Neutrinos and collider physics

In the traditional 'vanilla' seesaw mechanism [11–15], the L–R neutrino mixing is given by

$$\begin{eqnarray}&&{V}_{{\ell }N}\ \simeq \ \sqrt{\displaystyle \frac{{M}_{\nu }}{{M}_{N}}}\ \lesssim \ {10}^{-6}\sqrt{\displaystyle \frac{100\;\mathrm{GeV}}{{M}_{N}}},\end{eqnarray} \tag{ 5 }$$

due to the smallness of the light neutrino mass ${M}_{\nu }\lesssim 0.1\;\mathrm{eV}$ [57]. Thus for a low seesaw scale in the sub-TeV to TeV range, the experimental effects of the light–heavy neutrino mixing are expected to be too small, unless the RH neutrinos have additional interactions, e.g. when they are charged under $U{(1)}_{B-L}$. However, there exists a class of minimal SM plus low-scale type-I seesaw scenarios [36, 119–130], where ${V}_{{\ell }N}$ can be sizable while still satisfying the light neutrino data. This is made possible by assigning specific textures to the Dirac and Majorana mass matrices in the seesaw formula (4). The stability of these textures can in principle be guaranteed by enforcing some symmetries in the lepton sector [121, 122, 128, 131, 132]. We will generically assume this to be the case for our subsequent discussion on the collider signatures of low-scale minimal seesaw, without referring to any particular texture or model-building aspects. Also, unless otherwise specified, we will use a model-independent phenomenological approach, parametrized by a single heavy neutrino mass scale M_N and a single flavor light–heavy neutrino mixing ${V}_{{\ell }N}$, assuming that the mixing effects in other flavors ${\ell }\prime \ne {\ell }$ are sub-dominant. Although this assumption may not be strictly valid for a realistic seesaw model satisfying the observed neutrino oscillation data, it enables us to derive generic bounds on the mixing parameter, which could be translated or scaled appropriately in the context of particular neutrino mass models (see e.g. [133]).

Another natural realization of a low-scale seesaw scenario with large light–heavy neutrino mixing is the inverse seesaw model [31], where one introduces two sets of SM singlet fermions $\{{N}_{R\alpha },{S}_{L\rho }\}$ with opposite lepton numbers, i.e. $L({N}_{R})\;=\;+1=-L({S}_{L})$. In this case, the neutrino Yukawa sector of the Lagrangian is in general given by

$$\begin{eqnarray}-{{\mathcal{L}}}_{Y} & = & {h}_{l\alpha }{\bar{L}}_{{\ell }}\tilde{\Phi }{N}_{R\alpha }+{({M}_{S})}_{\rho \alpha }{\bar{S}}_{L\rho }{N}_{R\alpha }\\ & & +\displaystyle \frac{1}{2}\left[{({\mu }_{R})}_{\alpha \beta }{\bar{N}}_{R\alpha }^{C}{N}_{R\beta }+{({\mu }_{S})}_{\rho \lambda }{\bar{S}}_{L\rho }{S}_{L\lambda }^{C}\right]+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 6 }$$

where M_S is a Dirac mass term and ${\mu }_{R,S}$ are Majorana mass terms After EWSB, the Lagrangian (6) gives rise to the following neutrino mass matrix in the flavor basis $\{{({\nu }_{L{\ell }})}^{C},{N}_{R\alpha },{({S}_{L\rho })}^{C}\}$:

$$\begin{eqnarray}{{\mathcal{M}}}_{\nu }=\left(\begin{array}{ccc}{\bf{0}} & {M}_{D} & {\bf{0}}\\ {M}_{D}^{{\mathsf{T}}} & {\mu }_{R} & {M}_{S}^{{\mathsf{T}}}\\ {\bf{0}} & {M}_{S} & {\mu }_{S}\end{array}\right)\ \equiv \ \left(\begin{array}{cc}{\bf{0}} & {{\mathcal{M}}}_{D}\\ {{\mathcal{M}}}_{D}^{{\mathsf{T}}} & {{\mathcal{M}}}_{N}\end{array}\right),\end{eqnarray} \tag{ 7 }$$

which has a form similar to the type-I seesaw matrix (3), with ${{\mathcal{M}}}_{D}=({M}_{D},{\bf{0}})$ and ${{\mathcal{M}}}_{N}=\left(\begin{array}{cc}{\mu }_{R} & {M}_{S}^{{\mathsf{T}}}\\ {M}_{S} & {\mu }_{S}\end{array}\right)$. Here we have not considered the dimension-4 lepton-number breaking term $\bar{L}\tilde{\Phi }{S}_{L}^{C}$ which appears, for instance, in linear seesaw models [134–137], since the mass matrix in presence of this term can always be rotated to the form given by (7) [138]. Also observe that the inverse seesaw model discussed originally in [31] set the RH neutrino Majorana mass ${\mu }_{R}={\bf{0}}$ in (7). At the tree-level, the light neutrino mass is directly proportional to the Majorana mass term ${\mu }_{S}$ for $\parallel {\mu }_{S}\parallel \ll \parallel {M}_{S}\parallel$:

$$\begin{eqnarray}&&{M}_{\nu }={M}_{D}{M}_{S}^{-1}{\mu }_{S}{M}_{S}^{-{1}^{{\mathsf{T}}}}{M}_{D}^{{\mathsf{T}}}+{\mathcal{O}}({\mu }_{S}^{3}),\end{eqnarray} \tag{ 8 }$$

whereas at the one-loop level, there is an additional contribution proportional to ${\mu }_{R}$ [41, 139], arising from standard electroweak radiative corrections [36]. The smallness of ${\mu }_{R,S}$ is 'technically natural' in the 't Hooft sense [140], i.e. in the limit of ${\mu }_{R,S}\to {\bf{0}}$, lepton number symmetry is restored and the light neutrinos ${\nu }_{L{\ell }}$ are massless to all orders in perturbation theory, as in the SM.

