Cosmological Production of Dark Nuclei
CCosmological Production of Dark Nuclei
Michele Redi and Andrea Tesi
INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, ItalyDepartment of Physics and Astronomy, University of Florence, Italy
Abstract
We study the formation of Dark Matter nuclei in scenarios where DM particles are baryonsof a new confining gauge force. The dark nucleosynthesis is analogous to the formationof light elements in the SM and requires as a first step the formation of dark deuterium.We compute this process from first principles, using the formalism of pion-less effectivetheory for nucleon-nucleon interactions. This controlled effective field theory expansionallows us to systematically compute the cross sections for generic SM representations underthe assumption of shallow bound states. In the context of vector-like confinement modelswe find that, for nucleon masses in the TeV range, baryonic DM made of electro-weakconstituents can form a significant fraction of dark deuterium and a much smaller fractionof dark tritium. Formation of dark nuclei can also lead to monochromatic photon linesin indirect detection. Models with singlets do not undergo nucleosynthesis unless a darkphoton is added to the theory.
E-mail: [email protected], [email protected] a r X i v : . [ h e p - ph ] D ec ontents SU(2) L triplet 14 L triplets . . . . . . . . . . . . . . . . . . 155.2 Production of nuclei with A ≥ A.1 Details on the magnetic transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
B Bound State decays 24
The stability of Dark Matter (DM) on cosmological time scales strongly suggests the existence of newaccidental symmetries in Nature. In a minimalistic approach where DM is a representation of theStandard Model (SM) gauge group, the possibilities that DM is accidentally stable are very limitedand constrained [1]. A different class of models where DM is accidentally stable with a wide varietyof SM quantum numbers is the one where DM is a baryon-like object of some strongly interactingdark sector [2–4], in the framework of vector-like confinement [5]. In these scenarios, the presence ofelementary ‘dark’ fermions with an accidental U(1) symmetry guarantees the stability of the lightestdark baryons, as the stability of protons and nuclei is related to baryon number conservation in theSM.In such scenarios, dark nuclear forces are expected to give rise to dark nuclei. For example in [6, 7]it was shown that bound states with baryon number 2 exist and the absence of a Coulomb barrierimplies that states with large baryon number very likely exist in the spectrum. The possibility thatDM is confined into more complex structures is of obvious theoretical and experimental interest.2n this work we wish to quantitatively study the nucleosynthesis of the dark sector. In the SM,the formation of light elements during Big Bang Nucleosynthesis (BBN) depends upon a few oddcircumstances: for example on the fact that the deuterium binding energy is small and comparable tothe proton-neutron mass difference. We find that even for baryonic DM, the success of nucleosynthesisdepends on the formation of dark deuterium but, contrary to the SM, the different densities and bindingenergy of DM require a precise knowledge of the dark deuterium production cross-section.First, we establish the general features of dark nucleons and nuclei in models with a new confininggauge interaction and fermions in vector-like representations. Under broad assumptions the spectrumis dominated by the nuclear binding energies E B much smaller than the nucleon mass M N and electro-weak effects can be included perturbatively. Importantly the synthesis of nuclei requires release ofenergy which automatically allowed in models with electro-weak constituents where it is always possibleto radiate a photon, NN ( r , S ) (cid:26) D ( r (cid:48) , S (cid:48) ) γ, W, Z The key input to determine the abundance of nuclei is the deuterium cross-section. At first sightits calculation involves strongly coupled nuclear reactions that seem difficult to control. This is not thecase, however, in light of the smallness of E B /M N and a precise computation is possible for shallowbound states. We will get inspiration from effective field theory of nucleon interactions [8], applying thesame techniques to the case of dark nucleons with arbitrary quantum numbers ( r , S ). The cross-sectionfor bound state formation through electric (dipole) and magnetic interactions are found, σv rel (cid:12)(cid:12) electric = K E πα v × παM N (cid:18) M N E B (cid:19) v , σv rel (cid:12)(cid:12) magnetic = K M παv rel × παM N (cid:18) E B M N (cid:19) , where the first factor accounts for possible Sommerfeld enhancement (SE) and K E and K M are grouptheory factors. With these result in hand we can easily compute the abundance of deuterium and heaviernuclei in a given model by solving the relevant Boltzmann equations. For models with electro-weakcharges we find that a fraction of DM is bound in deuterium and heavier elements are not significantlyproduced. Formation of nuclei through photon emission can be tested in indirect detection experimentssuch as FERMI.The present work clarifies how in general the first steps of dark nuclei formation can occur, basedon first principle calculations. In this respect it provides a quantitative input for works that focus onthe asymptotic fusion of large nuclei [9, 10], and also for models where the dark baryon coupled todark photons as in [11], that we reconsider. On a more technical side, this paper extends the analysisof cosmological bound state formation of perturbative bound states [12, 13] to strongly coupled nuclei.The paper is organised as follows. After an introduction to the properties of nuclei in vector-likeconfinement scenarios in section 2, we derive in section 3 the Boltzmann equations for dark deuteriumin a few models. We compute the deuterium formation cross section with non-relativistic effective fieldtheory techniques in section 4 and the appendix. Section 5 discusses the case of dark nuclei made ofSU(2) L triplets, while in section 6 we consider dark nuclei charged under a dark photon. We concludeand outline future directions in section 7. 3 Dark Force
We frame our discussion of DM nuclei within the scenarios of vector-like confinement [5]. The SM isextended with a new non abelian gauge force and fermions charged under the SM and dark group,described by the renormalisable lagrangian, L = L SM − G aµν G a µν g + ¯Ψ( i /D − m Ψ )Ψ . (1)Such framework has special interest for DM as it automatically generates accidentally stable DMcandidates [2]. We will focus on SU(N) DC gauge theories with fermions in the fundamental of darkcolour and masses m Ψ smaller than the confinement scale Λ.Upon confinement the spectrum consists of dark pions and dark baryons with charges under the SMdetermined by their constituents. The theory features an accidental U(1) symmetry, the dark baryonnumber, under which the fermions Ψ transform with the same phase. This symmetry guarantees thestability of the lightest baryon, which for appropriate choices of SM representation can be a viable DMcandidate. The very same dark baryon number also guarantees the exact stability of the lightest statesin each charge sector. This leads to the stability or metastability of nuclei or more in general state ofmatter that carry baryon number. The quantum numbers under the SM are uniquely fixed as gaugeinteractions naturally select the smallest representation as the most bound.As an example we will consider the simplest models with SU(3) DC gauge group: • Ψ is a triplet under a SU(2) L . The lightest baryon V ≡ ΨΨΨ is also a triplet and has spin1/2. The lightest states are pions transforming as a triplet and quintuplet with mass splitting∆ M ∼ α / (4 π )Λ . The triplet is accidentally stable due to G-parity [15], but it can decaythrough dimension-5 operators. • Ψ is a singlet of the SM. In this context some heavier unspecified charge fermions are neededto guarantee thermal contact with the SM but play no other role in the dynamics. The lightestbaryon has spin 3/2 for one flavour of singlets or 1/2 for more flavours.Previous studies focussed on symmetric DM where the relic abundance is generated through thermaldecoupling, which leads to a baryon mass around 100 TeV to reproduce the known DM density. Herewill focus on asymmetric DM as this this maximises the formation of nuclei. This requires a suppressedsymmetric component so naturally M N <
100 TeV. Direct searches at the LHC place a bound of around1 TeV for the mass [16, 17], while the singlets just need to be heavier than about 10 MeV to avoidbound on number of species.
