Displaced vertex signature of type-I seesaw

A certain class of new physics models includes long-lived particles which are singlet under the Standard Model (SM) gauge group.A displaced vertex is a spectacular signature to probe such particles productions at the high energy colliders, with a negligible SM background. In the context of the minimal gauged $B-L$ extended SM, we consider a pair creation of Majorana right-handed neutrinos (RHNs) at the high energy colliders through the production of the SM and the $B-L$ Higgs bosons and their subsequent decays into RHNs. With parameters reproducing the neutrino oscillation data, we show that the RHNs are long-lived and their displaced vertex signature can be observed at the next generation displaced vertex search experiments, such as the HL-LHC, the MATHUSLA, the LHeC, and the FCC-eh.We find that the lifetime of the RHNs is controlled by the lightest light neutrino mass, which leads to a correlation between the displaced vertex search and the search limit of the future neutrinoless double beta-decay experiments.


Introduction
The last missing piece of the Standard Model (SM) is finally supplemented with the discovery of the Higgs boson at the Large Hadron Collider (LHC) in 2012 [1,2], which itself is not that surprising given the tremendous success of the SM to explain observed elementary particle phenomena. However, the SM is not complete in its current form, because, for example, the neutrinos are massless in the framework which is not consistent with the experimental evidence of the neutrino oscillation phenomena [3], indicating that the neutrinos have tiny non-zero masses and flavor mixings. Hence, we need to extend the current framework of the SM.
Unfortunately, ever since the discovery of the SM Higgs boson, no new signature of new physics beyond the SM has been observed. It may indicate that the current energy and luminosity of the LHC are not sufficient to directly probe new particles. If so, we can just hope for new particle signals to be observed at the future LHC after the planned upgrade, or at a future collider with energies higher than the LHC. However, there is another possibility: if new particles are completely singlet under the SM gauge group, it can naturally explain the null search results at the LHC because SM singlet particles cannot be directly produced at the LHC through the SM interactions. Such particles may be produced through new interactions and/or rare decay of the SM particles. At a first glance, it seems that such a scenario is even more challenging to test. However, if a new particle is long-lived, it can leave a displaced vertex signature at the collider experiments. Since the displaced vertex signatures are generally very clean, they allow us to search for such a particle with only a few events at the LHC or future colliders. For the current status of displaced vertex searches at the LHC, see, for example, Refs. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. The search reach will be dramatically improved in the future planned/proposed experiments, such as the High-Luminosity LHC (HL-LHC), the MATHUSLA [22], the Large Hadron electron Collider (LHeC) [23] and the the Future Circular electron-hadron Colliders (FCC-eh) [24].
In this paper, we first review the current theoretical and experimental studies focusing on the displaced vertex searches at the future collider experiments and express their results in a model independent form. As a concrete example, we consider a well-motivated simple extension of the SM, namely the minimal B − L (baryon minus lepton number) model [25][26][27][28][29][30], where the global B − L symmetry in the SM is promoted to the gauge symmetry. A minimal B −L model includes an additional electrically neutral gauge boson (Z boson) as well as a B − L Higgs boson which breaks the B − L symmetry. In addition, the model also includes three righthanded neutrinos (RHNs) to cancel all the gauge and mixed-gravitational anomalies. After the B − L symmetry breaking, the RHNs acquire Majorana masses, and the tiny neutrinos masses are automatically generated through the so-called type-I seesaw mechanism [31][32][33][34][35][36] after the electro-weak symmetry breaking. In this model context, we investigate the displaced vertex signature of the Majorana RHNs at the future high energy colliders through the production of the SM and the B − L Higgs bosons and their subsequent decays into RHNs. 4 Since the Majorana RHNs decay into the SM particles through small light-heavy neutrino mixings from type-I seesaw, the RHNs are likely to be long-lived. 5 4 The collider signatures pertaining to the RHNs pair production through the production of the Z boson and the Higgs bosons and their subsequent decays into RHNs have been studied before in the literature [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54]. 5 One can also consider a displaced vertex signature from RHN decay in type-III seesaw scenario [55].
