Displaced Heavy Neutral Lepton from New Higgs Doublet

In this section, we aim to search for the DV signature of heavy neutral leptons at 14 TeV LHC. It should be noted that the Yukawa matrix $y$ and mixing pattern $V_{\ell N}$ are sophisticated for the theoretical predictions of the $\nu\rm{2HDM}$. Following the spirit of the experimental groups, we focus on a simplified scenario, in which neutrinophilic scalars and heavy neutral lepton $N$ only couple to electron or muon exclusively. Due to much lower tagging efficiency, we do not consider the tau channel in this analysis. At the 14 TeV LHC, the signal process can be written as

where $\ell=e$ or $\mu$ exclusively. Corresponding signal processes are shown in Figure 4. The pair production of the charged scalar process $pp\rightarrow{H^{+}}{H^{-}}$ is mediated by the virtual $\gamma^{*}$ and $Z^{*}$, and the associated production processes $pp\rightarrow{H^{\pm}}{A/H}$ are mediated by the virtual $W^{*\pm}$. Due to the doublet nature of neutrinophilic scalars, the production cross sections at 14 TeV LHC only depend on their masses. For $m_{H^{+}}$ less than 1000 GeV, the cross sections are larger than 0.01 fb, which are hopefully to be detected Guo:2017ybk . Decay of charged scalar $H^{\pm}$ leads to a prompt charged lepton. In order to trigger the DV signal, we require at least one prompt charged lepton in the final states. Therefore, we do not include the contribution of process $pp\to HA\to\nu_{\ell}N\nu_{\ell}N$ in the following analysis.

The further decays of HNL generate the DV signature, which depends on its mass $m_{N}$ and mixing parameter $V_{\ell N}$ as shown in Figure 2. Theoretically speaking, there are two HNLs in the decay of neutrinophilic scalars, which could lead to two DVs signature. With higher tagging efficiency, the one DV signature would have a wider sensitive region than the two DVs signature in principle. However, the one DV signature corresponds to the inclusive process $pp\to\ell^{\pm}N+X$ with $X$ denoting the undetected particles, which has an additional contribution from $pp\to W^{\pm}\to\ell^{\pm}N$. The latter process could become the dominant channel for relatively large mixing parameter $V_{\ell N}$, which then makes it hard to distinguish various seesaw models with HNL. This ambiguity can be solved when $V_{\ell N}$ is small enough to make $H^{\pm}\to\ell^{\pm}N$ the dominant contribution, especially for the region as $m_{N}>m_{W}$. On the other hand, the two DVs signature with prompt charged lepton only arises in the $\nu\rm{2HDM}$, but the reconstruction processes for two DVs signal will lost most of the events.

The SM backgrounds of DV signature mainly stem from long-lived heavy flavor hadrons, e.g., $B^{0}$ meson, which can be effectively excluded by cuts on invariant mass and displacement Drewes:2019fou . The backgrounds may also be from the random crossing of tracks that compose a fake DV, from nuclear interactions with the detector material, or from fake photons, cosmic rays, and beam-halo muons, which are hard to simulate and estimate. The full SM background analysis is beyond the scope of this work, but is safe to consider it as one for an optimistic estimation Drewes:2019fou . In searching for one DV signature, we aim to figure out the contribution of neutrinophilic scalars, so the inevitable $pp\to W^{\pm}\to\ell^{\pm}N$ process is treated as an irreducible background. Since this new physics background is only sensitive within the region of $|V_{\ell N}|^{2}\gtrsim 10^{-9}$ and $m_{N}\lesssim 30$ GeV at LHC, we do not seek further cuts to suppress it.

Distributions of some related parameters for selection cuts are shown in Figure 5 . Three benchmark points are selected as $m_{H^{+}}=200$ GeV while $m_{N}=$30 GeV, 50 GeV, and 70 GeV with $|V_{\ell N}|^{2}=10^{-10}(\ell=e~{}\rm{or}~{}\mu)$ respectively. With a relatively clear environment for DVs search at LHC, here we do not include any SM background as a reference. Meanwhile, the cross section of new physics background $pp\to W^{\pm}\to\ell^{\pm}N$ is less than $10^{-3}$ fb with $|V_{\ell N}|^{2}=10^{-10}$, thus it is also negligible.