At zero temperature dark nucleons are expected to bind into larger nuclei due to residual stronginteractions. Based on nuclear physics examples, as well as lattice results [18], we consider as typicalbinding energies, 0 . < E B /M N < . . (2) In models with real reps such as fundamental of SO( N ) or adjoint of SU( N ) [14] baryon number is only conservedmodulo 2 so they do not support stable nuclei with A >
1. Models with pseudo-real representations such as fundamentalof Sp( N ) do not have stable baryons.
4n the SM due to the electro-static repulsion, only nuclei with atomic number A (cid:46)
100 are long livedor cosmologically stable. Moreover the presence of bottlenecks associated to the details of the nuclearspectrum implies that only the lightest nuclei are synthesised cosmologically.For accidental DM models with electro-weak constituents the situation is likely different. If the massof the dark baryon is larger than the electro-weak splittings, it is possible to exploit the approximateSU(2) L symmetry to classify nuclei into SM representations at least for the nuclei with small atomicnumber A . Actually, in the limit where we neglect SM interactions the theory has a larger accidentalglobal symmetry SU( N F ) corresponding to the dimension of the lightest nucleon representation. All thestates can thus be classified into multiplets of SU( N F ). Electro-weak interactions break this symmetrysplitting the lightest nucleon multiplet into SM multiplets by an amount,∆ M N ∼ I N ( I N + 1) α π M N (3)where I N is the isospin of the nucleon.The electro-weak splitting of nucleons induces a splitting between the dark nuclei. Since the shiftabove can be larger than the nuclear binding energies, the splitting of nuclei made of different nucleonrepresentation is dominated by the splitting of constituents over a large range of parameters. Oneconsequence is that as in the SM heavier baryons do not participate to nucleosynthesis. Cosmologicallythe bound state formation begins at temperatures of order E B /
20, where E B is the binding energy.Since the heavier nucleons decay into the lighter ones through strong interactions we can neglect thepopulation of heavier nucleons as long as, E B < ∆ M N −→ E B M N < α (4)This is the typical range expected for nuclear binding energies.Thus we can focus on the nuclei made of the lightest SM rep. The nuclei made of the lightestnucleon multiplet can be decomposed into SU(2) L reps split by nuclear binding energies ∆ E NB . Theelectro-weak correction to the binding energy is given by,∆ E WB ∼ λ α R nucleus (5)where λ = I N ( I N + 1) − I B ( I B + 1) / /a = √ M N E B ,see section 4. It follows that the nuclear binding energies dominate for E B /M N > − . Electro-weakcorrections can be included perturbatively in the relevant region of parameters, and for R nucleus < M − Z the SM gauge fields can be treated as massless.For large A a multitude of SM reps exist with isospin up to AI where I is the nucleon isospin.Of these the smaller representations are attractive making the nuclei more bound while the largestrepresentation have a Coulombian energy that scales as AI ( AI + 1) that unbind nuclei of arbitrarilylarge charges. In light of this, the valley of stability of dark nuclei will likely extend to very large to A , at least for small electro-weak charges.Only the lightest baryon in each baryonic number sub-sector will be stable so that at late timesall baryons produced cosmologically decay to a neutral state with isospin 0 or 1/2. This process iscontrolled by the decay rates among and within isospin multiplets induced by electro-weak interactions,analogously to de-excitation of hydrogen atom. In particular,5 Each isospin multiplet can decay to states with smaller isospin. Since we focus on the s -wavebound states the rate is dominated by emission of a photon through magnetic dipoles with∆ S = 1. As we show in appendix B the rate can be computed model independently as,Γ ≈ α ( E B − E B ) M N (cid:112) E B E B (6)where B , are the binding energies of two bound states with ∆ I = ∆ S = 1. • The splitting within each baryonic electro-weak multiplet is given by [1], ∆ M N = Q α M W sin θ W L triplet gives ∆ M N = 165 MeV. Since charged and neutral componentsremain in equilibrium, the abundance of charged nuclei is then approximately n V + ≈ n V for T > ∆ M .In what follows we will study the cosmological synthesis of dark nuclei. Cosmologically states withlarge A could be produced through fusion processes via aggregation of heavy elements. In [9, 11] it wasargued that at least for light DM the dark synthesis is very efficient and one ends up with a distributionof large nuclear states. These studies overlook the first step of formation of nuclei with baryon number2 (dark deuterium) that cannot take place through simple fusion processes but requires some energy toemitted. As we will show the deuterium abundance is often suppressed leading to a small abundanceof larger nuclei. As in the SM, we assume a separation of scales between the nucleon masses M N and the nuclearbinding energies E B . In this case the treatment of dark nucleosynthesis becomes relatively simple. Attemperatures below E B , when nuclear reactions can form nuclei, the nucleons are non-relativistic andalready decoupled from the SM plasma. The yield of dark matter number is then set by Y DM = n DM s = 4 . × − (cid:18) TeV M N (cid:19) , (8)where n DM is the DM number density and s = (2 π / g (cid:63) T the entropy of the plasma and g (cid:63) therelativistic degrees of freedom of the plasma at a temperature T .We will assume for simplicity that DM is asymmetric, since this maximises the production of boundstates. Given that thermal abundance indicates M N ∼
100 TeV we assume the DM mass to be belowthis value. This is also necessary to produce a significant fraction of nuclei.Dark BBN takes place during a period where the dark baryon number is conserved, that is1 = (cid:88) A, { i } AY A i Y DM ≡ (cid:88) A, { i } X A i , (9) The size of the nuclei scales as A / /M N , therefore they can be treated as elementary as long as the size is smallerthan 1 /M W . X A i = AY A i /Y DM of a nucleus with atomic number A i .Practically, we are only tracking how the total DM yield in eq. (8) gets redistributed among differentnuclear species.A successful nucleosynthesis of states with A (cid:29) • SM charged constituents, with production N + N → D + X , with X = W, Z, γ . • SM neutral constituents, with production N + N + N → D + N .In general bound state formation requires some energy to be released. We do not consider thepossibility of emitting a pion of the strong sector. At first sight this process could be favoured by thestrong coupling to nucleons but it is likely forbidden kinematically. Indeed, based on nuclear physicsexamples [18], we expect the following scaling (cid:112) M N E B ∼ . m π . (10)This is particularly significant for models with singlets where no other light states exist. When the constituents of DM have electro-weak charges, direct searches at LHC imply that the scaleof new states must be larger than about 1 TeV [16, 17]. For this value of the mass the numericaldensity in eq.