Contents
Abstract

The cosmological evolution of standard model Yukawa couplings may have major implications for baryogenesis. In particular, as highlighted recently, the CKM matrix alone could be the source of -violation during electroweak baryogenesis provided that the Yukawa couplings were large and varied during the electroweak phase transition. We provide a natural realisation of this idea in the context of Randall-Sundrum models and show that the geometrical warped approach to the fermion mass hierarchy may naturally display the desired cosmological dynamics. The key ingredient is the coupling of the Goldberger-Wise scalar, responsible for the IR brane stabilisation, to the bulk fermions, which modifies the fermionic profiles. This also helps alleviating the usually tight constraints from -violation in Randall-Sundrum scenarios. We study how the Yukawa couplings vary during the stabilisation of the Randall-Sundrum geometry and can thus induce large -violation during the electroweak phase transition. Using holography, we discuss the 4D interpretation of this dynamical interplay between flavour and electroweak symmetry breaking.

DESY 16-221

Cosmological evolution of Yukawa couplings: the 5D perspective


Benedict von Harling and Géraldine Servant


DESY, Notkestrasse 85, 22607 Hamburg, Germany

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


benedict.von.harling@desy.de, geraldine.servant@desy.de


1 Introduction

The origin of the flavour structure is one of the major puzzles of the standard model (SM). While many solutions have been proposed, the cosmological aspects of the corresponding models have hardly been studied. On the other hand, in many cases Yukawa couplings are dynamical and it is natural to investigate the possibility of their cosmological evolution, and whether this could help addressing open problems, like baryogenesis. Such questions were started to be addressed recently [1, 2, 3]. In particular, the CKM matrix can be the unique source of -violation for electroweak baryogenesis if Yukawa couplings vary at the same time as the Higgs is acquiring its vacuum expectation value (VEV) [4]. With these motivations in mind, we are interested in studying natural realisations of Yukawa variation at the electroweak (EW) scale.

In this paper, our aim is to investigate the possibility of varying Yukawas in Randall-Sundrum (RS) models [5]. One of the very attractive features of the RS model is that in addition to bringing a new solution to the Planck scale/weak scale hierarchy problem, it offers a new tool to understand flavour and explain the hierarchy of fermion masses [6, 7, 8]. The setup is a slice of 5D Anti-de Sitter space (AdS) which is bounded by two branes, the UV (Planck) brane where the graviton is peaked, and the IR (TeV) brane hosting the Higgs (which therefore does not feel the UV cutoff). Fermions and gauge bosons are free to propagate in the bulk. In this framework, the effective 4D Yukawas of SM fermions are given by the overlap of their 5D wavefunctions with the Higgs. Since the Higgs is localised towards the IR brane to address the Planck scale/weak scale hierarchy, small Yukawas are achieved if the fermions live towards the UV brane so that the overlap between the fermions and the Higgs is suppressed. On the other hand, heavy fermions such as the top quark are localised near the IR brane. This setup leads to a protection from large flavour and -violation via the so-called RS-GIM mechanism.

The key feature for flavour physics is therefore the localisation of the fermions in the AdS slice, which determines the effective scale of higher-dimensional flavour-violating operators. The profile of a fermion is determined by its 5D bulk mass. Because of the AdS geometry, modifications of order one in the 5D bulk mass have a substantial impact on the fermionic profile and therefore on the effective 4D Yukawa coupling. In fact, the 4D Yukawa couplings depend exponentially on the bulk mass parameter. Randall-Sundrum models are holographic duals of 4D strongly coupled theories. In this picture, the Higgs is part of the composite sector. The size of the Yukawa couplings is then determined by the degree of compositeness of the states that are identified with the SM fermions. Indeed fermions localised near the UV brane are dual to mainly elementary states leading to small Yukawas while fermions localised near the IR brane map to mainly composite states with correspondingly large Yukawas.

In the usual picture, the bulk mass parameter is assumed to be constant. On the other hand, it is quite well motivated to consider that this bulk mass is dynamical and generated by coupling the fermions to a bulk scalar field which in turn obtains a VEV. We can then expect a position-dependent bulk mass as this VEV is generically not constant along the extra dimension. In fact, the simplest mechanism for radion stabilisation, due to Goldberger and Wise [9], consists in introducing a bulk scalar field which obtains a VEV from potentials on the two branes. The most minimal scenario to dynamically generate the bulk mass is then to use this bulk scalar. Interestingly, during the process of radion stabilisation, the profile of the Goldberger-Wise scalar VEV changes. When the latter is coupled to the fermions, this induces a change in the bulk masses of the fermions which in turn affects their wavefunction overlap with the Higgs on the IR brane and thus the Yukawa couplings. The RS model with bulk fermions therefore naturally allows for a scenario of varying Yukawa couplings during the EW phase transition. Our goal is to study the cosmological dynamics of Yukawa couplings in this context.

