Search for Heavy Right-Handed Neutrinos at the LHC and Beyond in the Same-Sign Same-Flavor Leptons Final State

Search for Heavy Right-Handed Neutrinos at the LHC and Beyond in the Same-Sign Same-Flavor Leptons Final State

John N. Ng111misery@triumf.ca, Alejandro de la Puente222adelapue@triumf.ca, and Bob Wei-Ping Pan333two.joker@gmail.com
(1) Theory Group, TRIUMF,
4004 Wesbrook Mall
Vancouver BC V6T 2A3, Canada.

(2) Department of Physics
National Tsing Hua Univeristy
Hsin Chu 300, Taiwan, Republic of China
July 13, 2019
Abstract

In this study we explore the LHC’s Run II potential to the discovery of heavy Majorana neutrinos, with luminosities between and fb in the final state. Given that there exist many models for neutrino mass generation, even within the Type I seesaw framework, we use a simplified model approach and study two simple extensions to the Standard Model, one with a single heavy Majorana neutrino, singlet under the Standard Model gauge group, and a limiting case of the left-right symmetric model. We then extend the analysis to a future hadron collider running at TeV center of mass energies. This extrapolation in energy allows us to study the relative importance of the resonant production versus gauge boson fusion processes in the study of Majorana neutrinos at hadron colliders. We analyze and propose different search strategies designed to maximize the discovery potential in either the resonant production or the gauge boson fusion modes.

1 Introduction

There is convincing evidence for the existence of three active neutrino species Ade:2013zuv (); ALEPH:2005ab (), at least two are massive. Their mass pattern has been narrowed down to a normal or an inverted hierarchy or degenerate Homestake (); Gallex (); Sage (); Boxerino (); SuperK (); SNO (); however, the absolute mass scale remains unknown. In addition, we are now capable of measuring CP violation in the lepton sector, given that the last mixing angle, , has been measured by several reactor experiments Chooz (); Daya (); Reno () and the T2K accelerator experiment T2K (). The experimental status of neutrino physics suggests that the evidence for neutrino masses represents a clear motivation for new physics beyond the Standard Model (SM). In principle this is due to the fact that within the SM, neutrinos are massless and one may incorporate new degrees of freedom or effective interactions to generate Dirac or Majorana masses. The latter is an interesting possibility since a Majorana mass term violates lepton number by two units. While a neutrinoless double beta decay () signal would be a breakthrough giving us knowledge on the nature of neutrinos and with additional assumptions the scale of the active neutrino mass matrix, the observation of lepton flavor violating processes at the LHC can shake the foundations of the SM with additional mechanisms and with luck we may directly find the new particles associated with these interactions.

In the minimal Type I seesaw mechanism Seesaw (), the SM is extended with a single Majorana fermion, singlet under the SM gauge group. Within this framework, the Majorana fermion carries lepton number and couples to left-handed leptons through the Higgs field. In addition, a Majorana mass term for the Majorana fermion can be implemented consistent with the gauge symmetries of the SM. The Majorana nature of this fermion and its mixing with the SM active neutrinos may lead to interesting testable interactions at the LHC that violate lepton number such as the production of same-sign leptons in association with jets. This final state can be achieved through resonant production of a Majorana neutrino and through gauge boson fusion. There has been a number of theoretical studies aimed at examining the sensitivity that the LHC has to heavy Majorana neutrino masses and to set limits on the couplings to leptons that arise from mixing with the active SM neutrinos Datta:1993nm (); Almeida:2000pz (); Panella:2001wq (); Han:2006ip (); del Aguila:2007em (); Atre:2009rg (); Dev:2013wba (); Deppisch:2015qwa (). In particular, a 14 TeV LHC with fb may have sensitivity to resonant production of Majorana neutrinos with masses up to GeV using the and channels. However, the fusion can reach masses up to GeV in the channel Alva:2014gxa (). Furthermore, in the regime where the Majorana Neutrino has mass below the mass of the SM boson, the authors in Izaguirre:2015pga () have shown that the LHC can be sensitive to mixing angles in the range and Majorana neutrino masses below GeV. This is done by looking for a prompt lepton in association with a displaced lepton jet. For heavy Majorana masses, no significant reach can be achieved with the gauge bosons fusion process. These channels are collider analogue to () which are not available there. Moreover, the channel may be sensitive to electron-like Majorana neutrinos with masses between GeV and higher LHC energies Dicus:1991fk ().

