Cosmological constraints on light flavons

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1 Cosmological constraints on light flavons Matti Heikinheimo, Katri Huitu, Venus Keus, Niko Koivunen Department of Physics and Helsinki Institute of Physics, Gustaf Hällströmin katu, FIN University of Helsinki, Finland arxiv: v1 hep-ph] 8 Dec 018 December 31, 018 Abstract The Froggatt-Nielsen mechanism is a well-motivated framework for generating the fermion mass hierarchy. This mechanism introduces flavons, complex scalars which are singlet under the Standard Model gauge symmetry and charged under a new global family symmetry. We make use of a leptophilic flavon to produce the charged lepton Yukawa matrix. The real part of the flavon mixes with the Higgs boson and introduces lepton flavour violating interactions which are bounded by experiment. The imaginary part of the flavon,, is a long-lived light particle, whose abundance is restricted by cosmological observations. For m < m e where the decay of to charged leptons is kinematically forbidden, we identify allowed regions of m with respect to the vacuum expectation value of the flavon field where all experimental and cosmological constraints are satisfied. Matti.Heikinheimo@helsinki.fi Katri.Huitu@helsinki.fi Venus.Keus@helsinki.fi Niko.Koivunen@helsinki.fi

2 1 Introduction The origin of the fermion mass hierarchy is a long-standing problem of the Standard Model (SM). Amongst the many beyond the SM (BSM) scenarios aiming to explain this hierarchy, the Froggatt-Nielsen mechanism 1] offers a natural solution to this problem. This mechanism introduces a spontaneously broken global U(1) symmetry and a new scalar field, the flavon. The flavon, Φ, is a singlet under the gauge symmetry of the SM, but charged under the U(1) symmetry along with all other SM particles. The U(1) symmetry is spontaneously broken as the flavon field acquires a vacuum expectation value (vev), denoted by v φ. The real part of the flavon mixes with the SM Higgs field, resulting in two mass eigenstates. The lighter state is taken to be the scalar boson with a mass of approximately 15 GeV observed at the Large Hadron Collider (LHC), 3]. The imaginary part of the flavon field,, is a pseudo-goldstone boson. In general, the flavon has flavour violating couplings, which the SM-Higgs boson inherits due to their mixing. As a result, all three scalar mass eigenstates take part in flavour violating interactions. The resulting flavour violating processes, which are dominated by, are severly constrained by experiment, with the most stringent bounds coming from charged lepton flavour violation (CLFV) through the three body decay l i l j l k l l, the l i l j γ transition and the µ e conversion processes 4]-8]. With the couplings of inversely proportional to v φ, the non-observation of CLFV processes puts a lower bound on the vev of the flavon, v φ O(TeV). In this paper, we shall study the charged lepton sector and make use of a leptophilic flavon to produce the charged Yukawa mass structure 9, 10]. Other fermion Yukawa textures could be constructed with the introduction of extra flavons and family symmetries. We investigate cosmological implications of a very light, with m < m e so that its dominant decay channel is into two photons. The presence of the particles during the early epochs are felt via their contribution to the total energy density of the radiation dominated universe, and via the energy deposited by the decays of to the SM radiation bath during and after the processes of the big bang nucleosynthesis (BBN) and recombination. Thus the model faces severe constraints from the observed abundance of chemical elements 11] and the cosmic microwave background (CMB) 1, 13]. Moreover, with a mass below a few kev, is a hot relic, whose abundance is suppressed by cosmic structure formation at small scales 14]. The mechanism for production of varies by changing v φ ; for relatively small values, v φ O( ) GeV, will be produced as a relativistic relic through a freeze-out mechanism 15] with its abundance (had it been stable) well below the observed dark matter (DM) relic density, Ω DM,0 h = ± 0.00, (1) as measured by the Planck experiment 16]. Throughout this paper, we refer to Ω,0 h which is the abundance the flavon relic would have had today if it was stable. The true abundance is obtained by scaling this value by the decay factor. In this range of v φ, we show that increasing 1

