# Leptogenesis

Post-publication activity

Curator: Wilfried Buchmüller

Leptogenesis relates the matter-antimatter asymmetry of the universe to neutrino properties. Decays of heavy Majorana neutrinos generate a lepton asymmetry which is partly converted to a baryon asymmetry via sphaleron processes. Consistency with the observed baryon asymmetry implies lower and upper bounds on the light neutrino masses. During the past decade a variety of models have been developed, which make use of the leptogenesis idea to explain the cosmological baryon asymmetry. Moreover, significant progress has been made towards a full quantum mechanical description of leptogenesis.

$$\newcommand{\Mp}{M_{\mathrm{P}}}\def\mt{\widetilde{m}_1}\def\mb{\overline{m}}$$

## Matter-antimatter asymmetry

One of the mysteries of our universe is the observed density of baryons, i.e., protons and neutrons. Since there is good evidence that the universe is mostly made up of matter and no antimatter, the baryon density corresponds to the cosmological matter-antimatter asymmetry. It can be inferred from the ratio of the number density of baryons to photons in the universe, which has been determined most precisely by the measurement of the angular distribution of the temperature fluctuations of the microwave background radiation (Komatsu, 2010fb)

$\tag{1}\eta_B=\frac{n_B-n_{\bar{B}}}{n_{\gamma}} \simeq \frac{n_B}{n_{\gamma}} = 6.19 \pm 0.15 \times 10^{-10}.$

Figure 1: One of the 12-fermion processes which are in thermal equilibrium in the high-temperature phase of the standard model.

Why does this ratio have this value? Starting from a matter-antimatter symmetric initial state at high temperatures, as suggested by an inflationary phase of the very early universe, one would expect a much smaller baryon asymmetry. An attractive explanation of the observed value for ratio $$\eta_B$$ is the leptogenesis mechanism proposed by Fukugita and Yanagida (Fukugita, 1986hr), which relates the baryon asymmetry to properties of neutrinos, the lightest elementary particles.

The starting point are the general conditions for baryogenesis, which particle interactions and the cosmological evolution have to satisfy, as pointed out by Sakharov already in 1967 (Sakharov, 1967dj),

• baryon number violating interactions,
• $$C$$ and $$C\!P$$ violation,
• deviation from thermal equilibrium,

where $$C$$ and $$P$$ denote the discrete transformations charge conjugation and spatial reflection, respectively.

A further crucial ingredient of baryogenesis is the connection between baryon number $$B$$ and lepton number $$L$$ in the high-temperature, symmetric phase of the Standard Model of particle physics. Due to the chiral nature of the weak interactions $$B$$ and $$L$$ are not conserved (tHooft, 1976up). At zero temperature this has no observable effect due to the smallness of the weak coupling. However, as the temperature reaches the critical temperature $$T_c$$ of the electroweak phase transition, $$B$$ and $$L$$ violating processes come into thermal equilibrium (Kuzmin, 1985mm). The rate of these processes is related to the free energy of sphaleron-type field configurations which carry topological charge. In the Standard Model they lead to an effective interaction of all left-handed fermions (tHooft, 1976up) (cf. Figure 1),

$\tag{2} O_{B+L} = \prod_i \left(q_{Li} q_{Li} q_{Li} l_{Li}\right)\; ,$

which violates baryon and lepton number by three units,

$\Delta B = \Delta L = 3\;. \tag{3}$

The interaction (2) leads to processes such as

$\tag{4}u^c + d^c + c^c \rightarrow d + 2 s + 2 b +t + \nu_e + \nu_\mu + \nu_\tau\;.$

The sphaleron transition rate in the symmetric high-temperature phase has been evaluated by combining analytical resummation with numerical lattice techniques. The result is, in accord with early estimates, that $$B$$ and $$L$$ violating processes are in thermal equilibrium for temperatures in the range

$\tag{5}T_{\rm EW} \sim 100\ \mbox{GeV} < T < T_{\rm SPH} \sim 10^{12}\ \mbox{GeV}\;.$

