# Majorana vs pseudo-Dirac neutrinos at the ILC

## Abstract

Neutrino masses could originate in seesaw models testable at colliders, with light mediators and an approximate lepton number symmetry. The minimal model of this type contains two quasi-degenerate Majorana fermions forming a pseudo-Dirac pair. An important question is to what extent future colliders will have sensitivity to the splitting between the Majorana components, since this quantity signals the breaking of lepton number and is connected to the light neutrino masses. We consider the production of these neutral heavy leptons at the ILC, where their displaced decays provide a golden signal: a forward–backward charge asymmetry, which depends crucially on the mass splitting between the two Majorana components. We show that this observable can constrain the mass splitting to values much lower than current bounds from neutrinoless double beta decay and *natural* loop corrections.

## 1 Introduction

Extensions of the Standard Model that can explain neutrino masses [1, 2, 3, 4] are well motivated leads to the new physics realm. The mass scale of the neutrino mass mediators is unknown, and the possibility that they could be light enough to be produced and tested in laboratory experiments has been extensively discussed in the literature. Particularly interesting signals of this type of new physics are displaced vertices [5, 6, 7, 8, 9, 10, 11, 12, 13], since usually such light mediators are also very weakly coupled and have long lifetimes.

Present and future colliders have the opportunity to discover or constrain interesting regions of parameter space in these models [14], particularly in connection with low-scale leptogenesis [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. The possibility to test leptogenesis scenarios is however very challenging due to the large parameter space that can affect the generated baryon asymmetry. The putative discovery of the neutrino mass mediators and the measurement of their properties will be essential to achieve this goal. Particularly important questions are establishing the Majorana nature of the heavy neutral leptons expected in the Type I seesaw model and measuring their mass spectrum [33] and flavour mixings [11].

Determining the Majorana nature for on-shell particles is in principle straightforward, it is sufficient to observe their lepton number violating decays (LNV). However, light neutrino mass mediators with sufficiently large mixings require an approximate lepton number symmetry to avoid fine-tuning [34, 35, 36, 37, 38, 39]. This implies that mediators come in quasi-degenerate pairs (i.e. pseudo-Dirac particles) that interfere destructively to cancel the LNV decays [35]. We expect on general grounds that the cancellation of LNV decays will be effective provided the mass splitting, \(\delta M\), of the pseudo-Dirac pair is small compared to their decay width, \(\Gamma \), which typically requires a strong degeneracy. On the other hand, if LNV decays are not suppressed, i.e. \(\delta M \gg \Gamma \), it is interesting to understand to what extent their existence can be established.

In this paper we consider the production of pseudo-Dirac neutrinos *N* in a \(e^+ e^-\) collider, in processes such as \(e^+ e^- \rightarrow N \nu \), leading to a displaced semileptonic decay, \(N \rightarrow l^\pm j j\). The total number of positive and negative leptons in the displaced vertex is not a good discriminator of the Majorana nature, because the final light neutrino or antineutrino goes undetected. However, the pseudorapidity distribution of the final lepton changes drastically in the two cases [40] and therefore is a good discriminator. Other recent proposals to use angular information to test the Majorana nature have been recently discussed in [41, 42].

The paper is organized as follows. In Sect. 2 we review the minimal Type I seesaw model, where we define our notation and link the heavy neutrino mass splitting \(\delta M\) with LNV parameters. In Sect. 3, we review the current most stringent constraints on \(\delta M\) that come from neutrinoless double beta decay, and the requirement of no fine-tuning between tree and loop corrections to the light neutrino masses. In Sect. 4 we examine the process \(e^+e^-\rightarrow \nu N\rightarrow \nu l^\pm W^{\mp *}\), and show that the LNV contribution effectively vanishes when the mass splitting (and width difference) goes to zero. In Sect. 5 we study this process at the ILC, and quantify the forward–backward charge asymmetry of the lepton as a function of \(\delta M\). Backgrounds are avoided by requiring the heavy neutrino signature to include a displaced vertex. The putative observation of such an asymmetry implies strong bounds on \(\delta M\) of the order of decay width, that could useful to constrain resonant leptogenesis scenarios [43].

## 2 The minimal type I seesaw model

These textures are well known. Setting \(\mu =\mu '=0\) leads to the Linear Seesaw [45], setting \(\varepsilon =\mu '=0\) corresponds to the Inverse Seesaw [34, 44], while \(\varepsilon =0\) is sometimes called Extended Seesaw [38, 46]. A radiative model with \(\varepsilon =\mu =0\) has also been discussed in [47]. Any of the three terms imply the existence of LNV processes, and, as long as \(\varepsilon \ne 0\), one can explain the two measured light neutrino mass differences [36].

