Contents
Abstract

We present a comprehensive study of the electroweak phase transition in composite Higgs models, where the Higgs arises from a new, strongly-coupled sector which confines near the TeV scale. This work extends our study in Ref. [letter]. We describe the confinement phase transition in terms of the dilaton, the pseudo-Nambu-Goldstone boson of broken conformal invariance of the composite Higgs sector. From the analysis of the joint Higgs-dilaton potential we conclude that in this scenario the electroweak phase transition can naturally be first-order, allowing for electroweak baryogenesis. We then extensively discuss possible options to generate a sufficient amount of CP violation – another key ingredient of baryogenesis – from quark Yukawa couplings which vary during the phase transition. For one such an option, with a varying charm quark Yukawa coupling, we perform a full numerical analysis of tunnelling in the Higgs-dilaton potential and determine regions of parameter space which allow for successful baryogenesis. This scenario satisfies all experimental bounds and we discuss future tests. Our results bring new opportunities and strong motivations for electroweak baryogenesis.

DESY 17-229

Electroweak Phase Transition and

Baryogenesis in Composite Higgs Models


Sebastian Bruggisser, Benedict von Harling, Oleksii Matsedonskyi and Géraldine Servant


DESY, Notkestrasse 85, 22607 Hamburg, Germany

II. Institute of Theoretical Physics, University of Hamburg, 22761 Hamburg, Germany


sebastian.bruggisser@desy.de, benedict.von.harling@desy.de,

oleksii.matsedonskyi@desy.de, geraldine.servant@desy.de


1 Introduction

Models in which the Higgs boson is a composite pseudo-Nambu-Goldstone boson (PNGB) and the Standard Model (SM) fermions are partially composite offer a very popular alternative to Supersymmetry for solving the hierarchy problem and are prime targets at the LHC (see [Panico:2015jxa] for a review). These models feature a new sector with a strong dynamics which confines around the TeV scale. This sector possesses an approximate global symmetry , which is spontaneously broken at the condensation scale to a subgroup . The lightest Goldstone boson associated with this breaking is identified with the Higgs boson, which can then be naturally light. If one wants to ensure a custodial symmetry to suppress oblique corrections to electroweak precision tests, the minimal possible coset is , while larger groups can be viable too.

In these composite Higgs models, the Higgs potential is generated due to an explicit breaking of the Goldstone symmetry. This explicit breaking comes from an elementary sector consisting of fermions and gauge bosons which do not respect the global symmetry . It is communicated to the composite sector from which the Higgs originates through elementary/composite interactions present in the theory. These interactions are also responsible for the Yukawa couplings by inducing mixing between the elementary and composite fermions. The size of the Yukawa couplings is then determined by the degree of compositeness of the states that are identified with the SM fermions. The Higgs potential in composite Higgs models is thus intimately tied with the Yukawa couplings. This framework therefore appears to be an ideal laboratory to study the connection between electroweak symmetry breaking and flavour physics.

Despite the fame of these models, the electroweak phase transition in composite Higgs models has not been studied in much detail yet. In particular, the phase transition is relevant for the question whether composite Higgs models can allow for electroweak baryogenesis and thereby explain the baryon asymmetry of the universe. In the SM, this scenario fails because the electroweak phase transition is not first-order and the amount of CP violation is also not enough. As was shown in [Grojean:2004xa, Bodeker:2004ws, Delaunay:2007wb, Grinstein:2008qi], dimension-6 operators involving the Higgs which are expected to arise from the strong sector can make the phase transition first-order and also provide a new source of CP violation. Alternatively, this can be achieved if an additional singlet changes its vev during the electroweak phase transition. Such a singlet can arise as an extra PNGB in non-minimal composite Higgs models with global symmetry breaking patterns such as or . The former pattern was studied with regard to electroweak baryogenesis in [Espinosa:2011eu], while the latter was considered in [Chala:2016ykx]. In all these studies, however, the confinement scale of the strong sector has been taken to be constant which implicitly assumes that the confinement phase transition happens well before the electroweak phase transition. This assumption is not always justified. In [letter], we have studied the joint dynamics of the Higgs and the order parameter of the strong sector and found that both transitions can naturally happen simultaneously. This opens a rich range of possibilities for the nature of the electroweak phase transition and for electroweak baryogenesis in composite Higgs models. In this paper, we will extend this analysis.

In order to ensure a large separation between the UV scale (e.g. the Planck scale) and the confinement scale, the strong sector should be near a conformal fixed point for most of its evolution when running down to lower energies. This (nearly) conformal invariance is spontaneously broken when the strong sector confines. Provided that the explicit breaking of the conformal invariance is small, the spectrum of composite states then contains an associated light PNGB, the so-called dilaton. The vev of this field sets the confinement scale and this field can thus be thought of as an order parameter for the confinement of the strong sector.

The analyses in [letter] and in this paper are based on the combined potential for the Higgs and the dilaton. To this end, we rely on a four-dimensional effective field theory (EFT) describing the lowest lying degrees of freedom, such as the SM particles, the Higgs boson, and the dilaton. We present a universal simplified description, suitable to parametrise the IR physics resulting from different possible explicit UV complete constructions. Our ignorance about the details of the strong sector in the confined phase is parametrised by the minimal set of coupling constants and masses, in the spirit of [Giudice:2007fh, Chala:2017sjk]. The crucial ingredients of our framework are the fundamental symmetries which the strong sector is expected to feature, including the spontaneously broken global symmetry and its surviving subgroup , as well as an approximate conformal symmetry above the confinement scale. We also include temperature corrections to the potential and then study the phase transition in the two-field potential for the Higgs and dilaton.

The scalar potential for the Higgs arises at one-loop, and, because of the PNGB nature of the Higgs, is a trigonometric function of the ratio . It takes the generic form [Panico:2015jxa]

(1.1)

where the two terms are loop-induced by the same dynamics and is expected to be at least as large as  [Panico:2012uw]. However, in order to obtain the correct electroweak symmetry breaking scale and the correct Higgs mass, and have to fulfil the relations

(1.2)

This means that which in turn needs to be suppressed since electroweak precision tests and Higgs coupling measurements constrain [Grojean:2013qca]. Therefore, contrary to the generic estimate , has to be suppressed with respect to , which requires some accidental cancellation. This is the irreducible tuning of composite Higgs models [Csaki:2017eio].

