Impact of semi-annihilations on dark matter phenomenology –an example of Z_{N} symmetric scalar dark matter

Impact of semi-annihilations on dark matter phenomenology – an example of Zn symmetric scalar dark matter

Abstract

We study the impact of semi-annihilations where is any dark matter and is any standard model particle, on dark matter phenomenology. We formulate minimal scalar dark matter models with an extra doublet and a complex singlet that predict non-trivial dark matter phenomenology with semi-annihilation processes for different discrete Abelian symmetries . We implement two such example models with and symmetry in micrOMEGAs and work out their phenomenology. We show that both semi-annihilations and annihilations involving only particles from two different dark matter sectors significantly modify the dark matter relic abundance in this type of models. We also study the possibility of dark matter direct detection in XENON100 in those models.

LAPTH, Univ. de Savoie, CNRS, B.P.110, F-74941 Annecy-le-Vieux Cedex, France
Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy
National Institute of Chemical Physics and Biophysics, Rävala 10, Tallinn 10143, Estonia
Skobeltsyn Inst. of Nuclear Physics, Moscow State Univ., Moscow 119992, Russia

1 Introduction

The origin of dark matter of the Universe is not known. In popular models with new particles beyond the standard model particle content, such as the minimal supersymmetric standard model, an additional discrete symmetry is introduced [1]. As a result, the lightest new -odd particle, is stable and is a good candidate for dark matter. The phenomenology of this type of models has been studied extensively.

The discrete symmetry that stabilises dark matter must be the discrete remnant of a broken gauge group [2], because global discrete symmetries are broken by gravity. The most natural way for the discrete symmetry to arise is from the breaking of a embedded in a larger gauge group, e.g. [3]. The latter contains gauged as a part of the symmetry, and the existence of dark matter can be related to the neutrino masses, leptogenesis and, in a broader context, to the existence of leptonic and baryonic matter [4, 5, 6].

Obviously, the discrete remnant of need not to be – in general it can be any Abelian symmetry. The possibility that dark matter may exist due to is a known [7, 8, 9, 10, 11, 12, 13, 14, 15], but much less studied scenario.1 Model independently, it has been pointed out in Ref. [15] that in models the dark matter annihilation processes contain new topologies with different number of dark matter particles in the initial and final states – called semi-annihilations –, for example where can be any standard model particle. It has been argued that those processes may significantly change the predictions for the dark matter relic abundance in thermal freeze-out. Furthermore, an enlarged discrete symmetry group makes it possible to have more than one dark matter candidate. In this case, annihilation processes involving only particles from the dark sectors, leading to the assisted freeze-out mechanism, can also influence the relic abundance of both dark matter candidates  [16, 17]. The assisted freeze-out mechanism in the case of a symmetry was discussed in  [17]. However, no detailed studies have been performed that compare dark matter phenomenology of different models. This is difficult also because presently the publicly available tools for computing dark matter relic abundance do not include the possibility of imposing a discrete symmetry instead of a .

The aim of this work is to formulate the minimal scalar dark matter model that predicts different non-trivial scalar potentials for different symmetries and to study their phenomenology. In particular we are interested in quantifying the possible effects of semi-annihilation processes as well as of annihilation processes involving particles from two different dark sectors on generating the dark matter relic abundance. In order to perform quantitatively precise analyses we implement minimal and symmetric scalar dark matter models that contain one singlet and one extra doublet in micrOMEGAs [18, 19]. Using this tool we show that, indeed, the semi-annihilations and the annihilations between two dark sectors affect the dark matter phenomenology and should be taken into account in a quantitatively precise way in studies of any particular model.

2 Zn Lagrangians

2.1 Zn symmetry

Under an Abelian symmetry, where is a positive integer, addition of charges is modulo . Thus the possible values of charges can be taken to be without loss of generality. A field with charge transforms under a transformation as , where , that is .

