A Dark Vector Resonance at CLIC

One of the main problems in Particle Physics is to understand the origin and nature of Dark Matter. An exciting possibility is to consider that the Dark Matter belongs to a new complex but hidden sector. In this paper, we assume the existence of a strongly interacting dark sector consisting on a new scalar doublet and new vector resonances, in concordance with a model recently proposed by our group. Since in a previous work it was found that it is very challenging to find the new vector resonances, here we study the possibility of finding them at the a future Compact Linear Collider (CLIC) running at $\sqrt{s}=3$ TeV. We found two distinct scenarios: when the non-standard scalars are heavy, the dark resonance is intense enough to make its discovery possible at CLIC when the resonance mass is in the range $[2000,3000]$ GeV. In the second scenario, when the non-standard scalars are light, the new vector boson is too broad to be recognized as a resonance and is not detectable except when the mass of the scalars is close to (but smaller than) a half of the resonance mass and the scale of the dark sector is high. In all the possitive cases, less that a tenth of the maximum integrated luminosity is needed to reach the discovery level.


Introduction
The accumulated astrophysical evidence in favor of the Cold Dark Matter hypothesis is, currently, one of the main reasons to study extensions of the Standard Model (SM). A very popular construction in this area is to consider a second scalar field transforming as a doublet of SU (2) L , supplemented by an unbroken Z 2 symmetry. This is the so called "Inert Two Higgs Doublet Model" (i2HDM) [1][2][3] . Recently, motivated by the hypothesis of a dark matter belonging to a complex sector with its own set of non-Abelian interactions, our group studied a modified version of the i2HDM where the standard sector and the inert scalar doublet transform under different local SU (2) groups. 4 This setup may arise, for instance, if the dark matter is a composite state. In this case, the new SU (2) corresponds to a local hidden symmetry that dynamically emerges from an underlying (but not specified in our effective model) strong interacting sector. The gauge bosons of this effective SU (2) local group give origin to the dark vector resonances which are the next massive composite particles. This construction is a direct analogy of the description of the pions and the rho meson in usual hadron Physics.
As we will see in the next section, at low energy, the resulting model is just the i2HDM with extra massive vector resonances which couple weakly to the standard sector but strongly to the new scalars. Our previous study shows that the new resonances introduce important modification on observables like the predicted relic density of the Dark Matter candidate. 4 Moreover, we also learned that the eventual detection of the vector resonances at the LHC is a very challenging task even in the high luminosity regime. For this reason, it seems wise to study the detectability of these resonances in future accelerators, specially in the clean environment of the planned lepton colliders. However, from the future electron-positron colliders, only CLIC is planned to run at an energy high enough ( √ s = 3 TeV) to explore the mass range we expect for the vector resonance (overviews of CLIC and its possibility to search for New Physics can be found in [5][6][7][8][9][10] ). Additionally, the projected luminosity (2 ab −1 ) promises an adequate scenario for discovering subtle effects like the existence of a dark resonance.
In this work, we study the possibility of discovering the non-Abelian vector resonances at CLIC. The paper is organized in the following way. In section 2, we briefly describe our model emphasizing the new vector sector. Then, in section 3, we describe our analysis and our results. Finally, in section 4, we state our conclusions.

