Probing the Nature of Neutrinos with a New Force

We discuss the possibility to distinguish between Dirac and Majorana neutrinos in the context of the minimal gauge theory for neutrino masses, the B-L gauge extension of the Standard Model. We revisit the possibility to observe lepton number violation at the Large Hadron Collider and point out the importance of the decays of the new gauge boson to discriminate between the existence of Dirac or Majorana neutrinos.


INTRODUCTION
After the discovery of the Brout-Englert-Higgs boson at the Large Hadron Collider (LHC), we understand how the electroweak symmetry is broken in nature and how all the charged fermions should acquire mass through the Higgs mechanism. Unfortunately, the Standard Model (SM) does not provide an explanation for the origin of neutrino masses, and hence, the experimental evidence of neutrino masses calls for new physics beyond the SM. Thanks to the effort of the experimental community, the mixing angles and mass splittings in the neutrino sector have been measured with good precision, see Ref. [1] for a review about neutrino physics.
The nature of neutrinos remains unknown and it is a central open question in particle physics. The neutrinos can be Dirac or Majorana fermions [2], in the Dirac case the anomaly-free symmetry B −L is conserved or broken in a unit larger than two, while in the Majorana case B −L is broken in two units. The simplest way to distinguish between the existence of Dirac or Majorana neutrinos is to search for exotic lepton number (or B−L) violating processes at low energies such as neutrinoless double beta decays [3,4] or for signatures at colliders [5] if the B − L breaking scale is low. Clearly, the discovery of any of these processes will be crucial to understand the origin of neutrino masses and complete our understanding about mass generation.
The simplest mechanism for Majorana neutrino masses is the canonical seesaw mechanism [6][7][8][9]. In this context, by adding at least two copies of right-handed neutrinos it is possible to generate neutrino masses in agreement with the experimental observations. Unfortunately, the seesaw scale could be very large, M seesaw ≤ 10 14−15 GeV, and we might not have direct access to the mechanism behind neutrino masses. However, if the seesaw scale lies below or near the TeV scale, then we might test this mechanism in the near future. In the absence of any signature from neutrinoless double beta decays we need to investigate the possibility to observe lepton number violating signatures at colliders to establish the nature of neutrinos.
If the relevant scale (B − L scale) for the generation of Majorana neutrino masses is relatively close to the electroweak scale we can hope to observe signatures with same-sign leptons at colliders [5]. In the context of the canonical seesaw mechanism, there have been different studies of the production of right-handed neutrinos at particle colliders, see for example the studies in Refs. [10][11][12][13][14][15][16][17]. The main production channel considered in many studies is pp → W ± → N ± , which is generically suppressed by the active-sterile neutrino mixing. In the scenario where the right-handed neutrino masses are below the W -gauge boson mass, M N < M W , the right-handed neutrinos could be discovered using displaced vertices [18][19][20]. Furthermore, if M N < M h /2 the SM Higgs boson can decay into a pair of right-handed neutrinos and constraints can be placed by studying the properties of the Higgs [21][22][23][24]. For more details and a complete list of references see the review in Ref. [25].
The simplest gauge theory for neutrino masses corresponds to promoting B − L to a local symmetry. This is because the three right-handed neutrinos automatically cancel all gauge anomalies. In this context the neutrinos are Dirac fermions if the new Z BL gauge boson acquires mass through the Stueckelberg mechanism [26] or if B − L is spontaneously broken in more than two units. Alternatively, a new scalar with B − L charge equal to two can be introduced to break B − L spontaneously and generate Majorana masses for the neutrinos via the seesaw mechanism. In this scenario, the Z BL can mediate the pair production of right-handed neutrinos, and hence, this channel has the advantage of not being suppressed by the active-sterile neutrino mixing [27]. See Refs. [28][29][30] for other studies along these lines.
In this article, we discuss the possibility to distinguish between Dirac and Majorana neutrinos in the context of the minimal gauge theory for neutrino masses based on B − L. We revisit the possibility to observe lepton number violation at the LHC and point out the importance of measuring the decay branching ratios of the new gauge boson to discriminate between the existence of Dirac or Majorana neutrinos. Clearly, a future simultaneous discovery of the Z BL gauge boson and heavy right-handed neutrinos will be evidence for Majorana neutrinos. However, when M N > M Z BL /2 the production cross-section for a pair of right-handed neutrinos is highly suppressed and the prospects for observing lepton number violation are very small. We show how to distinguish between Majorana and Dirac neutrinos if a Z BL gauge boson is discovered even if there is no direct discovery of the right-handed neutrinos. The structure of our work is the following: in Section 2, we discuss the current bounds on the U(1) B−L gauge boson mass and its coupling to matter. In Section 3, we revisit the lepton number violating signals at the LHC through the process pp → N N → l ± l ± 4j, we show the predictions for the latter by performing the most general analysis. In Section 4, we demonstrate how to distinguish between Dirac and Majorana neutrinos by measuring the decay width of Z BL . We present our summary in Section 5.