A symmetry can arise as a discrete gauge symmetry from breaking a gauge group with a scalar, whose -charge is [2, 4]. For larger values of , the conditions the symmetry imposes on the Lagrangian approximate the original symmetry for two reasons. First, assuming renormalisability, the number of possible Lagrangian terms is limited and will be exhausted for some small finite , though they may come up in different combinations for different values of . Second, if the symmetry arises from some , the -charges of particles cannot be arbitrarily large, because that would make the model nonperturbative. If is larger than the largest charge in the model, the restrictions on the Lagrangian are the same as in the unbroken .

We shall see below that in spite of the large number of possible assignments of charges to the fields, the number of possible distinct potentials is much smaller.

2.2 Field content of the minimal model

In order to study the impact of different discrete symmetries on dark matter phenomenology, the example model must contain more than one neutral particle in the dark sector. The minimal dark matter model with such properties contains, in addition to the standard model fermions and the standard model Higgs boson , one extra scalar doublet and one extra complex scalar singlet  [5]. In the case of symmetry, as proposed in [5], those new fields can be identified with the well known inert doublet [20, 21, 22, 23] and the complex singlet [24, 25, 26, 27, 28]. The phenomenology of those models is well studied. However, when both the doublet and singlet are taken into account, qualitatively new features concerning dark matter phenomenology, electroweak symmetry breaking and collider phenomenology occur [6, 5, 29, 30, 31]. The field content of the minimal scalar model is summarised in Table 1.

2.3 Constraints on charge assignments

The assignments of charges have to satisfy

 XS>0,X1≠X2,−Xℓ+X1+Xe=0modN,−Xq+X1+Xd=0modN,−Xq−X1+Xu=0modN. (1)

The first and second conditions arise from avoiding the term and Yukawa terms for , respectively, and the rest from requiring Yukawa interactions between and standard model fermions. The choice of charges for standard model fermions, the standard model Higgs , the inert doublet and the complex singlet must be such that there are no Yukawa terms for and no mixing between and : only annihilation and semi-annihilation terms for and are allowed. While we will see below that there are many assignments that satisfy Eq. (1), in each case it was possible to find an assignment with the charges of standard model fields set to zero: .

All possible scalar potentials contain a common piece because the terms where each field is in pair with its Hermitian conjugate are allowed under any and charge assignment. We denote it by (the ‘c’ stands for ‘common’):

 Vc=λ1(|H1|2−v22)2+μ22|H2|2+λ2|H2|4+μ2S|S|2+λS|S|4+λS1|S|2|H1|2+λS2|S|2|H2|2+λ3|H1|2|H2|2+λ4(H†1H2)(H†2H1). (2)

2.4 The Z2 scalar potential

There are 256 ways to assign the possible charges to the standard model and dark sector fields. Of these, 8 satisfy Eq. (1); among them, there are 2 different assignments to the dark sector fields: and . Both give rise to the unique scalar potential

 V=Vc+μ′2S2(S2+S†2)+λ52[(H†1H2)2+(H†2H1)2]+μSH2(S†H†1H2+SH†2H1)+μ′SH2(SH†1H2+S†H†2H1)+λ′S2(S4+S†4)+λ′′S2|S|2(S2+S†2)+λ′S12|H1|2(S2+S†2)+λ′S22|H2|2(S2+S†2). (3)

2.5 Z3 scalar potentials and particle content

There are 6561 ways to assign to the fields. Of these, 108 satisfy Eq. (1); among them, there are 12 different assignments to the dark sector fields, giving rise to 2 different scalar potentials. The example potential we choose to work with (given by e.g. ) is

 VZ3=Vc+μ′′S2(S3+S†3)+λS122(S2H†1H2+S†2H†2H1)+μSH2(SH†2H1+S†H†1H2), (4)

which induces the semi-annihilation processes we are interested in. The second one is obtained from Eq. (4) by changing (with and ).

The following conditions are sufficient to have the global minimum of potential at electroweak vacuum with :

 λ1,λ2,λS,λS1,λS2 > 0, (5) λ3+λ4 > 0, (6) 4λS1λS2 > λ2S12, (7) μ′′2λS+μ2SHλ3+λ4 < 4μ2S. (8)

We use these conditions for our benchmark points.