In contrast with the standard seesaw, the smallness of neutrino masses does not require a large hierarchy \(m_D \ll M\), it is sufficient to suppress two LNV parameters: \(\mu , \varepsilon \ll M\). On the other hand, the splitting of the heavy states also depends on \(\mu '\), which has a negligible impact on light neutrino masses at tree level. If \(\mu '\) is large, one can have a sizeable splitting of the heavy states and can expect unsuppressed contributions to LNV processes. Nevertheless, in this case large loop corrections to light neutrino masses are also expected, which must be kept under control [38].

^{1}in all generality as:

*R*matrix, originally introduced in [48], is in this case a \(2 \times 2\) orthogonal complex matrix, fully defined in terms of the complex angle: \(\theta _{45}+ i \gamma _{45}\). Finally the \(H, {\bar{H}}\) hermitian matrices are defined by

*x*gives:

^{2}Thus, as was shown in [50], we find that \(\mu '\) can encode a large violation of lepton number without affecting significantly the light neutrino masses at tree level, and that this effect depends on the mass splitting between the heavy neutrinos.

## 3 Current constraints on mass splittings

The mass splitting of the heavy neutrinos is connected primarily to \(\mu '\) in the region of interest and LNV processes such as neutrinoless double beta decay (\(0\nu \beta \beta \)) are very sensitive to it [38, 51]. Thus, the non-observation of this process can set an upper bound on the splitting. In addition, loop corrections to the light neutrino masses have also been shown to be sensitive to \(\mu '\) [38, 50], so requiring no unnatural cancellation between tree level and one loop corrections to neutrino masses can also severely constrain the value of the splitting.

*ee*element as a result of gauge invariance. The nuclear matrix element (NME), \({\mathcal {M}}^{0\nu \beta \beta }\), is a function of the mass of the virtual neutrino. Finally, we have defined \(\Delta {\mathcal {M}}(M_a,\,M_b)\equiv {\mathcal {M}}^{0\nu \beta \beta }(M_a)-{\mathcal {M}}^{0\nu \beta \beta }(M_b)\).

*Z*and Higgs boson masses, respectively. In this case, due to the structure of \(m'_D\) in Eq. (8), the leading term again vanishes if the mass splitting goes to zero. Conversely, requiring the loop corrections to be sufficiently small gives an upper bound on the heavy neutrino mass splitting or on the heavy neutrino mixing for a fixed value of the splitting.

In Fig. 1 we show iso-contours of maximum splitting \(\mathrm{Max}(\delta M/M_4)\) on the plane \((M_4, |U_{e4}|^2)\), as can be derived from current constraints on \(0\nu \beta \beta \) [59] and loop corrections. To obtain the limits from \(0\nu \beta \beta \), we neglect the light neutrino contribution, since it is significantly below the present experimental limit in this model, in other words, we only consider the second term of Eq. (11). For each point, we require \(A_{\beta \beta }/{\mathcal {M}}^{0\nu \beta \beta }(0)<165~\hbox {meV}\) [59] , which becomes the most important constraint for \(M_4\lesssim 1\) GeV. Moreover, we require loop corrections to be at most of the same order of magnitude of the light neutrino masses. This becomes the most important bound for \(M_4>{\mathcal {O}}\left( {1\,\mathrm{GeV}}\right) \). These bounds are compared with present constraints for direct searches and lepton flavour violating processes. For example, we find that for \(M_4=10~\hbox {GeV}\) the mass splitting must be below 1 MeV for active-heavy mixing close to the experimental limit.

## 4 LNV at electron–positron colliders

In the previous section we have shown that strong limits to heavy neutrino mass differences and LNV exist. This points towards heavy neutrinos being pseudo-Dirac fermions in the region of the parameter space where this scenario can be tested in future colliders. It might seem hopeless in this situation to be able to determine the mass splitting by kinematical methods. However, the Majorana nature of the heavy neutrinos is essential to establish their connection to the light neutrino masses. This motivates the exploration of other collider observables that are sensitive to LNV processes, which can either determine if the heavy neutrinos are Majorana particles, and/or set stronger bounds on the heavy neutrino mass difference.