We denote the dilaton as . As the confinement scale is set by , we in particular replace (times a proportionality factor to be determined later) in Eq. (1.1) in order to derive the joint potential for the Higgs and dilaton. This part of the potential is then minimized for (however with , as currently ) which leads to a valley in the two-field potential along this direction. If the phase transition happens only at temperatures below the electroweak scale and temperature corrections to the potential are correspondingly small, this valley can attract the tunnelling trajectory and both and can obtain their vevs simultaneously. Since the confinement phase transition can be naturally very strongly first-order, this makes the electroweak phase transition first-order too and thereby solves the first problem that electroweak baryogenesis faces in the SM. In particular, we then do not need to invoke higher-dimensional operators or extra scalars for this purpose.

While the Higgs potential appears tuned today, it can become detuned in the early universe when the dilaton vev differs from its value today . In particular, the sizes of the elementary-composite mixings mentioned above depend on the confinement scale and thus on . Since they affect various loop corrections, we generically expect the tuned relations in Eq. (1.2) to be fulfilled only near the minimum of the potential. The resulting detuning away from the minimum can lead to an additonal valley either along the direction in the potential, or along . Again this valley can attract the tunnelling trajectory and affect the relation between and during the phase transition. For a deep valley along , the Higgs vev vanishes during most of the transition which is not suitable for electroweak baryogenesis. On the other hand, for the Higgs vev is on average larger than for the tuned case discussed above. This behaviour can, in particular, be helpful for generating enough CP asymmetry for electroweak baryogenesis as we discuss below.

We summarize the various ways in which the phase transition can occur in composite Higgs models in Fig. 1.

Figure 1: Sketch of different trajectories for the phase transition in composite Higgs models. The field sets the size of the strong-sector condensate and is the Higgs vev. The blue points correspond to (meta)stable vacua of the theory, and the black lines show possible phase transition trajectories.

The trajectory (1) corresponds to a two-step transition, where the confinement phase transition happens in the first step and electroweak symmetry is only broken subsequently in a second step. This chain of events was implicitly assumed in the previous works [Espinosa:2011eu, Chala:2016ykx, Delaunay:2007wb, Grinstein:2008qi, Grojean:2004xa, Bodeker:2004ws] (in [Espinosa:2011eu, Chala:2016ykx] the additional scalar also changes its vev in the second step). For the other two options presented in Fig. 1 the electroweak phase transition is instead tied to the confinement phase transition. The linear trajectory (2), where scales linearly with all the way between the two minima, can occur in models where the Higgs potential is not detuned away from the minimum of the potential. On the other hand, the composite Higgs models studied in this paper can have a trajectory like (3) for which during most of the phase transition.

Apart from affecting the tunnelling trajectory, the variation of the mixings during the phase transition can also provide additional CP violation. This can then solve the second problem for electroweak baryogenesis in the SM. Indeed, varying mixings lead to varying Yukawa couplings which have previously been shown [Bruggisser:2017lhc] to result in a new source of CP violation. This new source is not in conflict with bounds from e.g. electric dipole moments since it is active only during the phase transition. For instance, the CP violation can be large if at least one light quark mixing increases to the size of the top quark mixing when the dilaton vev is sent to zero. The CP violation then arises from the interplay between the top quark mixing and one light quark mixing. The dependence of the mixings on the dilaton vev is mostly determined by the scaling dimension of the composite operators to which the elementary fermions are linearly coupled. For constant scaling dimensions one can expect the mixings to either stay approximately constant too or decrease when the dilaton vev is sent to zero. The required growing behaviour instead can be obtained if the scaling dimension of the corresponding operator becomes energy-dependent. Such an energy-dependence can well occur in composite Higgs models. Alternatively, a sufficient amount of CP violation can be generated by the top quark mixings alone if they have a phase which varies with the dilaton vev.

Our approach enables to test a broad range of possible UV complete theories. Together, the analyses in [letter] and in this paper make the following progress:

  • First general analysis of the confinement and electroweak phase transitions in PNGB composite Higgs models.

  • First analysis of electroweak baryogenesis in PNGB composite Higgs models during the combined confinement and electroweak phase transition, including the analysis of the new CP-violating source from varying quark mixings.

  • Full numerical calculation of the two-field tunnelling for electroweak baryogenesis in PNGB composite Higgs models. This is especially important as the Higgs vev has a non-trivial dependence on the vev of the strong condensate.

Compared to[letter] which focussed on varying top mixings, we will in particular extend the analysis of the CP-violating source and systematically list the cases where varying quark mixings can generate the CP asymmetry. For one promising case, with varying charm mixings (which is both quantitatively and qualitatively different from varying top mixings), we will perform a scan of the parameter space, and determine the parameter region for which the confinement and electroweak phase transitions happen simultaneously and are first-order, and the amount of CP violation is sufficient for explaining the baryon asymmetry of the universe. Furthermore, we will present a detailed analysis of phenomenological implications of our scenario for collider physics, such as Higgs coupling measurements and dilaton searches. Overall, this analysis follows a series of papers on the impact of Yukawa coupling variation for electroweak baryogenesis [letter, Baldes:2016rqn, Baldes:2016gaf, vonHarling:2016vhf, Bruggisser:2017lhc].

The structure of this paper is the following. In Sec. 2 we review the basic ingredients needed to construct our EFT, including the generation of the SM fermion masses, the Higgs mass and conformal symmetry breaking. These key ingredients are combined together in Sec. 3 to provide a single framework for studying the phase transition. In Sec. 4 we add finite temperature effects. Sec. 5 is devoted to the analysis of the CP-violating source associated with the varying quark mixings. The numerical analysis of the phase transition and the resulting baryon asymmetry is described and the main results are presented in Sec. 6. Finally, we discuss experimental tests of our scenario in Sec. 7, including the current bounds on flavour-changing neutral currents, CP-violating Higgs couplings and gravitational wave signals, and conclude in Sec. 8. An appendix reviews the formalism to calculate the tunnelling path and action in potentials for one and more fields.

