Effective Field Theories for Dark Matter Pairs in the Early Universe

In this conference paper, we consider effective field theories of non-relativistic dark matter particles interacting with a light force mediator in the early expanding universe. We present a general framework, where to account in a systematic way for the relevant processes that may affect the dynamics during thermal freeze-out. In the temperature regime where near-threshold effects, most notably the formation of bound states and Sommerfeld enhancement, have a large impact on the dark matter relic density, we scrutinize possible contributions from higher excited states and radiative corrections in the annihilations and decays of dark-matter pairs.


INTRODUCTION
Complementary astrophysical observations strongly support the evidence that 80% of the present matter consists of dark matter (DM), and anisotropy measurements in the CMB determine precisely its relic density to be ΩDMh 2 = 0.1200 ± 0.0012. [1] Despite its nature being elusive, an extensive work has been put forward and a variety of models have been constrained to reproduce this density [2,3]. A prominent class of models involve heavy thermal DM particles, often referred to as Weakly Interacting Massive Particles (WIMPs), appearing in a typical thermal freeze-out scenario. In this proceeding, we focus on fermionic DM that experiences self-interactions through a long-range mediator within the dark sector. In particular, we consider a QED-like model and study the interactions within a thermal bath of dark photons. The proceeding is based on ref. [4]. In section 2 we show how to construct the DM non-relativistic effective field theories (EFTs). In section 3, we compute, in the EFTs, near threshold observables. We give the DM abundance by solving the rate equations in section 4 upon including the relevant processes. Finally, conclusions and outlook are in section 5.

NREFT DM
The Lagrangian density of a dark Dirac fermion X charged under an abelian gauge group reads where the covariant derivative is Dµ = ∂µ + igAµ, with Aµ the dark photon field and Fµν = ∂µAν − ∂ν Aµ. The dark fine structure constant is α ≡ g 2 /(4π). Additional interactions coupling the dark photon with the SM degrees of freedom (d.o.f.), such as kinetic mixing [5,6], are comprised in L portal , which are beyond the scope of this work and thus are omitted. We are interested in processes close to threshold, i.e., processes involving pairs of non-relativistic dark fermions with relative velocities v rel ∼ α 1. The dark photons form a thermal bath of temperature T that is weakly coupled to the DM. If the latter is thermalized then DM momenta scale like p ∼ √ M T . The scales are assumed to be hierarchically ordered as In a typical freeze-out scenario the decoupling from chemical equilibrium happens around M /T ≈ 25. The clear separation of different scales in (2) allows to build a tower of EFTs starting from (1), and extract the relevant interactions and corresponding observables around the decoupling time. Near threshold effects comprise the annihilation of DM pairs as well as electric transitions within the pairs. Higher multipole transitions will be suppressed at later times, i.e., smaller T . These processes play an important role for a quantitative treatment of the dynamics of the relevant d.o.f. in the early universe, and the corresponding observables appear in the evolution equations.
Integrating out hard modes leads to a non-relativistic EFT, here dubbed NRQED DM [7]. Hard processes such as annihilations, happening at a scale ∼ M , are encoded at leading order in the non-relativistic expansion in the matching coefficients of dimension-6 four-fermion operators that overlap only with S-waves. 1 At order α 3 their imaginary parts read [8,9] Im(ds) = πα 2 They originate from S-wave spin-singlet (XX → γγ) and spintriplet (XX → γγγ) annihilations, respectively. Next we integrate out modes associated to the soft scale M α. In order to enforce that the photon fields do not depend on the soft scale anymore, they are multipole expanded in the relative coordinate r ≡ x 1 − x 2 of the pair, i.e., the distance between a fermion located at x 1 and an antifermion located at x 2 . The effective field theory has the form of potential NRQED (pNRQED) [10,11] and we denote it as pNRQED DM . Its La- and S 1 = σ 1 /2 and S 2 = σ 2 /2 are the spin operators acting on the fermion and antifermion, respectively. At leading order the static potential is the Coulomb potential V (0) = −α/r. While the photon field depends only on the center-of-mass (c.o.m.) position R ≡ (x 1 + x 2 )/2, the bilocal field of the dark pair depends on both r, R and can be decomposed into a scattering and bound state part [12] φ (8) DM annihilations, inherited from the imaginary parts of the NRQED DM matching coefficients, induce the following local potential where S = S 1 + S 2 is the total spin of the pair and the dots comprise annihilations with non-vanishing orbital angular momentum. The case of pNRQED at finite temperature has been studied in refs. [13,14], whereas an application to DM models with scalar mediators can be found in refs. [15,16]. The matching is done in the weakly coupled regime order by order in α, although the EFT is suited to accommodate a non-perturbative framework as well [17,18]. The dynamics at the soft scale is encoded in the matching coefficients of pNRQED DM which are the potentials. The equations of motion are of the Schrödinger type, where the potentials distort the free wavefunctions into bound-state wavefunctions Ψn(r) S ij with discrete energies En, or into scattering wavefunctions Ψp(r) S ij with positive energies Ep. 2 By exploiting EFT techniques to separate the various scales being initially intertwined in (1), we arrive at a thermal field theory that describes dark fermion-antifermion pairs and dark photons of energy of the order of or below the Coulomb binding energy. 2 The spin wavefunction Sij accounts for the pairs being either in a spin-singlet or spin-triplet configuration,

