Heavy neutrino searches at future $Z$-factories

We analyze the capacity of future $Z$-factories to search for heavy neutrinos with their mass from 10 to 91 GeV. The heavy neutrinos $N$ are considered to be produced via the process $e^+e^-\to Z\to \nu N$ and to decay into an electron or muon and two jets. By means of Monte Carlo simulation of such signal events and the Standard Model background events, we obtain the upper bounds on the cross sections $\sigma(e^+e^-\to \nu N\to \nu\ell jj)$ given by the $Z$-factories with integrated luminosities of 0.1, 1 and 10 ab$^{-1}$ if no signal events are observed. On assumption of a minimal extension of the Standard Model in the neutrino sector, we also present the corresponding constraints on the mixing parameters of the heavy neutrinos with the Standard Model leptons, and find they are improved by at least one order compared to current experimental constraints.


I. INTRODUCTION
In the Standard Model (SM), only left-handed neutrinos are introduced and no mechanism is responsible for the generation of neutrino mass. However, the observation of neutrino oscillations [1,2] has given the evidence that neutrinos have tiny but non-zero mass, which may have opened a window towards the new dynamics beyond the SM. To explain the origin of neutrino mass and why they are much smaller than other fermion mass, different kinds of seesaw mechanisms [3][4][5][6][7][8][9][10][11] were proposed and work effectively as simple and straightforward methods. Among them, the Type-I Seesaw [3][4][5][6] is a quite natural extension of the SM by introducing gauge-singlet right-handed neutrinos without violating the SM gauge symmetries. Originally, it was proposed with the Majorana mass terms of the right-handed neutrinos at the scale of grand unified theories [12], which automatically lead to tiny neutrino mass. Later, it was found that a much lower Majorana mass scale, e.g. O(10) GeV, is also possible to explain the neutrino mass (see e.g. [13,14]), given small Yukawa couplings or some symmetry-protected cancellations in the neutrino mass matrix [15][16][17]. On the way to verify any of the Type-I Seesaw models, the most important evidence would be direct discoveries of heavy neutrinos. In this sense, the low-scale seesaw models are of special interest, because their particle spectra always contain heavy neutrinos with mass typically at O(10) GeV, within the reach of the colliders running now or in the near further. For example, extending the SM by three right-handed neutrinos with mass smaller than the electroweak scale [13,14], three heavy neutrinos beyond the SM spectrum are generated, one of which has a mass at keV scale and the other two at GeV to hundred GeV scale.
Another interesting property of this model is that the mixing between the light neutrinos and the heavy neutrinos can be quite large. Therefore, heavy neutrinos in such a model have good opportunities to be detected by collider experiments.
Experimentally, there have been direct searches for heavy neutrinos with O(10) GeV mass by the DELPHI Collaboration [18]. In their searches, the heavy neutrinos were considered to be produced via the e + e − → Z → νN process and decay into visible final states.
Unfortunately, no signals were observed and thus only upper bounds on mixing parameters were given. On the other hand, such heavy neutrinos also receive constraints from indirect searches like neutrinoless double-beta (0ν2β) decays [19][20][21], and can be explored in meson decays and τ decays [22]. Regarding future, there have been several lepton colliders pro-posed by different communities, including the Circular Electron Positron Collider (CEPC) [23], the International Linear Collider (ILC) [24], the FCC-ee (formerly known as the TLEP) [25] and the Super Z Factory. While the main target of these colliders is precision study of the Higgs boson properties, they will also be capable of searching for some new particles, new dynamics and even of studies of quantum chromodynamics and hadrons (see e.g. [26][27][28][29][30]). Most importantly here, they will be ideal facilities to search for heavy neutrinos. Such abilities of the CEPC with √ s = 240 -250 GeV and at the ILC with √ s = 1 TeV have been investigated by [31,32].
In this work, we will focus on searching for heavy neutrinos N with O(10) GeV mass at future Z-factories with integrated luminosities of 0.1, 1 and 10 ab −1 . Similar to DELPHI, we will also consider the signal production process e + e − → Z → νN of heavy neutrinos, and reconstruct the heavy neutrinos by their visible decaying final states each containing one charged lepton and two quark jets. The background mainly originates from the SM processes e + e − → τ + τ − and bb. Compared to a Higgs factory such as the CEPC with √ s = 240 -250 GeV, we find that future Z-factories are much more sensitive to heavy neutrinos with mass below 80 GeV, because the N production cross sections at the Z-mass pole is typically higher than those at 240 -250 GeV by orders. We have noticed that employing the technique of displaced vertex detection [33,34], a search for heavy neutrinos at a Zfactory is almost free of background and than can set even more stringent constraints to the mixing parameters between the heavy neutrinos and the SM leptons than a normal search.
However, the prerequisite of the displaced-vertex technique depends on a strong assumption that the studied heavy neutrinos have lifetimes long enough to fly a detectable distance before decaying, which is not true for many models. For example, if a heavy neutrino has decay channels with large decay widths, then it will have a very short lifetime so that its decay vertex is "not displaced". Heavy neutrinos in a model with a Majoron J [35,36], which was introduced to generate the Majorana scale in a ultraviolet-complete way, have such a feature. Typically, they can efficiently decay to the light Majoron via the invisible channels N → νJ and thus have short lifetimes. Therefore, while the constraints given by [33,34] do not apply to such heavy neutrinos, those given in this work are still valid. In other words, the results of our work without making use of the displaced-vertex technique are more model independent than [33,34].
The rest of the paper is orgnized as follows. In Section II, we will introduce the general setup of the new-physics scenario that we consider. The simulation of the production and decay processes of the heavy neutrinos and the corresponding background events at Zfactories will be described, and the event selection conditions will be explained in Section III. We will present the results in Section IV and conclude by Section V.