3 m leads to a larger initial abundance while reducing the lifetime of considerably, down to τ s. Lighter particles, m of a few mev, have much longer lifetimes, but their abundance is of order 10 5 with respect to the DM relic density. For higher values of the flavon vev, v φ 10 9 GeV, the couplings of are so small that does not thermalise with the SM bath and is produced through a freeze-in mechanism 17], with an abundance below that of the DM, Ω,0 h < Ω DM,0 h. The layout of the paper is as follows. We first review the Froggatt-Nielsen mechanism in Section, and introduce the scalar potential for the pseudo-goldstone field in Section 3. We discuss CLFV constraints in Section 4, and then study the thermal history of the model in Section 5. We present the resulting cosmological constraints in Section 6 and draw our conclusions in Section 7. The Froggatt-Nielsen mechanism The Froggatt-Nielsen mechanism is a well-motivated framework for generating the fermion mass hierarchy 1]. This mechanism introduces a complex scalar field, Φ, called the flavon, which is a singlet under the SM gauge group, but charged under a new global U(1) symmetry. All the SM particles are charged under this global symmetry. Consistent with the U(1) charges, the SM Yukawa interactions are generated through higher order operators of the form L c ij ( Φ Λ) nij fl,i Hf R,j + h.c., () where c ij are dimensionless order-one coefficients, Λ is the scale of new-physics, H is the SM Higgs doublet, and f L,R are the SM fermions. Conservation of the Froggatt-Nielsen symmetry requires the U(1) charges in Eq.() to add up to zero, resulting in n ij = 1 q Φ (q L,i + q R,j + q H ), (3) where q L,R, q H, q Φ are the charges of the SM fermions f R,L, the SM Higgs field H and the flavon Φ, respectively. As the flavon develops a vev, Φ = (v φ + φ)/, the effective operator in Eq.() generates the SM Yukawa interactions, L eff y ij fl,i Hf R,j + h.c. with y ij = c ij ( vφ Λ ) nij c ij ɛ n ij, (4) where ɛ is naturally a small parameter (v φ Λ). Note that the U(1) charge assignment determines the n ij power of ɛ which in turn determines the Yukawa matrix structure. This is the primary feature of the Froggatt-Nielsen mechanism which relates the fermion mass hierarchy to the U(1) charges of the fermions. In this paper, we shall only consider the leptonic sector and only allow the leptons and the flavon to transform under the U(1) symmetry 1. 1 Other fermions could be assigned extra flavons and transform under other symmetry groups.

4 When the Higgs field acquires a vev, H = 1 ( 0 v h + h the Yukawa Lagrangian becomes L eff y ( ij v h + h + v ) h n ij φ v φ ), (5) l L,il R,j + h.c., (6) where the primed leptons, l, are the gauge eigenstates. They are related to the mass eigenstates, l, by the unitary transformation U L,R, leading to a Yukawa Lagrangian of the form L eff l L v h y diag In general, the κ matrix, y diag = U L y U R, (7) l R + l L y diag l R h + l L v h κ vφ l R φ + h.c.. (8) κ = U L (n y) U R with (n y) ij = n ij y ij, (9) is not diagonal and sources the flavour violating processes in the model. In what follows, we parametrise our results in terms of the κ coupling where κ ij = 1 v h v φ κ ij. (10) Note that the flavon field is a complex field, explicitly deconstructed as whose leptonic couplings appear as 3 The scalar potential Φ = 1 (v φ + φ) where φ = σ + i, (11) L l,φ = l L κ l R σ + l L i κ l R + h.c.. (1) The Higgs-portal 18]-0] scalar potential is of the following form, V = µ h (H H) + λ h (H H) µ φ (Φ Φ) + λ φ (Φ Φ) +λ hφ (H H)(Φ Φ) µ φ 4 (Φ + Φ ), (13) 3

