Implications of dark sector mixing on leptophilic scalar dark matter

We propose a new viable outlook to the mixing between a singlet and a doublet leptonic dark sector fields. This choice relaxes the dark matter (DM) search constraints on the quintessential scalar singlet DM as well as presents new opportunities for its detection in the lab. The mixing produces an arbitrary mass difference between the two components of the extra doublet in a gauge-invariant way, without introducing any new scale of electroweak symmetry breaking in the theory. It also provides a useful handle to distinguish between the dark sector particles of different isospins, which is a challenging task otherwise. As the dark leptons coannihilate non-trivially, the mixing effectively enhances the viable parameter space for the relic density constraint. We also show its positive impact in various DM search prospects, e.g., indirect detection and collider searches. In low DM mass regime, our analysis shows that with a non-zero mixing, it is possible to relax the existing indirect search bounds on the upper limit of the DM-Standard Model coupling. In collider searches, from the analysis of the $3\tau + E^{miss}_T$ and $\ell\,\tau + E^{miss}_T$ channels, we show that one ensures the presence of the mixing parameter between the dark sector particles of the theory by looking at the peak and tail positions of the kinematic distributions. Even with a tweak in the values of other free parameters within the viable parameter region, the distinct peak and tail positions of the kinematic distributions remains a constant feature of the model.


I. INTRODUCTION
The dark matter (DM) constituting 27% of the energy budget of the Universe is now a settled fact. The cosmological considerations and astrophysical observations have put this matter beyond any doubt. The precisely measured value of the cosmological relic abundance by WMAP [1] and Planck [2] is Ω DM h 2 " 0.1199˘0.0027, h being the reduced Hubble constant. The search for a suitable DM candidate is still on as ever [3][4][5], and the most widely explored among all is the Weakly Interacting Massive Particle (WIMP) whereas the scalar singlet DM or scalar "Higgs-portal" scenario tops among WIMP paradigm [6][7][8]. It is also widely known that the parameter space of the Higgs-portal models has shrunk over the years [9,10] by stringent constraints from the direct detection (DD) [11][12][13][14], indirect detection (ID) [15][16][17] and invisible Higgs decay [18][19][20] searches.
Nonetheless, the Higgs-portal scenario is not entirely out of favour as a model. There are alternatives for evading the existing constraints, namely, by considering additional symmetries [21][22][23] or adding new particles [24][25][26] that give rise to new portals for DM annihilation. However, the possibility that we shall address in the present article is the so-called coannihilation [27][28][29][30][31][32][33][34][35][36][37][38] mechanism, the one where the DM annihilates with another dark sector particle, and the chemical equilibrium between the annihilating particles ensures the substantial depletion of DM number density. Previously, we have shown how we can successfully evade the existing stringent constraints on scalar singlet DM by introducing a vector-like dark lepton doublet [39]. This simple addition to the scalar singlet DM extension of the SM opens up new vistas of possibilities, namely (i) new dark leptons interacting with the SM via gauge interactions introduces new annihilation channels, and (ii) novel Yukawa structure in the Lagrangian enhances cross sections facilitating its search in colliders like LHC.
However, the SU p2q L gauge invariance mandates the degeneracy of the two components of the doublet at the tree level. To lift this degeneracy, in Ref [39], we introduced a Z 2 -even scalar triplet. The said scalar triplet otherwise does not play any role in the DM phenomenology, rendering the exercise ad hoc. Here we propose a better alternative to the earlier case, which is to add a Z 2 -odd singlet fermion instead. In principle, the same charge dark fermions can mix among themselves to give physical eigenstates, which in turn, gives rise to the mass splitting between the pair. This mixing gives rise to rich phenomenological implication, unlike the previous case. Hence, this exercise is not limited to generating finite mass splitting between the components of the doublet. In other words, far from being an ad hoc addition to the model, this addition dictates the outcome of the processes through the mixing angle. Here we are interested in the implications of mixing between the charged dark sector fermions as well as the distinguishability of the pure and the mixed leptonic states from the observations. sreemanti@iitg.ac.in : rislam@iitg.ac.in To put the matter into perspective, we would like to mention that the mixing between a singlet and a doublet in the context of DM studies in nothing new. Previously, several authors used this structure to explain various phenomena. In supersymmetric (SUSY) theories, the bino can mix with the Higgsinos to give rise to a neutralino [40][41][42][43]. Out from the realm of SUSY, this matter has been taken up by other authors to build up minimal DM models to address shrinking parameter space from relic density and direct detection measurements [44][45][46][47][48][49]. In addition to the context of DM, this extension has successfully explained small values of neutrino masses and the related phenomenology [50][51][52][53]. See Ref. [54], for a recent review and an exhaustive list of relevant papers. Now, where we differ from all these works is that they took a mixed dark lepton state as a DM candidate, which limits the number of decay channels. Here, the DM candidate is a separate scaler singlet, which phenomenologically is the same as the quintessential minimal extension of the SM. The dark leptons add an extra portal for pair annihilation of DM as well as provide coannihilation and mediator annihilation channels and thus enrich the DM dynamics. The mixing between the dark partners relaxes the existing experimental limits on the scalar singlet DM scenario. It also provides a handle to identify the viable parameter space dominated by dark partners of different isospins.