II. GENERAL SETUP OF THE SCENARIO
In this work, we consider a scenario generating the tiny neutrino mass via the Type-I seesaw mechanism but with a low Majorana mass scale (see e.g. [13,14]). It introduces n right-handed neutrinos R j (j = 1, ..., n), which are singlets of the SM gauge group SU(2), and their kinetic terms, mass terms and interaction terms with the SM fields are given by where H = iτ 2 H * , the lepton SU(2) L doublet L = (ν L , L ) T with = e, µ, τ , y is the 3 × n Yukawa coupling matrix and the n × n Majorana mass matrix M R is generated by some high-scale dynamics. After the spontaneous symmetry breaking of the Higgs field, we can diagonalise the neutrino mass matrix and obtain the 3 + n mass-eigenstate neutrinos, including 3 light neutrinos ν i and n heavy neutrinos N j 1 . Now, a flavor eigenstate is a superposition of the mass eigenstates, as and thus the neutrino-relevant weak interaction terms are given by From the above interaction terms (3), we read out that a heavy neutrino (with the mass smaller than the Z boson mass) can be produced associated with light neutrinos via 1 For clarification, here N j is not a Majorana neutrino by itself, but the left-handed component of a Majorana neutrino. It corresponds to N c j in [22]. e + e − → Z → νN (νN ), which is actually the dominant production process. 2 We also find that the produced heavy neutrino can decay weakly into one charged lepton and one on-shell or off-shell W boson up to its mass, N → − W +( * ) . Here, to reconstruct the heavy neutrinos,  To get a hint of the potential of a Z-factory in searching for such heavy neutrino signals, we compare the performances of a Z-factory and a Higgs factory with the electron-positron 2 Even when Z → N N is kinematically allowed, the cross section is doubly suppressed by the tiny mixing matrix elements |V j | 2 , which receive stringent constraints from previous experiments such as the DELPHI [18].
On the other hand, safely neglecting the difference between the charged lepton mass in N k decays, we have that Therefore, the two |V j | 2 factors get cancelled when we multiply σ(e + e − → νN k ) by B(N k → jj), and the relation σ(e + e − → νN k → ν jj) ∝ |V k | 2 is obtained. Here, we emphasize that unless specially noted, our analysis in this work will not depend on the assumption of this paragraph and last paragraph, which means that the validity of the results of this work is not limited to the specific model extending the SM with only the (1) terms.
The small peaks appearing where M N is slightly above the W boson mass are due to the opening of N decaying into an on-shell W boson. The decline of the cross sections at the mass close to 90 GeV results from the suppression of the phase space.

