The cosmological standard model successfully describes the evolution of the large-scale structure of our Universe. It requires the existence of a cold and collisionless matter component, called dark matter (DM), which dominates over the baryon content in the matter dominated era of our Universe. The Planck satellite measurements of the cosmic microwave background (CMB) temperature anisotropies have nowadays determined the amount of dark matter to an unprecedented precision, reaching the level of subpercentage accuracy in the observational determination of the abundance when combining CMB and external data [1,2], e.g., measurements of the baryon acoustic oscillation.

Interestingly, astrophysical observation and structure formation on subgalactic scales might point toward the nature of dark matter as velocity-dependent self-interacting elementary particles. On the one hand, observations of galaxy cluster systems, where typical rotational velocities are of the order v0∼1000  km/s, set the most stringent bounds on the self-scattering cross section to be less than σ/mDM≲0.7(0.1)  cm2/g in the bullet cluster [3] (in order to guarantee the production of elliptical halos [4,5]). On the other hand, a DM self-scattering cross section of the order σ/mDM∼1  cm2/g on dwarf-galactic scales, where velocities are of the order v0∼30–100  km/s, would lead to a compelling solution of the cusp-core and diversity problem without strongly relying on uncertain assumptions of modeling the barionic feedbacks in simulations. This velocity dependence of the self-scattering cross section can naturally be realized in models where a light mediator acts as a long-range force between the dark matter particles. For a recent review on self-interacting DM, see [6].

Generically, long-range forces can lead to sizable modifications of the DM tree-level annihilation cross section in the regime where the annihilating particles are slow. For the most appealing DM candidates, known as weakly interacting massive particles (WIMP) dark matter [7–9], such that the relic abundance in the early Universe is set by the thermal freeze-out mechanism when the DM is nonrelativistic, these effects can be sizable already at the time of chemical decoupling. Then the inclusion of the long-range force modification in the computation of the relic abundance is necessary to reach the required level of the accuracy set by the Planck precision measurement [1,2]. If the light mediators induce an attractive force between the annihilating DM particles, the total cross section is typically enhanced [10,11] which is often referred to as the Sommerfeld enhancement [12] or Sakharov enhancement [13]. Additionally, such attractive forces can lead to the existence of DM bound-state solutions [14–16]. This opens the possibility for conversion processes between scattering and bound states via radiative processes, influencing the evolution of the abundance of the stable scattering states during the DM thermal history. DM scenarios with Sommerfeld enhancement or with bound states have been widely studied in the literature [17–59] and it has been shown that the main effect of such corrections is to shift in the parameter space the upper bounds on the DM mass, otherwise the theoretically predicted DM density would be too large (overclosure bound).

Classic WIMP candidates with large corrections via Sommerfeld enhancement or bound states are particles charged under the electroweak interactions, like the Wino neutralino in supersymmetric models [60] or the first Kaluza-Klein excitation of the gauge boson in models with extra dimensions [61]. For the supersymmetric case it was realized very early on by [10,11] that the Sommerfeld effect reduces the Wino density up to 30% and pushes the mass of the Wino dark matter candidate to few TeVs in order to obtain the correct relic density. These studies have later been extended to the case of general components of the neutralino [29,30]. Similar and even stronger effects from the Sommerfeld enhancement and bound states were found in the case of coannihilation of the WIMP particle with charged or colored states [31–41]. If the electroweakly charged dark matter is sufficiently heavy, the Sommerfeld enhancement or the presence of bound states due to the exchange of electroweak gauge or Higgs bosons, see e.g., [42,43], are very generic as it was shown e.g., in the minimal dark matter model [40,44] and in Higgs portal models [45]. In these cases, long-range force effects play an important role also for the indirect detection limits [14,15,46–48] and especially for the Wino the Sommerfeld enhancement has lead to the exclusion of most parameter space [49–52]. Note that this effect can be important also when the dark matter is not itself a WIMP, but it is produced by WIMP decay out of equilibrium, like in the super-WIMP mechanism [62,63]. Indeed in such a scenario, the DM inherits part of the energy density of the mother particle and so any change in the latter freeze-out density is directly transferred to the superweakly interacting DM and can relax the BBN constraints on the mother particle [53].

