Baryogenesis from the weak scale to the grand unification scale

The current status of baryogenesis is reviewed, with an emphasis on electroweak baryogenesis and leptogenesis. The first detailed studies were carried out for $\mathrm{SU}(5)$ grand unified theory (GUT) models where $CP$-violating decays of leptoquarks generate a baryon asymmetry. These GUT models were excluded by the discovery of unsuppressed, ($B+L$)-violating sphaleron processes at high temperatures. Yet a new possibility emerged: electroweak baryogenesis. Here sphaleron processes generate a baryon asymmetry during a strongly first-order phase transition. This mechanism has been studied in detail in many extensions of the standard model. However, constraints from the LHC and from low-energy precision experiments exclude most of the known models, leaving composite Higgs models of electroweak symmetry breaking as an interesting possibility. Sphaleron processes are also the basis of leptogenesis, where $CP$-violating decays of heavy right-handed neutrinos generate a lepton asymmetry that is partially converted to a baryon asymmetry. This mechanism is closely related to that of GUT baryogenesis, and simple estimates based on GUT models can explain the order of magnitude of the observed baryon-to-photon ratio. In the one-flavor approximation an upper bound on the light-neutrino masses has been derived that is consistent with the cosmological upper bound on the sum of neutrino masses. For quasidegenerate right-handed neutrinos the leptogenesis temperature can be lowered from the GUT scale down to the weak scale, and $CP$-violating oscillations of GeV sterile neutinos can also lead to successful leptogenesis. Significant progress has been made in developing a full field-theoretical description of thermal leptogenesis, which demonstrated that interactions with gauge bosons of the thermal plasma play a crucial role. Finally, recent ideas on how the seesaw mechanism and $B-L$ breaking at the GUT scale can be probed by gravitational waves are discussed.

Figure 1

Sketch of the minimal field energy for a given value of the Chern-Simons number $N_{\mathrm{CS}}$. The energy approaches the minima with nonzero slope ([8]).

Figure 2

Temperature dependence of the expectation value of ${\varphi }^{†}\varphi$ in the standard model. $\varphi$ is the Higgs field. The points are the results of lattice simulations, and the full line is an interpolation of the data. The observable $⟨{\varphi }^{†}\varphi ⟩$ can become negative because it is ultraviolet divergent and additively renormalized. The shaded bands represent perturbative results for the broken and symmetric phases; their thickness estimates unknown higher-order contributions. From [146].

Figure 3

The standard model sphaleron rate computed on the lattice and the fit to the broken phase rate [Eq. (27)], shown with a shaded error band. At low temperatures the sphaleron rate is exponentially small, which requires a special multicanonical method for the simulation. The perturbative result ([100]) is the one-loop approximation to an expansion around the sphaleron solution. Pure gauge refers to the rate in hot $\mathrm{SU}(2)$ gauge theory. The sphalerons freeze out when $\mathrm{\Gamma }$ crosses the appropriately scaled Hubble parameter, which is shown as the almost horizontal line. From [147].

Figure 4

Effective potential giving rise to a first-order phase transition. The value at $\phi =0$ is subtracted.

Figure 5

Sketch of the bubble wall (between the dashed lines) in the rest frame of the wall. More particles are crossing the wall from right to left. The force on tops with spin in the $+z$ direction is smaller than the force on antitops with spin in the $-z$ direction.

Figure 6

EDM constraints for parameter benchmarks corresponding to different heavy-Higgs-boson masses in the type-II 2HDM. The dash-dotted line shows the electron EDM (eEDM) bound before the ACME experiment. The black dashed lines indicate the minimum $CP$-violating phase necessary for successful baryogenesis, with $m_{H^0}=200\mathrm{GeV}$ and varying $m_{A^0}=m_{H^{\pm }}$. The green area is excluded by neutron EDM (nEDM) bounds, and the blue area is excluded by the electron EDM bound from ACME (2014). From [148].

Figure 7

Large contribution to the electron EDM from top loop and singlet-doublet mixing. From [155].

Figure 8

Electron EDM vs the lightest chargino mass in the split NMSSM for two parameter sets (setup 1, red dots; setup 2, green dots). The dotted line denotes the ACME (2014) upper bound $|d_e|<8.7\times {10}^{-29}e\hspace{0.17em}\mathrm{cm}$. We have added the dashed line that indicates the ACME (2018) upper bound $|d_e|<1.1\times {10}^{-29}e\hspace{0.17em}\mathrm{cm}$. Adapted from [132].