III. EVENT SIMULATION AND SELECTION
For event simulation, we use MadGraph [37] as the event generator for both the SM background and the signal with the new dynamics implemented via FeynRules [38][39][40] as mentioned previously. After that, Pythia8 [41] and Delphes [42] are used for further hadronization and fast detector simulation, respectively. In the detector simulation, the eekt algorithm and exclusive search have been chosen to construct jets, which, compared to the default antikt algorithm, is more efficient for a lepton collider and mitigates energy peak drifts of jets.
The signal events are produced from the processes e + e − → νN → ν jj, and in this work we consider the charged lepton to be an electron or a muon. In both cases, the main background comes from the e + e − → bb and e + e − → τ + τ − processes, from which the bb and τ + τ − pairs further decay into final states containing one charged lepton, two jets and missing energy, / E jj, as shown in FIG. 3. As for the other qq events with q being a lighter quark, production at the Z-mass pole makes them highly boosted such that the decay products of the q or q are more likely to be as collinear as constituents of one jet. Therefore, it is much more difficult for a lighter quark pair qq to mimic a signal event than a bb pair.

FIG. 3: Sample Feynman diagrams of the background.
In the following, we will discuss how we choose the event selection conditions to suppress the background and increase the signal significance.
For the τ τ background as shown in the left panel of FIG. 3, the two τ leptons in an event fly back-to-back with a large boost. Therefore, the charged lepton and the total missing energy in the final state are almost collinear or reverse to each other, and the angle between the two jets decaying from the same τ are very small. These inspire some effective cuts on the angular distances ∆R jj and ∆R / E , where ∆R = ∆η 2 + ∆φ 2 with ∆η and ∆φ being the differences between the pseudorapidities and the azimuthal angles of the two involved particles, respectively.
In a bb background event as shown in the right panel of FIG. 3, the neutrino, the charged lepton and one of the two jets decay from the same bottom quark with a large boost, so the angular distances between the charged lepton and the neutrino and between the jet and the neutrino are both small. Therefore, we set cuts on ∆R / E and ∆R / Ej to reduce the background from the bb process.
Considering the signal process e + e − → νN , the energy of the resolved light neutrino is fixed owing to the momentum-energy conservation, as which can also provide an effective cut to suppress the background.
Inspired by the above considerations and some practical tests, we choose the following event selection conditions for the three categories depending on different heavy neutrino mass, the small-mass range (M N < 65 GeV), the middle-mass range (65 GeV < M N < 80 GeV) and the large-mass range (80 GeV < M N < 91 GeV).
• The event selection conditions for the small-mass range (M N < 65 GeV): -P j T > 3 GeV, |η j | < 2, M jj > 2 GeV, btag < 0.8, TauTag = 0, BTag = 0; -P T > 3 GeV, |η | < 1; -/ E T > 20 GeV; • the event selection conditions for the middle-mass range (65 < M N < 80 GeV): -P j T > 3 GeV, |η j | < 2, M j > 1.2 GeV, ∆R jj > 0.4, btag < 0.8, TauTag = 0, BTag = 0; -P T > 3 GeV, |η | < 1; -∆R / Ej > 1.0, 1.5 < ∆R / E < 5.0; • the event selection conditions for the large-mass range (80 < M N < 91 GeV): -P j T > 10 GeV, |η j | < 2, M j > 1.2 GeV, M jj > 55 GeV, ∆R jj > 0.4, btag < 0.8, TauTag = 0, BTag = 0; -P T > 3 GeV, |η | < 1; In general, for each event, we set cuts on the transverse momenta P T and the pseudorapidities |η| of the charged lepton and the jets, the transverse missing energy / E T , the invariant mass of each jet M j and of the two jets M jj , the angular distances between the two jets ∆R jj , between each jet and the missing energy ∆R / Ej and between the missing energy and the charged lepton ∆R / E . In addition, from the signal simulation we get the central value of the detected missing energies / E 0 and the half-height width Γ / E 1/2 of the distribution, and we require that the missing energy / E in each event does not lie beyond the region / E 0 ± Γ / E 1/2 . To further suppress the background, we also use the btag, BTag and TauTag, which gives information of the probability of a jet being a b-jet, whether or not a jet has been tagged as containing a heavy quark and whether or not a jet has been tagged as a tau, respectively.
For the small-mass range, we find that the most efficient cut is the / E T cut, which removes most of the background without touching most of signal events. In this case, the τ τ and bb processes have similar contributions to the background. For the middle-and large-mass ranges, the most efficient cut to suppress the τ τ background is the M j cut, and the most efficient one to suppress the bb background is the ∆R / Ej cut. In these two cases, the bb processes dominate the background.

