Search for a light $Z^\prime$ at LHC in a neutrinophilic $U(1)$ model

We consider a neutrinophilic $U(1)$ extension of the standard model (SM) which couples only to SM isosinglet neutral fermions, charged under the new group. The neutral fermions couple to the SM matter fields through Yukawa interactions. The neutrinos in the model get their masses from a standard inverse-seesaw mechanism while an added scalar sector is responsible for the breaking of the gauged $U(1)$ leading to a light neutral gauge boson ($Z'$), which has minimal interaction with the SM sector. We study the phenomenology of having such a light $Z'$ in the context of neutrinophilic interactions as well as the role of allowing kinetic mixing between the new $U(1)$ group with the SM hypercharge group. We show that current experimental searches allow for a very light $Z'$ if it does not couple to SM fields directly and highlight the search strategies at the LHC. We observe that multilepton final states in the form of $(4\ell + \slashed{E}_T)$ and $(3\ell + 2j + \slashed{E}_T)$ could be crucial in discovering such a neutrinophilic gauge boson lying in a mass range of $200$--$500$ GeV.


I. INTRODUCTION
The modern era of particle physics has seen an extremely successful period with the model accounting for three of fundamental interactions of nature via gauge symmetries, i.e., the standard model (SM) of particle physics. The SM successfully explains most phenomena involving the elementary particles in nature which have been corroborated through observations in dedicated experiments. The discovery of a 125 GeV scalar [1,2] viz. the Higgs boson has completed the hunt for all particles predicted in the SM. Despite the remarkable success of the SM, there still remain several unexplained observations from experiments that hint at the possibility of new physics beyond the SM (BSM). One such anomaly is the observation of nonzero mass and mixing of neutrinos from neutrino oscillation experiments [3][4][5][6][7].
The otherwise massless neutral fermion within the SM can, in competing BSM extensions, have either Dirac or Majorana type mass, which is something yet to be established. A large number of scenarios exist to explain observed neutrino masses and mixings [8][9][10][11][12] and these possibilities lead to interesting phenomenology of the resulting neutrino mass models [13].
Besides the neutrino mass puzzle, another curiosity that intrigues us is the true nature of the scalar that has been observed at the Large Hadron Collider (LHC). The complete confirmation of it being the SM Higgs will only be possible, once its interactions are precisely measured. Until then it does leave the possibility of new physics within the scalar sector as a vital area of interest. There are a vast number of BSM theories including some for neutrino mass models, which include an extended scalar sector beyond the SM Higgs doublet. Our focus would be on the type which is central to neutrino mass models.
The minimal extension of the scalar sector is usually done with or without a new gauge group, although an extended scalar sector is more natural in extended gauge models where the scalars are charged under the new gauge group and are responsible for the spontaneous breaking of the new gauge symmetry. All such extensions predict some new phenomena that are to be observed in ongoing and upcoming experiments. Extension of the SM with an additional Higgs doublet is one of the most popular extension of the SM and popularly known as the two Higgs doublet models (2HDM). In some models the second Higgs doublet is used to give Dirac masses to the light neutrinos by introducing new right-handed neutrinos. Such models are popularly called neutrinophilic 2HDM (ν2HDM) [14][15][16], which lead to interesting phenomenology and signatures at experiments [17][18][19][20][21][22]. Another popular extension of the SM is the extension with a new U (1) gauge group. The introduction of new gauge groups have a different type of consequence in terms of the signature of the model.
One immediate consequence is the prediction of a new massive gauge boson (Z ) after the symmetry breaking of the new U (1) symmetry.
We all know that Z bosons [23] are among the very well motivated new physics scenarios in the study of BSM physics. The fact that the all successful SM is a gauge symmetry begs the question for the BSM to belong to an extended gauge symmetry with the simplest being the addition of a U (1). There are numerous examples of models extending the SM gauge symmetry group by an additional U (1) factor, which can arise, for example, from grand unified theories where the group of higher rank is broken down to the lower rank SM gauge group, leading to an additional U (1) symmetry arising naturally, or in bottom-up approaches where the additional U (1) is added to alleviate problems in models of dynamical symmetry breaking, supersymmetry (for example the µ problem), extra dimensions, flavor physics, etc. and can also act as mediators for hidden sectors (for extensive reviews see Refs. [23][24][25]). There have also been proposals for neutrino mass generation, for example in U (1) B−L extension [26][27][28]. A discovery of Z and its decays could therefore lead us to an understanding of the underlying gauge charges the particles carry, which could give hints to the underlying physics BSM (as the conditions of the new symmetry being anomaly free leads to specific charge assignments). However, there is currently no experimental evidence of such a Z , which could have two possibilities. Z may be very heavy to be discovered at current energies and we need to go for higher energies in its search, or it may be light but couples very weakly to the SM particles (similar to the SM Higgs search). We consider the latter possibility in this work while also invoking the novelty of the model providing a solution to the neutrino mass puzzle, leading us to a twofold motivation to consider such an extension. As the LHC has not observed a signal for new physics, proposing a light Z in such extensions is quite difficult unless it weakly couples to the SM sector. In this model, which is trivially anomaly free, we can naturally have a light Z while ensuring a popular seesaw mechanism for neutrino mass. We also need not tune the gauge couplings to unnaturally small values for a light Z unlike for example in U (1) B−L models, as this extension allows the gauge couplings to be of similar strength to any SM gauge coupling.