The probe of the DM candidates in the controlled collider environment is always a challenging task. Here we will analyse the situation in the case of Large Hadron Collider (LHC). However, we do not address the usual problem of signal-background separation in a hadron collider here. We assume that this can be easily handled by the use of advanced modern techniques of analysis even for a small signal cross sections as we showed previously in Ref. [39]. We will focus on how the signatures of the mixing parameter from the kinematic distributions of the relevant observables can help us decipher the DM signals from our model. We will see that the addition of an extra singlet does not complicate the search strategies, but opens up new avenues. The mixing parameter gives an extra handle in tuning the kinematic distributions that will be easily accessible at the LHC. Apart from collider searches, we will see how does it affect the indirect search prospects of the DM.
We organised the paper as follows. Section II gives a detailed account of the model. The DM phenomenology, its formalism, and the observations from the relic density, direct and indirect detection calculation are discussed in Section III. Section IV addresses the features that will manifest itself in the collider environment. Finally, in Section V we present our conclusions.

II. MODEL DESCRIPTION
We discussed briefly in the introduction the motivation for the choice of our model content. Here we shall develop from that motivation a detailed description of the model parameters. As mentioned previously, ours is a leptophilic model that can evade all the existing bounds on scalar singlet DM. For that, we consider dark-sector partner(s) which can either be a doublet [39] or a singlet fermion [55]. Previously [39] we studied the case of a dark lepton doublet partner which coannihilates with the singlet scalar DM candidate as well as acts as a portal, depending on the parameter space, which results in relaxing the constraints considerably. As the gauge invariance does not allow non-degenerate mass states for the doublet partners at the tree level, we had to introduce a new scale of electroweak symmetry breaking. We added a, otherwise redundant, scalar triplet to generate finite mass splitting between them. This mass splitting plays a significant role in determining the DM signatures in the collider. However, the measurement of the ρ parameter [56] constrains the value of the mass splitting ď 10 GeV. The motivation to make it arbitrary leads us to add a Z 2 -odd singlet fermion instead of a Z 2 -even scalar triplet in the particle spectrum. We shall see in this section how this singlet fermion plays a vital role in lifting the degeneracy of the doublet states depending on the parameter space.
Moreover, in Ref [39], we had pointed out that none of the observations was useful to distinguish between the two coannihilating particles. Here, in principle, the physical states of the dark leptons can be a mixed state between the same charge component of the doublet and the singlet. Thus it gives us the most general minimal scenario where we can meet all our demands. We want to distinguish between these two dark leptons from the observations of the DM in experimental searches. In this light, we will show later on, that the pair annihilations are more useful in the collider analysis and indirect searches. Whereas in the relic density scenario, the coannihilation channels can better exhibit the mixing effects. With all these motivations clear in our mind, we give below the parameter content of our model. Table I shows the particle content and the quantum number assignments of our model. The Z 2 -odd dark sector contains a vector-like Dirac fermionic doublet, Ψ T " pψ 0 , ψ 1 q, a fermionic singlet ξ and a real scalar singlet φ. φ is our DM candidate which interacts with the SM Higgs via quartic as well as portal coupling and the Z 2 symmetry renders stability to it. The fields Ψ and ξ couples with the SM doublets L , H and the scalar singlet φ via three different kinds of gauge-invariant Yukawa interactions.
Z2```´´T ABLE I. Quantum number assignment of the relevent fields in our model. Electromagnetic charges are given by Q " t 3`Y .
Hence the resulting Lagrangian takes the form where L SM is the SM Lagrangian and a sum over the generation index α is implied. M Ψ " m ψ 0 " m ψ1 , is the degenerate bare mass term of the doublet and D µ " B µ`i g W t a W a`i g 1 Y B µ is the covariant derivative. The mass of the scalar singlet φ is given by The negatively charged dark fermions in the model mix among themselves through orthogonal transformation ψ " c α ψ 1`sα ξ , χ "´s α ψ 1`cα ξ ; to give the physical states ψ and χ, where the mixing angle α and the physical masses m ψ and m χ are given as The Yukawa coupling y and the mass m ψ 0 are dependent on the above free parameters of the model. These are given by The derivation of the above relations from the mass matrix diagonalisation and the resulting new vertices are in Appendix A. Apart from the those given in Eqs.
(2), (4) and (5), we have only two more free parameters in our model, namely, the third generation Yukawa couplings y D,S τ . We put the Yukawa couplings of light leptons at y D,S e " y D,S µ À 10´4 to conform the muon g´2 measurements [56]. All the Yukawa couplings are to remain in the perturbative regime such that y D,S τ , y ď 4π. A conservative choice of λ hφ À 10´4 is in place all through our analysis to keep the bounds from the direct detection searches and the invisible decay measurements at bay.

III. RELIC DENSITY ANALYSIS
Due to the presence of more than one dark sector particles, the DM number changing processes, in this model, are three-fold: (i) pair annihilation (φφ Ñ SM SM), (ii) coannihilation (φψ˘0 Ñ SM SM), and (iii) mediator annihilation (ψ˘0ψ¯0 Ñ SM SM). In Appendix B, we have shown the details of the involved processes in our model for all the three cases. Keeping in mind the assumption of thermal freeze-out, that the dark sector particles are in equilibrium with the thermal bath in the early Universe, whereas, in chemical equilibrium with each other, one can write the Boltzmann equation describing the number density, n of the DM as follows [27].
The effective velocity-averaged annihilation cross section, xσ eff vy can be written as where the indices i, j denote any of the dark sector particles φ, ψ˘0 or χ˘, and For our analyses, we have implemented the Lagrangian (1) along with all the above relations (Eqs. (2) to (7)) in FeynRules [57]. Using the resulting model file, we carried out all the following DM analysis with the help of micrOMEGAs [58].