Figure 9

Contours of ${\eta }_B/{\eta }_B^{\text{observed}}=1$ (solid lines) and 0.1 (dashed lines) along the $(m_{\tilde{H}},m_{\tilde{S}})$ plane. The orange area is excluded by the ACME (2014) bound. The orange dashed line corresponds to the anticipated sensitivity $|d_e|=1.0\times {10}^{-29}e\hspace{0.17em}\mathrm{cm}$, which essentially agrees with the ACME (2018) bound. From [167].

Figure 10

Schematic shape of the free energy as a function of the dilaton expectation value $\chi$. Red, hot region with $g_{\chi }\chi \lesssim T$; blue, cold region with $g_{\chi }\chi \gtrsim T$. From [78].

Figure 11

Contours of the $CP$-violating imaginary part of the top-Yukawa coupling in the $(m_{\chi },m_{\star })$ plane. The dashed lines correspond to a mesonlike dilaton, and the solid lines correspond to a glueball-like dilaton. From [78].

Figure 12

Tree-level and one-loop diagrams contributing to heavy-neutrino decays.

Figure 13

Decays and inverse decays of heavy neutrinos, $\mathrm{\Delta }L=2$ processes with virtual intermediate heavy neutrinos, and $\mathrm{\Delta }L=1$ scattering processes.

Figure 14

Top panel: decay, scattering, and washout rates normalized to the Hubble parameter at $z=1$ compared to the Hubble parameter $H(z)$. The two branches of ${\mathrm{\Gamma }}_W$ at $z\ll 1$ represent approximate upper and lower bounds. Bottom panel: evolution of $N_1$ abundance and $B-L$ asymmetry for both thermal and zero initial abundance. The neutrino parameters are $M_1={10}^{10}\mathrm{GeV}$, ${\tilde{m}}_1={10}^{-3}\mathrm{GeV}$, and $\overline{m}=0.05\mathrm{GeV}$. From [81].

Figure 15

Left panel: efficiency factor ${\kappa }_{\mathrm{f}}$ as a function of the effective neutrino mass ${\tilde{m}}_1$. The hatched region represents the theoretical uncertainty due to $\mathrm{\Delta }L=1$ scattering processes, the dashed lines indicate analytical results, and the circled line is a power-law fit. Right panel: upper and lower bounds on the heavy-neutrino mass $M_1$ (the weaker lower bound corresponds to thermal initial conditions). The dotted line is a lower bound on the initial temperature $T_i$; the gray triangle is excluded by theoretical consistency and the circled lines represent analytical results. In both panels the vertical lines indicate the range $(m_{\mathrm{sol}},m_{\mathrm{atm}})$ (see the text). From [84].

Figure 16

Evolution of the $B\text{−}L$ asymmetry for each lepton flavor as a function of $z=M_1/T$ for the two sets of neutrino masses and phases (left panel: $S_2$; right panel: $S_3$) listed in Table 1. From [246].

Figure 17

Resonant leptogenesis: lepton asymmetry ${\eta }^L$, with $L=L_{\tau }$, computed in a density matrix formalism for different initial conditions (solid lines) compared to ${\eta }^L$ obtained in a flavor-diagonal approximation (dashed lines). The model parameters are specified in the table. From [47].

Figure 18

Left panel: heavy-neutrino pair production and decay in an extension of the standard model with an additional $Z^{\prime }$ boson. Right panel: contours in the plane of neutrino mixing $|V_{lN}|$ and neutrino mass $m_N$. The dashed red lines correspond to different branching ratios $\mathrm{BR}(\mu \to e\gamma )$, and the blue lines correspond to two distances between displaced vertices at the LHC. From [133].

Figure 19

Constraints on mass and mixing for $N_1$ making up all of dark matter. The colored regions are excluded. A lepton asymmetry affects the proton-to-neutron ratio during big bang nucleosynthesis (BBN), which gives rise to an upper limit on the lepton asymmetry. The BBN limit given by the solid black line holds if all of the input lepton asymmetry is only in the muon flavor. The dashed line corresponds to the BBN limit if the lepton asymmetry is split equally among all three flavors. Large mixing angles are excluded because too much dark matter would be produced, or because x rays from the decay $N_1\to \nu \gamma$ would have already been detected. Additional constraints (not shown) come from structure formation ([291]). From [63].

Figure 20

Mixing $U_{\mu i}^2$ of heavy neutrinos $N_i$ with leptons $l_{\mu }$ as a function of the heavy-neutrino mass $M_i$. The gray region is excluded by direct searches of heavy neutral leptons, and the lines show the expected sensitivities for the ongoing T2K, NA62, Belle II, LHCb, ATLAS, and CMS experiments. The parameter f.t. measures the amount of fine-tuning for Yukawa couplings needed for successful leptogenesis. From [1].

Figure 21

Relative frequency for the leptogenesis parameters (left panel) ${\tilde{m}}_1$ and (right panel) ${ϵ}_1$. The solid lines denote the position of the median and the dashed lines are the boundaries of the 68% confidence region. From [85].

