Minimal Inert Doublet benchmark for dark matter and the baryon asymmetry

In this article we discuss a minimal extension of the Inert Doublet Model (IDM) with an effective CP-violating D=6 operator, involving the inert Higgs and weak gauge bosons, that can lift it to a fully realistic setup for creating the baryon asymmetry of the Universe (BAU). Avoiding the need to stick to an explicit completion, we investigate the potential of such an operator to give rise to the measured BAU during a multi-step electroweak phase transition (EWPhT) while sustaining a viable DM candidate in agreement with the measured relic abundance. We find that the explored extension of the IDM can account quantitatively for both DM and for baryogenesis and has quite unique virtues, as we will argue. It can thus serve as a benchmark for a minimal realistic extension of the SM that solves some of its shortcomings and could represent the low energy limit of a larger set of viable completions. After discussing the impact of a further class of operators that open the possibility for a larger mass splitting (enhancing the EWPhT) while generating the full relic abundance also for heavy inert-Higgs DM, we ultimately provide a quantitative evaluation of the induced lepton electric dipole moments in the minimal benchmark for the BAU. These arise here at the two-loop level and are therefore less problematic compared to the ones that emerge when inducing CP violation via an operator involving the SM-like Higgs.


Introduction and model setup
Thanks to the discovery of a resonance resembling the Higgs particle proposed in the 1960s [1][2][3], at ATLAS [4] and CMS [5] at the Large Hadron Collider in 2012, the minimal Standard Model of Particle Physics (SM) was completed.Subsequent studies of couplings of the Higgs particle to fermions and electroweak (EW) gauge bosons showed agreement with the SM predictions and thus demonstrated, once more, the powerful predictiveness of the theory.However, despite the success of gaining understanding of the properties of elementary particles and their interactions, it is well-known that the SM lacks in providing explanations for various phenomena, inter alia, the existence of dark matter (DM) and the observed baryon asymmetry of the Universe (BAU).
In this article, we attempt to address these questions via an effective-fieldtheory (EFT) approach for the Inert Doublet Model (IDM), minimally extended at a beyond-IDM energy scale Λ.The IDM has already been widely studied as a model for DM and in the context of the electroweak phase transition (EWPhT) as a first step towards explaining baryogenesis [6][7][8][9][10].Nonetheless, since the interactions between the additional inert scalars and the SM states preserve the CP symmetry, baryogenesis cannot be achieved in the non-modified 'vanilla' IDM.Adding an effective CP -violating operator allow us then to (quantitatively) accommodate the missing Sakharov condition and explain the BAU within the framework.Moreover, due to its minimal nature, this effective IDM could serve as a realistic economic benchmark extension of the SM that solves prominent shortcomings and -with its new scalars being preferably rather light -could be seen as the low energy limit of a larger class of viable completions residing at higher scales.
The existence of DM is well established through a wide range of observations [11][12][13], including colliding clusters (e.g. the bullet cluster), rotational curves of various galaxies, gravitational lensing, structure formation, big bang nucleosynthesis, and the cosmic microwave background.The energy density of the unknown DM component today is quantified by [14] Ωh with the critical energy density ρ crit = 3H 2 0 /(8πG N ) defined in terms of today's Hubble parameter H 0 and Newton's gravitational constant G N .To avoid an overclosure of the Universe, the model under consideration should not predict a larger DM relic abundance than the reference value given above.
Many candidates have been proposed to account for DM 1 , among which the weakly interacting massive particles (WIMPs) are of the most appealing.Their mass ranges between a few GeV and O(100) TeV and they interact only weakly with the SM particles.WIMPs are thermally produced via freeze out and their "final" comoving density can make out the entirety of the measurable DM relic abundance in Eq. (1.1).The IDM naturally features a WIMP DM candidate.
Similarly, H ± correspond to two new CP -even, electrically charged physical scalars, whereas H and A are two additional neutral scalars, the former being CP -even while the latter is CP -odd.We choose H to be the lightest scalar and therefore the DM candidate.Its stability is guaranteed by the aforementioned Z 2 symmetry, under which all SM fields are even but H 2 is odd.This is the prominent feature of the IDM and prohibits any interaction term between the inert doublet H 2 and SM fermions (and therefore perilous contributions to flavour-changing neutral currents [16]) at the renormalizable level.The scalar potential is given by where all the couplings are real2 and the masses of the scalars are given by with the short-hand notations The theoretical and experimental constraints on the model, e.g. from perturbative unitarity, vacuum stability, invisible SM Higgs decays into a pair of inert scalars, or electroweak precision tests, as well as the parameter space allowing for the correct DM abundance can be found in Ref. [10] and references therein.Since real couplings prevent CP violation, the IDM must be augmented in order to become a realistic model of baryogenesis.To this end, we focus mostly on the dimension-six operator which plays a rather unique role within the set of potential operators, as we will explain further below. 3Here, V µν V µν = W a µν W a,µν + B µν B µν represents the sum of products of the SU (2) L isospin and U (1) Y hypercharge field strength tensors and their respective duals.Lifting the assumption of equal coefficients does not change the result for the BAU, as this is governed only by the coefficient of the SU (2) L term.As we will see, considering the field strength coupling to the inert doublet has phenomenological advantages over the alternative involving the 'active' H 1 doublet, for example leading to suppressed electric dipole moments of leptons ( EDMs).Still, also the latter operator can lead to viable results for the BAU in corners of the parameter space, and we will analyze this below, too.However, the focus is on the role of the operator in Eq. (1.6) in baryogenesis and its impact on the DM relic abundance, which will be studied in detail in the next sections.
We point out that the IDM augmented by this operator delivers a minimal, yet versatile, benchmark model to study the simultaneous realization of DM and the BAU.Given stringent constraints on CP -violating operators from limits on EDMs, together with the modest required corresponding coefficients to realize the BAU that we find, it is very reasonable that CP violation is generated at a higher scale. 4This makes the implementation via effective operators particularly suitable, being able to describe the effect of a set of potential completions.
The structure of this article is as follows.After having introduced and motivated the model in Sec. 1, the baryogenesis mechanism is elaborated on in Sec. 2, including also a discussion on constraints from experimental limits on EDMs.Consecutively, the dark matter abundance for suitable parameters is investigated in Sec. 3. We finally present our conclusions in Sec. 4, while a series of appendices contains technical details, in particular a two-loop analysis of the EDM induced by the second Higgs.