5 where the U(1) symmetry is softly broken by the last term, which is responsible for the mass of the field. The minimisation conditions for the potential are µ h = v hλ h + v φλ hφ, µ φ = v φλ φ + v hλ hφ µ φ. (14) The mass eigenstates H 1 and H are given by ( ) ( ) ( H1 cos θ sin θ h sin θ cos θ σ H ), tan θ = λ hφv h v φ λ h v h λ φv φ. (15) The masses of the three physical scalar states are calculated to be m H 1, = 1 ( λ h vh + λ φ vφ ± λ hvh λ φv ) φ, m = µ φ, (16) cos θ where we take H 1 to be the SM-like Higgs boson with 15 GeV mass. Note that the mixing of h and σ, which is constrained by sin θ 0.3 1], introduces flavour violating couplings for the H 1 state, and allows for the H state to couple to SM fermions. The Yukawa couplings of the scalar mass eigenstate are explicitly written as ( L Y = l L cos θ y ) diag sin θ κ l R H 1 (17) ( + l L sin θ y ) diag + cos θ κ l R H + l L (i κ) l R + h.c.. For our analysis, we choose a representative values of m H = 500 GeV and sin θ = 0.1, which does not affect the qualitative behaviour of our model. The state is a pseudo- Goldstone boson and naturally is assumed to be light. For m < m e, the decay of to a pair of charged leptons is kinematically forbidden. As a result, decays predominantly to a γγ final state through a loop of charged leptons, as shown in Figure 1, with the leading order decay amplitude of Therefore, the lifetime of is ( M γγ = e4 κee 3π 4 m4 + κ µµ + κ ) ττ. (18) m e m µ m τ τ = 1 Γ γγ = 3π m M γγ, (19) which, depending on m and v φ, could be long enough to face severe constraints from cosmological observations, as will be discussed in detail in the following sections. 4

6 γ l i γ Figure 1: The long-lived state with m < m e decays predominantly to a γγ final state. 4 Constraints from CLFV processes We assign the Froggatt-Nielsen charges as shown in Table 1, to reproduce the correct charged lepton masses for ɛ = 0.1. The resulting Yukawa texture and κ matrix are of the form ɛ 6 ɛ 5 ɛ 4 y ɛ 5 ɛ 4 ɛ 3, κ 1 m e m µ ɛ m τ ɛ m µ ɛ m µ m τ ɛ, (0) ɛ 4 ɛ 3 ɛ v φ m τ ɛ m τ ɛ m τ with the precise values given in Appendix A. Particle e c L e R µ c L µ R τ c L τ R H Φ Charge Table 1: Our U(1) charge assignment for the SM fields and the flavon. We implement the current bounds from the three body decay l i l j l k l l, the l i l j γ transition and the µ e conversion processes, presented in Table. All three scalars take part in CLFV interactions. However, with m < m e m H1,H, the processes mediated by will be by far dominant to those of H 1 and H. Therefore, one can safely disregard these sub-dominant contributions, and only take into account the diagrams shown in Figure. 5

7 Observable Present limit 1 BR(µ eee) ] BR(τ eee) ] 3 BR(τ µµµ) ] 4 BR(τ µ e + e ) ] 5 BR(τ e µ + µ ) ] 6 BR(τ e + µ µ ) ] 7 BR(τ µ + e e ) ] 8 BR(µ eγ) ] 9 BR(τ µγ) ] 10 BR(τ eγ) ] 11 CR(µ-e, Au) ] Table : Current experimental bounds on the branching ratios of the three body decay l i l j l k l l, the l i l j γ transition and the µ e conversion processes. l i l j l k l l µ e γ N N l i γ l j Figure : The l i l j l k l l (left), µ e-conversion (center) and l i l j γ (right) processes mediated by the lightest scalar,. In the diagram on the right, the photon can be attached to the internal fermion propagator, or to the external fermion legs. The most constraining CLFV limit proves to be the µ eγ transition. The branching ratio of this process is ( α m µ BR(µ eγ) = κ ee κ 4096 π 4 Γ tot eµ + κ µe κ µµ + κ m µ τe κ µτ µ m τ ) + κ ee κ m µ µe + κ eµ κ µµ + κ eτ κ τµ, (1) 6 m τ