While a lot of effort has been made to compute quantitatively the effects of a long-range force on the DM relic density employing the classical Boltzmann equation method, it is still unclear if that is a sufficient description. Indeed, considering the presence of a thermal plasma on the long-range force leads on one side to a possible screening by the presence of a thermal mass, or on the other to the issue if the coherence in the (in principle infinite) ladder diagram exchanges between the two slowly moving annihilating particles is guaranteed. Moreover, from a conceptional point of view, there is yet no consistent formulation in the existing literature of how to deal with long-range forces at finite temperature, especially if the dark matter is, or, enters an out-of-equilibrium state (already the standard freeze-out scenario is a transition from chemical equilibrium to out-of-chemical equilibrium). The main concern of our work will be to clarify conceptional aspects of the derivation and the solution of the number density equation for DM particles with attractive long-range force interactions in the presence of a hot and dense plasma background, starting from first principles. From the beginning, we work in the real-time formalism, which has a smooth connection to generic out-of-equilibrium phenomena.

The simplified DM system we would like to describe in the presence of a thermal environment is similar to heavy quarks in a hot quark gluon plasma. For this setup it has been shown that finite temperature effects can lead to a melting of the heavy-quark bound states which influences, e.g., the annihilation rate of the heavy quark pair into dilepton [64]. For DM or heavy quark systems, the Sommerfeld enhancement at finite temperature has been discussed in the literature, where the chemical equilibration rate is (i) estimated from linear response theory [37,65,66] and (ii) based on classical rate arguments [67], is then inserted into the nonlinear Boltzmann equation for the DM number density “by hand.” Relying on those estimates, it has been shown that the overclosure bound of the DM mass can be strongly affected by the melting of bound states in a plasma environment [55–57]. However, strictly speaking, the linear response theory method is only applicable if the system is close to chemical equilibrium. Indeed the computation has been done in the imaginary-time formalism so far, capturing the physics of thermal equilibrium. One of our goals is to obtain a more general result directly from the real-time formalism, valid as well for systems way out of equilibrium.

Most of the necessary basics of the real-time method we use are provided in Sec. II as a short review of out-of-equilibrium quantum field theory. Within this mathematical framework, an exact expression for the DM number density equation of our system is derived in Sec. III, where the Sommerfeld-enhanced annihilation or decay rate at finite temperature can be computed from a certain component of a four-point correlation function. We derive the equation of motion for the four-point correlation function on the real-time contour in Sec. IV which becomes in its truncated form a Bethe-Salpeter type of equation. Since we close the correlation functions hierarchy by truncation, the system of coupled equations we have to solve contains only terms with DM two- and four-point functions. In Sec. V, we derive a simple semianalytical solution of the four-point correlator under certain assumptions valid for WIMP-like freeze-outs. Our result does not rely on linear response theory and it is therefore quite general applying also in case of large deviations from chemical equilibrium. The limit of vanishing finite temperature corrections is taken in Sec. VI, showing the consistency between our general results and the classical Boltzmann equation treatment. Here, we also compare to the linear response theory method and clarify the assumptions needed to reproduce those results. Our main numerical results for the finite temperature case are given in Sec. VII, both for a gauge theory and for a Yukawa potential, and discussed in detail in Sec. VIII. Finally, we conclude in Sec. IX.

The generalization of quantum field theory on the closed-time-path (CTP) contour, or real-time formalism, is a mathematical method which allows one to describe the dynamics of quantum systems out of equilibrium. Prominent applications are systems on curved space-time and/or systems having a finite temperature. In this work, we assume that the equilibration of DM in the early Universe is a fast process, and consequently, the initial memory effects before the freeze-out process can be ignored. This leads to the fact that the adiabatic assumption for such a system is an excellent approximation, motivating us to take the Keldysh-Schwinger prescription¹

In ordinary QFT the initial vacuum state Ω appearing in correlation functions ⟨Ω|inT[O(x,…)]|Ω⟩in is equivalent up to a phase to the final vacuum state. For this special situation the operators are ordered along the “flat” time axis ranging from tin=-∞ to tout=∞. By means of Lehmann-Symanzik-Zimmermann (LSZ) reduction formula it is then possible to relate correlation functions to the S matrix and compute cross sections. This in-out formalism breaks down once, e.g., the initial vacuum is not equivalent to the final state vacuum. An expanding background or external sources can introduce such a time dependence. In our work, there are mainly two sources of breaking the time translation invariance. First, since we have a thermal population, we consider traces of time-ordered operator products, where the trace is taken over all possible states. The many particle states are in general time dependent. Second, we have a density matrix next to the time ordering. The CTP, or, in-in formalism we adopt in this work can be, pragmatically speaking, seen as just a mathematical way of how to deal with such more general expectation values. The Keldysh description of the CTP contour applies if initial correlations can be neglected and we refer for a more detailed discussion and limitation to [68].