(8) is much smaller than the yield of baryonic matter, suggesting that the formation ofbound states less likely than in the SM.We consider the first step of formation of dark deuteron N + N → D + X , where X stands for anelectro-weak gauge boson γ/Z/W in equilibrium with the SM bath. For simplicity we assume that noother channels contribute, so that the Boltzmann equation for D takes the form˙ n D + 3 Hn D = (cid:104) σ D v (cid:105) (cid:20) n N − ( n eq N ) n eq D n D (cid:21) , (11)where n B and n D are the numerical densities of dark baryons and dark deuterium and we assumeradiation domination. During radiation domination, the Hubble parameter is H = (cid:114) g (cid:63) ( T ) π T M Pl , M Pl ≡ . × GeV . (12)The second term in the square brackets of eq. (11) becomes exponentially suppressed as e − E B /T , whenthe temperature drops below the binding energy. At this stage, dark deuterium can be formed, sothat it is convenient to introduce a new time variable z ≡ B/T . In terms of the mass fractions, theBoltzmann equation can be cast in the following form, dX D dz = 2 c √ g (cid:63) M Pl E B (cid:104) σ D v (cid:105) z Y DM (cid:20) (1 − X D ) − β (cid:0) g N g D g (cid:63) (cid:1)(cid:0) z / e − z Y DM (cid:1)(cid:0) M N E B (cid:1) / X D (cid:21) (13) We will focus mostly on electro-weak constituents that are more naturally compatible with direct detection bounds.See however [19] for counter-examples. c = 1 . β = 45 / (16 π / ) and g ∗ is the effective number of degrees of freedom. Only at atemperature significantly lower than the binding energy the bound state can be produced withoutbeing immediately dissociated. Given the smallness of Y DM , this happens at values of z f ≈
20, whenthe coefficient of the term linear in X D inside the square brackets becomes of O(1). At this time wehave the most efficient stage of deuterium synthesis starting with boundary condition X D ( z f ) ≈ X D approaches a constant value that is linearly sensitive to the product of the bindingenergy and the cross-section, given by X D − X D = 2 c √ g (cid:63) M Pl E B Y DM (cid:90) ∞ z f dz (cid:104) σ D v (cid:105) z . (14)Away from saturation, X D (cid:28)
1, we have the following for an s -wave cross section, X D = 5% (cid:18) M (cid:19) (cid:18) E B /M N . (cid:19)(cid:18) (cid:104) σ D v (cid:105) α/M N (cid:19)(cid:18) g (cid:63) . (cid:19) / (cid:18) z f (cid:19) . (15)Contrary to the SM the production of deuterium is far from saturation for heavy dark matter. Forthis reason the actual abundance depends on the precise value of the cross-section that we will computein section 4. When the fundamental fermions are SM singlets, the hadrons and mesons of the dark sector can bemuch lighter than the electroweak or even QCD scale, without occurring in constraints from LHC directsearches. As a consequence DM numerical densities comparable or even larger than visible matter (seeeq. (8)) become possible. Naively this is the most favourable situation for nucleosynthesis and evenvery large nuclei could be formed as advocated in [9].Nevertheless in this section we would like to argue that, in presence of SM singlets, nucleosynthesisis very unlikely to take place unless extra light degrees of freedom such as a dark photon [11, 20] ora scalar [21–24] are included (see also realisations in scenarios of mirror world [25]). In absence oflight fields external to the strong sector, such as a dark photon or light pions (see the discussion in theprevious section), the first step of nucleosynthesis cannot occur as a 2 → → N + N + N ↔ D + N reactions. Even at temperatures below E B , when the production reaction is theonly one that can occur, the fusion is suppressed as compared to the previous case by an additionalpower of Y DM . The Boltzmann equation in this case takes the form˙ n D + 3 Hn D = (cid:104) σ → v (cid:105) (cid:18) n N − ( n eq N ) n eq D n D n N (cid:19) . (16)where we have introduced a generalised 3 → −
5. The thermallyaveraged 3 → (cid:104) σ → v (cid:105) = n eq D ( n eq N ) (cid:104) σ → v (cid:105) = (cid:90) d p (2 π ) E d p (2 π ) E d p (2 π ) E e − E E E T |M NNN → DN | (17)For the analog of s -wave processes at low energy σv goes to a constant. Following [26] we estimate (cid:104) σ → v (cid:105) ∼ (4 π ) N M N . (18)8ntroducing the deuterium baryonic fraction the Boltzmann equation (16) can be written as dX D dz = a g / (cid:63) M Pl E B (cid:104) σ → v (cid:105) z Y (cid:34) (1 − X D ) − β (cid:0) g N g D g (cid:63) (cid:1)(cid:0) z / e − z Y DM (cid:1)(cid:0) M N E B (cid:1) / X D (1 − X D )2 (cid:35) (19)where a = 1 .
16 is a numerical coefficient arising from the evaluation of H and s . We can now comparethe third line of the above equation with eq. (13) that sets the abundance of dark deuterium from 2 → z (cid:38) z f the source term decouples as fast as z − , is suppressedby higher powers of the binding energy and especially by one more power of Y DM .We then obtain the estimate X D ∼ − (cid:18) g (cid:63) (cid:19) / (cid:18) E B /M N . (cid:19) (cid:18) z f (cid:19) (cid:18) (cid:104) σ → v (cid:105) /M N (cid:19)(cid:18) GeV M N (cid:19) , (20)which is utterly negligible in the relevant range of parameters. In this section we explain how the cross-section for formation of nuclei can be computed from firstprinciples exploiting universal properties of short range nuclear interactions. The main point is that,for shallow bound states such as nuclei where E B (cid:28) M N , it possible to write a general effective theoryof nucleons. This effective field theory (EFT) reproduces the effective range expansion of quantummechanics and allows us to compute the properties of nuclei such as their production cross-section, see[27] for a review. Here, we quickly outline the formalism and apply to the formation cross-section ofdark deuterium. The production of nuclei with A = 2 is a process entirely analogous to the deuteron formation in theSM, pn → dγ . This can be calculated in quantum mechanics with appropriate potentials, but, asnoticed long ago [28, 29], it does not depend much on the details of the potentials used, as long as theyare short range. As emphasised in [30, 31], the generality of this phenomenon is immediately capturedby the π -less effective theory of non-relativistic nucleons [8] that we briefly review. We refer the readerto the appendix and Refs. for more details. Since the energy scale relevant for nuclei formation is much below the pion mass it is useful todescribe nucleons with a non relativistic lagrangian where the pions are integrated out, the π -less EFT[8]. Such theory is extremely simple because it only contains contact interactions among nucleons andcouplings to SM gauge fields. Generalising the results of the Refs. above, the nucleons, in a genericisospin representation r , are described by the effective lagrangian L = N † (cid:32) iD t + (cid:126)D M N + D t M N (cid:33) N + L + κM N g N † J a ( (cid:126)σ · (cid:126)B a ) N , (21)where the covariant derivative D µ = ∂ µ − ig A aµ J a contains the minimal coupling to SM gauge fieldsand we have included 4-nucleons interaction and a magnetic dipole interaction where J a is generator In [32–34] this formalism was applied to Wino scattering and annihilation. In the case where the strong sector violates CP through a θ angle an electric dipole is also present producing similareffects for deuterium formation. We will neglect this term.