8 where we have neglected m e and m in comparison to m µ. Note that this result constrains κ (and consequently v φ ) independently of m, since in the studied mass range of 0 < m < m e, the BR(µ eγ) is independent of m. For our specific charge assignment in Table 1 and the resulting Yukawa texture in Eq.(0), we obtain BR(µ eγ) < v φ > 14.4 TeV. () 5 Production of in the early universe For a given mass, the determining factor in the thermalisation and abundance of is the flavon vev, since the interaction strength of with the SM particles is inversely proportional to v φ. A relatively small v φ, of order GeV, results in the thermal freeze-out of. In this vev range, increasing v φ leads to an earlier freeze-out. If the flavon vev is very large, v φ 10 9 GeV, never comes into thermal equilibrium with the SM bath, and is produced through the freeze-in mechanism. In the freeze-out scenario, the field is kept in thermal equilibrium with the SM plasma in the early universe primarily through the processes shown in Figure 3 and their corresponding u- and t- channel counterparts. γ l l l H 1, H 1, H 1, W, Z W, Z H 1, Figure 3: The most important number changing diagrams for in the early universe. We estimate the freeze-out temperature by comparing the Hubble rate, H, to the interaction rate Γ of the -production process αx y, where Γ αx y (T ) = n α vσ αx y. (3) Here, α represents the heaviest particle in the process (other than ) which is in thermal equilibrium with the SM heat bath, and x and y represent other particles involved, e.g. photons. The relativistic number density of the particle species α is n α = { 3 4π ζ(3)gt 3 1 π ζ(3)gt 3 fermions, bosons. (4) When T m α, the thermally averaged cross section scales as vσ αx y 1/s T. Hence the interaction rate n σv scales as T, while the Hubble rate scales as T as a function 7

9 of the temperature. Therefore, in this relativistic regime, the interaction rate becomes faster compared to the Hubble rate as the universe expands and the temparature decreases. When α becomes non-relativistic at T < m α, the number density and thus the interaction rate will be exponentially suppressed. Hence, if the process αx y is not in equilibrium at T = m α, it never was and it never will be. The following condition can thus be used as a rule of thumb in determining whether the process αx y thermalises or not: Γ αx y (T = m α ) < H(T = m α ) = no thermalisation. (5) Therefore, the freeze-out temperature is the temperature at which the last numberchanging process involving, falls out of equilibrium with the SM heat bath which we estimate to be T fo m α. This treatment results in a step-like abundance of as a function of the flavon vev. In Figure 4, we show this behaviour for different masses, and identify the thermalisation process in each v φ interval. For higher vevs, does not thermalise with the SM heat bath and is produced through a freeze-in mechanism. Note that the lower bound on v φ is imposed by the CLFV constraints ee γ μμ γ WW Ωh /ΩDMh m= 6 mev m= 3 mev m= 800 μev ττ γ vϕ GeV] Figure 4: Abundance of with respect to v φ, produced through the freeze-out process for different values of m. The gray dashed vertical lines indicate the v φ intervals and the corresponding main production process. In calculating the thermally averaged cross section, we use the standard approximation 15] where instead of integrating over the exact statistics of relativistic particles, we assume head-on collisions and approximate the initial energies of the incoming particles with their average thermal energies, given by E = { 7π 4 180ζ(3) T 3.151T fermions, π 4 T.701T bosons. (6) 30ζ(3) 8

