when discussing finite temperature effects. We employ the scheme, because dimensional regularization does not break scale invariance. To the lowest order the divergences can be removed by renormalization of , i.e. , and also by the shift , where , and is the scale introduced in dimensional regularization. The effective potential for can be straightforwardly computed: (10) where is so chosen that the loop correction vanishes at . with a term linear in included but without the Higgs doublet has also been discussed in [79–83]. The classical scale invariance forbids the presence of this linear term. To find the minimum of we look for the solutions of (11) The first equation gives , which has three solutions: (i) and ; (ii) and ; and (iii) and . One can easily convince oneself that the solution (i) implies if the second and third equations in (11) are used, where (12) Therefore, the solution (i) is inconsistent, unless we use the fine-tuned relation among the coupling constants. Next, we consider the solution (ii) and find that with . The third solution (iii) can exist if 0$]]> is satisfied, and we find (13) (14) Consequently, the solution (iii) presents the true potential minimum if 0$]]> is satisfied. The Higgs mass at this level of approximation becomes (15) In the small limit we obtain , where the first equation is the SM expression, and the second one is simply assumed in [53]. There will be a correction (7%) to (15) coming from the SM part, which will be calculated later on.

We would like to note that the effective potential in (25) has a flat direction, which corresponds to the end-point contribution of [82, 83]: If is satisfied, for any value of , so that (except for ) the symmetry is spontaneously broken in this direction. The origin that for the solution (iii) is the absence of a mass term in the effective Lagrangian (4); we have assumed classical scale invariance to begin with. A mass term in (4) would effectively generate in a term linear in . This linear term can lift the into a positive direction [80, 81], while remains in the flat direction [82, 83].

Finally, we would like to recall once again that we regard the Lagrangian (4) together with our approximation method as an effective theory for the condensation of the scalar bilinear, which takes place in the gauge theory described by (1). That is, we discard fundamental problems such as the intrinsic instability inherent in (4) [82, 83], because we assume that such problems are absent in the original theory described by (1).

We are now in a position to use the effective Lagrangian (8) to discuss DM. First, we replace and the Higgs doublet appearing in by and , respectively, where and are the would-be Nambu–Goldstone fields, and is given in (14). The linear terms in and in should be suppressed, because they will be cancelled against the corresponding tad pole corrections. We integrate out the constituent scalars to obtain effective interactions among , , and the Higgs , where and are defined in (5). Their inverse propagators should be computed to obtain their masses and the corresponding wave function renormalization constants. Up to and including one-loop order we find: (16) with , where is given in (15), is the SM correction given in (30), and (17) with . Note that we have included the canonical kinetic term for , but the wave function renormalization constant for is ignored, which is approximately equal to one within the approximation here. The DM mass is the zero of the inverse propagator, i.e. (18) and (which has a canonical dimension 2) can be obtained from (19) with . The Higgs and masses can be similarly obtained from the eigenvalues of the – mixing matrix (20)

The squared Higgs and masses, and , are zeros of . That is, the SM correction (30) and the correction from the mixing (20) are included in . This mixing has to be taken into account in determining the renormalization constants, which we will ignore in the the following discussions, because the effect is very small (as mentioned above). In contrast, the mixing can have a non-negligible effect on the masses. If 2 M_0$]]>, DM or would decay into two 's within the framework of the effective theory, because the effective theory cannot incorporate confinement. Therefore, we will consider only the parameter space with .

The link of to the SM model is established through the interaction with the Higgs, which is generated at one-loop as shown in Fig. 1, yielding the effective couplings (21) where² (22) , and  GeV. We have used the s-channel momenta for DM annihilation, because we restrict ourselves to the s-wave part of the velocity-averaged annihilation cross section . Similarly, we have used the t-channel momenta for the spin-independent elastic cross section off the nucleon . Fig. 1.

The interaction between DM and the Higgs arises at the one-loop level. The lower diagrams are , so that the upper diagrams are dominant, because in a realistic parameter space.

