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 of the CTP contour, as illustrated in Fig. 1. The time contour C in the Keldysh-Schwinger prescription consists of two branches denoted byτþ andτ−. The upper time contourτþ ranges from the
initial timeti¼−∞tot¼∞while the lower contourτ−is considered to go from∞back to−∞. Therefore, times on theτ− branch are said to be always later compared to the times onτþ. The time ordering of operator products onC can generically be written as
TC½O1ðt1Þ…OnðtnÞ
≡X
P
ð−1ÞFðpÞθCðtpð1Þ; tpð2ÞÞ…θCðtpðn−1Þ; tpðnÞÞ
×Opð1Þðtpð1ÞÞ…OpðnÞðtpðnÞÞ; ð2:1Þ where the sum is over all the permutationsPof the set of operators Oi and FðpÞ is the number of permutations of fermionic operators. The unit step function and the delta distribution on the Keldysh-Schwinger contour is defined as
θCðt1; t2Þ≡
Correlation functions, i.e., contourC-ordered operator prod-ucts averaged over all states where the weight is the density matrix of the system denoted byρˆ, are defined by
hTCOðx1; x2;…; xnÞi≡Tr½ρˆTCOðx1; x2;…; xnÞ: ð2:3Þ Let us introduce commonly used notations and properties of two-point correlation functions of fermionic or bosonic operator pairs relevant for this work. Because of the two-time structure, there are four possibilities to align the two-timesx0 andy0onCand hence four different components of a general two-point function denoted by Gðx; yÞ, where in matrix notation it can be written as
Gðx; yÞ≡hTCψðxÞψ†ðyÞi
FIG. 1. Keldysh-Schwinger approximation of the closed-time-path contourC, consisting of two time branches τþand τ−.
1In ordinary QFT the initial vacuum state Ω appearing in correlation functionshΩjinT½Oðx;…ÞjΩiinis 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 tin¼−∞ to tout¼∞. By means of Lehmann-Symanzik-Zim-mermann (LSZ) reduction formula it is then possible to relate correlation functions to theSmatrix 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].
DARK MATTER SOMMERFELD-ENHANCED ANNIHILATION… PHYS. REV. D 98,115023 (2018)
115023-3
Here,Gσxσy meansx0∈τσx and y0∈τσy withσi¼ for i¼x, yand the four different components of Gðx; yÞ are defined as
G−þðx; yÞ≡hψðxÞψ†ðyÞi; ð2:5Þ
Gþ−ðx; yÞ≡ ∓hψ†ðyÞψðxÞi; ð2:6Þ Gþþðx; yÞ≡θðx0−y0ÞG−þðx; yÞ þθðy0−x0ÞGþ−ðx; yÞ;
ð2:7Þ G−−ðx; yÞ≡θðx0−y0ÞGþ−ðx; yÞ þθðy0−x0ÞG−þðx; yÞ;
ð2:8Þ where −ðþÞ on the r.h.s. of the second line applies for fermionic (bosonic) field operators. From these definitions one can recognize that not all components are independent, namely the following relation holds:
Gþþðx; yÞ þG−−ðx; yÞ ¼Gþ−ðx; yÞ þG−þðx; yÞ: ð2:9Þ Furthermore, let us introduceretardedandadvanced corre-lators defined by
GRðx; yÞ≡θðx0−y0Þ½G−þðx; yÞ−Gþ−ðx; yÞ
¼Gþþðx; yÞ−Gþ−ðx; yÞ
¼−G−−ðx; yÞ þG−þðx; yÞ; ð2:10Þ
GAðx; yÞ≡−θðy0−x0Þ½G−þðx; yÞ−Gþ−ðx; yÞ
¼Gþþðx; yÞ−G−þðx; yÞ
¼−G−−ðx; yÞ þGþ−ðx; yÞ; ð2:11Þ as well as thespectral functiongiven by
Gρðx; yÞ≡GRðx; yÞ−GAðx; yÞ ¼G−þðx; yÞ−Gþ−ðx; yÞ:
ð2:12Þ From these definitions we can derive further useful properties:
GAðx; yÞ ¼−½GRðy; xÞ†; Gþ−ðx; yÞ ¼ ½Gþ−ðy; xÞ†;
G−þðx; yÞ ¼ ½G−þðy; xÞ†: ð2:13Þ In the case of free (unperturbed) propagatorsG0, the following semigroupproperty holds:
GR0ðx; yÞ ¼Z
d3zGR0ðx; zÞGR0ðz; yÞ; fortx > tz> ty: ð2:14Þ This equality can be verified by noticing that for those time configurations the correlators are proportional to on-shell plane waves in Fourier space. Note that there is no time integration in the above equation. Together with the relations in Eq.(2.13)further semigroup properties can be derived and all important ones are summarized for the use in subsequent sections in AppendixA.
