Probing the Dirac or Majorana nature of the Heavy Neutrinos in pure leptonic decays at the LHC

We propose a strategy for distinguishing the Dirac / Majorana character of heavy neutrinos with masses below the $W$ boson mass, using purely leptonic decays at the LHC. The strategy makes use of a forward-backward asymmetry of the opposite charge lepton in the $W^{+}\rightarrow l^{+}l^{+}l^{'-}\nu$ decay. In order to check the experimental feasibility of the model, we show, through a numerical analysis, that in the decay $W^{+}\rightarrow e^{+}e^{+}\mu^{-}\nu$ the two positrons in the final state can be distinguished for different ranges of the heavy neutrino masses. Finally, we estimated the number of events of $W^{+}\rightarrow e^{+}e^{+}\mu^{-}\nu$ for a Dirac and Majorana $N$ neutrino. For an integrated luminosity of 120 fb$^{-1}$ at LHC RUN II, signals can be found if heavy-to-light neutrino mixings are $ |U_{N \mu}|^2,|U_{N e}|^2 \gtrsim 10^{-6} $.


I. INTRODUCTION
Experiments in the last decades have confirmed that at least two of the three known neutrinos must have nonzero masses, and that all three have non-trivial mixings with respect to electroweak flavors. Ultimate confirmation of neutrino masses came from the results of atmospheric, solar and reactor neutrino experiments [1][2][3][4]. This evidence, compounded by the fact that the neutrino masses happened to be tiny with respect to the other Standard Model (SM) fermions, is currently an outstanding theoretical path for physics that goes beyond the SM. Most of the explanations of the neutrino mass smallness are based on the existence of extra heavy neutral fermions, which could be Dirac or Majorana [5]. In fact, the most widely accepted mechanism to generate small neutrino masses, is the seesaw mechanism [6][7][8][9][10], which involves extra heavy sterile neutrinos (henceforth denoted generically by N ). Moreover, in most of these scenarios the neutrinos are Majorana instead of Dirac fermions, and this discrimination about their nature is a crucial and challenging piece of information that experiments must elucidate.
Direct collider searches for sterile heavy neutrinos may provide a simultaneous probe of both their Dirac or Majorana nature, as well as their mixing with the active neutrinos. At hadron colliders, a strong signal of heavy Majorana neutrinos has been shown to exist and it is the same-sign dilepton final states, with two jets and no missing transverse energy: pp → W → N e ± → e ± e ± jj. This was originally proposed in [11] and further studied in [12][13][14][15][16][17][18][19][20]. Using this channel, both CMS and AT-LAS experiments at the LHC have set direct limits on the light-heavy neutrino mixing [21][22][23][24], for masses of * carolina.arbelaez@usm.cl † claudio.dib@usm.cl ‡ ivan.schmidt@usm.cl § juan.vasquezcar@usm.cl m N ∼ 100 − 500 GeV. For neutrino masses in the region m N < m W , the produced jets in the final state may not pass the cuts required to reduce the backgrounds, so the purely leptonic channels, such as pp → e ± e ± µ ∓ ν, in which ν could be either a neutrino or a anti-neutrino, may be more favorable [25]. Here we will focus on the signal W ± → e ± e ± µ ∓ ν − , which would indicate the existence of a sterile neutrino through the subprocess W ± → e ± N , followed by N → e ± µ ∓ ν. Notice that in the leptonic channel it is not clear a priori how to distinguish Dirac neutrinos from Majorana ones, since the final neutrino gets undetected in both cases (i.e, the observed final states are e ± e ± µ ∓ or µ ± µ ± e ∓ plus missing energy ). Hence, an obvious question is whether a Dirac or Majorana N can be distinguished at the LHC in these pure leptonic modes.
This has been answered in Refs. [26][27][28][29], where it was shown that the Dirac and Majorana cases could in fact be distinguished, depending of the relative size of the mixing between the e or µ flavors with the heavy neutrinos. In particular, they have have addressed the issue of Dirac/Majorana discrimination using the muon spectrum and a multi-variable analysis.
Here we present a simpler approach that makes use of the forward-backward asymmetry proposed in [23,30]. Complementary to the strategy presented in [27][28][29], we find that a suitable quantity to discriminate between Dirac and Majorana in the purely leptonic channels is the forward-backward asymmetry of the unlike-charged lepton [14]. This strategy works even when the mixing of the heavy neutrinos with the charged leptons are equal or of the same order of magnitude.