One such observable is the ratio of same-sign over opposite-sign dileptons [33, 60, 61], due to heavy neutrino oscillations. In this paper, we consider instead a possible asymmetry in the pseudorapidity distribution of a charged lepton coming from heavy neutrino decay. Specifically, we will focus on \(e^{-}e^{+}\rightarrow \nu N^{*} \rightarrow \nu \, \mu \, W^{(*)}\). In this section, we shall discuss this process in depth, without being concerned about the experimental signature nor the backgrounds, which will be taken into account in the following section.

*N*being Dirac or Majorana. In contrast, the diagram on the right is LNV, and can only occur if

*N*is Majorana. Both contributions cannot be told apart because of the presence of a light left- or right-handed neutrino in the final state, which is unobserved. However, we shall see in the next section that the pseudorapidity distribution of the lepton will be different [40] and therefore this observable is a discriminator of the LNV and LNC situations.

*W*is on-shell, it would be a polarization vector, otherwise it represents an additional

*W*propagator coupled to a fermion current. The propagator of each virtual heavy neutrino \(N_j\), with mass \(M_j\) and width \(\Gamma _j\) is:with \(q=p_3+p_4\). In our calculation, we have written the virtual

*W*propagator \(D_{\mu \nu }\) in the unitary gauge, which can depend on \(p_A=p_5-p_2\) or \(p_B=p_5-p_1\).

In the opposite limit, when \(\delta M \gg \Gamma _4\), the interference terms in Eqs. (26) and (28) are strongly suppressed due to the negligible overlap of the two Breit-Wigners peaked at \(q^2 = M_4^2\) and \(M_5^2\). Only the first and third terms contribute when \(q^2=M^2_4\) or \(q^2= M_5^2\) respectively and \(\Phi _B \simeq \Phi _A\) in this case.

We note that the total rate is no different in the two limiting cases, since \(\Phi _A^{LNV} \simeq \Phi _B^{LNV}\), while \(\Phi _A^{LNC} \simeq 2 \Phi _A^{LNV}\), and \(\Phi _B^{LNC} \simeq 0\).

## 5 Forward–backward asymmetry at the ILC

*W*exchange to the unpolarized differential cross section for the process \(e^+ e^- \rightarrow {{{\bar{\nu }}}} N_i\) (neglecting the electron and light neutrino masses) is given by:

*s*,

*t*are the Mandelstam variables. Changing variables to the pseudorapidity of the heavy neutrino:

*Z*-exchange contribution at this center of mass energy, \(e^+ e^- \rightarrow Z^* \rightarrow {{{\bar{\nu }}}}\,N_i\) also gives an asymmetry, but it is smaller.

The contribution of the process \(e^+ e^-\rightarrow \nu N_i\) via *W* exchange gives exactly the opposite distribution in pseudorapidity, so that the sum of the two contributions gives the result on the right panel of Fig. 3, that is, zero asymmetry.

Since the \(N_i\) has a significant boost, the decay products follow the same angular distribution. We expect therefore a significant asymmetry in pseudorapidity of leptons with a given charge in the Dirac case, where only one diagram contributes, versus the Majorana case where both do equally.

The heavy neutrino production is expected to have a very large background coming from the SM process \(e^-e^+\rightarrow W^+W^-\). In order to avoid this background, we require the heavy neutrino to be nearly on-shell, with a large enough lifetime in order to decay far from the interaction point. Experimentally, this leaves a displaced vertex signature, which has been studied extensively in the literature [5, 6, 7, 8, 9, 10, 11, 12, 13]. To be able to observe this signature at the LHC or future colliders, the lifetime needs to be large enough, which requires heavy neutrinos with masses between \(1-50~\hbox {GeV}\).

In order to study the prospects of measuring this lepton asymmetry at the ILC, we have implemented the model in SARAH 4.13.0 [63, 64, 65], with the calculation of the mass spectrum and decay widths carried out in SPheno 4.0.3 [66, 67]. The output of both programs was input into WHIZARD 2.6.0 [68, 69], which generated \(e^+e^-\) interactions at the ILC. The simulation included a polarization of \((0.80,\,0.30)\) for the initial state electrons and positrons, respectively, as well as ISR and beamstrahlung. Following the reports in [70, 71], the collisions were produced at a center-of-mass energy of 250 GeV, with a final integrated luminosity of \(2~\hbox {ab}^{-1}\). As our final state involves quarks, coming from \(W^*(p_4)\), the parton shower and hadronization of the jets was carried out with the built-in version of Pythia 6 [72]. For the detector simulation and reconstruction of events, we used DELPHES 3.4.1 [73, 74], with the DSiD card [75].