After the simulations, we first select events carrying at least one electron (muon) with transverse momentum greater than 20 GeV and pseudo-rapidity of $|\eta_{\ell}|<2.5$, which are the trigger cuts of the DV signature. For a lighter $m_{N}$, the prompt lepton tends to be more energetic. Distributions of $P_{T}^{e}$ are similar to those of $P_{T}^{\mu}$ for the same value of $m_{N}$, even though the multiplicity of muons is slightly greater than that of electrons due to the different acceptance at the LHC detector.

Then we select tracks by using a softer momentum cut $P^{trk}>5~{}\rm{GeV}$ to escape the magnetic field, as well as a large transverse impact parameter of $d_{0}>2~{}\rm{mm}$. The parameter $d_{0}$ is defined as

where $x^{trk},y^{trk}$ are track positions in the transverse plane from the primary interaction vertex, $P_{x}^{trk},P_{y}^{trk}$ are the $x$- and $y$-components of the track momentum, and $P_{T}^{trk}=\sqrt{(P_{x}^{trk})^{2}+(P_{y}^{trk})^{2}}$ is the transverse momentum of a track. With the fixed value of $|V_{\ell N}|^{2}=10^{-10}$, the maximum value of $d_{0}$ decreases as $m_{N}$ increases. In this sense, the case with $m_{N}=30$ GeV is more promising for DV signature than those with heavier $m_{N}$.

Considering that the dominant decay products of HNL always contain an electron (muon) as shown in Figure 2, we require that the reconstructed displaced vertex has at least one electron (muon) track. In this work, the long-lived HNLs are considered to decay before the muon chamber. Since tracks from the same HNL are expected to share the same origin point, we reconstruct the displaced vertex by requiring that at least two charged tracks satisfy $\Delta{x}<1{\rm{mm}},\Delta{y}<1{\rm{mm}},\Delta{z}<1{\rm{mm}}$, and tracks are isolated with the condition $\Delta{R}=\sqrt{{\Delta\eta}^{2}+{\Delta\phi}^{2}}>0.1$. In order to reduce backgrounds from long-lived SM hadrons, the reconstructed displaced vertex is required to satisfy a displacement cut of $5~{}{\rm{mm}}<l_{0}<3000~{}\rm{mm}$ and an invariant mass cut of $m_{DV}\geq 5~{}\rm{GeV}$. Since the DV signature originates from the HNL, $m_{DV}\lesssim m_{N}$ is expected. Here, the upper bound on $m_{DV}$ is not applied so as to extract the mass of HNL. We summarize all of the selection cuts in Table.1.

Assuming Poisson distribution, the significance for $n$ observed events is calculated as Cowan:2010js

where $P(n|b)={b^{n}}e^{-b}/{n!}$ is the Poisson probability, $b$ is the event number of the backgrounds, and $n=b+s$ is the total event number of background and signal. With one background event, we should have at least nine signal events for the discovery of $S(n|b)>5$. Meanwhile, the system uncertainty of signal and background estimation are overlooked. The significance of benchmark points is also shown in Table 1. For $m_{N}=30$ GeV and $m_{N}=50$ GeV, the cross sections after all selection cuts are similar. With 3 ab^-1 data, the significance could reach about 1300 for one DV pure electron(muon) channel. The significance for two DVs pure electron(muon) channels are smaller than one DV, which could reach about 600. For $m_{N}=70$ GeV, although it is less promising than the previous two benchmark points, the significance could reach 90 even for the two DV channels.

