# Fermion masses and mixing in a 4+1-dimensional domain-wall brane model

###### Abstract

We study the fermion mass and mixing hierarchy problems within the context of the 4+1d domain-wall brane model of Davies, George and Volkas. In this model, the ordinary fermion mass relations of grand unified theories are avoided since the masses are proportional to overlap integrals of the profiles of the electroweak Higgs and the chiral components of each fermion, which are split into different 3+1d hyperplanes according to their hypercharges. We show that the fermion mass hierarchy without electroweak mixing can be generated naturally from these splittings, that generation of the CKM matrix looks promising, and that the Cabibbo angle along with the mass hierarchy can be generated for the case of Majorana neutrinos from a more modest hierarchy of parameters. We also show that under some assumptions made on the parameter space, the generation of realistic lepton mixing angles is not possible without fine-tuning, which argues for a flavour symmetry to enforce the required relations.

## I Introduction

In the standard model (SM), three of the most open problems are how the fermion mass hierarchy is generated, the origin of small mixing angles in the Cabibbo-Kobayashi-Maskawa (CKM) matrix and near tribimaximal mixing in the lepton sector. With neutrino masses now known to be nonzero but under , the mass hierarchy has a spread of at least 14 orders of magnitude, given that the top quark has a mass of roughly . Amongst approaches used for solving these problems are grand unified theories (GUTs), higher dimensional operators, and flavor symmetries.

One of the most promising new theoretical frameworks for solving hierarchy problems that has emerged over the last decade has been extra-dimensional models, such as the Arkani–Hamed-Dimopoulos-Dvali (ADD) model originalbranepaper (), and the two Randall-Sundrum (RS) models randallsundrum2 (); randallsundrum1 (). The ADD and RS1 frameworks solve the hierarchy problem between the Planck scale and the electroweak scale, which is of a similar order of magnitude to that of the fermion mass spectra. For other papers on extra-dimensional models, see Refs. antoniadisnewdimattev (); newdimatmillimeter (); exotickkmodels (); gibbonswiltshire (); pregeometryakama ().

In RS2 models the gauge hierarchy problem is not solved by extra-dimensional physics, but the split fermion idea of Arkani-Hamed and Schmaltz splitfermions () can be used to generate fermion mass hierarchies from exponentially sensitive overlap integrals of extra-dimensional profile functions. Similarly, the RS1 setup can address this problem by allowing fermions to propagate in the bulk and thus acquire non-trivial profiles grossmanneubertbulkfermions (); gherghettapomarol2000 (). The idea is that the 3+1d fermion zero modes are in general localized around different locations along the extra dimension, with dimensional reduction then producing an effective 3+1d Yukawa coupling constant that is the product of the 4+1d Yukawa coupling constant and an overlap integral involving profile functions. When the profiles are split, the overlap integrals are suppressed, leading to small 3+1d effective Yukawa coupling constants. This fits in well with the fact that quark and lepton masses, except for the top quark, are suppressed with respect to the electroweak scale. Scalar bosons will also in general be split, a phenomenon we shall use to suppress colored-Higgs-induced proton decay (see Refs. mckellarpdecsuppaper (); twistedsplitfermions (); cpandtwistedsf (); realisticsplitfermions (); higgslocinsfmodels (); cpin5dchangng (); nnbaroscinlargeed (); coulombicintbwsplitfermions (); kakizakidoublettriplet (); kakizakimassfitting () for more on the use of the splitting of fermions and bosons in extra dimensions to generate fermion mass textures and to suppress proton decay and other baryon number violating processes).

