CP in the Dark
Abstract
We build a model containing two scalar doublets and a scalar singlet with a specific discrete symmetry. After spontaneous symmetry breaking, the model has Standard Modellike phenomenology, as well as a hidden scalar sector which provides a viable dark matter candidate. We show that CP violation in the scalar sector occurs exclusively in the hidden sector, and consider possible experimental signatures of this CP violation. In particular, we study contribution to anomalous gauge couplings from the hidden scalars.
DESY 18129
KATP192018
I Introduction
The LHC discovery of a scalar with mass of GeV Aad et al. (2012a); Chatrchyan et al. (2012) completed the Standard Model particle content. The fact that precision measurements of the properties of this particle Aad et al. (2015, 2016) indicate that it behaves very much in a Standard Model (SM)like manner is a further confirmation of the validity and effectiveness of that model. Nonetheless, the SM leaves a lot to be explained, and many extensions of the theory have been proposed to attempt to explain such diverse phenomena as the existence of dark matter, the observed universal matterantimatter asymmetry and others. In particular, numerous SM extensions consist of enlarged scalar sectors, with singlets, both real and complex, being added to the SM Higgs doublet McDonald (1994); Burgess et al. (2001); O’Connell et al. (2007); BahatTreidel et al. (2007); Barger et al. (2008); He et al. (2007); Davoudiasl et al. (2005); Basso et al. (2014); Fischer (2016); or doublets, the simplest example of which is the twoHiggs doublet model (2HDM) Lee (1973); Branco et al. (2012). Certain versions of singletdoublet models provide dark matter candidates, as does the Inert version of the 2HDM (IDM) Deshpande and Ma (1978); Barbieri et al. (2006); Lopez Honorez et al. (2007); Dolle and Su (2009); Lopez Honorez and Yaguna (2010); Gustafsson et al. (2012); Swiezewska (2013); Swiezewska and Krawczyk (2013); Arhrib et al. (2014); Klasen et al. (2013); Abe et al. (2015); Krawczyk et al. (2013); Goudelis et al. (2013); Chakrabarty et al. (2015); Ilnicka et al. (2016). Famously, the 2HDM was introduced in 1973 by Lee to allow for the possibility of spontaneous CP violation. But models with dark matter candidates and extra sources of CP violation (other than the SM mechanism of CKMmatrix generated CP violation) are rare. Even rarer are models for which a “dark” sector exists, providing viable dark matter candidates, and where the extra CP violation originates exclusively in the “dark” sector. To the best of our knowledge, the only model with scalar CP violation in the dark sector is the recent work of Refs. CorderoCid et al. (2016); Sokolowska (2017), for which a threedoublet model was considered. The main purpose of Refs. CorderoCid et al. (2016); Sokolowska (2017) was to describe the dark matter properties of the model. In Ref. Moretti (2017) an argument was presented to prove that the model is actually CP violating, adapting the methods of Refs. Grzadkowski et al. (2016); BéluscaMaïto et al. (2018) for the complex 2HDM (C2HDM).
In the current paper we will propose a model, simpler than the one in CorderoCid et al. (2016), but which boasts the same interesting properties, to wit: (a) a SMlike Higgs boson, , “naturally” aligned due to the vacuum of the model preserving a discrete symmetry; (b) a viable dark matter candidate, the stability of which is guaranteed by the aforementioned vacuum and whose mass and couplings satisfy all existing dark matter search constraints; and (c) extra sources of CP violation exist in the scalar sector of the model, but only in the “dark” sector. This hidden CP violation will mean that the SMlike scalar, , behaves almost exactly like the SM Higgs boson, and in particular (unless contributions from a high number of loops are considered) has couplings to gauge bosons and fermions which are exactly those of a scalar. This is all the more remarkable since the CP violation of the proposed model is explicit. The extra particle content of the model, as advertised, is simpler than the model of CorderoCid et al. (2016), consisting of two Higgs doublets (both of hypercharge ) and a real singlet (). This is sometimes known as the Nextto2HDM (N2HDM), and was the subject of a thorough study in Mühlleitner et al. (2017). The N2HDM version considered in this paper uses a different discrete symmetry than the symmetries considered in Mühlleitner et al. (2017), designed, as will be shown, to produce both dark matter and dark CP violation. The paper is organised as follows: in section II we will introduce the model, explaining in detail its construction and symmetries, as well as the details of spontaneous symmetry breaking that occurs when one of the fields acquires a vacuum expectation value (vev). In section III we will present the results of a parameter space scan of the model, where all existing constraints – both theoretical and experimental (from colliders and dark matter searches) – are taken into account; deviations from the SM behaviour of in the diphoton channel, stemming from the existence of a charged scalar, will be discussed, as will the contributions of the model to dark matter observables; in section IV we will discuss how CP violation arises in the dark sector, and how it might have a measurable impact in future colliders. Finally, we conclude in section V.
Ii The scalar potential and possible vacua
For our purposes, the N2HDM considered is very similar to that discussed in Ref. Mühlleitner et al. (2017), in that the fermionic and gauge sectors are identical to the SM and the scalar sector is extended to include an extra doublet and also a singlet scalar field – thus the model boasts two scalar doublets, and , and a real singlet . As in the 2HDM, we will require that the Lagrangian be invariant under a sign flip of some scalar fields, so that the number of free parameters of the model is reduced and no treelevel flavourchanging neutral currents (FCNC) occur Glashow and Weinberg (1977); Paschos (1977). The difference between the current work and that of Mühlleitner et al. (2017) consists in the discrete symmetry applied to the Lagrangian – here, we will consider a single symmetry of the form
(1) 
With these requirements, the most general scalar potential invariant under is given by
(2)  
where, with the exception of , all parameters in the potential are real. As for the Yukawa sector, we consider all fermion fields neutral under this symmetry. As such, only the doublet couples to fermions, and the Yukawa Lagrangian is therefore
(3) 
where we have only written the terms corresponding to the third generation of fermions, with the Yukawa terms for the remaining generations taking an analogous form. The lefthanded doublets for quarks and leptons are denoted by and , respectively; , and are the righthanded top, bottom and fields; and is the charge conjugate of the doublet .
