Bound states of WIMP dark matter in Higgs-portal models II: thermal decoupling
PPrepared for submission to JHEP
Nikhef-2021-004
Bound states of WIMP dark matter in Higgs-portalmodels II: thermal decoupling
Ruben Oncala and Kalliopi Petraki
Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies (LPTHE),UMR 7589 CNRS & Sorbonne Universit´e, 4 Place Jussieu, F-75252, Paris, FranceNikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands
E-mail: [email protected] , [email protected] Abstract:
The Higgs doublet can mediate a long-range interaction between multi-TeVparticles coupled to the Weak interactions of the Standard Model, while its emission canlead to very rapid bound-state formation processes and bound-to-bound transitions. Us-ing the rates calculated in a companion paper, here we compute the thermal decouplingof multi-TeV WIMP dark matter coupled to the Higgs, and show that the formation ofmetastable dark matter bound states via Higgs-doublet emission and their decay decreasethe relic density very significantly. This in turn implies that WIMP dark matter may bemuch heavier than previously anticipated, or conversely that for a given mass, the darkmatter couplings to the Higgs may be much lower than previously predicted, thereby alter-ing the dark matter phenomenology. While we focus on a minimal singlet-doublet modelin the coannihilation regime, our calculations can be extended to larger multiplets wherethe effects under consideration are expected to be even more significant. a r X i v : . [ h e p - ph ] J a n ontents Model-independent unitarity arguments [1], as well as model-specific calculations show thatviable thermal-relic dark matter (DM) scenarios in the multi-TeV regime feature interac-tions that generate long-range potentials and give rise to bound states. The formationand decay of metastable bound states alters the DM decoupling in the early universe [2]and contributes to its indirect signals [1, 3–8]. In fact, in a variety of models, bound-stateformation (BSF) can be faster than annihilation [9–15].This is particularly striking in models where DM couples to a light scalar boson chargedunder a symmetry. It was recently shown that the emission of such a scalar boson by a pairof interacting particles alters their effective Hamiltonian and can result in extremely rapidmonopole capture processes [13]. This may potentially have severe implications for multi-TeV DM coupled to the Higgs doublet. Moreover, it has been shown that the 125 GeVHiggs boson can mediate a significant long-range interaction between TeV-scale particles,despite being heavier than all other Standard Model (SM) force mediators [12, 16].These considerations impel investigating the role of the Higgs doublet in the cosmologyof multi-TeV Higgs-portal DM. The above effects can be potentially important in scenariosthat involve a trilinear coupling between the DM and Higgs multiplets, i.e. δ L ⊃ ¯ χ n Hχ n +1 +h.c., where χ n stands for a fermionic or bosonic n -plet under SU L (2), and DM is the lightest– 1 –eutral mass eigenstate arising from the mixing of χ n and χ n +1 after the electroweakphase transition (EWPT). These models have been among the archetypical scenarios ofDM coupled to the Weak interactions of the SM (WIMPs) [17–34].Here we will focus on a minimal singlet-doublet realisation of this class of scenarios. In acompanion paper, we have computed the non-relativistic potentials, the BSF cross-sections,the bound-state decay rates and bound-to-bound transitions [35]. In this paper, we employthe results of [35] to compute the DM thermal decoupling from the primordial plasma inthe early universe. Several aspects of this calculation are examined in detail, including theeffect of both radiative BSF and BSF via scattering on the relativistic thermal bath, theimportance of bound-to-bound transitions and the late recoupling of the DM destructionprocesses due to BSF via Higgs emission. As in [35], we carry out all computations assumingelectroweak symmetry; indeed, for DM heavier than 5 TeV, freeze-out begins before theEWPT, even though the complete thermal decoupling may occur much later. The validityof the approximation is discussed in detail.We note that our calculations are important broadly for scenarios that introduce newstable species in the fashion considered here, even if these species do not account for DM.Besides specifying the parameters for which the relic density of the lightest new masseigenstate matches the observed DM abundance, we show how the relic density is affectedby the new effects within a broader parameter space. As is standard, the relic density ofstable particles sets a cosmological constraint on new physics scenarios.This work is organized as follows. In section 2, we summarise the model following[35], and briefly review its basic properties in the broken electroweak phase. Moreover,we discuss the temperature dependence of the Higgs-doublet mass in relation to the BSFprocesses of interest. In section 3, we lay out the formalism for computing the DM thermaldecoupling, and discuss the interplay among bound-state formation, ionisation, decay andbound-to-bound transition processes. The results of this computation are presented anddiscussed in section 4. We conclude in section 5.– 2 – The model
We begin by specifying the model following ref. [35], before summarising the mass eigen-states and interactions in the broken electroweak phase.We consider a gauge-singlet Majorana fermion S = ( ψ α , ψ † ˙ α ) T of mass m S , and a Diracfermion D = ( ξ α , χ † ˙ α ) T of mass m D with SM gauge charges SU L (2) × U Y (1) = ( , / S and D are assumed to be odd under a Z symmetry that leaves all the SM particlesunaffected. Under these assignments, the new degrees of freedom (dof) allow to extend theSM Lagrangian by the following interactions δ L = 12 ¯ S ( i /∂ − m S ) S + D ( i / D − m D ) D − ( y L ¯ D L HS + y R ¯ D R HS + h.c.) , (2.1)where H is the SM Higgs doublet of mass m H and hypercharge Y H = 1 /
2, and D L ≡ P L D = ( ξ α , T and D R ≡ P R D = (0 , χ † ˙ α ) T , with P R , L = (1 ± γ ) / D µ ≡ ∂ µ − i g Y B µ − i g W aµ t a is thecovariant derivative, with t a = ( σ , σ , σ ) and σ being the Pauli matrices. The particlecontent of eq. (2.1) is summarised in table 1.field SU L (2) U Y (1) Z S − D − H Table 1 . Particle content and charge assignments
We take m S and m D to be real. This can be achieved by rephasing ψ and either ξ or χ . Rephasing the remaining spinor eliminates the phase of one of the Yukawa couplings.Thus the free parameters of the present model are 4 real couplings (two masses and twodimensionlesss Yukawa couplings), and a phase that allows for CP violation.We will focus on the regime where S and D can co-annihilate efficiently before theEWPT of the universe. This occurs if their masses are similar, within about 10%. Alarger mass difference would imply that the number density of the heavier species in thenon-relativistic regime becomes negligible, since it would be suppressed with respect to thelighter species by exp[ − ( δm/m ) x ] (cid:46) .
