Fat brane and seesaw mechanism in extra dimensions

In this paper we present a higher-dimensional seesaw mechanism. We consider a single, flat extra dimension, where a fat brane is localized and contains the standard model (SM) fields, similar to Universal Extra Dimension models. There is only one Dirac fermion in the bulk, and in four dimensions it results in two towers of Kaluza-Klein (KK) Majorana sterile neutrinos, whose mass mixing with the SM neutrinos is suppressed due to a brane-localized kinetic term. The interaction between the sterile neutrinos and the SM is through the usual coupling with the Higgs boson, where the coupling depends upon the compactification radius $R^{-1}=10^{-2}-1$ GeV and the width of the fat brane $L^{-1}=2$ TeV, where the latter value is chosen to avoid LHC constraints. Due to this suppression mechanism the mass of the lightest sterile neutrinos can be of order $\mathcal{O}(1-10)$ TeV while naturally explaining the small SM neutrino mass, which in turn is easily obtained for a large range of parameter choices. Furthermore, neutrino oscillations are not substantially influenced by the tower of sterile KK particles. Finally, leptogenesis is investigated in this setup, and it is viable for some values within the parameter space.

In a recent paper [65], an ED was employed along with a finite width 'fat' brane to explain the expected smallness of the coupling between SM and a dark mediator, where this mediator is either a vector or a scalar field. In this setup, a BLKT was used and the interaction between the SM and the mediators was found to be suppressed, when compared with the coupling between these same mediators and a dark matter candidate, confined in a separate thin brane. A similar result was obtained for a vector field in the bulk, in a model with two ED [66,67].
The small value of the SM neutrino mass can be explained if one uses the seesaw mechanism, where the diagonalization of the neutrino mass matrix leads to a massive mostly sterile neutrino and a very light mostly active neutrino. This mechanism is known to be possible using large extra dimensions [27] or warped geometry [28], so that a natural extension of previous works [65] would be to investigate if a fermion in the bulk, playing the role of a sterile neutrino, has its interaction with SM suppressed in such a way that the seesaw mechanism can be realized. This is the purpose of the present work. We find that the two towers of Majorana sterile neutrinos can indeed provide the explanation of the small SM neutrino mass through a higher-dimensional seesaw mechanism. The lightest sterile neutrino masses can be of order O(1−10) TeV because the mass mixing between the sterile neutrinos and the SM neutrino is naturally very suppressed for a wide range of parameter choices. Neutrino oscillations are not influenced by the tower of sterile neutrinos since the survival probability is practically equal to one. Finally, we investigate leptogenesis in this setup, showing that it can explain the observed baryon asymmetry of the universe for some values of the parameter space.
This paper is organized as follows. In Sect. II we consider a fermion in the bulk and derive the equations of motion and wave functions. The interaction with the SM is presented in Sect. III along with the seesaw mechanism and the survival probability of the SM neutrino.
In Sect. IV we investigate leptogenesis and Sect. V is reserved for conclusions.

II. NEUTRINO IN THE BULK
We consider a single, flat ED represented by an interval 0 ≤ y ≤ πR, with a fat brane localized between πr and πR, where the SM is confined, and we assume L ≡ R − r R. There is only one generation of a Dirac fermion in the bulk Ψ(x µ , y), with Dirac and Majorana mass terms. We consider the induced kinetic term in the fat brane, so that the corresponding action is [28,60] where A = 0 − 3, 5 is the 5-D index, m D is the Dirac mass, m M is the Majorana mass, Γ 4 = iγ 5 and Ψ c = C 5ΨT is the charge conjugated spinor, with C 5 = γ 0 γ 2 γ 5 . The BLKT in the action is given by [60] 1 1 For simplicity, we restrict our attention for the case where the BLKT parameter δ A is the same for the two components of the Dirac spinor ψ 1 , ψ 2 , i.e. δ 1 A = δ 2 A ≡ δ A , although different contributions might be possible, as presented in [60].