As an example, the time integration over the Schwinger-Keldysh contour C of products of correlators can be written as C++(x,y)=[∫z0∈Cdz0∫d3zA(x,z)B(z,y)]x0,y0∈τ+=[(∫τ+dz0+∫τ-dz0)∫d3zA(x,z)B(z,y)]x0,y0∈τ+=∫-∞∞dz0∫d3zA++(x,z)B++(z,y)+∫∞-∞dz0∫d3zA+-(x,z)B-+(z,y)=∫-∞∞d4z(A++(x,z)B++(z,y)-A+-(x,z)B-+(z,y)).(2.15)Equation (2.15) is called a Lagereth rule and it is straightforward to work out similar rules for, e.g., different components or double integrations of higher-order products of Keldysh-Schwinger correlators as they will appear later in this work. Let us move on and define Wigner coordinates according to T=(x0+y0)/2,t=x0-y0,(2.16)R=(x+y)/2,r=x-y.(2.17)In the second line all variables are 3-vectors. The Wigner-transformed correlators are defined as G˜(t,r,R,T)≡G(T+t/2,R+r/2,T-t/2,R-r/2)=G(x0x,y0y).(2.18)In all computations, the tilde will be dropped such that we can write for the Fourier transformation of G(x,y): G(ω,p,R,T)=∫dtd3rei(ωt-p·r)G(t,r,R,T).(2.19)One of the great advantages of separating microscopic (t, r) and macroscopic (T, R) variables according to the Wigner transformation is that Fourier transformations of integral expressions can be considerably simplified by using the gradient expansion. For example, Eq. (2.15) in Fourier space can be written as C++(ω,p,R,T)=A++(ω,p,R,T)GABB++(ω,p,R,T)-A+-(ω,p,R,T)GABB-+(ω,p,R,T)≃A++(ω,p,R,T)B++(ω,p,R,T)-A+-(ω,p,R,T)B-+(ω,p,R,T),(2.20)GAB≡e-i(∂TA∂ωB-∂ωA∂TB-∇RA·∇pB+∇pA·∇RB)/2≃1.(2.21)Throughout this work, such exponentials containing derivatives are always approximated as in the last line, defining the leading order term of the gradient expansion. For homogeneous and isotropic systems the correlators do not depend on R and thus for the spatial part the leading order term is exact. For a discussion of the validity of the leading order term of the temporal part we refer to [69]. Let us emphasize here, that for typical DM scenarios the leading order term is always assumed to be a very good approximation.

Next, important properties of two-point correlators in thermal equilibrium are provided. Under this assumption, different components of correlators become related which are otherwise independent. For a system being in equilibrium (here we consider kinetic as well as chemical equilibrium), the density matrix takes the form ρ^∝e-βH,(2.22)where H is the Hamiltonian of the system and β factor is the inverse temperature T of the system. The density matrix in thermal equilibrium can be formally seen as a time evolution operator, where the inverse temperature is regarded as an evolution in the imaginary time direction. Making use of the cyclic property of the trace it can be shown that under the assumption of equilibrium the components are related via G-+(x0-y0)=∓G+-(x0-y0+iβ).(2.23)This important property is called the Kubo-Martin-Schwinger (KMS) relation, where - (+) applies for two-point correlators of fermionic (bosonic) operators. Furthermore, in equilibrium the correlators should depend only on the difference of the time variables due to time translation invariance. Consequently, the KMS condition in Fourier space reads G-+(ω,p)=∓eβωG+-(ω,p).(2.24)From this condition, various equilibrium relations follow: G+-(ω,p)=∓nF/B(ω)Gρ(ω,p),G-+(ω,p)=[1∓nF/B(ω)]Gρ(ω,p),(2.25)G++(ω,p)=GR(ω,p)+GA(ω,p)2+[12∓nF/B(ω)]Gρ(ω,p),(2.26)where the Fermi-Dirac or Bose-Einstein phase-space densities are given by nF/B(ω)=1/(eβω±1). Thus in equilibrium, all correlator components can be calculated from the retarded/advanced components, where the spectral function Gρ is related to those via Eq. (2.12).