We consider an extra U (1) symmetry under which the SM particles are sterile. This is more in the line of a hidden extra U (1) considered before in another context by one of us [29,30]. Only new SM isosinglet fermions, an electroweak (EW) singlet scalar and a neutrinophilic Higgs doublet speak to this extra U (1). These new fields act as messenger particles between the U (1) and the SM sector. The extra U (1) symmetry is broken at the EW scale by the vacuum expectation value (VEV) of an EW singlet Higgs boson along with the second Higgs doublet. Thus the model predicts a heavy Z at the EW scale along with additional neutral fermions and scalar particles. We show through this work that the prediction of such an extension of the SM which can explain the light neutrino mass and with a particle spectrum that has minimal interactions to the charged fermions has its own set of challenges of observation and how such a scenario can be observed in the ongoing collider experiments.
The search for Z boson has been extensively studied at the LHC where most of the searches put strong limits on the mass of the Z based on its interaction properties [23,31].
The most popular channel to search for Z is usually the dilepton channel which gives stringent constraint on the production of Z at the LHC [32,33]. However, in our model, an interesting scenario arises where the Z can be significantly lighter than current limits and can evade bounds from the existing Z search. For such a Z we find that the multilepton channel proves much more promising. In this study, we mainly focus on Z from the viewpoint of its neutrinophilic nature. 1 The paper is organized as follows. In Sec. II we briefly discuss the framework of the U (1) gauged neutrinophilic model and calculate the mass and mixing parameters for the scalar, gauge and fermion sectors in the model. In Sec. III we discuss the relevant theoretical and experimental constraints before we move on to Sec. IV where we present the LHC analysis of the model in the 4 and 3 rich final states coming from the Z mediated heavy neutrino production. Finally we summarize and conclude in Sec. V.

II. THE MODEL
The model is an extension of the SM where the gauge group is augmented with an extra U (1) X gauge group and four new fields, viz. a second Higgs doublet (H 2 ), a scalar singlet (S), and two chiral sterile neutrinos (N L , N R ) added for each generation. All the new fields are 1 Similar models in the context of an ultralight mediator with cosmological implications and neutrino phenomenology have been studied before [34,35]. charged under the gauge group U (1) X while all the SM particles are neutral. The charge assignments of the new particles along with the first Higgs doublet (H 1 ), which is the SM Higgs doublet, are listed in Table I. Looking at the charge assignments, it is quite clear why we refer the model as a neutrinophilic one. The new isosinglet charge-neutral fermions are the only spin-1/2 fields which carry a U (1) X charge and therefore would lead to couplings of the new gauge boson with the neutrinos after symmetry breaking.
With the assigned charges, the most general gauge invariant Lagrangian that can be added to the SM Lagrangian, is given by Note that the last term in the Lagrangian breaks the U (1) X symmetry explicitly. This soft-breaking term is needed to give mass to the pseudoscalar after the symmetry breaking.
In addition, the singlet scalar S plays a crucial role in defining the mechanism for neutrino mass generation, notwithstanding the fact that it is also responsible for the mass of the U (1) X gauge boson. We shall now discuss the mass and mixings of the scalars, gauge bosons and matter fields following the spontaneous symmetry breaking of the gauge symmetries.

A. Masses and mixing of the scalars
The U (1) X symmetry is spontaneously broken when either the singlet S or the doublet H 2 acquires a VEV while the SM gauge symmetry breaks when either of the two Higgs doublets get a VEV. The Higgs doublets and the scalar singlet fields can be redefined by shifting with their VEVs in the usual way. Defining the VEVs for the Higgs doublets and singlet S as v 1 , v 2 , and v s , respectively, we can rewrite the fields as follows: In order for the potential to be minimum at the values of the VEVs, they should satisfy the following tadpole equations.