Benchmark scenarios considered in our study for a normal light neutrino hierarchy. We show masses, mixing, decay width and decay length

Name | Mass (GeV) | \(|U_{e4}|^2\) | \(|U_{\mu 4}|^2\) | \(|U_{\tau 4}|^2\) | \(\Gamma _4\) (meV) | \(c\,\tau _4\) (mm) |
---|---|---|---|---|---|---|

| 5 | \(1.5\times 10^{-6}\) | \(1.0\times 10^{-5}\) | \(1.3\times 10^{-5}\) | 0.02 | 10 |

| 20 | \(7.5\times 10^{-7}\) | \(5.0\times 10^{-6}\) | \(6.3\times 10^{-6}\) | 20 | 0.01 |

*light*and

*heavy*), which have a high probability of satisfying the previous constraints. These benchmarks differ by the heavy neutrino masses and mixing, with further details given in Table 1.

In Fig. 4, we present the pseudorapidity distribution of the \(\ell ^-\), for both benchmarks. We show results for different mass splittings, depending on the value of \(\Gamma _4\). The top row shows the distribution for \(\delta M\ll \Gamma _4\). Following our reasoning from Sect. 4, we expect the LNV contribution for this process to be negligible, such that the \(\ell ^-\) will be necessarily produced in association with a \({\bar{\nu }}\). As a consequence, the \(\ell ^-\) should follow the pseudorapidity distribution shown on the left panel of Fig. 3, which is what we observe. Similarly, the \(\ell ^+\) will be produced in association with a \(\nu \), and follows the opposite distribution. This means that, in this situation, two opposite forward–backward asymmetries can be expected, one for the \(\ell ^-\) and one for the \(\ell ^+\).

In Fig. 5, we calculate \(A_\eta ^\pm \) for several values of \((\delta M/\Gamma _4)^2\) and interpolate the results. The shaded regions indicate the \(3\sigma \) confidence intervals, evaluated by taking into account the expected number of events. We find that the behaviour on the LNV and LNC limits matches our expectations, that is, \(A^\pm \rightarrow 0\) when \((\delta M/\Gamma _4)^2\rightarrow \infty \), and \(|A^\pm |\sim 1\) for \((\delta M/\Gamma _4)^2\rightarrow 0\).

For the *light* benchmark, we have a relatively large enough number of events, so the asymmetry can be determined with good precision. At \(3\sigma \), \(A_\eta ^\pm \) is compatible with zero for \((\delta M/\Gamma _4)^2\gtrsim 20\), and with \(\pm 1\) for \((\delta M/\Gamma _4)^2\lesssim 1\). Therefore, (not) observing the asymmetry establishes upper (lower) limits on \(\delta M\), depending on \(\Gamma _4\). In addition, we have a region where \(|A^\pm _\eta |\) might be measured to be neither zero nor unity. In this case, the splitting could be constrained as \({\mathcal {O}}\left( {1}\right)<(\delta M/\Gamma _4)^2<{\mathcal {O}}\left( {10}\right) \). Such an observation would be particularly interesting in connection to resonant leptogenesis models.

On the other hand, the *heavy* benchmark has much less events and the precision is poorer. The asymmetry is compatible with zero for \((\delta M/\Gamma _4)^2\gtrsim 0.3\) and with unity when \((\delta M/\Gamma _4)^2\lesssim 1\). Here we can again place upper or lower bounds on \(\delta M\), provided we know \(\Gamma _4\). In this case there is not enough precision to measure \(|A_\eta ^\pm |\) to be different from both zero and unity at \(3\sigma \).

*light*benchmark, still quoting \(3\sigma \) errors, we find \(A_\eta ^{\mathrm{tot}}=0.94\pm 0.09\) when \((\delta M/\Gamma _4)^2=10^{-2}\), \(A_\eta ^{\mathrm{tot}}=0.75\pm 0.15\) for \((\delta M/\Gamma _4)^2=1\), and \(A_\eta ^{\mathrm{tot}}=0.01\pm 0.24\) if \((\delta M/\Gamma _4)^2=10^3\). Observing the asymmetry at \(3\sigma \) would mean that \((\delta M/\Gamma _4)^2\lesssim 0.16\), which implies \(\delta M\lesssim 8\) \(\mu \)eV. In contrast, a symmetric distribution would be observed for \((\delta M/\Gamma _4)^2\gtrsim 45\), which leads to \(\delta M\gtrsim 100\) \(\mu \)eV.