The last term in Eq. (4) induces a mixing between the down component of and . In terms of the mass eigenstates , , we have

 H2=(−iH+x1sinθ+x2cosθ),S=x1cosθ−x2sinθ. (9)

The dark sector of this model consists of 3 complex particles , , and with the charge of 1. Taking the masses of , and the mixing angle as free parameters of the model, we get the following relations

 μ2S = M2x2sin2θ+M2x1cos2θ−λS1v22, (10) μSH = −4(M2x2−M2x1)cosθsinθ√2v, (11) μ22 = −(λ4+λ3)v22+M2x1sin2θ+M2x2cos2θ. (12)

The and the mass of can be presented by formulas

 λ1 = 12M2hv2, (13) MH+ = √μ22+λ3v22. (14)

where is mass of SM Higgs.

2.6 Z4 scalar potentials and particle content

There are 65536 ways to assign to the fields. Of these, 576 satisfy Eq. (1); among them, there are 36 different assignments to the dark sector fields, giving rise to 5 different scalar potentials. Among those the only potential that contains semi-annihilation terms is

 V1Z4=Vc+λ′S2(S4+S†4)+λ52[(H†1H2)2+(H†2H1)2]+λS122(S2H†1H2+S†2H†2H1)+λS212(S2H†2H1+S†2H†1H2), (15)

invariant under e.g. the assignment of charges .

The following conditions are sufficient to have global minimum of potential at electroweak vacuum with :

 λ1,λ2,λS1,λS2 > 0, (16) λS−|λ′S| ≥ 0, (17) λ3+λ4−|λ5| > 0, (18) (|λS12|+|λS21|)2 < λS1λS2. (19)

Our benchmark points considered below satisfy these conditions.

The other four scalar potentials can formally be obtained from the -invariant potential Eq. (3) by setting all the new terms added to to zero, with the exception of the 1) , , 2) , , 3) , , , , , 4) , , , , , , terms.

The term in potential (15) splits the down component of into two real scalar fields with different masses,

 H2=⎛⎝−iH+H0+iA0√2⎞⎠. (20)

Note that the complex scalar does not mix with because these fields have different charges. As a result this model contains two dark sectors, the first one with the complex scalar (the charge is 1), the second one comprising the complex scalar and the real scalars and ( the charge is 2). Any of the neutral particles with a non-zero charge can be a dark matter candidate. We will consider the masses of the neutral scalar particles, , and , as independent parameters, then

 μ2S = M2S−λS1v22, (21) λ5 = M2H0−M2A0v2, (22) μ22 = M2H0−(λ3+λ4+λ5)v22, (23) MH+ = √M2A0+M2H02−λ4v22, (24) λ1 = 12M2hv2. (25)

3 Relic Density in Case of the Z3 Symmetry

3.1 Evolution equations

Consider the -symmetric theory. The imposed symmetry implies, as usual, just one dark matter candidate. This is because the charges and correspond to a particle and its anti-particle. The new feature is that processes of the type , where is any standard model particle, also contribute to dark matter annihilation. The equation for the number density reads

 dndt=−⟨vσxx∗→XX⟩(n2−¯¯¯n2)−12⟨vσxx→x∗X⟩(n2−n¯¯¯n)−3Hn, (26)

where we use , is the Hubble rate, and angular brackets mean thermal averaging. We define

 σv≡⟨vσxx∗→XX⟩+12⟨vσxx→x∗X⟩andα=12σxx→x∗Xvσv, (27)

which means that . Here and in the following we use the notation, . In terms of the abundance, where is the entropy density, we obtain

 dYdt=−sσv(Y2−αY¯¯¯¯Y−(1−α)¯Y2) (28)

or, using the entropy conservation condition ,

 3HdYds=σv(Y2−αY¯¯¯¯Y−(1−α)¯Y2). (29)

where is the equilibrium abundance. We use standard formulae for and [32] that allow to replace the entropy evolution with the temperature one. To solve this equation we follow the usual procedure [32, 18]. Writing we find the starting point for the numerical solution of this equation with the Runge-Kutta method using