After the spontaneous breaking of the EW and U (1) X symmetries, we are left with three physical CP -even neutral Higgses, a charged Higgs, and a pseudoscalar Higgs. Following the restrictions given by the above minimization conditions, the mass matrix for the pseudoscalars in (η 1 η 2 η s ) T basis becomes It is evident from the mass matrix that two pseudoscalars remain massless after the diagonalization to their mass eigenstates. These two massless modes are eaten up by the two neutral gauge bosons, viz. Z and Z , to acquire masses. The remaining pseudoscalar is a physical state with a mass m A = It is worth noting that if the soft-breaking term was absent, i.e., µ 12 = 0 in the Lagrangian given in Eq. (1), all the pseudoscalars would have been massless. This is expected since, in the scalar sector of the Lagrangian, one can recover a global U (1) symmetry, viz. φ → e −iθQ φ, where φ represents any of the scalars. This global symmetry remains intact even after both the SM and U (1) X gauge symmetries are spontaneously broken, leading to a massless physical scalar in the particle spectrum. The soft-breaking term is therefore needed to avoid this massless pseudoscalar.
The mass matrix of the charged scalars in (φ + 1 φ + 2 ) T basis is given by This 2 × 2 mass matrix can be easily diagonalized by rotating with an angle β, which is defined by the ratio of the VEVs of the two Higgs doublets given by tan β = v 2 v 1 . It should be noted that the same angle β also diagonalizes the pseudoscalar mass matrix. One of the charged scalar is massless and corresponds to the charged Goldstone, which is eaten up by the W ± gauge boson to get its mass. The remaining physical charged scalar is orthogonal to the massless one and is given by The CP -even scalar mass matrix in the (ρ 1 ρ 2 ρ s ) T basis is given by In general, the determinant of the mass matrix of CP -even scalar is nonzero, which tells us that there will be three massive CP -even scalars after the symmetry breaking. We identify the three CP -even mass eigenstates as h 1 , h 2 , and h 3 . They are linear combinations of the flavor states and can be written as where Z h ij represents the mixing matrix for the CP -even states. For our analysis, we hereafter denote h 1 , h 2 , and h 3 as the physical eigenstates in ascending order of their masses. For simplicity, we restrict our choice on the parameters in the scalar sector such that the lowest mass eigenstate among all scalars will be the 125 GeV scalar, identified as the SM Higgs boson observed at the experiments. As we do not consider a full analysis of the scalar sector in this work, it helps us to focus solely on the Z and heavy neutrinos of the model. The other two CP -even states are taken to be beyond 700 GeV.
As the properties of the lightest scalar must be similar to the SM Higgs boson, we choose the parameters such that h 1 belongs mainly to the first Higgs doublet H 1 . In terms of the mixing matrix components |Z h 11 | 2 1. This natural choice is easily achieved if the diagonal entries of mass matrix M 2 H are much larger compared to the off-diagonal entries. This choice also suggests that v 1 v, which implies that tan β 1. We discuss the choice of tan β further in Sec. II C. In this setup, the three diagonal entries are controlled by v 1 , µ 12 , and v s .
So, the mass of the heavy scalars will be given (to an approximation) by m h 2 √ µ 12 cot β, and m h 3 2λ s v 2 s . The mass for the charged scalar as well as the pseudoscalar will also be similar to the mass of h 2 .

B. Gauge kinetic mixing and masses of gauge bosons
The presence of two or more U (1) gauge group in a theory allows us to write a gauge kinetic mixing term between the two U (1) gauge bosons without spoiling the gauge invariance of the Lagrangian [36]. The kinetic term for the gauge bosons in the Lagrangian, after including the gauge kinetic mixing, then becomes whereg is the kinetic mixing parameter. The following field redefinitions make the kinetic term diagonal with the desired coefficient The field redefinition tells us thang should be less than 1 for the fields to be real. This is usually referred to as the "theoretical constraint' ong. After achieving the correct form for the gauge kinetic term with the above field redefinitions, we can now try to write the mass terms of gauge bosons arising from the kinetic terms of the scalars, where with the gauge covariant derivatives for the corresponding scalars defined as The U (1) X charges of all the fields are proportional to q x . In a gauge theory, the gauge coupling always comes with the gauge charges, i.e., the constant that we will see is g x q x .
This means that we can absorb q x in g x . So, we will take q x = 1 henceforth. With the VEVs as defined in Eq. (15), we get mass terms for the gauge bosons as follows: From Eq. (13), we see that C µ is always accompanied by the factor 1 1 −g 2 . Since the coupling g x always comes with C µ , and hence with C µ , we may absorb this extra factor inside g x . Also, from the above equation, we see thatg does not appear separately. Hence, without loss of generality, we do the following redefinitions in the coupling in order to get simplified expressions In Eq. (19), the last redefinition means that we replace g x by g x 1 −g 2 in each place in the Lagrangian. We should also note that there is no restriction on g x from theoretical constraint even though we had restrictions ong.
The mass matrix for the neutral gauge bosons, in the basis of B µ W 3 µ C µ T is given by The diagonalization of the mass matrix of the neutral gauge bosons can be done in the following way, (i) First we rotate W 3 µ and B µ to get A µ and X µ .
where tan θ W = g 1 g 2 . The mass term for neutral gauge boson then becomes where g z = g 1 2 + g 2 2 . The above expression does not have any mass term for A µ . This means A µ is massless, which can be identified as the photon. The angle θ W can be identified as the Weinberg angle as we get in the SM.
