Bound states of WIMP dark matter in Higgs-portal models I: cross-sections and transition rates
PPrepared for submission to JHEP
Nikhef-2021-003
Bound states of WIMP dark matter in Higgs-portalmodels I: cross-sections and transition rates
Ruben Oncala and Kalliopi Petraki
Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique et Hautes Energies (LPTHE),UMR 7589 CNRS & Sorbonne Universit´e, 4 Place Jussieu, F-75252, Paris, FranceNikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands
E-mail: [email protected] , [email protected] Abstract:
We investigate the role of the Higgs doublet in the thermal decoupling ofmulti-TeV dark matter coupled to the Weak interactions of the Standard Model and theHiggs. The Higgs doublet can mediate a long-range force that affects the annihilationprocesses and binds dark matter into bound states. More importantly, the emission of aHiggs doublet by a pair of dark matter particles can give rise to extremely rapid monopolebound-state formation processes and bound-to-bound transitions. We compute these effectsin the unbroken electroweak phase. To this end, we consider the simplest renormalisablefermionic Higgs portal model, consisting of a singlet and a doublet under SU L (2) that arestabilised by a Z symmetry, in the regime where the two multiplets coannihilate. In acompanion paper, we use the results to show that the formation of metastable bound statesvia Higgs-doublet emission and their decay decrease the relic density very significantly. a r X i v : . [ h e p - ph ] J a n ontents SS/D ¯ D bound states: ( , n(cid:96)m = { } D ¯ D bound states: ( , n(cid:96)m = { } DD bound states: ( , n(cid:96)m = { } DS bound states: ( , / n(cid:96)m = { } H exchange 434.2 B and W exchange 49 A.1 The 2PI kernel 52A.2 Wavefunctions 54
B Perturbative transition amplitudes: an example 57C Overlap integral for monopole bound-to-bound transitions 59D Scalar emission via vector-scalar fusion 60 – 1 –
Introduction
Particles coupled to the Weak interactions of the Standard Model (SM), known as WIMPs,have been arguably the most widely considered candidates for dark matter (DM) in thepast decades. Among the archetypical WIMP models are Higgs-portal scenarios where DMis a linear combination of the neutral components of electroweak multiplets that couple tothe Higgs doublet. The discovery of the Higgs boson and the measurement of its propertiesimpel the investigation of its implications for such scenarios.While most related research in the past focused on electroweak-scale WIMP masses, m ∼
100 GeV, the current experimental constraints motivate considering the multi-TeVmass regime more carefully. This is particularly important in view of the numerous existingand upcoming observatories probing high-energy cosmic rays, such as H.E.S.S., IceCube,CTA and KM3Net. The experimental exploration of the multi-TeV scale urges the com-prehensive theoretical understanding of the dynamics and possibilities in this regime.The hierarchy between the multi-TeV and electroweak scales implies the emergence ofnew effects. In particular, the Weak interactions between particles with multi-TeV massmanifest as long-range , since the interaction range l ∼ (100 GeV) − may be comparable orexceed the de Broglie wavelength ( µv rel ) − and/or the Bohr radius ( µα ) − of the interactingparticles, where µ = m/ (cid:38) TeV, v rel and α are the reduced mass, relative velocity andcoupling to the force mediators. The long-range nature of the interactions gives rise tonon-perturbative phenomena, the Sommerfeld effect and the existence of bound states.Bound states form invariably with dissipation of energy. It has been recently shownthat the emission of a scalar boson charged under a symmetry alters the effective Hamilto-nian between the interacting particles and gives rise to monopole transitions; this rendersbound-state formation (BSF) extremely rapid even for small couplings [1]. For WIMPs,this implies that BSF via emission of a Higgs doublet may be a very significant inelasticprocess. Moreover, it has been shown in simplified models, that the 125 GeV Higgs bosoncan mediate a sizeable long-range force between TeV-scale particles, despite being heavierthan all SM gauge bosons [2, 3].The phenomenological importance of the above is rather large. It has been long knownthat the Sommerfeld effect [4, 5] – the distortion of the wavefunction of interacting parti-cles due to a long-range force – affects the DM annihilation cross-sections at low relativevelocities [6]. This, in turn, alters the DM chemical decoupling in the early universe, andconsequently the predicted DM mass and couplings to other particles [7]. It also affectsthe radiative signals expected from the DM annihilations during CMB and inside galaxiestoday [8]. More recently, it has been realised that the formation and decay of metastable(e.g. particle-antiparticle) bound states in the early universe can decrease the DM den-sity [9], and contribute to the DM indirect detection signals [10–17]. Importantly, BSFcan be faster than annihilation in a variety of models [1, 3, 18–20], and it can also producenovel indirect signals even in models where annihilation is absent or suppressed [21–27].Collecting the above considerations, in this work we consider bound states in Higgsportal models, and in a companion paper [28] we compute their effect on the DM relicdensity. We are interested in scenarios that feature a trilinear coupling between the DM– 2 –nd Higgs multiplets, i.e. δ L ⊃ ¯ χ n Hχ n +1 + h.c., where χ n is a fermionic or bosonic n -plet under SU L (2), and DM is the lightest linear combination of the neutral χ n and χ n +1 components after electroweak symmetry breaking. Such scenarios have been consideredextensively in the literature, and appear also in supersymmetry, e.g. Bino-Higgsino orHiggsino-Wino DM [29–46]. For concreteness, we shall focus on the most minimal suchmodel with a Majorana singlet and a Dirac doublet [29–39].Even neglecting bound state effects during the DM freeze-out, many of these modelsare predicted to reside in the TeV scale (see [37] for a summary). BSF in the early universeincreases the DM destruction rate and, for a given set of couplings, pushes the predictedDM mass to even higher values [9]. If DM is heavier than 5 TeV, freeze-out begins before theelectroweak phase transition (EWPT), even though it may be completed much later. Herewe will assume electroweak symmetry throughout; this considerably simplifies the anyhowlengthy calculations, and is a practice that has been followed in past literature [37]. Wediscuss in detail the extent of validity of this approximation for the DM thermal decouplingin ref. [28]. Here we simply note that, by virtue of the Goldstone boson equivalence theorem,BSF via Higgs-doublet emission remains important in the broken electroweak phase, whereit occurs via emission of the Higgs boson and the longitudinal components of the Weakgauge bosons [47].In the epoch prior to the EWPT, if the two DM multiplets are nearly mass-degenerate(perhaps as a result of a larger symmetry group), the DM freeze-out is determined by alltheir self-annihilation and co-annihilation processes [48]. Our computations focus on thisregime. Then, in the class of models considered, the entire Higgs doublet contributes tothe (long-range) potential between the DM multiplets. Moreover, bound states can formvia emission of a Higgs doublet. As we shall see, the cross-sections for BSF via Higgs-doublet emission can exceed those for annihilation or BSF via vector emission by ordersof magnitude, and result in a late period of DM chemical recoupling [1]. This, in turn,opens the possibility that thermal-relic WIMP DM may be much heavier than anticipated,potentially approaching the unitarity limit [28], which in the non-relativistic indeed implies(or presupposes) the presence of long-range interactions [15].This work is organized as follows. In section 2, we introduce the DM model and derivethe long-range potentials between the DM multiplets in the unbroken electroweak phase.We identify the scattering and bound eigenstates of these potentials by appropriate spin andgauge projections, before computing the DM annihilation processes and bound-state decayrates. In section 3, we calculate all the radiative BSF cross-sections, while in section 4 weconsider BSF via scattering on a relativistic thermal bath. These results are employed inthe companion paper [28] to compute the DM thermal decoupling in the early universe. Weconclude in section 5. Several technical aspects are addressed in the appendices, including aproof of the (anti)symmetrisation of the two-particle-irreducible kernels and wavefunctionsin the case of identical particles (appendix A), and the analytical computation of monopolebound-to-bound transitions in the Coulomb limit (appendix C.)– 3 – Long-range dynamics in the unbroken electroweak phase
We introduce a gauge-singlet Majorana fermion S = ( ψ α , ψ † ˙ α ) T of mass m S , as well as aDirac fermion D = ( ξ α , χ † ˙ α ) T of mass m D with SM gauge charges SU L (2) × U Y (1) = ( , / S and D are odd under a Z symmetry that leaves all the SM particlesunaffected. Under these assignments, the new degrees of freedom (dof) allow to extend theSM Lagrangian by the following interactions δ L = 12 ¯ S ( i /∂ − m S ) S + D ( i / D − m D ) D − ( y L ¯ D L HS + y R ¯ D R HS + h.c.) , (2.1)where H is the SM Higgs doublet of mass m H and hypercharge Y H = 1 /
2, and D L ≡ P L D = ( ξ α , T and D R ≡ P R D = (0 , χ † ˙ α ) T , with P R , L = (1 ± γ ) / D µ ≡ ∂ µ − i g Y B µ − i g W aµ t a is thecovariant derivative, with t a = ( σ , σ , σ ) and σ being the Pauli matrices. The particlecontent of eq. (2.1) is summarised in table 1.field SU L (2) U Y (1) Z S − D − H Table 1 . Particle content and charge assignments
Here, we have taken the mass parameters m S and m D to be real. This can always beachieved by rephasing ψ and either ξ or χ . Rephasing the remaining spinor eliminates thephase of one of the Yukawa couplings. Thus the free parameters of the present model are4 real couplings (two masses and two dimensionlesss Yukawa couplings), and a phase thatallows for CP violation.We are interested in the regime in which S and D can co-annihilate efficiently beforethe EWPT of the universe, which occurs if their masses are similar, within about 10%.This is because the number density of the heavier species in the non-relativistic regime issuppressed with respect to that of the lighter species by a factor exp[ − ( δm/m ) x ]. Since x ≡ m/T (cid:38)
25 during DM freeze-out, if δm/m (cid:38)
10% the co-annihilations and the self-annihilations of the heavier species are typically subdominant to the self-annihilations ofthe lightest species, although their relative importance depends also on the correspondingcross-sections. Such a small discrepancy in the masses does not significantly affect (mostof) the cross-sections that we compute in the following; we shall thus take the masses tobe equal, which greatly simplifies the computations and allows to obtain analytical results, m D = m S ≡ m. (2.2)We will very often use the reduced mass of a pair of DM particles, µ ≡ m/ . (2.3)– 4 –oreover, in order to reduce the number of free parameters, we set y L = y R ≡ y, (2.4)which we take to be real. (The CP violation is anyway not important for our purposes.)Our computations can of course be extended to more general Yukawa couplings. As isstandard, we define the couplings α ≡ g π , α ≡ g π , α H ≡ y π . (2.5)Various aspects of its phenomenology and experimental constraints of the model (2.1)have been considered extensively in the past [29–38], and we shall not review them here.We only note that after the EWPT, S and D mix to produce three neutral mass eigenstates,the lightest of which is stable and can play the role of DM. In the following, we focus oncomputing the long-range effects in the unbroken electroweak phase. In the companionpaper [28], we briefly review the mass eigenstates and their interactions after electroweaksymmetry breaking for the choice of parameters denoted in eqs. (2.2) and (2.4), beforecomputing the DM decoupling in the early universe, which alters the predicted mass-coupling relation, thereby affecting all experimental constraints. The D , ¯ D and S fermions interact with each other via the B , W and H boson exchangesthat give rise to long-range potentials. The kernels generating these potentials are shownin fig. 1. To compute them, we decompose the incoming and outgoing momenta as p = P/ p, p (cid:48) = P/ p (cid:48) , (2.6a) p = P/ − p, p (cid:48) = P/ − p (cid:48) . (2.6b)For low momentum transfers, we find (see e.g. [1, 18–20, 49]) i [ K D ¯ D ↔ SS ] s s ,s (cid:48) s (cid:48) ij (cid:39) + i m ( y δ ij ) 12 (cid:34) δ s s (cid:48) δ s s (cid:48) ( p (cid:48) − p ) + m H − δ s s (cid:48) δ s s (cid:48) ( p (cid:48) + p ) + m H (cid:35) , (2.7a) i [ K D ¯ D ↔ D ¯ D ] s s ,s (cid:48) s (cid:48) ij,i (cid:48) j (cid:48) (cid:39) + i m ( p (cid:48) − p ) (cid:0) g Y D δ ii (cid:48) δ jj (cid:48) + g t ai (cid:48) i t ajj (cid:48) (cid:1) δ s s (cid:48) δ s s (cid:48) , (2.7b) i [ K DD ↔ DD ] s s ,s (cid:48) s (cid:48) ij,i (cid:48) j (cid:48) (cid:39) − i m (cid:16) g Y D δ ii (cid:48) δ jj (cid:48) + g t ai (cid:48) i t aj (cid:48) j (cid:17) δ s s (cid:48) δ s s (cid:48) ( p (cid:48) − p ) − (cid:16) g Y D δ ij (cid:48) δ ji (cid:48) + g t aj (cid:48) i t ai (cid:48) j (cid:17) δ s s (cid:48) δ s s (cid:48) ( p (cid:48) + p ) , (2.7c) i [ K DS ↔ DS ] s s ,s (cid:48) s (cid:48) i,i (cid:48) (cid:39) − i m ( p (cid:48) + p ) + m H ( y δ ii (cid:48) ) δ s s (cid:48) δ s s (cid:48) . (2.7d)In determining the sign of each contribution in the above, we have taken into accountthe number of fermion permutations needed to perform the Wick contractions. This is– 5 – s i ¯ D s j p p p (cid:48) p (cid:48) S s (cid:48) S s (cid:48) i K D ¯ D ↔ SS = 12 D s i ¯ D s j p p Hp (cid:48) p (cid:48) S s (cid:48) S s (cid:48) + D s i ¯ D s j p p Hp (cid:48) p (cid:48) S s (cid:48) S s (cid:48) D s i ¯ D s j p p p (cid:48) p (cid:48) D s (cid:48) i (cid:48) ¯ D s (cid:48) j (cid:48) i K D ¯ D ↔ D ¯ D = D s i ¯ D s j p p B, W a p (cid:48) p (cid:48) D s (cid:48) i (cid:48) ¯ D s (cid:48) j (cid:48) D s i D s j p p p (cid:48) p (cid:48) D s (cid:48) i (cid:48) D s (cid:48) j (cid:48) i K D ¯ D ↔ D ¯ D = 12 D s i D s j p p B, W a p (cid:48) p (cid:48) D s (cid:48) i (cid:48) D s (cid:48) j (cid:48) + D s i D s j p p B, W a p (cid:48) p (cid:48) D s (cid:48) j (cid:48) D s (cid:48) i (cid:48) D s i S s p p p (cid:48) p (cid:48) D s (cid:48) i (cid:48) S s (cid:48) i K D ¯ D ↔ SS = D s i S s p p Hp (cid:48) p (cid:48) S s (cid:48) D s (cid:48) i (cid:48) Figure 1 . The kernels generating the long-range potentials between pairs of D , ¯ D and S fermions.The double lines represent the SU L (2) × U Y (1) = ( , /
2) fermion D while the single lines standfor the gauge singlet S . The arrows on the fermion lines denote the flow of Hypercharge. Theindices i, j, i (cid:48) , j (cid:48) and s , s , s (cid:48) , s (cid:48) are SU L (2) and spin indices, respectively. The factors 1 / D ¯ D ↔ SS and DD ↔ DD , ensure that the resummationof the kernels does not result in double-counting of loops (cf. appendix A.) the origin of the relative minus sign between the t - and u -channels of the D ¯ D ↔ SS and DD ↔ DD interactions. The factors 1 / Momentum space.
The kernels of both the t - and u -channel diagrams depend onlyon the momentum transfer, which is however different in the two cases, K t ( p − p (cid:48) ) and K u ( p + p (cid:48) ). This implies that in position space, the u -channel potential depends on the– 6 –rbital angular momentum mode (cid:96) of the state under consideration [1, appendix A]. Specif-ically, the static potentials generated by t - and u -channel diagrams are [1, 18] V t ( r ) = − i m (cid:90) d q (2 π ) i K t ( q ) e i q · r , (2.8a) V u ( r ) = − ( − (cid:96) i m (cid:90) d q (2 π ) i K u ( q ) e i q · r . (2.8b)Inserting eqs. (2.7) into (2.8) yields the well-known Coulomb and Yukawa potentials. Spin diagonalisation.
The factors δ s s (cid:48) δ s s (cid:48) and δ s s (cid:48) δ s s (cid:48) arising from t - and u -channelexchanges respectively, can be written in matrix form in the basis {↑↑ , ↑↓ , ↓↑ , ↓↓} as t -channel: , u -channel: . (2.9)Clearly, the t -channel interactions conserve spin along each leg of the ladder, and the corre-sponding spin factor is simply the unity operator. On the other hand, the u -channel inter-actions conserve the total spin only. Indeed, the u -channel spin eigenvalues are {− , , , } ,and correspond to the eigenvectors of total spin1 √ , , − , T , (0 , , , T , √ , , , T , (1 , , , T . (2.10)In the following, we shall therefore project the asymptotic states of pairs of S, D, ¯ D fermionsonto eigenstates of total spin. Gauge diagonalisation.
Since the interactions respect the SU L (2) symmetry, this amountsto projecting on SU L (2) representations of the incoming or outgoing pairs. For two mul-tiplets transforming under the representations R and R of a gauge group, the Coulombpotential generated by the gauge-boson exchange in the configuration R ⊂ R ⊗ R of thepair is V R ( r ) = − α R /r with [50] α R = α C ( R ) + C ( R ) − C ( R )] , (2.11)where α is the fine structure constant of the group, and C ( R ) is the quadratic Casimiroperator of the representation R . For SU (2), ⊗ = + , and C ( ) = 3 / C ( ) = 2.Thus, the D ¯ D , DD and ¯ D ¯ D pairs appear in SU L (2) singlet and triplet configurations, with α = 3 α / α = − α / DD and ¯ D ¯ D ,the (anti)symmetry of the SU L (2) (singlet) triplet states in gauge space generates also thefactor ( − I +1 for the u -channel diagrams, where here I = 0 , U Y (1) potentials are of course not affected by the SU L (2) diagonalisation. All these results can be easily recovered by organising the gauge factors of eqs. (2.7b) and (2.7c) in4 × { ij, i (cid:48) j (cid:48) } and diagonalising them. – 7 –he remaining task is the H -mediated interaction of eq. (2.7a). This occurs only inthe SU L (2) singlet state. Projecting D ¯ D on the singlet, we obtain δ ij √ y δ ij ) = √ y . (2.12) Kernel (anti)symmetrisation.
