Dark Matter in scalar Sp(N) gauge dynamics

We consider a model with Sp dark gauge group and a scalar field in the fundamental representation, which leads to two co-stable DM candidates at the perturbative level thanks to a global U(1) accidental symmetry. After gauge confinement at low energy scale, only one of the two candidates is still stable. We compute the DM relic abundance by solving the Boltzmann equations numerically. The presence of light dark glueballs gives extra cosmological effects and can affect Higgs physics. We study the DM phenomenology, providing the predictions for direct and indirect detection (including the Sommerfeld enhancement). We show that the model predicts a slightly suppressed indirect detection cross section in comparison to the usual WIMPs paradigm.


Introduction
The possibility that Dark Matter (DM) originates from elementary scalar/gauge dynamics has been explored in [1] and [2]. In particular, [1] considered one new elementary scalar field S that fills the fundamental representation of a dark gauge group G = { SU(N ), SO(N ), Sp(N ), G 2 }. Every choice of G is characterized by an accidental symmetry, which leads to stable DM candidates with non-trivial features that are characteristics of each group. DM is always accompanied by light dark glueballs.
The dark group G can become strongly interacting ('confined phase') and/or get spontaneously broken by vacuum expectation values of S (Higgsed phase breaking G to a subgroup H that will also confine at low energy scale). A surprising equivalence between the confined and Higgsed phases for scalars S in the fundamental representation of the gauge groups G has been studied in [1]. In these models S has an unique self-quartic, that leads to an unique symmetry breaking pattern where the only surviving scalar is a Higgs-like singlet, named scalon.
It is interesting to study the DM phenomenology of these models. Since the Higgs and confined phases, thanks to the equivalence between them, contain the same asymptotic particles, one can perform all computations in the Higgs phase for weak couplings. The phenomenology can be very different for each gauge group. In the SU case there turns out to be one single DM candidate and the related phenomenology has been studied in detail in [1]. The situation for the Sp dynamics is more involved: at the perturbative level there are two co-stable DM candidates arising from the spontaneous symmetry breaking Sp(N + 2) → Sp(N ). After the confinement of the subgroup Sp(N ), one of two candidates forms bound states (mesons) that can decay, leaving only one DM candidate. Therefore, we expect the DM phenomenology for Sp case to be more intriguing and worthy of more detailed calculations.
In this paper we compute the DM relic abundance of the Sp DM model: we consider a Sp dark gauge group and a scalar field that fills the fundamental representation and we solve the related Boltzmann equations numerically. We make some simplified assumptions to take into account gauge confinement, and we also consider the presence of light glueballs, which can dilute the DM relic abundance.
It is well known that light mediators can affect DM annihilations cross section through an enhancement factor (Sommerfeld enhancement) and bound state formation [3,4]. The Sp DM model includes two kinds of light mediators: a scalar particle (the scalon) and massless gauge bosons, which arise from the spontaneous symmetry breaking of the gauge group. We also introduce how to compute Sommerfeld and bound state corrections and how to take them into account in the predictions for both the relic abundance and indirect detection signals.
We calculate the relevant cross sections for the direct and indirect detections and compare them with the current experimental limits as well as the future prospects, and we find that the Sp DM model is testable on the future indirect detection experiment like CTA [5]. We also investigate the phenomenology of the scalon and of light glueballs and their connection with Higgs physics. This paper is outlined as follows: in section 2 we introduce the main features of our model, including the Lagrangian, the breaking pattern, the perturbative mass spectrum, the accidental symmetries and the formation of bound states after confinement. In section 3 we discuss the Sommerfeld enhancement factors for annihilation cross section and the formation of perturbative bound states. In section 4 we compute the DM relic abundance by solving the Boltzmann equations. In section 5 we discuss the predictions and the experimental limitations of this model, including scalon production, direct and indirect detections. Conlusions and the summary of results are given in section 6. Appendix A gives the explicit expressions of the Feynman rules of the Sp DM model in the Higgs phase.

