Seesaw Neutrino Signals at the Large Hadron Collider
We discuss the scenario with gauge singlet fermions (right-handed neutrinos) accessible at the energy of the Large Hadron Collider. The singlet fermions generate tiny neutrino masses via the seesaw mechanism and also have sizable couplings to the standard-model particles. We demonstrate that these two facts, which are naively not satisfied simultaneously, are reconciled in the five-dimensional framework in various fashions, which make the seesaw mechanism observable. The collider signal of tri-lepton final states with transverse missing energy is investigated for two explicit examples of the observable seesaw, taking account of three types of neutrino mass spectrum and the constraint from lepton flavor violation. We find by showing the significance of signal discovery that the collider experiment has a potential to find signals of extra dimensions and the origin of small neutrino masses.
The neutrino property, in particular its tiny mass scale is one of the most important experimental clues to find new physics beyond the standard model (SM) neu_review (). From a theoretical viewpoint, the seesaw mechanism seesaw () has been known to naturally induce small neutrino masses by integrating out new heavy particles (right-handed neutrinos) which interact with the left-handed SM neutrinos. In this Type I seesaw scheme, the right-handed neutrinos have intermediate-scale masses for obtaining light neutrinos. The seesaw effect appears as higher-dimensional operators suppressed by their heavy mass scale and are usually negligible in low-energy effective theory. Alternatively, TeV-scale right-handed neutrinos are found from the seesaw formula to have very weak couplings to the SM sector and their signals would not be naively detected in future experiments such as the CERN Large Hadron Collider (LHC).
It seems therefore difficult to directly observe heavy states which are relevant to suppressing neutrino masses. In the previous work HMY (), it was pointed out that an observable seesaw mechanism can be implemented in a five-dimensional framework where all the SM fields are confined in a four-dimensional boundary while right-handed neutrinos propagate in the bulk of extra-dimensional space DDG (); ADDM (). The existence of extra dimensions is also one of the exciting candidates for new physics beyond the SM ExD_review () and related neutrino phenomenology has been extensively studied in the literature neuExD (). In the above framework, the right-handed neutrinos and their extra-dimensional partners exist around the TeV scale and have sizable SM gauge and Yukawa couplings in the low-energy effective theory, while the seesaw-induced masses are made small.
The SM neutrinos have tiny masses due to a slight violation of the lepton number. This fact implies that the events with same-sign di-lepton final states dileptons () may be too rare to be observed unless, e.g., some particular flavor structure is assumed in neutrino mass matrices. In this paper, as in our previous work, we analyze lepton number conserving processes, in particular, the tri-lepton signal with large missing transverse energy: . This process is expected to be effectively detected at the LHC because only a small fraction of SM processes contributes to the background against the signal. The LHC signatures are studied in typical two types of observable seesaw models in five dimensions and with three types of neutrino mass patterns allowed by the present experimental data neu_analysis ().
We also present various extra-dimensional approaches which provide the situation that TeV-scale right-handed neutrinos generate a proper scale of seesaw-induced masses and simultaneously have observable interactions to the SM fields. They include boundary Majorana mass terms, boundary conditions for bulk neutrinos, the AdS gravitational background, and their combinations. These scenarios do not rely on particular (singular or aligned) generation structure of mass matrices, and is available even in the one-generation case. For such TeV-scale particles with sizable couplings to the SM sector, the collider experiment will generally have a potential to find a signal of extra dimensions and the origin of small neutrino masses.
This paper is organized as follows. In Section II, we formulate the general five-dimensional setup for neutrino physics, discussing the seesaw operation and the electroweak Lagrangian. Several explicit models for the observable seesaw are presented for the collider study. In Section III, after the discussion of phenomenological constraints and representative points of model parameters, we numerically investigate the LHC signatures of the seesaw models given in Section II and illustrate the significance for the signal discovery. In Section IV, we further show that various different configurations for the observable seesaw are viable even in one extra dimension, giving the low-energy effective vertices of heavy neutrino fields. Section V is devoted to summarizing our results and discussing future work.