Baryogenesis at the electroweak scale
In this work we consider the scenario of baryogenesis during the EWPhT, an idea that has been extensively studied in the past (see, e.g., Refs.[30][31][32][33][34][35][36][37][38][39][40][41] and references therein).The BAU is quantified by where n B is the difference of baryon and anti-baryon number densities and n γ is the number density of photons.Assuming CP T invariance, the three vital ingredients for successful baryogenesis, elaborated by Sakharov [42] in 1967, must be fulfilled: in addition to violation of baryon number B and of charge conjugation symmetry C as well as of the combination CP of charge-conjugation and parity symmetry, the presence of an out-of-equilibrium process is a requisite.The first condition of this list is met by the fact that neither baryon nor lepton number is conserved in the SM because of the U (1) B+L anomaly, as shown by 't Hooft in 1976 [43].This violation is mediated by sphaleron processes which become effective at sufficiently high temperatures as later realized by Kuzmin, Rubakov and Shaposhnikov [30].In fact, for temperatures below ∼ 10 13 GeV [44], sphalerons are expected to be active and in thermal equilibrium, effectively preventing the creation of a net baryon number.Below the EW scale, on the other hand, sphaleron processes are Boltzmann suppressed.However, a baryon asymmetry can be created during the EWPhT and it will then remain, provided that the sphalerons are quickly turned off thereafter.This is the so-called wash-out condition which requires of a strong firstorder phase transition, generating the out-of equilibrium situation mentioned above, that we know is not provided by the SM since the Higgs mass of m h ≈ 125 GeV is too large.On top of that, the CP violation in the weak sector of the SM is too small to explain the measured BAU even if the other two Sakharov conditions were fulfilled (see Ref. [44] for instance).Hence, an SM extension must feature an additional source of CP violation and a strong first-order EWPhT.
The latter issue is addressed in the model considered here by the presence of the additional Higgs doublet H 2 , which also opens up the possibility for a multi-step EWPhT (see Fig. 1).Starting for example in the symmetric phase with vanishing vevs, the scalar potential can evolve either via one transition after which the SM Higgs doublet has developed a finite vev, i.e., H 1 , H 2 = (0, 0) → v 1 / √ 2, 0 , or via (multiple) intermediate steps.Note that the inert nature of H 2 is restored at zero temperature.We will consider a two-step EWPhT with one additional transition, proceeding as , 0 , as analyzed recently by two of the present authors in Ref. [10]. 5  To study the generation of the BAU, first we realize, following the analysis pioneered in Ref. [46], that the operator introduced in (1.6) can be written as (see 5 Although Ref. [45] found recently the possibility of two simultaneous vevs in a slice of parameter space, which would be interesting to examine further, we consider the case of only one finite vev at a time. one-step tw o -s te p also Refs.[47][48][49]) with the (dual) SU (2) L field strength tensor W a µν ( W a µν ), the beyond-IDM energy scale Λ absorbed in the Wilson coefficient c2 ≡ λ CP /Λ 2 , and the baryon current j µ B . 6The interaction term in Eq. (2.2) leads to an effective chemical potential, producing a shift in the energy levels of baryons with respect to antibaryons in the thermal distribution, and the sphaleron processes generate a BAU during a moderate temporal change of H 2 , like during a two-step EWPhT described above.
The mentioned shift in the free energy leads to a minimum associated to an equilibrium value for the baryon number density of [46] n eq B = c2 3) The evolution of baryon number then follows a Boltzmann-like equation of the form with the sphaleron rate given in terms of the weak coupling α W ≡ g 2 /(4π) as Following the estimate in Ref. [46] and considering a strong first-order EWPhT, the resultant BAU in terms of the vev H 2 = v c at the critical temperature T c reads where ∆t is the period of time needed by the transition to take place in a volume with a radius given by the correlation length ξ ∼ (α W T c ) −1 .The bubble expansion is assumed to occur with constant velocity v wall , so that ∆t = ξ/v wall .To quantify the BAU via Eq.(2.1), we recall that the photon number density is given by with the Riemann ζ-function and g * = 2 spin polarizations, respectively [44].The resultant dependence of the BAU on the critical vev, the bubble wall velocity7 and the Wilson coefficient c2 is shown in Fig. 2. Assuming the new coupling constant to be ∼ O (1), we can inspect that Λ ≈ 200 TeV results in the measured value of the BAU for a viable value of v c (gray band) and a wide range of bubble wall velocities.We note that the crucial ingredients of our setup, strong two-step EWPhT and sufficient CP violation, offer promising handles to further probe the framework.On the one hand, the sizable vevs of the intermediate transition might cause very characteristic gravitational waves signatures (see, e.g., Refs.[53,54]) whose study is, nonetheless, out of the scope of this work.On the other hand, additional sources of CP violation are in general constrained by null results in measurements of the electric dipole moment of elementary or composite particles like leptons ( EDM) or baryons (see, e.g., Refs.[55][56][57][58][59][60][61][62][63]).To ensure that the contribution of the operator defined in Eq. (1.6) to the EDM is below the sensitivity of ongoing experiments, we will focus on this aspect for the remaining part of this section.The current best upper bound on the eEDM, parametrized by d e , given in Ref. [64], and the projection of the ACME collaboration read [63,65] (2.8b) Similarly, the current limit on the µEDM set by the muon (g − 2) experiment at Brookhaven National Laboratory and the projected ones by J-PARC and PSI muEDM are [66][67][68] (2.9a) As a consequence of the Z 2 symmetry, the operator of the IDM EFT (IDMeft) contributes to the EDM only at two-loop level at leading order, whereas the dominant contribution of the related SM EFT (SMeft) operator c1 The details of the calculation for both operators are presented in Appendix A. The SMeft operator has been recently analyzed by Kley et al. [63] and it is considered here for the sake of comparison.Analogous to the analysis of the IDMeft operator, Fig. 3 illustrates the BAU obtained with the SMeft operator during the EWPhT associated with H 1 .As can be seen, a rather similar size of the CP -violating operator to the one involving the inert doublet, studied before, is required to arrive at the correct baryon abundance.Choosing a rather generic value of c1 = 15 PeV −2 at the energy scale µ = m h , that reproduces the correct BAU, and utilizing the publicly available Mathematica package DsixTools 2.0 [69,70] for accounting of the running dictated by the ) , bubble wall velocity v wall , and Wilson coefficient c1 .Like in Fig. 2, the bubble wall velocity and BSM coupling are indicated by line style and colour, respectively.The light-gray contour represents the range of possible critical vevs of an EWPhT found in Ref. [10].
renormalization group equations, the EDM as derived in Sec.A.1 reads which is already in tension with the bound of Eq. (2.8a) (though it could still be met in corners of the parameter space).In contrast, applying the result from Sec. A.2, the EDM d H 2 induced by the IDMeft operator reads where we have assumed a typical inert DM mass m H = 71 GeV in the low-mass regime and the other inert states being degenerate in mass (throughout the paper), here with the splitting ∆m ≡ m H ± ,A − m H = 410 GeV, and λ 345 = −0.002,c2 = 25 PeV −2 .These results suggest that the IDMeft operator can account for the BAU while generating an eEDM within the projected range of experimental sensitivity of ACME III, however safely below the current limit.The EDMs of the other leptons are considerably out of reach.
Before closing the analysis of baryogenesis, it is worth pointing out potential improvements for a more accurate calculation of the BAU.For instance, in addition to investigating the impact of different bubble wall profiles on the effective chemical potential and thus on the maximally achievable BAU, a more precise description of the dynamics of the PhT, including the latent heat driving the expansion of the bubble and the frictional force the bubble experiences while expanding in the plasma, would allow to quantify the sphaleron dynamics and thereby the resultant BAU more accurately.
Finally, we would like to mention that the operator (1.6) is in fact quite unique when seeking to add CP violation to the IDM involving the inert Higgs.As demonstrated for this operator, but holding more generally, this has the advantage that EDMs arise at higher loops compared to the case of similar operators featuring H 1 -the reason being that more lines, involving H 2 (that does not feature a zerotemperature vev), need to be closed.
Potentially alternative choices for CP -violating terms involving H 2 read where in particular the first operator is interesting since it allows for a Yukawa-like interaction of the inert Higgs with fermions, respecting the Z 2 symmetry.However, none of them is capable of injecting the sought CP violation at the phase transition in the H 2 direction, given that the background value of H 1 , entering the operators, vanishes there -and the same holds for the operators discussed below in Eq. (3.1).