 3Hd¯¯¯¯Yds=σv¯¯¯¯YΔY(2−α), (30)

where . This is similar to the standard case except that increases by a factor . Furthermore, when solving numerically the evolution equation, the decoupling condition is modified to

 Y2≫αY¯¯¯¯Y+(1−α)¯¯¯¯Y2. (31)

This implies that the freeze-out starts at an earlier time and lasts until a later time as compared with the standard case. This modified evolution equation is implemented in micrOMEGAs [19, 33]. Although semi-annihilation processes can play a significant role in the computation of the relic density, the solution for the abundance depends only weakly on the parameter , typically only by a few percent. This means in particular that the standard freeze-out approximation works with a good precision.

3.2 Numerical results with micrOMEGAs

Using the scalar potential defined in Eq. (4) we have implemented in micrOMEGAs the scalar model with a symmetry. The scalar sector contains an additional scalar doublet and one complex singlet. The neutral component of the doublet mixes with the singlet, the lightest component is therefore the dark matter candidate, while the heavy component can decay into , where is the standard model-like Higgs boson. Because can decay into light particles, is unstable even if the mass difference between and is small. Note that the doublet component of DM has a vector interaction with the . This interaction is determined by the gauge group and leads to a large direct detection signal in conflict with exclusion limits, for example from XENON100 [34]. The only way to avoid this constraint is to consider a DM with a very small doublet component, namely we have to assume that the mixing angle

 θ≤0.025. (32)

In the limit of small mixing, annihilation processes such as where X stands for , are dominated by the term. The semi-annihilation process is mainly determined by a product of and arising from the terms and in Eq. 2 and Eq. 4. To illustrate a scenario where semi-annihilation channels contribute significantly and which predicts reasonable values for the relic density and the direct detection rate, we choose a benchmark point with the following parameters

For this point, the relic density is . The dominant contribution to is from semi-annihilation (54% for ) while the annihilation channels give a relative contribution of 22%,13% and 10% respectively. Fig. 1 illustrates the dependence of the relic density on the DM mass as compared to the relic density when semi-annihilation is ignored, . Here all other parameters are fixed to their benchmark values. When  GeV, semi-annihilation with a Higgs in the final state is kinematically forbidden at low velocities. If increases, semi-annihilation plays an important role and decreases rapidly due to the contribution of the channel . Note that also decreases when is such that the channel is allowed. When approaches , falls again because the semi-annihilation channel is enhanced due to exchange near resonance.

The spin independent (SI) scattering cross section on nuclei as a function of the DM mass is illustrated in Fig. 1 (right panel). Here we average over dark matter and anti-dark matter cross sections assuming that they have the same density. The main contribution comes from the -exchange diagram because there is a coupling2. Furthermore, one can easily show that the scattering amplitudes are not the same for protons and neutrons, with . Since the current experimental bounds on are extracted from experimental results assuming that the couplings to protons () and neutrons () are equal and the same as the couplings of to protons () and neutrons (), we define the normalised cross section on a point-like nucleus [35]:

 σSIxN=2π(MNMx1MN+Mx1)2([Zfp+(A−Z)fn]2A2+[Z¯fp+(A−Z)¯fn]2A2). (33)

This quantity can directly be compared with the limit on .

4 Relic Density in Case of the Z4 Symmetry

4.1 Evolution equations

In the case of a symmetry all particles can be divided into 3 classes3 {0,1,2} according to the value of their charges modulo . We can choose SM particles to have . We will use the notation for the thermally averaged cross section for reactions where represent any particle with given -charge. Let and be the masses of the lightest particles of classes 1 and 2 respectively. The lightest particle of class 1 is always stable and therefore a DM candidate. The lightest particle of class 2 is stable and can be a second DM candidate if . Note that if , then will decay before the freeze-out of and the relic density can be computed following the standard procedure.