We close this discussion by noting the importance ofconsidering properly both the t - and u -channel diagrams when identical particles are presentin the initial and/or final states, as e.g. in the D ¯ D ↔ SS and DD ↔ DD interactions ofthe present model. Using the correct kernel, as derived in appendix A, and taking intoaccount the above yields the overall factor[1 − ( − (cid:96) ( − s +1 ( − I +1 ] / U Y (1) SU L (2) Potential Sign of the potential (cid:96) + s = even (cid:96) + s = odd D ¯ D ↔ SS − (cid:20) − (cid:96) + s (cid:21) √ α H e − m H r r attractive 0 D ¯ D ↔ D ¯ D − ( α + 3 α ) / r attractive attractive − ( α − α ) / r repulsive repulsive DD ↔ DD (cid:20) − ( − (cid:96) + s (cid:21) ( α − α ) / r (cid:20) − (cid:96) + s (cid:21) ( α + α ) / r repulsive 0 SD ↔ SD / − ( − (cid:96) + s α H e − m H r r attractive repulsive Table 2 . The static potentials generated by the W , B and H -exchange diagrams shown in fig. 1. (cid:96) and s denote the orbital angular momentum mode and the total spin respectively. – 8 – .3 Asymptotic scattering and bound states The potentials of table 2 determine the asymptotic states. For all gauge assignmentsexcept the singlet states, SU L (2) × U Y (1) = ( , D ¯ D ↔ SS interaction implies that a systemof coupled Schr¨odinger equations must be solved. We work out this case in detail insection 2.3.2, after we discuss the general properties of the wavefunctions and the underlyinghierarchy of scales in section 2.3.1. All the results on the wavefunctions of the scatteringand bound states are summarised in tables 3 and 4. The non-relativistic potentials of table 2 include both Coulomb and Yukawa contributions.The latter does not allow for analytical solutions. In order to obtain analytical expressionsfor the wavefunctions and ultimately the various cross-sections of interest, we shall neglectthe Higgs mass in the potentials of table 2, m H → , (2.14)although we will retain it in the phase-space suppression of BSF via H emission computed insection 3, as well as in the H propagator of BSF via off-shell H exchange with the thermalbath, computed in section 4. We discuss the range of validity of this approximation insection 2.3.3.In the approximation (2.14), we can express all wavefunctions in terms of those for aCoulomb potential V ( r ) = − α/r , which we shall denote as ϕ ( r ; α ) and we now summarise.For clarity, we denote by α S and α B the couplings of the scattering and bound states. Themomentum of each particle in the CM frame in the scattering states is k ≡ µ v rel , with v rel being the relative velocity. The Bohr momenta in the scattering and bound states are κ S ≡ µα S and κ B ≡ µα B . For convenience, we define the parameter ζ S ≡ κ S /k = α S /v rel (in section 3 we will also use ζ B ≡ κ B /k = α B /v rel ), and the variables x S ≡ kr and x B ≡ κr .The energy eigenvalues of the scattering and bound states are E k = k µ = µv , E n = − κ µ n = − µα B n . (2.15)The scattering state wavefunctions can be decomposed in partial waves, ϕ k ( r ; α S ) = ∞ (cid:88) (cid:96) S =0 (2 (cid:96) S + 1) (cid:20) ϕ | k | ,(cid:96) S ( x S ; α S ) x S (cid:21) P (cid:96) S ( ˆk · ˆr ) , (2.16)where ϕ | k | ,(cid:96) S ( x S ; α S ) = − (cid:112) S ( ζ S )(2 (cid:96) S + 1)! Γ(1 + (cid:96) S − i ζ S )Γ(1 − i ζ S ) × e − ix S x S (2 i x S ) (cid:96) S F (1 + (cid:96) S + i ζ S ; 2 (cid:96) S + 2; 2 i x S ) , (2.17)– 9 –nd S ( ζ S ) ≡ πζ S − e − πζ S (2.18)is the well-known s -wave Sommerfeld factor. The bound-state wavefunctions are ϕ n(cid:96)m ( r ; α B ) = κ / Y (cid:96)m (Ω r ) 2 n (2 (cid:96) + 1)! (cid:20) ( n + (cid:96) )!( n − (cid:96) − (cid:21) / × e − x B /n (2 x B /n ) (cid:96) F ( − n + (cid:96) + 1; 2 (cid:96) + 2; 2 x B /n ) , (2.19)where the normalisation of the spherical harmonics is (cid:82) d Ω Y (cid:96)m (Ω) Y ∗ (cid:96) (cid:48) m (cid:48) (Ω) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) .The emergence of the Sommerfeld effect and the existence of bound states emanatefrom the different scales involved in the interactions of the D , ¯ D and S particles, µv / (cid:28) µv rel (cid:28) µ < m and µα / (2 n ) (cid:28) µ | α | /n (cid:28) µ < m, (2.20)where here α = α S or α B . (Note that α S may be negative.) In computing the BSFcross-sections in sections 3 and 4, we make approximations based on these hierarchies.The hierarchies (2.20) imply that the couplings (2.5) must be evaluated at the ap-propriate momentum transfer in every occurrence. The average momentum transfers are(cf. section 3 for the last two)annihilation vertices: m scattering-state potentials: µv rel bound-state potentials: µα B emission vertices for BSF: ( µ/ α B /n + v )emission vertices for bound-to-bound transitions: ( µ/ | α B /n − α (cid:48) B | Here for simplicity we will neglect the running of the couplings, although it is easy torestore the scale dependence in all of our analytical results. SU L (2) × U Y (1) = ( ,
0) states
The H exchange mixes SS and D ¯ D gauge singlet states. The coupled resummation ofthe D ¯ D ↔ D ¯ D and D ¯ D ↔ SS potentials is shown schematically in fig. 2. We define thewavefunctions of the gauge-singlet statesΦ j ( r ) = (cid:32) [ φ ( r )] j SS [ φ ( r )] j D ¯ D (cid:33) , (2.21) In computing the DM freeze-out in [28], we adopt the values of the gauge couplings α and α at the Z pole, α ( m Z ) (cid:39) . α ( m Z ) (cid:39) . α H increases with Q , we consider the quoted valuesof α H to correspond to the highest relevant scale, Q = m , such that the theory remains well-defined at all Q (cid:54) m . For large values of α H , a Landau pole appears at fairly low energies (but larger than m ); howeverthis may be cured by other new physics around those scales. For the renormalisation group equations inthe present model, we refer to [38]. – 10 – (4) D ¯ DD ¯ D = + i K D ¯ D ↔ D ¯ D G (4) D ¯ DD ¯ D + i K D ¯ D ↔ SS G (4) SSD ¯ D G (4) SSSS = + i K SS ↔ D ¯ D G (4) D ¯ DSS
Figure 2 . The resummation of 2PI diagrams for the gauge-singlet states, SU L (2) × U Y (1) = ( , S and D respectively. where the component wavefunctions are[ φ ( x )] j SS ≡ (cid:104) Ω | T S ( x/ S ( − x/ |S j ( , (cid:105) x =0 , (2.22a)[ φ ( x )] j D ¯ D ≡ (cid:104) Ω | T D ( x/
2) ¯ D ( − x/ |S j ( , (cid:105) x =0 . (2.22b)Here, S j ( , denotes the gauge-singlet states. Along with their wavefunctions Φ j , they carryquantum numbers that define their energy, angular momentum and spin, and that we havehere left implicit. The superscript j aims to differentiate between states with the samequantum numbers but different boundary conditions. Ω here stands for the vacuum of theinteracting theory, T is the time ordering operator, S , D and ¯ D are the field operators.The resummation sketched in fig. 2 implies that Φ obey the Schr¨odinger equations (cid:20) − ∇ µ + V ( , ( r ) (cid:21) Φ( r ) = E Φ( r ) , (2.23)where V ( , is the 2 × V ( , ( r ) = − r (cid:32) δ (cid:96) + s, even √ α H exp( − m H r ) δ (cid:96) + s, even √ α H exp( − m H r ) ( α + 3 α ) / (cid:33) . (2.24)In the limit m H →
0, the system (2.23) easily decouples. We first define the couplings α R ≡ (cid:20)(cid:113) [( α + 3 α ) / + 8 α H δ (cid:96) + s, even − ( α + 3 α ) / (cid:21) , (2.25a) α A ≡ (cid:20)(cid:113) [( α + 3 α ) / + 8 α H δ (cid:96) + s, even + ( α + 3 α ) / (cid:21) , (2.25b)as well as the unitary matrix P ≡ √ α A + α R (cid:32) √ α A √ α R −√ α R √ α A (cid:33) , (2.26)whose columns are the normalised V ( , eigenvectors with eigenvalues − α R and α A . Wenote that α R , α A (cid:62)
0. Then, the rotated wavefunctions ˆΦ ≡ P † Φ obey the Schr¨odingerequations (cid:20) − ∇ µ + ˆ V ( , ( r ) (cid:21) ˆΦ( r ) = E ˆΦ( r ) , (2.27)– 11 –ith the potential ˆ V ( , = P † V ( , P = − r (cid:32) − α R α A (cid:33) . (2.28)We now seek scattering and bound state solutions to eq. (2.27). Scattering states.
The scattering state solutions of eq. (2.27) are given by eq. (2.16),albeit the normalisation of each component is allowed to vary and will be determined bythe boundary conditions on Φ j . Analysing eq. (2.27) in partial waves, the solutions areˆΦ j | k | ,(cid:96) ( x S ) = (cid:32) N j R ϕ | k | ,(cid:96) ( x S ; − α R ) N j A ϕ | k | ,(cid:96) ( x S ; α A ) (cid:33) , (2.29)where at x S → ∞ , the wavefunctions ϕ k ,(cid:96) ( x S ; α ) obey dϕ | k | ,(cid:96) ( x S ; α ) dx S − i ϕ | k | ,(cid:96) ( x S ; α ) = e − i ( x S − (cid:96)π ) . (2.30)We seek scattering-state solutions to eq. (2.23) that asymptote at r → ∞ to a pure SS or D ¯ D state. In terms of partial waves, this implies that at r → ∞ , SS -like state: d [ φ (cid:96) ( x S )] SS i dx S − i [ φ (cid:96) ( x S )] SS i = δ SS i e − i ( x S − (cid:96)π ) × √ δ (cid:96) + s, even , (2.31a) D ¯ D -like state: d [ φ (cid:96) ( x S )] D ¯ D i dx S − i [ φ (cid:96) ( x S )] D ¯ D i = δ D ¯ D i e − i ( x S − (cid:96)π ) , (2.31b)where i = SS, D ¯ D denotes the component. In eq. (2.31a), we included the antisymmetri-sation factor √ δ (cid:96) + s, even due to the identical particles in the SS -like state, with s = 0 or1 being the total spin (cf. appendix A.2).Since ˆΦ | k | ,(cid:96) ( x S ) = P † Φ | k | ,(cid:96) ( x S ), the asymptotic behaviours (2.30) and (2.31) imply (cid:32) N SSR N SSA (cid:33) = P † (cid:32) √ δ (cid:96) + s, even (cid:33) = √ δ (cid:96) + s, even √ α A + α R (cid:32) √ α A √ α R (cid:33) , (2.32a) (cid:32) N D ¯ DR N D ¯ DA (cid:33) = P † (cid:32) (cid:33) = 1 √ α A + α R (cid:32) − √ α R √ α A (cid:33) . (2.32b)We recall that α R and α A depend on (cid:96) + s , and note that while the D ¯ D -like state should not be antisymmetrised, the SS components of the symmetric (cid:96) + s = odd modes do vanish dueto the antisymmetrisation of the kernel ( α R = 0 for (cid:96) + s = odd). Combining eqs. (2.29)and (2.32), we obtain the wavefunctions Φ j | k | ,(cid:96) ( r ) = P ˆΦ j | k | ,(cid:96) ( r ). The results are summarisedin table 3. Bound states.
It is easy to see that eq. (2.27) has only one set of bound-state solutions,ˆΦ n(cid:96)m (r) = (0 , ϕ n(cid:96)m ( r ; α A )) T , with E n = − µα A / (2 n ). The unrotated wavefunctions areΦ n(cid:96)m (r) = P ˆΦ n(cid:96)m (r). The result is summarised in table 4.– 12 – Y (1) SU L (2) State Com-ponent Partial-wave wavefunctions0 SS -like SS δ (cid:96) + s, even √ α A + α R (cid:104) α R ϕ | k | ,(cid:96) ( x S ; α A ) + α A ϕ | k | ,(cid:96) ( x S ; − α R ) (cid:105) D ¯ D δ (cid:96) + s, even √ α A α R α A + α R (cid:104) ϕ | k | ,(cid:96) ( x S ; α A ) − ϕ | k | ,(cid:96) ( x S ; − α R ) (cid:105) D ¯ D -like SS √ α A α R α A + α R (cid:104) ϕ | k | ,(cid:96) ( x S ; α A ) − ϕ | k | ,(cid:96) ( x S ; − α R ) (cid:105) D ¯ D α A + α R (cid:104) α A ϕ | k | ,(cid:96) ( x S ; α A ) + α R ϕ | k | ,(cid:96) ( x S ; − α R ) (cid:105) D ¯ D D ¯ D ϕ | k | ,(cid:96) ( x S ; ( α − α ) / DD DD δ (cid:96) + s, odd √ ϕ | k | ,(cid:96) ( x S ; ( − α + 3 α ) / DD DD δ (cid:96) + s, even √ ϕ | k | ,(cid:96) ( x S ; − ( α + α ) / / DS DS ϕ | k | ,(cid:96) (cid:0) x S ; ( − (cid:96) + s α H (cid:1) Table 3 . The scattering states and their wavefunctions in the limit m H →
0. Here, ϕ | k | ,(cid:96) ( x S ; α )denotes the (cid:96) mode of a scattering state with momentum k for a Coulomb potential V ( r ) = − α/r ;the position variable is x S ≡ kr . The couplings α A and α R , defined in eqs. (2.25), depend on (cid:96) + s .For (cid:96) + s = odd, the ( , ) mixed states decouple. U Y (1) SU L (2) Bindingenergy Bound state/component Wavefunctions0 µα A n SS (cid:114) α R α A + α R ϕ n(cid:96)m ( r ; α A ) D ¯ D (cid:114) α A α A + α R ϕ n(cid:96)m ( r ; α A )1 µ [( − α + 3 α ) / n DD δ (cid:96) + s, odd √ ϕ n(cid:96)m ( r ; ( − α + 3 α ) / / µα H n DS δ (cid:96) + s, even ϕ n(cid:96)m ( r ; α H ) Table 4 . The bound states and their wavefunctions in the limit m H →
0. Here ϕ n(cid:96)m ( r ; α ) denotesthe bound state wavefunction with quantum numbers { n(cid:96)m } for a Coulomb potential V ( r ) = − α/r .The couplings α A and α R are defined in eqs. (2.25), and depend on (cid:96) + s . For (cid:96) + s = odd, the ( , )state becomes purely D ¯ D . – 13 – .3.3 Validity of the Coulomb approximation m H → The scattering states are listed in table 3. The DS states with (cid:96) + s = even are subject to an attractive Higgs-mediated potential, and the Coulombapproximation is good as long as [19] µv rel (cid:38) m H . (2.33)The condition becomes somewhat stronger for the DS scattering states with (cid:96) + s = odd,where the Higgs-mediated potential is repulsive. Moreover, it is relaxed or strengthenedin the presence of an additional attractive or repulsive Coulomb potential due to B or W exchange, as is the case with the SS -like and D ¯ D -like scattering states for (cid:96) + s = even [2,figs. 2, 3]. Bound states.
The bound states of the present model are listed in table 4. The DS states with (cid:96) + s = even are bound only by the Higgs-mediated potential; they exist if( µα H /n ) /m H > .
84, and become essentially Coulombic if this condition is strengthenedonly by a factor of a few [19]. (For example, the binding energy of the ground state exceeds80% of its Coulomb value if µα H /m H >
10 [19, fig. 13].) The mixed
SS/D ¯ D states arebound by the combined attraction of the B, W and H bosons, which removes the conditionof existence and relaxes the condition for the Coulomb approximation [3, fig. 6]. Thus, thestrongest condition for the Coulomb approximation to be satisfactory is µα H /n > few × m H . (2.34)We note that this condition is essentially satisfied everywhere BSF via Higgs emission isphenomenologically significant (cf. section 3.) Indeed, the energy available to be dissipatedmust exceed the Higgs mass, ( µ/ α H /n ) + v ] > m H , while BSF is most significantwhen v rel (cid:46) α H /n . Since α H <
1, this yields a stronger condition.Bound states can also form via B or W emission, in which case there is no phase-spacesuppression that ensures the validity of the Coulomb approximation. However, in [28] weshall see that these processes are less significant for the DM thermal decoupling in thepresent model.We further discuss the Coulomb approximation for the DM freeze-out in ref. [28].– 14 – .4 Annihilation The S , D and ¯ D fermions annihilate into SM particles via various processes that we list intable 5 together with their tree-level cross-sections and Sommerfeld factors. We consider s -wave contributions only. Because the non-relativistic potentials between the annihilatingparticles depend in many cases on their spin and gauge representations, we project theinitial states on eigenstates of total spin and Weak isospin. With the help of table 3, itis straightforward to obtain the Sommerfeld factors for all states except the spin-0 ( , SS and D ¯ D channels and we discuss in detail below.The Sommerfeld factors are expressed in term of the S ( ζ ) function defined in eq. (2.18),and the variables ζ ≡ α /v rel , ζ ≡ α /v rel , ζ H ≡ α H /v rel , ζ A ≡ α A /v rel , ζ R ≡ α R /v rel , (2.35)where the couplings α , α , α H , α A and α R have been defined in eqs. (2.5) and (2.25). Infig. 3, we present the total s -wave 2-to-2 annihilation cross-section, averaged over the dofof the incoming particles. Mixed ( ,
0) spin-0 states
The annihilation amplitudes of the SS -like and D ¯ D -like states (denoted by M ) are relatedto the perturbative amplitudes (denoted by A ) as follows i M SS -like → f ( k ) = (cid:90) d k (cid:48) (2 π ) (cid:104) [ ˜ φ k ( k (cid:48) )] SSSS i A SS → f ( k (cid:48) ) + [ ˜ φ k ( k (cid:48) )] SSD ¯ D i A D ¯ D → f ( k (cid:48) ) (cid:105) , (2.36a) i M D ¯ D -like → f ( k ) = (cid:90) d k (cid:48) (2 π ) (cid:104) [ ˜ φ k ( k (cid:48) )] D ¯ DSS i A SS → f ( k (cid:48) ) + [ ˜ φ k ( k (cid:48) )] D ¯ DD ¯ D i A D ¯ D → f ( k (cid:48) ) (cid:105) , (2.36b)where f stands for the final state, and [ ˜ φ k ( k (cid:48) )] ji are the momentum-space wavefuntions,with the j and i indices denoting the state and the component respectively, as in section 2.3.Since the perturbative s -wave SS annihilation vanishes, in our approximation there is nointerference between SS and D ¯ D channels. Taking into account that the perturbative s -wave annihilation amplitudes are independent of the momentum to lowest order in v rel ,we obtain the standard result,( σv rel ) SS -like → f = (cid:12)(cid:12) [ φ k ( )] SSD ¯ D (cid:12)(cid:12) ( σv rel ) D ¯ D → f , (2.37a)( σv rel ) D ¯ D -like → f = (cid:12)(cid:12)(cid:12) [ φ k ( )] D ¯ DD ¯ D (cid:12)(cid:12)(cid:12) ( σv rel ) D ¯ D → f , (2.37b)where [ φ k ( r )] ji are the position-space wavefunctions computed in section 2.3. Using theresults quoted in table 3, we find (cid:12)(cid:12) [ φ k ( )] SSD ¯ D (cid:12)(cid:12) = 2 α A α R (cid:104)(cid:112) S ( ζ A ) − (cid:112) S ( − ζ R ) (cid:105) ( α A + α R ) , (2.38a) (cid:12)(cid:12)(cid:12) [ φ k ( )] D ¯ DD ¯ D (cid:12)(cid:12)(cid:12) = (cid:104) α A (cid:112) S ( ζ A ) + α R (cid:112) S ( − ζ R ) (cid:105) ( α A + α R ) . (2.38b)– 15 – H = DD and SSDDDS - - - - - - - v rel σ a nn v r e l / ( π m - ) All channelswith Higgs potentialwithout Higgs potential α H ≲ α H = α H = - - - - - - - v rel σ a nn v r e l / ( π m - ) Figure 3 . The s -wave annihilation cross-sections, by initial state ( left ) and total ( right ), averagedover the dof of the incoming particles, with and without the Higgs-mediated potential (cf. table 5).Because the processes affected by the latter have either low multiplicity or small perturbative cross-sections, the Sommerfeld effect at low velocities arises mostly due to the B µ and W µ gauge bosons.Note that we have weighted the contribution of each annihilation channel with the number of DMparticles eliminated in each process as estimated upon thermal averaging (cf. ref. [28].) Note that eq. (2.38a) includes the symmetry factor of the spin-0 SS -like state, and that (cid:12)(cid:12) [ φ k ( )] SSD ¯ D (cid:12)(cid:12) = 0.We are ultimately interested in the reduction of the DM density via the various anni-hilation processes. For the spin-0 ( ,
0) states, the rate is( n S n S ) spin-0( , (cid:104) σv rel (cid:105) SS -like → f + 2( n D n ¯ D ) spin-0( , (cid:104) σv rel (cid:105) D ¯ D -like → f , (2.39)where n and (cid:104)·(cid:105) denote densities and thermal averages, and the factor 2 in the secondterm appears because D and ¯ D are not identical (cf. ref. [28].) In the limit m S = m D ,the densities are ( n S n S ) spin-0( , = ( n D n ¯ D ) spin-0( , , thus the DM density destruction rate can becomputed by regarding that the spin-0 ( , D ¯ D perturbative annihilation cross-sectionsare enhanced by the factor12 (cid:12)(cid:12) [ φ k ( )] SSD ¯ D (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) [ φ k ( )] D ¯ DD ¯ D (cid:12)(cid:12)(cid:12) = α A S ( ζ A ) + α R S ( − ζ R ) α A + α R . (2.40)We quote this result in table 5, but emphasise that ( n S n S ) spin-0( , and ( n D n ¯ D ) spin-0( , depend ex-ponentially on the corresponding masses, thus eq. (2.40) ceases to be a good approximationalready for fairly small mass differences | m D − m S | .– 16 –hannel U Y (1) SU L (2) Spin dof ( σ v rel ) / ( πm − ) Sommerfeld factor SS → HH † , − − α + 2 α H ) / S [( ζ + 3 ζ ) / − D ¯ D → HH † α + 2 α H ) / S [( ζ − ζ ) / α / α A S ( ζ A ) + α R S ( − ζ R ) α A + α R − − D ¯ D → W W α / S [( ζ − ζ ) / α / α A S ( ζ A ) + α R S ( − ζ R ) α A + α R − D ¯ D → BB , − , − α α S [( ζ − ζ ) / D ¯ D → W B − − Y L α / , (cid:80) Y L = 1 S [( ζ + 3 ζ ) / − D ¯ D → F L ¯ F L N L × α / N L = 12 S [( ζ − ζ ) / − Y R α / , (cid:80) Y R = 8 S [( ζ + 3 ζ ) / D ¯ D → f R ¯ f R , − − α H / S [( − ζ + 3 ζ ) / DD → HH , − − DS → W H α α H / S ( − ζ H )0 2 0 − DS → BH α α H / S ( − ζ H ) DS → f R ¯ F L , F L ¯ f R / , − Table 5 . Annihilation processes, their tree-level s -wave velocity-weighted cross-sections σ v rel and Sommerfeld factors. All σ v rel are averaged over the degrees of freedom of the corresponding projected scattering state (5th column). For DD , σ v rel includes the symmetry factor due to theidentical initial-state particles. For the gauge-singlet spin-0 D ¯ D channels, see text for discussion. S ( ζ ) and the various ζ parameters are defined in eqs. (2.18) and (2.35). – 17 – .5 Ground-level bound states and their decay rates Besides annihilating directly into radiation, the S , D and ¯ D fermions can form unstablebound states that decay rapidly into radiation, thereby enhancing the DM destruction rate.However, the DM annihilation via BSF is impeded by the inverse (ionisation) processes.The efficiency of BSF in reducing the DM density depends on the interplay of bound-stateionisation and decay processes and bound-to-bound transitions [9].The ionisation processes become inefficient as the temperature drops around or belowthe binding energy. This occurs earlier for the most deeply bound states, which also havethe largest decay rates. Thus, the tightest bound states have the greatest effect on the DMdensity, and for this reasons we shall consider the ground level of each bound state speciesonly, { n(cid:96)m } = { } , which has the largest binding energy. We list the ground-levelbound states in table 6.The BSF cross-sections and bound-to-bound transition rate are computed in sections 3and 4. Here we discuss the bound-state decay into radiation and the relevance of bound-to-bound transitions.