The paper is organized as follows. In Sec [II] we start by recalling the basic facts and some kinematical considerations involving the benchmark processes W ± → e ± e ± µ ∓ ν, followed by the subprocesses W ± → e ± N and N → e ± µ ∓ ν. We analyze a specific forwardbackward asymmetry for the lepton number conserving (LNC) mode W + → e + e + µ − ν e and the lepton number violating (LNV) mode W + → e + e + µ − ν µ . This asymmetry allows to distinguishing between Dirac and Majorana neutrinos in these purely leptonic decays at the LHC. In section [III] we discuss the distinction of the two same sign charged leptons in the final state at the LHC, which is crucial in order to extract the asymmetry from the experiment. In Sec [IV] we conclude with a discussion of our results.

II. THE DECAY RATES AND THE FORWARD-BACKWARD ASYMMETRY
In this section we present the theoretical expressions for the purely leptonic processes W ± → ± ± ∓ ν, the forward-backward asymmetry of the unlike-charge lepton ∓ and the analysis that indicates this asymmetry should be related to the Dirac/Majorana nature of the intermediate neutrino N .
The part of the lagrangian relevant for this process is given by where P L = (1 − γ 5 )/2, U lN is the mixing matrix of the charged leptons, l = e, µ, τ and N denotes the heavy neutrino. For simplicity, let us focus on the mode W + → e + e + µ − ν; the modes with other flavor and charge are similar. Fig. 1 shows the two possible processes for this mode, considering that the final neutrino,ν µ or ν e , is not observable. The upper and lower diagrams correspond to a lepton number conserving (LNC) and a lepton number violating (LNV) process, respectively.
The rates of these two processes, in the notation of Fig. 1 and neglecting the masses of the final leptons, can be given in the narrow width approximation for N as: and where dΦ 2 denotes the 2-particle phase space of the first vertex and dΦ 3 the 3-particle phase space of the N decay [31]. Since a Dirac N will only produce the LNC process while a Majorana N will produce both LNC and LNV, we can find ways to detect the nature of N by distinguishing between these two processes at the LHC. One cannot distinguish them by the rates because the mixings |U N e | and |U N µ | are not known a priori, nor can we distinguish them by the final particles, because the final ν e andν µ escape detection. However, due to the chiral character of the weak interactions and the fact that µ − is in a different fermion line (see Fig. 1), there will be a difference in the angular distribution of the muon momentum. Indeed, in the N rest frame, if we define our z axis along the initial W + momentum q, and we call θ the polar angle of the muon momentum k 1 along this axis, i.e. cos θ = q · k 1 /(|q||k 1 |) the angular distributions of the muon along the W + direction in the N rest frame are indeed different in the LNC and LNV processes. Calling x N ≡ m N /M W , from Eqs. (2) and (3) we get: and The global factor C is where we defined the function and where the neutrino width, Γ N , for m N 10 GeV can be estimated as [23,26]: where we have included the function F N (x N ) This function F N (x) is due to the momentum dependence of the W propagator, and is an improvement in the m N dependence of Γ N estimate of Refs. [23,26], where the W propagator was taken as a point interaction. In the limit In turn, the factor A(x N ) of the angle-dependent term in the LNC distribution is given by: and it is called the analyzing power, as it modulates the angular dependence of the distribution [14,30]. In the limit The difference between the LNC and LNV angular distributions of Eqs. (4) and (5) is essentially the analyzing power factor A(x N ) in the LNC expression. This difference can be explained as follows. The neutrino N produced in the decay W + → e + N must be of left handed chirality because of the weak current (in the same way, the produced positron in this decay must be right handed). Provided that m N is considerably smaller than M W , the helicity of N will be mostly left handed as well (in the N rest frame, the N spin will be polarized along the W momentum q -see Fig. 3). For the calculation of the primary process W + → e + N , where N has a polarization vector s µ (this vector satisfies s · s = −1 and s · k = 0, where k is the N momentum), one obtains the following squared matrix element: Given a neutrino momentum k = (E, 0, 0, k), the right and left helicities of N correspond to s = ±(k, 0, 0, E)/m N , respectively. It is then easy to show that the probabilities of the two helicities are in the ratio |M| 2 right : |M| 2 left = m 2 N : 2M 2 W , and so the probabilities to produce N with right handed or left handed hecility are Now, in the decay of a polarized N , the polarization vector sets a direction along which an anisotropic muon emission occurs (for an unpolarized N the muon emission is isotropic). Moreover, since the muon comes from a different weak current in the LNC and LNV processes (see Fig 1), the anisotropic distributions in these two processes are different. We must point out that the predominant left handed polarization of the produced N occurs regardless of its Dirac or Majorana nature. The latter has an effect in the subsequent decay of N . For a Majorana N , both decays are allowed, while for a Dirac N produced in W + → e + N , the latter mode is forbidden. Now, these decays, for a polarized N have the following θ distributions: and where θ is, as before, the angle between the momentum of the muon and the polarization vector of N at rest. Here it is clear that the two angular distributions are different precisely because the muon is attached to a different fermion line in the weak interaction. Now, to get to the distributions of the full LNC and LNV processes of Eqs. (4) and (5) we must weigh the polarized N decays by the probabilities in Eq. (12) of the two longitudinal polarizations produced in the primary decay W → e + N . The result coincides with the distributions of Eqs. (4) and (5).
