CP Violating Effects in Heavy Neutrino Oscillations: Implications for Colliders and Leptogenesis

Two of the important implications of the seesaw mechanism are: (i) a simple way to understand the small neutrino masses, and (ii) the origin of matter-anti-matter asymmetry in the universe via the leptogenesis mechanism. For TeV-scale seesaw models, successful leptogenesis requires that the right-handed neutrinos (RHNs) must be quasi-degenerate and if they have CP violating phases, they also contribute to the CP asymmetry. We investigate this in the TeV-scale left-right models for seesaw and point out a way to probe the quasi-degeneracy possibility with CP violating mixings for RHNs in hadron colliders using simple observables constructed out of same-sign dilepton charge asymmetry (SSCA). In particular, we isolate the parameter regions of the model, where the viability of leptogenesis can be tested using the SSCA at the Large Hadron Collider, as well as future 27 TeV and 100 TeV hadron colliders. We also independently confirm an earlier result that there is a generic lower bound on the $W_R$ mass of about 10 TeV for leptogenesis to work.


Introduction
Neutrino oscillation experiments have established that neutrinos have very small but nonzero masses. This begs for some new physics beyond the Standard Model (SM), since the SM predicts the neutrinos to be exactly massless to all orders in perturbation theory. While the nature of the underlying new physics is far from clear, a simple paradigm that provides a natural way to understand tiny neutrino masses is the seesaw mechanism [1][2][3][4][5]. The two key ingredients of the so-called type-I seesaw mechanism are the existence of right-handed neutrinos (RHNs) and their Majorana masses. In order to calculate the light neutrino masses using the seesaw formula, we need the Dirac Yukawa coupling matrix and the Majorana mass matrix of the RHNs. Understanding both the Dirac mass matrix and the RHN Majorana mass matrix is therefore key to understanding the origin of neutrino masses. Another important implication of seesaw paradigm is that it provides a natural framework for understanding the origin of matter-antimatter asymmetry via the mechanism of leptogenesis [6]. Typical scenarios considered are those with high-scale seesaw [7]. However if the seesaw scale is in the TeV range, successful leptogenesis requires that there must be at least two RHNs which are quasi-degenerate [8]. We investigate the collider signatures and leptogenesis in a TeV-scale left-right embedding of seesaw that includes quasi-degenerate RHNs and show that observations of Majorana signatures in pp-colliders for this model can provide a way to not only probe the mixings and CP violation in the RHN sector but also to test leptogenesis.
To see the advantage of left-right models in this discussion, note that in the case of minimal seesaw models, where one extends the SM by including the RHNs 'by hand', one can diagonalize the RHN mass matrix and absorb the required rotation matrices into the Dirac Yukawa couplings. The specific rotation angles and phases then lose their separate identity since they can be absorbed by redefining the Dirac Yukawa coupling (see Section 2). Similar thing happens when the SM is extended by a flavor universal U (1) gauge symmetry. On the other hand the situation is different if the minimal seesaw is embedded into the leftright symmetric model (LRSM) based on the gauge group SU (2) L × SU (2) R × U (1) B−L [9][10][11] or a flavor-dependent U (1) acting on the RHNs. Here we focus on the LRSM where any rotation V R of the RHN mass matrix (except for the overall phase) will manifest in the W ± R interaction and become in principle measurable if an on-shell W R is produced at the Large Hadron Collider (LHC) or future colliders. In this paper, we point out that observables which can help us to measure the mixing angles and phases in V R at colliders using on-shell W R production are the same-sign charge asymmetry (SSCA) A αβ defined in Eq. (3.27) and the ratio of SSCAs R ( ) CP defined in Eq. (3.28). Similar observables were previously considered in the context of minimal seesaw [12] and U (1) B−L models [13].
In pp-collisions, the same-sign (SS) dilepton final states, which accompany the oppositesign (OS) dilepton events, have been known to provide a 'smoking gun' test of the Majorana nature of the RHNs [14] and hence of the seesaw paradigm. In the hierarchical RHN case, the number of SS and OS dilepton states arising from RHN production and decay for each flavor combination are equal and the relative strengths of signal between different flavor modes depends on the right-handed (RH) lepton mixing angles [15]. However, as has been pointed out in Refs. [16][17][18], if the RHNs are quasi-degenerate enough so that the coherence condition is met [19], the RHNs can oscillate into each other. 1 In this case, the ratio of SS and OS dilepton signals due to Majorana RHNs becomes flavor-dependent and can become unequal for each flavor combination if there is CP violation (CPV) in the RHN mass matrix [18]. It is encouraging to note that recent CMS [23] and ATLAS [24] searches for RHNs have included both SS and OS dilepton events in their event selection.
In this paper, we study the CPV effect in the RHN mass matrix by focusing only on the SS final states, which have relatively less SM background, as compared to the OS final states. In particular, we consider the SSCA observables A αβ defined in Eq. (3.27) and R ( ) CP defined in Eq. (3.28) for specific charged-lepton flavor combinations (α, β) and point out how one can extract the RHN mass matrix mixing and phase information from collider observations. It is worth noting that the SSCA is also well-suited to study the CP violating effects in other situations in particle physics, such as meson mixing; see e.g. [25].
We focus on a simple but a realistic example of two nearly degenerate RHNs N e, µ in LRSM with TeV-scale W R . We assume that third generation RHN N τ is heavier so that it has no effect on the light or heavy neutrino masses. Our model can be easily generalized if the third RHN is also degenerate with the other two. We quantify the dependence of SSCAs on the CP phase δ R in the mass matrix of the two oscillating RHNs and discuss when this asymmetry is observable. We explicitly show that the charge asymmetry of SS dilepton states for different flavors can provide useful information on the phase as well as the mixing angles of the RHN mass matrix. It turns out that due to an intrinsic difference between the production rates for W ± R in pp collisions due to parton distribution function (PDF) asymmetries between the up and down quarks in proton, a non-zero A αβ is present even in the absence of CPV. However, the asymmetries can still be measured at the LHC for a sizable SS dilepton signal from W R boson decay and can provide a useful probe of the RHN mass matrix. At future higher-energy colliders, such as the High-Energy LHC (HE-LHC) [26] with center-of-mass energy √ s = 27 TeV and future 100 TeV collider, such as Future Circular Collider (FCC-hh) [27] or Super Proton-Proton Collider (SPPC) [28], the charge asymmetries can be measured for larger W R masses. On the other hand, we will see that in the ratios R ( ) CP , the PDF uncertainties cancel, and therefore, these are cleaner observables for probing the CPV in the RHN sector.
It is well known that if the RHNs are quasi-degenerate, the mixing angle and CP phase in the RHN sector could play an important role in resonant leptogenesis [8,[29][30][31]. We show how the SSCA observations in colliders can provide key insight into TeV-scale leptogenesis. In fact, in combination with other collider observations such as the W R mass, it can even rule out TeV leptogenesis for certain parameter ranges (i.e. CP phases and mixing angles) of the model. Our work is largely complementary to the falsification scheme for high-scale leptogenesis [32], as well as to other probes of CPV and low-scale leptogenesis at colliders [33][34][35].
To carry out our leptogenesis calculation, we choose a generic form of the Dirac Yukawa coupling matrix using the Casas-Ibarra parameterization [36]. 2 It is known in the LRSM that for resonant leptogenesis to provide adequate lepton asymmetry, the dilution and washout effect from RH gauge interactions must be small [38][39][40]. This puts an absolute 2 Our model has parity broken at higher scale leaving SU (2)R unbroken till the TeV scale and therefore we cannot use the Dirac mass matrix formula derived in Ref. [37] which is valid for C-symmetric LRSM. lower bound on the W R mass, at 9.4 TeV for normal hierarchy (NH) ordering of active neutrino masses and 8.9 TeV for inverted hierarchy (IH), as we will find in the scenario of this paper. These limits are close to an earlier result in Ref. [41] which uses a different form of the Yukawa texture. We also find that to make leptogenesis work in the LRSM the Yukawa couplings y αi of RHNs can not be either too small or too large, turning out to be in the range of 1.0 × 10 −6 |y| max 8.6 × 10 −4 . As a result, the leptogenesis constraints turn out to be stronger than the low-energy high-precision measurements such as neutrino-less double beta (0νββ) decays, the lepton flavor violating (LFV) decays such as µ → eγ and the electric dipole moment (EDM) of electron.
This paper is organized as follows: In Section 2, we discuss the physical significance of the RHN CP phase in the minimal type-I versus left-right seesaw. In Section 3, we provide the basic framework for the SSCAs, considering both three and two-body decays of RHNs. We also clarify the coherence conditions for RHN oscillation at high-energy colliders. In Section 4, we estimate the prospects of SSCAs at future high-energy hadron colliders, as shown in Figs. 2 to 7. In Section 5, we elaborate on the role of RHN mixing and CP phase in TeV-scale resonant leptogenesis. We also estimate the leptogenesis constraints on W R boson mass. In section 6, we discuss the constraints on the RHN sector from 0νββ decays, µ → eγ and electron EDM. We conclude in Section 7. The analytic formula for the square root of the RHN Majorana mass matrix M 1/2 N is given in Appendix A, and more details about heavy-light neutrino mixing are presented in Appendix B.