Decay into radiation
The decay rate of bound states with zero angular momentum into radiation isΓ dec B ( X X ) → f = ( σ v rel ) X X → f × | ψ n (0) | , (2.41)where X X represent the constituent fields of the bound state and f stands for the finalstate particles. ( σ v rel ) X X → f is the s -wave annihilation cross-section of an X X scat-tering state with the same quantum numbers (spin, gauge and global) as the bound state,averaged over the dof that correspond to those quantum numbers only (rather than all thedof of an X X scattering state.) For an attractive Coulomb potential of strength α , thesquared ground-state wavefunction evaluated at the origin is | ψ n (0) | = κ B /π = µ α /π ,where κ B = µα is the Bohr momentum.Taking into account the s -wave annihilation processes of table 5, we compute thetotal decay rates of the ground states and list them in table 6. The decay of the mixed SS/D ¯ D bound state occurs via its D ¯ D component, and the rate is computed analogouslyto the annihilation of the mixed scattering states described in section 2.4. For the DD bound state, a factor 2 due to the antisymmetrisation of the wavefunction has alreadybeen included in the corresponding σ v rel in table 5 and should not be included again whencomputing the decay rate of the bound state.As seen in table 5, the DS bound state cannot decay into two particles. It may decayinstead into three bosons, however the corresponding rates are suppressed by higher powersof the couplings, O ( α α H , α α H , α α α H , α H ), as well as the three-body final-state phase In addition to its ionisation becoming inefficient earlier, the cross-section for capture into the groundstate is typically larger than those of excited states, if BSF occurs via vector emission [9, 18–20]. Thisstrengthens the argument for considering only the ground states of each bound species. However, in ref. [1]it was shown that if BSF occurs via emission of a charged scalar (here the Higgs doublet), the capture intoexcited states can be comparable to the capture into the ground state. It is thus possible that excited stateshave substantial impact in the present model. We leave this investigation for future work. – 18 –oundstate ( B ) U Y (1) SU L (2) Spin dof( g B ) Bohrmomentum ( κ B ) Decay rate (Γ B ) SS/D ¯ D mα A mα A ( α + 3 α )16 (cid:18) α A α A + α R (cid:19) D ¯ D m ( α + 3 α )8 m ( α + 3 α ) [( α + 2 α H ) + 40 α ]2 · DD m ( − α + 3 α )8 m ( − α + 3 α ) α H · DS / mα H B ( SS/D ¯ D ):eq. (3.26),eqs. (4.12) and (4.19) to (4.21) Table 6 . The ground-level bound states, { n(cid:96)m } = { } , and their rates of decay into radiation,in the limit m H →
0. The decay rate of the DS bound state is suppressed, and we reference insteadthe formulae for its transition rate to the SS/D ¯ D bound state. In the first row, α A and α R arefound from eqs. (2.25) for (cid:96) = s = 0. The binding energy of each bound state is |E B | = κ B /m . space. The DS bound state instead decays much faster into the tighter SS/D ¯ D boundstate, as we shall now see. Transitions into deeper bound levels
Besides decaying directly into radiation, bound states may transition into lower-lying boundlevels via dissipation of energy. The bound-to-bound transition rates are computed in asimilar fashion to BSF processes, as we shall see in sections 3 and 4. Here we note that inradiative transitions (either scattering-to-bound or bound-to-bound), spin is conserved atleading order in the non-relativistic regime. Spin-flipping transitions can occur via emissionof a vector boson, but rely on spin-orbit interaction and are suppressed by higher powers ofthe couplings. Consequently, they are subdominant to the direct bound-state decay ratesinto two relativistic species.Considering the bound states of table 6, there are only two spin-conserving transitions,of which only one may occur with emission of a single boson contained in the theory. Notingthat α A (cid:62) α H (cf. eq. (2.25b)), this transition is B ( DS ) → B ( SS/D ¯ D ) + H. (2.42)In fact, much like BSF, bound-to-bound transitions may occur either radiatively or via scat-tering on the relativistic thermal bath. In sections 3 and 4, we compute the correspondingrates for the transition (2.42) and reference the results in table 6.– 19 – rad G (4)in G (4)out i A T Figure 4 . Radiative bound-state formation and bound-to-bound transitions.
We now compute the cross-sections for the radiative formation of the ground-level boundstates of table 6. We first outline the elements of the computation, explain the approxima-tions involved, and define some useful quantities. Then we proceed with the computationof the amplitudes and cross-sections. The final results are listed in tables 7 to 10 andillustrated in fig. 9. M k → n(cid:96)m and M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m . As depicted schematically in fig. 4,the full amplitudes consist of the radiative transition part A T computed perturbatively andconvoluted with the initial and final-state wavefunctions [18], i M k → n(cid:96)m = (cid:90) d k (cid:48) (2 π ) d p (2 π ) ˜ φ k ( k (cid:48) ) i A T ( k (cid:48) , p ) [ ˜ ψ n(cid:96)m ( p )] † √ µ . (3.1a) i M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m = (cid:90) d p (cid:48) (2 π ) d p (2 π ) ˜ ψ n (cid:48) (cid:96) (cid:48) m (cid:48) ( p (cid:48) ) √ µ i A T ( p (cid:48) , p ) [ ˜ ψ n(cid:96)m ( p )] † √ µ . (3.1b)Equations (3.1) can accommodate the possibility that the incoming and/or outgoing statesare superpositions of different Fock states, as is the case with the mixed SS/D ¯ D statesdiscussed in section 2.3. Then, the wavefunctions are vectors and A T becomes a matrix. Transition amplitudes A T . The radiative parts of the BSF and transition diagrams areshown in figs. 5 to 8. Following refs. [1, 18, 20], we compute the amplitudes A T , applyingthe standard approximations due to the hierarchy of scales and retaining only the leadingorder terms. Among else, we shall use the following approximate spinor identities valid forlow momentum changes, p (cid:39) p (cid:48) ,¯ u ( p, s ) u ( p (cid:48) , s (cid:48) ) (cid:39) +2 m δ ss (cid:48) , ¯ u ( p, s ) γ µ u ( p (cid:48) , s (cid:48) ) (cid:39) +2 p µ δ ss (cid:48) , (3.2a)¯ v ( p, s ) v ( p (cid:48) , s (cid:48) ) (cid:39) − m δ ss (cid:48) , ¯ v ( p, s ) γ µ v ( p (cid:48) , s (cid:48) ) (cid:39) +2 p µ δ ss (cid:48) . (3.2b)Moreover, we emphasise that the signs arising from the fermion permutations needed toperform the Wick contractions must be carefully taken into account, as they often differamong various diagrams contributing to the same amplitude. An example calculation ispresented in detail in appendix B. – 20 – verlap integrals. The scattering and bound state wavefunctions are listed in tables 3and 4. To express M k → n(cid:96)m and M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m compactly, we define the Coulombic overlapintegrals [1, 18–20] R k ,n(cid:96)m ( α S , α B ) ≡ ( µα B ) / (cid:90) d p (2 π ) ˜ ϕ k ( p ; α S ) ˜ ϕ ∗ n(cid:96)m ( p ; α B ) , (3.3a) J k ,n(cid:96)m ( α S , α B ) ≡ (cid:90) d p (2 π ) p ˜ ϕ k ( p ; α S ) ˜ ϕ ∗ n(cid:96)m ( p ; α B ) , (3.3b) Y W k ,n(cid:96)m ( α S , α B ) ≡ πµα (cid:90) d k (cid:48) (2 π ) d p (2 π ) k (cid:48) − p ( k (cid:48) − p ) ˜ ϕ k ( k (cid:48) ; α S ) ˜ ϕ ∗ n(cid:96)m ( p ; α B ) , (3.3c) Y H k ,n(cid:96)m ( α S , α B ) ≡ πµα H (cid:90) d k (cid:48) (2 π ) d p (2 π ) k (cid:48) − p [( k (cid:48) − p ) + m H ] ˜ ϕ k ( k (cid:48) ; α S ) ˜ ϕ ∗ n(cid:96)m ( p ; α B ) , (3.3d)and R n (cid:48) (cid:96) (cid:48) m,n(cid:96)m ( α (cid:48) B , α B ) ≡ (cid:90) d p (2 π ) ˜ ϕ n (cid:48) (cid:96) (cid:48) m (cid:48) ( p ; α (cid:48) B ) ˜ ϕ ∗ n(cid:96)m ( p ; α B ) , (3.3e)where ˜ ϕ k and ˜ ϕ n(cid:96)m are Fourier transforms of the Coulomb wavefunctions (2.16) and (2.19). In the Coulomb regime, the overlap integrals can be computed analytically, after Fouriertransforming into position space. The scattering-bound integrals of eqs. (3.3a) to (3.3d)have been computed in refs. [1, 18–20]. We compute the bound-bound integral (3.3e) inappendix C. For the ground-level bound states, { n(cid:96)m } = { } , the results are R k , ( α S , α B ) = (cid:34) π (cid:18) − ζ S ζ B (cid:19) (cid:18) ζ B ζ B (cid:19) S scl ( ζ S , ζ B ) (cid:35) / , (3.4a) J k , ( α S , α B ) = ˆ k (cid:20) πµα B (cid:18) ζ B ζ B (cid:19) S vec ( ζ S , ζ B ) (cid:21) / , (3.4b) Y W k , ( α S , α B ) = ( α /α B ) J k , ( α S , α B ) , (3.4c)lim m H → [ Y H k , ( α S , α B )] = ( α H /α B ) J k , ( α S , α B ) , (3.4d)and R , ( α (cid:48) B , α B ) = 8( α B α (cid:48) B ) / / ( α B + α (cid:48) B ) , (3.4e)where ζ S ≡ α S /v rel and ζ B ≡ α B /v rel (3.5)entail the coupling strengths α S and α B of the potential in the scattering and bound statesrespectively, and we have defined the functions [1, 9, 18–20] S scl ( ζ S , ζ B ) ≡ (cid:18) πζ S − e − πζ S (cid:19) (cid:34) ζ B e − ζ S arccot( ζ B ) (1 + ζ B ) (cid:35) . (3.6a) S vec ( ζ S , ζ B ) ≡ (cid:18) πζ S − e − πζ S (cid:19) (1 + ζ S ) (cid:34) ζ B e − ζ S arccot( ζ B ) (1 + ζ B ) (cid:35) . (3.6b) Note that R k ,n(cid:96)m , R n (cid:48) (cid:96) (cid:48) m,n(cid:96)m are dimensionless, while J k ,n(cid:96)m and Y k ,n(cid:96)m have mass-dimension − / – 21 –ote that S scl and S vec include the s - and p -wave Sommerfeld factors, S ( ζ S ) ≡ πζ S / (1 + e − πζ S ) and S ( ζ S ) = S ( ζ S )(1 + ζ S ), respectively. Indeed, eq. (3.4a) arises from the (cid:96) S = 0and eqs. (3.4b) to (3.4d) arise from the (cid:96) S = 1 modes of the scattering state wavefunctions.In eq. (3.4d) we took the limit m H → m H (cid:54) = 0 (but using the Coulomb wavefunctions.) BSF cross-sections.
The cross-sections for BSF via emission of a massless vector ( B or W ) or scalar ( H or H † ) boson are, respectively [18, 19] v rel dσ V k → n(cid:96)m d Ω = | P V | π m | M k → n(cid:96)m | (cid:20) − (cid:16) ˆ P V · ˆ M k → n(cid:96)m (cid:17) (cid:21) , (3.7a) v rel dσ H k → n(cid:96)m d Ω = | P H | π m |M k → n(cid:96)m | , (3.7b)where P V and P H are the momenta of the emitted bosons, which dissipate the kineticenergy of the relative motion and the binding energy, | P V | or (cid:112) P H + m H = ω k → n(cid:96)m , with ω k → n(cid:96)m (cid:39) E k − E n(cid:96)m (cid:39) ( m/ (cid:0) α B /n + v (cid:1) . (3.8)In eq. (3.7a), we have summed over polarisations of the emitted vector. As we shallsee in the following, to working order, the amplitudes for BSF via vector emission are M k → ∝ k , while the amplitudes for BSF via scalar emission are independent of thesolid angle Ω. Then, eqs. (3.7) simplify to [18, 19] σ V k → n(cid:96)m v rel = α B · πm (cid:18) ζ B ζ B (cid:19) | M k → n(cid:96)m | , (3.9a) σ H k → n(cid:96)m v rel = α B πm (cid:18) ζ B ζ B (cid:19) |M k → n(cid:96)m | h H ( ω k → n(cid:96)m ) , (3.9b)where h H ≡ | P H | /E H is the phase-space suppression due to the Higgs mass, h H ( ω ) ≡ (cid:0) − m H /ω (cid:1) / . (3.10) Bound-to-bound transition rates.