Based on the above selection cuts, we first analyze the events with at least one DV. The results for a fixed value of $m_{H^{\pm}}=200$ GeV are shown in the upper two panels of Figure 6. It is clear that the DV signature can be discovered within the parameter space of 3 GeV $<m_{N}<200$ GeV when $10^{-19}<|V_{lN}|^{2}<10^{-4}$. Since the production of HNL is not suppressed by the mixing parameter $V_{\ell N}$, we can explore the range of $|V_{\ell N}|^{2}$ below seesaw predicted value. Compared with the canonical $W^{\pm}\to\ell^{\pm}N$ channel, the viable parameter space of the charged scalar $H^{\pm}\to\ell^{\pm}N$ channel is much larger. For instance, the $W^{\pm}\to\ell^{\pm}N$ channel could only probe $|V_{\ell N}|^{2}\gtrsim 10^{-9}$ with $m_{N}\lesssim 30$ GeV at LHC Drewes:2019fou , while the $H^{\pm}\to\ell^{\pm}N$ channel could probe $|V_{\ell N}|^{2}\gtrsim 10^{-19}$ with $m_{N}<200$ GeV. This upper limit corresponds to the kinetic threshold of $H^{\pm}\to\ell^{\pm}N$ decay, since we fix $m_{H^{\pm}}=200$ GeV in the analysis. The displacement cut $l_{0}\in[5,3000]$ mm leads to the upper and lower bound on $|V_{\ell N}|^{2}$ for certain $m_{N}$. Qualitatively speaking, the larger the $m_{N}$ is, the smaller the $|V_{\ell N}|^{2}$ would be. For $m_{N}=10$ GeV, the promising range is $10^{-11}\lesssim|V_{\ell N}|^{2}\lesssim 10^{-5}$, which just reaches the seesaw predicted limit. For $m_{N}=100$ GeV, the promising region becomes $10^{-18}\lesssim|V_{\ell N}|^{2}\lesssim 10^{-10}$. There is a dip around $m_{N}\sim m_{W}$, because the two-body decays as $N\to\ell^{\pm}W^{\mp}$ is allowed when $m_{N}>m_{W}$, so a much smaller $|V_{\ell N}|^{2}$ is required to satisfy the displacement cut of $l_{0}$. The sensitivity region of electron mixing and muon mixing patterns are quite similar. Although the acceptance rates of electron and muon at the detector are slightly different, it is shaded by the relatively large uncertainty of the Monte Carlo simulation procedure.

We then tighten the selection cuts by requiring at least two DVs in the final state while keeping the other cuts in Table 1 unchanged. The results are shown in the lower two panels of Figure 6. Considering that the acceptance rate for DVs search at LHC detectors decreases linearly with length Alimena:2019zri ; Knapen:2022afb , we can simply assume that the reconstruction efficiency for two DVs signature search would decrease quadratically. Therefore, the overall discovery region of two DVs is $10^{-18}<|V_{\ell N}|^{2}<10^{-4}$ with HNL mass dependence, which is slightly smaller than that of one DV. In contrast to the one DV signature, the two DVs signal is free from the canonical $W^{\pm}\to\ell^{\pm}N$ background. For $m_{N}\sim$ few GeV with $|V_{\ell N}|^{2}\gtrsim 10^{-5}$, the two DVs channel is better than the one DV channel to probe the new contribution from neutrinophilic scalars. However, such a region is now already excluded by LHC direct search.

Besides the HNL mass $m_{N}$ and mixing parameter $V_{\ell N}$, the cross section of the DV signature also depends on the charged scalar mass $m_{H^{+}}$. In Figure 6, it is shown that the upper bound of $m_{N}$ for the DV signature corresponds to the charged scalar mass $m_{H^{+}}$ with the proper mixing parameter $V_{\ell N}$. By setting the mixing parameter $|V_{lN}|=10^{-10}$ and $|V_{lN}|=10^{-14}$, we then explore the promising region of the DV signature in the $m_{N}-m_{H^{+}}$ plane at the HL-LHC. The benchmark value $|V_{\ell N}|=10^{-10}$ is larger than the canonical seesaw prediction value $|V_{\ell N}|^{2}\simeq m_{\nu}/m_{N}\sim 10^{-12}$, meanwhile the benchmark value $|V_{\ell N}|=10^{-14}$ is smaller than the theoretical favor prediction. The results are shown in Figure 7. For one DV signature, the sensitive region of $m_{H^{+}}$ can reach 1200 GeV. Due to lower reconstruction efficiency, the sensitive region of $m_{H^{+}}$ for two DV signal would reach about 1100 GeV. A larger value of $m_{H^{+}}$ will lead to a smaller sensitive region of $m_{N}$. For example, the sensitive region is $m_{N}\in[10,80]$ GeV when $m_{H^{+}}=300$ GeV and $|V_{\ell N}|^{2}=10^{-10}$, which reduces to about $m_{N}\in[20,70]$ GeV when $m_{H^{+}}=700$ GeV. Considering similar tagging efficiency for electron and muon at LHC, the pure muon mixing won’t give essentially different results from the electron mixing pattern.

For the promising region of $m_{N}$, it heavily depends on the mixing pattern. A smaller mixing parameter $V_{lN}$ usually requires a larger $m_{N}$ to satisfy the displacement cut $5~{}{\rm{mm}}<l_{0}<3000~{}\rm{mm}$. When $|V_{\ell N}|^{2}=10^{-10}$, the promising regions always satisfy $m_{N}<m_{W}$ for both one DV and two DVs signature, which indicates that the three body decays of HNL $N$ are the dominant decay modes for mixing parameter $|V_{\ell N}|^{2}$ larger than the seesaw predicted value. On the other hand, when $|V_{\ell N}|^{2}(=10^{-14})$ is smaller than the theoretical favor value, most of promising regions of DV signatures would satisfy $m_{N}>m_{W}$, so two body decays of HNL $N$ become the dominant channels. In this scenario with $|V_{\ell N}|^{2}=10^{-14}$, one may probe $m_{N}$ up to $\sim 300$ GeV.