CP phase in the RHN sector
In this section we introduce the problem we are addressing and our goal for this paper. The type-I seesaw [1][2][3][4][5] generally has the Dirac mass matrix M D , as well as the RHN Majorana mass matrix M N , both involving CP phases. Of course, as is well known, the rotation angles and the phases can be redefined by choice of basis depending on the full theory. If we consider a basis where the charged lepton mass matrix is diagonal, the phases and rotation angles of both M D and M N are physical parameters and their measurement would provide useful insight about theories of neutrino masses. The question we address in this paper is: is it possible to measure the phases and rotation angles of the unitary matrix V that diagonalizes the M N matrix in the basis where the charged lepton mass matrix is diagonal? This is important since those phases not only determine the final leptonic mixing angles and phases of the light neutrinos but also play a role in explaining the origin of matter via leptogenesis.
Consider the type-I seesaw extension of the SM (we call this the 'SM seesaw'), where the leptonic sector of the Lagrangian can be written as: where − → W µ and g L are respectively the SM SU (2) L gauge fields and coupling constant, τ is the vector of Pauli matrices, ψ L , H and R are respectively the left-handed lepton doublets, Higgs doublet and right-handed lepton singlets in the SM, and H ≡ iτ 2 H * . If we choose a basis such that the charged lepton Yukawa coupling matrix h e is diagonal, the RHN mass matrix M N remains non-diagonal and this fixes the leptonic basis and all angles and phases in this basis are physical. We can next diagonalize the mass matrix M N by a unitary rotation V T M N V . The question we ask now is whether we can measure the angles and the phases that parameterize V . The answer in the SM type-I seesaw is that we can rewrite h ν as h ν = h ν V † and the matrix V simply redefines the Dirac mass term M D in the type I seesaw and does not have a separate identity and therefore cannot be measured separately. In other words, it is as if we had chosen the neutrino Yukawa matrix as h ν to start with. The resulting rotation angles and phases that arise after seesaw diagonalization with the new Dirac mass matrix appear in the weak interaction Lagrangian as RHN admixtures with the light neutrino states and can be measured in principle [12,42]. However, these effects always appear with a coefficient suppressed by the heavy-light neutrino mixing, which makes it in general difficult to observe, except in special cases [43,44] with special textures for Dirac mass matrix to realize large light-heavy neutrino mixing.
On the other hand, in the LRSM, there is an extra gauge boson interaction and we can write the leptonic part of the Lagrangian as: where − → W Rµ and g R are respectively the SU (2) R gauge fields and coupling constant, Φ and ∆ R are the bidoublet and SU (2) R triplet Higgs fields of the LRSM respectively, and Φ = τ 2 Φ * τ 2 . For definiteness, let us choose Φ = diag(κ, 0), so that h e connects only charged leptons and h ν connects ν L with the RHN N . Again as in the type-I seesaw case, let us choose a basis where h e is diagonal. In this basis in general, the RHN mass matrix given by M N = f R v R is non-diagonal, with v R the vacuum expectation value (VEV) of the ∆ 0 R field. We can now ask if V diagonalizes the RHN mass matrix, are the rotation angles and phases in this matrix V observable? We claim that in the LRSM, due to the presence of the W R interaction, the V matrix is unambiguously observable since it rotates the leptonic fields in the W R interaction. This also contributes to the CP phase in the leptonic sector as well as to leptogenesis. Our goal in this paper is to show how to measure the rotation angles and CP phases in V for the RHN sector at colliders. Since we also expect this phase to contribute to leptogenesis, we want to display the connection between collider information and leptogenesis requirements in the hope that we can test leptogenesis for this particular model at colliders.