Model detail 2.1 Lagrangian and mass spectrum
The group Sp(N ) is defined for even N as the transformations that leave invariant the tensor γ N ≡ 1 I N /2 ⊗ , where ij is the 2-dimensional anti-symmetric tensor. We consider a new Sp(N +2) gauge sector, with vectors G a µν in the adjoint and one scalar field S in the fundamental representation, which is pseudo-real 1 . We thereby introduce a complex (N + 2)-dimensional scalar S. The Lagrangian is given by with scalar potential where H is the SM Higgs field. The Lagrangian conserves an accidental global U(1) symmetry (under which S has charge 1), by virtue of S T γS = 0. The RGEs at one-loop level are where g is the gauge coupling of the new sector. We normalize the generators in the fundamental representation as Tr[T a , T b ] = 1 2 δ ab . It is interesting to consider dynamical symmetry breaking through the Coleman-Weinberg mechanism, obtained by setting M S = 0. Assuming that λ HS is negligibly small, the RGEs give that S can radiatively acquire a vacuum expectation value S(x) = (0, ..., 0, (w + s(x))/ √ 2) T , breaking the gauge group Sp(N + 2) to Sp(N ). At the same time, the accidental U(1) symmetry gets rotated to an unbroken global U(1) with generator diag(1, . . . , 1, 1, 1) + diag(0, . . . , 0, 1, −1) = diag(1, . . . , 1, 2, 0), obtained by combining the original U(1) with the diagonal generator of the broken Sp(2). We call this unbroken global U(1) dark-baryon number. If the Higgs mass term is absten too, this model can also generate the weak scale v 246 GeV as [6] v w λ HS 2λ H , 1 We remind the reader that the Sp(N ) generators can be written as where σ k (k = 1, 2, 3) are the Pauli matrices, T sym = {T real , 1 N /2 / √ N } are the symmetric generators of U(N /2), and T asym = T imag are the anti-symmetric ones [2]. assuming a small positive λ HS (more complicated expressions hold if λ HS is not negligibly small).
Writing the gauge bosons as the perturbative mass spectrum in the broken phase is: • the singlet scalon s with one-loop supressed mass M 2 s = β λ S w 2 ; • one real Z with mass M Z = gw/2 and zero dark-baryon number; • one complex W, with mass M W = M Z and dark-baryon number 2.
For N = 0 this is the Sp(2) = SU(2) model of [7,8,6] where W and Z are co-stable DM candidates. For N ≥ 2 the spectrum contains extra particles: • N complex massive vectors X in the fundamental representation of Sp(N ) with mass M X = M W / √ 2 and dark-baryon number 1; • N (N + 1)/2 massless vectors A in the adjoint representation of Sp(N ).
The Z boson decays into A's through loops involving the ZWW † and ZX X † gauge couplings. At perturbative level the W and X are DM candidates, co-stable thanks to accidental baryon number conservation.

Condensation of Sp(N )
The theory becomes strongly coupled and the unbroken Sp(N ) confines at the energy scale with α(M W ) = g 2 (M W )/4π, where g 2 (M W ) is the value of the dark gauge coupling at the scale M W . After confinement we get the following spectrum of asymptotic states: • The scalon s, the Z and W bosons, and dark glue-balls AA.
• Two kinds of dark mesons: the unstable X † X and X † D µ X , which have the same quantum numbers as s and Z, and M µ = X T γ N D µ X , with dark-baryon number 2 as W. Only one linear combination of M and W appears among the stable asymptotic states, while the other corresponds to a resonance. A similar situation holds for s and X † X , and for Z and X † DX .
• Dark baryons B (defined as states formed with one i 1 ···i N −2 tensor) are not stable because the tensor can be decomposed as i 1 ···i N −2 = γ i 1 i 2 · · · γ i N −3 i N −2 + permutations [9]. This means that B splits into N /2 − 1 mesons M.
Both the W and the mesons M carry charge 2 under conserved U(1) baryon number.