Given the fact that the LNC and LNV muon angular distributions are different, we propose to use the following forward-backward asymmetry to try to distinguish between the Dirac vs. Majorana character of the intermediate neutrino N that induces the events: where N (cos θ > 0) and N (cos θ < 0) denote the number events with the muon moving forwards or backwards with respect to the decaying W , in the N rest frame of the heavy neutrino. If N is a Dirac neutrino, only the LNC process occurs, in which case the analyzing power A(m N /M W ) determines directly the outcome of the experimental forward-backward asymmetry, regardless of lepton mixing (although the mixing determines the total number of events): Instead, if N is a Majorana neutrino, the rate of events W → e + e + µ − ν is the sum of the LNC and LNV rates given in Eqs. (4) and (5), and then the forward-backward asymmetry will depend not only on the analyzing power of the LNC component, but also on the relative lepton mixing elements |U N e | and |U N µ |: The Dirac asymmetry, Eq. (18), is independent of the mixings and, due to the function A(x N ) (see Fig. 2), is approximatively −1/6 for a wide range of m N , except if m N approaches M W , in which case it tends to vanish.
In contrast, the Majorana asymmetry, Eq. (19), does depend on the mixings and in general it is larger than in the Dirac case. For events e + e + µ − , if |U N µ | |U N e ||, it tends to the same value as in the Dirac case, because it is dominated by the LNC contribution, but in the opposite limit, namely |U N µ | |U N e |, dominated by the LNV contribution, it is larger: close to −1/2 for most values of m N , decreasing to −1/6 as m N approaches M W .
Consequently one can in principle distinguish the Dirac from the Majorana case using this asymmetry. However, notice that in the case |U N µ | |U N e |, for events e ± e ± µ ∓ the asymmetry is the same for both Dirac and Majorana N , because the Majorana asymmetry is dominated by the LNC contribution.
In this case the channels µ ± µ ± e ∓ are more appropriate for discrimination, because now the Majorana asymmetry is dominated by the LNV contribution. The angular distribution in this case is that of the electron flavor instead of the muon.

III. DISTINGUISHING THE SAME-SIGN LEPTONS AT THE LHC
In order to experimentally determine the asymmetries of Eqs. (18) and (19), one needs to identify which of the two equal-sign leptons originates from the first vertex and from the second vertex (see Fig. 1). From this identification one can do the appropriate boosts to the W rest frame or to the N rest frame, as required. In this section we study this issue of telling apart the two equal-sign leptons (see also Ref. [27]).
As an example, consider the LNC production and decay channel pp → W + → e + N → e + µ − e + ν e , in which two positrons are in the final state. In the LNV channel there is oneν µ in the final state, but as we shall see, the considerations we use apply to the latter case as well.
In Fig. 4 we show the invariant masses for the e + 1 µ − ν e and e + 2 µ − ν e , where e + 1 and e + 2 are the harder and softer positron sorted by energy, respectively. The events were generated at LO using MadGraph 5 [32], hadronized with Pythia 6 [33] and passed to Delphes 3 [34] for detector simulation. In the figures we may see that when the heavy neutrino mass is above 70 GeV the hardest positron reconstructs better its mass. This is in agreement with the expectation that in this case the heavy neutrino carries most of the energy.