Dark Matter results
In addition to the analysis on the possibility to attain the BAU in this model, it is also important to examine the impact on DM phenomenology.As found in the preceding section, the Wilson coefficient of the IDMeft operator must fulfil c2 ∼ 25 PeV −2 for a two-step EWPhT with a critical vev of v c ∼ 100 GeV for generating a BAU matching the measured one.Here we discuss the consequences of this operator for DM physics.Therefore, we calculate the relic abundance as well as the direct-detection (DD) cross sections with the public micrOMEGAs package [71].The details of the analysis on the impact on the relic abundance and DD cross section are presented in Appendix B.
Previous studies of the (original) IDM show that the interesting parameter space comprises the DM mass regimes of 55 GeV m H 80 GeV and m H 500 GeV (see, e.g., Refs.[16,72]). 8In contrast to mass spectra with a large DM mass, the low-mass regime also features a suitable parameter space with a strong first-order EWPhT either via one step or two steps as described before.Therefore, together with the CP -violating operator, the low-mass regime can in principle accommodate DM and baryogenesis, provided that the impact of the new operator on the DM relic abundance is not harmful.First, we demonstrate in Fig. 4 that the dimension-six operator contributes to the total thermally averaged annihilation cross section σv constructively, regardless of the sign of c2 .As long as |c 2 | 10 −1 TeV −2 , the annihilation 8 The first range can be even extended down to ∼ 44 GeV for a narrow BSM mass spectrum [22].cross section σv and thus the resultant relic abundance are virtually identical to the respective quantities in the vanilla IDM which means that the Wilson coefficients appearing in Fig. 2 clearly do not affect the DM relic abundance significantly.We emphasize that the c2 = 25 PeV −2 does not change the relic abundance and yet delivers the measured BAU.Accordingly, the viable parameter space in the low-mass regime is shown in the left panel of Fig. 5.Note that the mass splitting is sufficiently large, so that even larger mass splittings, as required for a two-step EWPhT, effectively do not change the surviving parameter set.The red lines represent the XENON1T DD bounds, indicating that only Higgs-portal couplings |λ 345 | 0.01 are experimentally allowed.
Looking at Fig. 4, one can anticipate that increasing c2 will lead to a shift of the viable colored area towards smaller λ 345 in the region of m H > m h /2.Interestingly, that could in principle enhance the possible DM parameter space, opening the region between 63 GeV and 70 GeV and thereby avoiding the necessity to sit in rather tuned regions, visible in the left plot of Fig. 5.However, as it turns out, the corresponding required size of c2 would lead to a significantly too large BAU.On the other hand, a UV completion that induces c2 |H 2 | 2 V µν V µν is generically also expected to generate the CP -conserving operator c 2 |H 2 | 2 V µν V µν (see Appendix C), which does not impact the BAU.To explore this possibility, we show in Fig. 5 the corresponding DM parameter space for c 2 = 6 • 10 −7 GeV −2 .We note that this would correspond to new particles not far above the TeV scale with O(1) CP -conserving couplings, while the respective CP -violating interactions would need to be some orders of magnitude smaller.Interestingly enough, there are completions where the CP -conserving operator receives additional contributions compared to the CP -violating one (see The red solid lines enclose the region in agreement with XENON1T DD bounds.The dependence of the relic abundance on the mass splitting has been studied in Ref. [10]. Appendix C for more details).We now inspect that the formerly excluded parameter space opens and a viable DM abundance can be achieved for a much broader range of masses.Fortunately, DD bounds are basically unaffected because the operator contributes to the DD cross section only at the loop level.In summary, our benchmark scenario provides successful baryogenesis together with a much broader range of viable DM masses of 55 GeV m H 70 GeV, compared to the original IDM.