Same-sign charge asymmetries
We first show how SS dilepton signals arise from the seesaw Lagrangian (2.3) in the LRSM. For this purpose, we start with the RH charged currents in the leptonic sector of LRSM, which originate from the second term in Eq. (2.3) and are explicitly given by where C is the charge conjugation operator. Note that the first term on the RHS is responsible for OS dileptons whereas the second term uses the Majorana condition for RHNs and is the reason why SS dilepton states appear. To present our discussion involving heavy RHN oscillation, we assume that there are two quasi-degenerate RHNs 3 carrying the lepton flavors α = e, µ, and the third one N τ to be much heavier so that it plays no role in our discussion. Thus, the LRSM scenario we are considering can be viewed as an effective theory with only two RHNs. The lighter flavor states N e, µ are related to the two mass eigenstates N 1,2 via Then we can write down explicitly all the terms in the Lagrangian (3.1), which dictate the production of RHNs from W R boson decay: We assume that the two RHNs N e, µ are lighter than the W R boson, i.e. M N 1,2 < M W R , 4 with M N 1,2 the mass eigenvalues for the two RHNs and M W R the W R mass. so that they can be produced on-shell from W R decay. As a result of the Majorana nature, the heavy RHNs N α decay into both positively and negatively charged leptons, when they are produced at colliders. This can happen either through an off-shell W R boson, i.e. the three-body decays N α → ± α qq (with q, q being SM quark jets), or through the heavy-light neutrino mixing, i.e. the two-body decays N α → ± β W ∓ . Note that in the latter case the flavor index β = e, µ, τ and it might differ from the flavor of decaying RHN, i.e. α = β. The SSCA, which is a CPV effect, arises from the propagation of RHN mass eigenstates, and can in principle come from either the three-body or two-body decays of RHNs. However, we will see in the following subsections that in the parameter space of interest, the contribution from the Yukawa coupling mediated two-body decays can be neglected, and we can only see the CP-induced SSCAs from W R -mediated three-body decays. 3 There is often a misconception that having two quasi-degenerate RHNs means that they are a pseudo-Dirac pair but this need not be so, depending on the relative CP phase between them. For example, consider a mass matrix of RHNs that have positive eigenvalues after diagonalization; in this case the RHN pair is not pseudo-Dirac. Only if they have opposite CP phases (i.e. one eigenvalue positive and one negative with similar magnitude), that means a pseudo-Dirac pair. 4 This choice is also preferred by vacuum stability arguments [45][46][47].
3.1 Effect of three-body decays N α → ± α qq In this section, we consider only the contributions from the W R -boson mediated threebody decays N α → ± α qq . Following the notation of Ref. [18], we denote by A( ± α ± β , t) the time evolution of the amplitudes for the SS dilepton events W R → ± α (N β → ± β qq ) at high-energy colliders. Then the flavor dependence is as follows: where Γ N 1,2 are the total decay widths of the two mass eigenstates N 1,2 , and E N 1,2 the energies of N 1,2 at colliders. Note that the dependence of the amplitudes in Eqs.
where Γ avg ≡ (Γ N 1 + Γ N 2 )/2 is the average total width of RHNs, and ∆E N ≡ E N 2 − E N 1 the energy difference of the two RHN mass eigenstates at high-energy colliders. Note that the SSCAs do not depend on the RHN energies E N 1, 2 but only on the energy difference ∆E N , which can be estimated depending on whether the RHNs N 1, 2 are non-relativistic or relativistic (see Sec. 3.5). Eqs. (3.7)-(3.9) can be simplified in the limit of Γ N 1 = Γ N 2 , which is a good approximation for a pair of quasi-degenerate RHNs in the parameter space of interest. Then the flavor-dependent SS dilepton event numbers are proportional to the factors The R factors are normalized to follow the sum rule 3.2 Effect of two-body decays N α → ± β W ∓ In the generic LRSM, the Dirac neutrino mass matrix M D matrix is correlated with the charged lepton masses. However, the neutrino sector and charged lepton sector can be decoupled from each other if we choose the VEV configuration for the bidoublet Φ fields to be Φ = diag(κ, 0) (see e.g. Refs. [48][49][50] for more details on the scalar sector of LRSM). This choice has the advantage that we do not have to worry about simultaneously fitting the neutrino and charged lepton masses. In the specific type-I seesaw case with only two RHNs, one of the active neutrinos is massless, i.e. m 1 = 0 for normal hierarchy (NH) and m 3 = 0 for inverted hierarchy (IH), and M D can be parameterized in the Casas-Ibarra form [36]: where U is the PMNS mixing matrix for light neutrinos, and m ν = diag{m 1 , m 2 , m 3 } is the diagonal mass matrix for the active neutrinos. The analytic formula of the square root of the 2 × 2 RHN matrix M 1/2 N can be found in Appendix A.1, 5 and O is an arbitrary complex matrix in the form of where ζ is a free parameter, either real or complex, and v EW is the electroweak VEV.
If ζ is complex, having large Imζ will significantly enhance the Yukawa couplings y = M D /v EW (see e.g. Refs. [51,52]), as required by leptogenesis in the TeV-scale LRSM (see Section 5). For simplicity, we assume ζ is purely imaginary, with sin ζ = i sinh(Imζ) and cos ζ = cosh(Imζ). If Imζ > 2, which is preferred by leptogenesis (see Fig. 8 Then the M D matrix elements in Eq. (3.13) for the NH case can be explicitly written as follows: where the flavor index α = e, µ, τ . Using the relation in Eq. (A.10), it is straightforward to prove that These relations are also true for the IH case, and imply that Defining the rescaled flavor-dependent BR for the two-body decays , (3.20) if the two-body decays of RHNs dominate over the three-body decays, then the SS dilepton event numbers for the flavor ± α ± β combinations are respectively proportional to the following factors: From this result, we can see that the CPV effects are cancelled out in the two-body decays of both N e and N µ , at least in the parameter space of interest in this paper.