Lagrangian in Higgs phase: Sp(N + 2)→ Sp(N )
After symmetry breaking the Lagrangian in eq.(1) can be further expanded as: The kinetic terms are The interaction terms between WZ and WX are The interaction terms between X Z are The kinetic and interaction terms for the scalar are The other interaction terms between W, Z, X and A are where · · · denotes WW † WW † and X X † X X † interactions and f abc N are the structure constants and T a N are the generators of the Sp(N ) group in the fundamental representation. Schematically, the interaction vertices can be summarized as: • Gauge-scalar vertices: sZZ, ssZZ, sWW † , ssWW † ,sX X † , ssX X † .
By using the eq.(9)∼eq. (14) we can get the desired Feynman rules for the interaction vertices, which are collected in Appendix A.

Sommerfeld enhancement and bound states
The DM relic abundance as well as indirect detection signals are determined by the nonrelativistic annihilation (and co-annihilation) cross sections among DM particles and with every particle that can interact with them. By using the Feynman rules listed in Appendix A, we compute explicitly these cross sections in section 4.2. Besides, the DM relic abundance and the indirect detection predictions are calculated in section 4.3 and section 5.3, respectively. The annihilation cross sections get enhanced if the particles in the intial states can exchange a light mediator, a contribution known in the literature as Sommerfeld enhancement [10][11][12][13][14]. This effect can be relevant for the computation of DM relic abundance as well as (and especially) for indirect detection. Generally speaking, it can be taken into consideration by multiplying the cross sections by an enhancement factor S.
In this section we summarize how to compute the enhancement factor for all the interesting cases that occur in our model. All the interaction processes that we are interested in can be divided into two categories: • X X † and X X annihilations: they can exchange a massless dark gluon A generating a Coulomb potential. If the state X X † (X X ) is in the representation J of Sp(N ), the cross section gets enhanced by [3]: with α eff = λ J α, where α = g 2 /4π and λ J is a group theory factor which is computed, as explained in [3], as λ J = C R − C J /2, where C J and C R are the Casimir for the J and fundamental reresentation, respectively. v rel is relative velocity among two particles X .
• all other processes: the two particles in the intial state are {WW † , WZ, WX † , ZZ}. For every case, they can exchange a scalon (with mass M s ) generating an attractive potential. In the non-relativistic limit, the potential can be expressed as: so the cross sections get enhanced by [3]: where κ ≈ 1.74, α eff = g 2 /16π, v rel is relative velocity among two particles of mass M . Under the thermal equilibrium assumption, v rel can be estimated as There is a second phenomenon, which is related to Sommerfeld enhancement, can affect both the relic abundance and the indirect detection cross section: two scattering particles can form bound states through the exchange of a light mediator [3,4,[15][16][17][18][19][20][21]. In particular, the interesting process is the formation of bound states via emission of the corresponding mediator (if the mediator is too heavy one can consider processes with emission of lighter particles, such as light glueballs).
The physics is very different for massive or massless mediators: if the mediator is massive, which is the case for the scalon exchange (with mass M s ) among W, X , and Z that we mentioned above, bound states can exist only if they satisfy the condition α eff M > κM s (for the ground state; for excited states the condition is even stronger) [3,4], where α eff = g 2 /16π and M = {M W , M X } is the mass of the particles involved in the scattering. However, this condition is never satisfied in our model, so there will be no bound states generated by potential (16).
If instead the mediator is massless, which is the case for dark gluons exchange between X X † (X X ), the corresponding potential is Coulombian, and therefore bound states are always allowed. In particular one gets the usual infinity of bound states with binding energies E n = α 2 eff M X /4n 2 , with α eff as in eq.(15).

DM relic abundance
In this section we compute the cosmological relic abundance of DM by solving the related Boltzmann equations. As first step we consider perturbative annihilations and semi-annihilations of co-stable W and X , which occur at high temperatures. Then we take into account the non-perturbative dynamics effects, i.e. the confinement of the Sp(N ) gauge group and the corresponding hadronization that become important at lower temperatures, in section 4.3. We recall that for N = 0 our model coincides with the Sp(2) = SU(2) model, which has already been studied in [7,8,6]. Therefore, we will focus on the original cases for N ≥ 2.