1, with x ≡ m/T ∼
25 during DM freeze-out. Sincethe masses must be similar for the processes considered here to be relevant, we take themto be equal for simplicity, m D = m S ≡ m. (2.2)It will be useful also to introduce the reduced mass of a pair of DM particles, µ ≡ m/ . (2.3)– 3 –n addition, in order to reduce the number of free parameters, we set y L = y R ≡ y, (2.4)which we take to be real. (The CP violation is not important for our purposes.) Ourcomputations can of course be extended to more general Yukawa couplings. As is standard,we define the couplings α ≡ g π , α ≡ g π , α H ≡ y π . (2.5)For later convenience, following [35], we also define the couplings α R ≡ (cid:104)(cid:112) [( α + 3 α ) / + 8 α H − ( α + 3 α ) / (cid:105) , (2.6a) α A ≡ (cid:104)(cid:112) [( α + 3 α ) / + 8 α H + ( α + 3 α ) / (cid:105) . (2.6b)The model of eq. (2.1) and various aspects of its phenomenology have been consideredfor general parameters extensively in the past [17–26]. Here we will only briefly review themass eigenstates and their interactions after electroweak symmetry breaking for the choiceof parameters denoted in eqs. (2.2) and (2.4). Mass eigenstates in the broken electroweak phase
At electroweak symmetry breaking, the neutral component of the Higgs doublet acquiresa vacuum expectation value. In terms of their SU L (2) components, the H , ξ and χ fieldsare H = φ + √ v H + h + i φ ) , ξ α = (cid:32) ξ + α ξ α (cid:33) , χ α = (cid:32) χ α χ − α (cid:33) , (2.7)with v H (cid:39)
246 GeV being the Higgs vacuum expectation value.We define the left-handed multiplet of the neutral states ˆ N α ≡ ( ψ α , ξ α , χ α ) T . Then,by inserting eq. (2.7) into the Lagrangian (2.1), we find the corresponding mass terms, δ L N , mass = −
12 ˆ N α ˆ M N ˆ N α + h.c. , (2.8)with ˆ M N = m yv H / √ yv H / √ yv H / √ myv H / √ m . (2.9)We diagonalise eq. (2.8) by setting N α = U ˆ N α , N α = ˆ N α U T , M N = ( U T ) − ˆ M N U − , (2.10)– 4 –here U is the unitary matrix U = − / √ / / i / √ − i / √ / √ / / . (2.11)The corresponding mass eigenvalues are m ≡ m − yv H , m ≡ m, m ≡ m + yv H . (2.12)In addition to the neutral states, there is a charged Dirac fermion of mass m . Interactions in the broken electroweak phase and constraints
The interactions among the neutral states, N ≡ ( N , N , N ) T , are described by the La-grangian [21] δ L N , inter = g c W Z µ N † ¯ σ µ i √ N + N ) − y h ( N N − N N + N N ) + h.c. , (2.13)where c W = g / (cid:112) g + g .Since the coupling to Z µ is non-diagonal, with the mass splitting being always muchlarger than ∼ O (100 keV) for the y values we will consider here (cf. section 3), the con-straints from direct detection experiments due to this interaction are evaded. On the otherhand, the coupling to the Higgs boson is expected to yield sizable DM-nucleus scatteringand potentially strong constraints. Existing analyses of the direct detection data for thismodel do not extend to the multi-TeV regime that is of interest here. Moreover, the directdetection constraints on the model (2.1) are generally significantly relaxed around the so-called blind spots where the coupling to the Higgs vanishes, roughly when y L = − y R (seee.g.[20, 21].) A detailed phenomenological analysis is beyond the scope of the present work.However, our results are important for interpreting the experimental constraints, since theyimply a different relation between the DM mass and couplings in order for the observedDM density to be attained via thermal freeze-out. The interactions of eq. (2.1) – in particular the exchange of B , W and H bosons – generatelong-range potentials among the S , D and ¯ D species that distort the wavefunctions ofscattering states and give rise to bound states. The long-range dynamics in the symmetricelectroweak phase has been discussed in detail in the companion paper [35], and shall notbe repeated here. However, since our focus is the effect of bound states on the DM thermaldecoupling, in table 2 we summarise for convenience the bound levels we consider. The cross-sections for BSF via H emission depend on the Higgs doublet mass [35]. Takinginto account the finite temperature 1-loop corrections to the effective potential (see e.g. [36,– 5 –ound state B U Y (1) SU L (2) Spin dof ( g B ) Bohr momentum ( κ B ) SS/D ¯ D mα A / D ¯ D m ( α + 3 α ) / DD m ( − α + 3 α ) / DS / mα H / Table 2 . The ground-level (principal and angular-momentum quantum numbers { n(cid:96)m } = { } )bound states and their Bohr momenta κ B in the limit m H →
0. The binding energies are |E B | = κ B /m . The couplings are defined in eqs. (2.5) and (2.6). The SS/D ¯ D , D ¯ D and DD bound statescan decay directly into radiation. The DS rate of decay into radiation is suppressed, howeverthe DS bound state can transit spontaneously into an SS/D ¯ D bound state via H emission. Allother bound-to-bound transitions are suppressed. The bound state decay and transition rates aresummarised in [35, table 6]. m H ( T ) ≈ − m h πT (cid:18) α + 3 α + 2 λπ + y t π (cid:19) , (2.14)where m h (cid:39)
125 GeV is the Higgs boson mass at zero temperature, λ = m h / (2 v H ) (cid:39) . V SM ⊃ − λ | H | , and y t (cid:39) .
994 is the top quark Yukawacoupling. The EWPT occurs as m H ( T ) →
0, i.e. at estimated temperature T EWPT ≈ √ m h (cid:112) πα + 3 πα + 2 λ + y t (cid:39)
151 GeV . (2.15)In computing the DM decoupling, we use eq. (2.14) at T (cid:62) T EWPT , and set m H → m h at T < T
EWPT while still using the annihilation and BSF rates computed under the assumptionof electroweak symmetry. We discuss this approximation in section 4.2.We may now estimate whether or when m H ( T ) implies that BSF via Higgs emissionis kinematically suppressed. In a thermal distribution, the energy dissipated during BSFaverages to (cid:104) ω (cid:105) = 3 T / |E B | (cf. eq. (3.23) and [35].) The first term suffices to providefor m H ( T ) for all T > T
EWPT since m H ( T > T
EWPT ) (cid:46) . T , as well as after the EWPT,down to temperatures T ∼ m h / (cid:39)
83 GeV. However, since the BSF cross-sectionsweigh preferentially low values of v rel , the kinematic suppression may become important atsomewhat larger T than this estimate implies, unless |E B | is sufficient to provide for m H .– 6 – Boltzmann equations for dark matter thermal decoupling