where the step-function is θ(y) = 0 for y < πr, θ = α for πr < y ≤ πR , with δ A > 0 and α being a positive constant with dimensions of energy. We define region I as 0 ≤ y < πr and region II as πr < y ≤ πR. Writing the bulk sterile neutrino as Ψ = Ψ 1 + Ψ 2 , 2 we can expand it as a tower of KK states Using this decomposition the 4-D action is found after integrating out the ED, where the wave functions f 1,2 (y) satisfy the following orthogonality relations In order for Eq. (6) to be true, the integration by parts gives the following coupled BC 2 (0) = 0, which should be satisfied for all m and n. With the decomposition (4) and using the Majorana condition for the 4-D fieldsψ 2 , we get the equation of motion for the two components of the wave function [28] (±∂ y − m D )f The first-order equations (7) can be transformed into a second-order equation for f (n) 2 , for example. 3 This procedure gives where m n = x 2 n /R 2 + m 2 D + m M and the roots x n will be determined by the appropriate transcendental equation. Having determined f 2 There are no chiral fermions in 5-D and chirality in 4-D is recovered through the Z 2 orbifold symmetry y → −y, where one spinor is taken to be even under this symmetry, while the second one is taken to be odd. As we shall see, the wave functions do not satisfy orbifold boundary conditions (BC) in the region πr < y < πR, therefore we do not obtain a chiral spinor. In addition, the Dirac mass term in 4-D would be canceled if one had used orbifold BC, which again, is not the case here. 3 The choice of which wave function would have a second-order equation is arbitrary. If we had chosen f The solutions of Eqs. (7) and (8) are found for the two different regions in the ED space, that is, inside the fat brane and outside it. The solution for the wave function in region I have the form f (n) 2 (y) = A n cos(x n y/R) + B n sin(x n y/R), but imposing that it vanishes at y = 0 in order to satisfy the coupled BC, we have the following solutions for this region f (n) 2,I (y) = Λ n sin where Λ n is the normalization constant found using Eq.
wherem 2 n ≡ (m n − m M + m n δ A αR) 2 − m 2 D and the prime is a derivative with respect to y. Using Eq. (5), the normalization constant is where A  .
The 4-D Lagrangian contains Dirac and Majorana mass terms, but we can form the linear √ 2, such that they diagonalize the mass matrix. The resulting tower of Majorana eigenstates have the corresponding physical masses given by m where a hierarchy between the mass parameters and a positive bulk Majorana mass is assumed to assure that the physical masses are always positive.

III. SEESAW MECHANISM
There is an interaction between the bulk fermion, the Higgs H and the SU (2) L doublet fermion L f given by (2) are the correspondent 5-D Yukawa couplings and h.c. stands for Hermitian conjugation. We will omit flavor indices. Since we are interested in the interaction with conventional SM particles, we will assume only the zeroth-KK mode for the SM fields. After expanding the sterile neutrino in a KK tower of states, the 4-D couplingsλ where λ 4,1(2) ≡ λ 5,1(2) Λ 0 is defined to be a 4-D dimensionless Yukawa coupling, (πL) −1/2 is the usual normalization of the UED SM fields and recall that L is the width of the fat brane.
We plot the couplingsλ presence of BLKT, therefore, makes the coupling smaller and more suppressed, shrinking the oscillatory pattern as the combination δ A α is increased. This is the same behavior found in [65] (where the coupling was proportional to L/R, for the lightest KK states), where here we can also see that larger compactification radius R decreases the couplings as well.
The interaction in the mass eigenstate basis is λ We are ignoring possible additional phases for the couplings, because it turns out that both couplings can be turned into real numbers by a phase adjustment, so that in what follows only the absolute value of them is important. After the spontaneous symmetry breaking via the Higgs mechanism the off-diagonal mass (2) v and v = 246 GeV is the Higgs vacuum expectation value. The mass term for the neutrinos can be written  Given the cutoff scale Λ, above which the theory becomes non-perturbative, it is possible to determine how many particles will contribute to the mass matrix. It is usually assumed ΛL = 20 for UED models [37], thus in our case, considering L −1 = 2 TeV to avoid LHC As can seen from the mass matrix, the smaller the couplingsm Since the sterile neutrinos are much heavier than the off-diagonal masses m We see that the standard seesaw expression is obtained if there is only one particle (λ ν ≈ −m 2 1 /m 1 ). From Eq. (19) we also understand why a very large number of particles would increase λ ν , jeopardizing the success of the seesaw mechanism.