Sphaleron processes have a profound effect on the generation of the cosmological baryon asymmetry in the hot early universe. In a weakly coupled plasma, one can assign a chemical potential $$\mu$$ to all fermionic and bosonic fields with the same gauge interactions. In the Standard Model, with one Higgs doublet $$H$$ and three quark-lepton generations one then has $$16$$ chemical potentials, for the Higgs doublet, the left-handed quark and lepton doublets $$q_i$$ and $$\ell_i$$, and the right-handed quark and lepton singlets $$u_i$$, $$d_i$$, and $$e_i$$$$(i=1,\ldots,3)$$. For a non-interacting gas the chemical potentials determine the asymmetries in the particle and antiparticle number densities. For massless particles the asymmetries at temperature $$T$$ are given by

$\tag{6} n_i-\overline{n}_i={g T^3\over 6} \left\{\begin{array}{rl}\beta\mu_i +{\cal O}\left(\left(\beta\mu_i\right)^3\right)\ , &\mbox{fermions}\ ,\\ 2\beta\mu_i+{\cal O}\left(\left(\beta\mu_i\right)^3\right)\;, &\mbox{bosons}\ , \end{array}\right.$

where $$\beta = 1/T$$. In the case of leptogenesis all chemical potentials are very small, i.e., $$\beta \mu_i \ll 1$$.

Quarks, leptons and Higgs bosons interact via Yukawa and gauge couplings and, in addition, via the nonperturbative sphaleron processes. In thermal equilibrium all these processes yield constraints between the various chemical potentials. The effective interaction implies the relation

$\tag{7} \sum_i\left(3\mu_{qi} + \mu_{li}\right) = 0\;.$

The Yukawa interactions, supplemented by gauge interactions, yield relations between the chemical potentials of left-handed and right-handed fermions, which hold if the corresponding interactions are in thermal equilibrium. In the temperature range $$100\ \mbox{GeV} < T < 10^{12}\ \mbox{GeV}$$, which is of interest for baryogenesis, this is the case for gauge interactions. On the other hand, Yukawa interactions are in equilibrium only in a more restricted temperature range that depends on the strength of the Yukawa couplings. In the simplest version of leptogenesis discussed below this complication is ignored, although this is not justified generically (see below).

Using Eq. (6), the baryon number density $$n_B \equiv g B T^2/6$$ and the lepton number densities $$n_{L_i} \equiv L_i gT^2/6$$ can be expressed in terms of the chemical potentials:

\begin{aligned}\tag{8} B &= \sum_i \left(2\mu_{qi} + \mu_{ui} + \mu_{di}\right)\;, \\ \end{aligned}

\begin{aligned}\tag{9} L_i &= 2\mu_{li} + \mu_{ei}\;,\quad L=\sum_i L_i\;.\end{aligned}

For massless neutrinos the asymmetries $$L_i-B/3$$ are conserved. Using the relations implied by the Yukawa interactions one obtains $$\mu_{li} \equiv \mu_l$$, $$\mu_{qi} \equiv \mu_q$$, etc. Demanding furthermore that the vacuum carries no hypercharge, one can express the total baryon and lepton asymmetries in terms of a single chemical potential, for instance $$\mu_l$$,

$\tag{10}B = -4\mu_l\;, \quad L = {51\over 7}\mu_l\;.$

This yields an important connection between the $$B$$, $$B-L$$ and $$L$$ asymmetries,

$\tag{11} B = c_s (B-L)\ , \quad L= (c_s - 1)(B- L) \ ,$

where $$c_s = 28/79$$. These relations hold in the high-temperature symmetric phase of the Standard Model.

The relations (11) between the quantities $$B$$, $$B$$$$-$$$$L$$ and $$L$$ suggest that violation of $$B$$$$-$$$$L$$ is needed in order to generate a baryon asymmetry. Because the $$B$$$$-$$$$L$$ current is conserved, the value of $$B$$$$-$$$$L$$ at time $$t_f$$, where the leptogenesis process is completed, determines the value of the baryon asymmetry today,

$\tag{12}B(t_0)\ =\ c_s (B-L)(t_f)\;.$

On the other hand, during the leptogenesis process the strength of interactions that violate $$B$$$$-$$$$L$$, and therefore $$L$$, can only be weak. Otherwise, because of Eq. (11), they would wash out any baryon asymmetry. As explained below, the interplay between these conflicting conditions leads to important constraints on the properties of neutrinos.