Model Reminder
Our model is an extension of the Inert Two Higgs Doublet Model and is based on the gauge group SU (2) 1 × SU (2) 2 × U (1) Y . Our construction has been completely described in 4 and here we only remind its most important aspects.
The standard sector (including a scalar doublet) is assumed to transform only under SU (2) 1 and U (1) Y while a second scalar doublet transforms under SU (2) 2 (and U (1) Y ). The breaking down of the SU (2) 1 ×SU (2) 2 to the standard SU (2) L is described by a non-linear link field Σ which transforms as a bi-doublet under SU (2) 1 × SU (2) 2 . The Lagrangian describing the bosonic sector of the model can be written down as where A 1µ and A 2µ are the gauge bosons of SU (2) 1 and SU (2) 2 respectively and The scalar doublets are denoted by φ 1 and φ 2 and their covariant derivatives are: and derivative of the link field is: The standard fermions, on the other hand, couples to the gauge bosons of SU (2) 1 and U (1) Y , and to φ 1 as in the SM .
After the original symmetry breaking process, which we assume to happen at a scale u larger than the electroweak scale v, the Lagrangian can be written down in the unitary gauge (Σ = 1) as: As usual, the electroweak symmetry is spontaneously broken when φ 1 acquires a vacuum expectation value (vev) φ 1 = (0, v/ √ 2) T . We assume that φ 2 does not get a vev and consequently the Lagrangian remains invariant under the Z 2 transformation φ 2 → −φ 2 .
In the limit where the coupling constant associated to SU (2) 2 (g 2 ) is much larger than the one associated to SU (2) 1 (g 1 ), the physical vector fields can be written in terms of the gauge eigenstates as follows: and where ρ 0,± µ designates the new vector resonances and as usual, In the same limit, the masses of the vector states can me expressed as: Notice that the interaction of the dark sector with the standard one is suppressed by a factor g 1 /g 2 . Consequently, large values of g 2 , and vector resonances with masses above 2 TeV, make the new sector invisible at current LHC searches. [11][12][13] In the scalar sector, the spectrum is straightforward since no mass mixing term arise due to the Z 2 symmetry. Consequently, near the minimum of the potential, the scalar doublets can be parameterized as: Notice that the Z 2 symmetry makes the lightest new scalar (which we assume to be h 1 ) stable and a dark matter candidate. 4 For this reason, we call the new sector (formed by ρ 0,± µ , h 1 , h 2 and h ± ) the "dark sector". The model described above has several new parameter such as the masses of the new vector and scalar states, the scale u and the parameters of the scalar potential. However, for this work, the only relevant free parameters of the model are: the masses of the vector resonances (M ρ ), the masses of the new scalars (m h1 , m h2 and m h± ) and a ≡ u/v. In what follows we will take a = 3, 4, 5 and M ρ ∈ [2, 3] TeV since in this way we keep g 1 /g 2 0.2 which is consistent with our level of approximation. The chosen values of a are the same already consider in our previous work on the dark matter phenomenology. 4 They are also representative of low, moderate and high composite scale in the dark sector, given the values od M ρ .
The model reproduces the observed relic density provided that M h1 < M ρ

Results
The aim of this work is to study the possibility of discovering the new vector resonance ρ µ at the future lepton collider CLIC. The basic idea is to take profit from its high energy mode ( √ s = 3 TeV), its expected high integrated luminosity (L ≈ 2 ab −1 ) and to use the effects of initial state radiation and radiative return to the resonance in order to scan the relevant range of possible ρ mass values: M ρ ∈ [2, 3] TeV. The upper limit of this interval is determined by the maximum center of mass energy available at CLIC while the lower limit makes us sure that the resonance has escaped detection at the LHC. 4,[11][12][13] We focus on the process e + e − → µ + µ − which, at leading order, is described only by the interchange of a photon, a Z-boson and a ρ 0 in the s-channel.
Our model was implemented in CalcHEP 14 using the LanHEP 15, 16 package. We used CalcHEP to generate events taking into account the initial state radiation with the accelerator parameters informed by the Particle Data Group 17 and listed in the Table 1. In this simulation, we considered the contributions of the standard sector as well as the production of the dark resonance. We smeared the momenta of the events generated with CalcHEP, using a Gaussian distribution, in order to take into account a finite momentum resolution of the detector. For this purpose, we use ∆p /p = 0.05. Then, we computed the invariant mass of the µ + µ − pair in an interval around the (expected) mass of the resonance. Finally we fit the spectrum with a Gaussian resonance and a quadratic background in order to obtain the number of resonant events and the number of background events. All our simulations were performed at leading order and we considered only the irreducible background. An additional limitation of our methodology is that we consider only a local statistical analysis of the resonance in the sense that the computed statistical significance of the signal refers only to the local significance and not to the global one.
As was already pointed out in, 4 the possibility of discovering the new vector bosons in a resonant process depends on whether they can decay or not into the new scalars. The reason behind this feature is that we are assuming that the coupling constant g 2 is large (in order to guarantee small interactions with the standard sector), making the "dark sector" strongly coupled. Consequently two kinematic regimes open up depending of whether the masses new scalars are larger or smaller than M ρ /2. We call them the Heavy Scalars and Light Scalars scenarios, respectively. In fact, the partial decay width of ρ 0 (which are relevant for the process we are studying) are given by: where f represents a standard fermion, N c is the number of colors and the last equation is given for M W M ρ . As an example, in Figure 1 we plot the total decay width of ρ 0 (with M ρ = 2500 GeV) for three different values of a and as a function of m h1 when all the non-standard scalars are degenerated. We clearly see how quickly the total width grows for m h1 < M ρ /2, originating the two kinematic regimes.