As discussed in the previous section, mixing affects only the charged dark fermions. Therefore xσ eff vy will be sensitive to mixing for the annihilation channels which involve these dark fermions in the initial state and/or the propagator. One such possibility is the pair annihilation of φ, where the charged dark fermions appear in the tchannel propagator (Fig. 10). But the mixing effects are most prominent if one considers the coannihilation channels. This is because in some of the coannihilation diagrams (Figs. 1a and 1b), in addition to the propagators, one of the initial state particles are directly affected by mixing. Therefore, the coannihilation channels of φ with ψ`and χà mply show the mixing effects. (Throughout this article, we denote the charged dark leptons by`sign. However, All the arguments are equally applicable for negatively charged particles as well.) On the other hand, ψ 0 not being a mixed state, φ ψ 0 coannihilation channel (Fig. 1c) is much less affected by mixing. As we are interested to study the mixing effects on the phenomenology, we will mostly concentrate on the coannihilation channels for our DM analysis. This implies that the mass splitting between DM and the dark fermions should be small throughout the study (À 30 GeV) and the dark sector-SM coupling is not very large (À 1). For fixed mass splittings between the dark sector particles, the DM-SM Yukawa coupling vs DM mass correlation is discussed amply in the literature [35,55]. In Ref. [39], the transition between the three kinds (pair, co-and mediator annihilation) of DM annihilation is depicted for a similar model with only a lepton doublet as the coannihilating partner of the scalar DM. In this work, this dynamics becomes more interesting in the presence of the charged singlet and doublet dark fermion mixing. In Fig. 2, y D τ is plotted against m φ for fixed δm and y S τ . In the absence of mixing, depicted by the red line in Fig. 2(a), the scenario is identical to Ref. [39,Fig. 4]. In this case, apart from the subdominant pair annihilation, only φψ`coannihilation contributes to relic density, φχ`channel being redundant due to y S τ =0. However, as mixing increases, both φψ`and φχ`coannihilation channels contribute. For our choice of parameters, δm χ ď δm ψ , hence φχ`coannihilation ( Fig. 1a) is more dominant than the φψ`counterpart for a non-zero mixing. This is because both φψ`and φχ`coannihilations are predominantly controlled by the channels which have W`in the final state. Now, for φψ`coannihilation case, these channel exists even for s α " 0, whereas, for φχ`case, this channel is realized largely through mixing. Therefore, the mixing effect is more prominent in φχc oannihilation scenario over the φψ`case.
With y S τ set to zero, as one goes for larger mixing angle, φχ`τ´coupling (Eq. (A10)) increases. This, along with the fact that both dark fermions now can non-trivially coannihilate, the total DM annihilation cross section effectively increases. Hence, for a fixed m φ , larger mixing corresponds to smaller coupling to be relic density allowed. Therefore, y D τ gradually decreases with increase in s α .
(a) δm ψ = 25 GeV The scenario changes substantially if the same correlation is drawn with a fixed non-zero y S τ . In Fig. 2(b), we assign a large value to y S τ . For such a large coupling, pair annihilation has a substantial contribution to the relic density. This is due to the fact that in this model, in the expression for annihilation cross section, the functional dependence on the DM-SM Yukawa coupling (λ, let us suppose) is λ 4 in pair annihilation, while for coannihilation, it is λ 2 . This is also in agreement with [39, p.5, p.6] where we showed that for a fixed δm and variable m φ , coannihilation is dominant for small m φ and large DM mass, pair annihilation takes over. In this plot, as y S τ is already large, y D τ is not required to be as high as that in Fig. 2(a). Now, for small m φ values, φχ`has the most dominant contribution in relic density. φχ`τ´coupling (Eq. (A10)) is the largest for s α = 0 and due to this, in order to achieve right relic, the required value of y D τ is the smallest, as we see in this plot for the red band around m φ ď 180 GeV. As the mixing increases, the above coupling decreases, therefore larger y D τ is required. However, as m φ increases further, pair annihilation takes over and the mixing effect is lost in the "tail" region of the correlation around m φ ą 500 GeV. As discussed above, this is because mixing does not play a significant role in the pair annihilation channels.