Let Y j ≡ n j /s and Y B ≡ n B /s be the number-density-to-entropy-density ratios of the freespecies j and the bound state B respectively. In our model, j = S , D , ¯ D and B = SS/D ¯ D , D ¯ D , DD , DS (cf. table 2.) We are ultimately interested in the total DM yield Y ≡ Y S + Y D + Y ¯ D = Y S + 2 Y D . (3.1)Note that the bound states are metastable and their abundance becomes eventually negli-gible, so we do not include them in eq. (3.1). As is standard, we will use the time parameter x ≡ m/T. (3.2)The entropy density of the universe is s = (2 π / g ∗ S T = (2 π / g ∗ S m /x . We denoteby g ∗ S and g ∗ the entropy and energy dof respectively, and define g / ∗ , eff = g ∗ S √ g ∗ (cid:18) − x g ∗ S dg ∗ S dx (cid:19) . (3.3)The evolution of Y j and Y B is governed by the coupled Boltzmann equations dY j dx = − λx (cid:88) i (cid:104) σ ann ji v rel (cid:105) (cid:16) Y j Y i − Y eq j Y eq i (cid:17) − λx (cid:88) i (cid:88) B (cid:104) σ BSF ji → B v rel (cid:105) (cid:18) Y j Y i − Y B Y eq B Y eq j Y eq i (cid:19) − Λ x (cid:88) i (cid:104) Γ j → i (cid:105) (cid:18) Y j − Y i Y eq i Y eq j (cid:19) , (3.4a) dY B dx = − Λ x (cid:104) Γ dec B (cid:105) ( Y B − Y eq B ) + (cid:88) i,j (cid:104) Γ ion B → ij (cid:105) (cid:32) Y B − Y i Y j Y eq i Y eq j Y eq B (cid:33) + (cid:88) B(cid:48) (cid:54) = B (cid:104) Γ trans B → B(cid:48) (cid:105) (cid:18) Y B − Y B(cid:48) Y eq B(cid:48) Y eq B (cid:19) , (3.4b)where λ ≡ (cid:114) π m Pl mg / ∗ , eff and Λ ≡ λs x = (cid:114) π m Pl m g / ∗ , eff g ∗ , S , (3.5)and the equilibrium densities in the non-relativistic regime are Y eq i (cid:39) π ) / g i g ∗ , S x / e − x and Y eq B (cid:39) π ) / g B g ∗ , S (2 x ) / e − x e |E B | /T , (3.6)where g i are the spin and SU L (2) dof of the free species, with g S = 2, g D = g ¯ D = 4. Thedof g B and the binding energies E B of the bound states we consider are listed in table 2.For later convenience, we also define the total DM dof g DM ≡ g S + g D + g ¯ D = 10, and theequilibrium density of (3.1) Y eq = 90(2 π ) / g DM g ∗ , S x / e − x . (3.7)– 7 –n the above, Γ dec B , Γ ion B → ij and Γ trans B → B(cid:48) are respectively the rates of B decay into radiation,ionisation (a.k.a. dissociation) to ij , and transition into the bound level B (cid:48) . The ratesΓ j → i describe the transitions between free particles, due to decays, inverse decays and/orscatterings on the thermal bath; overall, these processes do not change the DM numberdensity, but retain equilibrium among the dark species. Note that in eqs. (3.4) we mustuse the thermally averaged rates, (cid:104) Γ (cid:105) . The thermal average introduces Lorentz dilationfactors for decay processes – which however are insignificant in the non-relativistic regime– as well as Bose-enhancement factors in the case of transitions and capture or ionisationprocesses. We discuss this in more detail in section 3.3. The thermally-averaged rates andcross-sections of inverse processes are related via detailed balance that we have alreadyemployed in writing eqs. (3.4), (cid:104) Γ trans B → B(cid:48) (cid:105) = Γ trans
B(cid:48) → B × ( Y eq B(cid:48) /Y eq B ) , (3.8a) (cid:104) Γ ion B → ij (cid:105) = s (cid:104) σ BSF ij → B v rel (cid:105) × ( Y eq i Y eq j /Y eq B ) , (3.8b) (cid:104) Γ i → j (cid:105) = (cid:104) Γ j → i (cid:105) × ( Y eq j /Y eq i ) . (3.8c)The fractional relic DM density isΩ (cid:39) ( m − √ πα H v H ) Y ∞ s /ρ c , (3.9)where Y ∞ is the final yield, and we have included the mass shift of the lightest state thatarises after the electroweak symmetry breaking (cf. eq. (2.12)); this is significant only forthe lower end of the mass range we consider and for large couplings α H . In eq. (3.9), s (cid:39) . − and ρ c (cid:39) . · − GeV cm − are the entropy and critical energydensities of the universe today [38]. The system of coupled Boltzmann eqs. (3.4) is numerically difficult to solve. We shall thusadopt an effective method that reduces eqs. (3.4) to one equation for the DM yield (3.1).For convenience, we first define the total formation cross-section, ionisation rate andtransition rate of every bound state B , σ BSF B ≡ (cid:88) i,j g i g j g DM σ BSF ij → B , (3.10a)Γ ion B ≡ (cid:88) i,j Γ ion B → ij , (3.10b)Γ trans B ≡ (cid:88) B(cid:48) (cid:54) = B Γ trans B → B(cid:48) . (3.10c)We begin by assuming that the i ↔ j interactions are sufficiently rapid to ensure kineticequilibrium, such that Y i /Y eq i = w , where w is the same for all i = S, D, ¯ D . Due to theirrapid decays, inverse decays and transitions to other bound levels, the bound states aretypically close to equilibrium, thus dY B /dx (cid:39)
0. Under this assumption, eqs. (3.4b) yield a– 8 –ystem of linear equations for Y B that can be solved and re-employed in eq. (3.4a) [39]. Forbound states that do not participate in any bound-to-bound transitions, this simplifies to Y B = Y eq B (cid:104) Γ dec B (cid:105) + w (cid:104) Γ ion B (cid:105)(cid:104) Γ dec B (cid:105) + (cid:104) Γ ion B (cid:105) . (3.11)In the model under consideration and within our approximations [35], the spin-1 D ¯ D and DD bound states do not participate in any bound-to-bound transitions, while thespin-0 SS/D ¯ D and DS bound states can rapidly transit into each other via H absorp-tion/emission (cf. [35, table 6].) For the latter, eqs. (3.4a) read (cid:32) (cid:104) Γ dec SS/D ¯ D (cid:105) + (cid:104) Γ ion SS/D ¯ D (cid:105) + 2 (cid:104) Γ trans SS/D ¯ D → DS (cid:105) − (cid:104) Γ trans DS → SS/D ¯ D (cid:105)−(cid:104) Γ trans SS/D ¯ D → DS (cid:105) (cid:104) Γ ion DS (cid:105) + (cid:104) Γ trans DS → SS/D ¯ D (cid:105) (cid:33) (cid:32) Y SS/D ¯ D Y DS (cid:33) == (cid:32) (cid:2) (cid:104) Γ dec SS/D ¯ D (cid:105) + w (cid:104) Γ ion SS/D ¯ D (cid:105) (cid:3) Y eq SS/D ¯ D w (cid:104) Γ ion DS (cid:105) Y eq DS (cid:33) , (3.12)where we set (cid:104) Γ dec DS (cid:105) (cid:39) (cid:104) Γ trans SS/D ¯ D → DS (cid:105) = (cid:104) Γ trans DS → SS/D ¯ D (cid:105) ( Y eq DS /Y eq SS/D ¯ D ),due to detailed balance eq. (3.8a). The factors 2 in the first row account for transitions toand from the two conjugate bound states DS and ¯ DS .Next, we use eq. (3.11) for the D ¯ D and DD yields and the solution of eq. (3.12) forthe SS/D ¯ D and DS yields, in the Boltzmann eq. (3.4a). Summing over all free particlespecies, we find that the evolution of Y is governed by the Boltzmann equation dYdx = − (cid:114) π m Pl m g / ∗ , eff x (cid:104) σv rel (cid:105) eff [ Y − ( Y eq ) ] , (3.13)where the equilibrium density Y eq is given in eq. (3.7). The DM destruction cross-section (cid:104) σv rel (cid:105) eff receives contributions from direct annihilation and BSF processes, (cid:104) σv rel (cid:105) eff = (cid:104) σ ann v rel (cid:105) + (cid:104) σ BSF v rel (cid:105) eff , (3.14)with (cid:104) σ ann v rel (cid:105) ≡ (cid:88) i,j g i g j g DM (cid:104) σ ann ij v rel (cid:105) , (3.15)and (cid:104) σ BSF v rel (cid:105) eff = (cid:104) σ BSF