Similarly to the above, the rates for the radiativede-excitation of bound states are d Γ V n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m d Ω = | P V | π m | M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m | (cid:20) − (cid:16) ˆ P V · ˆ M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m (cid:17) (cid:21) , (3.11a) d Γ H n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m d Ω = | P H | π m |M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m | , (3.11b)where | P V | and | P H | are determined again by the amount of dissipated energy, which isnow the difference between the binding energies of the two states, ω n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m (cid:39) E n (cid:48) (cid:96) (cid:48) m (cid:48) − E n(cid:96)m (cid:39) ( m/ (cid:0) α B /n − α (cid:48) B /n (cid:48) (cid:1) . (3.12)For monopole transitions via H or H † emission, the amplitude is independent of the P H direction, and eq. (3.11b) simplifies toΓ H n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m = α (cid:48) B /n (cid:48) − α B /n πm |M n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m | h H ( ω n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m ) , (3.13)where the phase-space suppression h H is defined in eq. (3.10).– 22 – .2 SS/D ¯ D bound states: ( , n(cid:96)m = { } The BSF processes are listed in table 7, and the radiative part of the diagrams contributingto these processes are shown in fig. 5. We project the bound-state fields on the spin-0 statevia [ U − ] r r = (cid:15) r r / √
2, and the D ¯ D component on the SU L (2) singlet via δ i (cid:48) j (cid:48) / √ D ¯ D → B ( SS/D ¯ D ) + B The perturbative parts of the amplitude are i A s s ij (cid:104) D ¯ D → ( SS ) spin-0( , + B (cid:105) (cid:39)(cid:39) i δ ij (cid:15) s s √ y g Y H m (cid:20) k (cid:48) − p )[( k (cid:48) − p ) + m H ] + 2( k (cid:48) + p )[( k (cid:48) + p ) + m H ] (cid:21) , (3.14a) i A s s ij (cid:104) D ¯ D → ( D ¯ D ) spin-0( , + B (cid:105) (cid:39)(cid:39) i δ ij √ (cid:15) s s √ g Y D m p (2 π ) (cid:2) δ ( k (cid:48) − p − P B /
2) + δ ( k (cid:48) − p + P B / (cid:3) , (3.14b)In eq. (3.14a), the fermion permutations introduced factors ( −
1) and (+1) for the t - and u -channel diagrams. The projection on the antisymmetric spin-0 eigenstate alloted anotherfactor ( −
1) to the u -channel. The full amplitude (3.1a) is i M s s ij = 1 √ µ (cid:90) d k (cid:48) (2 π ) d p (2 π ) ˜ ϕ k (cid:18) k (cid:48) ; α + 3 α (cid:19) × (cid:20) i A s s ij (cid:104) D ¯ D → ( SS ) spin-0( , + B (cid:105) (cid:114) α R α A + α R ˜ ϕ † ( p ; α A )+ i A s s ij (cid:104) D ¯ D → ( D ¯ D ) spin-0( , + B (cid:105) (cid:114) α A α A + α R ˜ ϕ † ( p ; α A ) (cid:21) , (3.15)where only the (cid:96) S = 1 component of the scattering state wavefunction is meant to be kept,and here α A and α R should be evaluated from eqs. (2.25) for (cid:96) = s = 0. This becomes i M s s ij (cid:39) i δ ij (cid:15) s s (cid:18) πα α A + α R m (cid:19) / ×× (cid:26) √ α R Y H k , (cid:18) α + 3 α , α A (cid:19) + √ α A J k , (cid:18) α + 3 α , α A (cid:19)(cid:27) . (3.16)Note that we have neglected the ± P B / δ -functions of eq. (3.14b) thatgive rise to higher order corrections [18, 19]. Squaring and summing over the initial-stategauge indices and spins selects the ( ,
0) spin-0 D ¯ D scattering state, which has one dof.Using the overlap integrals (3.4), we find (cid:88) s ,s (cid:88) i,j (cid:12)(cid:12)(cid:12) M s s ij (cid:12)(cid:12)(cid:12) (cid:39) π (cid:18) α α A + α R (cid:19) (cid:18) α H α A (cid:114) α R α A (cid:19) (cid:18) ζ A ζ A (cid:19) S vec (cid:18) ζ + 3 ζ , ζ A (cid:19) . (3.17)The cross-section is obtained from eq. (3.9a) setting α B → α A , and is shown in table 7.– 23 – .2.2 D ¯ D → B ( SS/D ¯ D ) + W The perturbative parts of the amplitude are i ( A s s ) aij (cid:104) D ¯ D → ( SS ) spin-0( , + W (cid:105) (cid:39) i t aji (cid:15) s s √ y g m ×× (cid:20) k (cid:48) − p )[( k (cid:48) − p ) + m H ] + 2( k (cid:48) + p )[( k (cid:48) + p ) + m H ] (cid:21) , (3.18a) i ( A s s ) aij (cid:104) D ¯ D → ( D ¯ D ) spin-0( , + W (cid:105) (cid:39) i t aji √ (cid:15) s s √ g m × (cid:26) m πα k (cid:48) − p )( k (cid:48) − p ) (3.18b)+2 p (2 π ) (cid:2) δ ( k (cid:48) − p − P W / δ ( k (cid:48) − p + P W / (cid:3)(cid:9) , where the signs of the t - and u -channel diagrams in eq. (3.18a) are as in D ¯ D → SS + B above, and the first factor 2 in the first term of eq. (3.18b) is the quadratic Casimir of SU L (2) (see ref. [20] for details of this computation.) The full amplitude (3.1a) is i ( M s s ) aij (cid:39) i ( t a ) ij (cid:15) s s (cid:18) πα α A + α R m (cid:19) / × (cid:26) √ α R Y H k , (cid:18) α − α , α A (cid:19) + √ α A (cid:20) J k , (cid:18) α − α , α A (cid:19) + Y W k , (cid:18) α − α , α A (cid:19)(cid:21)(cid:27) , (3.19)where again α A and α R should be evaluated from eqs. (2.25) for (cid:96) = s = 0, and we haveneglected the ± P W / δ -functions in eq. (3.18b). Squaring, summing overthe initial and final state gauge indices and spins selects the ( ,
0) spin-0 D ¯ D state, whichhas three dof. Using eqs. (3.4) and (3.6), we find13 (cid:88) s ,s (cid:88) i,j,a | ( M s s ) aij | (cid:39) π (cid:18) α α A + α R (cid:19) (cid:18) α α A + α H α A (cid:114) α R α A (cid:19) × (cid:18) ζ A ζ A (cid:19) S vec (cid:18) ζ − ζ , ζ A (cid:19) . (3.20)The cross-section is obtained from eq. (3.9a) setting α B → α A , and is shown in table 7. DS → B ( SS/D ¯ D ) + H The perturbative parts of the amplitude are i A s s i,h (cid:104) DS → ( SS ) spin-0( , + H (cid:105) (cid:39) − i δ ih (cid:15) s s √ y m × (3.21a) × (2 π ) (cid:2) δ ( k (cid:48) − p − P H /
2) + δ ( k (cid:48) − p + P H / (cid:3) , i A s s i,h (cid:104) DS → ( D ¯ D ) spin-0( , + H (cid:105) (cid:39) − i δ ih √ (cid:15) s s √ y m (2 π ) δ ( k (cid:48) − p + P H / . (3.21b)where now the fermion permutations introduced factors (+1) and ( −
1) for the t - and u -channel DS → SS + H diagrams respectively, and ( −
1) for the DS → D ¯ D + H diagram.We note that there are two diagrams where an off-shell vector boson ( B or W ) emittedfrom one leg and an off-shell Higgs emitted from the other leg fuse to produce the final-state– 24 –iggs. In appendix D, we show that these diagrams are of higher order, thus we do notconsider them here. The full amplitude (3.1a) is i M s s i,h (cid:39) − i δ ih (cid:15) s s (cid:115) πα H α A √ α A + √ α R √ α A + α R R k , ( α H , α A ) , (3.22)where we have neglected the ± P H / δ -functions in eqs. (3.21). Squaringand summing over the initial and final state gauge indices and spins selects the ( , / DS state, which has two dof. Using the overlap integrals (3.4), we find12 (cid:88) s ,s (cid:88) i,h |M s s i,h | (cid:39) π α H α A (cid:0) √ α A + √ α R (cid:1) α A + α R (cid:18) − ζ H ζ A (cid:19) (cid:18) ζ A ζ A (cid:19) S scl ( ζ H , ζ A ) . (3.23)The cross-section is obtained from eq. (3.9b) setting α B → α A , and is shown in table 7. B ( DS ) → B ( SS/D ¯ D ) + H Using the perturbative amplitudes (3.21), we may now compute the rate of bound-to-boundtransition (2.42). Projecting on the spin-0 DS state, and taking into account the bound-state wavefuntions of table 4, we find that the full amplitude is, analogously to eq. (3.22),given by i M i,h (cid:39) − i δ ih m (cid:112) πα H √ α A + √ α R √ α A + α R R , ( α H , α A ) , (3.24)Squaring and averaging over the initial and final state gauge indices, and using the overlapintegral (3.4e), we find12 (cid:88) i,h |M i,h | (cid:39) πm α H (cid:0) √ α A + √ α R (cid:1) α A + α R ( α H α A ) ( α H + α A ) . (3.25)The transition rate is found from eq. (3.13) to beΓ DS → SS/D ¯ D = 2 m α H ( α A − α H ) (cid:0) √ α A + √ α R (cid:1) α A + α R ( α A α H ) ( α A + α H ) (cid:20) − m H m ( α A − α H ) (cid:21) / . (3.26)– 25 – ound state SS/D ¯ D : ( , , spin 0 , { n(cid:96)m } = { } Scattering state (spin 0) Radboson Cross-sectionState U Y (1) SU L (2) dof (cid:96) S ( σ BSF v rel ) / ( πm − ) D ¯ D B α α A α A + α R (cid:20) α H α A (cid:114) α R α A (cid:21) S vec (cid:18) ζ + 3 ζ , ζ A (cid:19) SS B SS scattering state) D ¯ D W α α A α A + α R (cid:20) α α A + α H α A (cid:114) α R α A (cid:21) S vec (cid:18) ζ − ζ , ζ A (cid:19) DS +1 / H α H α A (cid:0) √ α A + √ α R (cid:1) α A + α R (cid:18) − α H α A (cid:19) S scl ( ζ H , ζ A ) h H ( ω )¯ DS − / H † same as above Table 7 . Radiative processes and cross-sections for capture into the ground level of the
SS/D ¯ D bound states. Here, α A , α R are obtained from eqs. (2.25) for (cid:96) = s = 0. Each cross-section isaveraged over the dof of the corresponding scattering state (4th column.) S vec and S scl are definedin eqs. (3.6), and h H in eq. (3.10). Here, ω = m ( α A + v ) / s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP B i i (cid:48) j j (cid:48) P/ pP/ − pr r P/ − pP/ pr r s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W i i (cid:48) j j (cid:48) a P/ pP/ − pr r P/ − pP/ pr r s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H i i (cid:48) j (cid:48) h P/ pP/ − pr r P/ − pP/ pr r Figure 5 . The radiative parts of the diagrams contributing to the formation of
SS/D ¯ D boundstates. Top row: D ¯ D → B ( SS/D ¯ D ) + B . Middle row: D ¯ D → B ( SS/D ¯ D ) + W . Bottom row: DS → B ( SS/D ¯ D ) + H , which has also a conjugate counterpart (not shown.) Single and doublelines correspond to S and D fermions, while vector, gluon and dashed lines to B , W and H bosons.The arrows on the field lines denote the flow of Hypercharge. Wherever not shown, the momenta,spins and gauge indices of the external particles can be deduced from the other graphs. – 26 – .3 D ¯ D bound states: ( , n(cid:96)m = { } The BSF processes are listed in table 8, and the radiative parts of the diagrams contributingto these processes are shown in fig. 6. For all processes below, we project the bound-statefields on the SU L (2) singlet via δ i (cid:48) j (cid:48) / √
2. Moreover, since spin is conserved at working order,as already seen in section 3.2, we project both the scattering and the bound states on thespin-1 configuration by contracting the spin indices with [ U σ spin-1 ] s s [ U ρ † spin-1 ] r r , where theindices σ, ρ = − , , U spin-1 are well-known, we shall not need themexplicitly. We will instead only invoke that the operators [ U σ † spin-1 ] s s are symmetric in s , s , and [ U σ spin-1 ] s s [ U ρ † spin-1 ] s s = δ σρ . D ¯ D -like → B ( D ¯ D ) + B Besides the projections mentioned above, here we also project the D ¯ D component of thescattering state on the SU L (2) singlet configuration via δ ij / √
2. The perturbative parts ofthe amplitude are i A σρ (cid:104) ( SS ) spin-1 → ( D ¯ D ) spin-1( , + B (cid:105) (cid:39)(cid:39) − i √ δ σρ y g Y H m (cid:20) k (cid:48) − p )[( k (cid:48) − p ) + m H ] + 2( k (cid:48) + p )[( k (cid:48) + p ) + m H ] (cid:21) , (3.27a) i A σρ (cid:104) ( D ¯ D ) spin-1( , → ( D ¯ D ) spin-1( , + B (cid:105) (cid:39)(cid:39) + i δ σρ g Y D m p (2 π ) (cid:2) δ ( k (cid:48) − p − P B /
2) + δ ( k (cid:48) − p + P B / (cid:3) . (3.27b)In eq. (3.27a), the fermion permutations introduced factors ( −
1) and (+1) for the t - and u -channel diagrams. Upon projection on the symmetric spin-1 eigenstate, their relativesign does not change. The factor √ SU L (2) singlet final D ¯ D state. Using the wavefunctions listed intables 3 and 4, we find the full amplitude (3.1a) to be i M σρ (cid:39) i δ σρ α A + α R √ πα m ×× (cid:26) √ α A α R (cid:20) − Y H k , (cid:18) α A , α + 3 α (cid:19) + Y H k , (cid:18) − α R , α + 3 α (cid:19)(cid:21) + α A J k , (cid:18) α A , α + 3 α (cid:19) + α R J k , (cid:18) − α R , α + 3 α (cid:19)(cid:27) , (3.28)where here α A and α R should be evaluated from eqs. (2.25) for (cid:96) S = s = 1 (scatteringstate). As before, we have neglected the ± P B / δ -functions in eq. (3.27b).Next, we square, sum over the spins, and average over the three dof of the incoming spin-1state. Using the overlap integrals eq. (3.4), we find13 (cid:88) σ,ρ =1 | M σρ | (cid:39) π α α B α A ( α A + α R ) (cid:18) ζ B ζ B (cid:19) ×× (cid:20)(cid:18) − α H α B (cid:114) α R α A (cid:19) S / ( ζ A , ζ B ) + (cid:18) α R α A + 2 α H α B (cid:114) α R α A (cid:19) S / ( − ζ R , ζ B ) (cid:21) , (3.29)– 27 –here here α B = ( α + 3 α ) / ζ B = ( ζ + 3 ζ ) /
4. The cross-section isobtained from eq. (3.9a), and is shown in table 8. SS -like → B ( D ¯ D ) + B Using the perturbative parts (3.27), and the wavefunctions listed in tables 3 and 4, we findthe full amplitude (3.1a) i M σρ (cid:39) i δ σρ α A + α R √ πα m ×× (cid:26) − (cid:20) α R Y H k , (cid:18) α A , α + 3 α (cid:19) + α A Y H k , (cid:18) − α R , α + 3 α (cid:19)(cid:21) + √ α A α R (cid:20) + J k , (cid:18) α A , α + 3 α (cid:19) − J k , (cid:18) − α R , α + 3 α (cid:19)(cid:21)(cid:27) , (3.30)where again α A and α R should be evaluated from eqs. (2.25) for (cid:96) S = s = 1 (scatteringstate), and with the help of the overlap integrals (3.4),13 (cid:88) σ,ρ =1 | M σρ | (cid:39) π α α B α A α R ( α A + α R ) (cid:18) ζ B ζ B (cid:19) ×× (cid:20)(cid:18) − α H α B (cid:114) α R α A (cid:19) S / ( ζ A , ζ B ) − (cid:18) α H α B (cid:114) α A α R (cid:19) S / ( − ζ R , ζ B ) (cid:21) , (3.31)with α B = ( α +3 α ) / ζ B = ( ζ +3 ζ ) /
4. The cross-section is obtained from eq. (3.9a),and is shown in table 8. D ¯ D → B ( D ¯ D ) + W The perturbative part of the amplitude is (cf. eq. (3.18b)) i ( A σρ ) aij (cid:104) ( D ¯ D ) spin-1 → ( D ¯ D ) spin-1( , + W (cid:105) (cid:39) i δ σρ t aji √ g m ×× (cid:26) m πα k (cid:48) − p )( k (cid:48) − p ) + 2 p (2 π ) (cid:2) δ ( k (cid:48) − p − P W /
2) + δ ( k (cid:48) − p + P W / (cid:3)(cid:27) . (3.32)Using the wavefunctions listed in tables 3 and 4, and anticipating that the scattering statewill be an SU L (2) triplet, we find the full amplitude (3.1a), i ( M σρ ) aij (cid:39) i δ σρ t aji (cid:112) πα m ×× (cid:26) Y W k , (cid:18) α − α , α + 3 α (cid:19) + J k , (cid:18) α − α , α + 3 α (cid:19)(cid:27) . (3.33)Squaring and summing over gauge and spin indices, projects the scattering state on thespin-1 SU L (2) triplet that has 9 dof. Using the overlap integrals (3.4), we find19 (cid:88) σ,ρ (cid:88) i,j,a (cid:12)(cid:12)(cid:12) [ M σρ ] aij (cid:12)(cid:12)(cid:12) (cid:39) π α α B (cid:18) α α B (cid:19) (cid:18) ζ B ζ B (cid:19) S vec (cid:18) ζ − ζ , ζ B (cid:19) . (3.34)The cross-section is obtained from eq. (3.9a), and is shown in table 8. It agrees with theresults of refs. [3, 20] appropriately adjusted.– 28 – .3.4 DS → B ( D ¯ D ) + H The perturbative part of the amplitude is i A σρi,h (cid:104) ( DS ) spin-1 → ( D ¯ D ) spin-1( , + H (cid:105) (cid:39) i δ σρ δ ih √ y m (2 π ) δ ( k (cid:48) − p + P H / . (3.35)Using the wavefunctions listed in tables 3 and 4, we find the full amplitude (3.1a), i M σρi,h (cid:39) i δ σρ δ ih (cid:115) πα H [( α + 3 α ) / R k , (cid:18) − α H , α + 3 α (cid:19) . (3.36)and taking into account the overlap integral (3.4a),16 (cid:88) σ,ρ (cid:88) i,h (cid:12)(cid:12)(cid:12) M σρi,h (cid:12)(cid:12)(cid:12) (cid:39) π α H α B (cid:18) α H α B (cid:19) (cid:18) ζ B ζ B (cid:19) S scl ( − ζ H , ζ B ) , (3.37)with α B = ( α +3 α ) / ζ B = ( ζ +3 ζ ) /
4. The cross-section is obtained from eq. (3.9b),and is shown in table 8. – 29 – ound state D ¯ D : ( , , spin 1 , { n(cid:96)m } = { } Scattering state (spin 1) Radboson Cross-sectionState U Y (1) SU L (2) dof (cid:96) S ( σ BSF v rel ) / ( πm − ) D ¯ D -like 0 B α α B α A ( α A + α R ) (cid:20)(cid:18) − α H α B (cid:114) α R α A (cid:19) S / ( ζ A , ζ B )+ (cid:18) α R α A + 2 α H α B (cid:114) α R α A (cid:19) S / ( − ζ R , ζ B ) (cid:21) SS -like 0 B α α B α A α R ( α A + α R ) (cid:20)(cid:18) − α H α B (cid:114) α R α A (cid:19) S / ( ζ A , ζ B ) − (cid:18) α H α B (cid:114) α A α R (cid:19) S / ( − ζ R , ζ B ) (cid:21) D ¯ D W α α B (cid:18) α α B (cid:19) S vec (cid:18) ζ − ζ , ζ B (cid:19) DS +1 / H α H α B (cid:18) α H α B (cid:19) S scl ( − ζ H , ζ B ) h H ( ω )¯ DS − / H † same as above Table 8 . Same as table 7 for the D ¯ D bound states. Here, the bound-state coupling is α B =( α + 3 α ) /
4, and correspondingly ζ B = ( ζ + 3 ζ ) /
4. In the first two processes, α A and α R should be evaluated from eq. (2.25) with (cid:96) S = s = 1 for the scattering state. For the phase-spacesuppression h H , here ω = m (cid:2) ( α + 3 α ) /
16 + v (cid:3) / s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP B i i (cid:48) j j (cid:48) K/ k ( (cid:48) ) K/ − k ( (cid:48) ) s s i (cid:48) j (cid:48) K/ − k ( (cid:48) ) K/ k ( (cid:48) ) s s s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W i i (cid:48) j j (cid:48) a s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H i i (cid:48) j (cid:48) h Figure 6 . Same as in fig. 5, for the formation of D ¯ D bound states. Top: D ¯ D -like → B ( D ¯ D ) + B and SS -like → B ( D ¯ D )+ B . Bottom left: D ¯ D → B ( D ¯ D )+ W . Bottom right: DS → B ( SS/D ¯ D )+ H . – 30 – .4 DD bound states: ( , n(cid:96)m = { } The BSF processes are listed in table 9, and the radiative part of the diagrams contributingto these processes are shown in fig. 7. In all processes below, we project the bound-statefields on the SU L (2) singlet via (cid:15) i (cid:48) j (cid:48) / √
2. Moreover, as in section 3.3, we project both thescattering and the bound states on the spin-1 configuration via [ U σ spin-1 ] s s [ U ρ † spin-1 ] r r , andinvoke that [ U σ † spin-1 ] s s are symmetric in s , s , and [ U σ spin-1 ] s s [ U ρ † spin-1 ] s s = δ σρ . DD → B ( DD ) + W The perturbative part of the amplitude is i ( A σρ ) aij (cid:104) ( DD ) spin-1 → ( DD ) spin-1( , + W (cid:105) (cid:39) i δ σρ g m ×× (cid:26)(cid:20) m πα k (cid:48) − p )( k (cid:48) − p ) (cid:18) − i f abc t bi (cid:48) i t cj (cid:48) j (cid:15) i (cid:48) j (cid:48) √ (cid:19) + 2 p (2 π ) δ ( k (cid:48) − p ) (cid:18) t ai (cid:48) i (cid:15) i (cid:48) j √ − t aj (cid:48) j (cid:15) ij (cid:48) √ (cid:19)(cid:21) − ( i ↔ j, k → − k ) } , (3.38)where the last line accounts for the u -channel diagrams. The different number of fermionpermutations in the t - and u -channel diagrams introduces a relative ( −