## Lepton-number violation and neutrino masses

Lepton number violation is most easily realized by adding right-handed neutrinos to the Standard Model, which allow for an elegant explanation of the smallness of the light neutrino masses via the seesaw mechanism. The most general Lagrangian for couplings and masses of charged leptons and neutrinos is given by

\begin{aligned} \tag{13} \mathcal{L} =& {\bar l}_{Li} i \partial\llap{/}l_{Li} + {\bar e}_{Ri} i \partial\llap{/}e_{Ri} + {\bar \nu}_{Ri} i \partial\llap{/}\nu_{Ri}\\ & +\; f_{ij}{\bar e}_{Ri}l_{Lj}H^{\dagger} +h_{ij}{\bar \nu}_{Ri}l_{Lj}H - \frac{1}{2}M_{i}\nu_{Ri}\nu_{Ri} + {\rm h.c.}\ .\end{aligned}

The right-handed neutrinos have no Standard Model gauge interactions and can therefore have Majorana masses that violate lepton number. The vacuum expectation value of the Higgs field, $$\langle H\rangle=v_F$$, generates Dirac masses $$m_e$$ and $$m_D$$ for charged leptons and neutrinos, $$m_e=f v_F$$ and $$m_D=hv_F$$. Since the Majorana masses are not controlled by electroweak symmetry breaking, they can be much larger than the Dirac neutrino masses, $$M \gg m_D$$. The mass matrices $$m_D$$ and $$M$$ contain altogether 6 physical $$C\!P$$ phases, which lead to $$C\!P$$ violating decays and scatterings. Diagonalizing the $$6\times 6$$ neutrino mass matrix one obtains three heavy and three light neutrino mass eigenstates,

$\tag{14}N\simeq \nu_R+\nu_R^c\quad,\qquad \nu\simeq V_{\nu}^T\nu_L+\nu_L^c V_{\nu}^*\, ,$

with masses

$m_N\simeq M\, \quad,\quad m_{\nu}\simeq- V_{\nu}^Tm_D^T{1\over M}m_D V_{\nu}\, . \tag{15}$

In a basis where the charged lepton mass matrix $$m_e$$ is diagonal, $$V_{\nu}$$ is the mixing matrix in the leptonic charged current.

As an example, consider a hierarchical Dirac neutrino mass matrix, as in grand unified models (Langacker, 2012), with a third generation Yukawa coupling $${\cal O}(1)$$, as it is the case for the top-quark. Identifying the heaviest Majorana mass with the mass scale of grand unification, one obtains the heavy and light neutrino masses

$\tag{16}M_3 \sim \Lambda_{\rm GUT} \sim 10^{15}\ {\rm GeV}\ , \quad m_3 \sim \frac{v^2}{M_3} \sim\ 0.01\ {\rm eV}\;.$

Figure 2: Tree level and one-loop diagrams contributing to heavy neutrino decays whose interference leads to Leptogenesis.

It is very remarkable that the light neutrino mass $$m_3$$ is of the same order as the mass differences $$(\Delta m^2_{sol})^{1/2}$$ and $$(\Delta m^2_{atm})^{1/2}$$ inferred from neutrino oscillations. This suggests that, via the seesaw mechanism, neutrino masses indeed probe the grand unification scale. The difference between the charged current mixing matrices of quarks and leptons is a puzzle that can be explained in grand unified models. Like for quarks and charged leptons also right-handed neutrinos may have hierarchical masses. For instance, if their masses scale like the up-quark masses one has $$M_1 \sim 10^{-5} M_3 \sim 10^{10}$$ GeV.

The heavy neutrino $$N_1$$ is of particular importance for leptogenesis. Its tree-level decay width into lepton-Higgs pairs reads

$\Gamma_{D1}=\Gamma\left(N_1\to H + l_L\right) +\Gamma\left(N_1\to H^{\dagger} + l_L^{\dagger}\right) ={1\over8\pi}(h h^\dagger)_{11} M_1\;. \tag{17}$