Case 1: Heavy Scalars Scenario
In the case of heavy scalars, the ρ 0 appears as a narrow resonance (with a width of just a few GeV) in the di-muon spectrum (see Figure 2a). After the smearing procedure, a broader peak is obtained (Figure 2b). We fit this peak in the interval [M ρ − Mρ /10, M ρ + Mρ /10] using a Gaussian function (signal) plus a quadratic polynomial (background). Then we extract a number of events for the signal and a number of events for the background integrating the Gaussian function and the quadratic polynomial respectively in the interval defined above . The corresponding cross sections are shown in Figure 3. The error bars Figure 1: Width of the vector resonance (ρ µ ) as a function of the mass of the non-standard scalar in the case they are degenerated. Here we take M ρ = 2.5 TeV for the signal and the background reflect the uncertainties introduced by our Monte Carlo, the smearing of the momenta and our method of extracting the signal through a fitting procedure. On the other hand, although the signal and background differential cross sections decrease with the di-muon invariant mass (M µµ ), the energy distribution of the electrons increase with √ s reaching its maximum at the maximum nominal center of mass energy √ s max = 3 TeV. The reason for this behavior is the fact that the events with low energy electrons are due to the initial state radiation. Additionally, the interval where the resonance is fitted and integrated increase with M ρ . All these effects produce the M ρ dependence of the signal and background cross sections observed in Figure 3. Notice that in Figure 3b) the background obtained for different sets of data (different values of a ) almost coincide. This feature shows the self-consistency of our fitting procedure. We see that the signal (Fig. 3a) and the background (Fig. 3b) cross sections are of the same order of magnitude and the expected excess of events (for the projected luminosity) should be clearly observable. Indeed, using the usual definition for the statistical significance we estimate the luminosity needed to get the discovery criteria S = 5. The result is shown in Figure 4. Notice that the needed luminosity is always much smaller than the maximum luminosity expected at CLIC:

Case 2: Light Scalars Scenario
As we explained above, when the non-standard scalars are light the ρ 0 becomes broad and, in general it is difficult to identify it as a proper resonance. Indeed, after the smearing process, we could fit resonances only in the case of a = 5 and for this scenario we restrict only to this value of a. An example of such a resonance is shown in Figure 5. However, this region of the parameter space is worth to be explored. In doing so we found three important alternatives: 1) when only one neutral scalar (the DM candidate) is the only light scalar, 2) when both neutral scalars are light and the charged ones remain heavy and 3) when all the non-standard scalars are light. In the following paragraphs we will comment all these alternatives. For the convenience of the discussion we will define a new parameter (δm) as where m LS is the mass of the light scalar.
Similarly to the previous case, we fit the spectrum and we identify the resonant and background events. The only difference is that, this time, we fit the resonance (and the background) in the interval [M ρ − Mρ /5, M ρ + Mρ /5].

Only one light scalar
When only one scalar is light, the neutral vector resonance (ρ 0 ) still cannot decay into non-standard scalars since in our model there is no ρh 1 h 1 nor ρh 2 h 2 interaction terms but only ρh 1 h 2 . So this case is equivalent to case analyzed in the previous subsection.

Two neutral light scalars
When the two neutral scalars are light (but the charged ones remain heavy) the width of the vector resonances receives a moderate increment. For a given value of M ρ , we were able to identified resonances provided that δm 250 GeV. In Figure 6, we show the cross sections for signal and background for two situations: a) as a function of M ρ while keeping δm = 100 GeV and b) as a function of the mass of the (degenerated) scalars while keeping M ρ = 2500 GeV. Although the signal is systematically smaller than the background, the signal over background ratio takes acceptable values (0.1 S/B 0.5).