As an example, the time integration over the Schwinger-Keldysh contourCof products of correlators can be written as
Cþþðx; yÞ ¼Z
z0∈Cdz0 Z
d3zAðx; zÞBðz; yÞ
x0;y0∈τþ
¼ Z
τþdz0þ Z
τ−dz0 Z
d3zAðx; zÞBðz; yÞ
x0;y0∈τþ
¼ Z ∞
−∞dz0 Z
d3zAþþðx; zÞBþþðz; yÞ þ Z −∞
∞ dz0 Z
d3zAþ−ðx; zÞB−þðz; yÞ
¼Z ∞
−∞d4zðAþþðx; zÞBþþðz; yÞ−Aþ−ðx; zÞB−þðz; yÞÞ: ð2:15Þ Equation(2.15)is called aLagereth ruleand 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 defineWigner coordinates according to
T¼ ðx0þy0Þ=2; t¼x0−y0; ð2:16Þ
R¼ ðxþyÞ=2; r¼x−y: ð2:17Þ
In the second line all variables are 3-vectors. The Wigner-transformed correlators are defined as Gðt;˜ r;R; TÞ≡GðTþt=2;Rþr=2; T−t=2;R−r=2Þ
¼Gðx0x; y0yÞ: ð2:18Þ
BINDER, COVI, and MUKAIDA PHYS. REV. D 98,115023 (2018)
115023-4
In all computations, the tilde will be dropped such that we can write for the Fourier transformation ofGðx; yÞ:
Gðω;p;R; TÞ ¼Z
dtd3reiðωt−p·rÞGðt;r;R; TÞ: ð2:19Þ
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 thegradient expansion. For example, Eq. (2.15)in Fourier space can be written as
Cþþðω;p;R; TÞ ¼Aþþðω;p;R; TÞGABBþþðω;p;R; TÞ−Aþ−ðω;p;R; TÞGABB−þðω;p;R; TÞ
≃Aþþðω;p;R; TÞBþþðω;p;R; TÞ−Aþ−ðω;p;R; TÞB−þðω;p;R; TÞ; ð2:20Þ GAB≡e−ið∂AT∂Bω−∂Aω∂BT−∇AR·∇Bpþ∇Ap·∇BRÞ=2≃1: ð2:21Þ
Throughout this work, such exponentials containing deriv-atives are always approximated as in the last line, defining the leading order term of the gradient expansion. For homo-geneous and isotropic systems the correlators do not depend onRand 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 equilib-rium (here we consider kinetic as well as chemical equilibrium), the density matrix takes the form
ˆ
ρ∝e−βH; ð2:22Þ
whereHis the Hamiltonian of the system andβfactor is the inverse temperatureTof 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−þðx0−y0Þ ¼∓Gþ−ðx0−y0þiβÞ: ð2:23Þ This important property is called the Kubo-Martin-Schwinger (KMS) relation, where−(þ) 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−þðω;pÞ ¼∓eβωGþ−ðω;pÞ: ð2:24Þ From this condition, various equilibrium relations follow:
Gþ−ðω;pÞ ¼∓nF=BðωÞGρðω;pÞ;
G−þðω;pÞ ¼ ½1∓nF=BðωÞGρðω;pÞ; ð2:25Þ Gþþðω; pÞ ¼GRðω;pÞ þGAðω;pÞ
2 þ
1
2∓nF=BðωÞ
Gρðω;pÞ; ð2:26Þ where the Fermi-Dirac or Bose-Einstein phase-space den-sities are given bynF=BðωÞ ¼1=ðeβω1Þ. Thus in equilib-rium, all correlator components can be calculated from the retarded/advanced components, where the spectral function Gρis 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
hTCOðx1; x2;…; xnÞi ¼Z
dμiρðμiÞZ
μi
½dμOðx1; x2;…; xnÞ
×ei R
x∈CLðxÞ ð2:27Þ
and the action on the CPT contour is S¼R
x∈CLðxÞ.
μcollectively represents the fields, andρstands 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 ofμi at the initial timetithat we take to−∞in the Schwinger-Keldysh prescription of the contour, and the first one takes the average ofμi with the weight of ρðμiÞ. Now, to derive the EoM for two-point correlators, let us consider an infinitesimal perturbation O0†¼O†þϵ, satisfyingϵðtiÞ ¼0. By relying on the measure-invariance principle under this transformation, one obtains the EoM of the two-point correlators from
DARK MATTER SOMMERFELD-ENHANCED ANNIHILATION… PHYS. REV. D 98,115023 (2018)
115023-5
hO0†ðyÞi¼hO†ðyÞiþϵðxÞþi where δL represents a derivative acting from the left. The same procedure can be applied to the case ofO, as well as for deriving the EoM of higher correlation functions. The relation between the abstract EoM of correlators and differ-ential 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 equa-tions of two-point funcequa-tions 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.
III. EQUATIONS OF MOTION IN