(ii) Now, the mass matrix of X µ and C µ is given by The above mass matrix can be diagonalized by the orthogonal transformation between X µ and C µ as follows where After the diagonalization, the mass of the physical gauge bosons are , and the final mixing matrix becomes Note that the mixing between the Z and Z needs to be quite small such that it does not modify the Z boson couplings with the SM fields. In order to study the parameters that would be most relevant in establishing the Z-Z mixing, we look at Eq. (25) in more detail. We find that the kinetic mixing dictates that the coefficient g x appears with the SM VEV while the U (1) X coupling g x appears with the VEV of the second scalar doublet in the numerator of Eq. (25). Assuming that the kinetic mixing coefficient and the U (1) X gauge coupling are of the same order, one can approximate Eq. (25) depending on the choice of tan β. Note that for tan β 1, i.e., v 1 v, the dominant term in the numerator becomes Thus depending on the choice of tan β, we expect the mixing angle to vary for different ranges of g x and g x values.
To highlight the case where tan β > 1, i.e., v 2 > v 1 , we scan over a range of values for g x and g x as well as v s for 1 < tan β < 60 and calculate the mixing angle θ . In Fig. 1, we show the dependence of the Z-Z mixing angle (θ ), as a function of M Z along with its dependence on the variation of the gauge kinetic mixing (g x ). Note that for large values of M Z > 1 TeV the denominator term is significantly large and therefore the mixing angle is naturally small. However the numerator in Eq.
for tan β 1 and we find that even with the kinetic mixing vanishing, the mixing angle has values larger than O(10 −2 ) for M Z < 750 GeV. This is expected as the denominator (M 2 Z − M 2 Z ) becomes smaller, while g x is nonvanishing and constrained by the Z mass. This gives an interesting result that, even with vanishing kinetic mixing, if the Z gets a part of its mass from the scalar doublet, it leads to a substantial Z-Z mixing, which would disfavor the parameter space due to strong constraints from Z boson measurements. However it is still possible to obtain small θ < 10 −3 for the light Z case, provided there is a cancellation in the numerator term ∝ (2g x + g x ). These are the points highlighted in the figure with crosses (×) corresponding to negative values of g x . Thus it is possible to obtain small Z-Z mixing compatible with Z boson data even for tan β 1. The stronger constraint on such a scenario however comes from Higgs data and perturbativity arguments, which we discuss later along with the more favorable choice of parameter space where tan β 1.

C. Masses and mixing of the charged lepton and quarks
The Lagrangian responsible for the masses and the mixing of leptons and quarks is essentially the Yukawa terms.
The masses and the mixing can be arranged in the same way as it is done in the SM.
The only difference is that the mass of the SM fermions are proportional to the VEV of So, in order to achieve the correct mass, we need to choose Yukawa couplings 1, tan β should be such that cos β > 1 √ 4π from perturbativity consideration. This gives, although an approximate one, an upper bound of tan β < 3. With this bound in mind we shall restrict our study to values of tan β < 1 for further analysis. Recall that for any value of tan β > 1, there is significant increase in the couplings of the SM fermions with the scalars in the model. A critical scrutiny of its implications and phenomenology for the scalar sector in the current model is left for future work and we focus on the Z signal in this work.

D. Masses of neutrinos
In this model, we give Majorana masses to the neutrinos via inverse seesaw mechanism [37][38][39]. We rewrite the relevant part of the Lagrangian below.
We have added three generations of sterile neutrinos (N i R and N i L ) corresponding to the three generations of fermion in the SM, which renders all the Yukawa couplings (Y ν , Y L and Y R ) as 3 × 3 matrices. Note that the two chiral states N R and N L combine to form a vectorlike fermion (N ), which is a singlet under SM and carries the same U (1) X charge as its chiral components. After symmetry breaking, the mass term for the neutrinos are given by L mass The mass matrix in ν L N C R N L T basis is given by where Also, m L and m R are naturally small due to the so-called 't Hooft criteria [40]. Indeed, in the limit m L,R → 0, the lepton number is restored as a conserved symmetry.