MINIMAL GAUGE THEORY FOR NEUTRINO MASSES
The simplest gauge theory for neutrino masses is based on the local B − L gauge symmetry. The right-handed neutrinos needed to generate Dirac/Majorana masses for neutrinos are also the extra degrees of freedom needed to define an anomaly-free gauge theory based on B − L. In this context the new gauge boson, Z BL , has the following interactions: where the family index i = 1, 2, 3. In the above equation e i = e iL + e iR , u i = u iL + u iR , and d i = d iL + d iR are the Dirac spinors for the charged fermions.   [32]. The constraint from the bound on ∆N eff [33] is shown by the green line and applies only to the scenario with Dirac neutrinos. The predictions for the decay width of Z BL in the scenario with Dirac neutrinos are shown by the black lines.
In Fig. 1 we show the relevant bounds in the g BL − M Z BL plane. The blue line corresponds to the bound from LEP [31] (M Z BL /g BL > 7 TeV), while the orange line corresponds to dilepton searches at the LHC with √ s = 13 TeV and 36.1 fb −1 [32]. The black lines define the different values for the decay width of the Z BL gauge boson, and the bound from N eff [33] (M Z BL /g BL > 10.33 TeV) is shown by the green line that is only relevant when the neutrinos are Dirac fermions. Notice that the LHC bounds are the most relevant when the gauge boson mass is below 4 TeV. All these bounds are relevant to understand the predictions for the processes investigated in the next section.
In the minimal B − L gauge theory the Dirac Yukawa coupling for neutrinos reads as with L ∼ (1, 2, −1/2),H = iσ 2 H * , and H ∼ (1, 2, 1/2) is the SM Higgs doublet. As we mentioned above, the mass of the B − L gauge boson can be generated via the Stueckelberg mechanism [26] leaving the U(1) B−L gauge group unbroken. In this simple theory the neutrinos are Dirac particles, see Ref. [34] for a recent discussion of the different possibilities. Alternatively, a scalar can be introduced with B − L charge equal to two, and once this scalar acquires a non-zero vacuum expectation value it will give mass to the Z BL and the right-handed neutrinos. Therefore, the canonical seesaw mechanism for Majorana neutrinos can be implemented.

• Dirac Neutrinos
In the case when the neutrinos are Dirac fermions the decay width of the B − L gauge boson can be predicted as function of the gauge coupling and its mass. The branching ratio for the invisible decay can be quite large due to the fact that there is an extra contribution of the right-handed neutrinos, the invisible branching ratio is close to 38% as it is shown in the left panel in Fig. 2. This is a simple but important result because as we will discuss in the next sections, the branching ratios of Z BL can be used to distinguish between the Dirac and Majorana scenarios.

• Majorana Neutrinos
In the case with Majorana neutrinos and the canonical seesaw mechanism we can hope to observe lepton number violation at the LHC. The masses for Majorana neutrinos are generated after symmetry breaking through the canonical seesaw using the terms: The best way to observe lepton number violation at the LHC is through the pair production of right-handed neutrinos, i.e. pp → Z BL → N i N i , where N i correspond to the physical states associated to the right-handed neutrinos. In the next section, we will revisit the predictions for lepton number violation and discuss the possibility to observe these signatures.
It is important to mention that in this case the prediction for the neutrino branching ratio depends on the mass of the right-handed neutrinos.
The right panel in Fig. 2 shows the branching ratios of Z BL in the Majorana case with all three right-handed neutrino masses set to M N = 500 GeV. The neutrino branching ratio goes from 23% to 38% as the Z BL → N i N i decay channel become kinematically allowed. Notice that the latter is not an invisible decay since the N i 's can decay into visible states inside the detector.