This paper is organized as follows. In Sec. 2, we review the prospect of the search reach of displaced vertex signatures at the future high energy colliders. Employing the (2σ) search reach of the displaced vertex signatures obtained in various analysis, we present a model-independent formula for the search reach in terms of the production cross section of a long-lived particle as a function of its lifetime, mass and its mother particle mass whose decay products are the long-lived particle. In Sec. 3, we give a review on the minimal B − L extended SM. In Sec. 4, we consider the pair production of RHNs through the production of the Higgs bosons and their subsequent decays into RHNs. We apply the best reach values for the production cross section obtained in Sec. 2 to the RHNs production, assuming a suitable lifetime of the RHNs. For benchmark mass values of the B − L Higgs boson and RHNs, we determine the corresponding parameter space for RHNs Majorana Yukawa couplings and a mixing between the SM and the B − L Higgs bosons. In Sec. 5, we calculate the lifetime of the RHNs for realistic parameters to reproduce the neutrino oscillation data. Using this realistic value for the lifetime, we repeat the analysis in Sec. 4 to determine the parameter space corresponding to the search reach. In Sec. 6 we discuss the correlation between the displaced vertex search and the search limit of the future neutrinoless double beta-decay experiments. Sec. 7 is devoted to conclusions.
2 Displaced vertex search at the future colliders An electrically neutral particle with a sufficiently long lifetime (for example, its decay length is of O(1 mm) or larger), once produced at the colliders, displays a signature of the displaced vertex, namely the vertex created by the decay of the particles is located away from the collision point where the particle is produced. The final state charged leptons and/or jets from a displaced vertex can be reconstructed by a dedicated displaced vertex analysis. Since displaced vertex signatures from SM particles are very well understood, the signature from a new long-lived particle can be easily distinguished, making it a powerful probe to discover such particles.
Let us first review the search reach of displaced vertex signatures at the future colliders which have been investigated in Refs. [22,56,57]. In Refs. [22,56], the authors have proposed the MATHUSLA detector which is specifically designed to explore the long lifetime frontier; the plan is to build a detector on the ground, about 100 m away from the HL-LHC detector. The authors have also considered displaced vertex using the inner-detector of the HL-LHC. Similarly, in the Ref. [57], the authors have studied the prospect of a dedicated displaced vertex search at the future electron-proton colliders, such as the LHeC and the FCC-eh. In their analysis, they consider a pair production of a long-lived particle "X" created from the rare decay of the SM Higgs boson. They have shown the search reach for the branching ratio of the SM Higgs boson to a pair of X particles as a function of X particle's decay length (cτ ) ranging from sub-millimeter to 10 7 m.
We first summarize the results in Refs. [22,56,57] in Fig. 1. Here, for a fixed mass of the X particle (m X = 20 GeV), we show the search reach for the X particle pair production cross section (σ XX ) at the future colliders as a function of the lifetime of X particle cτ . The dashed and solid lines show the search reach for the displaced vertex signatures at the HL-LHC and the MATHUSLA experiments, respectively, with a 3 ab −1 luminosity. The dotted lines (almost degenerate) correspond to the discovery reach at the FCC-eh with a 3 ab −1 with electron beam Figure 1: The plot shows the discovery reach for a dedicated displaced vertex searches at the future experiments and newly proposed extensions to the current LHC experiment. The lines corresponds to the total production cross section to produce a pair of "X" particles in the final states as a function of the X particle decay length, where the mass of the X particle and its mother-particle are fixed to be 20 GeV and 125 GeV, respectively. The region above the dashed (solid) line corresponds to the search reach at the HL-LHC and the MATHUSLA experiements. The dotted curved lines correspond to the search reach of the various proposed electron-proton collider upgrade of HL-LHC. energies of 60 GeV (top) and the LHeC with a 1 ab −1 (bottom).
In Fig. 1, the process pp/ep → h → XX is considered. We generalize the process to pp/ep → S → XX, where the mother-particle (S) is a boson, but not the SM Higgs boson, with a mass m S . In order to make the results in Fig. 1 to applicable to this general case, note that the search reach shown in Fig. 1 is model-independent if the curves are plotted as a function of the lifetime of X in the laboratory frame. For the process pp/ep → h → XX, the lifetime of the particle X in the laboratory frame (τ ) is given by its proper lifetime (τ ) as because of the Lorentz boost. We then express the search reach of the cross sections in Fig. 1 as The model-independent search reach can be obtained as a function of cτ for the fixed values of m h = 125 GeV and m X = 20 GeV. Now it is easy to convert the search reach results in Fig. 1 to our general case: where σ XX (cτ ) represents different curves shown in Fig. 1. Hence, depending on the choice of masses for m S and m X , the curves shown in Fig. 1 shifts either to the left or to the right. 2). Here, we have fixed m X = 20 GeV. As we raise/lower m S for m X = 20 GeV, the line shifts to the left/right, since the created X particle is more/less boosted. In the following, we employ the generalized formula to investigate the search reach of long-lived RHNs at the future high energy colliders.