Notice that since the two doublets have the same quantum numbers and are not physical (only the mass eigenstates of the model will be physical particles), the potential must be invariant under basis changes on the doublets. This is a wellknown property of 2HDMs, which the N2HDM inherits: any unitary transformation of these fields, with a unitary matrix , is an equally valid description of the theory. Though the theory is invariant under such transformations, its parameters are not and undergo transformations dependent on . A few observations are immediately in order:

Since only has Yukawa interactions it must have a vev to give mass to all charged fermions^{1}^{1}1And neutrinos as well, if one wishes to consider Dirac mass terms for them.. In fact the Yukawa sector of this model is identical to the one of the SM, and a CKM matrix arises there, as in the SM.

The symmetry considered eliminates many possible terms in the potential, but does not force all of the remaining ones to be real – in particular, both the quartic coupling and the cubic one, , can be a priori complex. However, using the basis freedom to redefine doublets, we can absorb one of those complex phases into, for instance, . We choose, without loss of generality, to render real.
A complex phase on renders the model explicitly CP violating. Considering, for instance, the CP transformation of the scalar fields given by
(4) 
we see that such a CP transformation, to be a symmetry of the potential, would require all of its parameters to be real. Notice that the CP transformation of the singlet trivially does not involve complex conjugation as is real. In fact, this is a wellknown CP property of singlet fields Branco et al. (1999). One point of caution is in order: the complex phase of is not invariant under the specific CP transformation of Eq. (4), but by itself that does not prove that the model is explicitly CP violating. In fact, one could consider some form of generalized CP (GCP) transformation involving, other than complex conjugation of the fields, also doublet redefinitions: . The model can only be said to be explicitly CP violating if there does not exist any CP transformation under which it is invariant. So, conceivably, though the model breaks the CP symmetry defined by the transformation of Eq. (4), it might be invariant under some other one. The point is moot, however: As we will see ahead, the vacuum of the model we will be considering is invariant under the CP transformation of Eq. (4) (and the symmetry of Eq. (1)), but the theory has CP violation. Thus the CP symmetry was broken to begin with, and hence the model is explicitly CP violating.
Let us consider now the possibility of spontaneous symmetry breaking in which only the doublet acquires a neutral nonzero vev: . Given the structure of the potential in Eq. (2), the minimisation conditions imply that this is a possible solution, with all scalar components of the doublets (except the real, neutral one of ) and the singlet equal to zero, provided that the following condition is obeyed:
(5) 
Since all fermion and gauge boson masses are therefore generated by , it is mandatory that GeV as usual. At this vacuum, then, it is convenient to rewrite the doublets in terms of their component fields as
(6) 
where is the SMlike Higgs boson, with interaction vertices with fermions and gauge bosons identical to those expected in the SM (the diphoton decay of , however, will differ from its SM counterpart). The mass of the field is found to be given by
(7) 
and since GeV, this fixes the value of one of the quartic couplings, . The neutral and charged Goldstone bosons and , respectively, are found to be massless as expected, and the squared mass of the charged scalar is given by
(8) 
Finally, the two neutral components of the doublet , and , mix with the singlet component yielding a mass matrix,
(9) 
with and . There are therefore three neutral scalars other than , which we call , and , in growing order of their masses. This mass matrix can then be diagonalized by an orthogonal rotation matrix , such that
(10) 
and the connection between the original fields and the mass eigenstates is given by
(11) 
The rotation matrix can be parametrized in terms of three angles, , and , such that
(12) 
where for convenience we use the notation , . Without loss of generality, we may take the angles in the interval .
In the following we discuss several phenomenological properties of this model. The vacuum preserves the symmetry. As a result, the physical eigenstates emerging from and , i.e. the charged scalar and the neutral ones , and , carry a quantum number – a “dark charge” equal to – which is preserved in all interactions, to all orders of perturbation theory. In the following we refer to these four eigenstates as “dark particles”. On the other hand, the SMlike particles (, the gauge bosons and all fermions) have “dark charge” equal to 1. The preservation of this quantum number means that dark particles must always be produced in pairs while in their decays they must always produce at least one dark particle. Therefore, the lightest of these dark particles – which we will choose in our parameter scans to be the lightest neutral state, – is stable. Thus, the model provides one dark matter candidate.
The model indeed shares many features with the Inert version of the 2HDM, wherein all particles from the “dark doublet” are charged under a discrete symmetry, the lightest of which is stable. The main difference with the current model is the mixing that occurs between the two neutral components of the doublet and the singlet due to the cubic coupling , which can be appreciated from the mass matrix of Eq. (9). In what concerns the charged scalar, though, most of the phenomenology of this model is equal to the Inert 2HDM.
Iii Parameter scan, the diphoton signal and dark matter observables
With the model specified, we can set about exploring its available parameter space, taking into account all of the existing theoretical and experimental constraints. We performed a vast scan over the parameter space of the model (100.000 different combinations of the parameters of the potential of Eq. (2)), requiring that:

By construction, all treelevel interactions and vertices of the Higgs particle are identical to those of the SM Higgs boson. As a consequence, all LHC production cross sections for are identical to the values expected in the SM. Additionally, all decay widths of , apart from the diphoton case to be treated explicitly below, are identical to their SM values up to electroweak corrections. This statement holds as we require the mass to be larger than roughly 70 GeV, to eliminate the possibility of the decay (when this decay channel is open it tends to affect the branching ratios of , making it difficult to have be SMlike).