10 The yield of the relativistic particle, produced in a freeze-out process, at present-day is Y,0 = 45 π 4 ζ(3) g eff, g s (x f ), (7) where g eff, = 1. This is related to the abundance through the relation Ω,0 = m s 0 Y (,0 ) h. (8) 3H 0 8πG N This is the abundance the flavon relic would have today had it been completely stable, and is the number upon which the experimental bound is placed. The details of the calculation of the relevant production cross sections for are presented in Appendix B in case of the freeze-out production and in Appendix C in case of the freeze-in production. To clarify the discussion above, we treat the case of m = 1 mev, as an instructive example, to outline the thermal history of the field as a function of the flavon vev. For values of v φ > GeV, will be produced through the freeze-in mechanism and contributes to a fraction of the DM abundance. If v φ GeV, reaches thermal equilibrium with the SM particles. The freeze-out temperature reduces with decreasing v φ, as lighter SM particles are able to remain in equilibrium with. For v φ > GeV, the freeze-out temperature is T fo m W, for v φ > GeV it is T fo m τ, and for v φ > GeV it is T fo m µ. Below this value, remains in equilibrium during BBN, and therefore contributes to the effective number of neutrinos N eff, where the constraint N eff < 1 11] marginally allows the presence of one scalar degree of freedom. will then decouple at the temperature T fo m e. Figure 5 shows the abundance of with respect to v φ for m = 1 mev. Note that the apparent discontinuity represented by the gray dashed vertical line is due to the assumption that for v φ > GeV, interacts so feebly with the SM bath that it never thermalises. As a result, production of goes abruptly from a freeze-out to a freeze-in mechanism. The realistic treatment of this transition should be done by solving the Boltzmann equation numerically which is out of the scope of this paper. As it will be discussed in the next section, for applying the cosmological constraints, it is useful to present the abundance of in terms of its lifetime. Hence, for our example of m = 1 mev, we also show the lifetime of corresponding to the values of v φ in Figure 5 on the top horizontal axis. 6 Cosmological constraints While is relativistic, it contributes to the total energy density of the radiation dominated universe. Additionally, as decays to photons, it deposits energy to the SM radiation bath 9

11 LifeTime s] Ωh /ΩDMh freeze-out freeze-in Ω h =Ω DM h Freeze-in production Freeze-out production Apparent discontinutity vϕ GeV] (m =1 mev) Figure 5: Abundance of with respect to the flavon vev and the lifetime of for m = 1 mev. The particle is produced through the freeze-out mechanism for v φ GeV, and through the freeze-in process for v φ > GeV. The red band represents the region where the abundance of, had it been stable, would be equal to the observed DM relic density. during and after the processes of BBN and recombination. The abundance of is therefore constrained from the observed abundance of chemical elements and the CMB 11]-13]. Moreover, for values of m 3 kev, is a hot relic and can not constitute the majority of DM, as it would suppress cosmic structure formation at small scales 14]. Therefore, only a small sub-dominant component of hot DM, with a density of O(10 ) of Ω DM h or so, is feasible. For a direct comparison with the results of 13], in Figure 6, we present constraints on the abundance of normalised to the abundance of DM, Ω,0 h /Ω DM,0 h, evaluated at presentday, as a function of the lifetime of. The value Ω,0 h is the abundance the -relic would have had today if it was stable. The actual relic abundance of is lower than Ω,0 since it is a decaying particle. The relevant bounds in each mass range are as follows. For m 14 mev, the abundance of is very low and its lifetime is very long, so no cosmological process imposes any constraints on the parameter space regardless of the freeze-out or freeze-in production of. Note that in this mass range, the lower bound of v φ is only imposed by the CLFV constraints. In the intermediate mass range of 14 mev < m < 3 ev, both production mechanisms of, freeze-out and freeze-in, are important. For such small m values, is a hot relic and can only contribute a small fraction of the DM density. For values of 3 ev< m < m e, freeze-out production of leads to large abundance and 10

