Contents

Probing Exotic Fermions from a Seesaw/Radiative Model at the LHC

Kristian L. McDonald

ARC Centre of Excellence for Particle Physics at the Terascale,

School of Physics, The University of Sydney, NSW 2006, Australia

klmcd@physics.usyd.edu.au

There exist tree-level generalizations of the Type-I and Type-III seesaw mechanisms that realize neutrino mass via low-energy effective operators with . However, these generalizations also give radiative masses that can dominate the seesaw masses in regions of parameter space — i.e. they are not purely seesaw models, nor are they purely radiative models, but instead they are something in between. A recent work detailed the remaining minimal models of this type. Here we study the remaining model with and investigate the collider phenomenology of the exotic quadruplet fermions it predicts. These exotics can be pair produced at the LHC via electroweak interactions and their subsequent decays produce a host of multi-lepton signals. Furthermore, the branching fractions for events with distinct charged-leptons encode information about both the neutrino mass hierarchy and the leptonic mixing phases. In large regions of parameter-space discovery at the LHC with a 5 significance is viable for masses approaching the TeV scale.

1 Introduction

The Type-I [1] and Type-III [2] seesaw mechanisms offer a simple explanation for the existence of light Standard Model (SM) neutrinos. In these approaches the tree-level exchange of heavy intermediate fermions achieves neutrino masses with an inverse dependence on the heavy-fermion mass, , suppressing the masses relative to the weak scale. In the low-energy effective theory these masses are described by the non-renormalizable operator , which famously has mass-dimension  [3].

There exist generalizations of the Type-I and Type-III seesaws that can similarly explain the existence of light SM neutrinos. The basic point is that the Type-I and Type-III seesaws can be described by a generic tree-level diagram with two external scalars and a heavy intermediate fermion; see Figure 1. The use of different intermediate fermions allows for variant tree-level seesaws, where either one or both of the external scalars is a beyond-SM field.

Naively it appears that many variant seesaws are possible. However, the vacuum expectation values (VEVs) of the beyond-SM scalars are generally constrained by -parameter measurements to satisfy . Such small VEVs can arise naturally if they are induced and therefore develop an inverse dependence on the scalar masses, i.e. . Demanding such an explanation for the small VEVs greatly restricts the number of minimal realizations of Figure 1 [4]. Because the VEVs of the beyond-SM scalars are induced, these generalized seesaws generate low-energy effective operators with . It was shown that there are only four such minimal models that give effective operators with  [4]; namely the model of Ref. [5], the models of Refs. [6, 7] and Ref. [8], and the model proposed in Ref. [4].111We list the particle content for these models, and show the explicit nature of the associated Feynman diagrams, in Section 2; see Table 1 and Figure 4 respectively.

Figure 1: Generic tree-level diagram for a seesaw mechanism with a heavy intermediate fermion. The simplest realizations are the Type-I and Type-III seesaws, for which the external scalars are the SM doublet, , and is a Majorana fermion. In the generalized seesaws the fermion can be either Majorana or Dirac and the external scalars can be beyond-SM multiplets.

The generalized seesaws turn out to be more complicated creatures than their Type-I and Type-III cousins. Extensions of the SM that permit the tree-level diagram of Figure 1 automatically admit the term in the scalar potential [4]. This allows one to close the seesaw diagram in Figure 1 to obtain the loop-diagram in Figure 2. Thus, strictly speaking, the generalized models are not purely seesaw models, nor are they purely radiative models, but instead they are something in between. Both mechanisms are always present, with the tree-level mass being dominant in some regions of parameter space and the radiative mass being dominant in other regions. If one envisions the theory space for models with massive neutrinos, the generalized seesaws exist in the intersection of the set of models with seesaw masses and the set of models with radiative masses (see Figure 3). Put succinctly, these are seesaw/radiative models, and they are irreducible, in the sense that modifying the particle content to remove one effect necessarily removes them both. As we shall see, this is manifest in an identical flavor structure for the seesaw and radiative masses.

Figure 2: The generic loop-diagram present in any model that realizes the tree-level seesaw in Figure 1.