The quartic couplings of the potential cannot be arbitrary. In particular, they must be such that the theoretical requirements of boundedness from below (BFB) – that the potential always tends to along any direction where the scalar fields are taking arbitrarily large values – and perturbative unitarity – that the model remains both perturbative and unitary, in all scalar scattering processes – are satisfied. The model considered in the current paper differs from the N2HDM discussed in Ref. Mühlleitner et al. (2017) only via the cubic coefficient , so the treelevel BFB and perturbative unitarity constraints described there (in sections 3.1 and 3.2) are exactly the ones we should use here.

The constraints on the scalar sector arising from the PeskinTakeuchi electroweak precision parameters , and Peskin and Takeuchi (1990, 1992); Maksymyk et al. (1994) are required to be satisfied in the model. Not much of the parameter space is eliminated due to this constraint, but it is still considered in full.

Since the charged scalar does not couple to fermions, all physics bounds usually constraining its interactions are automatically satisfied. The direct LEP bound of GeV assumed a 100 % branching ratio of to fermions, so that this constraint also needs not be considered here.

Since all scalars apart from do not couple to fermions, no electric dipole moment constraints need be considered, this despite the fact that CP violation occurs in the model.
With these restrictions, the scan over the parameters of the model was such that:

The masses of the neutral dark scalars and and the charged one, , were chosen to vary between 70 and 1000 GeV. The last neutral mass, that of , is obtained from the remaining parameters of the model as explained in Mühlleitner et al. (2017).

The mixing angles of the neutral mass matrix, Eq. (12), were chosen at random in the interval and .

The quartic couplings and are constrained, from BFB constraints, to be positive, and were chosen at random in the intervals and , respectively. is chosen in the interval .

The quadratic parameters and were taken between 0 and GeV.
All other parameters of the model can be obtained from these using the expressions for the masses of the scalars and the definition of the matrix . The scan ranges for the quartic couplings are chosen larger than the maximally allowed ranges after imposing unitarity and BFB. Therefore, all of the possible values for these parameters are sampled. We have used the implementation of the model, and all of its theoretical constraints, in ScannerS Coimbra et al. (2013). N2HDECAY Engeln et al. (2018), a code based on HDECAY Djouadi et al. (1998, 2018), was used for the calculation of scalar branching ratios and total widths, as in Mühlleitner et al. (2017).
As we already explained, the treelevel interactions of are identical to the ones of a SM Higgs boson of identical mass. The presence of the charged scalar , however, changes the diphoton decay width of , since a new loop, along with those of the gauge boson and charged fermions, contributes to that width. This is identical to what occurs in the Inert model, and we may use the formulae of, for instance, Ref. Swiezewska (2013). Thus we find that the diphoton decay amplitude in our model is given by
(13) 
where the sum runs over all fermions (of electric charge and number of colour degrees of freedom ) and , and are the wellknown form factors for spin 0, 1/2 and 1 particles (see for instance Refs. Spira (1998, 2017)). The charged Higgs contribution to the diphoton amplitude in Eq. (13) changes this decay width, and therefore the total decay width, hence all branching ratios, of with respect to the SM expectation. However, the diphoton decay width being so small compared to the main decay channels for (, and ), the overall changes of the total width are minimal. In fact, numerical checks for our allowed parameter points have shown that the branching ratios of to , , and change by less than 0.05% compared to the corresponding SM quantities – therefore, all current LHC constraints for the observed signal rates of in those channels are satisfied at the 1 level.
As for the branching ratio into two photons, it can and does change by larger amounts, as can be appreciated from Fig. 1. In that figure we plot the ratio of the branching ratio of into two photons to its SM value as a function of the charged Higgs mass.
Comparing these results to the recent measurements of the signal rates^{2}^{2}2Notice that since in this model has exactly the same production cross sections as the SM Higgs boson, the ratio of branching ratios presented in Fig. 1 corresponds exactly to the measured signal rate, which involves the ratio of the product of production cross sections and decay branching ratios, between observed and SM theoretical values. from Ref. Sirunyan et al. (2018), we see that our model can accommodate values well within the 2 interval. The lower bound visible in Fig. 1 emerges from the present experimental lower limit from Sirunyan et al. (2018) at . The experimental upper limit, however, is larger than the maximum value of possible in our model. The latter results from the combination of BFB and unitarity bounds which constrain the allowed values of the coupling . The lowest allowed value for , which governs the coupling of , is about , and its maximum one roughly . Since the value of grows for negative , the lower bound on induces an upper bound of .
Thus we see that the model under study in this paper is perfectly capable of reproducing the current LHC data on the Higgs boson. Specific predictions for the diphoton signal rate are also possible in this model – values of larger or smaller than unity are easily accommodated, though they are constrained to the interval . As the parameter scan was made taking into account all data from dark matter searches, we are comfortable that all phenomenology in that sector is satisfied by the dark particles.
Let us now study how the model behaves in terms of dark matter variables.
Several experimental results put constraints on the mass of the dark matter (DM) candidate, and on its couplings to SM particles. The most stringent bound comes from the measurement of the cosmological DM relic abundance from the latest results from the Planck Collaboration Aghanim et al. (2018), . The DM relic abundance for our model was calculated with MicrOMEGAs Belanger et al. (2014). In our scan we accepted all points that do not exceed the value measured by Planck by more than . This way, we consider not only the points that are in agreement with the DM relic abundance experimental values but also the points that are underabundant and would need further dark matter candidates to saturate the measured experimental value.
Another important constraint comes from direct detection experiments , in which the elastic scattering of DM off nuclear targets induces nucleon recoils that are being measured by several experimental groups. Using the expression for the spinindependent DMnucleon cross section given by MicrOMEGAs, we impose the most restrictive upper bound on this cross section, which is the one from XENON1T Aprile et al. (2017, 2018).