Ii General framework
ii.1 Five-dimensional Setup
Let us consider a five-dimensional theory where the extra space is compactified on the orbifold with the radius . The SM fields are confined on the four-dimensional boundary at . Besides the gravity, only SM gauge singlets can propagate in the bulk not to violate the charge conservation DDG (); ADDM (). The gauge-singlet Dirac fermions () are introduced in the bulk which contain right-handed neutrinos and their chiral partners. The Lagrangian up to the quadratic order of spinor fields is given by
The conjugated spinor is defined as such that it is Lorentz covariant in five dimensions. The covariant derivative generally contains the contribution of spin connection given by the fünfvein. The bulk Dirac mass term involves the step function so that it is invariant under the reflection. Such an odd-function dependence could originate from some field expectation value. The bulk mass parameters , , and are -parity even and generally depend on the extra-dimensional coordinate which comes from the delta-function dependence (resulting in localized mass terms) and/or the background geometry such as the warp factor in AdS. We also introduce the mass terms between bulk and boundary fields:
where and denote the mass parameters after the electroweak symmetry breaking (the original Yukawa term will be given later). Throughout this paper, we take the fundamental scale of five-dimensional theory as the unit of mass dimension-ful parameters. The boundary spinors () contain the left-handed neutrinos . The parity implies that either component in a Dirac fermion vanishes at the boundary () and therefore either of and becomes irrelevant.111The exception is the generation-dependent parity assignment on bulk fields parity (). We do not consider such a possibility in this paper. In the following we assign the even parity to the upper (right-handed) component of bulk fermions
and will drop the term. In the above, while we only consider the boundary terms at , other boundary terms at can also be written down in the same fashion and have physical relevance on curved backgrounds and/or with complicated field configurations.
With a set of boundary conditions, the bulk fermions are expanded by Kaluza-Klein (KK) modes with their kinetic terms being properly normalized
The wavefunctions are generally matrix-valued in the generation space and we have omitted the generation indices for notational simplicity. After integrating over the fifth dimension, we obtain the neutrino mass matrix in four-dimensional effective theory:
where , and is composed of the boundary neutrinos and the KK modes . The zero modes of the left-handed components have been extracted according to the boundary condition. The neutrino mass matrix for is given by
where the Majorana masses , the KK masses , and the boundary Dirac masses are
In the expression of KK masses , the factor is related to the five-dimensional geometry, for example, for the flat background and for the AdS background with the curvature . It is noticed that becomes proportional to if are the eigenfunctions of the bulk equations of motion, and also becomes proportional to if the bulk mass parameters , are independent of the coordinate .
ii.2 Seesaw and Electroweak Lagrangian
We further implement the seesaw operation assuming and find the induced Majorana mass matrix for three-generations light neutrinos222In theory with more than one extra dimensions, this matrix product (the sum of infinite KK modes) generally diverse without some regularization bf ().
It is useful for later discussion of collider phenomenology to write down the electroweak Lagrangian in the basis where all the mass matrices are generation diagonalized. The original Lagrangian of four-dimensional neutrinos comes from (5) and the SM part. The kinetic and mass terms and the interactions to the electroweak gauge bosons are given in the mass eigenstate basis as follows:
where and are the electroweak gauge bosons and is the gauge coupling constant. The 2-component spinors are three light neutrinos for which the seesaw-induced mass matrix is diagonalized
and denote the infinite number of neutrino KK modes for which the bulk Majorana mass matrix is diagonalized both in the generation and KK spaces by a unitary matrix :
The lepton mixing matrix measured in the neutrino oscillation experiments is given by where is the left-handed rotation matrix for diagonalizing the charged-lepton Dirac masses. It is interesting to find that the model-dependent parts of electroweak gauge vertices are governed by a single matrix which is defined as
When one works in the basis where the charged-lepton sector is flavor diagonalized, is fixed by the neutrino oscillation matrix.
The neutrinos also have the interaction to the electroweak doublet Higgs . If assuming lives in the four-dimensional fixed point at , the boundary Dirac mass (2) comes from the Yukawa coupling
where . The doublet Higgs has a non-vanishing expectation value and its fluctuation . After integrating out the fifth dimension and diagonalizing mass matrices, we have
where means the -th mode of the right-handed component.