The emergence of the EW scale in RS models comes during the stabilisation of the size of the AdS slice. At high temperatures, the thermal plasma deforms the geometry and the IR brane is replaced by a black hole horizon. Going to lower temperatures, eventually a phase transition takes place and the IR brane re-emerges. This phase transition is typically strongly first-order and proceeds via bubble nucleation. The walls of these bubbles then interpolate between AdS with an IR brane at infinity and at a finite distance. In the dual 4D theory, this transition is described by the dilaton – which maps to the radion – acquiring a VEV. To realise a model where the Yukawas are larger during the phase transition (as needed if we want to use the SM Yukawas as the unique -violating source during EW baryogenesis [4, 1]), we ask for the Yukawas to become larger when the IR brane is pushed to infinity.

We will discuss two realisations of this. One way to induce varying Yukawas is to add an operator on the IR brane that effectively changes the value of the Yukawa coupling as the position of the IR brane changes. This mechanism enables variations of order one for the Yukawas and can be relevant for -violation if applied to the top quark. We discuss this option in sec. 6. The other possibility to implement large Yukawas during the phase transition is to have a bulk mass for the fermions which decreases towards the IR. Since smaller bulk masses make the wavefunctions grow faster towards the IR, this leads to fermions which become increasingly IR-localized when the IR brane is pushed to infinity. The wavefunction overlap with the Higgs near the IR brane and thus the Yukawas then grow too. This mechanism can be relevant for -violation for all quarks and enables a large variation of the Yukawa couplings, from values of order one to today’s small values of the light quarks. This realisation will be described in sec. 7.

The plan of the paper is the following. The motivations for this study are reported in sec. 2, where we summarise the key features of electroweak baryogenesis. The Goldberger-Wise mechanism and the description of the EW phase transition in RS models are reviewed in secs. 3 and 4 respectively. The derivation of 4D Yukawa couplings in RS models is reviewed in sec. 5. In sec. 6, we present a first possible mechanism for Yukawa coupling variation, which relies on a new contribution to the Yukawa coupling on the IR brane. Sec. 7 discusses a generic mechanism for modifying fermionic profiles. The main idea is presented through a simple model in 7.1. Its realistic implementation is given in sec. 7.2. In sec. 8, we discuss the implications of our constructions for flavour and -violating processes. Sec. 9 provides the interpretation of the models in the dual CFT. We conclude in sec. 10.

2 Electroweak baryogenesis with varying Yukawa couplings

Electroweak baryogenesis is an appealing mechanism for explaining the baryon asymmetry of the universe, which relies on electroweak scale physics only (see e.g. [10]). It occurs in the framework of a first-order electroweak phase transition, in the vicinity of Higgs bubble walls, separating the broken phase where baryon number is conserved from the symmetric phase where sphaleron transitions are unsuppressed. Because of -violating interactions in the bubble walls between particles in the plasma and the Higgs, a chiral asymmetry may be generated and converted into a baryon asymmetry by sphalerons in front of the bubble walls. Due to the wall motion, the baryon asymmetry diffuses into the broken phase, where sphalerons are frozen, and the asymmetry is not washed out. All models of EW baryogenesis postulate the existence of a new -violating source beyond the CKM phase, as needed to explain the baryon asymmetry. This is typically strongly constrained by measurements of electric dipole moments (EDMs), see e.g. [11]. However, as studied in depth in [4], if Yukawa couplings vary across the bubble walls, this provides a source of -violation which is active at early times only, and therefore not in tension with EDM experimental bounds. This source scales like

(2.1)

where is the fermion mass matrix, is the matrix that diagonalizes , the derivative is with respect to the coordinate perpendicular to the bubble wall and only the diagonal elements of the matrix in brackets are relevant. Such a term vanishes for the Yukawas in the SM as they are constant across the bubble wall. On the other hand, it is conceivable to use the CKM matrix as the -violating source for EW baryogenesis if the Yukawa couplings vary at the same time as the Higgs is acquiring its VEV . In fact, the correct amount of baryon asymmetry is naturally obtained if the Yukawa couplings varied from values of order 1 in the symmetric phase () to their present value in the broken phase () [4]. This observation is the driving motivation for this study and we are interested in providing a natural realisation of such Yukawa coupling variation during the EW phase transition.

There are two ways to get enough -violation from the source term (2.1). It is possible to rely on the top Yukawa coupling only, provided that its phase changes during the EW phase transition. Indeed, writing , we have

(2.2)

This can happen naturally in models where the top Yukawa coupling receives two contributions of order one,

(2.3)

with being constant, while is varying across the bubble wall and is some arbitrary initial complex phase. This setup generates an effectively varying phase . Using the profile of the Higgs VEV across the bubble wall, we can trade the coordinate for . This thus leads to a phase which effectively varies as the Higgs field is rolling towards the minimum of its potential:

(2.4)

As shown in [12, 13, 4], if the top Yukawa coupling had such a varying phase during the electroweak phase transition, this can explain the baryon asymmetry of the universe.

The other possibility is to have Yukawa couplings whose phases do not vary but whose absolute values change. As follows from eq. (2.2) with , in this case the source for one flavour vanishes and we thus need at least two flavours. These two situations are studied in depth in [4]. Although the full calculation is presented in [4], it was shown that the top-charm system gives the dominant contribution. Our two models I and II discussed in sections 6 and 7 of this paper illustrate these two cases. To introduce these new findings, we need first to review several basic features of the physics of Randall-Sundrum models.