Comments on High-Mass Regime
The analysis of the extended IDM has shown so far that the D = 6 operator can give rise to the measured DM relic abundance and the BAU with DM in the lowmass regime.In the remainder of this section we will pursue the question of whether corresponding parameter space exists also in the high-mass regime.Based on one of the findings in Ref. [10], this regime does not feature a two-step EWPhT and hence renders the operator in Eq. (1.6) futile for producing the BAU.Yet, one can consider the CP -violating SMeft operator c1 |H 1 | 2 V µν V µν for generating the BAU via a one-step EWPhT instead, see Fig. 3.However, regardless of the choice of the two D = 6 operators, the high-mass mass regime does not feature a strong firstorder EWPhT while creating a substantial fraction of the DM relic abundance, as the latter requires a fairly degenerate BSM mass spectrum [10,22].The reason for this is the increase of the cross section of DM annihilation into longitudinal gauge bosons for larger mass splittings, i.e. for ∆m 10 GeV [10], which consequently results in underabundant DM.Nonetheless, it is precisely for ∆m ∼ 200 GeV that one can attain a strong first-order EWPhT in this regime.One way to potentially cure this problem is introducing further effective operators which modify interactions between the DM particle and SM gauge bosons.The dimension-six operators that serve this purpose and that we will consider in the following read where we take the four C i to be real 9 for the sake of simplicity.They are promising, since they can contribute to annihilation into longitudinal gauge bosons (i.e., Goldstone modes).The contributions of each of these operators to the total cross section are investigated in Appendix B. We find that negative values of the Wilson coefficients lead to destructive interference and thus to an enhancement of the relic abundance.In fact, the behaviour of the total cross section is determined by an interplay between reducing the impact of the annihilations of two DM particles into EW gauge bosons and increasing the annihilations into a pair of either SM Higgs bosons or top quarks.A scan over possible values of the C i leads for example to a viable benchmark of As can be seen in Fig. 6, this set allows to reproduce the measured DM relic abundance for a large mass splitting of ∆m ∼ 120 GeV 10 GeV, while still respecting all experimental and theoretical constraints.However, it turns out that this is not enough to reach a strong first-order EWPhT, in particular because also the large required |λ 345 | weakens the transition.Anyways, the extension of the viable DM region to significantly larger mass splitting furnishes already a significant first step towards a realistic model of baryogenesis and DM also in the high mass regime.In fact, further operators that are expected in typical UV completions (including those presented in the appendix), such as |H 1 | 6 , also enhance the EWPhT (see Refs. [73][74][75][76]) and a combined effect could lead to a strong transition.Still, regarding the beauty of minimality, the low mass regime arguably furnishes a more attractive scenario of baryogenesis and DM in the IDM framework.