The equations for the number density of particles 1 and 2 read

 dn1dt = −σ1100v(n21−¯n21)−σ1120v(n21−¯n21n2¯n2)−σ1122v(n21−n22¯n21¯n22)−3Hn1, (34) dn2dt = −σ2200v(n22−¯n22)+12σ1120v(n21−¯n21n2¯n2)−12σ1210v(n1n2−n1¯n2) (35) −σ2211v(n22−n21¯n22¯n21)−3Hn2,

where we use to designate the equilibrium number density of particle . In all annihilation and coannihilation processes are taken into account. Here the semi-annihilation processes include all those, where 2 DM particles annihilate into one DM and one standard particle, specifically and . These two cross sections are also described by the same matrix element. However, there is no simple relation between these two cross sections because one process is in the -channel and the other in the -channel. In terms of the abundance, ,

 3HdY1ds = σ1100v(Y21−¯¯¯¯Y21)+σ1120v⎛⎜⎝Y21−Y2¯¯¯¯Y21¯¯¯¯Y2⎞⎟⎠+σ1122v⎛⎜⎝Y21−Y22¯¯¯¯Y21¯¯¯¯Y22⎞⎟⎠, (36) 3HdY2ds = σ2200v(Y22−¯¯¯¯Y22)−12σ1120v⎛⎜⎝Y21−Y2¯¯¯¯Y21¯¯¯¯Y2⎞⎟⎠+12σ1210vY1(Y2−¯¯¯¯Y2) (37) +σ2211v⎛⎜⎝Y22−Y21¯¯¯¯Y22¯¯¯¯Y21⎞⎟⎠.

Solving these equations we use standard formulas for entropy and the Hubble rate temperature dependence [32] that allow to replace the dependence on entropy with one on temperature. The thermally averaged cross section involving particles of different sectors can be expressed as

 σIJKLv(T) = T64π5s2¯¯¯¯YI(T)¯¯¯¯YJ(T)∫ds√sK1(√sT)pinpout (38) ∑a∈Ib∈Jc∈Kd∈Lpol.∫1−1|Mab→cd(√s,cosΘ)|2dcosθ, ¯¯¯¯YI(T) = T2π2s∑i∈Igim2iK2(miT), (39)

where is the matrix element for the process and are modified Bessel functions of the second kind. For reactions which are kinematically open at zero relative velocity, depends slowly on temperature. Otherwise there is a strong temperature dependence, where is the difference between the sums of the masses of outgoing and incoming particles. Equation (38) leads to relations between different cross sections

 YIYJσIJKLv=YKYLσKLIJv. (40)

In particular it implies that, , where the abundance of incoming SM particles .

Introducing , Eqs. (36) and (37) take a simple form

 3HΔYids=−Ci+Aij(T)ΔYj+Qijk(T)ΔYjΔYk, (41)

where

 Ci = 3Hd¯¯¯¯Yids, (42) A = ⎛⎜ ⎜ ⎜⎝2(σ1100v+σ1122v+σ1120v)¯¯¯¯Y1−(σ1120v+2σ1122v)¯¯¯¯Y21¯¯¯¯Y2−σ1120v¯¯¯¯Y1−2σ1122v¯¯¯¯Y12(σ2200v+σ2211v)¯¯¯¯Y2+0.5(σ1210v+σ1120v¯¯¯¯Y1¯¯¯¯Y2)¯¯¯¯Y1⎞⎟ ⎟ ⎟⎠, (43) Q1 = (σ1100v+σ1122v+σ1120v00−σ2211v), (44) Q2 = (−σ1120v−σ1122v12σ1210v0σ2200v+σ2211v). (45)

At large temperatures we expect the densities of both DM components to be close to their equilibrium values. In general in micrOMEGAs [36] the equation for the abundance is solved numerically starting from large temperatures. However, this procedure poses a problem for Eq. (41). The step of the numerical solution is inversely proportional to and as long as is not suppressed by the Boltzmann factor included in , the step is too small and the numerical method fails.