The minimal B − L extended Standard Model
Here we review the minimal B − L extended SM (the minimal B − L model). The particle content of the model is listed in Table 1. In this model, the global B − L symmetry in the SM is gauged, and in addition to the SM particle content, three RHNs and a complex scalar (B − L Higgs field) are introduced. While the B − L Higgs field spontaneously breaks the B − L symmetry by its vacuum expectation value (VEV), the three RHNs are necessary to cancel all the gauge and mixed-gravitational anomalies. The Yukawa sector of the SM is extended to include where the first and second terms are the Dirac and Majorana Yukawa couplings. Here, we where v BL = √ 2 ϕ is the VEV of the B − L Higgs field. A renormalizable scalar potential for the B − L Higgs field (ϕ) and the SM Higgs doublet (H) is given by where v SM = 246 GeV is the VEV of the SM Higgs doublet, and we take λ > 0 which introduces a mixing between the two scalar fields. In the unitary gauge, we expand the SM and B − L Higgs fields around their VEVs, H = ( v SM √ 2 0) T and ϕ = v BL / √ 2, to identify φ SM and φ BL being the SM and the B − L Higgs bosons in the original basis. The mass matrix for the Higgs bosons is given by where m H = √ 2λ H v SM , and m 2 ϕ = 2λv 2 BL . We diagonalize the mass matrix by where h and φ are the mass eigenstates. The relations among the mass parameters and the mixing angle (θ) are the following:

Displaced vertex signature of heavy neutrinos
Let us now consider the displaced vertex signatures of the heavy neutrinos N i in the minimal B − L model. 6 . For the main process for a pair production of N i , we can consider two cases: one is through the Z boson production [?, ?, ?] and its decay to N i s, and the other is through the production of the Higgs bosons (h and φ) and their decays. 7 The LHC results on the search for the Z boson resonance of the minimal B − L model severely constrain the B − L gauge coupling to be very small (see, for example, Ref. [62]), so that the heavy neutrino production cross section from the Z boson decay is expected to be small. Hence, we focus on the heavy neutrino production through the Higgs bosons in this paper 8 .
Because of its representation under the gauge groups, φ BL has no tree level coupling with the SM particles at the renormalizable level, and hence it cannot be directly produced at the high colliders. However, as described in Eqs. (3.5) and (3.6), the mass eigenstates are mixture of φ BL and φ SM and through the mixing, the Higgs boson φ can be produced through the same process as the SM Higgs boson. At the LHC, among a variety of the production processes of the SM Higgs boson, such as gluon-gluon fusion (ggF), vector boson fusion (VBF), and the productions associated with W/Z bosons (V h) and with tt (tth), the ggF channel dominates the production cross section. 9 For a small mixing, the production cross section of the SM-like Higgs boson (h) is given by where σ h (m h ) is the SM Higgs boson production cross section with the Higgs boson mass of m h = 125 GeV. In the limit of θ → 0, σ(pp → h) reduces to the SM case. Similarly, the production cross section for the B − L-like Higgs boson (φ) is expressed as where σ h (m φ ) is the SM Higgs boson production cross section if the Higgs boson mass were m φ As discussed before, we are interested in a small mixing, and hence for the remainder of this paper, we shall simply refer to the mass eigenstates h and φ as the SM(-like) Higgs boson and the B − L Higgs, respectively. Using Eq. (4.2), we show in Fig. 4 the contour plot for the production cross section of φ at the 13 TeV LHC (σ(pp → φ)) in the (m φ , sin 2 θ)-plane. Here, we also show the constraint obtained by the LEP experiments on the search for the SM Higgs boson through its production associated with the Z boson [65]. For a given Higgs boson mass, no evidence of the SM Higgs boson production sets an upper bound on the Z-boson associate production cross section, which is interpreted to be an upper bound on the anomalous SM Higgs coupling with Z boson as a