The interference between tree-level amplitude and one-loop vertex and self-energy corrections (see Figure 2) leads to a $$C\!P$$ asymmetry in the decays. For hierarchical heavy neutrinos, $$M_1 \ll M_2,M_3$$, one finds (Covi, 1996wh):

\begin{aligned} \varepsilon_1 &=\frac{\Gamma\left(N_1\to H + l_L\right) -\Gamma\left(N_1\to H^{\dagger} + l_L^{\dagger}\right)} {\Gamma\left(N_1\to H + l_L\right) +\Gamma\left(N_1\to H^{\dagger} + l_L^{\dagger}\right)} \nonumber\\ &\simeq {3\over16\pi}\;{1\over\left(h h^\dagger\right)_{11}} \sum_{i=2,3}\mbox{Im}\left[\left(h h^\dagger\right)_{i1}^2\right] {M_1\over M_i}\; . \tag{18}\end{aligned}

The $$C\!P$$ asymmetry is conveniently expressed in terms of the light neutrino mass matrix $m_{\nu}$,

$\varepsilon_1 \ \simeq\ - {3\over 16\pi} {M_1\over (h h^\dagger)_{11} v_F^2} \mbox{Im}\left(h^* m_\nu h^\dagger\right)_{11}\;, \tag{19}$

and satisfies the upper bound (Davidson, 2002qv; Hamaguchi, 2002b)

$|\varepsilon_1| \ \leq\ \frac{3}{16\pi} \frac{M_1m_3}{v_F^2}\;. \tag{20}$

For $$C\!P$$ phases $$\mathcal{O}(1)$$, values of $$\varepsilon_1$$ close to the upper bound are typical in models of neutrino masses. Using the seesaw relation for light and heavy third generation neutrinos one then has $$\varepsilon_1\sim\ 0.1\ M_1/M_3$$. Quark and charged lepton mass hierarchies vary from $$10^{-4}$$ to $$10^{-5}$$. A corresponding mass hierarchy for right-handed neutrinos implies a $$C\!P$$ asymmetry $$\varepsilon_1 \sim 10^{-5} \ldots 10^{-6}$$.

## Thermal leptogenesis

The lightest of the heavy Majorana neutrinos, $$N_1$$, is ideally suited to generate the cosmological baryon asymmetry. Since it has no Standard Model gauge interactions it can easily propagate out of equilibrium in a thermal bath of quarks, leptons and Higgs particles, thereby satisfying Sacharov's third condition. The Lagrangian (13) also violates lepton number and the discrete symmetries $$C$$ and $$C\!P$$, so that all Sacharov conditions are fulfilled for the generation of a lepton asymmetry. Sphaleron processes then partially convert the lepton asymmetry into a baryon asymmetry.

At high temperatures, $$T>M_1$$, the heavy Majorana neutrinos can efficiently erase any pre-existing lepton asymmetry. Once the temperature of the universe drops below the mass $$M_1$$, the heavy neutrinos are not able to follow the rapid change of the equilibrium distribution. Hence, the necessary deviation from thermal equilibrium ensues as a result of having a too large number density of heavy neutrinos, compared to the equilibrium density. Eventually, the heavy neutrinos decay, and a $$B$$$$-$$$$L$$ asymmetry is generated owing to the presence of $$C\!P$$-violating processes. The resulting baryon asymmetry is

\begin{aligned} \tag{21} \eta_B &= \frac{n_B - n_{\bar{B}}}{n_\gamma} \simeq -c_s\frac{n_L - n_{\bar{L}}}{n_\gamma} \equiv -\frac{c_s}{f}N_{B-L} \\ &= -\frac{3}{4}\frac{c_s}{f} \varepsilon_1 \kappa_f \simeq 10^{-2} \varepsilon_1 \kappa_f \;.\end{aligned}

Here $$c_s$$ is the fraction of $$B$$$$-$$$$L$$ asymmetry converted into a baryon asymmetry by sphaleron processes, and $$f$$ is a dilution factor accounting for the increase of the photon number density between the onset of leptogenesis and recombination. $$N_{B-L}$$ is the amount of $$B$$$$-$$$$L$$ asymmetry in a comoving volume element that contains one photon at the time of leptogenesis. The efficiency factor $$\kappa_f$$ represents the effect of washout processes in the plasma and is obtained by solving the Boltzmann equations. Typical values are $$\kappa_f = 10^{-1} \ldots 10^{-2}$$. Together with a $$C\!P$$ asymmetry $$\varepsilon_1 = 10^{-5} \ldots 10^{-6}$$ one then obtains for the baryon asymmetry $$\eta_B = 10^{-8} \ldots 10^{-10}$$, consistent with observation. It is remarkable that a heavy neutrino mass hierarchy comparable to the one of quarks and charged leptons, which leads to a small $$C\!P$$ asymmetry, together with the kinematical factors $$f$$ and $$\kappa_f$$ can explain the observed matter-antimatter asymmetry!