Alternatively, one can extend the SM to be left-right symmetric; partnering right-handed charged leptons with new right-handed neutrinos into an SU(2) doublet. Within this class of models, the gauge couplings are left-right symmetric and a new charged gauge boson, , connects right-handed charge currents Pati:1974yy (); Mohapatra:1974hk (); Mohapatra:1974gc (); Senjanovic:1975rk (); Senjanovic:1978ev (). The phenomenology of left-right symmetric models is very rich with the appearance of new scalar degrees of freedom and lepton violating process such as the production of same-sign leptons. The latter can be induced from the production of a right-handed charged gauge boson. A number of theoretical studies have examined the sensitivity that the LHC has to elements of the right-handed lepton mixing matrix, the mass of and Majorana neutrino masses Keung:1983uu (); Tello:2010am (); Chen:2011hc (); Aguilar-Saavedra:2014ola (); Heikinheimo:2014tba (); Deppisch:2014qpa (); Vasquez:2014mxa (); Das:2012ii (); Han:2012vk (); Chakrabortty:2012pp (); Chen:2013foz (); Gluza:2015goa ().

With the LHC running, both CMS and ATLAS have searched for same-sign dilepton final states using light- or -jets Chatrchyan:2013fea (); Chatrchyan:2012paa (); TheATLAScollaboration:2013jha (). In addition, a recent analysis by CMS Khachatryan:2015gha () is used to search for heavy Majorana neutrinos in the same-sign dimuon channel with an integrated luminosity of fb and 8 TeV center of mass energies with a sensitivity to a Majorana mass of GeV. Dedicated searches to probe same-sign leptons within the framework of left-right symmetric models have also been carried out. In particular, both CMS and ATLAS have set limits on heavy Majorana neutrino production assuming and identical quark and neutrino mixing matrices for the left- and right-handed interactions Khachatryan:2014dka (); ATLAS:2012ak (). The exclusion regions extending to TeV in the ( plane.

Our work compliments and extends the various analyses discussed above. In the present analysis we explore the LHC’s Run II potential to discover heavy Majorana neutrinos with luminosities between and fb the final state. Given that there exist many models for neutrino mass generation, even within the Type I seesaw framework, we use a simplified model approach and study two simple extensions to the Standard Model, one with a single heavy Majorana neutrino, singlet under the Standard Model gauge group, and a limiting case of the left-right symmetric model. We would like to emphasize that simplified models are not complete models. They are constructed to highlight specific points which in our case means LHC signals, and they can be mapped into more realistic models. They can also be viewed as truncations of more complete models. For the Type I Seesaw, a more natural framework would include at least two heavy right-handed neutrinos to explain the mass differences between the active neutrinos. However, in the mass eigenstate basis of these new heavy Majorana neutrinos the mixing angles and the masses can be treated as independent variables. Furthermore, more details regarding the model are necessary to determine the correlation between the new parameters and the mass differences of the active neutrinos. We will be conservative and assume that only the lightest of the new heavy sates are within reach of the LHC or future circular collider. Incorporating a second and third heavy Majorana neutrino simply leads to a duplication of our results; thus we ignore the heavier degrees of freedom and also the mass relation between active neutrinos. Thus, we can focus on the lightest heavy Majorana neutrino without loss of generality. Furthermore, the usual Type I seesaw model has very small mixing of light and heavy neutrinos. This can be circumvented by arranging structures in the heavy neutrino mass matrix. The details are highly model dependent; albeit important for fitting low energy neutrino data, see e.g. ZhZh (); Ibarra:2010xw (). In the simplified model this is captured by allowing the couplings of the heavy neutrino to the charged leptons to be free parameters. We then extend the analysis to a future hadron collider running at TeV center of mass energies. This extrapolation in energy allows us to study the relative importance of the resonant production versus gauge boson fusion processes in the study of Majorana neutrinos at hadron colliders. We analyze and propose different search strategies designed to maximize the discovery potential in either the resonant production or the gauge boson fusion modes. Our work is strictly phenomenological. We check consistency with the strongest model independent constraints that arise from low energy interactions such as rare decays of the muon, unitarity of the PMNS matrix, and the null evidence for (). A well known caveat to be added here is that other than unitarity tests, low energy constraints are obtained with the assumption that other new physics do not contribute significantly.

The summary of our study is as follows: In Section 2 we introduce the models and interpret current low energy and collider constraints. In Section 3 we calculate the leading SM backgrounds and propose various search strategies designed to maximize the discovery potential in either the resonant production or the gauge boson fusion modes of a heavy Majorana neutrino. In Section 4 we provide concluding remarks.