In Fig. 2, we considered constant mass splittings between the dark sector particles. Now let us discuss how different values of these mass splittings affect the DM dynamics through mixing. In Fig. 3, y D τ and y S τ correlation is plotted as a function of s α and for different δm's. Similar to Fig. 2, here also we have chosen δm χ ď δm ψ to facilitate the φ χ`coannihilation. As already discussed, large mass splitting between DM and dark fermions will suppress the coannihilation contribution towards relic density and larger coupling is required to compensate for that deficit. Hence, depending on the value of m χ and m ψ , the shift of the relic allowed points along y S τ axis is straightforward. Rather, the distribution of the points along the y D τ axis is an interesting feature. For low mixing, the dominant φχ`coannihilation channel is φχ`Ñ τ`Zpγq ( Fig. 1a) and its cross section is almost proportional to py S τ q 2 . This satisfies relic density even for y D τ " 0 as one can see from the red band in the subfigures of Fig. 3. As the mixing increases, initially, there is a contest between the two terms in φ τ`χ´coupling Eq. (A10). For low y D τ , as mixing increases, to compensate for the overall reduction of the coupling, we see a slight increase in y S τ . But as y D τ increases further, the second term in the expression for the coupling becomes gradually negligible, making the coupling almost proportional to s α y D τ for large mixing. Therefore we observe that to keep within the (a) relic bounds, for large mixing and large y D τ , y D τ decreases as s α gradually increases. It is clear in all the plots except Fig. 3(c) that the green and blue points shift to the left along y D τ axis as s α gradually increases. Now, we observed that in Fig. 3(a), where m ψ´mχ = 5 GeV, φχ`coannihilation is the most dominant channel. But as this splitting decreases to 2 GeV in Fig. 3(b), φψ 0 channel also contributes substantially. This is because of the coannihilation channels having W boson in the final state, which are available for φψ 0 coannihilation even without mixing but accessible largely through mixing for φχ`coannihilation. The presence of these channels suppresses the mixing dependence in the y D τ vs y S τ correlation, causing reduction in the spread of multi-colored points in Fig. 3(b) compared to Fig. 3(a). This spread becomes zero in Fig. 3(c), where m ψ and m χ becomes almost degenerate. As already argued, here φ ψ 0 coannihilation is the most dominant channel in this case with W˘in the final state and this makes the correlation completely independent of mixing.
For Fig. 3(d)-(f), δm ψ is large so that φψ`coannnihilation contribution is very small. This establishes φ χ`Ñ SM SM as the only active coannihilation channel. In absence of φψ`(hence also φψ 0 ) coannihilation and for low mixing, the only surviving φχ`coannihilation in xσ eff vy has little or no dependence on y D τ . The points corresponding to very low mixing, amply show this effect as the red band gradually flattens out with increasing δm ψ as we go from Fig. 3(d)-(f). Ultimately in Fig. 3(f), the red band widens and becomes parallel to y D τ axis and the blue band corresponding to large mixing becomes the thinnest.
Finally, let us see the effect of variable δm's instead of discrete intervals discussed so far. The two plots in Fig. 4 show two such correlations of the δm's as a measure of mixing. Fig. 4(a) focusses on the asymmetry of δm χ vs δm ψ correlation as a measure of mixing. Mixing plays a vital role here in the distinction between the dark fermions, unlike Ref [39], where such a discrimination was not possible. The two δm's vary over the same range and the Yukawa  Fig. 4(a) and δm ψ 0 vs δm ψ in Fig. 4(b) for relic density allowed points as a function of the mixing angle. Since all other parameters are the same for the dark leptons discussed in each plot, it is only up to the mixing parameter to determine the dominant relic density contribution between them. The DM mass varies in the range 65 GeV ď m φ ď 1 TeV.
couplings y D τ and y S τ are fixed at the same value. This implies that apart from the mixing parameter, χ`and ψs tand on equal footing in the context of coannihilation possibilities. However, on fixing different values for the mixing, we see that the plot shows a clear distinction between the two dark leptons of different isospins. In absence of mixing, ie, for the blue points, the relic density allowed region along δm χ axis broadens with the increase of s α whereas along δm ψ it narrows down. This is attributed to the fact that for s α " 0, there are more diagrams in φψ`coannihilation channel than the φχ`possibility. These extra diagrams (Fig. 1b) arise due to Gauge couplings, which are exclusive to φψ`Ñ SM SM coannihilation because in absence of mixing, ψ`is purely part of the SU p2q L doublet and χì s purely a singlet. It is this extra diagram that makes φψ`coannihilation stronger than the φχ`counterpart for s α " 0. This can also be verified from the distribution of the blue points, where we see that points at δm χ = 30 GeV correspond to δm ψ up to 15 GeV, whereas, points at δm ψ = 30 GeV corresponds to δm χ only up to 12 GeV.
We have explicitly checked the above for a few benchmark points and observed that for s α " 0, φψ`Ñ W`ν τ is stronger than any of the φχ`Ñ SM SM channels if the two yukawa couplings are equal. For s α " 1{ ? 2, on the other hand, φψ`and φχ`coannihilation contribution becomes equal even for these extra Gauge channels, so the red points show a symmteric distribution along both the axes. For the green points, ie, for s α " 1, we simply see the opposite of the s α " 0 case, because W`ψ´ψ 0ˇs α"0 " W`χ´ψ 0ˇs α "1 . This also agreed with our observation of benchmark point results, that the φχ`coannihilation is more dominant than φψ`counterpart for s α " 1. To sum up, we can conclude that the relic density allowed region for the pure leptonic eigenstates widens along δm χ axis, whereas the mixed states tend to widen gradually along δm ψ axis. Fig. 4(b) on the other hand, discusses δm ψ 0 vs. δm ψ correlation as a function of the mixing parameter. The DM mass varies in the same range as Fig. 4(a), and the Yukawa couplings are fixed at y D τ " y S τ = 0.5. For the two extrema of the mixing angle, ie, for s α " 0 and 1, m ψ0 = m ψ , as obvious from Eq. (7). Therefore, it is obvious that for these two s α values, the nature of the correlation will be a straight line with a 45 0 slope, which is exactly what the red line represents. However, for the intermediate s α values, we see distribution on both sides of the red line. As we can see, the spread of the points varies with s α . It becomes maximum for s α " 1{ ? 2 and gradually reduces as one approaches the extrema.
It is to be noted that if the relic density constraint is not imposed, then for a fixed s α , the points would be distributed symmetrically around the red line, maintaining it as the line of symmetry. This is obvious from the model description arguments of Section II. However, on adding the constraint, one can see from the plot that this symmetry is lost. For large δm's, we observe that except for the two extrema of the mixing angle, the relic allowed region shifts slightly to the left of the red line.