In the left panel of Fig. 2 we use the parameter scan previously described to compute dark matter observables. We show the points that passed all experimental and theoretical constraints in the relic abundance versus dark matter mass plane. We present in pink the points that saturate the relic abundance, that is the points that are in the interval between and around the central value, and in violet the points for which the relic abundance is below the measured value. It is clear that there are points in the chosen dark matter mass range that saturate the relic density. In the right panel we present the spinindependent nucleon dark matter scattering cross section as a function of the dark matter mass. The upper bound (the grey line) represents the latest XENON1T Aprile et al. (2017, 2018) results. The pink points in the right plot show that even if the direct bound improves by a few orders of magnitude there will still be points for the entire mass range where the relic density is saturated.
Thus we see that the model under study in this paper can fit, without need for fine tuning, the existing dark matter constraints. Next we will study the rise of CP violation in the dark sector.
Iv CP violation in the dark sector
As we explained in section II, the model explicitly breaks the CP symmetry defined in Eq. (4). Notice that the vacuum of the model which we are studying – wherein only acquires a vev – preserves that symmetry. Therefore, if there is CP violation (CPV) in the interactions of the physical particles of the model, it did not arise from any spontaneous CPV, but rather the explicit CP breaking mentioned above^{3}^{3}3Again, because this is a subtlety of CP symmetries, let us repeat the argument: The fact that the model explicitly violates one CP symmetry – that defined in Eq. (4) – does not necessarily mean there is CPV, since the Lagrangian could be invariant under a different CP symmetry. If, however, we prove that there is CPV after spontaneous symmetry breaking with a vacuum that preserves the CP symmetry of Eq. (4), then that CPV is explicit..
There are several eventual experimental observables where one could conceivably observe CPV. For instance, a trivial calculation shows that all vertices of the form , with , are possible. These vertices arise from the kinetic terms for where from Eq. (6) we obtain, in terms of the neutral components of the second doublet,
(14) 
where is the coupling constant and is the Weinberg angle. With the rotation matrix between field components and neutral eigenstates defined in Eq. (12), we easily obtain ( =1,2,3)
(15) 
Thus decays or production mechanisms of the form , , for any dark neutral scalars, are simultaneously possible (with the boson possibly offshell) which would clearly not be possible if the had definite CP quantum numbers – in fact, due to CP violation, the three dark scalars are neither CPeven nor CPodd, but rather states with mixed CP quantum numbers. The simultaneous existence of all vertices, with , is a clear signal of CPV in the model, in clear opposition to what occurs, for instance, in the CPconserving 2HDM – in that model or are possible because is CPodd and , are CPeven, but or are forbidden. Since in our model all vertices with occur, the neutral scalars cannot have definite CP quantum numbers. Thereby CP violation is established in the model in the dark sector. Notice that no vertices of the form are possible. This is not due to any CP properties, however, but rather to the conservation of the quantum number. Thus observation of such decays or production mechanisms (all three possibilities for , , would have to be confirmed) could serve as confirmation of CPV in the model, though the nonobservability of the dark scalars would mean they would only contribute to missing energy signatures. Both at the LHC and at future colliders, hints on the existence of dark matter can appear in mono or monoHiggs searches. The current model predicts cascade processes such as and , leading to monoZ and monoHiggs events, respectively. This type of final states occurs in many dark matter models, regardless of the CPnature of the particles involved. Therefore, these are not good processes to probe CPviolation in the dark sector.
However, though CPV occurs in the dark sector of the theory, it can have an observable impact on the phenomenology of the SM particles. A sign of CPV in the model – possibly the only type of signs of CPV which might be observable – can be gleaned from the interesting work of Ref. Grzadkowski et al. (2016) (see also Ref. BéluscaMaïto et al. (2018)), wherein 2HDM contributions to the triple gauge boson vertices and were considered. A Lorentz structure analysis of the vertex, for instance Hagiwara et al. (1987); Gounaris et al. (2000, 2002); Baur and Rainwater (2000), reveals that there are 14 distinct structures, which can be reduced to just two form factors on the assumption of two onshell bosons and massless fermions, the offshell being produced by collisions. Under these simplifying assumptions, the vertex function becomes ( being the unit electric charge)
(16) 
where is the 4momentum of the offshell boson, and those of the remaining (onshell) bosons. The dimensionless form factor is CP violating, but the coefficient preserves CP. In our model there is only oneloop diagram contributing to this form factor, shown in Fig. 3. As can be inferred from the diagram there are three different neutral scalars
circulating in the loop – in fact, the authors of Ref. Grzadkowski et al. (2016) showed that in the 2HDM with explicit CPV (the C2HDM) the existence of at least three neutral scalars with different CP quantum numbers that mix among themselves is a necessary condition for nonzero values for . Notice that in the C2HDM there are three diagrams contributing to – other than the diagram shown in Fig. 3, the C2HDM calculation involves an additional diagram with an internal boson line in the loop, and another, with a neutral Goldstone boson line in the loop. In our model, however, the discrete symmetry we imposed forbids the vertices and (these vertices do occur in the C2HDM, being allowed by that model’s symmetries), and therefore those two additional diagrams are identically zero. In Grzadkowski et al. (2016) an expression for in the C2HDM was found, which can easily be adapted to our model, by only keeping the contributions corresponding to the diagram of Fig. 3. This results in
(17) 
where is the electromagnetic coupling constant and the LoopTools Hahn and PerezVictoria (1999) function is used. The factor denotes the product of the couplings from three different vertices, given in Ref. Grzadkowski et al. (2016) by
(18) 
where the () factors, shown in Fig. 3, are related to the coupling coefficients that appear in the vertices (in the C2HDM they also concern the and vertices, cf. BéluscaMaïto et al. (2018)). With the conventions of the current paper, we can extract these couplings from Eq. (15) and it is easy to show that
(19)  
where the simplification that led to the last line originates from the orthogonality of the matrix. We observe that the maximum value that can assume is , corresponding to the maximum mixing of the three neutral components, , and . This is quite different from what one expects to happen in the C2HDM, for instance – there one of the mixed neutral states is the observed 125 GeV scalar, and its properties are necessarily very SMlike, which implies that the matrix should approximately have the form of one diagonal element with value close to 1, the corresponding row and column with elements very small and a matrix mixing the other eigenstates^{4}^{4}4Meaning, a neutral scalar mixing very similar to the CPconserving 2HDM, where and mix via a matrix but does not mix with the CPeven states.. Within our model, however, the three neutral dark fields can mix as much or as little as possible.