General out-of-equilibrium observables, like the dynamic of the number density or spectral information of the system, can be directly inferred from the equation of motions (EoM) of the corresponding correlators. Throughout this work, we derive the correlator EoM from the invariance principle of the path integral measure under infinitesimal perturbations of the fields. The equivalence of CTP correlators and the path integral formulation is given by ⟨TCO(x1,x2,…,xn)⟩=∫dμiρ(μi)∫μi[dμ]O(x1,x2,…,xn)⁢ei∫x∈CL(x)(2.27)and the action on the CPT contour is S=∫x∈CL(x). μ collectively represents the fields, and ρ stands for a state that could be either pure or mixed, as in Eq. (2.3). The second integral in Eq. (2.27) is a path integral with a boundary condition of μi at the initial time ti that we take to -∞ in the Schwinger-Keldysh prescription of the contour, and the first one takes the average of μi with the weight of ρ(μi). Now, to derive the EoM for two-point correlators, let us consider an infinitesimal perturbation O′†=O†+ε, satisfying ε(ti)=0. By relying on the measure-invariance principle under this transformation, one obtains the EoM of the two-point correlators from ⟨O′†(y)⟩=⟨O†(y)⟩+ε(x)+i∫x∈Cε(x)⟨TCδSδLO†(x)O†(y)⟩+O(ε2)(2.28)⇒0=δC(x,y)+i⟨TCδSδLO†(x)O†(y)⟩,(2.29)where δL represents a derivative acting from the left. The same procedure can be applied to the case of O, as well as for deriving the EoM of higher correlation functions. The relation between the abstract EoM of correlators and differential equations for observables will be part of the next section. In general, a correlator EoM depends on higher and lower correlators which is called the Martin-Schwinger hierarchy. For systems where the coupling expansion is appropriate, it might be sufficient to work in the one-particle self-energy approximation, where the EoM are closed in terms of two-point functions, and the kinetic equations can systematically be obtained by expanding the DM self-energy perturbatively in the coupling constant. The kinetic equations of two-point functions in the self-energy approximation are also known as the Kadanoff-Baym equations. For example, at next-to-leading order (NLO) in the self-energy expansion of the two-point correlators the standard Boltzmann equation is recovered.

Finite temperature corrections to nonperturbative systems, e.g., Sommerfeld-enhanced DM annihilations or bound-state decays, where a subclass of higher-order diagrams becomes comparable to the leading order (LO) in vacuum, are less understood in the CTP formalism. The strategy in the next section will be to address this issue by going beyond the one-particle self-energy approximation [69]. More precisely, we derive the exact Martin-Schwinger hierarchy of our particle setup in the CTP formalism by using Eq. (2.29) and truncate the hierarchy at the six-point function level. The system of equations is then closed with respect to two- and four-point functions. This approach allows us to account for the resummation of the Coulomb diagrams and their finite temperature corrections at the same time. Furthermore, we show how to extract the DM number density equation from the EoM of two-point functions and that it depends on the four-point correlator. The complication is that the differential equation of the four-point correlator is coupled to the two-point correlator and in subsequent sections we solve this coupled systems of equations.

Throughout this paper, we consider the following minimal scenario capturing important effects to study long-range force enhanced DM annihilations and bound states under the influence of a hot and dense plasma environment: L=χ¯(i∂-M)χ+gχχ¯γμχAμ+ψ¯(i∂-m)ψ+gψψ¯γμψAμ-14FμνFμν.(3.1)The particles of the equilibrated plasma environment with temperature T are the Abelian mediators Aμ and the light fermionic particles ψ with mass m≪T. Fermionic DM χ is assumed to be nonrelativistic, i.e., M≫T. All fermionic particles are considered to be of Dirac type. We assume the mediator to be massless; however, we provide the final results also for the case of a massive Aμ with mass mV≪M.