We numerically checked that for fixed Yukawa couplings, as mixing increases, the contribution of φψ`Ñ SM SM channels rapidly decreases, but φψ 0 Ñ SM SM remains much less unaffected, except a very small increase. Our relic density calculation for a few benchmark points within the allowed region confirmed that φψ 0 Ñ W`τ´channel is the most dominant channel in xσ eff vy.
This explains the shift at large δm when mixing is the range 0.0 ď s α ď 1{ ? 2. As φψ`contribution decreases, to maintain relic abundance in the observed range, one needs stronger φψ 0 coannihilation. This is why the region with a very large δm ψ 0 remains out of the relic allowed regime, being too much suppressed. It is also justified that we see the upper limit of allowed δm 0 decreasing gradually as s α increases from 0 to 1{ ? 2. Ultimately, for gray points (s α " 1{ ? 2), the shift from the red line becomes maximum at large δm's. On the other hand, we observed that φχ`Ñ SM SM contribution, which was also negligible so far, becomes substantial for large mixing, ie, in the range 1{ ? 2 ď s α ď 1.0. As already explained, φψ 0 coannihilation remains mostly unaffected by mixing. Therefore, as φχ`contribution gradually increases with mixing in the above range, it also relaxes the exclusion limit for large δm ψ 0 . So, again we see the relic allowed region gradually shifting right towards the red line as s α increases. This justifies why the shift for the black points is less than that for the gray points. Ultimately, for s α " 1.0, the correlation merges with the red line, implying zero shift.
The above features are model-independent and can be generalized for any singlet DM coannihilation scenario that involves a singlet as well a doublet coannihilating partner.

A. DM signatures at the LHC
The problem of finding signatures of DM in a collider environment is a very challenging task. Segregating the signals of DM from a multitude of invisible particles is notoriously hard. Several search strategies are there for this task which addresses the problem from the perspective of the hadron colliders (LHC and future hadron colliders) as well as lepton colliders which are for the very purpose where hadron collider has limitations. In hadron colliders, we do not have access to the longitudinal component of the missing momenta. Hence the task of finding the signature of DM is more difficult as we have to depend entirely on the observables constructed from the transverse components of momenta. One can segregate multilepton + missing energy signals from the backgrounds even for a small signal cross section through advanced techniques of Multivariate Analysis [39]. Here we will go beyond the signal-background efficiency and look for specific distributions through which we can decipher the DM signals in a collider environment. We discussed previously that the presence of a doublet along with a singlet dark fermions and the mixing between them plays a significant role in the phenomenology of our model. Here we point to the ways to find out those signatures in collider environments that will highlight this feature of the model and try to segregate the effects of each from the distributions. The peaks and end-points of a kinematic distribution can be associated with the masses of the mediating particles. The kinematic distributions of transverse momentum, p T , transverse mass, m T and invariant mass are a few very significant distributions to study.
In the following we will discuss the relevant distributions in a hadron collider environment like LHC. To perform the analysis for the LHC at the CM energy ? S " 13 TeV, we proceeded as follows: (1) FeynRules [57] has been used to generate model files. (2) Events have been generated using MadGraph5 [59] and showered with Pythia 8 [60].
(3) The detector simulation has been performed with the help of Delphes [61]. (4) The distributions were drawn with the help of MadAnalysis 5 [62].
The preliminary selection cuts used in the analysis are: • p T ą 10 GeV and |η| ă 2.5 for all charged light leptons, • p T ą 20 GeV and |η| ă 5 for all non-b-jets, and • ∆R ij ą 0.4 between all possible jets or leptons.
The distance between two objects i and j defined as ∆R ij " a pφ i´φj q 2`p η i´ηj q 2 , where φ i and η i are the azimuthal angle and rapidity of the object i, respectively.
For the effect of mixing angle α in the collider signatures, two kinds of processes will be useful (1) the processes that are predominantly W boson mediated, and (2) those that are also predominantly Z boson mediated.