At a certain finite temperature the condensation of the scalar bilinear will be dissolved, and above that temperature the EW symmetry will be restored. The nature of the EW symmetry breaking is crucial for baryon asymmetry in the Universe [68–71]. Here we investigate how the scale and EW symmetry breakings disappear as temperature increases from a low temperature.⁴ To this end, we integrate out the quantum fluctuations at finite temperature within the framework of the effective theory in the mean field approximation. As a result we obtain an effective potential at finite temperature consisting of four components [72–75]: (24) where is the effective potential given in (10) with and replaced by , and (the condensate) is defined in (6). Further, and are the one-loop contributions at zero and finite temperature , respectively, and is the ring contribution. The Coleman–Weinberg potential is normalized such that (25) where we use  GeV. This normalization ensures that the potential does not change given in (13) obtained from . It can be explicitly written as (26) where (27) (28) We work in the Landau gauge, in which the Faddeev–Popov ghost fields are massless even at finite temperature, so that they do not contribute to . The would-be NG bosons are massless only at the potential minimum. But we have neglected their contributions in (26), because they are negligibly small. The tedious expression for comes from the fact that the Higgs mass is generated from the condensation of the scalar bilinear: it is the second derivative of in (10) with respect to . Note that contributes to the Higgs mass correction⁵ (29) which is about 7% in . We follow [73] and find (30) where the thermal masses are (31) (32) the coupling constants , , and stand for the , gauge coupling constants and the top Yukawa coupling constant, respectively, and is defined in (9) with . The thermal functions and are defined as (33) (34) In the actual calculations we employ the idea [92] for approximating the thermal functions as (35) Finally, the ring contribution from the gauge bosons is [73]: (36) where (37) The critical temperatures of the scale phase and EW phase transitions (which we denote by and , respectively) can be different. If and are distant from each other, two phase transitions cannot influence each other much. In the case that they are close or equal, i.e. , two phase transitions can influence each other. In fact, depending on the choice of the parameter values, these different cases can be realized in our model. Below we consider some representative examples.

(i) Scale phase transition with ,

First we consider the case with , i.e., no connection between the hidden sector and the SM sector. We choose: (38) where we will use the same and as well as the same parameter values for and when discussing case (ii) with the SM connected. (If , only the linear combination is an independent coupling.) In Fig. 2 (left) we show against . We see from the figure that the scale phase transition is first order with . The right panel shows the form of the potential for (red-dashed), , and 6.9 (green dash-dotted). As we will see below, the strong first-order scale phase transition in the hidden sector can infect the EW phase transition. Fig. 2.

Left: The scale phase transition for case (i), in which the hidden sector is disconnected from the SM. The (dimensionless) critical temperature is . Right: The (dimensionless) potential against for (red dashed), , and 6.9 (green dash-dotted). The potential energy density at the origin is subtracted from so that the form of the potential for different temperatures can be compared.

We have considered the SM without the Higgs mass term, which is coupled through a Higgs portal term, the last term of (1), with a classically scale invariant hidden sector. The hidden sector is an SM-singlet and described by an gauge theory with scalar fields. At lower energies the hidden sector becomes strongly interacting, and consequently the gauge-invariant scalar bilinear forms a condensate (3), thereby violating scale invariance and dynamically creating a robust energy scale. This scale is transmitted through the Higgs portal term to the SM sector, realizing EW scalegenesis. Moreover, the excitation of the condensate can be identified with the DM degrees of freedom, which are consistent with the present experimental observations [59].

The nature of the scale phase transition in a non-abelian gauge theory is not yet known. By the scale phase transition we mean a transition between phases with a zero and nonzero condensates of the scalar bilinear. We have addressed this problem by means of an effective theory for the condensation of the scalar bilinear. Since the EW scale is (indirectly) created in the hidden sector, it is expected that there exists a nontrivial interplay between the EW and scale phase transitions. We have indeed confirmed this expectation and found that there exists a parameter space in our model in which both the EW and scale phase transitions can be a strong first-order phase transition. This is not the final conclusion, because our result is based on the mean field approximation in the effective theory. A more accurate calculation could change this result. It is well known that a strong first-oder phase transition in the early Universe can produce gravitational wave background [93, 94], which could be observed by future experiments such as the Evolved Laser Interferometer Space Antenna (eLISA) experiment [95, 96]. In our scenario there can exist two strong first-oder phase transitions, whose critical temperatures lie close to each other.

The nature of the EW symmetry breaking is crucial for baryon asymmetry in the Universe [68–71]. For a successful EW baryogenesis, there have to exist CP phases other than that of the SM. Unfortunately, there is no such phase in our model as it stands. We will come to an extension of the model so as to realize a successful EW baryogenesis elsewhere.

Open Access funding: SCOAP³.