Combining three-and two-body decays
After taking into account both three-and two-body decays of N α of the RHNs, we define the SSCA A αβ at proton-proton colliders as , (3.27) which depends on the factor where the first and second terms are respectively the three-body (gauge-mediated) and two-body (Yukawa-mediated) decay contributions, with the respective branching ratios (BRs): . (3.30) The extra factor of 1/2 in the second term of Eq. (3.28) and the factor of 2 in Eq. (3.30) account for the fact that RHNs decay both into charged leptons and active neutrinos through the Yukawa couplings and in the large RHN mass limit due to the Goldstone equivalence theorem. Although the twobody decays do not contribute to the CP-induced SSCA, they will affect the measurement of SSCAs at high-energy colliders, when the corresponding branching ratio BR y is sizable. In particular, in the limit of BR y BR g , the CP-induced SSCA will be highly suppressed. As long as the CP phase δ R = 0, π/2, π and three-body decay BRs are sizable, the SSCAs A αβ can be induced by CPV in the heavy neutrino sector. Furthermore, as long as the RHN mixing angle θ R = 0, we can have the e ± µ ± events from RHN decay, and the asymmetry A eµ does not depend on the RHN mixing angle θ R , as the factor sin 2 2θ R cancels out in the ratio (3.27). 6 In the limit of BR g BR y , i.e. when the three-body decay BR is much larger than the two-body decay BR, we have the relation If the two-body decays are sizable, the SSCAs A ee and A µµ might be different, as in general the ratios B ee = B eµ , depending on the Yukawa coupling structure. Any significant violation of the relation (3.32) and A ee = A µµ would imply sizable mixing of a third RHN N τ with N e, µ or imply that the two-body decays of RHNs are important. In the limits of δ R → 0, π/2, π or in the two-body decay dominated regime, the factors which is non-zero purely due to the proton PDF effects, and to some extent, similar to the pure PDF-induced charge asymmetry in the SM W ± production at pp colliders: which has been measured by both ATLAS [53] and CMS [54] collaborations in the pp collisions, as well as in Pb-Pb collisions [55,56] at the LHC. The flavor-dependent SSCA A αβ defined in Eq. (3.27) can be used to probe directly the mixing angles and CP phases in the RHN sector. For instance, if a significant deviation of A αβ from the pure PDF-induced A (0) is observed at LHC or future higher energy colliders, then it is expected that there are (at least) two quasi-degenerate RHNs and there is a new CP phase δ R = 0, π/2 and π in the RHN sector. The CP phase δ R can be directly determined by the asymmetry A eµ , up to a two-fold ambiguity. Then we can determine the RHN mixing θ R and remove the ambiguity of δ R from the measurement of A ee or A µµ , or at least narrow down its value to a limited range, as shown in Section 4.1.

Same-sign ratio R CP
As discussed above, the CP-induced SSCA effect can be potentially smeared by the PDF effects. Therefore, it is useful to define the following ratio of ratios [12], which depends only on the CPV effects: with = e, µ and R( ) defined in Eq. (3.28). In the definitions above, the dependence of ratios of production cross sections on the proton PDFs are cancelled out, and we are left with only the CP-induced asymmetries in the ratios R (e, µ) In the absence of CP violation, i.e. for δ R = 0, π/2 or π, the factors R( + β ) and the ratios R (e, µ) CP = 0. Furthermore, in the limit of vanishing two-body decay contributions, R(e ± e ± ) = R(µ ± µ ± ) (cf. Eq. (3.10)) and R (e) As long as we can collect enough SS events at the LHC and future high-energy colliders to have a sizable signal-to-background ratio, the ratios R (e, µ) CP can be used to directly probe the mixing angles and CP phases in the RHN sector. As R (e, µ) CP do not depend on the proton PDFs, unlike the asymmetries A αβ , any deviation of R (e, µ) CP from zero will indicate the existence of CP violation in the RHN sector.

Coherence conditions
If the heavy RHNs are non-relativistic at colliders, i.e. their energies E N 1, 2 M N 1, 2 , we need only to replace x = ∆E N /Γ avg by R = ∆M N /Γ avg in Eq. (3.11), with ∆M N ≡ M N 2 − M N 1 the RHN mass splitting. If the RHN masses are much smaller than the W R boson mass, the RHNs are relativistic at high-energy colliders and the energy difference ∆E N ∆p N with ∆p N the momentum difference for the two RHNs. To observe RHN oscillation from the SSCA signals at colliders, we have to make sure the coherence conditions are satisfied [18,19]: (i) the two RHNs are coherent when they are produced, i.e. the uncertainty in their mass square σ M 2 N is greater than their actual mass square difference ∆M 2 N , and (ii) the coherence is maintained until they decay into lighter particles, i.e. σ x /δv g > 1/Γ N , with σ x the RHN wave-packet size and δv g the group velocity difference of the two RHNs. These coherence conditions impose upper bounds on the RHN mass splitting [19], depending on how the RHNs are produced and decay at colliders. For the production of RHNs from W R decay, the first coherence condition leads to with Γ W R being the total W R decay width. For TeV-scale W R , the condition (3.37) requires that ∆M N O(100 GeV) at LHC. The second condition provides a more stringent limit which leads to the upper bound ∆M N O(GeV). Throughout this paper we assume the two RHNs are quasi-degenerate with a mass splitting ∆M N GeV and the coherence conditions are satisfied.

Prospects at future high-energy colliders
The smoking-gun signals of a heavy W R boson are the SS dileptons plus jets without missing energy, i.e. pp → W ± R → ± ± jj. The dominant SM backgrounds are mainly from diboson, Z + jets, tt processes, or "fake" leptons, i.e. jets misidentified as leptons [57,58] or lepton charge misidentified, with a fake rate of O(10 −4 ) or smaller depending on the transverse momentum p T of jets and leptons [59]. The dominant uncertainty for measuring SSCAs comes from the PDFs, which is more significant than the reducible backgrounds from fake leptons and other processes.
At pp colliders, the production cross section for W + is larger than that for W − , i.e. σ(pp → W + X) > σ(pp → W − X), due to different PDFs for the u and d quarks in proton. Based on the same logic, we can produce more W + R than W − R at the LHC and future pp colliders, which implies that even without any CPV in the RHN sector, we should expect  Figure 1: Parton-level, leading-order SS dilepton production cross section σ(pp → W ± R ) × BR(W ± R → ± ± jj) (with = e, µ) at the 14 TeV LHC (upper), 27 TeV HE-LHC (lower left) and 100 TeV FCC-hh (lower right), as a function of W R mass, with g R = g L . The solid red (blue) lines correspond to the central values for W + R (W − R ) production and the shaded bands are due to PDF uncertainties. equivalently on the parton energy fraction x 1 x 2 =ŝ/s ∼ M 2 W R /s (with s the center-ofmass energy) [58]. Adopting the NNPDF3.1lo PDF datasets with α s (m Z ) = 0.118 [60] Fig. 1, with the central values for W + R (W − R ) shown as the solid red (blue) lines, and the shaded bands due to the PDF uncertainties. 7 The higher-order QCD corrections depend on the centerof-mass energy s and W R mass, and an average NLO k−factor of 1.1 is included for the higher-order effects [58], which is assumed to be the same for both W + R and W − R . 8 For the sake of concreteness, we have assumed the gauge couplings for SU (2) L and SU (2) R to be the same, i.e. g L = g R . It turns out that the PDF uncertainties are more significant than 7 We repeated the simulations using CT14lo [62] and MMHT2014lo [63] PDF datasets, and found that although the production cross sections of W ± R vary to some extent, and there are uncertainties intrinsic to PDF extraction from data [64], these issues do not affect qualitatively the main results of this paper. 8 The k-factors might be different for W + R and W − R , and the difference might depend on the center-ofmass energy and WR mass [66]. In any case, we expect the difference to be small, just like the SM case, where k W + NLO = 1.17 and k W − NLO = 1.21 [67].
the reducible backgrounds from the fake leptons and other processes; in other words, the cross section uncertainties are expected to be dominated by the proton PDFs, in particular when the ratio M 2 W R /s is large, as shown in Fig. 1.