1. 3τ`E miss T channel From Eq. (A9), we can see that the couplings of W boson with the dark fermions are functions of α. Without the mixing, i.e., when s α " 0, ψ is exclusively a doublet and χ is purely a singlet. This fact immediately implies that when W Ñ ψ 0 χ channels are off, the resulting final states are a consequence of pure doublet contribution. Hence, by tuning the mixing, one can control the % of singlet contribution in the channels. The channels we can look into are as follows: (1) τ ν 2φ: s-channel processes via W Ñ ψ 0 ψpχq followed by ψ 0 decaying into ν φ which remains totally invisible and ψpχq Ñ φ τ . Since the only visible final state in this channel is a single τ and the missing energy can come from both ψ 0 and ψpχq, it is very difficult to conclude anything about the DM signature. (2) 3τ ν 2φ: Here, the signal processes can proceed through the following different modes: (a) W -mediated schannel processes through qs q 1 Ñ ψ 0 ψpχq channel; and (b) W -mediated s-channel as well as quark-mediated t-channel processes through qs q 1 Ñ W Zpγq and qs q 1 Ñ W h channels. We encounter the latter case in the context of the Unitarity problems of the gauge bosons and is very similar to the f s f 1 Ñ W Zpγq and f s f 1 Ñ W h. See for example Ref. [63][64][65] for some recent papers in this context and the references therein. The point is that as the CM energy increases, the cross section of these channels decreases which we can explain from the Equivalence Theorem of the gauge bosons. We see a similar trend in our case also. As a result, the dominant channels of 3τ ν 2φ process become mostly W -mediated s-channels mentioned in (a) above. See Fig. 5 for the complete diagrams with their decay channels. In the CM frame, the two particles split with equal and opposite transverse momentum (p T ), whose magnitude is, say p CM T . As we can see from Fig. 5, one of these two particles decays into two-body final states, whereas the other one splits into four-body final states, resulting into 3τ s and missing energy signal. If we distinguish these three τ leptons according to the decreasing order of their p T , the leading-p T τ will mostly come from the two-body decays, i.e., Fig. 5(b) and (c). On the other hand, the two subleading τ 's will always come from the four-body decays mentioned above. As the third τ is the least energetic one, it will be difficult to explain the kinematics of the third τ with multiple accompanying invisible particles coming from such four-body decays. Moreover, the statistics of the third τ will also be very less. Hence we will confine our discussions within the two leading-p T τ 's and their observables. It is obvious from Fig. 5 that, the two-body visible channel is ψpχq Ñ φ τ (Fig. 5(b) and (c)), whereas the invisible channel is ψ 0 Ñ ν φ (Fig. 5(a)). In Fig. 6, we show some of the relevant distributions of the 3τ channel for a set of benchmark points (BPs) which satisfy the required relic density. In Table II, we give those BPs and the corresponding cross sections for this channel. Here the relic density is satisfied through pair annihilation channels as we concluded for the doublet case [39]. From the distributions of Fig. 6, we can deduce the following inferences: (i) We see sharp "Jacobian peaks" in the p T distributions for leading and subleading-p T τ 's for BPs 5-7. These "Jacobian peaks" appear at m{2 in p T -distributions and at m in missing transverse mass distributions. Here m is the mass of the parent particle which decays to τ . From the peaks of Fig. 6(a) and (b) we can infer that τ 1 is coming from χ and τ 2 from W bosons for BP7. We can conclude that both χ and W boson are on-shell for this benchmark point. For BPs 5-6, the peaks are slightly towards the right. This fact signifies that here the leading and sub-leading τ 's do not entirely come from χ and W , but predominantly so. We see further confirmation for this from the distributions of the respective transverse masses which we have not included for the sake economy of space.
(ii) The BPs 1-4 and 5-7 can be similarly classified. While the earlier set gives a relatively flatter profile, the latter shows sharp peaks in the p T and E miss T distribution plots. From the discussion above, we can see that the reason for this is the off-shell-ness of the mediating particles. This point is also clear from the values of the parameters in Table II.
(iii) The p τ1 T distribution profile is similar to that of E miss T , whereas the peak of ∆φ τ1E miss T shows that τ 1 and E miss T are back to back in nature. We can infer that they are coming from the same parent particles and so they are equal and opposite in nature.
(iv) The invariant mass plot of τ 1 and τ 2 gives the signal that they are pair produced from the Z boson decay.
(v) The distinct features of BP 7 are evident from all the distributions. The very high values of m ψ 0˘indicate that these particles in the intermediate states will suppress the effects of the respective diagram very much to χ.

2.
τ`E miss T channel Now we shall look into the effects of mixing in the Z boson mediated signal processes. For this we focus on the τ 2ν 2φ channel. This channel can also have contributions from gg initiated s-channel processes which are very much suppressed for the chosen BPs. In Table III, we give those relic satisfied BPs and the corresponding cross sections for this channel. Similarly, the qs q mediated Higgs boson channels are also suppressed due to negligible hqs q couplings. We also have γ mediated processes, but they are independent of the mixing parameter s α at the production level, as can be seen from Eq. (A9) and hence they are not interesting for our purpose where we focus on the effects of the mixing parameter in the distributions. So our focus will only be on the Z boson mediated channels shown in Fig. 7. These channels can proceed through the following different modes as can be seen from Fig. 7: (a) Z boson mediated s-channel processes through qs q Ñ ψ ψ, χ χ, ψ χ and ψ 0 s ψ 0 channel; and (b) Z boson mediated s-channel as well as quark-mediated t-channel processes through qs q Ñ W W channel. As in the case for 3τ ν 2φ channel, here also the latter case is very similar to the f s f Ñ W W in the context of the Unitarity problems of the gauge bosons [63][64][65]. As a result, as the CM energy increases, the cross section of these channels decreases and can be explained from the Equivalence Theorem of the gauge bosons. Hence, the processes (a) is much dominant w.r.t. the processes (b) and it should be sufficient to focus only on the processes (a) given in Fig. 7 while explaining the features of this channel. In Fig. 8, we show the relevant distributions of the τ 2ν 2φ channel for the set of BPs which satisfy the required relic density.   Some interesting observations from the distributions of Fig. 8, are as follows: (i) If we closely look into the BPs for this channel in Table III, they are chosen such that the first BP is for s α " 1{ ? 2 where there is equal mixing between the doublet and singlet component of the dark fermions. We gradually change the value of s α to zero. The rest of the dependent and independent components are chosen just to keep the relic density within the allowed limit. (ii) For s α " 0, there is no mixing, and hence ψ is a pure doublet state whereas χ is purely singlet. Now since we see from Eq. (A9) that W χψ 0 and Zψχ vertices do not exist for such a case, we have only the doublet contribution for BP 0. As a result, the distributions are independent of the values of m χ and y S τ . (iii) From the numbers given in Table III we see that the role of mixing dictates the dominant channel and hence the trends of invariant mass distribution. Hence we conclude these numbers that, for all the BPs except BP3, Fig. 7(b) is the dominant channel. This can also be seen in the distribution of ∆φ τ in Fig. 8, where we see that both τ and is along the same direction, whereas ∆φ τ E miss T further establishes this conclusion where we will find τ and E miss T going in the opposite directions. We have not kept ∆φ τ E miss T plot here to avoid redundancy. (iv) The p T distributions of Fig. 8 justify the dominance of the channels Fig. 7 as a function of the mixing angle. In both p T and p τ T distributions, it is seen that towards the low p T region, Fig. 7(b) dominated channels are more probabilistic whereas the high p T region favours Fig. 7(a) dominated channels. This could be attributed to the feature that in Fig. 7(b) dominated channels, the energy share of both the light lepton and τ is less than the energy share in Fig. 7(a). In Fig. 7(b), both the lepton and τ are produced from the decay of a single ψ 0 whereas in Fig. 7(a), they are produced from different mediators. This feature is more prominent in p τ T rather than p T . This is because in Fig. 7(a), the branching of the decay chain producing τ is comparatively less than what it is in Fig. 7(b), making the difference in energy share of τ between Fig. 7(a) and (b) dominated channels more prominent. The lepton, on the other hand, is produced through more branching in both the figures, which makes the effect of reduced energy share less prominent as Fig. 7(b) takes over Fig. 7(a) with the increase of mixing angle in the chosen benchmark points.