These distinctions give an important difference relative to the Type-I and Type-III approaches. In the seesaw/radiative models neutrino mass is generated either by a tree-level seesaw described by the operator with , or by a radiative diagram generating a operator with additional loop-suppression. In either case the new physics is constrained to be much lighter than that allowed by the Type-I and Type-III seesaws; e.g. one can have which decreases with increasing . Collider experiments at the energy frontier will therefore explore the parameter space for these generalized models long before the full parameter space for the Type-I and Type-III seesaws can be investigated.

In the present work, we detail the nature of neutrino mass in the newly proposed model with , and study the collider phenomenology of the exotic fermions predicted by the model. These fermions form an isospin-3/2 representation of and contain a doubly-charged component. Collider production of the exotics is controlled by electroweak interactions and depends only on the fermion mass. The decay properties of the fermions, and therefore the expected signals at colliders like the LHC, have some sensitivity to model details and, in particular, depend on the mixing with SM leptons. However, because the model predicts a basic relation amongst VEVs (), some decay branching fractions can be largely determined; e.g. the total leptonic branching fractions can be determined with essentially no dependence on the neutrino mass hierarchy. However, the relative branching fractions for decays to different charged leptons have remnant dependence on the properties of the neutrino sector. We shall see that the number of light charged-lepton events (), relative to the number of tauon events, can encode information regarding the neutrino mass hierarchy and the mixing phases.

Figure 3: A Venn diagram for a portion of theory space with massive neutrinos.

A number of signals predicted by the model are reminiscent of those found in related seesaw models like the Type-III seesaw and the models [6, 8]. However, the branching fractions for lepton-number violating like-sign dilepton events is suppressed relative to that found in the Type-III seesaw [9]. In our analysis we discuss differences between the models and indicate strategies for searching for the exotic fermions; for example, the present model predicts an unobservable rate for lepton-number violating events like , whereas events like are expected in both the Type-III case [9] and the model of Ref. [6]. Such events, and others that we outline, lead to a host of multi-lepton final states. We shall see that the model also predicts a doubly-charged fermion that can be discovered at the 5 level for masses approaching the TeV scale in optimistic cases.

In our presentation we make efforts to follow the structure of Refs. [6] and [8], which detail the collider phenomenology of related exotic fermions, to allow for easier comparison. As shall be evident during our analysis, there are aspects of our work that are relevant for the related models. Our results suggest it would be interesting to undertake a detailed comparative analysis of the exotic fermions appearing in the models with , including the triplets , which appear in the model [5],222To date the collider phenomenology of these states have not been studied within the context of the model. A study of the bounds from flavor changing processes appeared in Ref. [10], and they were studied in different contexts in Ref. [11]. and the quintuplet fermions from the alternative models [6, 8]. All four models with contain doubly-charged exotic fermions; searches for these states would appear to provide a simple way to obtain generic experimental constraints for this class of models.

Many works have studied production mechanisms and detection prospects for the heavy neutrinos employed in the Type-I seesaw [12]. Both CMS [13] and ATLAS [14] have searched for the corresponding signals in the LHC data. Similarly, the triplet fermions in the Type-III seesaw are well studied [15], and Ref. [16] ([17]) contains an ATLAS (CMS) search for these exotics. A comparative study of LHC signals from the seesaws has appeared [9], and more general discussion of TeV-scale exotics related to neutrino mass [18], and right-handed neutrinos [19], exists in the literature. For the collider phenomenology of the exotic scalars in the model see Ref. [20] (also see [21]). Note that perturbative unitarity gives general upper-bounds on the quantum numbers of larger multiplets [22, 23], and that the quadruplet fermions of interest in this work were previously considered as dark matter [24], and in relation to radiative neutrino-mass [25]. Alternative models of neutrino mass realizing low-energy effective operators with exist [26], and an earlier work combined a seesaw model with a radiative model [27] (also see Ref. [28]).333The models of interest here differ from these earlier works. In Refs. [27, 28] distinct beyond-SM fields generate the tree- and loop-masses; one can modify the particle spectrum to turn off one effect while retaining the other. In the present models the same fields generate both the tree- and loop-masses, and experimentally viable masses can be achieved from either effect. Also, it was recently shown that some versions of the inverse seesaw mechanism can generate neutrino mass via a combination of both tree-level and radiative masses, similar to the models discussed here [29].