A αβ
The expected charge asymmetries A αβ (with αβ = ee, µµ, eµ) purely due to the PDFs and without any CPV in the RHN sector at the 14 TeV LHC, 27 TeV HE-LHC and 100 TeV FCC-hh are shown by the gray lines in Fig. 2, with the gray bands due to the PDF uncertainties. The CPV in the RHN sector induces extra SSCA, as given in Eq. (3.27). The special case with the largest CPV effect θ R = π/4 and δ R = π/4 is shown in Fig. 2 by the orange, blue and purple bands, which are respectively for A ee , A µµ , and A eµ . As a benchmark scenario, we have taken BR y = 0 in the left panels of Fig. 2, where the threebody decays of N α dominate over the two-body decays. In the right panels of Fig. 2 we show the asymmetries for a second benchmark scenario with BR y = 1/2, in which case the three-and two-body decays are comparable to each other. For the sake of concreteness, we have taken the Dirac CP phase in the PMNS matrix to be −π/2, which is favored by recent T2K [68] and NOνA [69] data, and the single Majorana phase is set to be zero, and x = 1 in Eq. (3.11).
When the current 13 TeV LHC constraints on W R mass of 5 TeV are taken into consideration [23,24,57], 9 the energy fraction x 1 x 2 =ŝ/s ∼ 0.1 and the PDF uncertainties are so large that the SSCAs A αβ can not be measured at 14 TeV LHC, as can be seen in the two upper panels of Fig. 2. At 27 TeV HE-LHC and future 100 TeV colliders, the ratio M 2 W R /s could be significantly smaller, even for larger W R boson masses, and we can distinguish the CP-induced SSCAs from the PDF-induced effects. In the optimal case with BR y = 0, the CPV in the RHN sector can be observed at HE-LHC if M W R < 7.2 TeV, as shown in the middle left panel of Fig. 2. At future 100 TeV colliders like FCC-hh and SPPC, one could probe CPV through SSCA up to a W R mass of 26 TeV. In particular, if M W R 11.5 TeV, the asymmetry A eµ < 0 at FCC-hh, which would otherwise be positive if there is no CPV in the RHN sector. If the BRs for two-body decays of RHNs are sizable, the CP-induced SSCAs are to some extent "diluted" by the second term in Eq. (3.28), as shown in the three right panels of Fig. 2. In the case of BR y = 1/2 with the three-and two-body decay widths being equal, the CP phase in the RHN sector can be probed up to a W R mass of 6.4 TeV at HL-LHC and improved up to 23 TeV at the future 100 TeV colliders.
Since the gauge interactions do not distinguish between electron and muon flavors, we expect the asymmetries A ee and A µµ to be similar in the case of BR y = 0 (assuming similar efficiencies for electrons and muons at future hadron colliders), as shown in the three left panels of Fig. 2. However, for BR y = 0, if the Yukawa couplings of N e, µ are different, then A ee and A µµ values differ, as shown in the three right panels of Fig. 2. They can be distinguished at future 100 TeV colliders, even after taking into account the PDF 9 The LHC limits on the WR boson mass depend on the RHN mass involved. Here we have taken the most stringent bound of 5 TeV from Ref. [24]. luminosity, as long as we can collect enough SS dilepton events to suppress the reducible SM backgrounds. Actually we need only O(100 fb −1 ) of data to have at least 100 events of both + + and − − at FCC-hh (HE-LHC) for a W R mass of 10 (5) TeV. Therefore, we expect the SSCAs to be a 'smoking gun' observable to reveal the existence of CPV in the RHN sector.
The dependence of A αβ on the CP phase δ R at HE-LHC and FCC-hh is shown in Fig. 3, for the special case of θ R = π/4. The ee, µµ with eµ channels are shown respectively in orange, blue and purple. Note that the eµ channel does not depend on θ R (the overall sin 2 2θ R term in Eq. (3.11) cancels out in the ratio for A eµ ), so the result for A eµ shown in Fig. 3 is applicable for arbitrary θ R . As in Fig. 2, the solid lines correspond to the central values while the shaded bands are due to the PDF uncertainties. The W R mass is set to be 5 TeV for the HL-LHC case (left) and 15 TeV for the FCC-hh case (right) as a benchmark choice. The latter choice also respects the leptogenesis constraints on W R mass (see Section 5). For comparison, the case without any CPV in the RHN sector is shown by the horizontal gray band in both panels. As expected from Fig. 2, the PDF uncertainties on the SSCA at 100 TeV collider are smaller than at HE-LHC, even though the W R mass is taken to be larger in the right panel of Fig. 3 than in the left panel.
More generic dependence of A ee at FCC-hh on the RHN mixing angle θ R and CP phase δ R is shown in Fig. 4. For the purpose of comparison and direct test of leptogenesis at future high-energy colliders (see Section 5), two benchmark scenarios are considered, which both respect the leptogenesis constraints on the W R mass. For the sake of concreteness, we have taken M N = 1 TeV and the ratio x = 1. In the left panel we have taken the W R mass to be 15 TeV and BR y = 1/2, which means that the three-and two-body decay widths are equal to each other. We show the values of A ee, µµ = 0.8, 0.6, 0.4, 0.2 and 0, which are respectively depicted in purple, blue, green, orange and magenta. In the right panel,  Fig. 10, which could generate successful leptogenesis; see Section 5.3 for more details on the leptogenesis constraints.
we have M W R = 20 TeV and BR y = 1/4, with the three-body decay width of RHNs being three times larger than that for the two-body decays. In this panel we show the contours of A ee, µµ = 0.75, 0.5, 0.25 and 0, respectively in blue, green, orange and magenta. In both panels, the solid lines correspond to the central values, while the shaded bands are due to the PDF uncertainties. If the W R boson can be observed at high-energy colliders, its mass can be fixed by the invariant mass of the SS dileptons and jets. Then by measuring the asymmetry A ee (and/or A µµ ), the values of θ R and δ R can be limited to a (narrow) band, as shown in Fig. 4, which can be further narrowed down by the measurement of A eµ , up to a twofold ambiguity, as exemplified in Fig. 3. As indicated by the brown shaded regions in Fig. 4, only certain regions of θ R and δ R could generate successful leptogenesis in the LRSM, therefore the SSCAs at future high-energy hadron colliders can be used to test leptogenesis; see more details on the leptogenesis constraints in Section 5.