2 Two Simplified Models

To study same-sign leptons production in association with jets at the LHC run II and future higher energy colliders, we examine two simplified models that give rise to this final state topology. The first one consists only of SM gauge interactions and the only added new degree of freedom is the singlet heavy Majorana fermion which mixes with the 3 active SM neutrinos. The mass of and its mixing with the active neutrinos are taken to be free parameters. The second model implements an additional SU gauge symmetry under which the right-handed leptons and quarks are charged unlike the left-handed chirality. now partners with the right-handed charged leptons. This can be part of a left-right symmetric model that can give rise to realistic masses for the active neutrinos. The details are not essential for our purpose. In what follows, we review different aspect of these two theoretical approaches and indicate how they can be related to more complete models.

2.1 Single Majorana Fermion Extension to the Standard Model

The following framework consists of an extension to the SM with a single Majorana fermion, singlet under the SM gauge interactions. Within this analysis, the Majorana fermion, , mixes with the three active neutrinos. We are aware that to get realistic active neutrino masses, two or more Majorana neutrinos have to be added. In the spirit of the simplified model we are considering we can assume that these additional states are either much heavier or have negligible mixings. Following the analysis detailed in Han:2006ip () where in terms of mass eigenstates, the gauge interaction Lagrangian is given by:

(1)

where and provide a relationship between the flavor basis, and the mass basis parametrized by such that

(2)

with . We refer the reader to Atre:2009rg () for more details regarding the mixing formalism. In our simplified approach, we parametrize the charged current interactions by introducing the parameters , where

(3)

For Type I seesaw models the mass of the Majorana neutrino is commonly assumed to be GeV to obtain active neutrino masses below eV AbsScale (). However, there exist well motivated models that can lower the seesaw scale and/or have a sizable coupling to the SM particles. Notable examples are structures or cancelations in the heavy neutrino sector that can lead to and larger mixings Ibarra:2010xw (). Approximately conserved lepton number has also been invoked GonzalezGarcia:1988rw (); Kersten:2007vk (). The inverse seesaw mechanism can also yield TeV scale masses with mixings at a few percent Chang:2013yva (). In our analysis we will take a model independent approach and vary the couplings freely to investigate the range that the LHC can probe.

Figure 1: Feynman diagrams for the -channel contribution to same-sign lepton production.

The signal that will be probed in this study is given by

(4)

where denote either an electron or muon. One important aspect of this reaction is that it contains no missing transverse energy at the parton level. Most of the missing energy will appear after a full detector simulation has been carried out, mostly due to the misidentification of jets. At the parton level there are two channels to consider:

  1. s-channel annihilation depicted in Figure 1:

    (5)

    with the decaying into light jets. The Majorana neutrino, can be on- or off-shell depending on the mass, . In this channel, the from the decay of can allow us to control the large SM backgrounds since one may reconstruct the mass using the two light jets for not too large .

  2. t-channel or fusion process depicted in Figure 2:

    (6)

    In this process, the Majorana neutrino is always off-shell. If the initial quarks have the same color, a contribution to the amplitude will arise by interchanging the two forward outgoing jets.

\thesubsubfigure
\thesubsubfigure
\thesubsubfigure
\thesubsubfigure
Figure 2: Feynman diagrams for the -channel contribution to same-sign lepton production. Diagrams (c) and (d) correspond to contributions where the final state patrons have the same color.

2.2 Gauge bosons beyond the Standard Model

The signal topology described above is not unique to models that generate mixing between a heavy ”right-handed” Majorana neutrino and the three active neutrino species. In fact, models that go beyond the SM and that include additional gauge degrees of freedom can give rise to same-sign lepton final states. Left-right symmetric models are primarily motivated to address the origin of parity violation in low energy physics. It assumes that the Lagrangian is left-right symmetric, with the observed asymmetry, such as in -decay, arising from a vacuum expectation not invariant under parity transformations. The SU(2)SU(2)U(1)P, at low energies, breaks down to the SM gauge group with new particles mediating new interactions arising at higher energies Pati:1974yy (); Mohapatra:1974hk (); Mohapatra:1974gc (); Senjanovic:1975rk (); Senjanovic:1978ev (), in particular, right-handed charged currents, . Below we summarize essential ingredients pertaining to left-right symmetric models in light of a same-sign lepton final state at the LHC and refer the reader to LeftRightReview () for a full review of the theory.

In our analysis we assume that new charged currents are associated with a gauged SU(2) symmetry and introduce three heavy Majorana fermions, , that together with the right-handed charged leptons transform under the fundamental representation of the new gauge symmetry. The matter content can be written as

(7)

Here by symmetry we have three right-handed neutrinos . Within this framework, the minimal setup consists of one scalar bidoublet and two complex scalar triplets that are given by

(8)

This is required to break the gauge symmetries down to U(1). The scalar phenomenology is very rich and recent studies have been devoted to the possibility that a doubly charged scalar can be detected at the LHC and future more energetic hadron colliders Bambhaniya:2015wna (); Bambhaniya:2014cia (); Mohapatra:2013cia (). In addition, production and decays of charged scalars could help determine the leptonic right-handed mixing matrix at the LHC Vasquez:2014mxa (). In principle, the heavy gauge bosons will mix with the SM gauge sector, but this mixing is naturally small due the V-A nature of the charged current observed at low energies. In our analysis we explore the limit where the mixing is negligible.