The freedom provided by the small LNV parameter ${\mu }_{S}$ in (8) is the key feature of the inverse seesaw mechanism, allowing us to fit the light neutrino data for any value of light–heavy neutrino mixing, without introducing any fine-tuning or cancellations in the light neutrino mass matrix (8) [141, 142]. In essence, the magnitude of the neutrino mass becomes decoupled from the heavy neutrino mass, thus allowing for a large mixing

$$\begin{eqnarray}&&{V}_{{\ell }N}\simeq \ \sqrt{\displaystyle \frac{{M}_{\nu }}{{\mu }_{S}}}\ \approx \ {10}^{-2}\sqrt{\displaystyle \frac{1\;\mathrm{keV}}{{\mu }_{S}}}.\end{eqnarray} \tag{ 9 }$$

The heavy neutrinos N_R and S_L have opposite CP parities and form a quasi-Dirac state with relative mass splitting of the order $\kappa ={\mu }_{S}/{M}_{S}$. All LNV processes are usually suppressed by this small mass splitting. For instance, in the one-generation case, the light neutrino mass in (8) can be conveniently expressed as ${M}_{\nu }\simeq \kappa {V}_{{\ell }N}{M}_{D}$, in contrast with ${V}_{{\ell }N}{M}_{D}$ in the type-I seesaw case (cf (4)). It should be noted here that the approximately L-conserving models with quasi-degenerate heavy Majorana neutrinos could provide a natural framework [143–146] for realizing the mechanism of resonant leptogenesis [147–149], where the leptonic CP asymmetry is resonantly enhanced when the mass splitting $\Delta {M}_{N}$ is of the same order as the decay width ${\Gamma }_{N}$.

As for the LNV signature at colliders, in a natural seesaw scenario with approximate lepton number conservation, the LNV amplitude for the on-shell production of heavy neutrinos at average four-momentum squared $\bar{s}=({M}_{{N}_{1}}^{2}+{M}_{{N}_{2}}^{2})/2$ can be written as

$$\begin{eqnarray}&&{{\mathcal{A}}}_{\mathrm{LNV}}(\bar{s})=-{V}_{{\ell }N}^{2}\displaystyle \frac{2\Delta {M}_{N}}{\Delta {M}_{N}^{2}+{\Gamma }_{N}^{2}}+{\mathcal{O}}\left(\displaystyle \frac{\Delta {M}_{N}}{{M}_{N}}\right),\end{eqnarray} \tag{ 10 }$$

for $\Delta {M}_{N}\lesssim {\Gamma }_{N}$, i.e. for small mass difference $\Delta {M}_{N}=| {M}_{{N}_{1}}-{M}_{{N}_{2}}|$ between the heavy neutrinos compared to their average decay width ${\Gamma }_{N}\equiv ({\Gamma }_{{N}_{1}}+{\Gamma }_{{N}_{2}})/2$. Thus, the LNV amplitude in (10) will be suppressed by the small mass splitting, except for the case $\Delta {M}_{N}\simeq {\Gamma }_{N}$ when it can be resonantly enhanced [147, 150]. This suppression for the inverse seesaw case is illustrated in figure 1 where we show the contours of ${\Gamma }_{N}/\Delta {M}_{N}$ for different values of the inverse seesaw mass parameters M_N and ${\mu }_{S}$. As we can see from the plot, to observe LNV with ${M}_{N}\gtrsim {\mathcal{O}}(100)$ GeV, one needs a large ${\mu }_{S}$, and therefore, small $| {V}_{{\ell }N}{| }^{2}$ (cf (9)), which will suppress the LNV signal. Its implications for collider searches of these scenarios will be discussed in section 2.3.

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By virtue of the last equivalence in (7), the inverse seesaw mechanism can be regarded as a variation of the type-I seesaw mechanism. Therefore, it is instructive to study the transition between the neutrino mass formulas given by (4) and (8). For illustration purposes, let us take a simplified version of (7) for a single generation case. The neutrino mass spectrum for this scenario is shown in figure 2 as a function of the LNV parameter ${\mu }_{S}$. As an example, we have chosen the Dirac mass term M_D = 1 GeV and the heavy neutrino mass term M_S = 100 GeV such that ${V}_{{\ell }N}\simeq {M}_{D}/{M}_{S}={10}^{-2}$ is consistent with the current upper limit set by the electroweak precision data (EWPD) [72]. First we consider the original inverse seesaw model with ${\mu }_{R}=0$ in (7). As shown in figure 2 (left panel), successful light neutrino mass generation occurs in this case only for $\mu \sim 1$ keV (cf (9)). As evident from (8), for ${\mu }_{S}\ll {M}_{S}$, the lightest neutrino mass is proportional to ${\mu }_{S}$, whereas the two heavy neutrinos form a quasi-degenerate Dirac pair with masses ${M}_{S}\pm {\mu }_{S}$. For ${\mu }_{S}\gg {M}_{S}$, the heaviest neutrino mass becomes equal to ${\mu }_{S}$ and the lighter ones form a quasi-degenerate Dirac pair with mass of order M_D. For ${\mu }_{R}\ne 0$, the situation remains unchanged for the general case ${\mu }_{R,S}\ll {M}_{S}$, as shown in figure 2 (right panel). However, for ${\mu }_{R,S}\gg {M}_{S}$, we recover the type-I seesaw with the lightest neutrino mass given by $-{M}_{D}^{2}/{\mu }_{R}$ (cf (4)), whereas the heavier ones form a Majorana pair with masses equal to ${\mu }_{R,S}$.