3 Review of the Goldberger-Wise mechanism

We now review a key aspect of RS models known as the Goldberger-Wise mechanism [9]. The general construction we consider is based on a slice of AdS with metric

(3.1)

and branes at (UV/Planck brane) and (IR/TeV brane). The theory could be defined on an orbifold or an interval. In either case, we restrict the coordinate to the interval here and below.111To calculate integrals over -functions on the boundaries, we first move the -functions away from the brane into the interval, perform the integral and then send to 0 (e.g. ). We assume that the Higgs is localized on the IR brane, whereas the fermions and gauge bosons live in the bulk. We also introduce a real scalar field in the bulk with potentials on the branes. Its action reads

(3.2)
(3.3)

All dimensionful parameters (in particular the AdS curvature scale ) are expected to be of order one in Planck units. The potentials cause the scalar to develop a VEV with a profile along the extra dimension given by (see e.g. [9])

(3.4)

where

(3.5)

The constants and are determined by the boundary conditions which read

(3.6)
(3.7)

where the warp factor at the IR brane,

(3.8)

defines the radion field. The aim is to stabilize the radion such that , which represents the effective cutoff scale on the IR brane (and therefore for the Higgs mass parameter). In the limit of large couplings and , the last term in eqs. (3.6) and (3.7) dominates and we get and . This in turn gives

(3.9)
(3.10)

where we have assumed that and are of comparable size. The leading corrections to this in and to zeroth order in are given by [9]

(3.11)
(3.12)

We see that is suppressed relative to by powers of the warp factor and is thus always negligible. Furthermore, can be neglected relative to for which we will assume in the following.

The contribution of the scalar VEV to the potential energy depends on the size of the extra dimension. Integrating over the extra dimension, the resulting potential for the radion is given by [9]

(3.13)

In the limit of large , the boundary conditions give and and the boundary potentials thus vanish. The corrections to this coming from eqs. (3.11) and (3.12) for finite are negligible for . Similarly, the corrections to the - and -dependent terms in eq. (3.13) are then negligible too. The potential has a minimum for . Expanding in , we find

(3.14)

A large hierarchy can thus be generated from an -ratio if . Note that an additional term in the potential can allow for a minimum also for negative [14, 15]. Such a term can arise from a detuned brane tension on the IR brane. However, we find that in the two models that we consider negative causes the Yukawa couplings to become nonperturbative when the IR brane is sent to infinity.222For model I, this can be anticipated from eq. (6.3). The new contribution to the Yukawa coupling remains proportional to for negative which causes it to diverge in the limit . Model II with negative can give growing wavefunctions for decreasing if for the modified profile discussed in sec. 7.2. For sufficiently small , this results in the fermions being IR-localized. Using eq. (7.9), we see from eq. (7.22) that the Yukawa coupling then is proportional to factors of for each of the two fermions. Again this diverges in the limit . We therefore focus on positive in this paper.

Note that the bulk potential in eq. (3.2) only contains a mass term for . In principle also higher-order terms in can appear. The leading such term, , was included in the analysis of refs. [16, 17]. The resulting profile for the Goldberger-Wise scalar was found to have -corrections compared to the profile for a bulk potential with only a mass term. Note that such a -term is in principle expected in model II discussed later because of the Yukawa coupling in the bulk, though it may be small. Nevertheless even if it is sizeable the profile for positive will still decay by an -factor when going from the UV to the IR. This is the crucial feature that we need for model II and we therefore expect this mechanism to work also if the bulk potential contains higher-order terms in . It is less clear, on the other hand, if the derivative of the Goldberger-Wise scalar at the IR brane is then still suppressed when the radion is at the minimum of its potential. This is the crucial feature which is needed for model I discussed later. However, as it has no bulk Yukawa coupling, the -term in model I can be forbidden by imposing a -symmetry. We leave a detailed study of our mechanism for this more general case to future work.

The non-constant piece of the potential (3.13) is of the dilaton type,

(3.15)

where is a very slowly-varying function since it depends on only. The cosmological dynamics of this very shallow potential was summarized in ref. [18]. We discuss it next.

4 The electroweak phase transition in Randall-Sundrum models

While there has been an extensive literature on the phenomenology of Randall-Sundrum models, little has been established on its early cosmology. On the other hand, the attractivity of this solution to the hierarchy problem also relies on whether it is cosmologically realistic. One of the very first aspects to be checked was that the Friedmann equation could in fact be recovered, as expected, since gravity is effectively 4-dimensional in this model, at energies below the EW scale when the radion is stabilized [19, 20].

Figure 1: Schematic depiction of the phase transition: A bubble of the Randall-Sundrum phase emerges from the surrounding AdS-Schwarzschild phase. In the transition region between the two phases, one sees the black hole horizon receding to infinity and subsequently the IR brane coming in from infinity.