IV. RESULTS AND ANALYSIS
In this section, based on the simulation of the signal and background events, we present the capacity of Z-factories to search for heavy neutrinos with three benchmark integrated luminosities L, 0.1 ab −1 , 1 ab −1 and 10 ab −1 . In practice, we estimate the expected upper bounds on the cross sections σ(e + e − → νN → ν jj) at 95% confidence level (CL), which can be approximately obtained by solving the equation with s ≈ 1.7. This equation is understood as follows. For convenience, we only perform the simulation once for each heavy neutrino mass, with a reference setup of a specific luminosity In principle, the measurement of the total width of the Z boson Γ Z = 2.4952±0.0023 GeV [43] also sets constraints to the e + e − → Z → νN cross sections, but we find that heavy neu- The upper bounds on the mixing parameters |V eN | 2 (left) and |V µN | 2 (right) given by Z-factories at 95% CL, compared to the upper bounds given by DELPHI [18], the 0ν2β decay experiments [19][20][21] and the CEPC as a Higgs factory [32]. The green, pink and red curves correspond to integrated luminosities of 0.1 ab −1 , 1.0 ab −1 and 10 ab −1 , respectively. See text for details.
Section II. For both |V eN | 2 and |V µN | 2 , we find a large improvement compared to DELPHI [18], the upper bounds being decreased typically by two orders of magnitude even with the lowest luminosity setup. For |V eN | 2 , the given upper bounds by the considered Z-factories are lower than that given by the 0ν2β decay experiments [19][20][21] by at least one order of magnitude in most of the mass range of the heavy neutrino. While for |V µN | 2 , the upper bounds given by the Z-factories are at least two orders of magnitude lower than that given by the CEPC as a Higgs factory [32] when M N < 70 GeV. One may worry that a heavy neutrino with mixing parameters as small as such bounds will be so stable that the detection of its decay is always out of reach by detectors. If this is true, the bounds given in FIG. 6 will not be valid, since all our analyses are based on the assumption that the signal events can be detected within detectors once they happen. To clarify that this will not be a problem, we estimate how far a 10 GeV heavy neutrino can typically fly before its decay, and the flying distances of the other heavier neutrinos are always shorter given the same relevant mixing parameters. Considering a detector having a diameter of O(1) meter, we find that as long as the mixing parameter |V N | 2 is larger than O(10 −9 ), the 10 GeV heavy neutrino is most likely to decay before it can fly out of the detector. Comparing O(10 −9 ) with the bounds in  FIG. 6, one can find that the bounds in the cases with L = 0.1 and 1 ab −1 are not affected by the limited size of the detector, and that the case with L = 10 ab −1 is also basically safe.

V. CONCLUSION
To conclude, we have presented a study of possible heavy neutrino searches at future Z-factories. For different heavy neutrinos with mass ranging from 10 to 91 GeV, we have obtained the expected upper bounds on the production cross sections of their discovery processes e + e − → νN → νejj and e + e − → νN → νµjj given by Z-factory with L = 0.1, 1 and 10 ab −1 , respectively. Under the assumption that the interactions between the heavy neutrinos and the SM particles are only induced by the neutrino mixing, the constraints on the cross sections have been translated to the constraints on the corresponding mixing parameters |V eN | 2 and |V µN | 2 , which, depending on the luminosity setup, are typically improved by two to four orders compared to the DELPHI constraints [18]. We also find that the future Z-factories will set much more stringent constraints on |V eN | 2 than the 0ν2β decay experiments [19][20][21] by one to three orders, and on |V µN | 2 than the CEPC as a Higgs factory [32] by two to three orders, given the heavy neutrino mass is smaller than 80 GeV.