Leptogenesis is a nonequilibrium process which takes place at temperatures $$T \lesssim M_1$$. For a decay width small compared to the Hubble parameter, $$\Gamma_1(T) < H(T)$$, heavy neutrinos are out of thermal equilibrium; otherwise, they are in thermal equilibrium. The borderline between the two regimes is given by $$\Gamma_1 = H|_{T=M_1}$$, which is equivalent to the condition that the effective neutrino mass

\begin{aligned} \tag{22} \mt = \frac{(m_D m_D^\dagger)_{11}}{M_1} \end{aligned}

is equal to the equilibrium neutrino mass

\begin{aligned} \tag{23} m_* = \frac{16\pi^{5/2}}{3\sqrt{5}} g_*^{1/2} \frac{v_F^2}{M_{\rm P}} \simeq 10^{-3}~\mbox{eV}\;,\end{aligned}

where $$H(T) = \sqrt{8\pi^3 g_*/90}\,T^2/M_{\rm P}$$ is the Hubble parameter, $$g_*=g_{\rm SM}=106.75$$ is the total number of degrees of freedom and $$M_{\rm P}=1.22\times10^{19}\,{\rm GeV}$$ is the Planck mass.

It is intriguing that the equilibrium neutrino mass $$m_*$$ is close to the neutrino masses suggested by solar and atmospheric neutrino oscillations, $$\sqrt{\Delta m^2_{\rm sol}} \simeq 8\times 10^{-3}$$ eV and $$\sqrt{\Delta m^2_{\rm atm}} \simeq 5\times 10^{-2}$$ eV. This encourages one to think that it may be possible to understand the cosmological baryon asymmetry via leptogenesis as a process close to thermal equilibrium. A necessary condition is that $$\Delta L=1$$ and $$\Delta L=2$$ lepton number violating processes are strong enough at temperatures above $$M_1$$ to keep the heavy neutrinos in thermal equilibrium and weak enough to allow the generation of an asymmetry at temperatures below $$M_1$$.

In general, the generated baryon asymmetry is the result of a competition between production processes and washout processes that tend to erase any generated asymmetry. Unless the heavy Majorana neutrinos are partially degenerate, $$M_{2,3}-M_1 \ll M_1$$, the dominant processes are decays and inverse decays of $$N_1$$ and the usual off-shell $$\Delta L=1$$ and $$\Delta L=2$$ scatterings. The leptogenesis process is quantitatively described by Boltzmann equations. Summing over the three lepton flavours and neglecting the dependence on Yukawa couplings, they can be written as (Buchmuller, 2002rq)

$\tag{24} {dN_{N_1}\over dz} = -(D+S)\,(N_{N_1}-N_{N_1}^{\rm eq}) \;,$ $\tag{25} {dN_{B-L}\over dz} = -\varepsilon_1\,D\,(N_{N_1}-N_{N_1}^{\rm eq})-W\,N_{B-L}\;,$

where $$z=M_1/T$$. The density $$N_{N_1}$$ is again defined as the number of heavy neutrinos in a portion of comoving volume containing one photon at the onset of leptogenesis, so that the relativistic equilibrium $$N_1$$ number density is given by $$N_{N_1}^{\rm eq}(z \ll 1)=3/4$$. Alternatively, one may normalize the number density to the entropy density $$s$$ and consider $$Y_X = n_X/s$$. If entropy is conserved, both normalizations are related by a constant.

Figure 3: The evolution of the $$N_1$$ abundance and the $$B-L$$ asymmetry for a typical choice of parameters, $$M_1=10^{10}\,$$GeV, $$\varepsilon_1=10^{-6}$$, $$\widetilde{m}_1=10^{-3}\,$$eV and $$\overline{m}=0.05\,$$eV. From (Buchmuller, 2002rq).