12 0.100 Ωh /ΩDMh Ω h =Ω DM h Ruled out by BBN Ruled out by CMB m= 14 mev m= 3 ev m= 1 Mev LifeTime s] Figure 6: The abundance vs. lifetime of for different m. The green and blue areas show the excluded regions from BBN and CMB observations, respectively. short lifetime, ruled out by the CMB data. However, the freeze-in production of allows for very small densities of to survive the cosmological bounds. In Figure 6, we show the abundance of for m = 14 mev, in orange, where all CLFVallowed values of v φ > 14.4 TeV, survive the cosmological constraints. The abundance of m = 3 ev is shown in brown where all v φ values that lead to a freeze-out production of are ruled out, but v φ values leading to the freeze-in production of are allowed, provided the abundance is below the CMB bounds and is less than 1% of DM relic density, which we take as an estimate for the allowed abundance of hot DM. A more detailed analysis on the allowed hot relic abundance would require numerical simulations of small scale structure formation with both hot and cold DM components, and is beyond the scope of this work. The abundance of m = 1 MeV < m e is shown in yellow where only very small densities of are allowed for very large values of v φ. The graph only shows v φ values up to GeV. In Figure 7, we show the allowed region in the m -v φ plane. As mentioned before, the lower bound on the flavon vev comes from the CLFV experiments, irrespective of m, represented by the green area. The blue area represents the region where the abundance of is constrained by CMB data and small scale structure observations. 7 Conclusions In this paper, we have used the Froggatt-Nielsen mechanism to generate the charged lepton Yukawa matrix with a leptophilic flavon whose real part couples to the SM Higgs field and its imaginary part,, is a light pseudo-goldstone boson. The resulting flavour violating couplings are constrained by the non-observation of the CLFV processes which put a lower 11

13 vϕ GeV] Ruled out by cosmological data Ruled out by CLFV data m GeV] Figure 7: Regions ruled out in the m -v φ plane, due to the constraints from CMB data and small scale structure observations, in blue, and from CLFV experiments, in green. bound on the vev of the flavon. The production mechanism for is determined by the value of the flavon vev where for relatively small values of v φ of order GeV, is produced relativistically through a freeze-out mechanism with its abundance below the observed DM relic density. In this range of v φ, we show that increasing m leads to a larger abundance while reducing its lifetime. Smaller masses, below the mev range, lead to longer lifetimes and much smaller relic densities. For high values of v φ, will be produced through a freeze-in mechanism. We study cosmological implications of such light particles with m < m e, whose dominant decay channel is into two photons. In this mass range, we show that the abundance of the is limited by BBN, CMB and small scale structure observations, and identify the allowed region in the m -v φ space. Acknowledgements KH and VK acknowledge the H00-MSCA-RISE-014 grant no (NonMinimal- Higgs). NK is supported by Vilho, Yrjö and Kalle Väisälä Foundation. References 1] C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, 77 (1979). ] G. Aad et al. ATLAS Collaboration], Phys. Lett. B 716 (01) 1 arxiv: hep-ex]]. 1

14 3] S. Chatrchyan et al. CMS Collaboration], Phys. Lett. B 716 (01) 30 arxiv: hep-ex]]. 4] U. Bellgardt et al. SINDRUM Collaboration], Nucl. Phys. B 99, 1 (1988). 5] Y. Amhis et al. Heavy Flavor Averaging Group Collaboration], arxiv: hepex]. 6] K. Hayasaka et al., Phys. Lett. B 687, 139 (010) arxiv: hep-ex]]. 7] F. Renga MEG Collaboration], Hyperfine Interact. 39, no. 1, 58 (018) arxiv: hep-ex]]. 8] W. H. Bertl et al. SINDRUM II Collaboration], Eur. Phys. J. C 47, 337 (006). 9] K. Huitu, V. Keus, N. Koivunen and O. Lebedev, JHEP 1605, 06 (016) arxiv: hep-ph]]. 10] V. Keus, J. Phys. Conf. Ser. 873, no. 1, (017) arxiv: hep-ph]]. 11] G. Mangano and P. D. Serpico, Phys. Lett. B 701, 96 (011) arxiv: astroph.co]]. 1] L. Verde, E. Bellini, C. Pigozzo, A. F. Heavens and R. Jimenez, JCAP 1704, 03 (017) arxiv: astro-ph.co]]. 13] V. Poulin, J. Lesgourgues and P. D. Serpico, JCAP 1703, no. 03, 043 (017) arxiv: astro-ph.co]]. 14] V. Iri et al., Phys. Rev. D 96, no., 035 (017) arxiv: astro-ph.co]]. 15] N. Fornengo, C. W. Kim and J. Song, Phys. Rev. D 56, 513 (1997) hep-ph/97034]. 16] P. A. R. Ade et al. Planck Collaboration], Astron. Astrophys. 594 (016) A13 arxiv: astro-ph.co]]. 17] L. J. Hall, K. Jedamzik, J. March-Russell and S. M. West, JHEP 1003, 080 (010) arxiv: hep-ph]]. 18] V. Silveira and A. Zee, Phys. Lett. B 161, 136 (1985); 19] R. Schabinger and J. D. Wells, Phys. Rev. D 7, (005) hep-ph/050909]; 0] B. Patt and F. Wilczek, hep-ph/ ] A. Ilnicka, T. Robens and T. Stefaniak, Mod. Phys. Lett. A 33, no. 10n11, (018) arxiv: hep-ph]]. 13