The layout of this paper is as follows. In Section 2 we introduce the model and discuss the symmetry-breaking sector. Section 3 details the origin of neutrino mass and Section 4 investigates the extent to which the parameters can be fixed by neutrino oscillation data. Collider production of exotic fermions is discussed in Section 5. The mass-eigenstate interaction Lagrangian is presented in Section 6 and exotic fermion decays are detailed in Section 7. Detection signals are discussed in Section 8 and the work concludes in Section 9.

2 A Seesaw/Radiative Model with

In this section we introduce the model of interest in this work and discuss aspects its scalar sector. As noted already, there are only four minimal models of this type that produce low-energy effective operators with  [4]. We list the particle content for these models in Table 1.444Note that model can also be implemented as an inverse seesaw mechanism [30]. Each model generates the tree-level diagram in Figure 1 and can thus achieve seesaw neutrino masses. However, because the VEVs of the beyond-SM scalars are induced the corresponding Feynman diagrams can be “opened up.” These “open” Feynman diagrams reveal the nature of the associated low-energy operators and are shown in Figure 4.

The new model appears as model in the table and is obtained by adding the following multiplets to the SM:

(1)

These permit the following pertinent Yukawa terms

(2)

whose explicit structure is

(3)

Here we write the fermion as a symmetric tensor , with components555Note that is not the anti-particle of : .

(4)

and similarly the scalar is represented by the symmetric tensor , with components

(5)

where one should differentiate between and . The matrix form of the real triplet is taken as

(6)

With the above, one can expand the Yukawa couplings to obtain the Lagrangian terms of interest for the seesaw and radiative diagrams. The explicit expansions appear in Appendix A. Note that flavor labels are suppressed in the above, so that with etc.

  Model Mass Insertion Ref.
Dirac [5]
Majorana [6]
Dirac [4]
Dirac [8]
Table 1: Minimal Seesaw/Radiative Models with  [4].

The scalar potential contains the terms

(7)

where is the SM doublet. The last term induces a VEV for after electroweak symmetry breaking:

(8)

The inverse mass-dependence in this expression shows that is naturally suppressed relative to the electroweak scale for  GeV. Similarly the terms666The expansion of the -term appears in Appendix A.

(9)

trigger a nonzero VEV for ,

(10)

These expressions show that is generically expected for , given that direct searches for exotic charged fields require  GeV. We work with throughout so that . This is a rather generic feature of the model; we shall see that it influences both the decay properties and collider signals of the exotic fermions. Note that in Eq. (8) one can consider to denote the full tree-level mass for , containing both the explicit mass-term for in Eq. (7) and the additional subdominant contributions from the VEVs of the other scalars. Similarly for in Eq. (10).

(a) (b)
(c) (d)
Figure 4: Feynman diagrams for the tree-level seesaws with . Model is the focus of this work (labels match Table 1).

The beyond-SM scalars and contribute to electroweak symmetry breaking and thus modify the tree-level value of the parameter. The SM predicts the tree-level value  [31], and the experimentally observed value is at the level [32]. Consequently beyond-SM scalars with isospin must have small VEVs. In the present model the tree-level parameter is given by

(11)

The constraint requires

(12)

which reduces to

(13)

Thus, we generically require  GeV. Such small VEVs arise naturally due to the inverse dependence on the scalar masses found in Eqs. (8) and (10). We plot the VEV as a function of for the fixed values of in Figure 5.

Figure 5: The VEV for the scalar as a function of the scalar mass , for fixed values of the dimensionful coupling . The solid (dashed, dot-dashed) line is for (), and the horizontal line is the upper bound on from the parameter constraint.

The nonzero VEVs for and induce mixing between these scalars and the SM Higgs. For the mixing of with the Higgs is subdominant to the - mixing, and to good approximation one can neglect the - mixing. Shifting the neutral scalars around their VEVs, and , the results of Ref. [33] allow one to approximate the neutral-scalar mixing as

(14)

where are the mass eigenstates. For the mixing angle obeys

(15)

giving . In what follows we denote the mostly-SM Higgs , with mass  GeV, simply as . Unfortunately the tiny mixing angle will not be discernible at the LHC for the parameter space of interest in this work (see Ref. [33] for details).