In this paper, we shall utilize the 4+1d domain-wall brane model devised by Davies, George and Volkas (DGV) firstpaper () to address the fermion mass and mixing angle problems. In this RS2-like model, the split fermion idea arises naturally, and thus the usual quark-lepton mass relations are avoided. It will be shown that the mass hierarchy problem can be solved using this method, and that the mass hierarchy and the Cabibbo angle can be accounted for in the two-generation case with Majorana neutrinos. We also explain why tribimaximal mixing cannot be accounted for without fine-tuning, and that the addition of a flavor symmetry therefore seems necessary. We are thus led to the view that extra dimensions provide an excellent way to qualitatively understand mass hierarchies, but they are insufficient to explain all the observed mixing angle patterns. The reason the flavor problem has proven to be so difficult may be because more than one ingredient is necessary: extra dimensions on their own, and flavor symmetry on its own, are only partially successful.

The following section reviews the DGV model and develops it further in several important ways: neutrino mass generation is examined and the see-saw mechanism implemented, and the dynamics of scalar-field localization is shown to be analytically tractable. Section III then analyses the parameter space of the model to produce the required mass and mixing angle hierarchies, with the aforementioned caveat for tribimaximal lepton mixing. Section IV is our conclusion.

## Ii The model

The DGV model is a specific extra-dimensional theory featuring the brane as a topological defect: a kink-like domain-wall configuration rubshapdwbranes (). Domain walls are stable classical solutions of suitable scalar field theories that exhibit a brane-like character, with energy-density peaked around the centre of the wall. Unlike fundamental branes, they have a finite width, and are most naturally used to replace the -function-like fundamental brane of the original RS2 model. Like RS2, a 3+1d graviton zero mode is dynamically localized. Unlike the original RS2 setup, all other degrees of freedom (fermions, scalars and gauge bosons) must be dynamically localized. In contrast to the fundamental-brane case, it is not possible to simply postulate that various fields are confined to a domain-wall: one must have dynamics to do it, and that is the main challenge in developing realistic models of this kind.

The dynamical localization of chiral fermion zero-modes is automatic when 4+1d fermions Yukawa-couple to a background scalar field in the form of a kink rubshapdwbranes (). Thus the chiral fermion structure of the SM can be naturally accommodated. Similarly, additional scalar bosons such as a Higgs doublet can be dynamically localized to a domain wall through a Higgs potential that couples those extra scalars to the background scalar field configuration modetower (). Such localized scalars can even obtain negative squared masses, thus triggering spontaneous symmetry breaking on the wall.

The most difficult issue is the localization of gauge bosons, with the need to maintain exact 3+1d gauge invariance to ensure gauge universality. A promising mechanism was proposed by Dvali and Shifman dsmech (), the physics of which is quite different from the localization of fermions and scalars. The idea is that a gauge group spontaneously breaks to a subgroup inside the wall, but is restored in the bulk. The bulk gauge theory is taken to be in confinement phase. The proposition is that the gauge bosons of are then dynamically localized to the wall as exactly massless states enjoying exact 3+1d gauge invariance. There are two heuristic arguments for why this should be the case. Dvali and Shifman themselves argued as follows: Take the case where and , and call the gauge boson of the “photon”. The photon is obviously free to propagate as a massless gauge boson in the plane of the wall. But in propagating transverse to the wall, into the bulk, the confinement regime is encountered, and the propagating states must be colorless and, importantly, massive glueballs. The photon must incorporate itself into a massive glueball to enter the bulk. But this mass gap makes this transition energetically disfavored, thus trapping the photon on the wall. Subsequently, Arkani-Hamed and Schmaltz dualmeisnerrplusdsmech () presented another heuristic picture: the photon field lines must be repelled from the bulk, because a confinement-phase region is by definition unable to support diverging electric fields. Thus the flux is channeled along the wall, effecting a dimensional reduction. At large distances within the wall away from the source, the field lines exhibit 3+1d Coulomb form. If the source is instead located in the bulk, then a flux tube leading to the wall is formed, with the field lines then diverging outward as if the source was located within the wall. Thus, the long-distance behavior of the field lines within the wall is independent of where the source is placed. If the source is smeared out along the extra dimension due to a profile function, then the corollary is that the asymptotic field line behavior is independent of the profile. Charge universality is thus maintained, no matter how the source is distributed along the extra dimension. These conclusions generalize to an arbitrary and , provided any glueballs associated with (e.g. QCD glueballs) are less massive than glueballs.