1) factor between thetwo, while the projection on the symmetric spin-1 state does not. Here we have neglectedthe ± P W / δ -functions already at the level of the perturbative amplitude.It is easy to check that the gauge factors in eq. (3.38) are symmetric in i ↔ j , asexpected, since the scattering state must be an SU L (2) triplet. Convoluting with thescattering state wavefunction and setting k → − k for the u channel renders the latteridentical to the t -channel up to the extra factor − ( − (cid:96) S = +1 since (cid:96) S = 1. Thus, the t and u channels add up, and we find the full amplitude (3.1a) to be i ( M σρ ) aij (cid:39) i δ σρ (cid:112) πα m × (cid:20) Y W k , (cid:18) − α + α , − α + 3 α (cid:19) (cid:18) − i f abc t bi (cid:48) i t cj (cid:48) j (cid:15) i (cid:48) j (cid:48) √ (cid:19) + J k , (cid:18) − α + α , − α + 3 α (cid:19) (cid:18) t ai (cid:48) i (cid:15) i (cid:48) j √ − t aj (cid:48) j (cid:15) ij (cid:48) √ (cid:19)(cid:21) , (3.39)where we also took into account the symmetry factors of the scattering and bound statewavefunctions, as stated in tables 3 and 4. Considering the relation (3.4c) between theoverlap integrals, the above simplifies to i ( M σρ ) aij (cid:39) i δ σρ (cid:112) πα m × J k , (cid:18) − α + α , α B (cid:19) G aij , (3.40)where here α B = ( − α + 3 α ) /
4, and G aij is the gauge factor G aij ≡ t ai (cid:48) i (cid:15) i (cid:48) j √ − t aj (cid:48) j (cid:15) ij (cid:48) √ α α B (cid:18) − i f abc t bi (cid:48) i t cj (cid:48) j (cid:15) i (cid:48) j (cid:48) √ (cid:19) , (3.41)with G aij G aij ∗ = 3 (cid:18) α α B (cid:19) . (3.42)– 31 –quaring eq. (3.40) and summing over gauge indices and spins projects the scattering stateon the spin-1 SU L (2) triplet configuration that has nine dof. Considering the overlapintegral (3.4b), we find19 (cid:88) σ,ρ (cid:88) i,j,a | ( M σρ ) aij | (cid:39) π α α B (cid:18) α α B (cid:19) (cid:18) ζ B ζ B (cid:19) S vec (cid:18) − ζ + ζ , ζ B (cid:19) . (3.43)The cross-section is obtained from eq. (3.9a) and is shown in table 9. DS → B ( DD ) + H † The perturbative part of the amplitude is i A σρi,h (cid:104) ( DS ) spin-1 → ( DD ) spin-1( , + H (cid:105) (cid:39) − i δ σρ (cid:15) ih √ y m (2 π ) δ ( k (cid:48) − p + P H / i δ σρ (cid:15) hi √ y m (2 π ) δ ( k (cid:48) + p + P H / , (3.44)where the fermion permutations alloted factors (+1) and ( −
1) to the t and u channelsrespectively. As seen from eq. (3.44), the resulting relative sign is canceled upon contractionof the bound-state fields on the SU L (2) singlet state. Convoluting eq. (3.44) with thescattering and bound state wavefunctions found in tables 3 and 4, and setting p → − p renders the u -channel contribution the same as t -channel, with the extra factor ( − (cid:96) = +1for (cid:96) = 0. Thus the t and u channels add up, and we find i M σρi,h (cid:39) − i δ σρ (cid:15) ih (cid:115) πα H α B R k , ( − α H , α B ) . (3.45)Here α B = ( − α + 3 α ) /
4, and we have included the symmetry factors of the scatteringand bound state wavefunctions. Squaring and summing over the gauge indices and spins,and using the overlap integral eq. (3.4a), we find16 (cid:88) σ,ρ (cid:88) i,h |M σρi,h | (cid:39) π α H α B (cid:18) α H α B (cid:19) (cid:18) ζ B ζ B (cid:19) S scl ( − ζ H , ζ B ) , (3.46)where we averaged over the six dof of the spin-1 SU L (2) doublet scattering state. Thecross-section is obtained from eq. (3.9b) and is shown in table 9.– 32 – ound state DD : ( , , spin 1 , { n(cid:96)m } = { } Scattering state (spin 1) Radboson Cross-sectionState U Y (1) SU L (2) dof (cid:96) S ( σ BSF v rel ) / ( πm − ) DD B DD scattering state) DD W α B α (cid:18) α α B (cid:19) S vec (cid:18) − ζ + ζ , ζ B (cid:19) DS +1 / H † α H α B (cid:18) α H α B (cid:19) S scl ( − ζ H , ζ B ) h H ( ω ) Table 9 . Same as table 7 for the DD bound states. Here, α B = ( − α +3 α ) / ζ B = ( − ζ +3 ζ ) / ω = m (cid:2) ( − α + 3 α ) /
16 + v (cid:3) /
4. All processes have conjugate counterparts. s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W i i (cid:48) j j (cid:48) a s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W i i (cid:48) j j (cid:48) a s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ − pP/ pP W i j (cid:48) j i (cid:48) a s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ − pP/ pP W i j (cid:48) j i (cid:48) a s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ − pP/ pP W s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H † i i (cid:48) j (cid:48) h s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ − pP/ pP H † i j (cid:48) i (cid:48) h Figure 7 . Same as fig. 5, for the formation of DD bound states. Top two rows: DD →B ( DD ) + W . Bottom row: DS → B ( DD ) + H † . – 33 – ound state DS : ( , / , spin 0 , { n(cid:96)m } = { } Scattering state (spin 0) RadBoson Cross-sectionState U Y (1) SU L (2) dof (cid:96) S ( σ BSF v rel ) / ( πm − ) DS +1 / B α H α S vec ( − ζ H , ζ H ) DS +1 / W α H α S vec ( − ζ H , ζ H ) SS -like 0 H † α A ( α A + α R ) (cid:20)(cid:18) (cid:114) α R α A (cid:19) (cid:18) α R α H (cid:19) S / ( − ζ R , ζ H ) − (cid:18) α R α A − (cid:114) α R α A (cid:19) (cid:18) − α A α H (cid:19) S / ( ζ A , ζ H ) (cid:21) h H ( ω ) D ¯ D -like 0 H † α A ( α A + α R ) (cid:20)(cid:18) α R α A − (cid:114) α R α A (cid:19)(cid:18) α R α H (cid:19) S / ( − ζ R , ζ H )+ (cid:18) (cid:114) α R α A (cid:19) (cid:18) − α A α H (cid:19) S / ( ζ A , ζ H ) (cid:21) h H ( ω ) D ¯ D H † (cid:18) − α − α α H (cid:19) S scl (cid:18) ζ − ζ , ζ H (cid:19) h H ( ω ) DD H DD scattering state) DD H (cid:18) α + α α H (cid:19) S scl (cid:18) − ζ + ζ , ζ H (cid:19) h H ( ω ) Table 10 . Same as table 10 for the DS bound states. For the SS -like and D ¯ D -like states, α A and α R should be evaluated from eq. (2.25) for (cid:96) S = s = 0. Here, ω = m ( α H + v ) /
4. All processeshave conjugate counterparts. DS bound states: ( , / n(cid:96)m = { } The BSF processes are listed in table 10, and the radiative part of the diagrams contributingto these processes are shown in fig. 8. We project the bound-state fields on the spin-0 statevia U s s spin-0 [ U − ] r r = ( (cid:15) s s / √ (cid:15) r r / √ DS → B ( DS ) + B The perturbative part of the amplitude is i A i,i (cid:48) (cid:2) ( DS ) spin-0 → ( DS ) spin-0 + B (cid:3) (cid:39) i δ ii (cid:48) g Y D m p (2 π ) δ ( k (cid:48) − p − P B / i δ ii (cid:48) g Y H y m k (cid:48) + p )[( k + p ) + m H ] , (3.47)where the fermion permutations alloted factors (+1) and ( −
1) to the t and u channels.Upon projection on the spin-0 states, the u channel acquired another factor ( − i M i,i (cid:48) (cid:39) i δ ii (cid:48) √ πα m (cid:2) J k , ( − α H , α H ) + 2 Y H k , ( − α H , α H ) (cid:3) , (3.48)– 34 – r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP B i i (cid:48) i i (cid:48) , s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP W i i (cid:48) a i i (cid:48) s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H † i i (cid:48) j h s s K/ k ( (cid:48) ) K/ − k ( (cid:48) ) i (cid:48) h s s K/ − k ( (cid:48) ) K/ k ( (cid:48) ) i (cid:48) hs r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H i i (cid:48) j h s r s r K/ − k ( (cid:48) ) K/ k ( (cid:48) ) P/ pP/ − pP H i i (cid:48) j h Figure 8 . Same as fig. 5, for the formation of DS bound states. Top row: DS → B ( DS )+ B (left)and DS → B ( DS ) + W (right). Middle row: D ¯ D -like → B ( DS ) + H † and SS -like → B ( DS ) + H † .The diagram on the left gives also D ¯ D → B ( DS ) + H † . Bottom row: DD → B ( DS ) + H . Next we square, sum over the gauge indices and average over the two dof of the spin-0scattering state. Using the overlap integrals (3.4), we obtain12 (cid:88) i,i (cid:48) | M i,i (cid:48) | (cid:39) π α α H (cid:18) ζ H ζ H (cid:19) S vec ( − ζ H , ζ H ) . (3.49)The cross-section is found from eq. (3.9a) for α B = α H and is shown in table 10. DS → B ( DS ) + W The perturbative part of the amplitude is i A ai,i (cid:48) (cid:2) ( DS ) spin-0 → ( DS ) spin-0 + W (cid:3) (cid:39) i t ai (cid:48) i g m p (2 π ) δ ( k (cid:48) − p − P B / i t ai (cid:48) i g y m k (cid:48) + p )[( k + p ) + m H ] , (3.50)where the signs are determined as in eq. (3.47). The full amplitude is i M ai,i (cid:48) (cid:39) i t ai (cid:48) i (cid:112) πα m (cid:2) J k , ( − α H , α H ) + 2 Y H k , ( − α H , α H ) (cid:3) , (3.51)– 35 –nd 12 (cid:88) i,i (cid:48) ,a | M ai,i (cid:48) | (cid:39) π α α H (cid:18) ζ H ζ H (cid:19) S vec ( − ζ H , ζ H ) , (3.52)where we used Tr( t a t a ) = 3 /
2. The cross-section is found from eq. (3.9a) for α B = α H andis shown in table 10. SS -like → B ( DS ) + H † The perturbative parts of the amplitude are i A i (cid:48) h (cid:104) ( SS ) spin-0 → ( DS ) spin-0 + H † (cid:105) (cid:39) − i δ ih y m (2 π ) δ ( k (cid:48) − p − P H † / − i δ ih y m (2 π ) δ ( k (cid:48) + p + P H † /
2) (3.53a) i A ij,i (cid:48) h (cid:104) ( D ¯ D ) spin-0 → ( DS ) spin-0 + H † (cid:105) (cid:39) i δ ii (cid:48) δ jh y m (2 π ) δ ( k (cid:48) − p + P H † / . (3.53b)In eq. (3.53a), the fermion permutations alloted signs (+1) and ( −
1) factors to the t and u channels. Upon projection on the spin-0 state, the u channel acquired another factor ( − D ¯ D component of the scattering state in eq. (3.53b), on the SU L (2)singlet configuration via δ ij / √ i A i (cid:48) h (cid:104) ( D ¯ D ) spin-0( , → ( DS ) spin-0 + H † (cid:105) (cid:39) i δ i (cid:48) h √ y m (2 π ) δ ( k (cid:48) − p + P H † / . (3.53c)Considering the wavefunctions of tables 3 and 4, the full amplitude (3.1a) is i M i (cid:48) h (cid:39) − i δ i (cid:48) h (cid:115) πα H (cid:18) α A α A + α R (cid:19) ×× (cid:20)(cid:18) (cid:114) α R α A (cid:19) R k , ( − α R , α H ) − (cid:18) α R α A − (cid:114) α R α A (cid:19) R k , ( α A , α H ) (cid:21) , (3.54)where α A and α R should be evaluated from eq. (2.25) for (cid:96) S = s = 0. Using the overlapintegrals (3.4), (cid:88) i (cid:48) ,h |M i (cid:48) h | (cid:39) π α H (cid:18) α A α A + α R (cid:19) (cid:18) ζ H ζ H (cid:19) × (3.55) × (cid:20)(cid:18) (cid:114) α R α A (cid:19) (cid:18) α R α H (cid:19) S / ( − ζ R , ζ H ) − (cid:18) α R α A − (cid:114) α R α A (cid:19) (cid:18) − α A α H (cid:19) S / ( ζ A , ζ H ) (cid:21) . The cross-section is found from eq. (3.9b) for α B = α H and is shown in table 10. D ¯ D -like → B ( DS ) + H † Starting from the perturbative parts (3.53a) and (3.53c), and considering the wavefunctionsof tables 3 and 4, the full amplitude (3.1a) is i M i (cid:48) h (cid:39) i δ i (cid:48) h (cid:115) πα H (cid:18) α A α A + α R (cid:19) ×× (cid:20)(cid:18) α R α A − (cid:114) α R α A (cid:19) R k , ( − α R , α H ) + (cid:18) (cid:114) α R α A (cid:19) R k , ( α A , α H ) (cid:21) , (3.56)– 36 –here again α A and α R should be evaluated from eq. (2.25) for (cid:96) S = s = 0. Using theoverlap integrals (3.4), (cid:88) i (cid:48) ,h |M i (cid:48) h | (cid:39) π α H (cid:18) α A α A + α R (cid:19) (cid:18) ζ H ζ H (cid:19) × (3.57) × (cid:20)(cid:18) α R α A − (cid:114) α R α A (cid:19) (cid:18) α R α H (cid:19) S / ( − ζ R , ζ H ) + (cid:18) (cid:114) α R α A (cid:19) (cid:18) − α A α H (cid:19) S / ( ζ A , ζ H ) (cid:21) . The cross-section is found from eq. (3.9b) for α B = α H and is shown in table 10. D ¯ D → B ( DS ) + H † The perturbative part of the amplitude is given in eq. (3.53b). We project it the D ¯ D scattering state on the SU L (2) triplet configuration via t aji / (cid:112) C ( ) = √ t aji , where C ( ) =1 / SU L (2) doublet representation, and obtain i A ai (cid:48) h (cid:104) ( D ¯ D ) spin-0( , ) → ( DS ) spin-0( , / ) + H † (cid:105) (cid:39) i √ t ahi (cid:48) y m (2 π ) δ ( k (cid:48) − p + P H † / . (3.58)Considering the wavefunctions of tables 3 and 4, the full amplitude (3.1a) is i M ai (cid:48) h (cid:39) i ( √ t ahi (cid:48) ) (cid:115) πα H R k , (cid:18) α − α , α H (cid:19) . (3.59)Squaring, summing over the final state gauge indices, and averaging over the three dof ofthe scattering state, we obtain13 (cid:88) i (cid:48) ,h,a |M ai (cid:48) h | (cid:39) π α H (cid:18) − α − α α H (cid:19) (cid:18) ζ H ζ H (cid:19) S scl (cid:18) ζ − ζ , ζ H (cid:19) , (3.60)where we used Tr( √ t a √ t a ) = 3. The cross-section is found from eq. (3.9b) for α B = α H and is shown in table 10. DD → B ( DS ) + H The perturbative part of the amplitude is i A ij,i (cid:48) h (cid:2) ( DD ) spin-0 → ( DS ) spin-0 + H (cid:3) (cid:39)(cid:39) − i y m (cid:2) δ ii (cid:48) δ jh (2 π ) δ ( k (cid:48) − p + P H /
2) + δ ji (cid:48) δ ih (2 π ) δ ( k (cid:48) + p − P H / (cid:3) , (3.61)where the fermion permutations alloted signs (+1) and ( −
1) factors to the t and u channels.Upon projection on the spin-0 state, the u channel acquired another factor ( − DD scattering state on the SU L (2) triplet configuration via the symbolicoperator ( U ) aij = ( U ) aji , which satisfies ( U ) aij ( U † ) bji = δ ab , i A ai (cid:48) h (cid:104) ( DD ) spin-0( , → ( DS ) spin-0( , / + H (cid:105) (cid:39)(cid:39) − i ( U ) i (cid:48) h y m (cid:2) (2 π ) δ ( k (cid:48) − p + P H /
2) + (2 π ) δ ( k (cid:48) + p − P H / (cid:3) . (3.62)– 37 –onsidering the wavefunctions of tables 3 and 4, the full amplitude (3.1a) is i M ai (cid:48) h (cid:39) − i ( U ) i (cid:48) h (cid:115) πα H R k , (cid:18) − α + α , α H (cid:19) , (3.63)where we included the symmetry factor of the DD wavefunction. Squaring, summing overthe final state gauge indices, and averaging over the three dof of the scattering state, andusing the overlap integral (3.4a), we obtain13 (cid:88) i (cid:48) ,h,a |M ai (cid:48) h | (cid:39) π α H (cid:18) α + α α H (cid:19) (cid:18) ζ H ζ H (cid:19) S scl (cid:18) − ζ + ζ , ζ H (cid:19) . (3.64)The cross-section is found from eq. (3.9b) for α B = α H and is shown in table 10.– 38 – - - - - - - v rel σ v r e l / ( π m - ) No DS bound states ifHiggs potential is neglected - - - σ v r e l / ( π m - ) No DD formation via B emission DD formation via W emission independent of Higgs potential - - - σ v r e l / ( π m - ) DD formation via W emissionindependent of Higgs potential B emission W emission H (†) emission α H = / Higgs potentialw / o Higgs potential - - - σ v r e l / ( π m - ) - - - v rel D S : ( , / ) , s p i n DD : ( , ) , s p i n DD : ( , ) , s p i n α H = α H = α H = / Higgs potentialw / o Higgs potential SS / DD : ( , ) , s p i n Figure 9 . The radiative BSF cross-sections vs relative velocity. The four rows correspond tocapture into the ground levels of the bound states marked on the right. In the DD and DS panels,we have included the capture into the conjugate bound states, and all BSF channels have beenweighted with the number of DM particles eliminated in each process as estimated upon thermalaveraging (cf. ref. [28].) Left column : The contributions of B , W and H ( † ) emission, for α H = 0 . v rel ∼ . H ( † ) emission) than annihilation, σ ann v rel / ( πm − ) ∼ − . Rightcolumn : The sum of the B -, W - and H ( † ) -emission contributions, for different values of α H . In bothcolumns, we show the cross-sections considering and neglecting the Higgs-mediated potential. Thevarious σv rel normalised to πm − are independent of the DM mass, except for the cutoff on BSFvia H ( † ) emission due the Higgs mass; for this purpose, we have used m = 50 TeV and temperature T = 300 GeV which sets m H (cid:39)
168 GeV. Note that we take the Higgs potential to be Coulombic(see text for discussion.) – 39 – .6 Unitarity and BSF via Higgs emission
The unitarity of the S matrix implies an upper limit on the partial-wave inelastic cross-sections, σ inel (cid:96) (cid:54) σ uni (cid:96) = (2 (cid:96) + 1) π/k , where (cid:96) is the partial wave and k is the momentum ofeither of the interacting particles in the CM frame. In the non-relativistic regime, k = µv rel with µ being the reduced mass, thus σ uni (cid:96) v rel (cid:39) (2 (cid:96) + 1) πµ v rel . (3.65)As already discussed in ref. [1], the high efficiency of BSF via charged scalar emission(here, the Higgs doublet) implies that the unitarity limit (3.65) may be saturated alreadyfor rather small values of α H . If the incoming particles interact via an attractive long-rangeforce, then this occurs for the continuum of velocities v rel (cid:46) α B , otherwise only for a finiterange or discrete values of v rel (cf. fig. 11.) The apparent violation of unitarity at larger α H by the computations of sections 3.2 to 3.5, whether it occurs for an infinite or finiterange of v rel , indicates that these computations must be amended. At small values of α H ,higher order corrections to the perturbative transition amplitudes A T are expected to beinsignificant. Restoring unitarity necessitates instead that the two-particle interactions atinfinity are resummed [15].The results of sections 3.2 to 3.5 already include the resummation of the (leading-order)long-range interaction between the incoming particles, computed in section 2.2. However,according to the optical theorem, all elastic and inelastic processes to which the incomingstate may participate contribute to its self-energy. Typically, contact-type interactionscan be neglected, as they do not distort significantly the wavefuctions of the interactingparticles. Nevertheless, if a contact-type interaction is very strong – as is the case whenthe corresponding cross-section approaches (or even appears to exceed) the unitarity limit(3.65) – then its contribution to the two-particle self-energy may be significant.The contributions to the 2PI kernels arising from inelastic processes that involve Higgsemission are shown in fig. 10. These diagrams include both scattering and bound interme-diate states of the S , D and ¯ D particles, and therefore include bremsstrahlung, BSF andbound-to-bound transitions. Note that in fig. 10 we have not included the resummation ofthe long-range kernels of section 2.2 in the incoming and outgoing pairs, as this would re-sult in double-counting; only 2PI diagrams must be included in the kernels that determinethe potential.The proper resummation of the diagrams of fig. 10 requires developing suitable for-malism, and is beyond the scope of the present work. To ensure that our cross-sectionsare consistent with partial-wave unitarity, we shall instead adapt the result of ref. [51] thatresummed the box diagrams arising from the perturbative part of s -wave annihilation intoradiation (hard scattering), to compute the effect on the scattering-state wavefunctions andultimately on the full cross-sections for s -wave annihilation into radiation. This procedure– 40 – K ⊃ i A T G (4) H i A T Figure 10 . The contributions to the 2PI kernels arising from inelastic processes that involveHiggs emission. The solid lines stand for any of the S , D or ¯ D particles, and G (4) includes theirscattering and bound states. The dashed line represents the Higgs doublet. A T are the perturbativetransition amplitudes with H ( † ) emission, computed in sections 3.2 to 3.5 regulates the annihilation cross-sections as follows [51, eq. (40)], σ reg s -wave ( i → f ) = σ s -wave ( i → f ) (cid:32) (cid:80) f (cid:48) σ s -wave ( i → f (cid:48) )4 σ uni s -wave (cid:33) , (3.66)where we have generalised the result of ref. [51] to multiple annihilation channels. Equa-tion (3.66) ensures that the unitarity limit is respected by the total s -wave inelastic cross-section, since it implies r reg = r/ (1 + r/ (cid:54)
