Leptogenesis and Dark Matter

Leptogenesis and Dark Matter Wilfried Buchmüller DESY, Hamburg, Germany DOI: http://dx.doi.org/10.3204/PUBDB-2018-00782/C3 We briey review the work carried out in project C3 of the SFB 676. The main topics are Quantum Leptogenesis, the treatment of thermal leptogenesis with nonequilibrium eld theory, and gravitino dark matter. In a minimal supersymmetric model with cosmological B−L breaking, a consistent picture is obtained of ination, preheating and the transition to the hot early universe by entropy production, including the generation of matter and dark matter. 1 Introduction During the past thirty years leptogenesis [1] has become one of the leading candidates for explaining the origin of the matter-antimatter asymmetry of the universe. For more than a decade electroweak baryogenesis was considered superior to leptogenesis, based on the hope to understand the baryon asymmetry within the Standard Model. But with increasing lower bounds on the Higgs boson mass and the discovery of neutrino masses in atmospheric neutrino oscillations consistent with leptogenesis, the connection between the baryon asymmetry and neutrino masses became the favoured mechanism. In fact, based on leptogenesis, the smallness of neutrino masses had been anticipated [2]. At the beginning of the project leptogenesis had already been established as a quantitatively successful theory of the baryon asymmetry, based on numerical and semi-analytical solutions of Boltzmann equations. However, this approach has severe conceptual deciencies. The Boltz- mann equations are classical equations for number densities. On the other hand, the employed kernels are zero-temperature S-matrix elements that involve quantum interferences in a crucial manner. Hence, one might worry that in the thermal plasma any generated baryon asymmetry is washed out. During the rst phase of the project these concerns led to a detailed study of leptogenesis based on nonequilibrium eld theory at nite temperature (Section 2). A second major topic has been the connection between leptogenesis and dark matter in supersymmetric theories (Section 3). For standard WIMP dark matter the high reheating temperature required for leptogenesis leads to a gravitino problem, the production of too much entropy from decays of unstable gravitinos after primordial nucleosynthesis (BBN). It was shown that this problem can be avoided for gravitino dark matter, if the gravitino is the lightest superparticle (LSP). In supersymmetric theories there is a natural connection between leptogenesis and hybrid ination. In the second phase of the project this led to a series of papers where this connection was studied in detail. It is intriguing that for typical leptogenesis parameters one obtains a reheating temperature after ination, which is consistent with gravitino dark matter. Moreover, the scale of supersymmetry breaking is important for the gravitino mass as well as for the SFB 676 Particles, Strings and the Early Universe 317 Wilfried Buchmüller Figure 1: Path in the complex time plane for non-equilibrium Green's functions. Reprinted from Ref. [5], with permission from Elsevier. scalar spectral index which is obtained from the cosmic microwave background (CMB). Hybrid ination, together with reheating and entropy production leads to a characteristic spectrum of primordial gravitational waves (Section 4). In the outlook we describe the prospects for testing leptogenesis with further observations in cosmology and at the Large Hadron Collider (LHC). It is impossible to give adequate references to all the work that motivated and stimulated the research carried out within the project C3. Reviews of basics and recent developments of leptogenesis are, for example [3,4] where also extensive references can be found. Beyond that we refer the reader to the references given in the papers described below. 2 Quantum leptogenesis Leptogenesis is an out-of-equilibrium process during the very early high-temperature phase of the universe. In its simplest version it is dominated by the CP violating interactions of the lightest of the heavy Majorana neutrinos, the seesaw partners of the ordinary neutrinos, with lepton doublets and the Higgs doublet. These standard model elds have gauge interactions, and therefore they are in thermal equilibrium. On the contrary, the weakly coupled heavy neutrino is out of thermal equilibrium. Its decay width is small compared to the interaction rates of particles in thermal equilibrium. This departure from thermal equilibrium, together with a CP violating quantum interference, generates a lepton asymmetry and, via sphaleron processes, also a baryon asymmetry. A rigorous description of this subtle eect requires nonequilibrium eld theory for which one can use the SchwingerKeldysh formalism. Here the basic objects are Green's functions on a path in the complex time plane, see