So, the way to develop potentially realistic domain-wall brane models is clear. One first postulates a scalar field theory that admits a suitable topological domain wall solution. This configuration must break gauge group to , and must contain or be the SM gauge group. Chiral fermion zero modes and additional scalars such as an electroweak Higgs doublet are then dynamically localized as sketched above. The DGV model uses the minimal gauge structure where and firstpaper ().

### ii.1 The field content

The scalar fields in the model are

(1) | |||

(2) | |||

(3) |

The fermion content of the theory consists of the SM fermions with a gauge singlet right-handed neutrino for each generation . The SM fermions are placed into the following representations,

(4) | |||

(5) |

while the right handed neutrinos are singlets,

(6) |

contains the charge conjugate of the right-chiral down-type quark and the left-chiral lepton doublet, and contains the left-chiral quark doublet and the charge conjugates of the right-chiral up-type quark and electron-type lepton for the generation .

In this model, matter is confined to a domain-wall brane formed from a solitonic kink configuration for the field. To implement the Dvali-Shifman mechanism, is broken inside the domain wall by the second background field which transforms under the adjoint representation. It attains a non-zero value for the hypercharge generator component inside the domain wall, so the gauge group respected on the domain wall is that of the SM.

Chiral fermion zero modes are trapped on the domain wall through Yukawa interactions with the and background configurations. Similarly, additional scalar fields are trapped by introducing quartic interactions between those scalar fields and the background domain wall.

For the purposes of this paper, we shall ignore gravity, although a similar analysis will have to be done with its inclusion in a later paper. It has already been noted that the RS2 graviton localization mechanism also works for a domain-wall brane. For a discussion of how gravity affects the dynamical localization of other fields, see Refs. rsgravitydaviesgeorge2007 (); firstpaper ().

### ii.2 The background domain wall configuration

The background domain wall configuration is formed from a self-consistent classical solution for the coupled fields and . The singlet scalar field forms the kink-like domain wall, while the adjoint breaks down to on the domain wall by attaining a bump-like configuration.

The relevant part of the action for describing the dynamics of the background is firstpaper (),

(7) |

where T contains all the gauge-covariant kinetic terms for all the fields. is the part of Higgs potential containing the quartic potentials for and , with

(8) |

We want to break to the SM on the domain wall, while having the bulk respect the original gauge symmetry. We do this in the standard way by giving the component associated with the hypercharge generator a non zero value on the brane, and having all the other components vanish. Thus the potential reduces to

(9) |

where .

To find the background configuration, we need to solve the Euler-Lagrange equations for and subject to the boundary conditions

(10) | |||

(11) |

which are degenerate global minima of . For the sake of simplicity, we choose to impose the constraints

(12) | |||

(13) |

yielding the analytic solutions,

(14) |

where , and . We should stress that the above conditions are not fine tuning conditions, and they are chosen simply so that the background fields obtain analytic forms. To find solutions, these conditions need not be imposed, and for a finite range of parameters we can always find numerical solutions which are kink-like for and lump-like for firstpaper (). The graphs of these solutions for and are shown in Figures 1(a) and 1(b) respectively.

The kink-like has its energy density localized about , forming the domain wall brane. The bump-like breaks to on the domain wall.

To preserve the topological stability of the domain wall, a spontaneously broken reflection symmetry must be introduced. Under this discrete symmetry transformation,

(15) |

where the 4+1d Gamma matrices for are defined

(16) |

For the entirety of this paper, 4+1d Lorentz indices will be denoted with upper case Roman indices, while ordinary 3+1d Lorentz indices will be denoted with lower case Greek letters as usual. Also, . The next step is to localize the fermions to this background.