9f SU(2) L in the nucleon rep. The coefficient κ is expected to be of order unity (in the SM the isovectornuclear magnetic moment is κ V = 2 . L include only operators without derivatives,that can be decomposed into spin and isospin channels as L = − (cid:88) r ,S C r ,S N [CG M r ⊗ P iS ] N ) † ( N [CG M r ⊗ P iS ] N ) , (22)where the matrices CG r and P S act on the isospin and spin space respectively. Remarkably, the lagrangian above also describes the non-perturbative bound state allowing tocompute for example the production cross-section. A quick way to derive the main result is thefollowing. The non-relativistic amplitude for the elastic scattering of two nucleon has the general formin each isospin/spin channel r , S , A r ,S = 4 πM N p cot δ r ,S − ip (24)where δ r ,S is the phase shift and p = √ EM N is the nucleon momentum in the center of mass frame.For s − wave scattering in the low velocity regime one can show that p cot δ r ,S = − /a r ,S + O ( p ), where a r ,S is the scattering length. This is know as the effective range expansion. When this is large andpositive it follows that the amplitude has a pole at negative energy. From this we can recover thegeneral relation between the scattering length of the binding energy of shallow bound states,1 a r ,S ≈ (cid:112) M N E B , (25)where E B is the binding energy of lightest s -wave bound state.The coefficients of the 4-Fermi interactions in eq. (22) must be fixed to reproduce the effectiverange expansion of nucleon nucleon elastic scattering. As shown in the appendix, to leading order inthe derivative expansion but to all order in the scattering length a r ,S , on finds C r ,S = 4 πM N a r ,S . (26)Once this matching has been performed, the above lagrangian can be used to compute other processes,such as the production of deuterium (see [30] and refs for the SM case).Indeed, the amplitude above determines also the coupling of two nucleons and deuterium as g NND = Res E = − E B [ A ] = 8 πγM (27)where γ = 1 /a . This effective coupling can then be used to compute the interaction between 2-nucleonsand the deuterium. For SU(2) the matrices CG can be identified with the Clebsch-Gordan coefficients, while the explicit expression ofthe spin projectors onto the spin 0 and 1 states are P = σ √ , P i = σ σ i √ . (23)The labels r and S identify the SU(2) L and spin representations, while M and i are the indices of such representations. N N ) scattering to infer all thequantities needed to perform the leading order calculation of the deuteron formation rate. Since thisprocess occurs cosmologically at energy much below the mass of the pions (see also the discussion insection 2.1), it is very reasonable to use an effective field theory of non-relativistic nucleons (with SMquantum numbers) without the pions.
The effective field theory in eq. (21) allows us to compute the short distance cross section for theprocess N + N → D + V a in terms of the binding energies and scattering lengths alone.The nucleus can be formed through emission of a SM gauge boson either through the electriccoupling (minimal interaction) or through the magnetic dipole interaction in eq. (21). For κ ∼
1, asexpected for strongly coupled baryonic states, the two processes can have similar size. Importantly,different selection rules apply to electric and magnetic transition so that ∆ L = 1 for the first and∆ S = 1 for the second. This implies a different velocity scaling of the cross-sections.The amplitude for the formation of bound state can be simply computed with Feynman diagrams ofthe non-relativistic effective theory (21) using eq. (21) for the overlap of final state with the deuteron. ⊗ A = ( r , s ) ( r (cid:48) , s (cid:48) ) ε λ a (28)The only subtlety arise for large scattering length of the initial state where an extra long distancecontribution must be taken into account enhancing the tree level magnetic cross-section. This effect isdiscussed in detail in appendix A. The amplitudes for bound formation can be conveniently decomposedin the basis of total spin and isospin of initial and final states ( N N, D ) using the projectors of eq. (22).In the limit v rel (cid:28) A mag (( N N ) M,i → D M (cid:48) ,i (cid:48) + V a ) = 2 g κM N | (cid:126)k | g NND r (cid:48) (1 − a r γ r (cid:48) )( (cid:126)k × (cid:126)ε ( λ a ) ) i + i (cid:48) C aMM (cid:48) J , (29) A ele (( N N ) M,i → D M (cid:48) ,i (cid:48) + V a ) = 2 g M N | (cid:126)k | g NND r (cid:48) (cid:126)p · (cid:126)ε ( λ a ) δ ii (cid:48) C aMM (cid:48) J . (30) M, M (cid:48) are the indices of the isospin representations, while i, i (cid:48) are the indices of the total spin repre-sentations of initial and final states, and so these expressions should be read taking into account theselection rule of the magnetic and electric transition. The group theory factor is [13] C aMM (cid:48) J = 12 Tr[CG M (cid:48) r (cid:48) { CG M r , J a } ] . (31) For Majorana bound states a factor √ We neglect here the effect of non-abelian interactions [13]. This is justified if the nuclei are dominated by the stronginteractions so that their size is smaller than the Coulombian Bohr radius a − = λα M N / σv rel = | (cid:126)k | π (cid:90) d Ω k |A| , E (cid:126)k = E B + M N v . (32)This gives the following magnetic and electric cross-sections: Magnetic cross-section
The averaged cross-section for the production of an s -wave bound state with energy E B and isospinquantum number ( I (cid:48) , M (cid:48) ) through the emission of a (massive) vector boson V a from an initial statewith ( I, M ) at low velocity is given by,( σv rel ) mag aMM (cid:48) = κ g N σ (cid:115) − M a E B (cid:18) E B M N (cid:19) (1 − a r γ r (cid:48) ) | C aMM (cid:48) J | , σ ≡ παM N . (33)Here g N = 2(4) d R is the number of degrees of freedom of the nucleon initial state for Majorana (Dirac)particles. If the initial state supports a weakly bound state with a i = (cid:112) E B i M otherwise a i is negativeand can be large. For example in the SM the second term is a i γ f ≈ σv rel is constant as this correspond to a an s -wave capture with ∆ L = 0 and∆ S = 1. The presence of a coherent contribution from the initial state scattering length can beunderstood by noticing that being a s -wave process, the initial state can be in principle have anunnaturally large coefficient in eq. (21) that need to be kept into account to all orders. Electric cross-section (as in dipole approximation).
The averaged cross-section for the formation of an s-wave shallow bound state through dipole emissionof a photon is found to be,( σv rel ) ele aMM (cid:48) = 2 S + 1 g N σ v (cid:115) − M a E B (cid:114) M N E B (cid:18) M a E B (cid:19) | C aMM (cid:48) J | (34)The velocity suppression follows from the fact that in dipole approximation ∆ L = 1 so that an s -wavebound state is produce from a p -wave. Note that the formula above differs by a numerical factor fromthe cross-section for the formation of Coulombian bound state [13]. This is because the energy levelsof 1 /r potentials cannot be treated as shallow bound states.From the formulae in each isospin channel for electric and magnetic transitions we can recoverthe component cross-section, relevant for example for indirect detection in 5.3 using the appropriateClebsch-Gordan coefficients. Let us note that the formulae above are actually general and also applyto bound state made of different representations and different global symmetry group. The discussion so far has neglected the long distance Sommerfeld effect due to electro-weak interactionson the initial state. In a given SU(2) L channel, the cross section is multiplied by the corresponding12actor depending on whether we deal with s or p -wave initial states, see for example [35]. For masslessmediators one finds,SE s − wave = 2 πα eff /v rel − e − πα eff /v rel ≈ πα eff v rel , (35)SE p − wave = (cid:34) (cid:18) α eff v rel (cid:19) (cid:35) πα eff /v rel − e − πα eff /v rel ≈ π (cid:18) α eff v rel (cid:19) . (36)Here α eff is the effective strength of the electro-weak forces in a given channel. Importantly, taking intoaccount this effect, both the magnetic and electric transition rates have the same scaling with velocity, σv rel ∝ /v . Note that such enhancement of electric transitions is not present for deuteron in the SMso that the magnetic transition dominates at very low velocities. This can be different for dark nuclei.The approximate scalings above are accurate for v rel (cid:46) α . The effectiveness of the SE, then cruciallydepends on the typical velocity during the nucleosynthesis. Dark deuterium forms at T ∼ E B /z f where z f ∼
20, which implies v rel ∼ (cid:112) E B /M N / L symmetric, as in this case the relevant coupling is α ,instead of α em .Let us now discuss domain of validity of the massless approximations. From the point of view ofthe initial state particles, the mediator masses can be neglected as long as the de-Broglie wave-length issmaller than the range of the electro-weak interactions M − W . Therefore Sommerfeld effects are maximalif the above two conditions are met M W M N (cid:46) v rel (cid:46) α . (37)For v rel (cid:46) M W /M N the vector boson masses must be taken into account. These has 2 effects, the firstto freeze the enhancement to a constant value corresponding to the critical velocity,SE → max (cid:20) πα eff M N M W , (cid:21) (38)This approximation gives results comparable to the analytic formulas derived using the Hulthen poten-tial [13]. In addition the mediator mass produces peaks in the cross-section. As well known the resonantbehaviour originates from bound states of zero energy supported by the potential at the critical mass.Around the peak the SE has the model independent form [36],SE ∼ V M N v / | E B | (39)where V is the typical energy and E B the energy of the bound state close to threshold in the initialstate.We also note that in the limit of large scattering length of the initial state the second term in (30)can be interpreted as SE due to the strong interactions. Indeed this is large when the initial statesupports a bound state with energy E B (cid:28) M N . Using 1 /a = √ E B M N and extending the computationof bound state formation to finite velocity of nucleons one can check that the formula for bound stateproduction through magnetic interaction ( s -wave) reduces to eq. (39) where E B is the energy of thebound state in the initial state while V is the binding energy of the produced bound state. Thesubleading terms in eq. (33) is associated to the failure of factorisation between short and long distanceeffects. 13ame r S λ
Constituents D V VD V VD V VT V D T a V D T b V D T c V D T a V D T b V D T V D r ↔ r (cid:48) (cid:80) aMM (cid:48) | C aMM (cid:48) J | C J C +01 J ↔ (cid:112) / (cid:112) / ↔ (cid:112) / − (cid:112) / Table 1.