On the other hand, the knowledge of what happened before radion stabilisation is less under control. Nevertheless the phase transition leading to the stabilisation of the radion can be understood as follows [14]: At high temperatures, the system is in an AdS-Schwarzschild phase with a UV brane and a black hole horizon in the IR (whose Hawking temperature matches the temperature of the system). In the dual picture, this corresponds to the strongly-coupled theory being in the deconfined phase and the free energy scales like as expected. Going to lower temperatures, eventually a phase transition happens and the black hole horizon is replaced by the IR brane. This phase transition is typically strongly first-order and proceeds via bubble nucleation. Both geometries – AdS-Schwarzschild and the Randall-Sundrum geometry with two branes – have different topologies. They can be smoothly connected, however, by sending respectively the horizon and the IR brane to infinity which gives pure AdS (cutoff by the UV brane). It is therefore expected that the bubble walls interpolate between the two phases as follows [14]: Going perpendicular to the bubble wall from the AdS-Schwarzschild phase outside towards the Randall-Sundrum phase inside, we first see the horizon receding until we arrive at pure AdS. Further towards the inside, the IR brane comes in from infinity until it arrives at its stabilized position as determined by the Goldberger-Wise mechanism. This is depicted in fig. 1. The radion thus varies from on the outside of the bubble wall to inside the bubble. This behaviour will be crucial for us as our models have Yukawa couplings which grow with decreasing and thus grow across the bubble walls.

This phase transition to the Randall-Sundrum phase is special due to the nearly conformal nature of the radion potential (3.13) whose cosmology is different from the one of usual polynomial scalar potentials. The tunneling action can be calculated by determining the bounce solution for the radion potential which was given in the previous section [14, 15, 21, 22, 18]. One finds that in the calculable region of parameter space, the phase transition may complete but is typically very strong and happens after significant amount of supercooling, see [18] and [23] for a recent updated status summary.333See however ref. [24] for an alternative solution changing this conclusion. Indeed the nucleation temperature can be parametrically much smaller than the scale associated with the minimum of the potential. An interesting signature of this scenario is the typical large signal amplitude of the stochastic gravity wave background peaked in the millihertz range inherited from the time of the phase transition, and observable at LISA [15, 23].

During the phase transition, also the electroweak symmetry gets broken. We assume that the Higgs is localized on the IR brane. The action for the Higgs then reads

(4.1)

where . In terms of the canonically normalised Higgs field , the Higgs potential reads

(4.2)

The Higgs VEV then scales like

(4.3)

where is the electroweak scale. In deriving eq. (4.3), we have assumed that the Higgs is always at the minimum of its potential during the phase transition to the Randall-Sundrum phase. This is an idealised situation, however, we can expect this description to be physically sensible. To derive the exact relation between the Higgs VEV and the radion VEV, one has to compute the bounce, something which we postpone to future work. The special features in the RS case are i) a nearly conformal potential along the radion direction, ii) no quadratic term for the Higgs if the radion vanishes. Therefore, electroweak symmetry cannot be broken unless the radion has a VEV. If electroweak symmetry breaking takes place only after the radion settled in its minimum, then there is no variation of the Yukawa couplings during the electroweak phase transition. It is therefore crucial that they both change at the same time, which is what we expect. Indeed, if the Higgs and the radion were on equal footing, i.e. both having similar potentials, then the path in the two-dimensional field space would be along the diagonal as both fields would follow the same tendency if they have similar masses. On the other hand, if the radion is much heavier than the Higgs, we expect the tunneling to proceed first along the radion direction and only then along the Higgs direction. We illustrate this schematically in fig. 2. Therefore, the optimal case will be for a relatively light radion. Determining the precise relation between the Higgs and radion VEVs as a function of the radion mass will be an interesting task in itself.

Figure 2: Left: Sketch of possible paths in the dilaton-Higgs field space. The red solid line reflects the linear relationship (4.3) between the two VEVs. Right: The corresponding profiles of the fields along the bubble walls. The red solid line corresponds to the dilaton and Higgs bubble walls precisely overlapping and thus the linear relationship (4.3). For the dotted and dashed lines, the dilaton reaches its minimum before the Higgs does, which tends to attenuate the variation of the Yukawa couplings during the EW phase transition.

The breaking of electroweak symmetry is thus tied to the radion cosmology.444Implications for cold baryogenesis were studied in refs. [26, 25]. Since the phase transition to the Randall-Sundrum phase is typically strongly first-order, the electroweak phase transition is then first-order too. This motivates the possibility of electroweak baryogenesis, provided that the bubble wall velocity is smaller than the sound speed (for larger velocities the baryon asymmetry vanishes as there is no time for -violating diffusion processes in front of the bubble walls where sphalerons are active). In fact, the danger for electroweak baryogenesis in strong first-order phase transitions is that the friction exerted by the plasma on the wall might not be sufficient to prevent the bubble wall from a runaway behavior in which case the wall keeps accelerating, towards ultra-relativistic velocities. The determination of the bubble wall velocity is a non-trivial calculation. It depends on the strength of the phase transition, i.e. the amount of latent heat released, as well as the amount of friction between the particles in the plasma and the bubble wall [27]. Friction is due to particles changing mass across the wall. In contrast with the SM or the MSSM, we expect that a large number of degrees of freedom become very massive during the RS phase transition. Since the precise theory in the CFT phase is unknown (in particular the number of CFT degrees of freedom), the friction is left as a free parameter. But we can expect that for a large number of CFT degrees of freedom, friction will be relevant. It is clear however that it will be effective only for not too low nucleation temperatures. As the nucleation temperature is typically smaller than the scale set by the radion VEV at the minimum of its potential [18], conditions for EW baryogenesis may not be satisfied for a generic choice of parameters. We leave the model-dependent detailed analysis of the EW phase transition for future work. Therefore it should be clear that the possibility of EW baryogenesis is based on the assumption that there exists a region in parameter space where the phase transition is moderately strong and the bubble wall velocity can be subsonic. We then show that the RS setup generically incorporates the variation of Yukawas during the EW phase transition and therefore enables to realise EW baryogenesis with the CKM matrix as the only -violating source.555Note that another paper, ref. [28], entertained the idea of varying Yukawas during the dynamics that stabilize fermion profiles in (unwarped) extra-dimensional models, however, at a scale much above the electroweak scale, and therefore for a baryogenesis mechanism requiring higher-dimensional - violating operators.