In Fig. 4 we show, for a random combination of dark scalar masses ( GeV, GeV and GeV) the evolution of normalized to ^{5}^{5}5For this specific parameter space point, we have . with , the 4momentum of the offshell boson. This can be compared with Fig. 2 of Ref. Grzadkowski et al. (2016), where
we see similar (if a bit larger) magnitudes for the real and imaginary parts of , despite the differences in masses for the three neutral scalars in both situations (in that figure, the masses taken for and were, respectively, 125 and 400 GeV, and several values for the mass were considered). As can be inferred from Fig. 4, is at most of the order of . For the parameter scan described in the previous section, we obtain, for the imaginary part of , the values shown in Fig. 5. We considered two values of (corresponding to two possible collision energies for a future linear collider). The imaginary part of (which, as we will see, contributes directly to CPviolating observables such as asymmetries) is presented as a function of the overall coupling defined in Eq. (19). We in fact present results as a function of , to
(a)  (b) 
illustrate that indeed the model perfectly allows maximum mixing between the neutral, dark scalars. Fig. 5 shows that the maximum values for Im are reached for the maximum mixing scenarios. We also highlight in red the points for which the dark neutral scalars have masses smaller than 200 GeV. The loop functions in the definition of , Eq. (17), have a complicated dependence on masses (and external momentum ) so that an analytical demonstration is not possible, but the plots of Fig. 5 strongly imply that choosing all dark scalar masses small yields smaller values for Im. Larger masses, and larger mass splittings, seem to be required for larger Im. A reduction on the maximum values of Im (and Re) with increasing external momentum is observed (though that variation is not linear, as can be appreciated from Fig. 4). A reduction of the maximum values of Im (and Re) when the external momentum tends to infinity is also observed.
The smaller values for Im for the red points can be understood in analogy with the 2HDM. The authors of Ref. Grzadkowski et al. (2016) argue that the occurrence of CPV in the model implies a nonzero value for the basisinvariant quantities introduced in Refs. Lavoura and Silva (1994); Botella and Silva (1995), in particular for the imaginary part of the quantity introduced therein. Since Im is proportional to the product of the differences in mass squared of all neutral scalars, having all those scalars with lower masses and lower mass splittings reduces Im and therefore the amount of CPV in the model. Now, in our model the CPV basis invariants will certainly be different from those of the 2HDM, but we can adapt the argument to understand the behaviour of the red points in Fig. 5: those red points correspond to three dark neutral scalars with masses lower than 200 GeV, and therefore their mass splittings will be small (compared to the remaining parameter space of the model). In the limiting case of three degenerate dark scalars, the mass matrix of Eq. (9) would be proportional to the identity matrix and therefore no mixing between different CP states would occur. With this analogy, we can understand how regions of parameter space with larger mass splittings between the dark neutral scalars tend to produce larger values of Im.
Experimental collaborations have been probing double production to look for anomalous couplings such as those responsible for a vertex Aaltonen et al. (2008, 2009); Abazov et al. (2011, 2012); Aad et al. (2012b); Chatrchyan et al. (2013); Aad et al. (2013); Khachatryan et al. (2015a); Khachatryan et al. (2015b). The search for anomalous couplings in those works uses the effective Lagrangian for triple neutral vertices proposed in Ref. Hagiwara et al. (1987), parametrised as
(20) 
where vertices were also considered. In this equation, is the electromagnetic tensor, and . The coupling above is taken to be a constant, and as such it represents at most an approximation to the of Eq. (17). Further, the analyses of the experimental collaborations mentioned above take this coupling to be real, whereas the imaginary part of is the quantity of interest in many interesting observables. With all that under consideration, latest results from LHC Khachatryan et al. (2015b) already probe the coupling of Eq. (20) to order , whereas the typical magnitude of (both real and imaginary parts) is . We stress , however, that the two quantities cannot be directly compared, as they represent very different approaches to the vertex. A thorough study of the experimental results of Khachatryan et al. (2015b) using the full expression for the vertex of Eq. (16) and the full momentum (and scalar masses) dependence of the form factors is clearly necessary, but beyond the scope of the current work.
The crucial aspect to address here, and the point we wish to make with the present section, is that is nonzero in the model under study in this paper. Despite the fact that the neutral scalars contributing to the form factor are all dark particles, CP violation is therefore present in the model and it can indeed be “visible” to us, having consequences in the nondark sector. We also analysed other vertices, such as – there CPV form factors also arise, also identified as “”, and for our parameter scan we computed it by once again adapting the results of Ref. Grzadkowski et al. (2016) to our model. In the C2HDM three Feynman diagrams contribute to this CPviolating form factor (see Fig. 17 in Grzadkowski et al. (2016)) but in our model the symmetry eliminates the vertices and , so only one diagram involving the charged scalar survives. From Eq. (4.4) of Ref. Grzadkowski et al. (2016), we can read the expression of the CPviolating form factor from the vertex, obtaining
(21) 
Interestingly, this form factor is larger, by roughly a factor of ten, than the corresponding quantity in the vertex (though still smaller than the corresponding C2HDM typical values). This is illustrated in Fig. 6, where we plot the imaginary part of as given by Eq. (21) for 450 GeV,
having obtained nonzero values. Therefore CPV also occurs in the interactions in this model, though presumably it would be no easier to experimentally establish than for the vertex. The point we wished to make does not change, however – if even a single nonzero CPV quantity is found, then CP violation occurs in the model.