2 Review of key concepts

In this section, we will summarize the main concepts which we will adopt for our description of the phase transition and electroweak baryogenesis in composite Higgs models. These concepts represent a typical (though not the only possible) picture of composite Higgs models and their flavour structure, and have been motivated in a large amount of literature, of which we will only point out a few representative works. For general reviews of composite Higgs models [Kaplan:1983fs], we refer the reader to [Contino:2010rs, Panico:2015jxa, Bellazzini:2014yua].

2.1 Standard Model masses from anomalous dimensions

We now review the basics of fermion mass generation in composite Higgs models and state the notations. Let us begin with the flavour-diagonal case, i.e. without mixing between different SM generations. The SM fermion masses are generated from couplings

(2.1)

of elementary fermions to operators from the strong sector . The dimensionless coefficients are assumed to be of order one in the far UV, where the couplings are generated. The renormalisation group (RG) evolution then changes them when running down to the confinement scale. This is driven mostly by anomalous dimensions of the operators , which remain sizeable over a wide energy range due to an approximate conformal symmetry. The RG equation reads (see e.g. [Contino:2010rs])

(2.2)

where is the scaling dimension of minus 5/2 (which is constrained as by virtue of the unitarity bound on fermionic CFT operators), is the typical coupling of the strong sector (we will give more meaning to it later), and the ellipsis stands for terms suppressed by higher powers of and . This running stops at the confinement scale,

(2.3)

where the CFT disappears and the operators can excite bound states of the strong dynamics, the fermionic partners. At energies below , Eq. (2.1) is mapped onto mixings between these composite fermions, which we denote as , and the elementary fermions ,

(2.4)

where is the Goldstone matrix and the couplings are now defined at the confinement scale. As the strong sector spontaneously breaks the global symmetry,

(2.5)

the Goldstones are introduced as a compensator between the elementary fermions transforming in and composite multiplets of (the elementary fermions generically do not fill complete multiplets of but can be given some transformation properties under in order to write down the original operator (2.1)). The Higgs multiplet comes as a part of these Goldstones:

(2.6)

Since the interaction (2.4) leads to the mixing of elementary and composite fermions, this mechanism is known as partial compositeness. By integrating out the partners, we obtain the Yukawa couplings

(2.7)

where and are the mixing parameters of left- and right-handed fermions, respectively, and we have parametrized the masses of the partners as

(2.8)

The SM fermion mass hierarchy is then explained by order-one differences in the anomalous dimensions of the operators in Eq. (2.1).

Figure 2: Examples of the running of the mixing parameters (2.9) (the effective Yukawa couplings follow similar trends), for positive and an order-one initial value (blue), a small negative and a small initial value (green), a negative and a small initial value (red), a negative and an order-one initial value (yellow). The green area shows the range of energies relevant for our analysis. The behaviour of the mixings outside the green region is not relevant for our analysis and can differ from what we show. For instance, all the initial values could be of order one, but the sign of the anomalous dimension could change with energy.

The RG equation (2.2) only defines the running down to the condensation scale, where the CFT description breaks down. We will later link this condensation scale to the vev of the dilaton , the PNGB of the spontaneously broken conformal invariance of the strong sector. When the dilaton vev changes, the condensation scale changes accordingly and the mixing parameters in turn change following the RG. This allows us to use the RG equation (2.2) to obtain the dependence of on the dilaton vev 111The general solution for the case where the anomalous dimension is itself scale-dependent was discussed in Ref. [vonHarling:2016vhf].

(2.9)

where we have fixed the integration constant by requiring that for the dilaton vev today the mixing is . More generally, throughout the paper sub/superscripts ‘’ will refer to the present-day values of the parameters. Let us discuss some special cases of this general solution. A positive anomalous dimension leads to a mixing which is decaying towards the IR. If then , the second term in the r.h.s. of Eq. (2.2) is never important and Eq. (2.9) simplifies to

(2.10)

This behaviour of the mixing parameter can be used to obtain small Yukawas for the light SM fermions. Small Yukawas are irrelevant for the purpose of our analysis, however, as they only contribute negligibly to the Higgs potential (see next section) or CP violation (see Sec. 5). A negative anomalous dimension, on the other hand, can make the mixing sizeable in the IR. Once is large, the second term on the r.h.s. of Eq. (2.2) becomes important. For being positive, this causes to run to the fixed point

(2.11)

In Fig. 2, we show several examples of the running according to Eq. (2.2). The behaviour corresponding to the red and yellow line is suitable to obtain respectively a varying or an almost constant top Yukawa at the energies of interest. The behaviour corresponding to the green and blue line, on the other hand, can yield respectively e.g. a sizeably growing or a decaying charm Yukawa. As will be shown later, a varying top or a growing charm Yukawa are essential for creating a sufficient amount of CP violation during the phase transition.

We will now turn to the more realistic case with flavour mixing, which as we will see later can be crucial for otaining enough CP violation. The couplings (2.1) in the UV and (2.4) in the IR then respectively generalize to

(2.12)

Differences in the anomalous dimensions of the operators lead in the IR to large hierarchies between entries in the matrix with different index . Aside from this, the structure of can be either arbitrary (“anarchic”) with different entries varying by order-one factors, or it can have a certain pattern dictated by flavour symmetries (see e.g. [Csaki:2008eh]). Integrating out the composite partners, Eq. (2.7) for the Yukawa couplings generalizes to

(2.13)

It can be shown that for a matrix with hierarchical entries with respect to the index , the approximate equality holds, where is a unitary matrix (see e.g. [Panico:2015jxa]). Therefore after performing unitary rotations with these on the left- and right-handed SM fermions, we obtain

(2.14)

Even if the matrix is anarchic, this leads to hierarchically different Yukawa eigenvalues

(2.15)

where now stands for a typical entry of the matrix. From Eq. (2.15), it is now clear that the Yukawa couplings of the SM fermions depend on the scale . This dependence will crucially impact the dynamics of the electroweak phase transition if it is to happen at the same time as the confinement phase transition.