ii.3 Models for Observable Seesaw
The heavy neutrino interactions to the SM fields are determined by the mixing matrix both in the gauge and Higgs vertices. The matrix is determined by the matrix forms of neutrino masses in the original Lagrangian . The matrix elements in have the experimental upper bounds from electroweak physics, as will be seen later. Another important constraint on comes from the low-energy neutrino experiments, namely, the seesaw-induced masses should be of the order of eV scale, which in turn specifies the scale of heavy neutrino masses . This can be seen from the definition of by rewriting it with the light and heavy neutrino mass eigenvalues
where is an arbitrary matrix with . Therefore one naively expects that, with a fixed order of and for the discovery of experimental signatures of heavy neutrinos, their masses should be very light and satisfy keV (this does not necessarily mean the seesaw operation is not justified as is fixed). The previous collider studies on TeV-scale right-handed neutrinos TeVRH () did not impose the seesaw relation (14) and have to rely on some assumptions for suppressing the necessarily induced masses . For example, the neutrino mass matrix has some singular generation structure, otherwise it leads to the decoupling of seesaw neutrinos from collider physics.
We here present two scenarios in which heavy neutrino modes are accessible at future colliders. The numerical study of these two models will be performed in the next section. It is noted that they are illustrative examples and there are many other possibilities for the observable seesaw with extra dimensions. We will comment on such various alternatives in a later section.
ii.3.1 Model 1 Particular Majorana Masses
A possible scenario for observable heavy neutrinos is to take a specific value of Majorana mass parameters. Let us consider the situation that the bulk Majorana mass and bulk Dirac masses are vanishing on the Minkowski background. The Lagrangian is
The equations of motion without bulk masses are solved by simple oscillators and the mass matrices in four-dimensional effective theory (5) are found
From these, we find the seesaw-induced mass matrix and the mixing with heavy modes:
The KK neutrinos have the mass eigenvalues and (). The effect of infinitely many numbers of KK neutrinos appears as the additional factor . An interesting case is that (the eigenvalues of) takes a specific value where contains half integers DDG (): the seesaw-induced mass is then suppressed by the tangent factor (not only by large Majorana mass), on the other hand, the heavy mode vertex is un-suppressed. This fact realizes the situation that right-handed neutrinos in the seesaw mechanism are observable at sizable rates in future collider experiments HMY () (see also BMOZ ()).
As an explicit example, we consider flavor-independent Majorana masses where is small () and denotes a deviation from massless neutrinos. A vanishing makes the light neutrinos exactly massless, where a complete cancellation occurs within the seesaw effects of heavy neutrinos. As we will see, the parameter takes a tiny value for giving the correct neutrino mass scale.333That seems a fine tuning of model parameters; the bulk Majorana masses must be fixed almost exactly. This tuning is ameliorated by considering a different extra-dimensional setup with the same neutrino mass matrix (see Section IV.2). In the KK-mode picture, the mass spectrum becomes almost vector-like and no chiral zero mode exists. The seesaw-induced mass and the KK Dirac masses are given by
We will consider for the LHC analysis of low-lying KK neutrinos. The neutrino Yukawa coupling is expressed as
ii.3.2 Model 2 Light Dirac Neutrinos
Another example of observable heavy states is realized by assuming no Majorana mass for bulk neutrinos, which leads to lepton number conservation while having sizable couplings to the SM neutrinos. The Lagrangian is
The solution to the bulk equations of motion in the presence of bulk Dirac masses are given by
The zero mode is massless at this stage and has a localized wavefunction controlled by the bulk Dirac mass . The -th excited modes have the squared mass eigenvalues . The mass matrices in four-dimensional effective theory (5) are found
While the excited modes are heavy (), the zero mode has no contribution from bulk and KK masses. Therefore the zero modes compose of Dirac particles with the SM neutrinos and obtain their masses from the SM Higgs field: . On the other hand, since the excited modes have KK Dirac masses and no lepton number violation, they do not give rise to the seesaw-induced mass (i.e. the contributions from and are cancelled to each other) and the right-handed components do not mix with the left-handed SM neutrinos. We thus find the light Dirac neutrino masses and the mixing with heavy modes:
The Dirac neutrino mass can be suppressed by the exponential wavefunction factor , while the heavy KK modes are kept observable. For example, if , the neutrinos are obtained for other parameters being on TeV scale. A negative value of means that the zero mode is localized away from the SM boundary (), which situation leads to the suppression of Dirac neutrino mass . The heavy modes have rather broad wavefunctions in the bulk and the couplings to the SM sector are independent of the exponential suppression. That allows the heavy modes to take sizable boundary couplings and to be observed.