There are four classes of processes that contribute to the different terms in the above equations: decays, inverse decays, $$\Delta L=1$$ scatterings of real heavy neutrinos and $$\Delta L~=~2$$ processes mediated by virtual heavy neutrinos. The first three processes all modify the $$N_1$$ abundance and try to push it towards its equilibrium value $$N_{N_1}^{\rm eq}$$. Denoting by $$H$$ the Hubble expansion rate, the term $$D = \Gamma_D/(H\,z)$$ accounts for decays and inverse decays, whereas the scattering term $$S = \Gamma_S/(H\,z)$$ represents the $$\Delta L~=~1$$ scatterings. Decays also yield the source term for the generation of the $$B-L$$ asymmetry, the first term in Eq. (25), whereas all other processes contribute to the total washout term $$W = \Gamma_W/(H\,z)$$, which competes with the decay source term. The dynamical generation of the $$N_1$$ abundance and the $$B-L$$ asymmetry is shown in Figure 3 for typical parameters and different initial conditions, $$N_{N_1}^{\rm in} = 3/4$$ and $$N_{N_1}^{\rm in} = 0$$, corresponding to thermal and zero initial $$N_1$$ abundance, respectively. For the chosen parameters the generated $$B$$$$-$$$$L$$ asymmetry is essentially independent of the initial conditions and entirely determined by neutrino parameters.

## Constraints on neutrino masses

The $$\Delta L=2$$ lepton number changing processes with heavy neutrino exchange generate a contribution to the washout rate that depends on the absolute neutrino mass scale, $$\Delta W \ \propto \ M_1\ \mb^2\ M_{\rm P}/v_F^4$$, where $$\mb^2 = m_1^2 + m_2^2 + m_3^2$$. As long as $$\Delta W$$ can be neglected, the efficiency factor $$\kappa_f$$ is independent of $$M_1$$. With increasing $$\mb$$, however, the washout rate $$\Delta W$$ becomes important and eventually prevents successful leptogenesis. This leads to an upper bound on the absolute neutrino mass scale (Buchmuller, 2002rq).

One can also obtain a lower bound on the heavy neutrino masses, because the upper bound on the $$C\!P$$ asymmetry $$\varepsilon_1$$, as well as the efficiency factor $$\kappa_f$$ only depend on $$M_1$$, $$\mt$$ and $$\mb$$. Since the rates entering the Boltzmann equations are functions of the same quantities, there exists for arbitrary light neutrino mass matrices a maximal baryon asymmetry $$\eta^{\rm max}_{B}(\mt,M_1,\mb)$$. Requiring this to be larger than the observed one, $$\eta^{\rm max}_{B}(\mt,M_1,\mb) \geq \eta^{\rm obs}_{B}$$, one obtains a constraint on the neutrino mass parameters $$\mt$$, $$M_1$$ and $$\mb$$. For each value of $$\mb$$ there is a domain in the ($$\mt$$-$$M_1$$)-plane, which is allowed by successful baryogenesis. For $$\mb \geq 0.20$$ eV this domain shrinks to zero, which can be translated into upper limits on the individual neutrino masses and a lower limit on $$M_1$$, the smallest mass of the heavy Majorana neutrinos. A quantitative analysis yields (Buchmuller, 2004nz)

$\tag{26}m_i < 0.1\,{\rm eV}\;, \quad M_1 > 4 \times 10^8~\mbox{GeV}\ ,$

where a thermal initial $$N_1$$ abundance has been assumed. For zero initial $$N_1$$ abundance one obtains the more restrictive lower bound $$M_1 > 2 \times 10^9~\mbox{GeV}$$. For $$\mt > m_*$$, the baryon asymmetry is generated at a temperature $$T_B < M_1$$. Hence the lower bound on the reheating temperature $$T_i$$ is less restrictive than the lower bound on $$M_1$$.

The results of a detailed analytical and numerical analysis are summarized in Figure 4. In the so-called weak washout regime, where $$\mt < m_*$$, the baryon asymmetry, and therefore the lower bound on $$M_1$$, depends on the initial $$N_1$$ abundance. On the contrary, in the strong washout regime, where $$\mt > m_*$$, the baryon asymmetry $$\eta_B$$ is independent of the initial $$N_1$$ abundance. Furthermore, the final baryon asymmetry does not depend on the value of an initial baryon asymmetry generated by some other mechanism. Hence, the value of $$\eta_B$$ is entirely determined by neutrino properties. In this way leptogenesis singles out the neutrino mass window