Conclusions
In this work, we investigated different effective operators to augment the IDM in order to fully account for baryogenesis without losing the DM candidate.We found that in the low-mass regime the IDMeft operator |H 2 | 2 V µν V µν allows to explain the  ) and two possible mass splittings ∆m with degenerate non-DM inert scalar masses.Since the CP -violating operator discussed in the previous section does not lead to a baryon asymmetry in the high-mass regime, it is turned off, i.e. c2 = 0.The parameter space is truncated at 60% of Ωh 2 ref and the red lines represent the XENON1T DD bounds such that the parameter space between both lines is not excluded.
measured BAU in addition to the DM abundance, indicating a beyond-IDM energy scale Λ ∼ 200 TeV (assuming an O(1) coupling) and avoiding stringent constraints from the eEDM (see Appendix A for the details of the two-loop calculation).We also pointed out that once adding the corresponding CP -conserving operator, the viable DM range gets significantly broadened to 55 GeV m H 70 GeV.
On the contrary, the high-mass regime needs a few more effective operators due to the mutual exclusion of a sizable fraction of the DM relic abundance and an appropriate nature of the EWPhT in the original IDM.Considering the CPviolating SMeft operator |H 1 | 2 V µν V µν for generating the BAU indicates a scale (c 1 ) −1/2 ∼ 300 TeV, while additional D = 6 operators, detailed above, can help to reconcile the DM relic abundance and a strong EWPhT when appearing at a scale of O (1 TeV).
In conclusion, our analysis demonstrates that the economic extension of the SM scalar sector by one inert SU (2) L doublet can in fact be a crucial first step towards a model that solves quantitatively some questions that the SM left open.Its augmentation with the advocated IDMeft operator delivers a simple and realistic benchmark that explains both the BAU and DM that can be investigated further.The EFT approach allows to cover a multitude of potential UV completions, with a couple of them being presented in Appendix C.

A Calculation of the lepton EDM
This appendix is dedicated to the explicit calculation of the EDM parameter d of the lepton for the two D = 6 operators involving the field strength tensors, i.e. the one in Eq. (1.6) and the similar operator featuring the SM-like Higgs instead of H 2 .The low-energy effective operator associated with the EDM reads with σ µν ≡ i [γ µ , γ ν ] /2 and the electromagnetic field strength tensor F µν (see, e.g., Refs.[63,77]).The second expression contains the usual chiral projections The following calculation is based on 'naive dimensional regularization', as discussed in Refs.[63,78,79], which retains the anti-commutation properties of γ 5 for any number of space-time dimensions.The γ 5 matrix can be expressed in terms of the other γ µ matrices and the Levi-Civita symbol ε µνρσ as γ 5 ≡ −iε µνρσ γ µ γ ν γ ρ γ σ /4! with ε 0123 = 1.In the following, we consider a lepton with mass m , electric charge Q in terms of the elementary charge e, incoming momentum p 1 , and outgoing momentum k 1 , as well as an incoming photon with momentum p 2 .