In this model, the light and KK neutrinos are all Dirac particles and their mass eigenvalues are given by
We will consider for the LHC analysis of low-lying KK neutrinos. The neutrino Yukawa coupling is expressed as
where is the unitary matrix which rotates the three-generation right-handed zero modes so that the light Dirac mass matrix is diagonalized to .
ii.3.3 Model 3 Small Lepton Number Violation
A slightly different model for observable heavy neutrinos is constructed by introducing small bulk Majorana mass into Model 2, which means the light neutrinos are Majorana particles. The Lagrangian is
With non-vanishing Majorana masses, the lepton number is broken and the seesaw-induced mass is generated by the integration of heavy modes as in Model 1;
The mixing with KK neutrinos has a similar expression to Model 2;
where we have assumed . The zero-mode contribution is suppressed if it is enough separated from the SM boundary with the localizing wavefunction, which implies . It is found from the above expressions that the seesaw neutrino mass and the couplings of heavy modes can be determined independently that makes the seesaw mechanism observable. The neutrino mass, i.e. the size of lepton number violation is controlled by the bulk Majorana mass . For example, if and a few % mixing of heavy mode,
for eV seesaw-induced masses. The fundamental scale of five-dimensional theory is irrelevant for this evaluation. The zero mode obtains a Majorana mass of the order of and is a light isolated particle with a negligible interaction to the SM sector.
In the end, the low-energy theory contains light Majorana neutrinos with the seesaw-induced mass , almost decoupled zero modes with mass around keV scale, and heavy KK Dirac neutrinos. The mass eigenvalues of these states are explicitly given by
The low-lying KK states would be observable at colliders for . The neutrino Yukawa coupling has a similar expression to that in Model 1.
Iii Seesaw Signatures at the LHC
The production of KK-excited neutrino states is the most important signal in our scenarios, since the signal enables us to explore the mechanism responsible for the generation of tiny neutrino masses. An immediate question is which processes we should pay attention to find out the signal at the LHC. As shown in the previous work HMY (), the tri-lepton signal with missing transverse energy is most prominent since only a small fraction of SM processes contributes to the background against the signal. This lepton number conserving processes, , dominantly occur through the diagrams shown in FIG. 1. In this section, we investigate such seesaw signatures in Models 1 and 2, which are presented in the previous section as typical examples of the observable seesaw. In the following simulation study, we assume, for simplicity, that the bulk mass parameters of right-handed neutrinos are common in flavor space and the complex phases vanish in the orthogonal matrix . With these assumptions, we perform the numerical analysis for the tri-lepton signal in various mass hierarchies of neutrino masses, that is, the normal, inverted, and degenerated patterns.
iii.1 Constraints on Neutrino Yukawa Couplings
Before going to discuss the simulation study in details, we summarize the neutrino mass and mixing matrices which are mandatory to investigate the collider signatures at the LHC. The two matrices are parameterized as
where means (). The Dirac and Majorana phases are denoted by and , respectively. Note that Majorana phases are not relevant in Model 2 since there is no Majorana mass term in the neutrino sector. The neutrino mass differences and the generation mixing parameters have been measured at neutrino oscillation experiments neu_analysis (). We take their typical values,
Since the scenarios we are studying also affect several physical observables such as the flavor-changing processes of charged leptons LFV (), it is important to consider the constraints on neutrino Yukawa couplings to have proper representative points. Integrating out all heavy KK neutrinos, we obtain the following dimension 6 operator in low-energy effective theory which contributes to the leptonic flavor-changing neutral current;
where the coefficient matrix ( and for Models 1 and 2) turns out to be
The operator receives phenomenological constraints as shown in Ref. lowene (), and each component of neutrino Yukawa couplings is thus restricted by comparing the model predictions of the coefficient with experimental data. In particular, the most severe limit is given by the 1-2 component, i.e., the search which puts on the upper bound more than 3 orders of magnitude stronger than the others. To weaken the bound on this operator, especially for the 1-2 component, we take representative values of lepton mixing matrix as shown in TABLE 2. As a result, new physics parameters in the coefficient such as in Model 1 and in Model 2 turn out to be constrained as follows for each pattern of neutrino mass hierarchy:
Notice that the coefficient in Model 1 is irrelevant to the compactification radius and so the bounds on are. For Model 2, the above bounds are obtained for GeV. The compactification radius is also limited by the LEP experiment through the masses of KK excited neutrinos. The lightest ones are for Model 1 and for Model 2, and these states have not be experimentally detected so far. We numerically checked that the constraint is not so severe if GeV. Finally, the SM Higgs mass is to be GeV in evaluating the decay widths of heavy KK neutrinos.