function of the Higgs boson mass. In our model, this upper bound is interpreted as an upper bound on sin θ as a function of m φ , which is shown as the dashed curve in Fig. 4. The gray shaded region in the figure is excluded by the current measurement of the SM Higgs boson properties by the ATLAS and CMS collaborations. The Higgs boson properties are characterized by the signal strengths defined as where σ i are the Higgs boson production cross sections through i-channel with i = ggF, V BF, W h, Zh, tth, and BR f are the Higgs boson branching ratios into final states f = ZZ , W W , γγ, τ + τ − , bb, µ + µ − . The cross sections and branching ratios with the subscript "SM " denote the SM predictions. The latest updates of the signal strengths from the ATLAS and the CMS experiments at the LHC Run II with a 37 fb −1 luminosity are listed in Refs. [64,66], along the authors have considered MATHUSLA, FASER and CODEX-b collider experiments 9 At the 13 TeV LHC, the SM Higgs boson with mass of around 125 GeV, the Higgs boson production cross sections through these channels are evaluated as [63,64]  with the references. The averaged signal strength is obtained as µ 0.925 ± 0.134, which is consistent with the SM prediction µ = 1. In our model, the couplings of the SM-like Higgs boson is modified by a factor cos θ from the SM case. Hence, the production cross section of h is scaled by cos 2 θ while its branching ratios remain the same as the SM one. Therefore, the signal strengths are given by µ i f = cos 2 θ × µ f . In anticipation of the analysis presented below, we also consider the constraint from the invisible SM Higgs decay modes, and the current upper bound on the branching ratio of the Higgs invisible decay mode is given by BR higgs inv < 0.23 [67]. Together with the averaged signal strength, we obtain an upper bound on the mixing angle as sin 2 θ ≤ 0.12 at 95% C.L and the excluded region is depicted by the gray shaded region in Fig.4. Hence, for the entire range of m φ shown in the figure, the mixing angle is small.
We now consider the decay of the Higgs bosons. In the presence of the mixing, the total decay width of the SM-like Higgs and the B − L Higgs into the SM final states are given by respectively, where Γ SM (M ) is the total decay width of the SM Higgs boson if the SM Higgs mass were M . The list of the partial decay widths of the SM-like Higgs boson is given in the Appendix. In Fig. 5 we show the total decay width Γ SM (m H ) as a function m H . The partial decay widths of φ and h into a pair of heavy neutrinos are given by respectively, where Γ N N (M ) is the total decay width of φ BL with a mss M into heavy neutrinos in the limit of θ → 0, which is given by Here, we have considered only one RHN with a Majorana Yukawa coupling Y and its mass m N , for simplicity. If m φ > 2m h , φ can also decay into a pair of SM-like Higgs bosons, and the partial decay width of this process is expressed as where For a small mixing angle θ 0.1 and v SM < v BL , C φhh can be approximated as where we have used Eq. (3.6). Hence, Γ(φ → hh) is determined by m φ and θ. Using the decay widths of φ and h, the branching ratio of φ and h into a pair of RHNs are given by respectively. Let us now consider the production cross section for the RHNs at the LHC from the φ and h productions and their decays. Using Eqs. (4.1), (4.2) and (4.10), the cross section formulas are given by respectively, and they are controlled by four parameters, Y , θ, m φ and m N . Throughout this section, we fix m N = 20 GeV, for simplicity. The representative diagrams of the RHN productions including their decays are shown in Fig. 6. We will discuss the decay of RHNs into the SM final states in details in Sec. 5. In the remainder of the analysis in this section, we fix the lifetime of RHNs to yield the best reach of σ XX in Fig. 1 for both the future HL-LHC and MATHUSLA displaced vertex searches, namely, σ min (HL − LHC) = 20.7 and σ min (MATH) = 0.3 fb, which corresponds to cτ = 3.1 and 58.4 m, respectively. Here, we identify X with the RHN while S is either h or φ.