SS/D ¯ D v rel (cid:105) eff + (cid:104) σ BSF D ¯ D v rel (cid:105) eff + 2 (cid:104) σ BSF DD v rel (cid:105) eff + 2 (cid:104) σ BSF DS v rel (cid:105) eff , (3.16)where the factors 2 in the DD and DS terms account also for the formation of the conjugatebound states. The individual contributions are found as follows. For the bound-states thatdo not participate in any bound-to-bound transitions, (cid:104) σ BSF D ¯ D v rel (cid:105) eff (cid:104) σ BSF D ¯ D v rel (cid:105) = (cid:104) Γ dec D ¯ D (cid:105)(cid:104) Γ dec D ¯ D (cid:105) + (cid:104) Γ ion D ¯ D (cid:105) , (3.16a) (cid:104) σ BSF DD v rel (cid:105) eff (cid:104) σ BSF DD v rel (cid:105) = (cid:104) Γ dec DD (cid:105)(cid:104) Γ dec DD (cid:105) + (cid:104) Γ ion DD (cid:105) , (3.16b)– 9 –hile for the coupled bound states (cid:104) σ BSF
SS/D ¯ D v rel (cid:105) eff (cid:104) σ BSF
SS/D ¯ D v rel (cid:105) = (cid:104) Γ dec SS/D ¯ D (cid:105)(cid:104) Γ dec SS/D ¯ D (cid:105) + (cid:104) Γ ion SS/D ¯ D (cid:105) + 2 (cid:104) Γ ion DS (cid:105) (cid:104) Γ trans DS → SS/D ¯ D (cid:105)(cid:104) Γ ion DS (cid:105) + (cid:104) Γ trans DS → SS/D ¯ D (cid:105) Y eq DS Y eq SS/D ¯ D , (3.16c) (cid:104) σ BSF DS v rel (cid:105) eff (cid:104) σ BSF DS v rel (cid:105) = (cid:104) Γ trans DS → SS/D ¯ D (cid:105) × (cid:104) Γ dec SS/D ¯ D (cid:105)(cid:104) Γ dec SS/D ¯ D (cid:105) + (cid:104) Γ ion SS/D ¯ D (cid:105)(cid:104) Γ ion DS (cid:105) + (cid:104) Γ trans DS → SS/D ¯ D (cid:105) + 2 (cid:104) Γ ion DS (cid:105)(cid:104) Γ trans DS → SS/D ¯ D (cid:105)(cid:104) Γ dec SS/D ¯ D (cid:105) + (cid:104) Γ ion SS/D ¯ D (cid:105) Y eq DS Y eq SS/D ¯ D . (3.16d)In eqs. (3.16), (cid:104) σ BSF B v rel (cid:105) are the thermal averages of the actual velocity-weighted formationcross-sections for every bound state, defined in eq. (3.10a); we discuss them further in thefollowing section. Note that if the transitions between the SS/D ¯ D and DS bound statesare very rapid, in particular when (cid:104) Γ trans DS → SS/D ¯ D (cid:105) (cid:29) (cid:104) Γ ion DS (cid:105) , then the branching ratios thatweigh their actual BSF cross-sections in eqs. (3.16c) and (3.16d) are equal. We now consider in more detail the contributions to the effective DM destruction cross-section in our model, based on the computations of ref. [35]. We begin with direct annihila-tion in section 3.3.1, and then discuss BSF in section 3.3.2. In figs. 1 and 2 we illustrate thecontributions to BSF, while in fig. 3 we compare all contributions to the DM destructioncross-section for a chosen set of parameters, showcasing the effect of the Higgs potentialand of BSF via Higgs emission.
In our model, the total annihilation cross-section is σ ann v rel = [ g SS ( σ ann SS v rel ) + 2 g D ¯ D ( σ ann D ¯ D v rel ) + 2 g DD ( σ ann DD v rel ) + 4 g DS ( σ ann DS v rel )] /g DM , (3.17)where the indices denote the two-particle scattering states, with dof g SS = 4, g DD = 16, g D ¯ D = 16, g DS = 8. The DD and DS contributions carry factors of 2 to account also forthe annihilation of the conjugate states, and D ¯ D and DS carry factors of 2 to account for– 10 –he two distinguishable particles annihilated in each process. From [35, table 5], we find g SS ( σ ann SS v rel ) / ( πm − ) = 0 , (3.18a) g D ¯ D ( σ ann D ¯ D v rel ) / ( πm − ) = 1 × (cid:18) α α (cid:19) × α A S ( ζ A ) + α R S ( − ζ R ) α A + α R + 3 × (cid:20) ( α + 2 α H )
12 + 10 α (cid:21) × S (cid:18) ζ + 3 ζ (cid:19) + 3 × α α × S (cid:18) ζ − ζ (cid:19) + 9 × (cid:20) ( α + 2 α H )
12 + α
12 + 2 α (cid:21) × S (cid:18) ζ − ζ (cid:19) , (3.18b) g DD ( σ ann DD v rel ) / ( πm − ) = 3 × α H × S (cid:18) − ζ + 3 ζ (cid:19) , (3.18c) g DS ( σ ann DS v rel ) / ( πm − ) = 6 × (cid:16) α α H α α H (cid:17) × S ( − ζ H ) . (3.18d)In the above, the s -wave Sommerfeld factor is S ( ζ S ) ≡ πζ S − e − πζ S , (3.19)where ζ S ≡ α S /v rel , with α S being the strength of the Coulomb potential of the scatteringstate. The various ζ S appearing in eqs. (3.18) are ζ ≡ α /v rel , ζ ≡ α /v rel , ζ H ≡ α H /v rel , ζ A ≡ α A /v rel , ζ R ≡ α R /v rel . (3.20)Each of the terms in eqs. (3.18) is the product of the dof, the perturbative annihilationcross-section and the Sommerfeld factor of a spin- and gauge-projected state. The D ¯ D cross-section includes contributions from both the SU L (2) singlet and triplet projections,which are characterised by different non-relativistic potentials and thus have different Som-merfeld factors. For the singlet states, the potential depends also on the spin. Indeed, thespin-0 SU L (2) singlet SS and D ¯ D states mix due to the Higgs mediated potential. Sincethe perturbative s -wave annihilation of the SS component vanishes, the contribution fromthe annihilation of the SS -like state has been included in the D ¯ D -like state, for simplicity.We refer to [35] for more details.The thermally averaged annihilation cross-section (3.15) is found from (3.17), (3.18)and (cid:104) σ ann v rel (cid:105) = (cid:16) m πT (cid:17) / (cid:90) d v rel e − mv / (4 T ) ( σ ann v rel ) . (3.21) The radiative BSF cross-sections have been summarised in [35, tables 7–10], and we shalldenote them here as σ rBSF B v rel [ xx ] with xx and B being the scattering and bound states. As is well known, for pairs of identical particles (here SS , DD , ¯ D ¯ D ), this factor is canceled uponthermal averaging by the factor 1/2 needed to avoid double-counting of the initial particle states [40]. – 11 –ound states can also form through scattering on the thermal bath, via exchange of anoff-shell mediator; the corresponding cross-section factorise in their radiative counterpartsand temperature-dependent functions [35, 41, 42]. Collecting these results, the velocity-weighted cross-sections σ BSF B v rel for the formation of the various bound-state species B receive the following contributions from the individual channels (cf. [35, tables 7-10]) σ BSF
SS/D ¯ D v rel = 1 g DM (cid:110) × × (1 + R B ) × σ rBSF SS/D ¯ D v rel [( D ¯ D ) spin-0( , ]+2 × × (1 + R W ) × σ rBSF SS/D ¯ D v rel [( D ¯ D ) spin-0( , ]+4 × × (1 + R H /h H ) × σ rBSF SS/D ¯ D v rel [( DS ) spin-0( , / ] (cid:111) , (3.22a) σ BSF D ¯ D v rel = 1 g DM (cid:110) × × (1 + R B ) × σ rBSF D ¯ D v rel [( D ¯ D -like) spin-1( , ]+1 × × (1 + R B ) × σ rBSF D ¯ D v rel [( SS -like) spin-1( , ]+2 × × (1 + R W ) × σ rBSF D ¯ D v rel [( D ¯ D ) spin-1( , ]+4 × × (1 + R H /h H ) × σ rBSF D ¯ D v rel [( DS ) spin-1( , / ] (cid:111) , (3.22b) σ BSF DD v rel = 1 g DM (cid:110) × × (1 + R W ) × σ rBSF DD v rel [( DD ) spin-1( , ]+2 × × (1 + R H /h H ) × σ rBSF DD v rel [( DS ) spin-1( , / ] (cid:111) , (3.22c) σ BSF DS v rel = 1 g DM (cid:110) × × (1 + R B ) × σ rBSF DS v rel [( DS ) spin-0( , / , B emission]+ 2 × × (1 + R W ) × σ rBSF DS v rel [( DS ) spin-0( , / , W emission]+ 1 × × (1 + R H /h H ) × σ rBSF DS v rel [( SS -like) spin-0( , ]+ 2 × × (1 + R H /h H ) × σ rBSF DS v rel [( D ¯ D -like) spin-0( , ]+ 2 × × (1 + R H /h H ) × σ rBSF DS v rel [( D ¯ D ) spin-0( , ]+1 × × (1 + R H /h H ) × σ rBSF DS v rel [( DD ) spin-0( , ] (cid:111) . (3.22d)In each term above, the