### ii.3 Localizing the charged fermions and the left-chiral neutrino

To localize the charged fermions and the left-chiral neutrino to the domain wall, we need to couple them to the background. The relevant Yukawa Lagrangian which localizes the fermions, for one generation, is firstpaper (),

(17) |

The resulting 5d Dirac equation, for the charged fermions, is

(18) |

As explained in firstpaper (), to find the zero modes, it is enough to look for solutions for each charged fermion of the form where the are 3+1d massless, left-chiral spinor fields. Substituting this into the above Dirac equations yields the solutions for the profiles,

(19) |

where the are normalisation constants, and

(20) |

These profiles have maxima at

(21) |

Hence, the charged fermions and the left-chiral neutrino, which reside in the non-trivial representations of , get split along the extra-dimension according to their hypercharges and Yukawa couplings to the background. A similar effect was used in kakizakidoublettriplet (); kakizakimassfitting ().

During the mass fitting sections, we will need to describe the theory in terms of non-dimensionalized variables and profiles, since we do not know the value of . The non-dimensionalized domain-wall Yukawa couplings have already been defined in Eq. 20 and so we just need to non-dimensionalise the profiles. Defining the non-dimensionalized extra-dimensional coordinate, , as

(22) |

and changing variables, we see that the normalisation condition for the profiles becomes

(23) |

Hence, in order to use functions which are normalised to one over , we define the non-dimensionalized profiles, , as

(24) |

Thus, the profiles scale as times a dimensionless function, which is not surprising since we know that the 4+1d field has mass dimension , while the 3+1d field has mass dimension as usual.

Since the factor of in Eq. 23 will always arise in the normalisation condition when changing variables from to , we will define the non-dimensionalised profiles for any field in the same way as Eq. 24, and they will be denoted with the same symbol used for the dimensionful profiles but with an overscript tilde.

In the case that we have generations of fermions, is generalized to

(25) |

where and are summed from to . Hence, in the general case, there can be intergenerational mixing between the quarks and leptons through the interaction with the background. The background couplings and have now become Hermitian matrices over flavour space and need not commute. To solve the equations, we look for zero mode solutions of the form,

(26) |

where the are massless left chiral 3+1d fields for . Putting this into the 4+1d Dirac equation results in the matrix differential equation for the profiles , which are now matrix valued functions of ,

(27) |

The case where and do not commute, which leads to a natural realisation of the split fermion scenario discussed in Refs. twistedsplitfermions (); cpandtwistedsf (), cannot be solved analytically, and so for the sake of simplicity we will only search the parameter space that obeys,

(28) |

Since both the matrices are required to be Hermitian as well, they are thus simultaneously diagonalizable, so that for some unitary matrices ,

(29) |

where the and are understood to be the eigenvalues of and respectively. Choosing to localize left-chiral zero modes for is then equivalent to demanding that all the eigenvalues of are positive definite. Solving the 5d Dirac equation then yields the general solution for the profiles,

(30) |

Here we have written the multi-generation solutions in terms of the solutions for the one generation case. The are normalisation constants, chosen such that the profile matrix satisfies the normalisation condition,

(31) |

The parameters and are the non-dimensionalized versions of and , and are defined in the same way as the non-dimensionalized constants from the one generation case were in Eq. 20. The are unitary matrices which are present since the solution is unique up to matrix multiplication. The , in fact, correspond to a choice of which 3+1d states are the domain wall eigenstates and thus localized to the wall. Unless otherwise stated, we will assume these to be the same as the weak interaction eigenstates.

### ii.4 Adding singlet right-handed neutrinos

To localize the right-chiral neutrinos, we need to couple them to the background. As they are gauge singlets, they cannot couple to the adjoint Higgs . Thus we can only add,

(32) |

to . The relative minus sign in front of the Yukawa interactions for the is introduced because for these fields we want localized right-chiral zero modes which represent the right-handed neutrinos in the effective 3+1d theory, as opposed to left-chiral zero modes. This allows us to treat in the same way as and .