Note that even if the bubble wall velocity is supersonic, our discussion is relevant since baryogenesis at the electroweak scale is still possible through a different mechanism, so-called “cold baryogenesis”, which does not rely on a transport mechanism, and is especially motivated in the context of the supercooled RS phase transition [26, 25]. The source of -violation that we find from Yukawa variation could be used also in this context.

We thus want to generate Yukawa couplings between the Higgs and the fermions that change in size when the IR brane is moved away from the minimum of the Goldberger-Wise potential. To this end, we consider in sections 6 and 7 two realisations, first through a new IR contribution from the Goldberger-Wise field to the Yukawa couplings and second through the bulk coupling of the Goldberger-Wise field to the fermions. Before doing that, we review how Yukawa couplings arise in RS models.

5 Review of fermions in Randall-Sundrum models

We now review fermions in RS models and how the fermion mass hierarchy arises. In this paper, we are mainly interested in the Yukawa couplings of the up-type quarks. We denote by and the bulk fields that give rise to the left-handed quark doublet and the right-handed up-type quarks, respectively. Including the kinetic term for completeness, the bulk action for the left-handed quark doublets reads (see e.g. [29])

(5.1)

and similarly for the right-handed up-type quarks .666On an orbifold needs to be odd, , since is odd. Alternatively we can define the theory on an interval and then impose the same boundary conditions as on the orbifold. is the inverse vielbein, is the spin connection and with is the bulk mass of the 5D fermion. For simplicity, we have suppressed the flavour indices. Note that we can perform unitary transformations such that the kinetic terms and the mass terms are diagonal in flavour space. We will use this basis throughout this paper. The Yukawa coupling reads

(5.2)

where has dimension -1 and is the brane-localized Higgs (whose kinetic term and potential are given in eq. (4.1)).

We decompose the bulk fermions and into left- and right-handed spinors and Kaluza-Klein (KK) modes. This gives and

(5.3)

and similarly for . The equations of motion for then read

(5.4)

where are the KK masses. Notice that we have allowed for the possibility that is a function of which will become important later. The wavefunctions fulfill the orthonormality conditions777For the general case of a position-dependent bulk-mass parameter , the equations of motion for the two chiralities can be combined and rewritten as where This has the form of a Sturm-Liouville equation (see e.g. eq. (13) in ref. [8]). The problem therefore has a discrete set of real eigenvalues . The eigenfunctions form a complete set and satisfy the orthonormality relation which gives eq. (5.5). This guarantees that the Lagrangian in terms of the KK modes is diagonal.

(5.5)

In order to ensure that the boundary terms vanish after the variation of the action, we can impose that either the left- or right-handed fermion is zero at the two branes (see e.g. [30]). This leaves one chiral massless mode, , which we identify with the SM fermion. We then choose the boundary conditions such that has a left-handed massless mode, whereas the massless mode from is right-handed. If the bulk masses and are constant, as usually assumed in the literature, the wavefunctions for the left-handed massless modes from then read

(5.6)

where

(5.7)

is a normalization constant. For later convenience, we redefine for the bulk fermions with right-handed massless modes. Their wavefunctions are then again given by eqs. (5.6) and (5.7) with replaced by . With this convention, both left- and right-handed massless modes are UV (IR) localized for ().

The effective 4D Yukawa coupling between the SM fermions and the Higgs is given by

(5.8)

where to obtain a canonically normalized kinetic term and

(5.9)

For , this becomes exponentially suppressed. This shows how large hierarchies between the 4D Yukawa couplings can be obtained in RS starting from bulk mass parameters and 5D Yukawa couplings of order one in units of the AdS scale . Notice that already in this setup the Yukawa couplings depend on the position of the IR brane. Since the light quarks are all localized towards the UV brane, however, their Yukawa couplings decrease when the IR brane is sent to infinity, . Correspondingly, they are small in a large portion of the bubble wall during the phase transition and -violation is suppressed. We will later see how modified fermion profiles can lead to increased Yukawa couplings during the phase transition.