Let us illustrate how we can get the DM nonrelativistic effective action in the thermal medium of light particles. It is obtained in two steps. First, hard modes of p≳M are integrated out. In this limit, the DM four-component spinor χ splits into two parts, a term for the particle denoted by the two-component spinor η and a term for the antiparticle denoted by ξ. Second, we assume that DM does not influence the plasma environment during the freeze-out process. This is typically the case since the DM energy densities are smaller than that of light particles at this epoch. And thus, we may also integrate out the plasma fields by assuming they remain in thermal equilibrium. The resulting effective action on the CTP contour for particle η and anti-particle ξ DM is given by SNR[η,ξ]=∫x∈Cη†(x)[i∂t+∇22M]η(x)+ξ†(x)[i∂t-∇22M]ξ(x)+∫x,y∈Cigχ22J(x)D(x,y)J(y)+iOs†(x)Γs(x,y)Os(y).(3.2)Dark matter long-range force interactions are encoded in the first term of the interactions in Eq. (3.2). This term includes the current and the full two-point correlator of the electric potential on the CTP contour which are defined by J(x)≡η†(x)η(x)+ξ†(x)ξ(x),D(x,y)≡⟨TCA0(x)A0(y)⟩,(3.3)respectively. The last term in Eq. (3.2) describes the annihilation part and we only keep the s-wave contribution Os(x)≡ξ†(x)η(x),Γs(x,y)≡π(αχ2+αχαψ)M2(δ4(x-y)02δ4(x-y)δ4(x-y)),(3.4)with the fine-structure constant being αi≡gi2/4π, and summation over the spin indices are implicit. Γs is shown in the matrix representation of the CTP formalism; see e.g., Eq. (2.4) in previous section. Hence the delta functions on the right-hand side are defined on the usual real-time axis. Similar to the vacuum theory, the annihilation part Γs can be computed by cutting the box diagram (containing two Aμ) and the vacuum polarization diagram (containing one Aμ and a loop of light fermions ψ), where now all propagators are defined on the CPT contour. Finite temperature corrections to these hard processes in Γs are neglected²

Assuming free plasma field correlators in the computation of Γs is a good approximation since the energy scale of the hard process is ∼M which is much larger for nonrelativistic particles than typical finite temperature corrections being of the order ∼gψT. Consequently, the dominant thermal corrections should be in the modification of the long-range force correlator D, where the typical DM momentum-exchange scale enters which is much lower compared to the annihilation scale.

In our effective action Eq. (3.2), we have discarded higher order terms in ∇/M (like magnetic interactions) and also interactions with ultrasoft gauge bosons,³

To fully study the out-of-equilibrium dynamics of the bound-state formation [21,22,42,70] at late times of the freeze-out process, it is necessary to include emission and absorption via ultrasoft gauge bosons, e.g., via an electric dipole operator. We drop for simplicity ultrasoft contributions and discuss in detail the limitation of our approach later in this work; see Sec. VI. Note here that at high enough temperature those processes are typically efficient, leading just to the ionization equilibrium among bounded and scattering states. As long as ionization equilibrium is maintained, the effective action we use is sufficient enough to describe Sommerfeld enhanced annihilation and bound-state decay at finite temperature. To estimate the time when the ionization equilibrium is violated concretely, we have to take into account these processes in the thermal plasma, which will be presented elsewhere. In vacuum, ionization equilibrium starts to become violated when the decay width of the lowest bound state exceeds the ionization rate.

In the next sections we proceed as in the following. First, we compute the finite temperature one-loop corrections contained in the potential term D explicitly. Since the number density of DM becomes Boltzmann suppressed in the nonrelativistic regime of the freeze-out process, the dominant thermal loop contributions arise from the relativistic species ψ. This implies that we can solve for D independently of the DM system since we assume DM does not modify the property of the plasma. The correction terms for the DM self-interactions are screening effects on the electric potential, as well as imaginary contributions arising from soft DM-ψ scatterings, derived and discussed in detail in Sec. III A. Second, in Sec. III B the kinetic equations for the DM correlators are derived. We show how to extract from these equations the number density equation, including finite temperature corrected processes for the negative energy spectrum (bound-state decays) as well as for the positive energy contribution (Sommerfeld-enhanced annihilation) in one single equation.