R CP
As shown in Fig. 2, the prospects of the asymmetries A ee, µµ, eµ are strongly affected by the PDF uncertainties. In contrast, the ratios R CP as functions of the RHN mixing angle θ R and CP phase δ R for the case BR y = 0. In this case, leptogenesis is not viable, since the RHNs do not have CPV decays to charged leptons. Here we have taken M W R = 10 TeV for concreteness, but the R CP contours are independent of M W R for BR y = 0.  depend on both three-and two-body decays, as expected. But the key point is that unlike the asymmetries A αβ , the ratios R (e, µ) CP are free from the PDF uncertainties, and thus provide a clean probe of CPV at future colliders and up to a higher W R mass. In particular, as long as the W R mass lies below the prospect of ∼ 6 TeV at the HL-LHC, the ratios R (e, µ) CP can be measured the CPV in the RHN sector can be deciphered, whereas we need a higher center-of-mass energy to unambiguously measure the asymmetries A αβ , as shown in Fig. 2. Furthermore, as in the case of A αβ , to probe R (e, µ) CP we need only O(100 fb −1 ) of data to effectively suppress the SM backgrounds at FCC-hh (HE-LHC) for a W R mass of 10 (5) TeV. As indicated by the brown shaded regions in Fig. 6 and 7, only certain regions of θ R and δ R could generate successful leptogenesis in the LRSM, and therefore, the measurements of SSCAs at future high-energy hadron colliders can be effectively used to test leptogenesis; see more details in Section 5.

Testing leptogenesis at future hadron colliders
In this section we study the implications of RHN mixing and associated CP phase for leptogenesis, where lepton asymmetry is generated from the CP violating decays of RHNs which is then transferred into the baryon asymmetry through electroweak sphaleron pro-cesses. In the type-I seesaw with hierarchical RHNs (the so-called 'vanilla leptogenesis'), the RHN masses are required to be 10 9 GeV [70,71] for successful leptogenesis. With fine tuning and flavor effects taken into account, the hierarchical RHN masses can be lowered down to 10 6 GeV [72]. In any case, these RHN masses are too heavy to be produced at foreseeable colliders. The situation can be alleviated in the framework of resonant leptogenesis [8,[29][30][31], where (at least) two RHNs are quasi-degenerate, thereby resonantly enhancing the lepton asymmetry even for TeV-scale masses [73,74]; see Ref. [75] for a review. Interestingly, this quasi-degeneracy is also a requirement for generating a non-zero SSCA [18]. Thus, measuring the SSCAs can not only provide important insight into the nature of the RHN mass matrix but also the origin of baryon asymmetry via resonant leptogenesis. In this section we will find that only in certain parameter space of θ R and δ R can the baryon asymmetry observed in the Universe be successfully produced with the right sign. The measurement of SSCAs A αβ and R (e, µ) CP would then help to test resonant leptogenesis directly at future high-energy hadron colliders.

Dependence on the RHN sector
In the minimal framework of resonant leptogenesis, the lepton asymmetry is generated from the on-shell decays of RHNs N i → Lφ (with i = 1, 2) via the Yukawa coupling y = M D /v EW , with L and φ respectively the SM lepton and Higgs doublets. Given the analytic form of M D in Eq. (3.13), it is straightforward to calculate the flavor-dependent lepton asymmetry from the interference of tree-and loop-level diagrams, which can be approximated as [39] is the washout factor in presence of both Yukawa and W R -mediated gauge interactions, with B iα the BR for the decay is the dilution factor due to the W R -mediated RH gauge interactions of RHNs. Here γ N i Lφ , γ N i Lqq and γ N i W R are the thermally-averaged rates due to the decays of N i through the Yukawa and gauge interactions and the 2 ↔ 2 processes mediated by the W R boson, respectively. The main dependence on the RHN mixing and CP phase is contained in the with i, j = 1, 2 but i = j.