It is possible to have an estimate for the upper limit ofm where the sum over n becomes the product of ΛR states. For R = 1 GeV, the number of states are ΛR ∼ 4 × 10 4 , therefore, for m 1 ∼ 10 TeV we should havem 1 10 −5 GeV.
Obviously this is just a rough estimate, but it gives an idea of how easy the seesaw mechanism could be satisfied within the present model, having the lightest sterile neutrinos with masses of order 10 TeV. On the other hand, larger compactification radii would lead to a much larger number ΛR of states, therefore requiring smaller couplings for the same neutrino mass.
We numerically solve Eq. (18) and gather in Table I Table I), while for R −1 = 1 GeV there are ∼ 10 4 states. Although the couplings are relatively more suppressed for larger compactification radii, they are not sufficiently small to compensate the additional number of KK states contributing to the neutrino mass in Eq. (19). Therefore, in order to compensate the eventually large number of KK states for larger R, smaller 4-D Yukawa couplings λ 4,1(2) are needed. We see from Table I that Table I down to smaller values, m D ∼ 1 − 10 TeV, respectively. The same reasoning also applies to obtain smaller values for the BLKT parameter δ A α. Notice that the roots x n are usually much smaller than m D , leading to a difference between one KK state and the next one of order (x 2 n+1 − x 2 n )/(2m D R 2 ). The mass difference between two neighboring KK states can be much smaller than the difference between the masses of the two sterile neutrinos within the same KK state, that is, m Finally, in order to understand the influence of the KK particles on neutrino oscillations we will evaluate the total probability of the SM neutrino oscillating into any other sterile neutrino KK state. It is convenient to work with the survival probability P ν L →ν L (t), as a GeV.
function of time, that the SM neutrino is preserved [27] where the energies E i are the mass eigenvalues in our case and U ν L i are the mass eigenvectors.
The gauge eigenstates are therefore written in terms of the mass eigenstates as where the mass eigenvectors U are Each row is the eigenvector correspondent to the eigenvalue E i = λ i , and it can be written where i = k. Using the parameters in Table I Table I. Different values of the parameters give quite similar results.
Due to the large sterile neutrino masses, experimental/observational constraints do not pose challenges to this model [68][69][70][71]. Furthermore, an interaction between the sterile neutrinos with the weak gauge bosons would arise from ν Lα = U αi ν i + Θ  [72]. For the values presented in Table I one can get Θ (n) α1(2) ≤ 10 −9 (and even smaller values for slightly smaller 4-D Yukawa couplings λ 4,1(2) ). Therefore the mixing of sterile neutrinos with active SM neutrinos is extremely small, not modifying significantly the weak currents.

IV. LEPTOGENESIS
In this section we investigate whether baryogenesis via leptogenesis is viable in the present model. Since the two KK towers imply that there is a large number of sterile neutrinos contributing to the seesaw mechanism, their individual couplings to Higgs bosons are small compared to the usual scenarios with no more than three sterile neutrinos. We recall from Section III that the number of KK sterile neutrinos is limited by the cutoff scale such that, The sterile neutrino decay rate is therefore the sum over the rates for all final states N (n) 1,2 → h (m) + ν (p) . The individual Yukawa couplings are no longer described by Eq. (15) because they should include, in the integral over the ED, the contribution of the excited KK states of the Higgs and the SM neutrino. Their corresponding wave functions are [30] It turns out, for the range of values of the parameters to be considered here (as presented below), the Yukawa couplings have practically the same order of magnitude over all of the KK spectrum, as depicted in Fig. 5.
In UED models the mass spectrum of the SM particles is given by m (p) SM = m 2 SM + p 2 /L 2 , where p = 0, 1, 2, . . . and m SM is the mass of the known SM particles. As it has been discussed before, for our choice of parameters, there are ΛL = 20 roots below the cutoff scale, and since L −1 = 2 TeV, the excited SM KK states have masses of approximately p/L( m SM ). The decay rate of the sterile neutrinos into SM KK states is of the same order of magnitude as for the corresponding zeroth SM-mode that is shown in Fig. 6.