B − L FORCE AND LEPTON NUMBER VIOLATION AT THE LHC
The observation of lepton number violation by two units at the LHC will shed light on the origin of neutrino masses. In the gauged U(1) B−L scenario, the right-handed neutrinos can be produced at the LHC through the B − L gauge boson: pp → Z * BL → N i N i [27], with i = 1, 2, 3. The cross-section for this process is given by where the partonic cross-section corresponds to and The parameter τ =ŝ/s, whereŝ is the partonic center-of-mass energy squared, s is the hadronic center-of-mass energy squared, τ 0 = 4M 2 N i /s is the production threshold, and µ is the factorization scale that is set to µ = M Z BL . The f -functions correspond to the parton distribution functions for which we use the MSTW2008 [35] set.
In Fig. 3 we show the predictions for the pp → N i N i cross-section as a function of the righthanded neutrino mass. This plot shows that when the decay Z BL → N i N i is kinematically closed, i.e. M N i > M Z BL /2, the cross-section drastically drops to very small values. This occurs because for these masses the cross-section never hits the Z BL resonance. Requiring the Majorana Yukawa coupling to be perturbative translates as an upper bound of M N < √ 2πM Z BL /g BL , we make sure this is satisfied in Fig. 3.
In order to study the lepton number violating signatures we need to calculate the branching ratios for the right-handed neutrinos. The decay widths for the right-handed neutrinos are given by where θ is the mixing angle between the SM Higgs and the scalar that breaks U(1) B−L . The matrix defining the mixing between the right-handed and left-handed neutrinos can be written as [37] where c ω i = cos ω i , s ω i = sin ω i and ω i are complex angles. The PMNS matrix can be written as  Table I. For our numerical evaluation we perform a scan over the lightest active neutrino mass and the Majorana phases α 1 and α 2 in the range shown in Table II. The scenario with normal hierarchy (NH) corresponds to ∆m 2 3 = ∆m 2 31 > 0, while the scenario with inverted hierarchy (IH) corresponds to ∆m 2 3 = ∆m 2 32 < 0. In Fig. 4 we show the branching ratios BR(N i → l − W + + l + W − ) as a function of the righthanded neutrino mass; for the plots in the left panel the R matrix is set to the identity matrix which corresponds to the simple scenario with all complex angles ω i set equal to zero. This means that the mixings V i depend only on low energy physics, and hence, the structure of the PMNS matrix is being reflected in these plots. Thus, by measuring the branching ratios of the N i 's we can learn whether the R matrix is close to the identity matrix, since in this case there is a clean prediction for each branching ratio. We focus on the N i → l ∓ W ± channels because these are the ones that lead to signatures of lepton number violation as we will see below.
For the plots in the right panel in Fig. 4 we perform a scan on the ω i complex angles in the ranges shown in Table II. The imaginary parts of ω i exponentially enhance the entries in the R matrix so we make sure that each entry in the Dirac Yukawa matrix Y D ν remains perturbative. This demonstrates that once the freedom in the R matrix is taken into account the predictions can change drastically. For example, the branching ratio for N 1 → µ ∓ W ± which is around 4.5% for R = 1 can become as large as 50% once the random scan is performed. We find that the branching ratios are not sensitive to whether we have normal hierarchy or inverted hierarchy in the active neutrino sector, so the plots have the same behavior for IH.
In Fig. 5 we show the decay width of N i as a function M N i for each right handed neutrino. As can be seen from Eqs. (7)-(9) the decay widths are proportional to the light neutrino masses, and hence, they depend on whether we have NH or IH. The plots on the left correspond to normal hierarchy while the ones on the right correspond to inverted hierarchy. For the top row we fix R = 1, which means the Γ N i are only dependent on the light neutrino masses m i . In the NH scenario this is the reason why Γ N 1 can vary over several orders of magnitude, while Γ N 2 and Γ N 3 are restricted to a small window. For the IH scenario, the decay width Γ N 3 is the one that has a large range since ν 3 is the lightest.
In the middle row of Fig. 5 we show the decay width for a scan of the R matrix taking only real parameters. The difference with R identity is that now there is more freedom in the decay width for N 2 and N 3 in the NH and N 1 and N 2 in the IH. Once the R matrix allowed to take on random values it becomes very difficult to distinguish between the NH and the IH scenarios. An