2.2 Higgs shift symmetry breaking

If we want to ensure a custodial symmetry in order to suppress oblique corrections to electroweak precision observables, the minimally sufficient global symmetry of the strong sector is  [Agashe:2004rs]. If this symmetry were only broken spontaneously, the Higgs potential would vanish as expected for an exact Goldstone boson. However the SM fermions do not fill out complete multiplets of , while only part of is gauged by the SM gauge bosons. The elementary-composite couplings of Eq. (2.1) therefore involve operators which transform under , and fermions which do not. This is an explicit breaking of which leads to a potential for the Higgs boson. Similarly the gauge interactions also explicitly break which gives an additional, though subdominant contribution to the potential (see below). The contribution from Eq. (2.1) can not be too large either, however, as the Higgs mass should be suppressed with respect to the masses of the rest of the strong-sector resonances. The general form of the Higgs potential therefore can be considered as an expansion in an adimensional quantity parametrising the relative strength of the explicit breaking,

(2.16)

where and are trigonometric functions. This expression is intuitively clear from the following arguments. The loop factor suppression follows from the fact that the effective potential is generated at one loop with an elementary state running in the loop. The factor reproduces the correct overall dimension of the potential (see Sec. 3.1) and is composed of the two characteristic parameters of the strong sector – the typical coupling and the value of the strong condensate. Finally, that the are trigonometric functions is expected from the fact that the Higgs enters the theory in the form of the Goldstone matrix (2.6).

The same mixings (2.1) and (2.4) which are responsible for the Yukawa couplings (2.7) thus also produce the Higgs potential. Given that the current value of the top quark Yukawa coupling dominates over the rest, we can expect that the top quark mixings set the size of the Higgs potential at present times. A more quantitative description of the potential is postponed to Sec. 3.3.

In the following, for the sake of decreasing the overall number of parameters, we will set the typical strong-sector coupling and the coupling determining the mass of the fermionic partners equal. The rational behind this is that the fermionic resonances are expected to give a sizeable contribution to the potential, which therefore by dimensional analysis has to depend on their masses and couplings. This assumption is however not completely flawless, as some of the explicit models show a preference for a sizeable mass gap between the top quark partners and the rest of the composite resonances implying  [Matsedonskyi:2012ym, Redi:2012ha, Marzocca:2012zn, Pomarol:2012qf]

(2.17)

We will comment more on this assumption in Sec. 5, as its discussion requires a dedicated analysis of the energy dependence of the mixings.

The value of the Goldstone decay constant determines by how much our model is deformed with respect to the SM and in particular sets the scale by which higher-dimensional operators are suppressed. It is therefore forced to be somewhat larger than the Higgs vev, with the currently preferred value being  TeV. However, the potential of the type (2.16) generically has a minimum at either or . This can be seen by taking a concrete example of the trigonometric function , e.g. , which does not allow for minima at  222This can be translated to the inability to generate a Higgs quartic coupling which is less suppressed than the quadratic term. Expanding the trigonometric functions appearing in the leading order of , we obtain both the quadratic term and the quartic term , with an overall coefficient of the same size. This generically gives in the minimum. More elaborate constructions can however alleviate this tuning, see e.g. recent attempts in [Batell:2017kho, Csaki:2017eio].. To obtain as required therefore necessitates a tuning, of the order (a more quantitative way to estimate the tuning will be given in Sec. 3.4).

The Higgs potential also receives additional contributions, in particular from interactions with SM gauge bosons. Their effect, though generically expected to be subdominant, can become important in the region where the leading contribution of the SM fermions is tuned to smaller values [Panico:2012uw]. We however do not need to consider this source of shift symmetry breaking separately. As we will discuss in more detail in Sec. 3.3, we will fix the current Higgs potential by the observed Higgs mass and vev, and will not distinguish the separate contributions to it. The Higgs potential at different energies, which we will instead consider in more details, is expected to become detuned, with the large fermionic contributions dominating over the contributions from gauge bosons.

The Higgs also induces the masses of the electroweak gauge bosons which are given by

(2.18)

where is the Higgs vev today,  GeV and stands for both the and boson. Notice that this differs from the corresponding expression in the SM (and similarly slightly differs from ). From this, we also see why the composite Higgs couplings deviate by order from the corresponding couplings in the SM. Varying the mass term for the gauge bosons with respect to the Higgs, we obtain a trilinear vertex

(2.19)

which deviates by a factor of from the tree-level result in the SM.

2.3 Conformal symmetry breaking

We will assume that the confinement phase transition and the formation of the strong condensate can be described as the transition of a single field from zero to some finite vev. All mass scales of the strong sector, including discussed in the previous section, will be linked to the vev of . In this section we will discuss the main factors determining the dynamics of in isolation, i.e. we will momentarily neglect the Higgs.

In order to reproduce the observed flavour structure of the SM using partial compositeness, large anomalous dimensions for the operators which mix with the fundamental fermions are required. Furthermore, these large anomalous dimensions need to persist over a wide range of energies from some high UV scale down to roughly the TeV scale, where the strong sector confines. This can be achieved if the strong sector exhibits an approximate conformal invariance and remains strongly coupled over this energy range (see e.g. [Contino:2010rs]). It is not possible to break the conformal symmetry purely spontaneously and we instead need to add an explicit source of conformal-symmetry breaking [cpr, Coradeschi:2013gda, Bellazzini:2013fga, Chacko:2012sy, Megias:2014iwa]. To this end, we introduce the strong sector operator

(2.20)

whose scaling dimension differs from four. This results in a scale dependence of the renormalized dimensionless parameter which to lowest order satisfies the RG equation

(2.21)

where is the scaling dimension of minus (which is constrained as by virtue of the unitarity bound on scalar CFT operators). In order to break conformal invariance in the IR, we choose negative. Provided that is small in absolute value and at the UV scale is somewhat small too, only slowly grows when running towards the IR. The (nearly) conformal invariance is then maintained for a large energy range. Eventually, however, and thus the distortion induced by grows so large that the strong sector condenses and conformal invariance becomes spontaneously broken. We further assume that remains small even at the condensation scale. In this case, the explicit breaking of the conformal invariance is weak compared to the spontaneous breaking by the non-vanishing condensate. There is then an associated light PNGB, the so-called dilaton 333Despite the fact that the dilaton is allowed to be rather light in our construction, it can not be used as the SM Higgs impostor [Megias:2016jcw]., which we identify with the field introduced earlier. The breaking of conformal invariance is reflected by a non-trivial potential for the dilaton given by