On the left quantum numbers of nuclear bound states (deuterium and tritium) for nucleons SU(2) L triplets. On the right group theory factors for the transition from an initial state with isospin r to a bound statewith isospin r (cid:48) . SU(2) L triplet In this section we compute explicitly the production of deuterium in the V model where the constituentsare triplet of SU(2) L . In this scenario, in the limit of vanishing SM couplings, the models enjoys anSU(3) F flavour symmetry and the lightest baryon multiplet is an octet. This decomposes under SU(2) L as = = V + , (40)where with abuse of notation we named the triplet nucleon V . SM gauge interactions split the twomultiplets as in eq. (3). The triplet is expected to be the lightest state in light of the smaller weakcharge. Since at the temperatures relevant for bound state formation the abundance of the quintupletis exponentially suppressed we can focus on the nucleon V .In absence of electro-weak interaction he nuclei with baryon number 2 belong to the product of twooctet × = + S + A + + + S , where S ( A ) refers to the symmetry of the isospin wave-function. Lattice studies indicate that all these representation actually form bound states [18] and alsoprovide the binding energies in different channels. As expected in the flavor symmetric limit the singlet(corresponding to the H − baryon in QCD) is the most bound. Following the discussion of section 2.1,we can work in a limit where we can classify the bound states according to SU(2) L representations,while neglecting the baryon in the whose abundance is Boltzmann suppressed. Therefore the darkdeuterium of this model belongs just to the product V × V = S + A + S . (41)Anti-symmetry of the full wave-function implies that singlet and quintuplet of SU(2) L ( D and D )are spin-0 while isospin triplet are spin-1 ( D ) (for s-wave bound states). The triplet and quintupletdeuterons are branches of A and S respectively so they are heavier than the singlet, with splittingdominated by the strong interactions.The classification can be generalised to larger nuclei. The dark tritium made of 3 triplets hasquantum numbers, ( D + D + D ) × V = S + 3 × A + 2 × S + (42)14 - - - -
110 10 - - - - = B / T 〈 σ v 〉 / σ 〈 σ v 〉 / σ Triplet of SU ( ) L E B / M N = E B / E B = M N = ( → ) M ( → ) E ( → ) M ( → ) E
20 40 60 80 10010 - - - -
110 10 - - - - = B / T 〈 σ v 〉 / σ 〈 σ v 〉 / σ Triplet of SU ( ) L E B / M N = E B / E B = M N = ( → ) M ( → ) E ( → ) M ( → ) E Figure 1.
Thermally averaged cross sections as a function of the temperature z = E B /T . The rates aredecomposed per channel and per type (magnetic M , and electric E ). where the symmetry of the wave-function determines the spin. We can estimate the electro-weakbinding energy by adding a nucleon to the deuterium and taking into account the reduced mass.We summarise the bound states up to dark Tritium in table 1. SU(2) L triplets To compute the abundance of deuterium we assume that the initial state is SU(2) L symmetric, i.e. weneglect the mass difference between V ± and V . Using (33) and (34) the cross-section for deuteriumformation through emission of
W, Z, γ , averaged over initial states, is approximately given by,( σv rel ) a → = σ a × (cid:115) − M a E B (cid:34) SE s κ (cid:18) E B M N (cid:19) (1 − a γ ) + SE p (cid:115) M N E B (cid:32) M a E B (cid:33) v (cid:35) ( σv rel ) a → = σ a × (cid:115) − M a E B (cid:34) SE s κ (cid:18) E B M N (cid:19) (1 − a γ ) + 14 SE p (cid:115) M N E B (cid:32) M a E B (cid:33) v (cid:35) ( σv rel ) a → = σ a × · (cid:115) − M a E B (cid:34) SE s κ (cid:18) E B M N (cid:19) (1 − a γ ) + SE p (cid:115) M N E B (cid:32) M a E B (cid:33) v (cid:35) ( σv rel ) a → = σ a × · (cid:115) − M a E B (cid:34) SE s κ (cid:18) E B M N (cid:19) (1 − a γ ) + SE p (cid:115) M N E B (cid:32) M a E B (cid:33) v (cid:35) (43)where σ a = πα a M N , α W,Z,γ = α × [2 , c W , s W ] (44)The first term in bracket corresponds to magnetic ∆ S = 1 transitions and the second the electricone ∆ L = 1. The triplet can be produced either from a singlet or quintuplet channels. The scatteringlength are given by 1 /a i ≈ (cid:112) M N E B i . If a state is unbound the formulas above still apply with anegative scattering length. This is the case of deuterium in the SM where nn is weakly unboundproducing a large scattering length.In Fig. 1 we show the numerical values of the electric and magnetic cross-sections 1 ↔ σ = πα /M for various choices of the binding energies. Due to the SE described in section 4.3 thep-wave electric cross-section can be larger than the magnetic one.15
000 4000 6000 8000 10 00010 - - - - - - M N [ GeV ] X D E B / M N = E B / M N = E B / M N = E B / E B = ( ) L w / Sommerfeld ( solid ) w / o Sommerfeld ( dashed ) - - - - - - M N [ GeV ] X D E B / M N = E B / M N = E B / M N = E B / E B = ( ) L w / Sommerfeld ( solid ) w / o Sommerfeld ( dashed ) Figure 2.