first factor accounts for the number of DM particles destroyed (uponthermal averaging), as well as the capture of the conjugate scattering state if applicable,in analogy to eq. (3.17) for annihilation. The second factor corresponds to the dof of thescattering state.The factors in the brackets sum the radiative and via-scattering contributions to BSF.The factors R H , R B , R W indicate the off-shell exchange of H , B and W bosons with thethermal bath, and depend on ω/T , where ω = µ ( α B + v ) / α B being the strength of thepotential of the corresponding bound state (cf. table 2 and [35, table 6].) R H depends alsoon m H /ω , and essentially replaces the phase-space suppression h H ( ω ) ≡ (cid:0) − m H /ω (cid:1) / (3.24)due to on-shell H ( † ) emission that is included in the radiative cross-sections. The R H , R B , R W factors can be found in [35].Next, we must thermally average eqs. (3.22). In BSF, the emitted boson carries awaya small amount of energy that can be comparable to the temperature of the primordialplasma during the DM decoupling. The Bose enhancement due to the final state bosoncan thus be significant, and must be included in thermal averaging the BSF cross-sectionsto ensure that detailed balance holds [2], (cid:104) σ BSF B v rel (cid:105) = (cid:16) m πT (cid:17) / (cid:90) d v rel e − mv / (4 T ) (cid:18) e ω/T − (cid:19) ( σ BSF B v rel ) , (3.25)As seen in eq. (3.16), the contributions of each bound level to the effective DM de-struction cross-section (3.14) have to be waited by the appropriate branching fractions thataccount for the portion of bound states that decay into radiation thereby reducing the DMdensity. The bound-state decay and transition rates needed to compute these branchingfractions can be found in [35, table 6]. In thermally averaging these rates, we may ne-glect the Lorentz dilation factor that is (cid:39) (cid:104) Γ dec B (cid:105) (cid:39) Γ dec B , (3.26a) (cid:104) Γ trans DS → SS/D ¯ D (cid:105) (cid:39) [1 + R H ( ω ) /h H ( ω )] (cid:18) e ω/T − (cid:19) Γ trans DS → SS/D ¯ D , (3.26b)where the dissipated energy here is ω = m ( α A − α H ) /
4, and h H ( ω ) is the phase-spacesuppression defined in eq. (3.24). Finally, the bound-state ionisation rates are computedusing the detailed balance eq. (3.8b), and summing over all ionised states as in eqs. (3.10a)and (3.10b); this yields (cid:104) Γ ion B (cid:105) (cid:39) (cid:104) σ BSF B v rel (cid:105) × g DM g B (cid:18) mT π (cid:19) / e −|E B | /T . (3.26c) Ionisation equilibrium
Equation (3.26c) implies that at T (cid:29) |E B | , the ionisation of the bound states tends tobe faster than their decays and transitions, i.e. (cid:104) Γ ion B (cid:105) (cid:29) (cid:104) Γ dec B (cid:105) , (cid:104) Γ trans B (cid:105) , provided that We recall from [35] that a factor of [1 + 1 / ( e ω/T − – 13 – σ BSF B v rel (cid:105) is sufficiently large. If so, the system reaches a state of ionisation equilibrium ,where the effective BSF cross-sections (3.16) become independent of the actual ones [43], (cid:104) σ BSF B v rel (cid:105) eff (cid:39) g B g DM Γ dec B (cid:18) πmT (cid:19) / e + |E B | /T , (3.27)where for the DS bound state whose direct decay into radiation is suppressed, we mustuse the effective decay rate (cf. eq. (3.16d)) (cid:104) Γ dec DS (cid:105) → (cid:104) Γ trans DS → SS/D ¯ D (cid:105) (cid:104) Γ dec SS/D ¯ D (cid:105)(cid:104) Γ ion SS/D ¯ D (cid:105) + (cid:104) Γ dec SS/D ¯ D (cid:105) + 2 (cid:104) Γ trans DS → SS/D ¯ D (cid:105) ( Y eq DS /Y eq SS/D ¯ D ) . (3.28)Since the bound-state decay rates are proportional to the annihilation cross-sections of thecorresponding scattering states (cf. e.g. ref. [35]), eq. (3.27) implies that at high tempera-tures and while ionisation equilibrium holds, the BSF contribution to the DM destructionrate is negligible in comparison to that of direct annihilation (cf. e.g. [13, eq. (3.20)].)However, as T approaches or drops below |E B | , the ionisation rates become exponen-tially suppressed and are overcome by the bound-state decay and/or bound-to-bound tran-sition rates. For the uncoupled bound states D ¯ D and DD , this implies that the effectiveBSF cross-sections increase exponentially until they saturate to their maximum values, theactual BSF cross-sections. For the SS/D ¯ D and DS coupled system, (cid:104) Γ trans DS → SS/D ¯ D (cid:105) > (cid:104) Γ ion DS (cid:105) ,occurs before the decay rates surpass the ionisation rates; in this interval, the effective BSFcross-sections (3.16) together with the detailed balance eq. (3.26c), imply that ionisationequilibrium holds for the sum of the SS/D ¯ D and DS contributions, (cid:104) σ BSF
SS/D ¯ D v rel (cid:105) eff + 2 (cid:104) σ BSF DS v rel (cid:105) eff (cid:39) g SS/D ¯ D g DM Γ dec SS/D ¯ D (cid:18) πmT (cid:19) / e + |E SS/D ¯ D | /T , (3.29)where again we neglected the SS/D ¯ D decay against ionisation rate. At even lower temper-atures, when ionisation becomes slower than decay, the effective BSF cross-sections reachtheir actual values.We illustrate the above in figs. 1 and 2, where we also compare radiative BSF and BSFvia scattering. Two observations are useful more generally for calculations of freeze-outwith bound states: • In some (but not all) cases, BSF via scattering dominates at early times; BSF viaHiggs exchange may also dominate at late-times over on-shell emission due to thephase-space suppression of the latter. Nevertheless BSF via scattering does notchange significantly the effective BSF cross-section with respect to considering ra-diative BSF only, because overall it becomes subdominant while the system is still inionisation equilibrium, or around the time it exits it. • For the D ¯ D , DD bound states, ionisation equilibrium ceases at T > |E B | (cf. fig. 1.)In contrast, the bound-to-bound transitions prevent the SS/D ¯ D and DS coupledsystem to reach ionisation equilibrium. However, it closely tracks it until much lowertemperatures, T (cid:28) |E B | , due to the largeness of the BSF cross-sections (cf. fig. 2.)– 14 – emissionW emission H (†) emission
10 10 - - -
10 x = m / T 〈 σ v r e l 〉 / ( π m - ) T = | ℰ DD | - - - 〈 σ v r e l 〉 / ( π m - ) T = | ℰ DD _ | Radiative BSFBSF via scatt.Ion equilibriumEffective BSF
10 10 x = m / T DD : ( , ) , s p i n T = | ℰ DD | D D : ( , ) , s p i n T = | ℰ DD _ | Figure 1 . Thermally averaged BSF cross-sections for the D ¯ D and DD bound states; the latterincludes the capture into its conjugate. We have used m = 20 TeV, α H = 0 . m H ( T ); the spikes in the radiative BSF occur at the EWPT, when the Higgsmass tends to zero before becoming m h (cid:39)
125 GeV. The vertical lines mark the temperaturesequal to the binding energies. Note that the cross-sections have been regulated according to [35,section 3.6.].