(2.22)

where all mass dimensions are set by which is the only mass source in the theory and is run down to the scale using the RG equation (2.21). The first term in the potential respects scale invariance, , and therefore is not -suppressed. For this reason we chose to normalise it with a generic dilaton coupling with a power that follows from dimensional analysis (see Sec. 3.1). The constant is of order one and the ellipsis stand for contributions in higher order of . The first term alone would not allow for a global minimum with a non-vanishing , which confirms the need for explicit conformal-symmetry breaking. Without loss of generality we assume that the leading term in the potential which breaks conformal invariance is proportional to the first power of . The non-trivial dependence of on the condensation scale, and hence on , allows for a global minimum of the potential. We denote this minimum as , the dilaton vev today. Minimizing the potential and using the RG equation (2.21), we can determine and , in terms of and the dilaton mass :

(2.23)
(2.24)

We can take as an independent breaking source (assuming that its microscopic description can be provided in the UV-complete theory) and fix the scaling dimension and boundary condition to generate the desired and . On the other hand, we can associate with the conformal symmetry breaking due to partial compositeness that is already present in our model. Indeed, loops involving composite and elementary fermions generate

(2.25)

where is the number of QCD colors. If no other type of conformal-symmetry breaking is allowed in the theory, the presence of a large seems then to be necessary to generate the non-vanishing condensate. This large mixing, however, could lead to a large breaking of the shift symmetry of the Higgs, which consequently would no longer be an approximate Goldstone boson. To solve this issue one may for instance consider a scenario where the large mixing is associated with the elementary right-handed top which is then chosen to transforms as a singlet under , so that its large mixing would not break the Higgs shift symmetry. We will leave a further study of this option for future work. However, the terms in Eq. (2.25) will give an additional, albeit generically subdominant, contribution to the dilaton potential as we discuss in Sec. 3.3.

The cancellation of an NDA-sized scale-invariant quartic, , requires the scale-invariance breaking sources to exit the perturbative region. From Eq. (2.23), we see that then . Since grows further towards , the description of the dynamics of the phase transition becomes less robust within our approach. In order to proceed, we will assume a moderate suppression of the scale-invariant quartic, , so that cancelling it does not require . To prevent from exiting the perturbative regime at lower energies we cut its growth by accounting for the next-to-leading order term in the RG equation,

(2.26)

with the order-one coefficient taken positive. As in Eq. (2.23), we can fix the integration constant by the requirement that the potential is minimized at . The solution to the RG equation then reads

(2.27)

As in Eq. (2.24), we could further trade the scaling dimension for the dilaton mass .

Note that this has a form reminiscent of the Goldberger-Wise potential [Goldberger:1999uk] which arises in certain 5D duals of confining theories. However, for the corresponding Goldberger-Wise potential has a barrier at zero temperature with crucial implications for the strength of the phase transition (see [vonHarling:2017yew]), while our dilaton potential (2.22) with (2.27) has no such barrier. We will show that a strong first-order phase transition can nevertheless follow.

3 Zero-temperature effective field theory

3.1 Scalar potential: Matching two descriptions

In the previous section, we have reviewed the effective potentials of the Higgs, in the one coupling-one mass scenario, and the dilaton. We will now combine both potentials, relating the mass scale of the problem to the dilaton vev. This will provide us with a unified framework allowing for the description of the confinement phase transition and the electroweak phase transition together. It is most natural to assume that the Higgs is a meson-like state of the underlying confining theory, in analogy to the QCD pions associated with chiral symmetry breaking. For the dilaton, instead, one can argue for it being either meson-like (see e.g. the lattice studies observing a light meson-like state [Aoki:2014oha, Appelquist:2016viq] in theories) or glueball-like (as expected in theories dual to a warped extra dimension such as [Randall:2006py]).

In the case where both the Higgs and the dilaton are meson-like states, we can consistently assume that they are characterised by approximately the same typical coupling, hence we set

(3.1)

A glueball-like dilaton would instead behave very differently. In large- confining theories, which we will use as a reference, an interaction involving glueballs and mesons scales like [Witten:1979kh]

(3.2)

This suggests defining the coupling of the glueball-like states as

(3.3)

and for the meson-like states as

(3.4)

The factors of are chosen to ensure that one recovers a generic strongly-coupled theory in the limit  444Notice, however, that for instance in explicit 5D constructions this normalization can differ by a factor of order a few.. Let us now find the general form of the effective potential involving simultaneously the Higgs and the dilaton , forgetting about Goldstone symmetries for the moment. In the absence of explicit mass scales, we can only construct the effective potential out of the fields and and the couplings and . A generic term in the effective potential can thus be written as

(3.5)

Relations between the powers and can be obtained by dimensional analysis. Keeping units of length , time and mass (i.e. not working in natural units where ), one concludes that the dimensions of the potential, fields, couplings and derivatives are (see e.g. [Panico:2015jxa, Chala:2017sjk])

(3.6)

where . From this, one finds the relations and . Imposing also the large- scaling from Eqs. (3.2), (3.3), (3.4) gives an additional relation and fixes the term in Eq. (3.5) to

(3.7)

This applies to both a glueball-like or a meson-like dilaton, depending on which -scaling we choose for .

We are now in a position to derive the dependence of the Higgs potential on the dilaton, restoring the full Goldstone symmetry. The dilaton vev is the only source of mass in the theory. We therefore need to replace the Goldstone decay constant , which balances the Higgs in the functions of Eq. (2.16), by the dilaton vev times a proportionality factor. In order to determine this factor, we can write the functions as power series in and match with Eq. (3.7). This gives

(3.8)

Note that this has still the right dimensions since after restoring factors of , the loop factor is which is dimensionless. From this expression we can in particular read off the relation between the current Goldstone decay constant and the current dilaton vev ,

(3.9)

In order to obtain the combined Higgs-dilaton potential, we then need to add Eqs. (2.22) and (3.8). Notice that the first term in Eq. (2.22) has the correct power of the coupling as follows from Eq. (3.7).