In the minimal setup, lepton masses are due to the following Yukawa interactions

(9)

In the absence of spontaneous CP violation we obtain a Dirac mass term mixing left- and right-handed leptons given by

(10)

where and are the vacuum expectation values () of the Higgs bidoublet, and . In addition both left- and right-handed neutrinos acquire a Majorana mass given by

(11)

with and . In contrast to the Type I seesaw models need not be zero at the tree level, although it is expected to be small compared to the Fermi scale. The details are model dependent. The diagonalization of the mass matrices proceeds as in the Type I seesaw, leading to charged current interactions that in the mass eigenstate basis are given by

(12)

where are the left- and right-handed mixing matrices respectively. In the above Lagrangian, and have been obtained in the limit where . In this limit and parametrizes the mixing of light neutrinos through charged current interactions leading to neutrino oscillations, while parametrizes new phenomena that can be probed at the LHC and through various lepton flavor violating processes Das:2012ii (); Vasquez:2014mxa ().

The goal of this study is to probe , in particular the signal topologies in Figures 1 and 2 with at the LHC with TeV center of mass energies and at a future TeV Collider. To accomplish this, we focus only on the structure given in Equation (12) to parametrize the charged currents in the lepton sector and assume that couples at leading order to quarks with a hierarchical structure similar to that of the SM CKM matrix. This is done to suppress the bounds on that arise from measurements of CP violating effects of the mass difference and meson parameters Beall:1981ze (); Barenboim:1996nd (); Bertolini:2014sua (). However we direct the reader to Zhang:2007fn (); Senjanovic:2014pva (); Senjanovic:2015yea () where a general study on right-handed quark mixings can be found.

2.3 Constraints

In the singlet Majorana extension to the SM introduced Section 2.1, the Majorana neutrino mixes with active neutrinos with a strength proportional to for . As a result, this framework is sensitive to model independent constraints that arise from lepton unitarity Langacker:1988up (); Bergmann:1998rg (). These are

(13)
Figure 3: Contribution to the decay rate in the presence of a heavy Majorana neutrino. In left-right symmetric models, the the SM gauge boson can be replaced by .

In addition, indirect constraints that are model dependent arise from rare decays of the muon and muon properties, such as and the muon anomalous magnetic moment, . The validity of these constraints assumes that is the only source of new physics entering the calculation of the matrix elements, such that there is no cancellations arising from unknown physics. The Feynman diagram is depicted in Figure 3 and we use the unitary gauge to avoid the need to implement the coupling of to the would-be goldstone modes. The amplitude is given by

(14)

where denotes the photon polarization vector with momentum . Charged lepton self-energy diagrams and the active neutrino diagrams that are necessary to cancel divergences are not shown. The Majorana neutrino contribution is given by

(15)

where can be determined from

(16)

where . The contributions from the active neutrinos can be obtained by taking the limit . These contributions can be subtracted out by using the unitarity of the neutrino mixing matrix. With the above two equations, the branching ratio for is given by

(17)

where the second equality is valid in the limit of very heavy compare to the gauge boson.

For the case of right-handed gauge interactions we use the unitarity of the right-handed neutrinos mixing matrix and take the limit where two of the three decouple from the low energy theory to obtain

(18)

where denotes the mass of the SM charged gauge boson and arises due to normalization. The current experimental bound is  Renga:2014xra ().

Similarly, the contribution from to the anomalous magnetic moment of the muon, if given by

where . The first limit in the above equation is consistent with the results appearing in Grau:1984zh (). The experimental limit for the deviation is . A few comments are in order:

  1. Neutrinos, heavy or otherwise, contribute negatively to and will not explain the alleged discrepancy with the SM.

  2. The calculation of does not make use of the unitarity condition of the lepton mixing matrix.

Neutrinoless double beta decay can be a sensitive probe for the absolute neutrino mass scale and/or the mixings of heavy neutrinos with the active ones. The caveat here is that the theoretical uncertainties involved in the calculation of the nuclear matrix elements are difficult to quantify. To compound this when heavy neutrino intermediate states are involved one is dealing with very short distances between the two neutrons involved, and the uncertainties are even larger. With this in mind, the best limit on this reaction comes from the decay of Ge which gives an upper bound on the Majorana active neutrino mass of meV KlapdorKleingrothaus:2000sn (). This bound translates to a bound on the light-heavy mixing element , Section 2.1, and it is given by

(20)

Within left-right symmetric models, Section 2.2, a similar bound can be obtained on the mixing matrix elements between the right-handed leptons, , where runs over the three heavy right-handed neutrinos, . In particular, one can constrain

(21)

Das:2012ii () using the current experimental limit of  Rodejohann:2011mu ().