The parameters that determine need to be chosen such that the measured masses and mixing parameters are reproduced. This still leaves a considerable freedom. For definiteness, we will use a benchmark point for these parameters from ref. [31]. We need to adjust the parameters, however, since for the benchmark point a hierarchy was assumed, whereas we choose in this paper.888For example for , this would give an IR scale . This would be consistent with electroweak precision tests even without a custodial symmetry (though it requires a cancellation of order 25% in the contributions to to be viable) [32, 33]. In addition, we reduce the 5D Yukawa couplings involving the left-handed top-bottom doublet by a factor compared to those of the benchmark point. This will ensure that the couplings do not become nonperturbative in the limit in the models that we consider later. After making these two modifications, we adjust the bulk-mass parameters such that the 4D Yukawa couplings are again reproduced. We will only list the parameters for the top-charm sector since it gives the dominant effect for the models that we consider later. In a basis such that the couplings in the Lagrangian are proportional to , the 5D Yukawa couplings read

(5.10)

and the bulk-mass parameters are

(5.11)

Here and below indices on the fields and denote the generation. The value for can be consistent with constraints from the -coupling [31]. Together with the other parameters for the benchmark point, these parameter values reproduce the measured quark masses and mixings when the running from an IR scale of to the electroweak scale is taken into account. Note that we assume a slightly larger IR scale. However, we expect the required adjustments in the parameters that we are interested in to be small and will neglect them in the following. Note also that the Yukawa couplings in the plots are thus given at and will change slightly when run down to the electroweak scale.

In the next sections, we will make some rather small but influential modifications to this commonly used picture.

6 Model I: A new IR contribution to the Yukawa couplings

The first model that we present involves a higher-dimensional coupling of the Goldberger-Wise scalar to the Yukawa operator on the IR brane. This gives an additional contribution to the Yukawa coupling. We then use the fact that the VEV of the Goldberger-Wise scalar changes when the IR brane is moved, leading to a change in the Yukawa coupling. The boundary potential keeps the VEV at the IR brane relatively constant, . A coupling therefore does not result in a sufficient change for our purposes. We instead consider a derivative coupling which can for example arise due to a finite thickness of the brane. For the up-type quarks, eq. (5.2) now becomes

(6.1)

We have again suppressed the flavour indices for the fields and the coupling constants and (which have dimensions and , respectively). Similar couplings can exist for the down-type quarks but it is enough to focus on the up-type couplings for our purposes. The nonvanishing derivative of the VEV (3.4) of the Goldberger-Wise scalar at the IR brane gives an additional contribution to the 5D Yukawa coupling which depends on the position of the IR brane:

(6.2)

where

(6.3)

Note that the contribution from the derivative coupling is suppressed by a factor if the radion is at the minimum of its potential, . This can be understood as follows: Both the bulk potential and the kinetic term of the Goldberger-Wise scalar in eq. (3.2) contribute to the radion potential. Since , the former is suppressed by . The minimum of the potential then occurs at a radion VEV for which the latter is suppressed by too. This leads to

(6.4)

near the stable position of the IR brane. This suppression can be seen in fig. 3(a), where we plot the VEV of the Goldberger-Wise scalar along the extra dimension if the radion is at the minimum of its potential (we choose and ). The suppression is lifted when the IR brane is moved to infinity, , and the Yukawa coupling correspondingly grows. This is visible in fig. 3(b) which shows the VEV for the same parameters as in fig. 3(a) but with the radion at .

Figure 3: (a) VEV of the Goldberger-Wise scalar in eq. (3.4) along the extra dimension if the radion is at the minimum of its potential, chosen as . (b) VEV for the same parameters as in (a) but for the radion at .

The resulting 4D Yukawa coupling is obtained from eq. (5.9) with the replacement . The new contribution to the effective 5D Yukawa coupling grows by a factor when is changed from to . Accordingly, this model enables variations in the Yukawa couplings of order one only. The Yukawa coupling receives two contributions like in eq. (2.3), on the other hand, and we can therefore still use it for the top quark as discussed in sec. 2. Note that since the top is localized in the IR, the prefactor from the wavefunction overlaps in eq. (5.9) depends only very weakly on (for the bulk mass parameters in eq. (5.11), it changes by about when is varied from to ). The dominant variation in the Yukawa coupling then arises from .

In order to reproduce the observed quark masses and mixings, we need to match the effective 5D Yukawa coupling evaluated at the minimum of the Goldberger-Wise potential with the values in eq. (5.10). This fixes the combination . In order to estimate the size of the remaining, free combination of and , we use naive dimensional analysis (NDA) [34, 35]. Assuming that all loop processes become strong at a cutoff scale , we write

(6.5)

where is the -dimensional loop factor and and are functions of the dimensionless ratios and . After canonical normalisation of the fields, this gives

(6.6)

where we have used that from NDA and the coefficients and are of order one. We next need to estimate the allowed sizes of and . The AdS curvature scale is limited by the requirement that higher-curvature terms in the action can be neglected so that the solution to the Einstein equation can be trusted. Using NDA, this gives [36]. Similarly, the VEV at the IR brane is limited by demanding that the backreaction of the Goldberger-Wise scalar on the geometry can be neglected. Since we want to ensure this also away from the minimum of the Goldberger-Wise potential, the resulting condition is somewhat more stringent than usual. Indeed, for the VEV is well approximated by in the IR. The contribution to the energy-momentum tensor from the kinetic term is then not suppressed by (contrary to the case ). In particular, near the IR brane we have

(6.7)

Demanding that this is negligible compared to the contribution from the bulk cosmological constant, , gives .