As an example of a possible experimental observable to which the form factors for the interactions might contribute, let us take one of the asymmetries considered in Ref. Grzadkowski et al. (2016), using the techniques of Ref. Chang et al. (1995). Considering a future linear collider and the process , taking cross sections for unpolarized beams for the production of two bosons of helicities and (assuming the helicity of the bosons can be determined), the asymmetry is defined as
(22) 
with the angle between the electron beam and the closest boson with positive helicity, denoting the velocity of the produced bosons and the function is given in appendix D of Ref. Grzadkowski et al. (2016). Choosing the two points in our parameter scan with largest (positive) and smallest (negative) values of for GeV, we obtain the two curves shown in Fig. 7.
Clearly, the smallness () of the form factor renders the value of this asymmetry quite small, which makes its measurement challenging. This raises the possibility that asymmetries involving the vertex might be easier to measure than those pertaining to the anomalous interactions, since we have shown that is typically larger by a factor of ten in the former vertex compared to the latter one. To investigate this possibility, we compared , considered above, with the asymmetry defined in Eq. (5.21) of Ref. Grzadkowski et al. (2016). A direct comparison of the maximum values of and shows that for some regions of parameter space the former quantity can indeed be one order of magnitude larger than the latter one; but that is by no means a generic feature, since for other choices of model parameters both asymmetries can also be of the same order. Notice that both asymmetries show a quite different dependence.
V Conclusions
We presented a model whose scalar sector includes two Higgs doublets and a real singlet. A specific region of parameter space of the model yields a vacuum which preserves a discrete symmetry imposed on the model – thus a charged scalar and three neutral ones have a “dark” quantum number preserved in all interactions and have no interactions with fermions. The lightest of them, chosen to be a neutral particle, is therefore stable and a good dark matter candidate. The first doublet yields the necessary Goldstone bosons and a neutral scalar which has automatically a behaviour almost indistinguishable from the SM Higgs boson. A parameter scan of the model, imposing all necessary theoretical and experimental constraints (including bounds due to relic density and dark matter searches, both direct and indirect) shows that the SMlike scalar state indeed complies with all known LHC data for the Higgs boson – some deviations may occur in the diphoton signal rate due to the extra contribution of a charged scalar to the involved decay width, but we have shown such deviations are at most roughly 20% of the expected SM result when all other constraints are satisfied, and this is still well within current experimental uncertainties.
The interesting thing about the model presented in this paper is the occurrence of explicit CP violation exclusively within the dark matter sector. A complex phase allowed in the potential forces the neutral components of the second (dark) doublet to mix with the real singlet to yield three neutral eigenstates, none of which possesses definite quantum numbers. Signals of this CP violation would not be observed in the fermion sector (which, by the way, we assume is identical to the one of the SM, and therefore has the usual CKMtype source of CP violation) nor in the interactions of the SMlike scalar – protected as it is by the unbroken symmetry, and by the mass ranges chosen for the dark scalars, will behave like a purely CPeven SMlike scalar, even though the CP symmetry of the model is explicitly broken in the scalar sector as well! Can the model then be said to be CP violating at all? The answer is yes, as an analysis of the contributions from the dark sector to the vertex demonstrates. Even though the dark particles have no direct fermion interactions and could elude detection, their presence could be felt through the emergence of anomalous triple gauge boson vertices. Though we concentrated mainly on vertices we also studied interactions, but our main purpose was to show CPV is indeed occurring. Direct measurements of experimental observables probing this CPV are challenging: we have considered a specific asymmetry, , built with production cross sections, but the magnitude of the CPV form factor yields extremely small values for that asymmetry, or indeed for other such variables we might construct. Direct measurements of production cross sections could in theory be used to constraint anomalous vertex form factors – and indeed several experimental collaborations, from LEP, Tevatron and LHC, have tried that. But the experimentalists’ approach is based on constant and real form factors, whereas modelspecific expressions for such as those considered in our work yield quantities highly dependent on external momenta, which boast sizeable imaginary parts as well. Thus a direct comparison with current experimental analyses is not conclusive.
The other remarkable fact is the amount of “damage” the mere inclusion of a real singlet can do to the model with two doublets. As repeatedly emphasised in the text, the model we considered is very similar to the Inert 2HDM – it is indeed simply the IDM with an added real singlet and a tweaked discrete symmetry, extended to the singlet having a “dark charge” as well. But whereas CP violation – explicit or spontaneous – is entirely impossible within the scalar sector of the IDM, the presence of the extra singlet produces a completely different situation. That one obtains a model with explicit CPV is all the more remarkable when one considers that the field we are adding to the IDM is a real singlet, not even a complex one. Notice that within the IDM it is even impossible to tell which of the dark neutral scalars is CPeven and which is CPodd – all that can be said is that those two eigenstates have opposite CP quantum numbers. The addition of a real singlet completely changes the CP picture.
The occurrence of CP violation in the dark matter sector can be simply a matter of curiosity, but one should not underestimate the possibility that something novel might arise from it. If the current picture of matter to dark matter abundance is indeed true and the observed matter is only 5% of the total content of the universe, then one can speculate how CP violation occurring in the interactions of the remainder matter might have affected the cosmological evolution of the universe. We reserve such studies for a followup work.