As mentioned above, m L , m R m D ,M N , thus the neutrino masses can be given, with a very good approximation, by It is worth mentioning that, in this scenario, the neutrino Yukawa coupling Y ν , can be of The light neutrino mass matrix in Eq. (32) must be diagonalized by the physical neutrino mixing matrix U PMNS [46], i.e., Thus, one can easily show that the Dirac neutrino mass matrix can be defined as : where R is an arbitrary orthogonal matrix. Accordingly, the (9 × 9) neutrino mass matrix M ν can be diagonalized by N , i.e., N T M ν N = M diag ν , which is given by [47]  where It is clear that the deviation of a nonunitary matrix N 3×3 from the standard U PMNS is measured by the size of 1 2 F 2 . Also, the muon g − 2 anomaly and the lepton flavor violating processes can be affected by the F size [48]. Consequently, that imposes upper bounds on F entries to be small [49][50][51], which is automatically satisfied in our model due to the smallness of v 2 (i.e., v 1 v).
In normal hierarchy scenario, i.e., assuming m ν 1 < m ν 2 < m ν 3 , the two mass square differences determined from the oscillation data [52] is given by ∆m

III. EXPERIMENTAL CONSTRAINTS
The extension to the SM considered in this model affects the three sectors of the SM, viz.
(i) scalar sector, (ii) neutrino sector, and (iii) neutral gauge boson sector. We therefore need to focus on each of these to evaluate the experimental constraints that affect the parameter space of the model.  coupling which allows the Z to decay dominantly to a pair of the heavy neutrinos (when kinematically allowed) while all other modes are suppressed. We will see that this also helps us evade existing collider limits on light Z .

B. Constraints from HiggsSignals and HiggsBounds
The introduction of another Higgs doublet and singlet modifies the scalar sector. The modifications are of the following two forms.
• Due to the mixing between scalars, the production and branching fraction of the observed 125 GeV scalar gets modified with respect to the SM Higgs. These properties are measured in terms of signal strength of Higgs which gives constraint on the parameters [53][54][55][56][57][58][59][60].
• The model predicts heavy scalars which may be observed at the LHC. However, the LHC did not observed any new scalar other the 125 GeV one. This gives another constraints on the production of any new scalars.
Note that the choice of small tan β leads to suppressed couplings of charged scalars and pseudoscalar to the fermions as can be seen from the couplings shown in Table II. As a result, the production of these scalars at a collider are significantly suppressed. This helps us to evade any bounds coming from the nonobservation of such scalars at the LHC. However, the coupling of CP -even scalars (h i ) to the fermions are not all suppressed due to the small values of tan β. These couplings are mainly dictated by the entries in the CP -even scalar mixing matrix given by Z h i1 . Since we demand that the 125 GeV scalar belongs mainly to the H 1 doublet, we restrict ourselves to Z h  Table III. The only parameter that we do vary in the scalar sector when we fix the benchmark points for our analysis would be the singlet VEV v s and the corresponding quartic term coefficient λ s , which will affect the Z and h 3 masses.

C. Search for new Z gauge boson
The phenomenology of Z in the model is quite different from that of the more traditional U (1) extensions. In the absence of gauge kinetic mixing, the coupling of Z to the SM fermions gets modified by an additive factor proportional to sin θ , which has to be small to be consistent with the measurement of Z boson properties. However, the introduction of kinetic mixing parametrized by g x , we have an additional part in coupling, which is proportional to g x cos θ . We have listed the expression for the coupling of the Z with the matter fields of the model in the Appendix for reference.
As none of the SM fields are charged under the new U (1), the Z couples to the SM charged fermions only via the Z-Z mixing. For tan β > 1 we found that the mixing angle was dependent on both g x and g x . A small θ 10 −3 for M Z in the range of 200-500 GeV required a cancellation such that g x −2g x . However, this choice would imply that the coupling of the Z with the SM fermions and the new heavy neutrinos would have somewhat similar strength. Thus, in order to have substantial production cross section, one also gets a substantial branching fraction of the Z decay into SM fermions. For a light Z , the strongest constraint from the LHC comes from its decay into the dilepton channel [32]. Evaluating this limit for the case tan β > 1, puts a strong limit on the values of g x and g x ∼ 10 −3 .
Thus the promising search channel, when tan β > 1, still remains the dilepton mode, even with the heavy neutrino decay modes available for the Z . In contrast, when we consider the more favorable option of tan β < 1, we find that the constraint on θ is much more easily satisfied by suppressing the kinetic mixing parameter g x (even for light Z ) while the decay modes of the gauge boson can be significantly tilted in favor of the new neutral fermions in the particle spectrum. However, a too suppressed g x would also suppress the production cross section of the Z at the LHC, as can be seen by looking at its coupling with the SM quarks (see the Appendix). We would therefore like to find a region of parameter space where the gauge boson is produced at the LHC and leaves an observable imprint in final states still allowed by the LHC data.