3 Neutrino Mass

Having described the model and introduced the scalar sector we now turn to the origin of neutrino mass. The Yukawa Lagrangian (2) mixes the SM neutrinos with the neutral fermions . In the basis , we write the mass Lagrangian as

(16)

The mass matrix is comprised of two parts; namely a tree-level term and a radiative term, . The most important radiative correction is the contribution to the SM neutrino mass matrix, which results from the Feynman diagram in Figure 2. One can therefore write the leading order loop-induced mass matrix as . We will detail the form of matrix in Section 3.2; for now it suffices to note that the entries of the loop-induced mass matrix must be on the order of, or less than, the observed SM neutrino masses. In what follows we consider the seesaw and radiative masses in turn.

3.1 Tree-Level Seesaw Masses

Extracting the mass terms from the Yukawa Lagrangian, one can write the mass Lagrangian as

(17)

where the Dirac mass-matrices are

(18)

In general, the heavy fermion mass matrix is an matrix for generations of exotic fermions. We consider the minimal case of , for which .777Results written in terms of also hold for the more general case of , for which one can always work in a basis where is diagonal, . Note that we do not include the effects of Yukawa terms like and . These are suppressed by the small VEVs, and for simplicity we assume small-enough Yukawa couplings so they can be neglected. The mass matrix can be partitioned into a standard seesaw form:

(19)

where the Dirac mass matrix and the heavy-fermion mass matrix are

(20)

For  TeV and  GeV, the entries of the distinct mass-matrices are hierarchically separated, which we denote symbolically as:

(21)

With this hierarchy, a standard leading-order seesaw diagonalization can be performed.

The mass eigenstates are related to the interaction states via , where the leading-order expression for the rotation is

(22)

The diagonalized mass matrix is

(23)

where is the PMNS matrix which diagonalizes the mass matrix for the light SM neutrinos:

(24)

The heavy neutrinos receive mass corrections of order , which split the would-be heavy Dirac fermion into a pseudo-Dirac pair. However, this splitting is tiny, being on the order of the light neutrino masses, and can be neglected for all practical purposes. To good approximation the heavy neutrinos can be treated as a Dirac particle.

The tree-level piece of the SM neutrino mass-matrix has a standard seesaw form:

(25)

giving

(26)

where label SM flavors. This expression has the familiar seesaw form of a Dirac-mass-squared divided by a heavy fermion mass. Using Eq. (10) one can manipulate the tree-level mass matrix to obtain

(27)

which will be useful in what follows. Furthermore, denoting the beyond-SM dimensionful parameters by a common scale and using Eq. (8) gives

(28)

This shows that the tree-level masses arise from a low-energy effective operator with , giving as expected.

3.2 Combined Loop- and Tree-Level Masses

In addition to the tree-level diagram one must calculate the radiative diagram in Figure 2. There are three distinct diagrams with different sets of virtual fields propagating in the loops. One diagram contains the neutral fields , and the other two have the singly-charged fields and , respectively.888Recall that is not the anti-particle of , and that . To good approximation one can neglect the splitting between members of a given multiplet when calculating the loop diagrams. The components of have degenerate tree-level masses that are lifted by radiative effects. We shall see in Section 7 that these mass-splittings are much smaller than the common tree-level mass. Similarly, the components of receive small radiative mass-splittings that can be neglected when calculating the loop-masses.999 The components of also receive tree-level splittings due to the VEVs of the various scalars. The only contributions that can be sizable come from . However, constraints from the parameter require  GeV [34], consistent with such splittings being small. With this approximation, the only differences between the loop-diagrams are the numerical factors from the vertices, and the total mass-matrix for the SM neutrinos is

The tree- (loop-) mass is the first (second) term in the curly brackets. Observe that both terms have an identical flavor structure; this is a signature feature of the seesaw/radiative models, and it means the structure of the matrix that diagonalizes does not depend on whether the tree- or loop-mass is dominant. Furthermore, the ratio is insensitive to both the Yukawa couplings , and the quartic coupling ; therefore the small- limit, which makes small, does not affect . We present various limits of the radiative mass in Appendix B.