Acknowledgements
We acknowledge the contribution of the research training group GRK1694 ‘Elementary particle physics at highest energy and highest precision’. PF and RS are supported in part by the National Science Centre, Poland, the HARMONIA project under contract UMO2015/18/M/ST2/00518. JW gratefully acknowledges funding from the PIER Helmholtz Graduate School.
References
 Aad et al. (2012a) G. Aad et al. (ATLAS), Phys. Lett. B716, 1 (2012a), eprint 1207.7214.
 Chatrchyan et al. (2012) S. Chatrchyan et al. (CMS), Phys. Lett. B716, 30 (2012), eprint 1207.7235.
 Aad et al. (2015) G. Aad et al. (ATLAS, CMS), Phys. Rev. Lett. 114, 191803 (2015), eprint 1503.07589.
 Aad et al. (2016) G. Aad et al. (ATLAS, CMS), JHEP 08, 045 (2016), eprint 1606.02266.
 McDonald (1994) J. McDonald, Phys. Rev. D50, 3637 (1994), eprint hepph/0702143.
 Burgess et al. (2001) C. P. Burgess, M. Pospelov, and T. ter Veldhuis, Nucl. Phys. B619, 709 (2001), eprint hepph/0011335.
 O’Connell et al. (2007) D. O’Connell, M. J. RamseyMusolf, and M. B. Wise, Phys. Rev. D75, 037701 (2007), eprint hepph/0611014.
 BahatTreidel et al. (2007) O. BahatTreidel, Y. Grossman, and Y. Rozen, JHEP 05, 022 (2007), eprint hepph/0611162.
 Barger et al. (2008) V. Barger, P. Langacker, M. McCaskey, M. J. RamseyMusolf, and G. Shaughnessy, Phys. Rev. D77, 035005 (2008), eprint 0706.4311.
 He et al. (2007) X.G. He, T. Li, X.Q. Li, and H.C. Tsai, Mod. Phys. Lett. A22, 2121 (2007), eprint hepph/0701156.
 Davoudiasl et al. (2005) H. Davoudiasl, R. Kitano, T. Li, and H. Murayama, Phys. Lett. B609, 117 (2005), eprint hepph/0405097.
 Basso et al. (2014) L. Basso, O. Fischer, and J. J. van Der Bij, Phys. Lett. B730, 326 (2014), eprint 1309.6086.
 Fischer (2016) O. Fischer (2016), eprint 1607.00282.
 Lee (1973) T. D. Lee, Phys. Rev. D8, 1226 (1973).
 Branco et al. (2012) G. C. Branco, P. M. Ferreira, L. Lavoura, M. N. Rebelo, M. Sher, and J. P. Silva, Phys. Rept. 516, 1 (2012), eprint 1106.0034.
 Deshpande and Ma (1978) N. G. Deshpande and E. Ma, Phys. Rev. D18, 2574 (1978).
 Barbieri et al. (2006) R. Barbieri, L. J. Hall, and V. S. Rychkov, Phys. Rev. D74, 015007 (2006), eprint hepph/0603188.
 Lopez Honorez et al. (2007) L. Lopez Honorez, E. Nezri, J. F. Oliver, and M. H. G. Tytgat, JCAP 0702, 028 (2007), eprint hepph/0612275.
 Dolle and Su (2009) E. M. Dolle and S. Su, Phys. Rev. D80, 055012 (2009), eprint 0906.1609.
 Lopez Honorez and Yaguna (2010) L. Lopez Honorez and C. E. Yaguna, JHEP 09, 046 (2010), eprint 1003.3125.
 Gustafsson et al. (2012) M. Gustafsson, S. Rydbeck, L. LopezHonorez, and E. Lundstrom, Phys. Rev. D86, 075019 (2012), eprint 1206.6316.
 Swiezewska (2013) B. Swiezewska, Phys. Rev. D88, 055027 (2013), [Erratum: Phys. Rev.D88,no.11,119903(2013)], eprint 1209.5725.
 Swiezewska and Krawczyk (2013) B. Swiezewska and M. Krawczyk, Phys. Rev. D88, 035019 (2013), eprint 1212.4100.
 Arhrib et al. (2014) A. Arhrib, Y.L. S. Tsai, Q. Yuan, and T.C. Yuan, JCAP 1406, 030 (2014), eprint 1310.0358.
 Klasen et al. (2013) M. Klasen, C. E. Yaguna, and J. D. RuizAlvarez, Phys. Rev. D87, 075025 (2013), eprint 1302.1657.
 Abe et al. (2015) T. Abe, R. Kitano, and R. Sato, Phys. Rev. D91, 095004 (2015), [Erratum: Phys. Rev.D96,no.1,019902(2017)], eprint 1411.1335.
 Krawczyk et al. (2013) M. Krawczyk, D. Sokolowska, P. Swaczyna, and B. Swiezewska, JHEP 09, 055 (2013), eprint 1305.6266.
 Goudelis et al. (2013) A. Goudelis, B. Herrmann, and O. Stål, JHEP 09, 106 (2013), eprint 1303.3010.
 Chakrabarty et al. (2015) N. Chakrabarty, D. K. Ghosh, B. Mukhopadhyaya, and I. Saha, Phys. Rev. D92, 015002 (2015), eprint 1501.03700.
 Ilnicka et al. (2016) A. Ilnicka, M. Krawczyk, and T. Robens, Phys. Rev. D93, 055026 (2016), eprint 1508.01671.
 CorderoCid et al. (2016) A. CorderoCid, J. HernándezSánchez, V. Keus, S. F. King, S. Moretti, D. Rojas, and D. Sokolowska, JHEP 12, 014 (2016), eprint 1608.01673.
 Sokolowska (2017) D. Sokolowska, J. Phys. Conf. Ser. 873, 012030 (2017).
 Moretti (2017) S. Moretti, Talk presented at Scalars 2017, Warsaw (2017).