We note that g x 10 −2 is sufficient to keep θ < 10 −3 . This choice allows us to enhance the production of Z at a collider by four orders of magnitude, compared to the case when g x = 0 where sin θ ∼ 10 −5 − 10 −6 (recall that sin θ depends on g x too). On the other hand, the coupling of Z with the heavy neutrinos is mainly governed by the choice of g x . where = e, µ. As one can clearly see, this interplay actually helps us to produce Z at a higher rate while being within the bounds from the LHC in Z → + − mode [32]. At the same time, we achieve a significantly high production cross-section of N N through the Z resonance. In

IV. COLLIDER ANALYSIS
We now look at the collider signatures for the new gauge boson Z at the LHC. The most obvious signal for a heavy Z is via the Drell-Yan channel. In our scenario, the Z couples to the SM sector mostly through the mixing parameter and g x . Therefore, the on shell production rates of the Z are crucially dependent on the θ , which is also dependent on g x . For the gauge boson in the mass range of 200-500 GeV, constraints indicate θ 10 −3 which provides a significant limit to the production cross section of σ(p p → Z ). However the cleanliness of the dilepton channel along with the resonant production of Z still provides a significantly strong constraint on Z mass. 2 This bound can be relaxed if the Z decay to the charged lepton pair is suppressed, as shown in Fig. 4. The decay to a pair of heavy neutrinos opens up an interesting channel to search for Z in this model. In addition we find that the upper bound on the production cross section σ(p p → Z ) in this channel can be larger than what would be allowed in the absence of the Z → N N decay. 3 Thus we focus on the Z signal through the pair production of heavy neutrinos via Z resonance [64][65][66][67][68][69][70][71][72][73]. Notably the pair production of heavy neutral leptons has also been looked at in the context of seesaw 2 The small Z width allows the use of Narrow-Width Approximation (NWA) in calculating the di-lepton cross-section using σ × BR. 3 Our choice of parameter space gives six heavy neutrinos (ν k , k = 4, 5, . . . , 8,9) of which four are taken to be heavier than M Z . The lighter ones are nearly degenerate in mass, which we identify as N (ν 4 , ν 5 ∈ N ) in our analysis.
scenarios for neutrino mass [74][75][76][77] and some classes of U (1) X extensions with alternative charges to the more popular U (1) B−L [78,79]. The production of heavy Majorana neutrinos in the context of same-sign dilepton and multilepton searches have been carried out at LEP by DELPHI [80] and L3 [81,82] Collaborations as well as at the LHC by CMS [83,84] and ATLAS Collaborations [85]. The searches look for heavy neutral lepton singly produced through the Z boson at LEP and W boson at the LHC, which then decays to a charged lepton and W . This mode translates into an upper bound on the mixing parameter V N 2 between the light neutrinos (flavor ) and the heavy neutrino. Note that in our case we can parametrize the off-diagonal V N 2 ∼ F 2 as given in Eq. (37). As our m D ∝ v 2 and • 4 + / E T .
Although these are all interesting channels to look for Z in this model, especially the same-sign dilepton with jets and missing transverse energy (MET), we mainly focus on for the three benchmark points, respectively. All these three points satisfy the constraints discussed in the last section.
Before discussing each specific analysis, we would like to mention the public packages that we have employed to perform the analysis. The model was implemented in SARAH [92] to get the Universal Feynman Object (UFO) [93] files. SPheno [94,95] was used to generate the mass for the particle spectrum as well as the mixing parameters and mixing matrices connecting the gauge eigenstates to their mass eigenstates. The UFO model files were then used to calculate the scattering process with Madgraph and generate parton-level events with the MadEvent event generator using the package MadGraph5@aMCNLO (v2.6.7) [96,97] at the LHC with 14 TeV center-of-mass energy. These parton-level events were then showered with the help of Pythia 8 [98]. Detector effects were simulated using fast detector simulation in Delphes-3 [99] using the default ATLAS card. The final events were analyzed using the analysis package MadAnalysis5 [100] to present our results.
The 4 final state is a relatively background free and clean event sample to study at the LHC. Some model dependent analysis has been carried out by experiments at the LHC to look for such final states [101,102]. We have checked that these analyses do not add any further constraints on our choice of the benchmark points. The four-lepton final state in our case occurs when both the W and Z bosons coming from each N , decay leptonically. In the case of N → W we expect MET from the neutrinos coming from the W decay while the N decays directly to neutrinos in the Z channel. Although the branching ratios of leptonic decay modes of W and Z is much smaller compared to their hadronic decay modes, higher charged lepton multiplicity in the final states are known to provide a cleaner signal with smaller SM background at a hadron collider. Thus the backgrounds for multilepton final states are manageable to negligible sizes at a hadron machine. This is one of the primary motivations behind the study of a 4 final state at the LHC.
The major SM background for the 4 + / E T final state comes from the following subprocesses [88]: All SM backgrounds were generated using the same event generator as in the case of the signal. We then scale the background cross section with their respective k factors to make up for the next-to-next-to-leading-order (NNLO) corrections for ZZ and NLO corrections for ttZ and V V V backgrounds. The k factors are taken to be 1.72, 1.38, 2.01, and 2.27 for ZZ [103], ttZ [104], W Z [105], and V V V [106,107], respectively.