15 A The Yukawa texture The charged lepton Yukawa matrix with the order-one coefficients is 3.81 ɛ ɛ ɛ 4 y = 1.36 ɛ ɛ ɛ 3, (9) 0.5 ɛ ɛ ɛ with the κ matrix as κ = GeV v φ (30) B Thermalisation of The H 1, H 1, process The diagrams contributing to the process H 1, H 1, are shown in Figure 8. H 1, H 1, H 1, H 1, H 1, H 1, H 1, H 1, H 1, Figure 8: The diagrams contributing to the H 1, H 1, process. With m < m e m H1,H, the H 1 H 1 cross section is calculated to be σ H1 H 1 = (31) { 1 1 (!)(!)λ 11 (!)λ 1(3!)λ 111 (!)λ ] (!)λ 11 64πE AH1 E BH1 v rel s m 1 s m + (!)(!)λ 11 (!)λ 1(3!)λ 111 (!)λ ] (!)λ 11 16λ ] 1 1 k1 + β 1 ln s m 1 s m s β 1 1 k 1 β 1 ( )] } +(!) 4 8λ4 1 s β1 (1 k 1 ) β 1 (1 k 1 ) ln k1 + β k 1 β 1!, where we have used the following notation λ ijk = λ Hi H j H k, λ i = λ Hi, λ ij = λ Hi H j, (3) 14

16 and k i = m H i s, β i = 1 4m H i, m i = m Hi, (33) s and E AH1 and E BH1 are the initial energies of the scalars. Similarly, the cross section for the H H is calculated to be σ H H = (34) { 1 1 (!)(!)λ (!)λ 1(!)λ 1 (!)λ ] (3!)λ 64πE AH E BH v rel s m 1 s m + (!)(!)λ (!)λ 1(!)λ 1 (!)λ ] (3!)λ 16λ ] 1 k + β ln s m 1 s m s β 1 k β ( )] } +(!) 4 8λ4 s β (1 k ) β (1 k ) ln k + β 1 1 k β!, and for the H 1 H process to be σ H1 H = (35) { 1 1 (!)(1!)λ 1 (!)λ 1(!)λ 11 (!)λ ] (!)λ 1 64πE AH1 E BH v rel s m 1 s m + (!)(1!)λ 1 (!)λ 1(!)λ 11 (!)λ ] ] (!)λ 1 16λ1 λ 1 k1 k + β 1 ln s m 1 s m sβ 1 1 k 1 k β 1 +(!) 4 8λ 1λ ( )] } s β1 (1 k 1 k ) β 1 (1 k 1 k ) ln k1 k + β k 1 k β 1!, where β ij = 1 (m H i + m H j ) s + (m H i m H j ) s. (36) The γl i l i process The process γl i l i proceeds through the diagrams presented in Figure 9. 15