The exact number density Eq. (3.24) depends on the Keldysh four-point correlation function Gηξ,s++--(x,x,x,x). In this section, we derive the system of closed equation of motion needed in order to obtain a solution for this four-point function, including the full resummation of Coulomb divergent ladder diagrams. The result will be a coupled set of two-time Bethe-Salpeter equations on the Keldysh contour as given by the end of this section, Eqs. (4.20)–(4.21). They apply in general for out-of-equilibrium situations and include in their nonperturbative form also the bound-state contributions if present. In order to arrive at those equations, a set of approximations and assumptions is needed. We therefore would like to start from the beginning in deriving those equations, which might lead to a better understanding of their limitation.

In the first simplification, we treat the annihilation term Γs as a perturbation and ignore it in the following computations, since the leading order term in the annihilation part is already contained in Eq. (3.24). The exact set of EoMs for two- and four-point functions in the limit Γs→0 are given by [i∂x0+∇x22M]Gη(x,y)=iδC(x,y)-igχ2∫z∈CD(x,z)⁢[Gηξ(x,z,y,z)-Gηη(x,z,y,z)],(4.1)[i∂x0+∇x22M]G¯ξ(x,y)=iδC(x,y)-igχ2∫z∈CD(x,z)⁢[Gηξ(z,x,z,y)-Gξξ(x,z,y,z)],(4.2)[i∂x0+∇x22M]Gηξ(x,y,z,w)=iδC(x,z)G¯ξ(y,w)-igχ2∫x¯∈CD(x,x¯)⟨TCη(x)J(x¯)ξ†(y)ξ(w)η†(z)⟩,(4.3)where we will work for the rest of this work with the conjugate antiparticle correlator G¯ξ and, here, the spin-uncontracted correlators are defined as G¯ξ(x,y)≡⟨TCξ†(x)ξ(y)⟩,(4.4)Gηξ(x,y,z,w)≡⟨TCη(x)ξ†(y)ξ(w)η†(z)⟩,(4.5)Gηη(x,y,z,w)≡⟨TCη(x)η(y)η†(w)η†(z)⟩,(4.6)Gξξ(x,y,z,w)≡⟨TCξ†(x)ξ†(y)ξ(w)ξ(z)⟩.(4.7)The EoMs for the two-point functions Eqs. (4.1) and (4.2) are equivalent to Eqs. (3.15) and (3.17) in the limit Γs→0, and we have just rewritten them in terms of the four-point correlators and conjugate antiparticle propagator, defined in Eqs. (4.4)–(4.7). In our notation, the spinor indices of the operators having equal space-time arguments in the four-point correlators are summed and J is the current as defined in Eq. (3.3). The different conventions for the ηη and ξξ four-point correlators are because η† and ξ are the creation operators. From the exact differential equation of the four-point correlator in Eq. (4.3), it can be seen that the correlator hierarchy is still not closed yet, since it depends on the six-point function. We close this hierarchy of correlators by truncating the six-point function at the leading order: [i∂y0+∇y22M]⟨TCη(x)J(x¯)ξ†(y)ξ(w)η†(z)⟩≃iδC(y,w)[Gηξ(x,x¯,z,x¯)-Gηη(x,x¯,z,x¯)]-iδC(x¯,y)Gηξ(x,y,z,w),(4.8)i.e., only the integral kernel of the six-point function containing the eight-point function was dropped. The set of correlator differential equations is now closed under this truncation procedure. A fully self-consistent solution requires in principle to solve the equations for the five correlators Gη, Gξ, Gηξ,Gηη,Gξξ simultaneously. This is beyond the scope of this work and we have to further approximate the system in order to obtain at least a simple semianalytical solution at the end.

The solution of our target component Gηξ can formally be decoupled from the solution of Gηη and Gξξ by approximating the latter two quantities as the leading order contribution (dropping the integral kernel, known as Hartree-Fock approximation). Then, one can recognize that [Gηξ(x,z,y,z)-Gηη(x,z,y,z)]≃[Gη(x,y)G¯ξ(z,z)-Gη(x,y)Gη(z,z)+Gη(x,z)Gη(z,y)]=Gη(x,z)Gη(z,y),(4.9)[Gηξ(z,x,z,y)-Gξξ(x,z,y,z)]≃[G¯ξ(x,y)Gη(z,z)-G¯ξ(x,y)G¯ξ(z,z)+G¯ξ(x,z)G¯ξ(z,y)]=G¯ξ(x,z)G¯ξ(z,y).(4.10)In both equations, the last step is a strict equality only if the DM particle Gη(z,z)=-nη and antiparticle G¯ξ(z,z)=-nξ number densities are equal. This is true if there is no DM asymmetry which we will assume throughout this work.