In addition to low energy direct and indirect constraints, collider searches can be used to probe the single Majorana extension of the SM, in particular flavor violating final states. The most recent analysis by the CMS collaboration targets the same-sign dimuon channel with TeV center of mass energies and fb of integrated luminosity Khachatryan:2015gha (). The limits obtained are on the mixing element and result on the following upper bounds: for Majorana neutrinos masses of and GeV respectively. However, for a and GeV Majorana mass the unitarity bound given in Eq. (13) is stronger, making the latter the primary constraint limiting the value of that we use in our analysis. This analysis equally constrains the left-right symmetric model discussed in the previous section. In addition, both CMS and ATLAS have reported limits on heavy Majorana neutrino production in the context of the left-right symmetric models Khachatryan:2014dka (); ATLAS:2012ak (). The limits are for heavy Majorana production with and identical left- and right-handed mixing matrices. The region excluded extends to TeV, with a region defined in the plane.

3 Same-sign leptons at the LHC and beyond

3.1 Background

Within the SM, lepton number violating processes are absent at zero temperature. However, there exist certain SM processes that can give rise to a same-sign lepton final state in association with jets. The two leading backgrounds discussed below are simulated at leading order using MadGraph 5 Alwall:2011uj (). We implement PYTHIA Sjostrand:2006za () for the parton showering and hadronization and a fast detector simulation is carried out using Delphes 3 delphes (). The detector simulator is used for jet clustering, track reconstruction and lepton identification and isolation. Light jets are reconstructed using the anti- algorithm with parameter . Furthermore, we calculate the lepton isolation criteria, , by summing the of all tracks within a cone of size around the leptons and require that

(22)

The two leading background processes at the LHC are the following:

  1. with low missing transverse energy, .

    This SM process is an irreducible background and dominates when the amount of missing transverse energy is small and comparable to the signal. The parameter space for heavy Majorana neutrinos within the SM was studied in Han:2006ip () using TeV center of mass energies and a luminosity of fb. The authors identified the following series of cuts that would suppress this background and enhance the significance of a :

    (23)

    However, a search performed by the CMS collaboration for same-sign muons and electrons using fb of data at TeV Giordano:2013xba () determined that the leading background for same-sign leptons in association with jets is from the QCD multijet component with misidentify jets as leptons. Consequently, the impact of the QCD background should be analyzed further.

  2. QCD multijet background.

    This background is mostly due to with two jets misidentified as leptons. In addition, leptons may arise from decays of heavy flavored jets. Although there are studies by ATLAS on muon fake rates ATLASfakemu (), the electron fake rate does not have a reference value. However, from the studies in Giordano:2013xba () and ATLASfakemu (), the electron fake rate can be anywhere between . In what follows we refer to the fake rate as and emphasize that the fake rate highly depends on the detector and energy, making it difficult to determine with a fast detector simulator such as Delphes. In our analysis, we will assume a given fake rate and determine the background suppression as a function of the misidentification efficiency. We extract an upper bound on this rate by comparing our simulated background after applying the following kinematic cuts that we find to significantly enhance our signal over the irreducible background described above:

    (24)

    where and denote the leading an subleading leptons, to the number of simulated events using the latest search for by the CMS collaboration Giordano:2013xba () with fb at TeV. The result is a muon misidentification rate (fake rate) of . We use a generous upper bound for the electron misidentification rate of .

Parton level (fb) 0.3927 0.7849 0.3927 1.79
1178 2354 1178 5370
Detector level + (24) (fb) 0.1187 0.2674 0.1187 0.471
356 802 356 1410

Detector level + (24)

+

(fb) 0.0094 0.0202 0.0119 0.0643
28.3 60.7 35.6 193

Detector level + (24)

+

(fb) 0.0047 0.0100 0.0059 0.0323
14.18 30 17.6 96.9
Table 1: SM backgrounds at TeV. The event number is shown at a luminosity of fb.
Parton level (fb) 3.686 7.370 3.694 44.2
11058 22110 11082 133000
Detector level + (24) (fb) 0.962 2.148 1.189 11.6
2887 6444 3568 35000

Detector level + (24)

+

(fb) 0.0597 0.1388 0.0784 1.59
179 416 235 4770

Detector level + (24)

+

(fb) 0.0291 0.0692 0.0395 0.798
87.2 207.6 118.4 2390
Table 2: SM backgrounds at TeV. The event number is shown at a luminosity of fb.