3.2 Canonical variables

To complete our EFT we need to include the kinetic terms of the Higgs and dilaton. For consistency of our description, these kinetic terms need to be invariant under the shift symmetries. It is precisely this invariance which allows us to fix the form of the Higgs potential in terms of trigonometric functions and suppression factors proportional to the symmetry breaking sources (2.16). In case of an isolated Higgs and a constant scale it is sufficient to choose the Higgs kinetic term as

(3.10)

to respect the symmetry (which is weakly broken by the scalar potential) and the smaller (gauge) symmetry (which is an exact symmetry of the full Lagrangian). The presence of the gauge symmetry follows from the fact that the Higgs parametrizes the phase of the Goldstone matrix, , and phases rotated by are not distinguishable. However, we can not trivially apply the same kinetic term in case of a dynamical scale . Although the scalar potential (3.8) does respect the symmetry , the kinetic term (3.10) does not. The simplest way to derive the invariant kinetic terms is to switch to the dimensionless Goldstone boson , which substitutes the argument in the trigonometric functions of the scalar potential (3.8). Using dimensional analysis as discussed in Sec. 3.1 555We recall that transforms as an electroweak doublet. Therefore at the level of dimension-four operators we can not write down a kinetic mixing term. Higher-order operators will be discussed in Sec. 3.4., we then find

(3.11)

which trivially respects the gauge symmetry . The constant prefactor of the kinetic term for has been chosen such that at a constant dilaton vev, we can switch to the field

(3.12)

and reproduce the Higgs potential (2.16). For later convenience, we also introduce the field

(3.13)

which has a canonically normalized kinetic term at . On the other hand, necessarily has a mixed kinetic term needed to insure invariance under .

In the following, when considering the phase transition, it will be convenient to use field variables which always remain canonically normalized,

(3.14)

so that

(3.15)

We can express the scalar potential (3.8) in terms of these new field variables by using the inverse relations

(3.16)

3.3 Parametric form of the scalar potential

We are now ready to add the final details to the zero-temperature potential given by Eqs. (2.22) and (3.8), namely to take into account the tuning which is necessary to obtain the observed Higgs mass and vev today. From this we will also see how the varying mixings affect the potential at values of different from the value today. Let us begin with discussing the Higgs potential today. For the functions , we choose a parametrisation which can be matched onto the most commonly used models [Panico:2012uw]

(3.17)

The coefficients and are fixed by the observed Higgs mass and vev as

(3.18)

As mentioned in the introduction, reproducing the SM Higgs parameters requires a certain amount of tuning. This manifests itself in the fact that and typically sizeably deviate from generic NDA estimates. However, the model parameters generically vary if we change from its vev today as happens during the phase transition. For example, we have seen in Sec. 2.1 that the mixings can substantially change with . We expect that such a variation leads to a detuning of the potential and that the potential becomes completely generic away from the current minimum. Choosing the trigonometric functions as in Eq. (3.17), the leading-order estimate in Eq. (3.8) for the Higgs potential generated by quarks reads

(3.19)

with the coefficients given by

(3.20)

Here is given by Eq. (2.9), and are free parameters of our EFT whose absolute values are expected to be of order one and is the number of SM QCD colors enhancing the quark loops. A non-zero means that the leading contribution to the coefficient is suppressed with respect to the naive estimate [Matsedonskyi:2012ym, Panico:2012uw]. In the known composite Higgs models one can have . For instance, contributions with arise from fermionic composite operators in Eq. (2.1) transforming in representations with dimension and is generated from operators with  [Panico:2012uw].

Altogether, we can approximate the Higgs potential for arbitrary as

(3.21)

with in terms of and given in Eq. (3.16). This interpolates between the tuned potential in Eq. (3.17) for the dilaton vev and the mixings today and the detuned potential in Eq. (3.19) for and . In this expression we have also accounted for the fact that at fixed scales as the fourth power of .

For the Higgs-independent dilaton potential, we will similarly include a -dependent term which we expect to be generated from the mixings as discussed around Eq. (2.25). Altogether this gives

(3.22)

where the coefficients and are generically of order one. The -dependent term is expected to be related to the Higgs potential. Since this dependence can only be extracted in explicit models which we do not discuss, however, we will limit ourselves to the assumption that there are no correlations between the -coefficients controlling different terms of the scalar potential. Note that the -dependent term can produce a barrier between and at zero temperature if , analogously to the Goldberger-Wise potential for the dilaton which was used in the studies of the phase transition of Randall-Sundrum models in [Creminelli:2001th, Randall:2006py, vonHarling:2017yew].

3.4 Key properties of the two-field potential and preliminary analysis

Figure 3: Examples of potentials with a valley along the direction (left plot) and with a valley along (right plot). The green line shows the -dependent minimum of the Higgs potential. Since the Higgs potential is loop-suppressed with respect to the dilaton potential, and we assume order-one values of the mixings , the valleys are not very pronounced.
Figure 4: Same as in Fig. 3 but in canonical variables .

We can now perform a preliminary analysis of the scalar potential. The Higgs potential away from is typically dominated by its detuned part, , as the NDA contributions are larger than what is needed to reproduce the potential today at and . This detuned part admits minima at or for fixed , depending on the signs of the coefficients and . Sketches of the two-dimensional potential for these two cases are shown in Figs. 3 and 4. Any other minimum between these two possibilities would instead require fine-tuning of potential coefficients and hence is not expected to be present when the coefficients vary sizeably with . The minima with respect to result in corresponding valleys in the two-dimensional potential which will attract the tunnelling trajectory during a first-order phase transition. If this attraction is large, which is expected for large mixings and a resulting deep valley, the tunnelling trajectory can come very close to this valley.