The most natural scenario is the seesaw predicted value $|V_{\ell N}|^{2}=m_{\nu}/m_{N}$. The results are also shown in Figure 7. As already shown in Figure 6, current DV searches via the $W^{\pm}\to\ell^{\pm}N$ channel can not probe such tiny mixing parameter. With an unsuppressed cross section, we find that the $H^{\pm}\to\ell^{\pm}N$ channel could probe $m_{H^{+}}\lesssim 1200$ GeV in this scenario. The promising range of $m_{N}$ heavily depends on $m_{H^{+}}$. Typically for $m_{H^{+}}=600$ GeV, we could probe $m_{N}\in[30,110]$ GeV in the one DV channel and $m_{N}\in[50,100]$ GeV in the two DVs channel.

The Compact Linear Collide (CLIC)ILC:2007oiw ; CLICdp:2018cto ; Brunner:2022usy is a proposed multi-TeV $e^{+}e^{-}$ linear collider. In this paper, we consider the 3-TeV collision energy stage with a high luminosity of $\mathcal{L}=5~{}{\rm ab}^{-1}$. According to the results in Figure 3, the cross section of $H^{+}H^{-}$ at CLIC is much larger than it is at LHC for TeV scale $m_{H^{+}}$. So CLIC is expected more promising to probe the heavily charged scalar region. In the $\nu$2HDM model, the long-lived HNL can be generated at CLIC through ${e^{+}}{e^{-}}\rightarrow{H^{+}}{H^{-}}\rightarrow{\ell^{+}}N\ell^{-}N$ process. As shown in Figure 8, there are two different production channels for $H^{+}H^{-}$. The $s$-channel process is mediated by virtual $\gamma^{*}$ or $Z^{*}$, which only depends on the mass of the charged scalar. On the other hand, the $t$-channel process is mediated by the HNL $N$, which is also determined by the Yukawa coupling $y$. In principle, large $y_{eN}$ can be obtained by tunning the structure of Yukawa matrix $y$. For simplicity, we consider $y\sim 10^{-2}$ to avoid tight constraints from the lepton flavor violation. In this way, the contribution of the $t$-channel process can be neglected. So the following results of CLIC in this study are conservative.

Similar to the LHC study, we also focus on the decay of charged scalar $H^{\pm}\to\ell^{\pm}N$ to distinguish the $\nu$2HDM from other HNL models in this section. Therefore, the DVs signature from $e^{+}e^{-}\to HA\to\nu_{\ell}N\nu_{\ell}N$ is not taken into account. It should be mentioned that the DVs signature also arises from process $e^{+}e^{-}\to NN$ via $t$-channel mediator of $H^{\pm}$. However, the cross section of $e^{+}e^{-}\to NN$ is also suppressed by small Yukawa coupling $y\sim 10^{-2}$ in our consideration. On CLIC, searches for long-lived particles through DVs signature have no irreducible SM background. There is one irreducible background from new physics as $e^{+}e^{-}\to W^{+}W^{-}\to\ell^{+}N\ell^{-}N$. However, the corresponding cross section is suppressed by the mixing parameter as $|V_{\ell N}|^{4}$, which is negligible under current experimental limits. So it is still safe to consider the total backgrounds as one.

To search for the possible existence displaced vertex signature on CLIC, we perform a similar analysis as on LHC. In Figure 9, we show distributions to use. As before, three benchmark points are selected as $m_{N}=30,50,70$GeV with $m_{H^{+}}=200$ GeV and $|V_{\ell N}|^{2}=10^{-10}(\ell=e~{}\text{or}~{}\mu)$. Considering the clean environment for DVs search at CLIC, we did not put any backgrounds as a reference. Distributions of electron mixing and muon mixing are quite similar to each other. Benchmark points can be distinguished by variables as $P_{T}^{\ell},d_{0},l_{0},m_{DV}$. As different detector geometries of CLIC and LHC, we have changed the selection cuts on displacement as $5~{}{\rm{mm}}<l_{0}<1100~{}\rm{mm}$, meanwhile other cuts are kept unchanged. We summarize all of the selection cuts for CLIC in Table 2.