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Various laboratory searches have put stringent constraints on sterile neutrino mixing with active ones in a wide mass range of M_N from eV to TeV. For M_N values well below 1 MeV, as e.g. in eV-seesaw models [151], the sterile neutrinos can be probed by neutrino-oscillation experiments. Assuming all sterile neutrinos to be of the same order of magnitude, current data rule out $1\;\mathrm{neV}\lesssim {M}_{N}\lesssim 1\;\mathrm{eV}$ [151–154]. For $10\;\mathrm{eV}\lesssim {M}_{N}\lesssim 1\;\mathrm{MeV}$, the mixing of sterile neutrinos with electron neutrino has been constrained by searches for $0\nu \beta \beta$ and precision measurements of β-decay energy spectra. For $1\;\mathrm{MeV}\lesssim {M}_{N}\lesssim 1\;\mathrm{GeV}$, the mixing with both electron and muon neutrinos have been constrained by peak searches in leptonic decays of pions and kaons. Sterile neutrino mixing with all neutrino flavors in the MeV–GeV mass range has also been searched for through their decay products in beam dump experiments. Upper limits on the active-sterile neutrino mixing elements have also been derived from cosmological bounds on sterile neutrino lifetimes as required for the success of BBN [155–157].

Here we summarize the current state-of-the-art sterile neutrino searches in the mass range $100\;\mathrm{MeV}\leqslant {M}_{N}\leqslant 500\;\mathrm{GeV}$, as relevant for collider experiments at the energy frontier and other planned experiments at the intensity frontier. Figures 3–5 show the current constraints and some future projections on sterile neutrino mixing with the electron, muon and tau neutrinos, respectively. In these plots, the (gray) contour labeled 'BBN' corresponds to a heavy neutrino lifetime $\gt 1$ s, which is disfavored by BBN constraints [155–157]. The (brown) line labeled 'seesaw' shows the scale of mixing as expected in the canonical seesaw (cf (5)). We should remember that both these limits may get substantially modified in presence of more than two heavy neutrinos [133]. Other limits shown in figures 3–5 are explained below.

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2.2.1. Neutrinoless double beta decay

The contributions of heavy Majorana neutrinos ${N}_{R\alpha }$ to $0\nu \beta \beta$ amplitude is described by the standard neutrino exchange diagram between two β-decaying neutrons, via a non-zero admixture of a ${\nu }_{{Le}}$ weak eigenstate parametrized by the mixing element ${V}_{{{eN}}_{\alpha }}$. The $0\nu \beta \beta$ half-life is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{{T}_{1/2}^{0\nu }}={\mathcal{A}}{\left|\displaystyle \frac{{m}_{p}}{\langle {p}^{2}\rangle }\sum _{i=1}^{3}{U}_{{ei}}^{2}{m}_{i}\;+\;{m}_{p}\displaystyle \sum _{\alpha =1}^{{\mathcal{N}}}\displaystyle \frac{{V}_{{{eN}}_{\alpha }}^{2}{M}_{\alpha }}{\langle {p}^{2}\rangle +{M}_{\alpha }^{2}}\right|}^{2},\end{eqnarray} \tag{ 11 }$$

where ${\mathcal{A}}={G}^{0\nu }{g}_{A}^{4}| {{\mathcal{M}}}_{N}^{0\nu }{| }^{2}$ and $\langle {p}^{2}\rangle ={m}_{p}{m}_{e}| {{\mathcal{M}}}_{N}^{0\nu }/{{\mathcal{M}}}_{\nu }^{0\nu }{| }^{2}$. Here m_p and m_e are the proton and electron masses respectively, ${G}^{0\nu }$ is the phase-space factor, g_A is the nucleon axial-vector coupling constant, U_ei is the light neutrino mixing matrix, m_i and ${M}_{\alpha }$ are respectively the light and heavy neutrino mass eigenvalues, and ${{\mathcal{M}}}_{\nu }^{0\nu },{{\mathcal{M}}}_{N}^{0\nu }$ are the corresponding nuclear matrix elements (NMEs). These NMEs are conventionally calculated for the limiting cases ${m}_{i}\ll {p}_{F}$ (light) and ${M}_{\alpha }\gg {p}_{F}$ (heavy), p_F being the characteristic momentum transferred via the virtual neutrino, which is $\sim 200\;\mathrm{MeV}$ corresponding to the mean nucleon momentum of Fermi motion in a nucleus. However, the interpolating formula (11) allows us to calculate the $0\nu \beta \beta$ half-life for arbitrary heavy neutrino masses using the NMEs ${{\mathcal{M}}}_{\nu }^{0\nu }$ (light) and ${{\mathcal{M}}}_{N}^{0\nu }$ (heavy) [158, 159].

Using the combined 90% C.L. limit on $0\nu \beta \beta$ half-life ${T}_{1/2}^{0\nu }{(}^{76}\mathrm{Ge})\geqslant 3\times {10}^{25}\;\mathrm{years}$ from GERDA+Heidelberg-Moscow experiment [160], we derive from the second term in (11) upper limits on $| {V}_{{eN}}{| }^{2}$ as a function of a generic heavy neutrino mass M_N. Our results are shown in figure 3, where the shaded (orange) region between the solid and dashed lines, labeled 'GERDA', shows the uncertainty due to NMEs [159, 161]. Here we have used the recently re-evaluated phase-space factors [162] and the NMEs from a recent calculation within the quasi-particle random phase approximation (QRPA) [159, 163]. Similar limits are obtained using the half-life limit ${T}_{1/2}^{0\nu }{(}^{136}\mathrm{Xe})\geqslant 2.6\times {10}^{25}\;\mathrm{years}$ from KamLAND-Zen experiment [164, 165] and the corresponding QRPA NMEs [159].

From figure 3, it seems that the $0\nu \beta \beta$ constraints are very severe, thus shadowing the future prospects of observing LNV in other processes involving the electron channel. However, one must keep in mind that the $0\nu \beta \beta$ limits may be significantly weakened in certain cases when a cancellation between different terms in (11) may happen [166], e.g. due to the presence of Majorana CP phases. In general, the Majorana nature of neutrinos does not guarantee an observable $0\nu \beta \beta$ rate in all models [167]. Also, in the inverse seesaw scenario with pseudo-Dirac heavy neutrinos, the $0\nu \beta \beta$ limits are usually diluted by the small LNV term $\kappa ={\mu }_{S}/{M}_{S}$. Therefore, it is still important to include the electron channel while performing an independent direct search for heavy neutrinos at colliders.