We also note here that the computation of the DM thermal decoupling (cf. section 4)shows that much of the BSF effect on the relic density arises after the system exitsionisation equilibrium. (This was also found in ref. [2].)The above imply that it is not safe to estimate the BSF effect by assuming ionisationequilibrium until T ∼ |E B | and neglecting any effect thereafter, an approach previouslyadopted in refs. [44–46]. Considering instead the BSF cross-sections is necessary foran accurate computation. – 15 – - - -
10 x = m / T 〈 σ v r e l 〉 / ( π m - ) - - - 〈 σ v r e l 〉 / ( π m - ) T = | ℰ D S | T = ℰ D S - ℰ SS / DD _ B emissionW emission H (†) emission - - - 〈 σ v r e l 〉 / ( π m - ) T = | ℰ SS / DD _ | T = ℰ D S - ℰ SS / DD _
10 10 x = m / T SS / D D + D S D S : ( , / ) , s p i n T = | ℰ D S | T = ℰ D S - ℰ SS / DD _ Radiative BSFBSF via scatt.Ion equilibriumEffective BSF SS / D D : ( , ) , s p i n T = | ℰ SS / DD _ | T = ℰ D S - ℰ SS / DD _ Figure 2 . As in fig. 1, but for the
SS/D ¯ D and DS states that transition into each other viaHiggs emission or absorption. The DS panels include the capture into its conjugate. In the bottomrow, we show the sum of the SS/D ¯ D , DS and ¯ DS contributions. The vertical lines mark thetemperatures equal to the binding energies and the energy splitting between SS/D ¯ D and DS . Thefeature around x (cid:39)
50 occurs when the Higgs doublet mass becomes lower than the energy splittingbetween the two bound states; this opens up the bound-to-bound transitions via on-shell Higgsemission (at higher T they occur only via off-shell Higgs exchange with the thermal bath), anddrives the SS/D ¯ D bound states somewhat away from ionisation equilibrium. – 16 – Results
Collecting all the above, we are now ready to compute the DM decoupling and relic density.We consider and compare the cases described in table 3, and recall that our calculationsalways assume electroweak symmetry. We discuss this approximation in section 4.2.
AnnS BW Annihilation with Sommerfeld effectdue to the
B, W -mediated potentials.
AnnS BW + BW -BSF BW Annihilation and BSF via B or W emission orexchange, including the B, W -mediated potentials.
AnnS
BW H + BW H -BSF
BW H
Annihilation and BSF via on-shell B , W or H ( † ) emission, including the B, W, H -mediated potentials.
Table 3 . The combinations of effects we compare in the following, in terms of their impacton the DM decoupling. When considering BSF, we always include both radiative BSF and BSFvia scattering. However, we have examined their effects separately, and found that the inclusionof BSF via scattering does not change the results obtained when considering radiative BSF only.Moreover, in the present model, considering the Higgs-mediated potential while omitting BSF viaHiggs emission, or the reverse, do not result in a significant effect on the relic density (cf. fig. 3),we thus do not present these cases separately.
In fig. 4, we present an example of the time evolution of the effective cross-section andthe DM density. For the parameters chosen, the exponential increase of (cid:104) σv rel (cid:105) eff due toBSF when the ionisation processes cease, gives rise to a second period of DM destructionthat decreases the DM density by two orders of magnitude! In fig. 5, we show the timelineof the DM thermal decoupling. We define the recoupling period of DM destruction dueto BSF as the interval between the two occurrences when d (ln Y ) /d (ln x ) = 0, and thechemical decoupling as the latest time when d (ln Y ) /d (ln x ) = 10%. In the same plot, wealso mark the EWPT, as well as the time beyond which the finite Higgs mass affects itslong-range effect. Since in part of the parameter space, the recoupling occurs after theEWPT and the full chemical decoupling occurs even later, the effect of BSF via Higgsemission is most important for the range of DM masses where the binding energies exceedthe Higgs boson mass, m h (cid:39)
125 GeV. These ranges are also marked in fig. 5. (Wediscuss the validity of various approximations, including that of electroweak symmetry, insection 4.2.)In fig. 6, we show the values of α H vs. m that reproduce the observed DM density,as well as the impact of the various processes on the relic density. As already seen infig. 4, at m (cid:38) few TeV, BSF via emission of a Higgs doublet is estimated to decrease therelic density by up to two orders of magnitude. The implications are twofold. For a fixedmass m , the coupling α H is predicted to be almost up to an order of magnitude smallerthan when neglecting BSF via Higgs emission. This should be expected to change (relax)experimental constraints very significantly. Conversely, for a given coupling, a much larger– 17 – ith BSFAnn with Sommerfeldperturbative Ann
10 10 - - - = m / T 〈 σ e ff v r e l 〉 / ( π m - ) violation of s - wave unitaritybound - - - 〈 σ e ff v r e l 〉 / ( π m - ) without Higgs emission violation of s - wave unitaritybound SS / D D onlyD D & DD onlyDS onlyAll bound states
10 10 x = m / T w i t h H i gg s po t e n t a il violation of s - wave unitaritybound with Higgs emission w i t hou t H i gg s po t e n t a il violation of s - wave unitaritybound Figure 3 . Contributions to the effective DM destruction cross-section. The solid lines includedirect annihilation with Sommerfeld effect plus BSF according to the colour legend. We have used m = 20 TeV, α H = 0 . m H ( T ). Note that the cross-sections have been regulated according to [35, section 3.6.]. The binding energy of the D ¯ D and DD bound states does not depend on the coupling to the Higgs, and their formation via H emission orexchange is always suppressed due the Higgs mass; their contribution is dominated by W emission.Both the SS/D ¯ D and DS binding energies depend on α H , which ensures that their formation via H emission is not suppressed when the Higgs-mediated potential is taken into account and providedthat α H is sufficiently large (bottom right panel.) The DS bound states do not exist when neglectingthe Higgs-mediated potential (upper row.) m is anticipated. In fact, DM masses almost up to the unitarity limit can be attained for α H <
1. (We discuss the unitarity limit in more detail in section 4.3.) This motivatesexperimental searches at very high masses.To understand better the effect of the various bound states, in fig. 7 we show the α H − m relation determined by considering direct annihilation plus each of the four bound statesseparately. The spin-1 DD and D ¯ D bound states have only a small effect because theirbinding energy is independent of α H and somewhat small. This implies that ionisationsinhibit the DM destruction via their formation until late, when BSF via Higgs emission iskinematically blocked, and BSF via B or W emission is not sufficiently fast to overcome– 18 – - - - x = m / T 〈 σ v r e l 〉 e ff / ( π m - ) F r ee z e - O u t E W P T C he m . de c . C he m . de c . AnnS BW AnnS BW + BW - BSF BW AnnS
BWH + BWH - BSF
BWH