Dark deuterium mass fraction X D as a function of the nucleon mass M N for several choices of thebinding energies (with and without the Sommerfeld effect). To compute the abundance of deuterium in principle one should write a different Boltzmann equa-tion for each bound state. We can however simplify the problem by noting that transition betweendifferent bound states are fast so that they are in equilibrium among them (see section 2 and appendixB). This implies that n D i /n D j = n eq D i /n eq D j . The abundance of deuterium is then determined by eq.(13) with the effective cross-section and degrees of freedom,( σv rel ) eff = (cid:88) i ( σv rel ) i , g eff D ( T ) = (cid:88) i g D i exp (cid:20) − E B − E B i T (cid:21) . (45)In figure 2 we present the total mass fraction of deuterium X D as a function of the M N for differentchoices of binding energies of the singlet ( D ) and triplet ( D ) deuterium. We assume for simplicitythat D does not play a role even though being the least bound isotope with baryon number 2 itcould enhance the production of D through a large scattering length. Solid lines correspond to theabundance including electro-weak SE effects while dashed lines are obtained with the short distancecross-sections. The SE enhancement has significant impact especially because it eliminates the velocitysuppression of electric transitions. For E B < M W only the photon can be emitted with smaller cross-section in light of the electric coupling and multiplicity. The transition between SU(2) L symmetricemission and photon emission originates the features in the plot.Differently from the SM most of the baryons can form deuterium only for large binding energies.An order 1 fraction of deuterium can only be obtained for TeV masses, around the experimentalcollider bound. The large mass scale associated with DM with electroweak charges is the principalobstruction to convert into deuterium and then heavier nuclei an O(1) fraction of DM. Notice thatthere is no reduction in the mass fraction of heavy nuclei caused by the reduction of V ± from the decay V ± → π ± V , which happens only T <
100 MeV. On the contrary in the SM the main limitation toform deuterium is the neutron decay. A ≥ The formation of dark tritium T is determined schematically by the following reactions D : V + V (cid:29) D + W (46) T : D + V (cid:29) T + W , D + D (cid:29) T + V , (47)16here we allow for weak and strong T formation processes, the latter without the emission of SMradiation. As for the deuterium, dark tritium is produced via exothermic reactions when E T − E D > E T − E D > E T − E D and E T − E D the dissociation of dark tritium is exponentially suppressed. The mostfavourable situation arises, as in the SM, for E T − E D > E D , such that at the time of deuteriumformation, the dissociation of tritium is already ineffective.These assumptions greatly simplify the discussion and they allow us to estimate the upper boundof dark tritium abundance. The Boltzmann equations including deuterium and tritium are given by X (cid:48) D ( z ) = 2 z z (cid:20) (1 − X D − X T ) − β ( M N E D ) / z / e − z g eff D Y DM X D − b − X D − X T ) X D − b X D (cid:21) , (48) X (cid:48) T ( z ) = 3 z z (cid:20) b − X D − X T ) X D + b X D (cid:21) . (49)Where we have introduced the following notation z ≡ c √ g ∗ M Pl E D Y DM (cid:104) σ D v (cid:105) eff , b ≡ (cid:104) σ T v (cid:105) eff (cid:104) σ D v (cid:105) eff , b ≡ (cid:104) σ T v (cid:105) strongeff (cid:104) σ D v (cid:105) eff . (50)In deriving the above Boltzmann equations we have written effective production rates including theeffect of nearby bound states with the same baryon number. In particular, (cid:104) σ D v (cid:105) eff is defined ineq. (45) and the reactions for Tritium are defined similarly, although we would like to distinguish theweak (cid:104) σ T v (cid:105) eff from the strong (cid:104) σ T v (cid:105) strongeff process. This inclusive approach allows us in principle totake into account all possible bound states of Table 1.When the dissociation rate of deuterium, D + W → V + V , becomes exponentially suppressed, wesee that the terms proportional to b and b tend to transfer a fraction of X D to X T with an overalldecoupling as fast as 1 /z (neglecting possible enhancements from Sommerfeld effect). As expected thestrong fusion reactions ( b -term) have smaller rates per deuteron since they are proportional to X D ,partially reducing the increase in the hard rate, b /b ≈ /α .Knowing the reaction rates one can simply solve numerically the above set of equations, howeverit is interesting to analyse it in the limit of small nuclei mass fractions (i.e. X D (cid:28) z ≈ O (1), therefore the non-trivial evolution happens when the overall rateis very small z /z (cid:28)
1. By contrast, in the BBN we have z ≈ , which would allow for a fast fusionof heavier nuclei if bottlenecks for the binding energy were absent [37]. The formation of dark Tritiumis then further suppressed and we have X D ≈ z z f , X T ≈ X D b + 18 X D b . (51)These expression are accurate as long as one can neglect the loss terms in the Boltzmann equation for X D and at leading order in b and b . This is reliable as long as X D (cid:28) min[1 /b , / √ b ]. The possibility of forming bound states is relevant for indirect detection as it would lead to the emissionof monochromatic photons of energy equal to the binding energy of the nucleus. This possibility isof great interest also because asymmetric scenarios typically do not produce indirect detection signals17ince the DM cannot annihilate into SM particles. It is also independent on whether dark nuclei aresynthesised cosmologically. We leave a detailed study to [38] and here outline the main results.In the V model at late times DM is made of the neutral component V of the nucleons plus a modeldependent population of deuterium, also in the neutral component. We will neglect the deuteriumpopulation for the purpose of this section. If the nuclear binding energy of deuterium is larger than M W one can form the triplet deuterium via the tree-level process V V → D ± W ∓ . The magnetictransition (33) including long distance nuclear effects (see below) gives (cid:2) σ V V → D ± W ∓ v rel (cid:3) hard = πα M N × (cid:115) − M W E B κ (cid:18) E B M N (cid:19) (cid:20) (1 − a γ ) + 12 (1 − a γ ) (cid:21) , (52)where we neglected the electric transition which is velocity suppressed in absence of SE. For E B < M W the W is off-shell leading to a suppressed cross-section.Another contribution arises from the SE due to SU(2) L gauge interactions and nuclear forces.Thanks to this effect, two neutral nucleons can form deuterium (in neutral component) through theemission of a photon. Physically this is possible because | V V (cid:105) (cid:96)S is not a mass eigenstate and canoscillate into | V + V − (cid:105) (cid:96)S . In the symmetric limit the mixing can be extracted from the Clebsch-Gordancoefficients, | V + V − (cid:105) = 1 √ | (cid:105) + 1 √ | (cid:105) + 1 √ | (cid:105) , | V + V (cid:105) = 1 √ | (cid:105) + 1 √ | (cid:105) , | V V (cid:105) = − √ | (cid:105) + (cid:114) | (cid:105) . (53) The following processes are then possible | V V (cid:105) sS =0 → | V − V + (cid:105) sS =0 → D + γ , (54) | V V (cid:105) pS =1 → | V − V + (cid:105) pS =1 → D + γ , (55)such that we have formation of neutral D from an initial state | V V (cid:105) in the singlet or quintuplet ofSU(2) L , either from p -wave spin-1 through an electric transition or from s -wave spin-0 via a magnetictransition. Indirect signals from Sommerfeld of
SU(2) L gauge interactions At the low velocities relevantfor indirect detection the electro-weak symmetry breaking effects can be important and a numericalsolution is required, see [39, 40]. For s -wave the SE due to electro-weak interactions is identical to theone of Wino DM χ studied widely studied in the literature, see [41]. In particular we are interestedin the long distance effects that allow the neutral Winos state χ χ to oscillate into χ + χ − . We defineas SE → + − the corresponding Sommerfeld factor of the Wino that can be derived from the followingpotential V S =0 Q =0 = (cid:18) + 0 − M − A −√ B −√ B (cid:19) , (56)where A = α em /r + α c e − M Z r /r , B = α e − M W r /r and ∆ M is the mass splitting produced byelectroweak symmetry breaking, equal to ∆ M = 165 MeV.Noticing that in our case we have the same SE of Wino DM, so that we can exploit the calculationperformed for the Wino and apply it to the dark deuterium. Therefore, the indirect detection signalcan be simply calculated as σ V V → D + γ = SE → + − (cid:2) σ V + V − → D + γ (cid:3) hard . (57)18
000 1500 2000 2500 300010 - - - - - - - - - - - - M N [ GeV ] 〈 σ D + γ v 〉 [ c m s - ] Triplet of SU ( ) L E B / E B = E B / M N = a γ = - LAT
NFW profile [ R4 ] - - - - - - - - - - - - M N [ GeV ] 〈 σ D + γ v 〉 [ c m s - ] Triplet of SU ( ) L E B / E B = E B / M N = a γ = - LAT
NFW profile [ R4 ] Figure 3.