In Tables 1 and 2 we show the generated backgrounds at parton level and after a full fast detector simulation for TeV center of mass energies respectively with fb of integrated luminosity. The detector level results are shown applying the cuts discussed in Equation (24) and the requirement that the leading two jets arise from the decay of a SM gauge boson. The latter is used to enhance the sensitivity to the single heavy Majorana neutrino extension to the SM discussed in Section 2.1.

3.2 Sm + : S-channel collider reach at and TeV

\thesubsubfigure
\thesubsubfigure
Figure 4: The - and -channel contribution using default parton-level cuts implemented in MadGaph 5 at (left) and (right) TeV using a flavor universal mixing angle of .

The presence of heavy Majorana neutrinos in either a structure only consisting of SM gauge symmetries or an extension of the SM with an additional non-abelian group structure will contribute to the process in two topological classes: - and -channels. The -channel topology is depicted in Figure 1 and it is dominated by quark anti-quark fusion, in particular, the first generation. The -channel topology is depicted in Figure 2 and has contributions from both quark-quark and quark-antiquark fusion. In the massless limit, for quarks, these two contributions have different helicity amplitudes. In this section we examine the sensitivity that Run II at the LHC and a future 100 TeV collider will have to the model discussed in Sections 2.1. For the left-right symmetric model, a combined - and -channel analysis is performed, since within this theoretical framework, the mass of the SU(2) gauge boson is also a free parameter. The simulation of the signal at the parton level is carried out using MadGraph 5 Alwall:2011uj (). In Figures 4(a) and 4(b) we show the - and - channel contributions to the same-sign electron cross section for the LHC running at TeV (left) and TeV (right) within the framework a single heavy Majorana neutrino extension of the SM using a coupling of . From the plots we can observe that the -channel contribution to the production cross section starts dominating for masses, , approximately above TeV.

In the above section we introduced the two most dominant backgrounds that must be suppressed to enhance the sensitivity that a hadron collider requires to probe a model with a lepton number violating same-sign lepton final state. Below we examine and introduce a search strategy aimed at extracting a statistical significant signal for Majorana masses below TeV. The constraints discussed in Section 2.3, in particular , narrow down the relevant final states to or , since in either case the cross section depends only on or , and we can use either coupling to suppress the constraint.

The simulation of the signal at the parton level is carried out using MadGraph 5 Alwall:2011uj () with model files generated with FeynRules Alloul:2013bka (). We use two input parameters in our simulation, the mass of the heavy Majorana neutrino, , and a universal mixing to the active neutrino species, . We separately generate and final states. In addition we focus on Majorana masses above GeV to avoid constraints from LEP Achard:2001qv (); Abreu:1996pa (); Akrawy:1990zq () that restrict to values below for Majorana masses between GeV. The partial widths of are thus given by

(25)
\thesubsubfigure
\thesubsubfigure
Figure 5: Survival rate for a same-sign lepton final state with at least two jets for TeV (left) and TeV (right) center of mass energies.

We begin the analysis by applying the cuts used to treat the backgrounds described in Equation (24). However, the full detector simulation misidentifies a number of leptons as jets. The survival rate for our signal is shown in Figure 5(a) for a TeV collider and in Figure 5(b) for TeV. A clear signature of this model is the that arises from the decay of a Majorana neutrino. Since we are looking at the hadronic decay mode of the gauge boson, the can be tagged using the invariant mass of two jets in the event, in particular the leading two jets. In Figure 6 we show the invariant mass for different masses of the Majorana neutrino, .

Figure 6: Invariant mass distribution of the two leading jets in our signal for various heavy Majorana neutrino masses, .

To analyze the reach at the LHC and in a future TeV collider to the S-channel signal of the simplified model, we define a significance variable by

(26)

where denotes the number of signal events and the number of background events. Since we are using an inclusive final state (in charge) with same-sign leptons, the production cross-section is given by

(27)

Given that is Majorana, and given the fact that they are independent of the momentum transfer, we can treat as a scaling factor to the total amplitude. Therefore, we choose to carry out the simulation for a fixed value of for all three lepton flavors. Thus

(28)

and since for the range of masses that we are considering, we can write

(29)

Similarly for the final states. In our analysis we include systematic and statistical uncertainties and model the distribution as Gaussian with an statistical error given by , for number of events. We find the confidence level fit to using a variable defined by

(30)
\thesubsubfigure
\thesubsubfigure
Figure 7: The expected limit at TeV and fb of integrated luminosity. On the left, the limits are shown without the QCD contribution while on the right, varying fake rates are introduced for the final state.

where , denotes the number of signal events with the benchmark value and denote the SM irreducible and QCD backgrounds respectively. The latter is scaled with the fake rate described earlier in the section. The systematic uncertainty is parametrized by and we assume it to be negligible compared to the statistical component. In Figures 7(a) we plot the C.L region of parameter space against the irreducible background using a center of mass energy of TeV and fb of data. In Figure 7(b) we show the effect the QCD background with different fake rates in the channel. In Figures 8(a) and 8(b) we show the results of the simulation for a TeV collider with fb of data.