Boltzmann Equations Notations
We use the following notations: • we define a dimensionless parameter z = M X /T , where T is the temperature of the universe; • H = (T 2 /M P ) 4π 3 g * s /45 is the Hubble constant (M P is the Planck mass); • s = 2π 2 g * s T 3 /45 is the entropy density; • g * s is the number of relativistic d.o.f. If freeze-out of DM happens before EW phase transition, g * s = 106.75 + N (N + 1) + 1 where the extra d.o.f. comes from the massless dark gluons A and the presumibly light scalon s; is the number density of the i species divided by the entropy density; • Y i,eq = Y i,eq (z) is the equilibrium distribution of the i species. Before DM thermal freezeout, all species are in thermal equilibrium. At freeze-out all species except DM (W and X ) keep staying in thermal equilibrium. DM is non-relativistic at freeze-out (which occurs roughly at z ∼ 25). We remind that the equilibrium distribution for a non-relativistic species is given by Y i,eq 0.145g i (M i /T ) 3/2 e −M i /T /g * s where M i is the mass of the particle and g i its number of d.o.f. (g W = 6, g X = 6N , g Z = 3); • σv (i → j) is the thermal-averaged cross section for the i → j process in the nonrelativistic limit; assuming s-wave annihilations we can take σv σ 0 .

Numerical factors
We use the following convention in computing the cross sections and deriving the Boltzmann equations: • a factor 1/2 for identical particles in the final states is included in the cross sections; • a factor κ = 1/2 for complex DM annihilations is included in the cross sections (more precisely κ = 1/2 if the inital states are both complex; κ = 1 if the intial states are both real or one is real and one is complex); • we put a factor 1/2 for identical particles in the initial state in the equations; • we put a factor ±∆ i if in the process the number of DM particles of kind "i" increases (decreases) of ∆ i units.
With the above notations and numerical factors, we can write the general form of the Boltzmann equations as Hz s where S = 1/2 if the inital particles are identical and S = 1 for other cases.
List of (semi-)annihilation processes All the kinematics permitted annihilation and semi-annihilation processes that can change the DM number of 1 or 2 units are listed below (Note that some processes, e.g. W → X X , are forbidden by energy-momentum conservation): • W † W → X † X , ss, Zs; • X † X → AA, ss, Zs, As, ZA; • WZ → X X ; • X † W → AX , sX ; • X X → AW, sW; • ZZ → X † X .

Equations
By using the eq.(18) and all (semi-)annihilation processes mentioned above, we can get the following coupled Boltzmann equations for the number densities of W and X : Hz s

Non-relativistic Cross sections
We compute the tree-level non-relativistic cross section at the leading order in the relative velocity v rel , i.e. taking into account only the s-wave contribution σ 0 (averaging over initial spin and gauge components, and multiplying by κ = 1/2 for 2 complex DM particles in the initial state or κ = 1 for other cases, as pointed out in the previous section). In this approximation, the thermal averaged cross section is simply σv = σ 0 . For the group theory factors, we use the following notations: The cross sections of all the relevant processes can be expressed as: σv σv (X † X → As) = As a check of our computations, we have verified that all the cross sections scale as 1/s in the ultra-relativistic limit 2 .