Along a valley , the electroweak gauge bosons generically are massless which leaves the sphaleron processes active during most of the phase transition and is thus not suitable for electroweak baryogenesis. We are therefore instead interested in a valley along . If the mixings are growing towards the IR, such a valley is present if

(3.23)

On the other hand, we need the SM fermions to be massive as well, in order to source the CP violation. For this, the condition (3.23) is necessary but not sufficient. Indeed, in many explicit composite Higgs models the fermion masses are proportional to and thus also vanish at . Therefore if the potential features a valley along which is deep enough to make the tunnelling trajectory follow it closely, electroweak baryogenesis can be spoiled as well. As we will see later, the exact tunnelling path depends on a number of parameters and does not necessarily follow the valley. If this does happen, on the other hand, we can use the fact that the cosine in the expression for the mass is absent when we couple the elementary fermions to composite operators in the following representations:  [Pomarol:2012qf]. Interestingly, this selection criterium disfavours the minimal embeddings with and representations only (taking into account that the allowed has difficulties in reproducing a realistic Higgs potential [Panico:2012uw]). The preferred models therefore can have a rich spectrum of composite fermions as a distinctive phenomenological feature [Pappadopulo:2013vca, Matsedonskyi:2014lla]. In summary of this discussion, we present in Figs. 3 and 4 two cases, with a valley along and the desired potential with a valley along .

Another important comment concerns the overall tuning of the model. As was already mentioned, the NDA estimates of the Higgs potential typically give too large a Higgs mass and either or for the Higgs vev. Obtaining then requires some tuning of the Higgs potential, which can be estimated to be of order

(3.24)

Current experimental constraints coming from various observables indicate that [Grojean:2013qca]

(3.25)

In order to estimate the overall tuning of the Higgs sector (which includes the tuning in ), we take the product of ratios of the required values of the Higgs potential coefficients over the values that are generically expected,

(3.26)

Here we only include the contribution to and from the top quark, as the other quark mixings are expected to be small at present times. We emphasize that the amount of fine-tuning that is required in the model with sizeable variation of the mixings is not different from the fine-tuning in the usual composite Higgs models.

We are now also in the position to discuss the important phenomenological question of Higgs-dilaton mixing, in the true minimum of the scalar potential. This mixing can alter the Higgs couplings compared to the SM predictions, in addition to the usual universal composite Higgs deviations proportional to (cf. Eq. (2.19)). We will work in the basis (,) where possible kinetic mixings are redefined away, therefore all the mixing effects are contained in the scalar potential (3.21) and are determined by the mixing mass

(3.27)

Note that the first term in the second line vanishes, since the first derivative of the scalar potential vanishes in the minimum:

(3.28)

Mixing between the Higgs and the dilaton can still arise from the second term in the second line which gives

(3.29)

where , is the -function in the RG equation (2.2) and we have neglected higher powers of . The mixing mass thus becomes the larger, the more the mixing parameters vary with energy. In the limit of a large dilaton mass , the Higgs-dilaton mixing angle is given by . As follows from [Chala:2017sjk], the effect of this type of mixing is most pronounced in the Higgs-photon and Higgs-gluon couplings, where it can easily outrun the universal composite Higgs effects of order . Several other operators relevant for Higgs physics can however also be affected, if the dilaton mass becomes small enough. Once observed, the deviations of the Higgs couplings can become an important test of the scenarios with sizeably varying Yukawa interactions as we discuss in more detail in Sec. 7. While performing the numerical analysis of the phase transition, we will keep track of the mass mixing and require that to ensure that experimental constraints on the Higgs couplings are fulfilled.

4 Description at finite temperature

We can distinguish two qualitatively different types of finite-temperature effects: those leading us out of the applicability of the EFT, for temperatures and dilaton vevs such that , and those which do not, for . The latter are simply accounted for by the standard thermal corrections. The former can only be properly accounted for in the complete UV description up to the energy scale , including the heavier bound states and eventually their deconfined constituents. In the following we will use a limited knowledge about this UV theory to determine the main relevant features of the phase transition. This discussion is in many aspects analogous to that in [Creminelli:2001th, Randall:2006py, Nardini:2007me, Konstandin:2010cd] for the phase transition in 5D dual models.

For , the strong sector that gives rise to the Higgs is in its deconfined and (nearly) conformal phase. By dimensional analysis and large- counting, it is clear that the free energy in this phase scales as , where the constant depends on the number of d.o.f. per color in the strong sector. For definiteness, we will use the result for super-Yang-Mills (including a factor due to strong coupling which can be calculated from the AdS dual [Gubser:1996de]). The free energy then reads

(4.1)

We expect that any realistic strong sector would require an approximately similar number of d.o.f. per color as super-Yang-Mills and that the free energy would thus not differ much from the relation that we use. An additional contribution arises from the elementary SM fields, with being the total number of d.o.f. of the SM. We will neglect this contribution for now assuming (it will be accounted for in our numerical study). We also assume that the conformal symmetry breaking effects are sufficiently small to be negligible.

We next need to determine the potential at finite temperatures in the regime . For , the thermal corrections to the potential from composite states are calculable and small. Again momentarily neglecting the thermal corrections from the SM, the free energy in this regime is well approximated by

(4.2)

where and are given in Eqs. (3.21) and (3.22). The relative energy of the two phases described by Eqs. (4.1) and (4.2) is fixed by the fact that both should match in the limit and . Given the absence of robust knowledge about the temperature corrections in the opposite regime , we could neglect this part of the potential and glue (4.1) to (4.2) at using a step-function transition. The tunnelling rate which is obtained for such an approximation to the free energy would of course only give an estimate. It is however expected not to be significantly different from the exact result. To see this, recall that we are interested in regions of parameter space where the nucleation temperature is below the weak scale, and therefore significantly below the scale of the true minimum. Whatever features the bounce action has from the region , they are characterized by the only relevant scale , and are therefore expected to be smaller than the contribution from the region under control . An additional factor making the latter region more important is the fact that the dilaton potential in this region is characterised by the behaviour with and slowly varying. The height of the potential barrier is therefore almost constant over the large interval of starting from the maximal point at . A sketch of the described behaviour of the free energy is shown in Fig. 5.

Figure 5: Schematic shape of the free energy as a function of , in the “hot” region with (in red) and in the “cold” region with (in blue).

The critical temperature at which the transition becomes energetically allowed is given by the temperature at which the free energies in the deconfined and confined phases become equal. Neglecting the Higgs-dependent part in (4.2), the relevant free energy in the confined phase is given by the global minimum of the dilaton potential at ,

(4.3)

Equating with (4.1), the critical temperature follows as

(4.4)

This in particular shows that the strength of the phase transition grows with increasing . The thermal corrections from the SM fields which we have neglected so far increase the relative depth of the minimum in the deconfined phase at and thereby decrease the critical temperature.