17 γ(q) l i (p ) γ(q) l i (p 1 ) l i (p ) l i (p ) l i (p 1 ) l i (p ) where Figure 9: Diagrams (a) and (b) contributing to the γl i l i process. The spin averaged amplitude squared for this process is: M = M a + M b = M a + M b + M am b, (37) M a = ( κ ii e) (p 1 q) m i (p 1 q p q p 1 p + m i ) + (p 1 q)(p q) ], M b = ( κ ii e) m (p q) i (p 1 q p q p 1 p + m i ) + (p 1 q)(p q) ], M ( κ ii e) am b = m (p 1 q)(p q) i (m i + p 1 p + p 1 q p q) +(m i + p 1 p + p 1 q)(p 1 p + p q m i ) ] (38) The l i l i γ process The process l i l i γ proceeds through the diagrams presented in Figure 10. l i (p 1 ) γ(q) l i (p 1 ) γ(q) l i (p ) l i (k) l i (p ) l i (k) Figure 10: Diagrams (c) and (d) contributing to the l i l i γ process. The spin averaged amplitude squared for this process is: M = M c + M d = M c + M d + M cm d, (39) 16

18 where M c = M d = M cm d = κ ii e m k p κ ii e m k p 1 ] ] 4(k p 1 )(k p ) m (p 1 p ) 4m i m, ] ] 4(k p 1 )(k p ) m (p 1 p ) 4m i m ( κ ii e) m k p ] m k p 1 ] ] 8(k p 1 )(k p ) 4m i m. (40) To calculate the thermally averaged cross section, we approximate the initial energies of the photon and the lepton l i with their average thermal energies, as shown in Eq. (6). Note that the incoming particles have different statistics and therefore different momenta. As a result, the center-of-momentum frame is of no use here. We will calculate the γl i l i cross section in the general co-linear frame where the incoming particles will collide head-on with non-equal momenta. The γl i l i cross section in the general co-linear frame The cross section for the process γ( q)l i ( p 1 ) ( p )l i ( p ) in the general co-linear frame is given by σ γli l i = 1 p dω E 1 E q v rel 16π ( E + E 1 ( p )] 1 1 q ) cos ω M γli li p. (41) where the incoming momenta, p 1 and q, are co-linear while p 1 q. The scattering angle ω is defined to be the angle between p 1 and p. The momentum p is then given by where p = β β 4αγ, (4) α α = 4 ( p 1 q ) cos ω (E 1 + E q ) ] β = 4( p 1 q ) cos ω m i m + E 1 E q + p 1 q ] γ = m i + E 1 E q + p 1 q ] + (m i m ) +m m i p 1 q E 1 E q ] m i E 1 + E q m i + E 1 E q p 1 q ]. (43) 17

19 C The freeze-in production of Consider the to annihilation process AB XY, where A and B are bath particles and the final state particles X and Y contain one or two particles. The Boltzmann equation for the freeze-in production of through this process is given by ṅ + 3Hn = dπ X dπ Y dπ A dπ B (π) δ (4) (p X + p Y p A p B ) (44) ] M X+Y A+B f X f Y (1 ± f A )(1 ± f B ) M A+B X+Y f A f B (1 ± f X )(1 ± f Y ), which could be written as the following one dimensional integral ṅ + 3Hn = T 3π 4 (m A +m B ) dsg A g B s (ma + m B ) ] s (m A m B ) ] 1 s σ AB XY (s)k 1 ( s T ). (45) Most of the freeze-in production occurs at low temperatures where we approximate the annihilation cross section to be σ AB XY (s) σ AB XY ((m A + m B ) ). (46) This will allow a further simplification of the Boltzmann equation, ṅ + 3Hn = T 8π g Ag 4 B (m A m B ) 3/ v rel σ AB XY (s = (m A + m B ) ) ] ds s (m A + m B ) 1 ( ) s K 1, (47) (m A +m B ) s T which can now be written in terms of the yield, y, The yield is then calculated to be ṅ + 3Hn = 1.66 π g s g ρ T 6 45m P L dy dt. (48) y today = 45 m P L g A g B (m A m B ) 3/ ( vrel σ π 6 AB XY s = (ma + m B ) )] (49) Tmax= 1 dt T min =0 g s (T ) ds s (m g (T ρ A + m B ) 1 ( ) s K 1. )T 5 (m A +m B ) s T 18