Relic abundance
If the dynamics is purely perturbative, the DM abundance is given by the sum of W and X abundances, and one can use the following expressions (the subscript 0 means quantities evaluated at the present time) : or Instead, in our case the Sp(N ) dynamics confines at the scale Λ DC , given by eq. (7). After the confinement, X can form two kinds of dark mesons: (1) the meson X † X (and X † D µ X ), which carry no dark-baryon number and decay into glueballs through the X X † AA vertex; (2) the mesons M ∼ X T γ N D µ X with dark-baryon number 2. The cubic vertex X X W † becomes a MW mass mixing. In the limit of Λ DC w, their masses are M M √ 2M W so that M will decay into W and glue-balls through the AX X W † vertex (we assume that decays are faster than M * M annihilations), leaving W as the only DM candidate.
We take into account the confinement of Sp(N ) dynamics and the corresponding hadronization under the following simplified assumptions: • confinement of Sp(N ) dynamics occurs after the perturbative freeze-out of W and X , i.e.
Λ DC T f ∼ M W /25 (this is roughly the correct order of magnitude for T f , and can be verified through the numerical solutions of Boltzmann equations as shown in figure 1). This gives an upper limit on g that is, using eq.(7): In the opposite regime confinement occurs before DM decoupling and invalidates our computations.
• about half X form charged mesons M: whenever two X meet, they can form either the meson X † X or the meson M ∼ X X , and each case has a 50% chance. However, only mesons M contribute to W abundance.
• all the mesons M decay to W (+ glueballs/scalon): these processes occurs through X X AW and other vertices. As we have already pointed out, we assume that M decays always faster than M * M annihilations.
Under these assumptions, the DM relic abundance is given by

Sommerfeld enhancement and bound states
In this subsection we dicuss the effects of the Sommerfeld enhancement factors and (perturbative) bound states formation on the relic abundance. Following the analysis in section 3, we compute the Sommerfeld enhancement factors for the (semi-)annihilation cross sections in 4.2.
Then we solve the Boltzmann equations to get the DM relic abundance as explained in section 4.3 and we compare the results obtained (i) with the enhancement factors and (ii) without the enhancement factors. We find that the difference between (i) and (ii) is neglible ( 3%). So we conclude that Sommerfeld enhancement gives only small corrections to the relic abundance and can be safely ignored (notice that the contribution of the Sommerfeld enhancement to indirect detection cross section is instead not neglibile as we explain in section 5.3).
In principle, the formation of Coulombian bound states from X X † (X X ) states through the exchange of dark gluons A (discussed in section 3) can also affect the relic abundance. However, the order of magnitude of this effect is roughly comparable to Sommerfeld corrections. Generally speaking, the bound states formation is relevant whenever Sommerfeld enhancement is significant. As we discussed above, this is not the case for our model. Therefore, we can reasonably neglect these Coulombian bound states in our computation.
As a final remark we notice that the effect of both the Sommerfeld enhancement factors and the perturbative bound state formation on the relic abundance is reasonably subleading if compared to the uncertanties due to the process of hadronization that occours at T Λ DC , when the theory confines, which we discussed in section 4.3.

Glueballs dilution
Dark glueballs (DG) can decay to SM particles through the Higgs portal. The one-loop effective interaction between the scalon s and the dark gluons A has been computed in [1] as: with α = g 2 /4π. The lifetime of glueballs can be estimated as τ DG = Γ −1 DG , which depends on the mass ordering. If the mass of the glueballs is larger than the weak scale, they decay into Higgs components as [1]: The glueballs life-time can instead become cosmologically large if glueballs are enough lighter than the weak scale; in this case we get [1]: where M DG ≈ 7Λ DC is the mass of the glueball [22], f DG ∼ M 3 DG is a dark matrix element, Γ h DG is the decay width of a SM Higgs with mass M DG , γ is the mixing angle that diagonalises the Higgs-scalon mass matrix, M S i are the mass eigenvalues. In more detail we have: If the lifetime is long enough (τ DG t Λ DC , where t Λ DC is the time at which confinement occurs, estimated by H(T = Λ DC ) = t −1 Λ DC ), glueballs can dominate the energy density of the Universe while decaying into SM particles, and the reheating effect dilutes the DM density as Y diluted = Y D [23]. The dilution factor is estimated as: with g SM = 106.75 and g DG = N (N + 1). This effect is taken into account into the computation by multiplying the r.h.s. of eq.(38) by a factor of D. However, it becomes irrelevant if the lifetime is very short. We define the mixing angle between the Higgs and the dark glueball from eq.(41): In figure 2 we show the predicted Higgs-glueball mixing angle 2 as a function of the glueball mass M DG with an assumption that some extra new physics gives fast glueball decays through different channels. We also show the current limits at LHC (shaded regions) and the future sensitivity of SHiP experimental proposal [24] (dashed line). The predicted DM relic density as a function of g 2 and M W is plotted in figure 1. In the red region the overall DM abundance turns out to be larger than the cosmological value (Ω DM h 2 = 0.11), while the green region indicates Ω DM h 2 < 0.11. The red-solid line is obtained by numerical integration of the Boltzmann equations and taking into account the dilution factor in eq. (43) due to long-lived glueballs. We also show, as the red dashed line, the result obtained by assuming that some extra new physics gives fast glueball decays (i.e. no dilution factor). Besides, the limitations of current (future) direct [25] and indirect [26,27] detections are also displayed with black solid (dashed) lines, respectively. Future perspective on indirect detection (CTA) [5] can explore a significant region of the parameter space especially in the case in which glueballs decay very fast.
In figure 1 (bottom right corner) we show the solutions of the Boltzmann equations for values of the parameters such that the DM abundance is reproduced. We see some modifications with respect to the conventional WIMPs behaviour. In particular at freeze-out (roughly z ∼ 25) the W abundance is initially highly suppressed with respect to X because of the mass gap between  TeV) such that the cosmological DM abundance is reproduced. We find that the freeze-out temperature is roughly T f ∼ M W /25 Λ DC , which is in agreement with our assumptions.
the two states, but it suddenly increases because of the conversion processes X X → sW and X X → AW.