For our analysis, we choose events which have exactly N = 4 isolated charged leptons ( = e, µ) in the final state. As basic acceptance cuts, we demand that all reconstructed objects are isolated (∆R ab > 0.4). In addition, • All charged leptons must have p T > 10 GeV and lie within the rapidity gap satisfying |η | < 2.5.
• We impose additional conditions to demand a hadronically quite environment by putting veto on events with light jets and b jets with p T b/j > 30 GeV and |η b/j | < 2.5.
This helps in suppressing a significant part of the background coming from tt(Z) production.
• We also demand a veto on any photon in the final state with p γ T > 10 GeV and |η γ | < 2.5. We list the signal and background cross sections after the basic acceptance cuts on the charged leptons and the veto on additional light jets, b jets and photons in the final state in To improve the signal to background ratio, one needs to exploit the kinematics of the signal events against that off the SM background. To achieve that, we must look at kinematic distributions of some relevant variables. In Figs. 6 and 7, we plot area normalized distributions for some of these important kinematic variables after detector simulation. In the left panel of Fig. 6, we note that the p T distribution of the leading charged lepton peaks around the heavy neutrino. We also note that with higher mass difference one expects to get the peak at a higher value of p T for the signal. Thus a stronger p T cut on the leading lepton would help remove the SM backgrounds with leading leptons on the softer side compared to the signal. However, the charged leptons in the SM background originate from the Z and W bosons and also show a peak around p T ∼ M Z/W /2 leading to a significant overlap with that of the signal events of BP1 and to some extent with that of the remaining two BPs too. to a great extent without affecting the signal too much. The plot in the right panel of Fig. 6 supports this expectation. Note that as the particle spectrum is light and the corresponding decay products do not carry too much p T we put an upper bound of 200 GeV on the p T of the leading lepton and / E T which helps in suppressing some SM background. The effect of the aforementioned selection cuts are shown in Table VI. The invariant mass of e + e − and µ + µ − are shown in Fig. 7. We note that the signal events would not show a peak around the Z boson mass unless N decays via the (ν Z) mode. For the backgrounds, the invariant mass of OSSF leptons peak at the Z boson mass.
A large fraction of the signal events comes from N → e W decay mode. Thus an invariant mass cut on the OSSF leptons of electron type should be more useful in removing that background. However, as the p T e ± of the signal events are not very hard, we observe an overlap of the Z peak with the signal events in the M e + e − distribution. So a cut of Z peak in the e + e − mode does not help a lot in improving the signal to background ratio.
On the other hand, we expect that the fraction of events for the signal that contain at least a µ + µ − pair will be much smaller [∼ (28 − 31)% for the 3 BPs] when compared to the full 4 mode (as evident from the branching fractions of N and Z). In contrast, the background is expected to be equally divided in the e and µ modes. So although the normalized distribution in M µ + µ − distribution shows a significant part of the signal in the mass bin of Z peak, we must realize that the distribution only corresponds to a very small fraction of the 4 + / E T events after cuts. Therefore a cut to remove the Z peak in the µ + µ − distribution (80 < M µ + µ − < 95 GeV) 4 helps in suppressing a significant part of the SM L = 100 fb −1

SM-background Signal
Cuts  background and improves the signal significance. To facilitate this we also demand that the four-lepton final state signal has at most a single pair of µ + µ − .
The result of the analysis and the respective selection cuts are presented in Table VI for an integrated luminosity of L = 100 fb −1 at the 14 TeV LHC.
We calculate the signal significance (S) by using the following formula.
where S and B are number of signal and background events, respectively. The signal significance for these three benchmark points are provided in the last column of We now focus on the final state with a larger production rate as compared to the 4 final state, viz. the 3 + 2j + / E T signal at the LHC [83][84][85]. However this channel has little advantage over the 4 mode since the background events also become larger in this channel.
The main SM background comes from the following subprocesses [88]: As before, we include k factors for the LO cross section for the SM background to account for the NNLO correction for W Z and tt and the NLO correction for V V V and ttZ backgrounds.
The k factor is 1.6 for tt [108].
The object reconstruction to identify the final state particles is similar to what was done for the 4 + / E T final state. The basic acceptance cuts considered for the 3 + 2j + / E T signal are that all reconstructed objects are isolated (∆R ab > 0.4) and satisfy the following requirements.
• We have exactly three charged leptons, N = 3 ( = e, µ) in the final state, each with p T > 10 GeV and lying within the rapidity gap |η | < 2.5.
• We have exactly two light jets, N j = 2 in the final state, each with p T j > 30 GeV and lying within the rapidity gap |η j | < 2.5.