2.2.2. Peak searches in meson decays

Peak searches in weak decays of heavy leptons and mesons are powerful probes of heavy neutrino mixing with all lepton flavors. The most promising are the two-body decays of electrically charged mesons into leptons and neutrinos: ${X}^{\pm }\to {{\ell }}^{\pm }N$ [168–170], whose branching ratio is proportional to the mixing $| {V}_{{\ell }N}{| }^{2}$. Thus, for a non-zero mixing and in the meson's rest frame, one expects the lepton spectrum to show a second monochromatic line at

$$\begin{eqnarray}&&{E}_{{\ell }}=\displaystyle \frac{{M}_{X}^{2}+{m}_{{\ell }}^{2}-{M}_{N}^{2}}{2{M}_{X}},\end{eqnarray} \tag{ 12 }$$

apart from the usual peak due to the active neutrino ${\nu }_{L{\ell }}$. For sterile neutrinos heavier than the charged lepton, the helicity suppression factor inherent in leptonic decay rate is weakened by a factor ${M}_{N}^{2}/{m}_{{\ell }}^{2}$ [169] due to which the sensitivity on $| {V}_{{\ell }N}{| }^{2}$ increases with M_N till the phase space becomes relevant. Peak searches have been performed in the channels $\pi \to {eN}$ [171–175], $\pi \to \mu N$ [176–180], $K\to {eN}$ [181] and $K\to \mu N$ [181–185]. The current 90% C.L. limits on $| {V}_{{\ell }N}{| }^{2}$ (for ${\ell }=e,\mu$) derived from these searches are shown in figures 3 and 4, labeled as '$X\to {\ell }\nu$' (with $X=\pi ,K$ and ${\ell }=e,\mu$). The limit from $\pi \to \mu N$ is not shown here, since it is only applicable in the mass range 1 MeV $\leqslant \;{M}_{N}\;\leqslant$ 30 MeV.

The peak searches could in principle be extended to higher masses with heavier meson/baryon decays [186–188]. For instance, the Belle experiment [189] used the decay mode $B\to X{\ell }N$ followed by $N\to {\ell }\pi$ (with ${\ell }=e,\mu$) in a data sample of 772 million $B\bar{B}$ pairs coming from ${\Upsilon }(4s)$ resonance to place 90% C.L. limits on $| {V}_{{eN}}{| }^{2}$ and $| {V}_{\mu N}{| }^{2}$ in the heavy neutrino mass range 500 MeV to 5 GeV, as shown in figures 3 and 4, labeled as 'Belle'.

Limits on the mixing parameter can also be set from the three-body decay of muons, where a sterile neutrino contribution may distort the spectrum of Michel electrons [169]. In case of τ-leptons, the two-body decays into hadrons $\tau \to {NX}$ are promising. If the hadronic system X hadronizes into charged pions or kaons, then its mass and energy can be reconstructed at high precision. Using future B-factories with a large dataset of τ decays like ${\tau }^{-}\to N{\pi }^{-}{\pi }^{+}{\pi }^{-}$, stringent limits on the mixing parameter $| {V}_{\tau N}{| }^{2}$ can be placed [190], as shown in figure 5, where the (red, solid and dashed) contours labeled B-factory are the conservative and optimistic projected limits at 90% C.L. from ∼10 million τ-decays.

We should note here that the bounds from peak searches are very robust since they use only the kinematic features and minimal assumptions regarding the decay modes of the heavy neutrino. Moreover, since the heavy neutrino is assumed to be produced on-shell, these limits are valid irrespective of whether the heavy neutrino is a Majorana or Dirac particle.

2.2.3. Beam dump experiments

2.2.4. Rare LNV decays of mesons

2.2.5. Z-decays

2.2.6. Electroweak precision tests

Heavy neutrinos with masses of the order of electroweak scale can be directly produced on-shell at colliders. Such a direct search was performed in ${e}^{+}{e}^{-}$ annihilation at LEP [221, 222], assuming a single heavy neutrino production via its mixing with active neutrinos: ${e}^{+}{e}^{-}\to N{\nu }_{L{\ell }}$, followed by its decay via NC or CC interaction to the SM W, Z or Higgs (H) boson: $N\to {\ell }W,\;{\nu }_{L{\ell }}Z,\;{\nu }_{L{\ell }}H$. Concentrating on the decay channel $N\to {eW}$ with $W\to$ jets, which would lead to a single isolated electron plus hadronic jets, the L3 collaboration put a 95% C.L. upper limit on the mixing parameter $| {V}_{{eN}}{| }^{2}$ in a heavy neutrino mass range between 80 and 205 GeV [222], as shown by the (red, solid) contour labeled 'L3 II' in figure 3. This search was mainly limited by the maximum center-of-mass energy $\sqrt{s}=208$ GeV at LEP. Future lepton colliders can significantly improve the sensitivity in this mass region, as illustrated in figure 3 by the projected limit labeled 'ILC', which is obtained assuming a $\sqrt{s}=500$ GeV ILC with luminosity of 500 fb⁻¹ [223].

In the context of hadron colliders, a Majorana heavy neutrino leads to the smoking gun lepton-number violating signature of same-sign dilepton plus jets with no missing transverse energy: ${pp}\;(p\bar{p})\to {W}^{*}\to N{{\ell }}^{\pm }\to {{\ell }}^{\pm }{{\ell }}^{\pm }{jj}$ [36, 150, 203, 224–229]. An inclusive search for new physics with same-sign dilepton signals was first performed in $p\bar{p}$ collisions at the Tevatron [230]. After the inauguration of the LHC era, the CMS and ATLAS collaborations have performed direct exclusive searches for the on-shell production of heavy neutrinos above the Z-threshold. The previous searches with 4.7 fb⁻¹ data at $\sqrt{s}=7$ TeV LHC set 95% C.L. limits on $| {V}_{{\ell }N}{| }^{2}\lesssim {10}^{-2}-{10}^{-1}$ (with ${\ell }=e,\mu$) for heavy neutrino masses up to 300 GeV [231, 232]. More recently, these limits were extended for masses up to 500 GeV with 20 fb⁻¹ data at $\sqrt{s}=8$ TeV [233, 234], and are shown in figures 3(ATLAS) and 4 (ATLAS and CMS). For ${M}_{N}\sim 100$ GeV, the direct limits in the muon sector are comparable to the indirect limits on $| {V}_{\mu N}{| }^{2}\lesssim {10}^{-3}$ imposed by the EWPD [69, 72] and LHC Higgs data [235, 236]. With the run-II phase of the LHC starting later this year with more energy and higher luminosity, the direct search limits could be extended for heavy neutrino masses up to a TeV or so. Also note that the LFV processes put stringent constraints on the product $| {V}_{{\ell }N}{V}_{{\ell }\prime N}^{*}|$ (with ${\ell }\ne {\ell }\prime$) [237–240], but do not restrict the individual mixing parameters $| {V}_{{\ell }N}{| }^{2}$ in a model-independent way. So the direct searches provide a complementary way to probe the light–heavy neutrino mixing in the seesaw paradigm.