Figure 22

Left panel: region of successful flavored leptogenesis in the ($m_1$,$M_1$) plane for zero and thermal initial $N_1$ abundance. Right panel: correlation between Dirac phase and Majorana phase yielding maximal baryon asymmetry (normal hierarchy). From [74].

Figure 23

Scatterplot in the ($m_1$,$m_{ee}$) plane where the washout of large initial asymmetries is required: $N_{B-L}^i=0.1(\mathrm{red}),0.01(\text{green}),0.001(\mathrm{blue})$; the vertical lines indicate the values of $m_1$ in the three cases, beyond which 99% of the scatter points are found. From [141].

Figure 24

Number of produced Majorana neutrino per unit time and unit volume as a function of the temperature for $M_N={10}^7\mathrm{GeV}$. The dotted curve is the result with thermal masses included but without any soft gauge interactions. The full line includes an arbitray number of soft gauge interactions, as illustrated in Fig. 25. From [20].

Figure 25

Self-energy diagram with soft gauge bosons for Majorana neutrino $N$, whose imaginary part contributes up to leading order to the $N$-production rate. From [20].

Figure 26

Number of produced massless Majorana neutrino per unit time and unit volume as a function of the temperature. All leading-order contributions are shown. $1\leftrightarrow 2$ are the inverse decay processes without and with multiple scattering mediated by soft gauge bosons displayed in Fig. 24. The $2\to 2$ scattering contributions involving gauge bosons are of similar size and dominate at high $T$, while the top-quark scattering is always subdominant. From [46].

Figure 27

Function $\mathcal{W}$ appearing in the washout rate in Eq. (135). Shown are the LO result including thermal masses (dotted line), the relativistic NLO result, and the NLO result in the nonrelativistic approximation (dashed line). Also shown is the LO LPM-resummed result that is valid in the ultrarelativistic regime ($M_I\lesssim gT$) (dash-dotted line). From [59].

Figure 28

Path in the complex time plane for nonequilibrium Green’s functions.

Figure 29

Transformation of the lepton self-energy diagram to a “double-blob” diagram amenable to resummation. From [135].

Figure 30

Comparison of the Kadanoff-Baym computation including LPM-resummed gauge interactions (full), without computing gauge interactions but instead parametrizing them with thermal widths (KB), with Boltzmann equation containing (inverse) sterile-neutrino decays using classical statistics ($B$) and quantum statistics (QB). The mass of the lightest heavy neutrino is ${10}^{10}\mathrm{GeV}$, the temperature is constant at ${10}^{11}\mathrm{GeV}$. This corresponds to the previously discussed ultrarelativistic regime $T\gtrsim M/g$. From [135].

Figure 31

Left panel: hybrid inflation. The time evolution of the inflaton field $\mathrm{\Phi }$ leads to a tachyonic mass of the waterfall field $S$ that triggers a rapid transition to a phase with spontaneously broken $B-L$ symmetry. Right panel: comoving number densities of the particles of the $B-L$ Higgs sector ($S$), the thermal and nonthermal (s)neutrinos $(N_1^{\mathrm{th}},N_1^{\mathrm{nt}})$, the MSSM radiation ($R$), the gravitinos $(\tilde{G})$, and the $B-L$ asymmetry. Values were obtained by solving the Boltzmann equations for $v_{B-L}=5\times {10}^{15}\mathrm{GeV}$, $M_1=5.4\times {10}^{10}\mathrm{GeV}$, and ${\tilde{m}}_1=4.0\times {10}^{-2}\mathrm{eV}$. From [94].

Figure 32

Viable parameter space (green) for hybrid inflation, leptogenesis, neutralino DM, and big bang nucleosynthesis. Hybrid inflation and the dynamics of reheating correlate the parameters $v_{B-L}$, $T_{\mathrm{rh}}$, $m_{3/2}$, and ${\tilde{m}}_1$ (black curves). Successful leptogenesis occurs outside the gray-shaded region. Neutralino DM is viable in the green region, corresponding to a Higgsino (wino) with mass $100\le m_{\mathrm{LSP}}/\mathrm{GeV}\le 1060$ (2680). From [97].

Figure 33

GW spectrum for $G\mu =2\times {10}^{-7}$. Different values of $\sqrt{\kappa }$ are indicated in different colors; the blue curve corresponds to a cosmic-string network surviving until today. The dot-dashed lines depict an analytical estimate. The lighter gray-shaded areas indicate the sensitivities of (planned) experiments SKA ([297]), LISA ([14]), LIGO ([3]), and ET ([238]). The crosses within the SKA band indicate constraints by the IPTA ([303]). From [97].