• We impose veto on events with a b jet having p T b > 30 GeV and |η b | < 2.5. This again helps in suppressing a significant part of the background coming from tt(Z) production.
• We also demand a veto on any photon in the final state with p γ T > 10 GeV and |η γ | < 2.5.
We list the signal and background cross sections after the basic acceptance cuts on the charged leptons, jets, and a veto on any b jet and photons in the final state in Table VII As we note that the signal is rich in e ± and the µ multiplicity peaks at one, it again seems beneficial to put a constraint on N µ ≤ 1 which should not affect the signal too much while suppressing the SM background. This can be seen from the cut-flow numbers presented in However, we still note that the / E T > 15 GeV cut will suppress the ZZ background as seen in p T of the leading lepton and / E T to suppress the SM background which has a longer tail in the distributions extending beyond 200 GeV.
In Fig. 9 we plot the p T of the leading jet and the invariant mass distribution in e + e − .
As the jets for the signal are not expected to be hard, we put an upper bound on them as p T j 1 < 200 GeV. The dominant suppression in the background comes from the invariant mass cut where we remove the Z peak. As we expect the electron or positron (e) to come from the decay of N for the signal, we expect no Z peak in the signal. Thus the invariant mass cut along with the constraint on µ multiplicity proves to be the most important condition that improve the S/B for the 3 + 2j + / E T final state.
The result of the analysis and the respective selection cuts are presented in Table VIII for an integrated luminosity of L = 100 fb −1 at the 14 TeV LHC. We can see that, as in the case of 4 + / E T , the signal for BP1 and BP2 again has quite large significance, albeit slightly smaller for the same integrated luminosity. The above analysis however shows that both the 3 and 4 final states show a promising discovery channel for light Z which does couple to the SM particles directly, with the higher lepton multiplicity case doing slightly better. The analysis can be extended to include heavier Z as well and consider the other final states available for the Z , which would be similar to the more traditional Z searches such as the U (1) B−L models for example [64,65].

V. SUMMARY AND OUTLOOK
We consider a neutrinophilic model as an extension of the SM by introducing a U To highlight the features of the model, we calculate the mass and mixing of the scalar, gauge and matter fields after symmetry breaking and look at the experimental constraints on the model parameters. We find that once the scalar sector is set to agree with the Higgs searches, by choosing the lightest CP -even scalar to be the 125 GeV SM Higgs boson, the Z phenomenology is only dependent on the Z-Z mixing and its coupling to the heavy neutral fermions. Following an examination of the allowed region for the mixing angle and the U (1) X gauge coupling we determine two regions of parameter space depending upon the value of tan β, the ratio of the doublet VEVs. For tan β > 1 we find an upper bound on the ratio v 2 /v 1 < 3 from the perturbativity requirement on the fermion-fermion scalar couplings. We also observe that g x and g x are of the same order when tan β > 1, which gives us a Z phenomenology driven by the Z-Z mixing angle sin θ with the dominant decay to SM fermion pair. A more interesting scenario emerges for tan β < 1 where the g x and g x are no longer required to be of the same order anymore. We find that the Z signatures are now dependent on the interplay of the Z-Z mixing as well as the U (1) X gauge coupling g x which is allowed to be large. Thus the Z can now decay dominantly to a pair of heavy neutrinos while the Z is produced through the Z-Z mixing parameter driven by g x . We analyze the signal for such a scenario at the LHC with √ s = 14 TeV in the 4 + / E T and 3 +2j+ / E T channels for a Z lying in the mass range 200-500 GeV. We find that although the dilepton Drell-Yan channel is much suppressed here, the discovery prospects of observing a neutrinophilic Z is significantly high in the above channels. We show the significance of the signal using an integrated luminosity of 100 fb −1 for three benchmark points. We conclude that multilepton final states could be crucial in discovering such a neutrinophilic gauge boson lying in the mass range of 200-500 GeV with even a very tiny gauge-kinetic mixing of the order O(10 −3 ).
We must point out here that other interesting signatures of the Z in such a model is being left for future work, which include flavor violating decays of the Z , a more detailed analysis of the scalar sector with the Z and implications of a very light Z , and a singlet scalar [109].

APPENDIX: COUPLING OF Z GAUGE BOSON WITH FERMIONS
Below, we list the coupling of the Z gauge boson with the fermions in the model. We define s W ≡ sin θ W and c W ≡ cos θ W where θ W is the Weinberg angle while s θ ≡ sin θ and c θ ≡ cos θ where θ is the Z-Z mixing angle. In addition, T 3 and Q f represent the isospin and electric charge of the fermions, respectively, while P L/R = 1∓γ 5 2 are the projection operators.
where N is the neutrino mixing matrix as defined in Eq. (36). We note that ν i for i = 1, 2, 3 are identified as the light neutrinos and rest are heavy neutrinos. These neutrinos are Majorana fermions written in four-component notation.