It is interesting to determine the regions of parameter space in which the seesaw-mass dominates. Eq. (LABEL:combined_nu_mass) shows that larger values of tend to increase the ratio , while larger values of tend to suppress this ratio. We plot as a function of the fermion mass and the scalar mass in Figure 6. The fixed values  TeV and  GeV are used. The plot shows the region in the plane of greatest interest for the LHC. The tree-level mass is dominant in much of this parameter space and the ratio satisfies for the entire region shown. As can be seen, keeping one of the masses small ( GeV), the other can be taken large () while retaining , while for the loop-mass becomes dominant for  TeV. Thus, for heavy the tree-level region of parameter space will be most relevant for colliders like the LHC.

We are interested in the collider phenomenology of the exotic fermion and take it as the lightest beyond-SM multiplet. We focus on the parameter space in which the triplet is the heaviest, namely

(30)

Within this range, the specific value of is not particularly important for the collider phenomenology of . The reader should keep in mind that, in terms of the mechanism of neutrino mass, values of in the lower range of this interval tend to increase the ratio , while larger values have the reverse effect.

Figure 6: Ratio of the tree-level mass to the loop-mass, , as a function of the fermion () and scalar () masses. The meshed (plain) region has (, and for the entire region.

4 Fixing the Yukawa Couplings

With three SM neutrinos, a generic mass matrix of the form

(31)

has a vanishing determinant. The mass eigenvalues therefore contain one massless and two massive (mostly) SM neutrinos in addition to a pseudo-Dirac heavy exotic fermion. The presence of a massless neutrino means the absolute neutrino mass scale is fixed by the observed mass-squared differences. Furthermore, up to an overall scale factor, the couplings can be largely expressed in terms of the oscillation observables.

Writing the PMNS mixing-matrix as

(32)

the matrix contains the Majorana phase, and can be taken as for our case with a massless neutrino [36]. The best-fit neutrino oscillation parameters are listed in Table 2 [35], and we use these for our numerics throughout. The CP phase and the Majorana phase are not experimentally known and can assume any value at the 2 level; we therefore treat these as free parameters.

We denote the ratio of mass-squared differences by

(33)

and write the Yukawa couplings as

(34)

Here are the magnitudes of the flavor-space vectors , so that and are complex vectors of unit norm:

(35)

In the following we obtain the form of these unit vectors for a normal hierarchy and an inverted hierarchy. This information influences the collider signals of the model.

Parameter Best fit ()
Normal Hierarchy Inverted Hierarchy
Table 2: Neutrino oscillation parameters [35].

4.1 Normal Hierarchy

Consider a region of parameter space in which the the seesaw mass is dominant and the radiative mass can be neglected. Defining the quantity as

(36)

the results of Ref. [36] allow one to write the mass eigenvalues as

(37)

Noting that

(38)

fixes the product of parameters appearing in Eq. (37) as

(39)

Due to the related flavor dependence of the seesaw and radiative masses, one can extend these results to the more general case where the radiative mass is important/dominant. Rewriting Eq. (37) to include the loop mass in Eq. (LABEL:combined_nu_mass) gives

(40)

Combining Eqs. (40) and (39) allows one to fix a combination of the parameters in terms of the overall scale of the neutrino masses in the general case.

The Yukawa unit vectors can be fixed in terms of the oscillation observables:

(41)

up to the dependence on the unknown phases and . For example, with , the PMNS best-fit values give

(42)

If the tree-level mass is dominant, or on the order of the loop-mass, one has

(43)

where we introduce the dimensionless vectors

(44)

and write their magnitude as . We shall see later that the quantities influence the decay properties of the exotic fermions. Eqs. (39) and  (43) give the relation

(45)

for the case of a normal hierarchy. This fixes the relative size of and in terms of the overall scale of the SM neutrino masses. Provided the radiative mass is not significantly larger than the tree-level mass, this relationship also provides a good approximation in the presence of the loop-mass.

4.2 Inverted Hierarchy

The same procedure can be followed for the inverted hierarchy; defining

(46)

the light-neutrino mass eigenvalues are

(47)

when the tree-level mass dominates. The nonzero mass-eigenvalues are fixed via the observed mass-squared differences, giving

(48)

Including the loop mass gives

(49)

allowing one to fix the overall scale via (48). The Yukawa unit-vectors are fixed to be

(50)

When the tree-level mass is dominant, or on the order of the loop-mass, one has