We first consider the case where h and φ masses are almost degenerate, m h m φ = 126 GeV. In this case, the total cross section σ XX is given by the sum of the productions from φ and h. 10 The best search reach of the displaced vertex signatures at the HL-LHC or the MATHUSLA are expressed as Note that along the dashed curve, the RHN pair production is dominated by the decay of h (φ) for sin θ < 0.02 (sin θ > 0.02). Similarly, the RHN pair production is dominated by the decay of h (φ) for sin θ < 0.002 (sin θ > 0.002) along the solid curve. In our model, once Y and m N are fixed, the relation between the B − L gauge coupling and the Z boson mass is determined by (see Eq. (3.2)) (4.14) The Z boson has been searched by various experiments, and the upper bound on the B − L gauge coupling as a function of its mass is obtained for a wide mass range of O(1) m Z [GeV] ≤ 5000. For a Y value chosen in Fig. 7, we examine the consistency with the current constraints from the Z boson search. For several benchmark Y values, we show in Fig. 8 the relation of Eq. (4.14) along with the current experimental constraints from the Z boson searches (the shaded regions are excluded from the result in Ref. [68], the LHCb results [69,70], and the resent ATLAS results [71]). In the left panel, we show the relation for Y = 0.0181 (dotted),  Fig. 7. From the results, we see that if the RHN pair production is dominated by the SM-like Higgs boson decay, the allowed parameter space is very severely constrained except for a window around m Z = 100 GeV. However, Fig. 8 also shows that we can avoid the constraints by lowering g BL .
Let us next consider the case that m φ and m h are well separated. In this case, we calculate the RHN pair productions from the decays of h and φ separately. For each case, we assume the lifetime of the RHN to yield the best search reach at the HL-LHC or the MATHUSLA shown in Fig. 1. The best search reach of the displaced vertex signatures at the HL-LHC or the MATHUSLA are expressed as σ min = σ(pp → S → N N ), where S is either h or φ. Once m φ is fixed, this equation leads to a relation between Y and θ as shown in Fig. 7 for m φ = 126 GeV. Using Eqs. (2.2), (4.10) and (4.11), we express σ min = σ(pp → S → N N ) for S = φ and h, separately, as where R is defined as .
The first equation in Eq. (4.17) indicates that for a fixed R(m φ ) < 1, Y 2 becomes singular for sin 2 θ = R(m φ ). Thus, there is a lower bound on sin θ to achieve the best reach cross section σ min . For sin θ ∼ 1, Y 2 becomes singular in both the equations. However, such a large mixing angle is excluded by the measurement of the SM Higgs boson properties at the LHC. In Fig. 9, we show our results for the case that the RHNs are produced from the SM-like Higgs boson decay. Here, we have fixed m φ = 70 GeV and m N = 20 GeV. In the left panel, the best reach cross section at the HL-LHC (MATHUSLA) is achieved along the dashed (solid) diagonal line with a negative slope. The four solid diagonal lines with positive slopes denote the relations between Y and sin θ to yield BR(φ → N N ) = 99.99%, 98%, 75%, and 25%, respectively, from top to bottom. The gray shaded region is excluded by the LHC constraint on the Higgs branching ratio into the invisible decay mode [67], which is simply given by for the present case. The right panel corresponds to Fig. 8. The three diagonal lines corresponds Y 1.90 × 10 −2 (dotted), Y 1.00 × 10 −2 (dashed) and 3.59 × 10 −3 (solid), respectively, which are chosen from the intersections of the diagonal line for BR(φ → N N ) = 98% with the solid, dashed and dotted lines with negative slopes in the left panel. Fig. 10 shows the results corresponding to Fig. 8, but for the case that the RHNs are produced from the B − L Higgs boson decay, with m φ = 70 GeV and m N = 20 GeV. Along the dashed and solid curves, the best reach cross section is achieved at the HL-LHC and the MATHUSLA, respectively. The four solid diagonal lines with positive slopes denote the relations between Y and sin θ to yield BR(φ → N N ) = 99.99%, 98%, 75%, and 25%, respectively, from top to bottom. The gray shaded region is excluded region by the SM Higgs boson invisible decays search. As we have discussed, the curves show the singularities for small sin θ values. In the right panel, the three diagonal lines corresponds Y Let us here summarize our results as m φ is increased. From the left panel in Fig. 10, the bottom-left panels in Fig. 11 and Fig. 12, we can see that the resultant curves and the diagonal lines are shifting upward to the right as m φ is increased. This is because (i) as m φ is increased, the partial decay widths of φ into the SM particles become larger and as a result, Y is increased to yield a fixed branching ratio to φ → N N , see Eq. 4.17; (ii) since σ h (m φ ) is decreasing as m φ is increased, the lower bound on sin θ (at which Y becomes singular) is increasing (see the discussion below Eq. (4.17). Hence, the LHC constraint from the invisible decay of the SM Higgs boson relatively becomes more severe. In fact, the bottom-left panel of Fig. 12 shows that the dashed curve appears inside the gray shaded region and thus the entire parameter region which can be explored at the future HL-LHC is already excluded. According to Eq. 4.14, the B − L gauge coupling becomes larger for a fixed m Z as Y becomes larger. Hence, the current constraints from the Z boson search become more severe as m φ is increased as can be seen from the right panel in Fig. 10, the bottom-right panel in Fig. 11 and the bottom-right panel in Fig. 12. Note that if we take g BL small enough, for example, g BL < 10 −4 , all the existing collider constraints from the Z boson search can be avoided.