Indirect detection bound from emission of photon lines of energy E D through magnetic coupling( κ = 1). Exclusion limits from FERMI are obtained by recasting the results in [43] to account for the differentratio of photon energy and DM mass. SE for the triplet has been extracted from [41]. The short-distance contribution to the cross-section of the charged components can be computed using(33) and (53), (cid:2) σ V + V − → D + γ v rel (cid:3) hard = κ πα em M N (cid:18) E B M N (cid:19) (cid:20) (1 − (cid:112) E B /E B ) + 12 (1 − a γ ) (cid:21) . (58)The annihilation rate of Winos into photon pairs, σ χ χ → γγ + γZ , has the property that both the γγ and γZ final states are reached from χ χ with the same SE → + − . This allows us to provide analternative form for eq. (57) in terms of Winos cross-sections [39]SE → + − = [ σ χ χ → γγ + γZ v rel ] full (cid:2) σ χ + χ − → γγ + γZ v rel (cid:3) hard , [ σ χ + χ − → γγ + γZ v rel ] hard = πα em α M N . (59)Due to SE the effective cross-section has peaks whose location is identical to the one of the Wino.The first peak appears for M N ≈ . E B > M N / Indirect signals from strong interactions
Beside the standard SE due to electro-weak interac-tions also the strong interactions can enhance the cross-section when the initial channel supports abound state at threshold. This effect is captured by the scattering length in eq. (33). We can computethe cross-section relevant for indirect detection in the SU(2) L symmetric limit. Since the quintuplet isexpected to be less attractive or even weakly repulsive the largest effect will be associated to the quintu-plet initial state. Using the Clebsch-Gordan coefficients of eq. (53) and neglecting further enhancementfrom electro-weak interactions we find, σ V V → D + γ v rel ≈ κ πα em M N (cid:18) E B M N (cid:19) × M N E B /a + M N v / . (60)19 - - M N [ GeV ] X D E B / M N = E B / M N = E B / M N = E B / E B = α D = -
200 400 600 800 100010 - - - - - - - - M N [ GeV ] X D E B / M N = E B / M N = ( w / magnetic ) dashed ( w / o magnetic ) E B / E B = α D = - Figure 4.
Dark deuterium abundance in models with 2 degenerate flavours and a dark photon.
In this section we briefly study bound state formation in a model with singlets. For simplicity weconsider the minimal scenario with SU(3) gauge group and 2 degenerate flavours. In this case thestrong dynamics is as in the SM and the lightest baryon is an SU(2) F doublet, Q . The dark deuteriumis an s -wave bound state in the singlet or a triplet rep of SU(2) F with spin 1 or 0 respectively, Q × Q = + . (61)Strong interactions favour the SU(2) F singlet as the lightest state (the analog of the SM deuterium)that will then be absolutely stable. Differently from the SM also the triplet could be bound. We willassume that the temperature of the dark sector is of the order of the SM bath which can be realisedintroducing heavy fermions charged under the SM.Singlets allow the mass scale to be much lower than TeV. As discussed in section 3.2 despite thelarger numerical density of nucleons deuterium cannot be formed because 3 → Q D = 2 J so that the nucleons have charges ±
1. This could be modified with different chargeassignments with minor changes to the nuclei formation.Neglecting model dependent SE due to the dark photon the cross-section for formation of deuteriumthrough emission of a dark photon is,( σv rel ) NN → D ( ) + γ (cid:48) = πα D M N × (cid:34) κ (cid:18) E B ( ) M N (cid:19) (1 − a ( ) γ ( ) ) + 14(12) (cid:115) M N E B ( ) v (cid:35) (62)where we used (cid:80) | C MM (cid:48) J | = 1 / m γ (cid:48) (cid:28) E B .In Fig. 4 we show the abundance of deuterium obtained integrating the Boltzmann equation andassuming that no other nuclei are produced. For masses in the GeV range, suggested by the coincidenceof DM and visible matter density, the production of deuterium is very efficient even for dark photoncouplings as small as 10 − . Note that electric transitions, even though velocity suppressed, are relevantfor small binding energies as the one of deuterium in the SM ( E B /M N = 0 . Conclusions
Big Bang Nucleosynthesis is a cornerstone of the cosmological history of the Universe and one mightwonder if a similar process could take place in the dark sector. In this work we have studied the synthesisof dark nuclei in theories where DM is a baryon of a new gauge interaction. Such models, motivatedby the idea of the accidental stability of DM, also predict the existence of stable and metastable nucleiwith SM charges that could be formed during the evolution of the Universe.The key step for Dark Nucleosynthesis is the formation of dark deuterium, a nucleus with baryonnumber 2. This process requires energy to be released and it is automatically possible through emissionof SM gauge bosons in models where the fundamental constituents have electro-weak charges. Remark-ably, the relevant cross-section can be determined from first principles in terms of the binding energiesof nuclei. To this aim, we have found it useful to employ pion-less EFT for the nucleons. This con-struction reproduces the effective range expansion of quantum mechanics with short range potentials[8] and it allows us to compute in general the cross-sections for the production of shallow bound statesexpected for nuclear interactions. We note that the cross-sections differ from the ones often used inthe literature, depending in a non-trivial way on binding energies and velocity. In particular electrictransitions grow for small binding energies while magnetic transitions decrease. Electric transitionsmoreover are velocity suppressed unless enhanced by Sommerfeld enhancement.Having determined the relevant cross-sections one can solve the Boltzmann equations for the abun-dance of deuterium and heavier elements. In the case of DM with electro-weak charges, for examplea baryon triplet of SU(2) L here studied in detail, one finds that only a fraction of DM binds intodeuterium unless binding energies much larger than in nuclear physics are assumed. The main reasonfor this is simply the numerical density of DM: direct searches imply that the DM mass is greater than1 TeV for electro-weak constituents. The small numerical abundance compared to the SM nucleonssuppresses the production of nuclei. Therefore dark BBN ends after deuterium formation leaving afraction of deuterium plus small traces of tritium. In minimal models with singlets, production of darkdeuterium is kinematically forbidden and nucleosynthesis cannot start. This conclusion changes com-pletely including a dark photon: for DM masses in the GeV range, deuterium is formed very efficientlydue to the large density and heavier nuclei can thus be formed through fusion reactions allowing topopulate nuclei up to large atomic numbers.While in this work we have focused on asymmetric scenarios where DM mass is a free parameter,our results can be extended to symmetric models, producing extra annihilation channels and nuclei-anti-nuclei. For the simplest scenarios however the critical abundance is reproduced for masses around100 TeV and no significant fraction of nuclei is produced.The formation of DM nuclei is interesting experimentally as it can lead to novel signatures inDM indirect detection, even in asymmetric scenarios and change the prediction for direct detectionexperiments due to the different composition of DM. The emission of monochromatic photons withenergy equal to the nucleus binding energy is a smoking gun of dark nuclei that could be searchedexperimentally [38].This work extends the computation of perturbative bound state formation in [13] to strongly coupledbound states. We have provided general formulae, that could be used in other contexts, for electricand magnetic interactions also taking into account important long distance effects associated to boundstates close to zero energy. It would be interesting to generalise this formalism to the fusion of stronglycoupled bound states as well studying perturbative bound states within this framework.21 cknowledgements We wish to thank Hyung Do Kim, Gordan Krnjaic, Maria Paola Lombardo, Filippo Sala and Juri Smirnov foruseful discussions. AT is partially supported by the grant “STRONG” from the INFN. We thank the GalileoGalilei Institute for Theoretical Physics for the hospitality during the completion of this work.