All the direct searches at the LHC so far have only considered the simplest production process for heavy neutrinos through an s-channel W-exchange [36, 150, 203, 225–229], as shown in figure 6. However, there exists another collinearly enhanced electroweak production mode involving t-channel exchange of photons: ${pp}\to {W}^{*}{\gamma }^{*}\to N{{\ell }}^{\pm }{jj}$ [241], cf figure 7, which gives a dominant contribution to the heavy neutrino production cross section for higher M_N values. This is mainly because of the fact that with increasing heavy neutrino mass, the production cross section for the t-channel process drops at a rate slower than that of the s-channel process, though the exact cross-over point depends crucially on the selection cut for the p_T of the additional jet associated with the virtuality of the t-channel photon [241, 242]. The photon-mediated process ${pp}\to {W}^{*}{\gamma }^{*}\to N{{\ell }}^{\pm }{jj}$ has two contributions: an inelastic part with t-channel virtual photon and an elastic part $p\gamma \to N{{\ell }}^{\pm }j$ with a real photon emission from one of the protons. The elastic part is calculated using an effective photon structure function for the proton [243, 244], whereas the inelastic part is computed for a non-zero minimum p_T of the jet associated with the virtuality of the photon [241]. A comparison of the production cross sections for the s-channel Drell–Yan (DY) process ${pp}\to {W}^{*}\to N{{\ell }}^{\pm }$ and the photon initiated processes is made in figure 8 for a representative value of ${p}_{T,\mathrm{min}}^{j}=20$ GeV which, alongwith a jet separation cut $\Delta {R}^{{jj}}\gt 0.4$, is sufficient to ensure that the photon-mediated processes are collinear safe. We find that the total photon-initiated contribution becomes dominant over the DY cross section for ${M}_{N}\gtrsim 600$ GeV. Here, we have not included the QCD corrections, which could further lower the cross-over point to the level presented in [241], but this requires a more careful analysis and will be presented elsewhere. Note that the numerical results shown in figure 8 are slightly different from those presented in [242], which can be mainly attributed to the choice of regulator used to treat the collinear behavior.

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In any case, including the collinear enhancement effect could further enhance the heavy neutrino signal sensitivity at the next run of the LHC [241]. As an illustration, we have shown in figures 3 and 4 projected conservative limits with 300 fb⁻¹ data at $\sqrt{s}=14$ TeV (blue, dashed contours labeled 'LHC 14'), assuming that the cross-section limits are at least as good as the existing ones at $\sqrt{s}=8$ TeV, as reported in [234]. The direct collider limits for ${M}_{N}\lt 100$ GeV are not likely to improve significantly with higher collision energy, due to the increased pile-up effects, thus obfuscating the low-p_T leptons produced by the decay of a low-mass heavy neutrino. Instead, a displaced vertex search might be useful to probe the low-mass range between 3 and 80 GeV for mixing values ${10}^{-7}\lesssim | {V}_{{\ell }N}{| }^{2}\lesssim {10}^{-5}$ [245].

For heavy Dirac neutrinos as predicted in theories with approximate L-conservation (cf (6)), the same-sign dilepton signal is suppressed. In this case, the golden channel is the trilepton channel: ${pp}\to {W}^{*}\to N{{\ell }}^{\pm }\to {{\ell }}^{\pm }{{\ell }}^{\mp }{{\ell }}^{\pm }+{\rlap{/}{E}}_{T}$ [246–252]. Using this trilepton mode and also taking into account the infrared enhancement effects [241], direct limits on the mixing of heavy Dirac neutrinos with electron and muon neutrinos were obtained [251] by analyzing the tri-lepton data from $\sqrt{s}=8$ TeV LHC [253].

Finally, we note that there exist no direct collider searches for heavy neutrinos involving tau-lepton final states. This is mainly due to the experimental challenges of τ reconstruction at a hadron collider. The situation is expected to improve in future with better τ-tagging algorithms and/or in cleaner environments of a lepton collider.

The simplest extension of the SM gauge group to explain the heavy Majorana neutrino mass in (2) is the inclusion of an additional U(1) gauge group along with an associated Z' gauge boson. This symmetry can be added by hand to the SM but it could also naturally arise from a UV-complete theory such as the Pati–Salam model, SO(10) or E₆ GUTs. For a review of various Z' models, see e.g. [290]. The mass and couplings of the Z' boson are strongly constrained by EWPD. Constraints from lepton universality at the Z peak puts a lower limit of ${M}_{{Z}^{\prime }}\gtrsim {\mathcal{O}}(1)$ TeV [291], whereas direct searches at the LHC exclude ${M}_{{Z}^{\prime }}$ below about 2 TeV [3]. Similarly, the mixing angle between Z' and the SM Z is limited to be less than ${\mathcal{O}}({10}^{-4})$.