All the scalars are light
In the following paragraphs, we explore the case where all the scalars are degenerated This degeneration is compatible with the dark matter phenomenology. Indeed our previous study shows that the model reproduces better the experimental information when the mass difference between h 1 and the other scalars is of the order of a few GeV's. Of course, such a small mass difference is not important for our collider simulations.
In Figure 7 we show the computed cross sections for the signal and the background for: a) δm = 100 GeV and M ρ ∈ [2, 3] TeV, and b) M ρ = 2.5 TeV and masse of the scalars in the range [1110, 1230] GeV. The signal is again systematically smaller than the background, but still comparable. At the maximum integrated luminosity a few thousands of signal events are expected. As in the previous case, we compute the integrated luminosity needed for reaching the discovery level. The results are shown in Figure 8.   In summary, there is some discovery potential (although reduced) in the channel e + e − → µ + µ − when all the scalars are light, at least when the scale of the vector resonance is high (i.e. when a = 5). However, for light scalars a more promising possibility arises through the e + e − → h + h − channel. In Figure 9 we plot the production cross section for the pair h + h − for different values of M ρ and assuming that the mass of the scalars is m h1 = m h2 = m h± = M ρ /2 − 100 GeV. The all the values of a considered in this paper, we found that the cross sections is significantly higher in this channel than in the di-muon one. Even the new vector resonance may be observable in the invariant mass distribution of the h + h − as seen in Figure 10. In this way, the pair production of the charged scalars appears an effective mean of discovering the dark sector at CLIC.

Mono-Z Production
Although in this work our main focus is the study of the di-muon channel as a mean for discovering the new dark resonance, in the following paragraphs we will briefly consider the mono-Z production. This channel is traditionally important for the search of dark matter at colliders. In our case the main mono-Z production processes are e + e − → Zh 1 h 1 and e + e -→ Zh 1 h 2 , being the latter the dominant one. For this analysis, we follow an intermediate scenario where the neutral scalars are supposed light (in the sense explained above, that is, m h1 = m h2 = M ρ /2 − 100 GeV ), while the charged scalars are supposed to be heavy. Interestingly, these two processes are weakly dependent on the a parameter ( in fact, the resonant production e + e − → ρ → Zh i h j is exactly independent of a ).
In Figure 11 we show the transverse momentum distribution of the Z boson (11 a) and the missing invariant mass distribution (11 b) for e + e − → Zh 1 h 1 and e + e -→ Zh 1 h 2 with M ρ = 2.5 TeV without any cut. For comparison purposes we include the SM distributions for e + e − → Zν eνe which is the main source of background. We see that although the main component of the signal (Zh 1 h 2 ) has a non-trivial structure due to the resonant contribution of the ρ 0 in the s-channel, the signal is in general several orders of magnitude smaller than the background. When we apply the cuts P T Z 100 GeV and M ρ − 100GeV M missing M ρ + 100GeV (where P T Z is the transverse momentum of the Z boson and M missing is the missing invariant mass) we obtain: σ e + e − → Zh 1 h i ≈ (1.6 ± 0.2) fb almost independently of the value of M ρ . On the other hand, the background cross section, under the same kinematic cuts, results to be: After the cuts, the signal is still two orders of magnitude smaller than the background. However with a luminosity of L = 2000 fb −1 the event excess will represent a statistical significance of S = 6. This significance, however, reflects only the statistical uncertainty. A more detailed analysis should take into account the systematic uncertainties coming form the particular characteristics of the detector and the reconstruction of the Z boson specially from its hadronic decay. An additional difficulty is represented by the invisible decay of the Z boson into neutrinos.

Conclusions
In this paper, we studied the possibility of using CLIC to discover a new vector resonance associated to a hypothetical strongly coupled dark sector. We focused on the di-muon production and we have strongly relied on the radiative return to the resonance. In the case where the non-standard scalars are heavy enough to forbid the decay of the vector resonances into particles of the dark sector, we predict an important excess of event for the resonance masses we considered,even when a moderate smearing of the momentum is taken into account. When the scalars are light, the new vector become broad and it is difficult to define a resonance. A remarkable exception is found when the mass of the scalars is close to (but smaller than) a half of the resonance mass and the scale of the dark sector is high. In all these positive cases, the high energy and high luminosity projected for CLIC are enough for leading to the discovery of the new dark resonance or severely constrain the model. However, for the light scalar case, the pair production of charged scalars arises as an attractive alternative for discovering the dark sector. In a complementary analysis, we studied the mono-Z production. We found that, although this channel is more challenging that the di-muon production, after the implementation of an adequate cut on the missing invariant mass, the high luminosity regime of CLIC can lead to an statistically significant excess of events. These facts suggest that CLIC would be an excellent environment for testing models with a complex dark sector and illustrate the importance of building such a machine for testing models beyond the Standard Model.