A Details on Dark Deuterium formation
In this appendix we summarize the derivation of [8, 44] that can be used to compute the dark deuteriumformation directly from eq. (21).Since the coefficients C r ,S of eq. (22) determine the scattering amplitude in each spin-isospin chan-nels, those associated with shallow bound states are large and should be treated as a relevant coupling[8, 44]. In an expansion in p but at all order in the coupling C r ,S , the amplitude for 2-nucleons elasticscattering can be written as A ([ N N → N N ] r ,S ) = − iC r ,S iC r ,S Σ( p ) = = C r ,S + C r ,S C r ,S Σ + · · · ,(63)which should be matched with eq. (24). Notice that this calculation requires a renormalisation of thetheory, since Σ( p ) is divergent. A useful scheme is the on-shell momentum subtraction (or the analogousPDS [45] scheme), which amounts to say that the 1-loop term plus the counter-term evaluated at p = iµ in dimensional regularisation should be equal to C r ,s ( µ ). In this schemeΣ( p ) = (cid:90) d q (2 π ) i ( q + E − (cid:126)q / M N ) i ( − q − (cid:126)q / M N ) = − i M N π ( µ + ip ) , p = (cid:112) M N E . (64)To leading order in the momentum expansion eq. (24) is reproduced with, C r ,S ( µ ) = 4 πa r ,S M − µa r ,S . (65)When a r ,S is large and positive it signals the presence of a bound state in the corresponding channel.The scattering length is related to the binding energy, since the pole in the amplitude corresponds toa momentum p = i √ M N E B , giving a r ,S = 1 / √ M N E B ≡ /γ r . Alternatively, one can solve for C r ,S requiring a pole in the amplitude for a given binding energy.Expanding the amplitude at the pole we can also determine the interaction of the bound state D r ,S with the two nucleons. Namely the coupling is the square root of residue of the amplitude at the pole.Therefore quite generally we can write as effective coupling g NND r = √ πγ r M N , γ r = (cid:112) B r M N ⊗ r , S = g NND (cid:18)
Res (cid:19) E = − B r g NND . (66)Once the coupling of the bound state to two nucleons is determined, one can compute with ordinaryFeynman diagrams the amplitude relevant for dark deuterium formation. This formalism is particularly22owerful as it allows to compute systematically the effects to higher orders and allows to resum effectsassociated to large scattering lenghts.In this work we are interested in all the processes of the type A ( (cid:2) N + N (cid:3) M,i r ,s → D M (cid:48) ,i (cid:48) r (cid:48) ,s (cid:48) + V aλ ) . (67)where a bound state is formed through emission of a gauge boson. This implies the existence ofselection rules: In the explicit case of SU(2) L , all the leading order amplitudes correspond to a ∆ I = 1transition. Instead, for spin and angular momentum there are two possibilities: i ) a magnetic dipoleinteraction of eq. (21) which corresponds to a ∆ S = 1 transition; ii ) an electric transition (in dipoleapproximation) with ∆ L = 1 that originates from the covariant derivative in eq. (21). A.1 Details on the magnetic transition
In the SM the deuteroun is an isospin 0 and spin 1 state. At low energy the formation is dominatedby the magnetic transition) from an initial spin singlet isospin triplet channel [31]. The selection ruleimplies that it proceeds from an s -wave initial state ( S ) so that σv goes to a constant for slownucleons. Moreover this process is significantly enhanced by the large scattering length of the S channel.Our scenario differs from the SM for the different group theory structure. To leading order for largescattering lengths of the initial state there are two diagrams that contribute, ⊗ A = ( r , S ) ( r (cid:48) , S (cid:48) ) A = ⊗ ( r , S ) ( r (cid:48) , S (cid:48) )Π (68)The crossed circle represents the deuteron coupling (66), while the filled circle corresponds tothe insertion of the resummed amplitude for the scattering in the ( r , S ) channel. The loop integralappearing in the amplitude A is finite giving,Π( k ) = (cid:90) d q (2 π ) i − q − (cid:126)q / (2 M N ) iq − (cid:126)q / (2 M N ) iq − | k | − (cid:126)q / (2 M N )= (cid:90) d q (2 π ) M N q + M N | (cid:126)k | q = M N π (cid:113) M N | (cid:126)k | + O ( E ) . (69)The ratio of the two amplitudes for small energy is A A = − a r γ r (cid:48) , (70)where we have used the fact that k = E B r (cid:48) . By virtue of this relation, the leading order term of themagnetic transition can be computed easily just focussing on A . Notably, A can be of the same orderand might dominate over the first one.In full generality, to leading order in the momentum expansion, the magnetic amplitude from aninitial state (cid:2) N + N (cid:3) r ,s to a final state with D r (cid:48) ,s (cid:48) + W aλ is given A ( (cid:2) N + N (cid:3) M,i r ,s → D M (cid:48) ,i (cid:48) r (cid:48) ,s (cid:48) + W aλ ) = 2 g κM N | (cid:126)k | g NND r (cid:48) (1 − a r γ r (cid:48) )( (cid:126)k × (cid:126)ε ( λ a ) ) i + i (cid:48) δ s + s (cid:48) , C aMM (cid:48) J (71)23hich is valid for any group and any representation. For non zero velocity of the nucleons a similarformula can be derived. B Bound State decays
The formalism above also allows to compute decay rates. Focusing on bound states with (cid:96) = 0 and spin0 , α via the magnetic interaction.The decay rate is given by Γ = | (cid:126)k | π |A D i → D j W | . (72)The amplitude for the transition between two bound states with ∆ I = 1 and ∆ S = 1, with the emissionof a gauge boson, is given by the 1-loop graph in eq. (68) evaluated at the difference of binding energiesbetween the two states, A = ⊗ ⊗ ( r , s ) ( r (cid:48) , s (cid:48) )Π (cid:126)ε λ a = 2 g κM N g NND r (cid:48) g NND r Π( | (cid:126)k | ) | (cid:126)k | ( (cid:126)k × (cid:126)ε λ ) i + i (cid:48) δ s + s (cid:48) , C aMM (cid:48) J , (73)By energy conservation | (cid:126)k | = E B , which gives Π = M N / (4 π √ M N E B ). Summing over photon polari-sations and spin indices, one findsΓ = κ (256 α ) g D r ( E B r − E B r (cid:48) ) M N (cid:113) E B r E B r (cid:48) (cid:88) aMM (cid:48) | C aMM (cid:48) J | . (74) References [1] M. Cirelli, N. Fornengo, and A. Strumia,
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