A.1 SM effective operator
Since the structure of the operator in Eq. (A.1) involves a chirality flip, the tree-level interaction between the photon and the lepton does not contribute to the EDM in the model at hand.At leading order in perturbation theory the present D = 6 operator connects the incoming photon via a loop (SM Higgs boson h and photon γ or Z boson) with the lepton, as shown in Fig. 7. Allowing for different coefficients for both field-strength terms in the following, i.e. cHW = cHB , the D = 6 operator becomes and gives rise to those two Feynman diagrams, considered in this calculation for the EDM for a general choice of the Wilson coefficients.For notational convenience, we define cγ ≡ cHW sin 2 θ W + cHB cos 2 θ W and cZ ≡ cHW − cHB .
Focusing on the left-hand diagram in Fig. 7 with a mediating photon and one specific chirality configuration, the matrix element reads Making use of the identity γ λ γ β = ({γ λ , γ β } + [γ λ , γ β ]) /2 = g λβ − iσ λβ , omitting the suppressed term proportional to the lepton mass in the numerator, and introducing the short-hand notation Ξ a,b ≡ (q +a) 2 −b 2 for the factors in the denominator coming from the propagators allow us to write As we will see later, the metric term does not contribute due to the anti-symmetry of the Levi-Civita tensor.The integral in Eq. (A.4) will appear frequently in the subsequent calculation and we will hence present its evaluation here.Recasting it by introducing the Feynman parameters x, y, z leads to10 α , and the shifted momentum q and momentum-independent remainder ∆ read (employing As the denominator of the integrand is symmetric in the integration momentum q upon sign flip, terms of the numerator linear in q will vanish after integration for symmetry reasons and only those terms containing either the product qλ qα or a qindependent numerator will remain.The former leads via dimensional regularization after a Wick rotation to Euclidean spacetime to in d = 4 − 2 spacetime dimensions.The latter (q-independent numerator), on the other hand, becomes Considering only the q-dependent numerator in the integrand, as the contributions from I 0 are further suppressed in m 2 /m 2 h 1, one gets in the MS renormalization scheme x,y,z In practice, the divergence and constant term arising from the dimensional regularization are absorbed by the SMeft counterterm operator L σ µν H 1 e R B µν + σ a /2 W a µν with the Pauli matrices σ a .Alternatively, one could just cut off the loop integral at the new-physics scale.
Hence, expanding in m m h ultimately leads to the EDM, reading Considering a mediating Z boson, coupling to the right-handed lepton in the first diagram, the matrix element reads 1) from Eq. (A.1) where we have, as before, introduced Feynman parameters, performed a Wick rotation, neglected the lepton mass, and considered the term proportional to the squared momentum in the integral (as shown explicitly above).Evaluating the integral for the massive mediator, and taking the 'mirrored' diagram into account, gives rise to where we have used c L + c R = T 3, − 2Q sin 2 θ W . Consequently, the full EDM ultimately amounts to and matches11 the one by Kley et al. [63] with cHB = cHW = c1 , reading

A.2 IDM effective operator
In addition to the SMeft operator in Eq. (A.2a), the IDM effective field theory (IDMeft) operator reads As the BSM Higgs doublet does not acquire a vev at zero temperature, the contribution to the EDM occurs at two-loop level for the first time: a loop involving H, A, or H ± connects the effective vertex to the SM Higgs boson.With the assignment of the particles' momenta given in the left-hand Feynman diagram in Fig. 8, the corresponding matrix element for a mediating photon and an H-loop for instance reads with the symmetry factor S (here S = 2) and the definition c γ ≡ c HW sin 2 θ W + c HB cos 2 θ W .
Introducing the Feynman parameters y 1 , z 1 for the first integral, the product of integrals becomes Analogously to the calculation in Sec.A.1, we keep only the leading term in the numerator that is quadratic in q1 and thus find with the first integral being over the Feynman parameters x 2 , y 1 , z 1 .Assuming neg- ligibly small ratios p 1,2 /m H gives rise to 3 γ 5 σ ρσ , (A. 27) where we applied the relation in Eq. (A.10).This integral can be evaluated in Euclidean space and reads (A.28) The present divergences can be eliminated by introducing appropriate counterterms L σ µν H 1 e R B µν + σ a /2 W a µν + h.c. as in the previous section, so that we can focus solely on the mass-dependent finite part F (m) of the integral, which is rather lengthy and thus not displayed here.The corresponding matrix element reads with the SMEFT Wilson coefficient c γ defined at the scale m H , being agnostic about its nature at this point.Note that we assume corrections involving lepton masses to be negligible, as they are considerably lighter than the (B)SM Higgs.
Taking the contributions of H, A, and H ± into account, together with their respective symmetry factors (S = 2 for H, A; S = 1 for H ± ), we find with degenerate BSM non-DM fields for the EDM parameter In the following, we will consider the dimension-six operators of (3.1), containing the SM gauge covariant derivative.As the center-of-mass energy of the annihilating DM particles is much larger than the masses of the SM gauge bosons involved, we apply the Goldstone boson equivalence theorem and therefore consider only the longitudinally polarized gauge bosons in the respective final states.Since CalcHEP does not take more than four fields for the computation of the cross sections into account, we keep only those -most important -terms in the following discussion.