We observe the same trend through all the distributions of Figs. 6 and 8. There is a gradual change in the peak and tail positions of the kinematic distributions with the variations of s α . This change is more prominent for the latter case as we have chosen the BPs as such. Here, apart from the mixing parameter, s α , all the other dependent and independent parameters of the model are tweaked to some extent so that they satisfy the required relic density. Despite these small tweaks, we can say from our observations that this feature of the distributions is solely dependent on s α and not anything else. We confirmed our assertion by keeping the rest of the independent parameters same and varying only the value of s α .

B. Indirect Search Prospects
Apart from the discovery potential at the collider discussed above, the dark sector mixing can affect other DM detection possibilities as well. For the leptophilic scalar DM discussed here, the dominant indirect detection channel is φφ Ñ τ`τ´( Fig. 9). We discussed the Fermi-LAT constraints on parameter space for the scalar DM and lepton doublet interaction in Ref. [39, p.7]. There, we showed that most of the parameter region is allowed by experimental bound, except for a small region at low DM mass. In m φ À 200 GeV, the region with large Yukawa coupling (y τ Á 2.0) is excluded by Fermi-LAT limits and this bound becomes more stringent in one considers small ∆m.
In this work, the velocity-averaged annihilation cross section, ie, xσvy τ τ will show dependence on the mixing parameter as the charged dark leptons appear in the t-channel propagator. In Table IV, we show that for fixed δm's and Yukawa couplings, a finite mixing can bring xσvy τ τ below the Fermi-LAT bounds which would otherwise be above the limits in [39].    IV. Relic density allowed benchmark points which are below the Fermi-LAT limit. Around m φ " 100 GeV, the Fermi-LAT bound is " 10´2 6 cm 3 /s. The bound on the upper limit of Yukawa coupling can be relaxed for a finite dark sector mixing. It is obvious that due to having two charged dark fermions in this work, there are two ∆mp" δm m φ ) parameters. On keeping one ∆m small and setting the other at a high value, it is possible to address both relic density and indirect search constraints. It becomes very interesting to apply this in low m φ and small ∆m scenario, because as said previously, this region is typically above the limits in similar models.
The benchmark points in Table IV are so chosen that for each point, DM-SM interaction takes place predominantly through the mixing. The relic density, on the other hand, is satisfied through various coannihilation and mediator annihilation channels which also have a strong mixing dependence. We observe here that for low DM mass of around 100 GeV, xσvy τ τ remains below the Fermi-LAT limit even for coupling as high as y D τ " 2.5 and one of the ∆m's sufficiently small.

V. CONCLUSION
We have proposed a viable model with a leptophilic singlet scalar WIMP DM. A Z 2 symmetry renders the stability to the dark sector. Apart from the DM candidate, the dark sector consists of a SU p2q L doublet and a singlet fermion. The presence of only a SU p2q L doublet in the dark sector which interacts through gauge as well as Yukawa couplings to the SM, adds new annihilation channels in the relic density calculation. However, in that case, one cannot distinguish one component of the doublet from the other by any means because of the degeneracy in their masses. Even if the degeneracy is lifted by the introduction of a new scale of EWSB through the extension of the scalar sector, both the doublet components interact through the same couplings and channels. This poses a serious problem in their distinction from one another. Introduction of a dark singlet in this scenario yields interesting features in the phenomenology. This is due to the fact that now, based on the electric charge of the new singlet, one component of the doublet will mix with it while the other component remains independent of mixing. Not only this additional degree of freedom automatically lifts the mass degeneracy of the dark fermions without making it an ad hoc proposition, but through the dark sector mixing, we can segregate their effects in an experimental setup. Hence, on one hand, our model revives the simplest model of scalar singlet DM from the clutches of the stringent bounds of DM search experiments. On the other hand, it opens new search possibilities in the controlled environment of non-collider as well as collider based experiments. We conclude our observation and inferences in the following: (1) For a better understanding of the mixing effects, in the relic density calculation, we choose the parameter region where the coannihilation channels are dominant. It is well known in the literature that the presence of coannihilation channels can boost up the relic density without adding to DM direct searches. For a similar model in Ref. [39] with leptophilic scalar DM and a fermionic doublet partner, we explored the viable parameter region thoroughly. In the present work, we show that in comparison with the previous study, here it is possible to relax the parameter region by a few orders of magnitude for appropriate tuning of the mixing parameter. This is due to the fact that as the dark fermions can non-trivially coannihilate now, effectively more channels are added to the total annihilation cross section. To compensate for this increase, we showed in the analysis that for larger mixing in the coannihilation regime, one needs to have a smaller coupling in order to be relic density allowed. This makes a larger parameter space viable over the full range of the mixing compared to the previous work.