In the last approximation, we perform a coupling expansion of Eqs. (4.1)–(4.3). After inserting the results of the two-point functions into the equation of Gηξ, by using the relations Eqs. (4.9) and (4.10) in the free limit, we obtain for the four-point correlator to the leading order in gχ: Gηξ(x,y,z,w)≃Gη,0(x,z)G¯ξ,0(y,w)+gχ2∫x¯,y¯∈CGη,0(x,x¯)G¯ξ,0(y,y¯)D(x¯,y¯)Gη,0(x¯,z)G¯ξ,0(y¯,w)+∫x¯,y¯∈CGη,0(x,z)G¯ξ,0(y,y¯)(-i)Σ¯ξ(y¯,x¯)G¯ξ,0(x¯,w)+∫x¯,y¯∈CGη,0(x,x¯)G¯ξ,0(y,w)(-i)Ση(x¯,y¯)Gη,0(y¯,z),(4.11)where in the last two terms, particle and antiparticle are disconnected and we have introduced the single-particle self-energies according to Ση(x,y)≡-igχ2D(x,y)Gη,0(x,y),Σ¯ξ(x,y)≡-igχ2D(x,y)G¯ξ,0(x,y).(4.12)The first integral term in Eq. (4.11) contains the ladder diagram exchange between particle and antiparticle. Similar equations for Gηη and Gξξ can also be obtained by applying the same steps. In order to obtain the spectrum of bound-state solutions as well as a fully nonperturbative treatment of the Sommerfeld-enhancement we have to resumm Eq. (4.11) somehow.

We define our resummation scheme of the four-point correlator by resumming the ladder exchange, as well as the self-energy contributions in Eq. (4.11) on an equal footing.

In other words, we are resumming the leading order terms in the coupling expansion of the four-point correlator Gηξ and similar for Gηη and Gξξ. On the one hand, this is a critical point, since this procedure is not exact and we cannot guarantee that other contributions do not play an important role as well, e.g., one of the limitations are systems with large coupling constants or large DM density where we cannot decouple the solution of Gηξ from Gηη and Gξξ. The latter limitation might not be a problem since when focussing on the freeze-out the DM density becomes dilute. On the other hand, as we will discuss in detail later in this work, the final result based on this resummation scheme behaves physically, fulfils KMS condition in equilibrium,⁵

Another resummation scheme for Gηξ we tested can be obtained directly after the steps of the truncation in Eq. (4.8) and Hartree-Fock approximation in Eqs. (4.9)–(4.10). After some algebra, this would result in the Bethe-Salpeter equation Gηξ(x,y,z,w)=Gη(x,z)G¯ξ(y,w)+gχ2∫x¯,y¯∈CGη(x,x¯)G¯ξ(y,y¯)D(x¯,y¯)Gηξ(x¯,y¯,z,w). Note that this equation would be equivalent to the original vacuum BS equation when naively extending the time integration in the latter equation towards the Keldysh contour C. The main difference here compared to our resummation scheme is that the two-point functions are fully interacting. The l.h.s. of the ++-- or --++ component of this equation fulfils the KMS condition in equilibrium, when taking the two-time limit. The right-hand side depends in general on three times. It can be reduced to only two times by assuming a static form for the mediator correlator D(x,y)=δC(tx,ty)V(x-y). Integrating over Keldysh-contour delta function leads to the fact that also the r.h.s. fulfils the KMS condition. However, this is a strong assumption on the form of the mediator correlation function, sending the off-diagonal terms D+- or D-+ to zero. The simplest possibility to take into account the off-diagonal terms and at the same time obtain a two-time structure also of the r.h.s. would be to assume that every component of D is proportional to a time delta function. A crucial observation we have made is that this type of approximation seems to violate the KMS condition through the off-diagonal terms, although the r.h.s. has a two-time structure. The reason might be in the resummation of different orders in the coupling, as caused by the interacting DM two-point correlators in the BS kernel. The coupling expansion and the resummation of terms of equal order in the coupling parameter (our scheme), seems to be essential in order to obtain our final BS Eq. (4.20), fulfilling the KMS condition.