\thesubsubfigure
\thesubsubfigure
Figure 8: The expected limit at TeV and fb of integrated luminosity. On the left, the limits are shown without the QCD contribution while on the right, varying fake rates are introduced for the final state.

It is worth mentioning that our results improve by a non-negligible amount if instead of doing our fit to the the whole range of Majorana masses, we carry out a fit to bins of the Majorana mass parameter, . In particular, we define a bin size of GeV and fit our simulated data to obtain the C.L regions; these are depicted in Figures 9(a) and 9(b) together with the results shown in the previous two figures for comparison. It is worth pointing out then that our analysis shows that a TeV machine will have sensitivity to couplings approximately an order of magnitude below those probed by the latest CMS same-sign muon search Khachatryan:2015gha ().

\thesubsubfigure
\thesubsubfigure
Figure 9: The expected limits with a fit to GeV bins of the invariant mass, . The plot on the left corresponds to TeV center of mass energies with varying fake rates while the plot on the right is a TeV machine. For comparison the unbinned fit is shown in red.

3.3 Sm + : T-channel collider reach at and TeV

We saw at the beginning of Section 3.2 that the -channel contribution to the production cross section decreases significantly for Majorana masses above TeV and the -channel contribution begins to dominate. This effect can be easily understood given that for quark-antiquark invariant masses below the mass of the heavy neutrino, the is closed while the is open; that is

(31)

Although the phase space contribution from a -particle final state is larger than a -particle final state (-channel), the -channel amplitude is greatly affected due to the PDF suppression at high momentum fraction of the quark-antiquark initial state. A naive estimate allows us to determine that a TeV machine with fb of data cannot extract a statistically significant signal for masses, , in the range TeV. However, a TeV machine with fb of integrated luminosity can yield approximately events with before a full detector simulation is implemented. In Appendix A we show an explicit analysis of the various contributions to the -channel amplitude that is used to determine a series of efficient cuts to extract a signal for Majorana masses above TeV. In particular, we show that the amplitudes are proportional to the jet-pair invariant mass with the strongest enhancement arising from contributions proportional to the Majorana mass. In addition, for momentum transfer, , below , the amplitudes are also proportional to the lepton pair invariant mass.

\thesubsubfigure
\thesubsubfigure
Figure 10: Survival rate for a same-sign lepton final state with at least two jets for and TeV center of mass energies.

In Figure 10 we show the performance of the detector simulation after implementing the default cuts introduced in Equation (24). For the range of masses that we are interested in we show the same-sign lepton reconstruction efficiency as a function of in both and TeV machines. In both cases, the overall drop in efficiency is mainly due to the large pseudorapidities associated with jets in the hard process since the track reconstruction efficiency drops significantly for values of above . In fact, from Figure 11 we can see how boosted jets are in comparison to the SM background.

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Figure 11: Pseudorapidity distribution of leading (left) and subleading (right) jets for the signal and SM backgrounds using a heavy Majorana neutrino with mass TeV.

The -channel dominated signal is especially interesting since in the limit, the distribution of kinematical variables does not depend on the mass of the heavy Majorana neutrino. In addition, in this limit, the leptons are highly isotropic with values significantly higher than those of the SM background. This is shown in Figure 12.

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Figure 12: distribution of leading (left) and subheading (right) leptons for the signal and SM backgrounds using a heavy Majorana neutrino with mass TeV.

Finally, the lepton-pair separation, , and invariant mass distribution, , can also be used to reduce the background. The former is due to the fact that both leptons are back-to-back in the transverse plane. This is shown in Figure 13.

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Figure 13: (left) and (right) distributions for the signal and SM backgrounds using a heavy Majorana neutrino with mass TeV.

Together with the default cuts introduced in Equation (24), we implement a series of additional cuts to extract a signal arising mainly from a -channel dominated amplitude labeled T(s)13 and T(s)100 for and TeV machines respectively:

T13
T100
Ts13
Ts100 (32)

While we emphasize the this strategy can yield large significances, it is very much dependent on the fake rate associated with the QCD background. That is, if the fake rate is higher, a tighter cut on the lepton must be used. Furthermore, the lepton separation, , is highly correlated with and the lepton’s transverse momentum; we observe that it only reduces the signal after all other cuts are applied.