Parameter
Scan Range NH (IH) exploration for different values of the R matrix is not commonly done in the literature.
In the third row of Fig. 5 we present the results for a random scan of the R matrix considering complex entries. We find that the decay width can be 6 orders of magnitude larger than when taking only real entries in the R matrix; this happens because imaginary parts of ω i exponentially enhance the entries in the mixing matrix V . We find that the maximal values for the decay widths are Γ max GeV. Therefore, the heavy neutrinos can decay more promptly than we might expect from the naive seesaw relation The decaying length of the heavy neutrinos can have a large range from 10 −9 mm to 10 5 mm, and hence, these heavy neutrinos can be searched for using different techniques. When the decay length is between 10 −2 mm and 10 3 mm then these appear at the LHC as displaced vertices. Additionally, when the lightest neutrino mass is taken to be very small, the decay length can be in the order of meters and detectors such as FASER [38] or MATHUSLA [39] can be used to search for them.
Lepton number violation can be probed by searching for the process pp → N i N i → l ± l ± 4j at the LHC. The expected number of events for this process is given by where the hadronic decay of the W boson is BR(W ∓ → jj) 2/3. In Fig. 6 we show the expected number of events at the LHC for center-of-mass energy of 14 TeV assuming L = 3000 fb −1 for the integrated luminosity. The points in black correspond to the simplified scenario with R = 1 and for the gray points we perform a scan on the free parameters in the range shown in Table II. See Ref. [27] where the authors discussed in detail the relevant SM backgrounds, ttW and multi-bosons, and how to distinguish between the signal and background imposing different kinematical cuts. The top row in Fig. 6 corresponds to the production and decay of N 1 . The left panel corresponds to the N 1 N 1 → e ± e ± 4j channel. Here, the case R = 1 is close to the largest number of events obtained from the random scan. The middle panel is for the N 1 N 1 → e ± µ ± 4j channel and in this case the random scan can increase the number of events by a factor of 4. The right panel is for the N 1 N 1 → µ ± µ ± 4j channel and here the R = 1 case predicts a much lower number of events than can be obtained from the random scan which can increase the number of events by a factor of 100. As can be appreciated, the predictions for the number of events is very sensitive to the form of the R matrix and can be quite different from the ones obtained using the naive seesaw relation V 2 ≈ m ν /M N . If these channels are discovered in the near future, this information can be used to learn about the R matrix and the seesaw relation. In Appendix A we present the results FIG. 6: Scatter plot of the expected number of events at the LHC for center-of-mass energy of 14 TeV assuming L = 3000 fb −1 for the integrated luminosity. The black points correspond to the case with R = 1, while the gray points correspond to a random scan on the entries of the R matrix. These plots correspond to the case with normal hierarchy and we scan over the lightest neutrino mass, the same pattern is observed for inverted hierarchy. of our scan for the active-sterile neutrino mixing.