With regard to heavy neutrinos, the main phenomenological advantage is the possibility that the heavy neutrinos are charged under the additional U(1). This would provide a new and potentially strong production channel at colliders. A Feynman diagram for resonant heavy neutrino production at the LHC via such a Z' portal with a final state of two leptons and four jets [292] is shown in figure 9 (left panel). An important point to note is that the total cross section of this process is independent of the mixing strength ${V}_{{\ell }N}$. This process could therefore be observed at the LHC as long as the total decay width of the heavy neutrino is large enough such that it decays within the detector. The relevant parameter range for this to occur is shown in figure 9 (right panel) [293]. Even the canonical type-I seesaw with small mixing $| {V}_{{\ell }N}| \lesssim {10}^{-6}$ for TeV-scale M_N can be potentially probed in this case, possibly through displaced vertices. This also includes the potential to observe LFV signatures, despite the unobservably small LFV rates for low-energy processes such as $\mu \to e\gamma$, as they are strongly suppressed by such a small mixing.

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The minimal LRSM which extends the SM gauge symmetry to ${SU}{(2)}_{L}\times {SU}{(2)}_{R}\times U{(1)}_{B-L}$ [24, 25, 295] provides a simple UV-complete seesaw model, where the key ingredients of seesaw, i.e. the RH neutrinos and their Majorana masses, appear naturally. The presence of RH neutrinos is a necessary ingredient for the restoration of L–R symmetry and is also required by anomaly cancellation, whereas the seesaw scale is identified as the breaking of the ${SU}{(2)}_{R}$ symmetry. It is worth noting that in the presence of three RH neutrinos, the $(B-L)$-symmetry which was a global symmetry in the SM becomes a gauge symmetry in the LRSM, as the gauge anomalies cancel by satisfying the condition $\mathrm{Tr}{(B-L)}^{3}=0$. Moreover, the electric charge formula takes a form similar to the Gell-Mann–Nishijima relation: $Q={I}_{3L}+{I}_{3R}+(B-L)/2$, where I_3L and I_3R are the third components of isospin under ${SU}{(2)}_{L}$ and ${SU}{(2)}_{R}$ respectively, and the SM hypercharge can now be understood as $Y/2={I}_{3R}+(B-L)/2$ [296, 297].

In the LRSM, leptons are assigned to the multiplets ${L}_{{\ell }}={({\nu }_{{\ell }},{\ell })}_{L}$ and ${R}_{{\ell }}={({N}_{{\ell }},{\ell })}_{R}$ (where ${\ell }=e,\mu ,\tau$ is the generation index) with the quantum numbers $({\bf{2}},{\bf{1}},-1)$ and $({\bf{1}},{\bf{2}},-1)$ respectively under ${SU}(2)\times {SU}(2)\times U{(1)}_{B-L}$. The Higgs sector of the minimal LRSM contains a bidoublet ϕ with quantum numbers $({\bf{2}},{\bf{2}},0)$ and two triplets ${\Delta }_{L,R}$ with quantum numbers $({\bf{3}},{\bf{1}},2)$ and $({\bf{1}},{\bf{3}},2)$ respectively. The VEV v_R of the neutral component of ${\Delta }_{R}$ breaks the gauge symmetry ${SU}{(2)}_{R}\times U{(1)}_{B-L}$ to $U{(1)}_{Y}$ and gives masses to the RH gauge bosons W_R, Z_R boson and the RH neutrinos N_R. The VEVs $(\kappa ,{\kappa }^{\prime })$ of the neutral components of the bidoublet ϕ break the SM symmetry and are therefore of the order of the electroweak scale.

The LRSM Lagrangian relevant for the neutrino mass is given by

$$\begin{eqnarray}&&-{{\mathcal{L}}}_{Y}=h\bar{L}\phi R+\tilde{h}\bar{L}\tilde{\phi }R+{f}_{L}{L}^{{\mathsf{T}}}C{\rm{i}}{\sigma }_{2}{\Delta }_{L}L+{f}_{R}{R}^{{\mathsf{T}}}C{\rm{i}}{\sigma }_{2}{\Delta }_{R}R+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 13 }$$

where $\tilde{\phi }={\sigma }_{2}{\phi }^{*}{\sigma }_{2}$ and $h,\tilde{h},{f}_{L,R}$ are 3 × 3 complex Yukawa couplings. After symmetry breaking, equation (13) leads to the Dirac mass matrix ${M}_{D}=h\kappa +\tilde{h}\kappa \prime$ and the Majorana mass matrices ${M}_{L}={f}_{L}{v}_{L}$ and ${M}_{R}={f}_{R}{v}_{R}$ for the light and heavy neutrinos respectively, where v_L is the VEV of the neutral component of ${\Delta }_{L}$. This leads to the neutrino mass matrix

$$\begin{eqnarray}{{\mathcal{M}}}_{\nu }=\left(\begin{array}{cc}{M}_{L} & {M}_{D}\\ {M}_{D}^{{\mathsf{T}}} & {M}_{R}\end{array}\right),\end{eqnarray} \tag{ 14 }$$

as compared to the type-I case given by (3). In the usual seesaw approximation $\parallel {M}_{D}\parallel \ll \parallel {M}_{R}\parallel$, diagonalizing (14) leads to the light neutrino mass matrix of the form

$$\begin{eqnarray}&&{M}_{\nu }={M}_{L}-{M}_{D}{M}_{R}^{-1}{M}_{D}^{{\mathsf{T}}},\end{eqnarray} \tag{ 15 }$$

where the second term on the RHS is the type-I seesaw contribution which is inversely proportional to v_R, whereas the first one is the type-II seesaw contribution which is directly proportional to v_L. It should be noted here that in the minimal LRSM, if charge conjugation is the discrete L–R symmetry, ${M}_{D}={M}_{D}^{{\mathsf{T}}}$ and ${M}_{L}=({v}_{L}/{v}_{R}){M}_{R}$. In this case, the Dirac Yukawa couplings are generically constrained by the light and heavy neutrino mass and mixing parameters [298]. However, there are exceptions, e.g. in A₄ symmetry-based models, where

$$\begin{eqnarray}{M}_{D}\ \propto \ \left(\begin{array}{ccc}1 & \omega & {\omega }^{2}\\ \omega & {\omega }^{2} & 1\\ {\omega }^{2} & 1 & \omega \end{array}\right),\end{eqnarray} \tag{ 16 }$$

with ${\omega }^{3}=1$ and ${M}_{N}={m}_{N}{{\bf{1}}}_{3}$. For such symmetry-based models, ${M}_{\nu }$ vanishes identically for v_L = 0, independently of the size of the Dirac Yukawa couplings⁷ . Likewise, in versions of LRSM where parity and ${SU}{(2)}_{R}$ gauge symmetry scales are decoupled [299], the Dirac Yukawa couplings could be sizable [132], while being consistent with the light neutrino data.