B.1 Impact on thermally averaged annihilation cross section σv
The first operator, associated with the Wilson coefficient C 1 , induces and the evolution of the thermally averaged annihilation cross section with respect to C 1 , as well as the contributions of the most relevant and interesting annihilation processes, are visualized in the left panel of Fig. 9.As expected for interference effects in the calculation of the cross sections, the sign of the Wilson coefficient significantly affects the annihilation cross sections for large values.In turn, the difference between the cross sections for opposite signs tends to zero as the effect becomes marginal for sufficiently small Wilson coefficients and the curve approaches the annihilation cross section governed by the renormalizable (vanilla) IDM.This expected effect is evident in each plot of Figs.9-10.The second dimension-six operator that we consider leads to Its impact on the DM annihilation cross section is depicted in the right panel of Fig. 9.The behaviour of the four processes is qualitatively the same for both effective operators.The minima of the annihilation cross sections for HH → ZZ and HH → W + W − are located at the same value of the Wilson coefficient, since the contributions of the annihilation channels (i.e.four-point interaction, s-and t-channel, and uchannel if necessary) for both processes are equal pairwise.
The third and fourth operator, corresponding to C 3 and C 4 , respectively, lead to and the corresponding cross sections in Fig. 10 exhibit a different behaviour than the previous ones.The operators presented above are almost identical: The differences appear in interaction terms involving the neutral Goldstone boson G. Hence, the Wilson coefficients C 3 and C 4 affect the cross section of HH → ZZ in an asymmetric way, whereas they influence the cross sections of the other annihilation processes in a symmetric way.As a result, the latter cross sections are invariant under an exchange of Wilson coefficients.While the cross sections for HH → hh can easily be understood from the four-point interaction, the particular behaviour of the cross sections of DM annihilations into gauge bosons requires the interplay of s-, t-and possibly u-channels to obtain the cancellation.due to momentum conservation for the contribution to the HHh-vertex factor, with p 1 and p 2,3 being the SM Higgs' and the DM particles' momenta, respectively.Since micrOMEGAs computes σ SI in the limit of vanishing square of the momentum transfer, i.e. q 2 ≡ p 1,µ p µ 1 = 0, contributions of the operators associated with C 3,4 are absent in our numerical results.This approximation is justified as the transferred momentum in DD scattering processes is O (1 − 100) MeV m h [14].Hence, the only contribution to the SI DD cross section arises from the first operator and the dependence of σ SI on the sign of C 1 and the Higgs portal coupling is shown in Fig. 11.

C Remarks on UV-complete models C.1 UV realization in the low-mass regime
To realize the effective CP -violating operator H † 2 H 2 B µν B µν (and similarly with W a µν ) in Eq. (1.6), crucial for the low-mass regime, there are various possibilities, in particular at the one-loop level, which is sufficient to generate the required magnitude of c2 ∼ 25 PeV −2 , found in Sec. 2. Examples of UV realizations are depicted in Fig. 12.
To generate the operator at tree-level, one can introduce a heavy spin-1 field V µ in the (1, 2) 1 representation of the SM (with Q = T 3 + Y /2).On top of this, one can for example add a scalar singlet S, which allows for various loop-generated contributions, or envisage vector-like fermions that also generate the operator at oneloop level (see below).We note that the operator in Eq. (1.6) captures all such UV completions via a single new parameter.Extending the results of Ref. [80] to the IDM, we find that it can be induced from the following bosonic terms (see Fig. 12) Note that the heavy vector must transform in the same way under a Z 2 transformation as the inert Higgs doublet H 2 to allow for tree-level generation.In addition, this UV extension naturally gives rise to the CP -conserving operator H † 2 H 2 W a µν W aµν (similarly for B µν ) through the first line or via loop-suppressed realizations via the other CP -conserving operators.Interestingly, this term receives contributions from an additional diagram, compared to the CP -violating one, being proportional to |γ V | 2 -which could motivate its larger size (see the discussion in Sec. 3). 12nother possibility is introducing vector-like fermions with appropriate Z 2 and weak hypercharge.As an example, we consider the vector-like fermions N (L,R) : (1, 1) 0 and ∆ (L,R) : (1, 2) −1 .The relevant part of the Lagrangian reads and the primed heavy vectors are in the same representations as their siblings above.Relevant diagrams for the matching are depicted in Fig. 13.