DIRAC vs MAJORANA: THE ROLE OF THE Z BL DECAY WIDTH
The discovery of the Z BL gauge boson does not guarantee the discovery of right-handed neutrinos, and hence, we might be unable to disentangle between neutrinos being Dirac or Majorana. In this section, we argue that a measurement of the Z BL total width, Γ Z BL , will suffice to distinguish between the scenario with Dirac or Majorana neutrinos. We expect the LHC to reach this precision [40]. For example, take the high precision LEP measurement of the Z boson in the SM to less than one percent Γ Z = 2.4952 ± 0.0023 GeV [41]. The decay width of Z BL for the  different channels is given by, where the last term is the contribution from decay into neutrinos. In the scenario with Dirac neutrinos we have while in a scenario with Majorana neutrinos we have Notice that Γ D ν ≥ Γ M ν . Consequently, the total Z BL decay width is different depending on whether neutrinos are Dirac or Majorana. However, we should point out that there is degeneracy Γ D ν Γ M ν in the scenarios with M N i M Z BL . All the results in this section are independent of the activesterile neutrino mixing.
In Fig. 7 we present a contour plot of the neutrino branching ratio for Z BL in the M N vs M Z BL plane, where we are assuming that M N 1 = M N 2 = M N 3 = M N for simplicity. This branching ratio is independent of the value of the coupling g BL . Whenever M N > M Z BL /2 the only decays into neutrinos are Z BL → ν i ν i and this branching ratio is equal to 23%. As the Z BL → N i N i channels become kinematically open the neutrino branching ratio starts to increase and goes to 38% for M N M Z BL . To quantify the difference between the neutrino width in the Dirac vs Majorana case we define the following quantity where Γ D ν corresponds to Dirac neutrinos given by Eq. (15)   Now, let us discuss the correlation between the lepton number violating processes and the decay width of the B − L gauge boson. In Fig. 9 we show in orange the production cross-section for a pair of right-handed neutrinos and in blue the δΓ ν parameter as a function of M N . These plots shows the complementarity between direct production of N i 's and the measurement of the Z BL neutrino branching ratio to distinguish between Dirac and Majorana neutrinos. These plots show that as the pair production cross-section goes down, the δΓ ν parameter becomes larger eventually becoming equal to one.
If a Z BL gauge boson is discovered and the right-handed neutrinos lie in the range 100 GeV < M N < M Z BL /2 then there is a good possibility to directly produce the right-handed neutrinos at the LHC. However, when M N > M Z BL /2 it becomes very difficult to produce the right-handed neutrinos. At this point we could either be in a scenario with Dirac or Majorana neutrinos. However, by measuring the neutrino branching ratio of Z BL we can discriminate between these two scenarios.
• BR(Z BL → neutrinos) 23%: Measuring the neutrino branching ratio of Z BL close to 23% implies that neutrinos are Majorana and that the channels Z BL → N i N i are kinematically closed. This corresponds to having δΓ ν = 1. Consequently, even if we are unable to directly produce N i N i we will have indirect evidence that neutrinos are Majorana fermions. • 23% < BR(Z BL → neutrinos) < 38%: A measurement of the neutrino branching ratio between 23% and 38% will mean that neutrinos are Majorana and that both channels Z BL → ν i ν i and Z → N i N i are open. The Majorana neutrino can be further confirmed by direct observation of the right-handed neutrinos at particle colliders. This corresponds to having 0 < δΓ ν < 1.
• BR(Z BL → neutrinos) 38%: If the neutrino branching ratio is measured very close to 38% then it becomes hard to disentangle the nature of neutrinos since we can either be in the case with Dirac neutrinos or the one with Majorana neutrinos and M N M Z BL . This corresponds to having δΓ ν 0.
In the Dirac scenario the branching ratio into neutrinos is invisible. In the Majorana case, the decay into light neutrinos Z BL → ν i ν i is always invisible. However, for the decays Z BL → N i N i , the heavy neutrinos N i can subsequently decay into visible particles inside the detector. In Table III we show the predictions for the Z BL decay into neutrinos and its total width for M Z BL = 3.4 TeV and g BL = 0.1. As this table shows, when the right-handed neutrino mass is below 10 GeV it is difficult to distinguish between Dirac and Majorana neutrinos because the difference in the decay width into neutrinos is smaller than 10 −4 GeV. However, above 10 GeV one can distinguish the two scenarios for neutrino masses.

SUMMARY
We have discussed how to distinguish between Dirac and Majorana neutrinos in the simplest gauge theory for neutrino masses; namely, the B − L gauge extension of the SM. Assuming that the B − L symmetry breaking scale is not far from the electroweak scale, we revisited the prospects for observing lepton number violation at the Large Hadron Collider. We performed a general random scan on the parameters in the R matrix that enters in the mixing between neutrinos and demonstrated that the lifetime of the right-handed neutrinos can span many order of magnitudes. Even for right-handed neutrino masses above the electroweak scale decaying lengths in the order of meters are possible. We have shown that a large number of events for the processes pp → N i N i → l ± l ± 4j can be observed in some cases and using these channels one can learn about the structure of the mixing matrix and the seesaw relation.
We have discussed how the measurement of the Z BL decay width and branching ratios can help to discriminate between Majorana and Dirac neutrinos. Three different scenarios are possible: i) BR(Z BL → neutrinos) 23% would mean that M N > M Z BL /2 in which the pair-production cross-section for right-handed neutrinos is highly suppressed; nonetheless, measuring this branching ratio will imply that neutrinos are Majorana. ii) 23% BR(Z BL → neutrinos) 38% means that the decay channels Z BL → N i N i are open and we will able to directly pair produce the righthanded neutrinos at colliders. iii) BR(Z BL → neutrinos) 38% would be a pessimistic scenario in which we have M N M Z BL , which makes the right-handed neutrinos hard to observe at particle colliders and also hard to disentangle between Dirac and Majorana neutrinos, since the prediction for Dirac neutrinos is BR(Z BL → neutrinos) = 38%. Our results could help uncover whether the neutrinos are Dirac or Majorana fermions and complete our understanding of the mass generation.

A. Neutrino mixings
In Fig. 10 we present the results for the neutrino mixing matrix V ; the black dots correspond to the simple scenario with R = 1 and the gray points correspond to the scan over the free parameters in the ranges shown in Table II. FIG. 10: Scatter plot of the elements of the active-sterile neutrino mixing. The black points correspond to the case with R = 1, while the gray points correspond to a random scan on the entries of the R matrix. These plots correspond to the case with normal hierarchy, for the inverted hierarchy the plots have a similar behavior with the exchange N 1 ↔ N 3 .