Writing down the 5d Dirac equation for the , and demanding that , where the are 3+1d right chiral zero modes, in similar fashion to the charged fermions, leads to the profile,

(33) |

where is again a choice of basis matrix for the 5d fields, is a change of basis matrix for the 4d fields, the are normalisation constants, and the are the positive definite eigenvalues of the Yukawa matrix .

### ii.5 Localizing the electroweak symmetry breaking Higgs boson

The electroweak breaking Higgs doublet is localized in a very similar manner to the fermions. The most general localizing Higgs potential which respects the and discrete symmetries is

(34) |

To find the profiles of the electroweak Higgs doublet, , and the colored Higgs triplet, , embedded in the quintet , we search for solutions of the form,

(35) |

where the are the respective profiles, and satisfy the Klein-Gordon equations,

(36) |

where are the masses of the lowest energy modes for . Substituting this ansatz into the 4+1d KG equation with the potential , one obtains the equations for the profiles

(37) |

where

(38) |

Changing variables to the dimensionless coordinate defined in Eq. 22, the potentials of the above Schrödinger equations can be rewritten as shifted hyperbolic Scarf potentials, that is we can write them in the form

(39) |

where

(40) |

and the non-dimensionalized Higgs parameters and masses are defined as

(41) |

and are the eigenvalues of the equations for the electroweak Higgs and the colored Higgs hyperbolic Scarf potentials.

The hyperbolic Scarf potential has been well studied castillohyperscarfpot () as it is a member of a class of potentials satisfying the shape-invariance condition in supersymmetric quantum mechanics (for more on shape-invariant potentials see shapeinvkharesukdab (); levaishapeinvariantpots ()). For , it is known to have a set of discrete bound modes for , with eigenvalues

(42) |

Combining this with the previous equations for , we see that the potentials have a discrete set of bound modes with masses given by

(43) |

The physical electroweak Higgs and colored Higgs fields in the effective 4d theory on the brane correspond to the modes, and they thus exist in the 4d theory if . Assuming this, the profiles for these Higgs particles, and respectively, have the same form as those of the zero mode profiles for the charged fermions,

(44) |

Hence, we can interpret and to be effective couplings of the Higgs fields to the kink and the lump respectively.

The effective couplings and depend on the hypercharges, and thus they are in general different for the two Higgs components. This has a number of consequences. Firstly, since the masses of the electroweak and colored Higgs depend on their respective , the masses of the two components are split. There exists a parameter region where the electroweak Higgs has a tachyonic mass, , while that for the colored Higgs (if a bound state exists) is non-tachyonic, thus inducing electroweak symmetry breaking on the brane while preserving , as is desired. Since we know the exact form of the masses, a straightforward analysis shows that this parameter region is

(45) |

Secondly, as there only exist discrete bound modes for a species if , there exist parameter regions where the electroweak Higgs component will have discrete bound modes localized to the domain wall while at the same time the colored Higgs will have only unbound continuum modes in its spectrum. This suggests that an alternate approach to suppressing colored Higgs induced proton decay may be possible, as the continuum modes propagate in the full 4+1d spacetime so that the partial width contributed to proton decay from these modes may be suppressed by further powers of . The analysis of this situation is beyond the scope of this paper.

Note that it is possible for more than one KK excitation of the Higgs doublet to have nonzero vacuum expectation values, thus naturally generating a multi-Higgs doublet model on the brane. However, for simplicity, we will choose parameters such that only the electroweak Higgs has a tachyonic mass, and not its KK modes, and we will also have a bound state for the colored Higgs. For the purposes of this paper, we will use three such choices.

For the first choice,

(46) |

the mass eigenvalues are , and . The graphs of the profiles are shown in Fig. 2.

For the second choice,

(47) |

the mass eigenvalues are , and . As can be seen in Fig. 3, this leads to profiles which are much more localized than those for the first choice of parameters. As we will see, this has important consequences for the spread of domain wall Yukawa couplings and for the suppression of some of the decay modes for colored Higgs induced proton decay.