For the three benchmark points of $m_{N}=$30 GeV, 50 GeV, 70 GeV, the significance could reach 290(292), 407(408), 211(211) for one DV pure electron(muon) mixing pattern, and 82(82), 193(190), 44(44) for two DVs pure electron(muon) mixing pattern. Qualitatively speaking, the electron channels have similar significance as the muon channels at CLIC. With $|V_{\ell N}|^{2}=10^{-10}$, the most promising benchmark point is $m_{N}=50$ GeV. Compared with LHC, the significance for benchmark points at CLIC are smaller, because of the smaller production cross section of $H^{+}H^{-}$ for $m_{H^{+}}=200$ GeV.

We then explore the parameter space for one DV signature with fixed charged scalar mass $m_{H^{\pm}}=200$ GeV. The results are shown in the upper two panels of Figure 10. In both the electron and the muon mixing pattern, we can probe $|V_{\ell N}|^{2}\gtrsim 10^{-18}$ when $m_{N}>3.5$ GeV. For seesaw induced mixing $|V_{\ell N}|^{2}=m_{\nu}/m_{N}$, the one DV signature could discover 20 GeV $\lesssim m_{N}\lesssim 130$ GeV. Compared with the sensitive regions of LHC, the regions of CLIC are slightly smaller, because the cross section of $H^{+}H^{-}$ at CLIC is smaller than it is at LHC for $m_{H^{\pm}}=200$ GeV. Therefore, if no DV signature is discovered at LHC, then CLIC can hardly have any positive DV signature for light charged scalar.

For the two DVs signature, scanned results are shown in the lower two panels of Figure 10. Compared with the one DV signal, the promising areas become smaller due to more strict selection cuts. For pure electron or muon mixing pattern, we can probe HNLs with the square of the mixing parameter as small as $|V_{\ell N}|^{2}\sim 10^{-16}$, and the mass as small as $m_{N}\sim 8$ GeV. In the narrow mass region between 40 GeV and 110 GeV, we may discover the two DV signature with seesaw induced mixing $|V_{\ell N}|^{2}=m_{\nu}/m_{N}$. We also find that the two DV signature at CLIC could not probe regions with $|V_{\ell N}|^{2}>10^{-6}$, which is different from LHC. According to Figure 6, $|V_{\ell N}|^{2}>10^{-6}$ requires $m_{N}\lesssim 10$ GeV for observable DV signature. Due to the selection cut on the reconstructed DVs mass of $m_{DV}>5$ GeV, the acceptance efficiency for the DV signal is relatively low in this region. Meanwhile, the cross section of the signal process on 3 TeV CLIC is much smaller than on 14 TeV LHC for $m_{H^{+}}=200$ GeV, so the two DVs signature is not sensitive for mixing parameter $|V_{\ell N}|^{2}>10^{-6}$.

To obtain the limitation of CLIC on $\nu$2HDM, we then scan the parameter space in the $m_{H^{\pm}}-m_{N}$ panel with the mixing parameter $|V_{\ell N}|^{2}=10^{-10}$ and $|V_{\ell N}|^{2}=10^{-14}$ separately. The results are shown in Figure 11. Both the one DV and two DVs signature could probe $m_{H^{+}}\lesssim 1490$ GeV, which are close to the threshold of $H^{+}H^{-}$ production at the 3 TeV CLIC. For $|V_{\ell N}|^{2}=10^{-10}(10^{-14})$, the promising region of HNL is $m_{N}\in[15,80]$ GeV ($m_{N}\in[50,400]$ GeV), with little dependence on the scalar mass $m_{H^{+}}$. This is because the cross section of $H^{+}H^{-}$ at CLIC is nearly a constant for $m_{H^{+}}$ below the TeV scale. Compared with LHC, the CLIC has a larger sensitive region for $m_{H^{+}}$ above TeV, mainly due to a larger cross section of $H^{+}H^{-}$. For the two DVs signature, the sensitive regions are smaller than the one DV signature. And the sensitive regions for $|V_{\ell N}|^{2}=10^{-10}$ and $|V_{\ell N}|^{2}=10^{-14}$ have no overlap. For the seesaw induced mixing $|V_{\ell N}|^{2}=m_{\nu}/m_{N}$, the promising region is about $m_{N}\in[20,160]$ GeV for one DV signal and $m_{N}\in[50,110]$ GeV for two DVs signal, with little dependence on the charged scalar mass.