^{3}

Similarly, for the *heavy* benchmark, we have \(A_\eta ^\mathrm{tot}=0.92\pm 0.08\) if \((\delta M/\Gamma _4)^2=10^{-2}\), \(A_\eta ^\mathrm{tot}=0.41\pm 0.17\) for \((\delta M/\Gamma _4)^2=1\), and \(A_\eta ^\mathrm{tot}=-0.08\pm 0.18\) when \((\delta M/\Gamma _4)^2=10^3\). The observation of \(A_\eta ^{\mathrm{tot}}\) compatible with unity gives \((\delta M/\Gamma _4)^2\lesssim 0.53\), or \(\delta M\lesssim 15\) meV. Moreover, an observation compatible with zero implies \((\delta M/\Gamma _4)^2\gtrsim 0.72\), meaning that \(\delta M\gtrsim 17\) meV.

It is clear that these hypothetical constraints, of the order of \(\mu \)eV (meV) for the *light* (*heavy*) benchmark, are significantly stronger than any of those obtained in Sect. 3. This test thus implies a powerful way to indirectly extract information on \(\delta M\), and with it probe the nature of the heavy neutrinos as well as the breaking of LNV in Nature.

## 6 Conclusions

The minimal Type I seesaw adds two heavy neutrinos to the SM, providing mass to two light neutrinos. By choosing the model parameters appropriately, one can have relatively large active-heavy mixing even if the heavy neutrinos have masses of the order of the GeV. This is possible due to specific textures within the full neutrino mass matrix, motivated by the small breaking of Lepton Number, with LNV elements linked to \(\delta M=M_5-M_4\), the mass splitting of the heavy neutrinos. Currently, constraints coming from neutrinoless double beta decay and loop corrections require, in the parameter space accessible to colliders, values of \(\delta M/M_4\) between \({\mathcal {O}}\left( {10^{-2}}\right) \) and \({\mathcal {O}}\left( {10^{-6}}\right) \).

In this work we have considered heavy neutrino production at the ILC, with the heavy neutrinos decaying into a charged lepton and jets. We have demonstrated that LNV channels vanish when \(\delta M\rightarrow 0\), consistent with expectations. In addition, we have shown that, in the pseudo-Dirac limit, the ratio between LNV and LNC contributions is proportional to \((\delta M/\Gamma _4)^2\).

The presence of the LNV diagrams can be tested by observing the pseudorapidity distribution of the charged leptons into which the heavy neutrinos decay. We have shown that, when the LNV contribution is absent, the distribution of a lepton with specific charge is asymmetric. In contrast, when \(\delta M\) is large and LNV cannot be neglected, the asymmetry is absent. Thus, the presence of an asymmetric pseudorapidity distribution can set bounds on \(\delta M\) as a function of \(\Gamma _4\).

The process we explored in this work has very large backgrounds. To reduce them, we are forced to demand the existence of a displaced vertex. At the ILC, this requires values of \(M_4\) between 1 and 30 GeV, with \(\Gamma _4\) ranging from \(\mu \hbox {eV}\) to meV. This means that the values of \(\delta M\) that we can probe are around this order. For \(M_4 = 5\) GeV, the critical region where the asymmetry disappears is around \(\delta M/M_4\sim {\mathcal {O}}\left( {10^{-14}}\right) \). For \(M_4 = 20~\hbox {GeV}\), the region is around \(\delta M/M_4 \sim {\mathcal {O}}\left( {10^{-12}}\right) \). Thus, the study of the asymmetry can determine if the mass splitting is smaller or larger than these scales, establishing bounds which are much more precise than those currently available.

## Footnotes

- 1.
- 2.
Evidently, if \(\mu '\) is very small, one cannot neglect \(\mu \) and \(\varepsilon \) in the calculation of \(\delta M\). This would lead to a lower bound on the mass splitting, since the latter parameters are fixed by light neutrino masses.

- 3.
Even though \(\delta M\) has a strict lower bound due to the contribution of light neutrino masses, this is significantly smaller than the limits we are obtaining for these test points.

## Notes

### Acknowledgements

P.H. acknowledges support from Grants FPA2017-85985-P, and the European projects H2020-MSCA-ITN-2015//674896-ELUSIVES and H2020-MSCA-RISE-2015. J.J.P. acknowledges funding by the *Dirección de Gestión de la Investigación* at PUCP, through grant DGI-2015-3-0026. O.S.N. received funding from CienciActiva-CONCYTEC Grant 233-2015-1, as well as the grant *Becas Jóvenes Investigadores 2017* from the *Programa de Cooperación 07 de la Universitat de València.*

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