For the third choice,

(48) |

The resultant squared-masses for the lowest energy modes are and .

As we can see in the graphs of the profiles in Fig. 4, for this parameter region, the electroweak Higgs is highly peaked near the brane at , while the colored Higgs is more delocalized and substantially displaced from the wall. This parameter choice exploits the property of the Higgs sector that effective kink and lump couplings and are not the same for the colored and electroweak Higgs. As we will see, this kind of parameter choice can lead to suppression of all decay modes for colored Higgs induced proton decay, and ensure that the partial lifetimes for these modes are all many orders of magnitude above the current lower bounds.

Note that the Higgs vacuum expectation value, , is not uniquely determined by the constants which determine the Higgs profile. By dimensional reduction of the action,

(49) |

one can show that the effective electroweak symmetry breaking potential is

(50) |

where

(51) |

Thus the VEV of the Higgs doublet is

(52) |

and so whatever we choose for the other constants, we can always adjust appropriately so that we get the correct VEV of 174 GeV.

### ii.6 Generating mass matrices for the charged fermions

The electroweak Yukawa Lagrangian, , from firstpaper () which generates masses for the charged fermions is generalized to

(53) |

for generations of fermions. Here, lower case Greek letters are indices and the lower case Roman letters indicate flavor.

The terms generate mass matrices for the down-type quarks and electron type leptons, while the terms generate a mass matrix for the up-type quarks. Extracting the components from each term which generate 3+1d masses and performing dimensional reduction, one finds the mass matrices to be

(54) |

where is the profile of the electroweak Higgs doublet which is embedded in , and is the vacuum expectation value of the electroweak Higgs field attained on the brane.

Converting to dimensionless quantities, and defining the non-dimensionalized electroweak Yukawa couplings by

(55) |

we see that these mass matrices can be rewritten as

(56) |

There are some important consequences of the above forms of the mass matrices, which depend on overlap integrals of the profiles for the left and right chiral fermions and the electroweak Higgs. Firstly, the overlap integral dependence means we avoid the usual incorrect mass relations like which are characteristic of ordinary 3+1d models with a Higgs quintet. This is also the reason why we do not need a Higgs belonging to the representation of containing an electroweak Higgs triplet to get the Georgi-Jarlskog relations georgijarlskog (). Thirdly, since the fermions are split according to their hypercharges, and the splittings are dependent on the background couplings, we can potentially generate the fermion mass hierarchy and mixings by splitting the fermions appropriately so that the overlap integrals are in the desired ratios. It will be shown in a later section that this can be done.

### ii.7 Generating Dirac neutrino masses

To generate Dirac masses for the neutrinos, we need to add Yukawa interactions involving the , which contain the left handed neutrinos, the , which contain the right handed neutrinos, and which contains the electroweak Higgs. The correct terms to add to which are both invariant and respect the reflection symmetry which preserves the topological stability of the domain wall are

(57) |

Reducing these terms to their SM components, and integrating out the extra-dimensional dependence, one finds the resulting Dirac mass matrix for the neutrinos to be

(58) |

Defining the dimensionless neutrino Yukawa couplings as

(59) |

and changing to non-dimensionalized quantities, we can rewrite the Dirac mass matrix for the neutrino as

(60) |

### ii.8 Generating Majorana neutrino masses

Let us consider one generation first. To generate a Majorana mass for the neutrino, we need to add terms to the Lagrangian that will dimensionally reduce to terms proportional to in the effective 4d theory. Thus, we might want to consider adding a term like

(61) |

This is obviously gauge invariant, and it turns out that it is also invariant under the discrete reflection symmetry as well. We first need to consider what implications the addition of this term has for the existence of solutions of the 5d Dirac equation. The relevant Lagrangian is

(62) |

and thus the 5d Dirac equation becomes

(63) |

Demanding the conditions that

(64) |

and noting that the parts proportional to and must be independent of each other as the corresponding spinors transform as right-chiral and left-chiral spinors respectively, we get two independent equations for ,

(65) |

The first of the equations above is exactly the same differential equation as before without the new term, and thus the must also have the same form as before,

(66) |

The second condition then implies that , and since any phase can just be absorbed into the definition of , we can take . Hence, instead of a right-chiral zero mode, we now have a right-chiral Majorana mode of mass localized to the domain wall.