We conclude this section by generalizing our analysis for the long-lived heavy neutrino to the case for an SM-singlet particle X produced through pp → S → XX with an SM-singlet scalar S at the LHC. Since S is SM-singlet, it is produced at the LHC through a mixing with the SM Higgs boson, just like φ. Hence, the total production cross section of the process, pp → S → XX, is given by where θ is the mixing angle, and σ h (m S ) is the production cross section of the SM Higgs boson if the SM Higgs boson mass were the mass of S (m S ). Let us assume the lifetime of X to 100%, the red shaded region is excluded by the LEP results on the search for the SM Higgs boson, while the gray shaded region is excluded by the LHC measurement of the SM Higgs boson properties. Assuming BR(S → XX) 100%, we can read off the search reach of m S from Fig. 13 for a fixed value of sin θ. Our results for three benchmark points are listed in Table 2.

Lifetime of heavy neutrinos
We assumed a suitable lifetime of the heavy neutrino in the previous section. In this section, we calculate the lifetime of the heavy neutrinos for realistic parameters to reproduce the neutrino oscillation data and investigate the prospect of searching for the displaced vertex signatures of the heavy neutrino productions. After the B −L and electroweak symmetry breakings, the neutrino mass matrix is generated to be . The charged current interaction of the neutrino mass eigenstates is expressed as where α are the 3 generations of the charged SM leptons, and P L = (1 − γ 5 )/2 is the left handed projection operator. Similarly, for the the neutral current interaction, we have where θ W is the weak mixing angle.
Let us now consider the decay of the heavy neutrinos into the SM particles. In our analysis, we set m N = 20 GeV, hence the heavy neutrino decays into the SM quarks and leptons via intermediate off-shell W and Z bosons. The expression for the decay width of heavy neutrinos into various final states are as follows: 1. Leptons in the final states: with the Fermi constant G F , U αβ M N S is a (α, β)-element of the neutrino mixing matrix, and β,κ |U βκ M N S | 2 = 3 = β,κ δ βκ . In deriving the above formulas, we have neglected all lepton masses. For a degenerate heavy neutrino mass spectrum, we obtain 3 α=1 |R αi | 2 = m i m N . For the lepton final states, we have an interference between the Z and W boson mediated decay processes: 2. Quarks in the final states: where N c = 3 is the color factor. Since we have set m N = 20 and 40 GeV in the following analysis, we only consider the first two generation of quarks in the final states and β,κ |V q βqκ CKM | 2 = 2. For the final state quarks, there is no interference between W and Z boson mediated processes.  The heavy neutrino lifetimes become shorter as m lightest is raised (see Eq. (5.13)). However, if we consider the cosmological bound on the sum of the light neutrino masses, m lightest = 0.23 MeV [75], we obtain the lower bound on the decay length to be cτ 20 m. In fact, this lower bound can be significantly reduced if we consider the complex orthogonal matrix O in the general parametrization. See, for example, Ref. [76].
In the previous section, we have shown the relation between Y and sin θ to achieve the best reach cross section at the HL-LHC/MATHUSLA, assuming the heavy neutrino lifetime to be the best point for each experiment. To conclude this section, we repeat the same analysis in the previous section but for various values of cτ determined by m lightest values. For this analysis, we set m N = 40 GeV and m φ = 150 GeV. The decay lengths of heavy neutrinos for this parameter choice are depicted in Fig. 17. In this case, the cosmological lower bound on cτ is found to be cτ 1.3 m.