Fig.1. To study the approach to thermal equilibrium, we rst considered a scalar eld theory [5,6]. The SchwingerDyson equation for a path ordered Green's function on the contour C reads Z 2 4 0 0 0 (1 + m )∆C (x1; x2) + d x ΠC (x1; x )∆C (x ; x2) = iδC (x1 x2) ; C − − where the self-energy ΠC describes the interaction with the thermal bath. It is convenient to decompose the path ordered Green's function into the spectral function ∆− and the statistical propagator ∆+, − + 1 ∆ (x1; x2) = i [Φ(x1); Φ(x2)] ; ∆ (x1; x2) = Φ(x1); Φ(x2) : h i 2hf gi ∆− carries information about the spectrum of the system and satises a homogeneous equation, whereas ∆+ satises an inhomogeneous equation, which encodes the interaction with the 318 SFB 676 Particles, Strings and the Early Universe Leptogenesis and Dark Matter Figure 2: Statistical propagator + for . Reprinted from Ref. [5], with permission ∆q (t1; t2) q = 0 from Elsevier. thermal bath. For a homogeneous system the Fourier transform + depends on two time ∆q (t1; t2) coordinates. The statistical propagator also depends on initial conditions. The general solution was obtained in [5]. Starting from vanishing mean values of Φ and Φ_ one obtains the result plotted in Fig.2. One sees damped oscillations in t1 t2 with a frequency given by the mass − m of Φ and an approach to equilibrium in t1 + t2, which is controlled by the width Γ of Φ. For t1 + t2 > 1=Γ the memory of the initial conditions is lost. For a dilute, weakly coupled gas the Boltzmann equation can be recovered as rst term in a derivative expansion. Neglecting memory eects one nds + t1 + t2 ∆ (t1; t2) f(t; q ) ; t = ; q / j j 2 where f(t; q ) is the Boltzmann distribution function for quanta of the eld Φ. Armedj withj the knowledge of the Green's function describing the departure from thermal equilibrium, one can attack the treatment of leptogenesis in nonequilibrium eld theory [79]. The starting point is the Lagrangian 1 µ ∗ T 1 T = Niγ @µN + lLiφλe N + N λi1ClLiφ MN CN L 2 i1 − 2 1 T 1 ∗ T + ηijl φClLjφ + η lLiφCe lL φe ; 2 Li 2 ij j which describes the interaction of a Majorana neutrino N of mass M with lepton doublets lLi and the Higgs doublet φ; C is the charge conjugation matrix. Integrating out the two heavier Majorana neutrinos with masses M2 and M3 yields the eective dimension-5 coupling X 1 T ηij = λik λkj : Mk k>1 SFB 676 Particles, Strings and the Early Universe 319 Wilfried Buchmüller Figure 3: Two-loop contributions to the lepton self-energies ±, which lead to non-zero lepton Πk number densities. Reprinted gures with permission from Ref. [7]. Copyright (2010) by the American Physical Society. Using this eective coupling has the advantage that vertex- and self-energy contributions to the CP asymmetry in the heavy neutrino decay are obtained from a single graph. For lepton doublets and Higgs doublet equilibrium Green's functions can be used whereas the heavy neutrino N is described by a non-equilibrium Green's function similar to the one described above. The generated CP asymmetry is obtained from the two-loop graphs shown in Fig.3. The calculated lepton asymmetry Lkii is the quantum analog of the dierence of Boltzmann distribution functions fLi(t; k) = fli(t; k) f¯ (t; k). Including also thermal damping widths γ − li for lepton doublets and Higgs doublet, the full expression for the quantum asymmetry Lkii signicantly simplies. In the zero-with approximation for the heavy neutrino, γ Γ, one nds Z k k0 γγ0 Lkii(t; t) = ii 16π ·0 2 2 0 0 2 02 − 0 kk !p ((!p k q) + γ )((!p k q ) + γ ) q;q − − − − 0 0 eq 1 −Γt flφ(k; q)flφ(k ; q )f (!p) 1 e ; × N Γ − where ii is the CP asymmetry in the heavy Majorana neutrino decay for avour i, and flφ(k; q) = 1 fl(k) + fφ(q). Compared to the result obtained from Boltzmann equations, − Lkii(t; t) contains corrections due to o-shell eects, thermal damping eects and quantum statistical factors. Amazingly, these corrections partially compensate each other, so that the conventional Boltzmann equations provide a rather accurate prediction of the lepton asymmetry. It is straightforward to incorporate washout eects into the presented formalism. A complete theory of leptogenesis would also require to include the more subtle eect of spectator processes. The connection between the SchwingerKeldysh formalism and kinetic equations for quasiparticles was studied in [10]. Leptogenesis crucially depends on the CP violating phases in the couplings of the heavy Majorana neutrinos to lepton doublets and the Higgs doublet. It is remarkable that leptoge- 320 SFB 676 Particles, Strings and the Early Universe Leptogenesis and Dark Matter Spatial distribution (124.7 133.4GeV) − 14 Fermi LAT excess Ann. NFW α = 1.3 Dec. NFW α = 1.3 Ann. NFW α = 1.7 12 Dec. NFW α = 1.7 10 8 6 (arb. rescaled units) 4 dJ/dE 2 0 -10 -5 0 5 10 Galactic longitude ℓ Figure 4: Decaying vs.

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