Since processes induced by the RH currents and particles have not been observed so far, v_R has to be sufficiently large. In particular, hadronic flavor changing neutral current effects restrict ${M}_{{W}_{R}}\gtrsim 3$ TeV [300–303], assuming that the ${SU}{(2)}_{R}$ gauge coupling g_R has the same strength as the ${SU}{(2)}_{L}$ gauge coupling g_L. Direct search limits from the $\sqrt{s}=7$ and 8 TeV LHC data put similar constraints on ${M}_{{W}_{R}}$, depending on the heavy neutrino mass [304, 305]. This translates into a lower limit on ${v}_{R}=\sqrt{2}{M}_{{W}_{R}}/{g}_{R}\gtrsim 6.5$ TeV. On the other hand, for the the left-triplet VEV v_L, the electroweak ρ-parameter constraints set an upper limit on ${v}_{L}\lesssim 5$ GeV [306].

Due to the presence of RH gauge interactions, the LRSM gives rise to a number of new contributions to both LNV and LFV processes; see e.g. [307–309]. In particular, there are several diagrams that contribute to the $0\nu \beta \beta$ amplitude: (i) standard light neutrino exchange with mass helicity flip [310, 311], (ii) RH neutrino and RH gauge boson exchange [12, 19], (iii) RH Higgs triplet exchange [312], and (iv) mixed LH–RH contributions [78, 309, 313–316]. The latter depend on the size of the L–R neutrino mixing ${V}_{{\ell }N}\simeq {M}_{D}{M}_{R}^{-1}$. In the type-II seesaw dominance [317, 318], neglecting the mixed L–R as well as the canonical light-neutrino contributions, the $0\nu \beta \beta$ half-life due to purely RH currents can be written as

$$\begin{eqnarray}&&\displaystyle \frac{1}{{T}_{1/2}^{0\nu }}={\mathcal{A}}{\left|{m}_{p}{\left(\displaystyle \frac{{M}_{W}}{{M}_{{W}_{R}}}\right)}^{4}\displaystyle \sum _{i}\displaystyle \frac{{V}_{{ei}}^{2}}{{M}_{i}}\right|}^{2},\end{eqnarray} \tag{ 17 }$$

where V_ei is the mixing matrix for the RH neutrinos N_i with mass eigenvalues M_i and ${\mathcal{A}}$ is defined below (11). Using (17) and the current experimental limits on ${T}_{1/2}^{0\nu }$, lower limits on the RH gauge boson and RH neutrino masses can be derived [161, 307]. This is illustrated in figure 10, where the excluded areas from $0\nu \beta \beta$ searches are shown. Similarly, low-energy LFV processes such $\mu \to e\gamma$ and $\mu \to 3e$ can be drastically enhanced in the LRSM, with a host of new contributions [319]. This is also illustrated in figure 10 using maximal $e\mu$ flavor mixing of the heavy neutrinos [307].

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As for the LHC phenomenology, the presence of RH gauge interactions could lead to significant enhancement of the LFV/LNV signal. There are several contributions to the smoking gun LNV signal of same-dilepton plus two jets, as summarized in figure 10 (left). Even if the L–R neutrino mixing is small, heavy RH neutrinos could be directly produced via s-channel W_R exchange and subsequently decay via the same W_R [224]. The potential to discover LFV and LNV at the LHC in this scenario has been analyzed in [307, 320–323]. Figure 10 compares the sensitivity of LNV searches at the LHC with the sensitivity of $0\nu \beta \beta$ experiments. The $\sqrt{s}=14$ TeV LHC might be ultimately able to probe RH gauge boson masses up to ${M}_{{W}_{R}}=6$ TeV [324], whereas a futuristic $\sqrt{s}=80(100)$ TeV hadron collider could probe up to ${M}_{{W}_{R}}=26.6(35.5)$ TeV [325].

In variations of low-scale LRSM with large L–R neutrino mixing [132], there will be new contribution to the like-sign dilepton signal due to mixed RH-LH currents [326, 327], in addition to the purely RH and LH contributions. Note that the amplitude for the RR diagram in figure 10 (left) is independent of V_lN, and hence, does not probe the full seesaw matrix (14). On the other hand, the RL diagram is sensitive to the heavy–light mixing [326], and in fact, the dominant channel over a fairly large range of model parameter space, as illustrated in figure 11. Thus, a combination of the diagrams shown in figure 10 (left) is essential to fully explore the seesaw mechanism at the LHC. In this context, it is also useful to distinguish the RH gauge boson contributions to the collider signatures from the LH ones using different kinematic variables [326, 328].

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Recently, the CMS collaboration analyzed the $\sqrt{s}=8$ TeV LHC data with 19.7 fb⁻¹ resulting in the most stringent direct bounds on the RH gauge boson masses up to ${M}_{{W}_{R}}=3$ TeV [305]. While the analysis finds no significant departure from SM expectations, the LHC data exhibit an intriguing excess in ee production with a local significance of $2.8\sigma$ for a candidate W_R mass of ${M}_{{W}_{R}}\approx 2.1$ TeV; no excess is observed in the $\mu \mu$ channel. The excess could be interpreted as a hint for W_R production, but with a smaller RH gauge coupling ${g}_{R}/{g}_{L}\simeq 0.6$ [329–332]. If the excess turns out to be statistically significant in future with more data and independent scrutiny from ATLAS, it might be an evidence for L–R symmetry with high-scale parity breaking [299].