Instead of using the step-function behaviour described above, we will model a smooth transition of the free energy in the regime between and . It is reasonable to expect that some order parameter of this hot phase, which characterises the size of the strong-sector condensate and the breaking of electroweak symmetry and which we will also call , experiences a continuous evolution from to , where it smoothly transits into the dilaton of our EFT. This assumption allows us to introduce the Higgs as a variable which defines the direction of the condensate in the space. Despite the fact that it is generally difficult to argue for the existence of composite states in the hot plasma, as at some point they get “dissolved” to their elementary constituents, the PNGB Higgs is rather a collective excitation of the condensate with a given quantum number and hence can be considered in the hot phase as well, once we assume the existence of in that regime. As in [Randall:2006py], we will model the behaviour of in the regime by adding to the zero-temperature potential ((4.2) extrapolated to ) the temperature correction from (besides the SM fields) CFT d.o.f. to which we assign a mass . The thus defined thermal correction reads

(4.5)

where the sum runs over both CFT and SM d.o.f., is the number of d.o.f. for each particle species and the masses depend on and/or . Furthermore, the functions and are given by

(4.6)

Since the difference in and is small, we will for simplicity set for the fermionic CFT d.o.f. in Eq. (4.5). For the bosonic CFT d.o.f., we then choose the normalization

(4.7)

which ensures that the free energy (4.1) in the deconfined phase at is reproduced. Moreover, the thermal correction from the CFT d.o.f. becomes strongly suppressed for , thereby resolving the previously assumed step function. This produces a barrier in the potential, whose height increases with . Corresponding to this, the critical temperature (4.4) decreases with . The strength of the phase transition is thus strongly dependent on . In order to ensure that electroweak baryogenesis is possible, the temperature at which the phase transition takes place needs to be somewhat below the electroweak scale. This also ensures that a valley along as discussed in the last section for the zero-temperature potential is not too strongly modified by the temperature corrections. From (4.4), we see that such a low phase-transition temperature can always be achieved by taking sufficiently large.

Figure 6: Left: Potential as a function of and for a meson-like dilaton with mass and and evaluated at the nucleation temperature GeV. The other parameters are as in Table 2. The solid black line shows the tunnelling path to the release point, while the red dotted line indicates the subsequent rolling trajectory towards the minimum of the potential. Right: Potential at, from bottom to top, the nucleation temperature GeV, the critical temperature GeV and . The potential is plotted along the straight line parametrized by connecting the minimum at the origin with the second minimum at , and , respectively.

We emphasize that while the procedure outlined above to estimate the potential in the regime between and carries a certain amount of speculation, we expect that this does not affect the reliability of our main results (the dynamics of the phase transition and the induced CP asymmetry). Note also that we do not include daisy resummation in our analysis. It is a subdominant effect in the region of validity of our description.

In summary, the potential that we will use for our study is

(4.8)

where , and are respectively given by Eqs. (3.21), (3.22) and (4.5). In the left panel of Fig. 6, we plot the combined potential as a function of the canonically normalized fields and (cf. Eq. (3.14)) for a meson-like dilaton with mass and and evaluated at the nucleation temperature GeV (for which ). The other parameters are as in Table 2. In the right panel, we plot the potential along straight lines connecting the two minima for three different temperatures (see caption for details). The aforementioned thermal barrier is clearly visible.

Before moving on with the analysis of the phase transition using this potential, we now discuss sources of CP violation arising in our construction which are relevant for electroweak baryogenesis.

5 CP violation from varying Yukawa interactions

In electroweak baryogenesis, the baryon asymmetry is produced during charge transport in the vicinity of the Higgs bubble walls that form during a first-order electroweak phase transition. In Ref. [Bruggisser:2017lhc], it was shown that a new CP-violating source arises if the Yukawa couplings vary across the Higgs bubble wall and that this new source can allow for enough CP-violation to generate the observed baryon asymmetry. The kinetic equations incorporating the variation of the Yukawa couplings across the Higgs bubble wall were derived and the induced CP-violating force was extracted. The resulting produced baryon asymmetry was predicted for a large set of parametrizations of the Yukawa variation. It was in particular shown that successful electroweak baryogenesis can be realised from the variation of SM Yukawa couplings using only the top and charm. In the present work, we will apply these results using the precise Yukawa variation obtained in composite Higgs models.

The CP-violating source due to varying Yukawa couplings across the Higgs bubble wall which can enable electroweak baryogenesis reads [Bruggisser:2017lhc]

(5.1)

where is the mass matrix of up- or down-type quarks (the leptons will not be important in the following), is the unitary matrix which diagonalizes , i.e. , the derivative is taken along the direction perpendicular to the bubble wall and the index stands for the diagonal elements of the corresponding matrix. Using this expression one can single out two distinct ways of sourcing CP violation which we discuss in the following.

5.1 CP violation with hierarchical quark mixings

Let us assume that the elementary-composite mixings remain hierarchical and that only the top Yukawa is of order one in the entire interval . In this case the dominant contribution to is expected to arise from the top mass:

(5.2)

In order for this to be non-vanishing, the top mass needs to have a complex phase which varies along the bubble profile. The top mass as a function of the Higgs vev is given by

(5.3)

where we have set as discussed in Sec. 2.2 and assigned the index for the top for definiteness. In general we expect the mixings and and the coupling to have constant phases so that obtaining a varying phase of the top mass is nontrivial. The subdominant corrections from the light quarks, on the other hand, are known to be insufficient to produce a large enough . We can consider two options to generate a varying phase of the top mass.

First, we can assume that one of the top quark chiralities couples to two different composite operators. This produces the elementary-composite mixing

(5.4)

An overall phase change of can then be caused by a relative change of versus , with constant but different phases. This can be sizeable if the two mixings have a comparable size.

A second possibility is that the phase changes because of an additional scalar field which enters the mixing,

(5.5)

and which undergoes a phase transition together with the Higgs. This option was proposed in [Espinosa:2011eu] but will not be considered further in this work as the nature of the electroweak phase transition in this case differs significantly from what we focus on.

5.2 CP violation with non-hierarchical quark mixings

Δ