Figure 4: The top Yukawa coupling eq. (5.9) with given by eq. (6.3), as a function of the Higgs VEV for and different values of .

For definiteness, we set and . We then fix for given and by the requirement that the 5D Yukawa coupling for the top in eq. (5.10) is reproduced. We also trade the radion VEV for the Higgs VEV via the relation in eq. (4.3). In fig. 4, we plot the top Yukawa coupling as a function of the Higgs VEV for and different values of (for all these values ). As one can see, the coupling varies with decreasing Higgs VEV. This corresponds to the fact that the derivative of the Goldberger-Wise scalar at the IR brane and its contribution to the Yukawa coupling changes when the IR brane is sent to infinity. In the limit , the top Yukawa coupling becomes for which is still in the perturbative regime.

In summary, this simple construction allows for Yukawa coupling variation of order one during the EW phase transition. When applied to the top quark, it can therefore provide sufficient -violation for EW baryogenesis. As discussed in sec. 8, implications of this model for flavour and -violating observables are rather minor. We next move to what we consider to be the most interesting aspects of our study.

7 Model II: Large Yukawa couplings from modified fermion profiles

As reviewed in sec. 5, the massless modes of bulk fermions with constant mass terms have profiles along the extra dimension which are localized towards either the UV or IR brane. For our second model, we consider a Yukawa coupling of the Goldberger-Wise scalar to the bulk fermions, giving rise to position-dependent mass terms for the fermions. These modify the profiles of the massless modes and allow for profiles which are localized in the UV and thus decay towards the IR but then ‘turn around’ at some point along the extra dimension and start growing again towards the IR. Fermions which are UV-localized if the IR brane is at the minimum of the Goldberger-Wise potential can then become IR-localized when the IR brane is moved to infinity. This increases the Yukawa couplings to the Higgs on the IR brane.

The fermionic action in the bulk is the same as eq. (5.1) except for the last term which we replace by

(7.1)

where has dimension . We consider a similar coupling for the right-handed up-type quarks . Note that we can again perform unitary transformations such that the kinetic terms and the new Yukawa couplings are diagonal in flavour space. The calculations will be performed in this basis here and below. Furthermore, note that we have assumed that any constant contributions to the bulk masses are negligible. We expect that, even if they are sizeable, our picture does not change qualitatively. Indeed below we study a Goldberger-Wise scalar with a constant contribution to the VEV. The more general case with separate constant and -dependent contributions to the bulk mass would require a -dependent diagonalization of the action. But we expect that the resulting diagonal bulk masses would then give similar wavefunctions as for the Goldberger-Wise scalar with the constant and -dependent contributions to the VEV. Nevertheless we leave a detailed study of the more general case to future work. In sec. 7.1, we work out the consequences of the above coupling for a Goldberger-Wise scalar with a profile as discussed in sec. 3. In sec. 7.2, we then consider a modified profile for the Goldberger-Wise scalar with the aforementioned constant contribution which leads to faster growing Yukawa couplings to the Higgs.

7.1 Using the Goldberger-Wise scalar

The profile of the Goldberger-Wise scalar in eq. (3.4) has two pieces, and . As can be seen in fig. 3, the first piece becomes important only close to the IR brane. In order to simplify the calculation, we therefore approximate the profile by the second piece:999This profile also arises for a vanishing potential on the IR brane, (though such a scalar no longer stabilizes the extra dimension). Indeed the boundary conditions eqs. (3.6) and (3.7) in this case give and in the limit of large . Comparing the resulting sizes of the two contributions to the profile, and , we find that it is everywhere well approximated by eq. (7.2).

(7.2)

Later we will check explicitly that this gives an excellent approximation to using the exact profile in eq. (3.4). The bulk equation of motion for is given by eq. (5.4) with

(7.3)

where the constants

(7.4)

are dimensionless. The wavefunctions of the left-handed massless modes of then are

(7.5)

with the modified normalisation constant

(7.6)

and is the exponential integral function. For the bulk fermions with right-handed massless modes, we redefine . Their wavefunctions are then given by eqs. (7.5) and (7.6) with replaced by .

Figure 5: From left to right, the IR brane is being pushed away from the UV brane with the hierarchies and respectively. Upper panel: The normalized wavefunction of the right-handed charm along the extra dimension. The solid curve is the wavefunction for the position-dependent bulk mass in eq. (7.3), whereas the dashed curve is for the usual case with constant bulk mass. Lower panel: The bulk-mass parameter of the right-handed charm along the extra dimension. The solid curve is again for the position-dependent case in eq. (7.3) and the dashed curve for the usual constant case. The red curve marks the value for which the wavefunction changes from decaying to growing towards the IR.