Similarly with three generations, the profiles are unaltered by the Majorana mass terms, and the 3+1d Majorana mass matrix after dimensional reduction is then

(67) |

We have thus successfully shown that both Dirac and Majorana masses can be generated with the addition of a right chiral singlet neutrino, and thus the see-saw mechanism can be employed. We will now demonstrate that the fermion mass hierarchy and small CKM mixing angles can be generated from split fermion idea.

## Iii Generating the flavor hierarchy and mixing angles

The fermion mass matrices depend on overlap integrals of the fermion profiles and the electroweak Higgs. Since the left-chiral and right-chiral components are naturally split according to their hypercharges, and since these overlap integrals are exponentially sensitive to these splittings, it seems we can employ the split fermion idea splitfermions () to account for the fermion mass hierarchy from a set of domain wall couplings which are all about the same order of magnitude in this model.

Throughout the rest of the paper, we will quote the dimensionless background Yukawa couplings to five significant figures. The reasons for this are the exponential sensitivity of the profiles to these couplings and the difficulty that was found in generating the neutrino mass squared differences (which are quadratic in overlap integrals of these profiles) to an acceptable and reasonable precision. Since this is also a classical calculation where quantum corrections are ignored, and since the quark and neutrino masses are not as precisely measured or well known as those for the charged leptons, we will quote the resultant masses of the quarks and neutrinos to two significant figures, neutrino mass squared differences to one significant figure, and the charged lepton masses to three significant figures.

### iii.1 The one-generation case with a Dirac neutrino and the suppression of colored-Higgs-induced proton decay

In this section we shall show that the mass hierarchy amongst the first generation of fermions can be generated from the split fermion idea splitfermions () which arises naturally in our model. We will start with looking for solutions with the Higgs parameter choices of Eq. 46.

Firstly, we must make the neutrino light. The right chiral neutrino is always localized at while the choice of Higgs parameters in Eq. 46 (and in fact for those in Eqs. 47 and 48 as well), the Higgs is localized to the right. Hence, the easiest way to induce a small Dirac neutrino mass is to shift the lepton doublet to the left. As the lepton doublet, , has hypercharge and the charge conjugate of has hypercharge , choosing to be negative will displace the lepton doublet as desired while placing the right-chiral down quark to the right, near the electroweak Higgs.

We now need to make the charged fermion masses significantly larger. Since the charge conjugates of and , and the quark doublet have hypercharges , and respectively, making the ratio positive will shift far to the left, towards the lepton doublet, to slightly to the left, and to the right.

We found the following solution by using this configuration, making the parameter choice and , plotting the contours along which the overlap integrals give the desired mass ratios,

and then finding where the two contours intersected. Doing this yielded the solution for the couplings for the multiplet, and . With the ratios now fixed, setting the 5d electroweak Yukawas , and setting the kink coupling for the right handed neutrino to gives the masses,

(68) |

Thus, we have generated an neutrino mass below the current most stringent upper bounds of roughly eV pdgquark () , the correct electron mass, and up and down quark masses within current constraints of MeV MeV, and MeV MeV pdgquark ().

Furthermore, it turns out we get significant suppression of some modes of colored Higgs-induced proton decay with this setup. The colored Higgs scalar can induce the decays and , for which the Feynman diagrams are shown in Figures 8(a) and 8(b) respectively.

For the process , the partial lifetime of each contribution is

(69) |

where and are replaced by the effective 4d couplings strengths of the operators inducing the vertices and respectively. The operators responsible for the vertex are and , and their respective coupling strengths are

(70) |

The operators responsible for the vertex are and , and the associated coupling strengths are