Therefore, for the sake of simplicity and without loss of generality, in the following we can use the expressions for the sterile neutrino decay into only SM zero-modes, but to take into account the final SM KK states as well, we multiply the results of the SM zero-mode by the number of kinetically allowed processes ∼ ΛL(ΛL + 1)/2. This estimate holds by order of magnitude since we assume that m D m The parameter characteristic for leptogenesis is the washout strength where the subscripts and superscript refer to the sterile neutrino N (n) 1,2 , Γ is the decay rate in vacuum, H is the Hubble rate, T is the temperature, g * is the number of degrees of freedom for relativistic particles, and M Pl is the Planck mass. Due to the couplings being weak here in comparison with the usual seesaw scenarios, ΛL(ΛL + 1)/2 K typical configurations in parameter space. This relation implies weak washout, that is, the sterile neutrinos N (n) 1,2 remain far from equilibrium before their distribution becomes Maxwell-suppressed, and each individual sterile neutrino only washes out a small fraction of the lepton asymmetry. Note that we assume here that the initial abundances of the sterile neutrinos vanish.
Some studies of leptogenesis with a large number of sterile neutrinos [73] were carried out before the relevant reaction rates for sterile neutrinos in the relativistic regimes were thoroughly investigated [74][75][76]. As a consequence, the dynamics of leptogenesis in scenarios with many sterile neutrinos should be reconsidered in detail, which is beyond the scope of the present work. To obtain an estimate of the asymmetry, we rely on the work [77]. Its main shortcoming when applied to the present scenario is the assumption of a hierarchical that are close in mass. While the resonant enhancement can be included in the appropriate factor describing the decay asymmetry this leaves an inaccuracy in the efficiency factor of order one. We show in Fig. 7 the mass hierarchy between the sterile neutrino KK excited states and its zero-mode, for one sterile neutrino and some representative parameter choices, although other values give similar behavior. Nonetheless, we can make a prediction of order one accuracy when considering only the lightest 2n of the sterile neutrinos such that these will not wash out of most of the produced asymmetry. That is, we setn by the condition ΛL(ΛL + 1) 2n which is to be understood as an estimate.
Returning to our original discussion, for each individual sterile neutrino, the decay asym- TeV.
metry is given by where and where we have again included the factor accounting for the enhancement due to the SM KK states. The expression for g(x) is valid in the limit |m 2 ) 2 ∼ sin(φ n 1 − φ n 2 ) = 0. Provided we can neglect the washout by the above assumptions, the efficiency factor in the weak washout regime is given by κ  1,2 , we arrive at where s is the entropy density s ∼ g * T 3 .
The fraction involving the Yukawa couplings in ε (n) 1,2 is very small, so that in order to compensate it, the function g(x) should be large enough to eventually give the observed baryon asymmetry [78]. The function g(x) is large for m (n) 1 ∼ m (n) 2 . However, since the sums are over k and n, there is a large contribution from the KK tower of states, and the particular combination of g(x) with the Yukawa couplings such as to attain the observed asymmetry requires a specific choice of parameters.
In order to illustrate the parametric dependence of the asymmetry, we let one parameter free, while the other ones are kept fixed, considering always R −1 = 1 GeV and L −1 = 2 TeV for simplicity, although other values give similar results. We evaluate the maximum number of KK sterile neutrino statesn which are not washed out, for a set of parameters, as a function of the 4-D Yukawa couplings and show this in Fig. 8. We take λ 4,1 = λ 4,2 , as λ 4,1 = λ 4,2 gives similar and interpolating results forn.
Figs. 9, 10 and 11 show the baryon asymmetry as a function of one free parameter.
It can be seen that, in order to explain the baryon asymmetry, the bulk masses should be m D ∼ 30 TeV and m M ∼ 10 −9 GeV, while the BLKT parameter can have various values. Finally, in Fig. 11 the baryon asymmetry is shown as a function of the 4-D Yukawa