1, with r (reg) ≡ (cid:104)(cid:80) f σ (reg) s -wave ( i → f ) (cid:105) /σ uni s -wave .We emphasise that the assumptions made in deriving eq. (3.66) are not strictly satis-fied in our case, for at least two reasons: (i) Reference [51] assumed that for the resummedinelastic processes (hard scattering), σv rel is independent of v rel . For BSF via Higgs emis-sion, the corresponding cross-sections can be found from tables 7 to 10 by setting ζ S → v rel for v rel (cid:38) α B . (ii) In the present case, thenew contributions to the kernel may affect both the initial (scattering) and final (bound)state wavefunctions, while only the former is relevant for annihilation into radiation in theanalysis of ref. [51]. Nevertheless, we shall adopt eq. (3.66) as a perscription that regulatesthe inelastic cross-sections in the velocity range where the base calculation violates unitar-ity, while leaving them essentially unaffected outside that range. We leave a more precisetreatment for future work.In fig. 11, we show how the prescription (3.66) adjusts the inelastic cross-sections forthe scattering states that may participate in BSF via Higgs emission. We include the totalinelastic cross-section (BSF plus annihilation) in the resummation, although only BSF issignificant. However, both BSF and annihilation are regulated by the same factor, whichimplies that the annihilation cross-sections are affected significantly even while themselvesbeing well below the unitarity limit. – 41 – iolation of s - waveunitarity bound unregulatedregulated ( DS ) spin - → ℬ ( ) spin - ( SS / DD ) + H - - - - - v rel σ v r e l / ( π m - ) violation of s - waveunitarity bound unregulatedregulated ( DD ) ( ) spin - → ℬ spin - ( DS ) + H † ( DD ) ( ) spin - → BW - - σ v r e l / ( π m - ) violation of s - waveunitarity bound unregulatedregulated ( SS - like ) ( ) spin - → ℬ spin - ( DS ) + H † ( SS - like ) ( ) spin - → BB, WW - - σ v r e l / ( π m - ) violation of s - waveunitarity bound unregulatedregulated ( DS ) spin - → ℬ ( ) spin - ( DD ) + H ( DS ) spin - → ℬ ( ) spin - ( DD ) + H † ( DS ) spin - → BH, WH - - - v rel violation of s - waveunitarity bound unregulatedregulated ( DD ) ( ) spin - → ℬ spin - ( DS ) + H violation of s - waveunitarity bound unregulatedregulated ( DD - like ) ( ) spin - → ℬ spin - ( DS ) + H † ( DD - like ) ( ) spin - → BB, WW
Figure 11 . The effect of the perscription (3.66) on the inelastic cross-sections. While only BSFvia Higgs emission approaches or appears to violate the unitarity limit, the regurarisation affects allcross-sections with the same intial state. We have used α H = 0 . m = 50 TeV and T = 300 GeVwhich corresponds to m H (cid:39)
168 GeV. – 42 – R , d R , u R L L , Q L , Q L L L , Q L , Q L e R , d R , u R P H q i q f G (4)in G (4)out i A T B µ , W a,µ HP H q i q f G (4)in G (4)out i A T H † B µ , W a,µ P H q i q f G (4)in G (4)out i A T Figure 12 . Bound-state formation or bound-to-bound transitions via exchange of an off-shellHiggs doublet with the SM particles. The arrows on the field lines denote the flow of Hypercharge.All processes have their conjugate counterparts that occur via H † exchange. The dissipation of energy necessary for the capture into bound states or transitions betweenbound levels, may occur via exchange of an off-shell mediator with particles of the thermalbath [52–56]. References [55, 56] showed, in the context of a U (1) model, that the cross-sections for BSF via scattering factorise into the radiative ones (with any phase-spacesuppression due to the mass of the emitted vector removed), and a factor that depends onthe thermal bath and the interaction that mediates the scattering.In the following, we consider BSF and bound-to-bound transitions via off-shell Higgsexchange. We derive a similar factorisation and then compute the BSF cross-sections andtransition rates. We also adapt the results of refs. [55, 56] to our model, for BSF andtransitions via off-shell B and W exchange. H exchange4.1.1 Factorisation of the effective BSF cross-sections and transition rates For simplicity, we lay out the discussion in terms of the BSF cross-sections only. Thederivation for bound-to-bound transition rates is analogous.The thermally averaged rate per unit volume for BSF via off-shell Higgs exchange is d (cid:104) Γ H ∗ - BSF n(cid:96)m (cid:105) dV = g g (cid:90) d k (2 π ) d k (2 π ) f + ( k ) f + ( k )[1 + f − ( ω k → n(cid:96)m )] σ H ∗ - BSF k → n(cid:96)m v rel , (4.1) Other rearragement processes have been considered in ref. [57]. – 43 –here we defined σ H ∗ - BSF k → n(cid:96)m v rel ≡ [1 + f − ( ω k → n(cid:96)m )] − k k (cid:90) d P (2 π ) P d q i (2 π ) q i d q f (2 π ) q f f ± ( q i )[1 ∓ f ± ( q f )] ×× (2 π ) δ ( k + k + q i − P − q f ) 1 g g |M H ∗ - BSF k → n(cid:96)m | . (4.2)Here, the indices 1,2 correspond to the two incoming DM fields, while q i and q f denotethe initial and final bath particle momenta. We consider scattering only on relativisticspecies, whose density in a thermal environment is large, and set q i = | q i | and q f = | q f | . f ± ( E ) = ( e E/T ± − are the phase-space occupation numbers for fermions (+) and bosons( − ), and in eq. (4.2) the upper and lower signs correspond to scattering on fermionic orbosonic dof respectively. ω k → n(cid:96)m is the energy dissipated by the DM fields in the capture process, given byeq. (3.8). When the DM particles are non-relativistic, it depends only on the relativemomentum of the incoming DM particles and the binding energy of the bound state, but isindependent of the momenta of the bath particles. (We note that this does not hold for the3-momentum exchange | q f − q i | along the off-shell mediator.) The factor [1 + f − ( ω k → n(cid:96)m )]in eq. (4.1) is compensated by the inverse factor in eq. (4.2); this definition ensures that thethermal averaging of the cross-section (4.2) is the same as that of its radiative counterpart,which includes a Bose-enhancement factor for the radiated boson [9] (cf. ref.[28].)Next, we conjecture that the amplitude for off-shell H exchange, M H ∗ - BSF k → n(cid:96)m , can befactorised into the corresponding amplitude with on-shell H emission, M H - BSF k → n(cid:96)m , and afunction of the momenta of the bath particles, as follows (cid:88) i,f dof |M H ∗ - BSF k → n(cid:96)m | (cid:39) |M H - BSF k → n(cid:96)m | × R ( q i · q f ) , (4.3)where the sum on the left side runs over the dof (spin and gauge) of the initial and finalbath particles. M H - BSF k → n(cid:96)m may depend only on the momentum exchange q ≡ q f − q i ratherthan on q i and q f separately. R must be Lorentz invariant since the amplitudes are. Itmay thus depend only on the 4-vector products q i = q f = 0 and q i · q f ; in eq. (4.3) we havedenoted its dependence on the latter. The R factors will be specified in section 4.1.2 forthe processes shown in fig. 12. Switching the integration from q f to q , eq. (4.2) gives σ H ∗ - BSF k → n(cid:96)m v rel = [1 + f − ( ω k → n(cid:96)m )] − (cid:90) d q i (2 π ) | q i |× k k (cid:90) d P (2 π ) P d q (2 π ) q (2 π ) δ ( k + k − P − q ) 1 g g |M H - BSF k → n(cid:96)m | × q | q + q i | f ± ( | q i | ) [1 ∓ f ± ( | q + q i | )] R ( q i · q f ) , (4.4) For BSF via vector exchange/emission, the amplitude-squared does not factorise as in eq. (4.3). How-ever, it is still possible to obtain a factorisation formula similar to eq. (4.9) below, for an effective cross-section, at leading order in the non-relativistic approximation [55, 56]. (See also footnote 8.) – 44 –here q ≡ q f − q i (cid:39) | q + q i | − | q i | . (4.5)The second line of eq. (4.4) would form the BSF cross-section via on-shell emission, exceptfor two complications: (i) The dispersion relation of the radiated momentum is given byeq. (4.5) rather than the on-shell condition of the radiated boson, and depends on thevariable q i . (ii) The last line of eq. (4.4) depends on q . These complications are resolvedwithin the non-relativistic approximation, where the cross-section of BSF via scatteringcan be shown to be proportional to that of radiative BSF, as we shall now see. Non-relativistic approximation
Similarly to radiative BSF, if the incoming particles are non-relativistic and the final stateparticles are weakly bound, we may neglect the recoil of the bound state. Then, the energy-momentum conservation implies q (cid:39) ω with ω given by eqs. (3.8) and (3.12) for BSF andbound-to-bound transitions. (Here we drop the ω indices for simplicity.) The dispersionrelation (4.5) yields | q | (cid:39) ω (cid:34)(cid:114) | q i | ω + | q i | τ ω − | q i | τω (cid:35) , (4.6)with τ ≡ ˆ q i · ˆ q . It is also easy to show that q i · q f = ( q − ω ) /
2. Using the δ -function toperform the integration over d P d | q | (as is standard in the computation of 2-to-2 cross-sections), eq. (4.4) gives σ H ∗ - BSF k → n(cid:96)m v rel (cid:39) [1 + f − ( ω )] − (cid:90) d q i (2 π ) | q i | ωω + | q i | f ± ( | q i | ) [1 ∓ f ± ( ω + | q i | )] × k k P ω (cid:90) d Ω q π q (cid:18) ω + | q i || q | + | q i | τ (cid:19) g g |M H - BSF k → n(cid:96)m | × R (cid:18) q − ω (cid:19) , (4.7)with P (cid:39) m and | q | given by eq. (4.6). Note that the second line of eq. (4.7) differs from( σ H - BSF k → n(cid:96)m v rel ) by the factor ω + | q i || q | + | q i | τ (cid:30) ω | q | due to the different dispersion relation of theradiated momentum q , here given by eq. (4.5).The radiative BSF amplitudes are typically computed by expanding in powers of theradiated momentum q [19]. As seen in section 3, the dominant contribution to the various The expansion is in effect in the dimensionless combination | q | / (cid:112) κ /n + k = | q | / √ µω . For BSFvia on-shell emission, the radiated momentum is limited by the available energy, | q | (cid:54) ω , thus the expansionparameter is always (cid:112) ω/ (2 µ ) (cid:28)
1. However, for BSF via scattering, | q | can be comparable to or largerthan √ µω , particularly at T (cid:38) κ/n , which puts in question the validity of the expansion. Nevertheless,for the purposes of DM freeze-out, BSF typically reaches ionisation equilibrium at high T , where the DMdestruction rate via BSF is independent of the BSF cross-sections [58] (cf. ref. [28].) At lower T , where themagnitude of the BSF cross-sections matters, the | q | expansion is a valid approximation. – 45 – H - BSF k → n(cid:96)m amplitudes arises from the zeroth order term [1], i.e. M H - BSF k → n(cid:96)m are independentof q (and therefore τ ) at leading order. Reshuffling the various factors, eq. (4.7) becomes σ H ∗ - BSF k → n(cid:96)m v rel (cid:39) π k k ω P (cid:90) d Ω q g g |M H - BSF k → n(cid:96)m | × (4.8) × [1 + f − ( ω )] − (cid:90) d q i (2 π ) | q i | f ± ( | q i | ) [1 ∓ f ± ( ω + | q i | )] | q | ω ( | q | + | q i | τ ) R (cid:18) q − ω (cid:19) . The integration over d q i in the second line of eq. (4.8) eliminates any dependence ofthe integrand on the orientation of the vector q , allowing us to identify the first line asthe cross-section for on-shell Higgs emission with the phase-space suppression removed(cf. eqs. (3.7b) and (3.9b).) Thus, the effective cross-section for off-shell Higgs exchangecan be factorised at leading order as follows σ H ∗ - BSF k → n(cid:96)m v rel (cid:39) σ H - BSF k → n(cid:96)m v rel h H ( ω k → n(cid:96)m ) × R H ( ω k → n(cid:96)m ) , (4.9)where h H ( ω ) is the phase-space suppression (3.10) of the on-shell emission due to theHiggs mass, with ω k → n(cid:96)m being the dissipated energy (3.8), and we restored the indicesfor concreteness. The dimensionless factor R H ( ω ) is R H ( ω ) ≡ (cid:90) d q i (2 π ) | q i | q ω ( | q | + | q i | τ ) f ± ( q i ) [1 ∓ f ± ( ω + | q i | )]1 + f − ( ω ) R (cid:18) q − ω (cid:19) . (4.10)We recall that | q | is given by eq. (4.6). Since the entire integrand is rotationally invariant(recall that R is Lorentz invariant), we perform the d q i integration by setting q on the z axis. Then the azimuthal angle is parametrised by τ defined above. Changing integrationvariables from | q i | and τ to u ≡ | q i | /ω and z ≡ q /ω −
1, eq. (4.10) simplifies to R H ( ω ) = ω π (cid:90) ∞ du f ± ( ωu ) [1 ∓ f ± ( ω (1 + u ))]1 + f − ( ω ) (cid:90) u (1+ u )0 dz R (cid:0) zω / (cid:1) . (4.11)We compute R H next. The final result can be found in eqs. (4.19) to (4.21).Following the same steps, we find that the bound-to-bound transition rate via off-shellHiggs exchange is related to the radiative one viaΓ H ∗ - BSF n (cid:48) (cid:96) (cid:48) m → n(cid:96)m (cid:39) Γ H - BSF n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m h H ( ω n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m ) × R H ( ω n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m ) , (4.12)where ω n (cid:48) (cid:96) (cid:48) m (cid:48) → n(cid:96)m is the dissipated energy (3.12). Note that eq. (4.9) holds for the effective cross-section of BSF via off-shell Higgs exchange as defined ineq. (4.2). Since M H - BSF k → n(cid:96)m is presumed to be independent of q and therefore q · k , it has not been necessaryto integrate over the angular variables of k in order to obtain a factorised expression for the cross-section.This is in contrast to the case of vector exchange, where the amplitude does not factorise as in eq. (4.3) andin fact depends on q i · k and q f · k . A factorisation similar to eq. (4.9) is obtained only for the cross-sectionaveraged over the k solid angle [55, 56] (cf. section 4.2.) – 46 – .1.2 Amplitudes Similarly to their radiative analogues, the amplitudes for BSF and bound-to-bound tran-sitions via scattering consist of the perturbative transition amplitudes that encode thescattering on the bath particles, convoluted with the initial and final state wavefunctions.Focusing again on BSF (cf. eq. (3.1a)), i M H ∗ - BSF k → n(cid:96)m = (cid:90) d k (cid:48) (2 π ) d p (2 π ) [ ψ n(cid:96)m ( p )] † √ µ i A H ∗ - BSF T ( k (cid:48) , p ) φ k ( k (cid:48) ) . (4.13)We now compute A H ∗ - BSF T ( k (cid:48) , p ) for the scattering processes shown in fig. 12, and deducefrom eq. (4.11) the corresponding R factors. Scattering on fermions
The Higgs couples to the SM fermions via the operators δ L = − y e ( δ ab ¯ L L a H b ) e R − y d ( δ ab ¯ Q L a H b ) d R − y u ( (cid:15) ab ¯ Q L a H † b ) u R + h.c. , (4.14)where the a, b superscripts indicate the SU L (2) contractions, while the family indices aresuppressed. These couplings give rise to the scattering processes shown in fig. 12 (top).The corresponding BSF perturbative transition amplitudes (projected on the desired spinand gauge, scattering and bound states) are i (cid:2) A H ∗ - BSF T ( k (cid:48) , p ) (cid:3) h = i (cid:2) A H - BSF T ( k (cid:48) , p ) (cid:3) h (cid:48) × i q − m H ¯ u f ( − i y ∗ F ) δ hh (cid:48) (cid:18) − γ (cid:19) u i , (4.15)where h, h (cid:48) are the SU L (2) indices of the left-handed SM fermion field and the exchangedHiggs, respectively. The scattering and bound states gauge indices, if any, are left implicit.We use y F to denote collectively the SM Yukawa couplings of eq. (4.14). Inserting eq. (4.15)into (4.13), squaring and summing over the bath particle spin and gauge dof, we arrive atthe R factors (cf. eq. (4.3)), R = 2 × | y F | q i · q f (2 q i · q f + m H ) , (4.16)where we introduced a factor 2 to account for the partner process controlled by the samecoupling, where the initial (final) fermion becomes the final (initial) antifermion. Scattering on bosons
The perturbative transition amplitude for scattering on gauge bosons (fig. 12, bottom left),projected on the desired spin and gauge, scattering and bound states, is i (cid:2) A H ∗ - BSF T ( k (cid:48) , p ) (cid:3) a,µh = i [ A H - BSF T ( k (cid:48) , p )] h (cid:48) × i q − m H × i g T ahh (cid:48) ( q f + q ) µ , (4.17) In eq. (4.15), the sign of the γ term and whether the Yukawa coupling should be y F or y ∗ F depend onthe exact process we are considering. For scattering on antifermions, the spinors u i , u f become v i , v f . Inaddition, for a scattering involving an up-type right-handed (anti)quark, δ hh (cid:48) should be replaced by (cid:15) hh (cid:48) .However, all these differences do not affect the R factors. – 47 –here h, h (cid:48) and a are the SU L (2) indices of the outgoing and exchanged Higgs bosons andthe incoming gauge boson respectively. T a and g stand for the generators and the gaugecoupling of the gauge group under consideration. Inserting eq. (4.17) into (4.13), squaringand summing over the bath particle polarisations and gauge dof, we find the R factors R = 2 × πα C ( R H ) 4 q i · q f (2 q i · q f + m H ) , (4.18)where, as before, we introduced a factor 2 to account for the partner process where H † is the incoming bath particle (fig. 12, bottom right). C ( R H ) is the quadratic Casimir ofthe Higgs representation under the gauge group considered; here, C ( R H ) = Y H = 1 / C ( R H ) = 3 / SU L (2). Both eqs. (4.16) and (4.18) depend only on q i · q f , as presumed in eq. (4.3), and in fact inthe same fashion. Inserting them into eq. (4.11), and carrying out the integration over z ,we find the contributions of scattering on fermions and bosons to BSF, R FH = 2 × | y F | π × R + ,R BHH = 2 × (1 / α × R − ,R WHH = 2 × (3 / α × R − , (4.19a)(4.19b)(4.19c)where R ± are dimensionless functions of two parameters, ω/T and m H /ω , R ± ≡ π (cid:90) ∞ du e u ω/T e u ( ω/T ) ± e ω/T − e (1+ u ) ω/T ± (cid:26) ln (cid:20) u (1 + u ) m H /ω (cid:21) − u (1 + u )4 u (1 + u ) + m H /ω (cid:27) . (4.20)The factor 1 / R H factor thatdetermines the BSF via scattering cross-section (4.9) is R H = (cid:88) F R FH + R BHH + R WHH . (4.21)Among the SM fermions, the top quark yields the largest contribution as long as it remainsrelativistic.The R ± factors (4.20) diverge at m H →
0, which during the DM thermal decouplingaround the EWPT (cf. ref. [28].) This divergence can be removed by a full next-to-leading-order calculation, as done in ref. [56] in the context of a U (1) gauge theory. Performing sucha computation for the model considered here is beyond the scope of this work. However,comparing the results of refs. [55] and [56] for a massive and massless vector mediatorrespectively, we find that, upon thermal averaging, the former approximates well the latterat temperatures higher than the binding energy if the screening scale (i.e. the mediator– 48 –ass) is set to 0 . ω . Considering this, in eq. (4.20) we shall do the replacement m H → max( m H , ω ) . (4.22)We present R ± and R H in fig. 13. It is clear that they are more significant for ω/T (cid:28) T (cid:29) ω n (cid:48) → n = |E n (cid:48) − E n | . For BSF via scattering, the R ± factors weigh preferentially the contributionof DM pairs with low relative velocity. We note that even though in a thermal bath (cid:104) ω k → n(cid:96)m (cid:105) = |E n | + (3 / T > T (cf. eq. (3.8)), lower values of ω k → n(cid:96)m may still incur in asizeable portion of the DM collisions while T (cid:38) |E n | .Even when the R H factor (4.21) is less than 1, BSF via scattering may potentially be(i) faster than radiative BSF, which is suppressed by the h H ( ω k → n(cid:96)m ) phase-space factor(3.10), becoming entirely inaccessible for m H /ω k → n(cid:96)m >