(2) We show in the analysis that mixing can be a very useful tool in discriminating between the dark sector particles of different isospins. This is because of the fact that for the two extrema of the mixing angle, one of the dark partners is purely a singlet while the other one remains a pure SU p2q L doublet. For the intermediate values of the mixings however, it is obvious that the charged dark partners are mixed states. For very low or high mixing, coannihilation of these two dark fermions with DM will be substantially different from each other. This primarily because of the channels involving W boson, which are available for the SU p2q L doublet dark partner for very low mixing and the singlet dark fermion for very high mixing. In our analysis, we discuss with correlation plots how these Gauge couplings help to clearly demarcate the parameter region w.r.t the contribution of the singlet and doublet dark partners towards total DM annihilation. However, as the mixing increases, the relative contribution of these dark sector particles accordingly vary. In the mixed scenario, it is interesting to observe how the viable parameter region evolves when the other parameters are fixed and it is only up to the mixing parameter to dictate the contribution of various dark sector coannihilation channels.
(3) The mixing can directly affect various DM search prospects, e.g., indirect detection and collider searches. In indirect detection, the velocity-averaged annihilation cross section has a dependence on mixing due to having mixed states in the propagator. We show in our analysis that for low DM mass, it is possible to relax the existing bound on the upper limit of Yukawa coupling in the presence of mixing.
(4) From the observations of the kinematic distribution of various observables for the 3τ`E miss T and τ`E miss T channels, we conclude that one can clearly distinguish the effects of the mixing parameter that remains unaffected by the change in other free parameters of the model. That this gradual change in the peak and tail positions of the kinematic distributions with the variations of mixing is independent of other free parameters, was further established by changing the mixing parameter and keeping the rest of the independent parameters fixed. We can ensure the presence of the mixing parameter between the dark sector particles of the theory by looking at the peak and tail positions these distributions. That this feature of mixing is not limited to the Dirac fermions only can be concluded from the other studies in the literature [40,44]. We conclude this article with the assertion that the mixing between a singlet and a doublet dark sector fields can turn the table in favour of a quintessential scalar singlet DM model. It evades the stringent experimental bounds from the DM detection experiments as well as presents new opportunities for its detection in the lab.

ACKNOWLEDGMENTS
RI thanks the SERB-DST, India for the research grant EMR/2015/000333. SC acknowledges MHRD, Government of India for research fellowship.

(A10)
Appendix B: Dark matter annihilation channels In addition to the DM candidate, if there are other dark sector particles which are in chemical equilibrium with each other, the thermal freeze-out of WIMP will occur through three basic processes, namely the pair annihilation, coannihilation and mediator annihilation. In this model, apart from the scalar DM φ, there are three dark additional fields, namely ψ 0 , ψ˘and χ˘. For small mass splits, the DM pair annihilation cross section is superseded by that of the other two processes.
Among the non-DM dark sector particles, ψ 0 and ψ˘belongs to a dark SU p2q L fermionic doublet and χ˘is a singlet dark fermion. From the mass diagonalization in Appendix A, we see that the two charged dark fermions mix, the mixing parameter being s α . The pair annihilation channels are given in Fig. 10. Here, the mixing effect will be visible only for the diagram that has the charged dark fermion in the propagator. But since there is no W boson coupling involved in this process, it is difficult to distinguish between the singlet dark fermionic contribution from the doublet towards the total DM annihilation. φφ Ñ τ`τ´also contributes substantially to the indirect detection cross section.  Fig. 11 discusses the possibilities for coannihilation. As one can see from the diagrams, there are mixed states appearing in the initial state as well as in the propagator. Also, unlike the pair annihilation process, coannihilation diagrams also involve the W couplings, which is useful for the distinction between the singlet and the doublet fermion contribution. All these factors together make coannihilation an ideal scenario for exploring the mixing effects. As well known in the literature, these processes become effective if the mass splitting between the DM and the dark sector particles is not very large. But the parameter space is not as constrained as the mediator annihilation scenario as we discuss below, so one can amply see the mixing effects within the viable parameter region. To achieve right relic through these processes, one needs a moderate dark sector-SM coupling (À 1 if dark sector mass varies around a few hundred of GeVs.).
Finally, the mediator annihilation possibilities are given in Fig. 12. These processes become effective for a very small mass splitting between the DM and the dark sector particles. Also, DM annihilation can address the right relic through these processes if the dark sector-SM coupling is minuscule [39]. As obvious from the diagrams, the mixed states can appear in one or both of the initial state as well as in the propagator. But since the viable parameter region is very much constrained for these channels to be effective, the mixing effects are not very much perceivable.