Figure 1 .
Figure1.Illustration of possible scenarios for the evolution of the EW vacuum.During a two-step EWPhT, a non-trivial intermediate field configuration is possible in principle (dashed line), as found in Ref.[45].As a characteristic feature of the IDM, H 2 disappears for T → 0 in each case.

6 Figure 2 .
Figure 2. Dependence of the baryon asymmetry parameter η on the critical vev v c = √ 2 H 2 (T c ) , bubble wall velocity v wall , and Wilson coefficient c2 .The bubble wall velocity and BSM coupling are indicated by line style and colour, respectively.The light-gray contour represents the range of possible critical vevs at the first stage of a two-step EWPhT found in Ref. [10].

7 Figure 3 .
Figure 3. Dependence of the baryon asymmetry parameter η on the critical vev v c = √2 H 1 (T c ) , bubble wall velocity v wall , and Wilson coefficient c1 .Like in Fig.2, the bubble wall velocity and BSM coupling are indicated by line style and colour, respectively.The light-gray contour represents the range of possible critical vevs of an EWPhT found in Ref.[10].

Figure 4 .
Figure 4. Thermally averaged annihilation cross section σv for various channels in terms of the BSM coupling parameter c2 .The DM mass is m H = 71 GeV, the mass splitting ∆m = 410 GeV, and the Higgs portal coupling was chosen to be λ 345 = −0.002.The arrows indicate those values for c2 considered in Fig. 2.

Figure 5 .
Figure 5. Parameter space for a sizable amount of the measured DM abundance in terms of the DM mass m H and the Higgs portal coupling λ 345 for fixed mass splitting ∆m and Wilson coefficients c2 , c 2 (c 2 = 0 if not specified otherwise).The inner boundary corresponds to the full relic abundance, i.e.Ωh 2 = Ωh 2 ref , and the parameter space is truncated at 60% of Ωh 2 ref .The red solid lines enclose the region in agreement with XENON1T DD bounds.The dependence of the relic abundance on the mass splitting has been studied in Ref.[10].

Figure 6 .
Figure 6.Relic abundance in terms of the DM mass m H and the Higgs portal coupling λ 345 for fixed Wilson coefficients (see Eq. (3.2)) and two possible mass splittings ∆m with degenerate non-DM inert scalar masses.Since the CP -violating operator discussed in the previous section does not lead to a baryon asymmetry in the high-mass regime, it is turned off, i.e. c2 = 0.The parameter space is truncated at 60% of Ωh 2 ref and the red lines represent the XENON1T DD bounds such that the parameter space between both lines is not excluded.

Figure 7 .
Figure 7. Feynman diagrams for processes contributing to d , including the momentum flow.The dotted vertices correspond to insertions of the D = 6 operator and a cross attached to a dashed line indicates the SM Higgs vev entering the vertex factor.The right panel shows the respective 'mirrored' diagram.

Figure 8 .
Figure 8. Two-loop contribution to the EDM from the BSM operator |H 2 | 2 F µν F µν .The incoming vector boson is (by definition) a photon, but the internal one can be either a photon or a Z boson.

Figure 9 .
Figure 9. Evolution of the thermally averaged cross section σv with respect to C 1 (left) and C 2 (right).The DM mass is m H = 490 GeV, the mass splitting between the degenerate inert scalars and the DM particle is ∆m = 120 GeV, and the Higgs portal coupling reads λ 345 = −1.3.

Figure 10 .
Figure 10.Evolution of the thermally averaged cross section with respect to C 3 (left) and C 4 (right).The model parameters are the same as for Fig. 9.

B. 2 (C 3 + C 4 ) (p 1 ,µ p µ 2 − p 1 ,µ p µ 3 ) ∝ p 1 ,µ p µ 1 (
Impact on the direct-detection cross section σ SI One can ask whether and to what extent these operators affect the spin-independent (SI) direct-detection (DD) cross section σ SI , which are mediated solely by an SM Higgs boson.Besides the vanilla IDM vertex factor and a term proportional to C 1 , we obtain

6 Figure 11 .
Figure 11.Dependence of the neutron SI DD cross section σ SI,n on the sign of C 1 for the DM mass m H = 490 GeV, the mass splitting ∆m = 120 GeV, and the Higgs portal coupling λ 345 as displayed.

Figure 13 .
Figure 13.Example diagrams for realizations of the effective vertices in Eq. (C.3), induced by a heavy vector singlet B ( ) or vector triplet W ( )a .