|Model 1||Model 2|
iii.2 Tri-lepton Signals at the LHC
Now let us investigate the tri-lepton signal of heavy neutrino productions at the LHC. Since the tau lepton is hardly detected compared to the others, we consider the signal event including only electrons and muons. There are four kinds of tri-lepton signals: , , , and . In this work, we use two combined signals which are composed of (the signal) and (the signal). Figure 2 shows the total cross sections for these signals from the 1st KK neutrino production at the LHC, which are described as the functions of their mass eigenvalues with fixed values of . It is found from the figure that the cross sections have the universal behaviour within extra-dimensional models; for the normal mass hierarchy, the cross section for the signal is about one or two orders of magnitude larger than the signal, and for the inverted and degenerate spectra, the signal cross section becomes larger than or almost equal to the one. Further, the cross section for Model 2 is found to be small compared with that for Model 1. This is due to a small wavefunction factor of low-lying KK neutrino mode, , which is suppressed by to have tiny neutrino masses. We have also evaluated the contributions of tri-lepton signals from heavier KK neutrinos and found that they are small by more than one order of magnitude and are out of reach of the LHC experiment. A high luminosity collider with clean environment such as the International Linear Collider would distinctly discover the signatures of KK mode resonances.
To clarify whether the tri-lepton signal is captured at the LHC, it is important to estimate SM backgrounds against the signal. The SM backgrounds which produce or mimic the tri-lepton final state have been studied cut (); cut2 () and for the present purpose a useful kinematical cut is discussed to reduce these SM processes cut2 (). According to that work, we adopt the following kinematical cuts for both Models: (i) the existence of two like-sign charged leptons , , and an additional one with the opposite charge , (ii) both of the energies of like-sign leptons are larger than 30 GeV, and (iii) the invariant masses from and and from and are larger than GeV or smaller than GeV. The last condition is imposed to reduce the large background from the leptonic decays of bosons in the SM processes. Figure 3 shows the total cross sections of signals after imposing these kinematical cuts. To estimate the efficiency for the signal events due to the cuts, we use the Monte Carlo simulation using the CalcHep code CalcHep (). Since the event numbers of SM backgrounds after the cuts are about 260 for the signal and 110 for the one with the luminosity of 30 fbcut2 (), the events are expected to be observed if the lightest KK mass is less than a few hundred GeV.
For Model 1, FIG. 4 shows the luminosity which is required to find the seesaw neutrino signal at the LHC as the contour plots on the parameter plane. The luminosity contours for 10, 30, and 300 fb are depicted in the figures. These contours are obtained by computing the significance for the signal discovery
where () denotes the total number of the 2 or 2 events (that of the corresponding SM backgrounds) after the kinematical cuts. Since both and are proportional to the luminosity, it is possible to estimate the required luminosity for, e.g. giving , which is plotted in the above figures. The luminosity for signal confirmation () are also found by rescaling the results, according to the formula (luminosity) . It is found that, if is less than a few hundreds GeV, the signals would be observed at an early run of the LHC, in particular, Model 1 with the degenerate mass spectrum will definitely be excluded or confirmed. A larger luminosity is needed for a smaller size of extra dimension to reveal its existence.
Model 2, as it stands, generally predicts too small production rates to be found out at the LHC. However the extra-dimensional framework has various options without introducing additional particles. That leads to simple modifications of the model and can make it observable, as explained explicitly in the next section.
Iv Other Configurations for Observable Seesaw
We have discussed the collider signatures of two typical models of observable seesaw. They are constructed in five-dimensional spacetime and utilize the mechanisms which are peculiar to the presence of extra dimensions; the specific value of bulk Majorana mass in Model 1 and the suppression factor from localized wavefunction in Model 2. As we mentioned before, while the Lagrangian is simple and common, the five-dimensional theory makes right-handed neutrinos observable in various ways which have their own physical meanings. Among them, we here present three possibilities; localized Majorana mass terms, boundary conditions of bulk fields, and curved gravitational backgrounds (and their combinations).
iv.1 Boundary Majorana Masses
We first consider the case that Majorana mass parameters for bulk fermions depend on the extra-dimensional coordinate . An interesting case is that Majorana masses are localized at the boundaries of fifth dimension (). The boundary Majorana masses may be natural if the lepton number symmetry is exact in the bulk and locally broken at the boundaries. For the flat background, the two fixed points are physically equivalent, and in the following we choose as an example. We consider the Lagrangian
If one also includes the term, is replaced with in the following formulas. The solutions to the bulk equations of motion have been given in Section II.3.2. The mass matrices in four-dimensional effective theory are found
One type of the Majorana masses, , vanishes since have the negative parity and the wavefunctions become zero at the boundary. Another Majorana mass matrix, , has the off-diagonal () entries since the KK momentum is not conserved at the boundary.