10 10 - - - - - x = m / T Y = n / s F r ee z e - O u t E W P T C he m . de c . C he m . de c . Figure 4 . The effective cross-section (cid:104) σv rel (cid:105) eff / ( πm − ) and the dark matter yield Y ≡ n/s , vsthe time parameter x = m/T . We also mark the time of freeze-out, the EWPT, and the chemicaldecoupling for the three cases in the legend. We have used m = 50 TeV and α H = 0 . m [ GeV ] x = m / T C h e m i ca l d ec oup li ng E W P T Freeze - Out H - BS F k i ne m a t i c a ll yc on s t r a i ned R e c o u p li n g r e g i o n s ( D D / SS ) B ( D S ) B ( DD ) B ( D D ) B Limit of Coulombapprox. in scatteringpotential m h > 〈μ v rel2 〉 m h < | ℰ DS | m h < | ℰ SS / DD _ | Figure 5 . Various timeposts in the DM decoupling. See text for discussion. The coupling α H is chosen as function of the mass m such that the observed DM density is reproduced by the fullcalculation (cf. cyan line in fig. 6.) For the recoupling intervals, we indicate the bound state thathas the dominant effect. the suppression due to the low DM density. Passing on to the SS/D ¯ D and DS boundstates, for the lower range of m and α H , their formation destroys DM efficiently after theEWPT. Thus the threshold for their effect being important is set by |E B | > m h (cid:39)
125 GeV,as the grey dotted lines in fig. 7 indicate. – 19 – nnS BW AnnS BW + BW - BSF BW AnnS
BWH + BWH - BSF
BWH - - [ GeV ] α H End @ EWPT → F u l l w i t h o u t r e g . AnnS BW AnnS BW + BW - BSF BW AnnS
BWH + BWH - BSF
BWH m [ GeV ] Ω / Ω D M Figure 6 . Left: α H vs m such that the observed DM density is attained via thermal decoupling,when different combinations of effects are considered, as described in table 3. Note that the DMmass is m DM = m − √ πα H v H with v H (cid:39)
246 GeV, and does not differ substantially from m along any of the lines. In grey lines, we show the result if the Boltzmann equations are integratedonly up until the EWPT ( dotted ), and if the cross-sections are not regulated according to [35,section 3.6] ( dashed .) Right:
The effect of the various processes on the relic density. For all lines, α H is determined as a function of m by the full computation on the left (cyan line), but for eachline here the Boltzmann equations include only the processes indicated in the legend. SS / D D onlyDD & D D onlyDS only - - [ GeV ] α H m h = | ℰ D S | m h = | ℰ SS / DD _ | A ll bound s t a t es Figure 7 . α H vs m such that the observed DM density is attained via thermal decoupling,considering the contributions of the various bound states separately. – 20 –ven away from the correlation of parameters that reproduces the observed DM den-sity, the BSF effect on the relic abundance of the stable species can be very large as seen infig. 8. The parameter space where the relic density is cosmologically insignificant is greatlyenlarged. This is important for scenarios that do not aspire to explain the DM density, butnevertheless predict the existence of stable particles. Figure 8 . Contours of log (Ω / Ω DM ) (with the values indicated in black), when consideringAnnS BW only ( left ), and AnnS BWH + BW H − BSF
BWH ( right .) The red and cyan lines mark Ω = Ω DM .Note that the DM mass, m DM = m − √ πα H v H , differs significantly from m only at the top leftcorners of the plots. – 21 – .2 Major approximations and their validity We now summarise the main approximations made in our analysis and comment on theirpotential effect on the estimated relic density.(i)
Considered only ground-level bound states.
BSF via vector or neutral scalar emission is dominated by dipole and quadrapole mo-ments respectively. In these cases, the capture into the ground state is the dominantBSF process [2, 9–11, 47], the reason being twofold: it is the most exothermic process,and the overlap of scattering and bound state wavefunctions is larger.In contrast, BSF via emission of a charged scalar is a monopole transition, and thecapture into excited states can be comparable to or faster than the capture into theground state, despite the latter being more exothermic [13]. This suggests that in thepresent model, capture into excited states via Higgs emission may be important.Nevertheless, independently of the BSF cross-sections, the relative effect of the excitedstates on the relic density is moderated by their smaller binding energy that renderstheir ionisation efficient until later. We thus anticipate that in the present modelexcited states may have a significant albeit not dominant effect that would furtherdiminish the relic density and alter the coupling-mass relation along the directionfound here. This is worth pursuing in more detail in the future.(ii)
Regularisation of inelastic cross-sections in parametric regimes where BSF via Higgsemission approaches or appears to exceed the unitarity limit.
The efficiency of BSF via emission of a charged scalar implies that the computedcross-sections may reach or violate the upper limit on inelastic cross-sections impliedby unitarity (cf. eq. (4.4)) even for small or moderate values of the coupling of theinteracting particles to the scalar [13]. The restoration of unitarity implies that re-summed higher order corrections, i.e. higher-order contributions to the non-relativisticpotential, must be considered [1]. In the context of the present model, the problemwas discussed in ref. [35, section 3.6], where the regularisation scheme of ref. [48] wasadopted as an effective method to ensure that unitarity is not violated. However, asdiscussed in ref. [35], this scheme is not entirely suitable for the case of interest.In fig. 6 we compare the α H − m relations with and without regularisation. Clearly, atlarge m the effect is significant; the regularisation of the cross-sections ensures that m does not exceed the unitarity limit on the mass of thermal relic DM [1, 49], which wediscuss in section 4.3. This suggests that working out a more accurate regularisationscheme that would address the issues discussed in ref. [35] may be important in orderto obtain more accurate results. We leave this for future work.(iii) Neglected the Higgs mass in the Higgs-mediated potential.
The validity of the Coulomb approximation for the Higgs-mediated potential has beendiscussed in [35, section 2.3.3], where the relevant conditions have been put forward.Here we discuss their validity during the DM thermal decoupling.– 22 –cattering states: In a thermal bath, the condition µv rel > m H for the validity ofthe Coulomb approximation implies (cid:112) mT / (cid:38) m H . Considering the Higgs doubletmass (2.14) before the EWPT, this becomes T (cid:46) m , which clearly covers all of therange of interest for the DM freeze-out ( T (cid:46) m/ m H → m h (cid:39)
125 GeV, the condition is satisfied until after the DM chemical decoupling, asshown in fig. 5. Therefore, the Coulomb approximation does not pose any problem.Bound states: The condition µα H > few × m H , becomes xα H > few before the EWPT,with x = m/T . This is satisfied for all relevant x and α H for which BSF has an effect( x (cid:38)
50 and α H (cid:38) .
1, cf. figs. 5 and 6.) It is easy to check that this condition is alsosatisfied below the EWPT for all relevant DM masses and couplings ( m > α H (cid:38) .
1, cf. fig. 6.) Note that this is not coincidental; BSF via Higgs emission doesnot have a significant effect for lower α H values because of the phase-space suppressiondue to the Higgs mass (cf. fig. 7.) The estimation here thus confirms the argument ofref. [35] that bound states are nearly Coulombic in the parameter space where theirformation is kinematically allowed and significant.(iv) Assumed electroweak symmetry.
In fig. 5, we see that the DM destruction via BSF may be efficient after the EWPT.The breaking of the electroweak symmetry has several important implications thatwe now discuss.a.