Scalon production
The scalon s behaves like an extra Higgs boson with couplings rescaled by sin γ, where the mixing angle γ is given by eq.(42). We show in figure 2 (left) the predicted value of sin 2 γ, which is equal to the production cross section for the scalon in SM Higgs cross section units, as a function of the scalon mass M s . We assume that some extra new physics gives fast glueball decays (if glueballs decay slowly the production cross section is supressed by many order of magnitude and not testable at current/future experiments). Various present and projected constraints from Higgs measurements and direct searches are also shown [28,29]. In particular, it can be seen that measuring Higgs couplings with a 10 −3 precision, which can be attained at future lepton colliders, would allow to probe a significant region of the parameter space.

Direct detection
As pointed out in [1] the spin-indipendent cross section for direct detection is the same as in the SU(2) model, and it can be expressed as where f 0.3 is a nuclear matrix element. In figure 3 (top left corner) we plot the DM direct detection cross section as a function of the DM mass under the assumptions that W reproduce the full DM abundance and some extra new physics gives fast glueball decays. We also show the constraints from Xenon1T [25] as well as neutrino floor. If glueballs decay very slowly, the direct detection cross section is supressed and not observable at current and future experiments, as we can observe in figure 1.

Indirect detection
After the confinement of the gauge subgroup Sp(N ) below the energy scale Λ DC , W becomes the only DM candidate. When W and W † annihilate with each other, they can produce indirect detection signals, which can be expressed as where the dark hadrons are glueballs (AA bound state) and the uncharged mesons (X † X bound state). The energy of the scattering processes is roughly M W Λ DC , so we can use the perturbative results in section 4.2. In particular σv (WW † → AA) is one-loop supressed, so the relevant cross section into dark hadrons is given by σv (WW † → X † X ). Annihilations products will further decay into SM particles. The total annihilation cross section relevant for indirect detection is: We take into account the Sommerfeld enhancement considering the scalon exchange among W and W † through the sWW † interactions. The cross section get enhanced as For v rel M s /M W , which is the relevant regime for indirect detection, the eq.(17) can be further simplified as [3] with κ 1.74 and α eff = g 2 /16π. The indirect detection cross section is not affected by formation of perturbative bound states: indeed, as we have pointed out at the end of section 3, the potential (16) doesn't allow for the existence of WW † bound states, for the values of the parameters of the model.
In figure 3 we plot the DM annihilation cross section as a function of the DM mass under the assumption that W reproduce the full DM abundance. The solid lines are obtained by taking into account the dilution due to long-lived glueballs, while the dashed lines assume that some extra new physics gives fast glueball decays. The blue lines have been obtained without taking into account the Sommerfeld enhancement, which is instead included in the red lines. The vertical dashed line is the boundary between the perturbative and condensed phase, which is estimated as the scale at which M s ≈ Λ DC . We compare the annihilation cross section with the HESS and FERMI-LAT limits on gamma-ray signals [26,27]. The resulting bounds are plotted in figures 1 and 3. We also show future prospects at CTA [5]. In the region excluded by indirect detection the predicted DM abundance is much smaller than the cosmological abundance. However, the CTA prospects can test a significant region of the parameter space, especially in the case in which glueball decays are very fast. We notice that the indirect detection cross section turns out to be slightly suppressed with respect to the standard WIMPs scenario (in which one gets approximately the value σv cosmo 2.2 × 10 −26 cm 3 /sec). The suppression factor depends on N : it is roughly a factor of 2 for small values of N (such as the ones plotted in fig. 3), while it becomes negligible for larger values of N . This is a characteristic prediction of our model, which is testable by future experiments at CTA. The suppression factor becomes smaller if one includes the Sommerfeld enhancement (red lines). We notice that if we take the glueballs dilution effects into consideration (solid lines), the indirect detection cross section is further supressed by one or more order of magnitudes in a certain range of the parameters space.