It seems difficult to diagonalize the heavy-field mass matrix which is composed of and the KK masses . However the mixing vertex between heavy modes and the SM sector can be evaluated by getting the inverse of that is given by
Notice that all the components but the 1st one are vanishing in the interaction basis. From this mixing matrix, we obtain the seesaw-induced neutrino masses and the heavy-mode mixing with the SM neutrinos:
The form of is apparently the same as the usual four-dimensional seesaw mechanism, but has an extra factor which originates from the extra dimension, i.e., the volume factor and the localization factor controlled by the bulk Dirac mass . These two factors are available in the case of boundary Majorana masses and are not obtained for bulk Majorana masses as in the previous section. This is because these factors appear twice in the boundary Majorana masses for bulk fields, but only once in the boundary coupling to the SM fields, which result in the cancellation only for the seesaw-induced masses. In the present case, the two factors can be used for enhancing the heavy-mode couplings to the SM neutrinos, while keeping the seesaw-induced masses un-affected and made tiny. The enhancement by the localization factor requires . If one introduced Majorana masses at another boundary , the bulk mass parameter should be replaced with in the formula.
In this way, the Majorana masses on the boundary realize an observable seesaw model with appropriate wavefunction factors. The SM fields (neutrinos, electroweak gauge bosons, etc.) interact with the bulk sector only through the zero mode . It is noticed that is not a mass eigenstate, and the cross sections for collider physics might be peaked at the mass eigenvalues of KK neutrinos via the mixing with . This is however not the case in a quantitative meaning: the KK-mode contamination of 1% mixing is found from (48) to imply . This in turn implies by the seesaw formula (47) that the Majorana mass parameter for is roughly given by and very small. Therefore the heavy KK neutrinos do not so much mix with such a light zero mode and cannot be detected at collider experiments.
The conclusion is that, in the extra-dimensional setup in this subsection, the zero-mode wavefunction factors enhance the heavy-mode couplings, keeping the usual seesaw formula, and play a key role for realizing the observable seesaw. However from a phenomenological viewpoint, only the light zero mode is found to be accessible. That depends on which elements are vanishing in the neutrino mass matrices and could be changed by some effects within the model or its extensions. For example, an additional boundary mass or interaction term would lead to a repulsive effect which makes -odd fields off from the boundary so that they obtain nonzero Majorana masses. Another option is that a singular boundary profile is regulated by introducing some scalar field, which generates a four-dimensional domain wall with a finite width along the extra dimension. That would lead to non-vanishing couplings of bulk fields in four-dimensional effective theory.
iv.2 Boundary Conditions
Another important option of five-dimensional theory is to choose boundary conditions for bulk fields. For a finite size of extra space, the boundary conditions determine the bulk profile, i.e. wavefunctions of higher-dimensional fields, and then fix their low-energy physics. We have so far discussed the standard boundary condition for a five-dimensional spinor on the orbifold, that is, the Neumann and Dirichlet type boundary conditions for the upper and lower components, respectively; at both and . In this section, let us consider another mixed-type condition:444An overall sign is fixed by assuming that the upper component of has non-vanishing wavefunction at the boundary where the SM fields reside.
i.e. the upper component has a positive (negative) parity under the reflection about the () boundary. The lower component has the opposite parity assignment. In terms of KK-mode wavefunctions, and , in the absence of extra boundary terms. Notice that this is equivalent to the Scherk-Schwarz boundary condition SS () where a non-trivial twist is imposed in circulating along the extra dimension: .