Leptogenesis constraints on Yukawa couplings and W R mass
Without any significant cancellation or fine-tuning in the M D matrix, the Yukawa couplings y are expected to much smaller than the gauge coupling g R . In this scenario, the W R -mediated gauge interactions in the LRSM, apart from giving an extra contribution to the total width of N i , lead to ∆L = 1 scattering processes N ↔ qq , which cause significant dilution and/or washout of the lepton asymmetry produced by the Yukawa coupling induced RHN decays. This sets a lower bound on the W R mass, which is typically higher than the TeV scale, depending on the M N mass, the mass splitting ∆M N and other parameters [38][39][40][41]. For TeV-scale RHNs, typically |y| ∼ |m ν M N | ∼ 10 −6 , and the W R mass is required to be larger than roughly 50 TeV, to keep the dilution factor in Eq. (5.3) not too small. This is even beyond the reach of future 100 TeV colliders [58,76]. 11 However, if the ζ parameter in the matrix O in Eq. (3.13) is complex and | sin ζ|, | cos ζ| 1, then the Yukawa couplings y can be significantly enhanced (at the cost of fine-tuning in the seesaw formula), and the leptogenesis bound on W R mass can be relaxed accordingly. On the other hand, the couplings |y| can not made too large, otherwise the ∆L = 0 scattering process L α φ ↔ L β φ, the ∆L = 2 process Lφ ↔Lφ † and/or the inverse decay Lφ → N i will induce significant dilution/washout effect and render leptogenesis ineffective.
To estimate the leptogenesis constraints on the ζ parameter and the resultant Yukawa couplings |y| and the W R mass, we vary the neutrino oscillation parameters within their current 2σ ranges, as shown in Table 1 [79], and the other parameters in the following ranges: To be consistent with the analysis in Section 3, we require that the W R boson is heavier than For the sake of simplicity, we have taken ζ to be purely imaginary, and the RHN mass splitting ∆M N = Γ avg /2 to have the maximal lepton asymmetry in Eq. (5.4). The lower bound for RHN mass is taken to be 150 GeV such that the decay N 1,2 → L + φ is kinematically allowed when the thermal masses of the Higgs and lepton doublets are taken into consideration [80]. 12 The scatter plots for the dependence of lepton asymmetry |η ∆L | on the parameter |ζ| = Imζ are shown in Fig. 8, with the left and right panels respectively for the NH and IH cases. In both panels the horizontal dashed gray lines correspond to the value of ∆η L = −2.47 × 10 −8 [74] implied by current observations of baryon asymmetry [82], and the red points are excluded by the 0νββ decays discussed in Section 6.1. It turns out that the limits from LFV decay µ → eγ in Section 6.2 and electron EDM in Section 6.3 do not provide any limits for the parameter space shown here. As stated above, the value |ζ| can not be either too large or too small, and numerical evaluations reveal that for the parameter ranges given in Eq. (5.5) we have, (5.7) 12 We do not consider the possibility that baryon asymmetry can also be generated from the Higgs doublet decay φ → N + L when the RHNs are lighter than the SM Higgs doublet, see Refs. [81]. The leptogenesis constraint on the W R mass is shown in Fig. 9, as function of the RHN mass, for the NH (blue) and IH (orange) cases. As stated above, the W R mass limit depends on the RHN mass and other parameters, which is mainly from comparing the twoand three-body decays of RHNs. In the large M N limit, the dependence is respectively For the two-body decays we have taken into account also the dependence M D ∝ (m ν M N ) 1/2 . When all other parameters are fixed and M N gets larger, the three-body width Γ(N → qq ) grows faster than that for the two-body decays N → Lφ, therefore the W R mass has also be larger to make sure that the two-body branching fraction BR y and the resultant lepton asymmetry is not highly suppressed. When all other parameters are fixed and the RHN masses get smaller, the washout factor in Eq. (5.2) and the efficiency factor in Eq. (5.4) become larger and the dilution factor in Eq. (5.3) get smaller; when all these factors are combined in Eq. (5.1), there is an absolute lower limit on W R mass from leptogenesis for the special case with only two RHNs: M W R > 9.4 TeV for NH , 8.9 TeV for IH , (5.9) which corresponds to the RHN masses at M N ∼ 500 GeV in Fig. 9. This is similar to the W R mass bounds found in a different version of LRSM [41], including all three RHNs. Furthermore, the scalings in Eq. (5.8) are not very sensitive to the neutrino oscillation parameters, thus the NH and IH limits in Fig. 9 are roughly the same, in particular when the RHNs are heavy.

Testing leptogenesis at colliders
The RHN mixing angle θ R and CP phase δ R play an important role in Eq. in the two-dimensional plane of θ R and δ R could produce the observed baryon asymmetry (in both magnitude and sign), as exemplified in Fig. 10. We have chosen two benchmark points: (i) M W R = 15 TeV and BR y = 1/2 (the two-and three-body decay widths are equal), which is shown in blue; (ii) M W R = 20 TeV and BR y = 1/4 (the three-body decay width is three times larger than that for the two-body decays), depicted in red. In both cases, the W R mass is consistent with the absolute lower bound from leptogenesis given by Eq. (5.9). To be concrete, the average RHN mass is set at 1 TeV in both cases, and the light neutrino mass ordering is chosen to be NH. For the case of IH, the parameter space does not change too much for the two benchmark points. Some of the empty regions in Fig. 10, for instance with θ R < π/4 and δ R < π/2, always generate a "wrong" sign for ∆η L , coming from the product of factors (M 2 N 1 − M 2 N 2 )Im[y * α2 y α2 ]Re[(y † y) 12 ] in Eq. (5.4). The viable parameter space in blue and red in Fig. 10 are also shown as the brown shaded regions in Figs. 4, 6 and 7. It is clear that TeV-scale leptogenesis could be falsified at future hadron colliders by measuring the mixing angle θ R and CP phase δ R once the W R is discovered. For instance, if it is found that θ R < π/4 and δ R < π/2, then leptogenesis will be excluded, at least in the case with only two quasi-degenerate RHNs. Furthermore, if the W R boson mass is found to be lower than the absolute leptogenesis limit given in Eq. (5.9), leptogenesis will also be excluded. One should note that to test leptogenesis directly at future high-energy hadron colliders, either the two-body (BR y ) or the three-body (BR g ) BRs of RHNs can not be too large or small: When the two-body decay BR is too large (N α being lighter and/or W R heavier), the SSCAs signals at collider will be highly suppressed, as they originate only from the three-body decays. On the other hand, when the threebody decays dominate (N α being heavier and/or W R lighter), the leptogenesis would be inefficient, as it is only from the Yukawa coupling induced two-body decays. However, we would like to stress that no matter whether leptogenesis is excluded or not, observation of mixing angle and CPV in the RHN sector at future hadron collider would be a significant step towards understanding the origin of neutrino masses.

Low-energy constraints
The RHNs N i and heavy W R boson contribute to some of the low-energy and precision measurements such as 0νββ decays, LFV decay µ → eγ and electron EDM, which could be used to set limits on the masses and couplings in the RH sector, in particular on the complex parameter ζ in the matrix O [51]. However, as we will see in this section, not all these low-energy precision measurements provide competitive limits on the parameter ζ for the parameter space of our interest.

Neutrinoless double beta decay
If the light neutrinos in the SM are Majorana particles, they will induce 0νββ decays through the diagram illustrated in Fig. 11a [84,85]; for simplicity we assume H ±± R is heavy such that its contribution is negligible. Then the dominant contributions to 0νββ decays in LRSM are all shown in Fig. 11, including the RHN-mediated diagrams in Fig. 11b, heavy-light neutrino mixing in Figs. 11c and 11d, and W − W R mixing in Fig. 11e. For a given isotope, the lifetime of 0νββ can be factorized to be of the form [86] 1 T 0ν where G 0ν 01 is the phase space factor and M 0ν ν, N, λ, η are the relevant nuclear matrix elements [87][88][89]. The η's are the dimensionless particle physics parameters obtained from the Feynman diagrams in Fig. 11: , and therefore, the RHN and W R contributions in Eqs. (6.3)-(6.6) can always be made small by taking large RHN and/or W R masses. Furthermore, with the assumption of κ = 0 in Section 3.2, the W − W R boson mixing parameter ξ = 0 in Eq. (6.6) (see e.g. Refs. [48][49][50] for more details). With this choice, the current most stringent 0νββ decay limits from KamLAND-Zen [90] and GERDA [91] could exclude some of the parameter space, as shown in Fig. 8, but they can not provide any robust limits on the ζ parameter in the O matrix.