$\tag{27}10^{-3}~{\rm eV} < m_i < 0.1~{\rm eV}\ .$

Figure 4: Analytical lower bounds on $$M_1$$ (circles) and $$T_{\rm i}$$ (dotted line) for $$m_1 = 0$$, $$\eta_B^{CMB} = 6\times 10^{-10}$$ and $$\sqrt{\Delta m^2_{\rm atm}} = 0.05\,{\rm eV}$$. The analytical results for $$M_1$$ are compared with the numerical ones (solid lines). Upper and lower curves correspond to zero and thermal initial $$N_1$$ abundance, respectively. The vertical dashed lines indicate the range ($$m_{\rm sol}$$,$$m_{\rm atm}$$). The gray triangle at large $$M_1$$ and large $$\mt$$ is excluded by theoretical consistency. From Ref. (Buchmuller, 2004nz).

What is the theoretical error on the lower and upper bounds for light neutrino masses due to leptogenesis? A rigorous answer would require a full quantum field theoretical treatment of the nonequilibrium leptogenesis process, which, despite much progress in recent years, is not yet available. It is known, however, that there is a significant dependence on the lepton flavours due to their different Yukawa couplings, which has been neglected in the treatment presented above. These effects can relax lower and upper bound by about one order of magnitude (see the reviews (Davidson, 2008bu; Blanchet, 2012bk)).

In view of the constraints on light neutrino masses imposed by leptogenesis, knowledge of the absolute neutrino mass scale is of great importance. Hence, a measurement of the neutrino mass $$m_\beta$$ in tritium $$\beta$$-decay and $$m_{0\nu\beta\beta}$$ in neutrinoless double $$\beta$$-decay, or the determination of the sum $$\sum_i m_i$$ from cosmology, consistent with the above neutrino mass window, would strongly support the leptogenesis mechanism.

## Further developments

Early studies of leptogenesis were partly motivated by trying to find alternatives to electroweak baryogenesis, which did not seem to produce a big enough asymmetry. As described above, the simple case of hierarchical heavy neutrino masses with $$B$$$$-$$$$L$$ broken at the unification scale $$M_{\rm GUT} \sim 10^{15}$$ GeV, light neutrino masses $$m_{1,2} < m_{3} \sim 0.1$$ eV, and a reheating temperature $$T_{\rm i}\sim 10^{10}\ {\rm GeV}$$, yields a successful description of baryogenesis. During the past decade much progress has been made in further developing this picture. This includes the connection to dark matter and the role of nonthermal leptogenesis, where the heavy neutrinos are produced in decays of other heavy particles (see the review (Buchmuller, 2005eh)). Moreover, important flavour and spectator effects have been studied (see the reviews (Davidson,2008bu; Blanchet, 2012bk)), and significant progress has been made towards a full quantum mechanical description of leptogenesis based on Kadanoff-Baym equations (see the reviews (Blanchet, 2012bk; Fong, 2013wr)).

One has to emphasize that the minimal leptogenesis scenario, as described above, is far from unique and that many interesting alternatives have been suggested. First of all, supersymmetric leptogenesis is as successful as ordinary leptogenesis (see the reviews (Buchmuller, 2005eh; Davidson, 2008bu)). The dominant contribution to the $$B$$$$-$$$$L$$ asymmetry may also be produced in decays of the next-to-lightest heavy neutrino, as in $$N_2$$-leptogenesis. A strong enhancement of the $$C\!P$$ asymmetry, and therefore a lower reheating temperature is possible for quasi-degenerate heavy neutrinos, as in resonant leptogenesis or soft leptogenesis (see the reviews (Davidson, 2008bu; Blanchet, 2012bk)). Finally, leptogenesis is possible without heavy Majorana neutrinos, as in Dirac leptogenesis, triplet scalar leptogenesis of triplet fermion leptogenesis (see the reviews (Davidson, 2008bu; Fong, 2013wr)).

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### Internal references

• [Blanchet, 2012bk] Blanchet S. and P. Di Bari, "The minimal scenario of leptogenesis," New J. Phys. 14 (2012) 125012
• [Buchmuller, 2005eh] Buchmuller W., R. D. Peccei and T. Yanagida, "Leptogenesis as the origin of matter," Ann. Rev. Nucl. Part. Sci. 55 (2005) 311
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