Let us consider the following Lagrangian
and evaluate the seesaw mass matrix under the boundary conditions (49). The wavefunctions for free bulk fields are given by
The mass matrices in four-dimensional effective theory are found
The only difference from the previous standard boundary condition is the KK mass spectrum . We find the seesaw-induced neutrino mass and the heavy-mode mixing with the SM neutrinos:
The light neutrino mass has the factor as a consequence of summing up the heavy-mode seesaw contributions. Notice that, for the standard boundary condition, this factor is . The difference is understood in the following two limits: For the large radius limit, , the two boundaries are so separated in the extra-dimensional space that the difference of boundary conditions at is irrelevant to the SM physics at , and two factors merge into the same value . The other case, , is the decoupling limit of KK modes. They become so heavy that the low-energy physics is determined by light modes only. It is the chiral zero mode in case of the standard boundary condition. For the present twisted boundary condition, the zero mode is absent and the limit leads to vanishing seesaw-induced masses. That is, the inverse seesaw suppression inverse_seesaw () is realized at each KK level and the total seesaw-induced mass is proportional (not inverse proportional) to heavy-field Majorana mass .
In this way, the boundary condition mechanism leads to the situation that no massless mode appears in the KK decomposition and therefore bulk Majorana masses can be made small without being conflicting with the heavy-mode integration. Let us consider the case of small Majorana masses (). The seesaw-induced mass and the mass eigenvalues of KK Dirac neutrinos become
This agrees with the spectrum of Model 1 discussed in Section II.3.1 with the replacement . The mixing with heavy modes also has the correspondence under this replacement and with a field rearrangement. Therefore the present model with the twisted boundary condition is observable and gives the same seesaw phenomenology, in particular the LHC signatures, as given in Section III. A difference of two models is the interpretation of small parameters and . The parameter in Model 1 is a tiny deviation from the fixed value of model parameter () and is hard to be determined in dynamical way. On the other hand, is a Lagrangian parameter itself and is easier to be suppressed and controlled with high-energy physics.
If one includes the bulk Dirac mass , the above formulas in low-energy effective theory are modified as
In the regime , the Dirac mass parameter is effective in suppressing the seesaw-induced masses , compared with (53): for small bulk Majorana masses, we obtain .
iv.3 Boundary Majorana Masses and Boundary Conditions
An interesting and physically different scheme is given by considering both of boundary Majorana mass and non-trivial boundary condition of bulk neutrinos, discussed in the previous two sections. This model is particular in that the seesaw-induced neutrino mass vanishes for any values of model parameters. Therefore the heavy-mode couplings to the SM sector are arbitrarily fixed so that the scenario is observable at collider experiments. The Majorana mass parameters do not appear in any place of low-energy effective theory at the leading order.
Let us consider the same Lagrangian as in Section IV.1
That is, the Majorana masses for bulk fermions are only on the SM boundary. Further we assume the twisted boundary condition as in Section IV.2:
Therefore the wavefunctions and KK masses are given by (51) as previously. The mass matrices in four-dimensional effective theory are found
The Majorana masses vanish since have the negative parity and the wavefunctions become zero at the boundary on which the Lagrangian mass term is placed. Another Majorana mass matrix takes the common value for all the matrix elements including the off-diagonal ones. The vertex matrix of heavy modes can be evaluated by taking the inverse of that is given by
Notice that the ()-th components are all vanishing in the interaction basis of KK modes. Further the non-vanishing elements do not depend on the Majorana mass parameter . From this mixing matrix, we find the seesaw-induced neutrino mass and the heavy-mode mixing with the SM neutrinos:
It is interesting that the light neutrino mass vanishes, irrespectively of model parameters. The heavy-mode mixing is governed by the compactification scale and the boundary mass . Their ratio can therefore be arbitrarily fixed and made sizable. In this model, the bulk Majorana mass does not join in any formula of the seesaw operation and only affects the mass spectrum of heavy modes. The spectrum is found to be roughly determined only by the compactification scale and may be corrected by Majorana masses which are suppressed by the cutoff scale of the theory.
The above result shows that the scheme in this subsection gives a natural realization of the observable seesaw in the zero-th approximation. Towards a phenomenologically viable model, nonzero neutrino masses are needed to be generated by some dynamics. Among various possibilities, a simple way is to put, as a correction, the Majorana masses in the bulk and/or on the other boundary :
Repeating the previous procedure with these terms, we obtain the seesaw-induced neutrino masses
Finally we briefly comment on other patterns of the model. There seems to exist 3 degrees of freedom: the boundary Majorana masses on