LFV decay µ → eγ
The RHNs and W R boson might also induce new contributions to the LFV decay µ → eγ, which is predicted to be [86,92] where m µ and Γ µ are respectively the muon mass and width, s w ≡ sin θ w is the weak mixing parameter, α w ≡ g 2 L /4π is the weak coupling strength, and the form factors G γ L,R are respectively [86] , and the loop functions G γ 1,2 (x) are defined as (6.12) In the case of W − W R mixing ξ = 0 as a result of κ = 0, the first and third terms in Eq. (6.9) and the second term in Eq. (6.10) are all vanishing, and the second term in Eq. (6.9) is proportional to V µ1 V * e1 + V µ2 V * e2 = 0 in the limit of M N 1 = M N 2 ; therefore, only the first term in Eq. (6.10) contributes to the LFV decay µ → eγ. As shown in Appendix B, the S αi elements are the heavy-light mixing angles to the leading order, and the |G γ R | 2 factor is highly suppressed by the heavy-light mixing angle to the fourth power. In addition, the prefactors in Eq. (6.8) are of order 10 −3 . Evaluating numerically for the scatter points in Fig. 8, it turns out that the LFV decay branching ratio BR(µ → eγ) 10 −17 , which is orders of magnitude smaller than the current limit BR(µ → eγ) < 4.2 × 10 −13 from MEG experiment [93]. Therefore in the LRSM scenario we are considering, the LFV decays do not set more stringent limits on the RHN sector than the leptogenesis and 0νββ decay limits.

Electron EDM
The W − W R mixing might lead to a beyond SM contribution to the electron EDM at 1-loop level, which reads [94] Although the RHN mixing matrix V is complex in the presence of CP phase δ R , the LRSM contribution to electron EDM is vanishing in the limit of ξ → 0, therefore the electron EDM limit from ACME experiment [95] does not set any limits on the LRSM we are considering.

Conclusion
The type-I seesaw is one of the most compelling scenarios for explaining tiny neutrino masses. It uses heavy RHNs with Majorana masses as the two key ingredients. However, in absence of any evidence for new physics beyond SM, our current knowledge of RHNs is very limited. In this paper we have shown how one class of RHN models where two of the RHNs are quasi-degenerate with mixings and associated CP violation, and are part of the TeV-scale LRSM framework, can be directly probed at future high-energy hadron colliders, by measuring the charge asymmetries in the same-sign dilepton final states, e.g. number of + + versus − − events (with = e, µ) and associated SSCA observables A αβ and R ( ) CP defined in Eqs. (3.27) and (3.28), respectively. We find that due to the large PDF uncertainties, the CP-induced SSCAs A αβ can only be measured at future higher energy colliders, such as the √ s = 27 TeV HE-LHC and the 100 TeV FCC-hh or SPPC, but not at the HL-LHC, as illustrated in Fig. 2. The e ± µ ± channel is particularly suitable for measuring the CP phase through A eµ , as it does not depend on the RHN mixing angle θ R . When combined with the e ± e ± (and/or µ ± µ ± ) data, we can use the measurement of A αβ to determine both the RHN mixing angle and CP phase. With the measurement of A αβ , the CPV in the RHN sector can be probed with W R mass up to 6.4 TeV at 27 TeV HE-LHC, which can be improved up to 26 TeV at future 100 TeV colliders.
On the other hand, the ratios R (e, µ) CP do not suffer from the PDF uncertainties and can be measured at both LHC and future higher-energy colliders. Combining all the e ± e ± , µ ± µ ± and e ± µ ± channels, measurement of R (e, µ) CP can determine both the RHN mixing angle and CP phase, as shown in Figs. 6 and 7. Furthermore, as the ratios R (e, µ) CP do not depend on the proton PDFs, they can be used to probe the RHN sector up to higher W R mass, as long as the W R boson can be produced with an observable rate in the same-sign dilepton channel. When the two-body decays of RHNs are sizable, both A ee and A µµ , as well as R The RHN mixing and CP violation also play a key role in generating the observed baryon asymmetry in the framework of TeV-scale resonant leptogenesis. Thus the SSCA measurements at future high-energy colliders can be used to directly test leptogenesis as the mechanism for origin of matter, as exemplified in Figs. 4, 6 and 7. This depends largely on the branching ratios of the two-and three-body decays of RHNs, as the SSCA signatures at collider can only be induced by the gauge coupling mediated three-body decays of RHNs, while baryon asymmetry is only generated from the Yukawa coupling induced two-body decays. We find that leptogenesis requires the ζ parameter in the Casas-Ibarra parametrization to be within the range 1.3 Imζ 7.8 for the NH case and 0.8 Imζ 7.7 for the IH case (see Fig. 8), corresponding to a maximum Dirac Yukawa coupling 1.3 × 10 −6 |y| max 7.2 × 10 −4 for the NH case and 1.0 × 10 −6 |y| max 8.6 × 10 −4 for the IH case. In the minimal LRSM leptogenesis we are considering here, there is a lower limit on W R mass, which depends on the RHN mass, as shown in Fig. 9. The absolute lower bound W R mass was found to be 9.4 TeV for NH and 8.9 TeV for IH ordering of active neutrino masses. Regardless of whether leptogenesis can be tested at the LHC, HE-LHC or FCC-hh, observation of mixing angle and CP phase in the RHN sector at future hadron colliders would be a significant step in understanding the origin of tiny neutrino masses for this particular scenario (quasi-degenerate RHNs) for the type-I seesaw paradigm. with the orthogonal condition O T O = 1 3×3 . One should note that in the three RHN case, the lightest neutrino mass is not vanishing any more.

B Heavy-light neutrino mixing
In the effective LRSM we are considering, there are three active neutrino and two heavy RHNs, and the full 5 × 5 neutrino mass matrix can be diagonalized by a unitary matrix: where m ν = diag(m 1 , m 2 , m 3 ) and M N = diag(M N 1 , M N 2 ). The unitary matrix V has an exact representation in terms of the matrix ϑ [96,97]: where ϑ * = M D M −1 N to leading order in a converging Taylor series expansion, U ν and V R are respectively the unitary matrices diagonalizing the light and heavy neutrino mass matrices. To the leading order of ϑ, we obtain the heavy-light neutrino mixing matrices S and T given in Eq. (6.7).