In order to fix the parameters , we again use the benchmark point from ref. [31]. By demanding that the wavefunction overlap with the IR brane of our fermion profiles agree with that for the fermion profiles with constant bulk mass terms, we can translate their values for to values for our . Choosing and the hierarchy in the minimum of the radion potential as , we find for the top-charm sector:

(7.7)

In the upper panel of fig. 5, we show the resulting wavefunction of the right-handed charm along the extra dimension (mulitplied by as this gives the function whose square is normalized to one, cf. eq. (5.5)). The three figures correspond to the hierarchies and so the sequence from left to right can be understood as going along the bubble wall profile where the IR brane is moved to infinity. As one can see, the wavefunction initially decays when going from the UV to the IR but then starts to grow again. This can be understood as follows: As reviewed in sec. 5, for a fermion with constant bulk mass , the massless mode is UV (IR) localized for . In our setup, the bulk mass is and depends on the position along the extra dimension. In the lower panel of fig. 5, we plot the bulk-mass parameter for the right-handed charm along the extra dimension. The three figures again correspond to the hierarchies and . Notice that near the UV brane and the wavefunction thus decays towards the IR in that region. This changes to near the IR brane, on the other hand, leading to a growing wavefunction towards the IR. Since is always smaller than sufficiently deep in the IR, we see that in our model all fermions eventually become IR-localized if the IR brane is moved to infinity. This is visible in the upper right plot in fig. 5. In fig. 6(a), we show all the wavefunctions from the charm-top sector for the case that the radion is at the minimum of its potential, . Note that for the right-handed top, everywhere and the wavefunction is thus completely localized towards the IR.

As before, we assume that the Higgs is localized on the IR brane. In order to simplify the discussion, we do not couple the Goldberger-Wise scalar to the Yukawa operator on the IR brane as in sec. 6. Both effects – from the coupling in the bulk and on the IR brane – could of course be present simultaneously and would then give even stronger -violation during the phase transition. The 5D Yukawa coupling of the bulk fermions and to the Higgs on the IR brane is then given by eq. (5.2), leading to the 4D Yukawa coupling in eq. (5.8) with

(7.8)

Let us study the above expression in some limits. Since , the exponential integral functions in the normalization constants are well approximated by the leading term in the expansion

(7.9)

for large argument [37]. The expression for the 4D Yukawa coupling then simplifies to

(7.10)

From this we see immediately that in the limit we have

(7.11)

This just reflects the fact that all fermions become IR-localized for as noted above so that there is no wavefunction suppression of the Yukawa coupling any more. In the limit we find

(7.12)

This agrees with the expression in eq. (5.9) for the 4D Yukawa coupling for fermions with constant bulk masses, as is expected since becomes constant for . Similarly, the profile of the massless mode (7.5) agrees with the profile (5.6) for the case of constant bulk masses in that limit.

For a fermion that is localized towards the UV brane, the normalization constant (7.6) depends only weakly on the position of the IR brane. We can then neglect the corresponding part in the expression. If both and are UV-localized, this gives

(7.13)

We see that for , the exponential decreases if becomes smaller. For a certain range of , this can offset the increase due to the factor of . However, eventually the latter effect starts to dominate and the Yukawa coupling keeps growing with decreasing . This change happens near a position of the IR brane where the wavefunctions turn from decaying to growing towards the IR. For very small , the approximation leading to eq. (7.13) then eventually breaks down because the fields become localized in the IR and the Yukawa coupling is better approximated by eq. (7.11).

Figure 6: (a) Profile along the extra dimension of the left- and right-handed charm (blue and yellow), and the left- and right-handed top (green and red) for the approximation (7.2) to the Goldberger-Wise profile. (b) Yukawa couplings of the charm (blue), of the top (yellow) and the off-diagonal Yukawa couplings (green) and (red). The solid curves were generated using the approximation (7.2) to the Goldberger-Wise profile, whereas for the dashed curves the exact expression (3.4) was used.

In fig. 6(b), we plot the Yukawa couplings for the top-charm sector using the parameters for the benchmark point in eqs. (5.10) and (7.7). We again trade the radion VEV for the Higgs VEV via the relation in eq. (4.3). We see that the Yukawa couplings grow with decreasing Higgs VEV (or radion VEV). In particular, the charm coupling and the charm-top coupling become of order 1 for Higgs VEVs less than about GeV. On the other hand, the top coupling remains almost constant. This is due to the fact that the right-handed top is highly localized in the IR for any position of the IR brane (cf. fig. 6(a)).

So far we have approximated the Goldberger-Wise scalar by the simplified profile in eq. (7.2). Let us now consider the exact profile in eq. (3.4). For the left-handed massless modes of , the equation of motion (5.4) is then solved by

(7.14)

where again and . The normalization constant does not allow for an analytic expression and needs to be evaluated numerically from the orthonormality condition (5.5). As before, we redefine for the bulk fermions with right-handed massless modes so that their wavefunctions are given by eq. (7.14) with .

In fig. 6(b), we plot the resulting Yukawa couplings for the benchmark point in eqs. (5.10) and (7.7) as dashed lines using the same colour code as for the approximate profile (7.2). As one can see, the difference between using the exact and approximate profiles is marginal (the charm coupling and the top-charm coupling differ by about at and it is even less for the other couplings). This can be understood as follows: At the minimum of the Goldberger-Wise potential, for , we have as follows from eqs. (3.9) and (3.14). The profile of the Goldberger-Wise scalar is then everywhere well approximated by the simple profile in eq. (7.2). For , on the other hand, we have