In Fig. 18, we show our results corresponding to the top-left and bottom-left panels in Fig. 11. The left panel corresponds to the the top-left panel of Fig. 11. Here, we consider the The diagonal dashed lines from left to right are results correspond to m lightest = 8.90 × 10 −2 , 10 −2 , 5.00 × 10 −3 and 10 −3 eV, or equivalently cτ = 3.10, 27.2, 54.0 and 273 m. The solid diagonal lines denote the relations between Y and sin θ to yield BR(φ → N N ) = 99.99%, 98%, 75%, and 25%, respectively, from top to bottom. The gray shaded region is excluded by the LHC constraint on the invisible Higgs decay. The right panel corresponds to the the bottom-left panel of Fig. 11. Here, we consider the heavy neutrino production from the B − L Higgs decay. The line coding are the same as the left panel. The parameter region for m lightest 10 −3 eV is already excluded.

Complementarity to neutrinoless double beta decay search
The neutrinoless double beta decay of heavy nuclei is a "smoking-gun" signature of the Majorana nature of neutrinos. In this process, two neutrons in nuclei simultaneously decay into two protons plus two electrons without emitting neutrinos, and hence the lepton number is violated by two units. The neutrinoless double beta decay process is characterized by an effective mass where U ej is the (e, j)-element of the neutrino mixing matrix U MNS , see [77] for review of neutrinoless double beta decay. Employing the current neutrino oscillation data, the effective mass is described by the lightest light neutrino mass. The left panel in Fig. 20 depicts the relation between m ββ and m lightest for the NH (red shaded region) and IH (green shaded region) cases. In this panel, the current upper limit by the EXO-200 experiment (upper horizontal shaded region) and the future reach by the EXO-200 phase-II and nEXO experiments (lower horizontal shaded region) are also shown. As we have investigated in the previous section, the decay lengths of the heavy neutrinos are controlled by m lightest . In the right panel of Fig. 20, we show the decay length of the heavy neutrino N 3 for the NH case as a function of m lightest . The dashed and solid lines correspond to the heavy neutrino masses of m N = 20 and 40 GeV, respectively. In Fig. 20, our benchmarks of m lightest = 0.1, 0.01 and 0.001 are depicted by the vertical lines. It is interesting to compare the two panels. If the displaced vertex from a heavy neutrino decay is observed and the heavy neutrino mass is reconstructed from its decay products, m lightest is determined. If the neutrinoless double beta decay is observed, m ββ is measured. However, if m ββ is measured to be around 0.03 eV or 0.002 eV, m lightest is left undetermined. Hence, the observations of the displaced vertex and the neutrinoless double beta decay are complementary with each other.  [77]. In the right panel, the solid line depicts the total decay length of RHN plotted against the mass of the corresponding lightest light-neutrino mass. The dashed (solid) line correspond to fixed RHN mass of 20 (40) GeV. In both the panels, vertical solid lines correspond to the three benchmark points for the lightest neutrino masses for the NH and the IH, namely, m lightest = 0.1, 0.01, and 0.001 eV.

Conclusions
It is quite possible that new particles in new physics beyond the SM are completely singlet under the SM gauge group. This is, at least, consistent with the null results on the search for new physics at the LHC. If this is the case, we may expect that such particles very weakly couple with the SM particles and thus have a long lifetime. Such particles, once produced at the high energy colliders, provide us with the displaced vertex signature, which is very clean with negligible SM background. In the context of the minimal gauged B − L extended SM, we have considered the prospect of searching for the heavy neutrinos of the type-I seesaw mechanism at the future high energy colliders. For the production process of the heavy neutrinos, we have investigated the production of Higgs bosons and their subsequent decays into a pari of heavy neutrinos. With the parameters reproducing the neutrino oscillation data, we have shown that the heavy neutrinos are long-lived and their displaced vertex signatures can be observed at the next generation displaced vertex search experiments, such as the HL-LHC, the MATHUSLA, the LHeC, and the FCC-eh. We have found that the lifetime of the heavy neutrinos is controlled by the lightest light neutrino mass, which leads to a correlation between the displaced vertex search and the search limit of the future neutrinoless double beta-decay experiments.
Note added: While completing this manuscript, we noticed a paper by F. Deppisch, W. Liu and M. Mitra [78] which also considers the displaced vertex signature of the heavy neutrinos.