To avoid this problem, we use the fact that at large temperatures one can neglect the term in Eq. (41) and write the explicit solution for the linearised equation. The approximate solution in the case of large is

 ΔYi(s)=A−1ij(s)Cj(s). (47)

One can use Eq. (47) to find the lowest temperature where and start solving numerically Eq. (41) from this temperature. In the general case it gives a reasonable step for the numerical solution , where is the variable of integration. This method can, however, lead to some numerical problems if the masses of the two dark matter particles are very different. Let us call the light particle and the heavy particle . We have to start the numerical solution at a temperature above the freeze-out temperature of the heaviest DM,

 Tfo\mathpzch≈M\mathpzch/25. (48)

At this temperature,

 Y\mathpzclY\mathpzch≈expM\mathpzch−M\mathpzclTfo\mathpzch, (49)

and the step in the numerical solution of the two component equations will be suppressed by a factor . This small step size is problematic when solving numerically the equation with the Runge-Kutta method. This occurs when . In this case the equation for the heavy component must be solved independently assuming that the light component has reached its equilibrium density. If , the Runge-Kutta procedure can be used to successfully solve the thermal evolution equations (41).

The abundances and will be modified by the interactions between the two dark matter sectors.4 Thus the new terms in Eq. (36) will simply add to the standard annihilation process with SM particles and will contribute to decrease the final abundance . After freezes-out, interactions of the type lead to an increase of . When , the evolution of will be strongly influenced by the first sector since at its freeze-out temperature is large. Following the same argument as above the new annihilation terms in Eq. (37) will contribute to a decrease in the final abundance . Furthermore, the semi-annihilation process which is always kinematically open means that acts as a catalyst for the transformation of into SM particles. Thus the light component forces the heavy one to keep its equilibrium value, resulting in a significant decrease of the relic density of . When both DM particles have similar masses, the interplay between the two sectors is more complicated, in particular the rôle of the interactions of the type will depend on the exact mass relation between the two DM particles. For example, this interaction can lead to an increase of the abundance of if is large enough for the reverse process to give the largest contribution.

4.2 Numerical results

The scalar model with a symmetry contains two dark sectors. In sector 1 the DM candidate is a complex singlet, S, the main contribution to comes from annihilation into Higgs pairs and is determined by the term . Sector 2 is similar to the Inert Doublet Model (IDM). The DM candidate can be either the scalar or the pseudoscalar . Annihilation of DM into SM particles is usually dominated by gauge boson pair production processes, while annihilation into fermion pairs as well as co-annihilation processes can also contribute. Furthermore, for a DM mass at the electroweak scale, it was shown in [37] that annihilation into 3-body final states via a virtual can be important below the threshold. To avoid this complication we will consider a DM with a mass above masses of the , , and . Under this condition, the DM annihilation into SM particles in sector 2 is driven by gauge interactions and leads typically to a value of , except for a DM heavier than about 500 GeV. The co-annihilation of , , states increases .

We will consider a benchmark point where both DM candidates and have a mass near 350 GeV. Other parameters are chosen so that semi-annihilation processes play an important role, while both components have comparable relic density and . In particular to have requires the contribution of coannihilation processes – we therefore impose a small mass splitting , meaning that will be small, see Eq. (22). Furthermore, a small value of also leads to a small mass splitting with the charged Higgs. Note that for small and the positivity condition on the potential, Eqs. (2,15) is easily satisfied.

The results of the calculation of the relic density when including different terms in Eq. (36,37) is presented in Table 4. When only (co-)annihilation into SM particles are taken into account, the relic density of is too high, while annihilation is much more efficient in Sector 2. Adding the interactions of the type of brings the value of and closer to each other. In our example the DM in sector 1 has weak interactions with SM particles, therefore is large when sector 2 is neglected. As a result of interactions with sector 2 particles the value for is significantly reduced. This effect was also observed for a DM model with a symmetry [17] and was called the assisted freeze-out mechanism. Finally, when semi-annihilation processes are included, both and decrease.