Conclusions
Motivated by the previous work [1], we have studied the phenomenology of Sp DM model, which contains a new dark gauge group Sp(N + 2) that is spontaneously broken to Sp(N ) by a scalar S in the fundamental representation. We further restricted the parameter space assuming that: • the cosmological DM abundance is reproduced thermally; • the Sp(N + 2) → Sp(N ) symmetry breaking occurs dynamically la Coleman-Weinberg; • the S vacuum expectation value also induces the observed Higgs mass.
Thanks to these extra assumptions, DM phenomenology is described by only one free parameter, the DM mass M W (or the dark gauge coupling g). We have taken into account the confinement of Sp(N ) dynamics and the corresponding formation of bound states. Of course, some of the above assumptions can be relaxed, giving more general phenomenology.
In section 2 we have summarized the main features of the model, including the Lagrangian before and after the symmetry breaking, the perturbative mass spectrum and the formation of bound states. The theory admits an accidental global dark-baryon number U(1) symmetry, which renders two co-stable vector DM candidates at the perturbative level: W (with darkbaryon number 2) and X (with dark-baryon number 1) with masses M X = M W / √ 2. After confinement of Sp(N ) dynamics, two X can either form neutral mesons that decay to SM particles or charged mesons that decay into W, leaving the W as the only DM candidate of the model. In section 3 we discussed how to compute the corrections to DM annihilation cross sections due to the exchange of light mediators (namely the Sommerfeld enhancement and bound state formation), and we focused on the relevant processes which occur in both cosmological evolution and indirect detection signals.
In section 4 we computed the DM relic abundance by solving numerically the Boltzmann equations of W and X abundances at the perturbative level, and we also took into account the gauge confinement under some simplified assumptions. DM is accompanied by light, unstable glueballs, which can dilute the DM abundance and can be potentially probed through their coupling to the Higgs. We discussed the impact of the Sommerfeld enhancement and perturbative bound state formation on the relic abundance. All these results are summarized in Figure 1. Besides, we also show the predictions of Higgs and glueballs phenomenology in the right panel of figure 2.
In section 5 we studied the signatures of this model. We calculated the scalon production cross section and the results are shown in the left panel of figure 2. Besides, we also computed cross sections for direct and indirect detection and compared them with present experimental data and future prospects. We predicted a non standard indirect detection cross section, which turns out to be slightly suppressed in comparison to the usual WIMPs paradigm, and this is a characteristic and testable prediction of our model. The region in which the DM abundance is allowed by experimental bounds can be further explored by the future direct and indirect detection experiments, in particular the CTA. The phenomenology for direct and indirect detection is summarized in figure 3.
As we sometimes relied on approximations, various aspects of the model can be more precisely computed. For example, if Λ DC T f (i.e. confinement occurs before DM thermal freeze-out), the computation of the relic abundance will be qualitatively different. Therefore, it can be interesting for a new work to study this case with more detailed computation.