NEAR-THRESHOLD PROCESSES
Though pair-annihilation is a process appearing at the hard momentum scale, it encompasses near-threshold effects induced by repeated soft dark photon exchanges. Resumming such multiple rescatterings for dark fermion-antifermion pairs above threshold, i.e. when they form a scattering state, results in a Sommerfeld enhanced spin-averaged S-wave annihilation cross section in the c.o.m. frame. The velocity-independent contribution from the hard scale is separated from the soft-scale dependent Sommerfeld factor (for S-waves) [19,20] with ζ ≡ α/v rel and p = M v rel /2. On the other hand, for pairs below threshold, the relevant observables for annihilation are the decay widths given by Γ n,para Γ n,ortho for spin-singlet and spin-triplet S-wave bound states. We call them paradarkonium and orthodarkonium, respectively. We remark that the soft dark photon resummation effects are embedded already at the level of the Lagrangian (5), and that the annihilation rates follow directly from (-2) times the imaginary parts of the expectation values of (9). Besides local interactions, the Lagrangian (5) contains an electric dipole term that allows for the formation of a bound state through low-energy photon emission of a scattering state and viceversa, i.e., the dissociation of a bound state by absorption of a dark photon from the thermal bath. We abbreviate the processes by bsf and bsd, respectively. Their thermal rates may be computed, using the optical theorem, from the selfenergy diagrams in pNRQED DM , see fig. 1. Using the real-time formalism we obtain for the bsf cross section where n B is the Bose-Einstein distribution and ∆E p n = Ep − En = (M v 2 rel /4) 1 + α 2 /(n 2 v 2 rel ) + . . . . The dots stand for higher order corrections in the energy spectrum. For the bsd width, we get the convolution integral where 2 are the dark photon polarizations and σ n bsd is the photo-dissociation cross section averaged over the photon polarizations, which reads σ n bsd (k) = 1 2 E b n is the binding energy: E b n = En − 2M . Further low-energy processes are the bound state-to-bound state de-excitation transitions, whose termal widths are and similarly for excitations with ∆E n n = E n − En = (M α 2 /4) 1/n 2 − 1/n 2 + . . . . We observe that each of the rates in (14), (15), (17) and (18) factorizes into a thermal part and an in-vacuum part involving the electric dipole matrix element squared. The thermal component can be further simplified, since the distribution functions n B (En), n B (Ep) vanish exponentially for the temperature region set by (2). In that limit our result for the bsf agrees with the outcomes in [21,22] and for bsd with the ones in [13] for the hydrogen atom in QED at finite T and in [23] for the case of gluo-dissociation of heavy quarkonium in QCD. Eventually the determination of the thermal rates reduces to evaluating the dipole matrix elements.
Finally, we comment on the coupling constant. Since in the fundamental theory (1) the dark photons couple only to the dark matter fermions, the coupling runs with one flavor at scales larger than M , while it is frozen at the value α = α(M ) at scales below M . Thus the coupling constant appearing in the thermal rates discussed so far is in fact a constant.

RATE EQUATIONS
Having presented the relevant low-energy processes and the corresponding thermal rates, we include them in the dynamical rate equations to derive the DM thermal abundance. Here we rely on coupled semi-classical Boltzmann equations, which under certain circumstances, namely for Hubble rates H(T ) much smaller than the bound-state decays, may be written as a single effective rate equation, given by [24] The effective cross section, thermally averaged over the velocities of the incoming unbound pair, is given by 3 Here Γ n ann is meant to be replaced by (12) or (13) when performing the summation over S-wave bound states. Equation (20) holds in the limit when bound state-to-bound state transitions are much smaller than Γ n ann and Γ n bsd , and may be neglected, which is the so-called no-transition limit. Otherwise it has to be replaced by a more general expression presented in ref. [25].
First, we solve the effective Boltzmann equation (19) numerically with the effective cross section in (20) for the DM pairs being in the ground state and with the decay width at leading order (LO) in the matching coefficient (3), i.e. just given by the paradarkonium width Γ 1S,para ann = M α 5 /2. This provides our reference energy density. Then, we solve the Boltzmann equation when including the 2S state in the notransition limit, i.e. considering (20) up to n = 2S, but still with the paradarkonium decay width at LO. The ratio of the two energy densities is given in fig. 2 by the brown-dotted line. Next, we include S-wave excited states up to 10S. In fig. 2, we plot the ratio of the obtained energy density with respect to the reference one as the orange-dotted line.
Furthermore, we consider the effective cross section beyond the no-transition limit approximation, i.e., as given in ref. [25], and solve the Boltzmann equation for the ground state including up to n = 2 states, but neglecting P-wave annihilation widths, and at order α 2 in the matching coefficient (3). The ratio with respect to the reference energy density is shown as the brown-dashed curve in fig. 2, where we see that 2P-to-1S transitions affect the energy density more drastically than just including nS-states in the no-transition limit.
Finally, we evaluate the impact of O(α 3 ) corrections on the 1S state. We include such corrections in the ground state annihilation width and in σannv rel . The black solid line in fig. 2 shows the ratio of the obtained energy density with respect to our reference density. The O(α 3 ) corrections in the matching coefficients result in a much larger effective cross section due to the additional annihilation channel in the orthodarkonium states. It therefore decreases the DM abundance quite significantly by about 14% for DM with mass of 1TeV and coupling α = 0.1.

CONCLUSIONS AND OUTLOOK
In this proceeding, we have summarized the findings of the recent work [4], where we use the language of NRQED and pNRQED to describe the evolution of thermalized heavy DM pairs in the early universe. Under the hierarchy of scales (2), in the EFT dubbed pNRQED DM (5), we compute the relevant thermal rates appearing in the evolution equations. We observe that for fermionic DM pairs the additional annihilation in the spin-triplet channel, starting at order α 3 in the matching coefficients, gives large contributions to the effective cross  (9), beyond the no-transition limit approximation. We recall that the uncertainty in the measured relic density is 1%. Results are given for α = 0.1.
section. Furthermore, also bound state-to-bound state transitions appear to play an important role when determining the relic density of DM. Finally, we remark that for large T the multipole expansion may break down, an issue that should be addressed in future works.