The Goldstone modes of the Higgs doublet are absorbed by the
Z, W ± bosons. • BSF via emission of a Higgs doublet in the unbroken electroweak phase cor-responds to BSF via emission of h or the longitudinal modes of the Z, W ± bosons in the broken electroweak phase. The Goldstone boson equivalencetheorem implies that the amplitudes for BSF via emission of the longitudinal Z, W ± components are the same as those for the corresponding processes inthe unbroken electroweak phase, in the limit that the energy of the emittedvector boson is much larger than its mass, m Z,W /ω (cid:28) m H → m h (cid:39)
125 GeVafter the EWPT, ensuring that m h /ω < m Z,W /ω < . • The potential mediated by the Higgs doublet in the unbroken electroweakphase is mediated by h and the longitudinal Z, W ± components in the brokenphase. To compute the non-relativistic potential generated by the latter, weneed their contribution to the vector boson propagators, i q − m V q µ q ν m V , (4.1)where q and m V = g V v H / V = Z, W ± , with g Z = (cid:112) g + g and g W = g . In general, the exchange of– 23 – , W ± between a pair of Z -odd particles may change the mass eigenstate oneach leg. (Indeed, in the model under consideration, the Z, W ± bosons coupleonly non-diagonally to the mass eigenstates, cf. eq. (2.8).) Considering (4.1),the contribution from the exchange of the longitudinal Z, W ± components tothe 2PI kernels (cf. ref. [35, section 2]) is proportional to K L ∝ [¯ u ( p (cid:48) ) i g V /qu ( p )][¯ u ( p (cid:48) ) i g V /qu ( p )] /m V = ( i g V ) ( m (cid:48) − m )¯ u ( p (cid:48) ) u ( p ) ( m − m (cid:48) )¯ u ( p (cid:48) ) u ( p ) /m V , (4.2)where q = p (cid:48) − p = p − p (cid:48) , and we used the Dirac equation /pu ( p ) = mu ( p ).Considering the mass splittings ∼ yv H , this becomes K L ∝ g V ( yv H ) (2 m ) /m V ∝ y m . (4.3)Equation (4.3) shows that the potentials generated by the exchange of thelongitudinal Z, W ± is indeed proportional to the coupling to the Higgs doublet.The range of the potentials are m − V > m − h , thus the arguments presented initem (iii) for the Coulomb approximation remain valid. An analogous resulthas been obtained in [50] for a broken U (1) model.Note that in eqs. (4.2) and (4.3) we omitted various numerical factors and signsfor simplicity, and focused on deriving the scaling of the 2PI kernel. Consid-ering these factors in detail reproduces the Higgs-doublet mediated potential(aside from the screening scale.)b. The Weak gauge bosons become massive.
The non-zero
Z, W ± masses curtail the range of the potentials generated by theexchange of both their transverse and longitudinal components, and introducephase-space suppression to the BSF processes occurring via their emission. Thevalidity of the Coulomb approximation for the Z, W ± bosons can be assessed asin the preceding discussion for the Higgs. However, in the present model, the B, W -generated potentials and BSF via B or W emission do not have a significanteffect, due to the fact that one of the dark multiplets is a gauge singlet and theother belongs to a small representation. We thus do not consider the transverse Z, W ± components further. The effect of the longitudinal Z, W ± components wasdiscussed above.c. The components of the DM multiplets acquire different masses.
After acquiring a mass splitting, the various pairs of Z -odd particles can oscil-late into each other according to the non-relativistic potentials computed in [35,section 2] provided that the kinetic energy of their relative motion exceeds theirmass difference. This necessitates mv / > yv H , which, upon thermal averaging,becomes T > (4 / √ πα H v H . This condition is not satisfied below the EWPT forthe α H values of interest ( α H (cid:38) . α H − m relation between our this and the result obtained by integratinguntil late times. We see that even when the integration stops at the EWPT, theHiggs effect is still very significant, even if it appears only for larger α H values.The impact on the relic density reaches up to a factor of a few. The unitarity of the S matrix sets an upper limit on the partial-wave inelastic cross-sections, σ inel (cid:96) (cid:54) σ uni (cid:96) = (2 (cid:96) + 1) πk (cid:39) (2 (cid:96) + 1) πµ v , (4.4)where (cid:96) is the partial wave and k is the momentum of either of the interacting particles inthe CM frame. The last approximation in eq. (4.4) concerns the non-relativistic regime,where k = µv rel with µ being the reduced mass.The upper limit (4.4) suggests that for very large masses, annihilations in the earlyuniverse may not suffice to reduce the density of thermalised particles to the observedDM value. It thus sets an upper bound on the mass of thermal relic DM annihilatingpredominantly via a finite number of partial waves in the early universe [49]. For self-conjugate DM in thermal equilibrium with the SM plasma, this is [1, 2] m DM ,(cid:96) (cid:46)
197 TeV × (cid:40) √ (cid:96) + 1 , solely (cid:96),(cid:96) + 1 , (cid:54) (cid:96) (cid:54) (cid:96) max . (4.5)Equation (4.5) is modified by 1 / √ σ uni (cid:96) on µ and v rel implies that the limit (4.4) can beattained down to arbitrarily low velocities — thus the upper limit (4.5) on the mass ofthermal-relic DM can be reached — only if there is an attractive long-range force betweenthe interacting particles, and provided of course that the relevant couplings are sufficientlylarge [1]. Attractive long-range interactions imply also the existence of bound states, whoseformation and subsequent decay may decrease the DM abundance more efficiently thandirect annihilation [2]. This means that BSF may essentially be the dominant process thatsaturates the unitarity limit (4.4), and that additional partial waves to those dominatingin the annihilation processes may become important, thereby increasing the upper limit onthe DM mass [1].In general, the Weak interactions of the Standard Model are not sufficiently strong togenerate cross-sections that approach the unitarity limit (4.4), unless perhaps the inter-acting particles belong to very large SU L (2) representations. However, BSF via emissionof a scalar charged under a symmetry can be very efficient even for small couplings [13].Here, we have seen that BSF via Higgs emission can raise the predicted WIMP mass verysignificantly, bringing it potentially close to the unitarity limit. If DM annihilates into a dark plasma that has different temperature than the SM plasma or includesmany relativistic dof during DM freeze-out, then this value may somewhat change. Moreover, departuresfrom thermal cosmology, such as episodes of entropy injection (see e.g. [51]), imply that larger m DM valuesmay be permissible. – 25 – Conclusion
Our DM searches are currently at the onset of the exploration of the multi-TeV regime witha variety of existing and upcoming telescopes observing high-energy cosmic rays. In thismass regime, within the thermal-relic scenario, the DM interactions are expected to man-ifest as long-range and give rise to non-perturbative effects, in particular the Sommerfeldeffect and the formation of bound states. These effects may operate in the early universeduring the DM thermal decoupling, as well as inside DM haloes today, and significantlyalter the DM phenomenology.In the present work, consisting of this and a companion paper [35], we have consid-ered the role of the Higgs doublet as a light force mediator, in the thermal decouplingof multi-TeV WIMP DM. We have shown that the Higgs-doublet-mediated potential be-tween DM particles and the formation of DM bound states via Higgs-doublet emission candramatically change the predicted relic density. This, in turn, alters the coupling-massrelation that reproduces the observed DM abundance. Moreover, it greatly expands theparameter space where the stable relics do not overclose the universe, even if they are asubdominant component of DM. In the former case, the modified coupling-mass relationimplies that on one hand, for a given DM mass, existing constraints may be significantlyrelaxed, and on the other hand, DM may be much heavier than previously anticipated,potentially approaching the unitarity limit.While the amplitude for BSF via Higgs-doublet emission can be quite large even forsmall couplings of the DM multiplets to the Higgs, the Higgs-doublet mass introduces akinematic suppression to the cross-section that renders this effect relevant for larger DMmasses and/or couplings to the Higgs. In the specific singlet-doublet scenario consideredhere, we found that the effect is significant for m (cid:38) α H (cid:38) .
1. However, inmodels involving larger SU L (2) representations, the gauge interactions contribute moresignificantly to the binding energy of the bound states, thereby rendering the phase-spacesuppression less significant. We thus expect that the effect on the relic density will beimportant even for lower couplings.Finally, we note that the capture into excited bound levels, which we neglected here,may also have a sizeable effect due to the monopole nature of the transitions occurring viaHiggs-doublet emission. On the other hand, we have found that including BSF throughscattering on the relativistic thermal bath via an off-shell Higgs doublet does not affect therelic density significantly. – 26 – cknowledgements This work was supported by the ANR ACHN 2015 grant (“TheIntricateDark” project),and by the NWO Vidi grant “Self-interacting asymmetric dark matter”.
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