1, and (ii) significant with respectto direct annihilation and BSF via vector emission, since these processes are suppressedby one (two) extra power(s) of couplings compared to BSF via H ( † ) off-shell exchange(on-shell emission.) Analogously, bound-to-bound transitions via scattering may dominateover their radiative counterparts and/or the direct bound-state decay into radiation.To assess realistically the impact of BSF and bound-to-bound transitions via scatteringwe must thermally average the cross-sections and rates of eqs. (4.9) and (4.12), and considerthe interplay of bound-state formation, decay, ionisation and transition processes in thethermal bath. This is done in ref. [28]. Here we only note that considering BSF viascattering does not increase the DM destruction rate proportionally, since at early times astate of ionisation equilibrium is typically reached where the DM destruction due to BSFis independent of the actual BSF rate provided that the latter is sufficiently large [58]. B and W exchange References [55, 56] showed in the context of a U (1) gauge theory that the effective cross-section for BSF via off-shell vector exchange, defined via the thermally averaged rate pervolume (cf. footnote 8) d (cid:104) Γ V ∗ - BSF n(cid:96)m (cid:105) dV = n n (cid:18) µ πT (cid:19) / (cid:90) dv rel v e − µv / (2 T ) (cid:18) e ω k → n(cid:96)m /T − (cid:19) ( σ V ∗ - BSF k → n(cid:96)m v rel ) , (4.23)is, at leading order in the non-relativistic regime, proportional to the cross-section for on-shell emission, σ V ∗ - BSF k → n(cid:96)m v rel = ( σ V - BSF k → n(cid:96)m v rel ) × R V , where R V = 2 × α × R U (1) . As in section 4.1,the factor 2 accounts for the partner processes related via exchanging the initial (final) bathparticle with the final (initial) bath antiparticle, and α is the fine structure constant of thegroup. The factor R U (1) depends only on ω/T provided that V is massless, and has beenderived in [56] via a next-to-leading order calculation where the colinear and infrared thedivergences are cancelled. For a massive V , a simple analytical formula that depends on Reference [56] found that using the binding energy as the minimum screening scale provides a goodapproximation. While this is indeed so, the above prescription ensures that the R factor depends only on ω/T as predicted by the full computation, besides being a somewhat better approximation. – 49 – + R - R H m H / ω = m H / ω = - - - - ω / T R U ( ) R B R W - - - - ω / T Figure 13 . Left : The R ± and R H factors of eqs. (4.20) and (4.21) that determine the ratio ofBSF via off-shell H ( † ) scattering over on-shell H ( † ) emission. Right : The R B and R W factors ofeqs. (4.25) that determine the ratio of BSF via off-shell B or W scattering on fermions over on-shell B or W emission. Also shown, the R U (1) factor from [56] (cf. footnote 11.) ω/T and m V /ω has been computed in [55]. We define R U (1) to correspond to scattering onone species of relativistic Dirac fermions with charge unity, and use the results of [56]. Adapting the result to the present model, the cross-sections for BSF via off-shell B and W exchange are related to those of on-shell emission as follows σ B ∗ - BSF k → n(cid:96)m v rel (cid:39) ( σ B - BSF k → n(cid:96)m v rel ) × R B ( ω k → n(cid:96)m /T ) , (4.24a) σ W ∗ - BSF k → n(cid:96)m v rel (cid:39) ( σ W - BSF k → n(cid:96)m v rel ) × R W ( ω k → n(cid:96)m /T ) , (4.24b)where R B ( ω/T ) = 2 × c B α × R U (1) ( ω/T ) , (4.25a) R W ( ω/T ) = 2 × c W α × R U (1) ( ω/T ) . (4.25b)The factors c B and c W account for scattering on the relativistic SM fermions. The contri-bution of a chiral fermion F transforming under the representation R F of a gauge group is c F = C ( R F ) /
2, where C is the Casimir operator. When all the SM fermions are relativistic, c B = (1 / (cid:80) F Y F = 5 and c W = (1 / C ( ) ×
12 = 3, where C ( ) = 1 / SU L (2).Note that eq. (4.25b) includes only scattering on SM fermions via off-shell W exchange.However, non-Abelian gauge bosons may also scatter on themselves due to the trilineargauge coupling. Estimating this effect necessitates a dedicated next-to-leading order com-putation that is beyond the scope of this work. We shall thus neglect this contribution.Formulae analogous to eqs. (4.25) hold for bound-to-bound transitions via off-shell B and W exchange, however no such transition is of interest here.We present the R B and R W factors in fig. 13. R U (1) is related to R . defined in ref. [59, eq. (4.13) and fig. 14] as R U (1) ≡ R . / (2 π ).We thank Tobias Binder for providing the numerical values of R . . – 50 – Conclusion
Renormalisable Higgs portal scenarios in which DM is the lightest mass eigenstate aris-ing from the mixing of two electroweak multiplets that couple to the Higgs, are amongthe archetypical WIMP DM models. Here, we have considered the role of the Higgs dou-blet in the non-perturbative phenomena — the Sommerfeld effect and the formation ofbound states — that take place during the thermal decoupling of multi-TeV DM from theprimordial plasma.We have shown that the effect of the Higgs doublet is two-fold: (i) it mediates along-range interaction that affects the wavefunctions of both scattering and bound states,and (ii) its emission precipitates extremely fast monopole transitions, including captureinto bound states and transitions between bound levels. In a companion paper [28], weshow that the above effects can reduce the relic density of stable species very significantly,thereby altering experimental constraints.These results build on the work of ref. [1] that showed the importance of bound-stateformation via emission of a scalar charged under a symmetry (see also [47]), as well as thework of refs. [2, 3] that demonstrated the long-range effect of the Higgs boson betweenTeV-scale particles. In the present first computation of such effects involving the Higgsdoublet, we have focused on the simplest model, comprised by two (nearly) mass degeneratefermionic SU L (2) multiplets, a singlet and a doublet. Our calculations can of course beextended to other Higgs-portal models.We have considered transitions – both BSF and bound-to-bound transitions – via ra-diative emission of an on-shell Higgs doublet, as well as via scattering on the thermal baththrough off-shell Higgs-doublet exchange. We showed that the rates for the latter fac-torise into the former times a temperature-dependent function. This parallels the resultsof refs. [55, 56] that considered capture via exchange of an off-shell gauge boson and founda similar factorisation. However, the temperature-dependent function depends on the mul-tiple mode of the transition, and is thus different for the monopole transitions occurring viaHiggs-doublet (or generally charged-scalar) exchange and the dipole transitions occuringvia gauge-boson exchange.We finish by commenting on two technical aspects. While neglected here, the captureinto excited states can be significant (even if not dominant) due to the monopole transi-tions occurring in this class of models, as pointed out already in [1]. Moreover, monopoletransitions can lead to the apparent violation of unitarity even for small couplings. Herewe have adopted an effective prescription to treat this issue, however a dedicated study isrequired as described in the previous sections. We leave these issues for future work.– 51 – ppendicesA Kernel and wavefunction (anti)symmetrisation for identical particles A.1 The 2PI kernel
For identical particle (IP) pairs, t -channel diagrams have u -channel counterparts. However,adding them up and resumming them double-counts the loop diagrams because it corre-sponds to exchanging identical particles in the loops. The proper resummation necessitatesusing an (anti-)symmetrised kernel, as we will now show. Note that this holds not onlyfor tree-level 2PI diagrams, but more generally for 2PI diagrams involving loops in t - and u -type configurations. s A s (cid:48) A P/ p P/ p (cid:48) P/ − p P/ − p (cid:48) s B s (cid:48) B G = s A s (cid:48) A P/ p = P/ p (cid:48) P/ − p = P/ − p (cid:48) s B s (cid:48) B ++ s A s (cid:48) A P/ p P/ p (cid:48) P/ − p P/ − p (cid:48) s B s (cid:48) B i A ++ s A r A s (cid:48) A P/ p P/ q P/ p (cid:48) P/ − p P/ − q P/ − p (cid:48) s B r B s (cid:48) B i A i A ++... s B s (cid:48) A P/ − p = P/ p (cid:48) P/ p = P/ − p (cid:48) s A s (cid:48) B ( − f + s B s (cid:48) A P/ − p P/ p (cid:48) P/ p P/ − p (cid:48) s A s (cid:48) B ( − f i A + s B r A s (cid:48) A P/ − p P/ q P/ p (cid:48) P/ p P/ − q P/ − p (cid:48) s A r B s (cid:48) B ( − f i A i A +... Figure 14 . Resummation of t -type ( left ) and u -type ( right ) 2PI diagrams for pairs of identicalparticles. Summation over r A , r B and integration over q is implied. The u -type diagrams carryextra factors ( − f with respect to their t -type counterparts, where f = 0 or 1 if the interactingparticles are bosons or fermions respectively, due to the different number of fermion permutationsneeded to perform the Wick contractions. – 52 –e consider the 4-point function of a pair of identical particles, G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) ,where the momentum and spin assignments are shown in fig. 14. To ease the notation, thedependence of G IP on the total momentum P is left implicit. Let A be a function of thesame variables that stands for the sum of either the t - or u -type 2PI diagrams. Clearly, A (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = A (cid:18) − p, s B , s A − p (cid:48) , s (cid:48) B , s (cid:48) A (cid:19) . (A.1a)Then, the sum of complementary 2PI diagrams ( u - or t -type respectively) is( − f A (cid:18) p, s A , s B − p (cid:48) , s (cid:48) B , s (cid:48) A (cid:19) = ( − f A (cid:18) − p, s B , s A p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) , (A.1b)where f = 0 or 1 if the interacting particles are bosons or fermions. This factor arises fromthe different number of fermion permutations needed in the t - and u -type cases, in orderto perform the Wick contractions.The 4-point function G IP includes the two ladders shown in the two columns of fig. 14.These ladders are related by exchanging the momenta and spins of the initial (or final)state particles, thus we may write G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) + ( − f G IP (cid:18) − p, s B , s A p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) . (A.2)Let S ( P ) ≡ i / ( P − m ). The unamputated function G is G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) (2 π ) δ (4) ( p − p (cid:48) ) δ s A ,s (cid:48) A δ s B ,s (cid:48) B + S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) i A (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) S ( P/ p ) S ( P/ − p )+ S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) (cid:88) r A ,r B (cid:90) d q (2 π ) i A (cid:18) q, r A , r B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) S ( P/ q ) S ( P/ − q ) × i A (cid:16) p, s A , s B q, r A , r B (cid:17) S ( P/ p ) S ( P/ − p ) + · · · . (A.3)Equation (A.3) can be re-expressed as a Dyson-Schwinger equation with A being the kernel, G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) ×× (cid:34) (2 π ) δ (4) ( p − p (cid:48) ) δ s A ,s (cid:48) A δ s B ,s (cid:48) B + (cid:88) r A ,r B (cid:90) d q (2 π ) i A (cid:16) p, s A , s B q, r A , r B (cid:17) G IP (cid:18) q, r A , r B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19)(cid:35) . (A.4)In eq. (A.4), we can change the integration variable q → − q and switch r A ↔ r B . Addingup the resulting equation with eq. (A.4), we obtain G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) × (cid:110) (2 π ) δ (4) ( p − p (cid:48) ) δ s A ,s (cid:48) A δ s B ,s (cid:48) B + 12 (cid:88) r A ,r B (cid:90) d q (2 π ) (cid:20) i A (cid:16) p, s A , s B q, r A , r B (cid:17) G IP (cid:18) q, r A , r B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) + i A (cid:16) p, s A , s B − q, r B , r A (cid:17) G IP (cid:18) − q, r B , r A p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19)(cid:21) . (A.5)– 53 –ombining eqs. (A.1), (A.2) and (A.5), we obtain the Dyson-Schwinger equation for G , G IP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) ×× (cid:104) (2 π ) δ (4) ( p − p (cid:48) ) δ s A ,s (cid:48) A δ s B ,s (cid:48) B + ( − f (2 π ) δ (4) ( p + p (cid:48) ) δ s A ,s (cid:48) B δ s A ,s (cid:48) B + (cid:88) r A ,r B (cid:90) d q (2 π ) i K (cid:16) p, s A , s B q, r A , r B (cid:17) G IP (cid:18) q, r A , r B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19)(cid:35) , (A.6)where we defined i K (cid:16) p, s A , s B q, r A , r B (cid:17) ≡ (cid:104) i A (cid:16) p, s A , s B q, r A , r B (cid:17) + ( − f i A (cid:16) − p, s B , s A q, r A , r B (cid:17)(cid:105) . (A.7)Evidently, eq. (A.7) is the average of the t - and u -type 2PI diagrams, i K = 12 ( i A t + i A u ) . (A.8)The factor 1 / i K (cid:16) p, s A , s B q, r A , r B (cid:17) = ( − f i K (cid:16) p, s A , s B − q, r B , r A (cid:17) , (A.9)which we use below in the discussion on the (anti-)symmetrisation of the wavefunctions.Finally, we note that if the interacting particles carry additional conserved numbers,e.g. non-Abelian (gauge) charges, then appropriate factors ensuring their conservation mayappear in the 0th order terms of eq. (A.6), as well as inside A and consequently K . However,eq. (A.8) remains generally valid as is. A.2 Wavefunctions
The 0th order terms of the Dyson-Schwinger equations determine the normalisation ofthe wavefunctions (see e.g. [18, 60].) The two contributions appearing in the second line ofeq. (A.6) ensure that the wavefunctions of identical particles are properly (anti)symmetrised,as we will now show. Instead of deriving the normalisation conditions from eq. (A.6), weshall deduce them by comparing to the case of distinguishable particles (DP), whose wave-functions are normalised as standard [18, 60].For DP with equal masses, and incoming and outgoing momenta and spins as in fig. 14,the Dyson-Schwinger eq. for the four-point function G DP is (compare with eq. (A.6)) G DP (cid:18) p, s A , s B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) × (cid:34) (2 π ) δ (4) ( p − p (cid:48) ) δ s A ,s (cid:48) A δ s B ,s (cid:48) B + (cid:88) r A ,r B (cid:90) d q (2 π ) i K (cid:16) p, s A , s B q, r A , r B (cid:17) G DP (cid:18) q, r A , r B p (cid:48) , s (cid:48) A , s (cid:48) B (cid:19)(cid:35) . (A.10)– 54 –e diagonalise eq. (A.10) in spin space. The factor δ s A ,s (cid:48) A δ s B ,s (cid:48) B is simply the unity oper-ator, with all its eigenvalues being 1. Thus, the contribution from the spin- s state is G DP s ( p, p (cid:48) ) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) (cid:20) (2 π ) δ (4) ( p − p (cid:48) ) + (cid:90) d q (2 π ) i K s ( p, q ) G DP s ( q, p (cid:48) ) (cid:21) , (A.11)where K s is the projected kernel. G DP s receives contributions from all energy eigenstatesthat schematically read [18, 60] G DP n,s ( p, p (cid:48) ) (cid:39) i ˜Ψ DP n,s ( p ) [ ˜Ψ DP n,s ( p (cid:48) )] (cid:63) P ( P − ω n,s + i (cid:15) ) , (A.12)where here n denotes collectively all quantum numbers characterising an eigenstate ofenergy ω n,s . For scattering states, these include a continuous variable that corresponds tothe relative momentum of the two interacting particles, while bound states are characterisedby a set of discrete quantum numbers. The (momentum space) wavefunctions ˜Ψ DP n,s ( p ) obeythe Schr¨odinger equation, and have the standard normalisation conditions that emanatefrom the first term in eq. (A.11).Now we return to IP and the Dyson-Schwinger eq. (A.6). For fermions, the operators δ s A ,s (cid:48) A δ s B ,s (cid:48) B and δ s A ,s (cid:48) B δ s B ,s (cid:48) A have eigenvalues 1 and ( − s +1 respectively, while for bosonsthe eigenvalues are 1. Collectively, this is 1 and ( − s + f . Thus, eq. (A.6) yields G IP s ( p, p (cid:48) ) = S ( P/ p (cid:48) ) S ( P/ − p (cid:48) ) × (cid:20) (2 π ) δ (4) ( p − p (cid:48) ) + ( − s (2 π ) δ (4) ( p + p (cid:48) ) + (cid:90) d q (2 π ) i K s ( p, q ) G IP s ( q, p (cid:48) ) (cid:21) . (A.13)The projected kernel is i K s ( p, q ) = [ U s ] s A s B i K s (cid:16) p, s A , s B q, r A , r B (cid:17) [ U s ] − r A r B , (A.14)where U s is the projection operator on the spin- s state, with the symmetry property[ U s ] s A s B = ( − s + f [ U s ] s B s A . Equation (A.14) combined with eq. (A.9) imply i K s ( p, q ) = ( − s i K s ( p, − q ) . (A.15)The contribution to the four-point function from the n th energy eigenstate is G IP n,s ( p, p (cid:48) ) (cid:39) i ˜Ψ IP n,s ( p ) [ ˜Ψ IP n,s ( p (cid:48) )] (cid:63) P ( P − ω n,s + i (cid:15) ) , (A.16)where ˜Ψ IP n,s ( p ) are the IP wavefunctions. We now make the conjecture˜Ψ IP n,s ( p ) = 1 √ (cid:104) ˜Ψ DP n,s ( p ) + ( − s ˜Ψ DP n,s ( − p ) (cid:105) , (A.17)– 55 –here ˜Ψ DP n,s ( p ) are the solutions to the DP Dyson-Schwinger eq. (A.11), assuming the kernelis the same as that of eq. (A.13). Plugging eq. (A.17) into (A.16), and considering (A.12),we re-express G IP n,s as G IP n,s ( p, p (cid:48) ) = 12 (cid:2) G DP n,s ( p, p (cid:48) ) + G DP n,s ( − p, − p (cid:48) ) (cid:3) + ( − s (cid:2) G DP n,s ( p, − p (cid:48) ) + G DP n,s ( − p, p (cid:48) ) (cid:3) . (A.18)It is now easy to see that, by virtue of the DP Dyson-Schwinger eq. (A.11) and the propertyof the IP kernel (A.15), the four-point function G IP n,s ( p, p (cid:48) ) of eq. (A.18) satisfies the IPDyson-Schwinger eq. (A.13). Therefore, the wavefunctions (A.17) are indeed the desiredsolutions. Expanding in modes of definite orbital angular momentum (cid:96) , for which˜Ψ DP n,(cid:96)s ( − p ) = ( − (cid:96) ˜Ψ DP n,(cid:96)s ( − p ) , (A.19)eq. (A.17) becomes ˜Ψ IP n,(cid:96)s ( p ) = 1 + ( − (cid:96) + s √ DP n,(cid:96)s ( p ) . (A.20)Note though that, as mentioned in appendix A.1, if the interacting particles carry addi-tional conserved numbers, then appropriate (anti-)symmetrisation factors may appear ineqs. (A.19) and (A.20) (cf. e.g. DD potential in section 2.2.)– 56 – Perturbative transition amplitudes: an example
We demonstrate the calculation of diagrams contributing to the perturbative part of thetransition amplitudes of section 3. We will work out in detail the diagram shown in fig. 15. s r s r k k p p P H i i (cid:48) j (cid:48) h Figure 15 . Example of diagram contributing to the perturbative parts of the amplitudes ofvarious transition processes considered in section 3.