Just as light neutrino masses can be generated in a seesaw extension of the SM, a supersymmetric generalization of the seesaw mechanism can accommodate massive neutrinos in SUSY models with conserved R-parity. The superpotential of the type-I seesaw extension of the MSSM is given by

$$\begin{eqnarray}&&{\mathcal{W}}={{\mathcal{W}}}_{\mathrm{MSSM}}+{h}_{{ij}}{\epsilon }_{{ab}}{\widehat{L}}_{i}^{a}{\widehat{H}}_{u}^{b}{\widehat{N}}_{j}^{c}+\displaystyle \frac{1}{2}{({M}_{N})}_{{ij}}{\widehat{N}}_{i}^{c}{\widehat{N}}_{j}^{c},\end{eqnarray} \tag{ 19 }$$

where ${\widehat{L}}_{i}$ represents the chiral multiplet containing a ${SU}{(2)}_{L}$ lepton doublet ${(\nu ,e)}_{{Li}}$ and its corresponding superpartner, ${\widehat{H}}_{u}$ represents the Y = 1 Higgs doublet and its Higgsino superpartner, ${\widehat{N}}_{i}^{c}$ is a RH neutrino superfield, $i,j$ are family indices, $a,b$ are SU(2) indices and ${\epsilon }_{{ab}}$ is the antisymmetric SU(2) tensor. After EWSB, the neutrino mass matrix is given by (3), as in the non-SUSY case. Moreover, the SUSY analogue of the Majorana mass term in the sneutrino sector leads to sneutrino–antisneutrino mixing, which could give rise to a same-sign dilepton signal at colliders [357–361]. The neutrino Yukawa couplings lead to additional LFV effects in slepton masses, $\mu \to e\gamma$ in particular, through renormalization group effects in high-scale seesaw models [362–368]. In low-scale SUSY seesaw models, new sources of LFV are present due to large neutrino Yukawa couplings and threshold effects from low-scale RH neutrinos and sneutrinos [369–372]. In such scenarios, the LFV rates of $\mu -e$ conversion and $\mu \to 3e$ could be sizable, even bigger than the $\mu \to e\gamma$ rate.

In low-scale SUSY seesaw models, the lightest superpartner of the RH neutrino with a small admixture of its LH counterpart could be another viable candidate for DM, if it happens to be the lightest supersymmetric particle (LSP) [354, 373–377]. The sneutrino LSP scenario leads to distinct collider signatures, such as long missing transverse energy tail and enhanced same-sign dilepton signal in gluino and squark cascade decays [378–382], unlike the neutralino LSP in the MSSM scenario.

Within the SM, the requirement of gauge invariance automatically guarantees B and L conservation for all renormalizable interactions. However, this is not the case in the general MSSM, and the following B and L violating terms are allowed in the superpotential:

$$\begin{eqnarray}{{\mathcal{W}}}_{\mathrm{RPV}} & = & {\varepsilon }_{i}{\epsilon }_{{ab}}{\widehat{L}}_{i}^{a}{\widehat{H}}_{u}^{b}+{\lambda }_{{ijk}}{\epsilon }_{{ab}}{\widehat{L}}_{i}^{a}{\widehat{L}}_{j}^{b}{\widehat{E}}_{k}^{c}+{\lambda }_{{ijk}}^{\prime }{\epsilon }_{{ab}}{\widehat{L}}_{i}^{a}{\widehat{Q}}_{j}^{b}{\widehat{D}}_{k}^{c}\\ & & +{\lambda }_{{ijk}}^{\prime\prime }{\epsilon }_{{lmn}}{\widehat{U}}_{i}^{l}{\widehat{D}}_{j}^{{cm}}{\widehat{D}}_{k}^{{cn}},\end{eqnarray} \tag{ 20 }$$

where $i,j,k$ are family indices, $a,b$ are SU(2) indices, $l,m,n$ are color indices, and the chiral multiplets $\widehat{Q},{\widehat{U}}^{c},{\widehat{D}}^{c},{\widehat{E}}^{c}$ respectively represent the ${(u,d)}_{L},{u}_{L}^{c},{d}_{L}^{c},{e}_{L}^{c}$ and the corresponding superpartners. Within the MSSM, these terms are forbidden by imposing an additional global symmetry that leads to the conservation of R-parity, given by $R={(-1)}^{3(B-L)-2S}$ (with S being the spin of the component field) [383]. The first term in (20) leads to BRPV, while the remaining three terms collectively give rise to trilinear R-parity violation.

The RPV models provide an alternative way to incorporate massive neutrinos with the minimal particle content of the MSSM (for a review, see e.g. [384]). The BRPV model is the simplest one [44–53], and has several interesting consequences that can be probed at collider experiments (see e.g. [385–387]). Since the distinction between the matter doublet superfields ${\widehat{L}}_{i}$ and the Higgs doublet superfields ${\widehat{H}}_{d},{\widehat{H}}_{u}$ is lost in these models, it allows the mixing of neutrinos and neutralinos, sleptons and Higgs bosons, charged leptons and charginos. One linear combination of neutrino fields develops a Majorana mass at tree-level via mixing with Higgsinos, while the other combinations can acquire masses at loop-level via the trilinear couplings. Note that the trilinear coefficients $\lambda ,\lambda \prime$ and $\lambda \prime\prime$ are constrained from data on various low-energy B- and L-violating processes [388].

The collider phenomenology of RPV models has quite distinct features from that of the MSSM [384]. In particular, the LSP is unstable, and hence, not all SUSY decay chains lead to a large missing energy at colliders. The phenomenology of pair-produced SUSY particles is also modified due to new RPV decay chains. In addition, SUSY particles can now be singly produced, e.g. s-channel resonant production of sneutrinos in ${e}^{+}{e}^{-}$ collisions [389–391] and charged sleptons in hadron collisions [392–394]. RPV models also lead to sneutrino–antisneutrino mixing [395] and other low energy effects, such as $0\nu \beta \beta$ [396–398].