 Grzadkowski et al. (2016) B. Grzadkowski, O. M. Ogreid, and P. Osland, JHEP 05, 025 (2016), [Erratum: JHEP11,002(2017)], eprint 1603.01388.
 BéluscaMaïto et al. (2018) H. BéluscaMaïto, A. Falkowski, D. Fontes, J. C. Romão, and J. P. Silva, JHEP 04, 002 (2018), eprint 1710.05563.
 Mühlleitner et al. (2017) M. Mühlleitner, M. O. P. Sampaio, R. Santos, and J. Wittbrodt, JHEP 03, 094 (2017), eprint 1612.01309.
 Glashow and Weinberg (1977) S. L. Glashow and S. Weinberg, Phys. Rev. D15, 1958 (1977).
 Paschos (1977) E. A. Paschos, Phys. Rev. D15, 1966 (1977).
 Branco et al. (1999) G. C. Branco, L. Lavoura, and J. P. Silva, Int. Ser. Monogr. Phys. 103, 1 (1999).
 Peskin and Takeuchi (1990) M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 (1990).
 Peskin and Takeuchi (1992) M. E. Peskin and T. Takeuchi, Phys. Rev. D46, 381 (1992).
 Maksymyk et al. (1994) I. Maksymyk, C. P. Burgess, and D. London, Phys. Rev. D50, 529 (1994), eprint hepph/9306267.
 Belanger et al. (2007) G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 176, 367 (2007), eprint hepph/0607059.
 Belanger et al. (2014) G. Belanger, F. Boudjema, A. Pukhov, and A. Semenov, Comput. Phys. Commun. 185, 960 (2014), eprint 1305.0237.
 Aghanim et al. (2018) N. Aghanim et al. (Planck) (2018), eprint 1807.06209.
 Aprile et al. (2018) E. Aprile et al. (XENON) (2018), eprint 1805.12562.
 Coimbra et al. (2013) R. Coimbra, M. O. P. Sampaio, and R. Santos, Eur. Phys. J. C73, 2428 (2013), eprint 1301.2599.
 Engeln et al. (2018) I. Engeln, M. Mühlleitner, and J. Wittbrodt (2018), eprint 1805.00966.
 Djouadi et al. (1998) A. Djouadi, J. Kalinowski, and M. Spira, Comput. Phys. Commun. 108, 56 (1998), eprint hepph/9704448.
 Djouadi et al. (2018) A. Djouadi, J. Kalinowski, M. Muehlleitner, and M. Spira (2018), eprint 1801.09506.
 Spira (1998) M. Spira, Fortsch. Phys. 46, 203 (1998), eprint hepph/9705337.
 Spira (2017) M. Spira, Prog. Part. Nucl. Phys. 95, 98 (2017), eprint 1612.07651.
 Sirunyan et al. (2018) A. M. Sirunyan et al. (CMS) (2018), eprint 1804.02716.
 Aprile et al. (2017) E. Aprile et al. (XENON), Phys. Rev. Lett. 119, 181301 (2017), eprint 1705.06655.
 Hagiwara et al. (1987) K. Hagiwara, R. D. Peccei, D. Zeppenfeld, and K. Hikasa, Nucl. Phys. B282, 253 (1987).
 Gounaris et al. (2000) G. J. Gounaris, J. Layssac, and F. M. Renard, Phys. Rev. D61, 073013 (2000), eprint hepph/9910395.
 Gounaris et al. (2002) G. J. Gounaris, J. Layssac, and F. M. Renard, Phys. Rev. D65, 017302 (2002), [Phys. Rev.D62,073012(2000)], eprint hepph/0005269.
 Baur and Rainwater (2000) U. Baur and D. L. Rainwater, Phys. Rev. D62, 113011 (2000), eprint hepph/0008063.
 Hahn and PerezVictoria (1999) T. Hahn and M. PerezVictoria, Comput. Phys. Commun. 118, 153 (1999), eprint hepph/9807565.
 Lavoura and Silva (1994) L. Lavoura and J. P. Silva, Phys. Rev. D50, 4619 (1994), eprint hepph/9404276.
 Botella and Silva (1995) F. J. Botella and J. P. Silva, Phys. Rev. D51, 3870 (1995), eprint hepph/9411288.
 Aaltonen et al. (2008) T. Aaltonen et al. (CDF), Phys. Rev. Lett. 100, 201801 (2008), eprint 0801.4806.
 Aaltonen et al. (2009) T. Aaltonen et al. (CDF), Phys. Rev. Lett. 103, 091803 (2009), eprint 0905.4714.
 Abazov et al. (2011) V. M. Abazov et al. (D0), Phys. Rev. D84, 011103 (2011), eprint 1104.3078.
 Abazov et al. (2012) V. M. Abazov et al. (D0), Phys. Rev. D85, 112005 (2012), eprint 1201.5652.
 Aad et al. (2012b) G. Aad et al. (ATLAS), Phys. Rev. Lett. 108, 041804 (2012b), eprint 1110.5016.
 Chatrchyan et al. (2013) S. Chatrchyan et al. (CMS), JHEP 01, 063 (2013), eprint 1211.4890.
 Aad et al. (2013) G. Aad et al. (ATLAS), JHEP 03, 128 (2013), eprint 1211.6096.
 Khachatryan et al. (2015a) V. Khachatryan et al. (CMS), Phys. Lett. B740, 250 (2015a), [erratum: Phys. Lett.B757,569(2016)], eprint 1406.0113.
 Khachatryan et al. (2015b) V. Khachatryan et al. (CMS), Eur. Phys. J. C75, 511 (2015b), eprint 1503.05467.
 Chang et al. (1995) D. Chang, W.Y. Keung, and P. B. Pal, Phys. Rev. D51, 1326 (1995), eprint hepph/9407294.