We first express the fields in canonical form, following [61], H j ( x ) = (cid:90) d q (2 π ) (cid:112) E H ( q ) (cid:104) a j ( q ) e − i q · x + b † j ( q ) e + i q · x (cid:105) , (B.1a) H † j ( x ) = (cid:90) d q (2 π ) (cid:112) E H ( q ) (cid:104) b j ( q ) e − i q · x + a † j ( q ) e + i q · x (cid:105) , (B.1b) S ( x ) = (cid:90) d q (2 π ) (cid:112) E S ( q ) (cid:88) s (cid:104) c ( q , s ) u ( q , s ) e − i q · x + c † ( q , s ) v ( q , s ) e + i q · x (cid:105) , (B.1c)¯ S ( x ) = (cid:90) d q (2 π ) (cid:112) E S ( q ) (cid:88) s (cid:104) c ( q , s ) ¯ v ( q , s ) e − i q · x + c † ( q , s ) ¯ u ( q , s ) e + i q · x (cid:105) , (B.1d) D j ( x ) = (cid:90) d q (2 π ) (cid:112) E D ( q ) (cid:88) s (cid:104) d j ( q , s ) u ( q , s ) e − i q · x + f † j ( q , s ) v ( q , s ) e + i q · x (cid:105) , (B.1e)¯ D j ( x ) = (cid:90) d q (2 π ) (cid:112) E D ( q ) (cid:88) s (cid:104) f j ( q , s ) ¯ v ( q , s ) e − i q · x + d † j ( q , s ) ¯ u ( q , s ) e + i q · x (cid:105) , (B.1f)where j is the SU L (2) index, and the various q inside the integrals are equal to thecorresponding on-shell energies, E H ( q ) = (cid:112) m H + q , E S ( q ) = E D ( q ) = (cid:112) m + q . (B.2)The creation and annihilation operators obey the (anti)commutation relations[ a i ( p ) , a † j ( q )] = [ b i ( p ) , b † j ( q )] = (2 π ) δ ( p − q ) δ ij , (B.3a) { c ( p , r ) , c † ( q , s ) } = (2 π ) δ ( p − q ) δ rs , (B.3b) { d i ( p , r ) , d † j ( q , s ) } = { f i ( p , r ) , f † j ( q , s ) } = (2 π ) δ ( p − q ) δ rs δ ij , (B.3c)with all other combinations being zero. The one-particle states are | H j ( q ) (cid:105) = (cid:112) E H ( q ) a † j ( q , s ) | (cid:105) , | H † j ( q ) (cid:105) = (cid:112) E H ( q ) b † j ( q , s ) | (cid:105) , (B.4a) | S ( q , s ) (cid:105) = (cid:112) E S ( q ) c † ( q , s ) | (cid:105) , (B.4b) | D j ( q , s ) (cid:105) = (cid:112) E D ( q ) d † j ( q , s ) | (cid:105) , | ¯ D j ( q , s ) (cid:105) = (cid:112) E D ( q ) f † j ( q , s ) | (cid:105) . (B.4c)– 57 –e now return to the diagram of fig. 15. Its contribution to an amplitude is(2 π ) δ ( k + k − p − p − P H ) i A [ DS → D ¯ DH ] (cid:39)(cid:39) (cid:104) D i (cid:48) ( p , r ) ¯ D j (cid:48) ( p , r ) H h ( P H ) | ( − i y ) (cid:90) d x ¯ S ( x ) H † ( x ) D ( x ) | D i ( k , s ) S ( k , s ) (cid:105) = ( − i y ) (cid:88) t S ,t D (cid:88) n (cid:90) d x (cid:90) d q S (2 π ) d q H (2 π ) d q ¯ D (2 π ) (cid:112) E D ( k ) 2 E S ( k ) 2 E D ( p ) 2 E D ( p ) 2 E H ( P H ) (cid:112) E S ( q S ) 2 E H ( q H ) 2 E D ( q ¯ D ) × (cid:104) | a h ( P H ) f j (cid:48) ( p , r ) d i (cid:48) ( p , r ) (cid:104) c ( q S , t S ) ¯ v ( q S , t S ) e − i q S · x + c † ( q S , t S ) ¯ u ( q S , t S ) e + i q S · x (cid:105) × (cid:104) b n ( q H ) e − i q H · x + a † n ( q H ) e + i q H · x (cid:105) × (cid:104) d n ( q ¯ D , t ¯ D ) u ( q ¯ D , t ¯ D ) e − i q ¯ D · x + f † n ( q ¯ D , t ¯ D ) v ( q ¯ D , t ¯ D ) e + i q ¯ D · x (cid:105) d † i ( k , s ) c † ( k , s ) | (cid:105) = ( − i y ) (cid:88) t S ,t ¯ D (cid:88) n (cid:90) d x (cid:90) d q S (2 π ) d q H (2 π ) d q ¯ D (2 π ) (cid:112) E D ( k ) 2 E S ( k ) 2 E D ( p ) 2 E D ( p ) 2 E H ( P H ) (cid:112) E S ( q S ) 2 E H ( q H ) 2 E D ( q ¯ D ) × e − i ( q S − q H − q ¯ D ) x × ( − (2 π ) δ ( k − q S ) δ t S s ¯ v ( q S , t S ) × (2 π ) δ ( P H − q H ) δ hn × ( − π ) δ ( p − q ¯ D ) δ t ¯ D r δ j (cid:48) n v ( q ¯ D , t ¯ D ) × (2 π ) δ ( k − p ) δ s r δ ii (cid:48) = ( − ( − i y ) δ s r δ ii (cid:48) δ hj (cid:48) ¯ v ( k , s ) v ( p , r ) (cid:112) E D ( k ) 2 E D ( p )(2 π ) δ ( k − p ) × (cid:90) d x e − i ( k − P H − p ) x , (B.5)where in the third step, we did the following contractions, in order, • c ( q S , t S ) with c † ( k , s ), • a † n ( q H ) with a h ( P H ), • f † n ( q ¯ D , t ¯ D ) with f j (cid:48) ( p , r ), • d † i ( k , s ) with d i (cid:48) ( p , r ),and accounted for the signs arising from the permutations of the fermion operators. Con-sidering that | k − p | = | P H | (cid:28) | k | , | p | , we may set ¯ v ( k , s ) v ( p , r ) (cid:39) − mδ s r .Moreover, we do the following manipulation (2 π ) δ ( k − p ) (cid:90) d x e − i ( k − P H − p ) x = (2 π ) δ ( k − p ) (cid:90) d x e − i ( k + k − P H − p − p ) x = (2 π ) δ ( k + k − P H − p − p )(2 π ) δ ( k − p ) . (B.6)Finally, setting E D ( k ) = E D ( p ) (cid:39) m , eq. (B.5) yields i A [ DS → D ¯ DH ] (cid:39) ( − i y ) δ s r δ s r δ ii (cid:48) δ hj (cid:48) m (2 π ) δ ( k − p ) . (B.7) Because the diagram of fig. 15 is disconnected when considered alone, the energy-momentum conser-vation on the vertex suggests that it vanishes. This is an artifact of having set the incoming and outgoing D , ¯ D and S particles on shell. In reality, because of their interactions along the ladders (cf. fig. 4), thepropagating fields are not exactly on-shell. A method for integrating out the virtuality of these particleshas been suggested in [18] and employed e.g. in [20, 49]. – 58 – Overlap integral for monopole bound-to-bound transitions
We want to compute the overlap integrals defined in eq. (3.3e), R n (cid:48) (cid:96) (cid:48) m (cid:48) ,n(cid:96)m ( α (cid:48) B , α B ) ≡ (cid:90) d p (2 π ) ˜ ϕ n (cid:48) (cid:96) (cid:48) m (cid:48) ( p ; α (cid:48) B ) ˜ ϕ ∗ n(cid:96)m ( p ; α B )= (cid:90) d r ϕ n (cid:48) (cid:96) (cid:48) m (cid:48) ( r ; α (cid:48) B ) ϕ ∗ n(cid:96)m ( r ; α B ) , (C.1)where the position-space bound-level wavefunctions for the potential V = − α B /r are ϕ n(cid:96)m ( r ; α B ) = (cid:18) κ B n (cid:19) (cid:20) ( n − (cid:96) − n ( n + (cid:96) )! (cid:21) e − x B /n (cid:18) x B n (cid:19) (cid:96) L (cid:96) +1 n − (cid:96) − (cid:18) x B n (cid:19) Y (cid:96)m (Ω r ) , (C.2)with κ B ≡ µα B being the Bohr momentum and x B ≡ κ B r . Inserting eq. (C.2) into (C.1), R n (cid:48) (cid:96) (cid:48) m (cid:48) ,n(cid:96)m ( α (cid:48) B , α B ) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) (cid:20) ( n − (cid:96) − n ( n + (cid:96) )! ( n (cid:48) − (cid:96) − n (cid:48) ( n (cid:48) + (cid:96) )! (cid:21) / (cid:18) κ B κ (cid:48) B nn (cid:48) (cid:19) (cid:96) +3 / ×× (cid:90) ∞ dr e − ( κ B /n + κ (cid:48)B /n (cid:48) ) r r (cid:96) +2 L (cid:96) +1 n − (cid:96) − (cid:18) κ B rn (cid:19) L (cid:96) +1 n (cid:48) − (cid:96) − (cid:18) κ (cid:48) B rn (cid:48) (cid:19) . (C.3)To compute the integral, ee will use the identity [62, section 7.414, item 4] (cid:90) ∞ dr e − ρr r a L aq ( λr ) L aq (cid:48) ( λ (cid:48) r ) = (C.4)= Γ( q + q (cid:48) + a + 1) q ! q (cid:48) ! ( b − λ ) q ( b − λ (cid:48) ) q (cid:48) b q + q (cid:48) + a +1 2 F (cid:18) − q, − q (cid:48) ; − q − q (cid:48) − a ; ρ ( ρ − λ − λ (cid:48) )( ρ − λ )( ρ − λ (cid:48) ) (cid:19) ≡ h ( a, q, q (cid:48) , λ, λ (cid:48) , ρ ) , where F is the ordinary hypergeometric function. Equation (C.4) holds for Re( a ) > − ρ ) >
0. The overlap integral (C.3) can be expressed in terms of the h function as R n (cid:48) (cid:96) (cid:48) m (cid:48) ,n(cid:96)m ( α (cid:48) B , α B ) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) (cid:20) ( n − (cid:96) − n ( n + (cid:96) )! ( n (cid:48) − (cid:96) − n (cid:48) ( n (cid:48) + (cid:96) )! (cid:21) / (cid:18) κ B κ (cid:48) B nn (cid:48) (cid:19) (cid:96) +3 / × (cid:20) − ddρ h (cid:0) a, q, q (cid:48) , λ, λ (cid:48) , ρ (cid:1)(cid:21) , (C.5)with a = 2 (cid:96) + 1 , q ( (cid:48) ) = n ( (cid:48) ) − (cid:96) − , λ ( (cid:48) ) = 2 κ ( (cid:48) ) B /n ( (cid:48) ) , ρ = κ B /n + κ (cid:48) B /n (cid:48) . (C.6)We find R n (cid:48) (cid:96) (cid:48) m (cid:48) ,n(cid:96)m ( α (cid:48) B , α B ) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) ( − n − (cid:96) − ( n + n (cid:48) − (cid:112) nn (cid:48) ( n + (cid:96) )!( n (cid:48) + (cid:96) )!( n − (cid:96) − n (cid:48) − (cid:96) − × (cid:18) α B α (cid:48) B nn (cid:48) (cid:19) (cid:96) +3 / ( α B − α (cid:48) B ) (cid:18) α B n − α (cid:48) B n (cid:48) (cid:19) n + n (cid:48) − (cid:96) − (cid:18) α B n + α (cid:48) B n (cid:48) (cid:19) − ( n + n (cid:48) +1) × F (cid:34) (cid:96) − n, (cid:96) − n (cid:48) , − n − n (cid:48) , (cid:18) α B /n + α (cid:48) B /n (cid:48) α B /n − α (cid:48) B /n (cid:48) (cid:19) (cid:35) . (C.7)– 59 –ote that eq. (C.7) vanishes if α B = α (cid:48) B and n (cid:54) = n (cid:48) , but is equal to 1 if α B = α (cid:48) B and n = n (cid:48) ,due to the orthonormality of the wavefunctions. Equation (C.7) is useful for calculatingmonopole transitions between bound states of different potentials, i.e. for α B (cid:54) = α (cid:48) B . Thisresult complements the computation of ref. [1] of scattering-to-bound monopole transitions.For n = n (cid:48) = 1 and (cid:96) = 0, we obtain R , ( α (cid:48) B , α B ) = 8( α B α (cid:48) B ) / ( α B + α (cid:48) B ) . (C.8) D Scalar emission via vector-scalar fusion
In many of the BSF processes considered in this work, the radiative parts of the amplitudesreceive contributions from diagrams where off-shell vector and Higgs bosons fuse to producethe on-shell radiated Higgs boson; one such diagram is pictured in fig. 16. These diagramsresemble the ones where an on-shell vector is emitted from an off-shell vector or scalarmediator exchanged between the interacting particles (cf. figs. 5 to 8.) This suggests thatthey may be significant. Here we show that the diagrams of the type of fig. 16 are of higherorder than those featuring emission of a vector from an off-shell mediator. Moreover, BSFvia vector emission is of higher order than BSF via emission of a charged scalar [1]. Thus,the Higgs emission diagrams of the type of fig. 16 are very subdominant. s r s r K/ k ( (cid:48) ) K/ − k ( (cid:48) ) P/ pP/ − pP H i i (cid:48) j (cid:48) h Figure 16 . Scalar emission via vector-scalar fusion.
The contribution from the diagram of fig. 16 is i A = ¯ u ( P/ p, r ) i g γ µ t ai (cid:48) i u ( K/ k (cid:48) , s ) (cid:20) − i g µν ( k (cid:48) − p + P H / (cid:21) (D.1) × i g t aj (cid:48) j (3 P H / − k (cid:48) + p ) ν i ( k (cid:48) − p − P H / − m H ¯ v ( K/ − k, s )( − i y ) v ( P/ − p ) . Applying the standard approximations due to the scale hierarchies, the above becomes i A (cid:39) i g y t ai (cid:48) i t aj (cid:48) j m ( k (cid:48) − p − P H / · [( K + P ) / k (cid:48) + p ]( k (cid:48) − p ) [( k (cid:48) − p ) + m H ] . (D.2)The 4-vector product in the numerator is of order ∼ m ( α B + v ), which renders eq. (D.2)of higher order than eqs. (3.14) and (3.18), and even more so that eqs. (3.21).– 60 – cknowledgements We thank Karl Nordstr¨om for collaboration at the early stages of this work. This workwas supported by the ANR ACHN 2015 grant (“TheIntricateDark” project), and by theNWO Vidi grant “Self-interacting asymmetric dark matter”.
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