Conversely, when the heavy neutrino mass is below 40 GeV, it is the softer positron that reconstructs the heavy neutrino mass -as can be seen in Fig. 4. Once again, this is in agreement with the expectation that the heavy neutrino carries less energy than in the more massive case, since in the massless limit is would have exactly half of the energy taken from the W decay. We find that in the region between 50-70 GeV both electrons show similar behavior in the m N reconstruction and hence cannot be distinguished with the above strategy. There is another interesting region, which is when m N ≤ 10 GeV. In this region the decay length of the heavy neutrino becomes visible in the form of a displaced vertex inside the detector. Clearly in this case the distinction of both positrons will be obvious. The background processes and the possibility of distinguishing both same-sign leptons have been extensively studied in [27][28][29]. Therefore we do not discuss this issue further here. In summary, for m N 10 GeV, the two leptons can be separated due to vertex displacement. For higher masses no vertex displacement is observable since N is too short-living. In those cases the separation of the two leptons is less certain. As shown, statistically the primary lepton tends to be the most energetic one if m N 50 GeV and the less energetic one if m N 70 GeV. In the mass region between 50 and 70 GeV no such distinction occurs.
Another important experimental issue here is the expected number of events. For a realistic analysis that involves angular distributions, one would need a rather large sample of these rare events. According to Ref. [35], at the end of the LHC Run II, with an integrated luminosity of 120 fb −1 , ATLAS is expected to have of the order of 2×10 8 leptonic W decays into eν and µν. Considering that the W width is Γ W ∼ 2 GeV and the branching ratio into each lepton is near 10%, using Eqs. (4)(5)(6) and (8), we can estimate the expected number of W → e + e + µ − ν or W → µ + µ + e − ν events. Without considering the impact of systematic errors or backgrounds, the expected numbers of events for the Majorana and Dirac scenarios are estimated to be: From these expressions one can deduce that the largest of the mixings |U N e | 2 and |U N µ | 2 must be at least of order 10 −6 for the LHC Run II to get a positive signal of these modes. Take for example the case |U N e | |U N µ | ∼ |U N τ |; then the above reduces to: implying that the expected number of these events for |U N e | 2 ∼ 10 −6 is 6 (0) for Majorana (Dirac) N .
Finally, if |U N µ | |U N e | ∼ |U N τ |, the number of e ± e ± µ ∓ events for both Dirac and Majorana cases are much smaller, in which case the µ ± µ ± e ∓ events should be considered. Therefore, for an integrated luminosity of 120 fb −1 at the LHC Run II, W → e ± e ± µ ∓ events can be found provided |U N e | 2 10 −6 while W → µ ± µ ± e ∓ signals can be found if |U N µ | 2 10 −6 . These are the limits in cases where m N < M W , but as m N approaches M W these theoretical rates are smaller and the lowest |U N | 2 for detection become larger. Moreover, in cases where |U N e | 2 ∼ |U N µ | 2 , the discrimination between Dirac and Majorana N requires the forward-backward asymmetry of Eq. (17). In order to build this asymmetry one needs more data than the minima deduced above, hence |U N | 2 must be above 10 −6 or the W → ν data sample must be larger than what is expected at the end of the LHC Run II. On the other hand, current upper bounds for |U N e | 2 and |U N µ | 2 for N with mass in the range 10 GeV < m N < M W are near 10 −5 [36], leaving about an order of magnitude in the range of the mixings to explore in the LHC Run II.

IV. CONCLUSIONS
We presented a new method to discriminate Dirac vs Majorana sterile neutrinos with masses m N ≤ m W that would induce W trilepton decays e ± e ± µ ∓ and µ ± µ ± e ∓ at the LHC. A simple strategy based on a forwardasymmetry is proposed. This asymmetry is different depending on the Dirac/Majorana character of the heavy neutrino. If the heavy neutrino is a Dirac particle, the forward-backward asymmetry depends on the heavy neutrino mass but not on the heavy-to-light mixing parameters. Instead, for a Majorana neutrino the asymmetry depends on the both the mass and the mixing matrix elements. The method presented here is complementary to the strategy proposed in [27][28][29], in the sense that it allows to distinguish the neutrino nature also in the case where the neutrino-lepton mixing elements are of the same order.
In order to construct the asymmetry, a crucial point is to identify the primary and secondary leptons of same charge in the final state ± ± ∓ . Using invariant mass analysis distributions we find ranges for m N where the LHC could distinguish among these two leptons.
If m N 50 GeV (m N 70 GeV) the first positron tends to be the most (least) energetic, while in the region 50 m N 70GeV, the order of the two same-sign leptons can not be distinguished.
We estimated the number of these events for the integrated luminosity of 120 fb −1 expected at the end of the LHC Run II, and found that both Majorana and Dirac signals could be found provided that at least one the mixings |U N µ | 2 and |U N e | 2 is 10 −6 or above.