Parton level (fb) 0.3927 0.7849 0.3927 1.79
1178 2354 1178 5370
Detector level + (24) (fb) 0.1187 0.2674 0.1187 0.471
356 802 356 1410

T13

(fb)

Ts13

(fb)
Table 3: SM backgrounds at TeV. The event number is shown at a luminosity of fb. On the last two words we show the background reduction after applying the T13/Ts13 cut selection in Equation (32).
Parton level (fb) 3.686 7.370 3.694 44.2
11058 22110 11082 133000
Detector level + (24) (fb) 0.962 2.148 1.189 11.6
2887 6444 3568 35000

T100

(fb)

Ts100

(fb)
Table 4: SM backgrounds at TeV. The event number is shown at a luminosity of fb. On the last two words we show the background reduction after applying the T100/Ts100 cut selection in Equation (32).
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Figure 14: The expected limit at TeV and fb of integrated luminosity with varying fake rates on the QCD background in the same-sign electron channel (left) and same-sign muon channel (right). The red solid line indicates the bound from unitarity discussed in Section 2.3.

In Tables 3 and 4 we show the SM backgrounds after implementing the T(s)13 and T(s)100 cut selections. The Ts13 and Ts100 correspond to an additional cut on the -particle final state invariant mass that reduces the QCD background by up to an order of magnitude in a TeV machine.

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Figure 15: The expected limit at TeV and fb of integrated luminosity with varying fake rates on the QCD background in the same-sign electron channel (left) and same-sign muon channel (right). The red solid line indicates the bound from unitarity discussed in Section 2.3.

Using the cut flow discussed above, appropriate to extract a -channel dominated signal; we observe that the SM irreducible background can be completely eliminated. In addition, implementing a fake rate of order further suppresses the QCD background. However, the signal is proportional to the mixing angle raised to the fourth power, and the event yield is very small in the regions of small mixing angle. The yield can be enhanced with an additional cut on the -particle final state invariant mass by one order of magnitude. Using the analysis introduced in Section 3.2 we find the confidence level fit using the variable defined in Equation (30) with varying mixing angle squared, . We scan over the of the leptons, the pseudorapidity of the subleading jet, the same-sign lepton invariant mass and the -particle invariant mass to extract the maximum value of leading to a expected limit on the mixing angle as a function of the Majorana neutrino masses. The scan is listed below:

The results are shown in Figure 14 for a TeV collider and Figure 15 for TeV. In order to start probing couplings allowed by unitarity, in the large region, one will have to wait for a TeV collider and this highly depends on the ability to reduce the fake rate for both and below . For very small fake rates, a TeV machine can probe Majorana masses up to TeV with couplings as low as in the same-sign electron channel.

3.4 Left-Right Symmetric Model: Collider reach at and TeV

The presence of a charged gauged boson in the spectrum, associated with the SU(2) gauge symmetry, leads to several phenomenological differences that makes a same-sign lepton analysis different from the basic extension of the SM. This is due to the fact that for gauge boson masses, , above TeV, the -channel contribution to the production cross section is strongly suppressed. This makes a -channel dominated signal difficult to extract even with a TeV machine and fb of integrated luminosity. Since we focus on the limit where , we can expect that , and are not able to use the SM charged gauge boson mass as a means to suppress the SM backgrounds. Thus, depending on the mass hierarchy between the Majorana neutrino and we can expect different processes to dominate the signal:

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Figure 16: Branching ratio of (left) and (right) with TeV.

The search strategy will strongly depend on whether can be produced on-shell. In particular, a same-sign lepton plus two jets final state will be suppressed by a -body phase space factor in the region where . We can see this in Figure 16(a) for TeV. However, from Figure 16(b), we can see that the branching ratio of into two jets is mostly unaffected. Therefore, for , the and of the final state particles will highly depend on the and of the on its rest frame. However, for , resonances can be reconstructed depending on the phase space:

  •  ,

  •  .

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Figure 17: cross section for varying at TeV (left) and TeV (right) center of mass energies. The top, middle and lower panels correspond to masses TeV respectively.

To see which resonances can be used to better enhance a same-sign lepton signal, we analyze how (a) and (b) in Equation (3.4) behave as a function of and where the on-shell production of the right-handed gauge boson dominates. This is shown in Figure 17 where on the left column we plot the production cross section at TeV as a function of for values of and TeV respectively while results for TeV are shown on the right column. In the former we observe a small region for where we cannot use the invariant mass to reconstruct the Majorana mass while in the latter, the (b) production mode always dominates. Thus, for a TeV machine the invariant mass of the system can always be used to reconstruct the Majorana neutrino mass for both